Analysis of Periodic Behavior of GPS Time Series
at Pacoima Dam, California
by
Monchaya Piboon
B.A. Physics, University of Chicago (1999)
Submitted to the Department of Earth, Atmospheric and Planetary
Sciences
in partial fulfillment of the requirements for the degree of
Master of Science in Geophysics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2002
© Monchaya Piboon, MMII. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
JUL
FpEMl
ES
LIBRARIES
Author.................
Department of Earth, Atmospheric and Planetary Sciences
May 24, 2002
Certified by................
Thomas A. Herring
Professor of Geophysics
Thesis Supervisor
A ccep ted by .........................................................
Ronald G. Prinn
Chairman, Department Committee on Graduate Students
UNDOREN
2
Analysis of Periodic Behavior of GPS Time Series at
Pacoima Dam, California
by
Monchaya Piboon
Submitted to the Department of Earth, Atmospheric and Planetary Sciences
on May 24, 2002, in partial fulfillment of the
requirements for the degree of
Master of Science in Geophysics
Abstract
High precision
limeter level accuracy.
changes such as crustal
structures such as dams
GPS receiver can yield position measurements within milThis accuracy provides an opportunity to observe subtle
movements. In addition,it allows the monitoring of large
for safety reason.
A GPS system was installed at Pacoima Dam in order to monitor potential instability. The GPS system consists of 2 receivers, DAMI and DAM2, which
are part of GPS network in Southern California. The displacement derived from the
GPS receivers indicated that the dam's arch moves periodically with an annual period. Because of the annual period of the time series, we investigated if the movement
of the dam arch could be due to thermoelastic deformation.
An estimation of the dam arch displacement change based on thermoelastic
deformation theory is derived. To analyze the observables, we performed spectral
analysis between the GPS displacements and the temperature variation. The spectral analysis results is in agreement with the magnitude derived from thermoelasticity
theory. The results show that Pacoima Dam's displacement is consistent with thermoelastic deformation. The water loading however may also have contribution to the
movement of the dam.
Thesis Supervisor: Thomas A. Herring
Title: Professor of Geophysics
4
Acknowledgments
First of all I would like to express my appreciation to my acedemic advisor
and thesis supervisor, Tom Herring for his guidance and support for the past 3 years.
He has provided patient guidance and help on this project from the beginning to the
end.
Professor Dara Entekhabi provided opportunity for me to work on my thermal remote sensing project at Parsons Laboratory.
Ludwig Combrinck and Jeff Behr provided useful discussions at the beginning of the project. Mr. Leo Davidian of the Water Resource Division, Department
of Public Works, Los Angeles provided water level record of the Pacoima Dam.
Thanks to all my colleagues and staff members from the GPS and Geodynamics Laboratory, and from the 6th floor of the Green building. Also thanks to
fellow graduate students from the department, who have enriched my acedemic experience.
Carol Sprague and Vicki McKenna of the Educational Office at EAPS have
provided crucial administrative help during the completion of this thesis.
Thanks to my friends from MIT , especially those from TSMIT. I will always cherish our friendship and good memories. My warmest regards to George for
his support and caring throughout good or tough times. I am forever grateful to my
family members, who have given all love and support.
My graduate assistantship is granted by NASA Earth Observing System.
6
Contents
1 Introduction
11
2 Time Series from the Measurements at Pacoima Dam
15
2.1
Description of Pacoima Dam . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
GPS System at Pacoima Dam . . . . . . . . . . . . . . . . . . . . . .
18
Details about GPS data processing . . . . . . . . . . . . . . .
18
Time Series from Pacoima Dam . . . . . . . . . . . . . . . . . . . . .
19
2.3.1
DAM 1 Time Series . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3.2
DAM 2 Time Series . . . . . . . . . . . . . . . . . . . . . . . .
19
Thermoelastic Deformation at Pacoima Dam . . . . . . . . . . . . . .
22
2.4.1
DAM1-DAM2 Time Series . . . . . . . . . . . . . . . . . . . .
22
Temperature Time Series . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.5.1
24
2.2.1
2.3
2.4
2.5
Burbank Air Temperature . . . . . . . . . . . . . . . . . . . .
3 Thermoelastic Deformation
3.1
Equations for Thermoelastic Deformation . . . . . . . . . . . . . . . .
27
3.2
Half-Space Solution...... . . . . . . . . . . . . . .
. . . . . . .
29
. . . . . . . . . . . . .
29
3.3
3.2.1
Heat Diffusion Equation in Half-Space
3.2.2
Strain and Displacement Change...... . . .
Thermal Loading at Pacoima Dam
3.3.1
4
27
. . . . ...
31
. . . . . . . . . . . . . . . . . . .
36
Average Temperature versus Frequency . . . . . . . . . . . . .
36
Spectral Analysis of GPS and Temperature Time Series
41
4.1
Time Series and Spectral Analysis . . . . . . . . . . . . . . . . . . . .
41
4.2
Linear-Time Invariant System . . . . . . . . . . . . . . . . . . . . . .
42
4.2.1
Welch's Method . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2.2
Windowing.
. . . . . . . . . . . . . . . . . . . . . . . . ..
4.3
Input and Output in this Study . . . . . . . . . . . . . . . . . . . . .
46
4.4
Spectral Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . .
47
. . . . . . . . . . . . . . . . . . .
47
. . . . . . . . . . . .
48
. . . . . . . . . . . . . .
51
4.5.1
Results from window = 2 years with 1 year overlap . . . . . .
52
4.5.2
Results from window = 1 year with 0.5 year overlap . . . . . .
55
. . . . . . . . . . . . . . . . . . .
58
. . . . . . . . . . . . . . .
58
4.5
4.4.1
Plot of TFE and Coherence
4.4.2
Computation of Expected Displacement
Different Window Length and Overlappings
4.6
Interpretation of Spectral Analysis
4.7
Effect of Water Level in Spectral Analysis
4.8
Comparison between Gains Derived from Theoretical Approximation
and from Spectral analysis . . . . . . . . . . . . . . .
5
44
Conclusions
5.1
Thermoelasticity at Arch Dam. . . . . . . . . . . . . . . . . . . . . .
5.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures
2-1
Top view of Pacoima Dam (source: Los Angeles County Department
of Public Works)
2-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time Series of DAMI GPS site the subplot represents each components, North, East and Up
2-3
. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
23
Time series of Burbank Air Temperature used to represent temperature
variation at the dam . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-1
21
Time Series of DAM1-DAM2 GPS represents relative motion between
the 2 G PS sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-5
20
Time Series of DAM2 GPS site the subplot represents each components, North, East and Up
2-4
17
24
Plot of average temperature versus frequency. The plot also show av. . . . . . . . . .
39
4-1
Plot of Hanning window with window length 35 units . . . . . . . . .
45
4-2
Input and Output used in the Spectral Analysis . . . . . . . . . . . .
46
4-3
Plot showing gain and phase of transfer function and coherence from
erage temperature for various thickness of the dam
. . . . . . . . . . . . . . .
47
4-4
Fitting of the gain of transfer function at the lower-frequency range .
49
4-5
Expected and Observed GPS displacement.
spectral analysis.... . . . . . . . . . .
The expected displace-
ment is computed from the convolution of regressed transfer function
and the input ........
4-6
...............................
50
Transfer function and Coherence; Window = 2 years with 1 year overlap 52
4-7
Fitting of the gain of transfer function at the lower-frequency range;
Window = 2 years with 1 year overlap . . . . . . . . . . . . . . . . .
4-8
Expected and Observed GPS displacement and the residual; Window
=
4-9
53
2 years with 1 year overlap
. . . . . . . . . . . . . . . . . . . . . .
Transfer function and Coherence; Window
=
54
1 year with 0.5 year overlap 55
4-10 Fitting of the gain of transfer function at the lower-frequency range;
Window
=
1 year with 0.5 year overlap . . . . . . . . . . . . . . . . .
56
4-11 Expected and Observed GPS displacement and the residual; Window
= 1 years with 0.5 year overlap
. . . . . . . . . . . . . . . . . . . . .
57
4-12 Comparison Between Water Level and Residual (Expected Displacement - Observed Displacement
. . . . . . . . . . . . . . . . . . . . .
60
4-13 Comparison between computed Gain of transfer function (TFE), the
fitted gain and the gain derived from theoretical average temperature
62
Chapter 1
Introduction
Global Positioning System (GPS) is widely used in geodetic measurements
of the Earth. GPS uses global satellite reference system that could determine positions of receivers on Earth's surface at anytime. High precision GPS receivers can
determine position changes of receivers at millimeter level. This level of accuracy is
sufficient to measure subtle signals such as crustal movements due to tectonic forces
[7]. In seismic active area such as California, high accuracy GPS receivers are installed in a network to monitor crustal movements and other geophysical processes.
The velocity field derived from GPS networks is a valuable tool for geophysical measurements such as interseismic strain accumulation co- and post seismic deformation.
In addition, high accuracy GPS recievers are increasingly used to monitor large engineered structure such as dams, bridges and high-rise buildings for safety reasons.
GPS provides opportunity to monitor these structures accurately and continuously
without being labor- intensive. The position changes of various geodetic GPS sites
are recorded daily and their time series are available to the public online.
Like many geodetic measurement tools, GPS receivers are installed by
mounting the receivers on a stable surface. When using the GPS time series measurement, it is often assumed that the GPS monuments is stable and the change of
displacements is due to geophysical signals such as crustal movements alone. However, in many cases these monuments are exposed to other factors such as water loads,
-W
solar radiation and seasonal changes. These environmental factors could cause problems about the authenticity of perceived geophysical signals. This local instability is
a concern in a dense GPS array such as Southern California Integrated GPS Network
(SCIGN) array in the Los Angeles Basin, where the SCIGN science team has set a
goal of 0.5 mm for long term monument stability. Such stability however is not easy
to achieve. Many external factors besides geophysical signal can cause changes in significant magnitude. For example, land subsidence due to fluid extraction can be on
the order of centimeters per year. Generally, this environment factor is large enough
to have an effect on geodetic monument site stability. Consequently, this perturbs
geophysical signals and cause GPS time series to have non-secular behavior.
One of the non secular behaviors often observed in GPS time series is the
periodic variations with an annual period. This periodic displacement changes have
also been observed in the past in other geodetic instruments such as strainmeters and
tiltmeters. Because the period of the time series of displacement change is annual,
this non-secular behavior of the time series are linked to environmental factors that
also have annual periods. One of the most obvious environmental factors with such
period is the solar radiation. Non-uniform temperature distributions of Earth surface
due to the solar radiation can cause thermal stress, expansion and hence change in
displacements and instability at the geodetic sites. Past studies have shown that this
thermal effect can cause thermoelastic deformation in the order of several millimeter
as measured by strainmeters and tiltmeters [1],[2]. For a measurement of the order
of millimeter, this thermoelastic deformation is large enough to cause perturbation
to the time series of displacement change for high precision geodetic measurements
such as GPS.
This thesis focuses on finding explanation for the periodic behavior of Pacoima Dam, Southern California. An earlier study by Hudnut and Behr [8] has
observed this behaviors and showed spectral analysis on the residuals of the time
series. Our study uses spectral analysis to analyze time series of the GPS and of
temperature variation in a longer period of time. Thermoelastic deformation theory
is used to estimate possible magnitude of arch dam displacement caused by thermal
effect. Chapter 2 gives explanation about GPS system at Pacoima Dam. The chapter
also presents the GPS time series. It is shown in this chapter that from the GPS time
series the dam arch is moving periodically with an annual period.
In chapter 3, the estimation of the magnitude and phase delay is derived
based on thermoelastic deformation theory. The magnitude is estimated at different
frequency. The plot of magnitude change versus frequency is shown.
In chapter 4, spectral analysis between dam arch displacement and temperature variation is performed. The system between input and output is assumed to be
linear and time-invariant. Transfer function and coherence are derived.
Chapter 5 contains the conclusive remarks and possible future work.
14
Chapter 2
Time Series from the
Measurements at Pacoima Dam
Pacoima Dam is located in a seismic active area in Southern California.
The stability of the dam is a great concern to the public. High precision GPS represents alternative way to monitor the dam instead of traditional survey technique.
The Department of Public Works, Los Angeles with the corporation of US Geological
Survey and SCIGN (Southern California Integrated GPS Network) began monitoring
the arch dam with the GPS receivers, DAM1 and DAM2. The time series from both
receivers were derived, it represents daily measurement of the offset position of the
receivers. The time series from the GPS system indicates that the dam is deforming
with periodic behavior with an annual period. The annual signal of the time series
leads us to assume that this dam movement is due to thermal effect, the periodic
heating by the Sun.
In this chapter, we discuss how the time series representing the observed
displacement in our analysis are derived.
Section
2.1 describes the location and
dimension of Pacoima Dam. The unedited time series of the receivers at the dam
were shown in section 2.3. In section 2.4, we shows the processed time series which
illustrates that the dam arch moves periodically. We also show the representative of
temperature variation, the temperature time series in section 2.5.
2.1
Description of Pacoima Dam
Pacoima Dam is located in the San Gabriel mountains, about 4 miles north-
east of San Fernanado. The dam type is concrete-constant angle arch-gravity abutments, with granite foundation. The purpose of the dam is for flood control and
water conservation of Pacoima Creek. The construction of the dam was completed in
February 1929. The dimension of Pacoima Dam (Los Angeles County Department of
Public Works, updated 1/4/00) can be describedas follow:
1. crest height above original streambed 111.3 meter
2. crest height above foundation 113.4 meter
3. crest length 195.1 meter
4. crest width 3.2 meter
5. base thickness 30.4 meter
The upstream side of the dam faces East and the downstream side West.
Figure 2-1: Top view of Pacoima Dam (source: Los Angeles County Department of
Public Works)
17
2.2
GPS System at Pacoima Dam
The County of Los Angeles began monitoring the dam using continuous
GPS in order to detect potential instability of the dam. Near the center of the dam's
arch a DAM2 GPS station was placed atop a 2 m tall steel-tube monument. Another
station DAMi was placed on the thrust block at the end of the dam, to the South
of DAM2. The displacement change of the dam arch is computed as DAM1-DAM2
time series (subtraction of the time series) to remove a tectonic effect.
2.2.1
Details about GPS data processing
The continuous GPS system at Pacoima Dam is part of SCIGN and are
operated by the USGS- SCIGN operation center. The receivers are dual frequency
P-code. They sample all GPS observables at a rate of 1 sample every 30 seconds.
Data are collected at the receivers' internal memory, then downloaded once a day.
The files are retrieved and immediately moved over the internet to the permanent
SCIGN archive[8].
2.3
Time Series from Pacoima Dam
The time series of each GPS site can be shown in the following subsection.
2.3.1
DAMI Time Series
The time series of the displacement of DAM1,the GPS site located at the
abutement of the Pacoima Dam can be viewed in Figure 2-2.
2.3.2
DAM2 Time Series
The time series of the displacement of DAM2, the GPS site located at the
center of the Pacoima Dam crest in the downstream side can be viewed in Figure 2-3.
DAM1 Time Series
20
10 0
-..
-10
-20
.
.
. ...-
. . ..
.
-
ft
-.-.
0
-.
-.
200
400
I
I
. ..
600
-.
.
..
..-
1000
800
.
..-.
1400
1200
.
..-.
1600
..
1800
.
.
-.
2000
T
10
0
-10
-20
. .
-
0
200
400
600
-40'0
200
400
600
800
..
1000
.
.
1200
.
.
.
.
. .
.
.
1400
1600
1800
2000
1400
1600
1800
2000
-20
1000
1200
800
number of days after 1/1 /1995
Figure 2-2: Time Series of DAMi GPS site the subplot represents each components,
North, East and Up
DAM2 Time Series
10L-
0 -....
-.
-20
..
-60
0
. .
.. . .
. ..
.
.............................
500
-50L
. ..
.. . . . . . .. . . . . . . . . .. . . . . . . . . . .. . . . . . . .. . . .
-. . . . . . .
-40
. ..
2500
2000
1500
1000
500
0
500
1000
1500
1500
1000
number of days after 1/1 /1995
2000
2500
2000
2500
Figure 2-3: Time Series of DAM2 GPS site the subplot represents each components,
North, East and Up
2.4
2.4.1
Thermoelastic Deformation at Pacoima Dam
DAM1-DAM2 Time Series
The time series of DAM2 was then subtracted from DAM1 GPS time series.
The time series represents the relative movement between GPS receivers at the crest
and at the abutement. The resulted time series represents the movement of Pacoima
Dam's crest. The displacement change show prominent periodic behavior with annual
period in the East-West (upstream-downstream) direction. The data was linearly
interpolated for missing data points. Because there was a large gap after day 1800
(after January 1, 1995) we truncated the data after that day. The time series therefore
represents the movement of the dam arch from day 235 to day 1800 (August 23, 1995
to December 4, 1999).
Time Series of DAM1 -DAM2
-10l-
-20'
0
200
400
0
200
400
0
200
1200
1400
1600
1800
1200
1400
1600
1800
1200
1000
800
600
number of days after 01/01/1995
1400
1600
1800
600
1
1
600
800
1000
800
1000
1
-10
-20
1
1
400
1
11
Figure 2-4: Time Series of DAM1-DAM2 GPS represents relative motion between the
2 GPS sites
2.5
Temperature Time Series
In order to estimate the effect of thermal heating on the dam crest, a rep-
resentation of temperature variation at the Pacoima dam is needed. Unfortunately
there was no operational thermal sensor to measure temperature at the dam directly
[8]. Air temperature from a nearby city, Burbank was chosen to represent the daily
temperature variation. A recorded temperature was retrieved from an archive of a
commercial website www.wunderground.com.
Similar to the GPS time series, the
data was linearly interpolated.
2.5.1
Burbank Air Temperature
Burbank Daily Temperature
35-
302510-
a)
c5c.1
~10.-
'I
-
0
200
400
600
800
1000
1200
# of days after 1/1/1995
1400
1600
1800
Figure 2-5: Time series of Burbank Air Temperature used to represent temperature
variation at the dam
2000
The time series of GPS displacement, DAM1-DAM2 and of air temperature
are used in spectral analysis in chapter 4. Theoretical estimation to estimate the
magnitude of the thermal effect on dam deformation is derived in chapter 3.
26
Chapter 3
Thermoelastic Deformation
In this chapter, the displacement of the dam due to thermal effect is estimated. Basically, we calculated the average temperature of the dam wall as a function
of forcing frequency. We analyzed how the temperature diffuses into the wall as a
function of frequency. Analytical solutions for thermoelastic deformation are derived
in section 3.1. Using information about this computed temperature distribution in
the dam, strain and displacement change of the dam arch can be derived. The magnitude of transfer function as a function of frequency can be estimated. This estimated
transfer function will be compared with our spectral analysis of the observed time
series, GPS displacement time series and temperature variation.
3.1
Equations for Thermoelastic Deformation
The displacement of the dam due to thermal effects can be modeled based
on the theory of thermoelastic deformation. There are 2 steps involved in estimation
of deformation of structures due to non-uniform temperature. First, the description
of temperature distribution in the structures must be derived. The second step is
to substitute this description of temperature into the stress-strain equations which
describe the elasticity of structures. The information about stress and strain is then
used to derive the displacement that can be measured by instruments such as GPS. A
good reference about temperature distribution due to heat conduction can be found
in Carslaw and Jaeger[4] and in Ozigik [10]. General references for thermoelasticity
are Boley and Weiner[3] and Timoshenko and Goodier[12].
(1) The stress-strain relations are:
ex -
aT
- v(o-r + o2)]
=
[o- - v(o-, +
ey - aT =
ez -
aT
=
I[o-
(3.1)
o-)]
(3.2)
(3.3)
- v(OX + ory)]
E is the Young modulus, typical values 30 - 40 x 109 N/m 2 for concrete and
66 x 109 N/m 2 for granite
p is the Poisson ratio, typical values 0.3 for concrete and 0.25 for granite
a is the thermal expansion coefficient, typical values 1.2 x 10- 5 K-1 for concrete and 4.7 - 9.0 x 10- 6 K- 1 for granite
(2) The temperature distribution satisfies the heat diffusion equation:
where
crust materials.
,
2
T
1 T
Oz
2
K at
is the heat diffusivity of materials.
(3.4)
K =
1mm2 s
1
for typical Earth
Half-Space Solution
3.2
This half space model gives us some idea about the magnitude and phase
shift of the displacement due to periodic heating from the Sun. We assume that the
domain of the problem is a semi-infinite, half-space in the region z > 0 whose surface
is defined by the plane z = 0. The surface temperature is a periodic function of time
T(t)
=
To cos(wt -
#),
where w =
2
represents annual variation of the periodic heat
from the Sun.
3.2.1
Heat Diffusion Equation in Half-Space
The heat diffusion equation
2
T
1OT
iz
2
r, Ot'
in
T(t) = To cos(wt -
0 < z < oo,
#)
at
t >0
z =0
(3.5)
(3.6)
was solved by method of separation of variables [4]. From equation (3.5),
we assumed a separation of variables of the form:
T(z,t) = Z(z)F(t)
A possible solution should have this form:
(3.7)
T(z, t) = Z(z)ei('t-0)
(3.8)
Substituting equation (3.8) in equation (3.7) yields:
d2 Z(z)
dz 2
=
Z(Z)
The solution of equation (3.9) that is finite as z
Z(z)
Ae-
=
(3.9)
-+
oo is
(3.10)
Z'
~
=Ae-(1+i)z
(3.11)
Therefore the solution of equation (3.5) is:
T(z,t)
=
Aex(-z+)[zs(w
= A exp(-k z)[cos(wt -kz-
where k = fV2K
(3.12)
-ikt-*)
#) + isin(wt -kz-
#)]
(3.13)
The solution that satisfies the boundary conditions and has the same value. T(t)
To cos(t -
#)
=
at z = 0 is:
T(z, t) = To exp(-kz) cos(wt - kz -
#)
(3.14)
where k =
The solution shows that the amplitude of the temperature variation is attenuated by
a factor exp(-kz). There is also a lag kz in the phase of the periodic temperature.
These 2 terms depend on the depth, z. This temperature distribution is then used
in stress-strain equation to calculate the possible strain and displacement (in z direction) change due to the temperature.
3.2.2
Strain and Displacement Change
From the stress-strain relations in all 3 components, one can assume that
there is no strain change in x and y direction, cx = ey = 0 and o-,
o-y. Assume
that the z component is free to expand, o-z = 0. Substitute these variables into the
stress-strain equations (3.1)- (3.3):
o-x =
o-
=-
ET
(3.15)
The strain on the z component can be calculated by substituting orand a-
z=
1
aT + [z
E
- v(o-x + o-y)]
(3.16)
=
1 2aT
E 1-v
(3.17)
2v )aT
(3.18)
(1 +
1 -v
(3.19)
aT
Ez =
Substitute the expression of temperature distribution obtained from solving the heat
equation.
z=
1+ VaAT exp(1-v
-W-z)cos(wt
2
-
rw
z-#)
(3.20)
Integrate over the strain to obtain the displacement:
00
u=
J ezdz
(3.21)
0
/
00
uz =
0
0
aAT exp(1 -v
With some algebra, uz becomes :
z) cos(wt - kz - #)dz
V2rZ osw
(3.22)
uz
-v
ilz)[cos(wt
exp(-
aT
z + sin(wt - #) sin(
#Cos(
j
-z)]dz
2K
0
(3.23)
(3.24)
uz =
00
+
aAT[cos(wt -
cos(
#) Jexp(-
-W-z)dz+
V2r
0
sin(wt - #)
zdz]
z) sin(
exp(-
Using table of integration:
eax(a sin bx - bcos bx)
f eax sin bxdx
J
-
=
eax cos bxdx
In our equations, a = -
(
(3.25)
a 2 + b2
eax (acos bx
+ b sin bx)
and b =
(3.26)
b2
a2 +
.
uz =
1
v aT
Icos(wt -
#) exp(-
z)
/-w,,Cos(
- 7z)+
2,
Vr-
- si.( v /-" Z)
2K
v f 2,
+ sin(wt -
#)
F-z)r-
exp(-
r W sin(
2.
v
2K
z)
)-
o
cos( rz)
uz
-
aAT
-
- #) sn w exp(-
1-
-
-sin(wt
q$) sin(
1-v aj
2z) Icos(wt
z) - sin(wt -
2Kr
" Texp(-
-
#) cos(
-
2ri
IWz)
V2KZ
(3.27)
-z)]
#) cos(
2K z
- cos(wt -
#) sin(V
-q)+
cos(wt
-VW-
00
2Kr
(3.28)
z=
=z
-
1 -v
-1+
1- v
aAT
2w
[sin(wt
- #) + cos(wt - #)]
&ATV -i-sin(wt - q5) sin(") + cos(wt
4
2w
-
0b) cos( 1r)]v/'2
4
(3.29)
(3.30)
which can be rewritten as:
UZ = - 1-v1 AdT
w cos'
2w
t -o# -W-)
4
(3.31)
Substitute constants for typical properties of Earth crust: v = 0.25, a =
3 x 10- 5 K- 1 and r,
=
1mm 2 s- 1, w
=
1.9924 x 10- 7 s- 1 and AT
=
20 K. Estimated
magnitude of displacement change due to periodic heating of half-space is : u ~ 2mm.
The phase delay of the displacement can be estimated from the cosine expression of uz,
cos(wt -
#)
cos(w(t -
=
(3.32)
The time shift can be define as t'
,I
7r/4
27r/365
~ 45
days
(3.33)
the phase delay I results in ~ 45 days of time shift. This estimation showed
that even in half-space case, the magnitude and phase lag of the thermoelastic displacement can be detected by high precision GPS.
3.3
Thermal Loading at Pacoima Dam
Thermal variation is an important issue in dam monitoring. In most cases
of dam monitoring, engineers must estimate the magnitude of thermal effect in order
to check if thermal stress can cause cracks in concrete dam. Therefore it is necessary
to understand the distribution of temperature inside the mass of concrete in an arch
dam. The temperature of the dam can be approximated with a sinusoide defined
from the average annual temperature. The temperature usually taken into account
are air temperature, reservoir water temperature and solar radiation. The range or
amplitude of concrete temperature arising from exposure to air and water can be
determined by a simplified analytical model or the finite-element method. Here we
focus our attention on the simplified analytical approximation. The temperature of
the reservoir can be approximated with sinusoidal variations.
We derived the magnitude of temperature versus frequency. Here we assumed that the strain and displacement change is proportional to the temperature
change. This temperature versus frequency plot will be used to compared with the
transfer function derived in chapter 4.
3.3.1
Average Temperature versus Frequency
From the expression:
T(z, t) = To exp(-kz) cos(wt - kz -
where k =F
The strain can usually be estimated as:
(3.34)
E= aAT
(3.35)
where AT describes temperature distribution. The displacement change
can be obtained by integrating the strain.
u=
J cdz
(3.36)
The average temperature of the dam arch is also computed by integrating
over the thickness.
Tavg
f T(z)dz
(3.37)
The displacement change due to thermal effect is directly proportional to
the average temperature. Therefore, plotting of average temperature versus frequency
should have the same shape as the gain of transfer function between displacement
change and temperature distribution.
The expression for the average temperature can be derived similar to the
equation (3.28).
Tavg=
-
TO
exp(-
z)[sin(wt -
z
-
#) + cos(wt -
z - #)]
(3.38)
Evaluate equation (3.38) at a particular depth z yields the magnitude of
the average temperature as:
=TO
-T)|[1 - exp(-
I -z)]
(3.39)
The plot of the ratio, Tg versus frequency can be shown in figure (3-1).
We assumed that z represents thickness of the dam. This plot should have similar
shape and behavior as the gain of the transfer function derived in chapter 4.
Average Temperature versus Frequency for Various Thickness
1
0.9
0.7
0.6
0 .7
.
.
..
-.
. ...
.
. ..
... .
.
.
.
.
.
. . .
. .
.
.
..
..
.
. . .
.m.
.
. . .
.
. .
.
.
.
.
.
.
.
. .
. .
. .
. .
.. . .
.U5
0.4
10
:15m
0.2
30m:
0.1
0-
104
10 3
10-2
101
frequency (cycle/day)
Figure 3-1: Plot of average temperature versus frequency. The plot also show average
temperature for various thickness of the dam
40
Chapter 4
Spectral Analysis of GPS and
Temperature Time Series
4.1
Time Series and Spectral Analysis
A time series may be a random or non-deterministic function of an indepen-
dent variable, time. A future behavior of a time series cannot be predicted exactly.
In time series analysis, the most important assumption is that the stochastic process
is stationary. A wide class of stationary processes can be adequately described by
the mean, variance, covariance function and the Fourier transform of the covariance
function, the power spectrum.
According to the Fourier transform relation, the knowledge of the spectrum
of the process is equivalent to the knowledge of the autocovaince function in the time
domain. In many physical problem, computation of spectrum at particular frequencies yeilds result directly related to the physical properties.
A good reference on
spectral analysis can be found in Jenkins [9].
In this chapter, we computed power spectrum and the cross spectrum of the
time series in our study, the GPS time series and the temperature time series. The
computed power spectrum are used in our frequency response study. We calculated
transfer function and the coherence between the input temperature and the output
GPS displacement.
4.2
Linear-Time Invariant System
We assumed that the system between temperature variation and displace-
ment is linear time-invariant. The temperature is the input and dam deformation is
the output. The output is a result of the system convolved with the input,
y(t)
=
g(t) * x(t)
(4.1)
where x(t) is the temperature, y(t) is the dam deformation, g(t) is the impulse response function and * is the convolution operator. From this relation, one
could find a transfer function between input and output. The transfer function is
defined as:
G(w) =
Pry(w)
PXX(w)
(4.2)
where Pxy(w) is the cross-spectrum between input and output. Px(w) is
the power spectrum of the input.
We then computed the magnitude and phase of the complex G(w) which describes how the system modifies amplitude and phase. We also computed a coherence
function which describes the correlation between the 2 time series. The coherence is
defined as:
C2Y =
*x
XV()
P2()
(w
PYY P)
(4.3)
Pyy(w) is the power spectrum of the output.
To compute the transfer function and the coherence, we first computed the
cross spectra and power spectra. The power spectra are estimated by the discrete
Fourier transform.
PY(w) = X(w)Y*(w)
(4.4)
P22(w) = X(U)|2
(4.5)
PYY (w) = y(W)|2
(4.6)
where X(w) and Y(w) are the discrete Fourier transforms of x(t) and y(t).
4.2.1
Welch's Method
We used Welch's method by applying a tapering window prior to the Fourier
transformation. The Welch's method breaks the time series into sections, takes a fast
Fourier transform to estimate spectra for each section and gives the final spectra by
averaging all sections. Applying a window to the time series generally degrades the
spectra resolution. It however reduces the spectral leakage caused by the finite length
of time series and yields more reliable spectral estimates.
We applied Hanning window to both of the time series. After averaging the
power spectra over all segments, the transfer function and coherence can be calculated
according to the formula.
Because of the periodic nature of the time series we chose the length of the
window to be a multiple of 1 year (365 days). We tried several options of window
lengths and overlaps, 1 year with no overlap, 2 years with 1 year overlap and 1 year
with half year overlap.
4.2.2
Windowing
Truncating the data used in spectral analysis is equivalent to multiplying the
time series by a rectangular window function. A rectangular window causes leakage
in a spectral estimate . To reduce the leakage effect, we applied a window function
whose Fourier transform has lower sidelobes, and less leakage from the main lobe.
The window function used in this study is Hanning window. The Hanning window or
von Hann window in a discrete form is a function of duration T with N terms indexed
from 0 to (N-1)t,, where t, = 1 [6]. The Hanning window is defined as:
w(nt,) = 0. 5 - 0.5 * cos( 27lt)
N- 1
=
0
0 < nt < (N - 1)ts
elsewhere
(4.7)
(4.8)
in term of index:
w[n] = 0.5 - 0.5 * cos(2r
N - 1
= 0
elsewhere
)
0 < n < (N - 1)
(4.9)
(4.10)
The hanning window function is illustrated in figure 4-1.
1
1
1
i
5
10
15
W 1
u
0.9-
0.8-
0.7-
0.6-
0.5-
0.4-
0.3-
0.2
0.1 0
0
20
25
Figure 4-1: Plot of Hanning window with window length 35 units
30
35
4.3
Input and Output in this Study
In this spectral analysis, the input of the system is the temperature varia-
tion. The output is the GPS displacement in East-West direction.
Burbank Air Temperature with Mean Removed
10
.)
00
0)
C)
-10
400
600
800
1400
1600
1800
400
600
1400
800
1000
1200
Number of Days after 01/01/1995
1600
1800
1000
1200
Displacement change
10
-10
Figure 4-2: Input and Output used in the Spectral Analysis
4.4
Spectral Analysis Results
4.4.1
Plot of TFE and Coherence
The figure shown here is a result of spectral analysis with hanning window
of 365 days and overlapping window of 0.
Gain of transfer function
2
..
.. . .
i
.
.
i
.. ... ..
1.5
1
0.5
10~3
10-2
Phase
10
10"
10
10"
10
1
200-
1000-
-100 -200
10 3
1
10-2
I
Coherence
''''''I
0.5
0
10
3
10-2
cycles/day
Figure 4-3: Plot showing gain and phase of transfer function and coherence from
spectral analysis
0
4.4.2
Computation of Expected Displacement
The magnitude transfer function was fitted and excluded the high-frequency
component of the transfer function. We constructed a new transfer function as a
complex variable based on the fitted magnitude.
The phase of the new transfer
function is derived from the lower component of the transfer function (approximately
30 degrees). This fitted complex transfer function was later convolved with the input
according to the formula (4.1).
y(t) = g'(t) * x(t)
(4.11)
where g'(t) is a fitted transfer function, x(t) is the temperature input.
The convolution yields the expected displacement computed based on the
results of spectral analysis. This expected displacement is compared with the observed
GPS displacement. We also computed the residual which is defind as the expected
displacement subtracted by the observed displacement.
Gain of TFE
1.5
-
1 -
0.5-
010-4
10-2
10-3
10
Computed and Fitted of Gain of TFE
0.5
00
.
..
.5
...... ..
. .
.. .. ...
.
.
.
.
.
.
..
..
.
.
..
..
--
O
10~4
10-3
O
10-2
Figure 4-4: Fitting of the gain of transfer function at the lower-frequency range
49
(jc
10~
Expected and Observed Displacement
10|
-10- -
200
400
600
800
1000
1200
1400
1600
1800
2000
Residual (Expected - Observed)
1OF
-5
-10-
-15'
200
I
I
I
400
600
800
I
I
1000
1200
Number of days
I
I
I
1400
1600
1800
Figure 4-5: Expected and Observed GPS displacement. The expected displacement
is computed from the convolution of regressed transfer function and the input
2000
4.5
Different Window Length and Overlappings
In addition to the 1-year windowing appied in section (4.4.1), we also per-
formed spectral analysis with window length of 2 year and 1-year overlap and with
window length of 1 year and 0.5 year overlap. We then followed the same procedure
as in section (4.4.2) to compute the expected output based on the fitted transfer
function. The results of spectral analysis with different windowing and overlapping
can be shown as follow:
Results from window = 2 years with 1 year overlap
4.5.1
Gain of transfer function
3
S................
2.......
E
E
2
.
.
.
.
......
.
.
.
.
* .I...
...
1...................................................................-.
10-2
103
10
Phase
200
100
-
.
.
.
.
.
.
-
-
-
-
-100
-200
10-3
.
10-2
Coherence
I
I
. I I
10
1 -
0.5--
0104
10-1
10-2
cycles/day
Figure 4-6: Transfer function and Coherence; Window = 2 years with 1 year overlap
10
Gain of TFE
1.5
1
0.5
0
10-4
10-3
102
10-1
Computed and Fitted of Gain of TFE
1.5
1
0.5-
0-
-4
10-3
102
Figure 4-7: Fitting of the gain of transfer function at the lower-frequency range;
Window = 2 years with 1 year overlap
10-i
Expected and Observed Displacement
15 '
200
400
600
I
800
I
1000
I1
1200
I
1400
1600
1800
J
2000
Residual (Expected - Observed)
-15 1
200
I
400
I
600
I
800
I
I
1000
1200
Number of days
I
I
1
|
1400
1600
1800
2000
Figure 4-8: Expected and Observed GPS displacement and the residual; Window
2 years with 1 year overlap
4.5.2
Results from window
-
1 year with 0.5 year overlap
Gain of transfer function
-
-.
.- .
-.
. .
.
..
.
0.5|
10-3
10-2
Phase
10~
200
100
.. . .
.1. . . .. ..
~ff
~
-100
- ..
fi......l~~
il
.....
. . .. .
.....
.. . . . . . .
...
-200
10 3
10-2
Coherence
10~
1-
0.5-
01-10
10-2
101
cycles/day
Figure 4-9: Transfer function and Coherence; Window = 1 year with 0.5 year overlap
10
Gain of TFE
1.5
..
..
.. .. ...
.a. . .....
. ...
I
1
0.5
104
10-3
10-2
10
Computed and Fitted of Gain of TFE
I.U
m0
-0
0
0O
0 0
.0
c
00
0
O0.
10 4
10-3
10-2
Figure 4-10: Fitting of the gain of transfer function at the lower-frequency range;
Window = 1 year with 0.5 year overlap
56
10'
Expected and Observed Displacement
15
105-
0-5-10-15-
200
400
600
800
1000
1200
1400
1600
1800
I
I
I
1400
1600
1800
2000
Residual (Expected - Observed)
15110-
.10- -15'
200
I
400
I
600
800
I
I
1000
1200
Number of days
Figure 4-11: Expected and Observed GPS displacement and the residual; Window
1 years with 0.5 year overlap
2000
4.6
Interpretation of Spectral Analysis
Considering all the residual plot from figures 4-5, 4-8 and 4-11, the spec-
tral analysis using window length of 1 year and no overlap yields the best result of
computed displacement. We used the results from figures 4-4 and 4-5 to interprete.
The spectral analysis results are coherent up to a frequency of approximately 1 cycle per 10 days. From the lower frequency range (0.001 to 0.1 cycle per
day), the gain of transfer function is about 1 mm/C and the phase lag of the displacement compared to the temperature variation is about ' or 30 degrees.
6
This phase lag can be translated to the time shift in days according to the
cosine expression:
cos(wt -
#)
cos(w(t - -))
(4.12)
The time shift can be define as t'
, r/6
27r/365
~ 30
days
(4.13)
The residual in figure 4-5 shows that the temperature variation may not
be the only cause of the displacement change of the dam. In the next section, we
interprete how the water level influences the results of spectral analysis.
4.7
Effect of Water Level in Spectral Analysis
The displacement change of dam arch can be due to static loading factor
such as water loading, thermal loading. It can also be due to dynamics, instanta-
neous loading such as earthquake-induced forces [5]. In our analysis, we ignored the
dynamic load and considered the loads caused by temperature change and by water.
Water level can have effect on deformation of the arch dam [11].
The Pacoima dam water level data was available from the Water Resource
Division, Department of Public Works, Los Angeles. A comparison between water
level, the residual and the observed GPS displacement is shown in figure 4-12
Water Level at Pacoima Dam
2000
C 1950
0
-
.
-
-
-
.......
.
... ... . ...:.
-....
.. .. -....
.. .....
-.....
...
-..-..1850 - .......... .......... ............................
c1900
'
1800
-.......................I.... .....................
200
400
600
-20'200
400
600
800
1000
1200
Residual
1400
1600
1800
2000
800
1000
1400
1600
1800
2000
1200
Expected and Observed Displacement
-20'
200
'
400
I
I
600
I
I
I
1000
1200
1400
Number of days after 01/01/1995
800
I
I
1600
1800
Figure 4-12: Comparison Between Water Level and Residual (Expected Displacement
- Observed Displacement
2000
4.8
Comparison between Gains Derived from Theoretical Approximation and from Spectral analysis
We assess the gain of the transfer function derived in our spectral anal-
ysis based on our theoretical estimation in chapter 3. In chapter 3, we derived 2
formula (3.36) and (3.37)
u =
=
Tavg
JEdz
(4.14)
aTdz
J T(z)dz
We assume from the 2 expressions that the average temperature should have
similar behavior to the magnitude of the displacement. In other words, the average
temperature when plotted against the frequency should have similar shape to the gain
of the transfer function we derived. A plot comparing the magnitude of the average
temperature and the gain of the transfer function is shown in figure 4-13.
Gain of TFE
1.5
-e- Compute
Fitted
-
. . .. . .
...O . . .
-
Theoretical
1
.~~~~~~~
~
.
. . . .
~o
0.5
U
'
10-3
10-2
10
Figure 4-13: Comparison between computed Gain of transfer function (TFE), the
fitted gain and the gain derived from theoretical average temperature
From figure 4-13, it is shown that the shape of the magnitude of theoretically estimated transfer function is indeed similar to the computed gain transfer
function. It is however not possible at this point to estimate the accurate magnitude
of the transfer function directly. From equation (4.14), the displacement is obtained
by intergrating strain, which is proportional to the temperature change. Integrating
the temperature yields the displacement change plus some constant. There is however
no direct way to obtain the constant of the integration.
The similar shape of the plots shows that our computed transfer function
has result that is in accord with that given by the theory of thermoelastic deformation.
62
100
Chapter 5
Conclusions
5.1
Thermoelasticity at Arch Dam
Pacoima dam is monitored by 2 GPS receivers, DAMi and DAM2. DAM2
is located at the center of the dam arch and DAMI at the end of the dam to the South
of DAM2. The time series derived by substracting the positions of the 2 receivers represents the movement or displacement change of the dam. It is found that the arch is
moving East-West (upstream-downstream) with magnitude of 20 mm peak-to-peak.
The movement is periodic with an annual period.
The annual period of the GPS displacement time series lead us to assume
that the displacement change of the dam is due to thermoelastic deformation. The
periodic heating from the Sun caused the dam to expand and deform. We derived
thermoelastic deformation equations which show that the magnitude of the displacement change due to the thermal effect is large enough to be detected by GPS. We also
computed the phase delay between displacement and temperatuer time series. More
detailed calculation needs to be done in order to understand phase delay of arch dam
related to the temperature variation.
Our spectral analysis results are coherent up to a frequency about 0.1 cycle
per day. It is found that the temperature variation of 1 degrees can cause the dam
arch displacement of 1 mm. The spectral analysis is verified by our average temperature calculation. It is shown that the gain of our transfer function is in agreement
with the behavior of average temperature versus frequency as derived from the thermoelastic theory.
It is concluded that the displacement of the Pacoima dam is caused by
thermoelastic deformation of the arch. However, our computation shows that there
was some discrepancy between computed (based on transfer function) and observed
displacement. This discrepancy or residual may be due to the water level at the dam.
The analysis shows that the GPS system can be used to detect a subtle displacement such as one that is due to thermal effect like in this case. The GPS system
at Pacoima dam provides a good example of using GPS as a monitoring instrument
for safety purposes.
5.2
Future Work
Our model in chapter 3 is a simple one-dimension model. More detail model
is needed in order to explain the behavior of the dam more accurately. One approach
is to model the dam arch as a slab, a temperature distribution due to periodic heating at the surface can be derived. The hydrostatic effect on the arch can also be
computed based on the water level data. The water level also has an effect on the
temperature of the dam arch. Using this water level data and a slab model for the
dam arch, one can construct a model that described the displacement of the dam in
a more accurate manner.
In order to monitor the dam more effectively, more advanced technique for
dam monitoring may be considered. A finite element model can be implemented
according to the geometry of the dam. One can monitor factors such as temperature
variation and water level electronically. A thermometer can be installed and the
temperature can be retrieved daily. The water level is already monitored by electronic
gauges at least daily. These measurements can be integrated to the displacement
change of the arch measured by GPS. All the measurement can be used to validate
the finite element model. This new system of monitoring can be used to monitor the
dam as a supplement to a more conventional survey method.
66
Bibliography
[1] D.C. Agnew and H.O. Johnson. Monument montion and measurement of crustal
velocities. Geophysical Research Letters, 22(21), 1995.
[2] J. Berger. A note on thermoelastic strains and tilts. Journal of Geophysical
Research, 80(2), 1975.
[3] B.A. Boley and J.H. Weiner. Theory of Thermal Stresses. Dover Publications,
Mineola, NY, 1997.
[4] H.S. Carslaw and J.C. Jaeger. Conduction of Heat in Solid. Oxford University
Press, Oxford, England, 1957.
[5] Federal Energy Regulatory Commission. Engineering guidelines for the evaluation of hydro-power projects.
[6] P. Denbigh.
System Analysis and Signal Processing. Addison-Wesley, Essex,
England, 1998.
[7] T.A. Herring. Geodetic applications of GPS. Proceedings of the IEEE, 87(1),
January 1999.
[8] K. A. Hudnut and J. A. Behr. Continuous GPS monitoring of structural deformation at Pacoima Dam, California. Seismological Research Letters, 69(4),
1998.
[9] G.M. Jenkins and D.G. Watts. Spectral analysis and its applications.Holden-Day
series in time series analysis. Holden-Day, San Francisco, CA, 1968.
[10] M.N. Ozi§ik. Heat Conduction. Wiley-Interscience, New York, NY, second edition, 1993.
[11] J.L. Serafim and R.W. Clough. Arch Dams. A.A. Balkema, Rotterdam, 1990.
[12] S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. Engineering Societies
Monographs. McGraw-Hill, New York, NY, third edition, 1970.