To test the resolution of the method I generated a synthetic dataset using the elastic finite differences algorithm described in Cunha (1992). Both the displacement and pressure wavefields were ``recorded" at the gridpoints associated with the recording cable. Figure shows the recorded pressure field corresponding to a shot gather. The model is structurally complex and the effect of dipping beds is clear from the figure, as well as the presence of diffractions and converted modes.
Figure compares the true (computed by the modeling algorithm) horizontal component of the displacement field with the horizontal component retrieved from the pressure field using the vectorizer operator. The same comparison for the vertical component is shown in Figure . The result from subtracting the true from the retrieved fields is shown in Figure . The differences are larger at the near and far offsets because of the missing information associated with the finite aperture of the data. The missing information affects not only the estimation of the pressure gradient in the decomposition scheme, but also the estimation of the divergence of the displacement field that the modeling algorithm uses to generate the pressure data. The direct wave, as well as other events with similar step-out, are also not correctly retrieved because of the taper that needs to be applied near the boundary with the evanescent region (). As one would expect, the fit between the true and retrieved wavefields is better for the horizontal component because it is much easier to compute the spatial derivative along the direction in which the data is collected than orthogonally to that direction.
It is evident that the frequency content of the retrieved vertical component is lower than that of its true counterpart. This ``side-effect" should also be expected, because the vertical component of the vectorizer operator clearly favors the low-frequency part of the spectrum. Figure shows the time-spectra of a near and far offset of the retrieved and true vertical components. This figure shows that the frequencies where the zeros of the spectrum are located were correctly estimated, but their amplitudes (i.e., how far they are from the unit circle) are slightly different from the correct ones.
The method was also tested on an offshore dataset from Brazil, which was recorded in an area with an irregular ocean floor, and with slightly structured subsurface geology. According to the observer's report, the cable depth ranged from 9 to 11 meters, with some additional oscillations of about 1.5 meters in the sea level. Figure shows the common shot gather used in the tests.
Figure shows the resulting vector field obtained with the method described. As with the synthetics, the dominant frequency is lower in the vertical component than in the horizontal component. Although a direct comparison with the true vector field is not possible here, there are some tests that can show at least the self-consistency of the result. One test involves the sensitivity of the method to errors in the correct depth of the cable.
In the space-time domain each point of the cable will ``see" at the same time an upcoming wavefield arriving at an angle and a downgoing wavefield arriving at an angle . If the subsurface has some degree of lateral smoothing, then and the amplitude of the downgoing wave will be a delayed version of the amplitude of the incident wave with reversed polarity. The resulting superposition will have an apparent arrival angle (as measured by the displacement field direction) that covers the full range from to . In addition total wavefield will have maximum amplitude at the apparent angle .
Figure shows the absolute apparent angle panels generated from the elastic wavefields estimated from the data in Figure using six different values of cable depth. The first two panels are predominantly dark with a few white sparks, which indicates very small apparent angles, that is, almost vertical arrivals. The last panel starts to lose lateral coherence, which indicates random apparent angles. The two panels corresponding to cable depths of 8 and 10 meters give the more coherent images, which is consistent with the in-site measured values.