Internship Proposal: Efficient smoothing preconditioners for multidimensional inverse problems Many systems studied in geophysics can be modelled by a set of linear or non-linear equations. Efficient solutions can be found by casting such systems as inverse problems. Geophysical inverse problems are often constrained by large quantities of multidimensional seismic data. For a given problem the aim is to invert for physical properties of the Earth from the recorded seismic data: ... (1) Equation (1) describes a linear inverse problem, with least squares solution: ... (2) The data vector, d, represents the recorded seismic data. The vector of model parameters, m, holds the desired physical properties of the Earth. The matrix operator, G, contains the equation coefficients which map the physical parameters to the recorded data. Seismic data is noisy and contains information only within a narrow frequency band compared to the desired physical properties. For these reasons it is often necessary add extra constraints to a seismic inversion in order to style the model parameters to meet our a priori expectations. One method of constraining such a problem is Tikhonov regularization: ... (3) The Tikhonov operator, R, styles the model parameters. For example, choosing R to equal the identity matrix allows control of the energy in m using the scalar λ. Seismic data provides a set of measurements for every sub-surface location of the coordinate system x, y and energy travel time, t. For many geophysical inverse problems it is computationally convenient to invert each spatial location independently. This approach turns a large multidimensional inverse problem into many small independent 1D inverse problems. However, this also restricts the styling of model parameters to 1D. It is now computationally feasible to invert the entire seismic data in a single system, enabling multidimensional regularization. The seismic data can be unwrapped into a single vector, and offdiagonal elements in the Tikhonov operator, R, used to force relationships between model parameters in any number of dimensions. This unwrapping process is referred to as a helical coordinate transform, in which multidimensional filtering becomes 1D.(1) For many geophysical inverse problems it is particularly useful to style the model parameters to be smooth, in which case R is a multidimensional difference operator. In helical space this operator is sparse, composed of one block per dimension each with only two non-zeros diagonals. Due to its sparsity, the construction and action of this operator is computationally efficient. However, it has been found that iterative solutions of equation (3), in which R is a difference operator, converge slowly to a smooth solution.
An alternative to inversion regularization is inversion preconditioning. This involves a change of variables: ... (4) Inserting (4) into (3) yields: ... (5) The preconditioning matrix, P, changes the model parameters from u to m and is constructed to ensure faster convergence of an iterative solver. For a conjugate gradient solver, this is achieved when P = R-1.(1) Substituting this into equation (5) gives: ... (6) When multidimensional smoothness is desired, the operator P in equation (6) performs multidimensional integration, whereas the operator R in equation (3) performs multidimensional differentiation. Iterative solutions of equations (3) and (6) converge to the same state, but traverse very different paths.(1) A few iterations will provide a smooth solution to equation (6), while many are required for equation (5). However, the operator R is very sparse, while the operator P is not, making equation (3) more computationally efficient to implement. This internship will work towards design and implementation of an efficient inversion preconditioning scheme in which we seek to enforce multidimensional smoothness. This requires coding the operator P, which in explicit matrix form is too large to hold in memory for seismic inversion problems. The convergence rate, convergence path, computational efficiency and solution of the method should be compared to an existing implementation of the equivalent regularized problem in equation (3). This comparison will be made using the geophysical inverse problem of inverting for the change in the compression-wave velocity given time-lapse seismic data.
(1) Claerbout, J.F., Image Estimation by Example, 2008, http://www.reproducibility.org/RSF/book/gee/book.pdf
