Modelling, migration, and inversion using linear algebra

John C. Bancroft, Rod Blais

Modelling migration and inversion can all be accomplished using Linear Algebra. Key to these processes is the diffraction array that is multidimensional. Two dimensional poststack migrations require a four dimensional diffraction array. Current processing practices for Least-Squares analysis require a diffraction array that is two dimensional (a matrix), with one dimensional vectors for the reflectivity and seismic data. These matrices and vectors can be derived from multidimensional data by helical unwrapping. The field of Multilinear Algebra may allow the data to retain their multidimensional arrays, but require defining processes such as a two dimensional transpose of a three, or higher dimensional array. Modelling, migration, and inversion are demonstrated using Linear Algebra with MATLAB software, along with a corresponding 2D transpose of a 4D diffraction matrix.