Fast velocity model evaluation with synthesized wavefields

Evaluation criteria

Since the goal of the steps outlined above is to test the accuracy of several different models, there must be a way to judge or evaluate the images corresponding to each. In some cases, a qualitative judgment may suffice; for example, if one model clearly focuses the data at zero subsurface-offset. However, a quantitative measure of image quality would also be useful. Since only isolated locations are being imaged, we can expect a ``perfect" velocity model to focus all an image's energy at zero subsurface-offset. Therefore, a simple measure of image quality calculates what proportion of the energy indeed resides at zero or near-zero subsurface-offset:

$$F = \frac{\sum_{i=\mathbf{p}}{\vert A_i\vert}}{\sum_{i=\mathbf{p}}{\vert A_i\vert \exp{(\alpha \frac{\vert h_i\vert}{h_{\mathrm{max}}}}})},$$ (5)

where $\mathbf{p}$ is the set of all image points, $A_i$ is the amplitude at a given point, $h_i$ is the subsurface offset at that point, and $\alpha$ is an optional user-specified weighting parameter. This idea is similar to the motivation behind some inversion schemes such as differential semblance optimization (Symes and Carazzone, 1991). Using this measure, a value of $F=1$ means that all energy is perfectly focused at zero offset; as $F$ decreases toward zero the image becomes progressively less focused. Ideally, a measure such as this one would allow a more rigorous comparison among possible models when a more qualitative comparison fails to yield an obvious result.

Fast velocity model evaluation with synthesized wavefields