Electronic Supplement to
Three dimensional attenuation structure around the SAFOD site, Parkfield, California

by Ninfa Bennington, Cliff Thurber, and Steve Roecker

 

The following supplementary text and figures discuss three separate topics: (1) comparison of the joint method versus the alternating method for determination of site response (2) synthetic and checkerboard tests carried out for the Qp and Qs models, and (3) insights from the ratio of the Qs to the Qp model.   

Determining Site Response - Joint vs. Alternating Method

Two possible methods, the alternating method and the joint method, were examined to incorporate the effect of site response on our spectra. In method one, the alternating method, we followed a process similar to that taken by Tsumura et al. [1996] who calculated the site effects for each station as the residual between the observed and estimated spectra. Initial estimates for W_o, t*, f_c were required for each station and event. The initial value chosen for S(f_i) assumed no site response (S(f_i) = 1). After solving for the initial parameters, the Levenberg-Marquardt method [Aster et al., 2005] was used to solve the nonlinear problem of equation (1) for the attenuation and source parameters. Next, the residual between observed and estimated spectra for each station in that event were calculated. This was done for every event in our set and the average residual across all events was taken for each station as an estimate of the site response at that station. These average site response terms were incorporated into equation (1) and the Levenberg-Marquardt method was used to invert for W_o and t* for each station and f_c for the event with site response fixed.  After this was done for every event in the set, the residuals for each station were recalculated and the average residuals used to update the site response estimate. The process then alternated between incorporating the site response estimates into equation (1) to invert for the attenuation and source parameters for each event, and determining the new average site response terms at each station until the solution for the model parameters converged. The solution typically converged in 4 iterations or less. Method two, the joint method, is discussed in the main body of the article.

To compare the two methods, the alternating and joint methods were run on one batch of events using the estimated spectra for the P wave data. When the t* values for all our stations were plotted for method one versus method two (Figure A1), a best fit line (forced through the origin) with a slope of 0.957 and a correlation coefficient of 0.89 was obtained. This slope indicated that t* values determined from method two were, on average, slightly larger than those of method one. Additionally, it was found that qualitatively many of the spectra estimated for stations in method two appeared to have a better fit to the observed spectra than those of method one. This observation is demonstrated in Figure A2. In this figure, six stations from three regions of the graph displayed in Figure A1 were examined. Of these stations, NOXV, POLE, POKE, and TANK displayed large site responses. Method two produced expected spectra able to fit these large site responses unlike that produced by method one. The determined site responses shown in Figure A2 demonstrate how the alternating method fails to adequately represent the site response effects that are readily visible in the raw spectra. This suggests that t* values and source parameters obtained using method two were less biased by site effects then those obtained by method one. The alternating method fails because solving for the source and attenuation parameters separately first and then solving for the site response (by equating it to the average residual) produces a bias in the site response. This process is analogous to separately solving for velocity and hypocenters in a tomographic velocity inversion. Roecker et al. [2006] explain that excluding the perturbation to hypocenters within the velocity inversion produces a perturbed velocity model that minimizes the arrival time residuals subject to the "hidden" constraint that the hypocenter changes are zero. In other words, the velocity solution is the one most consistent with the condition that the current hypocenter locations are correct. Thus, when the hypocenter perturbations are determined separately, the perturbed hypocenters will be relatively close to their original locations. In the alternating method, excluding the perturbation to site response spectra within the inversion for the source and attenuation parameters similarly produces perturbed model parameters that are most consistent with site response perturbations that are zero. Thus, each time new site response spectra are determined, these new spectra will be relatively close to their starting values. For this reason the t* values in this study were obtained using method two.

Synthetic Models (Input and Output)

Synthetic tests were carried out in order to determine the validity of features within the Q models. Checkerboard tests (Figures B1 (a-c)) were used to determine the extent of blurring as well as the recoverability of features within the Q models. The output synthetic checkerboard tests were also used to examine the possibility of a systematic bias in the ratio of Qs/Qp, but this bias was found to be insignificant. Additionally, synthetic tests modeling features within the Q models (Figures B2 (a-b) and B3 (a-b)) and features of opposite sign to those in the Q models (Figures B2 (a-b) and B3 (a-b)) were completed to examine the recoverability of features within the Q models.

Qs/Qp Model

As discussed previously, the ratio of Qs/Qp can give insight into the saturation state of individual features within the two models. The ratio of the Qs to the Qp model is shown in Figure C1, but lacks resolution contours due to differing diagonal element resolution between the two models. While it is not possible to accurately identify what regions are considered well resolved for Figure C1, these regions most likely reside in the areas considered well resolved in both the Qp and Qs models. Additionally, most individual features discussed in the paper lack a clear trend in the value of Qs/Qp and therefore, the Qs/Qp ratios for these features cannot be used to constrain their saturation states.