Joint Inversion of Seismic and Magnetotelluric Data in the Parkfield Region of California Using the Normalized Cross-Gradient Constraint

Ninfa L. Bennington1*, Haijiang Zhang2, Clifford H. Thurber1, Paul A. Bedrosian3

*ninfa@geology.wisc.edu1University of Wisconsin-Madison 2Laboratory of Seismology and Physics of the Earth’s Interior, University of Science and Technology of China3US Geological Survey

Revisions toPure and Applied Geophysics

March 9, 2014

Abbreviated title: Joint inversion of seismic and MT data at Parkfield

Summary

We present jointly inverted models of P-wave velocity (Vp) and electrical resistivity for a

two-dimensional profile centered on the San Andreas Fault Observatory at Depth (SAFOD).

Significant structural similarity between main features of the separately inverted Vp and

resistivity models is exploited by carrying out a joint inversion of the two datasets using the

normalized cross-gradient constraint. The joint inversion scheme uses a normalized cross-

gradient penalty function to achieve structurally similar Vp and resistivity images that adequately

fit the seismic and magnetotelluric (MT) datasets. The new inversion code, tomoDDMT, merges

the seismic inversion code tomoDD and the forward modeling and sensitivity kernel subroutines

of the MT inversion code OCCAM2DMT. TomoDDMT is tested on a synthetic dataset and

demonstrates the code’s ability to more accurately resolve features of the input synthetic

structure relative to the separately inverted resistivity and velocity models. Using tomoDDMT,

we are able to resolve a number of key issues raised during drilling at SAFOD. We are able to

infer the distribution of several geologic units including the Salinian granitoids, the Great Valley

sequence, and the Franciscan formation. The distribution and transport of fluids at both shallow

and great depths is also examined. Low values of velocity/resistivity attributed to a feature

known as the Eastern Conductor (EC) can be explained in two ways: the EC is a brine-filled,

high porosity region, or this region is composed largely of clay-rich shales of the Franciscan. The

Eastern Wall, which lies immediately adjacent to the EC, is unlikely to be a fluid pathway into

the SAF’s seismogenic zone due to its observed high resistivity and velocity values.

Key words: Inverse theory, Joint Inversion, Seismic tomography, Magnetotellurics

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Introduction

The San Andreas Fault Observatory at Depth (SAFOD) is a scientific deep drilling

project that penetrated the San Andreas fault (SAF) at seismogenic depths in an effort to study an

active fault zone and address key questions regarding fault mechanics and earthquake generation

(Zoback et al., 2010). The SAF marks the plate boundary between the Pacific plate to the

southwest (SW) and the North American plate to the northeast (NE). In the Parkfield region of

central California, the southern Coast Ranges, located SW of the SAF, are composed of granitic

rock and are inferred to be part of the Salinian block (Page et al., 1998). In the same region, the

Franciscan formation lies to the NE of the SAF. The Franciscan formation is composed of

greywacke sandstone, shale, chert, and volcanic rocks, and is intruded by serpentinized

peridotites (Ernst, 1970). Moving further NE of the SAF, unmetamorphosed sedimentary rocks

of the Great Valley sequence are observed to unconformably overlay rocks of the Franciscan

block (Page et al., 1998).

The SAFOD drill site, located just NW of Parkfield, is ~1.8 km SW of the SAF surface

trace (Figure 1). The SAFOD project has spurred numerous geophysical studies of the Parkfield

region in the past few years in order to better characterize the crustal structure (e.g. Thurber et al.

(2003, 2004), McPhee et al. (2004), Roecker et al. (2004), Unsworth and Bedrosian (2004), Hole

et al. (2006), Bleibinhaus et al. (2007), Zhang et al. (2009)). While the SAFOD drilling project

has provided unique in-situ information about the geologic setting of the SAF at Parkfield, a

number of key issues have remained unresolved. Franciscan rocks outcropping ~3 km NE of the

SAF trace were expected to be encountered in the SAFOD main hole (MH) marking the

borehole’s passage into the North American plate. Instead, rocks from the Upper Great Valley

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sequence were discovered (Zoback et al., 2010). This is surprising given that the Great Valley

Sequence at Parkfield is located to the NE of the Franciscan terrain and normally lies structurally

above it.

Eberhart-Phillips et al. (1995), Thurber et al. (2003), and Unsworth and Bedrosian

(2004) have inferred the existence of a fluid rich basin on the NE side of the SAF near SAFOD,

again where the Franciscan rocks would be expected. Adjacent to this inferred basin is an area

termed the Eastern Wall (EW) that Unsworth and Bedrosian (2004) have suggested provides a

pathway for fluids entering the fault zone from the east. Based on evidence from seismic

attenuation, however, Bennington et al. (2008) proposed that the EW is not a significant fluid

pathway to the SAF’s seismogenic zone. Alternatively, Becken et al. (2011) have shown possible

evidence that fluids migrate from upper mantle depths into the NE fault block in this region

thereby causing the mechanically weak fault here. Thus, uncertainty remains about the location

of the Franciscan rocks at depth and the distribution and transport of fluids in this region.

On the SW side of the SAF, granitic rocks of the Salinian block were encountered at

shallow depths (granite at ~760 m and granodiorite at ~1450 m) within the SAFOD MH (Zoback

et al., 2010). However, as drilling advanced to greater depths and deviated from vertical, a

package of sedimentary rocks was encountered. Prior to drilling, it was thought that this location

was composed of granitic rocks within the damage zone of the SAF.

Magnetotelluric (MT) and seismic models provide complementary insight into the

subsurface and have provided key information for SAFOD. However, due to limitations inherent

in each of these methods, separate inversions for resistivity and velocity models (the standard

approach) may not be optimal. Examples of these inherent difficulties include seismic imaging of

4

a low velocity body surrounded by higher velocity material or MT imaging below a highly

conductive layer. While it is difficult to reliably link these two geophysical methods with

empirical models, structural similarities have been observed in the main features of the

separately inverted models at Parkfield (Figure 2). For example, the velocity model of Zhang et

al. (2009) at SAFOD shows strong agreement in its main structural features compared to the

resistivity model of Unsworth and Bedrosian (2004). Features coincident in location include a

high velocity/high resistivity feature interpreted as Salinian granite SW of the SAF, a low

velocity/low resistivity feature corresponding to the San Andreas fault zone, and a low velocity/

low resistivity basin-like feature on the NE side of the fault.

Zhang et al. (2009) used cluster analysis to make lithologic inferences from trends in the

seismic and resistivity models at SAFOD and were able to distinguish several major lithologies.

We have taken this one step further by developing and applying a joint inversion scheme to

invert for models of P-wave velocity (Vp) and resistivity. Haber and Oldenburg (1997) first

developed a structural approach to the joint inversion of two geophysical data sets. The authors

imposed spatially coincident boundaries, denoted as structure, in the parameterizations of two

geophysical models as a means of linking the separate inversions. The penalty function accounts

for minimizing both the misfit to each data set and the difference in structure between the two

models. Since structure is identified as a change in a model with position, Haber and Oldenburg

(1997) suggested determination of the structural portion of the penalty function using local

gradients or curvature. Gallardo and Meju (2004, 2007) formulated the cross-gradient constraint

to seek two structurally similar geophysical models that minimize their respective data misfits.

This constraint penalizes the presence of non-parallel gradients.

5

Here we take advantage of the significant structural similarity between the main features

of the seismic and resistivity models at SAFOD by carrying out a joint inversion of the two

datasets using a modified version of the method of Gallardo and Meju (2004, 2007). We present

a joint inversion scheme that uses the normalized cross-gradient penalty function to achieve

structurally similar Vp and resistivity images that adequately fit the seismic and MT datasets,

respectively. The new inversion code, tomoDDMT, merges the seismic inversion code tomoDD

(Zhang and Thurber, 2003) and the forward modeling and sensitivity kernel subroutines

(Wannamaker et al., 1987; Lugao and Wannamaker, 1990) of the MT inversion code

OCCAM2DMT(Constable et al., 1987; DeGroot-Hedlin and Constable, 1990). Synthetic tests

carried out using the normalized cross-gradient constraint show improved recovery of the

synthetic model’s structural features relative to separate inversions. Using tomoDDMT, we

obtain jointly inverted images of velocity and resistivity for the region around the SAFOD site.

From these images, we are able to address key geologic issues at Parkfield that have, until now,

remained unresolved.

Seismic and MT datasets

The seismic dataset we use to invert for the Vp model at Parkfield (Figure 1a) is identical

to that of Zhang et al. (2009). While Gallardo and Meju (2004, 2007) use only shot data in their

inversion, our study uses both local earthquake and shot data. The majority of the seismic data

are collected from the UW/RPI Parkfield Area Seismic Observatory (PASO) array. However,

data are also included from the UC-Berkeley High-Resolution Seismic Network (HRSN) and

USGS seismic network stations in the region around Parkfield. Borehole data were also collected

6

from seismic strings deployed in the SAFOD pilot hole and MH in July 2002 (Chavarria et al.,

2003), May 2005 (by Paulsson Geophysical Services, Inc.), and December 2004 and November

2006 (by the SAFOD project). Shot data consist of two refraction/reflection survey lines running

perpendicular to the SAF and through SAFOD: a 5 km line of Hole et al. (2001) and Catchings

et al. (2002) and a 46 km line of Hole et al. (2006). Additional explosion data used were from

PASO “calibration” shots carried out in 2002, 2003, 2004, and 2006. The seismic dataset consists

of 574 earthquakes, 836 shots (shots of known origin time and location), and 153 blasts (shots of

known location but uncertain origin time). Arrival times include 65,376 P-wave arrivals and

489,000 P-wave differential times. Differential times between nearby event pairs observed at

common stations were calculated for the 574 earthquakes using the cross-correlation package

BCSEIS (Du et al., 2004).

The MT dataset at Parkfield (Figure 1b) consists of three MT profiles that cross the SAF

and are offset at distances of ~2-4 km along strike (Unsworth and Bedrosian, 2004). For this

study, we only used the MT data from line 1. Near the fault, continuous stations (electric field

dipoles laid end to end) were spaced 100 m apart with additional, more widely spaced stations

placed farther from the fault to constrain the regional structure. Observations include full tensor

impedance data in the frequency band 100-0.001 Hz. We used the transverse electric (TE) mode

and transverse magnetic (TM) mode data, and vertical magnetic field transfer functions (Tyz). All

data were oriented along the SAF strike (N41W). Following Unsworth and Bedrosian (2004), we

included the solution for static shifts within the inversion and applied the following inversion

error floors: ρTM 10%, ρTE 20%, phase 5%, and Tzy 0.02.

7

Normalized Cross-gradient Constraint

By carrying out a joint inversion of the two datasets with a modified version of the

method of Gallardo and Meju (2004, 2007), we can exploit the significant structural similarity

between the main features of the separately inverted seismic and resistivity models at SAFOD

(Figure 2a-b). Gallardo and Meju’s (2004, 2007) technique uses a weighted penalty function to

encourage the gradient fields of the two geophysical models to become parallel or structurally

similar. To measure the structural similarity, the cross-gradient value t is used:

t(x,z) = ∇m1(x, z) x ∇m2(x, z) (1)

where ∇m1(x, z) and ∇m2(x, z) are the gradients evaluated at a specified cell in the two

geophysical models. The two models are considered structurally similar when the sum of the

absolute values of the cross-gradients in all cells is minimized, balanced against fitting the data.

Figure 3 displays an example of Vp and resistivity anomalies in (a) the near surface and

(b) at depth. As represented in Figure 3, values of Vp and resistivity generally vary more rapidly

in the near surface than at depth. As a result, the cross-gradient values in the near surface are

observed to be an order of magnitude greater than those at depth (Figure 3). Thus, one possible

shortcoming of the cross-gradient algorithm in equation (1) is that, as constructed, it more

heavily weights cross-gradient values in the near surface.

We have addressed this by reformulating the penalty function based on the normalized

gradients of the geophysical models:

!

!

tN(x,z) =" ˆ m

1(x,z) #" ˆ m

2(x,z) (2)

where !

!

" ˆ m 1 (x, z) and !

!

" ˆ m 2 (x, z) are the normalized gradients. As shown in Figure 3, the

normalized cross-gradient penalty function removes the dominance of normally high cross-

8

gradient values in the near surface. When the normalized gradients at a co-located point in two

geophysical models are exactly parallel, tN is minimized, or zero, at that point. From equation

(2), we see that when gradients are zero in the x and z directions for one or both geophysical

models, tN is also zero, or minimized, at that point. Thus, when one geophysical model is absent

of change (gradient equal to zero) where the other model shows an anomaly (a non-zero

gradient), the region absent of change will not be forced to become structurally similar to the

other model.

To define a discrete version of equation (2), we follow Gallardo and Meju (2004) and use

central difference to estimate the gradients of seismic velocity and resistivity in the x and z

directions. The resulting gradients are then scaled by their respective vector lengths. This

modifies equation (2) to:

!

!

tN(x,z) "

(mSr#m

Sl)$z

(mSr#m

Sl)2$z2 + (m

Sb#m

St)2$x2

*(m

Rb#m

Rt)$x

(mRr#m

Rl)2$z2 + (m

Rb#m

Rt)2$x2

%

&

' '

(

)

* *

#(m

Sb#m

St)$x

(mSr#m

Sl)2$z2 + (m

Sb#m

St)2$x2

*(m

Rr#m

Rl)$z

(mRr#m

Rl)2$z2 + (m

Rb#m

Rt)2$x2

%

&

' '

(

)

* * (3)

where the capital subscript of m indicates the seismic (S) or resistivity (R) model and the lower

case subscript indicates the top (t), bottom (b), right (r), or left (l) of center node. Equation (3) is

linearized with respect to the model perturbations using a Taylor series expansion such that:

!

!

"mR

#tN

#mR m R0,S0

$

%

& &

'

(

) )

+ "mS

#tN

#mS m R0,S0

$

%

& &

'

(

) )

= *! t

N0

(4)

where ∇mR and ∇mS are perturbations to the resistivity and velocity models, respectively,

!

!

"tN

"mR mR0,S0 and

!

!

"tN

"mS mR0,S0 are the partial derivatives of equation (3) with respect to the previous

9

iteration’s resistivity and velocity models respectively, and !

!

! t

N0 is the vector of normalized cross-

gradient values from the previous iteration’s resistivity and velocity models calculated from

equation (3).

Joint Inversion Algorithm

The goal of our joint inversion scheme is to minimize an objective function consisting of

three parts: (1) the misfit between observed and predicted arrival times for the seismic data, (2)

the misfit between observed and predicted apparent resistivity and phase data for the MT data,

and (3) the normalized cross-gradient constraint. The complete system of equations representing

this algorithm is:

!

!

Gs 0

"sL

s 0

#sI 0

0 G

r

$

0 "rL

r

0 #rI

µGCG,s

µG

CG,r

$

%

&

' ' ' ' ' ' ' ' ' ' ' ' '

(

)

* * * * * * * * * * * * *

+m

s

+mr

(

) *

%

& ' =

ds

0

0

dr

0

0

,µˆ t 0

(

)

* * * * * * * * *

%

&

' ' ' ' ' ' ' ' '

(5)

where Gs, Gr, GCG,s, and GCG,r are the sets of partial derivatives related to the velocity and

earthquake relocation inversion, resistivity inversion, and the velocity and resistivity portions of

the normalized cross-gradient constraint respectively, LS and LR are the velocity and resistivity

model smoothing matrices, λs and λr are the velocity and resistivity model smoothing weights

respectively, εs and εr are the velocity and resistivity damping weights respectively, I is the

10

identity matrix, µ is the normalized cross-gradient constraint weighting term, β is the resistivity

inversion scaling term, and ds and dr are the seismic and resistivity residuals, respectively.

The joint inversion is carried out within the newly established code, tomoDDMT, which

merges the 3-D seismic inversion code tomoDD (Zhang and Thurber, 2003) and the forward

modeling and sensitivity kernel subroutines (Wannamaker et al., 1987; Lugao and Wannamaker,

1990) from the 2-D MT inversion code OCCAM2DMT(Constable et al., 1987; DeGroot-Hedlin

and Constable, 1990). The normalized cross-gradient constraint is applied along the 2D MT

model plane that is spatially coincident with one plane of the 3D Vp model. TomoDD is used as

the foundation of our joint inversion algorithm and necessary subroutines from

OCCAM2DMTare incorporated into the algorithm to solve the MT inverse problem. The

normalized cross-gradient constraint equations (3) and (4) are added to link the resistivity and

seismic velocity inversions.

In separate inversions, seismic and resistivity models are generally determined using very

different gridding schemes due to resolution differences. For seismic inversions, grid nodes are

preferably finest near areas of dense seismicity where grid nodes are sampled the most

frequently. For MT, depth of penetration scales nonlinearly with frequency, and hence inversion

cell size is preferably finest in the near surface centered on MT stations, and the size of the grid

cells increases both with depth and laterally away from the MT stations. Application of the

normalized cross-gradient constraint requires a common grid. The difference in gridding is

overcome by projecting both models onto a regular, finer grid using the nearest neighbor

algorithm, which permits a linkage of the two models without loss of resolution in either. Next,

the calculations of the normalized cross-gradient G matrix and residual vector are carried out.

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The resistivity and velocity portions of the normalized cross-gradient G matrix are then projected

back to their original, irregular grids using the nearest neighbor algorithm. A simultaneous

inversion is then carried out to determine the seismic and resistivity models subject to the

normalized cross-gradient constraint (equation 5). TomoDDMT employs the LSQR algorithm

(Paige and Saunders, 1982) for the simultaneous solution of both models.

The LSQR algorithm used to solve the inverse problem does not permit calculation of the

full resolution matrix. As an alternative, we estimate the Vp model quality using the derivative

weight sum (DWS) distribution. This parameter reflects the density of rays passing near a grid

node, where weighting is calculated based inversely on each ray’s distance to a particular grid

node (Toomey and Foulger, 1989). Based on prior experience, well-resolved areas of the Vp

model correspond to DWS values > 20% of the average DWS value. For our separate and jointly

inverted Vp models, this equates to DWS values >30. Using the same seismic dataset, Zhang et

al. (2009) compute the full resolution matrix and find that well resolved regions of the Vp model

correspond to the region we infer to be well-resolved via DWS values.

Synthetic Testing of Joint Inversion Algorithm

A test of the normalized cross-gradient algorithm, tomoDDMT, was carried out using

synthetic data generated from the Vp and resistivity models shown in Figure 4. Synthetic data

were generated for the actual seismic and MT station distribution at Parkfield, and the forward

and inverse problems were carried out using the model discretization shown in Figure 5.

Gaussian noise was added to both the seismic and MT (both resistivity and phase) data. The

velocity and resistivity models resulting from independent inversion in tomoDD and

12

OCCAM2DMTare shown in Figure 6a. Results from the joint inversion of synthetic data in

tomoDDMT are shown in Figure 6c.

Due to the large size of the seismic and MT datasets at Parkfield, the joint inversion

algorithm is computationally expensive. To minimize these computational time demands, we

generate starting resistivity and Vp models for the joint inversion problem that are near to, but

not at the models’ global minima of data misfits. The starting Vp and resistivity models are

obtained by applying a Gaussian smoothing function to the separately determined models of

Figure 6a. The resulting smoothed starting models contain the basic structure required by their

respective datasets and allow progressive iterations of the joint inversion to converge to a

minimum misfit solution while promoting structural similarity between the Vp and resistivity

models.

Experimentation with the synthetic dataset revealed that the jointly inverted models had

the most robust recovery of synthetic features and the lowest normalized cross-gradient misfit

when the cross-gradient weighting was structured as:

µ=wcg×ei (6)

such that µ increases exponentially over progressive iterations, i, and is weighted by the fixed

term wcg. As structured in equation (6), the weighting scheme prevents the normalized cross-

gradient constraint from dominating earlier iterations, but allows it to become increasingly

important over further iterations as the two geophysical models approach their minimum misfit

solutions. We tested wcg terms ranging from 0.001 to 1000 and found that a wcg=1 produced a

solution that best recovered synthetic model features. Two additional cross-gradient weighting

schemes were tested in this study but were found to do poorer jobs recovering the synthetic

13

features: 1) µ was kept at a constant value over all progressive iterations, and 2) µ was linearly

increased with each iteration. Further details on these alternative-weighting schemes are

discussed in E-supp A. Additional weighting terms for the inversion (λs, λr, εs, and εr in equation

(5)) were also examined for a range of values. These weighting terms were tuned for the inverse

solution that best recovered features of the synthetic velocity and resistivity models without

introducing artifacts to the models. This occurred for λs = 20, λr = 3.0, εs = 35, and εr = 30.

The normalized cross-gradient constrained Vp model has a root mean square (RMS) data

misfit of 0.01 seconds, which is a reduction in misfit relative to the separately inverted Vp

model’s misfit of 0.02 seconds. The jointly inverted resistivity model has a χ2 misfit of 0.96. A χ2

misfit of 1 would indicate that the resistivity model fits the MT data to within its uncertainties.

The separately inverted resistivity model has a χ2 misfit of 1.00. Thus, the separate and jointly

inverted resistivity models are fitting the MT data equally as well.

We compare the separately and jointly inverted resistivity and Vp models to the main

features of the input synthetic model (Figure 6). Main features of the input synthetic model

include: a) a low Vp, low resistivity cross feature at the center of the model; strong lateral

contrasts in velocity and resistivity at b) X=-3 km c) X=0 km and d) X=4 km; and horizontal

surfaces at e) Z=1.5 km, f) Z=5 km, and g) Z=7 km depth (indicated in Figures 4 and 6). In both

the separately and jointly inverted resistivity models (Figure 6a and c), feature (a) is recovered as

a compact, circular anomaly with a factor of 10 increase in amplitude relative to the input cross-

like feature. The separately inverted Vp model (Figure 6a) displays a diffuse anomaly of reduced

amplitude compared to the input feature. For the jointly inverted Vp model, the feature remains

diffuse, but approaches the amplitude of the cross-shape present in the input model..

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The lateral contrast in the velocity and resistivity of feature (b) is better recovered by the

joint inversion. At X = -6 to -3 km, the jointly inverted Vp and resistivity model begin to

converge toward the input synthetic model values. Also, the Vp model shows contours of

decreased velocity at X > -3 km aligning closer to vertical feature (b). The shape of the high

velocity/low resistivity “step” composed of features (c) and (f) is also well recovered in the

jointly inverted velocity model. The corresponding resistivity model (Figure 6c) does not

completely image this “step” feature (i.e. feature (c) is absent), but it is notable that low

resistivities between X = -3 and 1 km have migrated deeper in order to approach the horizontal

resistivity contrast, feature (f). The separately inverted Vp and resistivity models are unable to

recover this “step” feature. The lateral velocity and resistivity contrast of feature (d) is also better

recovered by the joint inversion (Figure 6a and c). Between X = 5.5 and 10 km, low resistivity

values in the jointly inverted resistivity model more closely resemble the synthetic model values.

Also, relative to separate inversion, the jointly inverted Vp model shows vertical contours of low

velocity approaching nearer to feature (d) at X = 3 to 4 km.

The horizontal resistivity/velocity contrast of feature (e) is well recovered in both the

separate and jointly determined resistivity models. Recovery of this feature in both Vp models is

less robust. Notably, neither the separate nor the jointly inverted resistivity models are able to

recover the horizontal surface at 7 km depth (feature (g)). Instead, both models recover high

resistivity values as deep as 5 km depth at this location. The resulting jointly inverted Vp model

shows a decrease in velocity by as much as 1 km/s in this same region resulting in an

improvement to the lateral contrast seen at feature (g) relative to the separately inverted Vp

model. In summary, these results demonstrate the utility of the normalized cross-gradient

15

constraint under the actual station distribution (both MT and seismic stations) and model

discretization used in the inversion of the Parkfield data (discussed in the following section).

With this constraint, we show the overall improvement in recovery of input synthetic model

features both in amplitude and shape.

Starting Models for Inversion of the Parkfield Dataset

To generate starting models for our joint inversion work at Parkfield, we applied the same

process as was used to create our synthetic starting models (discussed in the “Synthetic Testing

of Joint Inversion Algorithm” section). Using the Parkfield seismic and MT datasets, the

separately determined Vp and resistivity models were solved for using tomoDD and

Occam2DMT, respectively (Figure 7). The Vp model has a RMS misfit of 0.10 seconds and the

resistivity model has a χ2 misfit of 1.90. The resistivity structure was also separately inverted for

using tomoDDMT with the normalized cross-gradient constraint turned off (E-Supp Figure B1).

The magnitude and distribution of features for this resistivity model were quite similar to that

determined in Occam2DMT, however, the resulting χ2 misfit of 2.10 was slightly higher. For this

reason, we chose to use the OCCAM2DMT-derived resistivity model to create our joint

inversion starting resistivity model. A Gaussian smoothing filter was applied to the separately

determined resistivity and Vp models, with the resulting smoothed structures used as starting

models for the joint inversion.

Zhang et al. (2009) used the same dataset as this study to invert for Vp, Vs, and Vp/Vs

models using tomoDD. For this reason, we used their 3-D Vp model to obtain our starting

velocity model. The grid used to invert for the Vp model in Zhang et al. (2009) is interpolated

16

onto a slightly finer grid for the joint inversion in order to better balance the number of velocity

model parameters solved for relative to the number of resistivity model parameters. This avoids

changes to the resistivity model dominating the solution. A map view of the grid nodes used for

velocity inversion is shown in Figure 5a. In the depth direction, grid nodes were placed between

-0.5 and 10.0 km below sea level (b.s.l.) with 0.5 km spacing between all nodes. Figure 5b and c

display the grid cells used in the forward and inverse resistivity problems, respectively.

Determination of Optimal Weighting Terms for Joint Inversion

The G matrix representing the joint inversion, equation (5), contains four different

systems of equations: the velocity, earthquake relocation, and resistivity inversion, and the

normalized cross-gradient constraint that links them. Initially, the inversion was performed

without considering the relative scale of each system of equations. This yielded a solution where

perturbations to the velocity model were zero or the misfit to the seismic dataset was divergent,

while the resistivity model misfit decreased and the normalized cross-gradient misfit increased

relative to the starting models. Singular value analysis revealed that singular values from the

resistivity portion of the G matrix dominated the solution by as much as two orders of

magnitude. We tested β factors ranging from 1 to 10000 to down-weight the resistivity portion of

the G matrix. A value of β=10 produced a convergent solution where the resistivity, velocity, and

normalized cross-gradient misfits all decreased. Values of β below this yielded noisy or unstable

solutions, or a divergent resistivity model.

To determine optimal weighting terms for the inversion via trade-off curve analysis (wcg,

λs, λr, εs, εr equation (5)), the Parkfield seismic and MT datasets were jointly inverted and the

17

resistivity or velocity model norms versus resistivity or velocity misfits, respectively, were

examined (Figure 8). The model norm is calculated as:

!

M1"M

0= (M

1(i, j) "M

0(i, j))

2

j=1

n

#i=1

m

# (7)

where M0 and M1 are the initial and final models, respectively, and m and n are the number of

grid nodes in the x- and z-direction in the model space, respectively. While we seek weighting

terms that minimize the data misfit, models producing the lowest misfit are the most oscillatory

(i.e., the highest model norm) and can contain spurious features. Conversely, models with the

highest data misfit and lowest model norm are most similar to the starting model (i.e., they lack

model features necessary to fit the data). Thus, weighting values at the corners of these trade-off

curves are considered optimal (Aster et al., 2013). Figure 8a-d shows the trade-off curves

obtained for a suite of smoothing and damping values for the resistivity and velocity portions of

the inversion. These curves suggest optimal weighting terms of λs = 340, λr = 4.5, εs = 250, and εr

= 100.

Inversion of the Parkfield data using these smoothing and damping parameters yielded

velocity and resistivity models that were still quite similar to the starting models. We also

experimented with solving for the Parkfield resistivity and velocity structure using smoothing

and damping values determined from synthetic testing of the joint inversion algorithm. Using

these weighting terms, we obtained solutions for the resistivity and Vp structure that contained

erratic oscillations in the resistivity and velocity models suggesting that the models were under-

regularized.

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In solving for the regional-global velocity structure of Sumatra, the Andaman Islands, and

Burma, Pesicek et al. (2010) attempt to identify the optimal smoothing and damping terms for

their inversion using both trade-off curve analysis and the inversion of synthetic data. Similar to

our own experimentation, they find that trade-off curve analysis and synthetic testing reveal

weighting terms that, respectively, underestimated and under-regularized the resulting solutions.

Instead, Pesicek et al. (2010) note that the most reasonable velocity model was achieved when

they used middle ground weighting terms between those values determined via trade-off curves

and synthetic testing. A reasonable model is interpreted as the solution that minimizes spurious

features within the model space while achieving a model structure that is not highly similar to the

starting model. Using this approach, we identify the optimal smoothing and damping values: λs =

240, λr = 3.5, εs = 200, and εr = 40.

The weighting term wcg was determined by examining the resistivity and velocity data

misfits versus the normalized cross-gradient misfit over a suite of wcg values (Figure 8e). In

order to best observe the effect of the normalized cross-gradient constraint on the recovery of

features within the velocity and resistivity models, evaluation of wcg was carried out after 3

iterations. An increase in wcg produced a lower normalized cross-gradient misfit as well as an

increase in the misfits to the velocity and resistivity data (Figure 8e). The value of wcg was

selected at the corner of the curves in Figure 8e. This is where the resistivity or velocity misfit

and the normalized cross-gradient are both being reduced but neither is minimized to the

detriment of the other. For both the velocity and resistivity data, the optimal wcg = 10.

Results and Discussion

19

The jointly inverted Vp and resistivity models are presented as SW-NE cross sections

oriented approximately perpendicular to the SAF trace where the SAFOD site is located at X=0

km (Figure 7). The white shading overlain on the Vp model masks poorly resolved regions of the

velocity model. For comparison to these jointly inverted models, we display the velocity (Zhang

et al., 2009) and resistivity models determined separately with tomoDD and OCCAM2DMT,

respectively (Figure 7a). These models were determined using the same seismic and resistivity

datasets as the joint inversion. It should be noted that differences between the separately inverted

resistivity model we obtain (Figure 7a) and the resistivity model of Unsworth and Bedrosian

(2004) (Figure 2b) are due to the choice of inversion scheme (OCCAM2DMTvs. Rodi and

Mackie (2001)). We have found that the joint inversion for 3D Vp and 2D resistivity structure

along a coincident 2D cross-section (at Y=0 km) has minimal impact on the 3D velocity structure

away from this cross-section. The absolute differences in velocity between the separate and

jointly inverted models are at a maximum at Y=0 km (the coincident cross-section) and diminish

to <0.3 km/s by Y=±2 and <0.1 km/s by Y=±4 km (E-Supp Figure C1). Differences beyond the

Y=0 km cross-section are due to the global smoothing regularization, which smears out changes

due to the normalized cross-gradient constraint.

As noted previously, the separately inverted Vp has a RMS misfit of 0.10 seconds and the

resistivity model has a χ2 misfit of 1.90. Unsworth and Bedrosian’s (2004) resistivity model is

approximately the same with a χ2 misfit of 1.8 (Figure 2).The normalized cross-gradient

constrained Vp and resistivity models have misfits of 0.08 seconds and 1.20, respectively. The

normalized cross-gradient constrained resistivity model has a 37% decrease in RMS misfit

relative to the separate inversion, and we observe a visual improvement in the fit of the model

20

response (predicted apparent resistivity and phase) to the observed MT dataset at 10 stations (two

of which are shown in Figure 9). Of the 54 MT stations, the remaining 44 sites show neither an

improvement nor degradation of the model response’s fit to the MT dataset. Figure 7a and b

show that the normalized cross-gradient misfit calculated for the separately determined velocity

and resistivity models is 1892. Joint inversion under the normalized cross-gradient constraint

yields models with a greatly decreased normalized cross-gradient misfit value of 809. Thus, the

jointly inverted models have increased model similarity under the normalized cross-gradient

constraint while fitting the data better than the separate inversions.

For the jointly inverted models, there is an overall contrast in Vp and resistivity values

across the fault: the SW side of the fault is dominated by higher Vp and resistivity values and the

NE side by lower Vp and resistivity values (Figure 7b). These observations agree with the results

of Zhang et al.’s (2009) Vp cross section through SAFOD (Figure 2a and 7a), Unsworth and

Bedrosian’s (2004) 2-D resistivity model at Parkfield (Figure 2b), and the resistivity model

separately inverted for in OCCAM2DMT(Figure 7a). This change in values of resistivity and

velocity across the fault is inferred to reflect an overall change in geology. On the SW side,

granitic rocks inferred to be part of the Salinian block are present and on the NE side Cenozoic

sedimentary rocks and metasedimentary rocks of the Franciscan terrane, both seen in outcrop,

are inferred.

We examine features of the jointly inverted resistivity and Vp model cross-sections

through the SAFOD site moving from SW to NE through the models (Figure 7b). The first

feature encountered on the SW side of the fault is the moderate resistivity and velocity area

(ρ=50 Ωm and Vp=5.5 km/s). It extends from ~0.5 km NE to 6 km SW of SAFOD and from ~1

21

to 3 km b.s.l. In Figure 10, the lithologies intercepted during drilling at SAFOD as interpreted by

Bradbury et al. (2007) are overlain on this portion of the jointly inverted Vp and resistivity

models. The two upper intervals of the SAFOD borehole (interval b-c) correspond to this region

of moderate resistivity and velocity. At these intervals, heavily fractured granites and

granodiorites were intercepted. Figure 11a-b shows the geologic cross section we have

developed, overlain on the jointly inverted Vp and resistivity model cross sections through

SAFOD. The inferred location and spatial extent of the heavily fractured Salinian block granitic

rocks is displayed in the figure. Below this depth, velocity increases to as high as 6.5 km/s and

resistivity increases to a high value of 300 Ωm. This increase in Vp and resistivity with depth

suggests that the fracture content of these granitoids decreases dramatically below 2 km depth as

would be expected with increased overburden. The spatial extent of these granitoids is indicated

in Figure 11.

Immediately adjacent to these granitoids, we observe a lateral velocity drop and an

associated drop in resistivity from 0.5 to 2 km NE of SAFOD and from 0 to 2 km b.s.l. Similar to

Ryberg et al. (2012), we observe that this lateral contrast correlates spatially with the transition

from granitoids (interval b-c) to sedimentary rocks (arkosic sediments of interval d-f) within the

SAFOD drillhole (Figure 10). This interval of arkosic sediments has associated Vp and

resistivity values of 5 km/s and 30 Ωm, respectively. Portions of the model with these property

values are identified as arkosic sediments in Figure 11.

Centered immediately below the SAF trace at X=2 km is the fault zone conductor (FZC).

This feature extends as deep as 1.5 km b.s.l. in the resistivity and Vp models (Figure 7b).

Resistivity and Vp values are lower (3.5 km/s and < 10 Ωm) than those values associated with

22

the adjacent arkosic sediments. Thus, the FZC is likely associated with a different package of

sedimentary rocks, perhaps Tertiary sediments such as the Etchegoin formation, which are seen

in outcrop at this location (Figure 11). McPhee (pers.comm.) suggests that the FZC may be only

partly associated with the Etchegoin formation as this portion of the SAF shows only a subtle

magnetic anomaly, whereas Etchegoin outcrops in other locations show a significant magnetic

signature. Given the low Vp and low resistivity values, we infer this region to be heavily

fractured, which agrees with the results of Zhang et al. (2009) who suggest the area is composed

of heavily fractured sedimentary rocks.

Immediately below the FZC, the jointly inverted models show a feature of more moderate

Vp and resistivity (4.5 km/s and 30 Ωm). Figure 10 indicates that the portion of the Great Valley

sequence intercepted during drilling at SAFOD is located within this feature. For this reason, we

identify this feature as sliver of Great Valley sequence rocks (Figure 11).

NE of the FZC, there is a low velocity, low resistivity feature at X= 4 to 8 km termed the

Eastern Conductor (EC). The EC extends to 3 km depth with resistivity and Vp values of 1 to 10

Ωm and 3-4 km/s, respectively. Resistivity values decrease with increased salinity (Delleur,

1999). While resistivity of a formation can vary dramatically due to the type of fluid present (e.g.

brine versus fresh water), it is also strongly affected by the pore space and interconnectivity, or

effective porosity (Sharma, 1997). Avseth et al. (2005) note that changes in porosity can strongly

affect Vp. Thus, it is possible the EC contains an electrolytic fluid such as brine and has

substantially increased effective porosity as has been suggested by Eberhart-Phillips et al.

(1995) and Unsworth and Bedrosian (2004) (Figure 11).

23

Alternatively, the low resistivity/low velocity of the EC could represent a material change

in the rock composition. It is possible that the EC is a highly clay-rich shale zone within the

Franciscan formation (Figure 11). Shale, a rock type within the Franciscan formation, does

exhibit extremely low resistivity values, typically 5-50 Ωm (Palacky, 1987). Johnston (1987)

shows that the electrical properties of shale are controlled by the clay content and interaction

between the clay matrix and pore fluids. Tosoya and Nur (1982) look at a variety of rocks,

including shales, distinguished by pores with low aspect ratios and find that increased clay

content causes an overall decrease in Vp. Johnston and Christensen (1995) measure Vp in a

variety of shales for a range of pressures and directions relative to bedding. They find that

anisotropy in shales is large (~20-30% of Vp) and is mainly attributed to the alignment of clay

minerals. They find that between 10 and 100 MPa, the Vp of shales parallel and perpendicular to

clay mineral alignment is ~4.2-4.5 km/s and ~3.2-3.5 km/s, respectively. The value of Vp for the

EC falls between this range of parallel and perpendicular Vp values.

Contours of increased resistivity and increased velocity (40 Ωm and 5.0 km/s) extend to

X = 3 km and Z = 2 km separating the EC from local seismicity to the SW (Figure 7). This

feature corresponds to the region Unsworth and Bedrosian (2004) term the eastern wall (EW).

Unsworth and Bedrosian (2004) suggest that resistivity values are low enough in the EW to

indicate adequate porosity for the transport of fluids from what they infer is the high porosity,

brine-filled EC to the fault zone at seismogenic depths. To estimate the porosity of the EW based

on our jointly inverted resistivity model, we follow Unsworth and Bedrosian (2004) who assume

a pore fluid salinity of 30,000 ppm. This salinity taken with our observed EW resistivity values

yield an estimated porosity of 0.5 to 5.0% (Archie, 1942), which is more than a 50% decrease in

24

estimated porosity relative to Unsworth and Bedrosian (2004). Such low porosities suggest that

the EW is not a viable fluid pathway. Our jointly inverted velocity model shows EW Vp values

of ~4.5 to 5.5 km/s. We note that the Franciscan formation outcrops above the location of the

EW in the geologic map of this region (Figure 1c). Brocher (2008) shows the velocity of

Franciscan rock at Z = 3 km depth would be 5.5 km/s, which is in good agreement with the Vp

values we observe. Brocher (2008) explains that his velocity estimates reflect changes in Vp due

to an increase in overburden pressure only. Other factors such as porosity, consolidation,

induration, and lithology are not accounted for in his study. Thus, it is unlikely that high

porosities exist within the EW. Finally, the EW locates within a high Q (low attenuation) region

in the results of Bennington et al. (2008), which further suggests the EW is a region of lower

porosity, and thus an unlikely fluid pathway between the EC and the SAF at seismogenic depths.

Finally, we note that the tomoDDMT algorithm uses only first arriving P-waves to solve

for the velocity structure at Parkfield. Following the method of Bennington et al. (2013), our

joint inversion algorithm could be extended to include both fault zone head waves and direct

wave secondary arrival times identified at Parkfield. These data would provide additional

constraints on the San Andreas fault zone velocity structure. Such modifications to the

tomoDDMT algorithm are beyond the scope of this paper and will be pursued in future studies.

Conclusions

We present a joint inversion scheme that uses the normalized cross-gradient penalty

function to achieve structurally similar Vp and resistivity images that adequately fit the seismic

and MT datasets. The new inversion algorithm, tomoDDMT, merges the seismic inversion code

25

tomoDD (Zhang and Thurber, 2003) and the forward modeling and sensitivity kernel

subroutines (Wannamaker et al., 1987; Lugao and Wannamaker, 1990) from the MT inversion

code OCCAM2DMT(Constable et al., 1987; DeGroot-Hedlinand Constable, 1990). By showing

that application of the normalized cross-gradient constraint yields improved recovery of the input

synthetic model features relative to separate inversions, we demonstrate the utility of this

constraint.

Jointly inverted models of Vp and resistivity for a two-dimensional cross-section

centered on SAFOD are presented. We identify the distribution of several geologic units whose

distributions have previously remained uncertain. These include granites and granodiorites of the

Salinian block, Great Valley sequence rocks, and the Franciscan formation. The distribution of

fluids (both near the surface and at depth) is also examined. We suggest the EC could have

dramatically increased porosities and contain high salinity fluids. Alternatively, this portion of

the Franciscan formation could be composed largely of clay-rich shales causing the low

resistivities and velocities observed in the EC. Finally, we infer that the EW, which lies

immediately adjacent to the EC, is not a significant fluid pathway to the SAF’s seismogenic

zone, contrary to previous studies.

Acknowledgements

This material is based upon work supported by the National Science Foundation under

Award Number EAR-0838249 and by a Morgridge Distinguished Graduate Fellowship in

Geoscience at the University of Wisconsin-Madison. The instruments used in the Parkfield field

program were provided by the PASSCAL facility of the Incorporated Research Institutions for

26

Seismology (IRIS) through the PASSCAL Instrument Center at New Mexico Tech. The facilities

of the IRIS Consortium were supported by the National Science Foundation under Cooperative

Agreement EAR-0552316.

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Figure 1. (a) Map with location of seismic stations (blue triangles), seismicity (black dots), and shots (red stars). Figure adapted from Zhang et al. (2009). (b) Map with location of MT stations (black triangles). Red cross indicates location of SAFOD drill site. Line 1 indicates MT profile line used in this study. (c) Geologic map of area around SAFOD. Extents of this map plotted as green box in (a) and (b). Map modified from Bradbury et al. (2007).

Figure 2. Comparison of the separately inverted velocity and resistivity models at SAFOD showing strong structural similarities between the two. (a) Vp model of Zhang et al. (2009). (b) Resistivity model from Unsworth and Bedrosian (2004). SAFOD is located at X=0 km. SG=Salinian granite, FZC= Fault zone conductor, EC= Eastern conductor.

Figure 3. A hypothetical example of Vp and resistivity anomalies (a) in the near surface and (b) at depth. The example demonstrates that the cross-gradient algorithm more heavily weights cross-gradient values in the near surface whereas the normalized cross-gradient algorithm removes this preferential up-weighting of cross-gradient values in the near surface.

Figure 4. Input models of Vp and resistivity used to generate synthetic data for the testing of the normalized cross-gradient algorithm. The main features of the input synthetic models are indicated: a) a low Vp, low resistivity cross feature at the center of the model; strong lateral

32

contrasts in velocity and resistivity at b) X=-3 km c) X=0 km and d) X=4 km; horizontal surfaces at e) Z=1.5 km, f) Z=5 km, and g) Z=7 km depth.

Figure 5. Model discretization showing (a) the grid nodes used in the velocity inversion, (b) the grid cells used in the forward resistivity problem, and (c) the grid cells used in the inverse resistivity problem.

Figure 6. Synthetic testing results: (a) separate inversion results, (b) synthetic models (with main features indicated as (a)-(g)) (c) joint inversion results The overlain solid lines represent the input synthetic models of Figure 4, and the white lines overlain on the Vp models indicate the well-resolved regions.

Figure 7. Cross-section through SAFOD for 3D velocity model and 2D resistivity model showing: (a) separately inverted Vp model (left) resistivity model (middle), and CG misfit for separately inverted models (right), and (b) jointly inverted Vp model (left), resistivity model (middle), and CG misfit for jointly inverted models (right). X=0 km corresponds to SAFOD. White shading indicate well-resolved regions of the separately and jointly inverted velocity models.

Figure 8. Trade-off curves determined via joint inversion of the Parkfield seismic and MT datasets and used for choosing initial values of: (a) resistivity smoothing (λr), (b) velocity smoothing (λv), (c) resistivity damping (εr), (d) velocity damping (εv), and (e) normalized cross-gradient weight (wcg). Optimal values are denoted on plots by stars.

Figure 9. Apparent resistivity and phase curves as well as model response at station L1_53_3 and RR_64_14 where the resistivity model is obtained under (a-b) separate and (c-d) joint inversion.

Figure 10. Zoom in of jointly inverted resistivity and velocity models near the SAFOD borehole. The lithologies intercepted during drilling at SAFOD (measured depths shown in white) as interpreted by Bradbury et al. (2007) are overlain on the models. Lithologies in the drill hole are identified as: a = Quaternary/Tertiary sediments, b = Salinian granites, c = Salinian granodiorites, d = arkosic sediments, e = clay rich zone, f = arkosic sediments, and g = siltstone and mudstone of upper Great Valley Sequence.

Figure 11. The jointly inverted Vp and resistivity model cross sections through SAFOD with the geologic cross section developed in this study overlain. Geologic units are defined as: Kgv = Great Valley sequence, Kjf = Franciscan formation, Ksgr = Salinian block- granite, Ksgd = Salinian block- granodiorite, Ta = Tertiary Arkosic sediments, and Te = Etchegoin Fm. φe and EW denote effective porosity and eastern wall respectively. The dashed white line indicates the well-resolved region of the jointly inverted velocity model.

33