Multidimensional recursive filter preconditioning in geophysical estimation problems

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# Introduction

Regularization is a method of imposing additional conditions for solving inverse problems with optimization methods. When model parameters are not fully constrained by the problem (i.e. the inverse problem is mathematically ill-posed), regularization restricts the variability of the model and guides iterative optimization to the desired solution by using additional assumptions about the model power, smoothness, predictability, etc. In other words, it constrains the model null space to an a priori chosen pattern. A thorough mathematical theory of regularization has been introduced by works of Tikhonov's school (Tikhonov and Arsenin, 1977).

In this paper, we discuss two alternative formulations of regularized least-squares inversion problems. The first formulation, which we call model-space, extends the data space and constructs a composite column operator. The second, data-space, formulation extends the model space and constructs a composite row operator. This second formulation is intrinsically related to the concept of model preconditioning (Vandecar and Snieder, 1994). We illustrate the general regularization theory with simple synthetic examples.

Though the final results of the model-space and data-space regularization are theoretically identical, the behavior of iterative gradient-based methods, such as the method of conjugate gradients, is different for the two cases. The obvious difference is in the case where the number of model parameters is significantly larger than the number of data measurements. In this case, the dimensions of the inverted matrix in the case of the data-space regularization are smaller than those of the model-space matrix, and the convergence of the iterative conjugate-gradient iteration requires correspondingly smaller number of iterations. But even in the case where the number of model and data parameters are comparable, preconditioning changes the iteration behavior. This follows from the fact that the objective function gradients with respect to the model parameters are different in the two cases. Since iteration to the exact solution is rarely achieved in large-scale geophysical applications, the results of iterative optimization may turn out quite differently. Harlan (1995) points out that the goals of the model-space regularization conflict with each other: the first one emphasizes ``details'' in the model, while the second one tries to smooth them out. He describes the advantage of preconditioning as follows:

The two objective functions produce different results when optimization is incomplete. A descent optimization of the original (model-space) objective function will begin with complex perturbations of the model and slowly converge toward an increasingly simple model at the global minimum. A descent optimization of the revised (data-space) objective function will begin with simple perturbations of the model and slowly converge toward an increasingly complex model at the global minimum. ...A more economical implementation can use fewer iterations. Insufficient iterations result in an insufficiently complex model, not in an insufficiently simplified model.

In this paper, we illustrate the two approaches on synthetic and real data examples from simple environmental data sets. All examples show that when we solve the optimization problem iteratively and take the output only after a limited number of iterations, it is preferable to use the preconditioning approach. When the regularization operator is convolution with a filter, a natural choice for preconditioning is inverse filtering (recursive deconvolution). We show how to extend the method of preconditioning by recursive filtering to multiple dimensions. The extention is based on modifying the boundary conditions with the helix transform (Claerbout, 1998).

 Multidimensional recursive filter preconditioning in geophysical estimation problems

Next: Review of regularization in Up: Multidimensional recursive filter preconditioning Previous: Multidimensional recursive filter preconditioning

2013-03-03