Three-dimensional Seismic Data Regularization a Dissertation Submitted to the Department of Geophysics and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

This dissertation addresses the problem of interpolating irregularly spaced seismic exploration data to regular spatial locations. This problem arises in practice in three-dimensional seismic exploration in such applications as Kirchhoff prestack migration, multiple elimination, waveequation migration, and 4-D seismic monitoring. I formulate data regularization as a linear-estimation problem and solve it with an iterative optimization algorithm. I develop a general method of accelerating the iterative convergence using inverse recursive filtering and show that invertible multidimensional filters can be created by spectral factorization in helical coordinates. Different choices exist for the regularization operator in iterative data regularization. • Smoothly varying surfaces are regularized with recursive helical smoothers based on the tension-spline theory. • The local plane-wave model is suitable for characterizing different types of seismic data. Such data are successfully regularized with plane-wave destructor filters. • Seismic reflection data exhibit additional degrees of predictability because of multiple coverage. They can be regularized with finite-difference offset and shot continuation filters. To provide a theoretical basis for the latter choice, I introduce a special partial differential equation that describes offset continuation as a wave-like process. I study the properties of this equation theoretically and prove that the process described by it provides for a kinematically v and dynamically equivalent offset continuation transform. Kinematic equivalence means that in constant velocity media the reflection traveltimes are transformed to their true locations on different offsets. Dynamic equivalence means that, in the OC process, the geometric spreading term in the amplitudes of reflected waves transforms in accordance with the geometric seismics laws, while the angle-dependent reflection coefficient stays the same. Synthetic and real data tests demonstrate the performance and applicability of the proposed data regularization methods. I show that a local plane-wave continuity of the input data allows for an efficient regularization with plane-wave filters. Increasing the filter accuracy leads to more accurate results. Alternatively, seismic data can be regularized with offset-continuation filters, which operate efficiently in frequency slices and bring information from neighboring offsets. Offset continuation succeeds in structurally complex situations where more simplistic approaches fail. The dataset tested is 3-D marine from the North Sea.