Preserving amplitude information when imaging 3-D multichannel seismic data is a challenging task. The imaging process can be described as seeking the inverse for a large matrix that relates the data to a reflectivity model. Due to irregular coverage at the surface and uneven illumination of the subsurface, the matrix is often ill-conditioned and its coefficients are badly scaled. We present a new approach for regularizing the coverage of 3-D surveys by partial imaging before migration. After regularization, 3-D data become better suited for prestack migration using either “true amplitude” Kirchhoff methods or wave extrapolation techniques.
Posing partial stacking before migration as an inverse problem, we develop a new optimization technique named “inversion to common offset” (ICO) that solves for optimal partial stacks from multifold 3-D data. The inversion approach takes advantage of the redundancy of information present in multichannel recording to recover densely and regularly sampled common-offset common-azimuth volumes. The reconstruction of each of these volumes combines data from multiple incomplete experiments over a common earth model to simulate the ideal data that would have been recorded from a single common-offset common-azimuth experiment. The method is based on the inversion of a partial prestack migration operator that maps seismic data from a given acquisition geometry to an arbitrary one. This data mapping operator is the azimuth moveout operator (AMO). The main advantage of ICO is that the AMO operator is compact and consequently cheaper to apply than other imaging operators, such as full 3-D prestack migration.
We present a cost-effective implementation of ICO based on a log-stretch transformation of the time axis. After this transformation, AMO becomes time invariant, so the inversion can be performed in the Fourier domain on each temporal-frequency slice independently. To accelerate the convergence of the iterative solution, we precondition the system of linear equations by two diagonal transformations that scale the rows and columns of the modeling operator. A stable solution is achieved by adding a penalty function that constrains the spatial variability of the model.
Results of applying ICO to a field 3-D land survey show that we can recover the high-frequency features of the reflectivity function from a severely undersampled data set obtained by decimating a densely sampled prestack data set. The images produced by prestack migration after regularization are superior to those produced by direct migration of the decimated data.