Multidimensional recursive filter preconditioning in geophysical estimation problems

Model-space regularization

Model-space regularization implies adding equations to system (1) to obtain a fully constrained (well-posed) inverse problem. The additional equations take the form

$$\epsilon \mathbf{D m \approx 0} \;,$$ (2)

where $\mathbf {D}$ is a linear operator that represents additional requirements for the model, and $\epsilon$ is the scaling parameter. In many applications, $\mathbf {D}$ can be thought of as a filter, enhancing undesirable components in the model, or as the operator of a differential equation that we assume the model should satisfy.

The full system of equations (1-2) can be written in a short notation as

$$\mathbf{G_m m} = \left[\begin{array}{c} \mathbf{L} \epsi... ...bf{d} \mathbf{0} \end{array}\right] = \hat{\mathbf{d}}\;,$$ (3)

where $\hat{\mathbf{d}}$ is the augmented data vector:

$$\hat{\mathbf{d}} = \left[\begin{array}{c} \mathbf{d} \mathbf{0} \end{array}\right]\;,$$ (4)

and $\mathbf{G_m}$ is a column operator:

$$\mathbf{G_m} = \left[\begin{array}{c} \mathbf{L} \epsilon \mathbf{D} \end{array}\right]\;.$$ (5)

The estimation problem (3) is fully constrained. We can solve it by means of unconstrained least-squares optimization, minimizing the least-squares norm of the compound residual vector

$$\hat{\mathbf{r}} = \hat{\mathbf{d}} - \mathbf{G_m m} = \left... ...thbf{d - L m} - \epsilon \mathbf{D m} \end{array}\right]\;.$$ (6)

The formal solution of the regularized optimization problem has the known form (Parker, 1994)

$$<\!\!\mathbf{m}\!\!> = \left(\mathbf{L}^T \mathbf{L} + ... ...bf{D}^T \mathbf{D}\right)^{-1} \mathbf{L}^T \mathbf{d}\;,$$ (7)

where $<\!\!\mathbf{m}\!\!>$ denotes the least-squares estimate of $\mathbf{m}$, and $\mathbf{L}^T$ denotes the adjoint operator. One can carry out the optimization iteratively with the help of the conjugate-gradient method (Hestenes and Steifel, 1952) or its analogs (Paige and Saunders, 1982).

In the next subsection, we describe an alternative formulation of the optimization problem.

Multidimensional recursive filter preconditioning in geophysical estimation problems