SEMI ANNUAL REPORT

TO THE NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

NASA CONTRACT NAG5-459

"Plate Motions and Deformations from Geologic and Geodetic Data"

for the period

1 August 1984 - 31 December 1985

"-. | HAS A-CR-17.72:99) PLATE MOTIONS AND N86-26341

DIFOBMATIONS EBCH GEOLOGIC AND GEODETIC DATA

Semiannual Beport, 1 Aug.; 198<T - 31 .Dec.

1985 (Massachusetts Inst. ,of Tech.), 53 p Unclas

HC A04/MF A01 . CSCL 08G G3/46 43593

Principal mvestigator: Professor Thomas H. Jordan

Department of Earth, Atmospheric, (i/vv/JoO^

and Planetary Sciences VVJ lu

Massachusetts Institute of Technology

Cambridge, MA 02139
TABLE OF CONTENTS

1. SIGNIFICANT ACCOMPLISHMENTS 3

2. PROBLEMS AND RECOMMENDATIONS 4

3. DATA UTILITY , 4

4. FUNDS EXPENDED 4

5. APPENDIX 1: Relative Movement of Tectonic Plates

in California Observed by Satellite Laser Ranging 5

6. APPENDIX 2: Constraints on Western U.S. Deformation

from Satellite Laser Ranging 13

7. APPENDIX 3: Reduction of Space Geodetic Vector

Position Data: Rate Scaling of Global Plate Tectonic

Models 22

V
SIGNIFICANT ACCOMPLISHMENTS

This Semiannual Report covers research conducted under NASA Contract NAG5-459 for the

period 1 August 1984 through December 31, 1985.

Our effort on behalf of the Crustal Dynamics Project during that period focused on the

development of methodologies suitable for the analysis of space-geodetic data sets for the

estimation of crustal motions, in conjunction with results derived from land-based geodetic data,

neo-tectonic studies, and other geophysical data. These methodologies were used to provide

estimates of both global plate motions and intraplate deformation in the western U.S. Significant

accomplishments include:

1. Results from the satellite ranging experiment for the rate of change of the baseline length

between San Diego and Quincy, California indicated that relative motion between the North

American and Pacific plates over the course of the observing period during 1972-1982 were

consistent with esimates calculated from geologic data averaged over the past few million years.

2. This result, when combined with other kinematic constraints on western U.S.

deformation derived from land-based geodesy, neo-tectonic studies, and other geophysical data,

places limits on the possible extension of the Basin and Range province, and implies significant

deformation is occurring west of the San Andreas fault

3. A new methodology was developed to analyze vector-position space-geodetic data to

provide estimates of relative vector motions of the observing sites. The algorithm is suitable for the

reduction of large, inhomogeneous data sets, and takes into account the full position covariances,

errors due to poorly resolved earth orientation parameters and vertical positions, and reduces baises

due to inhomogeneous sampling of the data.

4. This methodology was applied to the problem of estimating the rate-scaling parameter of a

global plate tectonic model using satellite laser ranging observations over a five-year interval. The

results indicate that the mean rate of global plate motions for that interval are consistent with those

averaged over several million years, and are not consistent with quiescent or greatly accelerated

plate motions.

5. This methodology was also used to provide constraints on deformation in the western

U.S. using very long baseline interferometry observations over a two-year period. Motions

-3-
relative to a fixed North American plate frame were consistent with global plate tectonic models at

Vandenburg, but were less at Monument Peak, Califonia, suggesting significant deformation west

of the San Andreas may be occurring between those sites.

PROBLEMS AND RECOMMENDATIONS

None

DATA UTILITY

Not applicable

FUNDS EXPENDED

As of 31 December 1985, a total of $72,720 had been spenX out of the current fund

limitation of $ 136,500.

-4-
APPENDIX I

RELATIVE MOVEMENT OF TECTONIC E»LATES

IN CALIFORNIA OBSERVED

ITE LASER RANGING

Abstract: A satellite laser ranging experiment conducted by NASA since 1972

has measured the relative motion between the North America and Pacific plates in

California. Based on these measurements, the 896-km distance between San Diego

and Quincy, California, is shortening at 62 + 9 mm/yr. This geodetic estimate is

consistent with the rate of motion between the two plates, calculated from geological data to be 53 + 3 mm/yr, averaged over the past few million years.

Lasers capable of tracking near-earth orbiting satellites have been providing

important geodetic data for more than a decade. Beginning in 1972 with the San

Andreas Fault Experiment (SAFE), short-term tectonic motions along a baseline

spanning the San Andreas system have been monitored by satellite laser ranging

(SLR) in an effort to improve our understanding of earthquake hazards in California

(Fig. 1). This experiment, which continues as part of NASA's Crustal Dynamics

Project, has measured the variations in the 896-km distance between two tracking

systems located on opposite sides of the fault with an accuracy unachievable by

classical geodetic techniques.

Previous reports (1, 2, 3) have shown the technique to be capable of monitoring short-term (""10 yr) tectonic movements at the centimeter-per-year level. In

this paper and the one which follows, we analyze the entire data set collected

over the 11-year period 1972-1982 and discuss some of its tectonic implications. In

particular, we shall show that space geodetic techniques are beginning to place

more stringent constraints on large-scale deformation of the western United States

than those derived from classical geological and geophysical observations (4).

Mobile laser systems were used throughout the tracking campaigns, campaigns.

Both sites were reoccupied approximately every two years for a duration of several

months in order to satisfy certain criteria of data acquisition and. earlier on,

tracking geometry. The first results were obtained by tracking a low orbiting

spacecraft. Beacon Explorer-C (BE-C), launched in 1965 and equipped with an array

of laser retro-reflectors to provide, among other objectives, a spaceborne target for

laser engineering studies. In 1976, a dedicated Laser Geodynamics Satellite

(LAGEOS) was launched into a high earth orbit with the sole purpose of providing,

^ti?'

an optimal target for global laser tracking in the context of geodynamic investiga-'_;

tions.

-5-
Relative Movement of Tectonic Plates from SLR

OTAY MT.

Pi«. 1: Map of California showing Quincy-Otay Mt. and Quincy-Monument

Pk. baselines measured by satellite laser ranging.

-6-
Relative Movement of Tectonic Plates from SLR

In satellite laser ranging, the observed quantities are the nearly instantaneous

ranges from a ground station to a spacecraft and the times at which these distances are measured. If the evolution of the orbit and the variations of the earth

rotational parameters are known accurately, then the positions of the observing

stations (to within an arbitrary shift in absolute longitude) can be calculated in the

geocentric reference frame used to describe the satellite motion. Analysis of

satellite laser ranging data requires careful consideration of: \

(1) the accuracy with which the motion of a near-earth satellite subjected to

complex gravitational and non-conservative forces can be modeled.

(2) the accuracy of the observations themselves, which limits the available

approaches, and

(3) the requirements of globally distributed tracking support.

BE-C, a low-altitude spacecraft with a complex shape, posed a difficult trajectory modeling problem. Orbital accuracy was severely degraded if more than a

few revolutions were analyzed simultaneously; that is, analysis had to be restricted

to relatively short arcs. A special data reduction technique was consequently devised to accommodate data collected from two tracking sites only. In spite of its

intrinsic limitations for measuring absolute baselines, this technique is capable of

detecting changes in the inter-station distance by using uniform tracking geometry

and identical force models for each bi-yearly analysis (2, 5).

v

The launch of LAG EOS at high altitude provided a well-designed laser target

far less sensitive to poorly known perturbing forces such as short wavelength

gravity and atmospheric drag. With the deployment of a worldwide network of

more advanced laser systems, trajectory accuracies achievable for LAGEOS

approach the quality of the laser tracking itself, leading to improved estimates of

the gravity field at long wavelengths and to highly accurate earth rotational

parameters (6, 7). Furthermore, the SAFE line has been incorporated as part of

the several hundred lines measured annually in the context of NASA's Crustal

Dynamics Project, so that a special data reduction technique is no longer required.

Unlike the baselines obtained from BE-C tracking, those derived from LAGEOS are

believed to be accurate in an absolute sense (they do. however, depend on the

values of GMg and the speed of light used in the reduction procedure).

LAGEOS observations are analyzed in various ways. We have derived robust

three-dimensional station positions by combining twelve monthly orbital arcs into

annual solutions. In this case, errors arising from weaker data sets and force

model imperfections are reduced through averaging over an entire year.

In addition to these annual solutions, we also computed selected subset solutions with finer temporal sampling, which extend the capabilities of SLR to resolve

relatively short (<1000 km) baselines, using vastly smaller data sets ( from a few

-7-
Relative Movement of Tectonic Plates from SLR

weeks or less). These rely on a short-arc technique, called Baseline Estimation

from Simultaneous Tracking (BEST) and involve simultaneous ranging from several

sites. Although absolute station positions may be contaminated by unaveraged data

and force model errors, these errors cancel in baseline calculations and therefore

do not bias the rate-of-change estimates (8).

The use of two different spacecraft results in an inhomogeous set of baseline

estimates, which complicates the retrieval of an average rate-of-change for the

entire time period covered by the data. Another source of inhomogeneity stems

from changes in the locations of ground stations during the experiment. The

southern site was changed from Otay Mountain near San Diego. California, to

Monument Peak. ~50 km away, in 1981. and the northern site, located near Quincy,

California, was moved by about 500 m in 1980. An adequate ground survey tying

the two locations with an accuracy comparable to the SLR measurements is available in the latter case, but not in the former. We are therefore dealing with a

data set properly described as the juxtaposition of four subsets, each internally

homogeneous but having unknown relative offsets in baseline length. These subsets

are listed in Table 1, and the individual data points are plotted in Pig. 2. For

clarity of graphical presentation, the baseline for each subset is arbitrarily offset

by 0.4 m.

We have analyzed the observations presented in Fig. 2 under the assumption

that the baseline rate-of-change remained constant, for the entire 11-year period

covered by the data. The time-dependent baseline length for the ith data subset

(i-1 4) is therefore given by a modeling equation of the form

B.(t) - B - (t-t0) + Bf, (1)

Table 1: SLR data sets used to derive the rate-of-change of the Quincy-San

Diego baseline.

Set

No.

1

2

3

4

Satellite

BE-C

LAGEOS

LAGEOS

LAGEOS

Baseline

Quincy-Otay Mt.

Quincy-Otay Mt.

Quincy-Mon. Pk.

Quincy-Mon. Pk.

Solution

Type

Special

Annual

Annual

BEST

No.

Pts.

4

3

2

14

r.m.s.

(mm)t

89

37

<1

43

Cumulative 1

Importance

0.26

0.58

0.06

0.10

t Misfit with respect to constant-rate solution of Fig. 2.

-8-
Relative Movement of Tectonic Plates from SLR

0.0

-0.4

-0.8

a

c

<o

-1.2

-1.6

Quincy - San Diego

1972 1974 1976 1978

Y«ar

1980 1982

Fig. 2: Variations in the Quincy-San Diego baseline length from SLR measurements between 1972 and 1982. Constant-rate solution was fitted simultaneously to the four data subsets listed in Table 1 and shown here with

arbitrary 0.4 m offsets. Error bars are standard deviations obtained from

individual baseline estimates.

where the intercept B? is the absolute baseline length at the fiducial time tQ, and

B is a constant rate-of-change.

Because of the offsets in the observed baseline lengths, the B?'a as well as B

must be considered a priori unknown; hence, the data generate a system of 23

equations in 5 unknowns. The four intercepts are only of ancillary interest, so it

is not necessary to solve for their values directly. We therefore applied • an

appropriate projection operator to the linear system to annihilate the dependence

;;•';».•'• •

of the data on these parameters and then solved for B by least-squares. The

weights used in the least-squares criterion were'ichosen to be proportional to the

-9-
Relative Movement of Tectonic Plates from SLR

standard errors depicted on Fig. 2. These formal errors of estimation are

appropriate as relative weights within each homogeneous data subset, but there

remains a subjective choice regarding the weighting of different subsets with

respect to each other. In view of the earlier discussion, we down-weighted the

BE-C data relative to the LAGEOS data by a factor of two in order to reflect the

greater reliability of the latter. This has the practical effect of nearly equalizing

the weights of the most reliable BE-C points and the least reliable LAGEOS points

in the inversion.

The least-squares formulation permits calculation of "data importances", additive quantities which represent the information contributed by each datum or data

subset to the solution (9). In the present case, since we are estimating the single

scalar quantity B, the data importances of the projected system sum to unity. The

cumulative data importances for the four data subsets are listed in Table 1.

The rate-of-change estimate obtained by the least-squares procedure is

B - -61.7 i 8.7 mm/yr, (2)

where the uncertainty is the standard error of estimation. The r.m.s. residual for

this solution is 52 mm. and the weighted r.m.s. residual is 29 mm. As seen on

Fig. 2. this solution satisfies all but five observations at the one-sigma level, and

all but one of the BEST estimates at the two-sigma level.

The misfit to the 1972 BE-C measurement is no cause for concern, since it

was the first collected for the SAFE baseline and is the most uncertain. Its individual importance is only 0.04; that is, it contributes but a small fraction of the

information used to estimate B. In fact, completely removing this first point from

the data set merely changes the answer to -59.9 ± 8.9 mm/yr, which is not

resolvable from the value quoted above.

The absolute baseline lengths retrieved from the BE-C data and the LAGEOS

data for Quincy-Otay (subsets 1 and 2 of Table 1) differ by 0.76 m. While such a

discrepancy was not unexpected in view of the differences in data reduction techniques. its size indicates BE-C force-model errors that were somewhat larger than

anticipated. However, our claim that the rate is not contaminated by the bias in

absolute baseline length is supported by the consistency of the baseline changes

between 1976 and 1979 for both spacecraft.

As for the poorly fitted BEST points, it must be noted that this data reduction technique, while minimizing the effects of dynamic errors on the baseline estimate. is quite susceptible to intermittent anomalies which occasionally occur -in the

laser data. Its reliance on a limited set of strictly simultaneous observations (in

reality, manufactured laser "normal points" at two minute intervals) allows little

averaging of the errors arising from anomalous data, causing an occasional spurious

-10-
Relative Movement of Tectonic Plates from SLR

baseline estimate. Fortunately, including the three outliers evident in Fig. 2 does

not affect the slope estimate derived from the BEST subset. This subset

comprises the majority of the data points but does not dominate the estimation of

B; in fact, owing to its short time span, its cumulative data importance is only

0.10.

By far the most important group of observations is the set of three annual

LAGEOS solutions for the Quincy-Otay baseline between 1976 and 1982 {subset 2);

its cumulative importance is nearly 0.6. These data cover approximately 5 years of

observations, so they provide a good constraint on the slope. Subset 2 alone

yields a rate of -53.6 + 11.4 mm/yr, again not significantly different from Eq. 2.

Tapley et al. (7) produced an independent set of eleven estimates for the

Quincy-Monument Pk. baseline from bimonthly arcs between October. 1981 and

November, 1983. For this two year period, they deduce a rate-of-change of -64 +

9 mm/yr. Their data reduction technique is different from ours, and they extend

the coverage by a full year beyond the BEST solutions shown on Fig. 2. It is

encouraging, therefore, that their value is in good agreement with Eq. 2.

Assuming the Quincy site lies on a rigid North America plate (NOAM) and the

Otay Mt./Monument Pk. sites on a rigid Pacific plate (PCFC), we can compute the

baseline rate-of-change using the PCFC-NOAM instantaneous angular velocity vector

derived from plate-tectonic data. The eleven-plate model RM2. which satisfies a

globally distributed set of marine magnetic rates, transform-fault directions and

slip-vector azimuths (10), yields B ~ -52.9 + 2.7 mm/yr, consistent at the onesigma level with the SLR estimate of Eq. 2. This removes the discrepancy noted

by Smith et al. (2), who derived a preliminary SLR rate of -90 + 30 mm/yr from

the 1972-1977 BE-C observations. As shown in Fig. 2, the BE-C and LAGEOS

points prior to 1979, taken alone, do indeed suggest a higher value, which could

indicate some temporal variation in the apparent rate, but the hypothesis that the

deviations from a constant-rate model are random statistical fluctuations cannot be

rejected at even a low confidence level (11).

Demosthenes C. CHRISTODOULIOIS

David E. SMITH

Geodynamics Branch. Goddard Space Flight Center,

Greenbelt, Maryland 20771

J. Bernard MINSTER

Science Horizons, Inc., 710 Encinitas Blvd.,

Encinitas, California 92024

Thomas H. JORDAN

Dept. of Earth. Atmospheric and Planetary Sciences.

-n-
Relative Movement of Tectonic Plates from SLR

Massachusetts Institute of Technology.

Cambridge, Massachusetts 02139

References and Notes

(1) D. E. Smith. R. Kolenkiewicz. G. H. Wyatt, P. J. Dunn. M. H. Totrence, Phil.

Trans. R. Soc. London. A 284. 529-536 (1977).

(2) D. E. Smith, R. Kolenkiewicz, P. J. Dunn. M. H. Torrence. Tectonophvsics 52.

59-67 (1979).

(3) D. E. Smith, NASA TM 80642. 3.40-3.42 (1980); D. C. Christodouiidis and D.

E. Smith, NASA TM 88123. 1.2-1.5 (1984).

(4) J. B. Minster and T. H. Jordan, in Tectonics and Sedimentation along the

California Margin. J. K. Crouch and S. B. Bachman, Eds. (Pacific Sect. Soc.

Econ. Paleontol. Mineral. 38, 1984), pp. 1-16.

(5) R. W. Agreen and D. E. Smith, J. Geophvs. Res. 79. 4413-4417 (1974).

(6) P. J. Lerch, S. M. Klosko, G. B. Patel, Geophvs. Res. Lett. 9, 1263-1266

(1982); W. E. Carter, D. S. Robertson, J. E. Pettey, B. D. Tapley. B. E.

. Schutz, R. J. Eanes, Science 224. 957-961 (1984).

(7) B. D. Tapley, B. E. Schutz, R. J. Eanes, J.. Geophys. Res, 89. in press (1984).

(8) D. C. Christodoulidis and D. E. Smith. NASA TM 82133 (1982); D. C. Christodouiidis, D. E. Smith. R. Kolenkiewicz. S. M. Klosko, M. H. Torrence. P. J.

Dunn, .L Geophys. Res. 89, in press (1984).

(9) J. B. Minster. T. H. Jordan, P. Molnar. E. Haines. Geophys. J^ R^ astr. Soc.

36, 541-576 (1974).

(10) J. B. Minster and T. H. Jordan. J. Geophys. Res. 83\ 5331-5354 (1978).

(11) In a Wald-Wolfowitz run test on the distribution of normalized residuals above

and below the median, the observed number of runs is 10; under the null

hypothesis that this sequence is random, the expected number is 12.0 with a

variance of 5.3.

(12) This research was supported by the National Aeronautics and Space Administration, under NASA's Crustal Dynamics Project. We are grateful to P. J.

Dunn. S. M. Klosko, R. Kolenkiewicz and M. H. Torrence for their invaluable

assistance in the completion of this work.

-12-
Appendix 2

CONSTRAINTS ON WESTERN U.S. DEFORMATION

FROM SATEX-T-ITE LASER RANGING

Abstract: The rate of shortening between San Diego and Quincy, California,

measured by satellite laser ranging (SLR) is substantially greater than that implied

by the rate of slip measured on the San Andreas fault. When combined with other

kinematical constraints on western U.S. deformation, these comparisons limit the

extension rate across the Great Basin to be less than 9 mm/yr and imply significant deformation of the Pacific plate west of the San Andreas. Along the central

California margin, the integrated rate of deformation is estimated to be 18 + 5

mm/yr and entails compression perpendicular to the San Andreas, as well as rightlateral strike-slip motion parallel to it. Within the framework of our tectonic

model, the SLR data limit the integrated deformation rate across the Southern

California Borderlands to be less than 7 mm/yr.

The rate-of-change of the Quincy-San Diego baseline (Fig. 1) measured in

NASA's San Andreas Fault Experiment (SAFE) since 1972, and more recently by

NASA's Crustal Dynamics Project is (1)

B - -61.7 + 8.7 mm/yr. (1)

The value calculated from the plate-tectonic model RM2 (2) is -52.9 ± 2.7 mm/yr,

which assumes Quincy lies on a cigid North America plate (NOAM) and San Diego

on a rigid Pacific plate (PCFC). The geodetic estimate of the SAFE baseline

shortening rate observed over an eleven-year period is thus compatible with plate

motions averaged over a much longer interval (1-3 million years in the case of

RM2). However, neither is easily reconciled with the hypothesis that PCFC-NOAM

relative motion is localized on the San Andreas fault. . We consider two fiducial

points juxtaposed on either side of the fault zone, rp just to the west of the San

Andreas and r£ just to the east, and let Vg^ be the (steady-state) velocity vector

of the former with respect to the latter. In central California, where the fault

best approximates a simple transform, a range of geological and geodetic observations local to the fault zone provides the estimate (3, 4)

r 34 + 3 mm/yr

vs. - < ~ (2).

I N41"W + 2*

The RM2 PCFC-NOAM velocity vectorrcalculated at the fiducial location

rc - 36TJ. 120.6'W (Fig. 1) is

 :;

- V*T

 : ' - -v '* -%

-13-
0.3.

ORIGINAL PAGE-IS

OF POOR QUALfTY

NORTH AMERICAN PLATE

PACIFIC PLATE

Pic. 1: Oblique Mercator projection of the western United States about the

RM2 PCPC-NOAM pole of rotation, stawinc major Quaternary faults and

reference points used in oar vector calculations. C is the fiducial point on

the San Andrea* fault where the vectors in Bqs. 2-6 are specified. AB is

the line along which the expansion rate of the Great Basin is defined.

Triangles are SLR stations at Qtrincy (Q) and San Die«o (S). California.

Faults are from (5) and (6); lettered features are the Gariock fault (G),

Hosari fault (H), San Andreas fault (SA). San Grecorio fault (SG), Sierra

Nevada front <8N). Wasatch front (WF). and Walker Lane (WL).

-14-
Western U.S. Deformation from SLR

r 55.9 i 2.7 mm/yr

vpN - < " (3)

I N35.5-W * 1.9'

The discrepancy between Eqs. 2 and 3 is significant at the two-sigma level in both

rate and azimuth; it is thought to be a measure of the present-day tectonic deformation integrated across a broad zone of PCFC-NOAM interaction that extends from

the continental margin of California to the Rocky Mountains (2. 5).

 v

How the deformation represented by this discrepancy is partitioned east and

west of the San Andreas is a question of considerable geological interest and

seismogenic implications. If VCN is the velocity of the fiducial point r£ in the

reference frame rotating with NOAM and vcp is the velocity of r£ in the PCFC

frame, then the discrepancy vector Vp« - VQ\, by definition, must equal

VQJJ - VQp: the deformation distributed over a plate boundary zone must sum up to

the relative plate velocity. VPM thus summarizes the deformation occurring on the

NOAM side of the San Andreas, whereas VQD summarizes the deformation on the

PCFC side. We have formulated these vectors in terms of frame-independent

integrals of the velocity gradient tensor along paths across the zone of deformation

(3).

The distribution of Quaternary faulting shown in Fig. 1, as well as the

observed seismicity, defines an essentially rigid Sierra Nevada-Great Valley block

(SNGV) extending from the San Andreas eastward to the Sierra Nevada front (5).

The relative motion between this block and NOAM is taken up primarily by deformation in the Great Basin of Nevada and western Utah. Stress and strain orientation data pertaining to Quaternary deformation of the Great Basin constrain the

direction of this motion to be approximately N6CTW (7). These constraints have

been used to construct a one-parameter family of kinematical models parameterized

by the total rate of expansion across the Great Basin from south-central Utah to

northern California (along the line AB in Fig. 11(3). We define a normalized velocity vector Vg» by the equation

VCN{«) - « VCN, (4)

where a is a dimensionless parameter such that ex * 10 mm/yr is the opening rate

for this path. In other words, v,-.« is the unique velocity which would describe

the local SNGV-NOAM motion at rc if the Great Basin were expanding at a total

rate of exactly 10 mm/yr along AB. As documented in (3), the data compiled by

Zoback and Zoback (7) yield the conditional estimate

-15-
<5

f

Western U.S. Deformation from SLR

r 10.1 + 0.7 mm/yr

< (5)

I N63*W + 5*

The estimation procedure, which is based on a numerical search over possible

SNGV-NOAM pole positions, explicitly accounts for sphericity of the earth's surface (on a flat earth, the modulus of VQ« would be exactly 10 mm/yr).

The locus of v^ parameterized by or is compared with the discrepancy vector

Y

PN * TSA "* ^°e upper vector diagram of Fig. 2. Clearly, the opening of the

Basin and Range is not in the right direction to account entirely for the

discrepancy, regardless of the value of <x. This implies that some deformation

must be occurring west of the San Andreas fault.

Given the definitions embodied in Eq. 4 and the constraints imposed by the

plate-tectonic boundary conditions, the vector summarizing deformation west of the

San Andreas can be recast in the form

vcp(cc) - oc VCN • (vpN - VSA), (6)

and an estimate of v~p conditional on ex can be derived from Eqs. 2, 3 and 5.

This expression and the corresponding equation for the variance matrix of v^pfoc)

yield the lower vector diagram in Fig. 2. The locus of V£p(cx) is a straight line in

the direction of VQJJ«

 an<^ ^ two-sigma error ellipse sweeps a band approximately

12 mm/yr wide, broadening slightly with increasing oc. For values of oc near zero,

Vpp(oc) is subparallel to the San Andreas and could be accommodated by predominantly right-lateral faulting localized along, say, the San Gregorio-Hosgri fault system. which parallels the California coastline (Fig. 2). On the other hand, if the

Great Basin extension rate were as great as 20 mm/yr (ex - 2). as some students

of Basin-and-Range deformation have advocated (9), then v/-.p would be essentially

perpendicular to the San Andreas and would imply compression across the California

margin with very little strike-slip motion. Interpretations of recently published

seismic reflection profiles traversing the southern part of the Hosgri fault favor

this model (10).

The SLR observations discussed in (1) provide an upper bound on a that

excludes this latter possibility. Assuming the Quincy site lies on SNGV and the

San Diego sites (Otay Mt. and Monument Pk.) on PCFC. we can derive an estimate

of B conditional on a by the same procedure used to obtain Eq. 5. The result is

B - -(52.9 - 7.9oc) ± (7.29 + 0.49oc2)* mm/yr. (7)

-16-
U.S, Deformation from SLR

ORIGBSM. PAGE-IS

OF POOR QUALITY

36°N

34°N

122°W 120°W

2: Ifnar Lambert cooformai conic projection of central California showing major zones of Quaternary faulting- (8). Lettered fault zones are the

Big Pine (BP). Calaveras (C). Hosgri (H). Nachniento (N), Pilarcitos (P). Rinconada (R). San Andreas (SA). San Gregorio (SG). Santa Lucia Bank (SLB).

and Santa Yhez (SY). C is the fiducial point at 36*N on the San Andreas

used in vector calculations. Usvsr iaswt: Velocity diagram in the tangent

plane at C showing discrepancy vector TpN • TgA and vector VQ^OC)

describing Bastn-and-Range deformation; <x - 1 corresponds to an opening

rate of exactly 10 mm/yr across the Great Basin along line AB in Pig. 1'

Error ellipses are two-sigma. Lower sMet: Velocity diagram at C showing

TQp, which describes deformation west of the San Andreas. Dashed line is

locus parameterized by OL Error ellipses are two-sigma.

-17-
Western U.S. Deformation from SLR

As oc increases from zero, the total expansion of the Great Basin accelerates, and

the agreement between Eqs. 1 and 7 degrades. If oc were as great as two, for

example, then the value of the SAFE baseline rate-of-change would be -37.1 ± 3.1

mm/yr. which is not consistent with the SLR results.

A formal upper bound on oc can be obtained from Eqs. 1 and 7 by applying a

one-sided chi-square test: at the 95% confidence level, we find oc < 0.80. If

instead of Eq. 2 we use the estimate of -64 + 9 mm/yr given by Tapley et ah.

(11), this bound is reduced to 0.56. The chi-square test assumes the standard

deviations assigned a priori to the SLR data are exact; scaling these uncertainties

to the root-mean-square misfit provided by the constant-slope solution reduces the

standard error of B in Eq. 1 to 8.3 mm/yr, and applying the appropriate one-sided

P test gives oc < 0.89.

The SLR measurements thus limit the extension rate of the Great Basin to be

less than 9 mm/yr. This is consistent with, but more stringent than, the upper

bounds obtained from direct geological and geophysical observations of Basin-andRange deformation (12). Moreover, through Eq. -1, the SLR data constrain the

vector V£p to have a significant projection parallel to the San Andreas; the lower

bound on this component of motion is 5 mm/yr, and it is likely to be considerably

larger (Fig. 2). In fact, our results are consistent with strike-slip rates of 6-17

mm/yr inferred from post-Pliocene offsets along the San Gregorio-Hosgri fault system (13). Hence, a model with purely compressive deformation along the California

margin is excluded. We note, however, that ex is constrained by observations in

the Great Basin to be greater than 0.1 (12). requiring the component of v^p perpendicular to the San Andreas to be greater than zero, as implied by geological

observations of regional compression west of the fault (10).

We infer that the rate of deformation west of the San Andreas is large. For

a - 0.9, our best estimate of v^p has a magnitude of 16 mm/yr, and it is

increases to 21 mm/yr for oc - 0.1. These estimates are one measure of the

seismogenic potential of the central California margin.

We have assumed throughout this analysis that the San Diego sites are rigidly

attached to PCFC. Any geologically plausible deformation west of these sites --

accommodated, say, on offshore faults within the seismicaily active California

Borderlands— would imply a decrease in the magnitude of B relative to Eq. 7 and.

hence, would lower our upper bound on oc. For example, the block-tectonic model

of southern California developed by Bird and Rosenstock (14) predicts that the San

Diego sites move with respect to PCFC at ~"3 mm/yr in the direction ~N40"W;

• such a motion would lower our bound on a from 0.9 to about 0.6. Conversely, a

lower bound on oc can be used to place an upper bound on the rate at which the

San Diego sites translate with respect to PCFC. Assuming this motion is N40*W.

.

and adopting oc > 0.1. we find that this rate must be less than 7 mm/yr to be

-is-
Western U.S. Deformation from SLR

consistent with the SLR data. The integrated rate of deformation across the

Southern California Borderlands is therefore inferred to b<* less than that across

the central California margin. Presumably, the difference between these deformation vectors is taken up by the northward compression of the western Tranverse

Ranges (14).

The preliminary results in this report point the way to future applications of

SLR and other space-geodetic techniques in the study of regional deformation. In

particular, they illustrate the advantages of combining SLR measurements with

other geological, geophysical and geodetic data sets. When considered in the context of regional kinematical models constructed from such data sets, the SLR

observations from a single California baseline yield quantitative constraints on the

distribution of deformation in the western United States. Much more stringent

constraints can be expected from the large set of North America baselines

currently being measured under the auspices of NASA's Crustal Dynamics Project

(15). Space-geodetic observations should be especially useful in fixing the rates of

regional deformation that are difficult to measure with conventional techniques but

are critical to the assessment of seismogenic potential. . /

Thomas H. JORDAN

Dept. of Earth. Atmospheric and Planetary Sciences.

Massachusetts Institute of Technology,

Cambridge. Massachusetts 02139

J. Bernard MINSTER

Science Horizons. Inc.. 710 Encinitas Blvd..

Encinitas. California 92024

Demosthenes C. CHRISTODOUUDIS

David E. SMITH

Geodynamics Branch. Goddard Space Plight Center,

Greenbelt. Maryland 20771

References and Notes

(1) D. C. Christodoulidis, D. E. Smith, J. B. Minster. T. H. Jordan. Science, this

issue (1984).

(2) J. B. Minster and T. H. Jordan. J.. GeopKys. Res. 83, 5331-5354 (1978).

(3) J. B. Minster and T. H. Jordan, in Tectonics and Sedimentation along the

California Margin. J. K. Crouch and S. B. Bachman, Eds. (Pacific Sect. Soc.

Econ. Paleontoi. Mineral. 38. 1984). pp. 1-16.

-19-
Western U.S. Deformation from SLR

(4) K. E. Sieh and R. H. Johns, Geol. Soc. Amer. Bull. 95, 883-896 (1984).

(5) D. P. Hffl. ,L Geophvs. Res. 87, 5433-5450 (1982).

(6) L. Wright, Geology 4, 489-494 {1976).

(7) M. L. Zoback and M. D. Zoback, J. Geophvs. Res. 85, 275-284 (1980); M. L.

Zoback and M. D. Zoback. -L. Geophvs. Res. 85, 6113-6156 (1980).

(8) C. W. Jennings. Calif. Geol. Data Map Ser.. map 1. Calif. Div. of Mines and

Geol.. Sacramento (1975); San Gregorio-Hosgri Fault Zone, California. B. A.

Silver and W. R. Normark, Eds. ( Calif. Div. of Mines and Geol. Spec. Rep.

137, 1978), pp. 1-56.

(9) Thompson and Burke (16) showed that Dixie Valley has widened in a N55*W

direction at an average rate of **1 mm/yr over the past 12.000 years.

Assuming uniform strain and extrapolating this result to the entire province,

they infer a total Basin-and-Range expansion rate of "20 mm/yr. Integration

of the strain rate estimates of Greensfelder et aL (17) along line AB of Fig.

1 yields a total rate in excess of 40 mm/yr. although their tabulated seismic

moment rates for historic seismicity are consistent with 5-22 mm/yr (4).

(10) Crouch et al. (18) argued that offshore seismic reflection profiles near 35"N

indicate a dominance of reverse faulting, and that the parallelism of fold axes

- and fault strikes is inconsistent with large scale right-lateral motion.

Compression west of the San Andreas is also supported by geodetic and geological evidence (19), and by the distribution of compressive axes for earthquake fault-plane solutions (20).

(11) B. D. Tapley. B. E. Schutz, R. J. Eanes, J., Geophys. Res. 89, in press (1984).

(12) Late Cenozoic (10-17 Myr) estimates of the Great Basin expansion rate range

from 3 to 20 mm/yr (3, 21). Rates more appropriate for the study of Quaternary deformation can be derived from Holocene paleoseismic mapping; Wallace's

(22) data yield 1-12 mm/yr along the line AB of Fig. 1 (3). The lower bound,

which corresponds to or > 0.1. is set by Thompson and Burke's (16) rate for

Dixie Valley.

(13) About 100 km of right-lateral displacement on the San Gregorio-Hosgri fault

system has been suggested (23). By modeling the Santa Maria Basin as a

pull-apart structure. Hall (24) obtained a "generalized estimate" of 30-50 km

over the past 3 Myr. for an average rate of 10-17 mm/yr across the Hosgri

fault. Based on offsets of marine terraces over the last 200,000 years at Pt.

Ano Nuevo. California. Weber and Lajoie (25) argued for a rate of 6-13 mm/yr

across the San Gregprio fault. . . .

(14) P. Bird and R. W. Rosenstock. Geol. Soc. Amer. Bull. 95. 946-957 (1984).

•i"

t
Western U.S. Deformation from SLR

(15) NASA Technical Paper 1464. Application of Space Technology to Crustal

Dynamics and Earthquake Research (1979); R. J. Coates. H. Prey, G. D. Mead.

J. Bosworth, The NASA Crustal Dynamics Project, preprint (1984).

(16) G. A. Thompson and D. B. Burke. Geol. Soc. Amer. Bull. 84, 627-632 (1973).

(17) R. W. Greensfeider, F. C. Kintzner, M. R. Somerville, Geol. Soc. Amer. BulL

91, 518-523 (1980).

(18) J. K. Crouch. S. B. Bachman, J. T. Shay, in Tectonics and Sedimentation

alone the California Margin. J. K. Crouch and S. B. Bachman. Eds. (Pacific

Sect. Soc. Econ. Paleontol. Mineral. 38, 1984), pp. 17-34.

(19) R. 0. Burford. in Proc. 2nd Internat. Symposium on Recent Crnstal Movements

(Aulanko, Finland, 1965), pp. 100-110; R. 0. Burford, Strain Analysts Across the

San Andreas Fault and Coast Ranges of California. Ph.D. Thesis, Stanford

University. Stanford, California (1967).

(20) W. J. Gawthrop, in San Gregorio-Hosgri Fault Zone. California. E. A. Silver

and W. R. Normark Eds. (Calif. Div. of Mines and Geol. Spec. Rep. 137,

1978), pp. 45-56.

(21) J. H. Stewart, in Cenozoic Tectonics and Regional Geophysics of the Western

Cordillera. R. B. Smith and G. P. Eaton Eds., (Geol. Soc. Amer. Mem. 152,

* 1978), pp. 1-13; also G. A. Davis, in Continental Tectonics. (Nat. Acad. Sci.,

Washington. D. C., 1980). pp. 84-95.

(22) R. E. Wallace, IL S.. Geol. Sarv. Open-File Rep.. 78-94 (1978); also J.. Geophvs.

Rea. 89, in press (1984).

(23) C. A. Hall. Jr.. Science 190. 1291-1294 (1975); S. A. Graham and W. R. Dickinson. Science 199. 179-181 (1978).

(24) C. A. Hall. Jr.. J^ Geophvs. Res. 86. 1015-1031 (1981).

(25) G. E. Weber and K. R. Lajoie, Geol. Soc. Amer. Abstr. with Prog. 9. 524

(1977).

(26) This research was supported by the National Aeronautics and Space Administration, under NASA's Crustal Dynamics Project.

-21-
APPENDIX 3:

Reduction of Space-Geodetic Vector Position Data:

Rate Scaling of Global Plate Tectonic Models

by

Mark H. Murray

under the superivison of

Thomas H. Jordan

December 1985

-22-
Abstract

Satellite laser ranging observations to Lageos during 1978-1983 are used to

analyze the vector rate-of-change of annual station positions for comparison with

global plate tectonic models. We develop an algorithm suitable for the reduction of

large, inhomogeneous space-geodetic data sets, which accounts for the full position

covariances, and coordinate transformations between epochs. Tectonic motions are

isolated by reducing systematic errors in radial position determination and

contaminations due to errors in earth-orientation parameters. This is accomplished

by forming a linear system of equations which describe differenced station position

vectors as a combination of tectonic motion model parameters, and nuisance

parameters which comprise non-tectonic motions such as rigid-body rotations. The

system is normalized with the square-root of the covariance matrix, and then

rendered insensitive to the nuisance parameters by the application of projection

operators. Model parameters are estimated from the resulting reduced system by

generalized-inverse techniques. This methodology is applied to the problem of rate

scaling of global plate tectonic models. The model is parameterized by 7m 0, where mQ

is an a priori model of tangential relative velocity vectors, such as RM2; thus, y = 1

implies that the mean rate of motion of the observing stations is identically equal

to the RM2-predicted rates. We find 7 = 0.95 ± 0.16, indicating that the mean rate of

SLR-derived motions for a five-year interval are consistent with RM2, which is

averaged over a 2-m.y. interval. We find that the data do not place significant

constraints on possible errors in marine magnetic anomaly timescales, but are not

»

consistent with an interval of quiescent or accelerated plate motions.

-23-
Introduction

Satellite laser ranging (SLR) and very-long baseline interferometry (VLBI) are

space-geodetic techniques currently capable of determining relative locations of

widely separated positions on the earth's surface to within sub-decimeter accuracy

(Shapiro, 1983). Results from interferometry with the Global Positioning System

(GPS) are achieving similar accuracies (Bock et al., 1985). This capability raises the

possibility of detecting and monitoring relative plate motions and deformations.

Analysis of the rate of change of baseline length between many station positions are

in good agreement with predictions based on global plate tectonic models (e.g. Smith

et al., 1979; Christodoulidis et al., 1985; Tapley et al., 1985; Clark et al., 1985), although

some significant discrepancies do exist (e.g. Minster and Jordan, 1984). These

treatments of space-geodetic data concentrate on the scalar baseline lengths, which

are relatively insensitive to systematic errors due to coordinate transformations

between epochs. However, the data are intrinsically three-dimensional. Thus, if the

errors can be removed or reduced in a systematic manner, an analysis of the rate of

change of the geocentric position vecto r could provide additional insight, especially

with regard to directions of motion. As these dat a sets grow, the need for efficient

algorithms capabl e of handlin g large, inhomogeneous, multiple-epoc h data sets

becomes increasingly critical. In this paper, we presen t the nucleus of such an

algorithm, and then, using data derived from laser rangin g to the Lageos satellite, we

apply this algorithm to the problem of rate scalin g of global plate tectonic models.

We first examine the natur e of the satellit e lase r rangin g data and some of its

inherent difficulties. In May 1976, Lageos (Lase r Geodynamic s Satellite) was

launched into a nearly circula r orbit at an elevation of 5900 km and an inclination of

109.8° to the equatorial plane. The satellit e is a sphere 60 cm in diameter, weighing

-24-
411 kg, and is covered with 426 retroreflectors. The satellite's design and orbit

minimize the fore*"? due to solar radiation pressure, earth albedo, and atmospheric

drag, and thus, Lageos very nearly defines an artifical reference frame in earth

orbit. Ground-based laser systems measure the round-trip pulse propagation time

and, using an adopted value for the speed of light, calculate the range. In theory, if

the satellite's position is known perfectly, the location of the laser site may be

determined after a number of ranges have been obtained. In practice, of course, the

problem is considerably more difficult. Since ranges to Lageos, in general, are not

made simultaneously, the orientation of the earth with respect to the satellite

reference frame must be determined for the time of the observation. This requires

knowledge of a complex set of motions which include earth precession, nutation,

polar motion, and rotation, as measured by UT1. The satellite's position is controlled

by the external forces which act upon it; these include the earth's gravity field,

luni-solar and planetary gravity fields, solid-earth and oceanic tides, solar radiation,

and earth albedo. Range measurements are affected by refraction of the laser pulse

in the atmosphere, tidal displacements, pulse detection accuracy, and are scaled by

the adopted value for the speed of light. Thus, the accuracy of the station positions

depends critically upon the accuracy of the earth-orientation, force, and

measurement models. One such set of models and the procedure for reducing raw

range values to station position coordinates used by Goddard Space Flight Center is

called SL5.1AP, and is described in Christodoulidis et al. (1985), and Smith et al. (1985).

The data that we utilize is derived from an improved set of models, denoted by SL6, and

are solutions for annual geocentric vector positions and covariances derived from

observations of Lageos between 1978 and 1983.

-25-
Methodology

We now consider the problem of constraining plate tectonic motions and

deformations from space-geodetic networks. Ideally, the analysis procedures should

be capable of handling data derived from disparate techniques such as SLR, VLBI, and

GPS simultaneously, taking into account the coordinate transformations and scaling

parameters particular to each. For example, the natural coordinate system for SLR

has an origin at the earth center of mass, whereas VLBI uses an origin at the solar

system barycenter; also, SLR, unlike VLBI, is sensitive to the adopted value of the

earth gravitational mass. Any estimation of parameters by inversion techniques

should account for possible errors in the data, therefore the analysis must

incorporate the full covariance matrix corresponding to the data. Another potential

problem arises from non-uniform sampling of network stations from epoch to epoch.

Usage of homogeneous data sets, i.e. including only stations that are sampled during

every epoch, is often too restrictive and ignores important sources of information.

However, inhomogeneous sampling may introduce biases in the estimation of station

parameters due to changing network geometry, which result in unequal weightin g

between successive epochs. For example, one epoch may preferentially sample "fast "

plates. The approach we have taken is similar to that used by Jordan and Sverdrup

(1981) for the estimation of earthquake location, whereby we form a linear system of

equations which describe differenced station position vectors as a combination of

tectonic motion model parameters and nuisanc e parameters, whic h compris e

non-tectonic motions such as rigid-body rotations. The system is normalized wit h

»

the square-root of the covariance matrix, and then rendered insensitive to the

nuisance parameters via application of projection operators, a process which we term

"denuisancing" .

-26-
Let us consider a simplified algorithm tailored to the specific problems presented

by SLR observations. First, if all the data is processed in a consistent manner, such as

with the SL6 model, scaling of the network due to the adopted value for the speed of

light will be uniform and hence negligible. Similarity, the earth center of mass,

which is an implicit function of the assumed gravity model, is constant for all epochs.

Thus, representation of scaling or transformation of coordinate systems may be

safely ignored. Errors in earth-orientation parameters, such as polar motion and

UT1, will result in rigid-body rotation of the entire network about the earth center of

mass. However, absolute plate motions, which are relative plate motions defined in a

particular angular velocity reference frame, will tradeoff exactly with rigid-body

rotations. Therefore, we may only resolve relative plate motions from the data and

will treat rigid-body rotations as nuisance parameters. Some provision should be

made for differences between errors in site motion tangential and radial to the

surface of the earth, which arise from inhomogeneous sampling of ranges. For

example, lasers may range at all horizontal azimuths, but may not range at vertical

angles below the horizon; thus, systematic errors in satellite location and in the

atmospheric refraction model are expected to map primarily into the radial

components of position. Therefore, radial components may also need to be treated as

nuisance parameters. Finally , tectonic motions may bias the denuisancing of

rotations, since incomplete sampling of the plate system might introduce rotations of

tectonic origin. This subtle problem may be minimized if prior information about

the motions is known; the a priori tectonic model is subtracted from the data before

denuisancing, and then is added to the resulting model after the inversion has been

completed. This process, which we term "bias stripping" , causes the component of the

solution which is not constrained by the data to coincide with the a priori model, thus

care must be exercised in the choice of the a priori model.

With these considerations in mind, we may formulate a specific approach. We

-27-
define x to be a vector composed of the three-component geocentric vector station

positions for all stations sampled during all epochs under study. These positions will

A

*T*

have associated covariances defined by Vx = <(x-<x>)'(x-<x>) >, where the brackets

A

denote expected value, and the T transpose. x and V x are the quantities supplied by

the SL6 solution. This covariance matrix is not an accurate reflection of the true

errors associated with the station coordinates due to systematic errors in the normal

point ranges used to derive the SL6 solution. Normal points are formed by grouping

raw ranges to Lageos from a laser site into 2 minute bins, and then finding average

ranges from those samples; this data compression, which can typically be two orders

of magnitude, is useful computationally and reduces biases due to uneven sampling of

the orbit of Lageos. The precision of ranges is on the order of 2-9 cm, and thus the

formal precision of the normal points is generally on the order of 1 cm or less.

However, due to systematic orbit model errors, which propagate as the square of time,

the actual accuracy of the normal points is more on the order of 10-15 cm over a 30

day orbital arc. To reconcile this problem, SL6 uniformly assigns each normal point a

formal uncertainty of 100 cm, which means that the stated covariances are probably

an overestimation of the errors. A more realistic estimate of covariance might be

A

V = v

2

 V , where v

2

 is a scalar to be determined a posteriori by the

sampling variance, which is a function of the weighted error vector and the numbe r

of degress of freedom.

We define

d = E-x (la )

Vd = E-V x - E T (Ib )

where E is the operator composed of I's, O's, and -1's which difference vector

positions between successive epochs for all sampled stations. d therefore represents

-28-
the apparent vector motions of the stations from epoch to epoch. Based on the

previous discussion, we consider a decomposition of these motions into rigid-body

rotations, tectonic motions, such as rigid plate motions and internal plate

deformations, radial motions and errors. We construct a linearized set of equations of

the form:

A- m + B- n = d (2)

where m is a vector composed of model parameters corresponding to tectonic

motions, and n is a vector composed of nuisance parameters. The particular

representation of these vectors will depend on the desired parameterization. For

example, if rigid-body rotations are to be denuisanced from the system, n will be

composed of the three-component angular velocity vectors corresponding to the

rotations between successive epochs, and B will be composed of the 3x3

anti-symmetric infinitesimal rotation matrices times the epoch-time differences for

each station sampled. m may be the denuisanced vectors of motion, in which case A

will be the identity matrix; or, as discussed in the following section, m may be a

rate-scaling parameter and A the vectors corresponding to an a priori global plate

tectonic model. Explicit representations of these matrices will be given for the

specific application of rate scaling of global tectonic models; however, we wish to

emphasize that this is a completely genera l formulism capable of handling many

possible parameterizations, provided the system is linea r or can be linearized.

We normalize the system by weighting wit h the square-root of the covariance

matri x

A " f> A

A- m + B- n = d (3)

w

A A

her e A = V,," 172 ' A, etc.; note that the carat now indicates normalization by

d

-29-
A

 1/2 Vd . The system is denuisanced by applying

A A

QB = I- B'B' , (4)

where the dagger, denotes the Moore-Penrose generalized inverse (Penrose, 1955). Q B

A T

is the projection operator which projects (3) into the null space of B ' , thus

rendering the system insensitive to the nuisance parameters; it is both symmetric

and idempotent (QB = QB

T

 = QB' QB ). Multiplying both sides of (3) by (4), and making

A

use of the fact that Q B * B = 0 , we find

Q B -A' m = QB - d . (5)

The model parameters are estimated by talcing the generalized inverse of the left

hand side of the reduced system (5)

m = (QB- A)f- QB 'd, (6)

with the associated unsealed covariance matrix

m

= ( A T- Q B 'A) f. (7b )

We now estimate the covariance scaling paramete r v - by comparing the misfi t

between the model and the data; this is defined by the normalized error vector, which

is a random variable found by substituting (6) int o (5):

e = QB' d - Q B -A- m (8a )

= (I - (Q B 'A)-(Q B - A) f)-Q B - d. (8b )

-30-
The squared length of the normalized error vector divided by the covariance scaling

parameter is itself a random variable and is chi-squared distributed with N-M degrees

A

of freedom, where N is the rank of the reduced data space Q B • d, and M is the rank of

A A .

the projection operator (QB 'A)'(QB * A)' (e.g., Jordan and Sverdrup, 1981). Thus, it

has the expected value

A A

< eT> e> / v

2

 = N - M. (9)

The observed misfit is one realization of this error process, and can thus be used to

f\

estimate v . This estimate is called the normalized sampling variance:

A

?

 AT A

s

2

 = e1' e /(N-M). (10)

Application of bias stripping is a simple extension of the method outlined above.

Let m Q be the a priori model which is assumed to be known perfectly; that is, let m 0

have zero variance (m = < m0>). The a priori model is removed from the linear

system (2):

A' (m-m0 ) + B- n = d-A-m 0 . (11)

The equations are normalized with the square-root of the covariance matrix in (1),

since m 0 has zero variance , and applyin g the appropriat e denuisancin g and

generalized-inverse techniques, we find the estimate of the model parameters

\

A A A

m = nu +(Qn'A ) Qo-( d - A-m n ) O2 a

0 )

'

t

(QB" AV-QB " d + (I - (Q B 'A)-(Q- B - A))-m0 . (12b )

-31-
It is straightforward to show that the unsealed covariance matrix, the normalized

error vector, and the sampling variance are given by (7), (8), and (10) respectively;

this follows from the fact that m 0 has zero variance. The first term in (12b) is

equivalent to (6), the result obtained with bias stripping. The second term projects

m 0 into the model space which is unconstrained by the data. Thus, the component of

the model which is not constrained by the data is made to coincide with the a priori

model.

We have described a very general approach for the estimation of model

parameters and variance scaling in the presence of nuisance parameters. We now

apply the methodogy outlined above to the specific problem of the rate scaling of

global plate tectonic models.

Rate Scaling of Global Plate Tectonic Models

As an introduction to this problem, we briefly describe one successful model of

plate tectonic motions obtained by Minster and Jordan (1978), hereafter denoted by

RM2. RM2 is a relative motion model based on the assumption that the earth's surface

is composed of eleven plates which are rigid spherical caps moving tangentially to

the surface. Directions of motion are estimated from azimuths of transform faults and

earthquake slip vectors, and rates of motion are estimate d from marine magneti c

anomalies. No data are used from locations where the assumptions do not appear to be

valid, such as regions of deformation at continent-continent boundaries. Magnetic

anomalies 2, 2', and, occasionally, 3 are employed, thu s the obtained rates are

averaged over a 2-m.y. interval. One-sigma errors for the rates are on the order of 5

to 10%.

As an application of the algorithm decribed in the previous section, we estimate

-32-
from the SL6 solution the RM2 rate-scaling parameter y. That is, we assume that the

orientation of the relative angular velocity vector for each pair of plates is known

perfectly and that the rate of rotation is of the form yco, where CD is the RM2 angular

rate of rotation. Thus, the relative magnitude of the RM2 rates are assumed to be

known perfectly, but they may be scaled according to y. For example, y = 1 implies that

the relative velocities of the observing stations are identically equal to the

RM2-predicted values, whereas y < 1 implies that the observed mean rates are slower

than RM2. Thus, we seek to characterize the data by a single model parameter. Such

a characterization maximizes the resolution of the model parameter at the expense of

greater variance. As such, it represents a logical first assessment of the data, and will

help to ascertain the appropriateness of RM2 as an a priori bias-stripping model. The

rate-scaling parameter y is a useful description of mean global plate motion, and

facilitates comparison of mean rate estimates derived from geophysical and geodetic

techniques representing timescales over many orders of magnitude. The utility of

such comparisons will depend on the assumption that the relative directions of plate

motion do not change substantially over the averaging interval. For example,

estimations based on plate reconstructions over the last 80 m.y. (e.g., Davis and

Solomon, 1981) which include major reorganization of plate geometry, are possible,

but will be degraded due to poorly-matched motions. RM2 provides a reference

estimate of y based on marine magnetic anomalies averaged over a 2-m.y. interval,

whereas space-geodetic data offer independent estimates which sample the last 10

years.

We now adapt the methodology of the previous section to the problem of

estimating y from SLR data. Let d be the vector composed of the SL6-derived

• A

geocentric vector station motions, with associated covariance V d , a s given by (1). To

compare d to RM2-predicted tangentia l motions, we must reduce biases due to

rigid-body rotations and radial motions from the data. An infinitesimal geocentric

-33-
rigid-body rotation may be represented by p'li, where Q is a three-component

rotation vector, and p is the 3x3 anti-symmetric matrix

0 z -y

-z 0 x (13)

y -x 0

where x,y,z are the nominal cartesian station coordinates (e.g., Goldstein, 1980). We

define B to be composed of the p;'s corresponding to motion of the ith station, and n

be composed of the fljk's corresponding to motion between the jth and kth epochs; in

this way, the rigid-body rotations described by n may be treated as nuisance

parameters. The assumption of infinitesimal rotation is valid for the observed

centimeter motions; such rotations are additive, allowing data from inhomogeneously

sampled networks to be full y utilized in a simultaneous inversion over all sampled

epochs. For example, for a two station network where station 1 is observed during

three epochs, and station 2 is observed in the first and last epochs, we would define

B - n

PI o

0 Pl

- P2 P2 -

•

""12"

-"23-

(14)

To remove radial components of motion and all covariances between tangential

and radial motions, we convert from geocentric to local geodetic coordinates via the

transformation matrix J, which has jacobian block diagona l elements corresponding

to the nominal station positions, and isolate the local north and east components of

motion by removing the rows and columns correspondin g to vertica l components via

the tangential selection matri x T, which is simpl y composed of 1's and O's.

Thus, we define

d '= F- d (15a)

-34-
Vd , = F-V d ' F (15b)

B'=F' B (15c)

where F = T'J. Because of the linear relationship between local geodetic velocities

and angular rate of rotation (Minster et al., 1974), we may parameterize the model by

y m0 , where mQ is the vector composed of predicted RM2-derived velocities in local

geodetic coordinates multiplied by the time difference between the epochs. The RM2

velocities are calculated with respect to a fixed plate; the choice of the fixed plate is

arbitrary since it results in a rigid-body rotation the network which is removed by

application of Q B , as given in (5). Thus, we may define the system:

ym0 +B'- n = d'. (16)

Noting the equivalence of (16) to (2), where y = m, and m 0 = A, we find from (6):

which, being scalar, has unsealed variance, given by (7)

cry

2

= (m0

T

- QB ,-m0 )' (18)

and v

2

 is estimated from (8) and (10).

This method does not account for possibl e biase s in model estimation due to

tectonic motion; we previousl y suggested tha t suc h contamination s might be reduced

by bias stripping with an a priori model of tectonic motion. It is straightforward to

show, however, that this procedure is applied implicitl y when estimating the

-35-
rate-scaling parameter. Define y(a) = 8y + a, where a is a constant. We bias strip using

amQ as the a priori model; thus, a = 0 corresponds to no bias stripping, and a = 1 is

equivalent to" bias stripping with mQ. Then (16) becomes

5y m0+ B'1 n = d'-ctm0 (19)

It follows that

and

- QB/ mQ] (21)

A A

Note that (QB/' m0 )

l <

 QB/- m 0 is an idempotent scalar projection operator, and must

A _

equal either 0 or 1. QQ-' m 0 is the relative motion model in a no net-rotation frame;

for non-trivial cases this will be non-zero and therefore the projection operator

equals 1. The quantity in the square brackets of (21) vanishes and we find the

estimate of the rate-scaling parameter is not a function of a; hence, bias stripping is

implicitly contained in this particular formulation.

The nominal station positions (shown in Figure 1) and distribution of normal

points for the stations used in the estimation of the rate-scaling parameter are listed

in Table 1; the inhomogeneous sampling of network geometry which is evident from

this distribution provides some justification for the multiple-epoch denuisancin g

procedures described above. The data, which consist of 52 local geodetic motion

vectors corresponding to 23 stations sampled over 6 epochs (1978-1983), are listed in

Table 2; coverage includes 5 of the 11 RM2-defined plates. These vectors with their

associated tangentia l error ellipses (1 sigma, unsealed) are shown in Figure 2; this

figure provides some sense of the distribution and quality of the formal errors of the

data—it does not portray the often significant covariance between station motions.

Applying the above methodology to the complete five-year interval data set, we

-36-
obtain:

Y = 0.95 ± 0.16 (one sigma).

A A

The one-sigma level of confidence is obtained from s2 = 0.355, and cr

 2

 = 0.072 .

Additional estimations based on epoch-subsets are listed in Table 3; for example, using

all the stations with data in both 1982 and 1983, which we denote {5,6}, the one-year

interval estimate of y is 0.51 ± 0.50. These results are plotted as a function of the

number of data vectors utilized in each solution in Figure 3. In all but four cases, the

SLR-derived rate-scaling parameter is consistent with the RM2-reference value (y =

1) at the one-sigma level of confidence. One exception, {3,4,5}, is very nearly

consistent (1.275 ± 0.270), and the remaining exceptions ({1,2}, {1,2,3}, {1,2,3,4}) all

include data from the first epoch, 1978. This epoch was sparsely sampled, using

measurement techniques which have been subsequently refined; it is therefore

possible that these data may be subject to systematic errors not accounted for by SL6.

If the 1978 data is ignored, estimates of the rate-scaling parameter averaged over a

one-year interval vary between 0.44 and 1.31, and show less variability for longer

intervals. In summary, we find that the SLR-derived mean rate of motion among the

observing stations for a five-year averaging interval, and even for intervals as short

as one year, are consistent with RM2.

Discussion

This result supports previous scilar baselin e rate-of-change analyses indicating

that the direct measurement of plate tectonic motions is feasible using space-geodetic

techniques (e.g., Christodoulidis et al., 1985); however, we find that an analysis of the

-37-
vector motions associated with network station position provides additional

information and constraints on "instantaneous" plate motions. For example, the use

of RM2 as an a priori model of tectonic motion appears to be reasonably justifiable, as

the improvement in results for longer averaging interval indicates. It should be

noted that estimation of the rate-scaling parameter assumed that RM2 is known

perfectly with the exception of the absolute magnitude of the rates. In reality, RM2 is

a model with associated covariance due to errors in azimuths and magnetic anomalies;

however, the errors associated with RM2 are small compared to those of SLR data, and

therefore are not likely to make any substantial contribution to the estimation of the

rate-scaling parameter. Further experimentation with alternative a priori models

which may provide additional improvements, such as NUVEL-1 (Demets et at., 1985),

should, of course, be pursued. As data coverage becomes more extensive, global plate

tectonic models based solely on space-geodetic data may ultimately supplant those

based on less instantaneous data; these improved models will help to reduce biases in

the estimation of model parameters.

Another notable feature of this vector analysis is the estimation of the sampling

A

variance s2 = 0.355. Previous estimates of the sampling variance, which have been

on the order of 0.10 to 0.15 (D. Smith, private communication), have relied on the

analysis of single scalar baseline lengths, under the assumption that additiona l

variances corresponding to other baselines, even if they are larger, will not increase

the overall variance of the system. This assumption is valid provided the variances

are uncorrelated. This is not the case for the SLR data; use of the full covariance

matrix as well as a unifie d algorithm which considers all the data simultaneously

indicate that the previous estimates of sampling variance are too small by a factor of

2 or 3.

We now consider some of the broader geophysical implications of these results.

We have found that the SLR-derived mean global plate rates for a 5-year interva l

-38-
agree with the 2-m.y. average RM2 rates; this suggests possible limitations for the

temporal variability of mean global motion. Anderson (1975) proposed that observed

tectonic motions in the vicinity of plate boundaries, such as subduction zone

trenches, may be relatively stationary for long periods (30-100 yr), except for brief

(5-10 yr) "breaking cycles" in which the accumulated stress along the boundary is

released. During such a cycle, the plate motions near the boundary are accelerated.

Episodic motion at plate boundaries is, of course, well documented; Anderson, from

observations of seismic migration patterns such as the Aleutian sequence between

1899 and 1905, suggested these episodic motions are correlated and that the

compression of the oceanic lithospheric plate between breaking cycles may extend

well into the interior of the plate. One possible implication of this model is that global

plate motions may be "jerky"; that is, quiescent except for relatively short periods of

accelerated motions. The five-year SLR-derived mean rate of motion is not consistent

with either a quiescent, or accelerated interval of plate motions. One-year estimates

display graeater variability but are, with the exception of the 1978 data, also

consistent with RM2. Possible interpretations of systematic temporal variations must

be regarded with some skeptism due to the sparcity of data, and to the

previously-mentioned systematic improvements in data collection techniques which

may not be accurately modeled by SL6. Clearly, many additional years of spacegeodetic data collection are required before strong constraints may be placed on

temporal variation in mean plate motion.

The sampling frequency of the SLR dat a severely restricts efforts to make

interpretations regarding longer averagin g intervals; this poin t is illustrated in

Figure 4 which shows estimates of global rate-scalin g parameter as a function of the

sampling interval. Space-geodetic data provide the most recent estimates averaged

over the shortest intervals; it is easily possibl e to imagin e temporal variations of

motion over the entire spectrum of timescales out to 2 m.y. or more. For example,

-39-
Vogt (1985) suggests, from a detailed examination of the magnetic lineations at six

accreting plate boundaries, that opening rates have been up to 7-35% faster over the

last 1 m.y. than over the period 1-2 m.y. Other geological techniques may provide

crude estimates for timescales intermediate between space-geodetic and marine

magnetic anomaly intervals. Estimates of slip rate at plate boundaries based on

paleoseismicity, such as those obtained by trenching on the San Andreas Fault by

Sieh and Jahns (1984), sample averaging intervals of 1000-10000 years. A robust

estimate of y would require many such estimates of slip rate over the same time

period; the natural paucity of suitable sites will hinder such an effort. Furthermore,

the rates obtained by Sieh and Jahns (1984) are consistent with more recent geodetic

estimates but show a large discrepancy with RM2 rates; Minster and Jordan (1984)

suggest that distributed deformation both east and west of the San Andreas may

account for this discrepancy. Thus, care must be exercised when measurements

obtained at plate boundaries are used to estimate global plate motion. Despite such

obstacles, the variability of the rate-scaling parameter over different timescales

warrants further investigation, because such variations might be a reflection of the

dynamical processes which drive the observed plate motions.

As has been noted, RM2 rates are derived from observations of marine magnetic

anomalies; errors in the magnetic anomaly timescal e will therefore result in scaling

errors of the RM2 rates. Under the assumption that global plate motion has been

nearly uniform over the last 3 m.y., space-geodetic data offer an independent

constraint on possible errors in the magnetic anomaly timescale. RM2 uses the

magnetic anomaly timescale of Talwani et at. (1971) for the estimation of the age of

anomalies 2, 2' and 3. In a more recent compilatio n of anomaly timescales, Ness et al.

(1980) point out several disadvantages of the Talwani scale. First, it was obtained from

profiles over the Reykjanes Ridge, which is a slow spreading ridge; such profiles tend

to have poorer resolution due to contaminatio n oi normal and reversed blocks from

-40-
subsequent flows. Second, it is a hybrid scale, constructed by assuming the age of the

end of anomaly 5 (9.94 m.y.) of Heirtzler et al. (1968) to be correct, assigning residuals

from the Heirtzler scale to be due to errors in that scale, and then compromising on

the age of the 3.1' anomaly from considerations of magnetostratigraphy. Finally, it

does not incorporate the change in K-Ar decay and abundance constants suggested by

the International Union of Geological Sciences in 1977; this change alone would

result in a 2.68% decrease in RM2 rates (Mankinen and Dalrymple, 1979). Ness et al.

(1980) propose a new anomaly timescale which displays systematic differences with

the Talwani scale. Table 4 compares the chronologies, and the corresponding

rate-scaling parameter obtained by dividing the age from the Talwani scale by teh

age from the Ness scale. These estimates of y are shown in Figure 4. The variability of

these estimates provides some measure of the possible errors in the RM2 marine

magnetic anomaly timescale; we find that the variations lie well within the formal

error bars of the SLR-derived estimate. Although there are indications that the

anomaly timescale used to determine the RM2 rates may be in error, the quality of the

SL6 data set is not at present sufficient to place useful constraints on that error.

Conclusions

We have developed a general methodology for the analysis of space-geodetic data

with the goal of constraining plate tectonic motions. The generalized-invers e

analysis of the vector rate-of-change of the geodetic network accounts for the ful l

covariance of the station motions and reduces contamination s due to rigid-body

rotations and poorly-constraine d vertica l motions. We have demonstrated the

applicability of these algorithms to obtain the RM2 rate-scaling parameter, which we

believe to be a useful description of mean global plate motion worthy of furthe r

investigation via other geophysical and geological techniques. An estimate of the

-41-
rate-scaling parameter from a five-year interval of laser ranging to the Lageos

satellite is found to be near unity and consistent with RM2 rates. There is a

suggestion that the magnetic anomaly timescale on which RM2 is based predicts

faster rates than more recent timescales, but SLR data do not as yet place any

constraints on this error. In addition, the mean rate of motion of the plates argues

against the suggestion that the plates are in a period of quiescent, or accelerated

motion.

An obvious extension of this analysis would be the inclusion of VLBI data which

presently offer usefu l network stations in Europe. More complicated

parameterizations of the model are under consideration. Examination of vertical

motions would provide a further test of consistency, and might indicate areas of

active deformation. Grouping the stations by assumed plate might yield constraints

on the directions of tectonic motions, as well as the stability of the plates themselves.

Allowing individual stations to move with constant velocity might reveal anomalous

behavior at plate boundaries. Space-geodetic datasets have begun to place useful

constraints on global plate motions. It is clear, however, that efficient algorithms to

handle the rapidly accumulating space-geodetic data, which are capable of treating

the problems associated with each technique in a self-consistent manner, will be

crucial for the detection and monitoring of temporal variations in plate motions.

-42-
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-44-
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Enoch-Subset Results

Epochs

1,2

2,3

3,4

4,5

5,6

1,2,3

2,3,4

3,4,5

4,5,6

1,2,3,4

2,3,4,5

3,4,5,6

1,2,3,4,5

2,3,4,5,6

#Data

3

11

8

11

9

15

22

21

21

27

35

32

41

46

Y

-3.470

0.439

1.308

1.248

0.514

-0.226

0.905

1.275

0.969

0.587

1.184

1.120

0.963

1.061

CTY

0.450

1.090

0.673

0.372

0.501

0.816

0.423

0.270 ..

0.258

0.389

0.232

0.194

0.218

0.171

A

s

2

0.009

0.392

0.348

0.179

0.345

0.368

0.315

0.279

0.277

0.363

0.287

0.341

0.327

0.335

A

<•/

22.540

3.028

1.303

0.770

0.726

1.810

0.568

0.260

0.241

0.418

0.187

0.110

0.145

0.087

1,2,3,4,5,6 52 0.947 0.160 0.355 0.072

Table 3. Results from estimation of- rate-scaling parameter based on epoch-subsets of

the data. Epoch 1 = 1978, epoch 6 = 1983, etc; £Data = number of relative motion vectors

observed during interval; the other quantitie s are defined in the text; all stations used

for each estimation.
Table 4. Comparison of Talwani et al. (1971) and Ness et al. (1980) marine magnetic

anomaly timescales, with corresponding estimo'e of rate-scaling parameter.

Anomaly Talwanidn.v.l Ness(m.v.)

1 0.00 0.00 1.000

0.69 0.72 0.958

1' 0.89 0.91 0.978

0.95 0.97 Q.979

2 1.71 1.66 1.030

1.86 1.87 0.995

2' 2.43 2.47 0.934

2.84 2.91 0.976

2.94 2.95 0.997

3.04 3.07 0.992

3.10 3.17 0.978

122 3.40 . 0.976

3 3.78 3.86 - 0.979

3.88 3.98 0.975

4.01 4.12 0.973

4.17 4.26 0.979

4.31 4.41 0.977

4.41 4.49 0.982

4.48 4.59 0.976

4.66 4.79 0.973
ORIGINAL

OF POOR QUALftY

Fiur e Cations

Figure 1. Nominal SLR station positions used to estimate rate-scaling parameter.

Station locations corresponding to site number are listed in Table 1.

Figure 2. Data, consisting of apparent tangential vector motions of network sites

positions between successive epochs with associated tangential one-sigma

(unsealed) error ellipse, used to estimate rate-scaling parameter. Also listed in

Table 2.

Figure 3. Estimate of rate-scaling parameter as a function of total number of motion

vectors utilized in epoch-subset trials. Dashed line is the RM2 reference estimate.

Error bars are one-sigma (scaled by sampling variance). Values listed in Table 3.

Figure 4. Estimate of rate-scaling parameter as a function of the temporal averaging

interval. RM2 is the reference estimat e wit h 1 m.y. error bars. One- and

five-year SLR-derived estimates have errors as in Figure 3. Values listed in Table

3. NLC-80 are estimates derived from the comparison of (he Talwani et al. (1971 )

marine magnetic anomaly timescale, upon whic h the RM2 rates are based, and

Ness et al. (1980) timescale, as listed in Table 4. Variabilit y of NLC-80 about RM2 is

well within the five-year SLR-derived one-sigma leve l of confidence.
ORIGINAL PAGE-IS

OP POOR QUALfTY

Figur e 1
Figur e 2
ORIGINAL PAGE IS

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