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

Supporting the hybrid norm

The L2-norm is often a non-optimal choice both because of the non-Gaussian nature of noise in data and it's tendency to create smooth models with traditional regularization schemes. To improve model estimation, authors have suggested using either direct L1-norm approaches, or more commonly, Iteratively Reweighed Least-Squares (IRLS) to approximate different norms. IRLS attempts to find the best model at a non-L2 norm by a series of linearization s. Each non-linear iteration updates the weighting $\bf W$ of the residual. For example we can achieve L1 like behavior by updating the weighting operator through,

$$w(i)= \frac{1}{\vert r(i)\vert}.$$ (10)

IRLS methods tend to be difficult to use because the user must choose carefully the number of relinearizations and the numbers of steps between relinearizations carefully. (Claerbout, 2009) suggested an alternate approach that dynamically changes the weighting function every iteration and uses a Taylor expansion of the standard conjugate direction algorithm to update the solution. Further he suggests a different norm, the hybrid norm, that creates a smooth transition between the standard L2 problem and a L1 problem. Given an error function $E$ and a residual vector $r(i)$ the hybrid norm is defined

$$E=\sum_i R^2(\sqrt{1+r^2(i)/R^2}-1),$$ (11)

where $\bf R$ is a user supplied bad-data percentile.

Creating an inversion framework that supports a Taylor expansion approach to conjugate directions requires adding two additional features to our vector class description. First, we need to associate a norm to each vector. Second we need to be able to multiply a vector by another vector, element by element. Adding support for the hybrid norm requires more changes. A vector must now have a bad-data percentage associated with it, it must be able to find its ith percentile value, and create a vector with this value.

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