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UNIVERSITY OF TEXAS AT AUSTIN

Institute For Geophysics
EARTH STRAIN MEASUREMENTS

WITH THE TRANSPORTABLE LASER RANGING SYSTEM:

FIELD TECHNIQUES AND PLANNING

Yosio Nakamura

H. James Dorman

Thomas Cahill

The University of Texas at Austin

Institute for Geophysics

Galveston Marine Geophysics Laboratory

700 The Strand

Galveston, 'Texas 77550-2768

November 1982

Final Report for Period December 1979 - November 1982

NASA Contract NAS5-25897

Prepared for

GODDARD SPACE FLIGHT CENTER

Greenbelt, Maryland 20771

Technical Officer: Christopher C. Stephanides, Code 942

The University of Texas Institute for Geophysics

Contribution No. 532
ii

ABSTRACT

We have conducted a feasibility study to examine the potential of the

Transportable Laser Ranging System (TLRS) for monitoring the ground deformation around satellite ranging stations and other geodetic control points.

Emphasis has been placed on testing the usefulness of the relative lateration

technique. The temporal variation of the ratio of the length of each survey

line to the mean length of all survey lines in a given area is directly

related to the mean ;n,;ar strain rate for the area. The data from a series

of experimental measurements taken over the Los Angeles basin from a MRS

station at Mt. Wilson show that such ratios can be determined to an accuracy

of one part in 107 with a measurement program lasting for three days and

without using any corrections for variations in atmospheric conditions. A

numerical experiment using a set of hypothetical data indicates that reasonable estimates of the present shear strain rate and the direcilion of the

principal axes in southern California can be deduced from such measurements

over an interval of one to two years. Thus, the relative lateration from

the MRS appears to be a very economical way to monitor ground deformations,

although there has been no opporunity yet to measure the actual ground

strain by reoccupying the Mt. Wilson site.
iii

Table of Contents

I. Introduction ................. .................................1

II. TLRS Ground-to-Ground Ranging...., ............. .......3

Advantages and Problems....... ............................3

Relative Lateration ............................ .........4

Relationship between Relative Lateration and Strain ....... 4

III.Mt. Wilson Experiment... ........................ ........ ...9

Field Experiment...... ..................................9

The Data .............................. ...................12

Range Ratios to a Single Reference Line ...................12

Time-of-Flight Ratios to the Mean ................... ... 14

An Alternative Atmospheric Correction ..................... 14

A Test for Systematic Error Due to Atmospheric Conditions.17

Results............, ................................... ..19

IV. Shear Strain Determination using Hypothetical Data .............21

V. Conclusions and Recommendations ................................22

Conclusions............ ... ..... ......23

Recommendations ................ ........................23

Acknowledgements....... .....................................24

References .....................................................25.

Appendix ............................ ...........................26'
I. INTRODUCTION

With the recent development in ground-to-satellite laser ranging

and Very Long Baseline Interferometry (VLBI) techniques, it is now

possible to measure precisely distances between locations separated

by several hundreds to thousands of kilometers. This makes it possible to monitor relative movements of globally distributed points on

the earth for geodynamic studies. However, one question that must be

answered is how representative each of these positions thus occupied

is for the region in which it is located. If some of these locations

are experiencing localized movements which are not representative of

the region, the global measurement would give erroneous results. An

answer to this questions can be found by measuring regional, deformations around each location.

A conventional method for determining regional deformations is

to perform repeated survey using an electro-optical distance measuring (EDM) device (e.g. Savage et al. [19811). However, such surveys

are expensive, and are rather limited in range. We, therefore, have

looked for a better alternative. The development of the Transportable

Laser Ranging System (TLPS) for ground-to-LAGEOS (Laser Geodynamics

Earth Orbiting Satellite) ranging [Silverberg and Byrd, 1981] has given

us an opportunity to test such an alternative. Because of its high sensitivity, being capable of detecting single photon returns, the TLRS

can measure distances to small targets (retro-reflectors) at any visible

points much beyond the normal ranges of other EDM devices. Thus, this

system may provide economical measurements of strain fields in areas

more than 200 km in diameter. If successful, such measurements will be 

valuable not only in the immediate neighborhood of satellite ranging 

stations, but also in understanding the dynamic behavior of both plate 

boundaries and areas internal to plates. 

We have conducted a limited feasibility study to examine this potential. Although the TLRS is a powerful system, it also has certain

limitations when used for a ground-to-ground ranging. The most important is the uncertainty of measurement results due to variability in

atmospheric conditions. To :)ypass this problem and avoid the expense

of flying an aircraft to monitor the atmospheric conditions along the

path of the laser beam, we have examined the use of the relative lateration, or the ratio method, which was used earlier by Carter and

Vincenty [1978] in an experimental survey around the McDonald Observatory.

We originally planned repeated field experiments at several sites

in the western United States. However, because of many scheduling

conflicts and delays associated with the overall TLR -LAGEOS ranging

experiments, the only field experiment we could perform during the

0,,arrent contract was a four-day measurement at Mt. Wilson over the Los

2-j;geles basin in January, 1981. We have been unable to reoccupy this

site for an actual strain measurement.

1
f

2

The present study, however, has given us some very encouraging

results. Even with no atmospheric correction at all, the range ratios

could be detjrmined town accuracy of one part in 10 7 . This is sufficient for an order-of-magnitude estimate of incremental shear strain

in the southern California region if two measurements separated by one

to two years are available. Higher accuracies would be attainable with

repeated measurements.

In this report, we first describe the advantages and problems of

ground-to-ground ranging by a MRS, leading to the use of relative

lateration, or range-ratio method, and its relationship to the regional strain (Section II). Then, we present the data and analysis of the

Mt. Wilson experiment (Section III). This is followed by a short treatment of regional strain determination using hypothetical data (Section

IV). Finally, we present the conclusions from this feasibility study

and offer some recommendations. Some pertinent data are presented in

the Appendix.

I

F4
3

II. MRS GROUND-TO-GROUND RANGING

Advantanges and Problems

The MRS is a highly mobile satellite laser ranging system designed

to perform ground-to-LAGEOS range measurements. It is also highly sensitive, being capable of determining the range to a LAGEOS satellite with

return signals as low as one photelectron every 20 to 50 laser shots

[Silverberg et al. 1982). Used as a ground-to-ground ranging device, it

can measure the distance to any single 1 i;ich (25 mm) corner reflector

within sight at very low laser power level. The measureable range is

limited only by the curvature of the earth. The required power Level is

so low that, unlike some systems used for similar measurements, the laser

beam can be maintained wany orders of magnitude below the eye-damage

threshold.

The practical precision of the MRS range measurements is limited

to about 1.5 cm for a one minute average, which is somewhat worse than

those of conventional EDM devices using modulated laser beams. However,

the long range capability of the MRS reduces the relative error to well

within the limits of interest in conventional surveys. The MRS has an

automatic painting system, an automatic calibration system and other

features which lend themselves to providing many horizontal (ground-toground) line measurements on an operational basis. Thu gs it will be a

good device to use if the data it provides is sufficient to determine

the regional deformation at a high-enough accuracy.

The most serious problem in using the MRS for ground-to-ground

ranging is the atmospheric effect. The temperature, pressure, and to a

lesser degree water vapor influence the index of refraction of air, and

thus the speed of a laser beam through the atmosphere. To obtain the

absolute distance between two points from a time of flight measurement

through air, one must make corrections for these atmospheric variables.

Estimates of these atmospheric variables along the beam path may

be made based on measurements at the two end points. This, however, is

unsatisfactory for long lines. A more precise way is to measure directly the atmospheric condition along the beam path by flying an aircraft

during the ranging. This, though done in practice, is a costly operation. A third ,alternative is to use more than one wavelength for ranging. Using the dispersive characteristics of light in air, one can correct for the atmospheric effects [Huggett, et al. 19771.

The present MRS operates in a single color. Flying an aircraft,

we judged, is too costly for repeated measurements in many directions.

Thus we had to look for another alternative.
4

Relative Lateration

One way to improve the accuracy of range measurements without relying on expensive in-flight measurement of atmospheric conditions is

to use a relative lateration technique, or the "ratio method" (Robertson, 1972). Instead of attempting to measure the absolute length of

each survey line to high accuracy, this technique determines only the

ratios of distances. This method is based on a supposition that the

temporal changes of atmospheric conditions along several survey lines

within a given region are n:i.milar to each other. Therefore, even when

the time of flight of a laser beam in each line fluctuates with changing atmospheric conditions, the ratios of the times of flight along

different survey lines tend to vary little with time.

Carter and Vincenty [1978] used this method in an experimental

EDM survey around the McDonald Observatory in 1977. They obtained

sets of measurements, one month apart, consistent to one to two parts

in 10 7 . They have just repeated this experiment and the data is now

being analyzed. Since the results of Carter and Vincenty appear to be

quite promising, we have decided to try the same for our TLRS measurements.

Relationshi p Between Relative Lateration and Strain

Unlike absolute measurements of distances, the relative distance 

measurements repeated after a certain time period will not give all of

the components of deformation, or incremental strain, for the time

period unless at least one survey line is measured absolutely. How- 

ever, a clear relationship exists between the changes of relative distances and incremental shear strain.

Let us consider n survey liner radiating from a central station.

In the present case, the TLRS is located at the central station and a

retroreflector is located at the end of each radiating line. Assume

that r11 lines lie in a horizontal., plane, neglecting both the curvature of the earth ' s surface and topographic height differences. Choosing a coordinate system with the origin at the central station, positive x towards east and positive y towards north, the original length

of line i to the reflector at coordinates (x i , yi ) at the time of the

initial survey is given by

i

-^ si = ( xi2 + yi2)(1)

Now as"ume that between the initial survey and a subsequent survey the entire area of the survey undergoes a uniform deformation represented by incremental strain components exx , exy and eyy . Then, the

line length becomes
OF POOR QUALITY 

si , = ((xi + exxxi + exyyi

)2 

+ (yi + exyxi + eyyyi)21

= [xi 2 + yi2 + 2exxx12 + 4exyxiyi + 2eyyyi21

= s i [1 + 2e

xx

sin 2ai + 4exysin a cos a + 2eyyCos 2ai]

= si [1 + exx sin 2 ai + exysin 2a + eyy Cos 2 ai ) (2)

where ai = tan -1 ( xi/yi ) is the azimuth of the line i measured clockwise from north, and the higher order terms in strain have been neglected. Then, the range increment di is given by

di = s i ' - S. = Si [e sin 2 a. + exysin 2ai .+ eyy COS 2 ai ] (3)

Next define original mean range and range ratios to the mean, respective, as

n

S = iEl s i/n (4)

and

ri = si/s

(5)

Then, the subsequent mean range and range ratios are

n. _ n

s' = i El si '/n = s + i E1 Si/n (6) 

and 

ri ' =s %s' = (s i + 

8)/(7

 + il E d i/n)

= ri (1 + Si /si - iEl 8iA ) ( 7)

where the higher order terms are again neglected. The increment

of the range ratio is, therefore,

n

ri r ri(ai/si - i£1 si

/ns) (8)
6

OF POOR QUALITY

Then, range ratio increment normalized by the original range ratio

is given by

n

Yi = (r1 ' - ri)/ri = 61

/si - J1 6 i A (9)

Substituting eq. (3) into eq. (9), and using (5), we obtain

n

Yi = [sin2 ai - iEl(risinZal) /nl

Eyy

n

+ [sin 2a1 - iEl (ri sin 2ai ) /n] ExY

n

+ (Cos2 ai - 1 E1 (rlCos2 al )/n) eyy (10)

Equation (10) may gave one an impression that a set of measurements of

the normalized range ratio increments Yi would give the incremental

strain components exx, exy and eyy. However, this impression is incorrect because the coefficients of e xx and eyy are not independent of each

other, as their sum vanishes, and therefore exx and eyy cannot be determined uniquely.

Now let

0 = exx + Eyy (11)

and

exx - eyy (12)

Then,

exx z 

(0 + T) (13)

and

Substituting ( 13) and ( 14) into ( 10), we obtain

n

Yi = [sin 2ai - i El (ri sin 2a,)/n) exY

n

- 31 [cos 2ai

(r Cos2ai)/n) t)' (15)

The coefficients of exy and T are known quantities for the initial setup

of the survey lines. Thus, for a set of measurements of the normalized

range ratio increments 'y 1 , the incremental shear strain components exy

and T can be determined by a least-square inversion of eq. (15).

Finally, the maximum incremental shear strain S and the direction

of the principal strain axes 0 are given by

S [(2exy)2 + T2 )^ (16)

and

a = tan-1 (2exy/T) (17)
a OF POOR QUALIV

The dilation 0 of eq. (11) disappears from eq. (15), and thus canilQt be

determined. This is expected because any uniform comprAssion or expansion

of the entire area causes no change in range ratios.

The treatment above assumes uniform deformation of the entire region.

If for some reason, such as the existence of active faults within the area,

the regional deformation is not uniform, large residuals will show up in

the least-square inversion of eq. (15). Thus, any residuals significantly

larger than the measurement errorN will indicate heterogeneous strain.

The remaining question is how accurately we can estimate the normalized

range ratio increments Y i . Since the measurements are done in terms of

time of flight of light beams, the uncertainty in speed of light is the

determining factor. The average speed of light, ci, between the central

station and a reflector i may be expressed as the suns of four components:

C = co + 9 + we + W (18)

where co is the speed of light in standard air, which is constant for all

survey lines at all times; k i is the chi."rection attributable to the reflector local-ion; which is time invariant for a given reflector; we is a component of correction attributable to weather common to all reflectors at a given

time; and wi is the residual weather correction. The line length si is given

in terms of round-trip time of flight, t i , as

S. _ }(c + k. + w + w . (19) 1 o 1 c

the mean range as

n n

S = }(c t + E k.t./n + w t + E w.t,/n) (20)

o

i=1 i i ci=1 1 1

n

where t = L t i /n is the mean time of flight, and the range ratio to the mean

as i=1

n n

r. = u.[1 + (k. - E k.t./nt + w. - E w.t. /nt)/c ] (21)

i=1 i=1

where ui = ti /t is the time-of-flight ratio to the mean, and the higher-order

terms have been neglected. Finally, the normalized range ratio increment is

given as

n

^i = , i + Hwi ' - W. - E (wi ' - wi)ti/ntT/co

i=1

(22)
8

where n = (u ' - ui )/ui

 is the normalized time-of-flight ratio increment and

quantities wi& primes designate those at subsequent measurement as before.

The higher-order terms & e again neglected. Note that the common weather

component, w , is eliminated by taking the range ratio (21), and the location

specific components, !ff.'s, are eliminated by normalization (22), leaving only

the residual weather components wiIs.

The normalized time-of-flight ratio increment, rt,, thus approximates the

range ratio increments, y,, with a small error due to residualweather term.

The latter is not location specific, and is not common to all lines at a given

time. If this term is sufficiently small, then we can substitute fli for Y.i

in calculating the shear strain increment using (15).
9

III. MT . WILSON EXPERIMENT

Field Experiment

At the request of the NASA Crustal Dynamics project, the TLRS team

from the McDonald Observatory of the University of Texas, led by Dr.

Eric Silverberg, deployed the TLRS at Mt. Wilson, California, in

January of 1981. At the same time, two of us (H.J.D. and T.C.) scouted

the surrounding area for suitable target sites and selected the reflector locations. Then, a field party from the National Geodetic Survey

(NGS), which was dispatched at the request of NASA to help us, deployed

retroreflectors at the chosen sites. The survey lines selected for the

site are shown in Figure 1. Table 1 lists the nominal coordinates of

the base station (TLRS L ­-10 at Mt. Wilson and of the end points of the

lines, where the retroreflectors were installed. Also listed in Table

1 are the approximate look angles from Mt. Wilson and ranges as computed from the indicated coordinates using the IAG standard ellipsoid

Geodetic Reference System 1967.

Each reflector except the one at Cahuenga was a metal box containing an array of three 11 inch (38mm) corner cubes, supplied by the NGS.

The box was mounted on a tripod and placed directly over the station

mark using an optical blumb bob. This elaborate configuration made it

necessary to guard the reflector continuously for the entire duration

of the experiment. The reflector used at Cahuenga was designed by one

of us MIN',,) ,:ror unmanned operation. It contained a single 1 inch (25mm)

cor,^,:ir etnd was fastened to an outcrop with anchor bolts at a site

off the r.-Cation mark, thus concealed from public view.

The reference point of the TLRS, from which the raw time-of-flight

measurements were made, was slightly offset from the Mt. Wilson station

mark given in Table 1. The measured coordinates of the station mark

relative to the TLRS were:

x = -1.4873 m (west)

y = 0.5093 m (north)

z = -3.3709 m (below)

The resulting corrections, to be applied to the observed quantities to

reduce them to the reference mark, are listed in Table 2. The corrections can be applied at any stage of data reduction.

After the initial setup, which began on January 9, 1981, the horizontal ranging data were collected over the four-day interval January 23

through 26, 1981, in cooperation with the NOAA National Geodetic Survey. Each of the reflector sites except-Cahuenga was manned continuously

during the entire experiment to record the temperature, pressure and relative humidity at the site at about 30 minute intervals. The details

of the data acquisition are given in Silverberg et al. [1982).

w
10

rO

_J W

v

tS-' J

T

^ Q

V/

LLr

ORIGINAL

OF POOR QUALITY P^au.n

4

Jv

l^

w^

w`

i

dl

1 al

♦ 1 ^1/

1 QI

1 ^.

/x 0

1 '

ab

\ 1~ J/f

/ a 1^ +^ tea'/

z 1 -'^ ^/ za:

W 1 ^/ Q Q

1 \ '^^^ ^' ^' ♦// 

Ina

^ 1WS^•/^i

1M

1 r^

13

's

/ o

i

m

W

40 i

f yy=

N u

V ^ ~

.r

by

tr^

o

UI 3.^ 

b N

N 44

A O

N ^.

OV

4

a,

4-) a

O o

ro N

44

ol

4

N JW

Z

I
11

ORIGINAL. PA U 170

OF POOR QUAL17Y

Table 1. Stations Used in Mt. Wilson Lino Survey

Station Longitude Latitude Elevation Look Angle Range Note

m Azimuth Altitude m

Mt. W.'Ison 241'56'17.85" 34°13'21.58" 1722.0 1

Cosiro 241012'55.4011 34'05'08.57" 858.9 257.366 -1.030 68393 2

Cahuenga 241°40'29.81" 34°08'13.00" 554.1 248.687 -2.681 26104 3

Son Pedro 2410 39 1 55.73" 33"44'40.88 11 

447.1 205.506 -1.508 58729 4

Niguel 242°16.00.18" 330 30'44.03" 288.0 158.814 -1.353 84459

Santiago 242'28'00.10" 33°42'37.89" 1733.0 139.168 -0.329 74933

San Juan 242'15'45.99" 330 54'49.47" 543.0 138.751 -1.688 45536

1 New marker 9.408m from Mt. Wilson E10A 0345°17'

2 Solitice Canyon B2 Aux. 1, which is 14.107m ENE of Castro 1898

3 Reference marK 4.3 of Cahuenga 02, 13.329m @257°47' from Cahuenga *2

4 L7 Ecc. Son Pedro Hills, which is 12.576m @318°57' from San Pedro *3

Table 2. Corrections to be Applied to Observations to

Reduce to Mi. Wilson Gound MarKer

----------------- I ----------------------------------------

Round-Trip Range Ratio*

Station Time of Flight Range (1) (2)

ns m ppm

---------------------------------------------------------

Castro -9.344 -1.4003 -25.08 -26.78

Cahuenga -9.054 -1.3568 -23.35 -23.51

San Pedro -1.798 -0.2694 -5.91 -7.93

Niguel 6.222 0.9325 13.61 9.93

Santiago 8.931 1.3385 20.64 17.07

San Juan 8.432 1.2636 20.09 17.64

-^< (1) Ratio to mean range

(2) Same but excluding Cahuenga and Niguel
12

The Data

The raw field data were initially processed at, the University of

Texas at Austin by the McDonald Observatory group. As described in

detail by Silverberg et al. [1982], the processing of the raw data involved accumulation of individual photon returns into 200 psec bins,

smoothing of the coadded returns by three-bin (600 psec) running averages, cross-corretation with a reference standard to eliminate longterm drift in the calibration constants, adjustments to account for

certain measurement irregularities, a-_L removal of a 86.8 nsec constant calibration correction.

The calibrated round-trip time-of-flight data, shown in Figure

2 and listed in Table Al in the Appendix, have not been corrected for

the offset of the MRS from the ground marker (Table 2). The data for

Cahuenga were not used for the analysis because of certain processing

difficulties encountered for the data for this station.

The data gap during the second day of observation was due to an

interruption in data acquisition caused by rain which accompanied the

passage of a cold front. The meteorological data taken at Mt. Wilson

site and other stations are shown in Figures Al through A3, and are

listed in Table A2 in the Appendix.

As expected, the raw time-of-flight data show large fluctuations,

which are only partially correlated with the meteorological data. The

relative RMS deviations of the time-of-flight data (Table 3, colume 3)

range from 1.53 ppm for Niguel, which was surveyed only after the passage of the cold front, to 3.78 ppm for San Juan, which was the shortest line. The weighted average for all lines is 2.84 ppm.

Range Ratios to a Single Reference Line

Silverberg et al. [1982] calculated the time-of-flight ratios and

atmosphere-corrected range ratios to a reference line following the

procedure used by Carter and Vincenty [1978]. The reference line they

chose was a smoothed curve (a cubic spline) through the Santiago data.

Their results (Table 3, column 5) show relative RMS deviations of timeof-flight ratios ranging from 0.4 ppm for Niguel to 1.6 ppm for San

Juan. The weighted average for all lines is 1.0 ppm, which is about a

factor of three improvement from the fluctuation of the time-of-flight

data.

Their results for the range ratios with atmospheric corrections

based on end-point meteorological data did not fare as well. In fact

the relative RMS deviations increased typically about 40% from those

of uncorrected time-of-flight ratios [Silverberg et al. 19821.
► A1^V. G q 'il b'.Ih ^^ .. .l 4'.A

0^0 R iL.I.47 13

9

8

7

6

456365

8

7

6 N

z 5

391914

x

^i 4

U_0 3

2

1

a. 563580

a 8 Z

O 7

c^ 6

5

500004

50

9

8

7

383846

CASTRO + + t ++

++

+

SAN PEDRO +*+^ +

+

NI GUEL

t

SANTIAGO +*+*+

++

++

SAN .JUAN t+++ *+ +

+ + +

+*+ + ++

++

+

z

O

a

Ea.

a.

22 23 24 25

.JANUARY, 1981

Fig. 2. Round-trip time of flight from Mt. Wilson. The data are not

corrected for the marker offset.
14

The main reason for the poor performance of atmosphere-corrected

values iz the difficulty of making proper atmospheric corrections. A

comparison of the variations of the group index of refraction calculated from the temperature and pressure at end points (Figure 3; also

listed in Table A3 in the Appendix) with the time-of-flight variations

(Figure 2) clearly shows that long-term variations are fairly well

matched but shorter diurnal fluctuations are larger for the index of

refractions than for the times-of-flight. Thus the index-of--refraction correction per Carter and Vincenty [1978] over-compensates for

diurnal variations.

Time-of-Flight Ratios to the Mean

In order to be consistent with the range-ratio/strain relationship

of the preceding, section, we calculated the time-of-flight ratios to

the mean. Since the time-of-flight measurements to all targets were

not made exactly simultareously, the data were linearly interpolated

before the mean time-of-flight for a given time was calculated. (Higher order interpolations or a spline approximation might be better, but

we judged the difference would be small.) Also, since we had data for

Niguel only during the last half of the experiment, th,s station was

excluded from the mean time-of-flight calculation.

The resulting time-of-flight ratios to the mean (Figure 4; also

listed in Table A4 in the Appendix) show a further improvement in the

fluctuations of the results. The relative RMS deviations (Table 3,

column 5) now range from 0.36 ppm for Niguel to 1.24 ppm for San Juan,

with the weighted average of 0.71 ppm for all lines, a factor of four

improvement from the raw time-of-flight data.

An Alternative Atmospheric Correction

As stated earlier, the short-term, diurnal fluctuations in the index of refraction at end points exceed the observed fluctuations in the 

time-of-flight values. This is probably due to the larger fluctuation

of the atmospheric temperature near the ground than those in most of

kl the intervening air mass; a result of the base station and,most of the

target stations being located well above the intervening terrain. In

this situation, a standard correction procedure like that of Carter

and Vincenty [1978] is not really applicable, and some alternate prod cedures are needed.

An experimental procedure we tried was to estimate the average temperature of the air mass by low-pass filtering the mean of the temperatures measured at the end points. The filter we used was a simple one

of adding all previous temperature readings each weighted by a factor

proportional to a negative exponential of the elapsed time. After
OF POOR QUALITY, is

MT. WILSON

++ + * +^ +

CASTRO

+ +

+

SAN PEDRO

+ + ++ + +

+ +

NIGUEL

++fi1♦ +*+ + ++

++++ + 40+ +*++

SANTIAGO

4'iM^MM4h^

+++^f+

S. ++

V, ++4.

SAN .JUAN ++ 4:4 4v*

+ ++41+ + + + ++ +

4 1++.-* + + +

+ {t +

+

+

++

55

50

1.000245

a

75

70

65

1.000260

85

0 80

r

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w

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w

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80

75

70

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z

0

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a

a.

22 23 24 25

.JANUARY, 1981

Fgi. 3. Group index of refraction computed from atmospheric data

at end points.

ire
55, ^^yy yy

r

16

OF POOR ^_ UA ITY

CASTRO

+

+

*

'^F +

+++

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+ + + + +

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+ +

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SAN .JUAN

7

6

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z

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22 23 24 25

.JANUARY, 1981

Fig. 4. Time-of-flight ratios to the mean.
17

trials with such filters of several different time constants, a time

constant of 12 hours was found to give the best result. The resulting relative RMS deviations of the ranges thus corrected for atmospheric conditions ( Table 3, column 4) range from 0.96 ppm for Castro

to 2.05 ppm for San Juan with the weighted average of 1.38 ppm for all

lines. This is about a factor of two improvement from the raw timeof-flight data. However, the range ratios calculated from these corrected ranges do not show any significant improvement over those of

the uncorrected ratios. The relative RMS deviations of the corrected

range ratios (Table 3, last column) range from 0.52 ppm for Niguel

and Santiago to 1.21 ppm for San Juan, with the weighted average of

0.72 ppm for all lines.

A comparison of the relative RMS deviations of various quantities

in Table 3 reveals that the best result is obtained for the uncorrected

time-of-flight ratios to the mean. The atmospheric corrections did not

improve the RMS deviations at all when ratios were taken.

A Test for .Systematic Error Due to Atmospheric Conditions

The reliability of the relative lateration depends on the validity

of the assumption that the temporal changes of atmospheric conditions

are similar for all survey lines in the area so that their effects

cancel out when ratios are taken. If this assumption is incorrect, a

systematic error due to varying atmospheric conditions is introduced

into the measured time-of-flight ratios. The greatly different atmospheric conditions before and after the passage of a cold front during

that experiment gave us an opportunity to test this assumption.

The test we performed is the likelihood ratio test. We divided 

the time-of-flight ratios of Table A4 for each line into two subsets,

the first half and the last half, of equal size (the last half was

one greater than the first half if the total number was odd). If the

systematic error due to atmospheric conditions is significantly larger

the mean ratio, ui , for the first subset will be significantly different from that, u2, for the second subset. Setting up a null hypothesis H0 :pl = u2, if it is true, then the likelihood ratio statistic

t = [nln2/ (n1+n2 ) ] ;I (u1 - u2 ) / [ (n1012 + n2022 )/(n 1 +n 2 - 2)] !^ (23)

has a t distribution with n +n -2 degrees of freedom, where nl and n

are the sample sizes of the ltwo subsets and }1l,u2 , 012 and U2 2 are used to

designate the sample means and the :ample variances of the first and

the second subsets, respectively, for convenience.

At 90% significance level, the t distribution has values of 1.69

for 34 degrees of freedom and 1.80 for 11 degrees of freedom, while

the t values computed from the data, Table 4, are much smaller. Therefore, the null hypothesis cannot be rejected at this level of significance.
18

ORIGINAL PAGE x

OF POOR QUALITY

Table 3. Comparison of Relative RMS Deviations in ppm

Number of Uncorrected Corrected Uncorrected Uncorrected Corrected

Station Data Points Time of Range(a) T-o-F Ratio T-o-F Ratio Range Ratio

Flight to Santiago(b) to the mean to the mean

Castro 36 2.68 0.96 0.7 0.56 0.54

San Pedro 24 2.91 1.12 0.8 0.522 0.54

Niguel 13 1.53 1.81 0.4 0.36 0.52

Santiago 36 2.45 1.03 (0.2) 0.54 0.52

Ean Juan 27 3.78 2.05 1.6 1.24 1.21

All 13G 2.84 1.38 1.0 0.71 0.72

(a) Corrected by using.12-hour low-pass filtered temperature

(b) From Silverberg et al. 119827. The deviation for Santiago is from the smoothed

curve, and is not included in calculating the average for all stations.

Table 4. Likelihood Ratio Test for Non-equality of Means

Subset 1 Subset 2 Degree

Station---------------------------- --------------------------- of t

n1 }11 C71 n2 112 172 Freedom

Castro 18 1.104914282 0.000000513 10 1.104914438 0.000000G94 34 0.745

San Pedro 12 0.948870911 0.000000356 12 0.948870818 0.000000598 22 0.445

Niguel 6 1.364486713 0.000000572 7 1.364486460 0.000000365 11 0.889

Santiago 18 1.210566249 0.000000774 18 1.210566130 0.000000515 34 0.531

San Juan 13 0.735648159 0.000000330 14 0.735648288 0.000600983 25 0.353
19

In :;the y words, no significant difference is found between the mean

time-of-flight ratios in the first and second halves of the experiment for any of the lines surveyed.

Results

Since there is no evidence for systematic errors caused by atmospheric conditions, the most likely estimates of the mean time-of-flight

ratios and their variances (and standard deviations) can be calculated

from the entire data set. The results are shown in Table 5. Also

listed in this table are the mean time-of-flight and the mean distances. The latter were calculated using atmospheric corrections based

on the low-pass filtered temperatures described earlier and pressures

interpolated to the average height of the beam from the end-point

measurements (extrapolation in case of Santiago because the average

height of the beam was lower than either end point). A group index.of

refraction of n 1.00028975 at the wavelength of 0.5320 pm, calculated

from the formulg given in American Institute of Physics Handbook [1972,

p. 6-1111 for standard dry air with 0.03% carbon dioxide at 15°C and

760 mm Hg, is used. No other corrections have been applied to the calculated distances; thus they are subject to minor systematic errors.

The estimated relative standard deviations of the mean time-offl?Lght ratios are approximately 1 x 10 -7 except for San Juan, which is

the .shortest line. In comparison, Savage and Prescott [1973] estimate

that the standard deviation of their Geodolite measurements of distances

are 3 and 8 mm for lengths of 1 and 37 km, respectively. Thus, the precision of the present time-of-flight ratios is at least a factor of two

better than that of their distance measurements. Furthermore, their

distances had to be corrected for temperature and humidity readings

made with an aircraft flying along the line of sight, while the present

time -cF:-flight ratios required no atmospheric correction at all.

Multiwavelength measurements of distances are definitely better 

than the above two in terms of relative accuracy. Huggett and Slater

[:.975) and Slater and Huggett [1976] show the standard deviation of

individual distance measurements to be less than 1 x 10 -7 on a 10.1 km

line. By taking the mean of many measurements, which is practical In

this case, the accuracy can be improved further. The ranges attainable

with the multiwavelength system, however, are quite limited compared

with the MRS measurements.
p

20

r

.. ORIGINALPACE 11

OF POOR QUALITY

Table 5. Mean Time of Flight, Distance and Ratios*

Mean Moan Mean S.D, of Relative

Station Time of Flight Distance .T-o-F Ratio Moan S.D.

ns m T-o-F Ratio ppm

Castro 456358.75 68388.52 1.10488758 9.00000010 0.09

San Pedro 391914.94 58730.88 0.94886293 0.00000010 0.11

Niguel 563587.77 04456.72 1.36449651 0.00000014 0.10

Santiago 500014.83 74931.62. 1.21058326 0.00000011 0.©9

San Juon 303856.63 45535.87. 0.77,366587 0.00000018 0.24

* These rosults have been corrected for the TLRS/ground-marKer offset.

Table 6. Increment in Normalized Ratio

Due to Hypothetical Strain Increment

and Rounded Values for Testing

-------------------------- w ---------

Normalized Rounded

Station Ratio Increment to

ppm 0.1 ppm

Castro

---------------------------------------

0.052 0.1

San Pedro -0.158 -0.2

Niguel -0.030 0.0

Santiago 0.Q68 0.1

' San Juan 0.070 0.1

------------------ -------------------
21

IV. SHEAR STRAIN DETERMINATION USING HYPOTHETICAL DATA

We have been unable to reoccupy the Mt. Wilson site for a repeat

measurement, which would allow a testing of the ratio method for a

shear strain determination in the region. This section, thezefore,

describes an exe raise we have conducted to see how well we can determine the regional shear-strain increment using a set of hypothetical

data.

We assume a hypothetical strain increment described by

e1 = 0.1 x 10 -6 maximum extension

e2 = -0.2 x 10 -6 maximum compression

el -e2 = 0.3 x 10 maximum shear

S

0

= 110 azimuth of maximum positive principal

axis (extension) measured clockwise

from north

This strain increment is approximately the annual strain increment in

southern California observed by Savage et al. (1981]. The resulting

increments in normalized range ratios for the five survey lanes used

in the Mt. Wilson experiment are listed in the center column of Table

6. Since we will not be able to measure these ratio increments at

this accuracy, we use the values rounded to 1 x 10 7 , as given in the

rightmost column of Table 6.

Substituting these rounded ratio increments into eq. (15), and inverting it in a least-squares sense, we obtain the following results:

el - e2

0.40 x 10-6

S = 109.5°

The result describes the original hypothetical shear-strain increment

reasonably well. A trial with a rounding to 1 x 10 8 results in almost complete duplication of the hypothetical strain increment.

The likelihood ratio test of the preceding section can be used to

estimate the required number of measurements to achieve a given level

of accuracy at a given confidence level. We use the standard deviation

of individual range ratio measurements of 5 x 10_ 7 as estimated from

the present data (Table 3, excluding San Juan). Thus, substituting

Q1 = Cr = 10 -6 and Iul - u I= 10-7 into eq. (18), we find that n1 =

n = 200 will give t = 1.99, which exceeds the value of t distriLtion,

1.97, for 198 degrees of freedom at 95% confidence level. Thus a variation in the range ratio of 10 7 found by averaging 200 ratio measurements

is significant at 95% level of confidence.

At a rate of one measurement every hour, it will take slightly more

than a week to complete this many measurements. Two such series of
22

measurements one year apart is sufficient to determine the shear strain

increment in southern California.

For a given set cf t and U's, n's are approximately inversely proportional to the square of the difference in }1 1 9 in equation (18).

Thus doubling the measurement interval, thereby doubling the expected

ratio variations, approximately quarters the required number of measurements. For example, a pair of 50-measurement sets two years apart will

give the shear strain rate in southern California.
V. CONCLUSIONS AND RECOMMENDATIONS

23

Conclusions

Even though the field experiment we performed during this contract

was quite limited compared with our original plan, we obtained several

interesting and important results. The following is a list of conclusions drawn from these results;

1. The increment of the ratio of the length of a survey line to the

average of several survey lines in a region is directly related

to the incremental shear strain in the region. Thus, the shear

strain rate can be calculated from observations of temporal variatLons in such ratios.

2. Using the TLRS, the time-of-flight ratios could be determined to

an accuracy (one standard deviation) of 1 x 10 -7 by averaging

measurements over a four day period. This accuracy was obtained

without using any atmospheric corrections at all. No improvement

was obtained when atmospheric corrections based on end-point

measurements were applied.

3: A calculation using a hypothetical data simulating the observed

strain field in southern California indicates that two sets of

TLRS ratio measurements separated by one to two years will be

sufficient to determine the direction and rate of shear strain

in the region.

4. Thus relative lateration using the TLRS has been demonstrated to

be a good method for monitoring the regional shear strain field

around satellite ranging stations. The TLRS operates successfully

over long distances. The ratio method is extremely economical.

It requires no environmental measuremei s and can be performed

with small unattended retroreflectors distributed over a wide

area. Thus these techniques greatly surpass the capability of

conventional EDM techniques.

Recommendations

1. The results of the present ouperiment are thus very encouraging.

However, they are based on only one experiment. Before this technique is put to a practical use, further demonstration is needed

to confirm the above results. Therefore, it is recommended that

this feasibility study be continued at least to include reoccupation of the Mt. Wilson site and two measurements at another properly selected site, preferably with a different meteorological

environment.

e-.-. - 'X. ..
24

2. Relative lateration is not "limitra to the data taken by the MRS.

The data reduction procedure iinc ' in the present study can be applied to other data from di.stane;e measurements. Therefore, it is

recommended that we reanalyze some of existing ranging data to see

if improvements in determination of shear strain rate can be

achieved. This can be done without further field measurements.

3. Additional feasibility test measurements similar to the Mt. Wilson

experiment may be obtained from fixed satellite ranging stations.

It is therefore, recommended that this possibility be examined.

4. Horizontal ranging to distant targets on the ground does not raquire all the sophistication of the MRS system. Therefore, when

the capability of the present technique is fully demonstrated, a

smaller, more portable single-photon ranging unit should be developed for this purpose.

5. Finally, the technology is advancing in other fields also. Such

techniques as miniature interferometer terminals (Counselman and

Shapiro, 19791 may someday be more useful in surveys of regional.

extent. Therefore, development in these other technigL,'S should

be reviewed while developing the present technique.

Acknowledgements

The initial data reduction of the Mt. Wilson experiment was done

in the Astronomy Department of the University of Texas at Austin. We

are grateful to Dr. Eric C. Silverberg for supplying us the processed

data on a computer tape. Dr. Cliff Frohlich kindly reviewed a draft

of this report, his constructive comments are greatly appreciated.
25

Referencas

American Institute of Physics Handbook, 3rd ed., McGraw-Hill, New York,

19 72 .

Carter, W. E. and T. Vincenty, Survey of the McDonald Observatory radial

line scheme by relative lateration technique, NOAA Technical Report

NOS 74 NGS 9, 1978.

Counselman III, C.C. and I.I. Shapiro, Miniature interferometer terminals

for earth surveying, Bull. Geod. 53, 139-163, 1973.

Huggett, G. R. and L. E. Slater, Precision electromagnetic distancemeasuring instrument for determining secular strain and fault movement, Tectonophysics, 29, 19-27, 1975.

Huggett, G. R., L. E. Slater and.J. Langbein, Fault slip episodes near

Hollister, California: Initial results using a multiwavelength distance-measuring instrument, J. Geophys. Res., 82, 3361-3368, 1917.

Robertson, K. D., The use of line pairs in trilateration and traverse,

Survey Rev., 21, 290-306, 1972.

Savage, J. C. and W. H. Prescott, Precision of Geodolite distance measurements for determining fault movements, J. Geophys. Res., 78, 6001-

6008, 1973.

Savage, J. C., W. H. Prescott, M..Lisawski and N. E. King, Strain accumulation in southern California, 1973-1980, J. Geophys. Res., 86, 6991-

7001, 1981. 

Silverberg, E. C., and D. L. Byrd, A mobile telescope for measuring continental drift, Sky and Telescope, 61, 405-408, 1981.

Silverberg, E. C., T. Cahill and J. Dorman, Relative lateration across the

Los Angeles basin using a satellite laser raning system, Bull. Geod., 

(in press), 1982.

Slater, L. E. and G. R. Huggett, A multiwavelength distance-measuring

instrument for geophysical experiments, J. Geophys. Res., 81, 6299-

6306, 1976.

w
26

x

APPENDIX
1 OI"i I 1 O I

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t,J N N M M M %Y V V V nt %T %T v '? sr 7 to to to m mto to to p In In In m to in in in tD N NNNNNNNNNNNNNNNNNNNNCJNC+1C%jN(%JNNNC.JNNN (NJ N 1

1 v

I

U

in

9

.0
20

10

0

20

10

30

20

0

w D: 10

F-' Q

30

w

20

10

20

10

0

30

20

10

36

MT. WILSON

}♦

t^ASTRO

♦

++

♦ NIMMIMq^^ ^M^iy111^M*^

SAN PEDRO

+*+*+* 4+*+*++

NIGUEL

#IqF *MMyL + ^++

SANTIAGO

,6 +

+

SAN .JUAN

♦

"Y +*+

+

*'♦ + + + #++

+ +4*.

*+

++

22 23 24 25

.JANUARY, 1981

Fig. Al. Observed atmospheric temperature.

• - - ^ _.^.^ sue•.
37

8 MT. WILSON

6 +

4 *M}-- +* +

2 ice# + %e *++

0 4HH"U. ♦^►

8 CASTRO

6

4

2 4W.

+

♦41*

4****mow♦++♦♦

6 '—

SAN PEDRO

4

2

D

+ 

++

+

+

+++ + +***+*

+

3 + + *i+*+$♦

1 NIGUEL

4M400+ * +

4(

SANTIAGO

MIF

tNF {msμ♦

SAN JUAN

NIF + +

UJC+1'ar^^iv

of POOR

6'd

65

s 2

E 71

w

o_

N

w

o: 4

73

f

4

2(

61f

2

2a

8

6

714

zz 23 24 25

JANUARY, 1981 

Fig. A2. Observed atmospheric ,pressure.
t!) 

::J:: 

~ 

~ 

'\ 

UJ 

0:: 

:::I 

(/) 

(/) 

UJ 

0:: 

0-

0:: 

0 

n. 

e( 

> 

c:::a 

UJ 

Ie( 

0:: 

:::I 

le( 

U) 

38 ----.----------------------J. f1 T. WI.LSON 

10 ~ 

5 t1 5 -

1 Ot5t1 5~ 

10 t51-

15 -

10 t51-

15 I10 t5tCAS'TRO 

I 

SAN PEDRO 

+ 

+ 

++++ 

I 

I 

SANTIAGO 

15 - SAN JUAN 

10 t5t-

, 

22 

+ + ,........ .Jf'+ ,+ .... + ~, •. ' ................ . 

I I I 

+ 

++ + "+o4t+. "'+ ~.+ .. 

I • I 

J I I 

t~ ++ + 

+ I ......... + ++ + !' ..... +++*+ I 

I I I 

23 24 25 

JANUARY I 1981 

Fig. A3 • Saturated vapor pressure. 

-

-

• 

. 

-. 

-

-

-

-

-

-

-

-

-

-

-

-