Two-Dimensional Solution

1.3. 7 Modeling the Temporal Evolution of the Stress Field

Chapter 2 Chapter 2 Stress Magnitude at Seismogenic Depths

2.2 Two-Dimensional Solution

2.2.1 Method

A general two-dimensional solution can be found for the relationship between the near- field rotation of the stress tensor and the ratio of the earthquake stress drop, !:iT, to the background deviatoric stress magnitude, T. I follow a procedure similar to that used by Sander [1990] for the stress rotations associated with linear density anomalies.

This is preferable to the solution obtained by Yin and Rogers [1995] because they make two assumptions which may not generally hold. The first assumption is that the mainshock fails in accordance with the Coulomb failure criterion, which may not be the case, for instance for a weak fault. The second assumption is that the magnitude of the deviatoric stress does not change, which is clearly not true since the mainshock relieves a portion of the shear stress. The second assumption is approximately correct if the ratio !:iT/Tis small, but if the stress drop is on the order of the deviatoric stress, as it appears to be in the Landers example, the assumption breaks down.

The post-mainshock stress tensor equals the pre-mainshock stress tensor plus the stress change tensor due to the mainshock (Figure 2.1). The pre-mainshock deviatoric stress tensor is:

CJpre = ( T 0 ) 0 - T

(2.1)

where T

=

(cr₃ - cr1)/2 is the deviatoric stress magnitude (tension is positive). The earthquake occurs on a fault plane oriented at an angle of () to the cr1 axis orientation (sign convention shown in Figure 2.1). The stress change tensor due to the earthquake is:

(2.2)

where ~Tis the earthquake stress drop. The post-mainshock stress tensor is therefore:

(2.3)

Solving for the eigenvalues and eigenvectors, one finds that the post-earthquake stress tensor is rotated from the pre-earthquake stress tensor by an angle of ()*, where

* ( 1 -

~r

^{sin 2() - [ (}

~r)

²

⁺

^{1 - 2}

~r

^{sin 2()}

^J ~ ⁾

() = atan b. () .

_I. cos 2

T

(2.4)

Note that the rotation depends on only two parameters: (), the orientation of the fault relative to the pre-earthquake stress field; and ~T

jT,

the ratio of the earthquake stress drop to the background deviatoric stress level. ()* versus () is shown for various values of ~T

jT

in Figure 2.2. This figure is very similar to that for a linear density anomaly [Bonder, 1990], the major difference being a 45° shift of the x-axis because of the difference between a shear source and a tensional or compressional source.

A potential problem with using Equation 2.4 to estimate ~T

jT

is apparent in Figure 2.2. As the orientation of the fault relative to cr1 goes to ±45°, the rotation angle goes to 0, for any j~T/TI <1. This is because at ()=45°, the two stress tensors

are collinear, so adding them will change the magnitude of the two axes, but will not cause a rotation. Where the results for all I~T /TI <1 start to converge, it becomes more difficult to infer ~T f'r from the stress rotations. An angle of rv45° to a₁ is also a favorable orientation for fault failure, so many earthquakes may fall into regions where very high-precision stress rotation measurements are necessary to determine

~T

/ T

even to within an order of magnitude.

2.2.2 Results

To estimate ~T

/T

using the solution given above, () and ()* must be observed. I find stress orientations before and after the Landers mainshock on four segments of the rupture (Figure 2.3, Table 2.1), similar to the segments used by Hauksson [1994].

The pre- and post-Landers stress tensors along each fault segment are different at the 95% confidence level of the inversion, strongly suggesting that the Landers mainshock caused a rotation of the stress field. The significance of the difference between the pre- and post-Landers stress fields can be seen using a technique for identifying stress changes [ Wyss and Lu, 1995]. Changes in the misfits of individual focal mechanisms with respect to a given stress tensor, here the bestfit pre-Landers stress, signal changes in the stress field. For earthquakes near Landers, the slope of the cumulative misfit curve, which represents the average misfit, changes from rv25° /event to rv44° /event at the time of the Landers mainshock (Figure 2.4). The high misfit of the aftershocks indicates that they are poorly fit by the pre-event stress tensor, implying a post- mainshock stress state which is different form the pre-mainshock state.

The stress rotations observed in Figure 2.3 are shown along with the analytic so- lution in Figure 2.2 and are used to constrain ~TjT. The Johnson Valley, Homestead Valley and Emerson segments provide both lower and upper bounds. For the Johnson Valley segment, 0.25s; ~T /T <1; for the Homestead Valley segment, 0.4s; ~T jT <1;

and for the Emerson segment, 0.35s; ~T

/T

^s;0.85. The Camp Rock segment expe- rienced a rv21±14° rotation, qualitatively suggesting high I~T /TI, but the rotation appears to be in the wrong direction, i.e., the direction which would be expected for

left-lateral slip. There is not very much slip on this segment [ Wald and Heaton, 1994], so the stress rotation here may be dominated by the stress changes due to the other segments. Additionally, there is little pre-Landers seismicity along this segment, so the pre-event stress orientation may not be well constrained.

The deviatoric stress magnitude, T, can be inferred from i:1TjT if the earthquake stress drop, i:1T, is known. The average stress drop for the Johnson Valley, Homestead Valley and Emerson segments, estimated from the mapped surface slip (Table 2.1), is rv80 bar. Assuming a value of i:1T /T ;:::::0.65, which fits the rotations of the three segments very well, ^T ;:::::120 bar. This is nearly an order of magnitude less than would be consistent with the crustal strength predicted by laboratory experiments.

Conservative error estimates (Table 2.1) constrain ^T to be less than rv250-320 bar. The estimated magnitude ofT could be incorrect if there was afterslip on the fault, i.e., if i:1T is an underestimate of the true total stress drop. However, if Tis to be on the order of 1 kbar, the afterslip would have to be very large, as large as the coseismic slip. It seems highly unlikely that such large afterslip could have gone undetected for a well-studied event such as Landers.