Aseismic crustal deformation in the Transverse Ranges of Southern California

Tectonophysics, 144 (1987) 159.. 180 Elsevier Science

Publishers

159

B.V.. Amsterdam

- Printed

in The Netherlands

Aseismic crustal deformation in the Transverse Ranges of southern California ABE CHENG I, DAVID D. JACKSON ’ and ~~TSUHIRO

~ATSU’U~

2

I Department of Earth and Space Sciences, UCLA, Los Angeles, CA 90024 (U.S.A.) ’ Geophysical Institute, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo I13 (Japan) (Received

January

7.1986;

revised version accepted

April 2.1986)

Abstract Cheng. A., Jackson, California. Geodetic We model

D.D. and Matsu’ura,

data at a plate boundary these displacements

blocks. The frictional We then estimate employs

M., 1987. Aseismic

In: R.L. Wesson (Editor),

the block

can reveal the pattern

are represented

velocities

and

based on geological

using prior data from Bird and Rosenstock VLBI measurements. the geodetic excellent.

data

However,

while the geologic

Our model consists has the same order the geodetic estimates

considerably accumulation;

fault

motion

of subsurface motion

by uniform

that accompany

effects

as the geologic

data.

Our

Bayesian

the method

fault patches,

estimates,

rate of about

fault;

this is about

is a major Ranges

drag

inversion

procedure

to the Transverse networks

The block motion

and for many

20 mm yr-’

between

fault patches.

across

faults

inferred

from

the agreement

the San Andreas

of crustal

by the prior model.

Ranges,

and NASA

shortening

is

fault, normal

Most of this shortening

and the White Wolf fault systems. contributor

from one fault to another. in the Transverse

half the rate implied

of southern

plate motion.

of frictional

data from the USGS trilateration

of 12 blocks and 27 rectangular

Ranges

on each of several rectangular

data. We apply

(1984) and geodetic

data imply a displacement

displacements

from geodetic

and seismological

in the Transverse

~eer~~~~~~~;e~, 144: 159-180.

and the elastic

dislocation

fault parameters

of magnitude

on the Sierra Madre-Cucamonga

Aseismic

deformation Faulting.

exceed 30 mm yr -I. The final model implies about 6 mm yr-’

to the trend of the San Andreas occurs

crustal

of Earthquake

as the sum of rigid block

interactions

prior estimates

Mechanics

to plate motion,

The geodetic

data

and the thickness

can help to identify

those faults are the San Andreas,

faults

San Gabriel,

of the frictional that are suffering Sierra Madre,

surface rapid

varies stress

Pleito, and White

Wolf faults.

Introduction Since 1971 the U.S. Geological Survey (USGS) has carried out precise length measurements on baselines near the San Andreas fault system in California (Savage et al., 1981). The National Aeronautics and Space Administration (NASA) has been making accurate measurements by Very Long Baseline Interferometry (VLBI) since 1980. These measurements tell us much about the details of plate motion, and the process of stress accumulation leading to earthquakes. ~40-19s1/87/$03.50

0 1987 Elsevier Science Publishers

In this paper we address the following questions: (1) How wide is the plate boundary, and can the plate motion be blamed on specific known faults? (2) Do the geodetic data agree with conclusions based on geologic observations and plate tectonic models? (3) Which faults are accumulating stress most rapidly? King and Savage (1984) analyzed trilateration data for the Transverse Ranges using a simple dislocation model. They showed that the strain rate is relatively low in this region, compared to other locations along the San Andreas fault. This

B.V.

implies Andreas

that the deep displacement

is lower there than elsewhere,

San Andreas

is locked

possibly

displacement

that

but

significant the data

model

the Garlock. yr

on faults

preferred

lation.

However,

fault

surface

with no shear

friction

restricts

stress

accumu-

on the upper part of the motion

and

causes

stress.

into an upper “brit-

King and Savage

tle zone” and a lower “ductile

zone”. The depth of

in strain

rate,

well by a

the San Andreas model

and

had 20 mm

slip on the San Andreas,

’ left lateral

or

slide freely

We divide each fault segment

be fit reasonably

Their

there,

would

out-

variations

with only two faults:

yr- ’ of right lateral mm

occurs

network.

spatial

could

or that the

to a great depth

side of the trilateration found

rate on the San

slip on the Garlock,

the boundary “locking

between

depth”:

measured

these zones

the width

in the fault plane,

width”.

For vertical

and 8

fault width

with

displacement

faults

are identical. rate

is called

of the brittle is called

the “fault

the locking

depth

In the ductile

across

the fault

the zone, and

zone, the

is simply

the

separated

by

both faults locked to a depth of 15 km.

velocity

We can include many more faults than previous investigators because we use a new nonlinear inversion procedure incorporating prior estimates of

the fault. In the brittle zone, the net displacement rate is the difference between the relative block

all the parameters.

rate is introduced

to represent

tional

and is assumed

The prior estimates

are based

on geology and seismology. By including more faults, we model the plate boundary more realistically. of many previously Our dislocation oughly

in a separate

reporting a similar central California.

many region

and we can test the importance neglected faults. model is described

more thor-

paper (Matsu’ura

et al., 1986)

analysis of the Hollister area of Jackson (1979) and Jackson

and Matsu’ura (1985) method in more detail.

describe

the

inversion

difference

velocity

of the two blocks

and the dislocation interaction

rate. This dislocation the effects of fricto be constant

over each dislocation patch. The dislocation rate is essentially a displacement deficit. likely to be repaid in the form of earthquakes or other episodic displacements at a later date. A schematic view of a single fault segment and the notation describe it are shown in Fig. 1.

we use to

If the entire surface separating

two blocks were be no dislocation moand the surface dis-

sliding freely, there would tion or stress accumulation.

Dislocation model Assumptions We assume that geodetic displacements, and the block and fault motions that cause them, are constant in time over the period 1971-1984. We use rates of change of line length data, and velocities and displacement

as our basic rates as our

primary unknown’parameters. By using geological estimates of block velocities in our prior model, we implicitly assume that blocks move steadily over periods of many thousands of years. The latter assumption is a working hypothesis; by using prior estimates with large uncertainties, we avoid strong dependence on this hypothesis. We represent the crust by an elastic half-space divided into a finite number of blocks. The fault surfaces separating these blocks are divided into segments from 10 to 100 km long based on mapped fault traces. In the absence of friction, the blocks

Fig. 1. Fault geometry fault

width

treated

(W),

as unknown

edge (d) calculated

comer

parameters.

The dislocation and

slip angle

The depth

east and north components The vector

block (the hanging

the other

(8).

is fixed, and the displacements

at the surface. nearer

and notation.

dip angle

block.

corresponds

fault segment.

b

rate (b), (X) are all

to the upper

(a,

and

fault

uY) are the

of the displacement

indicates

relative

motion

rate of the

wall block as shown) with respect

For each fault

segment

to the more northerly

the upper

to

left-hand

end of the mapped

161

placement

would

be rigid block

motion

the upper fault patch were completely the dislocation tion,

and

pending

motion

stress on

location

would

to determine of the upper

depth

and

the

The block

of the

in the

stress and its unloading

are

temporary

each

and the

“block term,

fault the

in our geodetic

seg-

the short

fault surface will cause some temporary of the block:

at the upper

it is this distortion

part

In

of the

distortion

that we model

the

we assume

are

effects of block motion from the distortion near the block boundaries, we need to observed displacement both close to and far from the boundary.

traces

of

faults.

In order

to the model.

from

with the actual

the dislocation.

geological

be compared

The

and the

friction

by

by earthquakes

aberrations; should

zone.

zone

with

are estimated that

motion

should average to

upper

frictional

block boundaries

boundaries

in Fig. 2, along

loading

motion”

and dip angle of each fault patch,

shown

de-

the

steady

of each fault

fitful

zone of any fault segment

Thus,

relative

fixed in our analysis.

the

the same rate as that of the lower ductile

rates

width data.

time,

displacement

These

vector,

geologic

et al. (1975) fault map of corners

are held

at a rate

ment.

rate

brittle

locked, then

accumulate

the locking

dislocation

Over

If

would equal the block mo-

block velocity. We use the Jennings California

only.

to distinguish

the

FAULTS, BLOCKS, AND MONUMENTS

NORTH

!.

PACIFIC

Fig. 2. Map correspond monuments,

showing

faults

to block codes

(as mapped

in Table

and circles represent

for the Palmdale

and San Fernando point)

K

nTHU

OCEAN

as idealized

for this study),

correspond

NASA VLBI monuments.

they are too close to other monuments; (fixed reference

and

1, and numbers

7

blocks,

and

to fault codes in Table

A few monuments

trilateration 2. Triangels

monuments. represent

Capital

USGS

letters

trilateration

used in the study are not shown on the map because

PA1 and PA2 are close to PAC. SAE is close to SAW, PIN to MTP, and TE3 to TEI. Detail nets are shown in Figs. 4 and 5. VLBI monuments

are off the map.

at Vandenburg

(block C) and Owens Valley

162

For many faults in the study area this condition satisfied, motion

and we can resolve theoretical

first computing

fault

is a combination

deformation.

motion

is simply

which

a monument

compute

theoretical

rivatives

with

the

method

The

then project-

from

and block

of the block

resent

we

tion.

average

values

using

summarized that

slips.

values

Similarly

earth

is not

and

a homogeneous

displacement

rates probably

at rectangular

over the fault the

patch),

slip rate.

appropriate

shallow

In other

displacement

the dislocation

The fault dislocation

averages

de-

(1977)

the

and fault

boundaries.

lies. For the dislocations, to parameter

the fault

do not change discontinuously

displacements

and partial

(below

Of course

specific

briefly in Matsu’ura et al. (1986). We assume the earth is a uniform elastic half-space.

“creep

parameters zone,

patch rep-

rather

than

for any specific

loca-

rate”

displacement

represents

inferred

detic baselines a few kilometers not be the same quantity that

the

from geo-

long, an it might is observed with

short baseline creepmeters across some faults. For very long strike slip faults, the rate of stress accumulation can be calculated from the dislocation parameters; roughly, it is proportional to the

Primary parameters

The primary parameters and northward velocity block; and the dislocation

consist of the eastward components for each rate (b), fault width

(IV), dip angle (6) and slip angle (X) for each fault patch. In the Transverse Ranges we assume 12 blocks and 27 fault segments, so we have 24 block parameters and 108 fault parameters, for a total of 132. The fault parameters and their measurement

at depth

halfspace,

of each

of block motion

contribution

Matsu’ura

and

slip is the tangential

creep rate is the shallow

by

on

respect of

the block block

rate

rates

The velocity

the velocity

words,

up on the fault.

for two monuments

them.

between

length

velocities,

differences

the line joining

monument

line

monument

ing the velocity onto

well the dislocation

that causes stress to build

We calculate

is

conventions

are illustrated

in Fig. 1.

dislocation rate and inversely proportional to the fault width. Assuming some fixed value for stress drop in a large earthquake, one can then calculate the recurrence rate and the characteristic displacement

for such earthquakes.

We give the relevant

equations and estimate values for faults in the Hollister Region in Matsu’ura et al. (1986). We do not report such calculations for the Transverse Ranges, because the required assumptions are questionable in a region so complicated by many fault intersections

and dip-slip

faults.

Derived parameters

Inversion method We also compute estimates of several derived parameters that are functions of the primary parameters.

The

“block

slip”

is the

tangential

Our observational data are rates of change of line length, while the model parameters are the

component (parallel to the block boundary) of the relative velocity between two blocks. It depends on the block velocities and the orientation of the

east and north components and the dislocation rate,

boundary. Positive block slip denotes right lateral motion. The “block convergence” is the component of relative motion normal to the boundary, measured such that convergence is positive. “Fault slip” and “fault convergence” are the components of the horizontal projection of the dislocation vector that are parallel and perpendicular to the fault trace, respectively. A positive fault slip value indicates a right lateral displacement deficit, and a positive fault convergence value indicates a convergence deficit. “Creep rate” is the difference

length

and slip angle

of each block velocity; fault width, dip angle,

of each fault patch.

rates are essentially

linear

The observed

functions

of the

block velocities and dislocation rates, but they are nonlinear functions of fault width, dip angle, and slip angle. Because of the large number of potentially active faults, the data may be insufficient to resolve all the relevant unknown parameters. Thus we confront a nonlinear, possibly underdetermined inverse problem. However. we know a fair amount about the expected values of the parameters, independent of the geodetic data. For example, the block velocities should be the same

163

order

of magnitude

rates,

and

the dip and

with geological quakes

method

the relevant

Jackson

and

only

unknown

prior

Matsu’ura

parameters

here.

It is then

Bayesian

by Jackson (1985);

Suppose

the inversion, differently putation. could

The primary

be combined

it is summ

equations

linear

and

reduced

a large enough

the probable

scribed

by the asymptotic

(ATE-‘A

+ D-’

of observed

data,

x is an

m-vector of unknown parameters, j is a vector of known but possibly nonlinear functions, and e is an n-vector of residuals. Suppose also that we have prior estimates x0 for each of the primary parameters, and let:

to a set of m but here we

for later convenience. j(x)

that the functions

within

them

in corn-

prior estimates

prior estimates,

and g(x)

neighborhood

estimation

are

of the

errors

covariance

are de-

matrix:

+ BTC’B)-’

where A is the (n X m) partial

where y is an n-vector

equations

and derived

solution,

X=

e=_v-f(x)

independent

Assuming

stabilize

but there is no need to treat

write them separately

(1979)

The prior estimates

from the observation

The

of the form:

derivative

matrix

of

j(x) with respect to x, and B is the (q x m) partial derivative matrix of g(x) with respect to x. The relative influence of each dataset is indicated by the partial

resolution

matrices

(Jackson,

1979)

given by: R s = XATE-‘A

d=xo-x be an m-vector of residuals to the prior estimates. Finally, let us have q prior estimates z of quantities derived

from the primary

unknown

where c is a q-vector of residuals to the prior estimates of the derived quantities z, and g(x) are known functions relating z to x. If the errors in the observations, primary prior estimates, and derived prior estimates can be described as random variables with mean equal to zero and covariance E, D, and C respectively, then variance estimate of x minimizes:

T2 = eTEp’e + dTD-‘d Minimizing

e’=

Fy-

d’=

Gx-

the

the final estimated values x = P; the covariance matrix X and the partial resolution matrices should be considered asymptotic estimates. The partial resolution matrices sum to an identity matrix, and their traces rV, cY:,,and rZ indicate the influence of each data subset on the final solution. The expected

values of the sums of squared

resid-

uals are given by: (eTE-‘e)

= n - r,,

+ cTC-‘c

T2 is equivalent

sum of squared equations:

R, = XBTC-‘B For nonlinear functions j and g, the partial derivative matrices A and B should be evaluated at

c = 2 - g(x)

matrices minimum

R,=XD-’

parame-

ters x by the equations:

c’=Jz

BMDP3R.

linearly

we have

and n observation

procedure

pro-

information.

in detail

briefly

agree

that earth-

zone, we can estimate

a nonlinear

is described

marized

should

at least approximately.

to employ

using

observed

Assuming

occur in the brittle

appropriate

and

slip angles

observations.

the fault width cedure

as the geologically

residuals

to minimizing

to the combined

the set of

(CTC’c)

= m - r, = q - rz

so that the quantities on the right may be interpreted as the “degrees of freedom” in the respective datasets. The total degrees of freedom in all data is n+q, because ~,.+rx+rZ=m.

Fj(x) Gx,

-Jg(x)

where FTE-‘F= I, GTD-‘G I. The matrices F, G, and observed and prior data, so above may be combined and accessible computer programs.

(dTD-‘d)

= I, and

JTC-‘J= J standardize the that the equations solved using readily We used the BMDP

The reported error estimates for each parameter are the standard errors, or square roots of the relevant diagonal elements of the modified covariante matrix X’, defined as: X’ = u2x

164

where

11’=

tion factor”. tainties

whose purpose

to match

residuals. D, and

T2/( n + q). u2 is the “ variance

We did not modify

parameters

error

sum

the uncer-

multiplied

by 100 to convert

of squared

resolution

is close to loo%,

the covariances estimates

E,

for some

are larger than the prior uncertainties.

In a simple combinations resolved

parameter

is to adjust

the observed

C, so the final

infla-

regression

without

of variables

from one another

ness. singularity

that

prior estimates, cannot

be fully

may cause nonunique-

or near singularity

of the normal

matrix, and large variances for the estimated parameters. For this reason it is usually recommended to test each parameter nificance. and eliminate (that

for statistical sigis, constrain to a

is the corresponding

determined

almost entirely

it is close

to zero,

dominated

If the

the estimate

by the geodetic

then

we

focus

the

our

final

is

data. If

estimate

attention

Some parameters

solved yet not significantly

is

by the geodetic

(or near the prior estimate.

on

those

significant

and

may be well re-

different

from their prior estimates).

prior

to percent. then

which are statistically

well resolved.

confined

element.

by the prior estimate.

Generally parameters

diagonal

from zero (or

These parameters

are

data to a value near zero

estimate)

A parameter

independently may

of the

be poorly

re-

fixed value) the insignificant parameters. Constraining an insignificant parameter is in fact

solved yet differ significantly from zero: its significance rests largely on the prior data. Finally, some

supplying

parameters

prior

information,

since the constraint

does not follow from the observations.

By supply-

ing prior data beforehand, we obviate the need for the traditional analysis of variance and significance testing. Instead, we shift the emphasis to the final estimate and its statistics. Of course, it is instructive to examine the partial resolution matrix R,. to learn the influence of the observed data, and to examine the residuals to the prior and observed data. Also, we may use the statistics of the final estimate to perform significance tests. Assuming Gaussian data errors and sufficiently

may be neither

well resolved

nor sig-

nificant. These could reasonably be omitted from the model. However, there is no harm in retaining them because the prior estimates a reasonable range of values.

confine

them to

Data Triluteration data

We use line length data provided by Dr. James Savage and his group at USGS in Menlo Park,

linear data equations, the parameter estimates are also Gaussian variables. The final estimates and

Calif. For each line, we determined the rate of change of line length and its standard error by

standard errors reported below serve as estimates of the mean and standard deviation of these vari-

linear

ables. We may test the statistical significance of any parameter (either primary or derived) using the standard “f-test”. The number of degrees of freedom is large enough that the half width of the

regression

deviation separately

of length

on time. The standard

of line length observations is estimated for each line from the residuals to the

linear trend, and this standard deviation becomes a multiplicative factor in the standard error of the trend. We adjusted the standard deviation for

95% confidence interval is approximately two standard deviations. Thus a parameter is different from any hypothetical value (such as zero, or its prior estimate) at about the 95% confidence level if the absolute difference exceeds two standard deviations. When we refer to “statistically signifi-

each line, rather

cant” parameters below, we mean those for which the absolute value of the final estimate exceeds twice the final standard deviation.

this assumption was quite good, so that the errors described by their formula are dominated by random measurement errors. In our case we may have other “errors” such as local episodic deformation or monument instability that are not fully represented by the formula. Data cover the time span

The diagonal elements of the partial resolution matrix RY must lie between zero and one. In the tables that follow, the quantity “Res” for each

than using the formula

of Savage

and Prescott (1973), because for our purpose even real fluctuations in line length represent errors in the secular rates. In fact, Savage and Prescott also assumed that line length changes are linear in time to derive their formula. They used data for which

165

from

1971 to 1984, although

been

surveyed

general,

over

the standard

portion

We assume

lated,

so that

diagonal

error that

VLBI

in pro-

The National

by the measure-

errors

covariance

tration

are uncorre-

line

for

160 lines

Tehachapi,

assumed (except

fault

San

networks. segments

for those

nets) is shown

from

Gabriel.

the Los

The relation and

the

USGS

the base station

between

Observatory)

measured

of the Palmdale

and

the stable

San

used

in Fig. 3. It is clear that

(VLBI). (near

to measure

baselines

Sites

in

at JPL

(near

and

VAN

Pearblossom).

are within

our study area, and

OVR (at the Owens Valley Radio

is presumed North

many

States using Very Long Base-

for this study

American vector

plate.

positions

to lie on

VLBI and

can

be

velocities,

rather than just line lengths. The “raw” data, consisting of relative vector positions of the vari-

blocks H and L,. and fault segments 6, 8, 21, 22, 23 and 27 are not adequately sampled by geodetic

ous sites at the time of each measurement, are published in the report by Kroger et al. (1984). We estimated the trend and seasonal variation for

observations, but the other blocks and faults are well covered. Figures 4 and 5 show the measured lines and the monuments for the Palmdale and San Fernando

Interferometry PEA

and Space Adminis-

has measured

United

(near Vandenburg)

Padres,

Aeronautics

(NASA)

the western

E is a

matrix

data

In

Pasadena),

and San Fernando

Fernando

have

period.

will decrease

data

the data

all lines

time

matrix.

We use data Palmdale,

lines

not

entire

to the time span covered

ments.

the

the

each vector component

nets.

of each baseline

by linear

regression. As data in our study we use measured eastward and northward velocities of the stations JPL, PEA, and VAN. We omit the vertical compo-

TRILATERATION

Fig. 3. Measured Fernando

nets.

trilateration

lmes and their relation

to idealized

LINES

faults.

Figures

4 and 5 give more detail

for the Palmdale

and San

166

PALMDALE NET

0

4

2

6

8

1Okm TEN

Fig. 4. Detail of Palmdale

trilateration

net showing

measured

lines, faults, and monuments.

SAN FERNANDO

Fig. 5. Detail of San Fernando

net showing

measured

Monument

NET

lines, faults, and monuments.

HAE is close to HAU.

167

nent

which

determining

is less accurate

and

the parameters

of interest.

the VLBI measurements yr time interval relatively do not

for

Because

we used span at most a 4

(1980-1984)~

large uncertainties. yet influence

uncertainties

less useful

the velocities Thus

the results

will shrink

suffer

the VLBI data much.

But the

as time goes by, and the

longer

baseline

data provided

quite

important

for defining

by VLBI should the total

be

displace-

ment across the plate boundary.

equivalently

115’ rt: 20”

to the north.

The

San Gabriel, Monica

Sierra Madre,

are assumed

for strike slip faults,

dip

with

slip

faults,

the model model

of Bird and Rosenstock

was adjusted

(1984). Their

to fit geologically

observed

displacement rates on major faults in southern California and the plate motion estimates of Minister and Jordan (1978). We make a few minor adjustments to their model: we modify some block boundaries slightly, and we combine their Chino, San Pedro, Santa Anna, and Vallecito blocks into one because they found negligible displacement between them. We assume rather generous prior to uncertainties (20 mm yr - ’ in each component) allow for temporal changes from the geological average

displacement

uncertainties

10 o to 30 o depending

our from

rates.

The

prior

estimates

inversion

primary

on the geological

fault parameters.

were used as input We used

estimates:

a total

22 block

Table 1, and 104 fault parameters We also input prior estimates

of 126 in

parameters: block slip, block convergence, and creep rate. The 78 assumed values and their uncertainties are shown in Table 3. The block slip and block convergence estimates were adapted from Bird and Rosenstock (1984). Creep values come either from Louie et al. (1985) or from the assumption that creep is unlikely to exceed 10 mm yr-’ (two standard deviations) on faults with no s creep observations. Table 3 also contains prior estimates for the fault slip and fault convergence rates. These quantities were not input as data;

parameters.

We assume prior estimates of the fault dislocation rate to offset the block displacement at the surface, except for a few faults such as “Garlock

rate

except for the Santa Ynez, Big Pine and Pine Mtn faults which we give a larger uncertainty. For dip slip faults we assume a dip of 65 o _t 20 O, or

to

in Table 2. of these derived

low.

faults, and 5 or 10 mm yr-’ for a few less active faults. We assume that the fault width is 10 + 5 km for all faults, based on the observation that most earthquakes, assumed to occur in the brittle zone, have focal depths in this range. Strike slip faults we assume to have a dip of 90’ + 10”

data

parameters

rather, the estimates and uncertainties deduced from the prior estimates

movement would equal the observed creep rate. We assume generous prior uncertainties for the dislocation rate as well; 20 mm yr-’ for most

from

complex-

2 shows the prior estimates

of the primary program.

prior

and 90 o for ranging

and their uncertainties appear in Table 1 along with our final estimates, which are discussed be-

E” where creep is observed. There, we choose a smaller dislocation rate, so that the difference between block slip and strike slip dislocation

and Santa

to the north,

angle are 180”

These prior estimates of block velocities

Cucamonga

down Susana,

while the White Wolf and Pleito dip down to the south in our prior model. Prior estimates of slip

and uncertainties

We take prior estimates

dipping Santa

to dip down

ity of the area. Table

Prior estimutes

for faults

San Cayetano,

equal

For

dip-slip

to zero

and

faults slip

with

angle

shown were of primary dislocation of 90 O, the

reported uncertainty in fault slip is zero. This quirk is caused by the fact that the theoretical fault slip is insensitive to small perturbations in b when b = 0 and X = 90 O. A similar quirk occurs for the fault convergence of strike slip faults when Ij = 0. Because

fault

slip and

fault

are quite nonlinear functions parameters, their error estimates when fi is small.

convergence

of the primary are not reliable

Results

Data influence and qualit>>of fit A summary

of the influence

of and residuals

to

each subset of the data is given in Table 4. In the table, the observational data are further broken

168

TABLE

1

Block parameters Block/model

East

Res

North

Res

Magnitude

Direction

(mm yr-‘)

(%)

(mm yrY’)

(%)

(mm yrV’)

(“)

36.8

-43

30.5

-40

A Cuyama prior

- 25.1 f 20.0

final

-19.7*

2.5

26.9 * 20.0 98

23.3 + 2.3

98

B Big Pine Mtn prior

- 26.4 f 20.0

final

- 19.6 f

2.2

29.3 k 20.0

39.4

-42

99

22.7 k

2.2

29.9

-41

31.4 * 20.0

45.2

-46

98

23.7 k

2.1

28.9

-35

31.4 * 20.0

45.2

-46

97

24.1 + 2.0

31.7

-40

31.4 * 20.0

45.2

-46

98

23.9 k

1.9

28.1

- 32

42.3 k 20.0

57.8

-43

25.6 k

1.7

30.4

-33

27.7 + 20.0

46.0

-53

19.9 + 0.8

27.2

-43

C Cachuma prior

- 32.5 + 20.0

final

-16.5

*

2.2

D Frazier prior

- 32.5 ? 20.0

final

-20.5

*

2.5

E Piru prior

- 32.5 * 20.0

final

-14.8

f

2.3

F Malibu prior

- 39.4 + 20.0

final

-16.4

+

1.8

99

G San Gabriels prior

- 36.7 k 20.0

final

-18.6

*

1.9

99

H San Joaquin prior

- 3.6 k 20.0

final

-6.3

i

2.7

4.6 rf- 20.0 97

8.0 +

5.8

-38

3.1

10.2

-39

I Pleito Hills prior

- 0.7 * 20.0

final

-1.2*

3.0

20.1 * 20.0

20.1

-2

97

19.7 f

1.8

19.7

-3

17.3 & 20.0

18.0

- 16

99

12.1 *

13.6

-27

J Tehachapi prior

- 5.0 * 20.0

final

-6.2

i

prior

-2.6

+ 0.0

final

-2.6

* 0.0

1.8

1.0

K Mojave 0

13.5 + 0.0

13.7

-11

13.5 * 0.0

13.7

-11

42.3 + 20.0

57.8

-43

25.8 + 3.8

32.0

- 36

L Santa Monica prior

- 39.4 * 20.0

final

-18.9

lir 3.6

0

down into trilateration and VLBI. The trace, r, of each partial resolution matrix measures the relative influence of the corresponding data subset. The sum of r for trilateration and VLBI corresponds to r,, above, and r for primary prior data

VLBI corresponds to eTEp ‘e above. while the SSR for primary prior and derived prior correspond to dTD-‘d and c’C_‘c. The final model fits the trilateration data with an rms of 1.5, meaning that

corresponds to r, etc. The trilateration and primary prior data provide almost the same amount of information (42% and 46%, respectively), with the derived prior data contributing a modest 11%. The contribution from VLBI is negligible for this model. The combined SSR for trilateration and

sumed standard deviation by that factor. Our error estimate for individual lines is based on the constancy of the rate of length change with time, while the rms error in Table 4 is based on the

on average

the absolute

residual

exceeds

the as-

spatial pattern of displacement rate. Thus the observed rms residual is not cause for alarm. The

169

TABLE

2

Primary

fault parameters

Fault/model

i,

Res

6

Res

h

Res

W

Res

(mm yr-‘)

(%)

(“)

(S)

(“)

($)

(km)

(%)

1 Ozena 3.0 * 20.0

prior

-0.1

final

f

3.4

93

90*

10

90*

12

90*

10

92+

12

180 k 10 0

180 + 12

lo+ 0

5

10 k 6

0

2 Big Pine W 6.0 f 20.0

prior

-4.7

final

f

3.1

97

180 * 10 2

183 + 12

1oi 2

5 2

7&6

3 Big Pine C prior

6.0 + 20.0

final

0.8 + 4.7

180 + 30

90 f 30 89

88 + 36

93

116 + 24

17

111 + 31

0

180 k 36

lo& 5 0

IO k 6

0

4 Big Pine E prior

9.0 + 20.0

final

1.2 + 4.1

18Ok20

115 f 20

lo*

0

182 k 24

13

180 k 33

0

181 i 36

0

5 0

lo&6

5 Pine Mtn W prior

-5.8

f

3.2

180 * 30

90 * 30

0.0 * 10.0

final

lo+ 9

5 5

11 i6

6 Santa Y nez W prior

0.0 f 10.0

final

1.8 f

4.8

180 * 30

90 * 30 47

91 + 36

1oi

5

1

10 k 6

3

10 k 6

0

7 Santa Ynez E -2.o*

final

3.4

180 * 30

90 * 30

0.0 + 10.0

prior

78

89 + 36

2

177 * 35

lo*

5 0

8 Pine Mtn E -3.3

final

f

5.6

180 i 30

90 * 30

0.0 f 10.0

prior

34

94 + 36

98

115 + 24

94

112 + 10

1

176 * 36

lo* 2

5 0

10+6

9 San Cayetano -0.1

final 10 Santa Susana

k

2.9

90 * 24

lo+ 0

11 Santa Susana

4.9

90 + 20

115 & 20

0.0 * 20.0 10.3 i

final

5

lo*6

0

81

89i

9

10 f 5 75

12 + 6

12

E

prior

0.0 f 20.0

final

9.0 *

3.3

115 * 20 98

71 * 10

90 * 20 82

6Oi

14

10 f 5 56

13*5

18

11 *6

27

N 0.0 f 20.0

prior

11.9&

final 13 San Gabriel

0

W

prior

12 San Gabriel

90 * 20

115 + 20

0.0 * 20.0

prior

9.6

90 + 20

115 + 20 76

128 i_ 14

94

110+

44

104 * 19

lo*5 9

W 0.0 k 20.0

prior

-29.3

final

of- 5.2

115 + 20 8

90 * 20 88

78k

94

1181

9

10 * 5 71

4*1

96

14 Sierra Madre prior

12.5 f

2.8

90 + 20

115 + 20

0.0 f 20.0

final

98

32+

6

7

10 * 5 89

7+1

98

IS Cucamonga 0.0 f 20.0

prior

-14.9

final 16 San Andreas

+ 9.8

149*

9

90*

10

90 f 20 84

75 f 13

10 f 5 65

13 + 4

42

A

prior

31 .o * 20.0

final

20.0 +

17 San Andreas

115 + 20 82

3.6

96

95*11

92

90+

180 + 10 7

185 + 10

8

184 + 11

10 2 5 14

15 * 5

22

B

prior

35.0 f 20.0

final

19.5 f

4.0

90 + 10 11

180 * 10

10 i- 5 5

11 i-5

27

170

TABLE

2 (continued)

Fault/model

18 San Andreas

i,

Res

s

Res

x

Res

W

Res

(mm yrY’)

(a)

(“)

(%)

(“)

(%)

(km)

@)

13

12 + 6

C

prior

35.0 * 20.0

final

21.5 + 4.4

19 San Andreas

35.0 * 20.0

final

16.6 f

2.7

lOi_ 5

180 f 10

11

5

178?

10

22

172+

78

170*

4

90 + 10

180 * 10

98 & 10

98

8

10 _t 5 62

X&3

56

E

prior

35.0 * 20.0

final

17.7 *

21 San Andreas

99*

D

prior 20 San Andreas

90 * 10 83

2.1

90 * 10 99

108k

1oi5

180 * 10

6

5

13*2

84

80

F

prior

35.0 f 20.0

final

16.9 f

90 + 10 1

3.1

90+

IO i 5

180 + 10

12

0

180 + 12

0

lo?

6

17

lo+

5

lo*

5

0

22 White Wolf W prior

- 0.2 f 20.0

final

15.0 f 11.7

65 f 20 61

69k

10 i 5

90 * 20

19

6X * 19

16

10

23 White Wolf E prior

- 0.2 * 20.0

final

- 13.5 + 12.5

90 + 20

65 f 20 61

65 f 11

96+

70.

15

6

8i6

54

24 Pleito prior

0.0 f 10.0

final

21.8 + 5.4

25 Garlock

51*

7

90*

10

90 * 20 85 * 10

89

10+ 5 14 * 4

58

44

W

prior

3.0 + 20.0

final

6.8 k

26 Garlock

65 k 20 74

5.0

91

10 * 5

180 f 10

91*12

180 * 12

11 &6

1

2

E

prior

- 3.0 * 20.0

final

-2.7

?

2.7

99

90+

10

180 * 10

90*

12

181 + 12

115 * 10

90_+ 10

109 t 12

94*

10+

5

9+6

3

4

27 Santa Monica prior

1.9 + 5.0

final

- 12.3 + 4.2

46

by number

in Fig. 2; “Res”

No/e: Faults percent.

final

are identified

Parameters

model

b,, 6, X, and W are described

fits the prior

data

almost

is the corresponding

in Fig. 1. Uncertainty

within

the

assumed uncertainty. The rms for all data, 1.2, is the square root of the variance inflation factor, c2, defined above. The SSR for the prior model is 14,873, of which 14,794 is due to misfit of the trilateration data. Taking 379 as the degrees of freedom of the prior model, and 247 as that of the final model, the F ratio for the improvement in variance

exceeds

75. Assuming

linear,

Gaussian.

uncorrelate’d data, the model is significant at well over 99% confidence. We do not need all these assumptions to see that the final model fits the data. especially the trilateration data, much better than the prior model does. Observed length rates, their standard errors,

diagonal estimates

predicted

10 * 5

11

element

13+6

7

of the resolution

are one standard

matrix,

11 converted

to

deviation.

rates, and residuals

are shown

for each

trilateration line in Table 5. Sixteen lines, or lo%, have absolute standardized residuals greater than 2. These lines are numbered 11, 28, 29, 32, 39, 51, 67, 109, 113, 118,139,143,144,147.152, and 157. Lines 28, 39, and 157 have large negative residuals, less than - 3.0. For some lines, such as 11, 28, 29, 32, 39, 118, and 147, the standard errors are very small. We may have underestimated the standard errors because the observed lengths were anomalously linear in time. Other lines such as 109, 139, 152, and 157 have relatively large observed length rates, yet they cross no faults and join stations that are at least a few km away from block boundaries. It should be mentioned that

171

TABLE

3

Derived

fault parameters

Fault/

Block slip

Block

Fault slip

Fault

Creep rate

model

(mm yr-‘)

convergence

(mm yr-‘)

convergence

(mm yr-‘)

(mm yr-‘)

(mm yr-‘)

-

1 Ozena prior

2.4 f

final

-0.5

5.0

f

2.6

6.3 k

5.0

1.3 * -0.4

3.0

+ 0.9

0.0 *

0.0

-0.6

+ 5.0

3.4

0.0 *

0.0

-0.4

k 5.6

6.0 k 20.0

3.0 * 20.0 -0.1

*

2 Big Pine W prior final

-3.0

+ 2.2

1.4 *

3.0

1.4*

1.1

-4.7

f

0.0 *

0.0

0.3 * 5.0

3.1

0.0 *

0.2

1.8 * 5.0

3 Big Pine C prior

5.0 +

5.0

4.0 f

3.0

6.0 + 20.0

0.0 f

0.0

- 1 .o + 5.0

final

0.4 + 2.3

1.7 f

1.6

0.8 k

4.7

0.0 f

0.0

- 0.5 f 6.2

prior

4.5 _t 5.0

7.4 *

5.0

9.0 * 20.0

0.0 +

1.3

- 4.5 * 5.0

final

0.3 k

1.1 ?z 1.3

1.2 f

0.0 *

0.2

- 0.9 k 6.2

0.0 *

0.0

0.0 i

0.0 +

1.2

1.8 f 5.3

4 Big Pine E 2.3

4.1

5 Pine Mtn W prior

-4.0

0.0 f 10.0

5.0

0.0 *

5.0

2.4

0.9 *

1.2

5.0

0.0 t

3.0

0.0 * 10.0

0.0 + 0.0

0.0 + 5.0

& 2.3

0.4 f

1.5

1.8 f

0.0 + 0.1

- 3.4 ? 6.4

0.0 *

3.0

0.0 + 5.0

final

i

-5.8

f

3.2

6 Santa Ynez W prior

0.0 +

final

-1.6

4.8

7 Santa Y nez E prior

0.0 f

final

-1.7

5.0

f

2.4

0.0 f

5.0

-0.2

+

1.3

0.0 *

3.0

0.0 + 10.0

0.0 i

0.0

0.0 * 5.0

3.3

0.0 *

0.2

0.4 + 5.6

0.0 f 10.0

0.0 i

0.0

0.0 + 5.0

0.0 + 0.4

- 2.3 + 7.1

-2.0

*

8 Pine Mtn E prior final

-5.6

-0.9+

1.3

k

2.5

prior

7.1 *

5.0

10.8 + 10.0

final

1.6 ?

1.0

1.7 * 0.4

prior

9.7 *

5.0

8.5 f 10.0

final

2.0 & 0.9

-3.3

k

5.6

9 San Cayetano

10 Santa Susana

0.0 i

8.5

7.1 * 5.0

0.0 *

0.1

0.1 5

2.9

1.6 f 0.5

0.0 *

0.0

W

11 Santa Susana

1.2 f

0.6

-0.2

*

1.6

3.9 *

5.0

12.3 k 10.0

final

1.1 f

1.0

2.0 * 0.3

0.0 _+ 0.0 -4.5

*

1.6

8.5

9.7 f 5.0

f

4.9

2.2 f 1.8

0.0 k

8.5

3.9 i 5.0

2.6 + 3.1

5.6 + 1.6

0.0 * 20.0

1.1 * 5.0

- 7.0 i_ 9.0

1.7 i 3.5

N

prior

1.1 *

5.0

5.5 + 5.0

0.0 & 0.0

final

4.6 i

2.0

0.4 *

2.3

3.0 f

3.5 0.0

13 San Gabriel

0.0 * -3.8

E

prior 12 San Gabriel

0.0 * 0.0

W

prior

-0.9

& 5.0

5.5 + 10.0

0.0 f

final

-0.4

*

1.7

5.5 + 2.2

6.3 * 4.6

9.9 * 15.1

prior

11.3 *

5.0

9.6 f 10.0

0.0 *

0.0

0.0 f

8.5

11.3 i 5.0

final

1.9 *

1.4

5.8 f

1.8

5.8 k

1.6

9.4 & 2.7

- 3.8 f 2.3

prior

2.7 f

5.0

14.6 k 10.0

0.0 f

0.0

0.0 *

8.5

2.7 + 5.0

final

0.3 *

3.3

3.9 f

3.2

12.3 i

9.1

- 3.6 k 4.7

0.0 *

8.5

-0.9

f 5.0

- 6.7 + 4.5

14 Sierra Madre

15 Cucamonga

16 San Andreas

3.7

A

prior

31.0 f 15.0

final

20.3 k

17 San Andreas

5.9 f

3.5

*

3.0

31.0 F 20.0

0.0 f

0.0

0.0 + 2.0

0.2 k

2.7

19.9 f

0.1 *

0.4

0.4 + 6.1

-1.1

3.6

B

prior

25.2 + 15.0

2.7 k

3.0

35.0 * 20.0

0.0 It 0.0

- 9.8 f 2.0

final

18.9 + 3.7

0.5 F

1.3

19.4 *

0.0 + 0.3

- 0.5 * 7.5

4.0

172

TABLE

3 (continued)

Fault/

Block slip

Block

Fault slip

Fault

Creep rate

model

(mm yrr’)

convergence

(mm yrr’)

convergence

(mm yrr’)

(mm yr-‘) 18 San Andreas

(mm yr-‘)

C

prior

33.2 * 15.0

6.0 k

3.0

35.0 + 20.0

final

19.8 *

1.2*

1.1

21.5 + 4.4

19 San Andreas

3.7

0.0 i -0.1

0.0

ml.8 i_ 2.0

& 0.6

~ 1 .I * 7.7

D

prior

36.9 k 15.0

1.7 +

3.0

35.0 * 20.0

final

17.2 k

0.5 *

0.4

16.4 k

20 San Andreas

2.0

2.7

0.0 * -0.3

0.0

1.9 + 2.0

+ 0.4

0.8 + 4.3

E

prior

36.9 k 15.0

-1.8

+ 3.0

35.0 & 20.0

final

17.2 + 2.0

-1.1

f

0.5

17.4 + 2.0

prior

36.1 * 15.0

-x.0+

3.0

35.0 + 20.0

0.0 f

0.0

1.1 + 2.0

final

16.7i

-3.9

f

0.7

16.9?

0.0 f

0.0

~ 0.2 * 4.5

8.5

- 10.8 i 5.0

4.9 f 11.1

- 5.0 & 6.0

21 San Andreas

0.0 * -1.o+

0.0

1.9 + 2.0

0.5

- 0.2 * 3.7

F 1.9

3.1

22 White Wolf W prior

- 10.8 + 10.0

final

-10.7

*

11.5 + 10.0

3.1

7.1 *

3.4

0.0 -5.7

&

0.1

_t 5.0

-0.1

f

23 White Wolf E prior

- 5.7 * 10.0

final

-2.4*

11.4 * 10.0

0.0 *

0.1

-0.1

+ x.5

- 5.7 * 5.0

3.0

3.4 i_ 3.2

- 1.5 *

3.4

-5.7

* 12.5

- 0.9 * 5.1

5.0

3.9 * 10.0

0.0 f

0.0

3.0

8.6 + 2.0

24 Pleito prior

-3.4*

final

-2.8

25 Garlock

k

-2.0

i

3.8

0.0 k

~ 3.4 i 5.0 - 0.8 + 3.6

W

prior

5.3 f 10.0

-4.4

*

5.0

- 3.0 + 20.0

final

4.6 + 2.8

-4.4

f

2.1

6.8 + 5.0

prior

- 0.1 * 10.0

-4.5

*

5.0

final

-3.8

-0.6

+ 0.7

26 Garlock

4.2

13.6 + 5.4 0.0 + 0.0 0.0 *

0.0

8.3 i 5.0 - 2.1 5 6.9

E +

1.9

0.0 f

5.0

- 3.0 * 20.0

0.0 + 0.0

-2.7

0.0 f

+ 2.7

0.1

2.9 k 5.0 -1.1

* 4.5

27 Santa Monica prior final

-2.5

Nofe: Prior estimates were calculated

lines

109,

& 3.2

0.0 * -0.5

of block slip, block convergence.

from primary

parameters

120, 128, and

in Tables

*

3.0 3.4

0.3

+ 2.3

-0.8

*

5.0

4.0 k

4.2

and creep rate were input as data with uncertainties

0.0 f 5.0 -1.7k4.1 shown. Other parameters

1 and 2. See also notes to Table 2.

136 are consistent

in

showing north-south shortening of about 3 mm yr-’ within block G. Our model explains part of this shortening, essentially by thrusting at depth on the San faults, but predicted. may cause

0.0 * -0.8

Gabriel, Sierra Madre, and Cucamonga the observed shortening surpasses the Meandering of the Garlock E Fault the large residual for line 51. Monu-

ment DIO is very close to this fault, and some surface displacement may occur south of DIO, contrary to our assumption. Data for line 113 and other lines from station LIT are highly questionable; the extremely large uncertainties reflect a very short time span (1 month) for observations.

Stations HAU. TEN, VER, MIN, and PE2 are each involved in two or more lines with large residuals, and the three lines joining HAU. TEN, and VER are all aberrant. Displacements local to these sites may contribute to the large residuals. Figure 6 shows a histogram of standardized residuals for the trilateration data. The distribution is clearly not Gaussian, because of the outliers discussed above. The outliers account for fully half of the sum of squared residuals, eTEp ‘e, in Table 4. With the few exceptions noted, the model provides a satisfactory fit to the trilateration data, and we see no reason to question the general validity of the data, the uncertainty estimates, or the assumptions of the model.

173

TABLE

Figure

4

Influence

and quality

gence. Referring

of fit

Data type

n

df

SSR

RMS

Trilateration

160

56

104

232

1.5

6

1

5

10

1.3

VLBI

8 shows the block

r

Primary

prior

132

61

71

17

1.0

Derived

prior

81

14

67

41

0.8

velocities tions:

to Table

are resolved

the Mojave

1, we see that the block

block is tightly

its prior

estimate,

no monuments.

and

Note:

379 n = number

matrix;

132

of data;

241 r = trace

df = n - r = degrees of freedom;

standardized

residuals;

360

1.2

of partial

the Malibu

resolution

Block

RMS = root mean square

block,

west. According

residual.

the Malibu

Block velocities 7. The block

are shown in Table

velocities

are resolved

parallel

the net motion

to the prior estimate,

travels

to the souththe Malibu

in the direction (all azimuths are In the final model

at 18 mm yr-’

towards

at 126” east of north. This huge residual, comparable in magnitude to the final estimate, corre-

(block slip) and perpendicular (block convergence) components for each block boundary in Table 3.

L

block

for all blocks

- 49” with respect to Mojave. The vector difference between the final and prior estimates of the Malibu block velocity is 28 mm yr-‘, directed

1 and Fig. into

block differ

it is furthest

block should move at 47 mm yr-’

Block velocities and missing slip on the San Andreas

Monica

block K. Block F,

best represents

-52” with respect to Mojave measured clockwise from north).

by

velocities

from the prior estimates

across the array because

SSR = sum of squared

constrained

the Santa

except H, I, and the constrained Total

conver-

very well with two excep-

contains significantly

slip and block

HISTOGRAM

OF RESIDUALS

0 -4.2

-3.6

-3.0

-2.4

-1.8

-1.2

-0.6

0

STANDARDIZED

Fig. 6. Histogram

of standardized

residuals

for 160 trilateration

0.6

1.2

1.8

2.4

3.0

3.6

11.2

RESIDUAL

data. Residuals

are standardized

using the prior standard

deviations.

174

the San Andreas fault. Faults 19, 20 and 21 (the southernmost segments of the San Andreas) show statistically significant negative residuals of about 20 mm yr-‘. Other faults with large, statistically

sponds to a shortfall of geodetically observed motion compared to the geological observations. Inspecting the block slip rates in Table 3 reveals that much of that shortfall occurs as missing slip on

TABLE

5

Observed and predicted line length changes No

From

Blk

To

Blk

Observed

Std. err.

Predict

8 9 10 11 12 13 14 15 16 17 18 19 20

AND AND AND AND BEN BEN BUR BUR BUR BUR BUR BUR BUR DEN DEN DES DES DES GRA GRA

K K K K G G G G G G G G G K K K K K G G

BEN BUR GRA LEO LEO POR DES GRA LEO LOC MIN PLN POR RIT TEN GRA LEO MIN LEO LOC

G G G K K K K G K K G G K K G G K G K K

~ 0.123 0.059 - 1.525 0.228 0.811 - 0.348 1.430 - 0.223 0.686 1.739 0.009 - 1.063 - 1.323 0.645 - 2.052 - 0.165 -0.215 - 0.925 - 0.605 1.940

0.274 0.376 0.289 0.430 0.318 0.466 0.268 0.199 0.244 0.292 0.373 0.162 0.199 0.529 0.320 0.454 0.296 0.550 0.318 0.386

0.401 0.287 - 1.036 0.403 0.814 -0.191 1.368 -0.163 0.839 1.523 - 0.752 - 1.068 - 1.021 1.494 -1.844 - 0.066 0.131 - 1.438 - 0.988 1.336

-1.909 -0.607 -1.693 -0.407 -0.010 -0.338 0.232 -0.299 - 0.630 0.740 2.038 0.032 -1.514 - 1.605 -0.650 -0.219 -1.167 0.932 1.206 1.565

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

GRA GRA GRA GRA HAU HAU HAU HAU HAU LEO LEO LOC LOC LOC MIN MIN RIT RIT TEN AVE

G G G G G G G G G K K K K K G G K K G K

MIN PLN POR RIT LOC MIN RIT TEN VER MIN POR MIN RIT TEN PLN RIT TEN VER VER LEO

G G K K K G K G K G K G K G G K G K K K

- 0.356 0.716 - 1.581 2.224 - 1.262 0.750 0.215 - 0.680 0.105 -0.851 0.315 0.901 0.478 - 1.633 0.729 1.534 - 1.406 - 0.201 - 2.054 0.417

0.336 0.192 0.217 0.404 0.203 0.323 0.622 0.259 0.445 0.831 0.567 0.245 0.221 0.270 0.355 0.491 0.269 0.263 0.300 0.429

- 0.561 0.481 - 1.847 1.646 - 1.474 0.197 - 0.869 0.278 1.206 - 1.817 - 0.379 0.229 0.128 - 1.714 0.956 1.217 - 1.708 - 0.391 -0.718 0.227

0.608 1.226 1.229 1.429 1.047 1.710 1.743 - 3.697 - 2.471 1.162 1.224 2.748 1.584 0.301 -0.641 0.645 1.123 0.722 -4.451 0.442

41 42 43 44 45 46 47 48 49 50 51 52

AVE AVE AVE BAL BLA DEN DIO DIO DIO DIO DIO DOU

K K K

PLN RIT SOL SOL LIR TOM GNE POL SAW TE4 THU SOL

G K K K K K J I G K K K

0.258 - 2.383 - 1.602 - 1.675 4.696 2.826 1.618 3.588 - 5.938 1.717 2.169 -2.117

0.750 0.681 0.477 0.532 1.257 0.662 0.248 0.653 0.739 0.630 0.799 2.564

-

0.728 1.577 -0.998 0.110 1.499 0.410 0.014 0.872 - 0.658 0.613 2.861 - 1.760

4 6

-

0.288 3.457 1.126 1.734 2.811 2.555 1.615 3.019 - 5.451 1.330 -0.118 2.395

Std. raid.

From

Blk

TO

Blk

Observed

Std. err.

Predict

Std. raid

53 54 55 56 57 58 59 60

FAI GNE GNE GNE PAJ POL POL POL

K J J J J I I I

SAE PO1 TE3 WHE SOL TEC TE4 WHE

G

0.495 0.948 - 2.438 2.292 0.338 1.536 - 0.635 - 1.927

7.987 0.501 1.301 1.388 0.894 0.706 0.401 1.209

3.890 1.200 ~ 1.885 2.065 0.563 1.290 -0.418 - 1.562

- 0.425 ~ 0.503 - 0.425 0.164 -0.252 0.347 PO.541 PO.302

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

SAW SAW SOL TEC TEI TEJ TEN APA APA APA FRA FRA FRA MAD MAD MON MON MON MTP MTP

G G K 1 J J G A A A D D D 0 B C C C A A

TE4 THU THU TE4 WEE WHE WAR REY SEC YAM MTP TEC THO SEC YAM NOR REY SEC PAT REY

K K K K H

- 2.433 0.921 0.521 2.349 - 5.008 ~ 2.568 4.297 ~ 0.663 - 1.132 1.422 0.633 - 0.669 0.630 0.362 - 1.229 1.096 0.572 - 0.059 - 1.209 - 0.707

0.651 0.789 0.358 0.585 1.111 0.920 1.261 0.583 0.376 0.76 I 0.453 0.626 0.737 1.263 0.450 0.758 0.609 0.832 0.526 1.066

~ 2.109 1.014 0.547 1.978 ~ 4.175 - 1.212 1.691 -0.917 ~ 1.024 0.829 0.726 - 1.169 0.519 - 0.945 - 0.728 0.936 0.354 -0.195 ~ 1.431 - 0.915

- 0.498 -0.117 - 0.073 0.635 - 0.209 - 1.473 2.066 0.436 - 0.287 0.780 - 0.207 0.798 0.151 1.035 -1.113 0.212 0.357 0.163 0.422 0.195

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

MTP MTP MTP MTP NOR NOR NOR NOR NOR PAT PAT PEL PIN PIN PIN REY REY SAL SEC ANT

A A A A E E E E E H H A A A A D D B C G

TEC THO WH2 YAM PAA REY SAN SUL THO PEL TEM YAM REY THO YAM SEC YAM YAM YAM BAD

B F D E F D A H B D D B C B B B G

2.365 - 2.078 1.511 2.620 - 0.409 - 0.663 ~ 0.286 - 2.448 - 0.563 - 1.601 - 2.107 - 2.235 - 0.707 - 2.078 2.620 1.654 -0.177 1.321 - 2.541 -2.518

0.544 0.964 1.979 0.762 0.369 1.032 0.456 0.788 0.807 0.303 0.465 0.483 1.066 0.964 0.762 0.621 0.649 0.460 0.826 0.833

2.186 - 2.244 0.450 2.824 ~ 0.624 - 1.332 - 0.626 -2.244 ~ 0.875 - 1.557 ~ 2.532 - 2.261 PO.917 ~ 2.241 2.819 1.505 0.122 0.967 - 2.554 - 2.749

0.330 0.172 0.536 - 0.267 0.582 0.649 0.746 - 0.259 0.387 -0.145 0.913 0.054 0.196 0.169 - 0.260 0.241 - 0.461 0.770 0.015 0.277

101 102

ANT BAD

G G

DIS DIS

G G

- 2.093 -2.106

1.079 2.740

0.993 1.168

- 1.019 - 1.195

No -

K K

K D C B A

D r J

175

TABLE No

5 (continued)

From

Blk

To

Blk

Observed

Std.

Predict

Std.

From

No

Blk

To

Blk

Observed

raid.

err.

Std.

Predict

Std. raid.

err.

IO3

BAD

G

PA2

G

-0.415

2.465

1.536

- 0.792

132

PIC

F

WAS

G

- 5.300

1.501

- 5.083

-0.144

104

BAD

G

TOM

K

-7.111

2.273

- 7.970

0.378

133

PIC

F

WHI

E

~ 4.898

3.667

- 3.178

- 0.469

105

DIS

G

PA2

G

- 2.368

0.546

- 2.004

-0.665

134

POT

G

SIS

G

-

1.259

1.039

- 1.493

106

DIS

G

SIS

G

-

1.172

1.409

0.082

-0.890

135

POT

G

SYL

F

~ 0.064

0.081

-0.021

- 0.536

107

HAE

G

MTH

G

- 2.950

0.594

- 2.167

-1.319

136

SAE

G

WAS

G

- 3.217

2.908

-0.958

- 0.776

108

HAE

G

PAC

G

- 2.713

0.925

~ 3.091

137

SIS

G

SYL

F

- 3.197

2.188

- 0.575

-

109

HAE

G

PAR

G

~ 2.994

0.909

- 0.972

138

WAS

G

WHI

E

1.140

5.393

- 0.931

0.384

0.409 -2.224

110

HAE

G

POT

G

0.979

1.453

- 0.347

111

HAE

G

TOM

K

4.063

1.968

5.260

-0.609

112

HAU

G

PLN

G

0.240

1.132

1.245

~ 0.888

113

LIT

K

SAE

G

274.644

114

LIT

K

WAS

G

- 4.041

115

LIT

K

WHA

E

78.722

124.920

-

116.215 1.403

0.913

0.225

1.198

139

5Nl

F

PE2

F

4.188

1.531

0.684

2.288

140

5Nl

F

PLI

F

2.415

1.142

0.287

1.863

- 2.304

2.383

141

5N1

F

TUJ

F

0.764

0.177

0.674

~ 5.084

0.744

142

6Pl

F

PLl

F

0.228

0.570

1.182

- 4.586

0.667

143

6Pl

F

TUJ

F

- 2.749

0.422

~ 1.796

- 2.254 ~ 2.386

116

MAY

F

PLN

G

1.203

0.909

- 1.196

144

BLU

F

PE2

F

- 2.904

0.569

-

117

MAY

F

PIC

F

0.154

2.549

~ 2.086

-0.007 0.879

145

BLU

F

SF8

E

- 0.659

0.991

- 0.888

118

MAY

F

POT

G

1.207

0.457

-0.021

2.687

146

MAY

F

PLl

F

-

1.421

0.075

-

119

MAY

F

SIS

G

~ 0.244

1.274

- 0.575

0.260

147

MAI

E

PLl

F

- 0.609

0.492

~ 2.120

120

MAY

F

WAS

G

~ 3.029

4.341

-2.516

148

MA1

E

SF8

E

-

1.181

0.910

0.409

149

MES

E

PLl

F

- 3.329

0.199

-

-0.118

1.547 1.403

0.505 -

1.677

0.231 -0.240 3.074 -

1.748 0.158

3.360

121

MTG

G

PAC

G

0.614

1.223

0.509

0.086

150

MIS

E

PE2

F

-4.130

0.498

- 4.364

122

MTG

G

PAR

G

- 0.245

1.267

~ 0.995

0.592

151

MIS

E

SF8

E

~ 1.201

0.464

- 0.85 I

152

P2

F

PE2

F

3.414

1.228

- 0.028

153

P2

F

PLI

F

- 2.803

0.918

- 1.090

-

1.867

154

P2

F

SF8

E

-3.161

0.165

- 2.980

-

1.091

155

P2

F

TUJ

F

- 0.247

1.045

0.329

-0.849

0.471 ~ 0.754

123

MTG

ti

POT

G

0.127

1.023

0.996

124

MTG

G

SIS

G

- 2.512

0.929

- 2.843

125

PA1

G

TOM

K

- 1.785

1.202

- 1.509

126

PA2

G

SIS

G

- 0.703

0.377

- 0.923

127

PAR

G

POT

G

- 0.365

0.558

0.482

-1.518

156

PE2

F

PLl

F

0.552

0.858

0.171

0.444

128

PLN

G

POT

G

- 3.545

2.828

- 2.482

-0.376

157

PE2

F

RES

F

- 5.353

1.252

- 0.239

- 4.084 - 0.035

0.357 -0.230 0.585

129

PLN

G

SYL

F

- 3.721

2.293

- 1.196

-1.101

158

PLI

F

SF8

E

- 2.485

0.320

~ 2.474

130

PLN

G

WAS

G

0.904

1.775

1.427

-0.295

159

RES

F

SF8

E

- 0.942

1.080

0.140

131

PIC

F

SYL

F

- 2.860

7.041

- 2.086

-0.110

160

SF8

E

TUJ

F

2.861

2.785

- 0.971

Now:

Std.

Raid.

significant

= (Observed

residuals

- Predict)/Std.

between

2.803

-0.551

-

1.001 1.376

Err

final and prior

esti-

mates are faults 2, 8, 9, 10, 11, 14, 16, and 18 (the Big Pine W, Pine Mtn E, San Cayetano, Santa Susana W, Santa Susana E, Sierra Madre, San Andreas A, and San Andreas C). This discrepancy between geodetic and geologic estimates contrasts

Trough

(Savage et al., 1979; Cheng,

1985)

and on

the close agreement between geodesy and geology for Hollister (Matsu’ura et al., 1986). But of course this evidence is circumstantial, and our assumption of constant block velocities over many thou-

sharply with our results for Hollister (Matsu’ura et al., 1986) where the estimated block velocities and

sands of years may fail. Errors in the geodetic model (hypothesis b) could result from erroneous data, or from a mis-

block slips agree to within a few mm yr-‘. Possible explanations for the discrepancy

be a serious problem

take in modeling. be-

tween geodesy and geology in the Transverse Ranges are (a) the geodetic and geologic rates are not comparable because of temporal variations in displacement rate, (b) the geodetic estimates are in error, or (c) the geologic estimates err. We do not believe that time dependent block motion, hypothesis (a) above, provides the answer. While the near surface fault motions vary with time, the deeper block motions appear quite steady. This assertion is based on the close agreement between

geodesy

and plate tectonics

in the Salton

Data errors are very unlikely in this analysis.

to

While some

systematic errors may cause annual or other short period variations (Jackson et al., 1983), such errors would have very little effect on the secular rates of line length change (Cheng, 1985). The trilateration data for the Transverse Ranges and for Hollister were collected by identical methods, and no discrepancy occurs at Hollister. A possible modeling error could result from our assigning a prior fault width of 10 &-5 km. The estimated fault width is poorly resolved (see discussion below), so that the final fault width estimate is strongly influenced by

176

BLOCK VELOCITIES

//// b L\

-----+ --

/

PRIOR FINAL

20

Fig. 7. Prior and final estimates

‘??

6Ommlyr

of block velocities.

Final estimates

indicate

a displacement

across

the San Andreas

fault slower than

in the prior estimates.

BLOCK SLIP AND CONVERGENCE

-2

OIprniV’ 4 15mm’vr

Fig. 8. Estimated arrow

block slip and convergence.

shows the full slip or convergence

Only statistically

rate, not the half-rate.

significant

(two sigma) values exceeding

5 mm

yr

C’ are

shown.

Each

the prior. If the San Andreas a much greater

depth,

strain would occur outside block

motion

observed

would

the network,

be required

displacements.

calculations

assuming

the San Andreas,

We

block slip could match

and greater

to match

performed

much greater

and we found

if the fault is locked thickness

fault were locked to

then some of the resulting the some

fault width on

that the estimated

the geological

observations

to 25 km or more (about

the

of the crust). This seems much too deep

to us, because San Andreas

earthquakes rarely

on this section

exceed

of the

15 km depth

yet to be written ceivable better

with the geodetic

In summary, listed

on this question,

that future geological

logical prior model. parable

because

rule

to explain

tween our estimated

will agree

values.

we cannot

hypotheses

and it is con-

estimates out

block velocities Perhaps

any

the two are not com-

block rates vary, or we may badly

underestimate

the

locking

Andreas,

or the geological

by lo-20

mm yr-‘.

depth

of

estimates

Our results confirm the assertion of Bird Rosenstock (1984) that substantial shortening curs across the Transverse Ranges, although estimated rates are about half theirs. Table 3

Rosenstock (1984) are not unthinkable. They designed their model to match plate tectonic dis-

White

placement Andreas unknown

with motion causing

them

San

Convergence across the Transverse Ranges

Fig. 8 show about 6 mm yr- ’ of block gence across the Sierra Madre-Cucamonga

rates

the

could be off

1986). Nevertheless, we cannot completely reject the idea that the San Andreas is presently locked to the base of the crust within the Transverse Ranges. Errors in the geologic rates of Bird and

possibly

be-

and the geo-

(Webb

and Kanamori, 19X5), and because the estimated fault width rarely exceeds 15 km in areas where it is well resolved (Cheng, 1985; Matsu’ura et al.,

faults,

of the

the discrepancy

system,

with comparable Wolf W faults.

onshore

is statistically

to blame

the San

level for the San Cayetano,

and Humphreys (1986) differs substantially from Bird and Rosenstock in many respects, but both models agree on about 35 mm yr-’ on sections A-E of the San Andreas (faults 16-20). Direct observations used in these models include 35-59 mm yr-i on fault 16 (Dickinson et al., 1972; Sieh

converFault

values for the Pleito and The estimated

on known

for motion that actually occurred on faults. An alternate model by Weldon

and ocour and

significant

convergence

at the 95% confidence Santa

Susana

W and

E, San Gabriel W, Sierra Madre, San Andreas E and F, White Wolf W, Pleito, and Garlock W faults. For statistical significance, we use the approximate criterion that the absolute value of the parameter estimate should exceed twice the standard error. We do not calculate resolution estimates of derived parameters. But it is clear that the estimates of block convergence come almost entirely from the trilateration data because such

and Jahns, 1984; Clark et al., 1985) 35-60 mm yr-’ on fault 19 (Rust, 1982) and 35 mm yr-’ on fault 20 (Weldon, 1984). Thus our geodetic model disagrees with all the geologic data we know about

convergence is derived from the well resolved block velocities, and because the final uncertainty is

for

dicted

faults

16-20.

On

fault

21,

Weldon

and

much smaller

than the prior uncertainty.

divergence

across San Andreas

The preF is suspect

Humphreys call for about 25 mm yr-‘, approximately 10 mm yr-i less than predicted by Bird and Rosenstock. The Weldon and Humphreys version is preferred by our final model. To make up for the 10 mm yr-’ difference from Bird and

because it depends greatly on the assumed unity of block L. Bird and Rosenstock (1984) found no geologic evidence for significant displacements between several smaller blocks, so we combined them into block L. The motion of this block is

Rosenstock on the southern San Andreas, and another 10 mm yr ’ difference on the San Jacinto, Weldon and Humphreys invoke 15-20 mm yr-’ right lateral displacement on offshore faults. Unfortunately we cannot test this hypothesis with presently available geodetic data. The last word is

poorly resolved, so we discount the predicted divergence and omit it from Fig. 8. The divergence across the Garlock W is small and probably not reliable, even though it is formally significant. The Pleito block (K) is small, it only has two monuments, and the complex faults around it may be

178

poorly

represented

boundaries. vergence

and

possible

error causing

in line

such

di-

length.

rejected

sub-

convergence where

It is not im-

San Andreas,

possible

sys-

studied

would cause local con-

gence. However,

our estimated

true tectonic convergence

(section

F) so

and poorly

and

difference

between

the fault

divergence

Pleito

exhibit

deficit

Fault

the

motions

convergence

on both faults,

like

measuring

the block and surface

trace.

sig-

Fault convergence,

exceeds

implying

a net

near the fault trace. The implied

diver-

gence

is not

significant

in either

case,

faults

could

be locked

at the surface.

so these For

the

Pleito fault both the block and fault convergence depend very heavily on just two trilateration lines (numbers

54 and 60 in Table 5) so these estimates

are not very robust even though tainty is relatively small. Some unforgettable

the formal

uncer-

faults

converrate is

relatively modest, and it is not significant on the Cucamonga fault. Thus our results disagree only slightly with Unfortunately,

Madre

fault slip, is a displacement

block convergence

secular

However, this hypothesis inadequately the observed widespread convergence. We indicate

the Sierra

fault convergence.

an apparent

explanation is that end effects outside the array (on the central

believe our estimates

Only nificant

near

However,

for example)

of the San Andreas rate there is uncertain

in our model results

(Savage and Prescott, 1983; Jackson et al., 1983; Savage and Gu, 1985) and none has been identified that would cause a spurious secular dilatation.

traction. explains

in our study were close to the south-

section

resolved.

and eastern

errors have been quite exhaustively

Another possible from fault motion

monuments ernmost

the dislocation

accumulation

is observed.

that the convergence

decrease tematic

(1986)

crustal

thickening

block

estimated

W from Fig. 8.

because

cause massive

from systematic

idealized the

across the central

Ranges

only modest

our

omit

Humphreys

convergence

Transverse should

we

on the Garlock

Weldon stantial

by

Thus

those of Weldon and Humphreys. the lack of data for monuments

south of the Cucamonga fault precludes an accurate estimate of convergence there. Similarly, we lack the data to confirm the convergence near

The following faults have cant final estimates of block

statistically slip, block

gence, or both: the Pine Mtn E, the San Cayetano, Santa Susana W and E, San Gabriel N and W, Sierra Madre, San Andreas A to F, White Wolf W, Pleito, and Garlock W and E (faults 8-14. 16-22, and 24-26). All fault segments with statistically significant fault slip or fault convergence

Ventura (in block F), predicted both by Weldon and Humphreys (1986) and by Bird and Rosenstack (1984).

are in the above list. These faults clearly

Dislocation

Some forgettable

rates, fault slip, and fault convergence

The dislocation rates, listed in Table 2, are generally well resolved. The Santa Susana W and E, San Gabriel W, Sierra Madre, Pleito, Santa Monica, and all segments of the San Andreas have statistically significant dislocation rates. The rates are projected into their tangential (fault slip) and normal (fault convergence) components in Table 3. Significant fault slip occurs on the Santa Susana E, Sierra Madre, and all branches of the San Andreas. Afterslip of the 1971 San Fernando earthquake may contribute to the negative fault slip deficit, and the positive creep rate, estimated for the Santa Susana E. None of the geodetic

significonver-

ute

to

Transverse

observed

geodetic

deformation

contribin

the

Ranges. faults

Faults not mentioned in the paragraph above are unimportant in explaining geodetic deformation. They could have been left out of the model with no adverse consequences except that we would have had to assume, rather than demonstrate, their insignificance. The forgettable faults are the Ozena; Big Pine W, C, and E; Pine Mtn W and E; Santa Ynez; Cucamonga; White Wolf E; and Santa Monica. We do not claim that the forgettable faults are geologically or seismologically unimportant, or that they have no seismic risk. They might become significant if we had more extensive or accurate geodetic data. However, except for the

179

Cucamonga, gence

their

block

are limited

slip and

block

by the presently

geological

conver-

available

data

If we neglected and

the forgettable

another,

and

another,

leaving

a total

vantage

of our

Bayesian

that

faults, we could

blocks A, B, C, and D, into one block, H

J into

we do not have

advance:

observable

similarly of seven inversion

to make

We can propose

and later simplify

F and

L into

blocks.

An ad-

technique

such decisions

a complicated

it as justified

is in

model

geodetic

Madre

trilateration

ingly detailed

picture

southern

poorly

resolved

estimate

and seldom

by more

than

2, is generally

differs from the prior

twice the standard

error.

Only for the San Gabriel W, the Sierra Madre, and the San Andreas D and E does the resolution exceed strongly

50%. The calculated length rates are nonlinear functions of the fault width for

shallow faults, so the reported standard deviation for the San Gabriel fault may not reflect its true uncertainty.

75 km in extent.

unambiguous

data

crustal

provide

a surpris-

of crustal deformation California

region.

of the same kind would and earthquake

fault segments listed in Table

about

show

the

wider

and White Wolf faults.

tectonic

The fault width,

network:

data

Available

data Fault width

and

shortening of about 6 mm yr-r normal to the San Andreas system, mostly on the San Gabriel, Sierra

complex

by the data.

faults,

region must be considerably

than the geodetic The

In any case there is geodeti-

slip on diverse

plate boundary

to a few mm yr-‘. combine

cally

estimates.

in the

Additional

have great value in

prediction

studies.

have uncertainties

These

of more than

mm yr -’ in block slip or block convergence, they should

be targets

for future

geodetic

3

and

studies:

Cucamonga, San Andreas A to C, White Wolf W and E, Pleito, and Santa Monica. VLBI data should strengthen

our knowledge

of cumulative

tion across the Transverse available as of November

deforma-

Ranges, but the data 1984 do not provide

sufficiently accurate velocity estimates observations span too little time.

because

the

Acknowledgements Conclusions We thank

the U.S. National

Aeronautics

and

Estimated block velocities are well resolved. They imply significant strike slip at depth on the Pine Mtn E, Santa Susana W, San Gabriel N, San

Space Administration who provided support via grant NAG 5447. We thank Dr. James Savage, Dr.

Andreas A to F, and White Wolf W, with significant convergence on four frontal faults (San

supplying

the data

collection

and

Cayetano, Sierra

Santa

Madre)

Susana,

San

Gabriel

as well as the Pleito

W, and

and

White

Will Prescott,

and Ms. Nancy

King for generously

explaining

reduction.

Prof.

details Peter

of its

Bird

pro-

vided many valuable suggestions during the course of this project. Dr. Ray Weldon corrected an error

Wolf W. In contrast, the Ozena, Big Pine, Pine Mtn, Santa Ynez, White Wolf E, and Santa Monica

in an earlier draft.

have insignificant displacement rates amounting to no more than a few mm yr- i. The role of the Cucamonga is uncertain, but it could converge as much as 10 mm yr i. Cumulative motion across the geodetic array is revealed in the relative block velocity between the Malibu and Mojave blocks, which amounts to only 18 mm yr-’ in the direction N49O W. This estimate is approximately half of the geologically determined rate. The apparent

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