A probabilistic approach to seismic diffraction imaging

Conclusions

We formulate and apply a probabilistic approach to seismic diffraction imaging. By treating the weight functions in path-integral imaging as diffraction likelihood, we are able to emphasize wavefield components in seismic images output by OVC that are the most likely to correspond to a properly migrated seismic diffraction image and suppress the wavefield components that are not likely diffractions, improving the signal to noise ratio. The toy model example illustrated how the probabilistic imaging process attenuated reflection energy and amplifies diffraction energy without explicit separation of the two, but rather as a result of applying weights to partial images. Thus, the method may be used in tandem with data-domain diffraction separation techniques to suppress any remnant reflection energy, as was done in the Nankai trough field data example. The process may attenuate some high frequency diffraction information, making it better suited for imaging diffractions in noisy environments. The synthetic experiment in this paper illustrated how the probabilistic method can be robust in such environments, creating an image featuring strong diffractions and suppressing noise with a RMS value 16 times greater than that of the noiseless diffraction energy. That synthetic experiment also demonstrated how the method has greater success imaging strong diffractions than weak ones and may have difficulty imaging diffractions whose moveout tails are superimposed by the tails of stronger overlaying diffractions. Diffractions imaged by the probabilistic process are laterally coherent in slope gathers because the weights are built from gather semblance. Correctly imaged diffraction energy is typically laterally coherent, but this assumption may be violated when energy from other diffractions becomes superimposed on a the slope gather centered above a correctly migrated diffraction. Therefore, this method should not be thought of as seeking to image every single diffractor within a seismic volume, or being superior to deterministic diffraction imaging, but rather as a supplementary tool to conventional diffraction imaging methods, outputting a image that identifies features that we can say with some certainty are correctly migrated diffractions. This can aid in the process of identifying geologically interesting features like the seafloor, faults, décollement or the transition from sedimentary to crystalline rock, seen in the field data example from the Nankai trough.

Creating a probabilistic diffraction image through the proposed process does not require a migration velocity as an input, but rather generates one as an output. This means that direct comparison of the diffractions resolved by the deterministic image and the probabilistic image in the noiseless synthetic experiment as a measure of the probabilistic method's utility is not particularly meaningful, as that deterministic image in a noiseless environment with perfect apriori information of the subsurface velocity is effectively the best diffraction image one could hope to achieve in a single migration (although least-squares or sparse inversion techniques could yield a better resolved diffraction image iteratively (Merzlikin et al., 2020)). Instead, the probabilistic imaging method determines the most likely velocity field, as well as a measure of velocity uncertainty, as it operates. That output migration velocity may also be determined using a single offset of data, as was the case in the experiments presented in this paper. The expectation velocity produced in the synthetic experiments presented here tracked the correct migration velocity to within one standard deviation, and for the Nankai Trough field data example output reasonable velocities that resolved a velocity inversion. The process of finding the expectation velocity requires diffraction information, so the method's ability to determine correct migration velocities may be limited where diffractions are not present, as seen in the toy model example and in the Nankai Trough example below the transition to crystalline rock.

The challenges encountered in the studies featured in this paper introduce some promising directions for future inquiry. The difficulties faced by the probabilistic method at resolving a large dynamic range of diffractions could be mitigated by decreasing the dynamic range of the weight functions by using, for example, powers of the combined weights, although that would also make the method more susceptible to noise. Finding the powers of weight functions that best balance highlighting diffraction signal and noise could improve the method. Similarly, finding more sophisticated ways of normalizing the weights to preserve weak diffraction and exploring other weights tied to the likelihood of diffraction could yield improved results. Particularly interesting is exploring different measures of diffraction ``flatness,'' or diffraction likelihood, than semblance in the slope-gather domain. Such measures may not be ideal in the presence of edge diffractions, as seen in the field data example. Although the probabilistic imaging process using semblance was able to resolve those edge diffractions, a better method may exist. Applying the weighted imaging process to 3D Oriented Velocity Continuation should also enhance the output diffraction images, as that would be able to account for the possible effects of out of plane diffractions and reduce the effect of intersecting diffraction tails on semblance. The concept of probabilistic weighting for path-integral imaging can be used for collections of images where the wavefront is parameterized by more variables than just migration velocity, making application to sets images output by other time migration parameters with a continuation operators a fascinating direction of inquiry. As an anisotropic continuation operator already exists, pursuing a similar framework to determine weights based on both velocity and anisotropy, as well as using the method to determine conditional probability for anisotropy for each migration velocity, seems promising.

The ability of the method to suppress noise, highlighted in the noisy synthetic experiment, suggests that applying this method to passive seismic data could be beneficial and makes it a promising direction of future inquiry. Geophone station signals could be used as input data for the continuation process with an extra event time, $t_o$ variable added. This variable would correspond to earthquake event time in the seismic record. This extra variable could be applied as a shift on $t$ prior to performing OVC and the probabilistic imaging process. Weights could be treated as conditional probabilities given event time $t_o$ . Parallelization in $t_o$ would be a relatively straightforward process, making the expensive required computations feasible.

Probabilistic path-integral diffraction imaging does not require advance knowledge of the migration velocity, which it generates as an output. Additional weights may be used to improve the results, and the method could be modified to emphasize different portions of the wavefield. Theoretically, this approach should function for not just velocity, but any time migration parameter with a continuation operator, so application to anisotropic imaging is a promising direction for future work.

A probabilistic approach to seismic diffraction imaging