WEMVA Examples

I exemplify the simple application of the backprojection operator in Equation (17) on a North Sea dataset Sava (2000); Vaillant and Sava (1999). Figure  is the velocity map used to compute the background wavefield, and Figure  is the image obtained after split-step migration with the background slowness model using three reference velocities.

The image is extracted from common-image gathers at a selected value of the offset ray-parameter Prucha et al. (1999), which is approximately equivalent to the image for a given incidence angle at the reflectors Sava and Fomel (2000). As expected, the geologic structure is not perfectly defined by one single incidence angle, although this is not a problem for these examples, since I use the image at a given incidence angle only to create the image perturbation, but use the entire prestack image as background during backprojection.

 

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Figure 4 Velocity map for the North Sea dataset.

 

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Figure 5 The image at a selected offset ray parameter.

Figure  is a simulated image perturbation ($\Delta {\bf R}$) localized under the salt flank. I create this perturbation by cutting a small window in the target region, shifting it down so that the phase difference between the two images doesn't violate the Born approximation, and taking the difference.

Next, Figure  is the result of applying the backprojection operator in Equation (17) to the image perturbation in Figure . The backprojection operator turns the image perturbation into a bundle of ``fat'' rays ($\Delta {\bf S}$) emerging from the region of perturbation. The rays follow various trajectories, in accordance to the velocity model and with the background image.

 

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Figure 6 Image perturbation situated under the salt flank, overlaid by a pair of specular rays.

 

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Figure 7 Slowness perturbation obtained by backprojecting the image perturbation in Figure . Overlaid are a pair of specular rays.

For comparison, I superimpose on both images in Figures  and a pair of specular rays, shot at roughly the same angle with respect to the normal to a hypothetical reflector in the perturbation region as the angle given by the offset ray parameter at which I selected the image perturbation. The rays overlap well over one pair of ``fat'' rays. In fact, these images graphically illustrate the band-limited character of wave-equation migration velocity analysis, which is its most important property.

 

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Figure 8 Image perturbation situated away from the salt flank, overlaid by a pair of specular rays.

 

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Figure 9 Slowness perturbation obtained by backprojecting the image perturbation in Figure . Overlaid are a pair of specular rays.

The backprojection in Figure  corresponds to just a particular selection of the incidence angle at the reflector. Perturbations at other angles backproject over other regions of the slowness model. When all backprojections are put together, we obtain a smoother version of slowness perturbation in comparison to that obtained using ray tomography Sava and Biondi (2000). Ray tomography requires a significant amount of model regularization Clapp and Biondi (1999) to control the shape of the inverted model. However, given its intrinsic band-limited nature, wave-equation migration velocity analysis requires less regularization, or model-styling, applied on the slowness model. The net result is that we need less a-priori information about our slowness model, and we can extract more information from our data.

Unlike in the first example, Figures  and , where part of the wavefield propagates through the salt and, therefore, some of the fat rays get significantly distorted, in a second example the wavefield propagates through a much simpler part of the velocity model, and so the fat rays are less distorted (Figures  and ).

As pointed out by Sava and Biondi (2000), the Born approximation is the biggest limitation of the method, since stability requires us to ensure that we do not violate the small-perturbation assumption. Also, the frequency content of the images is not the same, therefore we can obey the Born approximation in some regions, but violate it in others. Better ways to control the Born approximation await for future research.