Hybrid-norm and Fortran 2003: Separating the physics from the solver

Operator-based object-oriented solvers

There are at least two different approaches one can take to solving the typical geophysical inversion problem. Harlan (1996), among others, takes a Bayesian approach to inversion. The Bayesian approach allows a natural inclusion of a priori statistical properties of the model. SEP (Claerbout, 1999) has traditionally taken an approach which is described as either classical, traditional, or deterministic. The classical approach attempts to find the model $\bf m$ that minimizes the data misfit. Given a recorded dataset $\bf d$ , and a linear operator $\bf L$ , we attempt to minimize the residual vector $\bf r$ which is defined as

$$\bf0 \approx \bf r = \bf d - \bf L \bf m.$$ (1)

In the simplest case where we are using steepest descent to solve the linear least squares inversion, we estimate $\bf m$ by mapping the initial residual (in this simple case $-\bf d$ ) back into the same space as the model to form a gradient vector $\bf g$ by applying the adjoint of $\bf L$ . We then map the gradient vector back into data-space by applying $\bf L$ to form $\bf rr$ . Finally, we find the scaling factor of $\bf rr$ that will make $\bf r + \bf rr$ as small as possible. We then repeat this procedure until $\bf r$ is suitably small. More complex inversion approaches are normally built on this basic concept.

Hybrid-norm and Fortran 2003: Separating the physics from the solver