Test with synthetic data (wave equation)

Now, we test our imaging condition with a more realistic model. The data are modeled with a wave equation-finite differences program. Figure  shows the velocity model. Note the low velocity anomaly at 300 m depth. Also, a flat, constant-impedance contrast interface is located at 700 m. The low velocity anomaly creates multipathing that we want to include during the imaging.

From the whole receiver ${\bf u}(z,x,t)$ and source ${\bf d}(z,x,t)$ wavefields we extracted two constant depth planes: one at the reflector depth (Figure ) and one shallower (Figure ). In both figures we can see two reflectors, one due to the direct arrival and the other due to the second wave arrival produced by the low velocity anomaly.

Four traces were extracted for this test at two different offsets (1000 m and 2000 m). Figures a and a show the source wavefield and Figures b and b the receiver wavefield at reflector depth. Figures a and a show the source wavefield and Figures b and b the receiver wavefield at a shallower depth.

Figures  and show the deconvolution of the receiver by the source wavefield at the reflector depth. As we expected, there is a maximum at zero lag. The result is very similar using both deconvolution methods. Convolution of the source wavefield with the reflectivity seems to be a good modeling operator, at least at the reflector depth, since the data residual in the least squares inversion is small.

Figures  and show the deconvolution of the receiver by the source wavefield at a depth shallower than the reflector depth. As we expected, the value at zero lag is an order of magnitude smaller than the value at zero lag at the reflector depth. The result is very similar using both deconvolution methods.

For an offset of 2000 m and a depth of 500 m, the data residual in the least squares inversion is small, indicating that the convolution of the source wavefield with the reflectivity fits the receiver wavefield. But for an offset of 1000 m and a depth of 500 m, the data residual in the least squares inversion is not small. The explanation for this result can be found looking at Figures a and b. We can note that the separation between the two arrivals in the source and receiver wavefields is not the same. Thus the source wavefield convolved with a simple delta shifted from the zero lag (as reflectivity) is not enough to explain the receiver wavefield. The impact of this issue in the final image quality needs further investigation.

Comparing the deconvolution with the cross-correlation imaging condition in Figures , , , and , we conclude that deconvolution imaging condition effectively attenuates the image artifacts and handles the amplitudes better. In addition, the deconvolution imaging condition does a better job than the cross-correlation in preserving the amplitudes through the offset.