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Short Note
Steering filters in 3-D: Tubes rather than planes

Robert G. Clapp and Marie L. Clapp,

INTRODUCTION An operator composed of non-stationary plane-wave destruction filters, called a steering filter, has practical applications to many problems. Clapp et al. (1997) demonstrated their use for the missing data problem. Fomel (2000) showed how they could be used for signal-noise separation. In several papers Clapp and Biondi (1998, 2000); Clapp (2001), they are used for regularizing tomography. Prucha and Biondi (2002) used them to better handle the null-space in wave equation migration.

Several different methods have been suggested for constructing the 2-D representation of these filters. Clapp et al. (1997) proposed three methodologies: approximating a given dip by interpolating a triangle centered at a given dip or doing Lagrange and Pade interpolation for a given dip. The first method suffered from significant dispersion while all three were severely limited in the angular range that they could accurately describe. To address these and other issues, Fomel (2000) suggested building the filters by doing a Taylor series expansion around zero frequency. The resulting filters were not limited in the angles they could represent and had significant better dispersive properties but the filters were not causal, therefore they could not be used to precondition a model to speed up convergence Fomel et al. (1997).

Constructing a 3-D steering filter poses additional problems. Clapp (2000) suggested cascading two 2-D steering filters. The resulting response had approximately the right dip in both the x-z and y-z directions but was fairly inaccurate at other angles Fomel (2001). Fomel (2001) suggested that a better method was to form the autocorrelation of the combined filter then use Wilson-Burg factorization Sava et al. (1998) to obtain a causal representation of the filter. Combining these filters produced a 3-D steering filter. This methodology is effective but extremely computationally intensive.

In this paper, we suggest another way to construct a specific class of 3-D steering filters, ones with a significant angular bandwidth. These filters are ideal for tomography or other slow-varying functions. The key idea behind our approach is to transform our filter's Cartesian coordinate system to its spherical equivalents. We then estimate the filter coefficients given our desired dip and azimuth bandwidth. The resulting filters are able to handle almost any dip-azimuth combination, with minimal dispersion and angle errors. In this paper we begin by showing how to build these classes of filters in 2-D and 3-D. We then use the filters in a simple missing data problem.

METHODOLOGY The purpose of a non-stationary dip filter is to describe the covariance of a field that has a single dip at each location but may vary spatially. We can think of the dip at each location in terms of its normal and describe that normal in our choice of coordinate systems. An obvious choice for our coordinate system is the Cartesian mesh of our model. Another choice is polar coordinates (2-D) or spherical coordinates (3-D).

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