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

Appendix B: A Primitive, Semblance-Like Volume for Horizon Picking

Suppose we have a trace representative of a seismic horizon, apply cosine tapering to its edges, and pad it with zeros. This ideal horizon reference trace is called $h(t)$ . To create a measure of how well this ideal waveform matches with other traces throughout a seismic volume, the crosscorrelation is calculated throughout the volume over shift $\tau$ ,

$$\gamma(\tau,\mathbf{x}) = \int h(t+\tau)d(t,\mathbf{x}) d\tau,$$ (9)

where $d(t,\mathbf{x})$ is the seismic image. In order to ensure that the volume we generate is non-negative, let

$$\alpha(\tau,\mathbf{x}) = \gamma(\tau,\mathbf{x}) - \min_{\left(\tau,\mathbf{x}\right)} \left[ \gamma(\tau,\mathbf{x})\right] .$$ (10)

Automatic picking may be performed on the semblance-like volume $\alpha(\tau,\mathbf{x})$ to determine the shifts defining the horizon corresponding to reference trace $h(t)$ . Those shifts may be converted back to time in the image domain by adding a reference time for the ideal horizon trace, taken here to be that trace's midpoint.

This is a basic method for creating a horizon probability volume, which is intended for use in this paper as part of a proof of concept for automatic horizon interpretation using the variational picking algorithm. It could encounter difficulties in areas where significant lateral wavelet variation or faulting occurs.

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