The WEMVA flow presented above is actually the one for the procedure named Target Image Fitting (TIF) WEMVA. In a more radical inversion-theory approach, called Differential Semblance Optimization (DSO) WEMVA, steps 2 and 3 are combined into a single step, with the image improvement being performed by a weight operator during the inversion. In theory, DSO WEMVA will not exhibit the same Born-related problems as TIF WEMVA Sava and Symes (2002). However, it assumes that the quality of the image-improvement operator that is embedded as a weight in the inversion can be trusted. I am currently just trying to develop such an image-improvement operator specific to the FEAVO problem, and I need to be able to isolate its effects, so for the time being I will use TIF WEMVA on velocity anomalies inside the Born approximation.
In the context of the FEAVO problem, the image perturbation is the difference between the FEAVO-free image and the original one. I therefore need to eliminate the FEAVO effects from the image. To do so, I have to discriminate between AVO due to focusing and AVO due to actual changes in reflectivity with the incidence angle. I will proceed by first obtaining an estimate of the AVO due to the properties of the reflector.
The variation of amplitudes due to changes in reflectivity with incidence angles lower than has been modeled by Shuey (1985) as follows:
At each midpoint-depth location, I will compute the most plausible reflector-caused AVO as a linear trend in the (,amplitude) space, and I will obtain FEAVO by subtracting this trend from the image.
The challenge is now fitting to the data at each midpoint-depth location a linear trend in the (,amplitude) space. The trend should be as close as possible to that of the reflector-caused AVO, even when focusing effects are present. I will consider only the angles up to . After that, the accuracy of the angle-gather amplitudes decreases, unless the offsets are extremely densely sampled, which would increase the costs of the migrations in WEMVA too much. Also, after ,the two-term Shuey approximation of reflector-caused AVO stops working. I have experimented with his three-term formula for larger angles. The extra degree of freedom allowed for a curve that fitted everything, including the FEAVO effects at angles lower than . I have therefore to decided to use only the two-term formula on angles under .
The best way to fit the linear trend would be to formulate the fit as a 2-D inverse problem, penalizing nonphysical variations of AVO along the midpoint or along the reflector. While I intend to do so in the future, I experimented for the time being with a 1-D approach mapped in Figure 2.
The extracted anomalies are closer in morphology to the true FEAVO in the optimal image perturbation. Also, the extraction worked up to , in contrast with the previous image-processing-based technique proposed by Vlad et al. (2003), which only worked up to .
A 2-D extraction procedure in which the intercept and gradient would vary more smoothly along the midpoint would make it possible to eliminate in the future the low-pass filter step, which is responsible for the reverse-polarity ``borders'' around the extracted anomalies. However, the results of the 1-D extracting procedure may still provide some valuable insight into the effectiveness of FEAVO extracting procedure, given that the ``unfair'' advantage of the lack of noise is compensated by the simplicity of the 1-D extraction.
The slowness update produced by inverting the extracted image perturbation is displayed in Figure 4.
The method presented in this paper discriminates FEAVO on the basis of a local property--nonlinear variation with the squared sine of the angle. FEAVO anomalies, however, are correlated in tent-like shapes throughout the prestack volume, as shown in Figures 3, 4 and 9 in Vlad (2002). That paper proposed a ``focus-filter-spread'' approach to discriminate FEAVO from uncorrelated amplitude effects. The shapes of the anomaly surfaces in the prestack volume can be precomputed as functions of the known velocity. Summation along them would highlight the anomalies; only the ``bright stars'' would be kept. Then, the de-noised Radon-space image would be spread along the surfaces, to keep only the FEAVO anomalies in the image perturbation. Vlad et al. (2003) identified a major difficulty with this approach: the alternating polarities in the wavelet would lead to summed values canceling each other. The obvious solution--taking the unsigned values--would, however, lead to the inability to discriminate between positive and negative velocity anomalies, and to the inversion process never converging if the data contains FEAVO-generating velocity anomalies of more than one sign.
The conundrum of FEAVO extraction based on its global properties can however be broken by using a procedure that I present below. Instead of a ``focus-filter-spread'' approach, I will use a ``discriminate-focus-filter-mask'' one. I will first process the prestack volume through a FEAVO discriminator based on the local properties of the anomalies. Then, as in the ``focus-filter-spread'' approach, I will square or take the modulus of the values in the prestack volume, sum along precomputed velocity-dependent surfaces, filter to keep only the ``bright stars'' in the Radon domain, then spread back. But instead of feeding the result of spreading into the inversion, I will use it as a mask that I will apply to the result of local property-based FEAVO discrimination. That will filter out all areas with amplitude variations that are nonlinear, but are not FEAVO (because they do not correlate vertically and laterally in the known tent-shape). Since the synthetic dataset used in this paper does not have non-FEAVO amplitude departures from the Shuey (1985) model, it represents a good benchmark for the accuracy of the inversion, even in the absence of the de-noising ``focus-filter-mask'' step.