INTRODUCTION

Processing seismic data for amplitude inversion has many applications in AVO analysis, reservoir monitoring, detection of anisotropy, estimation of fracture density and orientation and other related applications. Such detailed exploration objectives have led to the careful design of seismic surveys. Unfortunately, during the acquisition stage, obstructions, cable feathering, environmental objectives, economic constraints and many other factors cause seismic data to be sampled in sparse and irregular fashion. These irregularities are often observed in the form of variations in fold coverage, which can manifest itself as an acquisition foot-imprint on prestack data or even the stacked image. If not accounted for, irregular sampling can affect prestack data analyses and may introduce noise, cause amplitude distortions, and even structural distortions in the final image. Beasley (1994); Black and Schleicher (1989); Chemingui and Biondi (1996a); Gardner and Canning (1994). Many techniques with varying accuracy and cost have been proposed for processing irregularly-sampled data; among them equalized DMO Beasley and Klotz (1992), geometrically calibrated DMO Ronen et al. (1995), spatial dealiasing Ronen and Liner (1987) and fold normalization Chemingui and Biondi (1996b). The goal has been always to avoid aliasing, interpolate missing data and normalize the imaging process for the effects of fold variations.

Chemingui and Biondi 1996a have demonstrated that the effects of irregularly-sampled data on seismic amplitudes can be substantial and have proposed a method for processing wide-azimuth 3D surveys that can largely overcome these problems. The technique is based on applying the AMO transformation Biondi et al. (1996) in order to organize the data into common-azimuth common-offset cubes and, therefore, to allow interpolation to a regular grid before imaging. In a subsequent SEP report, Chemingui and Biondi 1996b proposed an additional development in their technique to compensate for the effects of irregular fold distributions. The method extends the multiplicity concept to wave equation processes and uses a normalization procedure to correct the imaging operator for the effects of irregular coverage. The normalization presents an approximate solution to the problem of fold variation by normalizing each input trace in the prestack process according to the local fold of its corresponding bin.

In this report we present a new technique to invert for reflectivity models while properly handling the irregularities in spatial sampling. The technique is based on the method of least squares and consists of a two-step solution to the imaging problem. In the first stage an inverse AMO filter is computed to account for the interdependencies between data traces, then an imaging operator is applied to the filtered data to invert for the final reflectivity model. In the next section we show the relationship between irregular sampling and inverse theory and present a formalism for the normalization filter. The computation of each element of the filter requires the evaluation of an inner product in the model space. We show that each inner product corresponds to an AMO transformation between two data elements. We explore the effectiveness of the method in the 2D case for the application of offset continuation and partial stacking and compare our results to conventional processing techniques.