Wavefield Reconstruction
Authors: Duren Richard E. , Eriksen Erik A. , Trantham E. Clark 
Published: October 2008

Abstract
Wavefield reconstruction is a deterministic method for spatially interpolating pre-stack seismic data. It is robust in the presence of high amplitude coherent noise and it properly handles dipping reflectors, curved reflectors and diffractions. Wavefield reconstruction can be used to de-alias seismic data that is much sparser than we normally acquire. It can also be used to selectively filter out wavefields from seismic data (i.e. it can be used to remove multiples from pre-stack marine data).

Introduction
Because of economic and time constraints, multi-source marine 3-D seismic surveys are common. In single-source marine acquisition with the shot interval equal to the group interval, the common midpoint bins (CMP) contain every other available offset. While that is only half the offsets required to form a full stack array, it is usually sufficient so that the coherent noises (typically water bottom multiples) are not aliased. However, in dual or triple source acquisition, only one fourth or one sixth of the available offsets are contained in each CMP gather. That coarse sampling often results in the multiples being undersampled and aliased, even after accounting for the mute of the far offsets. The aliased portion of the multiple bleeds through the stack array and, if the multiple is strong enough relative to the primaries, it contaminates the stack. A simple example is shown in figures 1 and 2. Figure 1 is a stack of synthetic data, simulating single source acquisition, for a model that contains a single planar interface. The shallow event is a water-bottom multiple while the deeper event is the primary reflection. Figure 2 contains the same as Figure 1 except that, in this case, triple-source acquisition IS simulated. Note that, in addition to the multiple and primary, high-amplitude sloping events are visible on the stack. Those events are the result of the aliased portion of the multiple bleeding through the stack array. In this simple example, that "noise" contaminates the stacked image. There are many examples of marine seismic data where the multiple energy is strong enough to hinder pre-stack data analysis (i.e. AVO) regardless of whether or not the data are spatially aliased. Wavefield reconstruction can be used in a predict and subtract mode to significantly reduce the multiple energy.

Theory and method
Our wavefield reconstruction technique overcomes the spatial aliasing difficulties associated with 3-D marine data. The technique is based in part on a paper by Ronen (1987). He demonstrated that DMO could be used to connect data collected at different offsets with sampling theory to unravel aliasing due to under sampling of the individual offsets. Ronen's theory (1987), however, deals only with de-aliasing of the final stacked image, and it uses a DMO operator that does not preserve amplitudes. Our wavefield reconstruction technique extends his theory by deriving an operator that preserves amplitudes and applying the technique to prestack data. By also drawing on the work of Black et al. (1993) we were able to develop a reconstruction technique that preserves amplitudes. Black et al. (1993) treated the theory of true amplitude DMO while assuming that each of the offsets is perfectly sampled. But Black et a!. (1993) did not address any aliasing, which can result from sparse sampling.