In the previous chapter, I showed that in presence of adequate sampling one can use an asymptotic ``true-inverse'' or a properly scaled adjoint to get a useful image. Problems arise in 3D reflection seismology where, driven by economic constraints, 3D surveys typically have sparse geometry that results in spatial aliasing.
In this chapter, I introduce a new dealiasing technique named ``inversion to common offset'' (ICO) that takes advantage of the abundance of seismic traces in multi-fold 3D data to interpolate beyond aliasing. The matrix that relates the data to the model is the AMO operator where the model simulates a regular common-offset experiment. The technique can be viewed as a generalization of the inversion to zero offset (IZO) discussed by Ronen 1985. The main advantage of ICO, is that the modeling operator, AMO, is very compact and consequently cheaper to apply than other wave-equation processes. The new inversion also enables prestack analysis of the reflectivity function since the output models are partial stacks at non-zero offset. The partial models can be migrated separately and, either stacked together to form the final image, or, individually analyzed for amplitude and velocity variations.
I developed ICO for the application of regularizing the geometries of 3D surveys before imaging. After regularization, 3D data become handy for prestack migration using any wave extrapolation methods including finite-differencing and wave-number domain techniques. Another promising application demonstrated in this chapter is to reduce acquisition costs by collecting seismic data with fewer offsets. Posing partial imaging as an optimization process, one would use AMO as a modeling operator to generate data with a range of azimuth and offsets, compare them to the field data, and iteratively use the differences to improve the partial image.
The first section of the chapter presents the theory and the formulation of the least-squares solution for multichannel seismic data. Since the inversion of irregularly sampled data is an ill-conditioned problem, the second section discusses two formulations for a pseudo-inverse which I refer to as data-space and model-space inverse. The third section addresses practical issues of the implementation of the inversion, mainly cost efficiency in the log-stretch Fourier domain, proper preconditioning, and regularization of the iterative solution. Finally, the last section of the chapter demonstrates the application of the inversion to reduce the costs of 3D acquisition. It illustrates an example of applying ICO to regularize the coverage of a real 3D land survey and improve the quality of the image by regularizing the data before migration.