The diffraction limbs are interfering with each other in a nonlinear way. One such source of many crossing events is the complex diffraction pattern of many point scatterers close together. Our strategy is to identify geometric complexity when there are many crossing events in the unmigrated data.

They mitigate the problem but can create non-geologic looking shapes. Usually these methods cannot comprehend the geometry of the problem. For tomography, common regularization method would be to use multi-scale method or to change damping parameter. Regularization methods are usually used to mitigate these problems. Tomographic artifacts are also evident as V-shaped artifacts in 2-D profile of the velocity model. It is also common to see “bull’s eye” in tomography when we view depth slices. These are detrimental artifacts when we try to interpret detail features at the reservoir level. They look like little smiles or frowns in a 2-D section and they look like little circles in a time or depth slice. Introduction It is common to see migration artifacts especially when we zoom in close to the reservoir of interest. This gives a geometric criterion for migration or inversion methods like tomography. The B0 (Betti number of zero homology group) measures connectivity and B1 (Betti number of first homology group) measures one-dimensional holes and B2 (Betti number of second homology group) measures two-dimensional holes. The larger the Betti numbers are in an area, the more complex the seismic response would be for migration. In this paper, we will use the Betti numbers from computational topology to describe geometric simplicity. Geometric constraints are usually difficult to define for migration or inversion since geometry is in general a global concept. Most migration optimization or tomography optimization methods are based on amplitude like RMS measurements or semblance or correlations. We introduce a new criterion for migration based on geometric simplicity. Too little preprocessing could cause migration artifacts when the migration operator is presented with data which does not fit the assumptions. Too much signal processing could remove signal which used to be called “noise” since previous noise was an incomplete understanding of signal based on the wave equation. Summary With more sophisticated algorithms like reverse-time migration and waveform inversion, it requires a closer examination as to how we do preprocessing of the input seismic data. In general, the slowly varying aspects of the signal (especially the reflections) are not part of the measured 5D leakage so most important aspects of the signal appear to be reliably interpolated. Using the MWNI algorithm, we show with real data examples that 5D leakage can contain complex (quickly varying) aspects of the data such as noise, diffraction patterns and acquisition footprint. 5D leakage is the noise, and possibly the signal, that 5D interpolation is not able to correctly interpolate due to the fact that the data do not completely conform to the simplicity constraints used by the interpolation algorithm. To address this concern, we present a simple, general method of measuring 5D leakage. However, in general the interpolation methods work by making some type of assumption about the simplicity or sparsity of the underlying model that describes the seismic data, so there is a legitimate concern about whether any information or resolution is being lost during the interpolation process. Summary 5D interpolation methods have proven to be capable of overcoming the constraints of actual data acquisition in a wide range of situations.

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