Seismic sparse-layer reflectivity inversion using basis pursuit decomposition Rui Zhang 1 and John Castagna 2 ABSTRACT A basis pursuit inversion of seismic reflection data for re- flection coefficients is introduced as an alternative method of incorporating a priori information in the seismic inversion process. The inversion is accomplished by building a dic- tionary of functions representing reflectivity patterns and constituting the seismic trace as a superposition of these patterns. Basis pursuit decomposition finds a sparse number of reflection responses that sum to form the seismic trace. When the dictionary of functions is chosen to be a wedge-model of reflection coefficient pairs convolved with the seismic wavelet, the resulting reflectivity inversion is a sparse-layer inversion, rather than a sparse-spike inversion. Synthetic tests suggest that a sparse-layer inversion using basis pursuit can better resolve thin beds than a comparable sparse-spike inversion. Application to field data indicates that sparse-layer inversion results in the potentially im- proved detectability and resolution of some thin layers and reveals apparent stratigraphic features that are not readily seen on conventional seismic sections. INTRODUCTION In conventional seismic deconvolution, the seismogram is con- volved with a wavelet inverse filter to yield band-limited reflectiv- ity. The output reflectivity is band limited to the original frequency band of the data so as to avoid blowing up noise at frequencies with little or no signal. It has long been established (e.g., Riel and Berkhout, 1985) that sparse seismic inversion methods can pro- duce output reflectivity solutions that contain frequencies that are not contained in the original signal without necessarily magnifying noise at those frequencies. It is well known (e.g., Tarantola, 2004) that applying valid constraints in seismic inversion can stably in- crease the bandwidth of the solution. However, incorporation of the a priori information in the reflec- tivity inversion of seismic traces can be problematic. A common way of incorporating prior knowledge is to build a starting model biased by that information and to let the inversion process perturb the initial starting model and converge to a solution (e.g., Cooke and Schneider, 1983). The individual layers represented in the starting model can have hard or soft constraints assigned. This kind of meth- od can work very well when the starting model is close to the correct solution. Typically, the starting model is obtained by spatially inter- polating well logs along selected horizons. Unfortunately, these hor- izons must be picked on the original seismic data. If waveform interference patterns change laterally, horizon picks on a constant portion of a waveform (typically chosen to be peaks, troughs, or zero crossings) can be in error, resulting in an incorrect starting model and a potentially erroneous inversion. Similarly, if velocities and/or impedances for the inversion interval change laterally in a manner different from that resulting from the interpolation proce- dure, interpolated well logs may again be significantly in error, and the inverse process may converge to the wrong minimum. These problems may be ameliorated with a Monte Carlo approach, but such an approach cannot correct the fundamental nonuniqueness of the process that may cause minimums other than the correct one to have similar errors. A means of biasing the results toward expected reflectivity patterns is needed without relying on possibly erroneous manual interpretations or spatial interpolations. Nguyen and Castagna (2010) used matching pursuit decomposi- tion (MPD) to decompose a seismic trace into a superposition of reflectivity patterns observed in and derived from existing well con- trol. Matching pursuit decomposition (1) correlates a wavelet dic- tionary against a seismogram and finds the location, scale (i.e., center frequency), and amplitude of the best-fit wavelet, (2) sub- tracts the best-fit wavelet and records its characteristics in a table, and (3) repeats the processes on the residual trace until the residual energy falls below a selected threshold. For spectral decomposition, Manuscript received by the Editor 14 March 2011; revised manuscript received 24 July 2011; published online 22 December 2011. 1 University of Texas, Jackson School of Geosciences, Institute for Geophysics, Austin, Texas, USA. E-mail: rzhang@mail.utexas.edu. 2 University of Houston, Department of Earth and Atmospheric Sciences, Houston, Texas, USA. E-mail: jpcastagna@uh.edu. © 2011 Society of Exploration Geophysicists. All rights reserved. R147 GEOPHYSICS. VOL. 76, NO. 6 (NOVEMBER-DECEMBER 2011); P. R147–R158, 15 FIGS. 10.1190/GEO2011-0103.1 Downloaded 12/22/13 to 131.243.227.247. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/