1. Developed a method of nonlinear inversion and modeling for the complete waveform. This method can be used to estimate the source parameters of earthquakes for one or multiple 3 component stations over a range of magnitudes and source-receiver distances.

2. Applied regional and local waveform modeling and earthquake relocations on the 1993 Eureka Valley, eastern California earthquake sequence. Resolved that the earthquake sequence did not have any active low angle faulting component.

3. Performed a detailed analysis on the way uncertainty is handled in general probabilistic seismic hazard analysis. We provide a reason why hazard maps overestimate the predicted ground motions for future earthquakes.

Nonlinear Waveform Inversion and Modeling

We have developed a method to obtain fault geometry and seismic moment by nonlinear iterative waveform optimization without matrix inversion. This process minimizes the misfit between the complete waveform and synthetics generated using a fast $f-K$ summation technique (Zeng and Anderson, 1995; Zeng, 1997). The vector displacement field, $ u sub i $ due to an arbitrarily oriented double couple source can be written as the sum of the product of the directional cosines $A sub {ij}$, average slip, $ u bar $ and the 8 Green's function terms for the fundamental orientations.

where $r$ is source receiver distance and $t$ is time. The directional cosines $A sub ij$, which are non-linear functions of fault strike, dip, and rake, are determined by nonlinear optimization following the downhill Simplex procedure originally developed by Nelder and Mead (1966). The complete waveform and synthetics are first aligned by the lag with the maximum cross-correlation. The objective function some of the squared raw amplitude difference for each station. We modify the downhill Simplex procedure using Simulated Annealing by adding a small, exponentially decreasing fluctuation to the variables at each iteration. This small fluctuation, controlled by the temperature parameter, allows the optimization process to find the global minimum. We set an annealing schedule where the "temperature" is decreased over time. For one 3 component station, a solution can usually be found in several thousand iterations which takes about 30 seconds on a Pentium II PC. This is an improvement over grid search techniques requiring hundred of thousands iterations for a fine grid. The process is inefficient relative to generalized inverse but the process can be watched and the sensitivity of the solutions can be explored.

The Green's functions are computed in a layered elastic medium for double couple and explosion source, using the generalized R-T coefficient technique (Luco and Apsel, 1983) and an imaginary frequency and discrete wavenumber summation (Bouchon, 1979). Zeng and Anderson (1995) reorganized their program to replace the most time consuming computation of the $f-K$ dependent generalized R-T matrices which results in a 3 fold gain in computation speed.

The 1993 Eureka Valley, California Earthquake Sequence

The 1993 Eureka Valley earthquake sequence occurred within the Eastern California Shear Zone (Dokka and Travis, 1990). This earthquake may represent the transfer slip from the Owens Valley fault zone to the Fish Lake Valley-Furnace Creek-Northern Death Valley fault zones by strike-slip and normal faulting across Eureka Valley and Saline Valley (Dixon et al., 1995; Savage et al., 1990). Initial first motion focal mechanisms indicated a possible shallow west dipping fault plane (Ken Smith, personal communication) and earlier earthquake relocations using 3D velocity models were not able to resolve any fault planes (Asad et al., 1999) perhaps because of clock errors in the data. The main objective of our study was to resolve the focal mechanism using waveform modeling to investigate the possibility of active low angle faulting.

We first merged 2833 earthquakes from the Caltech/SCSN, and University of Nevada, Reno catalogs with P- and S-wave phase picks from 5 UNR portable sites. Timing errors were corrected using clock drift corrections. We then selected 350 well recorded earthquakes and relocated them using a 1D layered velocity model. The travel time residuals from this subset were then averaged and used as stations delays for the final relocations of earthquakes from January 1993 to July 1998 (Figure 1). The final time residuals were less than $+-$0.2 seconds and the horizontal errors were less than 1 km.

The source parameters of ninety Eureka Valley, California aftershocks ranging in size from $mw$ 1 to 4.9 are modeled using a nonlinear inversion process. The Green's functions and data are bandpassed filtered over a range depending on distance and magnitude. For regional earthquakes with ( M > 4.5 ), the bandpass is 100 to 20 seconds, 50 to 20 seconds period for ( 3.5 < M < 4.5 ) earthquakes, and 0.5 to 3 Hz for ( 0.5 < M < 2.5 ) local earthquakes recorded by short-period sensors. The local velocity models were modified from the model used in the earthquake relocations and regional models are constructed from simple crustal 2 to 3 layer models calibrated using nearby events. The seismic moments are in agreement within 10% of the UNR duration magnitudes and the source orientations generally agreed with the distribution of relocated seismicity and sometimes with sparse first motion data. Figure 1 shows the focal mechanisms of $mw$ > 3.7 aftershocks determined from inverse modeling using regional and portable stations. Figure 2 shows the three component fits of an M 3.7 aftershock using Berkeley Digital Network, Terrascope broadband, and UNR portable short-period stations.

The $mw$ 6.0 mainshock ruptured along the western side of the Last Chance Range fault along a S27$deg$W striking normal fault with a 51$deg$W dip. The aftershock relocations in the cross sections show that the rupture may have started at the epicenter at 12 km depth, and only reached an up-dip depth of 8 km along a slip direction of -83$deg$. The mainshock triggered hanging-wall deformation in the overlying Saline Range along a steeply east dipping range front with a slip vector varying from normal to right-lateral. These results are in good agreement with Peltzer and Rosen (1995) geodetic model. There is no evidence of low angle faulting seen in the cross sections (Figure 3) or west or east dipping planes of the focal mechanisms. To this date no one has found conclusive evidence for active low angle faulting which has implications for the frictional mechanics of faulting.

Probabilistic Seismic Hazard Analysis Without the Ergodic Assumption

An ergodic assumption is commonly made in probabilistic seismic hazard analysis (PSHA) where space and time can be interchanged in a random process. Some PSHA maps make this assumption and predict what seem to be high values of maximum ground motion for long repeat times. For example, Frankel et al. (1996), WGCEP (1995) and Ward (1995) give near fault accelerations of 1.2 and greater for an average repeat time of 1 in 2475 years (2% probability in 50 years). These values cannot be checked with accelerograph evidence but can be found unreasonably high by the distribution of precarious rocks (Brune, 1996). The distribution of precarious rocks in southern California is reasonably consistent with hazard maps given by Wesnousky (1986). The controlling difference between PSHA maps of others and Wesnousky is that he used only the mean value for attenuation of peak ground acceleration with distance, whereas, others added statistical uncertainty to the ground motion regressions. The later difference is the ergodic assumption.

The problem in general PSHA can be explored by a thought experiment for a characteristic-ground-motion-earthquake. After one repeat time the ground motion for all future earthquakes are known and the same. The uncertainty is zero. In other words, the ground motion at the site is always the same in characteristic-ground-motion-earthquakes and its most likely value is in the vicinity of the mean value predicted by the regression equations. The general PSHA fails to agree and exceeds the expected value for repeat times larger than one recurrence interval because the uncertainty in the regression analysis cannot be assumed to be interchangeable with space and time.

There are two types of uncertainties aleatory and epistemic. Aleatory uncertainty is due to the inherent unpredictable nature of future events. Epistemic uncertainty is due to incomplete knowledge and data about the physics of the earthquake process which can be reduced by the collection of additional information. The squared total uncertainty is the sum of the aleatory and epistemic uncertainties squared. In the general development of PSHA there is no distinction between these two different types and all of the uncertainty is assumed epistemic. The approach to correct PSHA is to distinguish clearly between aleatory and epistemic uncertainties. Make all of the uncertainty in general PSHA as aleatory and account for epistemic uncertainty by development of multiple models of PSHA. This approach seems most reasonable in the presence of large repeatable earthquakes.

There are profound implications for PSHA when no distinction is made between aleatory and epistemic uncertainties commonly made in the majority of hazard maps. We propose that many hazard maps may end up with different ground motions at low probabilities if the uncertainties are properly distributed between aleatory and epistemic. The economic consequences for sensitive structures that are designed for low probability levels are potentially huge.

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Anderson, J., G. and J. N. Brune (1999). Probabilistic Seismic Hazard Analysis without the Ergodic Assumption, Seism. Res. Lett. 70, 19-28.

Asad, A. M., S. K. Pullammanappallil, R. Anooshehpoor, and J. N. Louie (1999). Inversion of travel time data for earthquake locations and three-dimensional velocity structure in the Eureka Valley area, eastern California, Bull. Seis. Soc. Am., in press.

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Zeng, Y. (1997). Synthetic seismogram computation in a layered halfspace with source and receiver located at close depth ranges, Geophys. J. Int., submitted.