Tag Archives: inversion

Brief overview of Occam's inversion

This section is a brief overview of the Occam's inversion approach used by MARE2DEM. It may be helpful for understanding the various parameters in the input files. More detailed descriptions of Occam can be found in Constable et al. (1987) and deGroot-Hedlin and Constable (1990), and a specific application to 1D marine CSEM is described in Key (2009).

Occam's inversion seeks to minimize the following unconstrained regularized functional:

U = \| \mathbf{R m} \|^{2} + \| \mathbf{P( m - m_{*})} \|^{2} + \mu^{-1}\left [ \| \mathbf{W} (\mathbf{d} - F(\mathbf{m}) ) \|^{2} - \chi_{*}^{2} \right ] .

The first term is a norm of the model roughness and is computed by applying a differencing operator \mathbf{R} to the elements of the model vector \mathbf{m}. The model vector \mathbf{m} contains the \log_{10}\rho values for all the free model parameters. \mathbf{R} is chosen to be a matrix of first-differencing operators so that \mathbf{R} approximates the spatial gradient of \log_{10}\rho integrated over the parameter. The parameterization with respect to  \log_{10} \rho ensures that resistivity remains positive during the inversion. If jumps in resistivity are desired at certain boundaries, the corresponding elements of \mathbf{R} can be relaxed or  set to zero. The second term is a measure of the difference of \mathbf{m} from an a priori preference model \mathbf{m_{*}}. The diagonal matrix \mathbf{P} contains scaling parameters that determine the relative weighting between the preference and the model roughness. Preference model values, if used at all, are typically desired for a only few model layers and the corresponding diagonal elements of \mathbf{P} will be nonzero. This can be useful for constraining the anisotropy ratio for anisotropic inversion. The roughness and preference terms in the above equation are regularizers that serve to stabilize the inversion and keep it from producing wildly oscillating resistivity structure (see the regularized/unregularized comparisons in Constable et al. (1987).

Finally, the third term is a measure of the misfit of the model's forward response F(\mathbf{m}) (i.e., the electric and magnetic fields or MT impedances for model \mathbf{m}) to the data \mathbf{d}. There is no restriction on the data vector \mathbf{d}. For example, it can contain data from multiple transmitters, receivers and frequencies, and can include magnetotelluric (MT) data. \mathbf{W} is a data covariance weighting function and is here selected to be a diagonal matrix with elements corresponding to inverse data standard errors. In other words, \mathbf{W} weights the relative contribution of each datum to the misfit based on its uncertainty. Thus, data with large errors are scaled to limit their influence, while data with small errors will have a bigger impact on the misfit budget.

\chi_{*}^{2} is the target misfit and its inclusion illustrates that minimizing U does not necessarily find the best fitting model, but rather a smooth model that is within the specified target misfit. The target misfit \chi_{*}^{2} is usually chosen so that the root mean square (RMS) misfit x_{rms} is equal to unity:

 x_{rms} = \sqrt{\frac{\chi^{2}}{n} }= \sqrt{\frac{1}{n} \sum_{i=1}^{n} \left [ \frac{d_{i}- F_{i}(\mathbf{m})}{s_{i}} \right ]^{2} },

where n is the number of data and s_{i} is the standard error of the ith datum.

The Lagrange multiplier \mu serves to balance the trade-off between the data fit and the model roughness and model preference. The nonlinear minimization of equation {U} is described in Constable et al. (1987).  One of the main innovations of the Occam method is the automatic selection of \mu. So that's Occam described in a single page of text.

References

Constable, S. C., R. L. Parker, and C. G. Constable (1987), Occam's inversion - A practical algorithm for generating smooth models from electromagnetic sounding data, Geophysics, 52(03), 289–300.

deGroot-Hedlin, C., and S. Constable (1990), Occam's inversion to generate smooth two-dimensional models from magnetotelluric data, Geophysics, 55(12), 1613–1624.

Key, K. (2009). 1D inversion of multicomponent, multifrequency marine CSEM data: Methodology and synthetic studies for resolving thin resistive layers. Geophysics, 74(2), F9–F20. doi:10.1190/1.3058434