A variational approach for picking optimal surfaces from semblance-like panels

Conclusions

We propose a variational method for picking velocity surfaces from semblance-like volumes. When coupled with a continuation approach, this method is able to avoid many local minima. Using a discretized version of the variational statement with a $\ell$ -BFGS algorithm enables the method to converge more rapidly.

The variational method outlined here for determining optimal surfaces from semblance-like volumes is more computationally expensive than existing methods for determining optimal lines from semblance panels. However, the ability of the method to incorporate spatially adjacent information without explicitly imposing smoothing on the model justifies the added expense. Direct application of the method suffers from the multimodality and non-convexity of the proposed objective function and, unless a particularly well-informed starting model is used, the method is likely to converge to a local minimum. Such local minima may differ significantly from the global minimum. This difficulty may be overcome by applying the variational picking scheme with a continuation approach. Although continuation increases the cost of the method through the creation of additional smoothed semblance volumes and the application of the picking scheme to each smoothed volume, the ability of the method to avoid local minima and find a superior final model justifies the expense.

Applying the proposed approach to field data sets from the Viking Graben and the Gulf of Mexico shows how it is able to determine geologically plausible velocity models. The use of continuation both enables the scheme to behave more like a global minimizer, and allows models output by the approach to vary substantially from the starting model. Using the lowest-cost output model for each data set in seismic processing workflows produces quality images and further shows how the method may be incorporated as a tool for seismic processors.

The versatility of the variational method is demonstrated by deploying it to automatically pick a seismic horizon from the Heidrun Field. Because smoothing is not explicitly imposed on the model, and is only used during the continuation process, the approach is able to determine a seismic horizon that changes rapidly in space to follow the reflector.

This method solves for minimizing surfaces in $H^1$ , a space of smoothly varying functions. Minimizing surfaces are guaranteed to exist in this space because of the presence of a strictly positive $\epsilon$ in Equation 4, which is essential for demonstrating the existence of minimizers and proving that iterative schemes converge to those minimizers in an infinite-dimensional setting. Because RMS velocities are, in a sense, the integral of interval velocities, which exist in $L^2$ , or equivalently are square integrable functions, all RMS velocities will exist in $H^1$ . Therefore, this is not a limitation. However, if one is interested in solving for best-fit surfaces which are not smoothly varying but may feature discontinuities, they would want to set $\lambda$ to a high value and $\epsilon$ to 0 . Although having $\epsilon = 0$ violates the assumptions used in proving the existence of and convergence to minimizers of the variational method, in practice, minimizers are found to exist and can be iteratively determined.

The variational statement can be thought of as a modified soap bubble problem. It determines minimal area surfaces using a weight which rewards tracking highs in the semblance-like volume. If intervals within the semblance-like volume without high values exist, the method should output a minimal-area, or constant-gradient, surface connecting the high values bounding the region. This situation corresponds to intervals in reflection velocity scans without reflection energy. The method assumes that highs in the semblance-like volume correctly track the value of the parameter. Thus, the largest source of manual intervention in the examples shown in this paper comes from defining the functions to mute artificially high values from the semblance-like volumes where this assumption does not hold. Developing methods to automatically generate those mutes is a promising direction for future study.

Beyond the demonstrated applications shown here for processing 2D seismic lines and automatically interpreting seismic horizons, numerous promising directions for future study involving the application of this variational approach exist. Examples include using the method to generate starting models for full waveform inversion, extending the software to picking 3D velocity volumes from 4D semblance hypervolumes, calculating time-shifts to match time-lapse seismic volumes, or applying the method to other situations where one wishes to determine a laterally continuous surface. These extensions could be accomplished using the same variational statement featured in this paper. Modifying that variational statement could enable the creation of a method for simultaneously determining surfaces for multiple parameters, which would be useful for generating anisotropic models.

A variational approach for picking optimal surfaces from semblance-like panels