Phase Multipath Estimation for Global Positioning System (GPS)
Using Signal-to-Noise-Ratio (SNR) Data
by
Francesca Scir6 Scappuzzo
B.S. and M.Eng., Summa Cum Laude , Electrical Engineering
University of Catania, 1989
Submitted to the Department of
Earth, Atmospheric, and Planetary Sciences
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Earth and Planetary Sciences
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1997
@ 1997 Massachusetts Institute of Technology
All rights reserved.
Signature of Author
Department of Earth, Atmospher(icand Planetary Sciences
May, 1997
Certified by
Thomas A. Herring
Associate Professor of Geophysics
Thesis Supervisor
Accepted by
ARCHIVES
, ,-tj8AV(-'.-,.7
Thomas H. Jordan
Department Head
This thesis is dedicated to
Prof. Sebastiano Barbarino
Associate Professor of Electromagnetics
University of Catania, Italy
...perche' il Suo amoreper lo studio e per la scienza,
la Sua encomiabile dedizione all'insegnamentoe
la Sua grandeumanita'
siano da esempio ai Suoi studenti e colleghi
ed ispirino le generazionia venire ...
Phase Multipath Estimation for Global Positioning System
(GPS) Using Signal-to-Noise-Ratio (SNR) Data
by
Francesca Scire Scappuzzo
Submitted to the Department of Earth, Atmospheric, and Planetary Sciences
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Earth and Planetary Sciences
Abstract
A method for the estimation of GPS multipath phase error is proposed. Because
the Signal-to-Noise-Ratio (SNR) of the GPS measurements is strictly related to the
multipath phase error, the algorithm implemented manipulates the SNR of the
received GPS signal itself to perform the estimation. No additional information
about the geometry of the surroundings of the receiving station or the weather
conditions at the time of acquisition are needed. The scattering phenomenon is
generally due to the composition of several single multipath contributions, each one
characterized by a specific frequency. The algorithm takes advantage of this
property to decompose the variations of the SNR into individual components in an
iterative procedure. The contributions due to each frequency are then superimposed
to obtain the total multipath phase error.
The algorithm was applied to several data sets affected by strong multipath to test
the performance of the method. Spectral and statistical analyses were performed to
compare the observed phase residual obtained from the GPS measurements and the
estimated multipath phase error retrieved by the algorithm.
In most of the cases analyzed, a broad-band coherence function between the
observed phase error and the estimated multipath phase error was found in presence
of strong multipath. This suggests that the effects of multiple reflectors are correctly
retrieved by the proposed algorithm.
The set of data which show the highest coherence have multipath frequencies
between 0.001 and 0.01 Hz. Therefore, the estimation algorithm presents good
performances when the reflectors are located at a distance between 5.50 m and 55.0
m from the receiving antenna, for an elevation angle of about 10' and a rate of change
of the elevation angle of 0.1 mrad/sec. The high coherence is confirmed by the
statistical analysis, which shows a significant decrease of the RMS of the phase
residual after the multipath correction.
At lower frequencies it was found that the performance of the algorithm depends
on the receiver type, the antenna gain and the weather conditions.
Thesis supervisor: Thomas A. Herring
Title: Associate Professor of Geophysics
Contents
Introduction
i.1 Objectives
9
..................................................
1.2 Outline .....................................................
2
II
MultipathModeling in phasor space. Relationship between
Multipath Phase Errorand SNR
2.1 Introduction .................................................
13
13
2.2 Multipath representation ......................................
5
. 5
2.2.1 Multipath Modeling in the Phasor Space ..................
2.2.2 Relationship between the signal strength and the multipath
18
phase error ...........................................
21
2.2.3 The SNR and the measured signal strength, Ac ...........
2.2.4 Multipath frequency ................................... 22
2.2.5 Influence of the Reflection coefficient on the multipath
phase error ........................................... 24
2.2.6 Antenna Gain contribution to the Signal strength ........... 31
2.3 GPS receivers in presence of multipath .........................
35
2.3.1 GPS receivers ......................................... 35
36
2.3.2 The DLL and PLL in the presence of multipath ............
2.3.3
3
L2 signal processing in the presence of AS (Anti-spoofing) .... 37
Description of the algorithm for the Multipath Phase Error
Estimation from the SNR data
Introduction ................................................
3.2 Errors in the GPS Phase Measurements .......................
3.3 Multipath Phase Error Estimation Algorithm .....................
3.3.I Antenna Gain removal ..................................
3.1
3.3.2
3.3.3
Data Segment Selection .................................
Spectral Decomposition of the SNR ......................
39
39
40
42
42
46
46
4
Statistical and Spectral Analysis of estimated and observed
GPS phase errors
4.1
4.2
4.3
4-4
6
53
Introduction ................................................ .53
Data Preprocessing .. ..................................... -54
.55
Spectral Analysis ........................................
4.3.1 Multipath Frequency ............................... .55
4.3.2 Power Spectral Density ............................. .55
4-3.3 Auto Correlation and Cross Correlation ................... .56
4.3.4 Auto and Cross Spectral Density Function ................. .58
5.3.5 Coherence Function .................................. .59
Statistical Analysis of GPS Phase Residual ...................... .60
Applying the method ofmultipath estimation to GPS data
5.1 Introduction ..........................................
.63
5.2 Description of the GPS stations and the data sets analyzed .........
.64
5.2.1 The Tien-Shan data ....................................
5.2.2 The LIGO data .. ....................................
5.2.3 The IAP data ........................................
5.3 Multipath Estimation and Results of Spectral Analysis ............
5.3.1 KKAU and MANA ..................................
Station 0036 and 0028 (TRIMBLE 4oooSSE) ..............
5.3.2
5.3.3 COSE and GS29 ....................................
5.4 Multipath Estimation and Results of Statistical Analysis ............
.64
Conclusions
79
.65
.65
.67
.67
.71
.74
.77
REFERENCES ....................................................
83
APPENDICES ....................................................
A. Plots of spectral analysis
B. Tables of statistical analysis
C. Matlab Algorithms
90
Chapter 1
Introduction
Over the last decade the Global Positioning System (GPS) has become a very
useful tool for several scientific and civilian applications, playing a leading role in
geodynamic studies, from a local to a global scale. High accuracy GPS is being
extensively used in large scale dynamics of the Earth, such as plate tectonics, Earth's
rotation studies, sea-level change, post-glacial rebound, tidal effect and sea-surface
topography.
The extensive utilization of GPS in geosciences is mainly due to the recent
significant improvements of the techniques involved in high-accuracy GPS
positioning. The relatively low costs of post processing software and new GPS
receivers and their availability and portability during field work, make the GPS a very
convenient and useful instrument. The international collaboration in the GPS
scientific community is also sensibly contributing to such a wide diffusion of the
system.
However, even though research and technology on high accuracy GPS positioning
have rapidly developed, there is still a broad range of critical issues that scientists and
engineers need to exploit. Most of the new questions arise from the fact that the
fields of application and the accuracy required today for positioning instruments,
have greatly overcome the purposes for which GPS was originally designed,
producing a series of new challenging riddles.
One of the main problems to be faced when performing high-precision GPS
measurements is the error in positioning due to the occurrence of multipath. This
phenomenon is caused by the interference of multiple reflections with the direct
EM signal transmitted by the satellite, and represents a considerable source of error
in the GPS carrier phase observations (fig. 1.1).
The level and characteristics of the noise deriving from multipath depends upon
the geometry of the surrounding of the receiving antenna (presence of buildings,
trees, towers, and the mount itself), the reflectivity of nearby reflectors, the elevation
angle of the satellite at the time of acquisition, and other parameters which might
sensibly change with weather conditions and time.
GPS satellite
rface
GPS receiver
Fig. 1.1: GPS multipath phenomenon
Due to the variability of the parameters involved, it is very difficult or even
impossible to predict multipath with sufficient accuracy using a deterministic
approach such as electromagnetic modeling or analog techniques.
1.1 Objectives
The objective of the study presented in this thesis is to estimate the multipath
phase error from the SNR (Signal-to-Noise-Ratio) data, without using any a priori
information about the location of the receiving station, the weather conditions and
the geometry of the surroundings. This method manipulates the SNR of the
received signal, which in GPS data processing is usually only used as an indicator of
the data reliability and is not used for the positioning itself.
In order to test the performance of the estimation algorithm for different sites and
different GPS receiver types (Ashtech Z-12, Trimble SSE and SSI, and TurboRogue SNR8000), the method was applied to a large number of data sets, chosen for
the strong multipath in the phase error measurements.
1.2 Outline
In chapter 2 a mathematical representation of the multipath phenomenon is
derived. The dependency of the multipath phase error on the SNR, the antenna
gain, the multipath frequency and the reflection coefficient of the reflecting surface
is discussed.
After a brief introduction on the basic operation of GPS receivers, chapter 3
focuses on how a GPS signal corrupted by multipath is processed by different
receivers.
The algorithm for the estimation of the multipath phase error from the SNR data
is described and discussed in chapter 4.
In order to test how well the estimated multipath phase error matches the phase
error observed in the GPS measurements, statistical and spectral analyses were
performed. The theory of these analyses is presented in chapter 5.
The estimation algorithm was applied to several data sets affected by significant
multipath effects. Its performance was studied comparing the results obtained in
the different cases. The results of this analysis are discussed in chapter 6 and the
conclusions are drawn in chapter 7.
The plots relevant to the estimation and spectral analysis of the 42 data sets
studied are collected in appendix A. In appendix B, the tables containing the
results of the statistical analysis are shown. The matlab programs implemented for
the analysis can be found in appendix C.
Chapter 2
Multipath Modeling in phasor space.
Relationship between Multipath
Phase Errorand SNR
2.1
Introduction
The correction of GPS multipath phase errors is one method of improving the
accuracy of GPS measurements. In literature several approaches for GPS multipath
correction can be found. Hajj (1990) developed an algorithm to produce maps of the
multipath for a given environment around the receiving antenna. Unfortunately,
multipath errors are very sensitive to parameters such as the geometry or the
reflection coefficient of nearby reflectors which strongly depend on the particular
weather condition and season. This limits the efficiency of this method in
predicting the multipath effects with time.
An easy solution to the multipath problem consists in selecting a proper site, far
from possible reflectors. However in many applications it is not possible to choose
the site for the GPS positioning. Other ways to minimize multipath consist in the
design of antennas which reduce back-plane gain. Receivers manufacturers are
trying to correct multipath using particular selections of the cross-correlation
function. A branch of research on GPS multipath, is now oriented toward the
analysis of the data itself for the error estimation in post-processing mode
(Georgeadu and Kleusberg, 1988). One new promising method is to compute the
multipath error by using the SNR, as in Axelrad et al., (1994).
The circumstances in which the measurements are acquired are usually recorded
on the log sheets of the GPS measurements, which report the geometry of the site,
the weather conditions during the data acquisition, etc. But to gather the correct
information about the GPS receiver surroundings and relevant measurements can be
extremely difficult and inefficient. The possibility to correct for multipath
contribution to the phase error without a priori information is particularly important
for applications that require networks of GPS stations distributed over large areas on
the globe, such as geosciences.
The multipath phase error estimation algorithm proposed in this thesis makes use
of the SNR of the data itself, without the need for additional information about the
geometry and conditions of the surrounding of the receiving antenna. The method
for the estimation of 50, is based on the spectral analysis of the SNR information
provided by the GPS receiver. We assume that the multipath effects are due to
multiple reflectors, which will each have a particular frequency. Through spectral
decomposition of the SNR we can isolate the contributions of each reflector.
Inverting for the phase error of all the single contributions and superimposing the
results together, we obtain the total multipath correction (total multipath phase error).
This chapter is devoted to the definition of the theoretical model of the multipath
phenomenon used in the multipath phase error estimation algorithm. The important
relationship between the multipath phase error to be estimated , 80, and the SNR
data, used as input of the estimation algorithm, is discussed in section 2.2.2. The
dependency of the signal strength (A) and the multipath phase error (&0) on the
phase of the multipath signal, on the multipath frequency and on the reflection
coefficient of the reflecting surface is derived in sections 2.2.4 and 2.2.5. The
influence of the antenna gain on the signal strength is shown in sections 2.2.6.
2.2
Multipath representation
2.2.1
Multipath Modeling in the Phasor Space
In this section the mathematical model of the multipath phase error used for the
estimation algorithm is derived.
The multipath phase error, 80D, is defined as the angle between the direct signal,
Sd(t), and the total received signal, Sc(t), which is the composition of the direct signal
and the multipath signal, Sm(t). The signals can be written as:
Sd (t)= Ad (t).ei(Co't + Od)
[2.1]
[2.2]
't
Sm (t) = Am (t).e
i
t't+mi
Sc (t) = Ac (t) -ei(
O
A (t)
== Ad
e'i( t t +
d)
+
mi
(t .
where:
Ad = Amplitude of the direct signal
Ami = Amplitude of the i-th multipath contribution
kA= Amplitude of the total measured signal
Od = Phase of the direct GPS signal.
4mi= Phase of the multipath signal due to the i-th reflector.
[2.3]
[2.3
OC = Phase of the total measured GPS signal.
o = 2 7 f and f is the carrier frequency (LI or Lz)
In figure 2.1 the total signal measured by the receiver, the direct signal and the
signal due to the composition of several multipath reflections are shown in the
phasor space.
im
Figure
2.1: Phasor diagram of the GPS signal. 80 is the multipath phase error
in presence of a single reflector. It depends on the amplitude
(Am=Rf Ad) and phase (Om=Od+fmt) of the multipath phasor.
In the phasor space the signals of fig. 2.1 can be written as:
A = Adl
Amm
Am =
ei
d
Ami.
i-
A = AcI e
[2.4]
ie
Om,
=
Od
+
Pi =
d
+
(fm, t)
[2.5]
[2.6]
where:
i = Phase of the i-th multipath signal with respect to the direct GPS
signal
fni = Multipath frequency relevant to the i-th contribution, such that
Pi =fmi t +0oi
0oi = Phase of the i-th multipath signal with respect to the direct GPS
signal at the initial time t=0.
80- Total multipath Phase Error
Rfi= Reflection coefficient for the i-th reflector. IAmil = Rf IAdl.
The composite signal is given by:
c = d +Ami = Ad.ed
++
R
Ade(
d +
i)
[2.7]
The multipath phase error depends on the amplitude of the multipath signal
and its phase. Because 3 changes with time as a function of the multipath frequency
fm, the multipath phasor spins around the tip of the direct signal, producing positive
and negative values of the phase error 80.
If we rotate the diagram of figure 2.1 as in figure 2.2, so that 4d -0 and Ad is along
the Real axis, the total strength of the received signal and the phase error can be
written as:
lAc = Ad (1V+ýi
+ Rfifi cosi) 2 +(.Ri,sini
,,
tcp =
)2
[28
[2.8]
[2.9]
Im(Ac
Re
Figure 2.2: Phasor diagram 2.1 for Od =0 and Ad real.
2.2.2
Relationship between the signal strength and the
multipath phase error.
The amplitude of the composite signal Ac is related to the total phase error, 80
through the angle P3, phase of the multipath vector Am with respect the direct signal
Ad (fig. 2.1, formulas [2.8] and [2.9]).
To describe the phenomenon, we consider the simple case of a single reflector
(i=1). (The same theory is applicable to multiple reflections, considering the
superposition of the multipath contribution).
Because the multipath signal rotates around the tip of the direct signal , with
frequency fim, the amplitude of Ac oscillates between a maximum value Ad +Am and a
minimum value, Ad -Am. The change in amplitude of Ac due to the rotation of Am
can be easily deduced from 2.3. Ac is plotted as function of time for different values
of 1. Every time A, presents a maximum ( for 3=0' ) or a minimum ( for 1=180" ), the
multipath phase error is zero. The maximum phase error occurs for P=90" and
1=270'. (see fig. 2.1 and formula [2.9] ).
In table 2.1 the values of A, and 6Q as functions of P are summarized . It is
important to notice that Ac can assume the same amplitude for different values of 1
(for example for P=90" and 1=270'). This produces an uncertainty in the
determination of the sign of 86, which represent a very critical problem in the
multipath phase error estimation algorithm.
Ac =SNR
t
=0°
1-1800
13=900
Figure 2.3: Variation of Ac with the time, as function of the angle
3.
,=
fl t = 00
/= Phase error
SNR
0 = f t = 90 0
=* Phase error
SNR
= f.t = 180 0
= Phase error
SNR
/ = fQt = 2700
= Phase error
SNR
30
= 00
A =A
c
+
d
+A
=tan - 1
A =
c
A
2
d
(Max SNR)
[m
(Max 8I)
+A 2
m
50 = 0
(Min SNR)
Acc = Add - A m
m
- 31 = -tan
AA
A
c
=
- 1
Ad 2 +A
d
(Min &0)
m
2
Table 2.1 : Signal strength, Ac and multipath phase error, 80, as functions of 3.
In section 2.2.3 it is shown that Ac and the Signal-to Noise-Ratio (SNR)
actually represent the same quantity in the GPS receiver processing.
The relationship between the SNR and 80, is crucial for the phase
error estimation algorithm presented in chapter 4.
2.2.3
The SNR and the measured signal strength, Ac
In table 2.1 the amplitude of the received signal is assumed to be equal to the
value of the SNR computed in the receiver. This section shows why Ac and SNR in
the receiver represent the same quantity.
From formula [2.3], considering the PRN code function, the signal received by
the GPS ground antenna can be expressed as:
-4("t+
Sc (t) = ad(t) -P(t) i e
d)
+ .Ami (t) -P(t + 8) e
i(j -t+
mi)
[2.10]
where:
P(t) is the PRN code (±1)
8- multipath time delay
The signal internally generated in the receiver is given by:
S (t) = P(t + ).-e(
t+ )
[2.11]
where:
oo = Local Oscillator frequency
, = Tracking error of the Delay Lock Loop (DLL)
0 = Tracking error of the Phase Lock Loop (PLL)
In the receiver the GPS signal [2.10] is down-converted, cross-correlated with the
signal internally generated, and filtered. The result of this sequence of operations
gives the equations [2.4], [2.5] , and [2.6] introduced at the beginning of this
chapter.
In particular, A, represents the normalized average correlation coefficient, which is
the ratio signal/noise. In formulas:
AC = SC (t)-SLO (t))
=
normalized corr. coeff.
= SNR
[2.12]
where:
SNR =
Signal Strength
Noise
[2.13]
However, the precise normalization used in the receiver is not always the same, but
depends on the particular manufacturer. We can assume that the values of the
correlation coefficient are normalized by the expected correlation coefficient when
only noise is present.
It follows that the SNR is equivalent to the signal strength of the total composite
signal, Ac. Therefore, all the properties we found for Ac are valid also for the SNR.
In table 2.1 the values of the SNR and 601 as function of P were summarized.
2.2.4
Multipath frequency
The multipath phase error depends on the multipath frequency. The implications
of this dependency are very important in the spectral analysis performed for the
multipath phase error estimation algorithm described in the following sections.
As already shown in section 2.2.1, the phase of the i-th multipath signal Ami is
given by [2.5] :
[2.14]
Om, = d +f
where
[2.15]
fi= i, "t
Substituting 2.15 in 2.9 we obtain an expression of 8(1 as a function of fm:
(f tan-1
)=
m+R
+Rfco
-cos(fmt)t)
[2.16]
The frequency of the i-th multipath error depends on the carrier frequency, the
distance of the i-th reflector and the receiving antenna, the rapidity of the change in
elevation angle of the satellite and the elevation angle.
For an horizontal reflector at distance hi below the antenna, the multipath
frequency is (Leich, 1995):
i, = -. sin y-
[2.17]
where:
h = Distance between the antenna phase center and the reflector
S= Wavelength
Nr = Zenith angle = (90' - Elevation angle)
One of the consequences of equations [2.16] and [2.17] is that 80 varies with time
due to changes of the elevation angle, as the satellites rises or sets.
Moreover, the multipath phase error, 80, presents a particular frequency
component fmi for each reflector located at a certain height below hi from the
receiving antennas. From [2.8] and [2.9] we can deduce that Ac (=SNR) and 8o
present the same spectral content, being both functions of 13=fmt.
Therefore, the multipath phase error 50i due to a particular reflector can be
identified by the spectral analysis of the total signal strength Ac (SNR).
The method for the estimation of the total composite 8I, is based on the spectral
analysis of the SNR information provided by the GPS receiver. We assume that the
multipath effects are due to multiple reflectors, which will each have their own
frequency. Through spectral decomposition of the SNR we can isolate the
contributions of each reflector. Inverting for the phase error of all the single
contribution and adding them together, we obtain the total multipath correction, as it
will be shown in more detail in chapter 4.
2.2.5
Influence of the Reflection coefficient on the multipath
phase error
The multipath phase error 80i is a function of the Reflection Coefficient Rf, as
shown in formula [2.9]. The reflection coefficient depends on the electromagnetic
characteristics of the reflecting surface, such as permittivity (e), permeability (g), and
conductivity (d). It depends also on the frequency of the electromagnetic signal and
the elevation angle of the satellite 0.
In this section we derive the expression of the reflection coefficient Rf for one
single reflector as a function of E, gt, and a, and discuss how these electromagnetic
properties of the reflector influence the multipath phase error.
The geometry of our problem is shown in figure 2.4, where Einc is the incident
electromagnetic wave transmitted by the GPS satellite, Et is the EM wave refracted
into the material and Er is the EM wave reflected by the surface.
I A--J:.
&A--J: . -
z
C
i.
Figure 2.4: Electromagnetic plane waves propagating in medium 1 (air) and
medium 2 (reflector).
The incident electromagnetic plane wave emitted by the satellite, propagating
in medium 1 toward medium 2, can be written as:
E
= Eo e- i(t- •.r )
[2.18]
The plane wave reflected by the surface, propagating in medium 1 away from
medium 2, can be written as:
Er = Ero .e- i(• - v)
[2.19]
where :
k = Wave number
[2.20]
F=/3+ia
k2 = ie(-ieW)
= o 2E + iaUT
Substituting k( [2.20]) in [2.18] and [2.19] we obtain:
Ei, = Eo -(e-a-r, ). e-i('r,
[2.21,)
Er = Ero (e-a.r,). e-i(Otpr,)
where:
a = Attenuation Coefficient (Np/m)
p = Propagation Coefficient (rad/m)
e-(ar) = Attenuation factor
e- (r) = Propagation factor
Solving [2.20] for a and 0, the Attenuation Coefficient and the Propagation
Coefficient can be written as (Balanis, 1989):
S= o
0_
2
1+ 22
E2)2
[2.22]
1
a2 +
[2.23]
The term e - (ar) represents the attenuation of the electromagnetic wave in the
material, and the exponential e -j(r)represents the propagation term. Note that for
air (medium 1 where the GPS signal propagates), a 1= 0, and therefore e - (ar) s 1.
The electric fields of the incident and reflected wave can be decomposed in a
component perpendicular to the plane of incidence (horizontal polarization) and a
component parallel to the plane of incidence (vertical polarization). In formulas:
Horizontal Polarization:
-h
Eh
[Eh e-i',(xsinO,+zcosO,)
[2.24]
E h -i,(xsin,-zcos)[2.2
Vertical Polarization:
-. = [E . e-iPj(xsin°j+zCs
cos]
O-
,sin 0)
Er = [E',0 e-iP,(xsin,-zcos,)] (ycos Or +isin
2.25
Or)
where:
0i = Incidence angle
Or = Reflection angle
The Fresnel Reflection Coefficient for the horizontal (perpendicular)
polarization is given by:
Erh =2
COS Oi - 071
COS 8 t
772 COS Oi 771 COS Ot
EihN,
cos 8-
1
172
cos ei + .1
[2.26]
072
The Fresnel Reflection Coefficient for the vertical (parallel) polarization is given
by:
S=
Er
Ei
-cos Oi + _2
171
-171cOS 8i + 72 COS Ot
71 cos Oi + 772 COS Ot cos90 + 02
[2.27]
where:
Ot = Refraction angle
= Intrinsic impedance of the medium.
The angles of Incidence, Reflection and Refraction, of fig. 2.4 are related by
the Snell's laws:
O0 = Oi
k1 sin Oi = k2 sin 8,
Snell's law of reflection
Snell's law of refraction
[2.28]
[2.29]
The Intrinsic Impedance of a medium is given by:
E
H1=-=
H
{
iouM
0'+ioe
for dielectric materials
[2.30]
for conductive materials
In our case, the medium 1, where the incident and reflected signal propagate, is
air. For air the intrinsic impedance can be considered as the one for dielectric
materials.
Depending on the electromagnetic characteristics of the reflector, we can
distinguish 3 cases, which represent 3 different categories of materials.
Case I: Good Dielectrics
If the reflector (medium 2) is a good dielectric at the frequency of the GPS
signals (Microwaves) we obtain:
-cos ei -
n_ •E2
cos
cILo~
E
Inh =
= •1 =71
'22
2 2 << 1
E2 02
72
and
1
C2
Rf =
f
2
-I
R
COS 0 ++
62 cos
12COS
re,2
i+
Scos o,
Case II: Quasi-conductors
If he reflector is a quasi-conductor at the frequency of the GPS signals,
it follows that:
t cos
iW 2 o C/ s
7"+ i0E2 OS
Rh 2
2
2
E22
,22(02
1
-1i =
and
772
a 2 + itoE2
osO
i
2
Scos
Oi + /1 cos8
2___
Rf = <
-cos O+
cos O
i92
a 2 + i(82
RV=
cosBi +
Ei
i
" 2 + i082
cos O,
In this case the reflector is not a good dielectric, nor a good conductor, having
inter medium characteristics. The reflection coefficients (horizontal and vertical)
can be determined using the general formulas [2.26] and [2.27] , respectively.
Case III: Good Conductors
If the reflector is a good conductor at the frequency of the GPS signals,
considering the approximation of formulas [2.26] and [2.27] we obtain:
rRh -1
>> 1
>1
6220)2
2
72
>> 1 cos9.
and
Rf =
R, +1
Thus, for a very good conductor the magnitude of the reflection coefficients for
horizontal (perpendicular) and vertical (paralleD polarization approaches unity. This
implies that Am Ad and, therefore, the multipath phase error, 80, is very high.
It can be noticed that for a very good conductor the reflection coefficients
become essentially independent of the angle of incidence.
In fig. 2.5 a summary of the electromagnetic properties of some materials at
different frequencies is shown. (Kraus, 1992).
At microwave frequencies (at which GPS operates) sea water and wet rural ground
are quasi-conductive, while dry ground and urban ground are in the dielectric region.
It is interesting to notice that because wet ground is more conductive than dry
ground, it reflects the GPS signal better, producing a stronger multipath, even if the
geometry of the antenna surroundings is maintained constant.
In general fresh water has a o =10-3 mho/m while salt water has a much higher
conductivity a = 4.0 mho/m. Depending on the soil conditions (i.e. how much salt is
in the soil the ground presents different conductivity after rain.
Quartz (which is a good approximation to sandy soil) has a dielectric constant of
4 and, therefore, the reflection coefficients persent values between 0.3 and 0.6,
depending on the incidence angle. At low elevations an increase of moisture
content might increase the reflection coefficient of 60%, so that the reflection
coefficient is almost unity. (R 1).
This example suggests that the weather conditions might influence the
performance of GPS measurements, because of the different multipath environment
they can produce.
M =6
5
4
3
2
1
0
-1
-2
-3
-4
-4
-5
-5
.V= 1
2
3
4
5
6
7
8
9
10
11
12 13 14 15 16 17
v
Frequency = 10 Hz
Fig. 2.5 Ratio ao/w as a function of frequency for some common media.
At Microwave frequencies the most of the media shown are in the
quasi-conductive region (urban ground, rural ground, sea water), with the
exception of copper, which, being a metal, is a very good conductor even
at higher frequencies, up to the Visible spectra. (Kraus, 1992)
2.2.6
Antenna Gain contribution to the Signal strength
In formula [2.8], Ac is described as a function of Ad ,Rf and P. We already
discussed its dependency on Rf and P. To complete this theoretical analysis of 80
and Ac we need to study how the amplitude of the direct signal, Ad is influenced by
Ac. The direct signal sent by the transmitting antenna, Ad, is a function of the GPS
satellite antenna pattern, as will be discussed in section [2.2.6.1]. The total signal
measured by the receiver, Ac, depends on the receiving antenna pattern (section
[2.2.6.2] ).
22.6.1 GPS Transmitting antenna gain and its influence on Ad
The purpose of a GPS transmitting antenna is to illuminate the Earth surface in
the field of view of the satellite with a uniform strength.
The GPS transmitting antennas present patterns properly shaped to
compensate for the path loss occurring at low elevation angles. The path loss of the
signal is a function of the distance between the antenna phase center and the point
on the surface of the Earth where the GPS receiver is located. The path loss is
minimum when the satellite is at 90 ° elevation angle (at zenith) and is maximum
when the satellite is at 0" elevation angle (at the horizon). A comparison between the
pattern of an ideal antenna pattern and a typical GPS antenna is shown in fig. 2.6.
The edge of the Earth in view subtends an angle of approximately 27.74 * from
the GPS satellite altitude. The satellite antenna pattern extends behind the edge of
the Earth, as shown in fig. 2.7. The transmission power from the satellite is not
uniform with the angle of bore-sight of the satellite. The beam transmits more
power toward the Earth's limbs, so that low elevation angle measurements on the
ground will see more power.
The direct signal Ad is a function of the signal strength transmitted by the
satellite, Atrans , and also of the transmitting antenna pattern Gt(O,0), where 0 and 0
are elevation angle and azimuth of the satellite, respectively:
Ad = Ad(8,0) = Atrans Gt(0,
)
[2.31]
-2.1 dB
.
-an•
DEAL ANTENNA
PATTERN
Figure 2.6: Comparison between the antenna patterns of an ideal GPS
transmitting antenna and a typical one (Parkinson, 1996).
72
f a 1227 MHz: a = 45", 225*
L2 FREQUENCY
36
0
36
72
f = 1575 MHz: w = 450, 2250
L1 FREQUENCY
Figure 2.7: Antenna patterns of a GPS transmitting antenna at LI and Lz.
Block II satellites (Parkinson, 1996).
22.6.2 GPS Receiving antennagain and its influence on Ac
The signal strength received by the ground antenna ( A) is also a function of
the receiving antenna gain Grec(O,4).
GPS receiving antennas present a low directive gain, to acquire data
simultaneously from as many visible satellites as possible. This implies that multipath
signals scattered from any positive elevation cannot be rejected during the data
acquisition.
The fact that GPS antennas have a non zero gain below the horizon, represents
a trade-off between the need of rejection of reflected signals and the attempt to
receive direct signals at low positive elevation angles with a reasonable gain.
But why are data incoming from low elevation angles so important? Data from
low elevation angles are useful to reduce the correlation between estimates of the
vertical component of the site position and estimates of the corresponding zenith
propagation delay.
Because of this dependency of A, on the receiving antenna, the pattern of the
signal strength follows the antenna gain pattern, showing lower values for low
elevation angles and higher values for high elevation angles.
The performance of each particular GPS user antenna has a great effect on the
accuracy and precision of the GPS measurements. The antenna patterns of some of
the most commonly used receiving antennas are shown in fig. 2.8 and 2.9 (Shupler
and Clark, 1994). These measurements were performed in an anechoic chamber
and represent the results for the azimuthally averaged LI amplitude patterns of the
antennas, measured as the antenna zenith angle was varied.
-10
-20
-30
-I()
-180
-120
-60
0
120
60
180
Zenith Angle (deg)
-
ar-r
-Dor-aIgolan
,OKm lowrafy
-
A&Me
-
.
Astech
wrh aM 1ry
gaWQJIa'
Figure 2.8: LI relative amplitude patterns of GPS user antennas: Dorne-Margolin
and Ashtech. (Shupler and Clark, 1994).
0
-10
-20
-30
-40
- r-0%80
-180
2
-120
-60
0
60
120
18C
Zenith Angle (deg)
-
Wkrrwco
-
-
-
Iat.
1,
kt~vWo4
Figure 2.9: L2 relative amplitude patterns of GPS user antennas: Trimble, Texas
Instruments, Macrometer and Wilmanco. (Shupler and Clark, 1994).
2.3
GPS receivers in presence of multipath
2.3.1
GPS receivers
Multipath reflections affect both, the signal amplitude (SNR) (section 2.2) and
the carrier phase measured by the Phase Lock Loop (PLL). In this section a brief
discussion on GPS receivers is presented, with particular attention to how multipath
noise is processed in the DLL (Delay Lock Loop) and the PLL.
A GPS receiver is constituted of several components (hardware and software) that
process the incoming signal from a GPS satellite.
The GPS carrier signal is biphase modulated with C/A and P-codes. When it is
received by the antenna, it is first filtered by the receiver (in preprocessing mode) in
order to eliminate unwanted RF (Radio Frequency) environment from the desired
GPS signal band.
Because GPS signals are spread spectrum signals, it is necessary to up or down
convert them in order to reallocate their spectrum. The Down-Converter in the
receiver mixes the LOs (signals from the Local Oscillator), generated by the
frequency synthesizer, with the amplified RF input to obtain a shift of the frequency
spectrum of the signal to IF frequencies.
The IF signal obtained after down conversion, is then converted to a baseband
signal composed of in-phase (I) and quadraphase (Q) signals.
State of the art GPS receivers, after sampling and quantization, split the signal
processing into multiple channels for simultaneous tracking of multiple satellites, by
using the CDMA (Code Division Multiple Access) method.
Each satellite is tracked by the Delay Lock Loop (DLL) and the Phase Lock Loop
(PLL). These devices assure that the incoming codes and carrier phases are
matched (or locked) to the receiver-generated codes and phases, and remain locked
throughout the tracking of continuously received signals. Code matching directly
generates the pseudorange and the phase matching generates the ambiguous carrier
phase observable.
GPS satellites transmit highly coherent radio signals with right-hand circular
polarization over two L-band channels, LI (1575.42 MHz) and L2 (1227.60 MHz).
The LI carrier signal is modulated by two pseudo-random (PRN) sequences: the Pcode (precision code, 10.23 MHz) and the C/A code (coarse code, 1.023 MHz). L2
carrier signal is modulated by the P-code, only.
When Anti-spoofing is activated, the GPS signal is intentionally corrupted. AS is
implemented through a modification of the P-codes: authorized users are equipped
with a decryption devise, in order to lock on the P-codes.
If the P-code is available, both carriers can be reconstructed by the code
correlation technique.
If only the Y-code is available (i.e. the encrypted P-code), the LI carrier can still
be reconstructed by code correlation using the C/A-code, but a codeless technique
needs to be applied to reconstruct the L2 carrier.
Some of the codeless techniques commonly used are the L2 squaring
(Macrometer), the Cross-correlation of L2 with LI (Trimble and Rogue), and the ztracking mode (Ashtech).
2.3.2
The DLL and PLL in the presence of multipath
The main operations performed by a receiver are the tracking and the locking. A
receiver is code-locked when the code internally generated by the receiver has been
aligned with the received code from the satellite. The tracking consists in shifting
continuously the receiver code to match the incoming signal. This shifting of the
locally generated receiver code is controlled by the Delay Lock Loop. The
discriminator function (which is defined to identify the maximum of the correlation
between the internally generated signal and the GPS signal) is zero when the time
tracking error is zero. In the presence of multipath the zero crossing of the
discriminator function value is shifted in time by 8, the time delay of the reflected
signal with respect the direct signal.
Moreover the discriminator function is
modulated by the reflection coefficient of the scattering surface and by cos(p-80),
where p and 801 were defined in [2.4] and [2.5].
After removing the PRN code in the DLL, the unmodulated carrier wave is passed
to the PLL, where the phase measurement is performed. The result is the phase
offset between the received signal and the reference signal generated in the
receiver. In the presence of multipath the carrier tracking error is the multipath
phase error 80.
2.3.3
L2 signal processing in the presence of AS
(Anti-spoofing)
AS can severely limit the use of GPS for high accuracy positioning for unauthorized
users. However there are several solutions proposed for dealing with AS, which are
implemented by the GPS receivers.
Because L2 is biphase-modulated (by either the P-code or the C/A code), it is
possible to recover the Lz carrier without knowledge of the P-code, by simply
squaring the signal or cross-correlating Lz with LI. There is noise degradation added
in this codeless carrier recovery, because of the non-linear squaring operation or the
cross-correlation with a noisy signal (L). Other new techniques, such as z-tracking,
are used in new receivers, which presents better characteristics of the SNR.
2.3.3.1 L2 signal squaring
Signal squaring is one of several methods used to recover the pure carrier from the
biphase-modulated signal. Autocorrelating the received signal (squaring) all
modulations are removed, because a 180' phase shift during modulation is equivalent
to a change in the sing of the signal. As a result of this processing , the signal
frequency increases by a factor of two and the effective wavelength is half-wavelength
carrier phase observation L2. A serious drawback of this method (not any longer in
use in the new receivers) is also that the SNR is reduced by 30 dB with respect to
the code correlation technique..
2.3.3.2 Cross correlationof L2 with Li
This technique is based of the fact that both, LI and L2 carriers are modulated
with the same P(Y)-code, although the Y-code is not known. Due to the dispersive
nature of the ionosphere, the propagation of the electromagnetic signal is different:
the Y-code on L2 is slightly slower than the one on LI. The time delay necessary to
match the LI and L2 signals is a measure of the travel time difference between the
two signals.
The result of the correlation process between LI and L2 are the range difference
between the two signals and the phase difference between the beat frequency
carriers. With this method the SNR decreases of 27 dB with respect the code
correlation technique, reporting a 3 dB improvement with respect L2 squaring.
2.3.3.3 Z - tracking
This quasi-codeless technique is the most recent and provides the best SNR
performance in the presence of AS. This technique is based on the fact that the Ycode is the module-2 sum of the P-code and is a substantially lower rate encryption
code. The LI and L2 signals are separately correlated with locally generated P-codes.
Since there is a separate correlation on LI and Lz, the W-code on each frequency can
be obtained. The encryption code is estimated for each frequency and removed
from the received signal to allow locally generated code replicas to be locked with
the P-code signals of Li and Lz.
With this method the full-wavelength carrier phases of L and L2 is obtained. The
output pseudoranges and phases of the Z-tracking technique, are directly those
which would be obtained by a code correlation without antispoofing, but with a SNR
degradation of 14 dB.
Chapter 3
Description of the algorithm for the
Multipath Phase ErrorEstimation
from the SNR data
3.1
Introduction
Among other sources of noise, multipath phase errors represent a significant
limiting factor in high precision GPS positioning [Parkinson and Spilker, 1996]. By
using differential techniques in dual frequency mode it is possible to eliminate the
most of the GPS errors (clock errors, ionospheric effects, etc. ...). However, the
multipath phenomenon is characteristic of the surroundings of the receiving antenna
and needs to be corrected with other more local techniques.
Multipath reflections affect the SNR (section 2.2) and the carrier phase
observable measured by the Phase Lock Loop (PLL) (section 2.3).
In this chapter we describe the algorithm implemented for the estimation of the
multipath phase error from the frequency and amplitude of the oscillations of the
SNR data.
3.2
Errors in the GPS Phase Measurements
The dominant source of error in phase measurements is the clock error between
the transmitter (satellite) and the receiver (ground station).
To eliminate the effects of bias or instabilities in the satellite clock, differential
GPS methods can be applied. One method consists in computing the singledifferences, by differencing the phases of signals received simultaneously by two
ground stations. If the receivers are closely spaced (short baseline) the computation
of the single difference phase residual also reduces the effects of tropospheric and
ionospheric refraction on the propagation of the radio signal.
The effects of variations in the station clock can be corrected by computing the
double-differences phase residual. This observable is obtained by subtracting the
single-difference phase residuals relevant to the two satellites from one other
(double-differences).
Apart from clock errors, other significant sources of errors in GPS positioning are
due to orbital errors, phase center models, ionospheric and tropospheric effects,
multipath, and noise in the receiver. It is usually difficult to distinguish among
different errors, especially between ionospheric and multipath effects, because they
can produce similar oscillations in the carrier phase.
The spectrum of the ionospheric variations is a function of the elevation angle of
the satellite, the time of the day, the season and solar activity. The dispersive nature
of the ionosphere produces a different effect on the signal at different frequencies.
By combining the two carrier beat phase observations, LI and L2, it is possible to
remove the ionospheric contribution. This is the reason why GPS satellites transmit
in two different frequencies, f1 (1575.4 MHz) and f2 (1227.6 MHz).
One way to filter the ionospheric effects is to apply the so called dual-band
processing. It consists in computing a linear combination of LI and Lz (LC):
LC = S~
= 2.546.- L - 1.984- L2
[3.1]
In the following chapters we will denote LC with QDobs and we will refer to it as
the observed phase residual.
By computing IQoBs the effects due to other sources of error (different than the
ionosphere) are magnified. This is the reason why, for short baselines, where the
ionospheric errors cancel out with single differencing, it is preferable to treat LI and
L2 as two independent observables. For long baselines (more than a few kilometers
apart) the ionospheric errors are uncorrelated and it is necessary to use the dual-band
processing (computing LQC) to eliminate all the effects of the ionosphere.
In general we would like to eliminate both, multipath and ionospheric effect.
For short baselines LI and L2 can be treated as two independent observables, thus
eliminating the need for calculating LC, which in return would amplify the multipath
error.
For long baselines, the ionospheric effects are not strongly correlated and it is
necessary to use LC in order to eliminate them. The ionosphere does not
decorrelate after a few kilometers. It is in fact correlated for many hundred's of
kilometers. But how large is the decorrelated part? An approximate rule (discussed
in many of the early papers on GPS) is that the differential ionospheric delay is
between 1-10 parts-per-million of the separation between the sites, where the exact
amount depends on latitude, time of day, time of year and solar activity. Using this
approximation, for baselines of 1 km, the ionospheric contribution is likely to be
between 1 and 10 mm. 1 mm is small compared to the phase noise, but 10 mm (0.05
cycles) is large. Therefore, when the ionosphere is active, dual frequency may be
needed on baselines as short as 1 km.
When only distant receivers are available, it is usually necessary to compromise
between the amplification of other types of noise and the need to eliminate the
effects of the ionosphere by dual band processing.
Either case (short or long baselines) the multipath error cannot be filtered with
differential GPS techniques, because it is a distinct characteristic of the environment
around one particular receiver.
One technique for the estimation and correction of the multipath error is
proposed in the following sections.
3.3
Multipath Phase ErrorEstimation
Algorithm
The algorithm proposed in this thesis manipulates the variations of the SNR for
the estimation of the total multipath phase error 80EST, computed as the
superposition of several single reflections.
With reference to fig 2.1, the amplitude of the direct and multipath signals are
assumed constant for the time T, in which Am rotates around the tip of Ad. During
this time the angle j3 varies of at least 360" and the composite signal Ac shows an
oscillation of period T. This implies that the amplitude of the direct signal arriving
to the antenna and the multipath frequency fm are constant over T.
3.3.1 Antenna Gain removal
The first step of the estimation algorithm consists in removing the variations of
the SNR due to change of position of the satellite. This changes are due to the fact
that the direct GPS signal reaches the receiving antenna under different angles (0,
4)which are associated to different values of the antenna pattern.
In order to remove the antenna pattern contributions and the differences in the
transmission power from the satellites, we fit a 4-th order polynomial in sin(elev) to all
of the SNR data at each station. In this fit we include a zero-order term (i.e. a
constant offset) for reasons to be discussed below. Separate fits are made for LI and
L2 data. With L2 data a distinction is made between data processed in crosscorrelation mode and those data that are directly tracked. The latter occurs for the 1
or 2 GPS satellites that do not have Anti-spoofing turned on. This fitting processes
also normalizes the gain-corrected SNR data so that they are near unity. The
polynomials estimated with the procedure previously described look similar to the
anechoic chamber gain curves but tend to be larger in the mid-elevation angle range.
This discrepancy is probably be due to the effects of the transmission power from
the satellites.
For all receivers, except the Ashtech z12, the measured SNR and the gain
curves tend to approach zero as the satellite elevations tend to zero. (See fig. 3.2) .
For Ashtech z12 the SNR near zero degrees elevation angle is still about 90% of its
value at zenith. This behavior is not consistent with the gain pattern of the DorneMargolin antennas used with these receivers. Therefore, it can only be explained by
increased integration time at lower elevation angles (thus increasing the SNR) or by
the SNR values being biased. After comparing Ashtech and Trimble SSE data
collected on the same site (and showing similar multipath signals), we concluded
that latter was more likely explanation. This is the reason why we include an offset
term in gain fitting process and simply remove this offset from all the SNR values.
In the algorithm a problem with the gain removal is represented by the
dependency of the SNR on the elevation and azimuth angle, as well. The main
dependency is the elevation angle, but not taking in to account the azimuthal
dependency an error is introduced in the estimation, which influences the
performance of the algorithm. The main dependency is with elevation angle, but for
Trimble SST and SSE antenna there is a sizable azimuthal dependence that we have
not accounted for. The impact of this neglect has not been evaluated and is likely to
influence the lower frequency multipath signal estimation.
The impact of the antenna gain on the SNR depends on the particular signal
processing procedure performed by each receiver. For receivers which use the L2
squaring techniques, the SNL.Lz is a function of the elevation angle and decreases
as the square of the L2 gain pattern of the antenna. In receivers which use the crosscorrelation between LI and Lz, the SNRL2 decreases as the product of LI and Lz
gain patterns.
In receivers which perform the standard tracking the SNRLL2
decreases simply as the gain pattern.
In the algorithm, the SNRL2 derived from the receivers which performs L2
squaring and standard tracking, can be processed directly. But the SNILLz deriving
from an operation of cross correlation needs to be either divided by SNRILI (so that
SNRILz / SNR1LI is the input for the algorithm for the estimation of the L2 phase
error) or considered without modifications. In the first case the estimated phase
error is directly 80 2 and, in the second the estimated phase error is given by the
difference between the phase error estimated from SNRL2 (8D'2) and the one
estimated from SNR-LI (8•1). In other words: 802 = (8D' 2) - (801) . For the
estimation algorithm proposed the first technique was adopted.
In figure 3.2 the SNR for LI for the station MANA (Turborogue) and the
satellite PRN 26, is show, as a function of time. As it can be noticed comparing fig.
3.1 with fig. 3.2, the SNR as obtained from the receiver is strongly dependent on
the elevation angle of the satellite. After the gain removal and normalization, the
estimation algorithm is ready to manipulate the new SNR for LI shown in fig. 4.3.
station: MANA
1.5
2
2.5
3
satellite: PRN 26
3.5
Time (sec)
4
Figure 3.1: Elevation angle for the satellite PRN 26
4.5
5
5.5
6
x 10'
SNR Li as received bythe antenna
800
700
soo -
C400
300oo
C 200 -
100
II
1.5
1
2
2.5
3
3.5
4
4.5
5
5.5
time lsec]
6
x 10'
Figure 3.2: SNRILL for the station MANA (Turborogue) and the satellite
PRN 26.
7
I
I--
i
4:00
l
6:00
3:00
10:00
12:00
|
14:00
16:00
Figure 3.3: SNRLI for the station MANA after gain correction and
normalization.
3.3.2 Data Segment Selection
After the gain correction, the SNR is divided into segments. The length of these
segments is based on gaps that the data might have, or on the rate of change of the
elevation angle, so that portions of data relevant to a rising or setting satellite are
treated separately.
The SNR might have missing data in the sequence due to problems with the
reception of the signal or to obstruction of the satellite. The algorithm presents a
gap tolerance of about 10 epochs. If the missing data are less than 10 points, the
program linearly interpolates before the computation of the spectra. However, the
algorithm does not interpolate when fitting the sine and cosine components,
because the interpolation introduce false data which might produce a wrong
estimation of the phase error. On the other hand, not having enough data for the
estimation of the sine and cosine components, can cause anyway an inaccurate
estimation of the phase error. If the gap is too large, separate analyses are done for
the two segments of data. Because the length of the segment of data determines the
frequency resolution, also a too short time series can lead to a wrong phase error
estimation.
3.3.3 Spectral Decomposition of the SNR
As already discussed in section 2.2.4, the scattering phenomenon is generally
due to the superposition of several single multipath contributions, each characterized
by a specific frequency fm. Multipath is not a stationary phenomenon because its
frequency varies with the rate of change of the satellite elevation angle (formula
[2.5]). Therefore, spectral estimation must be performed on a span of data long
enough to provide the required frequency accuracy, but short enough to allow the
hypothesis of constant frequency.
The purpose of the spectral estimation performed by the algorithm is to
decompose the total multipath phase error in several components and isolate them
to compute the GPS phase error due to each component. The multipath
46
contributions are determined sequentially and the correspondent phase error are
1 EST is created from the sum of the complex
computed. The total phase error BQ
components of all of the contributions.
We decompose the SNR variations into individual components in an iterative
procedure. At each iteration, we compute the power spectra of the current SNR
residuals, which in the first iteration are simply the values of the SNR after we
remove the gain corrections. We select the frequency with the largest amplitude: if
this amplitude (sqrt of the power spectral density) is larger that 1/500 of the mean
SNR strength (obtained from the zero frequency component in the PDS) then the
iteration is continued. The 1/500 limit is set so that the phase contribution from the
remaining components will be less than 0.4 mm (190 mm/ 500).
For each period of the maximum frequency through the data segment, we estimate
a mean value and sine and cosine components at this frequency. Since the
frequencies are obtained from a FFT, we are assured that there will be an integer
number of these periods in the data segment. As can be seen in fig. 3.4, for
example, the frequency of the multipath is reasonably constant but the amplitude
changes with time. Our algorithm estimates separate sine and cosine components
for each full wavelength through the data segmant. The sine and cosine
components are saved and the contribution to the SNR computed and removed
from the SNR values. These residual SNR values are then used in the next iteration.
In addition to the 1/500 amplitude limit, the iteration is continued for no more than
50 iterations although this limit is rarely reached. At the end of the spectral
decomposition, we have a series of sine and cosine components, each applicable to
different times in the data depending on the frequency they represent. For each
data epoch, there are N applicable component pairs, where N is the number of
iterations. The compex sum of the N pairs (with their time arguments dependent
on the frequency they each represent) is formed and the phase error and amplitude
computed from the sum. The N multipath contributions estimated with the
algorithm are superimposed in figure 3.6, to provide the total multipath phase error
There are limits on the spectral components estimated. When the maximum
spectral components are found, we search frequencies for 2/T to n/(2T), where T is
the total duration of the segment, and n is the number of measurements in the span.
Because the high frequency end is half of the maximum frequency in the spectrum,
the algorithm will tend to smooth the SNR as well. This smooting can be observed
in fig. 3.4.
As explained in section 2.2.2, the determination of the sign of the multipath
phase error estimate remains ambiguous. In the algorithm the sign attributed to the
multipath phase error is based on the sign of the rate of variation of the elevation
angle (dO/dt). The sign actually depends on the rate of change of distance to the
reflector. Because we never try to compute the location of the reflectors, we use the
rate of change of the elevation angle instead. If dO/dt is positive, 86EST=atan2(Imag,Real); if If dM/dt is negative, 8&EST=+atanz(Imag,Real). The assumption
made is that, as the satellite rises, the distance to the reflector decreases (at least for
reflections from a flat Earth). This is verified in most of the cases, but it is not true
in general.
This ambiguity in the sign of the multipath phase error strongly
influences the performances of the estimation algorithm in those case when an
abrupt change of sign of the estimated phase error is observed.
The total multipath phase error estimate 8IDEsr is shown in figure 4.7 and is
indicated with LCGEST. The observed phase residual measured from the GPS
measurements 8G0OBs is shown in the same figure (LCOBS) and compared with the
estimated phase error due to multipath. In this example, the visual correlation of the
two curves is striking: the frequency content and the amplitude of the two curves
seems to agree very well. In the following chapters, quantitative analyses (spectral and
statistical) are performed to compare estimated and observed phase residuals.
KKAU SNRL1 PRN 16 ASIA1995 6 9
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1.8
1.6
1.4
..................
................................
1.2
..
. .
.
.. .... . ..
-
. .
.
.
.
.
.
.
. .
i.....
. .
. .
.
. .
.
.
... . . .
1
0.8
4. 3
....
4.4
·
·
4.5
4.6
4.7
time (sec)
Figure 3.4: Curve fit for SNRLI satellite PRN 16. Station KKAU.
4.8
4.9
x 104
KKAU SNR L2 PRN 16 ASIA 1995 6 9
2c
-
1.8
,4
I•'
1.0
1.4
1.2
1
INI
0.8
43
.
4.4
4.5
4.6
time (sec)
4.7
4.8
4.9
x 104
2
. ..... .
1.8
..
.
1.6
F,
1.4
.· . . . .
--
-
.'
·.
--
. .. . . . . ..
1.2
0.8
6
4. 3
I
I
I
4.4
4.5
4.6
~I
'
4.7
time (sec)
Figure 3.5: Curve fit for SNRIL2 satellite PRN 16. Station KKAU.
I
4.8
3
II ~ I
4.9
x 104
Im
Re
Figure 3.6: Composition of multipath single components.
GPS "OBSERVED" Phase Residuals for: KKAU PRN 16
0.2
0.1
..
...
..
...
...
...
..
.. ...
0
-.. .10
... .
.
. ...
..
..
I.
.
......... - --
-0.1
-0.2
:................
................
:...
..
,............... . ................
4
•I
4.7
4.8
4.6
4.5
4.4
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation
4. 3
0.2
v/
..
...
. .............
. ..
. .. .
........................
..
4. 3
0.2
1
· '~'1
0
-0.2
x 104
. . ... .. .. .. . .. . .
... . ... ......
0.1
-0.1
4.9
.
.........
4.4
...... ... .. ..
......
...
...............
· `''~
.........
........ .
. .......
.........
4.5
4.6
4.7
Comparison LC OBSERVED vs LC ESTIMATED
..
...
....
. ..
. .
. ..
. ..
. ..
. . . .. . .
. .
..
..
..
. .
· ·--··
........
.........
.
4.8
4.9
xo10
..
..........
0.1
0
-0.1
.
. .. . . .. .... . . .....
...........
...
.. . . . .. .. .. ...
. ...........
......... .....
-0.2
I
4. 3
4.4
....
4.5
4.6
Time (sec)
4.7
4.8
4.9
x 104
Figure 3 .7: The multipath phase error ( DEST = LCEST) estimated by the algorithm is
compared with the observed phase residual (6 0oBs =LC.OBS) computed
from the measurements.
Chapter 4
Statistical and Spectral Analysis of
estimated and observed GPS phase
errors
4.1 Introduction
The aim of the spectral and statistical analysis described in this chapter is to test
how well the retrieved multipath phase error from the Signal-to-Noise-Ratio (SNR)
agrees with the phase error observed in the GPS measurements. To this purpose
two time series, the observed phase residual (4OBs = 8sOBs) and the estimated
multipath phase error (EST = B EST), were studied and compared. This analysis
provided important information about the applicability, limits and performance of the
GPS multipath estimation method described in chapter 3. The results of this study
applied to actual GPS measurements are discussed in detail in chapter 5.
4.2 Data Preprocessing
With data preprocessing we indicate the procedure for the selection and
formatting of the GPS data before applying the analysis. The first goal of this
operation was to choose data sets containing a strong multipath error in the phase
observable.
As mentioned in chapter 2, the occurrence of multipath produces large regular
oscillations in the SNR. However, the simple analysis of the SNR might not provide
enough information for evaluating the characteristics of the data, as ionospheric
effects can also cause wave-like patterns in the signal received by the antenna. A
discussion on the errors in the GPS phase measurements can be found in section
3.2.
Because the observed phase residual, 4 0BS, does not contain ionospheric effects,
the combined analysis of SNR and oBas gives enough information for the selection of
the data. Whenever large structured oscillations were found in both SNR and 0oBs,
because in the observed phase residual the ionospheric effects are filtered out, we
could be confident in having GPS measurements affected by multipath, and the data
were considered appropriate for the analysis.
0oBs was computed by using the GAMIT GPS analysis software package
implemented at MIT (King and Bock, 1993). The data chosen for this analysis were
selected by using the CVIEW option of the program.
The data sets chosen were then divided into subsections having well defined
statistical properties and multipath interference. This operation permitted us to
isolate the phenomena without averaging out the effects of multipath over a large
number of samples.
4.3 Spectral Analysis
4.3.1
Multipath frequency
The multipath phenomenon is caused by the interference of electromagnetic
reflections with the direct electromagnetic signal transmitted by the satellite. The
level and frequency of the GPS noise deriving from multipath depends upon the
distance, geometry and reflectivity of nearby objects, the elevation angle of the
satellite and the rapidity of change of the elevation angle with respect the receiver,
and the value of the carrier frequency (section 2.2).
Because of the dependency of multipath on these parameters, each multipath
component due to a particular reflector have a distinct frequency. Therefore, the
multipath noise caused by the surrounding of a receiving GPS antenna can be
studied and identified by spectral analysis (section 3.3.3).
Spectral analysis is used here to compare the estimated multipath phase error
(~Esr)and the observed phase residual (oBs).
4.3.2 Power Spectral Density
In spectral analysis we try to represent a stationary process in terms of a sum of
sinusoids, in order to define its spectrum. We applied this analysis to the observed
and estimated phase errors, to determine their spectral content and how this was
related to the multipath phenomenon.
The two time series 1( oBs j and {ESTr J, are assumed Stationary Ergodic Random
Processes. In other words, we infer that their statistical properties do not vary with
time (Stationary) and that the average values computed over the ensemble at a time
t0 , 0x i (t) [ for x =OBS, EST), will equal the corresponding average values computed over
time for a single time history record, X(t)[ for x =OBS, EST} (Ergodic).
This important assumption allows us to perform our statistical and spectral analysis
within one time series, without the need of independent realizations of the process.
Let us consider a stationary ergodic random process:
{,t} = f{(n. At)}
for n = 1,2,3,...,N.
[4.1]
where the time resolution interval between the sample values is At , the total number
of samples is N and the total length of data analyzed is T=N At.
The Discrete Fourier Transform (DFT) of {1j is defined as:
{)K } = {(k -Af)} = Atm4(e('N •
[4.2]
for k = 1,2,3,...,N.
The Nyquist cutoff frequency of the sequence is f, = 1/(2 At ), and the minimum
frequency-resolution bandwidth is Af = 11T =1/N At.
Therefore, the Power Spectral Density of {1,} is given by:
2
[4.3]
2
N. AAt
For each couple of time series, {OBSn)} and {ESTnrj, we computed the Power Spectral
Densities Sos , SES, and SDIFF, power spectral density of their difference {ODIFF).
Each time series was previous normalized by subtracting its mean. The DFT was
computed via FFT (Fast Fourier Transform).
The Power Spectral Density was computed by using a Matlab algorithm
(FFTobsest.diff.m) which can be found in Appendix C.
4.3.3 Auto Correlation and Cross Correlation
An other way to compare f{oBsJ and {(ESr) is to compute the Auto-Correlation of
each of the two time series (Roo and REE) and the Cross-Correlation between them
(RoE)
To define Roo, REE and ROE between two time series {0OBSn) and {4
introduce the Cross-Covariance Function, CoE.
ESTn,
we first
The Cross-Covariance Function between two processes I{OBsnj and {ESTr)j for any
time delay t, is given by :
COE(r)=
[4.4]
EST (t + '))
(OoBSn -OBS)(OES(t+r) -
If we assume again that the two time series are stationary ergodic processes, their
means •ROBs and grsr are constants and independent of t. Therefore, COE can be
written as:
COE(
(O=BSn~
)(ESTn(t + )
We define the Cross-Correlation function between {JoBsJ and {•EST)
•E) =
[4.5]
OBSMEST
I oBSnX)(ESTn (t + r))
as:
[4.6]
and it follows that:
COE (
= ROE()-
[4.7]
OBSuEST
The equivalent Auto-Covariance Functions for the time series are:
1F
[4.8]
CEE()-
t(t (tES-Tn
-fr
and the Auto-Correlations for (os~s
a)
S)(OTt +
-EST
and {•--J are:
+
Roo(T) =
[
XOBSn)(OBSn(t +)
[4.9]
RE(') =
SY- ESr,(t +)
(ErXrn)(
Therefore:
Coo(T)= Roo(
) -oss
CEE() = REE()
[4.10]
2
If the mean of either measurement is zero (p=0,
=
=0 or p.,=C=0 ) itfollows that:
Coo(r)= Roo(r)
CEE() = REE(
[4.11]
)
COE( )= ROE(r)
The Matlab program implemented for computing the Auto and Cross-Correlation
of {(OBs) and {(ESTn) is called Auto-CrossCORR.m, and is presented in Appendix
C.
4.3.4 Auto and Cross Spectral Density Function
The spectral density function between two stationary ergodic random processes
can be defined as the Discrete Fourier Transform of the Correlation function
between those records:
N
SOE(k)= At
2 kn
ROEe
n=l
N
for k=1,2,3,...,N.
[4.12]
The two correspondent Auto-Spectral density functions for {oBs and {(-r})are,
therefore:
Soo (k) = At
Rooe
N
for k = 1,2,3,...,N.
n-1
[4.13]
SE(k) = At XREe-JN)
for k= 1,2,3,...,N.
n-1
The Matlab algorithm for the computation of the spectral density via correlation
function is shown in Appendix C and is called SPCORIRobs-est.m.
4.3.5 Coherence Function
The frequency relationship between {loBSn) and
computing their coherence function,
72 0E.
can be also studied by
The computation of Y2 oE provides
{IEST.)
information on what fraction of the power in 4 EST can be explained by OBS i.e.
which fraction of the total Phase Residual is due to multipath.
The results of this analysis allow us to recognize the range of frequencies for which
the two sets of data under analysis present an high correlation. In general this test is
used to verify and study the results obtained from previous Spectral Analysis.
The Coherence function computed was defined as:
2
YOE =
S0E
[4.14]
[4.14]
The algorithm implemented for the computation of the coherence function
(Coherenceobsest.m, Appendix C) divides the total time interval of the two
processes in subsections. The number of these subsections depends on the
particular data set considered. Several numbers of subsections are examined (2 to
12) and after several trials the more appropriate number is chosen. Because of the
sampling, the maximum value does not necessarily indicates the maximum of the
coherence function. The only information we can extract from the coherence
function computed in this analysis is the range of frequencies where the two time
series correlate better.
4.4 Statistical Analysis of GPS Phase Residuals
In this section we compute the statistical properties of the variance estimates of
GPS phase residuals. The procedure followed in the statistical analysis presented in
this section is analogous to the one derived in vanDam and Herring, 1994.
For each set of data, the observed phase residual from the GPS measurements,
the estimate of the phase error due to the multipath effect, and the phase residual
after correction, can be written as functions of the noises affecting the GPS
positioning. In order to assess the statistical properties of the phase error before and
after the correction for multipath, we can define:
'OBS
= WMPN + VMN
PEST =
WMPN + VEN
ODIFF = OOBS - 'PEST = VMN - VEN
[4.15]
where:
OOBs
= Phase residual observed directly from the GPS measurements
= Multipath phase error ("true" multipath phase contribution)
vMN = Noise in the measurements due to other sources different from multipath
OEsr= Multipath phase error estimated from the SNR data
WMPN
vEN =
Noise in the retrieval of the multipath phase error
PDIFF
= Phase residual after the correction for multipath
In this analysis it will be assumed that the time series under study are stationary
ergodic, with zero mean and standard deviation o.
Of all the quantities previously defined, only 0oBs, mEST
and PDIFF are known, and
the unknown phase errors vMN, VEN and
4
MPN
are assumed to be independent.
A very useful method of statistical analysis is the Chi-square test, where the Chisquare distribution (with N-1 degrees of freedom) is given by the estimates of the
variance normalized by the expected variances. However, because OBS, EST and
and the estimates of their variances are correlated, a simple Chi-square test
cannot be used to determine the statistical significance of the changes in the
variance estimates, due to the correction for the multipath phenomenon.
PDIFF
In order to perform an appropriate statistical analysis, we computed the
expectation and variance of the difference between the variance estimates of the
phase error, before and after applying the corrections for the multipath contribution.
If N is the number of samples and a the standard deviation of the time series, the
estimates of the variances of {oBs J, {4ESTn }, and {fDIFFn) are given by :
&2BS
=
'V (OBSn)
n=1
sr=$2EST
n=1
2
N-1i
n=1= N1
N-
N (VMN+MPNA
=
=1
2.
1
2
N-1
2(0MPn+EN1)
=
L
N-1
N(0
&rIF
=
ln=l
N-1
2
[4.16]
MN + ENn 2
N--1
It should be noticed that in these formulas the processes have been normalized by
subtracting their mean before computing the variances.
The expectation of the estimates of the variance can be written as:
(UE2T)1= OMpN +
[4.17]
ENM
(&2",)= acN + al.
The expectation of the difference of the variance estimates of OOBS and DIFF iS
given by :
(BS &D
=
-[4.18]
The variance of the difference in the variance estimates is :
var(BS -
F)
s
s-
DIFF
[4.19]
FF)
If we assume that:
(vNolM)
=
0 and (vM, vENi) =0
(OMPNiOMPNj)= ( vMN vMN)
= (vENi
EVj)
for each i
=0
for each i
j
then the variance of the difference in the variance estimates can be written as:
&2r2cS
=4. qFN var(OSS
&DN-1
1 + 6
[4.20]
where oMN, aEN and (MPN are unknown, and can be retrieved by inverting the
system:
(
S)=-2MN
(sr)=
+ 2PN
oPN + o
(&DIFF) =
+U
[4.21]
Chapter 5
Applying the method of multipath
estimation to GPS data
5.1 Introduction
In this chapter we discuss the results obtained from the phase error estimation
algorithm applied to GPS measurements. In particular, three GPS data sets were
chosen for this study and will be discussed in the next paragraph: the Tien-Shan
data, the LIGO data and the IAP data. Among these three collections, 42 data sets,
acquired between June 1995 and January 1997, were selected for multipath
estimation and analysis.
Although these measurements were not originally intended for studying
multipath effects on GPS measurements, they are suitable for testing the
performance of the method for their strong multipath contribution to the phase error
observable.
A brief description of the data sets selected, their characteristics and their
significance in this study are provided in section 5.2.
5.2 Description of the GPS stations and
the data sets analyzed
5.2.1 The Tien-Shan data
This data set was acquired during the 1995 -1996 Tien-Shan field project. The
original purpose of this campaign was to define a north-south deformation across
Tien-Shan and to study the interaction of the strike-slip faulting and thrust faulting
along the Talaso-Ferghana fault.
The two stations selected for the multipath analysis, KKAU and MANA, are part
of a network of approximately 95 sites. The network stretches across three of the
former Soviet Republics, a region extending from Uzbekistan east to the Chinese
border, and from Taldy-Kurgan (Kazakhstan) south to Tajikistan.
The station KKAU is located in Kazakhstan, at 41.695248" lat., 72.893497* long.,
and 1423.2 m elevation and uses a Trimble antenna as a receiving antenna.
The station MANA was located at a seismic observatory in the town of Manas, 30
km east of Talas, in Kyrgyzstan Oat. 42.493067, long. 72.497956%, elev. 1423.2 m).
The receiving antenna is a Dome Margolin antenna.
MANA was one of three permanent sites, besides POL2 and TALG. In the fall
of 1995, the receiver was moved to a new site near the town of Talas. This was a
better location for the receiver, because it was far enough from strong reflectors and
could be more easily mantained as a continuously operating site.
These two stations were chosen for the strong multipath effect present in the
phase measurements. Every time a GPS satellite was in a particular region of the sky,
a similar clear strong multipath was observed. This fact leads to the conclusion that
large reflecting surfaces were located near the receiving antennas.
However, the different frequencies of the multipath relevant to the two stations,
suggests that the reflectors were located at different distances from the receivers.
Therefore, from the comparison of the results of the statistical and spectral analysis it
was possible to gain useful information about the performance of the estimation
algorithm at different multipath frequencies and for different geometries.
KKAU and MANA use different types of antennas, Trimble and Dorne-Margolin,
respectively. The performance of the algorithm for data sets acquired with different
antennas was studied.
5.2.2 The LIGO data
This data set was chosen for the particular geometry of the surroundings of the
receiver 0036 ( fig. 5.1)
A strong multipath was observed when a satellite was at about 15' elevation and 45'
azimuth. Among the LIGO measurements two data sets, relevant to the stations
0028 and 0036, had the same high sampling rate (At = 5 sec). These two stations are
located at close distance from each other. As mentioned in chapter 3, the
ionospheric errors can be filtered using single differencing when dealing with short
baselines. Therefore, in this case, it is possible to treat LI and Lz separately as two
independent observables, rather than their linear combination LC.
Both estimations were performed for the dual-frequency phase residual, LC, and
for the frequencies LI and L2 individually.
5.2.3 The IAP data
This data set was acquired in January 1997, in California, during a two week field
work carried out as part of a course on High Accuracy GPS held at MIT. The same
measurements were repeated at the same site in two successive days, but using two
different receiving antennas: Trimble (station COSE) on January 23, 1997 and
Ashtech (station GS29) on January 24, 1997. The measurements are relevant to the
same satellites at the same time of the day, shifted by 3.5 minutes to account for the
daily change in geometry.
Moreover, the two antennas present a different gain at low elevation angles.
Because of the antennas characteristics, the multipath error is perceived differently
by the two receivers, even though the actual multipath environment is maintained
the same during the two sets of measurements.
However, it should be mentioned that the weather conditions during the second
day of measurements (performed by the Ashtech antenna) were different. On the
second day of measurements (January 24) the ground was wet, due to persistant rain.
Rain usually increases the reflection coefficient of reflecting surfaces, which can
produces an higher multipath error in the phase observable.
To study how the multipath effects can change due to weather conditions, the
estimation algorithm was tested using data acquired on two different days, one sunny
and dry, and the other gray and rainy. Comparing the measurements performed with
the same receiver in the same location under these weather conditions, yielded very
interesting results.
GPS satellite:
PRN 09
ceiver
1.5m
Fig. 5.1: Geometry of the surroundings of the receiver 0036 (LIGO data).
5.3
Multipath Estimation and Results of
Spectral Analysis.
5.3.1 KKAU and MANA
5.3.1.a) KKAU (TRIMBLE 4000SSE)
The data relevant to the GPS measurements acquired by KKAU on June 10, 1995
were analyzed. Large oscillations were observed in the SNR every time a satellite
passed through a specific region in the sky (low elevation angle and 50" azimuth).
An example of this phenomenon is shown in fig. A.1.1 and A.1.2. The SNR
presents large fluctuations when the satellite PRN 16 is at an elevation 0 = 10" - 20"
and azimuth 0 = 50" - 60", at the time t=4.5 - 5.0 10 4 sec.
The observed phase error determined from the measurement, OBss, and the
multipath phase error estimated from the SNR, 4 EST, are plotted and compared in
figures A.1.4 and A.1.7. In the plots, 0oBs and OEsr are referred to as LC OBS and
LC EST, respectively. The residual phase error given by the difference of these two
phase errors, DIFF, is plotted in fig. A.1.8.a) and is called LC DIFF. The observed
phase error shows very large values in correspondence of the structured oscillations.
The amplitude of ODIFF decreases significantly after applying the correction for
multipath.
To study the characteristics and frequency content of the three time series,
(observed, estimated and residual phase error) the Power Density Spectra (PSD) was
computed and plotted in fig. A.1.8.b). The PSD of OBs and 4 ESr are very consistent
for amplitude and frequency content. They exhibit three narrow distinct peaks at
frequencies between 0.001 Hz and 0.01 Hz, and one large peak at a lower frequency.
The frequency content of ODIFF is not as rich as it is for 4 0BS and OEsr.
This result suggests that the contribution due to the multipath effect is filtered out
and the remaining phase error, which does not present a clear spectral content, is
due to other sources of GPS random noise in the carrier phase observable. In this
particular case, where the multipath effect has a predominant role in the observed
phase error, the estimation algorithm and multipath filtering is efficient in reducing
the phase residual, as well as the error in the final GPS positioning. However, a low
frequency residual component is evident in the PSD of 4 DIFF. This low frequency
component might be caused by reflectors located around 1 - 2 m distance from the
antenna phase center of the receiving antenna. An other explanation for this low
frequency component could be the atmospheric refraction errors, that produce a very
similar pattern. Therefore, with only one example, it is difficult to judge if this low
frequency residual error after correction should be considered a limitation of the
performances of the estimation algorithm or just an error due to atmospheric effects.
The auto correlation functions of o0Bs and EST-r shown in fig. A.1.8.c) are very
similar and recall the auto correlation of wide-band random noise (the envelope
decreases very rapidly, with a first zero crossing at z=1/(2B), were B is the bandwidth).
The cross correlation function between 0oBs and OESr is nearly symmetrical and is
well centered at zero, again confirming the strong affinity of the two time series
being compared. Additionally, the cross-spectral density and the coherence
function of fig. A.1.8.d) corroborate this conclusion. The two time series show a very
high coherence function (about 0.8) for frequencies between 0.001 Hz and 0.01 Hz.
Another good example of how well the estimation algorithm works in presence of
strong multipath is shown in fig. in fig. A.2.1 and A.2.2. The SNR shows
distinguished oscillations whenever the satellite PRN 20 is observed by the receiver
at an elevation 0 = 10' - 24' and azimuth 0 = 40 ° - 80° , at the time t= 7.0 - 7.5 104 sec.
The observed phase error determined from the measurement, 0OBs, and the
multipath phase error estimated from the SNR, •EST, are plotted and compared in
figures A.2.3, A.2.4 and A.2.7 (segment 2.b). The residual phase error given by the
difference of the two phase errors, ODIFF, is depicted in fig. A.2.8.a).
In the PSD of 40Bs and E)sr (fig. A.2.8.b) two main peaks at the same frequencies
are predominant , but a significant difference in amplitude can be noticed.
The auto and cross correlations of fig. A.2.8.c) deserve special attention. The auto
correlation function of a random process can be directly interpreted as a measure of
how well future values can be predicted from past observations. The key information
is how large in time is the envelope of the auto correlation function. An envelope
which decays rapidly to zero as in fig. A.1.8.c) resembles the auto correlation ofwideband random noise. If the auto correlation decays very rapidly, the knowledge of the
exact time history from zero to time t will not significantly help us to predict future
values beyond the very near future, i.e. beyond 1/(2B) past the end of the observed
record.
A very large envelope which does not decay rapidly, but which remains almost
constant over the near future (central peak 1/(2B) ), as in fig. A.2.8.c), resembles the
auto correlation of a sin/cos wave + random noise. This suggests that it is possible to
predict future values of the data, based on past observations, assuming that the data
remain stationary. This comparison of the auto correlations of the two time series
with the auto correlations of functions equal to the sum of a sinusoid and random
noise, confirms the influence on o0ss and 4~Eof the multipath (sinusoidal in shape)
and of the contributions due to other GPS noises (random noise).
Auto and cross spectral power density and high values of the coherence function
(fig. A.2.8 d)), confirm the strong affinity of estimated multipath phase error and
observed phase error. As will be discussed in section 6.4, the multipath estimation
algorithm is very effective when multipath is the main source of phase error in the
GPS measurements, as for the receiver-satellite pair KKAU-PRN 20, segment 2.b.
This is confirmed by another interesting example relevant to KKAU -PRN 22,
segment #1. The SNR (fig. A.3.2) presents an evident sinusoidal pattern when
the satellite PRN 22 is at an elevation 0 = 100 - 30' and azimuth 4 = 50", at t=0.0 - 5.0
103 sec.
The observed phase error determined from the measurement and the
multipath phase error estimated from the SNR are presented in figures A.3.3 and
A.3.4. The residual phase error 4 DIFF is plotted in fig. A.3.5.a). For the Power
Spectral Density (fig. A.3.5.b)), the auto and cross correlation (fig. A.3.5.c)), the auto
and cross spectral density and the coherence function (fig. A.3.5.d)) were
computed. Analogous observations than those for KKAU-PRN 20 can be applied.
The examples reported so far are relevant to cases in which a strong, clear
multipath phenomenon at frequencies between 0.001 Hz and 0.01 Hz is observed in
In these cases the estimation algorithm showed good
performances, as determined by spectral analysis. But how does the estimation
algorithm work in the presence of other sources of noise or in the presence of a
lower frequency multipath? To answer this question segment #2 of the data set
the phase residual.
relevant to KKAU-PRN 22 was studied. In fig. A.3.6 the observed phase error
shows a low frequency component which is not present in the phase error estimated
from the SNR. As discussed above, this low frequency undulation might be caused
by multipath due to close reflectors or by atmospheric refraction. In fig. A.3.7 a)
4
DIFF presents a large amplitude, and a low frequency component.
The PSD of fig. A.3.7b) shows the very different frequency content of the
observed and estimated phase error.
In fig. A.3.7.c) the two auto correlation
functions show very different properties, and the correlation between the two series
presents low amplitude and evident asymmetry with respect to the zero. A very low
coherence function is found in fig. A.3.7.d) for all the frequency spectra.
5.3. .b) MANA (ROGUE SNR-8000)
When the SNR data relevant to the measurements acquired by MANA on June
10, 1995 were analyzed, large oscillations were observed every time a satellite was
found at low elevation angle and about 300" azimuth.
It appears evident in fig. A.4.1 and A.4.2 that the SNR exhibits large fluctuations
when the satellite PRN 26 is at 0 = 18" - 22* and 0 = 250* - 320', at t=1.0 - 2.0 104
sec. 0oBs and OEST are shown in figures A.4.3 and A.4.4. #DIFF, is plotted in fig.
A.4.5.a). Even though the large oscillations of 4oas are well matched by OESr, the
frequency content of the two time series is quite different. OBss presents a broader
spectra (Af=0.0003 - 0.003 Hz), while OEsr
has a sharp peak at about 0.0015 Hz,
therefore a frequency component at 0.00085 Hz is still present in
ODIFF,
as shown in
fig A.4.5.b).
The auto and cross correlations are shown in fig. A.4.5.c). The coherence function
(fig. A.4.5.d) is about 0.6 in the interval of frequency between 0.001 and 0.01 Hz.
Large oscillations of the SNR were also observed for PRN 28 (fig. A.5.2) when the
satellite was at an elevation 0 = 15" - 35" and azimuth 0 = 250" - 310", at t=4.5 - 5.3 104
sec (fig. A.5.1). OoBs and OESrare presented in figures A.5.3 and A.5.8. ODIFF is
depicted in fig. A.5.9.a). Even though the statistical analysis shows a very strong
decrease of the phase error after the correction for multipath, the frequency content
of Ooss and OEST is quite different, and a large lower frequency component is found in
vDIFF,
as observed in fig. A.5.9.b).
The auto and cross correlations of fig. A.5.9.c) suggest a predominance of random
noise on the multipath effect in the phase error. The coherence function (fig.
A.5.9.d) is close to 0.7 in the interval of frequency between 0.001 and 0.01 Hz.
5.3.1.c)
Discussion and Comparison
The previous analysis shows that the multipath frequency for KKAU is usually
higher than the one observed for MANA.
In figures A.1.8.b) , A.2.8.b) and A.3.5.b) it is evident that the multipath
frequency for KKAU is usually in the range 0.002-0.004 Hz.
For MANA, the multipath frequency is lower than for KKAU and it is in the range
0.0009 - 0.0010 Hz.
This indicates that the reflecting surface producing the
multipath in this case (MANA) is closer to the receiving antenna and its effect is
also more intense. The relationship between the distance of the reflector and the
multipath frequency is given in [2.17].
5.3.2 Stations 0036 and 0028 (TRIMBLE 4000SSE)
5.3.2.a) 0036: Dual-band processing
Some a priori information relevant to this site were available and were used to
better interpret the results of the spectral and statistical analyses. The geometry of
the surroundings of the receiver 0036 is depicted in figure 5.1, where the incoming
signal from the satellite PRN 09, located at about 15" elevation and 45' azimuth,
reaches the receiver after multiple reflections. The SNR relevant to this satellitereceiver pair is plotted in fig. A.6.4. The SNR is affected by very distinct large
oscillations when the satellite is at an elevation 0 = 14" - 22* and azimuth < = 40* - 50",
at t=7.3 - 7.5 104 sec (fig. A.6.3).
OBs and OEsr are compared in figures A.6.5 and A.6.6. The difference between
the two phase errors is plotted in fig. A.6.7.a).
The PSD of 0oBs and OEsr (fig. A.6.7.b) exhibit one narrow distinct peak at about
0.001 Hz. This information suggests that in this case the multipath is not produced
by a complex scattering from several reflectors, but from one distinct reflector
producing one distinct frequency component.
Because of the coherence of the two time series, the frequency content of ODIFF
does not present any strong predominant frequencies. This implies that the main
contribution to the phase error is due to multipath and is filtered out by the
correction. The remaining phase error is due to the other sources of GPS random
noise, which cannot be corrected with this method.
The auto correlation functions of 0oBs and OEST shown in fig. A.6.7.c) present very
interesting patterns. The envelope amplitude diminishes slowly and recalls the auto
correlation of a narrow-band random noise, where the first zero crossing is at t=1/B
(B is assumed small). The bandwidth is smaller in EsTr, causing its auto correlation
function to decay slower than the auto correlation of OoBs. It can be concluded that
Oos contains more random noise and a larger bandwidth scattering.
The cross-spectral density and the coherence function of fig. A.6.7.d) reinforce
the conclusion that a distinct frequency is dominant in the signal, with very narrow
bandwidth. The two time series show a high coherence function (about 0.7) for
frequencies between 0.001 Hz and 0.01 Hz.
Another example of strong multipath is shown in fig. A.8.5 and A.8.6. The SNR
relevant to the satellite PRN 25 shows large oscillations when the satellite was at an
elevation 0 = 15' - 20' and azimuth 4 = 164' - 168', at t= 7.5 - 7.7 104 sec. The
observed phase error determined from the measurement and the multipath phase
error estimated from the SNR are plotted and compared in figure A.8.7 (segment
oDIFF is shown in fig. A.8.8.a). In this case the oscillations are not as distinct as
#1).
for PRN 09.
The PSD of
0oBs
(fig. A.8.8.b) show two narrow peaks, respectively, at frequencies
of 0.001 Hz and 0.002 Hz. OEsr presents a narrow main peak at 0.002 Hz and a broad
band low frequency peak between 0.0001 Hz and 0.0005 Hz. As a result of this
spectral difference, the PSD of ODIFF has a predominant low frequency spectral
content.
The auto and cross correlations of fig. A.8.8.c) present an envelope rapidly
decreasing to zero, as for wide band random noise. This suggests that the multiple
phase error does not come from a single reflector, as in the previous case for PRN
09, but might be due to a more complex scattering phenomenon from close and
distant surfaces. The cross-spectral density and the coherence function of fig.
A.8.8.d) reinforce this conclusion. The two time series show a coherence function
of about 0.6 for frequencies between 0.001 Hz and 0.01 Hz.
An interesting example where there is a noticeble difference of amplitude
between 4os and 4 EST, rather than of frequency content, is given by LIGO 0036,
PRN 15. (segment #4). ODIFF is computed in fig. A.7.10.a). The observed phase
residual presents small oscillations similar to the ones found in the estimated phase
error, but with a different amplitude: 0oss is always negative while ESTr is shifted
upward and presents some positive and negative values.
The Power Spectra Density (PSD) of this segment of data is shown in fig.
A.7.10.b). The PSD of OBs and EsrT are consistent at high frequencies. They both
exhibit one peak close to 0.001 Hz.
The auto correlation functions in fig. A.7.10.c) are very different and the cross
correlation is asymmetric and small in amplitude.
5.3.2.b) 0036 and 0028: Single Differences for short-baseline LI and L2
As discussed above, for short baselines the ionospheric effects can be filtered
using single differencing. In this case it is possible to study the effect of multipath
at each of the two GPS frequencies and the performance of the estimation algorithm
for LI and L2.
In this section we show the results of the analysis of LI and Lz separately, as two
1lOBS, slEST, and
independent observables.
4 1DIFF indicate respectively, the
observed phase residual, the multipath phase error and their difference, respectively,
for LI.
OloBs and IlEST are compared in fig. A.9.1 and A.9 2. I1DIFF is shown in fig.
A.9.3.a). PSD of loBs and O'EST are presented in fig. A.9.3.b).
The spectral content of 01OBs and $IEST is very similar at higher frequencies,
showing a predominant peak at 0.002 Hz, but does not agree at lower frequencies.
Therefore, the PSD of OIDIFF shows a spectral content only at lower frequencies
(below 0.001 Hz).
The auto correlation of ~'EST (fig. A.9.3.c) recalls the one of a narrow-band
random noise. The cross correlation Roe is well centered but slightly asymmetric. A
good coherence is found at f=0.001 Hz.-0.01 Hz (fig. A.9.3.d). In this case the
frequency content matches very well, but the shift in amplitude causes (1DIFF tO stay
large also after the correction. A very different result is obtained for L2 (fig. A.9.4
and A.9 5), where the amplitude of 02DIFF is well reduced (fig. A.9.6.b) ), even
though a low coherence is observed (fig. A.9.6.d) ).
5.3.3 COSE and GS29
5.3.3.a) COSE (TRIMBLE 4000SSE)
The data considered for this analysis were acquired on January 23, 1997 using a
Trimble antenna . Any time a satellite was at a low elevation angle ( 0 = 10" - 20" )
and azimuth 0 = 50' - 70", the SNR showed large oscillations at LI and L2. An
example of the multipath phenomenon occurring in proximity of COSE is provided
in fig. A.11.1 and A.11.2.
The satellite PRN 25 passed through the critical region in the field of view of
COSE at t=5.6 - 6.2 104 sec. OBas and ES~rare plotted and compared in figure A.11.3
(segment #3). ODIFF iScomputed in fig. A.11.6.a).
The observed phase error presents only one very large oscillation at the end of the
segment, probably due to reflections from the ground because the antenna was
located on a large slope. The satellite was setting and this strong reflection can
therefore take place at a very low elevation angle (0= 10").
The Power Spectral Density of this segment of data is shown in fig. A.11.6.b).
The PSD of 4OBs and OEST are very consistent for amplitude and frequency content.
They both exhibit large peaks at the same frequency, about 0.001 Hz, but with very
different amplitudes. This is the reason why ODIFF still carries a considerable
frequency content. In this particular case, it is not the frequency content, but the
amplitude of the multipath phase error estimated by the algorithm that produced the
critical result. An overestimation of the amplitude of the oscillation increases the
total phase error after the multipath correction.
The auto correlation functions of Oos and EST shown in fig. A.11.6.c) recall the
auto correlation of a wide-band random noise. The cross-spectral density of fig.
A.11.8.d) confirm the previous results of the spectral analysis. The two time series
show a very high coherence function (about 0.8) for frequencies between 0.0005 Hz
and 0.01 Hz.
An interesting case to study is COSE PRN 01, segment lb., In chapter 3 the
technique for the multipath estimation was described. When the fit of the multipath
model and the SNR curve is performed, some problems can rise, due to the method
used for the fit (section 3.3.3). ODIFF presents sudden changes of sign deriving from
(fig. A.9.6.a)). Correlation and spectral analysis show a poor agreement between
mESr
O0sS
and OEsr.
5.3.3.b) GS29 (ASHTECH Z-XII3)
The same set of GPS measurements discussed in section 5.3.3.a) were repeated
by using an Ashtech antenna, placed at the same location of COSE, the day after (on
January 24, 1997). As for COSE, the satellite PRN 25 was considered, and the
analysis on the station GS29 was performed (fig. A.14.1 and A.14.2 )
O0Bs and OEsr are shown in figure A.14.3 and A.14.4 (segment #3). ODIFF is
computed in fig. A.14.7.a). The observed phase error presents only one very large
oscillation at the end of the segment, as was observed for COSE.
The PSD of this segment of data is shown in fig. A.14.7.b). The PSD of o0s and
mEST
are very consistent for amplitude and frequency content. They both exhibit one
large peak of similar amplitude at about 0.001 Hz. DIFF does not present a
considerable frequency content, because in this case the frequency content and the
amplitude of the multipath phase error estimated by the algorithm match very well
the observed phase error. Therefore, as will be discussed in the statistical analysis,
the application of the algorithm produces a strong decrease of the phase error after
the multipath correction.
The auto correlation functions of 04s and Omr presented in fig. A.14.7.c) is
similar to those found for wide-band random noise. The cross-spectral density of fig.
A.14.7.d) confirms the previous results of the spectral analysis. The two time series
show a very high coherence function (about 0.8) for frequencies close to 0.002 Hz.
In the conclusion of section 5.3.3.a) and 5.3.3.b), it was found that the estimation
algorithm applied to data acquired at the same location and the same time of the day
with two different antennas presented different performances. The algorithm seems
to work better for Ashtech antennas. However, because the weather conditions
during the two successive days were not the same, it is difficult to assess if the results
are actually due to the antenna type, or to the higher multipath effect observed on rainy
days. We already noticed that the multipath estimation algorithm is very efficient
when strong multipath is observed, especially if it is predominant with respect to other
kinds of noise. It is therefore possible that a different wetness of the ground,
producing a higher reflectivity of the reflecting surfaces, might have caused a better
performance of the algorithm. One way to answer this question is to consider two sets
of measurements acquired with the same antenna, during these two days of
observations and compare the results.
COSO(Ashtech).
This is done in the next section, for
5.3.3.c) COSO (ASHTECH Z-XII3)
For this analysis we selected the measurements relevant to the satellite PRN 30,
because of the large oscillations observed in SNR and 0oBs (fig. A.15.1 and A.15.2).
The phase residual DIFF is shown in fig. A.15.3.a). The PSD of o0Bs and rESr (fig.
A.15.3.b) ) present the same peaks at 0.0005 Hz, but 90Bs also has peaks at lower
and higher frequencies (0.0001 and 0.001), which are found again in the PSD of
'DIFF.
The correlation of the two series (fig. A.15.3.c) ) as well as the coherence (fig.
A.15.3.d) ) is very poor.
The same measurement was repeated the next day, in weather conditions favorable
for strong multipath (rainy day). The phase residuals relevant to these measurements
are plotted in fig. A.16.1 and A.16.2. The PSD of 0 Bs and 4ESr (fig. A.16.3.b) )
present a very similar spectral content, showing a predominant peak at 0.0004 Hz.
The correlation (fig. A.16.3.c) ) and coherence (fig. A.16.3.d) ) are higher than in
figures A.16.3.c) ) and (fig. A.16.3.d) ). This result confirms that the wetness of the
ground increases the multipath effects, and therefore the performance of the
estimation algorithm improves.
5.4
Multipath Estimation and Results of the
Statistical Analysis
The following is a summary of the results obtained from the statistical analysis of
the GPS phase residuals, described in Chapter 4. The study focused on 6 GPS
stations, MANA, KKAU, 0028, 0036, COSE and GS29, because of their strong
multipath environment. The analysis includes 42 sets of data, collected and sorted
in tables B.1.a) and b) through B.5.a) and b).
The variance of the observed phase residual is referred to as
GOBS 2
, and denotes
the total phase error due to the combined effects of different sources of noise for
the GPS carrier phase observable. ausr 2 is the variance of the estimate of the
multipath phase error retrieved by using the SNR information. The difference
between oBs 2 and GOr 2 is the corrected phase error, and its variance is denoted
with ODIFF2 . The difference of variance of the phase error before and after the
correction for multipath was computed along with the statistical uncertainty of the
estimate of the change. This computation allows us to determine how well the
correction improves the GPS positioning by quantitative means.
Whenever a negative value is obtained for (GooBs 2 - GDInr 2), the correction for the
multipath phase error does not reduce the variance of the total phase error. A
positive value, on the other hand, would imply a decrease of the variance of the GPS
phase error, and, thus, an improvement in the final GPS positioning. When such a
positive value is obtained, the percentage decrease of variance was computed, to
make it easier to quantify the improvement of the variance after correction.
There is a noticeable decrease of the variance after the multipath correction any
time the variance of the "true" multipath noise oMPN 2 is higher or at least comparable
to the variance of the GPS measurement noise, oMN2, due to other sources of noise
different from multipath. In other words, the algorithm for multipath phase error
estimation shows very good performance whenever the multipath effect is the main
source of phase error. Good examples of this observation are given in MANA 26 lb
and MANA 28 2a (Table B.l.a and b), KKAU 16 2c (Table B.2.a and b), LIGO
0028-0038 09 1 (Table B.3.a and b) and GS29 25 3 (Table B.5.a and b). In these
cases the decrease of variance of the phase error after correction is more than 35%.
The value yEN 2 is a measure of the error due to the estimation process and gives
information about the accuracy of the procedure for the phase error estimation.
The examples previously discussed present a small GEN2 value with respect to the
multipath noise variance oMPN2. This result confirms that the estimation algorithm is
most accurate when a strong multipath is present.
However, an exception to the previous cases is COSE 01 ic (Table B.4.a and b), a
measurement performed with a Trimble antenna. Even though the multipath noise
variance is more than 3 times larger than aMN
2, the multipath estimation noise is of
2
the same order of magnitude as oMPN and the decrease of variance after correction
is only 4%. This measurement was repeated the following day , in rainy weather,
exactly at the same location, using an Ashtech antenna (GS29 01). It turns out that
OMN 2 drops to 1/10 the value of ouCPN 2 and the decrease of variance after correction
improves to 16 % (Table B.5.a and b).
Good examples of poor multipath in an otherwise very noisy measurement
( oMPN2 << MN 2 ) are MANA 26 2c (Table B.l.a and b) , KKAU 22 2 (Table B.2.a
and b) and GS29 25 1 (Table B.5.a and b). In these cases there is no decrease of
variance after correction and the estimation noise variance aEN 2 is larger than the
multipath noise variance ouPN2 itself.
Chapter 6
Conclusions
The algorithm for the estimation of the GPS multipath phase error presented in
this thesis was applied and analyzed, to test its performance and limits. The
estimated multipath phase error and the observed phase residual were compared by
spectral and statistical analysis.
In most of the cases analyzed, a broad-band coherence function between the
observed phase error and the estimated multipath phase error was found in presence
of strong multipath. This suggests that the effects of multiple reflectors are correctly
estimated by the proposed algorithm. In other words the technique of estimating
each multipath phase error iteratively from the spectral analysis of the SNR and of
adding each contribution to obtain the composite value, produces a correct
estimation of the total multipath phase error.
In particular, the set of data which show the highest coherence (MANA and
KKAU and LIGO 0036) have multipath frequencies between 0.001 and 0.01 Hz.
Because the multipath frequency depends on the distance of the receiving antenna
from the reflecting surface, knowing the elevation angle of the satellite and its
variation with time, we can derive the distance of the reflecting object. The spectral
analysis suggests that the estimation algorithm presents good performances for
reflectors located at distances between 5.50 m and 55.0 m from the receiving
antenna, for an elevation angle of 10* and a rate of change of the elevation angle of
0.1 mrad/sec. The high coherence is confirmed by the statistical analysis, which
shows a significant decrease of the RMS of the phase residual after the multipath
correction.
When the multipath effect is due to reflections from the ground (about 1.5 - 2.0
m) the multipath frequencies are lower (COSE and GS29). At lower frequencies it
was found that the algorithm performed better for data sets acquired in the same
location by an Ashtech receiver (GS29, January 24,1997), rather than a Trimble
(COSE, January 23,1997). This result can be due to the different receiver types, the
different antenna gains or/and the weather conditions.
As explained in section 2.3, Trimble and Ashtech receivers use different
techniques for the signal processing of L2 in presence of antispoofing. Trimble
receivers cross correlate L2 with LI and the available SNR limits the bandwidth of
the L2 carrier. A narrower bandwidth corresponds to a longer integration time,
which makes it more difficult to track large oscillations of the SNR, especially at low
frequencies. Ashtech receivers use the Z-tracking technique, which provides a
better SNR compared to other techniques.
Because the estimation algorithm
proposed in this thesis manipulates the SNR information to retrieve the multipath
phase error, the errors introduced by each receiver in the determination of the SNR
strongly influence the performance of the algorithm.
GPS measurements acquired by the station COSO in the same days, one sunny
and dry (January, 23 1997), the other rainy and gray Ganuary, 24 1997), suggests that
also the weather conditions considerably change the performance of the algorithm.
A higher water content in the ground increases its reflection coefficient, causing the
multipath effects to be stronger in rainy more than in dry weather. On the other
hand, the performance of the algorithm is good when strong multipath is observed,
because the main contribution to the observed phase residual is due to the multipath
phase error. In other words, when multipath is the main error contribution, the
estimated multipath phase error is highly correlated to the total observed phase
residual.
Low frequency oscillations of the SNR might also be produced by the ionosphere
(scintillation) but these effects are likely to be small. Low frequency SNR variations
are most likely to be due to the gain pattern of the antenna, especially for the
Trimble antennas. These antennas are known to have azimuthally dependent gain
patterns, which are not accounted for in the estimation algorithm. The algorithm
might interpret these oscillations as multipath, and, therefore, estimate a false phase
error.
We already pointed out that different antennas might influence the performances
of the algorithm. The Dorne-Margolin antennas have a higher gain pattern at very
low elevation angles with respect Trimble antennas: this suggests that more
multipath contributions from lower elevation angles are received with a higher
amplitude.
The study of short baselines (LIGO 0036-0028, PRN 09) allows as to consider the
single differences for LI and L2 separately: the results of the spectral analysis show a
high broad-band coherence for the multipath phase error at LI but a very low
coherence at L2. However, in both cases the multipath estimation produces a
decrease of the RMS of the total phase error after correction, as it is shown by the
results of the statistical analysis. The best performance in terms of both, spectral
and statistical analysis was obtained using the estimation algorithm for the dualfrequency mode (LCEST). Because the square root of the PSD (power spectral
density) of the observed phase error is almost equal to the sum of the square root of
the estimated phase error and the difference between observed and estimated for
the most of the cases, we can deduce that the two curves we are comparing
(LCLOBS and LC_EST) are in phase, and there is only a difference in amplitude
between them. This is consistent with having a very high correlation function and
with L2 having the correct phase but the wrong amplitude.
One problem of the estimation algorithm which is important to mention is related
to the sign of the phase error. As it was pointed out in chapter 2, the same SNR can
be due to opposite values of the phase error, (positive and negative). Therefore, the
sign of the phase error cannot be uniquely determined. In some cases this
indetermination might cause an abrupt change of the sign of the estimated phase
error (COSE PRN 01, segment lb), producing a considerable degradation of the
performances of the estimation procedure.
In summary, the proposed algorithm shows good results when multipath is the
main contribution to the total phase error and it frequency is in the range 0.001-0.01
Hz. At lower frequencies, the performance depends on the receiver-antenna type
and the weather conditions. The sign of the phase error and the nature of the low
frequency oscillations are critical for the results of the estimation.
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Appendix A
90 90
60
.
1
., 30
21
240
300
270
GPS DATA: receiver location: CENTRAL ASIA
station: KKAU
satellite: PRN 16
acquisition date: 10 6 1995
station: KKAU
satellite: PRN 16
40C
30C
200 I·I,
-
, !
100
0
0
Time (sec)
Fig. A.1.1 :
x 104
1
2
3
Time (sec)
Satellite visibility chart for PRN 16 (skymap).
Elevation and azimuth of PRN 16 with respect to
the station KKAU.
90
4
5
x 10'
KKAU
80
_I
·
·
I
I
1.5
2
PRN 16
10 6 1995
·
·
I
r
I
1
I
I
I
4
4.5
-c60
........
o40
_ 20
---
Uj
..
· ·----.-
g
I
0.5
0
1
2.5
3
3.5
5
x 10
Signal to Noise Ratio for L1 and L2
·
I
I
20
,.
.
.....·........
.·.....
!.........
· ·...
- .·. . .......
I
I
I
I
t
.
~
I
I
I
~
I
I
|
|
·
·
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
I
0
Time (sec)
v n4
A
Fig. A.1.2 :
Station: KKAU - Satellite: PRN 16 Data acquisition date 10 6 1995.
Signal-to-Noise-Ratio (SNR) for Li and L2.
III,
GPS "OBSERVED" Phase Residuals for: KKAU PRN 16 - ASIA 1995 6 10
I
I-
0.2
0.1
0
-0.1
-0.2
-
.
|
-.
...
...
-..
.
.''
..".""_.
........
......
...
.;
..
.
......
..
!.
..
.
..
.
.
.•
:I
ii•'
i
........
..
.....-...
.:
::.
.......
.
,....
..
...
.....
.....
......
....
.:
.-..
.
.•
.. I
-
:
:
-III
-
.
" "1"
..."
...
. ... ."
...................
..................
.
•;·
-
I
"
I·--
·
·-
· I·
·
·
· : · I·--
·
· I ..-
-
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation x10
0.2
0.1
0
-0.1
-0.2
........... .......... .........
........................ ...
.... . ...
0.5
0.2 .....
0.1
0
-0.1
-0.2 r- .....
1
..
1.5
2
2.5
3
3.5
Comparison LC OBSERVED vs LC ESTIMATED
................. . . .......................
I--r
1: ......
~·. ,.....
· ..................
- ...
i
..............
ii,
0.5
Fig. A.1.3 :
i
1
..........
i
1.5
..........
..........
i
2
...
4
4.5
x10
......
....
..
I
...........
........
.........
;
;
;
;
2.5
Time (sec)
3
3.5
4
......
;
4.5
Station: KKAU - Satellite: PRN 16
Data acquisition date 10 6 1995
The observed phase residual (LCOBS) is compared with the
multipath phase error estimated from the SNR (LC..EST).
Unit: [cycles).
92
5
I
(
5
x 104
KKAU PRN 16 - ASIA 1995 6 10 Reduced set of data
3
3.2
3.4
3.6
3.8
4
Time (sec)
4.2
4.4
F'
4.6
4.8
5
x 10'
.
0.1
0
-0.1
-0.2
-
3
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
Time (sec)
Fig. A.1.4 :
Station: KKAU- Satellite: PRN 16 -
Data acquisition date 10 6 1995
LCOBS and LC..EST compared for t=3.0 - 5.0 10 4 sec
Unit: [cycles].
5
x 10
"RESIDUAL" = difference LC Observed - LC Estimated
: segment 2.a)
0.2
0.1
-0.1
-0.2
3.1
3.15
3.2
3.25
3.3
3.35
3.4
3.45
3.5
3.55
o = LC OBSERVED
,,
3.6
x 10
Time (sec)
* = LC ESTIMATED
U.z
-0.1
-0.2
3.1
3.15
3.2
3.25
3.3
3.35
3.4
3.45
3.5
3.55
Time (sec)
Fig. A.1.5.a)
:
KKAU - PRN 16 (10 6 1995) - [segment 2a]
Phase residual LC.DIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
3.6
x 104
•0
0
o
v
I
b
........
-A
......
. ...
.•°...
I3
! . .
o ...
• ..
CJ
PSD of LC DIFF
o>
4
o)
C.
;.
.
·..
.
.. I . .
....
;. ..
......
....
. .. ..
.... .......
• , ,,•,.
...{..... •
...
• ...
.ii
......
........ . .
C0
CDto
C.
=i
CA)
0
LC DIFF
0
I
7•
o
C
I
,
-L
rO
G)
PSD of LC EST
LC EST
4
N)
oa
7•
-o
_
CA)
C',
b)
C03
CA)
-T
a
I
: :
C>
LC OBS
Qo
--
N
CA)
PSD of LC OBS
CA,
.....
.....
•.,......
CA)
CA)
io
C
I
.
4.1
ioo-
KKAU PRN 16 ASIA 1995 6 10 segment 2a
U
m
...
I
w
C)
-.J
o
0I
C)
2000
-2000
-4000
-6000
4000
6000
CROSS-CORRELATION between LC OBS and LC EST
·
C13
. .. . . . . .......
.
. ...
........
·
..I............
I.0
c,,
o
o0
I
C,,
o.
o
-0.5
.. . .. . .. .... .. . .. . .
. .........
°•.•..•.°.
..................
..
. ..
w
1I
-6000
-4000
2000
-2000
4000
time delay: Tau (sec)
Fig. A.1.5.c)
:
KKAU - PRN 16 (10 6 1995) - [segment 2a]
Auto Correlation of LC_OBS (Roo) and LCEST (Ree)
Cross Correlation between them (Roe)
Unit: (cycles]2
6000
CD
-.
. . .... . . .
r
o
,,
L.
-L
".."
.......
...
.,
. ...
....
·........ ·
o
w
a
o
i
(D
.
o
o
o
..................
.
""..
................
om
o
0.
Soo(f) and S ee(f)
0"-
.... ......
...........
..
.
;...........
.
...
=
•.•
........ :...........: .......... ;
LC
'-Ia
CA)
01
C0
..- CA)
=1
c)
o
LC OBS / LC EST
0i
.. .. ..
................................
:: : i
. ... ... .. . .. : ..
............... ,,., ·....
...
,...,
................................
· · · · I
........
..........,....-
....
•oo o.o°.
:..........
o o
.. ....
..........
.......... ...........
....
...................
. .. .-..........
.
.......
-.
..
....................
...
%o.o.o.o.
,
.
.ooo.
oo
....
. ...
... . .ooo.
.
..
......
.........
. .
. .o.o.o.oo.
.
. .......
o
o!
ul
AMPLITUDE of SO(f)
.
.
...
...
...
bo
.......
.
..........
.
.............
. ........ . ............ ............
C
......... . .
bo:
..................
N O
C:)
< ......................
3:
CD
0
:3
;)
I
-I..
,.C
Coherence
"RESIDUAL" = difference LC Observed - LC Estimated
: segment 2.b)
A, ..
U.
0.05
-0.0%
_1n1
3.6
3.7
3.8
3.9
4
Time (sec)
o = LC OBSERVED
-
i3.6
3.7
Fig. A.1.6.a)
3.8
:
3.9
4.1
4.2
4.3
4.4
x 10
* = LC ESTIMATED
4
Time (sec)
4.1
4.2
4.3
KKAU - PRN 16 (10 6 1995) - [segment 2b]
Phase residual LCQDIFF, determined as the difference
between LCOBS and LCEST.
Unit: (cycles]
4.4
x 10
......
•
.....
......
::::::**::::::*::::::
i
cU
.,,ol
:
•
.
*Sd
:-4
.
O•
-q
'I
|7
I
I.p
U
o
.......
......
...
...... .
....
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i
'
0 :
CCo
CC..
Sii
$80 0 11o
I
...... . .......
.......
.....
...... . .......
CD
s
•
....
... : . ,,,"
• ,L.
" ;. " ..
I
•1
:
o,,
..... .....
|.
,
,
...... ? .....
.
} ......
....... : .....
....... • ...... }......
......
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i
.. .'•
. **
.° .#..
" ***
***
***
**
.. . . .....
l j
T1
C)
0*
.
.....
...
..
o
* ,
.. . .*. .,.
.
......
..'.*
.......
....
...
...... ....... . ......
.
,, ,. ....
,
.
: :1 :: 1
. . .
. -.
.
...
,, . :....,,
I::::: :.:
N
I
'
CD
C6m
oT
******···
...... ......... ......
!
I
r0
1110 017
O
C)
JJ1 01107
lo Sd
q
...... , . . ........
*...,.o*.o,.
*....,*
**********
CN
0d
co
***
***
***
****
***
...
*
******
****
***********
****
-
7o C
IS3 0110 aSd
S
**
**
***
*I**
*
...... .
D
IS3 01
U
C)
E
F:
KKAU PRN 16 ASIA 1995 6 10 segment 2b
CI,
O
o_1
o
o0
0
,,,.
-8000
-6000
I
-1.
-- g
-8000UU -6000
-4000
0
-2000
I
2000
I
-4000
.
.
2000
-2000
4000
.
.
6000
8000
I
I
4000
6000
8000
6000
8000
CROSS-CORRELATION between LC OBS and LC EST
-8000
-6000
-4000
-2000
0
2000
4000
time delay :Tau (sec)
Fig. A.1.6.c)
KKAU - PRN 16 (10 6 1995) - [segment 2b]
Auto Correlation of LC_OBS (Roo) and LC_EST (R,)
Cross Correlation between them (Ro)
Unit: [cycles] 2
100
o
.. ... .
..........
..
.....
....
.
. ..........
........
......
........;.......
:........
~.......
...................
..*.... *.*.. .
s.
.....--···. * *
..
-... ......
i.......... .......
.............
,
.,,2..
. .... . ......
...
2..s...·
·
· ·
·.
0
0
0
·
·
·
·
·
·
$9,
I
3r
I
d
c
~:~~~...=..........,..... ....
......
.....
........
0
o
0
I
d
()•S pue (lWOS
.5.·r·····.
.5.'"
-9.--·r
o
IS3 01/880 01
o
.
.....
.......
..
..,..,,,
..........
*
........
......
,..
,,,, ....
*
e
C
I
,-
,
:..0,o
..
*
o
... . .. .. ...
*
o, oo
*
o
•.....
,....,
o
...............
........
*
*
**
.o,.
o
o
O
o0
o
.. .** ........
* *
**
.. .. ... ... .. .. . ... ...
• I....... I ..,....
.......
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.
...
*
*
a
--------
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.o.o..
#*
C)
o
o
**
o
..........
............
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(i) S10 30nll~d•V~
......... 2*
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·.8oo~..
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............
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***
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o
I
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|
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o
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: : * ***?*********
****
··~,.....,.........,
:::::::
..........
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. ...
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......
*..............
..........
.
..........
%......
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o.,o·,o
,.
,,,,o,o%
,o,
IICI ::::::-:::::::::::::
*:::::.
::: :
III
0
E C')
-
0w
w
8
o
0
o
o)
0
4)
,0
C,
p4
U)
uU)v
KKAU PRN 16 - ASIA 1995 6 10 - Segment 2.c)
I
I
I
·
·
r
-
-
L
I
I
I
I
I
I
I
I
4.4
4.45
4.5
4.55
4.6
4.7
4.75
4.8
0.2
0.1
C,)
m
0
0
6o
S-0.1
-0.2
4.65
Time (sec)
0.2
4.85
x 104
-I
IL
w
Cl
0
\./ I
0
S-0.1
.....
...
....... .......... .
......
........
. . ...
.....
. ." I...
.i
-0.2
4.4
4.45
Fig. A.1.7 :
4.5
4.55
4.6
4.65
Time (sec)
4.7
4.75
!
!
4.8
4.85
Station: KKAU - Satellite: PRN 16 Data acquisition date 10 6 1995
LC.OBS and LC.EST compared for t=4.40 - 4.85 10 4 sec
Unit: (cycles}
102
x 104
"RESIDUAL" = difference LC Observed - LC Estimated
4.4
4.45
4.5
4.55
4.6
4.65
Time (sec)
o = LC OBSERVED
ý 0
4.4
4.45
Fig. A.1.8.a)
4.5
:
4.55
4.6
4.7
: segment 2.c)
4.75
4.8
x
104
*= LC ESTIMATED
4.65
Time (sec)
4.7
4.75
4.8
KKAU - PRN 16 (10 6 1995) - [segment 2c]
Phase residual LCDIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
103
4.85
4.85
x 104
Power Spectra Density: Segment 2c
KKAU PRN 16: Phase Error
:
0.1
·-
0
..
I..
.
,;;
:i:
. .
.
•
0.5
-0.1
r
S i~i::;::::
:::i! i :1 li"Iiii.
···
.
.
.
..;f.
,...°
I\
:"...
·
4.4
.
·
4.6
4.7
4.8
4.9
10-4
..
.
t
•
~·
"
'
··
··
··
··
··
·
·
Fig. A.1.8.b)
:
I
4.9
10 -4
.
.
• ... 2·~ .
.
.
o
.
•
•
°
.
.
.
":::
:··
···-·---
10-2
-1
10
.
. .
. .
• .
o
.
..
•..
.
.
.
°
.
°.
.
.
....
.
.
--
10-3
4
x 10
KKAU - PRN 16 (10 6 1995) - [segment 2c]
PSD of LCOBS, LCEST and LC_.DIFF
Unit: [cyclesl 2
104
:;; "
i ·I·~ B·1--·-
•
•
ril'
'
R
.'
•
o •
°
•·
•· • ° ,• o .
•· • .• •°°
•
° •
•··
• ° ° ,•
,
•
-1
10
· · ·· · ·
· · ·· ·· ·
; i
•·
•·
4.8
-.
·
·
10
•······ .
_
10-2
······::liii~\
0
4.6
4.7
Time [sec]
: '
10-
: ::
4.5
.
...
L,#
,.; L
" · \·········;Z·
: :;;:5;
4.4
.
· ____
I.
-V..
.
"
_
|
4.5
.
:. :€,:
:" "\ :•i ::::
. ....
n 93
.
..
10-2
10-1
0
k
0
0
seament 2c
KKAU PRN 16 ASIA 1995 6 10
cn
co
0
0
0o
I
-5000
-4000 -3000
-2000
-1000
II
I
0
1000
i
2000
3000
4000
5000
I
-I
1
I
___
i
I
I
I
I
0
1000
2000
3000
4000
I
-0uuu
-4000
-3000
-2000
-1000
5000
CROSS-CORRELATION between LC OBS and LC EST
o.5
S...................
.....................
r.
0
0
o
.......
2 -0.5
-
.
. .
o •
,Q
I
I
I1
-5000
-4000
Fig. A.1.8.c)
-3000
:
-2000
|
I
-1000
0
1000
time delay :Tau (sec)
2000
3000
4000
KKAU - PRN 16 (10 6 1995) - [segment 2c]
Auto Correlation of LC.OBS (Roo) and LC_.EST (Ree)
Cross Correlation between them (Roe)
Unit: [cycles]2
105
5000
2
..
0
0
CP
o
o
o0
w
*c
O
re
d
-C
0
I.
'
-LC
.
b.
o
°
. . .....
.. .,
....
. .
0o.
... • ...... :......
......
:. ............
.J
. .
..
o......,
° ....
.
.1.
~~..
...
,......•
.
oto3
...
..... ..
. .....
.... .
..........
..... ;.
..
0
oC
......
00
0
o. ,... ..
00)
a
AMPLITUDE of Soe(f)
...I ... ...... i ......
i.. ... ........
i ......
... .... ....
....
......
......
*..
* ....
. ......
.......
:......
•......
*,, .. ,
,,.
.•,
o,... •......
.......: .....
.°
,
... . ....
.. ..
..... . ..............
:
...................
... . .
. .. .. . .. ..... %...... :.......
o
Coherence
0,
v,
cD
u
b
SC:)
bo
bi
46
•
.
o
0
-&
o
Oo
O
::
. .1
.
.,,
::::: ::::::: :::::::
-.
•
I'
.......
....
*., =,
, u,,
,
. •..,1•,,,,o,,
••
• ,..
I•
| ·
•
o, ,
·. ,
1'''''$'''''$'
~~ ~ ~ ~ ~
.......... ....
......
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o
o
. .
... ....
" "."-.
" ' "-'.""
'""'' " "''''W'''''"'
~·~.f
Soo(f) and See)
I
o
LC OBS / LC EST
. . .....
. . . . .....
. . . . .-r .· .· ..
·
... ..:
.
I
o
90 90
12
-
45..
60
30
1 8....:-..
.i...
; ...
GPS DATA:
station: KKAU
300
270
receiver location: CENTRAL ASIA
acquisition date: 10 6 1995
satellite: PRN 20
station: KKAU
satellite: PRN 20
400
300
200
100
2
4
-........
.........:...........
0
6
Time (sec)
. ... .
Time (sec)
x 10'
Fig. A.2.1 : Satellite visibility chart for PRN 20 (skymap).
Elevation and azimuth of PRN 20 with respect to
the station KKAU
107
x 10'
KKAU
601~
I
'
PRN 20 106 1995
_
I
-
o
c
> 2C
0)
Ej
I
I
i
I
I
I
I
i
i
I
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
x 10
6
6.5
7
7.5
x 10
Signal to Noise Ratio for L1 and L2
i
ci 2
z
CI,
Cl
a:
z
u,
2.5
3
3.5
4
4.5
5
Time (sec)
5.5
Fig. A.2.2 : Station: KKAU - Satellite: PRN 20
Data acquisition date 10 6 1995
Signal-to-Noise-Ratio (SNR) for LI and L2.
108
GPS "OBSERVED" Phase Residuals for: KKAU PRN 20 - ASIA 1995 6 10
I·
.
....
I
I
1
--
-- I---
I
I
I
...............
•o•r.-
-- v=.•
3
3.5
4
3
3.5
4
3
3.5
4
4.5
_~_
_
4.5-- I
4.5
^
I
I·
·
I·
·
5
5.5
6
6.5
7
7.5
5
5.5
-. -I~rrI
6.5
7
7.5
6.5
7
5
5.5
Time (sec)
^ r- ^e
6I&
6
A W-r r
i
Fig. A.2.3 : Station: KKAU - Satellite: PRN 20
Data acquisition date 10 6 1995
The observed phase residual (LCOBS) is compared with the
multipath phase error estimated from the SNR (LC.EST).
Unit: [cycles]
109
i
i
·
x
7.5
104
KKAU PRN 20 - ASIA 10 6 1995: Reduced set of data
0.2
I
I
I
I
I
I
6.6
6.7
6.8
6.9
7
Time (sec)
7.1
0.1
1
-0.2
0,3
-
-
-
6.5
*I
0.15
SI
7.2
j
I
7.3
7.4
·
·
A1.
0.05
!..
... 1.
..... i!
.....
..........
0.1
:
II
. ........-
-0.05
.•!.. . ·-..-- -. . i ;.
t- ..· ·. ·.
-
-0.1
n0
I
; .i'iIl
i';
;I' I : I
-0.15
.~...
'11
-
7.5
x 104
.J
".I
-
6.5
6.5
I
6.6
I
6.7
-
r
6.8
6.9
7
Time (sec)
7.1
72
7.3
Fig. A.2.4 : Station: KKAU - Satellite: PRN 20
Data acquisition date 10 6 1995
LCOBS and LC_EST compared for t=6.5 - 7.5 10 4 sec
Unit: (cycles]
110
7.4
7.5
104
x
KKAU PRN 20 - ASIA 10 6 1995 : Segment 2.a
0.2
_
1
1
1
I
6.75
Time (sec)
6.8
6.85
6.9
6.95
6.75
Time (sec)
6.8
0.1
0
-0.1
F
-
02
6.5
6.55
6.6
6.65
6.7
7
x 10
0.2
-0.1
-n 9
6.5
6.55
6.6
6.65
6.7
6.85
6.9
Fig. A.2.5 : Station: KKAU- Satellite: PRN 20 - [segment 2a]
Data acquisition date 10 6 1995
LCOBS and LC_..EST compared for t=6.5 - 7.0 10 4 sec
Unit: [cycles]
111
6.95
7
x 10
"RESIDUAL" = difference LC Observed - LC Estimated
,,
: segment 2.a
U.1
0.05
u.
0
-0.05
-0.1
nf 1
6.55
6.5
6.6
6.65
6.7
6.75
6.8
6.85
6.9
6.95
6.9
6.95
Time (sec)
o = LC OBSERVED
6.5
6.55
Fig. A.2.6.a)
6.6
6.65
6.7
7
* = LC ESTIMATED
6.75
Time (sec)
6.8
6.85
KKAU - PRN 20 (10 6 1995) - [segment 2a]
Phase residual LCDIFF, determined as the difference
between LC..OBS and LCEST.
Unit: (cycles]
112
x 10
7
x 104
t
.
*•**
..
***•...
.
.......
•, .. ...
•... t...r
... .
:::::::::
. .• ..
.....
* ****.
.. . . .
...
.
::::
"1JOOSd
... :...:....
. . ...
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.".*.
,.*. .. .... *...
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uCo
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. . ..........
jN
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I
c6
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C .....
c,.,....
=
I
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C
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•
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a
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•
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c
1
. .. . ..
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...:.
.:::.:.:.::::.:::.:..
1...:..1
.:
....
:...-•....
i
o
JJ1 07 l0 GSd
oC)
IS3 01 o OSd
1=la 101
1
IS3 0"
KKAU PRN 20 ASIA 1995 6 10
cl)
seament 2a
0
0 0
0
I
I
I
I
I
I
I
I
I
0
I
-5000
-3000
I
I
_I
-I
-4000
r~AA
I
I
IAAA
-:UUU
I
I
~AAA
-4UUU
-2000
~AAA
-jUUV
-iUUU
0
-1000
I
I
-1UUU
i
i
"
1000
2000
3000
4000
I
I
I
I
I
I
!
I
I
I
0
1000
2000
3000
4000
5000
t
5000
CROSS-CORRELATION between LC OBS and LC EST
I .
..... . .. .
I
o•.
0.5
2
. ...
....
. .....
...
..
...
...
0
t,,
o
In
U)
0
C')
-0.5
...........
2···..
.. ·
I
I
-5000
-4000
-3000
Fig. A.2.6.c)
-2000
-1000
0
100 0
time delay : Tau (sec)
|
2000
I
I
3000
• m
KKAU - PRN 20 (10 6 1995) - [segment 2a]
Auto Correlation of LC.OBS (Roo) and LC_.EST (Ree)
Cross Correlation between them (Roe)
Unit: [cycles] 2
114
I
I
4000
5000
<
<
.'.
......
..
..... ..
.
V.-
~
..
°...;........
.
.
.
:7
. .
.o...... ...
... ... .......
..
0
(i) "s pue (OO
t-
... ..
......
o...... •%.•
.•. . ° %.•...
· .. · •............
••°°
°
... ° ...........
..°°°....
.%°,
..° ...........
°...°
. .;
°
0
IS3 71 S80o 31
S
or
c
.....
.
•·
°
°°°°.... °°
°.....o
.°.............
.
°..... ...
,
•··
0
i
U)
g
o... o.
.
°... .
.. . . . .
7°°........
.......
!i
..
!....
!! !.. .. ... .
..
.... .
m
..............
C)
°...............
°o.
,o.°°o
.°°
°°.~o°
° ..
. . .o . .
.
o...............°
.
o
......... ....... ........ ..... ..
i
..o
~
. •..
..
|...... °......
(I) S 0o:ifi.LIdV4W
.
.°°
.................. ..........
.°
..
.
..
..
..
..
..
°
°,
•
...
,
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......
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o.........
%......
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.
|
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... . ......
N
co
c
I
-
T-
>0
'o
>
O
0
=) V
c
::::::i:::::~:::::::::::: ::::::::
.
..
°°o
,o. ,°°
...
..
...... ......
..
. ..
•
0co
OOuOJLoo
0)=
0r
KKAU PRN 20 - ASIA 10 6 1995 :Segment 2.b
X
A
U.2
0.1
w
0
0 -0.1
-0.2
-0.2
7
0.15
0.1
7.05
.I
7.1
7.15
I..
. . ..
... .. ...
0.05
...
..
....
.. .
7.2
7.25
Time (sec)
7.3
7.35
7.4
7.45
7.5
x 104
I...
.
.........
......
......... .
...
....
...........
0
-0.05
-V
....
.. .........
..
...................
..........". .!..
. .. . . . .... ....
. . ... .:.
......
-0.1
-0.15
-0.2 7-
7.05
7.1
7.15
7.2
7.25
Time (sec)
7.3
7.35
7.4
Fig. A.2.7 : Station: KKAU - Satellite: PRN 20
Data acquisition date 10 6 1995
LCOBS and LCEST compared for t=7.0 - 7.5 10 4 sec
Unit: [cycles]
116
7.45
7.5
x 104
A
"RESIDUAL" = difference LC Observed - LC Estimated
A
: segment 2.b
0.1
-0.1
-0.2
7
7.05
7.1
7.15
7.2
7.25
Time (sec)
o = LC OBSERVED
7
7.05
Fig. A.2.8.a)
7.1
7.15
7.2
7.3
7.35
7.4
7.45
x 104
* = LC ESTIMATED
7.25
Time (sec)
7.3
7.35
7.4
7.45
KKAU - PRN 20 (10 6 1995) - [segment 2b]
Phase residual LC.DIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
117
7.5
7.5
x 10,
i-t
CL
0
ala
ý4C
0
'Ia
CD
0 Z4
3CD ila
C0
r)
w
I ....
..
.. · .
:::::::::::::...::::::
... .....
:
.......
.............
.....
•..... •..... .
°°,.,•°,,..•.°.
.•I,
...
. . .. •......,...
I
.....•......•...... .
-&
PSD of LC DIFF
Oo
o
00C
LC DIFF
L
-4
-4
-4
-
.
..
1......
t)
O
......
~CA)
..........
r"
..... I ....
I ...... i
)···:ii·····i?:i
··
.....
r°. .°°
.. °°.
....°e . #,
· . .. .•°.
· ·,,·.·,.,..·.·,·5··,
S.....
.....
..........
,.,..•.,.··· · ...
°
•,,,,,io,,o%,
,,,,.,
•..•• .. . .. .. . ...
:..
.. ,.......•.........
.
..........
.......
.
I
1
-I,
"
00
...
.
... ..
r
--
:--"'7"--'·
:
LC OBS
.:
.
!N)
........
|
....
2•·
a-.
•.
..
.....
°
•.~..~..
I:i:::::::::i::::::;:::
.o,,,•,.,,,•.,.,,,.,.
.
•··5···5
·° · · · · ·· 2:
·•·t··
•
°.......
........
..
°,°,,•..,,.•.,,,.°•,
·
-.--.
.. ,°
.I,
.,,
0 ::::::::::
CJ
PSD of LC OBS
..
I
.... ...............
;
I
00
...
C)
.......
"I
Osh-4,
C
ila
-4
-4
-4
-4
........
~3..
....
0-'
LC EST
I
PSD of LC EST
....
io
I
KKAU PRN 20 ASIA 1995 6 10
seament 2b
O
o
-J
0
I-
I
I
SI
-500C
-4000
-3000
I
I
I
_I
-I
,LeaAA
h
-30UU
I
,
AA
·
A•l•l
-SUUU
-4UUU
I
-2000
i
I
I
I
I
A
-zUUu
I
I
I
I
1000
2000
3000
4000
I
I
I
·
AA
I
0
I
-1000
-1 UU
I
l
I•
.
0
1000
I
I
lI .
_
2000
5000
- I
.
.I
3000
4000
-_
5000
CROSS-CORRELATION between LC OBS and LC EST
r
.-
-0.5
..
o...
..
.
I
.
o
.. .
.
I
.....
.
.
·
·
_t
.
.
·
.
•··~~
·
·
.
.
.
.
.
I
.
.
.
· ·
-4000
-3000
.
.
. •..
..
.
........
..
°··
if
-2000
..........
.
.~~~~~~~·
i
-5000
.
.
(
·
·
·
·
·
i
-1000
I
0
1000
.
·
''
.
..
o,
.
~
o~. '
.
·
..
..
..
.
..
i
I
2000
3000
KKAU - PRN 20 (10 6 1995) - [segment 2b]
Auto Correlation of LC..OBS (Ro,) and LC_EST (Ree).
Cross Correlation between them (Roe)
Unit: [cycles] 2
119
..
.
I
4000
time delay : Tau (sec)
Fig. A.2.8.c)
.
°
5000
0,
0
ci.
0
C) 0
(b
0
:r
cA
L
C
0L
'o
oo
.LC :)
....
.
bo
. ..........
~~............
b)
o9.
r
.·.........
C
-9
0
C
............
0
O
C
....
.........................
................... ...... ..
:: :::::::::::::: ::::::~::::::::
"......
:...... .............. ......
0
O
-L
AMPLITUDE of S (f)
......
. ....
°...........°........°..,.......
o
Coherence
.A
L
.
.
-
..................
.
--
.
No
C)
C)
.
oC
C
..
..............
o
o0o
....
.... .... . . . .
**- --
.. A
o
S (f) and S (f)
.
S.
0
...........i...............
C>
C)
.
.. *.....
,--
-L
0
I
LC OBS / LC EST
..
i
•
•.-
o
o
.........
...... * ..
~~~......:......~...:
..... .............
::: :::: :: ::::::::::.:
.
::::::
::::
... ;..~.;...............~~
............
.·...'..'·.
: : : :::.....: :
:....
: : 'r'............
1::::::::::::
-LC )
0O
.".
- .--
0
hr)
I
.X
9090
60
45*-3
1
30
.............
18 . ..
.....
.-...: " .
21
240
GPS DATA:
300
270
receiver location: CENTRAL ASIA
station: KKAU satellite: PRN 22
acquisition date: 10 6 1995
station: KKAU satellite: PRN 22
40C
o 30C
20C
10C
r
2
4
6
Time (sec)
Fig. A.3.1 :
8
10
0
2
4
6
Time (sec)
x 104
Satellite visibility chart for PRN 22 (skymap).
Elevation and azimuth of PRN 22 with respect to
the station KKAU
121
8
10
x 104
KKAU
80
.......
,
PRN 22
I
10 6 1995
I
!
I
r
..
. 60
...
........
....
.......
.....
K
.o40
S20
...........
0
0
1
2
.
.
3
I..
5
4
6
I
!
7
8
9
x 10
Signal to Noise Ratio for L1 and L2
40SIi
I
I
S......................................I...................I............
0
|
-I
-
A
--
L-
2
1
0
-
- -
3
..
- -,
4
i
.
6
5
7
,
8
60
40
.. . .
.
.
.
20
0
I
.
.
....
....
4
0
. ...
. ...
.. ..
. .. .
5
Time (sec)
Fig. A.3.2 :
Station: KKAU - Satellite: PRN 22
Data acquisition date 10 6 1995
Signal-to-Noise-Ratio (SNR) for LI and L2.
122
.... . --9
x 10'
ft ^
GPS "OBSERVED" Phase Residuals for: KKAU PRN 22 - ASIA 1995 6 10
..
I r% rf
C
%XAnr%\•
I
^
I•%•IIl& A 11ll
C 0.1
Ca,
0O
U -0.1
-0.2
-0.2 0
1
2
3
4
5
Time (sec)
6
7
8
Fig. A.3.3 : Station: KKAU - Satellite: PRN 22
Data acquisition date 10 6 1995
The observed phase residual (LCOBS) is compared with the
multipath phase error estimated from the SNR (LC..EST).
Unit: [cycles]
123
9
x 10'
KKAU PRN 22 - ASIA 1995 6 10 :segment #1
-0.1
-0.2
500
1000
1500
2000
Time (sec)
2500
B
I
3000
3500
I
1*
\.
* •
-... ,
--. "' .
".
" .•-
'
:
f
. \
'~
.1.
/
*
I
.
S
*
I~
i
.......... ...........
......
..................
....
..li
*
..
- !-
-.....
:/ ........'
'.: ....
-0.1
I
-0.2
500
Fig. A.3.4 :
1000
1500
2000
Time (sec)
\IPi
I·
2500
3000
Station: KKAU - Satellite: PRN 22
Data acquisition date 10 6 1995
LC_OBS and LCEST compared for t=0.0 - 3.5 10 3 sec
Unit: [cycles)
124
3500
"RESIDUAL" = difference LC Observed - LC Estimated
0.15
-
-
- -
-
-
-
-
.
.......................
0.1
: segment #1
I
.
.
.
........................
.
_
0.05
.......-..... ....
......
0
...
,
..............
-0.05
....
. ..
.......
-0.1
-0.15
-0.2
-
C
500
1500
1000
2500
2000
3000
3500
4000
3000
3500
4000
Time (sec)
o = LC OBSERVE D
* = LC ESTIMATED
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-n 9
0
500
Fig. A.3.5.a)
1000
:
1500
2000
Time (sec)
2500
KKAU - PRN 22 (10 6 1995) - [segment #1]
Phase residual LCLDIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
125
Power Spectra Density: Segment # 1
KKAU PRN 22: Phase Error
0.2
............. .....
0.1
.....................
0
_n
2
0
........... .....
2000
1000
.
.
.
.
.. .,•
.,.
.. ....,•
4000
.
*
*
*.
.
.
.......
. o. .
~S~"
•..
,.,. •.· ·
..........
3000
,
.•.....
• •
•i-i-~.iii
•
• .
-.. .........
-0.1
-
i
.
· ·
·
•
•
......
*
.
*
*****
. .
•. ....
. .. .-..-. • . . • . ..• ...
.....
........
••
•
• .
•.
•
..
.,i.•
10 -
10
*
10-
102
'"'
0.2 -
- --o
.....
0.1 ..
:
:
-
::::::.:.
•, .o,., . ·
• o. ,
,
.
•
•
,
rrr
··
··
····
··
··
····
····
~·L
. ....
..
..
.
.
...
. .
. .
0
0.5 --
-0.1
·
·
·
·
,o,
02'9 I
.
1000
2000
3000
10
10
4000
r
•··
..
.
................................
-A',. )
0
•
1000
•
•
2000
Time [sec]
3000
Fig. A.3.5.b)
•·
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. • o• .
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0
,,
4
010
104
4000
10-2
10
0.2'
..
.
10 -
PSD of LC.OBS, LCEST and LC..DIFF.
Unit: [cycles] 2
..
..
.· ·.
..
.
..
" ..
"
...
..
.
." •
.o " .
..
10
,
o
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,
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,
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.
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.
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···r--rr··· ·.......
rrr
KKAU - PRN 22 (10 6 1995) - [segment #1]
126
·
·
·
·
I· ·
0
i
0
-0.1
•
10-2
·•
,
,
10-1
KKAU PRN 22 ASIA 1995 6 10 segment #1
MJ
mi
0
O
o
--oI
o
1
-4000
--
-3000
-2000
-3000
-2000
4000
3000
2000
1000
0
-1000
6
40( 00
2000
-1000
0
between
CROSS-CORRELATION
1000
LC
OBS
2000
LC
and
3000
EST
1
I
I
I
I
I
I
I
-0.5
1
-4000
I
-3000
Fig. A.3.5.c)
I
-2000
1000
time delay : Tau (sec)
-1000
2000
3000
KKAU - PRN 22 (10 6 1995) - [segment #1]
Auto Correlation of LCOBS (Roo) and LC_EST (Ree).
Cross Correlation between them (Roe)
Unit: [cycles] 2
127
40400
n
t-'
V0
c/o
0
td'
0o
C,
0
h
:)
L
·
·
·
:·
o
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m
0
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...............
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.:
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0
.
AMPLITUDE of Soe(f)
.
. ...... • .
......
:........... I
0
·
.............
...........
o~o•
.•......
bo
__
Coherence
1
°...
o°•.
·
..
.. ......
. . °°.o
...
. ..
o
.......
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0
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0
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.
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.... ·...
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....
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..
.....
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0
Soo(f) and See(f)
......;.....
.··:
)
.•
.".
-
LC OBS / LC EST
KKAU PRN 22 - ASIA 1995 6 10 :segment #2
4.4
4.5
4.6
4.7
4.9
4.8
5
Time (sec)
I
I
I
-
I
I
.........................
.. .
.
.
.I
...
-
.
I
-0. r
I
-
I
.
.
. '.2J
i . .........
°I.
°
•
. ........
,. . ...
.. ...............
.°
.:I
••I
(
Fig. A.3.6 :
~
.
:A
..
4.4
\"
*..*
1 ...... :•i
-
"
.......
.. :.
. . ...
.1
-
|•J'
. Ai.
.' ....
- .. ... ... ...
... ... ...
-0.
I
I
.'.
I""I
1
x 104
°
,
.......
:..................
I
4.5
4.6
4.7
Time (sec)
4.8
Station: KKAU - Satellite: PRN 22
Data acquisition date 10 6 1995
LC.OBS and LCEST compared for t=4.3 - 5.1 10 4 sec
Unit: [cycles)
129
x 104
"RESIDUAL" = difference LC Observed - LC Estimated
4.4
4.6
4.5
4.7
Time (sec)
Ii
4.9
5
x 104
* = LC ESTIMATED
o = LC OBSERVED
0.3
4.8
: segment #2
II
i1
0.2
-0.1
-0.2
I
i
4.5
Fig. A.3.7.a)
4.6
4.7
Time (sec)
4.9
x 10'
KKAU - PRN 22 (10 6 1995) - [segment #2]
Phase residual LC..DIFF, determined as the difference
between LC..OBS and LC.EST.
Unit: [cycles]
130
S
(1
.
..
.
t
T
OO
C
04
I~
c;
-.•
oS
....
.....
"1Gdo OS¶l
O
· •·
....
..........
..
C
C;
M
..... . . . ...
.
.. .
. .. ".
"....
. . .. . .
•··i·
'
$80
CO
..
*1
t .d•• •...
C;
•I
oo
$80:3"i
o 0
··
· · ·, •I··~
·
.'....
o.
•
..
.
. . . .°.
°.....°.............
. °.
. .
.
.
.
°. . .°. . .•
.
....
.
..
. .....
.,.,:.. . .. .* . .. .,. .
. ....
. .'. . . . . . .' .
. .,. ,
. ..
..
. ... .. .. .. ..... . .
...
.
,..
•!.... .. .....
.... ......
ILS3 0"10 (oSd
oo3
3
Sr Ov*
0
I C.
-- O
.- IO 010 GO lSd
N
6o
C
to
9
U
"o
e
cqo0 1CCq%,
o
KKAU PRN 22 ASIA 1995 6 10
segment #2
ml
0
0
-.J
to 0
0-
I
-80
-80( 00
c
I
-60
-6000
I
I
-20000
-40
I
400
I
I
600
-4000
-2000
2000
4000
6000
8000
-4000
-2000
2000
4000
6000
8000
uJ
O
0
o
I?
0..
4q.
- a
--
-8000
-6000
CROSS-CORRELATION between LC OBS and LC EST
0.5
2
....
......
..........
....
.
.
.......
0
u
O
W
•
°
°
(I,
O
(I
0o
C-) -0.5
-1
"''
.,
,·-
·,·
-
-8000
-6000
Fig. A.3.7.c)
-4000
-2000
2000
time delay : Tau (sec)
4000
6000
8000
KKAU - PRN 22 (10 6 1995) - [segment #2]
Auto Correlation of LC_..OBS (Roo) and LC_EST (R,).
Cross Correlation between them (R0e)
Unit: [cycles] 2
132
>,
U)
iri
3
-.
O
0
(L
(1)
0
oQ.
0
C,
.....
.
......
:.. .....
:"
." ":: :
. . . . . . .:
°.
°...°
o...• . . . . . ... . . . .
.
............
..
..
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. ..
*":. .
. ..
... . .O. . ... .
.° ....
° . . . . ...
° . . . . . %.
-
c
..
...
. ......
...
I
,v:
..
........
... . .,
: .......
....... *... ...... ...............
..................
........
... . •
0
..
:.
. .. .. .. .. .. ..
.. °
. .
°
... .. °..
.,.%
.
o
0
0
o
0
. ...
(1)"spue (1)00°
..........
.. .......
• .....
°
Cl
cC
u
o
C
.........
N
-
0Q
I
T-
I
1s:: o"Is8031"
.
.....
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. .. ...
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C
. .. ...
°.°.°.
..
o)
0
o
..
..
...............
..
......
.
..
......
.
U
o
°
.....
° °...
.............
...
*.. ·****.......* .....
...........
.....
°...
.............
o
o
.......
(I)C°S Jo 3GflllYdVW
.......:......o
...
.
·
::::::·::::::~::''::::::::::::::::
.....................
?...... ......
,°**oooe***o.*eoo..eoo·o
. .. .....
... . . .
.
...............
... .. ......
.. .......
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..
:::::·::::::: :::::: ::::::::::::
. .. . .....
.
.
,
....... :......
·
..........
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d
C
..:: ....
. .. . " ...
• . .
~ . .....
....
:::::::::::::::'::::::!:::
:~
. ..
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Uo o
e0ue~ey00
·
....... :...... %......
.............
•·
OD
0
t4.4
O
C/)
oa
14
0
t 6-)
I.)
0
4-
KKAU PRN 22 - ASIA 1995 6 10 : segment #3
0.2
0.1
1-
....
!
I
!
7.6
7.8
I
-0.1
I
-0.2
7..2
0.1
S .
7.4
...
.
....
..
.............
...........
8
Time (sec)
8.2
8.4
8.6
x 10
...........
• ",
.. .......
.....
....
. ....
I-
I
-n o
7. 2
-..
Vr
I/
.. ... . ....
" \... .. .... ...
-0.1 .,...-.-- ". =.'... . .... .. ... ......
,- -
--
...... . ...
'".:.
.. .
,
..
.
.
-
8.2
8.4
Time (sec)
Fig. A.3.8 :
Station: KKAU -Satellite: PRN 22
Data acquisition date 10 6 1995
LC_OBS and LC__EST compared for t=7.2 - 8.7 10 4sec
Unit: [cycles]
134
x 10
"RESIDUAL" = difference LC Observed - LC Estimated
7.2
7.3
7.4
7.5
Time (sec)
o = LC OBSERVED
7.6
: segment #3.a)
7.7
7.8
x 104
7.7
7.8
* - LC ESTIMATED
0.05
0
-0.05
'-1
7.2
7.2
7.3
Fig. A.3.9.a)
7.4
:
7.5
Time (sec)
7.6
KKAU - PRN 22 (10 6 1995) - [segment #3.a]
Phase residual LC_.DIFF, determined as the difference
between LC.OBS and LCJEST.
Unit: [cycles]
135
x 10'
"RESIDUAL" - difference LC Observed - LC Estimated
7.8
7.8
7.85
7.9
7.85
Fig. A.3.10.a)
7.9
:
7.95
7.95
8
Time (sec)
8.05
8
Time (sec)
8.05
: segment #3.b)
8.1
8.15
x 10'
8.1
8.15
KKAU - PRN 22 (10 6 1995) - [segment #3.b]
Phase residual LCDIFF, determined as the difference
between LCOBS and LC.EST.
Unit: [cycles]
136
8.2
8.2
x 104
"RESIDUAL" = difference LC Observed - LC Estimated
: segment #3.c)
Time (sec)
8.2
8.25
Fig. A.3.11.a)
8.3
8.35
8.4
8.45
Time (sec)
x 10
8.5
8.55
8.6
KKAU - PRN 22 (10 6 1995) - [segment #3.c]
Phase residual LC-DIFF, determined as the difference
between LCOBS and LC.EST.
Unit: [cycles]
137
8.E5
x 10'
R.
ct
o
-AC
C1)
0c~O
-D
3
<c
iýn
-4
S0
-&
-
0
o1
l
LC DIFF
b
"
c•o
...,
i•a
o0
PSD of LC DIFF
m
I
I
I
--
o
-~
r.
mr
o0
·
....
..
•"
''
"
" ' .........
i...... . . ''
"'"
I
.. ....
i4 't
PSD of LC EST
z.....~c~
m
1
o
....
00
0 .............
......
..*:..
.... ...... ..
..............
aj
c
-
LC EST
o
.0
!!!
.·;
,F·
o
0
0
... . . . .
•..
•
I::::::::·::::;:::;
. ·I. ...···
:...
.
..
: : : :
•...· ·
. ..::.
.. .::::.:.
. . .. . . ::.
. . ::::.::
. . . .. .
....
""
.......
• ... . ...... ..•.. ...
.•. .
. . .
"•"":-'L"":~~~
""
,,
CI .
PSD of LC OBS
r
c'
r
rC1~~
Y
:JE.
·,.
L~
'h
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r
~'-L·
-r -Ze
a: -~
-~L~-
··
4c·
c·
0
a CoCoC
C>
0,
L1
I
Po
LC OBS
KKAU PRN 22 ASIA 1995 6 10
cn
segment #3
0
o
0
-1
Q-1.5
<;-1.5
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
c-)
-J
0
o,1
Q
-'
-1.5
1.5
x 10'
CROSS-CORRELATION between LC OBS and LC EST
1
I
0.5
I
I
I
L°°°
2 -0.5
°
.....
.....
...--
0.
_4
-1.
I
I
_
_.·I
L
-0.5
5
Fig. A.3.8.c)
:
0
time delay :Tau (sec)
I·
I
0.5
KKAU - PRN 22 (10 6 1995) - [segment #3]
Auto Correlation of LC_OBS (Roo) and LCEST (Re).
Cross Correlation between them (Ro)
Unit: [cycles] 2
139
1.5
x 10
0
0.
O3
oa.
C'
C
C
It
J
...
....
.......
..
°...
.....
...
°..
.
............
::::::::::::: ::::::::::::::: ::
6
r········
r
r········
·
. . . ......
(
. .
.
....
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%..
. ....
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o
.
.................
....
. .. . ... ..
.............
o
·
0
11 SO
8001
C
o)
o
to C
(1) S pue (0)00S
1
o-
oo
C;
.......
i
w
ci
IS3
.........
............
i-.
--
CO )
E
iL
r-
..
.
... . . . ......
.. . ..
....
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. .... ..
. ...... . ......
.
::.:.:::
........ ...... ...... .......
......
;............
. ..... . ... . .. . ... . .. . . .. . ... ... .. ...
.............
. .. .....
... .. . . . ..... . . .. . ... ... . .. .
...
L)
0
..
...
0
...
°..
....... .°.%.. ..
°..
0. ....
o..
..................
I
....
o
C)
nr~-
to
•
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T
11
C3-
...... . ..
.......0.. ...
. . .. . ..... .. ...
•...
...
. . . .
o
o
c'J 30~
.. *.
:......
·..
.............. :... ,.
. ..........
..
.... :......
......
'
~~~~'.............
..
o... .......
.....
.
eg eO lO
o
··
.............
c........
•~r·~·
,
.
o
o
%4-4
0
0
o
14
o4)
0
'a;
90 90
21
0
240
_
.300
270
GPS DATA:
receiver location: CENTRAL ASIA
acquisition date: 10 6 1995
station: MANA satellite: PRN 26
station: MANA
satellite: PRN 26
40C)
..
....
.....
...
............
300
E
200
..
........:.... .....*......
100
L
1
:
2
_
4
3
Time (sec)
Fig. A.4.1 :
5
I
'
I
6
x 10'
=
1
2
l
i
|
3
4
5
Time (sec)
Satellite visibility chart for PRN 26 (skymap).
Elevation and azimuth of PRN 26 with respect to
the station MANA
141
6
x 104
MANA
PRN 26 10 6 1995
60
i 40
0
> 20
0
I
I
I
I
I
1.5
2
2.5
3
3.5
i
4
4.5
5
5.5
6
x 10,
Signal to Noise Ratio for L1 and L2
1000
500
SI
I
!
L
I
I
2.5
3
3.5
4
4.5
5
I
I
I
I
I
I
I
i
4.5
5
5.5
n
1.5
1
60
40
20
nV
I
t
1
2
I
.
I
I
1.5
2
2.5
-
3
3.5
4
Time (sec)
Fig. A.4.2 :
Station: MANA - Satellite: PRN 26
Data acquisition date 10 6 1995
Signal-to-Noise-Ratio (SNR) for Li and Lz.
142
5.5
6
x 10'
GPS "OBSERVED" Phase Residuals for: MANA PRN 26 - ASIA 1995 6 10
I
I
I
.. ..................
-0
I
I
....... .......... ..... .....
.........
Ii
.
...
. ...-
1
I
1.5
2
2.5
3
3.5
4
4.5
5
5.5
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation
U.' 4
I
....
..
...........................-
............ . ....... ...........
-0.2
I
-n04
1
1.5
2
0
H
li
1
1.5
Fig. A.4.3 :
I
I
I
2.5
3
3.5
4
4.5
Comparison LC OBSERVED vs LC ESTIMATED
.........
.
0.2
I
I
6
x 10"
I
I
I
I
I
0.2
-0.4
.
i
i
.e1
-0.2
I
I
.2..........................
............
..............
4
-n
I
...
I
I
5
5.5
6
x 10'
...................................................................
...
...
..
.....
.
....-
2
2.5
3
/l
3.5
Time (sec)
Ii
4
4.5
5
5.5
6
x 10'
Station: MANA - Satellite: PRN 26
Data acquisition date 10 6 1995
The observed phase residual (LCOBS) is compared with the
multipath phase error estimated from the SNR (LC_EST).
Unit: [cycles)
143
MANA PRN 26 - ASIA 1995 6 10 :segment 1
0.3
0.2
0.1
0
-0.1
-0.2
1.15
,,
U.J I
1.2
1.25
0.2
co
/
0
.:
1.5
1.55
1.6
x 104
.... ........
** ^
-----------
....... .. . .......
....
*"
..I.----I ; :.
-- .......
'-- ---- :
***--
4 .................
.
i .... . :- .....
. -..........
..
/....,/*
.......... !
0
1.45
_
0
u 0.1
1.35
1.4
Time (sec)
1.3
---
:
. .......
-·-
I.. . ..
**--:d--------*
-
-0.2
-0.2
I
1. 15
1.2
1.25
I \I
1.3
II
I
1.35
1..4
1.45
1.5
Time (sec)
Fig. A.4.4 :
Station: MANA - Satellite: PRN 26
Data acquisition date 10 6 1995
LC_OBS and LC.EST compared for t=1.1 - 1.6 10 3 sec
Unit: [cycles]
144
1.55
1.6
x 104
"RESIDUAL" = difference LC Observed - LC Estimated
1.2
1.25
1.3
1.35
1.4
1.45
:segment 1.a
1.5
1.55
Time (sec)
x 10
MANA PRN 26 : o = LC OBSERVED
* - LC ESTIMATED
1.2
1.25
1.45
Fig. A.4.5.a)
:
1.3
1.35
1.4
Time (sec)
1.5
1.55
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Unit: [cycles]
145
1.6
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MANA - PRN 26 (10 6 1995) - [segment 1.b]
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time delay :Tau (sec)
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MANA - PRN 26 (10 6 1995) - [segment 1.b]
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151
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Station: MANA - Satellite: PRN 26
Data acquisition date 10 6 1995
LCOBS and LCEST compared for t=4.1 - 5.8 10 3 sec
Unit: (cycles)
153
5.6
x 10'
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: segment 2.a
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:
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4.35
Time (sec)
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4.45
4.5
4.55
MANA - PRN 26 (10 6 1995) - [segment 2.a]
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4.6
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time delay : Tau (sec)
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MANA - PRN 26 (10 6 1995) - [segment 2.a]
Auto Correlation of LCOBS (Roo) and LC..EST (Re).
Cross Correlation between them (RO)
Unit: [cycles] 2
156
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4.7
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5.1
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Fig. A.4.9.a)
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MANA - PRN 26 (10 6 1995) - [segment 2.b]
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Unit: [cycles]
158
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MANA - PRN 26 (10 6 1995) - [segment 2.b]
Auto Correlation of LC..OBS (Roo) and LCEST (Ra).
Cross Correlation between them (Roe)
Unit: tcycles] 2
160
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.
.
MANA PRN 26 ASIA 1995 6 10 segment 2c
0
..
.......
·
0
0I
o
00
I
I
-3000
i
-2000
-1000
-3000
-2000
-1000
I
I•
I
•
•
°
•
-40_
7
~~~·~~
~··~ ~
.1
.
t O
°
I
1000
2000
3000
4000
0
1000
2000
3000
4000
01
L-
0
o
I
-1
-40c 00
CROSS-CORRELATION between LC OBS and LC EST
I
3.
0.5
S............
I
............
..........
I
...........
.
I
.........
i
...
..
...........
........
.
.........
o
o
I
0
,.....~~.......
-0.5
1
4
-4000
--.
400
-3000
Fig. A.4.10.c)
·
·
·
~
·
·
·
·
·
·
·
~
· ·.
I
_t
-2000
~
1
1000
-1000
time delay :Tau (sec)
.
.·· .....
I
I
2000
3000
MANA - PRN 26 (10 6 1995) - [segment 2.c]
Auto Correlation of LCOBS (Roc) and LCEST (
Cross Correlation between them (Roe)
Unit: [cycles]
2
164
.
••..--···-~
4000
0\
Hr
laC.
0.
Co
S0
O
n
C)
0
O0
0n
C,)
E
uQ
U'
C,)
C,
0
o
0
U'
r1
cc,
o
Co
0,
Ob
0
k
***
***
...... :. . .
**...*... *....
*
.,..°-*
,
*
.............*......
... ,.o ....
. .
.
.
~r~~~~~~~~
..... %
.. .
..... °
°
°°°
.
·
...........%
.·
.·
...
....................
0>
0.
0
-A
.°
..
o....
..
......
....
0
mr
01
............. ,.......
°,°°°.°..o
,.........
•
,
.,
.....
,€...........
......
0
ro
0
...... 2......
•.
............
:....'
..
...
~.........
*.....
....................
.=...
.... ....
.
...... :.......:......
.........
?:!?![:ii??!
:•!?!!!•!• ?:: :::.
.
°
°°
...............
o
..
(1
0
0-A
AMPLITUDE of SO(f)
.......:...... .......•...... .......
°.......
···
.......
""·· "..... ...... ......
.................................
:
***
0P0
..
. ."...... •.......
o
0
rP,.P
.............
-LA :)
-A
Coherence
cn
CD
0
CD
U0=3
h
(0
-u
C)
m
0
0
0
C)
b)
U.,.
-LC
0**
--i
C1
0
-A
9
...
:.:
.. .......... ............
2.
.
.....
...... .......
. ...... ...... :.....
...... ....... ....... .......
: .......
..
.........
.
....... ...
.... .. :,.. .......
C0
01
0Tý
-L
0
:'"::...:::: .
.
...... ......
C)
01
0
-&
SO(f) and See()
0
LC OBS/ LC EST
90 90
60
12
130
...
....,... ..*
18
21
0
240•300
GPS DATA:
station: MANA
270
receiver location: CENTRAL ASIA
acquisition date: 10 6 1995
station: MANA
satellite: PRN 28
satelite: PRN 28
·IAl
4UU
o 300
I
200
100
n
0
2
4
6
Time (sec)
Fig. A.5.1 :
8
0
10
v n
4
2
4
6
Time (sec)
4% IV
Satellite visibility chart for PRN 28 (skymap).
Elevation and azimuth of PRN 28 with respect to
the station MANA
166
8
10
x 10'
MANA
,,
PRN 28
10 6 1995
I
.o
Cca
.2
w
0
1
2
3
4
5
6
7
8
9
x 10'
Signal to Noise Ratio for L1 and L2
4UU
-J
cc 200
z
CO,
0
__
'4UU
200
t
N
........
I
. .. ..
.. ..
............ ............ ...........
.
.
.
.
.
..
.......
........
.......
I
I
I
I
I
1
I
9
Time (sec)
Fig. A.5.2 :
.
Station: MANA - Satellite: PRN 28
Data acquisition date 10 6 1995
Signal-to-Noise-Ratio (SNR) for LI and Lz.
167
x 104
GPS "OBSERVED" Phase Residuals for: MANA PRN 28 - ASIA 1995 6 10
0.4
L
0.2
. -0.2
-0.4
0
9
8
6
7
5
3
4
1
2
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation x 104
~1
U.4
B
I
0.2
w
I
B
B
S.
0
-0.2
0
.
B
I
..
!.
1
I
I
2
5
6
7
4
3
Comparison LC OBSERVED vs LC ESTIMATED
9
8
x
10
U.4
C
C/)
O
0.2
0
0
-. 1
--
- -0.2
04J
0
1
, ·- i · ·--- i
i
I
,·-.-
2
3
4
5
Time (sec)
6
7
8
Fig. A.5.3 : Station: MANA - Satellite: PRN 28
Data acquisition date 10 6 1995
The observed phase residual (LCOBS) is compared with the
multipath phase error estimated from the SNR (LCEST).
Unit: [cycles]
168
9
x 104
MANA PRN 28 - ASIA 1995 6 10 :segment #1
0.2
,,
-U.'
n
02
0
0.2
1000
I.
2000
II .
.
3000
i~
4000
5000
6000
7000
8000
II
I
I
i
I
I
Time (sec)
-
9000
..........
0.15
a
w 0.1
a 0.05
.................
V...
00
6 -0.05
It'.
-0.1
01C
c
0I
i
1000
Fig. A.5.4 :
-
I
2000
3000
4000
5000
Tine (sec)
6000
7000
Station: MANA - Satellite: PRN 28
Data acquisition date 10 6 1995
LCOBS and LCEST compared for t-0.0 - 0.9 10 4 sec
Unit- [cycles]
169
8000
9000
"RESIDUAL" = difference LC Observed - LC Estimated
1000
2000
3000
4000
5000
Time (sec)
MANA PRN 28 : o= LC OBSERVED
0
1000
2000
3000
4000
5000
6000
: segment #1
7000
9000
8000
9000
*= LC ESTIMATED
6000
7000
Time (sec)
Fig. A.5.5
MANA - PRN 28 (10 6 1995) - [segment #1]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LCEST.
Unit: [cycles]
170
--
8000
"RESIDUAL" = difference LC Observed - LC Estimated
0
1000
2000
3000
Time (sec)
MANA PRN28 : o= LC OBSERVED
-,.
: segment 1.a
4000
5000
* = LC ESTIMATED
I
0
1000
Fig. A.5.6.a)
:
2000
3000
Time (sec)
4000
5000
MANA - PRN 28 (10 6 1995) - [segment 1.a]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LCEST.
Unit: [cycles]
171
I
.°. :::
o....° ::::::
.
,. .o.
::::::. . :::
. . .o...
....
,...,....·.......
•.
- ..~···~·
. ....
:..
- ....-.:*
.
.
........ ."..
.........
•...........
•·,,••••
·
·
:··
· %,...
·
•'
·
· · ·
.,···
.
,.·I...
:· ·
. °°
.,
· ·..
....
...:...
. . ..
•
•
,
*Iz . .
".•
."
•
o!
|,·U~c
.
I
o
0
0
Co
T
..i
.-..
."
.....
..-.:....:....
•....
~..
*•
o:
I) i
:,. '--";
S80 01 lo0 aSd
o
8O 0"I
,.......Z..,...........
::::i:::j::::·::::::::::
..
. ...
........
.... ... ...
•
....
..
...•...•... · · · ·
: . •··
...................
•.
:
.....
•....
....
:. . .
oo
0In
0
IS3 010o OSd
o
C>
183 0"
I
,-
... • .........
....
..
.........
.. ....
... ....
.. . ..
. . . ..· :
Ln
S 11la 0"I
o
I
JAM 0110 GSd
0 00
. ..; . .. . ... ..
1.1 . .•..
•
o
6y-
7'
0
"
C4
U~
MANA PRN 28 ASIA 1995 6 10
Cl,
segment la
O
0
0
0
I
I
V-~·
~··
·I· ·
· · \-
l-V
I
·
~··
~
~
IL·
~
.
·
0
O
SO
<-1
-6000
--1
-4000
-2000
I
4000
2000
II
6000
4000
I
I
2000
4000
L
-2000
-4000
-0UUU
6000
CROSS-CORRELATION between LC OBS and LC EST
0.5
.
-........ ...
..............
: !-
'': ~r~y· ~~I '~ '' I ''I'.' \'''~/
-0.5
.
.........
.
-
-
-
-
-
~'''''r 'r ''/ ~''Y''''''L.'''
.. ..,, °• °_................_
,,,,
____________I--
_1
-6000
-
-4000
-2000
2000
_________________
4000
time delay: Tau (sec)
Fig. A.5.6.c)
:
MANA - PRN 28 (10 6 1995) - [segment 1.a]
Auto Correlation of LC..OBS (Roo) and LC.EST ().
Cross Correlation between them (R,)
Unit: [cycles] 2
173
6000
m
o
o
S
...........
o
0
...
.(...............
.......
00
0
I
o
() S pue () S
0
IS3 0" / SOO 31
o
:)r-
c
0O
8.........
...........
,,% .
......
0
O
¶
!...
...
...
..
..
C
C
(
o
I
w
IV
..........
...............
. . . .
. . . . ...
..........
......
..... .....
.
...........
.. .. . .... .8..
..........
€...
. .......
..
.. ..
............................
00O
3C]rLl.l-ldiAIV
0
0
0
CW
(1) SCl)Io
...
~........ ...... '.
......
IV
00O
0
.,
0 S1o 3aflhndV4dV
(09
........: . o.
•o
...
..
•o, .
d;
....... :......
.... o.......
.
......
.....o.o
$
.
.,,,,°*o.•............o
.
O
0eGaJeq00
:......
.o.
oo.
.......
d
(I)
'I-
0%
C'
00
o
oo
(•!
"RESIDUAL" = difference LC Observed - LC Estimated
I
I
I
I
1
I
1
1
: segment 1.b
1
1
I
I0-
-0.1-09
..
6000
6500
7000
7500
8000
8500
Time (sec)
MANA PRN 28 : o = LC OBSERVED
6000
6500
7000
7500
* = LC ESTIMATED
8000
8500
Time (sec)
Fig. A.5.7.a)
I
MANA - PRN 28 (10 6 1995) - [segment 1.b]
Phase residual LCDIFF, determined as the difference
between LC.OBS and LC.EST.
Unit: [cycles)
175
-
• .,..
a'~
In
..
. -.
..-
I
C
I...,,
...
..
o.
. ... . .;
,
:':.,,-
•
:
:,
:t.•
..
.
: •.
:
N
·
T-
I
:"_..
•'
.:_
•.
1
O
...
'.......
..................
.
. .. . .. .. . .. ... .: .. ..
....... ,. ...............
..
······i······:······
.. ....
.. .... •. . .. . .. .. ..
,. .-...
. ... .o..
T-
• •.,o.
· -C '•
Y
•
SBO 01 oOSd
0
i. .....
o::•
...... ..
. ,..,
:. ......
--- -- --A
LO
I:
N
I
.-
...
.............
.I
.. ... ......
...
., .2.o•.
.oo
.....
.
.......
o
°
•.,
......
...... * .............
;,......
......
............. ......
.......
.....
7
Ln
. ..+
.S ..
..
. . .,. .
o..
. +
. ....
......
o
.1
JJCI 010o Sd
P
OSdIS3 0110OSd
J-4l0 001
....
. . .. . . .
...... ........
-
IS3 01
,
I
0
0
cu1
0
1r
MANA PRN 28 ASIA 1995 6 10
U,
mo
0O
segment lb
tO
k 0
-J
I
1
o
-4000
-3000
-2000
-1000
-3000
-2000
-1000
0
1000
2000
3000
4000
1000
2000
3000
4000
LU
w
o
to
0I
Q- I
-400UU
CROSS-CORRELATION between LC OBS and LC EST
I
0.5
-0.5
I
I
I
I
S
I
.°.°o°°..
.
......
°°
;.'''''
·; · · · r···
· · ;.·
· · , · · ·· ·
· c·
;''''·
1000
time delay :Tau (sec)
2000
· ·
_1
-40C )0
-3000
Fig. A.5.7.c)
-2000
-1000
3000
MANA - PRN 28 (10 6 1995) - [segment 1.b]
Auto Correlation of LC.OBS (Ro) and LC.EST (Re).
Cross Correlation between them (R,)
Unit: [cycles] 2
177
4000
c-
00
0
:
0
:3
:31
c)
0
O
o.
0
co
0
b
,..o.,,
......
'
..
.
... ,.
.
i.....
.
I ..... ; :..~~... ...
...
. . . .. : ......
.
:. . ......
. . . . ... .. ..
. . . . * .. . . . .
...
°..t......°*.°,°°,o
.....
......
.......... ;. .........
.o°.
. ...:...,.. •......
.. ... ....
o..°..,
,.
...........
6
l (n C
-
.....
. . ......
. ...........
........ .°..
..............
6
-L
.
,
.,*...
"....··
....
.......
......
...
•
,
o,,
.
.....
..
......
.
...... *,.
C) (n
0A
*,
...
• .............
...
..
AMPLITUDE of SO (f)
*....
**
"'*'**,**
.... . .. ...... .
'···"y·_···.
......
..
............ *
.......
....
...................
0o
=i,
...... ..
C :>
=%
-L
a
-A
L
•c
V
Coherence
o\
-.
I
o
r
.
0(1
0
0
.......
0J
0
.
. .
. ..
..
°
:
•.
·
o.oooeo...•o•..
............
::::::::::::
oolO.oalO.oeol...,
...
...
...
:** I ***I*:*I ::
. . ...
..
: : :::: : ::: :: :: : ::::
:
....
' .....
..........
0
0
II: ::::::::::::::::::
,.
_
So(f) and S (1)
,,
........... ...........
01
C)
.
LC OBS / LC EST
MANA PRN 28 - ASIA 1995 6 10 :segment #2
0.3 r
0.2
..
0.1
I
I. .
.
.
.
:
.....
.
.
.
.
.
.
.
it
/-.A
0
•
-0.1
-0.2
°~
...... ...
11t
tt
*1r
___
5
5.2
5.4
5.6
5.8
6
6.2
Time (sec)
x 10
A,-4
0.2
...................... ......................... ....................
. ...
...........". ..
.... .t
.i......
0.1
0
I
...i •l. ....
.......... ...... ...... .......... .. ....... ........... ....... .. .......
i..
...
-0.1
-0.2
.
.
.........
.
...
.
......
Fig. A.5.8
·
·
5.2
5.4
·
:,.
fr
·
5.8
: Station: MANA -Satellite: PRN 28 - [segment #2]
Data acquisition date 10 6 1995
LC_OBS and LCEST compared for t=4.8 - 6.3 10 4sec
Unit: [cycles]
179
I'
. .. . . . . . . 1.1
. .. . . . . .. . . . ...
5.6
Time (sec)
i
..
r
.. . . . . . . . . . .. . . .. . . . ...
'J• I •I..Ii_:'"l
IlI.
· ·
6.2
x 10'
"RESIDUAL" = difference LC Observed - LC Estimated
,,
: segment 2.a)
m
4.95
5
5.1
5.05
5.15
Time (sec)
MANA PRN 28 : o= LC OBSERVED
5.2
x 104
*= LC ESTIMATED
0.3
0.2
Co 0.1
0
y -0.1
-0.2
4.95
5
5.05
5.1
5.15
Time (sec)
Fig. A.5.9.a)
:
MANA - PRN 28 (10 6 1995) - [segment 2.a]
Phase residual LC.DIFF, determined as the difference
between LCOBS and LC..EST.
Unit: [cycles]
180
5.2
xl0'
........,......,......
--
- ." .. . .
.". . ..
.°........ °...
1
*
.
*
.
O)
.......:,.......?....."..
. .... ° .
............
.
......,.'..............
C
0
-I
S80"1 o OSd
.
,
• • ..
.. ..... .:..
' . ..:..
• • ,,,•
I
-
...............
.
.....o..
. . ...
. . . . . ...
..
...
...
...
..
S....
...
. .s
..
:
:
.
......
.....
.
.....
. ....
..
aSc
..............
3r-o
.
:
IS~30110 Sd
..
I I,
.... . .........
.... .
.......
e
.
.o.
......
.
. . . . ..
.............
•
. .............
-lO 0
."
:
I
:
.
10o
OSd
.
1IG0 301
•
. ........
,.
. . ..
I
°
.. .
...... .. ... . · ·°......r
· ·...
.. . ·. .· .
. . . . . .· ·. ;.....
......... . ;......
:
,·
· · ~·
:
· \··
;
\··
:
2·
•
ci
00C
IS3
07
No Va
c'-0
I
i)
uiF:
C4-
z
[-,
9
u
04
n,
jr
4-4I
0;
MANA PRN 28 ASIA 1995 6 10
I-
fn
V)
co
segment 2a
-J
I-
U)
1
r-
0
-1J
L-
I
I
-2000
-30( 00
I
-1000
I
..
1000
2000
3000
2000
3000
CROSS-CORRELATION between LC OBS and LC EST
4 -
1
I
....
7
0.5
... .......
...
o
o
2 -0.5
I...............
o
_1
-30( 00
-1000
-2000
1000
time delay :Tau (sec)
Fig. A.5.9.c)
:
MANA - PRN 28 (10 6 1995) - [segment 2.a]
Auto Correlation of LCOBS (Roo) and LCEST (Ree).
Cross Correlation between them (Roe)
Unit: [cycles] 2
182
U/)
E0
C,
aC
o
0.
00
ao
0
C,)
c
..
1....••,....
... . ........
O
*
.....
..=
......
~
-
*-
.
.
I
C OS
......
...
. ...
.....
I
*-r..-
... . ... ...
:....
0()s pue (1)S
................
...... ... :....... .... :...........
......
S....%.. •.I.•...••.
•. •..:.••
u'
r
~ ~ ~
· · ·· ·
)o uc
I
iS3 011S80 0i
d
-
L_
-
u
x W
S
U
0
E
P
w
O
cr
0
0
.....
""!""·
*..
.
......
...
n
.....
..............
·........
""'1"""""'
·.*.....
:::::X:::::::: :::::::
::
::: ::::
::
I
0
I
*...
..
. .
0
I
I
1j/n-d
0
..........
t
1
--
1r
0
4?7
DoT
c
. .,.::.
.:::
........
:: o :: o o::
. . ..
..
4...
•.........
n
C
C
0)
N
U;
(..4
...
...
................
.
.
::
:::::::::::::*****..
*......
WOO S1o3anfl.ltdYW
0
to
(ID0 o
II
....................
.......
. ...
..
..
•.°••
....
•
I.
.°....I•
••..
•
********.*****t*****.*****t*****.****
.......
::::
o
**
..
...
:::::::::::::::::::::::::::::::::::
.....
......
. ..... .. ... . ..... .. ......
......
......
......
*......
*. %**
••.
.
...
***
:::::::::::: ::::::
.
:: :::::::;::::::;:fiC':::
: : ": :
I
C
. o-- *......
....
..
..
...
..
•
. .. .....
... .. . ." ....... •...... :......
i
eougieqoo
"RESIDUAL" = difference LC Observed - LC Estimated
: segment 2.b)
0.1
I
I
I
5.25
5.3
5.35
Time (sec)
I
I
AA
0.0
-0.0
-0.1
iI
5.2
MANA PRN 28 : o = LC OBSERVED
5.4
5.45
5.5
x 10'
* = LC ESTIMATED
I
5.2
5.25
Fig. A.5.10.a)
5.3
5.35
Time (sec)
5.4
5.45
MANA - PRN 28 (10 6 1995) - [segment 2.b]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LCEST.
Unit: [cycles]
184
5.5
x 10'
'-
..
d
... .'
...
... .
.....
........
.......
*...
...
.
L)
CU
ui
I_
..
....
.
I......
.....
.. .... .... . ...
...
...
.
110o OSd
. . ..
... .
S0
.
S---. .........
0
...........
C
o0
880 0"I
.....
:::::
.. .........
o..
..
...
I....,
....
"::1:: ... .
~~~~~~1.::::1
.... · ........
.,.....,.oL*********
.... ...........
...............
c
IS3 3110 OSd
Cd
1S3 01
::
.
.
Ii-
c'J
U)
*.***.. ..
.
.*.
-oo-
. . .
o
*::::*:::::
:*: **:::::
**.******:*****r*****
***.**
.
•:...... ? .....
**.*...
**
. *****····:······
...
***********·
*****?*****
*****r*****
***********··
*
·
*****L*****·I·
..........
* .. ..
**-*****
**:******
**
**.******
********
**
=l1O 30 lo OSd
=1-110 01
70
7-
MANA PRN 28 ASIA 1995 6 10 segment 2b
I-
fn
V)
O
o
.0
o
2
Or
<r
I.--
-o
LU
o
0
o!
-3000
-2000
-1000
0
1000
2000
3000
CROSS-CORRELATION between LC OBS and LC EST
............
.
o
...............
.................
0
0
2 -0.5
_1
.. ............... ... . ...........
......
.
..
..
................
....
......
..
L-
-3000
-3000
-2000
Fig. A.5.10.c)
-1000
I
I
1000
2000
time delay :Tau (sec)
MANA - PRN 28 (10 6 1995) - [segment 2.b]
Auto Correlation of LCOBS (Roo) and LCEST (R).
Cross Correlation between them (Roe)
Unit: [cycles] 2
186
3000
C
0
U)
0I
a.
0
u,
o
....
..
.
..
.......
.o.,,.o
.
........
..
,.
.....
I
:
.
......
...........
.......
....
. .... ....
0
I
::::::.:::::.:::::
I
0
in
ue
I_
....
.
•
*....
•
.
·
..
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.
•
00
0
.
·
.
.
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;............
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...
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0
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1
m
Y
o
.
....................
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.
.....
N
C
1r
1v
2-
o
C C>
on
T-
I Co
Co0
..
:......
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........
!.
:.
......
..........
.
(I)"S 0o3OflhIndV4V
oo
Co
I•
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*.
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..
.
O,-
in
in
......
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ui
----ui
--t
e ueCeq
in
a
o
Of)'s pue (WOOS
o (Ul
O
o0
I
.
............
..
.......
tn
-
GO~uojiOLO
io
I
IS3 0 / SBO 01
"RESIDUAL" = difference LC Observed - LC Estimated
5.5
5.55
5.6
5.65
5.7
5.75
5.8
5.85
: segment 2.c)
5.9
5.95
6
x 10'
5.95
6
Time (sec)
*= LC ESTIMATED
MANA PRN 28 : o= LC OBSERVED
5.5
5.55
Fig. A.5.11.a)
5.6
:
5.65
5.7
5.75
Time (sec)
5.8
5.85
5.9
MANA - PRN 28 (10 6 1995) - [segment 2.c]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LC..EST.
Unit: [cycles]
188
x 104
:::::::::::....................
... o
...................
.............
................
.................
7'
.". .
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":...
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4.-.
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d
oo
u
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18~3 0
C
I
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....................
6
R.d
0n
:
cR
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Sd
............ * ***
........ *..........*.
V-
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Voci
Lei
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U)..
ui
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co
ui
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:.........
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IS3 01 lo Sd
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.......... ***.*..
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0
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,?A,,.
•. -,
. -:.'
..
. .... ..
r..
80 30110 OSd
o
....................
.........
....................
.. o
.......... %..........
-
d
S80 31
MANA PRN 28 ASIA 1995 6 10 segment 2c
a
0o
roI
-4000
-3000
-2000
-1000
0
1000
2000
.
.
I
0
3000
4000
. ...
..........
0
-e
-4000
-3000
-2000
0
-1000
1000
2000
3000
4000
3000
4000
CROSS-CORRELATION between LC OBS and LC EST
0
C.,
U,
U)
0
C-
-4000
-3000
-2000
-1000
0
1000
2000
time delay :Tau (sec)
Fig. A.5.11.c)
MANA - PRN 28 (10 6 1995) - [segment 2.c]
Auto Correlation of LC.OBS (Ro) and LCEST (Re).
Cross Correlation between them (R,)
Unit: [cycles] 2
190
0......
..
.
.
...
.
.
........
.................
.
,....,...............
.. .. .
* ".,0,o, .
.......
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0
o
1?
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C
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IoD
C
S
0o
% ......
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0
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0
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00
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o
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4
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0
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5 ·
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ci
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dd
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i
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oo
o
o)
C)
...... . ......
3flnid'YVN
(IWs joWo
.............
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....
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...... .:....... .
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......
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..............
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-CD
d
"RESIDUAL" = difference LC Observed - LC Estimated
6
6.05
6.1
6.15
Time (sec)
MANA PRN 28 : o = LC OBSERVED
6
6.05
Fig. A.5.12.a)
6.1
6.15
Time (sec)
: segment 2.d)
6.2
6.25
x 10'
*= LC ESTIMATED
6.2
6.25
MANA - PRN 28 (10 6 1995) - [segment 2.d]
Phase residual LC.DIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
192
x 104
.................
....
.
....
.
i
r.
.....
I
·.
. . . ... .•
.. ...
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I I
1S3 01
0
7
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,
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o...'
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2·········
,-
·
I
r·········
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I
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,
,
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01o0o OSd
cm
.,...............,
::::.:::::::::::.
·
Cl
1=I
·
=110 01
I'.,
c'JV
C6~
co
C6
.o.
.S,..Io,°,..'.o0
.····~·····
MANA PRN 28 ASIA 1995 6 10
U,
segment 2d
0ml
O
0
I-
-10 00
-30
I
I!
g
I
.. ..
.
...
.
.I
.
.
I
-1000
-2000
.
.
..
.
.
..
i
i
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1000
2000
2000
....
3000
w
..
...
..
.
...
. ..
.
. ..
.
.
.
CO
0
0
30400
-2000
1000
-1000
2000
3000
CROSS-CORRELATION between LC OBS and LC EST
1
·
I
t
0.5 . ..
..... ........ "......
-0.5 . ...............
°
........
°
......
........
°.......
.
...
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°
°
°I•
,
-1
............
...
. .. ..............
. .....
°... °°°o......
. .
.
I
;
..........
....
....o':-: .
...
..
.
.
.
.
I
i
-
-3000
-2000
1000
-1000
2000
time delay :Tau (sec)
Fig. A.5.12.c)
MANA - PRN 28 (10 6 1995) - [segment 2.d]
Auto Correlation of LCOBS (Ro) and LCQEST (Rd).
Cross Correlation between them (Roe)
Unit: [cycles] 2
194
3000
...
....
:.. ..
.
..
.
.......
...
0
0
..
,
........... .,
I i
-
I
0
..
..
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0
a)
.......
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cN
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r-
I
I
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•
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:........
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..
,................
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Ci
0
•
0
It)
.......
,....,..................o.
CV)
0
JS3 01 / S0 0-1
I
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cx
.
... . . ......
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=
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(........
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:. .... 5.*
21
240
18
GPS DATA:
I
300
270,*******I*******
********;*******,
24~0
270
receiver: LIGO
300
acquisition date: 25 9 1996
station: 0028 satellite: PRN 09
mA
4t6
.*......
-
f
> 16
W
3000
station: 0028 satellite: PRN 09
I
z 18
90
-60
.
...
.......... ........ .....................
42 ..............- ....... ..................
-. . . .. . .
- ...
- - - - ...
40
o 44
....
........
...........
...
.. .
38
IA
7.45
7.5
7.55
Time (sec)
Fig. A.6.1 :
xl
4
...
'
7.45
7.6
...
7.5
7.55
Time (sec)
Satellite visibility chart for PRN 09 (skymap).
Elevation and azimuth of PRN 09 with respect
to the station 0028
196
7.6
•A
x IU
LIGO 0028
PRN 09
7.5
(,
618
I
C
25 9 1996
.527.5
= 16
S14
7.46
=
=
i
I
I
7.48
7.5
7.52
7.54
7.56
Time (sec)
7.58
x 10'
Signal to Noise Ratio for L1 and L2
I
.... ...
........ .... ..
.......
.......
........
.............................. ...............................................
.. .........
14
12
..............................
10
8
7.416
7.48
··.
·· ·· ·· ·· ·· r · ·· · · · ·I· · · ·. ..............
...
:................
. ....
·V4....
~·/··.............
· · ·· ·· r · ·· ·· ·· ··
7.52
7.54
7.56
7.58
58
Time (sec)
Fig. A.6.2 :
Station: 0028- Satellite: PRN 09
Data acquisition date 25 9 1996
Signal-to-Noise-Ratio (SNR) for LI and L2.
197
x 10
90
12
*
90
60
'. ... 4..5.- 1
15
30
....
,....
. *:
e...
18.....
0O
********:******* 0)**************
18
21*
****
240
GPS DATA:
*30
*
300
270
acquisition date: 25 9 1996
receiver: LIGO
station: 0036 satellite: PRN 09
station: 0036 satellite: PRN 09
0)
120
50
0 18
45
116
40
z
I.-
·--
w
A
53~
7.35
7.4
7.45
7.5
Time (sec)
Fig. A.6.3 :
7.55
7.35
7.6
x104
. . . . . . ..-..............
...
....
ml
7.4
n I
I
7.45
7.5
Time (sec)
Satellite visibility chart for PRN 09 (skymap).
Elevation and azimuth of PRN 09with respect to
the station 0036
198
..
...................
.. ......
I
7.55
7.6
x104
LIGO 0036
o22
I
PRN 09 25 9 1996
I
I
7.42
7.44
7.46.7.487.5.7
7.42
7.44
7.46
I
I
I
I
i
I
7.52
7.54
7.56
o 20
C
7.4
> 16
W 14
7.4
7.48
7.5
Time (sec)
x 104
Signal to Noise Ratio for L1 and L2
I
I
I
I
.
I
I
I
I
!
7.4
7.42
7.44
7.46
7.48
7.5
7.52
7.54
7.56
x 104
S...........
oo
L
...................
Ii
7.4
.
.
I
I
I
7.42
7.44
7.46
Fig. A.6.4 :
...............
I
.
I
7.48
7.5
Time (sec)
Station: 0036- Satellite: PRN 09
Data acquisition date 25 9 1996
Signal-to-Noise-Ratio (SNR) for LI and L2.
199
I1
7.52
7.54
I
7.56
x 104
GPS "OBSERVED" Phase Residuals for: LIGO PRN 09 - ASIA 1996 9 25
I
I
i
jI
I~.~ I
%
-0. 5
7.47
0.5
A
I
I
|
:·
. ."
......
I..
" ".'°.-.7 .... ..
"
%
I
7.53
7.54
7.55
7.56
7.57
7.48
7.49
7.5
7.51
7.52
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation x 10
A.•
I! I
-..
•-..:•'•
_n
7.47
7.48
7.49
7.47
7.48
7.49
.I
ii i
7.5
7.51
7.5
7.51
i 7.52i
i
ii
?SH
i J
7.53
7.54
7.55
7.56
7.57
7.52
7.53
Time (sec)
7.54
7.55
7.56
7.57
0.
Fig. A.6.5:
Stations: 0028- 0036 Satellite: PRN 09 Data acquisition date 25 9 1996
The observed phase residual (LC.OBS) is compared with the
multipath phase error estimated from the SNR (LCJEST)
(differential mode)
Unit: [cycles]
200
x 10
0036 - 0028 PRN 09 - LIGO 1995 9 25
0
-0
5
7.47
7.49
7.48
7.5
7.51
7.52
7.53
Time (sec)
7.54
7.55
7.56
7.57
x 104
0.
o
I
-0.-0.
7.47
_v
I
/5
I
A.
I
%I
I
:.
7.48
7.49
Fig. A.6.6 :
....
......;'
... . . "..
......"..
":
:•I.
7.5
i
7.51
i I
7.52
7.53
Time (sec)
7.54
7.55
7.56
0028- 0036 Satellite: PRN 09
Data acquisition date 25 9 1996
LC.OBS and LC.EST compared for t=7.47 -7.58 10 4sec
Unit: [cycles).
201
7.57
x 104
"RESIDUAL" = difference LC Observed - LC Estimated
- 05.• E
I
I
I
·
I
·
·I
7.47
7.48
7.49
7.5
L
o
-
I
L
7.51
·
_
7.52
7.53
Time (sec)
LC OBSERVED
_
_
·
·
7.54
7.55
7.56
7.57
x 10
7.55
7.56
7.57
x 10
* = LC ESTIMATED
_-A0
7.47
7.48
Fig. A.6.7.a)
7.49
7.5
7.51
7.52
7.53
Time (sec)
7.54
0028- 0036- PRN 09 (25 9 1996)
Phase residual LC.DIFF, determined as the difference
between LCOBS and LC.EST.
Unit: (cycles]
202
t3
o
c.
r?
I-r
-o
-
U'
0)
I
o
..
.
··
e
.°
o
.....
o
....
....
o
o
······ (··
... ..... .....
...
..................
%.....
o
PSD of LC DIFF
LC DIFF
o
LC EST
o
in
.
....
o
o
. ......
.. ...
.
.
..
.. .
. .t.
.
....
8.
.
.
. . .. t .....
:"....
.....................
t . . ..
.. .. . . . . .. . . .. ..
.................
.*
..······
· ·r· · · ·
.··
.···
. . .. •.. . .. . .. .. •. ..
. ...
.. ...
.." ....
..%. . ... . . . .. .. .. ..
."
o
PSD of LC EST
.
.........
)·..........
....
... ....
...
-L
-'0o
C)
sin
oI
o
-I •-
...
,
0L
•
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i
.,
.
....
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.ia
0
o
i
.........
..
..
.. .........
o
PSD of LC OBS
-L
r
CJ
..
.........
-L
01
-4
0)
Ni
LC OBS
.
.
...
LIGO PRN 09 269 96
o)I
segment 1
0
-J
0
u)
ILl
0
w
o-.J
o
o
oI0
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
6000
8000
CROSS-CORRELATION between LC OBS and LC EST
3
I-
U,
Cd,
0
--
9
-8000
-6000
-4000
-2000
0
2000
4000
time delay : Tau (sec)
Fig. A.6.7.c)
0028- 0036- PRN 09 (25 9 1996)
Auto Correlation of LCOBS (Ro) and LC.EST a(R).
Cross Correlation between them (Ro)
Unit: [cycles]2
204
U)
C
0)
*.
...
......
o.........
. . . . . .,.. .
.. . . .. . .
.
.....
...
..
.....
. . .
....
..
. . ....
o........ .
........
.. .......
.. ... .. . .. . . ... ... .
..
, ..............
oo.oo....
:
a.
o
o
o
'I
4>V
,-
. ..
. o..
.. .
S..............
.....
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(i)Os pue (I)OOS
o
.
. . ...... .:....
.. ...
...... · I ...... . !·......
.i.
....
. ..... . ....
.. . .........
. .. .. .. ............
. . . ... . ... "
......
......
a,.
CL
t3O
O
0.
C)
......
C
CC
UaC
C
.....
....
"
iiiiiiiiiiiii
0
S3 0"1 /SBO
c
U)
0)
.....
.
...... .
........
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. ......
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......
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.
.. o...
.
o.....
..
.....
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. ..
.
i
.
. .... .... .
i
.
%.. . .. o.
t
I-
0
U)
A
o. . .. . . .
0
U)
0
0
0
o
..........
i orololli
CsoN V-
......
(o
o
.........
.. .
........
..................
.. .··. . . .. ... .
.
-
IT
Co
.·....**.. ...
. ..·*· ·.
. .... ...--... ..
. .. ...... ...... .... ....
....
. . ....
..*
·:··*****·
··--.......
.
..
........
******.****** ·***
·· **r ****.****
:::::::::::~:::::::::::::::
CC
0
a."
L
U?
CO)
cl)
o r)
0
cc
OC
Co
U)
Cm
......
......
· ·.
:::::::::::::::::::
............ ...
. ** . ::::::::::::
... ......
. ..... .··~······
****:****oo:-***,**.******
*******
.
......
.......
....... . .*.... ..*.*.*.*
r····..
0:
II
(
o>
.....
...
•
...
. ...
.....
II
.....
,0
I
r
.......
......
..
...
•
.....
. . . . . . I ......
... :
I . I . . .:.......
•........
..... .. .... . . ..
.
Go
0
%4-0
0
r'.
o0
0
C,)
N
rm
'4-a
90 90
12-
60
0
.... 43
15
18 ....... ...... :0
Z.
21
.
240
GPS DATA:
....... 0*
30
'
300
270
site: LIGO
acquisition date: 26 9 1996
receiver : 0028 satellite: PRN 15
wAhi •
receiver: 0028 satellite: PRN 15
CD
"O
I 310
I-305
2 305
300A
7.4
Time (sec)
Fig. A.7.1 :
7.5
7.6
7.7
Time (sec)
x 104
Satellite visibility chart for PRN 15 (skymap).
Elevation and azimuth of PRN 15 with respect to
the station 0028
206
7.8
7.9
xlO1
LIGO 0028
50
PRN 15 26 91996
..
...
....
...
....
.....
......
..
...
...... ....
so
40
0
.1
> 30
0
Mw
920
I
7.5
7.45
7.55
7.6
7.7
7.65
7.75
7.8
7.85
x 10I
Signal to Noise Ratio for L1 and L2
..................
in
-
7.5
7.45
7.55
7.6
7.65
7.7
7.75
7.8
I
I
I
I
I
7.7
7.75
7.8
7.85
_4
I
I~
20I-
....................................................
I
I
VI
t
I
____
.1n I
7.45
7.5
7.55
7.6
7.65
Time (sec)
Fig. A.7.2 :
Station: 0028- Satellite: PRN 15
Data acquisition date 25 9 1996
Signal-to-Noise-Ratio (SNR) for Li and L2.
207
7.85
x 10'
90 90
60
-
12
30
.... 45.-
15
18 .......... . .. ,***I *** 0
r. *
.
21.
'. .
'
.
'* 30
300
240
270
GPSDATA: site: LIGO acquisition date: 26 9 1996
receiver: 0036
receiver : 0036 satellite: PRN 15
satellite: PRN 15
315
40
a0
·'
1 310
r..
..
..7
********************f -
30
......
******
**********
2 305
20 · '
Q0nn I
in
7.2
7.4
Fig. A.7.3 :
7.6
Time (sec)
7.8
7.2
8
x 104
7.4
7.6
Time (sec)
Satellite visibility chart for PRN 15 (skymap).
Elevation and azimuth of PRN 15 with respect to
the station 0036
208
7.8
8
4
x 10
LIGO 0036
I
I
I
PRN 15 26 9 1996
I
II
I
1
I
46............. .. ... .......... ...... ·.. ....... .........
30 ..... r......... r..........s....'..~....~...........s...
20 ...
...
...~~.·..
7.35
7.4
'''
'''......................
s··....
...
....... e*......
7.45
7.5
7.55
7.6
7.65
7.7
7.8
7.75
7.85
x 10
Signal to Noise Ratio for L1 and L2
20
.
-
7.4
7.45
.
.
.
.
.
.
.
.
.
.
.
.
l*
7.55
7.6
7.7
7.65
.
..
......
.....-
I
*............
I
7.5
.
7.75
--
I
20
.•..............
.
•
.
....
.....
7.4
Fig. A.7.4 :
7.45
7.5
.... ;..
..
..
7.55
....
7.6
Time (sec)
....
....
7.65
Station: 0036- Satellite: PRN 15
Data acquisition date 25 9 1996
Signal-to-Noise-Ratio (SNR) for LI and L2.
209
.
...
7.7
....
....
7.75
x 10
GPS "OBSERVED" Phase Residuals for: LIGO PRN 15 - ASIA 1996 9 25
0.2
ccrr"
0.1
uJ
...
w
·C-C, 0 --------0 -0.1
0
uJ
......
,-0.2
...............
....
I
I
I
............
-------- -- --
------ -----
......
-s··
I1
,
I
7.8
7.75
7.7
7.65
7.6
7.55
7.5
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation
7.45
x
104
S0.2
0.1
~
0
S-0.1
U -0.2
****.*
......
.
r
I_
7.5
Fig. A.7.5:
*
. . .. ......
.
. . •. ..........
.........
.• ...........
.........
....
.....
.........
7.75
7.7
7.65
7.6
7.55
Comparison LC OBSERVED vs LC ESTIMATED
....
.
'.....
. ..............................
:
......
_
I
I
I
7.45
.
.........
7.5
7.45
0.2
0.1
0
-0.1
-0.2
...
*.***----
............
......
7.55
7.65
Time (sec)
i
**
7.8
x 10
......
I
I
7.75
7.8
Stations: 0028- 0036 Satellite: PRN 15 Data acquisition date 25 9 1996
The observed phase residual (LC.OBS) is compared with the
multipath phase error estimated from the SNR (LC..EST)
(differential mode)
Unit: [cycles]
210
...
.• ....
x10
0036 - 0028 PRN 15 - LIGO 1995 9 25
^^
U.3
0.2
0.1
0
-0.1
-0.2
7.45
7.5
7.55
7.6
7.65
7.7
7.75
7.8
Time (sec)
x 10
0.3
0.2
0
W
S•
I
I ........
l.....A.........................
0.1
Uj . ........... ..............
m
CO
0
.
I ..
.. ............ :
.
.
-'.
.
.
•.
.
.
••
.
.
.
......
.
.
.
.......
0
-0.1
.........
-0.2
. ..........................
.
.•."* r!j,,:,
:_'u
IP•...,.-
.......
" -"
-0.2
7.45
7.55
7.6
7.65
7.7
7.75
Time (sec)
Fig. A.7.6 :
0028- 0036 Satellite: PRN 15
Data acquisition date 25 9 1996
LC..OBS and LCEST compared for t=7.45 - 7.85 10 4se
Unit: [cycles]
211
7.8
x 10'
"RESIDUAL" = difference LC Observed - LC Estimated
7.47
7.48
7.49
7.5
7.51
Time (sec)
o = LC OBSERVED
7.47
Fig. A.7.7.a)
7.48
:
7.49
7.5
LIGO PRN 15 segm #1
7.52
7.53
7.54
7.55
x 10
7.53
7.54
7.55
* = LC ESTIMATED
7.51
Time (sec)
7.52
0028- 0036- PRN 15 (25 9 1996) [segment #1]
Phase residual LCDIFF, determined as the difference
between LCOBS and LC.EST.
Unit: [cycles]
212
x 10
-
.
....
..
........
..
*****
o . *..... ..
..o
::
:::::::::::::::::
:
..-.
.
.
.
. . . o.
..
.
.....
.
.....
....... ......
..,
.
.,
. . . ... .. :.''.;. ,.
..........
S.
•..
r,....r.
........
...
:............. :>:.
·
·
·• ·
·.
°o°°oo
·
·
·
C0 C~1
•r
-
0
I
;. .'.-.....
:....
.. \.r.~,.r...
"
"
· ,·° · s,,••,·
·
·
S)
*.
...
. .'
S80"31o C]Sd
..
.... "•;"-•.
. ..
•
•... ,
....
. '-.:
-.
· :... ...
:..
.. , ... . ..
:..
:'>
," ,,,,,,
:·
•
·
Cr- -i
· ·
,
·
0^
.
.
.
....
...
o.
..
....
o
...
1
..
°·.
-
,,ooo..
.......
...
...........
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••
CJ
IS3 01
IS3 31 lo 0Sd
0C')
0
..
...
....
.. .'. o.*ooo,,
,***e
*oo.*oo°
*oo
i
ll#.,,•,.,.,.,.,•.o
I)
-i:
-:...
.
•
0M
SGO 30
1g~
....
...
:::
......
....
. ..-.....:.... . •...
:
.: .....
M~
c
0N
. -. . . ...· A
.. -····
.
S:i
:.....
.......
...:
Lqr
1I-13 1o QSd
-I.I 0"
f
--
,,
C,
-
wE
4
tp 9
4.
M
,C..
u,
0 P,
4J~L 4
•
0-
04O'U
00
oo•
Q',I
",,
LIGO PRN 15 26 9 96 segment 1
CC8
00
I
o
<3
I
-4000
I
. . I..
.I
-3000
-2000
rr
•
I
-1000
0
I
I
1000
II
I
2000
.I
3000
4000
I7
.
0 0
.I..... .. .. .... . ......
0
<1
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
CROSS-CORRELATION between LC OBS and LC EST
S. . ..........................
0
o0
..... ............ ..
......... .......... ...........................
.
............. .....
0
02
S-0 .5 ............................. ............................ v ......................................
-1
,
-4000
Fig. A.7.7.c)
-3000
:
-2000
-1000
0
1000
time delay : Tau (sec)
2000
3000
0028- 0036- PRN 15 (25 9 1996) [segment #1]
Auto Correlation of LC..OBS (Ro) and LCEST (R).
Cross Correlation between them (R)o
Unit: [cycles] 2
214
4000
0.
0
0
U)
*1
O
0
C,
0
o
.,
Cr
C
~
m
C)
o
-'
O
v
VL
to
I.
..
,...
..
.......
b
.............
bo
........o...:::::::
::*:::::
. .... .. .... . .
. ... . . ... .
.................
. ....... .
· · ·
C)
--
0
00
0
0
%a
ft.
()
%......
··
....
. ....·
······
·. .............
~···,......
·.· ··
· ·~............
...........
··
o
C>,
11
40
:1 = 0
C)
CD
(D
=3
cD
'1-9
C)/
>0
a
.......
.....
>...... ... .. ...................
. ... .. .... .. ~~
............ ...... ............. 0o=
m
.. ................. ............. 0CO
0
C0
AMPLITUDE of SO(f)
0
0
..................... ........ .....
)
CD
So
S000
.
LC OBS / LC EST
io
c
.
)
b0
0
D
0
SOO(f) and S (f)
00
0
0
0
o.. O..o
..
...... .. ..
~................
.
'''"
.. ... ... .....
......
......... · · '
......
..
' ,+.,'
....
S'
..........................
·····~~
................................
. ... . .. . ..
.............
~~~~~'~ ~
.
...... ......
o...
... .........
·
''*''t=''''
· ·
~1
..................................
o,"·.
. .. . . ...-. . ,- . .-.
. .- .. --......
- . .. .. .
r
o0
.+
3.;
.....
i~......
i ....
i *.......
...
*...................
•
:.
0
*......*.........*......
.......
io
0o
Coherence
............ ....... ...... *......
aC
S0
"RESIDUAL" = difference LC Observed - LC Estimated
LIGO PRN 15 segm #2
0.3
0.2
0.1
0
-0.1
..........
-0.2
7.55
I
I
7.56
7.57
I
I
7.58
7.58
7.59
7.59
I
I
7.6
7.61
Time (sec)
x 10,
*= LC ESTIMATED
o = LC OBSERVI ED
0.3
0.2
-
·--
·-----------
-
-------
***----
---
I---------
-
0.1
0
......
.-............ ...............
:.......
-0.1
-0.2
I
7.55
I-7.57
7.56
Fig. A.7.8.a)
:
I
7.58
Time (sec)
I
I
7.59
0028- 0036- PRN 15 (25 9 1996) [segment #2]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LCJEST.
Unit: [cycles].
216
I
7.61
x 10
.
...
... ....
.....
..
.........
..... o
.....
.....
......... *
S....
.....
-.
•......
. ..
•
•·
·
s
·
c··r~~~~~·
..
•
·
*~J
·
.··
· ·
5
CU
..
·
.
..
T
0r
•. .... .. . . . . . . . . ....
• .
°..-. . °.
• '°
° •°°
.•.·
... •
.......
.·5
CD
2:
/
I.5
.. .
.
o
6
6
o80 "1ijoGSd
o.
',.
SBO 01I
-
... ....
::'::::::':
. .
.
.
.
o. . •:
::
...
....
::
" ''•
..
· ...
.....
. ...
..........
.'.
......
.. ...... ....
S.........
. .........
.. ' ...
......
::.,:::
"""
.....
..,....
:; I"
' S.
.....
:. .-.
..:----.
...............
c..
:N
•
0
........
. .
•...........
.....
.
Co
..
....
.
.
•...
.:. .. .. . ... . .. ... ...
I
.................
-...--.--.-.--..-.
•.: ..... . .........
.
.. ...
Co
CD
o
So Io
zw00oJo aSd
==110 0"1
IS3 0" 1o GSd
..
..BO
18 01
IsaO C
.
t
co
tD
E~
"-.
U)
-l=
CD
u)
~4Q
ON
V-
u
01
P,10
ul
8e P4:
LIGO PRN 15 26 9 96 segment 2
I
o4
-3000
-2000
-1000
0
1000
2000
3000
-3000
-2000
-1000
0
1000
2000
3000
o
0
I
OT
CROSS-CORRELATION between LC OBS and LC EST
a: 0.5
0
o
0
-0.5
-1
-3000
Fig. A.7.8.c)
-2000
:
-1000
0
1000
time delay :Tau (sec)
2000
3000
0028- 0036- PRN 15 (25 9 1996) [segment #2]
Auto Correlation of LC.OBS (Roo) and LC.EST (RE).
Cross Correlation between them (Ro)
Unit: [cycles] 2
218
U)
a
Q)
0
0-
a.
0~
C',
0
* .....
................. °.....
:::::::::::::::::::::::::::::::::::
.....
o··.·
-......
...
.
...
· 0
r··
So
0
()S
,
0
0
r,c0
PU (i) s
o
I
03O
C)
I
........
.
.....
3
.
·-
..
. ..... .....
· ·
·8.···· ·
· · · · · · · · · ···
·.
·
.....
:.....
...........
. :.....
.. . • ..........
..
.:. . .
.......
..
.... :.....
•(.......
· · · ·.
· 8···
3
i· (· ··. : .....
8.··
· ·:.....
·
· ·
·· 8
·.
.··
· ··
tJ
,•
0
0
C')
:
0
0
V --
S oIClnIld 4'JV
::
....
...
..
.
......
.. ..
....
*.....°..
.....
I
•o.,..,•,,..,,.,,
o
..
.... .
.. o.
o
o
m
0
I_
Sr-
I
0
C)
Iv
C>
7
0,-
.. °...°
...
....
:::::::::::::::::::::::::::::::::::
°.°.°°°..°•.°.°°°°.
. ............
....... .... :
.·o..'..,..,,o'o,,.,.O.,.,°,,,
···
0
o
)
(l)
:
..
°....°.°.°.°l.....°oo..°°°oo.o..°
.
.,.o.,0,,
**.
***o*....
...........
o
.
::::.**.......*....
.:::: l:*:::::i::::.
:::........
M::::
......
..........
.......
........ ......
:: . .:
::
. ... .-.
: :
0
x
co
pO
cu
O
0
0
0
W
cc
w
z
EE Ou-
U)
0
o0
Co
U)
to
r0
Co
co
cj
1r
..... :.....:.....:.....:.....:.....
.. .....
.
CJ
0
0
......... . .............
c
CD
(A
C')
0
1S3 31 / SBO 01
2)
0%
a0
o
0
%4.4
I
8-
0
Cf)t/
o
ob
8
oo
io
ua
t-
"RESIDUAL" = difference LC Observed - LC Estimated
LIGO PRN 15 segm #3
0.3
0.2
0.1
0
I
I
I
7.62
7.63
7.64
I
I
I
I
7.65
7.66
I
i
I
.
I
-0.1
-0.2
7.461
7.67
7.68
Time (sec)
o = LC OBSERVED
x 10
* LC ESTIMATED
U.%3
0.2
I-.
- ---
)0.1
~ ~~~
...-...-..-..---
-
S-o.2
-0.2
I
7.31
I
I...........
.. ... .. .. .. .. . . .
7.62
Fig. A.7.9.a)
7.63
:
II
I
. . . . .I . . . .
7.65
7.64
Time (sec)
. ..
7.66
=
7.67
0028- 0036- PRN 15 (25 9 1996) [segment #3]
Phase residual LC.DIFF, determined as the difference
between LCOBS and LC..EST.
Unit: [cycles].
220
. .
.
7.68
x 10
Cl.
E
CO
Q)
Cu
o
0.
C,)
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2000
3000
4000
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2000
3000
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CROSS-CORRELATION between LC OBS and LC EST
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Fig. A.7.9.c)
-2000
-1000
1000
time delay: Tau (sec)
2000
3000
0028- 0036- PRN 15 (25 9 1996) [segment #3]
Auto Correlation of LCOBS (Roo) and LC_EST R).
Cross Correlation between them (Roe)
Unit: [cycles] 2
222
4000
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LIGO PRN 15 segm #4
0.3
0.2
0.1
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.
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o
·
:
•
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I
0
-0.1
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I
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7.68
;I
7.7
7.72
o
=
7.74
7.76
Time (sec)
LC OBSERVED
7.78
7.8
7.82
x 10"
*= LC ESTIMATED
0.3
0.2 ...........-........
...............
.-
: ...............--
---
---.............
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0.1
*1
0
-0.1
-0.2
I 7.68i
Fig. A.7.10.a)
7.7
:
7.72
7.74
7.76
Time (sec)
7.78
i
7.8
0028- 0036- PRN 15 (25 9 1996) [segment #4]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LCEST.
Unit: [cycles]
224
i
7.82
x 10
..
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:.....
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segment 4
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0
0.2
0.4
0.6
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CROSS-CORRELATION between LC OBS and LC EST
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t-
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°
.1
-4000
-4000
I!
-3000
-2000
1000
-1000
2000
3000
time delay :Tau (sec)
Fig. A.7.10.c)
:
0028- 0036- PRN 15 (25 9 1996) [segment #4]
Auto Correlation of LCOBS (Roo) and LC._EST (Ree).
Cross Correlation between them (Roe)
Unit: [cycles] 2
226
4000
c)
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o
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0.
C,
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30
:
21
300
240
GPSDATA:
270
acquisition date: 26 9 1996
site: LIGO
receiver: 0028 satellite: PRN 25
receiver: 0028 satellite: PRN 25
170
7.5
7.6
Fig. A.8.1 :
7.7
Time (sec)
7.8
165
..
160
.°°
-1
gzr%
7. 5
7.9
x 10'
%
......
.. ....
........
..
.......
......
.....
.
7.6
7.7
Time (sec)
Satellite visibility chart for PRN 25 (skymap).
Elevation and azimuth of PRN 25 with respect to
the station 0028
228
7.8
7.9
x 10'
LIGO 0028
PRN 55
26 9 1996
35
-. 30
I
. 25
g 20
I
I
7.7
7.75
. .
7.6
7.55
7.65
.
7.85
x 10
Signal to Noise Ratio for L1 and L2
II
III
AAA
................
................
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...........
....
...
...
...
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20
15
10
5
7.55
25
20
15
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7.7
7.75
I
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7.85
I
..........................
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...
10
55
7.5
7.6
Fig. A.8.2 :
7.65
I
7.7
Time (sec)
g
7.75
Station: 0028- Satellite: PRN 25
Data acquisition date 26 9 1996
Signal-to-Noise-Ratio (SNR) for LI and Lz.
229
7.8
7.85
x 104
90
-
12
90
60
45....... 30
15
*0*.............* 0
..
18 ....... ; *....
30
21
240
270
GPS DATA: site: LIGO
300
acquisition date: 26 9 1996
receiver: 0036 satellite: PRN 25
receiver: 0036 satellite:PRN 25
9
Time (sec)
Fig. A.8.3 :
x 104
Time (sec)
Satellite visibility chart for PRN 25(skymap).
Elevation and azimuth of PRN 25 with respect to
the station 0036
230
x 104
LIGO 0036
PRN 25 26 9 1996
35
CM
........... ......... ~.....
.........................
......... .........
` 30
I
......... .........
................................
. 25
& 20
. ... ..
. .. .. ... ..
.... . ..
.......
...
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... . .. ...
LU
I
1_
ýj
55
•
7.a
•
.
.
7.65
7.7
7.75
.
7.85
x 10'
Signal to Noise Ratio for L1 and L2
25j
....
...
...
..................
zu
15
.....
.......
.....
...
...
...
...
....
...
...
....
........
....
...
...
...
....
...
...
...
10
7.65
7.6
7.55
7.75
7.7
7.85
25
20
-I--
I·~~···~~~
I~-'''''
i········--······
15
o7.
7.5
I
|
5
Fig. A.8.4 :
II
___I
|
|
|
7.6
7.65
7.7
Time (sec)
•
7.75
Station: 0036- Satellite-. PRN 25
Data acquisition date 26 9 1996
Signal-to-Noise-Ratio (SNR) for LI and Lz.
231
i
7.85
x 10'
GPS "OBSERVED" Phase Residuals for: LIGO PRN 25 - ASIA 1996 9 25
0.2
0.1
0
I
S......
I
I
I
........
..................
I
. ............. ...........................
-0.1
-0.2
. ...
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7. 55
0.2
. ...
...
I
I
......
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. '..... o........................... .
I
I
I
I
I
I
o......
....
.... .....
I
I
-.......
7.6
7.65
7.7
7.75
7.8
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation x 104
...
...
...
· ...
_
...· · ...· · ....·
I:··
·.................
; ..·
..
..
..
..
. . . ..
0.1
.....
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......
..
. . ..
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..
. ...
.
.
0
-0.1
-0.2
II
I
7. 55
0.2
II
I
7.6
II
II
7.65
7.7
7.75
Comparison LC OBSERVED vs LC ESTIMATED
7.8
x10 4
................ ·..!· ·..· · .· .· ·..· ·...
.:I ..,··· ..· ·.. ... ..t....
·
. .. .. . .... ... .. . ... I .. .. .
.
0.1
..
0
....... .....
-0.1
-0.2
5
7.5
Fig. A.8.5:
I
I
I
I
|
g
7.6
7.65
7.7
Time (sec)
I
..
..
"''
I
I
I
•
•
7.75
7.8
Stations: 0028- 0036 Satellite: PRN 25Data acquisition date 26 9 1996
The observed phase residual (LC.OBS) is compared with the
multipath phase error estimated from the SNR (LC..EST)
(differential mode)
Unit: [cycles]
232
xlo4
0036 - 0028 PRN 25 - LIGO 1995 9 25
I
0.2
I
I
I
1
I
-0.1
1
-0.2
7.55
7.6
7.65
7.7
Time (sec)
7.75
7.7
Time (sec)
7.75
7.8
x 10
0.2
u
I,,I
cc
Cn
0
0
o -0.1
-0.2
-0.2
7.55
Fig. A.8.6 :
7.6
7.65
0028- 0036 Satellite: PRN 25
Data acquisition date 26 9 1996
LCOBS and LC..EST compared for t=7.56 - 7.9 10 4 sec
Unit: icycles]
233
7.8
x
104
0036 - 0028 PRN 25 - LIGO 1995 9 25
[
I
I
I
I
I
I
1
II
7.64
7.66
~
-0.1
-0.2
I
7.56
I
7.6
7.58
7.62
7.68
Time (sec)
x 10
...
I
. .
i
I~~~~
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.
.
i
I
i
Ir.
I'r··
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'1::.: . . ..
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I I .........
-0.2
. .... . . !. . . . . . .. . . . . . .. i i . . .•- -..
-'.. . . .
• '-
7.56
7.58
7.64
7.62
.
7.66
Time (sec)
Fig. A.8.7 :
0028- 0036 Satellite: PRN 25
Data acquisition date 26 9 1996
LC_OBS and LC_EST compared for t=7.56 - 7.7 10 4 sec
Unit: [cycles]
234
7.68
x 10
"RESIDUAL" = difference LC Observed - LC Estimated
7.56
7.58
7.6
7.62
Time (sec)
o = LC OBSERVED
LIGO PRN 25 segm #1
7.64
7.66
7.68
x 10
7.66
7.68
* = LC ESTIMATED
0.2
S0.1
o')
0
o -0.2
-0.2
7.56
7.58
Fig. A.8.8.a)
7.6
:
7.62
Time (sec)
7.64
0028- 0036- PRN 25 (26 9 1996) [segment #1]
Phase residual LC.DIFF, determined as the difference
between LCOBS and LC..EST.
Unit: [cycles).
235
x 10
..........
........
::::::::::::::
.
:.......... ....
.....
.
..
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LIGO PRN 15 26 9 96
-8000
-60UU
-4UU0
-zUUU
U
segment 1
zUUU
4UUU
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8000
CROSS-CORRELATION between LC OBS and LC EST
1
I
1
0.5
~
o
0
I
02 -0.5
1
I
-8000
Fig. A.8.8.c)
I
I
-6000
-4000
:
-2000
I
2000
time delay :Tau (sec)
I
I
4000
6000
0028- 0036- PRN 25 (26 9 1996) [segment #1]
Auto Correlation of LC.OBS (Roo) and LC.EST (t,).
Cross Correlation be.tween them (R
Unit: [cycles] 2
237
8000
L
(D
0
aL
o
n.
0,
0
::::
......
1,
U)
In,
co
CR
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c
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0
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0
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. . . .•
. ..
. .
. ........
·.......
......
. ..
...
.. ..
.. .. ....
0
l0
0o
0
0
0
0
o0
t.............
.............
0
....
o.......
rlll~dliV
o...
·. · :.·;..........,,,,,
(1) s 10 ::
...
.........
......
....
..
. . .. ..........
*......
.....
......
...... :•......
....
o0U
c;
..........
***~
•**:
*
Lc;
o3
(3
br-
.
C
N
0-r
Sci3
0)
0u
t.
o
~o
0
11O
I
rSUt~,
41U
0i
15
u,
0
Cl
cn
tn 0
00
..
.......
%......
.....
.....
****
........ .•......
......
.............
.*....... ........
......
......
....................
ii ****:"•
0o
··· :00
OOu
"RESIDUAL" = difference LC Observed - LC Estimated
7.7
7.72
o
7.7
Fig. A.8.9.a)
7.72
:
7.74
-
7.76
Time (sec)
LC OBSERVED
7.74
LIGO PRN 25 segm #2
7.78
7.8
7.82
x 104
7.8
7.82
x 104
*= LC ESTIMATED
7.76
Time (sec)
7.78
0028- 0036- PRN 25 (26 9 1996) [segment #2]
Phase residual LC..DIFF, determined as the difference
between LC.OBS and LC.EST.
Unit: [cycles]
239
rt
r(A"
0)0"0
0
ttJ~
L
o
oC>
-
.....
....
·· ·
·
·
. .....
...
* *:.:.....
.....
*
• .....
·
· ,
.
·
.
......... :....::
:;..
........
C> ýn -· in
0
PSD of LC DIFF
oC
LC DIFF
0
0
-A
0
-L
1·
I.
C)
0
...
.....
...
t
.···
: *:.:*:
**:.
* •:•:
.... •:::
**********,*****|****
...............
....
•:...:.....
.....
.....
..
...
:
***..
: :::
..
;.
...
........ ...
I···
'
-1
.......
......*...
. ...
.....
in
0
PSD of LC EST
LC EST
AO
i1n
0
:
.
:
,.•-" :
...
-.
.....
... ...
..
.
..
.
.
. *** ....****I**....
**
.
..
....*
:
-
·
~i:::iiiiiiir
•... :........... = ...
*****,*****,*****4****
**
.... ... ,
i.n
-4
T7
...
:... .....
.... *** **.
,e
....
......
/
.
0
.
I,. I
.
PSD of LC OBS
·: :·S
··
:
C7T
0
: .. :
-...
.L.•.
oO0
LC OBS
~..•
:: .•
.- ....
~'': .,' .. . ...
:''''~
I
I
00C
Oo
:
........
....
.....
LIGO PRN 25 269 96 segment 2
m 1
I
I
I
I
I
I
I
2000
4000
6000
0I-1
0
-80
-80 00DO
-4000
-4000
-6000
-6000
I .I
- -
r---
-8000
-6000
-4000
-2000
-2000
2000
I .
.
2000
-2000
4000
6000
BC
80,00
6000
80 00
6000
8000
I
4000
CROSS-CORRELATION between LC OBS and LC EST
0.5
H
-0.5
R-°
1
-8000
-8000
....................................
-6000
Fig. A.8.9.c)
-4000
:
-2000
0
2000
time delay :Tau (sec)
4000
0028- 0036- PRN 25 (26 9 1996) [segment #2]
Auto Correlation of LC..OBS (Roo) and LC.EST (R).
Cross Correlation between them (Ro)
Unit: [cycles] 2
241
>8
C
0
a.
O
C.
O
C
C
C
8o
c
....
•.
.
...
-.
....
.........
6
O
o
d
....
... .
....... ..
- .-.
... .,.
- -.. .
O
0
.
* .
...
wC
!
.........
...... .o......
.. . .....
.. .....-..
o
O
o
oI
01
oI
....
;..• ... ..:..........
........
........
...
.....
6
N
IS3 01/80
1,.
DV
U)
ui
*
...
...
........ .. ...
..........
.***** : *
..
. ......
. .. .
....
....
.. ..
.....
.~~~
·
i
8.··
· · ·
·
·
8.::I:::i::I:::i::
.**...
...*.. .........
o
0
9
0
.
~
................
...
. ·
· · 8
; · · .:
.
8
.
:·:
· ·
,,,i,.1
8
I)ou,
.
C)
C
o
0
o
o
•W0
LO
o,
(;) SJo~OflhIYdtW
,. ...*.
. .
c
e
.... . . . .
. . . . .,o. . . .,
. . ..
•... ....... ....... ..
.. ...
........
...
.••.°o
..
. . . . . ,.o. . . . . ., .. . . ., .
.-.
00i
A
-
........... ...o..........
--.
ý-
u)
-
cO0,-
7,0...
....
....
00
o--
P
.".::..."
..",'"
....
.
....:.".":.:...
'' ':
cu ::.......
•.
.
"
..
:o
:
. .....
C)
-i
*0
C)
LL
0
w
z
w
w
x
0
0L)4
e6eqo
eougioqoo
4-,
-4)
U)
L.J
v,
0:
c0
.'
GPS "OBSERVED" L1 :LIGO PRN 09 - Single Difference 0028 0036
0.1
0.05 -...
0
-0.05
-0.1
7.46
0.1
0.05
0
-0.05
-0.1
.........
............
""'......."......
-
...............
-..............................
."
S
. ..
!
. . .I.-
I
.".....'.
......
\
..............
.........
.............
....
-....................
. . . . ..
..
!
I
I
.
.......
.......
.
.........
......
. . . .. . .
I
I
I
0.1
.
',
w 0.05 -.
WI 0
..
. .. ..
.. ..
0 -0.05
" -0.1 ................................
I
Fig. A.9.1 :
7.56
......
7.58
x 104
I
I.
.............. :
................
7.48
I
I
I
I
7.5
7.52
7.54
Comparison L1 OBSERVED vs L1 ESTIMATED
7.48
................
7.46
7.58
x 10'
7.56
7.54
7.52
7.5
7.48
GPS "ESTIMATED" L1 computed from the analysis of SNR variation
S................
7. 46
.
.
.................
:.
...
..
..
.........
..
..................................
.............
7.5
.....
I
Ii
7.52
7.54
I
....................
Time (sec)
0028- 0036 Satellite: PRN 09
Data acquisition date 25 9 1996
LLOBS and LilEST compared for t=7.46 -7.58 104 sec
Unit: [cycles]
243
7.56
7.58
x 10'
LIGO 1995 9 25 PRN 09 - Single Difference 0028 0036
·
0.15
·
·
c
0.1
i.
·
·
0.05
0
-0.05
-0.1
-0.15
I
I
. . !..
. .. . . . . .
I
I
7.52
7.54
7.56
I
I
·
-0.2
7.48
7.4
46
Time (sec)
0.15
-
0.1--
I
..............
I
I
......
.............
..
7. 58
1
x 10
........
.
w
w
a -0.05
..............................
.....
......... ....
.........................
. ..........
-0.15 -0.2
I.......1
I.........
1-········
7.48
Fig. A.9.2 :
1-··---f.~
1...~·...
•
•
|
7.46
,..........
.....
•. ".;t................
....... •./ .....
.........
.....
7.5
7.52
Time (sec)
7.54
7.56
0028- 0036 Satellite: PRN 09
Data acquisition date 25 9 1996
L..OBS and LiEST compared for t=7.47 - 7.58 10 4 sec
Unit: [cycles]
244
7.58
x10
I.
0.15
0.1
"RESIDUAL" = difference L1 Observed - L1 Estimated
I
LIGO PRN 09
I1
0.05
0
-0.05
1
-0.1
7.46
7.48
7.
7.5
7.54
j
7.5
7.52
7.54
7.56
Time (sec)
o
0.1,,
=
7.58
x104
* = L1 ESTIMATED
L1 OBSERVED
,I.
1
0 .1 .....
0.0
5 ......
0.
-0.C5 ......
-0
.1
-
iI
-.....
I.
7.46
Fig. A.9.3.a)
7.48
:
I
I
7.5
7.52
Time (sec)
i
7.54
7.56
0028- 0036- PRN 09 (25 9 1996)
Phase residual LL.DIFF, determined as the difference
between LiOBS and LLEST.
Unit: [cycles]
245
7.10
58
x10"
Power Spectra Density
LIGO PRN 09: Single Difference
4
0.1
0.05
0
-0.05
-0.1
.....
9
.....
S...... ..
_
_____:r
_
·
· · · I··
-J
0
.
.
. .. ......
.........
..
... ....
0
,I._
a
I..:::
Cl
,.
n
7.46
7.48
7.5
7.54 7.56 7.58
x 104
7.52
I
. . .
.
.
.
..-.
.
.
.
10-
10l
10-2
10
10"
10"
10-2
10'
Xr
0.1
0.05
Co
w
O
" -0.05
fAf
i
......-...... .....:
-.
..
....
-0.1 ..
...
7.46
7.48
.
i
7.5
7.52
. .. . .
. .,
.
.
.. . . . . ..
*
1
.
7.54
.,,
.
......
0.05 ·- ·.
.... .
0
..
......
...
..
-0.05
-0.1 ........
7.46
7.48
7.5
Fig. A.9.3.b)
.
7.56 7.58
x 104
I
...........
7.52 7.54
Time (sec]
:
7.56 7.58
x 10
0028- 0036- PRN 09 (25 9 1996)
PSD of LLOBS, LL.EST and LIDIFF.
Unit: [cycles]
2
246
LIGO PRN 09 26 9 96 :Single Difference
mA
=41
0
-,, 3
6000
-4000
-2000
-6000
-4000
-2000
-
0
2000
4000
6000
2000
4000
6000
w
cIn-J
0
C.,J
0
r<•
o
CROSS-CORRELATION between L1 OBS and L1 EST
t=.
O
I
eA
e -0.5
.
.
.
.
.
. .
.
.
. .
.
... . . . ....
. ....
.
.
.
......
.. . .
4
-I
.
I
-6()00
-4000
-2000
2000
4000
time delay: Tau (sec)
Fig. A.9.3.c)
0028- 0036- PRN 09 (25 9 1996)
Auto Correlation of LCOBS (Roo) and LCEST (Ree).
Cross Correlation between them (Roe)
Unit: [cycles] 2
247
6000
Auto-Spectral Power Density
LIGO PRN 095:L1
0.15
0.1
I
3000
i-
2500
0.05
-0.1
7.46
i.
7.52
7.5
7.48
I
Time (sec)
*1
.IF.
7.56 7.58
x 10'
. " ..
• .. ."
. ....
. . . •..
.
.'
...
'.
.
.
.
.
.
L
10l
1.~
.~...
-·
.
i"iiii iiiil
.\..--
.-
10
.j
10-2
10-1
frequency [Hz]
CROSS-SPECTRAL Power Density
COHERENCE OF L1 OBS and Li EST
.
--: .::::::: : : : :::::
: :: : : : : : : :
~iiriiii
500C
f.
~.
7.54
i1
81000
-0.05
300C
..
..
..
.
.
. . ...
. 250C
0.8
0.6
: -:--.:::::
:::. :::::
150C
0
-)200C
0
'
150C
0.4
0.2
a. 100(C
....
:...
.
.
...
..
"''
C50
0 L.
10 -4
--
-Z
10e
10
-1
10
:
10-
10-2
frequency [Hz]
frequency [Hz]
Fig. A.9.3.d)
10-
0028- 0036- PRN 09 (25 9 1996)
Auto Spectral Power Density of LC_OBS (Soo) and of
LC_EST (See). Cross PSD between them (Soe)
Unit: [cycles] 2
Coherence function of LC_OBS and LC..EST
248
10 1
GPS "OBSERVED" L2 : LIGO PRN 09 - Single Difference 0028 0036
0.:2
.. ....
0. 1
0 . ......
. .\.
.1
. :..
. . .. . . .....
.....
...
......
..
. .. ,:....... . ... . . ...
:;i
S'•
;.
i
...
: I
-0. 1
I
-0.:2
7.46
I
I
I
7.56
7.54
7.52
7.5
7.48
GPS "ESTIMATED" 12 computed from the analysis of SNR variation
0. 2
7.58
t
xiO
0.1
0-
-0.
--0.
1
2' 46
7.
·
7.48
·
0.2
7.5
7.52
7.54
Comparison L2 OBSERVED vs L2 ESTIMATED
·
·
·
.
.
7.5
7.52
Time (sec)
7.56
·
7..58
4
x 101
0.1
0
-0.1
i
-0.2
7. 46
Fig. A.9.4:
I
7.48
~
Il
I
7.54
7.56
0028- 0036 Satellite: PRN 09- Data acquisition date 25 9 1996
Lz_.OBS and Lz.EST compared for t=7.467-7.58 10 4 sec
Unit: [cycles]
249
58
7.
x 104
LIGO 1995 9 25 PRN 09 - Single Difference 0028 0036
0.2
......................
................... .............
......... ..
..
............... ......
-0.1
I
I
I
7.54
7.56
0_
7.46
7.48
7.5
7.52
Time (sec)
Time (sec)
Fig. A.9.5:
0028- 0036 Satellite: PRN 09
Data acquisition date 25 9 1996
Lz.OBS and LEST compared for t=7.47 - 7.58 10 4 sec
Unit: [cycles]
250
7.58
x 10'
x 10
"RESIDUAL" = difference L2 Observed - L2 Estimated
LIGO PRN 09
0.15
0.1
F
.
.
B
.
0.05
-0.05
-0.1
-0.15
-'..
-
..
.
_n
7.'46
I
I
7.48
7.5
7.52
7.54
7.56
7.58
x 10'
7.56
7.58
x 10
Time (sec)
o = L2 OBSERVED
0.2
* = L2 ESTIMATED
0.1 I-
-
-0.1 i_.....
-n o
7.46
S
7.48
.............
.............
7.5
7.52
7.54
Time (sec)
Fig. A.9.6.a)
0028- 0036- PRN 09 (25 9 1996)
Phase residual L2zDIFF, determined as the difference
between Lz.OBS and LzEST.
Unit: [cycles]
251
LIGO PRN 09: Single Difference
Power Spectra Density
cn 6
m
O
.. ....
..
..4..
...
i
0.1 ·· ·
-0.1
s,
...
.....
. ...
'
j:
''
IJ
o
·
o..
~..:::J
•°° • ° ...
•
a.
_A'#
.
i ii
•
: ::::::::l=
n 2
'''V
...-.
· · · ···· :
o0
i I··lj "
i I.
t
:...: . ........
:
:
o.
:.
.
. °...
.
i......
i'ii'iii
o
•
.oo,'
I
7.46
7.48
7.5
7.52
7.54
7.56
7.58
x 10'
10
4
10-2
10
10-'
6
"· " "
:
,-+A-VV- -- -- - V-- \
-0.1
0n
I
*
7.46
i
7.48
I
7.5
I
7.52
l
7.54
0.2
;1 " .
; ::'•
..' "
i::;
" "
' '
: : : ::
0
co2
CL.
r
i
7.56 7.58
x 10'
u-
'
10-2
10
10l
10-4
u- 6
0.1
' ";
/
:
:::::: :
:
:::::
10
:
:
:::
*1
94
0
o
02
-0.1
-n o
@
7.46 7.48
I i
7.5
Fig. A.9.6.b)
I
I
n
7.52 7.54 7.5( 6 7.58
Time [sec]
x 10'
10-4
0028- 0036- PRN 09 (25 9 1996)
PSD of LC.OBS, LC..EST and LCDIFF.
Unit: [cycles] 2
252
10-2
10-'
LIGO PRN 09 26 9 96 :Single Difference
O
0o
-6000
-4000
-2000
0
2000
4000
6000
-6000
-4000
-2000
0
2000
4000
6000
-t
't
0
I
CROSS-CORRELATION between L2 OBS and L2 EST
i
I
I
I
I
I
i
i
i
i
i
i
I
I
0.5
o
I
0
U
-0.5
-1
I
I
--
I
I
4000
6000
-
-6000
-4000
-2000
2000
time delay : Tau (sec)
Fig. A.9.6.c)
:
0028- 0036- PRN 09 (25 9 1996)
Auto Correlation of LC..OBS (Roo) and LC.EST (Re).
Cross Correlation between them (Roe)
Unit: [cycles] 2
253
LIGO PRN 095 :L2
Auto-Spectral Power Density
0.2
.,,...tI-I.
0.1
0
-0.1
·
0A
7.46
7.48
7.5
7.52 7.54
Time (sec)
7.56
7.58
frequency [Hz]
x 10'
COHERENCE of L20 sB and L2EST
0.8
0.6 '
0.4 ·'
0.2 ·'
At
10
4
······
··
····
··
···
····
··
····
: · ··' `·'·` =-·~i
'·':''·'
····
··
······
··
··
··
·
CROSS-SPECTRAL Power Density
0%k
r.f P£•f%
·· ·· ··
··
·······
·· ·· ··
··
····
·
··
··
·· ·· ···· ······
·· =`·';·'·::
· ·· ··
··
··
···
·· ··· ··
··
· ··
·
··
··
·· ·· ·· ·
~·.·..·······.·2\
· · · 1····t··.
·. · · · ~~·
·· ·
·
· · · ·· ·· ·
·· ·
·
· · · ·· ·· ·
· · ··
·
· · ·· ·· · ·
""
·
· · ·· ·· · ·
·· · ·
·
· · ·· ··· ·
·· ·
·
· · ·· · ·· ·
· · · -··' ""'
· · '·'···-···~L··~
·· · ·
·
· · ·· ·· ·
·· · ·
·
· · ·· ··· ·
·· · ·
·
· ·· · ·· ··
···
·
· · ·· ·· · ·
""
·
· · · · · ·
"·
....
"·
'
' """'
"'·
''
"
io-4
'V''
UY"
102-
10 210
frequency [Hz]
Fig. A.9.6.d)
:
10- 1
10.
1-0
10- 2
frequency [Hz]
0028- 0036- PRN 09 (25 9 1996)
Auto Spectral Power Density of LC_OBS (Soo) and of
LC..EST (Se). Cross PSD between them (Soe)
Unit: [cycles] 2
Coherence function of LC.OBS and LCEST
254
10-I
90 90
GPS DATA:
270
receiver location: SOUTHERN CALIFORNIA
station: COSE (trimble) satellite: PRN 01
acquisition date: 23 1 1997
station: COSE (trimble) satellite: PRN 01
100)
200
• 150
............ ............. ..............
*
100
!
......... .....
.....---
........
.....
.
0
Time (sec)
Fig. A.9.1 :
6
Time (sec)
x 104
Satellite visibility chart for PRN 01 (skymap).
Elevation and azimuth of PRN 01with respect to
the station COSE
255
x 104
1 (trimble)
COSE
I
50
PRN
01 23 11997
·
COSE (trimble)
·
100
I
PRNOI 2311997
I
I
|
I
. ..............-
I
I
I
I
i
5.5
4.5
Time (sec)
3.5
x 104
Signal to Noise Ratio for L1 and L2
320 I-
~10
I
0
I
. . .. IIIII
. .... .
. . ..
. ... .
..
... .
...
..
.. . .. . .. . .. .
. . .. .
4
3.8
3.6
4.6
4.4
4.2
..
. . .. .
. .. . .. ·.
5
4.8
..
. . . II
.l .· .`---~'
... ....
. . .
.
..
. .··
5.2
5.4
. .. .I.. .
5.6
_r
I
20
-
-
3
3.6
I---
.6.8..
3.8
Fig. A.9.2 :
:
--
-I
.. .. .
4
4.2
--
I -:I:I
.
4.4
4.6
Time (sec)
I
4.8
I
. .. .
........
Station: COSE - Satellite: PRN 01
Data acquisition date 23 1 1997
Signal-to-Noise-Ratio (SNR) for Li and Lz.
256
S
5
5.2
.
5.4
5.6
x10
GPS "OBSERVED" Phase Residuals for: COSE (tr) PRN 01 - CALIFORNIA 23 11997
0.2
0
....
-....
...
.
.........
.........................
....
...
;.
......... ......................
.
.
....
-0.2
-0.4
......
....................
................
......... .. -
.... ....................
.....
.......
,
•
'
.
-0.6
-0.8
3.6
0.2
5.2
5.4
5.6
4
4.2
4.4
4.6
4.8
5
3.8
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation x 10
,... ........... ,I......r.....l..... ........
J........
!..
0
-0.2
.-.~......t....................--:
-0.4
........ ........
:~ ..
..
..
..
..
..
..........
*.. ....
-0.6
-0.8
3.6
3.8
- ··!· · ·
0.2
4
5
5.2
4.8
4.2
4.4
4.6
Comparison LC OBSERVED vs LC ESTIMATED
5.4
··!!·· · ··~··
· · ··....
a· · · ·· · l· · ·
........
. ..
....
.....
... .. .. ... .
-0.2
...
. ..
. ... .. .. ... .
5.6
x 10
... .. . ... .. ..
O-0.4
-
-0.6
-0.8
3.6
3.8
Fig. A.9.3 :
4
4.2
4.4
4.6
Time (sec)
4.8
5
5.2
5.4
5.6
x 10
Station: COSE - Satellite: PRN 01- Data acquisition date 23 1 1997
The observed phase residual (LC.OBS) is compared with the
multipath phase error estimated from the SNR (LC..EST)
Unit: [cycles]
257
COSE (trimble) PRN 01 - SOUTHERN CALIFORNIA 23 1 1997
0.2
a
-0.2
-0.4
-0.6
-0.8
5.6
5.4
5.2
5
4.8
4.6
4.4
4.2
4
3.8
3.6
Time (sec)
0.2
.
I
.
i
........
•o
I
......
.,
I
,,
..•
,.
.
I
o,
I
, o
x 10'
o
,
i
o-,, ................
I
•o
.....
.. o
I
I" '
.........
o'o
0
.........
-0.2
-0.4
-0.6
y.o
........ ............. .
. ...
3.6
3.6
.
,..
..
3.8
3.8
Fig. A.9.4 :
..
.,
.
..
.
4
4
..
.
,.
...
• ..
.
.
.,,.
.
.
.•.
4.2
4.2
.......
•
:''''~'''' i~''`'
.
4.4
.....
..._.....
...
· · ·· · ·
•·
~~~·~····:·
....
.
4.6
4.8
.
.
· · ·· ·.··
·''~'·''
·
.,
.
5
,.
.
.
.
,
5.2
Time (sec)
Station: COSE - Satellite: PRN 01
Data acquisition date 23 1 1997
LCOBS and LC_.EST compared for t=3.5 - 5.7 10 4 sec
Unit: [cycles)
258
t'''
:'
.
°
.
5.4
..
.
.
5.6
x 10'
"RESIDUAL" = difference LC Observed - LC Estimated
3.5
3.55
3.6
3.65
3.7
3.75
Time (sec)
o - LC OBSERVED (COSE 01 trimble Part la)
3.5
3.55
3.6
3.65
3.7
3.8
3.85
: Part la
3.9
3.95
* LC ESTIMATED (COSE 01 trimble Part 1a)
3.75
3.8
3.85
3.9
3.95
Time (sec)
Fig. A.9.5.a)
:
COSE - PRN 01 ( 23 1 1997) - [segment la]
Phase residual LCJDIFF, determined as the difference
between LC.OBS and LCEST.
Unit: [cycles]
259
4
x 10'
4
x 10
I
...
.
.
%.t..........
2.
..:
Ile
:/
•
:
1
,•,
. ,.,
I
t
•..-•
·
......
.. . -S··
*.A
I:···
.···'··'·
·- .· . * .
i'
%
. ......
. .• . .....
. . ..
. .. .. ...
.
. . 2.
..
.
2
•
· ~···~·-····-·-
· · ·
4r·····s·5·
o
SGO 31
_r-
N
I
· · · · II
.........
•-............
:
S00 01
Sd
SBO 0"I lo (•Sd
.
·· · · ·i ·
........
ii
..
•.
..
........
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~
I
I
0o
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H~H
I~····I KŽ
l
N
0o
:........J............
::::{::ii:::•"
.
::
%..........
. .:..
o
r-
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C
.....
:::::
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S......
.....
... !....z.....
N
3•.
It
; .... ;. ...
(D
r
IS3 110
lo Sd
....
00
N
oSo 31
c
c
....
............
o ..........
C%
C)
......... :
...............I
....
................
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.. .
ot
o
o
o
=1i031o
0
CISd
cooD
,6
1-0la 01
*
2.J
vjE P,
1=·
,
0
,1
.U
eeb
o-4
C/)
u9QO'
I..z1o
0c
COSE (trimble) 23 1 1997
C,
segment la
0
oo
-1
-60 00
-4000
-2000
-4000
-2000
0
F·v
w
0
O
0
I
-6000
2000
4000
60400
2000
4000
60 00
CROSS-CORRELATION between LC OBS and LC EST
I
S..
.....
0.5
..............
..........
o.......................
0
2o
o0
I
U,LO)
00
.. . .. .
0
-0.5
...
.. ..
..
..
.. ..
. ..
.. .. .
..
.
.
.. .. . .. . .
.
.
...
•
.
.
.
..
.. . . . .
.
. . . . . . . .
.....
. . . . . .
_.
-6000
-4000
-2000
2000
4000
time delay :Tau (sec)
Fig. A.9.5.c)
COSE - PRN 01 ( 23 1 1997) - [segment la]
Auto Correlation of LCOBS (Roo) and LC_EST a(t).
Cross Correlation between them (Ro)
Unit: [cycles] 2
261
6000
n.
01
o-'-a*
0
a
0
0
0
-L
>
ii
0
..........
I............
: :
m
o
oo
0O
o
...........
.
.
.
.........
.
. ... .a
.......
.....:
.
..* ...
I..:
.I....*
.......
......
.........
. .
X
......
......
:.
°
.....
. .. ... ..
...........
... . . . .............. . ..
..*. .. .
...........
.. ......... ... . . . ...........
Co
0
AMPLITUDE of S,(f)
(
01o
I
JE
o
0
I
(D
%..
eut
CA
o0
-L
h
in
RCA)
L"
CA)
...
.
.
' ..
.. ...
....
"
01
0
.'
01
0
....
0
C
..
..
.....
01
0
.
.
.
.
''
i
Ko
0
''
.. .......:, ........
.., ."".....
*..... .. ..
.....
S,-...
....
o
0
Soo(f) and S (f)
-&
o0
LC OBS / LC EST
.*... ......
*.......*%......
I::::
I:::: :
O
..........
1134
-LI
-L
"1
-L
Coherence
"RESIDUAL" - difference LC Observed - LC Estimated
A
: Part lb
=
0.1
0.05
-0.05
04
4
4.1
4.2
4.3
4.5
4.4
4.6
4.7
4.8
4.1
4.2
4.3
4.5
4.4
x 10
* LC ESTIMATED (COSE 01 trimble Part Ib)
o - LC OBSERVED (COSE 01 trimble Part Ib)
4
4.6
4.7
4.8
COSE - PRN 01 (23 1 1997) - [segment Ib]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LC.EST.
Unit: [cycles]
263
5
4.9
Time (sec)
Fig. A.9.6.a)
5
4.9
Time (sec)
x 10
Ct,)
oo
10
03
g-
C
-.4
'.
CO
o
0L
-&
0
o°
o.....
0
.
....
,.
o
l
e
... ol.....
e
.... .. .... ... .. .. .. .
...
,,
.A.
0
r
%..
.....
:
.....
...
°.o .....
m
PSD of LC DIFF
0
LC DIFF
.
.
• ..... • ....
...
...
. ....
...
t
... ....
....
... .,o
• . ...:::::::::::::
.......
. i .. i ..•.. ,. .
:::::::i
...
'b1.
Do
-0&
io
o
I
0
.4
Co
0
O
......
o
•..... ...... •. .
.'····C'''"
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.... ,i.....
...
| ... I. .
....
.iiii
iiiii.i...i iii
....... . ..
....
. .. ...
. ...
....
.....•..... •..... 0. .
....
.
... .. ....
..
. ... ... .........
.
....... ........ ...
tr cJ .0
u
o
PSD of LC EST
o
LC EST
u
L
O
:.......
°.
I..
. . ., .
..
.,.
.....
i ...
~
. ..
i ...
~I.. ~
.....
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.. . .. .
. .. .
.,
... . . . .
, , •
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. • :. . :.
. .... .. . ..
%..,
C#
3
....°°• .
rA
.°e••.. °....
..
........
•o
....
.
01
0
PSD of LC OBS
0
LC OBS
COSE (trimble) 23 1 1997
segment lb
m 1
03
O
t-
0
-1
-0.
___
-1
-0.8
-0.6
-1
-0.8
-0.6
0.06
I ___
0.4
0.6
0.8
0.4
0.6
0.8
0I
0.I
0.I
-0.2
0
0.2
-0.4
-0.2
0
0.2
I
~
0.t
I
-0.2
-0.4
-.
0I
I-.. I
O
CROSS-CORRELATION between LC OBS and LC EST
1
x 10
o
O
I
-1
-0.8
Fig. A.9.6.c)
-0.6
:
-0.4
-0.2
0
0.2
time delay :Tau (sec)
0.4
0.6
COSE - PRN 01 (23 1 1997) - [segment lb]
Auto Correlation of LC.OBS (Roo) and LCEST at,.
Cross Correlation between them (R
Unit: [cycles] 2
265
0.8
1
x104
U,
0
o
CL-
0
0
CL
0..
0)
0.
I
0
..... ..... .
I
o
r0
it
ks
( 3·.
%
....
(C
.. .*.*.
....
........
...
... ... . ....
0
. ........
......
0
(f)l0S pue ()°s
0
..........
:::::::::::::::::::............
o
....................
ccc
CC
Cc
o
IS3 01 / S0O 31
c
co
m
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wo
C)
00
C/)
o
o
z
cr
0
......
% .°,,
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o0
.
*......
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...
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:::::::::
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c)
p-.I:
,-
a
(0
cr
T-
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....
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.
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(I) S lO 3EinldnidVV
,:......
::: :::::::::::::::
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.
|
o
eouo.otoo
o
*...
....
•
I
.. .
I
c
r--
0
'44
04
C
0
C/
0.=
-e
--
U
"RESIDUAL" = difference LC Observed - LC Estimated
5
5.1
5.2
5.3
5.4
5.5
: Part lc)
5.6
Time (sec)
o - LC OBSERVED (COSE 01 trimble Part Ic)
5
5.1
5.2
5.3
5.4
5.5
5.6
COSE - PRN 01 ( 23 1 1997) - [segment 1c]
Phase residual LC..DIFF, determined as the difference
between LC..OBS and LC.EST.
Unit: [cycles}
267
An
IU
* - LC ESTIMATED (COSE 01 trimble Part 1c)
Time (sec)
Fig. A.9.7.a)
5.7
vA
5.7
x 10'
0D
w
U)
-c
a-
z
cc
a-
w
Cl)
a
oO
L". . • t •• • •
.. . .
..:..:::::::
.. ......
...
.
. ..
5
0
Cl
0
In
Cul
gh
17
Cl
T-
::.
o
....
" ........
..............,
......
....
:...............
D 3o
o
7 o cm
SGO 31o0 aSd
o
J~.,
4
I
$'
/"~""
*1r
...
o
O
0 Bco
to
O;
I~o
.
...................
S
. .
I•g
Co
m C0
. . . . . .
:::::.::::.:::::.::::
.1.e
oo
1S3 0110 OSd
IS3 01
%
..
..............
.........
................
E
1=
a)
.0)
I..
..........~.....,
-oCD IV OC
IOla31 1OGSd
31110 31
9
u
,..g
0C/)
eU9
°\P1'
COSE (trimble) 23 1 1997
0e 0
segment 1c
0
co
O
o
0
-•-000 -6000
I--
-4000
-2000
0
2000
4000
6000
8000
-4000
-2000
0
2000
4000
6000
8000
6000
8000
U)
o
-I
o
I
.2
-8000
-6000
CROSS-CORRELATION between LC OBS and LC EST
L.
0
co
O
o
0,
o
- -0.5
-I
4
-8000
-6000
Fig. A.9.7.c)
-4000
-2000
0
2000
time delay :Tau (sec)
4000
COSE - PRN 01 ( 23 1 1997) - [segment Ic]
Auto Correlation of LCOBS (Roo) and LC_.EST (Ree).
Cross Correlation between them (Roe)
Unit: [cycles] 2
269
C
..
t-
L.
0)
a,
0.
(I
0
2
.
X :: :*: : ::ý :: -::::
oo0
{:
C)
0
N
X
,
Son
0)
:3
0*~
0
CD
X
VN
It
ui
I
I
CD o
0
0
I
d
I
I
d
0
o. .. .o . ... . .
..........
0
..
o .. . o
. ...
. ...
.. . ....
.
. ...
. . . .. . ... o.
X..
. . o.. ..
. .. .. ..
..
...... .; ...
.
................
·...........
.
......
lo
o
o
Sco
o
................................
....·..... ~.... ............. .
c>
C
ii >
LU
0
U)
a.
CYu
(D
E
0,
0a
a,
B,
2,
w
C'
0e
0J
1
IS3 01 /80 01
.• .., .. °.....
°
.
•........
°
..
. .. . .. .
.. . .: .
..
0a*
c12
T-
g
cuI
.,.....
..... ..:......
.... .
......
......
:......
.
......
.
......
.
......
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..
..
o
.Z...,,..
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..
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.
o,....·.
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.: ...
...
. . .. . ... . .. . .. .
. ..
: ..
. .. ..
0
. . . . .... . . .
...
..
..
..
I
.
..
:
li
0
09 0
0
...
..
·.
0w
0
|
....
...
. ...
.. .. ..
0
:.. .::::......
..
....
::::'::.:::
..
:
(I) Slo :l(l'lldVV
W
_IV
OW
.
o0
I
0 ,-
I
o
0
c I 00
4j1
'o4
03
u-eCi
U48
..
.....
...
·....
......
,.....
......
.............
......
D
eo
I
..
..
.. ..
i
CO
i
...... *...........................
cm
)
....... ..
0
LI
0
0
ii
-
7
.
...............
90 90
0
270
receiver location: SOUTHERN CALIFORNIA
GPS DATA:
station: COSE (trimble)
400
0,
z
......
.......... ...............
0
100
4
6
Time (sec)
8
e000010
····
~····(····s~~~~·
0
0
A
10
x 104
†
..
200
w
2
4
·
·
6
8
Time (sec)
Fig. A.10.1 : Satellite visibility chart for PRN 07 (skymap).
Elevation and azimuth of PRN 07 with respect to
the station COSE
271
1
I
S300
2
satellite: PRN 07
station: COSE (trimble)
satellite: PRN 07
1
0
acquisition date: 23 1 1997
10
x 104
COSE
PRN 07 23 1 1997
100
3
2
1
8
7
6
5
4
x 104
Signal to Noise Ratio for L1 and L2
30
I
SI
20
.
..
......
.
.
...
...
...
..
..
...
...
..
I.
I
i
I
...
..
...
...
. .... ...
..
Z10
0
1
2
3
4
I
I
I
I
--
5
A
6
m
7
A
8
Ci
z 20
I
Time (sec)
Fig. A.10.2 : Station: COSE - Satellite: PRN 07
Data acquisition date 23 1 1997
Signal-to-Noise-Ratio (SNR) for LI and L2.
272
x1o
GPS "OBSERVED" Phase Residuals for: COSE (tr) PRN 07 - CALIFORNIA 23 1 1997
0. 5
o
cO
w
0
S -o.
0
14000
2000
4000
6000
8000
10000
12000
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation
..........
. ............
. . .............. ..............
. . ............
. .............
...........
I
0
2000
...... ...... . * ......... .. :..
0.5 . ............
0
Qb'·······
.
......
4000
6000
8000
10000
Comparison LC OBSERVED vs LC ESTIMATED: part 2
I
'5······I
..
........
2000
4000
·! ·
•.
·
· ·
·
·
'2·
I
........
.... *..
........
.
· · ''·····
l
14000
12000
.......
.
.....
. ...
I
.... *...
-0.5
6000
8000
Time (sec)
10000
12000
Fig. A.10.3 : Station: COSE - Satellite: PRN 07- Data acquisition date 23 1 1997
The observed phase residual (LC.OBS) is compared with the
multipath phase error estimated from the SNR (LCESIT)
Unit: [cycles]
273
14000
COSE (trimble) PRN 07 - SOUTHERN CALIFORNIA 23 1 1997
...........
..........
..........
.........
.........
...
:.........~.............=...
........... ......
0.6
0.4
....
...
...
...
...
..
I .I.I. II.IIIII .I I..I.
..
I III I..
III III III
0.2
ii
0
II .I
.
I
I
I II II II II II IIII I
I
I
I
I
I
I
I
-0.2
I .I..I
.
... IIIIII
.. I..
.
.. I.
I.
.
......
IIII..I..
-0.4
-0.6
)
2000
4000
6000
8000
Time (sec)
10000
12000
141000
data set : part2
0.6 . ....
0.4
......
. . .
......
.:-•.
............. :.............. .............. . .............
.
............
0.2
IIP
•,.
0
-- :** - -- -- - :--- - -- - - :--- - -- -
:
-
- - "--
.*"--,"\- 9- - -
-0.2
I
-0.6
0~4)
1 '
IIII III II.
III III .II
..
IIII III IIII IIIIIIJIII III III IIIIII .II
2000
I
4000
I
I
8000
6000
Time (sec)
10000
12000
Fig. A.10.4 : Station: COSE - Satellite: PRN 07- Data acquisition date 23 1 1997
LCOBS and LCEST compared for t=0.0 - 1.4 10 4 sec
Unit: [cycles]
274
14( 00
"RESIDUAL" = difference LC Observed - LC Estimated
: Part la
rrr
Time (sec)
o - LC OBSERVED (COSE 07 tr Part la)
0
1000
Fig. A.10.5.a)
2000
* = LC ESTIMATED (COSE 07 tr Part la)
3000
4000
Time (sec)
5000
6000
COSE - PRN 07 ( 23 1 1997) - [segment la]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LCEST.
Unit: [cycles]
275
7000
.1
.
: .'
..... .....
... ° ......
F7T..T~ThT1
..
.... ,.....
%
.... ;...............
-
If
w
o
f-%
o
oa
CD
0C
6)
N
..
.•.... ........ :
•
.
.•..f...~·~·
N IL
SO80 01 JO0Sd
S8O 0
1S3 0 j0o aSd
IS3I
18S1
01
0
0
0
Co
0
0
o
o0o
Oj
O%
1J10 017 lo 0Sd
OJIO 01
0
0
0
0
0-
E
F
O en
I4
0
COSE (trimble) 07 23 1 1997
1
in
0
03
ca
segment 2a
0
-j
t0 0*
0I
.2
I
I
I
I
I
I
: -1
-80( 00
LU
-6000
-4000
4000
4000
2000
-2000
I
6000
6000
8000
6000
8000
i-
-- I
t
·
I
-80O0
I
-6000
I
-4000
I
-2000
I
I
2000
4000
CROSS-CORRELATION between LC OBS and LC EST
...
.
.
.
.........
I°............
..
-0.5
.
-8
..........
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.
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.
...
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L
-8000
-6000
Fig. A.10.5.c)
-4000
-2000
2000
time delay : Tau (sec)
4000
6000
COSE - PRN 07 ( 23 1 1997) - [segment la]
Auto Correlation of LC.OBS (Roo) and LC.EST (Re).
Cross Correlation between them (Roe)
Unit: [cycles] 2
277
8000
u)
0
a.
CL
0
L-
0. o
....o.......
. .,
%.
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.
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: Part lb
"RESIDUAL" = difference LC Observed - LC Estimated
.I
0.4
I
. ............
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.....
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I
%......
I
.............. .......
o.•.%.
..
..
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O0
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..............
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-0.6
I
I . . .
I
II
0.7
.... ......
...........
..
*..............
......
I
•
•
•
|
Time (sec)
0.4
0.2
'' ............. ............
.............
..
...........
-0.2
..
.. . . .
I
I
...............................
............. ............................
-·········(····
x104
* LC ESTIMATED (COSE 07 tr Part Ib)
o - LC OBSERVED (COSE 07 tr Part Ib)
.... . .I
1..4
1.3
1.2
0.9
.%~·
· ··· ···· ··· ile·
··· ··· ··· ··· ···· ·· f · ···· ··
.**
. . . . . ...
. . . . .. . . .
-0.4
-0.6
. . .. . .. .. . .. . .. ..
I
nI
i
I
.
..
..
.
.
I
.
.
.
.
.
.
.
I
.
.
I
1.2
0.9
0. 7
I
Time (sec)
Fig. A.10.6.a)
:
COSE - PRN 07 ( 23 1 1997) - [segment Ib]
Phase residual LC.DIFF, determined as the difference
between LC..OBS and LCEST.
Unit: [cycles]
279
1.4
x 10
.
.....
I....,....
.
0o
.....
.I.........
.
......
..... •.. .
)
I
ci
S80 031 o OSd
:...
. ....
...,.. ....:.. ..
................
..................
Il
c
880 07
I
............· · · ·
:::::::~:::::·:::::
..................
.....
....
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oo..e...............
:
....
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.........
...
co 0"
I
tu
IS3 01Oc1Sd
0
5
g
Cm o>
.........
....
...
.....
.....
..
..
co
In
c;
IS3 07
..
7....·..·;........
.....................
·. . . .. .. .. .. ..
01
(5
.
g
i=...
co
ci
aO
"ki
uII
4)
uuF
..........
······
... .. .
....:.
o
3
J11 01 10OSd
... ..
. . ...... .. ..... .. ...
...
. .. .. ...
. . ..
.
. ..
to
rl
COSE (trimble) 07 23 1 1997 segment 2b
P'
O
(a
o
o0
b:
Q-8000
-6000
-4000
-2000
-6000
-4000
-2000
0
2000
4000
6000
8000
2000
4000
6000
8000
tc/)
LU
0
i._I
-8000
CROSS-CORRELATION between LC OBS and LC EST
I
*1I
I
0.5
0
I:
-0.5
i
*I..''·....
C----j...-
--------- -------L
_1
-8000
-6000
-4000
2000
-2000
4000
6000
time delay :Tau (sec)
Fig. A.10.6.c)
COSE - PRN 07 (23 1 1997) - [segment Ib]
Auto Correlation of LC.OBS (Roo) and LC.EST (Re.
Cross Correlation between them (Re)
Unit: [cycles] 2
281
8000
0
0
.
0
·I
dI
_1
dI
·
8.····~·
..........
. . ..
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12
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15
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: 45,
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. .........
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21
30
.
240
.
300
270
receiver location: SOUTHERN CALIFORNIA
GPS DATA:
station: COSE (trimble) satellite: PRN 25
^^A
.........
acquisition date: 23 1 1997
station: COSE (trimble)
satellite: PRN 25
2
C)
40
-
.......... Ile
........
I
I
I
I
4.5
5
5.5
6
Time (sec)
I
~
6.5
x 104
4
4.5
5
5.5
Time (sec)
Fig. A.11.1 : Satellite visibility chart for PRN 25 (skymap).
Elevation and azimuth of PRN 25 with respect to
the station COSE
283
6
6.5
x 10
COSE (trimble)
PRN 25 23 1 1997
80
.60
····
-------- -------· ·· - · 1/--**--.******* ********----·
9I 40
; ··-- - -- - * ** * -* - - -- * ** *
--------
-w 20
0
4I
6.5
xlo'
5.5
4.5
Time (sec)
Signal to Noise Ratio for L1 and L2
w,.r
------------ *
S
-
-
I
-I
iiiii
2C
C
4
6C
........II
40 -.
B
I
................
4I
n
5.5
.......... I .........
I
ii
ii
iiii
20
0
i
4.5
6.5
I..
.
ii
ii
5.5
4.5
Time (sec)
Fig. A.11.2 : Station: COSE - Satellite: PRN 25
Data acquisition date 23 1 1997
Signal-to-Noise-Ratio (SNR) for LI and L2.
284
x10
GPS "OBSERVED" Phase Residuals for: COSE (trimble) PRN 25 - SOUTHERN CALIFORNIA 23 1 1997
0.2 ;...................:...................................................................
0
4t.:.3
;t"*."%.
.oT:....-.
.. ... . .. ........
......
..
j'
.
4
p·,:.iS
..........
.........
\
.......... ...... , .. -
-0.1
.
.1........
-0.2
,
4.6
4.4
GPS
...........
I......... *I
4.8
5
. .
...
I ..........................
TI I*
5.6
5.4
5.2
5.8
·
.·
A LC·
6
_L
6.2
"ESTIMATED
"
n
.......
* .........
..............................
Phase
Residuals
computed
.. ..
. .....
from
........
the
analys
E
S0
CO
. .
- -0.5
..
.....
..........
II
4.8
4.6
4.4
.............................
Ii
• - - - -°
I
4.4
,
I
--
--
!
•
.. . ...........
I
4.8
!
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-
•
5
6
6.2
x 10'
.
.
•
I
I
.
..........
..
..
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*
4.6
.
I
5.8
5.6
5.4
5.2
5
Comparison LC OBSERVED vs LC ESTIMATED
....................
•
-0.5
....
........
5.4
5.2
Time (sec)
-
.
i
•
•····r·····~·····
I
I
II
5.6
5.8
6
6.2
Fig. A.11.3: Station: COSE - Satellite: PRN 25- Data acquisition date 23 1 1997
The observed phase residual (LC..OBS) is compared with the
multipath phase error estimated from the SNR (LC_.EST)
Unit: [cycles]
285
x 10
"RESIDUAL" = difference LC Observed - LC Estimated
4.4
4.5
4.6
4.7
Time (sec)
o = LC OBSERVED (COSE tr)
4.8
: Segment #1
4.9
5
x 10
* = LC ESTIMATED (COSE tr)
•ldL"
U.1
I
I
I
I
I
I
I
I
I
I
4.8
4.9
0.1
0.05
-0.05
-0.1
Ii
4.6
Fig. A.11.4.a)
4.7
Time (sec)
COSE - PRN 25 ( 23 1 1997) - [segment #1]
Phase residual LCDIFF, determined as the difference
between LC.OBS and LC.EST.
Unit: (cycles]
286
xlo'
0
oo°
.|
•.
oo:°.....
%.... . .. ,...
7....
.,
..
.. .........
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% . . . . .o.. .•
.. . .
. ... .. . . ... . . ...
0.
..........
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=...
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04
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u
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COSE (trimble) PRN 25 - California 23 1 1997 segment 1
C,,
0
I
°°°.°°•
° °
..
0
0
n
I
2000
4000
6000
8000
2000
4000
6000
8000
......
-8000
-6000
-4000
-2000
-6000
-4000
-2000
in
o
-.J
o
_j
I
o
t=
-8000
CROSS-CORRELATION between LC OBS and LC EST
I
1
''''''
0.5
....
0
-0.5
E
.4
-8000
....
....
..
..
..
li b
-6000
Fig. A.11.4.c)
I
.......................
..... ............ ...........
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.
.
.
................................
I
-8000
0
.
.
.
.
.
.
.
.
.
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.
....
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1~~ II
B
I
-4000
-2000
m
2000
time delay :Tau (sec)
I
m
4000
I
6000
COSE - PRN 25 (23 1 1997) - [segment #1]
Auto Correlation of LC_..OBS (Ro) and LC..EST (Ree).
Cross Correlation between them (Roe)
Unit: [cycles] 2
288
I
I
I
1
8000
>1
U)
0
CL
U)
0
0
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124
0
"RESIDUAL" = difference LC Observed - LC Estimated
I
I
I
.
.............
: Segment #2
I
...........
I
...........
.
.
.".
.......... ..
...
.....
.......
.....
.......
.......
.....
......
----....
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.....
.
-0.1
5
o-*o
oo ·.- ·.
o.··.··.····.oo-oo.
1.o
oo
oo
,
.o
5.1
Fig. A.11.5.a)
o
.-
o
5.5
5.3
Time (sec)
5.1
o = LC OBSERVED (COSE tr)
5
o-
5.2
x10
* = LC ESTIMATED (COSE tr)
5.3
Time (sec)
5.4
5.5
5.6
COSE - PRN 25 ( 23 1 1997) - [segment #2]
Phase residual LC.DIFF, determined as the difference
between LC.OBS and LCEST.
Unit: [cycles]
290
x 10
N
I
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GPS DATA:
00
270
receiver location: SOUTHERN CALIFORNIA
station: GS29 (Ashtech)
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satellite: PRN 01
acquisition date: 24 1 1997
station: GS29 (Ashtech)
^^
satellite: PRN 01
i
200
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Time (sec)
Time (sec)
x 10
Fig. A.12.1 : Satellite visibility chart for PRN 01 (skymap).
Elevation and azimuth of PRN 01with respect to
the station GS29
298
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PRN 01
GP29 (Ashtech)
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0
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Time (sec)
x 10
Signal to Noise Ratio for L1 and L2
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180
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3.5
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4.5
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Time (sec)
Fig. A.12.2 : Station: GS29 - Satellite: PRN 01
Data acquisition date 24 1 1997
Signal-to-Noise-Ratio (SNR) for LI and L2.
299
.......
5.5
......
6
x 10'
GPS "OBSERVED" Phase Residuals for: GS29 (ash) PRN 01 - CALIFORNIA 24 1 1997
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Fig. A.12.3 : Station: GS29 - Satellite: PRN 01
Data acquisition date 24 1 1997
The observed phase residual (LCOBS) is compared with the
multipath phase error estimated from the SNR (LC_ES'I)
Unit: [cycles]
300
x 10
GS29 (Ashtech) PRN 01 - SOUTHERN CALIFORNIA 24 1 1997
4
3.5
4.5
5
5.5
Time (sec)
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3.5
4.5
Time (sec)
Fig. A.12.4 : Station: GS29 - Satellite: PRN 01
Data acquisition date 24 1 1997
LC.OBS and LC.EST compared for t=3.5 - 5.7 10 4 sec
Unit: (cycles]
301
5.5
x 104
"RESIDUAL" = difference LC Observed - LC Estimated
3.3
3.4
3.5
3.6
3.7
3.8
: Part la
o - LC OBSERVED (GS29 01 Ash Part la)
4
3.9
Time (sec)
x 10
* = LC ESTIMATED (GS29 01 Ash Part la)
0.2
0.1
MO 0
o -0.1
-0.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Time (sec)
Fig. A.12.5.a)
:
GS29 - PRN 01 ( 24 1 1997) - [segment la]
Phase residual LC..DIFF, determined as the difference
between LC.OBS and LC..EST.
Unit: [cycles]
302
4
x 104
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I
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-1000
0
1000
time delay : Tau (sec)
2000
3000
GS29 - PRN 01 (24 1 1997) - [segment la]
Auto Correlation of LC..OBS (Roo) and LCEST (R).
Cross Correlation between them (Roe)
Unit: [cycles] 2
304
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4.1
4.2
4.3
4.4
4.5
Time (sec)
o - LC OBSERVED (GS29 01 Ash Part Ib)
-W
4.6
4.7
4.8
5
4.9
x 10
*= LC ESTIMATED (GS29 01 Ash Part Ib)
t
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4.1
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4.2
:
4.3
4.4
4.5
Time (sec)
4.6
4.7
4.8
4.9
GS29 - PRN 01 ( 24 1 1997) - [segment lb]
Phase residual LCDIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
306
5
x 10"
3
C
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.
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.
PSD of LC DIFF
........
....
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LC DIFF
*cn
Cil
k
a
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0
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GS29 PRN 018 Califomia 2411997
segment lb
c8
0
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.
-2000
I
I
0
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2000
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4000
6000
I
I
8000
0
o
0I
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IIII
-8000
-6000
I
-4000
0
-2000
2000
4000
6000
8000
CROSS-CORRELATION between LC OBS and LC EST
0.5
0
to
0
U,
0
U..
UO
-0.5
I
I
I
-8000
Fig. A.12.6.c)
-6000
I
-4000
I
I
-2000
0
2000
time delay :Tau (sec)
I
4000
I
I
6000
8000
GS29 - PRN 01 (24 1 1997) - [segment Ib]
Auto Correlation of LCOBS (Roo) and LC..EST cR).
Cross Correlation between them (Ro)
Unit: [cycles] 2
308
0
0a-
U)
0.
U)
33
00
Hill
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5
5.1
5.45.
5.25.
m
5.1
5.2
5.3
5.4
5.5
5.
5.
5.6
5.7
Time (sec)
-0.
o = LC OBSERVED (GS29 01 Ash Part Ic)
.
5.8
x 10
* = LC ESTIMATED (GS29 01 Ash Part Ic)
0.e
2 .......
0.9
2 - --· --· · ·
4
·-
......
...... .......
.......
-
g
:------------·
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·
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·
·
·
·
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·
·
·
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i
I
Fig. A.12.7.a)
i
5.2
5
:
i
i
5.4
Time (sec)
5.5
5.6
5.7
GS29 - PRN 01 (24 1 1997) - [segment ic]
Phase residual LC.DIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
310
5.8
x 10'
....
,
.....
5.. . .·..
..........
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.
..
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GS29 PRN 018 Califomia 24 11997
segment 1c
-6000
-4000
-2000
0
2000
4000
6000
-6000
-4000
-2000
0
2000
4000
6000
0
0
0
CROSS-CORRELATION between LC OBS and LC EST
I-
0
.. .. . .. .
-0.5
..........
.. . .. . ..
. .. .
.... . .. . ...
I
-6000
-4000
2000
-2000
4000
time delay : Tau (sec)
Fig. A.12.7.c)
GS29 - PRN 01 ( 24 1 1997) - [segment Ic]
Auto Correlation of LC.OBS (Roo) and LC=EST (Ra.
Cross Correlation between them (Ro
Unit: [cycles] 2
312
6000
-
H
C.)
0
C),
-I
0t
0
C..
'.3
E.
V)L0r
0ij1
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o
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c:m C,,0
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o
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Ib
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..
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.
...........
··
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GPS DATA:
.
0
300
270
receiver location: SOUTHERN CALIFORNIA
station: GS29 (Ashtech)
satellite: PRN 07
acquisition date: 24 1 1997
station: GS29 (Ashtech)
100
satellite: PRN 07
400
300
i--·--
................
· c ~·-
200
· ··· · ·---
··
· · ·· ·
...............
100
0
2
4
6
Time (sec)
8
0
10
x 10
0
2
4
Fig. A.13.1 : Satellite visibility chart for PRN 07 (skymap).
Elevation and azimuth of PRN 07 with respect to
the station GS29
314
6
Time (sec)
8
10
x10o
GP29 (Ashtech)
PRN 07 23 1 1997
1
0
I
.L
0
1
2
3
4
5
Time (sec)
6
7
8
9
x 104
Signal to Noise Ratio for L1 and L2
250
a: 200
150
............ ......-.--
.. . . ..-......... .......... ..........
..........
.....-
-- -- - -...... ..........
.......... -.........
-.......... -.........-0
2
1
3
4
5
6
7
8
250
......
200
I\j.. .
150
.·-·...
..
°,......
...
--- --- --- -- : --- --- - -- --- --- --- ---- -------- --:
. -:-
100
50
*--.
.........
..........**
***......... ..
.....:*
..........**---
°° . ...
0
Time (sec)
Fig. A.13.2 : Station: GS29 - Satellite: PRN 07
Data acquisition date 24 1 1997
Signal-to-Noise-Ratio (SNR) for LI and L2.
315
...........
t -
9
x 104
GPS "OBSERVED" Phase Residuals for :GS29 (Ashtech) PRN 07 - California 24 1 1997
0.2
0
-
-0.2
-
-
I
..........
............................... .....
.............
\.'.:::''''''
........
-0.4
-
-0.6
151000
10000
5000
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation
0.2 .
0
...... ........ .. ...
. ..
.. ..
.. ..
.. ..
..
.
..
-0.2
.
...
. ...
.........
... ...
..
.
..
. ... ....
.. ..
.. ..
.. ..
..
.
..
.
.. . ... ... ... ... ... ... ...
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
...
-0.4
.
.. .........
..
.
.
...
..
...
.. ..
...
.. ..
...
.. ..
...
..
...
...
.. .. .. .. .. .. .. ..
-0.6
10000
5000
Comparison LC OBSERVED vs LC ESTIMATED I
154000
0.2
0
.,.............................-
-0.2
-0.4
-0.6
....
..
..
..
....
....
..
..
..
..
....
..
....
..
...
..
....
..
....
.........
..
.........
.....
...
..........
...
..
..
..
..
..
..
10000
5000
Time (sec)
Fig. A.13.3: Station: GS29 - Satellite: PRN 07- Data acquisition date 24 1 1997
The observed phase residual (LC.OBS) is compared with the
multipath phase error estimated from the SNR (LC_.EST)
Unit: [cycles]
316
15( 000
GS29 (Ashtech) PRN 07 - California 24 1 1997
I
1
i
0
-0.2
-0.4
-0.6
I
I
i
i
10000
5000
15000
Time (sec)
. ..
.
.
.
.
..
.
°..
.
.
.
....
................................
............
..................
.r•
.H '••..
,....
:
.,.,(,,.',.*,,:.-,.
.., ,.,,.,., . .',-,..,.+•,..-,,.,,.,.,,-.
,'-:
"',.":)~:2·+··~·~···
•u • ,,I.
,'
..
•..."¥
.;a......
4x?
-0.2
...... ......
....--- ·.......
o -0.4 ................
I I
..... ...............................
...........................
|
.°..•........°% °..........
...............
°°.....-
. ....
-0.6
-0.6
10000
5000
Time (sec)
Fig. A.13.4 : Station: GS29 - Satellite: PRN 07- Data acquisition date 24 1 1997
LC.OBS and LCEST compared for t--0.0 - 1.4 10 4 sec
Unit: [cycles)
317
15000
"RESIDUAL" = difference LC Observed - LC Estimated
v.
II
0
1000
2000
3000
4000
Time (sec)
o = LC OBSERVED - GS29 (Ashtech) 07
-%.
5000
: seg 2a
6000
7000
* = LC ESTIMATED - GS29 (Ashtech) 07
I
0
1000
Fig. A.13.5.a)
2000
3000
4000
Time (sec)
5000
6000
GS29 - PRN 07 ( 24 1 1997) - [segment la]
Phase residual LC-DIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
318
7000
@@@@
o@
..
. o ..
Of
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0
U).
,~~.ooo
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d
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0,
c
P
o
oo
Cl
0
0
o0
Cd
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4
4)o
0
C3O4
''
GS29 (Ashtech) PRN 07 - California 24 1 1997
(A.
segment 2a
m1
0
-j
t:0 0
C.)
-8000
-6000
-4000
-2000
-6000
-4000
-2000
0
2000
4000
6000
8000
2000
4000
6000
8000
c)
-J
o
01...
0
oo
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2Q*
0U
0-Uu
CROSS-CORRELATION between LC OBS and LC EST
- ..
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I
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0.5
0
•"•""••"-
.
--... ..... r
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†
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0
I-
0
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I
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i
. .
_1
-8000
-6000
Fig. A.13.5.c)
-4000
-2000
2000
time delay :Tau (sec)
4000
6000
GS29 - PRN 07 (24 1 1997) - [segment la]
Auto Correlation of LCOBS (Roo) and LC..EST (Ree).
Cross Correlation between them (Roe)
Unit: [cycles] 2
320
8000
U)
C
a-
t3
0
o
0
III
a-
cliCU
.
....... .... .....
.....
.
.......
.
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..
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0
c'J
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0
0
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: :
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0
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1S3 01 / SaO 01
0
(I)aS pue(I)OOS
0o
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a-
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o
6
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,-,-
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...... ·.
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52 "t.
...
.
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....................
.....................
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o
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a
o
U
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4.'
(.4
Ui
L
"RESIDUAL" = difference LC Observed - LC Estimated
·
1
I
i
i
: seg 2b
1
,~~..~.~~.~..~r.~.......~...~...........
..
-I
_ A
L,
I
0. 7
I
i
1.2
0.9
Time (sec)
o = LC OBSERVED - GS29 (Ashtech) 07 seg 2a
0%
0.7
0.8
0.9
x 10'
* = LC ESTIMATED - GS29 (Ashtech) 07 seg 2a
1.1
1
1.2
1.3
Time (sec)
Fig. A.13.6.a)
GS29 - PRN 07 (24 1 1997) - [segment lb]
Phase residual LCDIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
322
xl 0
I
...
.....
:.....
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......
J··
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*. . * *l.......... ..
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.
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i
i
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*
011 oSl
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•
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1..
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o
...... ::......
-191
=1110 07
o
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P
EE
P
I
O- •J
Io
GS29 (Ashtech) PRN 07 - California 24 1 1997 segment 2b
S .,
0
C.)
-j,O
o
to
oL-I
0
o,
0
2
L-
-8000
-6000
-4000
-2000
-6000
-4000
-2000
0
2000
4000
6000
8000
2000
4000
6000
8000
o0
c-)
0o
-8000
CROSS-CORRELATION between LC OBS and LC EST
I
B
B
0.5
0
o
2C)
0
-0.5
........
.
m.l
..
..
.
.
.
.
............
,
.
I
-8000
-6000
Fig. A.13.6.c)
-4000
-2000
2000
time delay: Tau (sec)
4000
6000
GS29 - PRN 07 ( 24 1 1997) - [segment lb]
Auto Correlation of LC..OBS (R.) and LCEST E).
Cross Correlation between them (R)
Unit: [cycles] 2
324
8000
o
....
..
..
°.....
°
......
.
°
. ...
..
°.......
................
..........
. . .......
o
·
.
·
.......
.
.....
o
.. . . .
:::
::::::::::::::
.......
"..°.................
......
°......................°.......
..........
..
. ..
...
........
. . ..
..........
..
. ..
. . .·
... :
°
.• :r::
.°...
..:::::::
................
.
.. ..
.
.
O
(1) S puL8(W)OS
IS3 01 / S0 3'1
0
C.)
C
0a'
cl
o
SI
C ov.
S)
-)
n-
. . . . . . . ... . . . .
. . ..
.
" """""
I - -.. -
"
. . . . ..........
. .. . . . .....
......
::::::::::::::::::::::::::::::::::
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....
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....
.
. . ...
.
. . ... ...
.. .
f
i
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.
.......
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YY
c'
......
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0)
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.
.
.
.
.
.
.
S10o3anlndY
JO
...
....
..··· .. ..
...
..
..
......
.. .. ..
· · · · ·.
.... .. ... ..
i
(C)
O
o
•
Do0o
GougJOeLoo
c
a
c
a
I
v.-
.....
.
.... . . .. . * . . . . ...o
o
....
. ..
...... :......
>
aw .... .. . ........
..
..
..
..
.
.
.
C
C
On
Co
C0
0
m
a
v,
(1)
-- r
.....
c
co ...........
0)
Z
L
0U
w
wc
cz
0
0
'-
-o
o
o
o
I
O
Ci,
4.
'-4
U)
90 90
12
-60
240
GPS DATA:
300
270
receiver location: SOUTHERN CALIFORNIA
station: GS29 (Ashtech) satellite: PRN 25
acquisition date: 24 1 1997
station: GS29 (Ashtech)
satellite: PRN 25
400
300
200
=......................
.......
............. .....
......
.......
100
0
2
4
Time (sec)
0
6
4
Time (sec)
x 104
Fig. A.14.1 : Satellite visibility chart for PRN 25 (skymap).
Elevation and azimuth of PRN 25 with respect to
the station GS29
326
6
x 10
PRN 25 23 1 1997
GP29 (Ashtech)
,,
I
I
I
XIM
S60
-
°°-
---.°
.
°
--
----
°--°-
°-
.
.°.-. . . .°
.
°--
°-
°-*
° °---
°
°-- .
°-
° °--*-°
°,
,...........
..
I
.o 40
20
°
.......
U %
..
.
.
°....°..
..
....
°
°..
...
....
;·· · · · ·; · · ·
3
2.5
2
1.5
.
.°.,. . .°.. . .°....
°. . . ... .°°.
.°
°.
;
3.5
........
.
----------------
.
·
----
.
,.......!!
.....
*
-- ...........
- - --
--· ·-----
.
· ···
;···
4
Time (sec)
6
5.5
5
4.5
6.1
x 104
Signal to Noise Ratio for L1 and L2
25
-j
M
z 20
15
2
1.5
3.5
3
2.5
5
4.5
4
6
5.5
6.'
_4
n
94
S200
.......
.......... . ......
.................
a: 150
z
-
CO 100
50
.........
...............
......
....
1.5
-------- -----
·. ·
~· .~
~
*
m
m
m
2
2.5
3
·
~.·~
.8
.
i
m
3.5
1-----·C
·
4
· · ·
·
·· ··--:·
m
4.5
Time (sec)
Fig. A.14.2 : Station: GS29 - Satellite: PRN 25
Data acquisition date 24 1 1997
Signal-to-Noise-Ratio (SNR) for LI and Lz.
327
· -· ·
m
5
-
-
-
- · I-
m
5.5
-
--
-
-
m
6
6.5
x 10'
GPS "OBSERVED" Phase Residuals for: GS29 (Ashtech) PRN 25 - Califomia 241 1997
I
0.2
0.1
0
-0.1
-0.2
I
I
·
|
r
I
I
I
·
·
·
I
II
r
r
.,j ..'
...
i.. "."...
. ,. ..
...
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation
x 10
.....
... ... . . , . . . ........... .. . .
0.2
0.1
0
-0.1
-0.2
I
0.2
0.1
0
-0.1
-0.2
.
4.4
4.6
I
I
. ..
..... . . ...
.
. . . .
4.6
. .
.....
......... ........ ...
..............
.
4.8
5
5.2
5.4
5.6
5.8
Comparison LC OBSERVED vs LC ESTIMATED
I
i
.. .. .. ...,,
. .... .... ...
. . .. .
4.4
..
4.8
...
I
"
I
.....
...................
'''' ''~''''''`
~ '''''~''''''''
'''''
I..
. ....
... ...
.....
5
I
5.4
5.2
Time (sec
~
6
I
...............
'~~'''''''''''
....
....
5.6
....
.....
5.8
I
xI 104
.. ..'~J4'
....
6
Fig. A.14.3: Station: GS29 - Satellite: PRN 25- Data acquisition date 24 1 1997
The observed phase residual (LC-OBS) is compared with the
multipath phase error estimated from the SNR (LC...EST)
Unit: [cycles)
328
6.2
x
6.2
xl0'
"RESIDUAL" = difference LC Observed - LC Estimated
'-
.-
4.4
o
-
4.5
4.6
4.7
Time (sec)
LC OBSERVED - GS29 (Ashtech) 25
4.8
: seg 1
4.9
x 104
* - LC ESTIMATED - GS29 (Ashtech) 25
0.2
-0.1
-0.2
4.4
Fig. A.14.4.a)
4.5
4.6
4.7
Time (sec)
4.8
4.9
GS29 - PRN 25 ( 24 1 1997) - [segment #1]
Phase residual LCDIFF, determined as the difference
between LC.OBS and LC.EST.
Unit: [cycles]
329
5
x104
GS29 (Ashtech) PRN 25: Phase Error
Power Spectra Density: Segment 1
0.2
S•:
.
*
I
(
.:
*
,
.!
. ... . ................
" " :1::::.
*
"................
-0.1
-0.2
·-
:;
4.4
=
4.8
10-4
0.2
"
' ' """ '
"'
r··.··~···~.-.rr
0.5
-0.1
···
·· ·
·
··
·--· ·
····
·· · ·
··
·
··
-0.2
4.4
4.6
4.4
4.6
4.8
Time (sec]
Fig. A.14.4.b)
4.8
5
·
·
· ··
,.rrr·r·
· · · ~···2.-.\r· ·· · · ··
·
'··-·~-~-~-~-··~r
··
·· ··~
· ·.r ·
~..~·
· rr·r······
·· · ·
·· · ·
·· ··
·· ·
·· · ·· ··
·· ·
.
. · ·
· · · · ·- · ·
·
· ·· ·
·
· ·· ·
·
· ·· ·
·
·
·
.
.
·
·
·
·
·
·
·
···
·
·
·
·
~·····
.·r··~
.. ·. · ·
r · ·· rr·
· ·· · · ·
· · · · · ·
· ·· · · ·
· · r···
0
f-4
10C
10l
10-2
101
10"
10-
10-"
10-'
x 10'
GS29 - PRN 25 (24 11997) - [segment #1]
PSD of LCOBS, LCEST and LC..DIFF.
Unit: [cycles] 2
330
101
10-2
le
GS29 (Ashtech) PRN 25 - Califomia 24 1 1997 segment 1
e/
m 1
0
0
-J
" 0
o
I
-8000
-
-6000
-4000
-6000
-4000
-2000
0
2000
0
2000
4000
6000
I
-M
1
-
a
-2000
-
CROSS-CORRELATION
between
0.5
... .. .. ..........
-0.5
-..................................... . . . . . .
LC
4000
a
OBS
and
LC
6000
8000
8000
EST
...........
...........
_1
-8000
-6000
Fig. A.14.4.c)
-4000
-2000
2000
time delay :Tau (sec)
4000
6000
GS29 -PRN 25 ( 24 1 1997) - [segment #1]
Auto Correlation of LCOBS (RPo) and LC.EST (Ree.
Cross Correlation between them (Roe)
Unit: [cycles] 2
331
8000
0.2
GS29 (Ashtech) PRN 25: Phase Error
Auto-Spectral Power Density :seg 1
-0.1
-0.2
4.4
4.8
4.6
Time (sec)
5
x 10
frequency [Hz]
CROSS-SPECTRAL Power Density
COHERENCE OF LC OBS and LC EST
-1
10
10 "2
frequency [Hz]
frequency [Hz]
Fig. A.14.4.d)
10"
GS29 - PRN 25 (24 1 1997) - [segment #1]
Auto Spectral Power Density of LC..OBS (So) and of
LC.EST (See). Cross PSD between them (Soe)
Unit: [cycles] 2
Coherence function of LC..OBS and LCEST
332
10 -1
"RESIDUAL" = difference LC Observed - LC Estimated
: seg 2
Time (sec)
o = LC OBSERVED - GS29 (Ashtech) 25
5
5.1
5.2
x 10'
* = LC ESTIMATED - GS29 (Ashtech) 25
5.3
5.4
5.5
5.6
5.7
Time (sec)
.,
A
Fig. A.14.5.a)
GS29 - PRN 25 ( 24 1 1997) - [segment #2]
Phase residual LC...DIFF, determined as the difference
between LCOBS and LC.EST.
Unit: [cycles]
333
IJ
Power Spectra Density: Segment 2
GS29 (Ashtech) PRN 25: Phase Error
0.2
........... ....... .....
.
:A,•:.~i ,I
..
. . . .
. . . .
. .
..
..................
o
CL
0CD 1
I
5.2
U-4
5.6
x 10'
5.4
0.2
0.1 .. ..
. .
. . .. i ... ..
o
¶i
:,
L
5
-
0-j2
.....
%
......
-0.1
Mo3
-1
10-2
10
10
...
. . ..
.
..
...... ..................
....
..
-0.1
1..
.
... ,
:............
.......
::
".
5.
5
,~
5.2
,
5.6
x 104
·
5.4
0.2
I
'
'
"""'
'
"
";;'1
i >1111ii
I
~· \I~· -·---·--·-·-····-·---·--·--·····--··
0
I·\V···Ui·
•
I::
.. ... ...
...........................-
.
-0.1
.
""""
10
I Vi
0.1
0',
'
-1
10-2
10.
10
1Lf ~
:. :. :..
: :::
.....
:.:.:.:.:: ..
.
.
:11:111:
:::
""
5
I
5.2
Fig. A.14.5.b)
5.4
Time [sec]
:;
3
10
5.6
"'
10- 2
x 10'
GS29- PRN 25 (24 1 1997) - [segment #2]
Power Spectral Density of LCOBS, LCEST and LC..DIFF.
Unit: [cycles] 2
334
-1
10
M
GS29 (Ashtech) PRN 25 - California 24 1 1997
.
M
O
segment 2
0
0
I
0
00
-4000
-6000
0
-2000
2000
4000
6000
80 00
2000
4000
6000
80400
w-J
0
S.
-80
K0
-4000
-6000
-2000
CROSS-CORRELATION between LC OBS and LC EST
0.5
...........
....
....
.....
....................
iii
I
I
I
I
0
........ .
....
.. ... ...
-0.5
_I
-8000
-8000
-6000
Fig. A.14.5.c)
-4000
2000
-2000
time delay :Tau (sec)
4000
6000
GS29 - PRN 25 (24 1 1997) - [segment #2]
Auto Correlation of LC..OBS (Ro~ and LC.EST (Re).
Cross Correlation between them (Ro)
Unit: [cycles] 2
335
8000
GS29 (Ashtech) PRN 25: Phase Error
5
5.2
5.4
Time (sec)
Auto-Spectral Power Density :seg 2
5.6
x 10'
frequency [Hz]
CROSS-SPECTRAL Power Density
COHERENCE OF LC OBS and LC EST
0.8 '
o 0.6
'
·"'
'''
""
' '
· "'
'''
·"
··· · ·
;.'.::'··· :··.·:-.'
1··.··. ·.·
·· ··
···
· ·
·····
r . .'.
~..·..·.·. r ~·~~...
· '··
o-
5 0.4
· · ·-
U
0.2
• o
-::·
""
· · · I··:·:-:·-:·:i;
'
' """'
V-
10-4
'···· ~.·.:;
-"··'·
~.'.:'.·rt
· ··
··
· ·· ·
·
· '····
· ·
· ··
· '·····
·
· ·· ·
· ·· c-·-·
· · · · · · · · ~···~····~
·
'···
·
· "····
·
····
'···
·
· "'····
····
~a~t,
n
"""'
:
10
V V
---V
···
V
10-2
)-
10- 1
frequency [Hz]
Fig. A.14.5.d)
frequency [Hz]
GS29 - PRN 25 ( 24 1 1997) - [segment #2]
Auto Spectral Power Density of LC_OBS (Soo) and of
LC..EST (Se). Cross PSD between them (Soe)
Unit: [cycles] 2
Coherence function of LC.OBS and LCEST
336
:seg 3
"RESIDUAL" = difference LC Observed - LC Estimated
0.3
0.2
0.1
0
-0.1
-0.2
I.I
'""'~"..
5.7
5.85
5.8
5.75
" ..I
......
I
5.9
I. .......
.....
.....
...... ......I
5.95
6
6.1
6.05
.....
..
..
I
6.25
6.2
6.15
Time (sec)
o = LC OBSERVED - GS29 (Ashtech) 25
X
U.;3
..
I
I
x 10
* = LC ESTIMATED - GS29 (Ashtech) 25
.
.
.
.
.
0.2
ALNY'
0.1
-0.1
· · · ·· ·
·
-·.....
-0.2
5.7
5.75
Fig. A.14.6.a)
5.8
:
5.85
5.9
..............................
5.95
6
6.05
6.1
6.15
6.2 I
Time (sec)
GS29 - PRN 25 (24 1 1997) - [segment #3]
Phase residual LCDIFF, determined as the difference
between LC.OBS and LC.EST.
Unit: [cycles]
m
337
I
6.25
x 104
Power Spectra Density : Segm 3
GS29 (Ashtech) PRN 25: Phase Error
2
0.2 .................... ..... .
, 00
-
a
V
x 10
-
"
x 10
.
0.2
0
···
1.5
0.1
W
*
C1 1.5
n 0.1 ........
..............
........
... .
0
I
0
--
· ·--
·--'
·-
·
---· · · · ·- · ·-----------
·-- ·
·
·
· · · · ··
"'· ·
-·'---··
-··
·
-J
~..·..·.===·~rl·\
·:··'·~-`-~-~·'-·
\ '
' `~~""
"'·
. · '··'·~·'·'·
·'
'
""'
'
""'
'
"·
-J -0.1
· ·. ·· ·-··- ·- ·
CO 0.5
......~~'
.........
· · · · ·-
···
..
''-r~-~-r---;---;r-- · ·
.
· I·---···
· ·. ·
: · -. · s-~;;.-J;
-0.2
--
'·'-'·
'
I··.-;·.-.
·
· · · ·· ··
·
· · · · · ·
·
a
5.7
5.8
5.9
6
6.1
6.2
x 10'
. . ..
.:... . .. . ... .. . . . . . .. . . .:. . . .. . ....
.
j
10"-
LL
0.2
u- 0.1
10- 2
10 - 3
10-4
-0.1
1.5
.
......
-'
....
·-··: ·· ·- -.·
··.
· ······
............
.
. . . . ... .
.
,.
.
1-'
. .
a.
-0.2
5.7
5.8
6
5.9
Time [sec]
Fig. A.14.6.b)
:
6.1
6.2
10-4
· ..
.=.·.
·...
··.-.-..
·-i;.ic;--l-.
'...
.:.
...
10"3
·....
:..
.. .
10-2
x 10
GS29 - PRN 25 (24 1 1997) - [segment #3]
Power Spectral Density of LC.OBS, LCEST and LCDIFF.
Unit: [cycles] 2
338
.
101
GS29 (Ashtech) PRN 25 - Califomia 24 1 1997
IM
segment 3
0
O
0
0
0
o
I
-6
-6000
-
I
-bUUU
-4000
-2000
0
2000
4000
6000
-4000
-2000
0
2000
4000
6000
CROSS-CORRELATION between LC OBS and LC EST
1
,
I
I
I
&
I
·I
·I
·I
1I
0
time delay :Tau (sec)
2000
4000
0.5L..
0
o
02 -0.5
-I
.9 L
-6000
-4000
Fig. A.14.6.c)
-2000
GS29 - PRN 25 (24 1 1997) - [segment #3]
Auto Correlation of LC.OBS (Roo) and LCF.ST (Re).
Cross Correlation between them (Roe)
Unit: [cycles] 2
339
6000
Auto-Spectral Pow Density :seg 3
GS29 (Ashtech) PRN 25 : Phase Error
40C
0.3
0.2
. . . .
...
. . .
.
... .
. .
. .
....
.
..
...
. .
. . .
.
. .
--30(
0i
0
0
i. ..1.
` S-
W-0.1
.. . .. . ..
.
. . .
. . .
. . .
. .
. .
"20(
. ..
."
°
'
8
W 10(
-0.2
.. . .. .. .
. . . .
. .
. . .
. . ..
. . . . .
. . i.
. . .
0...
1104
6.2
x 10'
6.1
6
5.9
Time (sec)
5.8
5.7
-.. .
.-;;-
0.8
• °
0.6
I-
.
,
,
.
.
:•ii. :::.1::::i
.. ... .~..
."
:I :\::::
!
: :;::. :
Ij
i
zi
· !· !· · ·i·
i, !i
iii
:
10-
!;
i i i:ii
::.iii:.. ...f. .:.i.i
10-2
10
frequency [Hz]
CROSS-SPECTRAL Power Density
COHERENCE OF LC OBS and LC EST
. .. • °
. . . ..
. 1
. .
...
. . ....
.; .; .. ';'. . .
0.4
[
0.2
• 1. . . .
n
O.
U10
1. .
103
10-2
.
.. .
10"
10-1
frequency [Hz]
Fig. A.14.6.d)
10-2
10frequency [Hz]
GS29 - PRN 25 (24 1 1997) - [segment #3]
Auto Spectral Power Density of LC..OBS (Soo) and of
LCEST (S). Cross PSD between them (Soe)
Unit: [cycles] 2
Coherence function of LC_OBS and LC_EST
340
10- 1
GPS "OBSERVED" Phase Residuals for: COSO (Ashtech) PRN 30 - California 23 1 1997
I
I
i
o 0.1
w
c- 0.05
w
Un
.- .
.
i
a
a
.....
.
.
.
. .
i
a
.t
.
.
.
0
O -0.05
-
a
""..
..!.1,
...
.
.
...
-0.1
I
•.
r
I
I
r"I
I
n
......
I
I .
I
3
2.8
2.4
2.6
1.8
2
2.2
1.4
1.6
1.2
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation x 10'
I
0.1
0.05
0
-0.05
-0.1
I
I
I
I
I
··
·
'
··
··~
·
·
·
···
·
·
....
...........
i.....
..
...................................................
.........
I..................
.................
I
1.4
.
:.....
2.2
2.4
2.6
1.6
1.8
2
Comparison LC OBSERVED vs LC ESTIMATED
... . .................................
--.
------
1
i.....
..............
5...
·· ·
1.2
0.1
0.05
0
-0.05
-0.1
I
I
I
I
I
i
1.2
1.4
1.6
1.8
...............
.
...
i
I
2.2
2
Time (sec)
2.8
x 10
i..
..
..
.
i
,
2.4
2.6
3
I
2.8
Fig. A.15.1: Station: COSO - Satellite: PRN 30
Data acquisition date 23 1 1997
The observed phase residual (LCOBS) is compared with the
multipath phase error estimated from the SNR (LC...EST)
Unit: [cycles)
341
I
3
x 10'
COSO (Ashtech) PRN 30 - Califomia 23 1 1997
1
1.1
1.3
1.2
1.4
1.5
Time (sec)
1.6
1.7
[
0.05
0
-0.05
-0.1
.
S.
.
1.8
1.9
I
I
I.
I
I
I
I
1.5
1.6
1.7
1.8
I
1.1
1.2
1.3
1.4
Time (sec)
Fig. A.15.2 : Station: COSO - Satellite: PRN 30
Data acquisition date 23 1 1997
LCOBS and LC.EST compared for t=1.0 - 2.0 10 4 sec
Unit: [cycles)
342
-1
.
.
i
F
1
2
x 10'
1.9
I
2
x 104
"RESIDUAL" = difference LC Observed - LC Estimated - 23 1 1997
0.15
0.1
0.05
0
-0.05
-0.1
1
1.1
1.2
1.3
1.4
1.5
Time (sec)
o = LC OBSERVED - COSO (Ashtech) 30
1.6
1.7
1.8
1.9
2
x 10
* - LC ESTIMATED - COSO (Ashtech) 30
0.15
0.1
S0.05
w
o.
-J
-0.05
-0.1
1
1.1
Fig. A.15.3.a)
1.2
1.3
1.4
1.5
Time (sec)
1.6
1.7
1.8
1.9
COSO - PRN 30 (23 1 1997) - [segment 1]
Phase residual LC..DIFF, determined as the difference
between LCOBS and LC.EST.
Unit: [cycles]
343
2
x 10'
Power Spectra Density
COSO PRN 30 (23 1 1997): Phase Error
l
0.1 ..........l....... .......... ....... .........
0.05
.......................
.........
"; 5'
"
'"v
i ;:?'
'
"
"\
0
-0.05
-0.1
t
le
-- -------
n
10-
1.8
1.6
1.4
1.2
x 10'
........... ...... ,......... ....
0.05
0
-0.05
-·--.
. -Li- ! !!•!ii?·
:/: :::
:•(-,.:t•:,:-.·.
....
i......... ..
........................
.
::
"""i'.: .::......
" " " '1"
10-2
10
10
100-1
x
·'
. . . .
.:..
:
.
.
.
.. .
··
..
. ...... ... ...... ..
...
.. .. .. ....
··
... ,
·......
.......
-0.1
......:...................
·.
·.
0 i-
1.2
1.2
1.8
1.8
1.6
1.6
1.4
1.4
10
--
4
·
--
---
10-3
~-
-·
x 10'
'
z
.... ..
.
.
. .
- .
.
0.2
"
'
"'
·.-. ·. ·. I·
0.4
.
i
"'
x
0.1
0.05
0
-0.05
-0.1
-1
10
10-2
··
··
···
·.--- ···
..'AA
··
··
I
1
1.2
1.6
1.4
Time [sec]
Fig. A.15.3.b)
1.8
2
x 10*
loe
-3
COSO- PRN 01 (23 11997) - [segment 1]
PSD of LC.OBS, LC_..EST and LC.DIFF.
Unit: [cycles] 2
10 344
-2
~ -2
10
10Frequency [Hz]
10- 1
segment 1
COSO (Ashtech) PRN 30 - California 23 1 1997
4
0.5
0
-
-
I .
. . ..
I
I
I
I..
-n r.
-
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-1
-_0.
-U.b
-U.4
-U.2
U
U.Z
U.4
U.b
U.0
1
1
,,
~
P,·r
Rt•R-R3RRIAT(3
_I
I
I
I
CROSS-CORRELATION
h~·~
twl~-·n LC
b
·
CI='"x
-1
-~ n
x
10
T
I--
t
-
0.2
a:
0
0
0 -0.2
0
CA
0-0.
Ac
-1
i
-0.8
Fig. A.15.3.c)
·
-0.6
·
I
-0.4
·
·
I
I
·
-0.2
0
0.2
time delay : Tau (sec)
·
·
·
0.4
0.6
0.8
COSO - PRN 30 (23 1 1997) - [segment 1]
Auto Correlation of LC..OBS (R) and LC..EST (Re.
Cross Correlation between them (Ro
Unit: [cycles] 2
345
1
x 10'
Auto-Spectral Power Density
COSO (Ashtech) PRN 30 (23 1 1997): Phase Error
0.1
1
0.05
8
Uo
0
c')
"1
-0.05
-0.1
1
1.2
1.6
1.4
Time (sec)
1.8
2
x 104
frequency [Hz]
CROSS-SPECTRAL Power Density
COHERENCE OF LC OBS and LC EST
-1
10"
frequency [Hz]
Fig. A.15.3.d)
10"
10-"
frequency [Hz]
COSO - PRN 30 (23 1 1997) - [segment 1]
Auto Spectral Power Density of LC_OBS (Soo) and of
LC...EST (See). Cross PSD between them (Soe)
Unit: [cycles] 2
Coherence function of LCOBS and LC..EST
346
-I
10'
GPS "OBSERVED" Phase Residuals for: COSO (Ashtech) PRN 30 - California 24 1 1997
....
0.1
0.05
0
-0.05
-0.1
........
.....
.
I
.
. . . I:.......
......
.. ......... ............
r I
..................
-
s It
I
......
I
.
..
...
.
. ...
III
I
I
I
I
I
I
I
I
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
GPS "ESTIMATED" Phase Residuals computed from the analysis of SNR variation x 10
c
I
0.1
0.05
0
-0.05
-0.1
..
...................
.......... ......
..........
. ..
...
I
__
I
.
.
..
F
.. ..
.
..
..
..
..
.
...........
...
...
...
.........
I..
..........
..........
..... ...
.....
......
.... . ..... ......
.....
.....
!
I
1.2
1.4
1.6
1.8
2
2.2
2.4
Comparison LC OBSERVED vs LC ESTIMATED
I
0.1
0.05
0
-0.05
-0.1
I
S...
I
I
1.2
1.4
2.8
I
I
3
x 104
.....
.....
.....
..
........... ..
..................
1
2.6
1.6
1.8
2
2.2
.... .... .
2.4
2.6
2.8
Time (sec)
Fig. A.16.1: Station: COSO - Satellite: PRN 30
Data acquisition date 24 1 1997
The observed phase residual (LC.OBS) is compared with the
multipath phase error estimated from the SNR (LCEST)
Unit: [cycles]
347
3
4
x 10
COSO (Ashtech) PRN 30 - Califomia 24 1 1997
1
1.2
1.4
1.6
1.8
2
Time (sec)
2.2
2.4
2.6
2.8
3
x 10
0.15
-----
0.1
0.05
0
. ..........
........ .:......... .......... .......... .......
_"
.... ..... .
IIII
... :.
II. II..
-0.05
-0.1
I
1
1.2
-
1.4
1.6
1.8
2
Time (sec)
2.2
2.4
2.6
Fig. A.16.2 : Station: COSO - Satellite: PRN 30
Data acquisition date 24 1 1997
LCOBS and LCEST compared for t=1.0 - 2.0 10 4 sec
Unit: [cycles]
348
2.8
3
x 10
"RESIDUAL" - difference LC Observed - LC Estimated - 24 1 1997
0.15
_
·
·
_·
·
·
0.1
0.05
0
-0.05
-0.1
*...
1.2
1.1
1
I
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Time (sec)
0.15
I.
o = LC OBSERVED - COSO (Ashtech) 30
I
I
x 10'
* = LC ESTIMATED - COSO (Ashtech) 30
II
I
I~
I
I
I
0.1
0.05
0
0 -0.05
-0.1
-0.1
1
1
1.1
1.1
1.2
1.2
1.
1.3
1.4
1.4
1.
1.5
1.6
1.6
1.7
1.7
1.8
1.8
1.9
1.9
Time (sec)
Fig. A.16.3.a)
COSO - PRN 30 (24 1 1997) - [segment 1]
Phase residual LC...DIFF, determined as the difference
between LCOBS and LCEST.
Unit: [cycles]
349
i
2
2
x 104
:
..
COSO PRN 30 (24 1 1997): Phase Error
0.1
cn 0.05
0
0
Power Spectra Density
· ·
..
I:
r.3
. .......4-.
I..I ..
.
.
. . . . . . . . . .. . ...........
:,......
:1:'
·'
t .·.
.
.............
.:
..............
......
... ·..
. Lir
.. .. . . . . . . . . . . .
-3 -0.05
-0.1 I
· r..~.~
·.·.
h ·
··
··
··
···
.
··
1
1.2
1.4
1.6
1.8
103
10 -4
0- 1
10-2
x 10'
X
0.1
0.05
0
-0.05
-0.1
:"
.. . . ... . . ... . . . .. . . .
..
...
.
.
.
.......
.
..
.
.
.
.
..
.
.
.
..
.
.
..
.
.
.
..
.
..
.
..
....
.........
...
... .
.
•
. ...
..
.
...
.
:'
:'::: ''
.
..
.
.
1.4
1.6
1.8
10- 4
.
..
.
'
' "':':
..
.
.
1...:....
10-3
....... =
.
.
.
...
.
::.;L~
.....
n
1.2
.
':
·- t..
10-2
10
..
·....
10- 1
x 10'
0.1 .. .
u. 0.05
5
0
0
S-0.05
.
.
x
.
.
. .. . . . .. . . .
.,
..
........
........................
.. .... .... . . . . . . . . . . . . . . .
-0.1
1
1.2
1.4
1.6
Time [secJ
1.8
-1
2
Frequency [Hz]
x 10'
104
Fig. A.16.3.b)
COSO-PRN 01 (24 1 1997) - [segment 1]
PSD of LC.OBS, LC.EST and LCJDIFF.
2
Unit: [cycles] 104
350
segment 1
COSO (Ashtech) PRN 30 - Caifomia 24 1 1997
et
S1
O
-J
0
o
I
0
<-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
I1
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x 10'
CROSS-CORRELATION between LC OBS and LC EST
0.5
I
I
I
I
I
I
II
I
I
I
I
-0.5
I1
-1l
1
I
I
~
l~I
-
-0.8
Fig. A.16.3.c)
-0.6
-0.4
0
0.2
-0.2
time delay :Tau (sec)
0.4
0.6
COSO - PRN 30 ( 24 1 1997) - [segment 1]
Auto Correlation of LC.OBS (Roo) and LC..EST
Cross Correlation between them (Ro)
Unit: [cycles] 2
351
0.8
.
1
x 10'
COSO (Ashtech) PRN 30 (24 1 1997): Phase Error
Auto-Spectral Power Density
I
1
1.2
1.4
1.6
Time (sec)
1.8
2
x 104
frequency [Hz]
COHERENCE OF LC OBS and LC EST
CROSS-SPECTRAL Power Density
1500
t' 1000
0
"
CL
500
n
-1
frequency [Hz]
Fig. A.16.3.d)
10"
10- 2
10
frequency [Hz]
COSO - PRN 30 (24 1 1997) - [segment 1]
Auto Spectral Power Density of LC.OBS (So) and of
LC..EST (Se). Cross PSD between them (Soe)
Unit: [cycles] 2
Coherence function of LCOBS and LC.EST
352
101
Appendix B
353
Station
Code
Satellite Set
Number
tIA
Number
of
Samples
site:
Observed Phase
Residual
Variance
A2
GOBS
ASIA
PRN
MANA
26
b'
MANA
28
-ii
N
Estimated
Multipath
Phase Error
Variance
Corrected
Phase Residual
Variance
A2
2
GEST
(TDIFF
Difference of Variance
before and after
Multipath Correction
A2
A2
TOBS -
NDIFF
ASIAA(5)PRN
10^(5)
1A(-)
10A(-.5)
[cycles]'
[cycles]
[cycles]
[cycles?)
2
Decrease of
Variance after
Correction
[%]
1%
la
132
877.42
1021.73
699.35
178.07 ± 93.05
20.30
lb
151
387.43
312.33
218.61
168.82 ± 34.97
43.57
2a
175
93.67
49.15
130.88
-37.22 ± 9.95
2b
232
40.90
20.75
40.38
0.52 ± 3.30
2c
133
202.96
119.46
243.15
-40.19 + 24.32
la
190
89.90
30.48
89.04
0.86 + 6.92
0.96
lb
106
269.83
211.86
218.56
51.27 + 33.40
19.00
2a
100
489.41
404.55
301.72
187.70 ± 56.21
38.35
2b
101
78.29
43.92
48.96
29.33 ± 8.56
37.47
2c
144
82.45
82.84
73.01
9.44 ± 9.17
11.45
2d
101
812.35
882.72
535.67
276.68 ± 90.63
34.06
i1
1.28
Station
Code
Satellite
Number
Set
Samples
t•
t•
H
t.
Number
of
site:
ASIA
MANA
N
PRN
26
Measurement Noise
Variance
^
Multipath Noise
Variance
^
Estimation Noise
Variance
GMN
GMPN
A 2
GEN
10^(-5)
10^(-5)
10^(-5)
[cycles]'
2
[cycles]
[cycles]
2
2
la
132
227.51
599.90
421.83
lb
151
146.86
240.58
71.75
2a
175
87.70
5.97
43.18
2b
232
30.26
10.64
10.12
2c
133
163.32
39.64
79.83
la
190
74.23
15.67
14.81
lb
106
138.26
131.56
80.29
2a
100
193.29
.. 296.13
108.43
2b
101
41.67
36.63
72.91
2c
144
36.31
46.14
36.70
2d
101
232.65
579.70
303.02
U'
MANA
28
2
,-3
Station
Code
Satellite Set
Number
0
Number
of
Samples
Observed Phase
Residual
Variance
Estimated
Multipath
Phase Error
Variance
Corrected
Phase Residual
Variance
Difference of Variance
before and after
Multipath Correction
Decrease of
Variance after
Correction
site:
A
ASIA
0v
KKAU
KKAU
KKAU
PRN
16
20
22
N
2
ooas
A2
Esr
A
2
OCDIFF
A 2
^2
COBS - (DI
10^(-5)
10^(-5)
10^(-5)
10^(-5)
[cycles]'
[cycles]'
[cycles] 2
[cycles])
2a
171
471.22
55.94
547.85
-76.62 ± 25.18
2b
267
188.44
42.20
131.44
57.01 ± 9.39
30.25
2c
154
320.32
392.58
206.61
113.71 ± 26.26
35.50
2a
160
101.39
93.30
183.08
-81.69 ± 14.98
2b
158
287.34
505.17
176.52
110.82 ± 16.30
38.57
1
118
207.93
403.47
141.21
66.72 ± 19.36
32.09
2
206
561.25
242.98
755.63
-194.38 ± 50.46
3
452
29.34
44.01
47.07
-17.73 .± 2.52
W
E.
tb3
0,
Station
Code
Satellite
Number
Set
Number
of
Samples
site:
GPS
Measurement Noise
Variance
^2
N
PRN
ASIA
KKAU
16
Multipath Noise
Variance
A2
Multipath
Estimation Noise
Variance
A2
MPN
(EEN
10A(.5)
10^(-5)
10A(-5)
[cycles]'
[cycles]'
[cycles]'
GMN
2a
171
481.57
-10.34
66.28
2b
267
138.84
49.60
-7.40
2c
154
67.17
253.14
139.43
2a
160
95.59
5.80
87.50
2b
158
-20.65
307.99
197.17
1
118
-27.16
235.09
168.37
2
206
536.95
24.30
218.68
3
452
16.20
13.14
30.87
rl.
KKAU
KKAU
-
20
22
-i
,m
Station
Code
Satellite Set
Number
Number
of
Samples
Observed Phase
Residual
Variance
site:
LIGO
L0n
PRN
0028
0036
09
0028
0036
15
0028
0036
N
^2
Estimated
Multipath
Phase Error
Variance
A2
Corrected
Phase Residual
Variance
A
2
Difference of Variance
before and after
Multipath Correction
A 2
2
Decrease of
Variance after
Correction
[%]
GE~r
ODIFF
GoBS - GDIFF
10^(-5)
10^(-5)
10^(-S)
10^(-5)
[cycles]'
[cycles]'
[cycles]'
[cycles]'
208
2084.25
2459.69
1089.03
995.22 + 130.22
47.75
160
530.79
306.73
413.08
117.71 + 49.58
22.18
2
126
128.23
95.13
124.95
3.29 ± 15.51
2.56
3
124
77.28
83.64
165.37
-88.10 ± 14.71
4
286
260.59
160.31
192.40
68.19 ± 18.15
26.17
1
256
397.59
597.31
297.77
99.82 + 21.43
25.11
2
254
162.60
255.62
105.95
56.66 ± 5.11
34.84
I1
25
IA
Station
Code
Satellite
Number
Set
Number
of
Samples
site:
LIGO
0028
GPS
Measurement
Noise
^
N
PRN
2
OMN
Multipath
Noise
^
2
MP-N
Multipath
Estimation
Noise
A
2
(EN
10^(-5)
10A(5)
10^(-5)
[cycles]'
[cycles]'
[cycles] 2
09
1
208
356.79
1727.45
732.24
15
1
160
318.57
212.22
94.51
2
126
79.02
49.21
45.92
3
124
79.50
-2.23
8.59
4
286
146.34
114.25
46.06
1
256
49.03
348.56
248.75
2
254
6.46
156.14
99.48
0036
0028
0036
LA'w0
cr
0028
0036
25
Station
Code
Satellite
Number
Set
Number
of
Samples
Observed Phase
Residual
Variance
Estimated
Multipath
Phase Error
Variance
Corrected
Phase Residual
Variance
Difference of Variance
before and after
Multipath Correction
Decrease of
Variance after
Correction
data:
A
LIGO
PRN
N
2
A 2
A2
OoBS
GEST
10^(-5)
10^(-5)
[cycles]'
[cycles]
2
A 2
^2
O'DIFF
OOBS - ODIFF
10^(-5)
10^(-5)
[cycles]?
[cycles] 2
[%]
LI
09
1
208
280.00
200.00
210.00
70.59± 2.35
25.38
L2
09
1
208
420.00
81.89
280.00
140.00± 2.20
34.18
Station
Code
Satellite
Number
Set
Number
of
Samples
GPS
maesurement
noise
Multipath
Variance
noise
Multipath
estimation noise
Variance
LIGO
PRN
A2
A2
N
(MN
(MPN
A2
EN
Ll
09
1
208
140.00
130.00
63.33
L2
09
1
208
310.00
110.00
-30.80
Station
Code
Satellite Set
Number
Number
of
Samples
Estimated
Multipath
Phase Error
Variance
Observed Phase
Residual
Variance
Corrected
Phase Residual
Variance
Difference of Variance
before and after
Multipath Correction
Decrease of
Variance after
Correction
site:
^
South
Calif
COSE
PRN
01
N
^
2
2
^
2
A
^2
2
GOBs
(EST
10^(.5)
10^(-5)
10^(5S)
10^(5)
[cycles]'
[cycles]'
[cycles])
[cycles]'
(DIFF
GOBS -
(DIFF
la
170
81.25
96.49
210.63
-12938 + 14.94
lb
332
32.14
115.13
139.01
-167.88 + 6.24
Ic
224
547.12
1576.49
525.23
21.89 + 84.43
[%]
4.00
-I
COSE
COSE
07
25
2a
233
86.71
195.98
352.02
-26531 + 20.21
2b
215
772.21
1910.36
695.44
76.73 ± 88.90
1
211
66.46
75.87
153.24
-86.78 + 10.17
2
212
48.01
192.05
176.02
-128.01 + 7.63
3
179
238.26
880.10
394.16
-155.86 + 47.53
9.94
ip
H
0\
Station
Code
Satellite
Number
Set
it
Number
of
Samples
site:
South
Calif
COSE
t,•oC
COSE
COSE
GPS
Measurement Noise
Variance
^
N
PRN
Calif
01
07
25
2
Multipath Noise
Variance
^
2
Multipath
Estimation Noise
Variance
A
2
(MPN
GEN
^
10 (-5)
10 ^ (-5)
[cycles]'
[cycles]2
[cycles] 2
GMN
^
0 (-5)
la
170
97.70
-16.45
112.94
lb
332
28.01
4.12
111.01
Ic
224
252.07
799.19
777.30
2a
233
121.38
-34.66
230.64
2b
215
-222.26
1008.77
929.60
1
211
71.91
-5.46
81.33
2
212
15.99
32.02
160.03
3
179
-155.86
362.12
517.88
Station
Code
Satellite Set
Number
Number
of
Samples
Observed Phase
Residual
Variance
Estimated
Multipath
Phase Error
Variance
Corrected
Phase Residual
Variance
Difference of Variance
before and after
Multipath Correction
Decrease of
Variance after
Correction
site:
A 2
South
Calif
GS29
GS29
GS29
PRN
01
07
25
N
Goas
A
2
OGEST
A 2
DIFF
A
A 2
GOBS
-
2
GDIFF
10A(.5)
10A(-5)
10A(.5)
10A(-5)
[cycles]'
[cycles]'
[cycles]'
[cycles])
la
173
50.94
56.28
90.32
-39.38 + 7.46
lb
332
106.06
2.95
114.65
-8.59 ± 1.97
Ic
219
635.20
1046.25
529.37
105.83 + 33.70
2a
234
30.19
7.75
43.50
-13.32 + 2.10
2b
211
797.58
1221.09
631.62
165.96 ± 48.85
1
208
123.95
59.26
176.77
-52.82 ± 11.76
2
212
178.50
4.62
192.37
-13.87 ± 4.01
3
173
319.50
297.42
130.28
189.22 + 22.96
[%
16.66
20.81
59.22
Station
Code
Satellite
Number
Set
Number
of
Samples
site:
South
^2
GS29
OS29
-
N
PRN
Calif
GS29
GPS
Measurement Noise
Variance
07
(MPN
A
2
(EN
10^ (-5)
10^ (-5)
[cycles] 2
[cycles]'
[cycles] 2
la
173
42.49
8.44
lb
332
108.88
-2.82
5.77
Ic
219
59.16
576.03
470.21
2a
234
32.97
-2.78
10.53
2b
211
104.05
693.53
527.56
1
208
120.73
3.22
56.04
2
212
183.24
-4.62
9.25
3
173
76.18
243.32
54.10
25
--
^2
Multipath
Estimation Noise
Vairance
10 (-5)
^
01
Multipath Noise
Variance
-
C,
Station
Code
Satellite Set
Number
Number
of
Samples
Observed Phase
Residual
Variance
Estimated
Multipath
Phase Error
Variance
Corrected
Phase Residual
Variance
Difference of Variance
before and after
Multipath Correction
Decrease of
Variance after
Correction
site:
^
0w
South
Calif
N
PRN
^
2
Goss
A
2
EST
MGDIFF
10^(-5)
[cycles]
2
2
A
A
2
(OBS
-
2
ODIFF
10^(-5)
10^(.5)
[cycles]'
)
[cycles]
[cycles]'
10^(-5)
COSO
30
1
300
64.53
18.22
68.98
-4.46 ± 3.75
30
1
300
98.09
15.68
82.52
15.56 ± 4.16
GPS
maesurement
noise
Multipath
Variance
noise
Multipath
estimation noise
Variance
^2
[%J
23 1 97
COSO
24 197
U.
Station
Code
South
Calif
COSO
Satellite Set
Number
Number
of
Samples
^2
PRN
N
GM N
GOMPN
2
OEN
A
30
1
300
57.65
6.88
11.34
30
1
300
82.46
15.62
0.06
23 1 97
COSO
241 97
15.86
366
Appendix C
367
~--·-IIPPIII~IPIl~··~IPPPIPPPIIIIIIIPIIIP
FFTobs est diff.m
t%
S
This program computes the Power Spectral Density of the time series
LC_OBS, LC_EST and LC DIFF
clear
load SAT EST
load SAT_OBS
Nl-length(SAT OBS(:,1));
N2-length(SAT_EST(:,1));
index-0;
hours-18;
minutes-45;
epoch - SAT OBS(:,i);
time _sec= (epoch-l)*30+hours'3600+minutes*60;
SAT_0BS(:,l)-time sec;
for j-i:Nl
for i-1:N2
if
SAT0OBS(j, 1)-SATEST (i, 1)
index-index+l;
SAT D 3(index,1)-SAT OCBS(j,1);
SAT D 3(index,2)-SAT OBS(j,4)-SAT_EST(i,7);
SAT 0 1(index,1)-SAT OBS(j, 1);
SAT E 2(index,1)-SAT EST(i,1);
SAT 0 1(index,2)-SATC BS(j,4);
SAT E 2 (index, 2)-SAT EST (i, 7);
end
end
end
SAT 1-SAT 0 1;
SAT 2-SAT E 2;
SAT 3-SAT D 3;
n-length(SAT 2(:,l));
dt-30;
m-n/2;
SATE (r,l)-SAT 2(1,1);
SAT_O (r, 1)-SAT 2(1,1);
SAT D(r,1)-SAT 2(1,1);
SAT 0(r,2)-SAT 1(1,2);
SAT E(r,2)-SAT 2(1,2);
SAT D(r,2)-SAT 3(1,2);
368
for ind-2:n
r=r+1;
SAT 0(r,1)-SAT 2(ind,1);
SAT E (r, 1)-SAT 2 (ind, 1);
SAT D(r,1)-SAT 2(ind,1);
SAT O(r,2)-SAT_ (ind,2);
SATE (r, 2) -SAT 2 (ind, 2);
SAT D(r,2)-SAT 3(ind,2);
if SAT 2(ind,1)>SAT 2(ind-1,1)-+dt
t miss-SAT 2(ind)-SAT 2(ind-1);
n miss-t miss/dt-1;
for i-1:nmiss+l
SAT O(r,1)-SAT 2(ind-1,1) +dti;
SAT E(r,1)-SAT 2(ind-1,1)+dt*i;
SAT D(r,1)-SAT 2(ind-, 1)+dt*i;
SAT 0(r,2)-SAT 1(ind-1,2)+i*(SAT_1 (ind,2)-SAT 1(ind-1,2))/ (n_miss+1);
SAT E(r,2)-SAT-2(ind-1,2)+i*(SAT 2(ind,2)-SAT 2(ind-1,2))/(nmiss+l);
SAT D(r, 2) -SAT 3(ind-i,2)+i*(SAT_3(ind,2)-SAT 3(ind-1,2))/ (n_miss+l);
if i<n miss+l
r-r+l;
end
end
end
end
N-r;
dt-30;
Ny-n/2+1;
fNy - 1/(2*dt);
T - n*dt;
df-fNy/(Ny-1);
f-[0 :df: fNy];
norm E -( SAT E(:,2)-mean(SAT E(:,2)) );
norm 0 ( SAT_0(:,2)-mean(SAT_0(:,2)) );
norm DIFF-( SATD(:,2)-mean(SAT D(:,2)) );
FTEST - fft(norm E);
FTOBS - fft(norm 0);
FT DI-F - fft(norm DIFF);
? FT EST-FT EST.*conj(FTEST)./n;
P FT OBS-FT OBS.'conj(FT_OBS)./n;
./n;
P_FTDIFF-FTDIFF. *conj (FTD-)
figure (1)
clf
subplot (3,2,1)
plot(SAT0O(:,1), SAT_O(:,2),'-.')
hold
grid
ylabel(' LC OBS ')
title(' COSE (Trimble) PRN 25: Phase Error')
369
subplot (3, 2, 2)
se.milogx(f,PFTOBS(l:Ny)*dt,'-.');
hold
grid
ylabel(' PSD of LC OBS')
title ('?ower Spectra Density: Segment 3')
subplot (3, 2, 3)
plot (SATE(:, 1l), SATE(:,2))
hold
grid
ylabel('LC EST ')
subplot (3, 2, 4)
semilogx(f,PFTEST(1:Ny)
hold
grid
ylabel(' PSD of LC EST')
subplot (3,2,5)
plot (SAT D(:, 1),
hold
dt);
SAT D(:,2))
grid
ylabel('LC DIFF')
xlabel('Time [sec]')
subplot (3, 2, 6)
semilogx(f,PFT_DIFF (l:Ny)dt);
hold
grid
ylabel(' PSD of LC DIFF')
return
370
Auto Cross CORR..m
%
%
This program computes the Auto correlation of the two time series
LCOBS, LCEST and their Cross correlation
clear
load SAT EST
load SAT OBS
clear
load SAT EST
load SAT OBS
Nl-length (SAT OBS (:,));
N21length(SAT EST(:,1));
index-0;
hours-18;
minutes-45;
epoch - SAT_OBS(:,l);
time sec= (epoch-) *30+hours*3600+minutes*60;
SAT OBS (:, 1)-timesec;
for j-l:Nl
for i-L:N2
if SATOBS(j,l)-SAT EST(i,1)
index-index+l;
SATD_3(index,l)-SAT_OBS(j,1);
SATD_3(index,2)-SAT OBS(j,4)-SAT.EST(i,7);
SAT 0 _l(index,l)-SATOBS(j,1);
SAT E 2(index,l)-SAT EST(i, 1);
SAT 0 1 (index,2)-SAT OBS (j, 4);
SAT E_2 (index, 2) -SAT EST (i, 7);
end
end
end
SAT 1=SAT 0 1;
SAT 2=SAT E 2;
SAT_3-SAT D 3;
n-length (SAT2 (:, 1));
dt-30;
m-n/2;
371
r-1;
SAT E(r,1)-SAT 2(1,1);
SAT-0 (r, 1)-SAT 2(1,1);
SAT D(r, 1)-SAT 2(1,1);
SAT O(r,2)-SAT 1 (1,2);
SAT E (r,2)-SAT 2 (1,2);
SAT D(r,2)-SAT 3 (1,2);
for ind-2:n
r-r+1;
SAT O(r,1)-SAT 2(ind, 1);
SAT E(r, 1)-SAT 2(ind, 1);
SAT D (r, 1)-SAT 2 (ind, 1);
SAT 0(r, 2)-SAT1 (ind, 2);
SAT E(r,2)-SAT 2 (ind,2);
SAT D(r,2)-SAT 3(ind,2);
if SAT 2(ind,1)>SAT 2(ind-1,1)+dt
t miss-SAT 2(ind)-SAT 2(ind-1);
n miss-t miss/dt-1;
for i-l:n miss+l
SAT_ (r,l)-SAT 2(ind-1,1)+dt'i;
SATE(r,1)-SAT 2(ind-!,l)+dt*i;
SAT D(r,1)-SAT 2(ind-1,1)+dt*i;
SAT 0(r,2)-SAT 1(ind-1,2)+i*(SAT_1(ind,2)-SAT_ (ind-1,2))/(n miss+1);
SAT_ (r, 2)-SAT 2(ind-1,2)+i(SAT_2 (ind, 2)-SAT_2 (ind-1, 2))/(nmiss+l) ;
SATD (r,2)-SAT3 (ind-1,2)+i"(S-AT 3(ind,2)-SAT_3(ind-1,2))/(nmiss+1);
if i<n miss+1
r-r+i;
end
end
end
end
N-r;
figure (I)
clf
subplot(4,1,1)
x-SAT 0(:,2);
c-xcorr (x,'coeff');
Tau-N*30;
L--Tau+30 :30 :Tau-30;
plot(L',c)
hold
grid
ylabel('Auto-Corr LC OBS')
title('COSE (trimble) ?RN 25
- California 23 1 1997
subplot (4, 1,2)
y-SAT_E (:,2);
c-xcorr (y, 'coeff');
Tau-N * 30;
372
segment 1')
L--Tau+30 :30 :Tau-30;
plot (L', c)
hold
grid
ylabel('Auto-Corr LC EST')
subplot (2, 1, 2)
x-SAT 0(:,2);
y-SAT E(:,2);
c-xcorr (x, y, 'coeff');
Tau=N*30;
L--Tau+30: 30: Tau-30;
plot (L', c)
hold
grid
ylabel(' Cross-Corr : R_o_e(Tau)')
xlabel(' time delay : Tau (sec) ')
title(' CROSS-CORRELATION between LC OBS and LC EST ')
return
373
SP CORR obs est.m.m
%
This program computes the AUTO Spectral Density of the two time series
LC OBS, LC EST and their CROSS Spectral Density from the correlation
functions.
clear
format long
load SAT EST
load SAT_0BS
N1-length(SAT OBS (:,1));
N2-length(SATEST (:, ));
index-0;
hours-18;
minutes-45;
epoch - SATCBS(:,I);
time sec- (epoch-1)*30+hours*3600+minutes*60;
SAT _OBS(:,l)-time sec;
for j-l:N1
for i-I:N2
if
SAT OBS(j,1)--SAT EST(i,l)
index-indexi1;
SAT D 3(index,1)-SAT OBS(j,1);
SAT D 3(index,2)-SATOBS (j,4)-SAT
SAT 0 1(index, 1)-SATOBS(j, 1);
SAT E 2(index,1)-SAT_ ST(i,1);
SAT 0 1(index,2)-SAT OBS(j,4);
SAT - 2(index,2)-SAT-EST(i,7);
end
end
end
SAT 1-SAT 0 1;
SAT 2-SAT E 2;
SAT 3-SATD_3;
n-length(SAT_2(:,1));
dt-30;
m-n/2;
r-l;
SAT E(r,1)-SAT 2(1,1);
SATO (r, 1)-SAT 2 (1,1);
SAT D(r,1)-SAT 2(1,1);
SAT 0(r,2)-SAT 1(1,2);
SAT _(r,2)-SAT 2(1,2);
374
ST(i,7);
SAT D(r,2)-SAT_3(1,2);
for ind-2:n
r-r+1;
SAT O(r, 1)-SAT 2(ind,1);
SAT E(r, 1)-SAT 2(ind, 1);
SAT D(r, 1)-SAT 2(ind, 1);
o(r,2) SAT_ (ind,2);
SATO
SAT E(r,2)-SAT 2(ind,2);
SAT D(r, 2)-SAT 3(ind,2);
if SAT 2(ind,1)>SAT 2(ind-1,1)+dt
t miss-SAT 2(ind)-SAT 2(ind-1);
n miss-t miss/dt-1;
for i-l:nmiss+1
SAT 0(r,1)-SAT 2(ind-1,1)+dt*i;
SAT E (r, 1) -SAT 2 (ind-1, 1) +dt*i;
SAT D(r,1)-SAT 2(ind-1,1)+d:*i;
SAT 0 (r,2)-SAT_1(ind-1, 2)+i' (SAT 1(ind, 2)-SAT 1(ind-1, 2))/(n_miss+l);
SATE(r,2)-SAT 2(ind-1,2)+i*(SAT_2 (ind,2)-SAT 2(ind-1,2))/(nmiss+1);
SAT_D(r, 2) -SAT_3(ind-1, 2)+i' (SAT3 (ind, 2)-SAT3 (ind-1, 2) )/(nmiss+1);
if i<n miss+l
r-r+1;
end
end
end
end
N-r;
x-SAT 0(:,2);
y-SAT_E(:,2);
xc-xcorr (x, y, 'coeff');
aco-xcorr (x, 'coeff');
ace-xcorr(y, 'coeff');
n-index;
dt-30;
Ny-n/2+1;
fNy - 1/(2*dt);
T - n*dt;
df-fNy/ (Ny-1);
f-[0:df:fNy];
FT OBS - fft(aco);
FT EST - fft(ace);
FT CROSS-fft (xc);
P FT OBS-FT OBS.'conj(FT_OBS);
P FT EST-FT EST.*conj(FTEST);
P-FT-CROSS-FT CROSS.*conj (FT_CROSS);
Rft-real (FT CROSS);
Ift-imag(FTCROSS);
375
figure (1)
clf
subplot (2,2,1)
plot(SAT O(:,l), SAT O(:,2),'-.')
hold
plot(SATE(:,l), SAT_E(:,2))
grid
ylabel(' LC OBS / LC EST')
xlabel(' Time (sec) ')
title(' COSE (trimble) PRN 25 - segm 3: Phase Error')
subplot (2,2,2)
semilogx(f,PFTOBS(1:Ny),'-.');
hold
semilogx(f,P_FT EST(1:Ny));
grid
ylabel(' So_o(f) and S_e_e(f)')
title('Auto-Spectral Power Density')
xlabel('frequency [Hz]')
subplot(2,2,4)
semilogx.(f, PFT_CROSS(1:Ny));
hold
grid
ylabel('AMPLITUDE of S_o_e(f)')
title('CROSS-SPECTRAL Power Density')
xlabel(' frequency (Hz]')
subplot (2,2,3)
space for : COHERENCE
return
376
IP1IILI11IPIIIIIPIPIPIIPIPPIII~PPPIIIIII
Coherence obs est.m
~I~III~LIIPIIIIDI-PIUIIIPPIIPIII~I~IICII
This program computes the Coherence Function of the time series
LC OBS, LC EST.
PPIPIIP~PIPIIPIIPIIIIIIIPOIIP~·PIIIIIIII
clear
format long
load SAT EST
load SAT-OBS
N1-length(SAT OBS(:,1));
N2-length(SATEST(:,1));
index=O;
epoch - SAT OBS(:,1);
time sec- (epoch-l)*30+hours*3600+minutes*60;
SAT OBS(:,l)-timesec;
for j-i:Nl
for i-1:N2
if
SATOBS(j, 1)-SAT EST(i,1)
index-index+l;
SAT D 3(index,1)-SAT_ OBS(j,1);
SAT D_3(index, 2)-SAT.OBS(j, 4)-SAT..EST(i, 7);
SAT 0 1(index,1)-SAT OBS (j,1);
SATE 2 (index, 1)-SAT EST (i, 1);
SAT_O_1(index,2)-SAT_OBS(j,4);
SAT E_2 (index, 2) -SAT EST (i, 7);
end
end
end
SAT 1-SAT 0 1;
SAT 2-SAT E 2;
SAT_3SAT D 3;
n-length(SAT2(:,1));
dt-30;
m=n/2;
r-1;
SAT E(r, 1)-SAT 2(1,1);
SAT O(r,1)-SAT 2(1,1);
SAT D(r, 1)-SAT_2(1,1);
SAT 0(r,2)-SAT 1(1,2);
SAT E(r,2)-SAT 2(1,2);
SAT D(r,2)-SAT 3(1,2);
377
for ind-2:n
r-r+1;
SATO(r, 1) SAT 2(ind, 1);
SAT E(r,1)-SAT 2(ind,l);
SAT D(r, 1)-SAT 2(ind, 1);
SAT_O (r,2)SAT_ (ind,2);
SAT E(r,2)-SAT_2(ind,2);
SAT_D(r,2)-SAT 3(ind,2);
if
SAT 2(ind,1)>SAT 2(ind-1,1)+dt
t miss-SAT 2(ind)-SAT_2(ind-1);
n miss-t miss/dt-1;
for i-l:n miss+l
SAT_O(r,l)-SAT 2(ind-1,1)+dt*i;
SAT E(r, l)SAT 2(ind-1,1)+dt*i;
SAT D(r,1) SAT 2(ind-1,1)+dt*i;
SAT_O(r,2)-SAT 1(ind-1,2)+i*(SAT_1(ind,2)-SAT 1(ind-1,2))/(n miss+1);
SAT E(r,2)-SAT 2(ind-1,2)+i (SAT 2(ind,2)-SAT 2(ind-1,2))/(nmiss+1);
SATD (r,2)-SAT3 (ind-1,2)+i*(SAT_3(ind,2)-SAT3 (ind-1,2))/((nmiss+1);
if i<n miss+l
r-r+1;
end
end
end
end
m-r;
dt-30;
clear SATEST
clear SAT OBS
SAT OBS-SAT 0;
SATEST-SAT E;
ns-input('NUMBER OF SUBSECTIONS
nps-round ((m-1) /ns);
x-[0:dt: (nps-l)'dt];
df-l./((nps-l)*dt);
f-df* [0:round(nps/2)-1];
=')
lambda(2:nps/2)-1./f(2:nps/2);
lambda (1)-2*lambda (2);
logla-logl0(lambda);
T - (nps-1)*dt/2;
nstart-1;
nstop-nps;
normOBSnormESTnorm OBS
norm EST
(SATOBS (:,2)-mean(SAT_OBS( :,2)) )./std( SATOBS(:,2));
(SAT EST(:,2)-mean(SAT_EST(:,2)) )./std( SAT EST(:,2));
- SAT OBS(:,2);
- SAT EST(:,2);
for j-l:ns
378
0-norm_0BS(nstart:nstop);
FT OBS- fft(O);
R FT OBS-real(FT OBS);
I FTOBS-imag(FT_0BS);
AOBS(:,j)-R FT OBS.*R FT OBS/T+I FT OBS.I_FTOBS/T;
E=nor mEST(nstart:nstop);
FT EST - fft(E);
R FT EST-real(FT EST);
I FT EST-imag(FT_EST);
AEST(:,j)-R FT EST.*RFTEST/T+IFTEST.I FTEST/T;
A OBS_EST(:,j)-R_FT_EST.*R FT OBS/T+I FT EST.*I FT OBS/T;
nstart-nstop+l;
nstop-nstop+nps;
end
if ns -- 1
A OBS bar-A OBS';
A EST bar-A EST';
A 0BS EST bar-A_BS EST';
else
A OBS bar-sum(A OBS')/ns;
A ESTbar-sum(A EST' )/ns;
A OBS EST bar-sum(AOBS EST')/ns;
end
coherence-(A OBS EST bar.*A OBS EST bar)./(A OBS bar. *AEST bar);
for i-l:nps
if coherence(i)--l;
coherence(i) -- 0.9999999;
end
end
figure (1)
subplot (2,2,3)
pf(2:nps/2)- f(2:nps/2);
pf(1)= pf(2)/2;
semilogx(pf,coherence(1 :nps/2),'d');
hold
grid'
semilogx(pf,coherence (:nps/2));
xlabel(' frequency (Hz]')
ylabel('Coherence ')
title('COHEPELNC OF LC OBS and LC EST')
return
379
I~Ll·rPPIIPPPPIIII~~~~~I~1~~PPIlIII~PII~~~
Statistical Analysis.m
This performes the statistical analysis of the time series
LC_OBS, LCEST and LC DIFF.
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clear
format long
load SAT EST
load SATOBS
N1-length(SATOBS (:,1));
N2-length(SAT EST(:,1));
index-0;
hours-18;
minutes-45;
epoch - SAT OBS(:,l);
time sec- (epoch-l)*30+hours*3600+minutes*60;
SAT OBS(:,l)-time_sec;
for j-1:N1
for i-1:N2
if
SAT OBS(j,l)-SAT EST(i,1)
index-index+l;
SAT D 3(index,l)-SAT OBS(j,1);
SAT_
D 3(index,2)-SAT OBS(j,4)-SAT EST(i,7);
SATO (index, 1)-SATOBS(j, );
SAT E 2(index, )-SAT EST(i, );
SAT_O_1(index,2)-SATOBS(j,4);
SAT E 2(index,2)-SAT EST(i,7);
end
end
end
SAT 1-SAT_0O1;
SAT 2-SAT_ 2;
SAT_3-SAT D 3;
n-length (SAT_2 (:,1));
dt-30;
m-n/2;
r-1;
SAT E(r,l)-SAT 2(1,1);
SAT- (r, 1)-SAT 2 (1, 1);
SAT D(r, 1)SAT 2(1,1);
SAT 0(r,2) SAT 1(1,2);
SAT E(r,2)-SAT 2(1,2);
SAT D(r,2)-SAT_3(1,2);
380
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