The redatuming approach, often referred to as up-down deconvolution, is well-known and is 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 (MDD), in which the integral equation solution is achieved by introducing the point-spread function concept. We have developed 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 MDD. In the latter case, we have developed 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 MDD within the range of sampling scenarios considered so far. We apply this methodology to synthetic data, and we discuss the potential to extend up-down deconvolution to a broader range of geological conditions.

中文翻译：

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复杂地质场景中的上下反褶积

重定基准法，通常称为上下反褶积，是众所周知的，用于去除海底地震调查中的水层和震源特征效应。上行波场可以表示为下行波场与地球反射率的多维卷积。因此，从上行波场中解卷积下行波场可以得到地球的反射率响应。解卷积过程需要求解一个多维积分方程，但是，在横向不变的介质中，在波场被分解成平面波分量之后，如果在傅立叶或氡域中作为频谱划分来执行解卷积，则可以极大地简化解卷积。实验观察到，一次执行一个平面波分量的去卷积会产生良好的结果，即使存在复杂的地下结构，前提是海床相对平坦。当不满足这种地质条件时，同样的问题可以用多维反褶积 (MDD) 的干涉测量法来表述，其中通过引入点扩展函数概念来实现积分方程解。我们开发了一种基于数值模拟的方法，以确定何时可以在平移不变波场的假设下求解与上下反卷积问题相关的积分方程，以及何时需要 MDD。在后一种情况下，我们开发了一个正则化的逆过程，它减轻了由于反演的典型不适定性质而导致的数值问题，并且结合了下行的插值策略，使 MDD 在目前考虑的采样场景范围内应用成为可能。我们将这种方法应用于合成数据，并讨论了将上下反褶积扩展到更广泛的地质条件的潜力。