Compression of seismic data using ridgelets

Abstract

Sergio E. Zarantonello*, 3DGeo Development Inc. and Santa Clara University, and Dimitri Bevc, 3DGeo Development Inc. Summary Ridgelets are wavelet-like bases developed by Donoho et al. ([1], [2], [3]) for the sparse representation of data with directional feature. The purpose of this study is to assess the effectiveness of ridgelet bases for the compression of seismic data. This work is a continuation of [4] where preliminary results using 2-D ridgelets to compress prestacked 3-D data were presented. Multidimensional ridgelets are standard multidimensional wavelets defined in a multidimensional pseudo-Radon domain. The 3-D ridgelet transform operates on an entire 3-D volume, instead of on a set of 2-D slices of the data as it’s 2-D counterpart, thus allowing for data coherency in three dimensions to be exploited. Our results suggest that compression ratios much higher than what can be attained with a 2-D procedure can be attained using 3-D ridgelets. In this pap er we show results of ridgelet compression on a shallow marine prestacked dataset, and give an outline of a compression system based on ridgelets in 3-D. Introduction Wavelets are an ubiquitous component of any state-of-theart image compression system. The success of wavelets is due to the following: • Wavelets are functions localized both in the physical and frequency domains (but contingent to the Heissenberg Uncertainty Principle). • The wavelet coefficients of smooth functions decay steeply with increasing resolutions. This allows for their sparse approximation and efficient compression. • Wavelets give an efficient representation of point singularities in the data. • The wavelet transform can be implemented using a fast algorithm. • Wavelet bases can be chosen to match certain characteristics of the target data. This allows for adaptive compression strategies using wavelets. A set of 2-D biorthogonal wavelet basis functions using the optimized filters of Donoho and Ergas [5] are shown in Figure 1. Figure 1. Tensorial biorthogonal wavelets in 2-D Wavelets in higher dimensions, as those shown in Figure 1, are usually the tensor products of one-dimensional wavelets. The disadvantages of tensorial wavelets is that they exhibit a directional bias in the directions of the coordinate axes, and that because of this, provide inefficient representions of line singularities. These shortcomings are overcome with directional wavelet systems such as the ridgelet basis functions shown in Figure 2.. Ridgelets provide sparse coding of images with edges. In addition to being localized in the physical and frequency domains, they are aligned along a wide range of directions, and thus give compact representions of line singularities.