Up and down deconvolution in complex geological scenarios

The redatuming approach, often referred to as up-down deconvolution, is well-known and applied to remove water-layer and source-signature effects in seabed seismic surveys. The upgoing wavefield can be expressed as the multidimensional convolution of the downgoing wavefield with the earth’s reflectivity. Consequently, deconvolving the downgoing wavefield from the upgoing wavefield, gives us the earth’s reflectivity response. The deconvolution process requires solving a multidimensional integral equation but, in a laterally invariant medium, after that wavefields are decomposed into plane-wave components, deconvolution can be enormously simplified if performed as a spectral division in the Fourier or Radon domain. It has been experimentally observed that deconvolution carried out one plane-wave component at a time gives good results, even in the presence of complex subsurface structures, provided that the seabed is relatively flat. When this geological condition is not satisfied, the same problem can be formulated in terms of interferometric redatuming using multidimensional deconvolution, where the integral equation solution is achieved by introducing the point-spread function concept. We present a methodology based on numerical simulations to determine when the integral equations associated with the problem of up-down deconvolution can be solved under the assumption of shift-invariant wavefields and when it requires multidimensional deconvolution. In the latter case, we propose a regularized inverse procedure that mitigates the numerical problems due to the typically ill-posed nature of the inversion and that, combined with an interpolation strategy for the downgoing, enables the application of multidimensional deconvolution within the range of sampling scenarios considered so far. We apply this methodology to synthetic data, and we discuss on the potential to extend up-down deconvolution to a broader range of geological conditions.