Pseudo Green's Functions And Waveform Tomography
PROGRESS REPORT
by Xi Song and Don Helmberger

Retrieving source characteristics from moderate-sized earthquakes, directivity, source complexity, etc. require better Green's function than usually provided by 1-D models, Song and Helmberger (1996). Thus, we developed a procedure to provide such functions based on perturbation theory, Song and Helmberger (1997).

We begin with a 1-D model which is divided into blocks. The velocities in the blocks are allowed to vary, which shifts the arrival time of the individual rays generated from the 1-D model similar to conventional tomography. The amplitudes of the rays are perturbed independently to accommodate local velocity variations in the structure. For moderate-sized earthquakes with known source mechanism and time history, the velocity variation in each block and the amplification factor for individual rays can be optimized using a simulated annealing algorithm. The resulting mod)fied Green's functions, Pseudo Green's functions, can be used to study the relative location and characteristics of neighboring events. The method is also useful in retrieving 2-D and 3-D structure, which is essentially waveform tomography.

One of the applications, or usefulness, of point-source parameters is in fine-tuning the propagation models involved and in further investigating structure heterogeneity. The method of Zhao and Helmberger [1994], further improved by Zhu and Helmberger [1996], is a natural approach towards this goal. This method matches observed seismograms against synthetics over discrete wave-trains and allows relative time shifts between individual wave-trains, e.g., the Pnl wave-train and the Rayleigh wave (hence the name "cut and paste"). This allows a better correlation between data and synthetic waveforms. One example of this method as applied to the 28 June, 17:00, 1991 Sierra Madre aftershock is shown in Figure 1. The upper panel shows the source inversion using long-period records from stations GSC, PFO, and ISA. Synthetics at station SBC are predictions with the resulting source mechanism. The lower panel displays the predicted broadband synthetics at all four stations. The waveforms are well-matched, especially the long-period records at stations GSC, PFO, and ISA. The SV waves are used together with the Rayleigh waves and the latter dominates the comparison between the data and synthetics, as seen in the complete radial and vertical components. However, it seems that a small timing perturbation would bring the SV synthetics into alignment with the observation (e.g., station ISA). This is more clearly seen on the vertical components (lower panel). These time shifts, together with those for the Pnl waves, the Love waves, and the Rayleigh waves, are indicative of further adjustment of model velocities, as discussed in Song et al. [1996]. However, all these time shifts may not be satisfied with adjustments to a 1-D model.

The method we developed resembles the conventional travel-time tomography in that travel-times of individual rays are connected to a slowness model, which consists of discrete, constant slowness cells. Besides modeling the travel-time of the first arrival, all important pulses on a seismogram are taken into consideration. These travel-times (to's of the generalized rays) are fit by matching the waveform data with the total synthetics S(t ), which is the sum of all the individual ray responses R(t ), after being shifted by dt and amplified by a factor A:

In the above equation, i is the index to the rays and "*" denotes time-domain convolution. Convolution with the o function in (1) corresponds to a time shift. The ray response Ri(t ) is
computed from a reference 1-D model. The time shift dti is formulated as in conventional traveltime tomography:

where dsj the velocity perturbation to block j and ljj is the length for which ray i travels in block j. To minimize the effect of the amplitudes of the ray responses on fitting the travel-time, Aj's are allowed to change freely over a restricted range. This freedom in the parameterization also serves to obtain practical Green's functions, or the pseudo Green's functions.

Figure 2 shows an example of the pGf technique as applied to the 1991 Sierra Madre mainshock and the big aftershock (17:00). In this experiment, the standard Southern California model of Dreger and Helmberger [1991] is used as the reference model. Green's functions are combined with the source mechanism, source-time function and seismic moment of Zhao and Helmberger [1994] for both events to generate the original 1-D ray responses. In this figure, individual ray paths and responses are shown only for the most important rays, while the total synthetics contains a total of 14 rays. These rays comprise a aufficient ray set to match the complete synthetics as generated by the reflectivity method [Saikia, 1994]. All seismograms are plotted in absolute time with the same amplitude scale. In column (a), each layer of the original 1-D model (SC) is perturbed as a single block. This setup has the minimal freedom in terms of fitting the travel-times. As a result, the onset of the simulation, that is, the sum of the rays after shifting and amplification, is slightly off, as compared to the data. The overall waveform, however, is well matched. If we take the resulting model, shown at the bottom, and compute l-D synthetics for it, the new synthetics would be the dotted trace shown superimposed on the perturbation result in the lower panel. This comparison of the two synthetics demonstrates the goodness of the time-shift approximation. The strength of the pulses on these two synthetics does not match exactly and we picture the amplification effect as a result of many other factors, including source complexity and very local variation in the model. Column (b) displays another test on the aftershock. This time, the SC model is divided into 10 blocks, the slowness of which can vary independently. As a result of more freedom, the resulting simulation match the data better than in column (a). In column (c), the same experiment is conducted using the mainshock data recorded at station GSC. As can be seen in the comparison of columns (b) and (c), the time shifts for most of the rays are consistent for the two events. However, the corresponding slowness models show a substantial difference. This brings up the non-uniqueness problem in the process of producing the required travel-time shifts.

In Figure 3, a set of experiments showing a spectrum of parameterization schemes, are displayed. In general, as the starting model is divided into more blocks, the waveform fits is improved, but the corresponding models for different events show greater difference. The dotted traces show a case where the time-shifts of the individual rays are not connected by any physical model. The simulations fit the data almost exactly, but there is no way of transporting the optimal parameters from event to event. In the rest of our experiment, we choose to divide the model into 6 blocks since this setup seems to generate reasonable synthetic fits as well as model stability. We applied the above procedure to paths from Sierra Madre to stations GSC, ISA, PFO, and SBC, using the tangential component of the broadband displacement data. Figure 4 shows the comparison between the data, the simulations and the original 1-D SC synthetics. The simulations fit the data well for all stations, both in waveshape and amplitude. Station SBC shows the most improvement over the original 1-D synthetics. All the models display lowvelocity in regions near the source, which could be real or caused by an error in origin time, see Song and Helmberger (1997).

References

Ding, X., High resolution modeling of deep earth structure, Ph.D. thesis, California Institute of Technology, Pasadena, California, 1997.

Dreger, D. and D. V. Helmberger, Source parameters of the Sierra Madre earthquake from regional and local body waves, Geophys. Res. Lett., 18 (11), 2015-2018, 1991

Helmberger, D. V., Theory and application of synthetic seismograms, in Earthquakes: Observation, Theory and Interpretation, edited by H. Kanamori, pp. 173-222, Soc. Italiana di Fisica, Bologna, Italy, 1983.

Song, X. J. and D. V. Helmberger, Source estimation of finite faults from broadband regional networks, Bull. Seism. Soc. Am., 86 (3), 797-804, 1996.

Song, X. J. and D. V. Helmberger, Pseudo Green's functions and waveform tomography, Bull. Seism. Soc. Am. (in press), 1997.

Song, X. J., L.-S. Zhao, and D. V. Helmberger, Broadband modeling of regional seismograms; the Basin and Range crustal structure, Geophys. J. Int., 125, 15-29, 1996.

Stead, R., Finite differences and a coupled analytic technique with applications to explosions and earthquakes, Ph.D. thesis, California Institute of Technology, Pasadena, California, 1990.

Zhao, L.-S. and C. Frohlich, Teleseismic body-waveforms and receiver structures beneath seismic stations, Geophys. J. Int., 124, 525-549, 1996.

Zhao, L.-S. and D. V. Helmberger, Regional moments, energy levels, and a new discriminant, Pure Appl. Geophys., 146, 281-304, 1996.

Zhao, L.-S. and D. V. Helmberger, Source estimation from broadband regional seismograms, Bull. Seism. Soc. Am., 84 (1), 91-104, 1994.

Zhu, L. and D. V. Helmberger, Advancement in source estimation techniques using broadband regional seismograms, Bull. Seism. Soc. Am., 86 (5), 1634-1641, 1996.