APPLICATIONS OF PLASMA DENSITY MEASUREMENTS
TO SPACECRAFT RADIO TRACKING
by
Thomas Marshall Eubanks
B.S.,
Massachusetts Institute of Technology
(1977)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS OF THE
DEGREE OF
MASTER OF SCIENCE
at the
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
June 1980
\c, Massachusetts Institute of
Technology
Signature of Author
Department of
Earth and Planetary Sciences
May 26, 1980
Certified by
Irwin I.
Shapiro
Thesis Supervisor
Accepted by
Chairman,
Departmental
U)ndgrer
MASSACHUSETTS INSTITUTE
OF TECHNOLO Y
JUN 19 1980
LIBRARIES
Graduate
Committee
-2-
APPLICATIONS OF PLASMA DENSITY MEASUREMENTS
TO SPACECRAFT RADIO TRACKING
by
Thomas Marshall Eubanks
Submitted to the Department of Earth and Planetary Sciences
on May 26, 1980 in partial fulfillment of the requirements
for the degree of Master of Science.
ABSTRACT
S-band plasma delays are estimated as part of a test of
the general relativistic time delay effect conducted during
the Viking Mission to Mars. The processing of radio tracking
data taken with the Viking Orbiters and Landers is discussed.
The statistical properties of Viking Orbiter dual-frequency
It was condelay and Doppler measurements are described.
cluded that the plasma delay can be adequately modeled as a
random walk. The implications of this model on estimation of
The results of
Viking Lander plasma delays are discussed.
use of the random walk model for Viking lander plasma delay
correction are compared with the results from other plasma
models, and it is concluded that this model is sufficient for
estimation of Viking Lander plasma delays.
Thesis Supervisor:
Title:
Dr.
Irwin I. Shapiro
Professor of Physics and Geophysics
-3-
TABLE OF
I.
INTRODUCTION
II.
THE
A.
B.
C.
D.
E.
CONTENTS
OBSERVABLES
Introduction
The Propagation of Electromagnetic
Waves in a Plasma
of the Group and
1.
The Definition
Phase Velocity
2.
Group and Phase Velocity in the
Coronal Plasma
Group Delays and Doppler Shift caused
by the Solar Wind
1.
The Thin Screen Model of the
Solar Plasma
The Measurement Apparatus
Terrestrial
Propagation Effects
9
10
11
12
17
18
24
27
III.
COMPUTER DATA PROCESSING
A.
Introduction
B.
Data Collection
C.
Data Editing and Calibration
1.
Range Calibrations
2.
The SX Bias
3.
The PRA Demod
D.
Data Editing
32
32
32
35
35
39
40
41
IV.
THE STATISTICAL NATURE OF THE PLASMA DELAY
A.
Introduction
B.
The Plasma Autocorrelation
The Autocorrelation of the SX Delay
1.
2.
The Autocorrelation of the SX Doppler
C.
Estimation and Smoothing of a Random
Walk Process
46
46
46
48
51
V.
LANDER PLASMA CORRECTIONS
A.
Introduction
B.
Computer Processing of Plasma Corrections
C.
Lander Residuals and the Plasma
Corrections
Experimental Tests of Our Conclusions
D.
60
60
60
CONCLUSIONS
73
VI.
64
70
-4-
Chapter I
Introduction
advances in many fields have
In the past twenty years,
made possible an enormous increase
in the accuracy of meas-
urements of positions and velocities of objects in the inner
solar system.
An important factor in this progress has been
the placement
of
the
program of
probes throughout the
solar system during
interplanetary exploration
initiated
by
the
United States in the early 1960's.
Radio tracking using spacecraft transponders makes possible accurate measurements
of delays
and Doppler shifts of
signals propagating between a ground station and the spacecraft.
With probes in interplanetary space, the solar system
can be used as a vast laboratory for gravity research,
cluding
research
on
general
properties of the planets.
relativity
dynamical
the
and
in-
The solar system, not being under
the control of the experimenter, is poorly designed for such
experiments.
One major complication to the interpretation of
present day radio
tracking data is the effect of the inter-
planetary medium on propagating radio waves.
of
the
total
plasma
spacecraft given
delay
from
downlink
another
problem addressed
delay
in
for delay observations from
only
spacecraft
this
The estimation
measurements
is
thesis.
the
of
major
the
one
plasma
experimental
This work was
done
in
connection with a test of general relativity conducted during
the Viking mission to Mars (reference 1).
-5-
At present, the solar system is used for the most definAs was first shown by I.
itive tests of general relativity.
I.
the mass of the sun causes
Shapiro in 1964 (reference 21),
an -increase in the radio propagation delay over that expected
from Euclidean
delay
occurs
geometry.
at
superior
directly between the
close
to the
The
sun.
maximum
conjunction,
earth and
Mars and
For Viking,
about 250 microseconds
excess
when
the
the
the delay
relativistic
sun
moves
raypath passes
at
that time
(Usec) while the corresponding
is
total
Unfortunately, the
round trip delay is about 2500 seconds.
plasma effect is also at a maximum at superior conjunction,
with the greatest measured
plasma delay being on the order
of 100 Usec at a radio wavelength of 12 cm.
mates
of
the
delay
are
spacecraft
and
plasma
thus
Accurate esti-
vital
the
to
general
relativity experiment.
The
Viking
under good conditions,
possible,
round
trip
radio
propagation
ground
equipment
to measure
time
with
an
the
make
it
Earth-Mars
uncertainty
of
about 10 nanoseconds (nsec) and the carrier frequency Doppler
shift
with
an
uncertainty of
propagation medium makes
it
on the
order
impossible to
of
1 mHz.
infer
The
the vacuum
range and line-of-sight velocity to that accuracy, and currently constitutes the largest source of error in the interpretation of interplanetary radio tracking data.
The
propagation
effects
from
the
medium
between
Earth and Mars are dominated, at radio frequencies, by the
the
-6-
contribution from the interplanetary plasma in the solar co-
atmosphere
terrestrial
the
parts of the propagation medium are
major
The other
rona.
and
dual-fre-
The
ionosphere.
quency data includes a contribution from the ionosphere but
not
from
the
from
contribution
non-dispersive
the
Earth's
neutral atmosphere.
solar
The
plasma
is
a highly
is proportional to the
excess plasma delay or Doppler shift
inverse
square of
plasma density.
the
carrier
The
dispersive medium.
frequency,
and
to
the
local
The solar wind is very complex, with density
It is impos-
fluctuations at times of the order of the mean.
sible to adequately estimate the solar plasma delays purely
from
a time
terrestrial
solar
wind
difficult
averaged
density
atmosphere
from
first
than modeling
as
model,
propagation
principles
the weather
is
delays.
would
on
with
the
model
the
done
To
probably
Earth.
It
be
more
is thus
necessary to consider statistical models of the plasma delay
and delay rate, similar in spirit to the models discussed in
references 13 and 14.
This thesis is concerned with Viking Orbiter and Lander
radio tracking data taken between July 20, 1976 and September
3,
1977.
Viking Lander
(VL)
1 and Viking Orbiter
1 were
(VO)
launched as a single spacecraft on 20 August 1975 and were
inserted
into. Martian orbit
on 19 June
1976.
VL2
and V02
were launched together on 9 September 1975 and were inserted
into Martian orbit on 7 August 1976.
VLl landed on the
-7-
surface
1976.
able
of
Mars
on
20
July
As of January 1980, V02
of
communicating
The Landers are
at S-band
by
VL2
on
3
September
is inactive and VL2 is incap-
directly
take part in these radio
ing
followed
and
Earth
with
is
tracking experiments.
equipped for interplanetary radio
(12 cm) only.
The
Orbiters have,
track-
in addition,
a coherent dual-frequency downlink, at S-band and X-band
cm).
Differenced
measurements
can
to
unable
dual-frequency
provide
(S-band minus X-band or
estimates
of
the
time
delay
(2.3
SX)
and
Doppler shift contributions from the interplanetary plasma.
On November
earth
and
through
Mars
25,
1976 and
again
(together with
superior
conjunction.
on January
the
Viking
The
21,
1979,
spacecraft)
plasma
and
the
passed
relativistic
effects are at a maximum near superior conjunction, while the
signal-to-noise ratio
tion
and
solar
is
radio
lowest there
(due to plasma attenuaDespite
interference).
this,
it
was
possible to track the Viking spacecraft to within 2 or 3 days
before and after superior conjunction.
The
SpaceCraft
designation
(S/C)
numbers
are
used
as
an
alternate
for the Viking spacecraft:
VLl
= S/C
VO1
= S/C 27
26
VL2 = S/C 29
V02 = S/C 30
The Orbiters
are. subject to unmodeled accelerations such
as gas leaks from the attitude control system, which compli
-8-
cate
the
interpretation
The Orbiter
Orbiter,
of
Orbiter
SX data are not
since the range or
range
and
affected by the
Doppler
data.
motion of
the
Doppler shift to the spacecraft
The
cancels out in the differencing.
Landers, which cannot
make dual-frequency measurements, are fixed on the surface of
Mars, which is nearly free from stochastic accelerations.
is
thus
necessary
to
plasma
estimate
Lander delay measurements
for
both the
corrections
uplink
for
It
the
and downlink
from Orbiter downlink dual-frequency measurements.
Chapter
differenced
II
will
observables
discuss
the
that can
and how the measurements are made.
computer processing
basic
observables,
be constructed
the
from them,
Chapter III discusses the
required before the dual-frequency
ter data can be used to obtain plasma delays.
Orbi-
Chapter
IV
gives the results of a statistical study of the plasma data,
and Chapter V describes the application of these results to
Lander range measurements.
-9-
Chapter II
The Observables
A.
Introduction
Radio tracking of interplanetary spacecraft provides two
observables, the round trip propagation (group) delay and the
Doppler shift of the carrier frequency.
The group delay and
Doppler shift are a function of the group and phase velocities,
respectively,
in
the
propagation
In a tenuous plasma,
signal path.
medium
along
the
such as the solar corona,
the phase and group velocities are displaced by opposite and
nearly equal
amounts
c, the
from
velocity of
light.
phase velocity is greater than the group velocity, which
The
is
the velocity of energy and information transfer in the medium
(reference 6).
In a tenuous plasma,
the group and phase velocities can
be approximated by
aN
v
phase
= c(l +
e)
2f
(2.1a)
and
aN
v
group
= c(l
-
2f2e)
(2.b)
where c is the velocity of light in a vacuum, f is the carrier frequency (Hz), a is a constant equal to 8.1'10 , and N
is the electron density in electrons cm-3
e
-10-
On
Earth,
good
vacuum,
usec
to
tion.
model
solar
but the
the
total
The
coronal
and
corona
range
plasma
delay
density
fluctuations
difficulty
is
Orbiters
ranging
taken
have
the
code
near
especially
ability
frequency
measurements
using the
Viking
to
at
of
the
Orbiters are
the
a
very
100
conjunchard
mean
severe
for
in
the
to
some
Viking
sensitive to delay range
conjunction.
The
Viking
a
received
S-band
and
X-band.
retransmit
both
superior
notoriously
exceeding
superior
coherently
plasma delays, and
considered
near
is
relativity experiment, which is most
measurements
be
would
coronal plasma can contribute up to
S-band
shows
This
cases.
the
S-band
group
delay
and
Dual-
Doppler
shift
used to estimate Viking
Lander
improve estimates of the true range to the
Landers.
In this
chapter
I will
first
discuss
the
nature
of
the
observables, both the delay and Doppler shift and the differenced
dual-frequency
between
second
the
part
observables,
observables
of
this
and
the
chapter,
I
including
the
connection
propagation medium.
will
discuss
the
In
the
techniques
and equipment used to make observations with the Orbiters and
the
Landers.
sing
required
The
to
next chapter concerns
handle
available from the Viking
mass
of
computer
proces-
dual-frequency
data
experiment.
The Propagation of Electromagnetic Waves in a Plasma
B.
In
of
the
the
an
a
vacuum,
there
electromagnetic
is
only one
signal,
c.
In
velocity
a
of
propagation
neutral medium, such
'as
the
Earth's
somewhat
atmosphere,
less
than
c.
In
the
a
velocity
of
tenuous
propagation
plasma,
there
is
are
(at
least) two velocities of propagation, the group and the phase
The group velocity is
velocity.
of
wave
packets
or
of
velocity of
the
modulation
of
the
carrier;
delay is the round-trip propagation time.
is
the
velocity
wave crest
is
of
propagation
indistinguishable
ured
modulo
only
the change in the
The
val.
of
21r),
the
concepts
total
propagation
The phase velocity
wave crests.
(the phase
phase
delay
the group
Since
can only be
cannot
be
group
and
phase
meas-
measured,
phase delay over a measurement
of
each
inter-
and
velocities
the
expressions derived for them are approximations, which become
less
valid
strength
as
the
plasma
increases.
In
density
Section
or
field
I
these
B.2
external magnetic
will
show
that
approximations are precise enough for the Viking radio propagation experiments.
(Section 1 is adapted
from
references 6
and 7.)
1.
The Definition of the Group and Phase Velocity
Assume
u(x,t
that
at
time
to wave packet
can be described
) which has Fourier transform of
A(k)-
1
/u(x,t
)e
-ik.x
3
- -dx
-I
Here
k is
tude
of
the wave vector,
k,
and
the
in units of cm
wavelength,
solution to the Helmholtz wave
is
by
X,
is
equation
1
, k
2ir/k.
for
a
(2.2)
is
the
The
magnigeneral
traveling wave
-12-
u(x,t)
where
w
is
number k.
the
-
A(k)e
/2w
angular
is equal
a
tenuous
vphase < c.
c,
plasma,
vphase
with
to ck, but,
the
wave-
in general, w
Sw(k)
k(2.4)
c
>
and,
(2.4)
in
a
neutral
medium,
Although the phase velocity can be greater
information is propagated
less than c,
(2.3)
The phase velocity is defined as
Vphase
In
k
frequency associated
In a vacuum, w
is a function of k.
-
O
than
the group velocity, which
at
is
and the postulates of special relativity are not
violated.
If
compact
the
wavenumber
and
centered
A(k)
distribution
some
about
value
of
some
k 0 , then
w(k)
can
dw
(k
w(k) = w(k ) + dw
-
(k -
+ higher order
k
(2.5)
terms
terms, the
integral
in Equation 2.3
can be performed to give
- w(k ))
u( ,t)
- e
--
0
be
k
0
it(k
is
expansion in k:
expressed as a Taylor series
Ignoring the higher order
signal
u(x
dw
d-w
dke k 0 t,to) 0
(2.6)
-13-
A comparison with
the original
signal
u(x,t ) shows
that,
to
within a phase factor, the wave packet travels undisturbed at
the group velocity, which is thus defined to be
dw
Vgroup
If the higher order terms
tant,
then
the
wave
d
(2.7)
in the expansion of w(k) are impor-
packet
will
change
shape
as
it
travels
and the group velocity may lose much of its meaning.
with
2.
Group and Phase Velocity in the Coronal Plasma
The
solar
about
4%
corona
consists
ionized
helium
of
largely
by
weight
ionized
and
hydrogen,
essentially
no
(about 10
This plasma is so tenuous
component.
-3
electrons cm 3 at 1 A.U.) that interactions between particles
neutral
in
the gas can be
Although the
free particles.
the
is
10 6K, it
of
order
it can be treated as a sea of
and
ignored,
wind has
solar
not
temperatures
relativistic medium,
a
on
since
thermal velocities are on the order of only 4000 km sec-1 for
electrons
ics
can
corona.
at that
be
used
The
temperature.
to
treat
following
Thus
radio
classical electrodynam-
propagation
discussion
is
adapted
in
from
the
solar
reference
7, page 210 and following.
Let
r
describe
the
position
coordinate system centered upon the
position
of
p.
Under
the
of
is
particle,
p,
in
a
instantaneous equilibrium
influence
wave, the equation of motion of p
a
of
a
propagating
radio
-14-
m dL
-eB
dt 2
m
where
and
tively,
B
is
of
direction
E
is
e
the
are
the
propagation
(2.8)
-eE
mass
particle's
magnetic
external
transverse
the
x dL
dt
c-
field
field of
respec-
charge,
parallel
to
constant),
the
and
be
a
the
propagating wave.
to
(assumed
electric
and
the transverse magnetic field of
(Notice that we are ignoring
interactions.)
the radio wave, as well as particle
Equation 2.8 can be solved to yield
e(mw(w
r
=
wb
where
ejB)/(mc)
is
the
1
w)E
(2.9)
frequency
of
particle in the external magnetic field,
frequency.
precession
of
the
called the cyclotron
The displacement of all charged particles
in the
plasma gives rise to a net dipole moment and thus to a macroscopic dielectric constant of
2
w
=
1-
w(w
P
+
(2.10)
wb )
where
w
=
(Tne2)
1/2
=
5.64'10
4
/n radians sec
is called
density
refers
the plasma
(particles cm
to the
(2.11)
(for electrons)
n
frequency, and
-3
).
(The
±
in
is
the
particle
Equation
2.10
two senses of circular polarization.)
number
and
2.11
-15-
kc
w -
the
Using
expanding
given
definitions
in a power series
in
Vphas
e
k
phase
v
1/2- 2
-
1/2
2
wb
w
w
w
+
.)
2
w
and
2.7
2
w2
(12--+3-b...)
2
w
4
1/4--
+
4
w
2
w
and
2.4
Equation
(_- + wb
w
c(l
group
+
(2.12)
in wp and wb, we get
2
w
c+
c(1
w
the wavenumber k is
E, and
relation between w,
The
...
(2.13a)
w
4
-
1/8
w
4
+
...
(2.13b)
w
and
dw
d
-
=
w
+
order
first
The
in
terms
the distortion
that
is a measure
and
expansion of w(k),
term
in w
rotation.
Faraday
to
I will
a propagating wave packet.
in
second-order
the
correspond
wb
T is the next term in the
of
(2.13c)
cw
dk
and
dominates,
that
show
T,
the
distortion term, is negligible.
Clearly the higher order
the
radians
solar
the
sec-1 and
radii
Under
wind,
solar
trons/cm3
(R ),
and
these
the
At
increases.
density
plasma
terms will be more important as
Ak
is
which
the
equal
is
conditions,
2"10 7 radians sec -1 .
w
Thus,
Thus,
4 *10 -
as deep as
average
average
to
S-band,
=
210
7
is
density
field
is
radians
considering
considering
2.3109
2r
radians cm
-
At
is
105
about
1
sec
5
data probes
the Viking
plasma
magnetic
4
w
-1
the electrons
and
elecGauss.
wb
only,
the electrons only,
=
-16-
2
w
P
-
1. 510-6
(2.14a)
1.2"'10
(2.14b)
w
and
TAk = 10
Only the first term in w
p
-9
<< v
group
- c
(2.14c)
need be retained.
To first order, therefore, the group and phase velocity
differ by equal but opposite amounts from c.
group velocity, the velocity of
information
tenuous plasma, is less than c, as expected.
Note that the
transport
in a
For the rest of
this thesis, we will assume that the group and phase velocity
in the solar wind are given to sufficient accuracy by Equation 2.1a and Equation 2.1b.
In this part of this chapter, we have used CGS units.
From now on we will use natural units in which c = 1.
w/27
If f
=
is the radio carrier frequency, then
Vphas e
= 1 +
4.03038'10 72
(2.15a)
v
= 1 -
4.03038'10 7 ne
(2.15b)
and
group
f2
-3
with n in electrons cm-.
e
C.
Group Delays and Doppler Shifts Caused by the Solar Wind
If
will be
R is
the
true
range
delay,
the
measured
delay,
r,
-17-
R
m
R
=
ds
S0 Vgroup
-
0
2
ds(1 +
2w
(2.16)
)
Le t
(e
)
7
R
P =
2m ff/
O
n ds = 0.03038
R
107
e
e
ds (2.17)
O
P is proportional to the integrated columnar content (the4R
ne ds) thus
t
-= R + P/f
2
(2.18)
R is independent of f, and therefore P can be estimated if
m
is measured for two different frequencies.
The
derivation
of
the
Doppler
phase delay
proceeds
much the same fashion as that of the group delay.
in
The total
phase delay cannot be measured, only the relative delay from
the start of the observation session or pass.
Since
it is
not possible to continuously monitor the phase delay from the
start of the mission, the total phase delay is unknown.
If
r p is the phase delay at the ground station, then the Doppler
shift measured, Dm, is
m
From
dual-frequency
ferred.
dr
dR
dt
adt
dP
1
w
2 dt(2.19)
Doppler measurements,
It is possible in theory to use
ments to find the
initial value for
P,
dP/dt
can be
in-
SX range measure-
and to use the more
-18-
accurate
Doppler
This process
is
provides
most
the
to
measurements
called
Range
accurate
estimate
Integrated
estimates
the cycle slip problem, which
the
change
Doppler
of
P.
is discussed
(or
in
P.
RID)
and
Unfortunately,
in Chapter III.
D,
has prevented use of the Doppler measurements, and the potential accuracy of RID measurements has yet to be achieved.
delay
1.
The Thin Screen Model of the Solar Plasma
The
Viking
over
the
corrected
for
Orbiters
downlink
plasma
can
be
used
only.
delays
The
for
to
measure
measured
both
the
the
range
plasma
must
uplink
and
be
the
downlink, and thus the uplink plasma delay must be calculated
from downlink
=
Pdown)
the
to
model assumes that
same as
find
plasma delay measurements.
a
consider
the
the
The
approximation
some model
of
the
(or Pup
uplink plasma delay,
(measured) downlink delay,
better
static
to
solar
Pdown"
P
In
,upis
order
Pup'
it
is
necessary
corona
and
of
the
to
measure-
ment.
Experimental
based
term
and
studies
(reference
20)
using
both
ground-
in situ spacecraft measurements show that the
average
coronal
electron
number
density
can
be
long
modeled
by
n
where
r
is
model of the
(r) =
in
1.55 10
solar
8
+
3.0 10
r
r
radii.
If
we
6
electrons cm
use
Equation
local plasma density, it is clear
contribution to
the
integrated
-3
2.21
(2.20)
as
a
that the major
plasma density will come
from
-19-
near the thin-screen point, the point on the raypath closest
Figure 1 shows
to the sun.
the observation geometry.
The
vacuum delay is
R. = R
i
P 2 are
P 1 and
R
+
up
down
called
(A. + B ) + (C. + D.)
=
i
1
1
(2.21)
1
P2 ' is
thin-screen points.
the
the
thin-screen point for the downlink matched to the ith uplink.
Each thin-screen point has a location in time as well as in.
space, and the ith thin-screen time is the time at which the
signal passes
and
the
through
P1 and P 2 ' are matched
Pi .
separation
spatial
between
in time,
points
thin-screen
is
The thin-screen model assigns all of Pup to P1 and
ignored.
Pdown to P2 and ignores contributions from other parts of the
This model is most realistic near superior conjunc-
raypath.
tion, when the
Far
from
raypath nearly grazes the
conjunction,
superior
irrelevant
at
since
such
the
times the
limb of
model
thin-screen
plasma
sun.
the
and
delay
is
delay
rate are small and changing slowly.
In
assume
our
that
implementation
P1
is
of
coincident
the
thin-screen
P2''
with
which
model,
we
amounts
to
ignoring the distance between P 1 and P 2 ' and any asymmetry in
the
plasma
contributions
away
from
the
thin-screen
point
(which would contribute. at different times to Pup and Pdown ) .
Ri
is
an
estimate
conjunction,
we
of
can
the
use
delay at
Ri
to
time
ti .
approximate
Near
the
superior
thin-screen
-20-
delay
Atts.
Earth and
If
Re
(=
is the distance between
1 A.U.)
the Mars-Sun
is
(= 1.5 A.U.)
the sun at t. 1 and Rm
the
distance at ti,*then
1
C.
1
2Ri
(R 2 + R 2 - R 2 )
i
Assume that Ci = Bi; if Atts is
m
(2.22)
e
the thin-screen delay then
1
in seconds and 998
tts is
r is measured in A.U.,
where
is
twice the conversion factor.
The spatial separation between the Orbiters and Landers
is small, about 5 104 km at most, and the spatial separation
The velocity
between the thin screen points is even smaller.
of
the
solar wind
is
about
400
km sec -
1
1 A.U.,
at
which
implies that the spatial separation between the thin-screen
points
introduce
With
plasma
a
timing
delay
rate
these timing errors would
35 nsec, which
of
errors on the order
of
1
psec/hour
introduce
is not negligible.
(a
2 minutes.
value),
large
a plasma delay error
We
found
of
from numerical
studies that our plasma estimates were remarkably insensitive
to thin-screen errors on the order of an hour or less.
It
was decided to ignore the spatial separation between measurements and desired corrections for the present.
In
ponder.
two-way
ranging,
the
spacecraft
acts
as
a
trans-
It receives the uplink ranging signal, and amplifies
-21-
and
rebroadcasts
talk between the
the
craft,
frequency.
to
The
the
and
Rup
is
and
by
turn-around
carrier
by b
k
(equal
(equal
ratios
to
and
is
P
P
up
-~f+
f
-
down
2
(kf)
(2.24)
P
P
up
down
r x = Rup +R down +u 2 + (bf) 2
(2.25)
up
down
and
Rdown
are
the true
uplink and downlink
range,
frequency independent effects have been ignored.
and
an
total
r
are measured.
Thus,
Sx
-
estimate
of
delay
Pdown
tion.
down
f2
-
only,
which can be found
2
2
b
as
1
1
(
)
(2.26)
(2.26)
k
indicated
if
earlier.
The
S-band correction is
1
=
f
the
x ),
(rs -
be equal to
SXCOR
In
these
+R R
the
is multiplied
range measured at S-band
= R
cross-
is
Let SXdelay
I
multiplies
downlink,
Using
to avoid
receiver on board the space-
coherently
S-band
X-band.
s
and at X-band
In order
received frequency
for
for
Earth.
transmitter and
Equation 2.19, the
where
to
transponder
240/221)
880/221)
it back
static
model,
2.25 becomes
which
(P
up
+
assumes
1
Pwn
-7 down
b
that
P
(2.27)
=
P down
Equa-
-22-
Pdown
f2
(1 +
SXCOR
f
with
= 2.3544
)
standard convention
any
range
is
If
reception at the Earth.
P2
from
P
refers
i)
measurement
with
to
down
(ti
ik
the
S-band
ti,
then,
time-tag
signal
influence
(which will
t
2
SXdelay (ti)
SXCOR(t
of
the time
is
measurement on the ground at time t + Di)
If
associated
tag
Pdown (t) denotes the plasma delay
time
at
time
the
that
Doppler measurement
or
(2.28)
SXran
b
The
contribution
1
--
-
the
then
2
Di)
(
-b
2
22
correction
for
Equations
from
(2.29)
)
a
range
2.26
and
2.28
SXCOR(ti)
-
f 2
(P
up
(t
i
-
(B. + C.
i
1
+ D))
1
(2.30)
+
an
Given
estimate,
P
f+
2down
estimate
SX(t),
of
(t. -
D.))
1
the
1
time,& ~,
thin-screen
and
an
of the SXdelay then the estimate of SXCOR,
SXCOR is
SXCOR(ti)
2
=
k
k2
A2
2(SX(t i)
- b
for an S-band delay correction.
for b and k,
A
A
+ b SX(t
i
-b&s
))
(2.31a)
Using the numerical values
we get
A
A
SXCOR(t ) = 1.080357 (SX(t ) + 1.179337 SX(t.-ts))
(2.31b)
-23-
the approximations used in Equation
It might be wondered if
2.18
are
calculated
I compared t
justified.
by
Equation
2.32 with delays determined from an accurate ephemeris.
comparison
is
shown
in
I.
Table
The maximum timing
(a large value),
error
With a plasma
near superior conjunction is about 30 seconds.
delay rate of 1 psec/hour
This
this timing error
would cause a 9 nsec error in the plasma delay correction.
TABLE I
t (seconds)
s
Approximation
Date
MM/DD/YY
Accurate
Julian Day
Error
11/04/76
11/14/76
2443087
2443102
1541
1507
34
1522
1501
21
11/27/76
2443110
1511
1497
14
12/09/76
2443122
1496
1488
8
From an ephemeris, the spatial separation between P 1 and
P2 ' can be calculated explicitly.
ally take
P2'
account of
the
It is planned to eventu-
spatial separation
between P 1 and
by multiplying the matched downlink delay by
r(P 1) 2.4
(r(P2'))
where r(Pi) is the Sun-P
D.
i
distance.
The Measurement Apparatus
Range
Deep Space
and
Doppler
Tracking
are measured
Network
(DSN).
at the
stations
of the
The DSN maintains three
-24-
tracking station complexes, spaced so that any interplanetary
spacecraft is visible from at least one of them at any time.
One station at each complex
is responsible
for most of the
DSN 14 at Goldstone, Califor-
dual-frequency radio tracking:
nia, DSN 43 near Canberra, Australia, and DSN 63 near Madrid,
Spain.
Each
of
these
mounted,
paraboloidal
is
fully
a
steerable,
The
in diameter.
64 meters
antenna
azimuthally
26-meter diameter antenna at DSN 12, on Goldstone, California, has been used for SX measurements since mid-1978.
Since about 1970, range delay at the DSN has been measured
by
ranging
(reference
technique
wave
or
sine
wave
transmission, and
correlated
square
wave
to
which
machines
2-5).
estimate
period.
modulates
transmitted
the
To
a
and
ranging,
the
the
received
delay
a
carrier
range delay modulo
resolve
ranging
sequential
In sequential
sequence
the
use
square
before
signals
the sine
ambiguity,
are
or
the
period of the modulation sequence is doubled and the correlation repeated, yielding the
range modulo the longer period.
This process is repeated until the codelength is larger than
the a priori range uncertainty.
Each sequence with a given
period is said to determine a ranging code component.
The synthesizer frequency used, fs,
(, 22 MHz) is multi-
plied by 96 to yield the transmitted carrier frequency, fc,
of 2.1 GHz.
For the Planetary Ranging Assembly (PRA), which
uses square wave modulation only, the transmitter range coder
output has a period of
-25-
64-2
tn=
3f
n
2 n+ll
f=
(2.32)
where n, a positive integer, is called the order of the code
Note that the transmitter code period is a func-
component.
The MU-2 machine, which can
tion of the carrier frequency.
use
sine
wave
with
operate
essentially
n
It can be seen from a graph of the correlator
equal to zero.
outputs
can
modulation,
3),
(Figure
the
that
must
correlation
be
done
in
quadrature to completely resolve the range code phase.
Two types of ranging machines, the PRA (also called the
PLanetary OPerational ranging machine, or PLOP)
are now used to make dual-frequency
The ,MU-2
measurements.
(only one exists) and
is
an
and the MU-2,
range. and Doppler
experimental
shift
ranging machine
is the only machine capable of making
unambiguous SX range measurements.
The PRA machine
is
the
standard DSN ranging machine, and is used at every DSN staFrom the start of the mission
tion.
1977, the MU-2 machine was at DSN 14.
until mid 1979,
(1975)
until April 15,
From November 15, 1978
this machine was at DSN 43.
While the MU-2
was at DSN 14 it was used exclusively with square wave modulation, but after it was moved to DSN 43 it was used exclu(A. Zygielbaum,
sively with sine wave modulation.
private
communication).
The bandwidth of the spacecraft transponder is about 3.5
MHz,
which,
lengths
together
(Equation
2.34),
with
the
choice
limits the
of
smallest
code
component
code
component
-26-
to a period
and
(or length)
tions
(references
shift
is
1,
which
using
the
shows
a
a
The
Phase
Locked
of
diagram
the
(PLL)
Loop
the
in
a phase
as
frequency
carrier
phase
of
signal
transmitted
block
5).
cycles
counts
an
about 10 ns or less under good condi-
4,
by
measured
has
and
level,
sub-nsec
measured
The
modulation.
wave
sine
experimental scatter of
ceiver
for square wave modulation,
is discretized at the
delay
range
for
that
half
to
of 2 psec
S-band
Doppler
signal
received
Figure
reference.
ranging
re-
the
in
2
system
(after
range
compo-
reference 3).
The
PRA machine
can
(S-band
or
one
nents
of
other
frequency code
be
ponent.
Thus
be
resolving
the
PRA
respect
SX
the
of
to
the
be
by
other
ambiguity
are
The
X-band
code)
code com-
(2 usec)
shortest
S-X
the
measure
code.
range
means.
modulo
range
This SX range
(see Equation 2.34).
resolved
must
The
machine
2 usec
ambiguity
chapter,
the
all
received
(usually chosen
PRA
the
is about
which
X-band)
correlated with
can only
ti,
correlate
The
discussed
details
the
in
of
next
in Section C.3.
spacecraft
to
the
will,
ground
in
general,
A change
station.
be
in
in
motion
with
during
range
a
range measurement causes a change in the phase of the returning
modulation.
frequency shift
To
is
increase
used to
integration
times,
the
produce a corresponding
the phase of the range correlation template.
carrier
change
in
-27-
E.
Terrestrial Propagation Effects
The
Earth's
the
to
contributes
ionosphere,
of
for
the
each
plasma
of
delay
tracking
every
60
this
model
stant
and
the
resulting
use
by
is
ionosphere
and
of
data
the
at
a
rotation
from
thin
to
shell
be
a
constant
each
in
sampled
mapped
are
the
Satellite
continuously,
assumed
density
improve
Technology
integrated
available
are
to
used
Faraday
is
direction
the
be
Applications
an
orbit
thickness
spherical
to
the
model.
In
shell
height
of
cona
above
Earth.
be the
Let E(t)
an
to
seconds
spacecraft's
The
correction.
geosynchronous
can
stations,
also
rotation
Faraday
which
delay,
ionospheric
DSN
station
plasma delay.
measured
the
measurements
a dispersive medium,
being
of
estimate
the
spacecraft elevation angle and D
delay,
ionosphere
zenith
craft ionospheric delay estimate
(t)
be
the space-
then
is
D (t)
ION E(t) =2
2
(rl/r
(1 -
where
r1
is
the
of
radius
the
2
r2 -
r1 is
equal
station, a
tion of
fifth
time
entire pass.
order
rather
E,
is
(2.33)
and
is
r2
mean
the
then
(r
1
/r
2
)2
is
0.89866,
For each pass at each ground
polynomial,
than
1/2
E)
If r 1 is assumed to be 6378
to 350 km,
which is the value used at JPL.
cos
Earth,
radius of the ionospheric shell.
km and
)
considered
fitted
to
ION
These polynomials, prepared under
to
be
E(t)
a
func-
over
the
the direction
-28-
of
H.
Dr.
Royden
at
JPL,
will
be
available
made
to
MIT
at
some time in the future.
Each DSN station uses one
several
in
measured
was
data could
concluded
delays
by
Dr.
from
observing
Thus
the
Faraday
times)
be
and
(the
from
assumed
zenith
delay
delay, about
the
bias
to
delays.
that,
satellites
sufficient
to
for
only
use
that
last
With
interrupted
change in
due
to
the
delay
satellites
geosynchronous
sometimes
for
equipment
failures.
contain
measurements
delay
ionosphere
comparison
there
is
estimates
on
be
the
one
cannot
measurements
infefinitely,
rotation
2 nsec
It
mapped
a
many
consistant
An attempt was made to estimate the bias
unknown bias.
consistency
ATS
addi-
between
comparison
rotation
session.
only
are
could be
This
mapped
are
(H. Royden, private communication).
Faraday
sessions
and
from
the
plasma delay, only the
measure the total
months,
improve
is
it
satellites
simultaneously.
different
polynomials,
Unfortunately,
observing
to
the local
ionospheric delay
the
Royden
satellite per station
an
and
used
several
cases,
directions
be
obtained
production
over
some
from one station,
visible
tional
In
delay.
ionospheric
satellite to measure
at
same
a
still
S-band.
at
be
positive
ionosonde data,
with
the
must
bias
order
There
each
of
in
the
is no
station,
the
but
at
from
all
it must
ionospheric
nighttime
zenith
reason to expect
which
means
that
there probably would be systematic errors on the order of 2-5
nsec
between polynomials
from different stations.
Note that
-29-
the zenith delay causes nonconstant error in the
a bias in
delay mapped to the spacecraft.
general,
the
be
Thus the bias will not, in
the
when comparing
same
delay
ionospheric
estimate from one station at different times.
The ionosphere data can be used to improve the interpretation
Lander
of
delay
First,
measurements.
and
Lander
Orbiter tracking data are often taken at different stations,
and these stations do not share the same ionospheric contribution.
polynomials
The
to
contribution
the
can
plasma
be
used
from
the
thus
delay
replace
to
the
station
Orbiter
ionosphere with that from the Lander station ionosphere.
Second the thin-screen model does not properly model the
The uplink
ionospheric delay contribution.
is
contribution
before
the
made
at
the
matched downlink
polynomials
can
be
used
to
uplink
the
replace
some
time
contribution.
The
time,
send
ionospheric
ionospheric
receive
iono-
time
spheric contribution in the matched observable by the appropriate send time ionospheric delay.
These corrections will be
most important far from superior conjunction, when the solar
plasma delay is small, and the ionosphere contributes a large
fraction of the plasma delay.
The zenith ionosphere delay can be modeled
fied diurnal
sine wave with
a peak
S-band
by a
recti-
zenith delay of
about 10 nsec at local noon, and a fairly constant night-time
value of about
2 nsec.
At
an elevation
angle
of
100,
the
S-band ionospheric delay is therefore approximately 30 nsec.
-30-
Only differences in the ionosphere between the stations cause
however,
errors,
and
the
delay
ionospheric
could
be
the
dominant cause of error only for lander observations at low
elevation angles.
The terrestrial atmosphere does not contribute to the SX
neutral atmosphere delay is indepen-
since the
observables,
The atmospheric delay must therefore be
dent of frequency.
The atmopshere can also be treated
estimated by other means.
by
the
slab
model
in
2.33)
which
r 2,
the
slab
This im-
the radius of the earth.
to rl,
is equal
radius,
(Equation
mediately gives a cosecant law mapping between the spacecraft
atmopsheric delay and the
typically
about
7
averages
from monthly
This
station.
ground
nsec
at
of
zenith delay.
is
frequencies,
radio
the
zenith delay,
The
estimated
pressure and humidity
correction
is calculated
in
at the
PEP and
stored in CAL(1) (see Appendix III).
Data from both the Orbiters and the Landers are necesfor
sary
the
general
Landers
are
are
the
delay
and
the
used to determine the motion of Mars.
To
use
system dynamics.
used
to
estimate
the
plasma
Orbiter plasma delay measurements to estimate
delays
many
In this experiment,
studies of solar
Orbiters
for
and
experiment
relativity
Lander plasma
requires an extrapolation across space and
an inter-
polation in time.
The
temporal
separation
between Lander
and
measurements is not a negligible source of error.
Orbiter
SX
A scheme
-31-
for
temporal
extrapolations
tion of the statistical
application
of
range measurements.
processing
and
required
devised
after
an
investiga-
properties of the plasma delay.
investigation is described
the
was
plasma
The
delay
Chapter 5 discusses
interpolations
to
Lander
next chapter discusses the computer
before
Doppler measurements.
in Chapter 4.
This
use
can
be
made
of
the
SX
delay
-32-
Chapter III
Computer Data Processing
A.
Introduction
Extensive computer processing is required before use can
be made of the data collected at the tracking stations of the
Deep Space Network
Chapter
II.
editing, and
(DSN).
Computer
The observables wre described in
processing
reformatting
at
JPL involves merging,
the raw data
The
tapes.
results
are then copied and mailed to MIT, where the computer processing
The
completed.
is
processing
done
at
MIT
includes
applying calibrations, merging, editing and reformatting the
data.
This
Orbiter
chapter
radio
is concerned
tracking
data.
with
First,
the
processing
I discuss
of
the
the
trans-
ferral of data from JPL to MIT and the problems of obtaining
a complete data set.
Second,
I discuss the nature and type
of the various range calibrations and the method of removing
PRA SX range ambiguities.
Finally, I describe the algorithms
used in data editing.
B.
Data Collection
Range and Doppler data were recorded
Tracking
Tapes
(PTT),
which
also
contain
at JPL on Project
engineering
from both the spacecraft and the ground station.
At JPL the
PTT are read and processed by the JPL Orbit Data Editor
program.
data
(ODE)
The resulting ODE tapes, called ODFILES at JPL, are
edited and used for orbit determination at JPL.
Originally,
-33-
SX data
I used
of
ODE
the
to
modified
that
more
20%
about
tapes.
Therefore,
process
the
they would contain
MIT
good
(the PTT,
not
data
the missing data.
turn
contained
missing data
The
but
resolved,
satisfactorily
been
This did not
two.
other
available on the
problem has never
hope
the JPL printout)
and
ODE tapes,
the
the
in
MIT
the three data sets available at
Each of
out to be the case.
were
to
sent
tapes
PTT
programs
computer
our
MIT copies
the
from
available
than- were
data
tapes mailed
ODE
the
used at JPL contained
edited SX data
print-out of
A
to MIT.
copies of
from
obtained
at
least
part of the problem seemed to be the use of different editing
The
algorithms.
data
PTTs
the ODE data not on the
be
can
As
sources.
three
all
from
consisted
experiment
this
in
used
data
in
seen
Table
III,
of the
SX
superseded
by
about 20%
contributed
of
data used.
April,
In
the
tion
must
Data
types
metry,
be
tion
are
(MDA).
blocked
Each
a
from
telemetry from the
and
range
and
is
of
station.
formatted
data block
the
sent
by
to
variety
ground
data,
Doppler
variety
ground
A
ATDF's.
from
transmitted
include
itself
the
for
flow stream
the data
Figure
(ATDF).
File
Data
Tracking
Archive
was
format
data
PTT
the
1978,
the
the
describes
informato
Metric
JPL.
tele-
spacecraft
at
star
of
station
sources,
Data
4
and
engineering
the
ground
sta-
Data Assembler
switch
ler, which creates packets from the data.blocks.
control-
-34-
is
switching
Packet
in packet switching there
intermittently, and
pre-allocate data rates.
This
overhead,
since
cost
some
of
(data
information
data
is no need to
flexibility is obtained at the
packet
each
time,
source,
send
sources
many
since
used
needed
etc.)
contain
must
to
the
reconstruct
the data stream at JPL.
The packets are
and
JPL,
are
the
and
for
indicates,
ODFILE's
program
ODFILE's
The
are
at
used
of
production
storage
from
JPL
are
the
of
The
for
much
in
ephemerides,
orbit
name
data.
The
PTT)
(ODE).
Editor
determination
same
the
two
the
as
from the
the NDP,
about
for
tracking
(or
Data
Orbit
kept
designed,
are
ATDF
the
the
at
The output from
(IDR)
tapes
ATDF
called
JPL
are
Record
archival
created
(NDL).
Logs
(NDP) is a program which separates the
Data
JPL.
at
Data
Network
reconstructs the data.
Intermediate
weeks
the
onto
written
Network Data Processor
packets
over satellite data links to
transmitted
manner
by
a
The
and
the
as
the
OBSLIB data type is used at MIT.
The processing up to
tion
of
stream, and
human
times
on
The ATDF are
editing.
is designed
intervention.
the
the NDL resembles a data communica-
NDL
in
case
to transmit data with a minimum
NDP
The
of
is
run
several
reading
errors.
available data,
without
sometimes
suspected
supposed to contain all
tape
The ATDF tapes have never been checked
appropriate
ODFILE
data
data problem is solved.
tapes
to
ensure
that
against the
this
missing
-35-
tapes
The data
from JPL
in a
onto
use
for
reformatted
OBServation
at
Upon receipt of
LIBrary
chosen to
store Orbiter
described
in Appendix
in time order.
MIT,
was
data
used
to
later modified
as
a
this
In
processing
lar,
the
to
range
Orbiter
range
is
SX CALibration
way
The
the
the
from
program
ATDF format
used
at
MIT
to
read by
written
(SXCAL),
tapes
data on OBSLIB
it
was
to
possible
into
use
the
ODFILES.
Lander
of
Orbiter
Lander
data
format which can be
and X-band
dual-frequency data from the
similar
is
(then with the Radio Science Group at
S-band
format.
very
format
SXFILE, and
X-band
ODEPEP,
program,
program,
Robert Goldstein
convert
and
MacNeil to process
by Dr.
by Dr.
The
The
SXFILE containing the data.
into the OBSLIB
modified
SXFILE
tapes.
is called the
S-band
remove
I
the
data
placed
SX data are placed upon the SXFILE
II.
PEP.
to
SX data
Another
well.
convert ODE tapes
MIT),
(OBSLIB)
is
data
A program, SXDIFF, written by or Paul MacNeil
PTT and to write an
was
tracking
Lander
MIT.
convenient
format
information within is
ODFILE tapes, the
the PTT, ATDF or
the
not
use by the MIT data processing programs.
for
of
are
range
S-band
data
calibrations
calibrations,
and
range
is
processing.
have
the
the
conceptually
In
same
particuformat
as
computer subroutines
that were written to apply the Orbiter calibrations have been
adopted
to
S-band data
Dr.
apply
for
Goldstein.
the programs
Lander
range
calibrations.
The
Lander
this experiment were calibrated and edited by
He put them onto OBSLIB tapes with the use
ODEPEP and SXCAL.
of
-36-
C.
Data Editing and Calibration
in processing
I wrote a number of programs to assist
SX
the
data,
and
(MFE) program
Edit
the
important of
most
SX
LISTSX
data:
pair
a
and
of
PLOTSX.
The
range calibrations and
as
SX
and
LISTSX
from
data
tapes.
The
the
data,
inspect
to
used
are
plot
as well
data,
SXFILE
and
actually
MFE program
input
Fix
and
list
deletes bad
several
programs
PLOTSX
Merge
to
programs
applies the
merging
the
which were
the
especially in connection with data editing.
There are
These
data.
four
of
no
require
calibrations
2
the
and
PRA demod)
(called the
of
removal
data
do
but
require
SX
the
BIAS
ambiguity
Doppler
The
editing.
the RANge
range
PRA
Psec
SX range
raw
use,
in order of
application,
resolution
(SXBIAS),
are,
four operations
(RANCAL)
CALibration
of
in the processing
stages
data
data
edit-
in their
order
extensive
ing.
These' operations
although
of
use,
In
particular,
it
this
is
will be
order
often
discussed below
is
not always strictly observed.
necessary
to
the
iterate
various
steps, especially the data editing.
Range Calibrations
1.
Range
geometry
measurements
delays
that
contain
must
be
and
instrumental
estimated
and
removed.
RANCAL provide an estimate of this excess range.
is
the
Craft
sum
(S/C),
of
three
and the
the
station-
DSS,
components,
called
Z-Correction
(Z-Corr) delays.
The
Each RANCAL
the
Space-
The DSS
-37-
ranging
at
hardware
by
the
the
and
DSS
The
is
delay
delay
the
delays
three
Delay
antenna
Device
The
surface.
by
a
device
used
on
only
are made mostly at
SX measurements
the
inside
after
(for
the
called
mounted
transponder
a
on
26 meter
the
test translator,
I will discuss only the
diameter antennas.
since
is
is
ZDD
delay
the test translator
or
which
(ZDD),
the
of
is measured before and
It
stations)
diameter
meter
64
the
all
that
the
in
spacecraft,
of
estimate
Note
ranging pass by a device called
Zero
the
on board
an
estimate
an
ground station electronics.
each
delay
the
to round-trip delays.
refer
the
is
delay
geometry.
site
of
estimates
station
Z-Corr
The
respectively.
caused
are
delays
spacecraft
and
64 meter
three
the
stations.
The
the
of
end
test
translator
which couples the
(Figure
end
appropriate
S-
and
the
By
shows
intermediate
are
scatter
due
typically
contain
being
definite
about
outliers,
typically many tens of
it can
with the
or
nsec.
bad
mixer
receiver
front
with
signal
noise
local
The
S-
and
The
DSS
the
thus
and
standard
DSS
the
both the
independently.
measurement
5
a
is
simulate
and
ratios,
to
front
translator
transmitted
measured
occasional equipment changes,
tion
test
frequency,
turn-around
X-band
a
the
mixing
the
at
transponder
transmitter klystron and the
calibrations
X-band
delay
6).
a
The
system.
ranging
as
acts
due
to
devia-
measurements
measurements,
which
standard deviations away from the
are
-38-
to delete any point greater
chose
away
the
from
to edit the
It was thus necessary
local mean.
We
5 standard deviations
than
Figure
mean.
local
RANCAL's.
shows
5
a
of
plot
RANCAL's with deletions.
The
spacecraft
delay
is
an
estimate
of
the
from telemetered spacecraft
turnaround time and is calculated
Before launch,
temperature and signal strength measurements.
the
delay
spacecraft
was
spacecraft
temperatures
construct
a
was
delay
ments by a table
During
delay.
calculated
these
table
calibration
internal
versus
at
measured
and
from
were
measurements
of
spacecraft
mission,
the
internal
variety of
a
used
to
temperature
the
spacecraft
measure-
temperature
telemetered
lookup and
transponder
interpolation.
range delay measured
The Z-Corr calibration converts the
by the electronics to the delay that would have been measured
if
ranging
the
location
had
machine
at
been
the
station
reference
The Z-Corr includes the propagation
(see figure 6).
delay from the antenna aperture plane to the test translator,
as
as well
r , the
delay between
antenna
the
aperture
plane
and the site reference location, both of which are calculated
from station geometry.
the
waveguide
which
is
itself
calculated,
which
station
mission.
between
The Z-Corr also includes the delay in
the
and
is measured.
feedhorn
the
The
delay
and
the
test
translator
in
the
test
translator
Z-Corr delay estimate
is constant at the 0.1 nsec level
(The Z-Corr
for
the
the path length to the ZDD.)
26 meter
for any
throughout the
stations must
include
-39-
in this
used
The RANCAL's
the direction of Tom Komarec at
bration
pass
each
for
called
the
the cali-
RANCAL cards, with
These cards
band.
under
prepared
The value of
JPL.
is punched onto cards,
one card per
are
experiment
are mailed
to
MIT where they are edited and stored on disk.
Dr.
Goldstein
package,
USeR
and
others
CALibrations
at
MIT
(USRCAL) ,
wrote
which
a
subroutine
finds
the
RANCAL
for a particular pass, performs data conversions, and returns
the calibration
frequency.
in seconds of delay scaled to the appropriate
is
USRCAL
data,
dual-frequency
called
and
the
once
SX
for
each
is
calibration
the differenced S-band and X-band calibrations.
exists
for
some
From
used.
spacecraft,
set
the
ground
the
selects
another
RANCAL
of
RANCAL's
pass,
all
and
station
RANCAL
from
the
band,
pass
value
for
the
is
If no RANCAL
and
selected
appropriate
the
in
the
from
formed
algorithm
selection
closest
for
band
time
to
the
target pass.
2.
The SX Bias
Immediately after
within
a
few million
was negative.
team
at JPL
average
estimate
km of
It was
that
systematic errors
ured during
launch,
this
spacecraft
Earth,
the measured
the
concluded by the
spacecraft
negative bias was
in the
ranging system.
caused
The SX
was
the
bias
from
in
the
all
SX
SX
range
range
is
SX delay
by unmodeled
range measthis
measurements.
called
still
navigation
the early cruise phase was averaged, and
was subtracted
of
when the
the
The
SXBIAS.
-40-
The estimated
added
to
still
unknown.
SXBIAS is
tributions
of
ignored
inspire
this
followed
in
currently
is
estimating
the
The plasma delay con-
ionosphere
averaging,
(these values are
SXBIAS
the
of
confidence.
terrestrial
the
in
cause
procedure
The
SXBIAS does not
were
The
SX delay).
Table II
given in
so
and
plasmasphere
that there
is
an
tional bias on the order of -10 nsec of SX range still
addiin the
SX data.
TABLE II
STATION
SXBIAS
nsec
DSN 14
before JD 2443300
20
DSN 14
after JD 2443300
26
DSN 43
26
DSN 63
26
3.
The PRA Demod
The PRA ranging machine can measure the SX range only to
within modulo tl,
ponent
the period of the shortest PRA range com-
(about 2 usec -
see Chapter
II.D and Equation
2.29).
An appropriate integer multiple of tl, determined from nearby
MU-2 data,
must be
added
to
each
called "demodding" the PRA DATA.
PRA
SX
range,
a
process
Demodding is reliable if
the total SX delay can be estimated with an error much less
than tl from nearby MU-2 SX measurements.
-41-
The
Further
time.
of
difficulty
conjunction,
is
demodding
conjunction,
the
20
than
to
rapid
Within
delay
plasma
before or
is
delay
unnecessary.
depends
demod
days
30
SX
total
the
PRA
2
upon
strongly
superior
after
and
less
than
1
isec
to
days
of
superior
3
fluctuations
make
demodding
unreliable.
PRA
cards,
demod
and
input
the demod.
until
values
to
are
determined
onto
MFE program, which actually applies
the
PRA SX range data are adjusted by multiples of
slope and
they match nearby MU-2 SX in both
the demod
typed
manually,
value is
not clear
from the data,
tl
If
level.
PRA datum is
the
deleted.
Lander delay
superior
residuals have a scatter
the
If
conjunction.
pends on PRA SX range data,
used
in
a
which
to test
delay
provides
an
demod values used
calibration
plasma
Lander range
PRA demod.
is
in lander
nsec
near
de-
correction
residual
An
immediately
independent check upon
can be
error of t
detectable,
the validity of
range calibrations
PRA
(of course, we
in just those data).
are especially interested
D.
the
of the
the validity
Lander
Lander
100
of
Data Editing
Approximately
2
104
SX
delay measurements
and
5
105
SX
Doppler shift measurements are available for this experiment.
The sheer amount of data to be processed made data editing an
important
part
of
this
experiment.
edited semi-automatically.
The
SX delay data
The SX Doppler data would
were
require
-42-
automatic
or
subset of
the
interactive
data
were
data
to
editing
be
used.
if
more
The
than
Doppler
a
small
cycle
slip
problem considerably complicates automatic Doppler editing.
SX
first
delay
editing
program
data
editing
pass,
was
used
immediately
to
done
was
delete
after
all
iteratively.
the
SX
PRA demod,
delay
points
In
the
the
MFE
in
a
data
segment for which
SXMIN eSXrang e
SXMAX
range
did
not
data
hold,
in order
segment
SXMIN and
cessed
near
typically
delete
covered
SXMAX were chosen,
data,
results
to
for
to
lie
the
a
segment
data.
Each
of
data.
look at the
unpro-
month's
from a first
just outside
data
obviously bad
of
the
worth
range
of
processed.
being
reasonable
For
example,
superior conjunction, SXMIN = -100 nsec and SXMAX = +100
Psec were used.
After
the
first
pass
of
data
editing,
the
data
were
reviewed and edited manually, with the help of the LISTSX and
PLOTSX programs.
to
be
Delete cards
deleted) were
then
(which specify a
prepared
for
input
to
data
editing
span of
data
the MFE
pro-
gram, which actually deleted the data.
Table
4070
III
SX delay
gives
a
of
summary
measurements
(or
19%)
delay measurements were deleted.
(or
16%)
came
from
the
delete
rest were manual deletions.
the
on
of
the
total
process.
21924
Of the 4070 deletions,
SXRMIN
and
SXRMAX
and
SX
636
the
-43-
Table
III
SX Delay Data
Sources
JD 2442950 - 2443434
01n
PTT
ODE
ODE
b ut no t
Total
Total
Total
Deletions
Good Points
Data
Data
PTT
Data
VO1
9083
9905
1199
10282
VO 2
9391
10560
2251
11642
4--
Total
18474
20465
3450
21924
4070
Spacecraft
17854
JPL print227
out data
18081
Grand Total
Two
categories
of
bad
SX
delay
data
can
be
guished, isolated bad data and groups of bad data.
distinWithin a
bad data group, the SX delay estimate are typically scattered
between SXMIN and SXMAX, with no apparent correlation between
adjacent measurements.
All observations
within a bad data
group were deleted, not just the points away from the local
mean
(Figure 8).
Bad data groups typically, but not always,
occur. at the beginning
or end of a pass.
It is known that
the ranging system collected "data" at times when the spacecraft was not above the local horizon.
At least some of the
-44-
bad data groups can be attributed
to this cause.
Other bad
data groups are caused by equipment problems at the tracking
station (A. Zygielbaum, personal communication).
An isolated bad datum (Figure 9) is one bad point in the
The cause of such bad data is
midst of a good data sequence.
unclear.
For isolated bad data, the local standard deviation
was estimated
and
the
if it was more than 5
point deleted
standard deviations from the local mean.
Isolated bad data
are hard to catch by eye, and this stage of the data editing
might be possible to automate
It
was iterated several times.
the deletion of isolated bad data in the future.
The
SX Doppler
data
this for this experiment.
included
in the
craft, which
of
improve
a multiple of
phase counting.
multiplication up to
Lander
1 cycle
of
noise
is
Uplink phase
X-band at
the signal-to-noise
reduces
to
The Doppler data suffer from cycle
errors
are phase
phase in the Doppler
used
Data editing problems have prevented
plasma delay estimates.
slips, which
be
could
of
the
the
space-
signal
re-
At times of rapid spacecraft accelera-
ceived on the ground.
tion (such as near periapse) the X-band Doppler phase changes
too fast for the Doppler cycle counter at the ground station
to maintain lock.
For
reliably
SX Doppler
detect
conjunction,
cycle slips.
the
editing
cycle
slips
scatter
in
it
in
is necessary to
the
data.
be
Near
able
to
superior
the Doppler data masks possible
Figures 10 and 11 illustrate the problem.
Two
-45-
integrated
overlapping
Doppler
sequences
overlapping
min
hr
40
data, which diverge due to cycle slips at about 23
spacecraft
are
compared.
Figure
10
shows
from different
the
It would be impossible to reliably detect these cycle slips
without overlapping Doppler data.
Figure 11 shows the over-
lapping data after the removal of the cycle slips.
After processing, the Orbiter SX data must be applied to
Lander
range
data.
Chapter
IV
describes
the
statistical
nature of the plasma delay, and Chapter V the application of
the plasma data to the Lander observable.
-46-
Chapter IV
The Statistical Nature of the Plasma Delay
A.
Introduction
As was shown in Chapter II, the solar plasma contributes
to the measurements of the delay and Doppler shift made with
the Viking
spacecraft.
To
ments,
is
to
it
necessary
make
full
estimate
use
and
of these
remove
measure-
the
plasma
delay contribution to the measured delay to the Lander.
The
plasma
density
wild fluctuations
ties
to
model
in
the
solar
corona
is
subject
to
(reference 8) and is beyond our capabili-
adequately
from
first
principles.
In such
circumstances it is natural to consider statistical models of
the
plasma
delay.
It
was
decided
separation between measurements, and
to
ignore
the
spatial
to treat the SX plasma
measurements as a function of time only.
In this chapter,
I will discuss a statistical study of
the SX plasma measurements.
I will define the autocorrela-
tion of a random sequence, and will describe the results of
autocorrelations of the SX data.
Given the available data, I
conclude that the appropriate model of the plasma delay is a
then discuss the
implications of that
conclusion for plasma delay estimation.
In the next chapter
random walk.
I will
I will discuss results obtained from applying these conclusions to the estimation of plasma delay corrections from the
Lander S-band delay measurements.
-47-
B.
The Plasma Autocorrelation
A random process,
xt,
called weakly
is
(or wide sense)
stationary (reference 19, pp 55-56) if it has a time invariant probability density function, and if
- Expectation (xtxt+r) = <xtxt+ >
c(T)
exists and is a function of r only.
(4.1)
In this case, the func-
tion c(r) is called the autocovariance function of xt , and
p(
is
function
the autocorrelation
Let
process,
x(t)
be
a
starting at time t .
+ At,
...
,
t
zero
mean,
(ACF)
(4.2)
of x(t).
stationary,
weakly
At
for
random
N observations
(Thus, the observation times are t , to
An unbiased
(N-1)At.)
+
c(O)
intervals
at equal
sampled
)
minimum variance) estimator for c(nAt),
(but not necessarily
e(nAt)
is:
N-n
c
N-n
t +iot
t +(i+n)At
i=0
This
estimator
processes;
is
indeed,
not
generally
the
unbiased
autocovariance
for
nonstationary
as defined
above may
not even exist for nonstationary random processes.
The
epochs of
the plasma delay and delay-rate
measure-
ments are not in general evenly spaced and thus Equation 4.3
cannot be used directly.
The standard modification of Equa-
tion 4.3 for unevenly spaced data is
-48-
); n > 0
S(nAt)
n >
N(n)
all pairs x t * x t * w(n,t 22-t );
with tlt
1
2
2
(4.4a)
(4
with
(4.4b)
2 -tl)
w(nt
N(n) = all pairs
with t -t
and
In
=
2 -tl)
w(n,t
Equations
really mean
0
t 2 - tl
1
(n-1/2)At
0
(n + 1/2)At
(4.4a)
all
and
pairs
which tt
tstart -Ct t 1
< (n - 1/2)At
(4.4b),
t
is
due
program,
to
Parzen,
see
"all pairs with
tfinish
times
at
wheret
tfinish
reference
I
24.
the Plasma Autocorrelation
wrote
Program
tl
t2
t
tl,
2
for
tis
start is
to Equation 4.3
This extension
called the integration time.
(4.4c)
t2 - t
-2
1
observations
of
t2 -
t2-t 1 < (n+1/2)At
(PAP),
a computer
to imple-
ment this algorithm.
1.
The Autocorrelation of the SX Delay
The
plasma
delay
is
weakly
not
stationary,
as neither
its mean nor its variance are constant with respect to time.
(See Figure
superior
12 for
SX plasma delay measurements taken near
conjunction.)
The
long-term
non-stationary
over
trend
time
in
spans
the
plasma
much
longer
delay makes
it
than a day.
This non-stationarity must be removed before PAP
can give meaningful autocorrelation estimates for lags on the
-49-
a
of
order
of
tenth
in
nonstationarity
a
We
primary
The
more.
SX delay
the
raypath geometry.
or
day
measurements
the
of
changing
that this nonstationarity
hypothesized
could be removed by normalizing
is
cause
the integrated
plasma density
with a geometrical correction factor, obtained by integrating
a static plasma density model
A
simple,
commonly
over each ray path.
used
model
of
the
plasma density
is
(reference 20):
p(r)
r
is
the
radius,
R0
and r0
Here
Numerous
trons
The
cm
distance
have
best
the
from
(4.5)
in electrons cm
Sun,
in
units
of
the
fit
shown
the
that
coronal
P 2.4
and
P
P 7.5
plasma density at
raypath given
the
over
density integrated
n
solar
P215 R O.
is generally taken to be 1 A.U. or
studies
-3
- 3
-
=
elec5RO
0
r
Chapter
in
II
is
H(z)
ar
=
(r)dx
(4.6)
Earth
where
It
is
function of
nistic
tive
since
useful
a
the
to
assume
that
the measured SX delay
stationary random process,
function
function
impact parameter.
raypath
t is the
of
such
SX'
variance
as
H(t).
and
of
H(Z),
the
SX
If
SX'
the
not be
a
a determi-
and
SX delay was
could
delay
SX',
is
an
addi-
stationary,
fluctuations
increases
-50-
We hypothesized that the SX delay
near superior conjunction.
was a multiplicative function of SX' and H(t), or that
SX' = SX/H(a)
(4.7)
is the assumed stationary process.
From an analysis of the Viking Orbiter SX delay measurements I found that
(A) SX varied
from 10 nsec to 50 usec over
of observations,
the
region
or by over 3 orders of magnitude.
The local variance was roughly proportional
to the
mean.
(B)
SX' varied
the
with
from 0.2 to
local
variances
SX'
with time.
1.6 over
varied
being
over
the same period,
roughly
less
than
constant
1 order
of
magnitude.
(C) There
is
still
conjunction,
a
but
long-term
the
closer
to
stationarity
data.
As
an
SX'
some
Figure 13 is
than
before
do
near
data
normalized
illustration,
2 days
trend
the
come
unnormalized
superior
conjunction.
and Figure 14 is
a plot of the corresponding SX' vs time.
the average of
much
compare SX and
I will
a plot of SX vs time
superior
the SX ranges shown
Let SX be
in Figure
13,
and let ASX be the peak-to-peak variation in Figure
13,
then
ASX/SX
= 0.14.
In
an
analogous way we
-51-
constructed
The
ASX'
still
trend
local
of
account
the
With
obtained.
simply
in
as
lag,
a
trend
(or white)
of
(Figure
to
inade-
the
take
into
Equation
partly
removed,
show that
ACF
due
in
the
(see
ACF decreases
the
noise
as
0.05.
SX delay could
random walk
the
and
to
to
not
r,
=
be
might
does
be
detrended
15).
Figure
linearly with
the
first
differ-
looks like uncorre-
SX delay measurements
ences of
16).
The Autocorrelation of the SX Doppler
Continuity implies that the
become correlated as
t]i+ 1
ured more often than
the
higher
or
terms
appear
random walk
lated
SX'
long-term
behaves
increasing
2.
the
ASX'/SX'
in
which
order
They
SX range
a
model,
that
autocorrelations of the
long arc
For
visible
higher
2.21.
found
fluctuations
plasma
quacy
and
frequency
plasma
measurements should
t
must
first difference of xt
.
The Doppler shift
range, and thus provides a
The
fluctuations.
also
be
more
accurate
plot
of
SX delay
SX
is measprobe
shift
Doppler
than
the
of
SX
delay
measurements.
Figure
17
the
delay and
and
that
the
With
respond
to
a
and
integrated
Doppler
Doppler
sampling
this
in
mind,
ACF's
of
the
agree
rate
ACF's
range
of
20
on
the
20
times
is
the
Doppler
integrated
16 days after superior conjunction.
vs time some
range.
is
Doppler
differences)
that
Notice
nsec
that
level
of
(which
were
the
cor-
obtained
-52-
(Figure 18).
Doppler
The average
sequences
data
and
used
the
in
root mean square of
Figure
18
shown
are
the
in
SX
Table
IV, the average of each SX Doppler data sequence is less than
the corresponding
rms
in all
thesis that the SX Doppler
cases, which
supports
the
hypo-
are a zero mean random process.
TABLE IV
Study
Statistics of SX Doppler Data Used in Autocorrelation
t ,
Jul an
Date
Spacecraft
t ,
Jul an
Date
RMS of SX
Doppler
Data, Hz
Average of
Doppler
Data, Hz
Number of
Observations
VO2
2443095.353
2443095.595
349
-0.000491
0.1397
VO2
2443101.080
2443101.347
383
-0.2927
0.3516
VO2
2443117.685
2443117.800
166
0.0209
0.10587
V01
2443124.839
2443124.9577
177
-0.00135
0.19388
VOl
2443160.885
2443161.317
606
0.00281
0.00937
Since
the
epochs
estimate
provided
Equation
4.3,
known.
mates
In
of
the
by
to
where
the
included
values
N
is
(This
4.4
ACF
4.3
is
known
number
estimate
pp.
34-36).
with
the
plots
of
Lines
the
a
white
to be
of
of
are
noise
a
at
is
due
the
Doppler
esti-
process
equal
approximately
observations
5a
ACF.
to
in
better
of
a,
ACF
that
much
deviation,
of
the
to
identical
is
standard
the
14,
equispaced,
properties
of
the
are
data
statistical
the
Equation
estimate.
reference
whose
this
Equation
particular,
provided
1//N,
by
of
included
see
Bartlett,
level
have
Several
in
of
been
the
-53-
sample ACF's
at
correlations
cant
ACF's,
an
with
19b,
be
best
An
AR(n)
and d,
c,
modeled
a
as
process
suggest
of
nature
the
with
decay
the
that
order
be
modeled
difference equation of the
show signifisample
increasing
in
possible damped oscillations
low
can
The
exponential
e, and
19a and
and c)
19a
level.
this
apparently
lag in Figures
Figures
Figures
(in particular
Doppler data
SX
by
an
nth
process.
(AR)
Autoregressive
order
may
stochastic
form
n
=
x
zero
is
8
where
III).
chapter
i-k
mean
white
noise
be
expanded
x,
can
sequence.
,
xt.
the
If
Bt
B
the
zero mean,
in
terms
is
a
of
14,
a
linear
x
mean
random
multiplied
by
zero
are
4.8
if the expected value is then taken and
normalized
equations
resulting
(reference
sequence
Equation
of
sides
1,Aand
(4.8)
i
.
are
both
for m =
+ B
t
k=l
combination of past
Since
K
i
by dividing
by
c(O),
the
so-called Yule Walker or Normal Equations result
k lkm-k
kilkPm-k
p=
m
These
equations
Estimates
for
Equations
the
ignore
Pi.
best
solved
be
the model
estimates of the
necessarily
can
m =
to
parameters,
The
$.i
estimate
possible
Yule Walker equations are
errors
4
give
$j
obtained
k
in
^,
in
terms
of
can be obtained
in
this manner
fi, since
of
(4.9)
1, n
but
the
solutions
Yule
are
pm.
from
not
Walker
to
the
frequently used as a guide in model
-54-
identification.
value
with
From the
rapid decay of the SX Doppler ACF
only
lag,
increasing
order
low
AR
are
models
suitable.
For an AR(1) process, Equation 4.8 reduces to
xt
=
1xt
+
ti
(4.10a)
0 < 1 for stationarity
the ACF decays exponentially
1€
Pk
(4.10b)
kI
and the Yule Walker Equations become the trivial
(4.10c)
P1 = 01
An
AR(1)
process can be
interpreted
samples of
as
a first
order Markov process
dx
dt
- x(t) + 8(t)
y
(4.11)
Under this interpretation, Equation 4.10 becomes
-At/y
x t=
(4.12)
+ Bt.
xt
and y can be identified with the correlation or
Since At is known, y can be estimated by
an AR(1) process.
y = -At/ln$
In
testing
a suspected
a white noise
data
results
is
shown
AR(1)
1'
of
in
1
a1 ,
the
Table
that,
the
the
(4.13)
1
null
hypothesis
estimate of
the
is
stan
is just 1//N.
analysis
V.
= -At/lnp
process,
Under
process.
dard deviation of
The
"l/e" time of
At
of
the
selected
least
two
sequences
SX
Doppler
have
sta-
-55-
tistically significant values for
The Correlation times
*1.
are all very short though, on the order of a minute or less.
imply that
The short correlation times of the SX Doppler
and
the -Doppler
random
to
minutes.
measurement
Lander
nearest
SX
practical
modeled
as
versus
21b)
delay
adequately
typically
the
time
modeled
measured
of
to
shows
that
are
further
measurement.
every
the
number
the
nearest
82.4%
than
The
of
a
four
Lander
SX
of
all
58
minutes
delay
(usable)
invervals of longer than a few minutes.
As
integral
of
a
the
SX
white
Doppler
from
are
therefore,
SX
as
data
matter,
the
is
be
histogram
measurements
uncorrelated over
a
The
(Figure
delay
can
rate
SX delay
observations
delay
the
The
walk.
eight
delay
delay
noise
data
process,
can
or
as
be
a
random walk:
Table V
for SX Doppler
Estimates
Parameter
Data
Fract
N
3095.353
3101.080
3117.685
3124.837
3160.885
349
383
166
171
606
The
integrated
data,
$
Time
JD-2440000
estimation
Doppler
however,
multiples
of
of
to
contain
one
cycle
$
1
1
min.
.054
.051
.078
.076
.041
.599
.286
.199
.210
.310
SX delays
extend
cycle
of
could
range
slips
Doppler
be
improved
sequences.
(errors that
phase
1.95
0.80
0.62
0.64
0.85
11.2
5.60
2.56
2.74
7.63
at
the
The
are
by using
Doppler
integral
ground
sta-
-56-
tion),
especially
when
the Doppler
phase
counter
stressed (i.e., when the Doppler rate is high).
about
106 SX Doppler
observations,
Doppler
is
highly
As there are
validation
is
a
critical problem, one that I have chosen not to tackle so far
(see Chapter III.D).
C.
Estimation and Smoothing of a Random Walk Process
A random walk can be modeled by a stochastic differen-
tial equation:
dx
=
w(t)
(4.14a)
where w(t) is a white noise or Wiener process.
correlated
random
back" term linear
in X(t)
tially
has
variable
an
An exponen"feed
additional
on the right hand side.The
equiva-
lent discrete version of Equation 4.10a is
x
+ Bt.
= xti
1
1-i
(4.14b)
1
where the t i are assumed equally spaced and St.
is a zero
i
mean white noise sequence, assumed to have a symmetric uni2
2
modal distribution and a variance of a2 and a variance of a .
unimodal symmetrical distribution.
In that case x
The
1
first difference of a random walk is
Vx
t
x
t
x
= B
ti-1
t
A random walk can be written as a sum
(4.11)
-57-
k-1
2
+
= xt
x
(4.12)
t
=0
The Maximum Likelihood Estimate (MLE) of a random variable with a symmetric unimodal distribution is the conditional mean of the variable, which is also the Minimum Mean
Square Error
157).
xt,
(MMSE) estimate
(reference 19, p. 149, pp.156-
Under these assumptions, the MLE of the present state,
given knowledge of some past state, x t
<
> = <x
Ixt
i
k-
+
i-k
,
> = x
(4.13)
i-,
i-k
is
i-k
this is called the forward state estimate.
The mean square
error is
2
2
f
= <(Xt
x i-
x
1
where Atf is (ti - tik)
i-k
2
> = At fa
=
2
f
(4.14)
and is called the forward interpola-
tion time.
In a similar fashion the MLE of xt. given knowledge of
,
some future state, x t
is
i+m
<xt
> =
x
1
i+m
(4.15)
1+m
-58-
This
is
called
the
backward
state
estimate
and
the
expecta-
tion of the mean square error is
2
where
Atb
is
ti+
-
m
t i and
2 2
(4.16)
= Atb
ob
is called
the backward
interpola-
tion time.
The
optimal
linear
(MMSE)
smoother
is
the
weighted
average of
the two estimates of x.
(reference 15, chapter 5):
xxt Ix
=
t
Using
1
Equations
1
1 +
1
-+2
2
f
ob
4.14
X t Ixt
i-k
1
and
a
a
f
4.16,
i+m
1
2
the
(4.17)
2
b
optimal
linear
smoother
for a random walk process can be simplified to obtained
pxt
t.
1i
where
a
p
is
Atb/(Atf +
Atb).
i-k
(4.18)
(l-p)xt
+
i+m
The optimal
random walk is thus just a linear
linear smoother
for
interpolation between the
nearest data before and after the smoothed point.
Given
based
a model
the
upon
corrections.
cation
of
of
the
model,
In
plasma
it
plasma
is
possible
the next chapter,
delay
behavior,
to
and
an
to
plasma
calculate
I will discuss
measurements
estimator
the
the
appli-
Lander
range
-59-
corrections, and the results of that application.
smoother derived
estimators.
The linear
in this chapter will be compared
to other
-60-
Chapter V
Lander
A.
Plasma
Introduction
In
the
previous
smoother
for
required
to
dent
the
of
the
Chapter,
SX
delay
calculate
estimator
was
Lander
possible
used,
delay
The
compare
to
I
chapter,
of
part
plasma
I
solutions
optimal
of
the
corrections
and
was
it
processing
is
indepen-
to
decided
write
a
In this way,
several estimators.
estimators
linear
effects
from their
on
residuals.
first
Lander
Most
the
plasma
this
chapter,
processing necessary to convert
into
derived
data.
Lander
program which could realize
it
Corrections
will
the
using
In
the
Lander
will
describe
the
Orbiter SX range measurements
corrections.
report
I
the
results
delay
second
of
data,
part
multiple
and
of
this
parameter
the
finally
be-
havior of the postfit Lander residuals is explored.
B.
Computer Processing of Plasma Corrections
As
observations,
Library
of applying
a means
wrote
I
Tapes)
which
a
program,
reads
to the
plasma corrections
an
UPOLT
(UPdate
OBServation
Lander
Observation
LIBrary
(OBSLIB)
tape and writes a new OBSLIB tape with the needed corrections
stored
on
converting
SX
corrections,
lators
as
functions:
UPOLT was designed
it.
range
and
they
to
were
measurements
serve
as a
found.
to
provide
into
Lander
flexibility in
plasma
framework for better
UPOLT
performs
the
delay
interpofollowing
-61-
A.
Read
B.
For
in an OBSLIB tape and an SXFILE.
each
Lander
for
C.
Various
subsets
of
One for the downlink and
the SX data are selected:
one
two
measurement,
range
uplink.
the
linear
fits
are
made
to
each
the
se-
uplink
and
of
lected sets of SX data.
D.
The
Lander
downlink
correction
is
correction,
average of the
the
each
sum
of
of
the
which
is
a
weighted
linear fits to the data.
Each of these functions is described more fully below.
A.
Part of the
buffer containing
As
Then the OBSLIB tape
SX delay.
ured
is
the buffer
needed,
observation and the meas-
time of each
the
used to fill a
SXFILE is read and
input
is scanned sequentially.
filled with later SX information or
the SXFILE tape can be rewound and the buffer refilled.
last
is
option
observations
The
desired.
as
the
OBSLIB
tape
does
not
contain
in time order.
time
necessarily
needed
This
tag
the
for
time
any
for
The measured
which
the
observation
is
not
plasma delay estimate
Lander delay is
gation delay over both uplink and
downlink only.
delay
Lander
sum of
the
downlink.
The
the
is
propa-
SX data are
Thus, each Lander range correction is the sum
of two SX interpolations, one corresponding to the uplink and
one corresponding to the downlink part of the signal.
using
the
the
entire
so-called
plasma
"thin-screen"
effect
occurs
model,
at
the
which
point
We are
assumes
where
the
that
ray
-62-
path passes closest to the sun
The
4).
screen
thin
is
(Chapter II C and
probably
a
good
reference
near
assumption
superior conjunction, where the plasma density increases as a
high
power
of
the
Far
to the sun.
distance
from superior
conjunction, it may not be a good assumption, but there the
effect of
timing
(Only near su-
is much decreased.
errors
perior conjunction does the plasma show a rapid
time varia-
tion.)
Both the SX delay time tag and the Lander delay time tag
are the time of reception or, more precisely, the time of the
As discussed in Chapter II,
start of the range correlation.
time for a Lander delay calibration
the downlink thin-screen
is
just the Lander delay reception time,
and the uplink thin
screen time is the reception time minus the round-trip propagation time between Mars and the thin screen point.
be
possible to
given
that
in
plasma
timing
particular,
detected
delay
errors
an
after
residuals.
thin-screen delay approximation
improve the
Chapter
on
II.
However,
estimates
the
error
order
of
run AP-43C,
It would
numerical
are
of
remarkably
many
several
causes
tens
hundred
show
experiments
of
insensitive
to
minutes.
In
seconds
negligible
in
changes
Atts,
in
the
It is doubtful that much would be gained by using
more accurate thin-screen times.
B.
For each of the two thin-screen times, the follow-
ing SX data were selected.
-63-
1.
two
The
SX
nearest
(before and
points
after
the thin-screen time) were found.
All
2.
SX
points
within
certain
a
called
AT,
SPAN, of those two points were used.
C.
A
from the
linear
was
fit
of
made
Each
selected data.
squares
least
fit
a
to
three
linear
straight
fit
is
to
line
data
a
the
sets
chosen
(non-weighted)
selected
data.
The data sets fit to are:
selected
All
1.
SX
epoch
of
the
epoch
of
the
the
before
data
Lander observation.
All
2.
selected
SX
the
after
data
Lander observation.
3.
The
D.
tion of
All
the
selected
plasma
SX data.
delay
is
estimate
by means
separate estimates
three
obtained
by combina-
of
a
weighted
average
SX/w
S=
/w
i=l
In Equation 5.1, SX
and
A
SX
is
the
i
is the value of
interpolated
(5.1)
i=l
SX
the SX from the
value.
The
weighting
used was
W2
c
i
(a 2 + alt
i
S2
cii.
n
-t
1)
ts
ith fit,
i
=
1,2)
i
=
3
scheme
-64-
where w i is the weight given to the ith fit, a i is the formal
regression,
from the linear
error
SX
the time of the
tn is
datum that was nearest in time to tts and in the ith data set
(see above),
screen
time
and the c i
for
Equation 2.23).
c
is the thin
observation
(see
In the standard parameter set, cl = 1, c 2 =
(seconds of range)
1, c3 = 0.1, and a = 0.25 x 10
i,
range
Lander
current
the
tts
and a are constants.
/day
2
3
The value of SPAN in the standard parameter set was
1 hour
Our working hypothesis was that SPAN should be
12 minutes.
set to the correlation time of the plasma measurements.
One
hour was the initial guess of this correlation time and SPAN
equal to 0.05 day, or 1.2 hours, was the value chosen for the
<1 minute would be
turns out,
As it
standard parameter set.
a more appropriate choice of AT.
Given
SX
for
each
of
the
two
thin
times,
screen
the
Lander plasma correction at S-band (SXCOR) is given by
SXCOR(t)
=
A
SX(t)
b2 A
+ b2 SX(t-)
)
k
2/
(k
2
2)
(5.3)
where k is the X-band turnaround ratio, and b is the S-band
turnaround ratio
C.
(see Equation 2.31a).
Lander Residuals and the Plasma Corrections
Lander
residuals
provide
an
independent
test
of
our
model of the plasma correction, which is not usually available in time series analysis.
Residuals are defined by
-65-
r(t)
-
= O(t)
(5.4)
C(t,?)
where
O(t)
= the measured roundtrip range to the lander.
C(t,2)
= the
computed
value
of
the
observation
given
the
parameter set a.
y
= the best available estimate of the true parameter
set vector y.
= the time of reception of the lander range measure-
t
ment.
The
residuals
Program
(PEP),
are
computed
which
is
by
the
Planetary
described elsewhere
Ephemerides
(reference
18).
PEP can calculate the theoretical observable and the partial
derivatives of the observable to selected parameters from a
parameterized model of motions of bodies in the solar system.
PEP can use the resulting
tials,
value.
to
do
a
residuals,
least-squares
The partials,
together with the par-
solution
once calculated,
for
the
parameter
can be stored on the
OBSLIB tape with the observables, thus reducing the cost of
each parameter solution.
In the work reported
in
this chapter,
PEP was
used
to
fit a model of the motion of the earth and Mars to the Lander
delay data,
and
the
sensitivity
residuals was investigated.
earth and
of
the
resulting
postfit
The model of the motion of the
the location of the ground
stations were obtained
-66-
the
be
the
of
model
used
parameters
orbits
and
teh
of
at
and
six
On
seconds.
PEP,
ables with
additional
some
postfit
axis
runs
of
rotation
Jupiter
period of
Mars,
of
Landers on
Mars
unit in light-
relativity
the
the
parameter
for.
are
residuals
predicted
from
and
the
partial derivatives of
to
the
parameters.
respect
than
cheaper
earth, the
the
The
for
each
Mercury,
of
and the value of the astronomical
adjustments
meter
the
26-parameter
observations.
coordinates of the
the
"RELDEL" was also solved
In
mass
the
a
conditions
initial
direction of
the
to
rotation phase of
a given epoch,
(three each)
fit
Mars,
and
earth
the
is
system
were:
the Moon, the
rotation
all
solar
parameters)
UPOLT
the
from
distinguished
(these parameters must
solution
parameter
a standard
In
of
17.
in reference
rotation of Mars is described
model
The
observations.
radio
and
optical
previous
from
reintegrating
the
This
equations
is
the
para-
the observconsiderably
of motion using
the
new parameter values.
estimate
An
provided
lander
by
the
range are
of
the
UPOLT
plasma
Lander
The
program.
updated
with
the
SXCOR,
correction,
residual
new
plasma
and
is
computed
corrections
as
is described in Appendix III.
We
are
driven
to
consider
the
lander
residuals
by
two
facts:
A.
PEP uses a Weighted Least Squares
y.
The optimum weight for
Estimator to find
each observation is the
-67-
error associated with the observation.
Appropriate
weights can best be determined by a consideration
of the residual scatter.
B.
independent check on the
The residuals provide an
validity
of
the
Given
plasma model.
any
plasma
model, we can derive the sensitivity of the plasma
estimate
to
external
conditions.
By
actually
finding the sensitivity of the residual scatter to
the same external conditions, a check of the validity of the plasma model can be made.
I wrote a program, the HistOGram (HOG) program, to
1.
Group the residuals by external conditions.
2.
Find the rms scatter within each group.
3.
Plot the rms spread of the residuals vs the external conditions.
Even a cursory glance at a plot of lander range residuals
versus
time
shows
an
increase
in the
(Figure 19).
near superior conjunction
residual
scatter
Thus the sensitivity
of the Lander delay residuals to the temporal separation between the measurement time and the time of superior conjunction was investigated.
Let
6Tsc =
IT
- Tscl
(5.6)
T = time of range measurement, in Julian days
T
sc
= time of superior conjunction, T
sc
=
(244)3108.
-68-
HOG was run with all residuals
for which
nAt < 6Tsc
were grouped
together where
are
Figure
shown
error
in
20,
which was
used
The results
in deriving
16T
were chosen to make the root mean square
weighted
least
residuals be
squares
slight changes
sigmas,
the
solution
-
estimate
parameter
in
They were used
1.
in
postfit parameter
rms of
the
PEP,
equal
now
on,
we
will
consider
the
(residual/a).
An attempt was made to find
the
residuals
weighted
possible
that
dependence upon the
the
time from the
range
20.
These
(rms) of the
in a new weighted
and
estimates.
weighted were
(5.8)
to
caused
With
only
the
new
.9727 for
one
(Appendix 1i, Run AP-41R).
From
thought
formal
The formal errors used were
This corresponds to the heavy black line on Figure
sigmas
a
200 days
I
sc
< 200 days
sc60-0.2"-T sc )nsec; 16T ssc
(20 nsec;
) =
(5.7)
At is the bin size.
(a) for each observation.
o(6T
(n + l)At
<
point),
or
lander
upon
upon
the
fit,
fit
(which is
the
external
weighted
residual
the sensitivity of
It
conditions.
residuals
might
range observation to the
the
number
statistical
a measure
of
of
SX range
error
how well
the
associated
a
(i.e.,
nearest
points
SX data
was
show
interpolation time of the estimate
or
the
on
weighted
used
with
agree
SX
in
the
with
-69-
the
curve
these
fit
three
Figures
to
them).
possibilities,
21,
22,
and
no
and
and
23.
Note
relationship
apparent
the
other
was
run
to
the
results
that
parameters.
the
investigate
each
between
shown
the
residuals
rms of
This
the
can
in
There
residuals
weighted
be
of
are
residuals were used.
1.0, as weighted
now center about
is
HOG
explained
by
the
random walk model as follows.
For
the
a random walk,
is
From
that
(Figure
this
data
far
measurement
SX point contains
associated with
from
noise
has
rms
reality,
each
range
of
P20
it
nsec
improved
plasma delay interpolaLander plasma
the estimate of the
In this case,
is not
delay
an
of
Near superior conjunction this measure-
19 and 20).
error.
all
conjunction,
superior
ment noise is probably swamped by the
tion
In
extrapolation.
random measurement noise
measurement.
seems
for
available
information
there
the nearest
Far
by including more SX points.
from
superior conjunction, this condition might not be valid.
The
with
autocorrelation
increasing
lag
of
(Figure
SX
sitive
to
the
error, which
tion
time).
observations
interpolation
is dominated
One
and
might
by terms
lander .points
very
times
slowly
of
the
extrapolation will be insen(or
time
expect
over
Thus,
17).
order of one tenth of a day, the
decreases
range
the
containing
that
was
to
if
on
the
the
statistical
the
gap
order
interpolabetween
of
days that some relationship could then be detected.
easily test this by deliberately deleting SX data.
SX
several
We could
-70-
D.
Experimental Tests of Our Conclusions
We have hypothesized that the random walk is a suitable
Under the random walk hypo-
model for the SX.plasma delays.
thesis the total information content of a sequence of plasma
measurement is contained in the last measurement.
of the
information available for
in
two
the
plasma values
thin screen time.
Thus, all
interpolation is contained
before and after
immediately
the
From this, two further hypotheses can be
derived:
1.
Any linear
plasma delay
from extrapolated
contribution
will
have
estimator
larger
a
mean
includes
that
a
plasma delay rates
square
error
than
the
optimal linear smoother and the more it relies upon
the
extrapolated
the
slopes,
larger
will
be
the
mean square error.
2.
The plasma delay
tive
to
the
interpolation
number
of
SX
be
should
data points
insensiused
(see
Equation 4.23).
Now, if C 3 is
set to o
estimate will
include
slopes.
the plasma delay
a contribution from
the extrapolated
We hypothesized
plasma interpolator.
RMS
106),
(in
O -C -
error
practice,
that this should be an
unreliable
We found that
1.2 using C 1 =1, C2 =1,
C 3 =0.1 (AP-48F)
5.3 using C =1,
C3 =10
C2 =1,
with SPAN = 0.05 day.
6
(5.9)
(AP-48E)
-71-
This large
If
increase in the rms error supports our hypothesis.
SPAN
is
measurements,
set
to
less
then only two
than
interval
between
measurements, one before
and one
after, will be selected for use
If,
in
addition,
=
c
10
5
C
the
time
in the plasma delay estimate.
=
10
6
,
C3
=
1 or
1, C 3 = 106 is used, then the estimate lies
line
through
the
optimal linear
4.18.
2
data
points.
smoother for
This
C1
=
1, C 2
upon the
straight
corresponds
a random walk derived
=
to
the
in Equation
Using these parameter values, we found that
1.2 using SPAN
C1
C
C
-
RMS
1.0 using
= 0.05 day
= 1.0
= 1.0
= 0.i
SPAN =
C
=
C2 =
1 minute
1.0
(AP-48F)
(5.10)
(AP-48C)
1.8
C3 = 10
Run AP-48F is a standard parameter set run, which uses about
20 SX data points for each Lander plasma delay calibration,
and for which the plasma delay estimate is a weighted sum of
three
4.22
linear
and
to
fits
4.24.
Run
the
selected
AP-48C
SX
data,
implements
the
as
in
Equation
optimal
linear
smoother of Chapter IV, which uses only two SX measurements
per
more
plasma delay
estimate.
complicated
smoother
This does
implemented
as well as the much
in
the
first
run.
Under the random walk hypothesis, the extra data used in the
first run contains no extra information on the desired delay
estimates,
and can
only degrade
the plasma delay estimate.
-72-
Thus
performance supports the
the actual smoother
random walk
hypothesis.
As
an
using
smoother
linear
optimal
As
can
does
be
not
plasma
test,
additional
seen
the
smoother
in
is
thin screen model.
using
I,
the
compared
screen
Appendix
model
the
use
(Run AP-48C)
model
static
of
optimal
the
thin
linear
with
(Run
the
AP-48G).
screen model
in
the
probably dominating any benefits from
the
the
improve
estimate
thin
I
postfit
residuals.
The
error
-73-
Chapter VI
Conclusions
I have tried to describe the processing
In this thesis
of plasma delay corrections for the Viking General Relativity
Experiment, as well as a statistical study of dual-frequency
measurements of plasma delays
IV,
I hypothesized
the
that
and delay rates.
plasma
delay
In Chapter
can
be,. for
the
purposes of Lander plasma corrections, adequately modeled by
a random walk, and in Chapter V, I tested this hypothesis on
actual Lander plasma corrections.
Much work remains to be done with the Viking data.
in
engaged
currently
processing
Viking
Lander
I am
Orbiter
and
data covering the period from JD 2443420 to JD 2443804, which
is not as straight forward as it might seem.
of the data span exposes
Lander motions used
data
is
Lander
required
data.
to
inadequacies
in PEP.
break
Another
in the model
Use of S-band Orbiter
degeneracies
four
The extension
hundred
by
exposed
days
of
of
the
Doppler
the
Lander
new
data,
including a second superior conjunction, remains to be processed.
Other data, such as lunar laser ranging data, should
be included in the solutions.
possibly
a
program
Doppler data.
"bug",
in
There is currently a problem,
the
interpretation
of
Lander
It is intended to commit a major effort in the
near future to this problem.
Some thought must also be given
to plasma corrections for the Lander S-band Doppler data.
-74-
The
plasma
General
Relativity
corrections
possible
to
near
improve
experiment
superior
Lander
is
most
sensitive
conjunction.
plasma
delay
superior conjunction by hand validation of
It
to
might
corrections
be
near
the appropriate SX
Doppler data.
Unfortunately,
tional
phenomenon do
to
discriminate
is
generally
sion
of
needed
before
the
tests
the
three
an
tracking
would
(the
down
in
be
of
Optical
tests
have
theories.
in
a
preciwill
magnitude
direct
increasing
required
the
the
amount are
It
be
impact
precision
dim.
At
Lander
MU-2 machines would need to be placed at
station
required
RANCALs
of
in
the
Chapter
would be
general
POINTS experiment).
measurement
of
III).
the
program,
With
it
supported soon.
relativity
Interferometers
Considerable
complexes.
the American space
experiment
tests
precision needed
improvement
of
gravita-
round trip dual-frequency tracking of a
plasma corrections.
tions
the
of
orders
chances of
by
level
system
required to dramatically improve the accuracy of the
would be
ment
experimental
solar
gravitational
an
that
several
The
of
the
various
by
test
delay
the very least
all
not have
between
theorists.
time
tests
acknowledged
these
upon the
of
current
in
are
likely
earth
improve-
range
calibra-
current
slow-
is unlikely
that
such
Future high prcision
to
orbit
be
conducted
(such
as
the
from
MIT
-75Appendix
I:
Results of Parameter Solutions using
Corrections*
Run ID
Root Mean
Square
Residual/
Errors
Number of
Lander Delay
Observations
Used
Plasma
Comments
AP-51
.489114
327
Standard parameter set
plasma delay extrapolation
AP-37D
.350473
330
Corrected
SX Bias
AP-42A
.347055
421
Uses old formal errors,
uses points recovered
by
correcting
range
code errors
AP-42R
.346925
421
AP-42A
to
Identical
except that the RELDEL
parameter was included
in the fit
AP-41C
.991057
330
Uses new
scheme
AP-41R
.97272
330
AP-41C
to
Identical
except that the RELDEL
parameter was included
in the fit
AP-43C
1.00984
410
AP-42A
to
Identical
except
that
the
new
formal errors are used
and some lander points
were deleted
*The Standard Parameter
in
the above,
and consists
Set
(see equation 5.2)
of
SPAN =
0.05 day
C,
= 1.0
C2
=
1.0
C 3 = 0.1
for
lack
formal
of
error
is the default
-76screen
(see
0.1 day
AP-48B
1.02357
410
thin
ERROR in
corrected
times
page 6a) SPAN =
AP-48C
1.00984
410
Optimal 1 near smoother
1
=
SPAN
10 ,
=
C3
minute
AP-48E
5.30030
410
Average of extrapolated
10
slopes with C
SPAN = 0.05 da
AP-48F
1.20653
410
Standard parameter set
plasma delay interpolation
AP-48G
1.00984
410
Optimal linear smoother
plasma
static
using
model
-77The SXFILE Format
Appendix II:
The
SXFILE
and Doppler data.
the
is
format chosen
to store the SX range
Each SXFILE consists of an 80 byte header,
used as an 80 character identifier, followed by an indefinite
number
of
Range
Each data block
observation,
mation.
or
SX data
Doppler
blocks
in
time
order.
is 96 bytes long and contains the epoch of
the observable
itself,
and various other
infor-
Each SX range data block is followed by another 96
byte data block which contains the original S-band and X-band
delay observables, the calibration values applied to the SX
observable,
the
ranging
code
from the ranging correlators.
lengths
used,
and
the
output
-78Information
Name of Variable
TIME
Storage
Inside
Each SXFILE Block
Type
Purpose
R*8
Epoch of Observation
TIME = Julian
Date -
2440000
KIND
1*2
KIND
NMOD
1"*2
Number of modifications to this
SX datum
SCNUM
1*2
S/C number
IQUAL
1*2
"Quality" of observationrelates
to the nature of its
calibrations
FREQ
R*8
Receiver synthesizer
- Hz -
SX
R*8
S-X range
g(-
1 for range
3 for Doppler
frequency
observable
S-3/11X
Doppler
RUNID
GENDTE
R*8
R*8
MODDTE
R*8
CLOCT
R*8
Range
Doppler -
NCOMP
1*4
Range
Used as 8 character
tifiers
-
Doppler -
run
Maximum Code
COUNT TIME
iden-
Length
number of ranging
Components
not used
IEE
1*4
Receiving station
number
DSNAME
1"*8
8 character
name
NDOPTP
1*2
Doppler ground mode
= 0 for range
receiving station
= 1 for one way OoppieP= 2 for two way
= 3 for three way
"
= 4 for three way coherent
14 trailing zero bytes
-79If data is range
Name of Variable
RANGE (2)
(KIND=1), a second
Tvpe
~
Purpose
R*8
S range
record follows
seconds
X range
SHORTC(2)
R*8
S shortest codelength length X
seconds
RANCAL(3)
R*8
S RANCAL values for this X
observation
S-X
IVOLTS(2)
1*4
IVOLTX(2)
1*4
24 trailing zero bytes
In phase
highest frequency
Quadrature
code phase - S-band
In phase
highest frequency
Quadrature
code phase - X-band
-80-
Appendix
Library Tape and the Plasma
The Observation
III:
Correction
OBServation
LIBrary
(OBSLIB)
tapes
are
used
to
store
delay, Doppler, and other measurements in a form suitable for
use
by PEP.
Information
of records.
beginning
The Type
of
OBSLIB tape
an
on
OBSLIB
Type
OBSLIB
are grouped
series,
each
III
record
and
Type
the
IV
tape.
into
series,
occurs
at
the
A
For
includes
the
information
time
the
residual
respect
to
various
IV
time
of
on
each
order,
although
the
data
etc.
data
one
for
within
a
series themselves can be
in
and
the
other
records
tag
of
carried
in
series
partials
parameters,
propagation
the
scaler variable are used
CAL
and
of
number
corrections
the
Type
IV
information
observable
observable
with
corrections.
The
the
plasma
vectors.
each
5, this
observable,
the
(O-C),
on
is
in Chapter
the PREDICT runs
itself,
Type
in
series are numbered,
of
variety
and
stored
the
identification.
record.
plasma
occur only at
beginning
follow
series must be in time order,
is used for
types
which contain data with
the
that,
Data
on 4
station, observable type,
measurements
record.
kept
Measurements
Note
any order.
is
I and Type II records
the same observed body, ground
One
tapes
are
Three
stored
vectors
in
the
and
in storing the corrections:
vectors to be placed on the
NCAL
The length of the
CAL(I)
Stores the actual correction
tape
a
-81-
SCAL(I)
Stores an error estimate of the actual correction
ICAL(I)
Stores the rank of the correction;
which correction to use
The
types
of
corrections
and
their
used
in deciding
positional
assign-
ments within the new vectors are as follows:
I
CORRECTION NAME
OBS.
RANK
TYPE
STATIC NEUTRAL ATMOSPHERE
STATIC NEUTRAL ATMOSPHERE
TMDLY
DOP
ACTIVE TERRESTRIAL
NEUTRAL ATMOSPHERE
TMDLY
DOP
PASSIVE TERRESTRIAL
IONOSPHERE
TMDLY
DOP
ACTIVE TERRESTRIAL
IONOSPHERE
TMDLY
DOP
1
1
TMDLY
DOP
STATIC SX FROM RNGNS
CALIBRATIONS FROM DOPNS
20
20
TMDLY
DOP
13
14
time)
EXTRAPOLATED (in
PLASMA CORRECTIONS (SX)
17
17
TMDLY
DOP
15
9
10
SOLAR PLASMA
CALCULATED IN MEDIA
11
12
INTEGRATED
TMDLY
16
RANGE CAL. VIA
S-X DOPPLER
Not used
17
RANCAL RANGE CALIBRATIONS
TMDLY
19
EXTRATERRESTRIAL ATMOSPHERE
TMDLY
21
SX CORRECTIONS FROM UPOLT
TMDLY
The
total
elements
propagation
correc tion
in the CAL vector.
is
a sum
C
A
L
C
U
L
A
T
I
N
P
E
P
over appropriate
This total correction
is stored
in a variable named SUMCOR.
The plasma correction, for the work reported
thesis,
is
provided
by
the
UPOLT program.
In
on in this
general,
the
-82-
OBSLIB
tape
already
on
case,
that
new plasma
input
it,
to
UPOLT
generally
the old
in
will
have
either
a
plasma
CAL(15)
or
plasma correction must be
correction,
the
residual
and
calibration
CAL(21).
replaced by the
theoretical
observ-
able must be changed, and NCAL also may need to be changed
reflect the increased size of the
If
UPOLT
SXCORne w
(see
correction,
is
from
the
to
CAL vector.
the new plasma correction, calculated
Equation
In
4.24),
and
CAL
vector,
SXCORol
then
is
d
the
the
old
update
in
plasma
equations
are
Computedne w
= Computedol d
+ SXCORne
Residual
= Residual
-
SUMCOR
The
new
new
new
plasma
CAL vector,
= SUMCOR
correction
which
is
old
old
SXCOR
+ SXCOR
is
stored on
new
placed
w
new
-
in
-
SXCORol
+ SXCOR
SXCOR
the
old
old
21th
the new OBSLIB
d
slot
tape.
of
the
-83-
Appendix
IV:
Conversion between Julian Date
and Civil Date
Comments
Julian Date
Civil Date
2442779
1 January 1976
2442931
1 June 1976
2442980
20 July 1976
2443025
3 September
2443108
25 November 1976
2443145
1 January 1977
2443296
1 June 1977
2443420
3 September
VL1
1976
1977
Lands on Mars
VL2 Lands on Mars
Superior Conjunction
End of Data Span used
in this Thesis
2443510
1 January 1978
2443667
1 June 1978
2443875
1 January 1979
2443895
21 January
2444026
1 June
1979
1979
Superior Conjunction
-84-
References
1.
I.,
I.
Shapiro,
Goldstein, J.
Sygielbaum, W.
Viking
F.
Relativity
Research, Vol.
2.
Brenkle, D.
P.
Goldstein,
M.,
R.
82,
Experiment,"
#
28, pp.
"Ranging with
Martin, W.
L.,
pp. 72-81,
4329-4334, 1977.
Sequential
and T.
II,
J.
Jackson,
and Sons,
8.
J.
Jokipii,
R.,
planetary
42-36, pp.
Pizzo,
in
"A
V.,
the
Vol.
II,
35-40,
Space
Deep
Network
1975.
Chapter
7,
John
1975.
6,
Chapter
John
Wiley
1973.
"Turbulence and Scintillations
Plasma,"
83, pp.
Wind,"
of
in the
Inter-
Astronomy
and
pp 1-28, 1973.
Three-Dimensional
Solar
Review
Annual
Astrophysics Volume 11,
9.
JPL
Astrophysical Concepts,
M.,
Harwit,
Rang-
"Terminology of Ranging Measure-
Classical Electrodynamics,
D.,
Wiley and Sons,
7.
pp. 46-49, 1968.
Sequential Acquisition
Calibrations,"
Progress Report
6.
JPL
JPL Viking R013, 1976.
Otoshi,
DSS
and
ments
Components,"
"system Performance of the Dual Channel MU-II
Martin, L. L.,
Komarek, T.,
Geophysical
1969.
Sequential Ranging,"
5.
Coded
I.
"The
JPL Space programs Summary 37-57, Vol.
ing System,"
4.
Binary
"A
of
Journal
Space programs Summary 37-52, Vol.
3.
Jr.,
H. Michael,
Cuddihy, and W.
B.
Komarek, A.
Cain, T.
L.
R.
MacNeil,
E.
P.
Reasenberg,
D.
R.
Model
Journal
5563-5572, 1978.
of
of
Corolating
Geophysical
Streams
Research,
-85-
10.
van
de
C.,
H.
Hulst,
Corona,"
Astron. Soc.
Bull.
of
Density
Electron
"The
Nether.
Vol.
the
XI,
Solar
# 410,
pp.
135-150, 1950.
11.
L.,
G.
Tyler,
P.
J.
"The
Zygielbaum,
Geophys. Res.,
12.
Kraus,
13.
Nash,
and
A.
Engineering
Box,
G.
E.
Vol.
82, # 28,
S.
K.
Jordan,
Corona
Experiment,"
J.
1977.
5, McGraw-Hill,
"Statistical
Proc.
Perspective,"
I.
and
Geodesy
IEEE, Vol.
and
P.,
M.
G.
An
Time Series Analysis-
Jenleins,
Applied Optimal Estimation,
editor,
A.,
Gelb,
-
66, # 5, pp.
Forecasting and Control, revised ed, Holden Day,
15.
1966.
1978.
532-550,
14.
Solar
Viking
A.
Komarek,
A.
Radio Astronomy, Chapter
J. D.,
R.
T.
Brenkle,
1976.
Press,
MIT
1974.
16.
Eubanks,
M.,
T.
tions,
17.
and
and R.
D.,
R.
Reasenberg,
Vol.
W.
84,
"Determination
E.,
M.
Ash,
Memo
on
Correc-
Plasma
2 February 1977.
Geophys. Res.,
18.
Goldstein,
R.
The
King,
# Bll,
Rotation of Mars,
10,
Oct.
1979.
satellite
Earth
of
J.
Orbits,"
Lincoln laboratory Technical Note 1972-5, 1972.
19.
Jazwinski,
A.
Stochastic Process and Filtering Theory,
H.,
Academic Press,
20.
Tyler,
G.
L.,
Zygielbaum,
P.
J.
The
Geophys. Res.,
21.
Shapiro,
I.
I.,
Rev. Lett.,
1970.
Brenkle,
Viking
T.
Solar
A.
Corona
Vol. 82, 4335-4340,
Fourth
Test
13, 789-791,
of
I.
Experiment,
J.
and
1977.
General
1964.
A.
Komarek,
Relativity,
Phys.
-86-
22.
Martin,
Ranging
Sequential
Binary
Layland,
W.
and J.
L.,
W.
with Sine Waves, JPL Deep Space Network Progress Report
42-31, pp.
23.
30-40.
Goldstein,
Signal
234:
24.
Dunsmair,
Series
1979,
MacNeil,
R.
B.
Breidenthal,
J.
P.
Brenkle,
D.
L.
P.
M.
Dec.
15,
by
Komarek,
A.
T.
Relativity
Retardation
L219-L221,
W.,
E.
Viking
Zygielbaum,
of
P.
Kaufman,
M.
T.
Cain,
C.
J.
Shapiro,
I.
I.
D.,
R.
Reasenberg,
Experiment:
Solar
Gravity,
A.
and
I.
Verification
Ap.
J.,
Vol
for
Time
1979.
Robinson,
Asymtotic
Theory
Containing Missing and Amplitude Modulated Data,
to be published.
-87-
Figure 1) The Observation Geometry
j4 .
Eh
5',.,
0-.
c9~;
FEe.
L~C
f
1
E~r3
-88-
Figure 2) Simplified Ranging System
Block Diagram
-89-
Figure 3) Idealized Ranging Receiver
Correlator Output
COyul_
~r
o4;p3
Coo#.stdo.
04-4Ir"ep
Co'Y't-~k L"W,e
Os".h.
C044
1
Recei"e
C.&s.
reI
I
Data Flow Stream
Figre 4,ATDF
r ic, ur e 4
2
e/ a~t
Re cc
Oopp(et
6 f~?C~SY~.AO,
e'4w
st46Of
P0
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id.
4
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o
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Atekue TeaOIG;A
(x
Aro P)
( ArP
R.
t cA~
-91Figure 5) 111 7 ncals
-
-
-
-
-
-
-
4.5
Iec
II
-
-
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-----
-
--
III
II
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3170
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3190
3200------
-92-
Figure 6) The Geometry of the DSS Correction
- sk.6 #e~ttafde
P la
Fs
Al tea.4a.
rofC f-0 $A
4&
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---
Re cei Le
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r
S~c~j.r'Cc~ Pe +e~elZy,
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-93-
Figure 7)
SX Delay Data Processing at M.I.T.
ATOrF or
rT
00ODL
PEP
X
03J
L
3
SXcAL
WE - X.ml44 EJLf
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Figure 9)
Plot of isolated bad datum
a
a
500
I
I
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I
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I-
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a
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450
40o
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a
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3390
Time, JD - 2440000
a
I
a
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a-
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1o)
Overlapping Integrated SX Doppler
%fore
Deletions
1 - VO-1
'- V0-2
1.0
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Time, JD - 2443200
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Overlapping Integrated SX Doppler
After Deletions
1 - VO-1
2 - VO-2
0 - Overlap
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Time, JD - 2443200
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1.003
-98-
SX range vs. time near Superior Conjunction
Superior Conjunction = JD 2443108 or Nov. 25, 1976
1 - DSN 14
4 - DSN 43
6 - DSN 63
Figure 12)
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18.0
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Nov. 27-28, 1976, 2 days after Superior Conjunction
Figure 13)
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-
I
I.........
*-
II
-
I
I ........-
-1--
--
I
I ........
---
t
-
K
.........
-I
-1-----I
---
I
I ........
- -
I
I ....
I
... -
44
I
...... .........
I I-
-
I J
II
I
I
1---------1 I- - ----------
,I
-I-i
I-----.. . I......... i .........
l
--------------------------- 1
I.... ... I ... ... I ... ... I........ I.... ... I........ I........
I-------I-1---- --------
I ------ t
I
I
I
S---------I------I
----- I
-------~cU~--------------I------1.-----oxg
E
t
o
',
0
o
-101-
Figure 15)
SX' = SX/H(L) Autocorrelation
t I = 2443058
t2 = 2443158
at - 0.1 day
Mean subtracted from SX' sequence
Straight line provided for reference only
-
---_
-- -------------.
--
--
---
,
- - - -----
_
_
So.A
-
r-
0
1"67
*
o"
--
e
--------
--
-.
.
Lag --- days
4
r
-102-
Figure 16a)
Correlation of First Differences of SX' - SX/H()
t i = JD 2443058
t=
dt = 0.1 day
(the mean was subtracted from the Differenced Sequence)
----------------------------7
------1.0
0.2
-
-I
I
0.8
I
I
II
*
-0.0 *
*
Note:
I
*
.I
I
I
I
1.o0
I
I
I
I
data usedI in Figure IS.
This Iis over the same
I
I
0.6
JD 2443158
0.4
*
I
I
I
I
I
I
I
I
0
I
I
I
'0
0
I
a
0.0
*Goo
I
0
a
to
*
I
a
-0.2
" II*
I
I
I
-0.4
•
I
I
I
I
I
II
I
I
I~~
I
I
I
II
I
I
Lag, days
I
I
I.
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
..I
I
ooo
I
*
I
6-o
o1
ILg
Noe
10
2.
3.
days
Thi
aedtaue
isoe
nFgr
-0.8,
-1.0
I
I
I
I
I
I
I
I
1.01
-0.6--
--
0
I
*,
~ooa
coc
aoI
I
I
- -....-- "
- -
--
5
-103-
Correlation of First Difference of SX' - SX/H(J)
Figure 16bi
t
-= JD 2443350
t2 = JD 2443400
At - 0.01 day
5107 Range Differences
The mean was not subtracted from this sequence
-
-e
--
---
0.6.
I
I
aI
0
0 0.
0 0 .. '
0.
41
0
a
a
a
o~~
a
a
:
a
0.1
0.2
a
.
o
11
0
0
0.3
a
a
a
a
a
a~
a
a
a
a
U -0.4.--'
7
o
Ia
a
0.0
a
I
,o
0a
0
o~ a
-o)...
a
11
a
-1
Lag, days
-104Figure 17)
Plot of Integrated SX Doppler
Circled points are Delay
.-------------------------.
.
.----
-
T
TT
.
.
--
.
.
T
i_'
,4 -
1
--
7
i
--
7
SX
1Time,
1*3
--
I
--
-
7
D-
2443124
_
psec
7
7
7
.84
1,
7
77
7
78
.6877
7
7
7
77
7
7
7
7
7
7
7
788
77
±
-
--
-
-
,8
--
-
-
--
Tit
-
--
-
.586e7
-
JD-
--
-
-
--
--
--
4412
-
-
--
-
-
--
-
-
8
-105-
Figure 18a)
Autocorrelation of SX Doppler
t1 I JD 2443095.353
t2 = JD 2443095.595
At = 1 minute
349 V02 SX Doppler Observations
The mean was subtracted from this data sequence
---1.0 -o
---------------------------------I
I
0I
UI
-S0.2
I
1
I
aa
a
a)
a
a
a
a
a
0.6 M-a
a
a
4) 0 110
a
a
o 04
-0.4 a
O. ' - -
(
II
0
--
o
0.6
0.
a
I
a
a
a
a1
a.
ao
a
a
a
a1
I
I~
a
a
a
a
a
ra
(a
~
-- --
-0.6---Lr(
02a'-0
4)I
-0.8riI
00't
-I .....
a
Ia
-- --
in t
Ia
a
an-r
ao
a
a
a
a
aa
a
aa
a
Lag ~~
Ia
20(
a
a04
a
a
a
- - - - -I - - - -
1a
aa
ao
a
a
a
a
Ia0
a-
,
a
aa
a
Ia
30
a
a
a
a
--- --
a
a
a
a
a
a
40
a
a
-- --
aI
Oo:II
oaI
a
00II
o
a
.
04
a
I0
0
a
0
a
08...
a
a
a
3
a
4
ate
~
-106-
Autocorrelation of SX Doppler Data
tI = JD 2443101.080
Figure 18bI
t2 - JD 2443101.347
At
= 1 minute
383 SX Doppler Observations - all V02
The mean was subtracted from this data sequence
1.0
lo
0.8
"
0.6
-
0.4
0
0,
0.2
-
0
-4
o
II..
'U
a
a
O0
a
a
I
0.0
I
a
O
a
0a
,oo
Sao
ao
a
o
so
_
,,
I
01i
a
a------
a
0
a
aa
a
aJ
a
a6a
I
-0.2
- -go%
0
I
-0.4
I
I
I
a
a
a
I
a
a
a
a
-*
a
-0.6
I
I
SI
--
-0.- -II
a
a
a
a
-084.
i
.
a
Ia
a.
a
aa
a
a
a
a
a
a
a1.
a
a
a
a
i
.2
Lag, minutes
a
a
-107-
Figure 18 c)
Autocorrelation of SX Doppler Data
t
= JD 2443117.685
t 2 = JD 2443117.800
&t = I minute
166 Observations from VO-2
1.0
-
o
0.8 -
0,6 -
-
0.
.L -
0.2
0
Correlation
0
Coefficient
-0.2 ..aa
aI
aI
0
I0
a
0
a
0
0a
ca
o
a
a o
a
0
LAG,
0
a
a
o
n0
o
I
8
n
a
minutes
co
a
aC
C
a
a
a
a
10
20
30
a
a
-0.6
-0.8 -
-1.0a.a..
0
LAG,
minutes
40
-108-
Figure 18 d)
Autocorrelation of SX Doppler Data
tI = JD 2442124.839
t2 = JD 2443124.9577
at =
minute
171 ObserLations from V0-1
1.0
o,
08
,
,
;
.
0.
i
I
I
7I
I
I
I
I
I
,
I
7,
7s
I
I
'
I
I
I
7
-7
7
,
a
,
7
7
7
a.a
a ,
a
i
I
0.6
0.2
-0.2cin --
SI
a
a
a'0
T
T
I
T
I
-
..a
0
(O
aI
0.
Correl
a1
a
a
SI
Coeffi
-I_
I
a
I~s
I I
SI
000
Sa
a
a
a
-0.
a
a
a
a
a
a
a
4
0
0"aaOI
-0.
-o.6
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
10
a
a
a
I
2.
2
2
2
-L
-086
00
a
a
a
a
,
a
a
a
aa.
a
00
l
a
Go,, ...
a
a
a
a
a
a
O aIO
0
O
0
00
a
a
a
a
a
a
a
a
a
S
,a
a
a
a
a
a
a
a
I
Ia
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
0
00
0
O OOIa
I a
a
a
a
0
a0
a00
I
0
I
,
000
I
0
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
30
aa
a
a
a
40
a
I
I
2.
a
a
.
a
a
a
a
a
a
a
aaa
a
20
aa
a
a
a
0
02
3
-1.0 .....
LAG, minutes
,
-
"
-109-
Figure 18 e) Autocorrelation of SX Doppler Data
ti = JD 2443160.8851
t2 = JD 2443161.317
at = 1 minute
606 Observations from V0-1
1.0
0.8
-
-
-
-
0.6
0.4
a
1
Correlation
Coefficient 0.0
ac
*
a
*
--
I|
/
0
a 0
£ 0
0
a
Y
a
!-
,
Oi
a
-
-0.4
-0.6
0a
10
a
-
-
20
a
I..................
-
-
-o0.8
-1.0
LAG,
minutes
30
S
-110-
Residual Plot from Run AP-48F
Standard Parameter Set Plasma Delay Interpolation
Figure 19al
400-
300
200
100L
44.
4
I
4; x,
ft
0tP
a
fg
UI
-
ioz
E:z
~_ ___
e
$
z"-
-V%
ee
*
z
Y
z
-100
-200
-300
-400
3000
3100
3200
3300
Time
Julian Date
2440000
3400
-111-
Figure 19b)
Residual Plot from run AP-48C Optimal Linear Smoother with
Thin Screen Plasma Model
400
300
200
10
z
1*
SA
A
ia.
r
- "
W
z
=
~-
- --
z
&
a
,
i
SI
. r
•
z x
-100.
z
U
0
U
-2001
0
-300 7
-400.
,
2980
3000
.
I
3200
3100
Time
Julian Date - 2440000
3300
-
:
.3420
3400
-112-
Figure 19c)
Residual Plot from run AP-48E Plasma Correction
in Average of Extrapolated Slopes
400 -
3001
z
z
200 1
0
x
6
0
z
0
4
+
S1~
I
z
Xf
a
.
a~
£
t
00
I
A
O0
- I II
J
*Z
c
I
Y
z
z*C
-200
-3001
-400
.
2980
3000
.
.
. .
..
.
"
3100
L
3200
3300
Time
Julian Date - 2440000
S'3420
3400
-113-
Optical Linear Smoother
Residual Plot from run AP-48G
)
Using Static (Pup = Pdown plasma model.
Figure 19d)
400T
Too
300oo
2001
100"
z
z+
i'=
3*
z
1
*
0
C
iI
z
Z
+z
0
-
i
I
-
,
-
zz
4
a
AA
z
ry
I
z
z~+
I
-200
o
o oo
-
-300
-400
2980
;,3000
3100
3200
Time
Julian Date - 2440000
3300
-3420
3400
-114-
Figure 20 0) HOG Output
Residual Scatter versus 6Tsc
410 Lander Range Residuals
Bin Width = 10 days
Date of Superior Conjunction:
or November 25, 1976
Bin No.
10
0
o
JD 2443108
20o
-
- - -
I
6
a
a
440
0
0
T
a-
0
a
IJ
i1
o
80
,
1
0
TSC
100
SC,
20
6
Tsc
IT
-
200
days
2I
TSCf, days
300
0
300
-115-
Figure 20 b) Histogram of Number of Lander Observations
versus Time to Superior Conjunction for
Run AP-48C.
Bin Width = 10 Days
50
L
gg
r
I
Number of
Lander
Observations
in Bin
30
2•
1I
.i
-rr
iI
1
12
r
Ii
r
II
1i
ii
Il
ii
ii
rr
i*
rr
rr
Il rr i
,irI i
rr
rrr;
r
i*ri
*rixi
rrrr
Bin
r
i
1
i
iumber
i
x
*
i
r
111
Izi
s
s
-116-
Figure 21 a) HOG Output
Residual Scatter vs. Smoother Interpolation Time
Interpolation time- IThin Screen time - time of nearest
SX observationj
410 Lander Range Residuals
Bin Width = 0.01 day (14.4 minutes)
Run AP-48F
Bin No.
S
o
1.2
1.0
a0
00
0.4- 7
0
~.0
a
C
7
a
7
7
-
0.1- -- - - 0.2
-.-7 -- - - 0.3
0 0- -- - - -6.
Interpolation Time
Days
-
--- -
-117-
Figure 21 b)
Histogram of number of Lander Obervations
versus Smoother Interpolation Time for
Run AP-48F.
Bin Width = 0.01 Day
60 -
50 40 rr
rI
rr
rr
rr
rr
rI
r
rr
rr
Et
111
II )
Lander
30
Observations
in Bin
20
10
0
I
*
Il
II
II
II
II
II
II
II
1:
~
I
Ii
..
r rrr
II
rrrr -r Irri II
11 rr
1111
rr
rrnrrrrr
r
rir
rr
rrn
Irirrrrir
rIIrrII
nrr
~r
rirrrrrr rr
rrr
rrr*rrr+rxrr
rrr
rrrr*r
rr
r
rrr
~ a~ll 11~1
rr*- II*l
II*1IILII
n
"
o0
Bin Number
I=
I
Sjo
!
-118-
Figure 22a) HOG Output
Residual Scatter vs. Number of SX Range
Observations Used in Interpolations
410 Range Residuals
Bin Width = 2 SX points
Bin No.
0
10
20
30
7
7
Ci
"4
2.0 M
7
0
7,
C..
I
,
16'
w
1.21.0
0
0.8
.4
Ci
o
I
,
,
C,)
44.
0
'0
0.4._:
10'
0
0.0
Cn
o
10
0
I
0
20
0
0
40
Total Number of Points Used
in SX Interpolation
60
-11ii9-
Histogram of number of Lander Observations
Figure 22 b)
versus number of SX delay observations used
in interpolation for Run AP-48F.
Bin Width = 2 SX delay observations
100
r
90
I
L
z
L
z
s
,I
60
"I
Lander
Observations
50
11
in Bin
1
40
z
z
r
.i
r
1
11*
t
30
,,fL
ilI
11
III
11
'11I
II
iI1
1I
;L
10
0
II
J '/I I)
11
1I
II
II1
1
11III
1IfllXI
11
fII11
rt
I 1
I
!
t
I
111 1x, 1 t
*,r
1 1
CJI
ILllt
r
Bin Number
i