Seismic data interpolation using plane-wave shaping regularization

Introduction

Choosing the most appropriate interpolation scheme to cope with insufficient seismic data can be challenging. Often the simplest direction in which we choose to interpolate is the inline or cross-line directions. This strategy can be improved by interpolating along seismic horizons to preserve structural information.

Previous work has been done in the area of image-guided interpolation. Blended-neighbor interpolation was developed by Hale (2010,2009). This method interpolates borehole data across seismic data to a 3D grid and can be extended to the image domain (Naeini and Hale, 2014). Structure-oriented filters can be applied to a seismic image to improve interpretation (Liu et al., 2010; Fehmers and Höcker, 2003). Previous work taking geologic structure into account for tomography applications was done by Clapp et al. (2004) by using space-varying steering filters.

Regularization is a technique to constrain model parameters for inversion. Solving ill-posed seismic inverse problems offers several choices in the form of regularization (Engl et al., 1996; Zhdanov, 2002). The well-known and widely used Tikhonov regularization (Tikhonov, 1963) is reliable but can be slow to converge because it conflicts with the main goal of the data misfit term in the objective function (Harlan, 1995). For seismic events with a dominant local slope, the corresponding operator of this is plane-wave destruction (PWD) where traces are predicted from their neighbors and subtracted (Fomel, 2002). Model reparameterization, another regularization style, encourages a certain behavior by applying a preconditioning operator (Fomel and Claerbout, 2003). The analog for local plane-wave events is plane-wave construction (PWC) which is the mathematical inverse of the PWD operator (Fomel and Guitton, 2006). A simpler form of PWC is the steering filters of Clapp et al. (1998).

In this paper, we investigate a different form of regularization: plane-wave shaping (Fomel, 2007) and demonstrate the power of plane-wave shaping (PWS) regularization on a 2-D synthetic example, on a 3-D synthetic example, and on 3-D field data. We heavily decimate the data and test the interpolation schemes. By comparing the data misfit as a function of iteration number we show that using shaping regularization along structure achieves an accurate solution in fewer iterations than the alternative regularization methods, PWD and PWC.

Because of the generality of using plane-wave shaping to approach regularization, this method may have utility in many areas of geophysics. Estimating a trustworthy velocity model in reflection seismology is one such inverse problem (Woodward et al., 2008; Clapp et al., 2004).

Seismic data interpolation using plane-wave shaping regularization