Preliminary Resuls of CMG Research: Statistical Seismic Imaging
This page contains some preliminary results of progress made for this project. Please note that these results are not conclusive, and are subject to change.
Caption Figure 2
A checker-board test. A stochastic model of the Earth's crust is shown in the top figure, where blue corresponds to 6.0 km/sec and red corresponds to 6.3 km/sec. The parameter ax is the characteristic length of the velocity heterogeneities, and for this test, the Hurst exponent is kept constant at 0.3. The model dimensions are 30km by 20 km deep. Synthetic data was generated through the velocity model by numerically solving the acoustic wave equation via finite differences. There were 200 shots, with a shot interval of 100m, a trace interval of 25m, with a symmetric 8km cable spread (yielding a maximum offset of 4km). The middle figure shows the synthetic data after processing (DMO, CMP stack, post-stack time migration, conversion to depth). Note that the scale of the data is similar to the velocity model. Using a stochastic Von Karmon correlation function model, I inverted the data in the middle figure to retrieve the horizontal characteristic lengths. Note that the inversion estimates correlation lengths (the bottom figure) that are quite close to those used in the original velocity model. The scale in the color bar goes from 200m (dark blue) to 900m (dark red). Note that in this simulation, the "data" are in the weak, or single, scattering regime.
Caption Figure 3
Stochastic parameter estimation on real data. The top figure shows a 25km line of stacked, migrated marine data from the North Sea. Note the band of bright, coherent reflectors at about 10-11 seconds, which are likely the arrivals from the Moho. The middle figure is the inversion results for the horizontal characteristic length parameter, ax. Note that for this result, I held the value of the Hurst exponent constant at .29, which is a reasonable value for the crust. If I allow the Hurst exponent (nu) to vary within reasonable values (.2-.3), the resolved values of ax are relatively unchanged and the inversion takes much more computational effort. So for this example I just held nu constant. The colorbar on the middle figure goes from 50m (dark blue) to 500m (dark red). The bottom figure is the value of the slowness (dip) for the maximum ax. The values of slowness go from -0.1 sec/km (dark blue) to +0.1 sec/km (dark red). Real data can often have a maximum correlation length which is not horizontal but rather dips slightly. For this reason, I invert simultaneously for the maximum dip direction and the ax.
Caption Figure 4
A synthetic example on estimating the vertical characteristic length (az). In order to invert for az, one must deconvolve the data, threshold to some percentage, and then numerically integrate the result. The idea is to completely remove the effects of the wavelet (by deconvolution) and recover the original velocity model (by thresholding and integration). This process all hinges on an accurate estimate of the seismic source, which is the cornerstone of any deconvolution problem. The figure to the top left is a stochastic velocity model with the shown ax and az values. The top middle figure shows the results after inverting for az after deconvolving with a KNOWN wavelet (i.e. the wavelet that was used in the finite difference scheme to produce the synthetic data). The figure on the top right shows the inversion results without using the known wavelet. That is, the wavelet is estimated directly from the data via tapered Fourier transforms. The results of the wavelet estimation are shown immediately above the top left figure. Note that the results (although reasonable) of estimating az are not nearly as good when the wavelet is estimated. The known wavelet is shown above the top center panel, and the acoustic, synthetic data is shown in the bottom panel. The scales on the colorbars are indicated by the numbers adjacent to the colorbar, and are in meters.