4.6  Checks on Inversion Results

Kamini Singha; Timothy C. Johnson; Frederick D. Day-Lewis; Lee D. Slater

4.6 Checks on Inversion Results

Tomographic inversion results are strongly affected by selected inversion parameters and regularization criteria, especially in the presence of large measurement errors. It is instructive, therefore, to run multiple inversions to gain insight into the effects of different software settings, which is the philosophy of the depth of investigation (DOI) analysis seen earlier. Rarely are default inversion settings appropriate and the inversion should be guided by prior information. Prior information that may be useful includes past geophysical results, (hydro)geologic maps, and drillers’ logs. If inverted electrical conductivity cross sections are inconsistent with such prior information, this could indicate that settings are suboptimal or that assumptions (e.g., 2-D heterogeneity) are violated. Table 1 lists some common problems and their associated symptoms and solutions. We emphasize that Table 1 is by no means exhaustive in terms of the symptoms, problems, and solutions and relationships between them. Rather, this is meant as a starting point for practitioners to begin thinking about the roles of various inverse settings.

Table 1 – Common problems with inversion settings, and the associated symptoms and solutions.

Symptom	Possible problems	Solution
Minimum/maximum estimated electrical conductivity too low/high compared to expected values	The inversion may be overfitting the data	Stop inversion at an earlier iteration or increase the assumed measurement error
	Non-random outlier data may be present	Check the dataset for outliers and edit Try the L1 norm for data misfit
Tomogram is speckly or looks like a checkerboard	The inversion may be overfitting the data	Stop inversion at an earlier iteration or increase the assumed measurement error in an Occam’s inversion
Tomogram is speckly or looks like a checkerboard	Non-random outlier data may be present	Check the dataset for outliers and edit Try the L1 norm for data misfit
The inversion cannot match the data to within the reciprocal error	The optimization algorithm may be caught in a local minimum	Change optimization tolerances Increase number of iterations Update the Jacobian more frequently Change the starting model
	The finite-difference or finite-element grid may be too coarse	Refine the grid or mesh
	The inversion grid may be too coarse	Increase the number of inversion parameters
	Non-random outlier data may be present	Check the dataset for outliers and edit Try the L1 norm for data misfit
The tomogram does not look like expected geology	Regularization criteria may be smoothing/blunting the tomogram too much	Try robust model misfit instead of L2 model misfit Use anisotropic regularization Try different regularization criteria
	Electrical conductivity may not correlate with lithology	Another geophysical technique may be needed
	Resolution may be above the scale of the pertinent heterogeneity.	Another geophysical technique may be needed or array type and/or electrode spacing used may need to be reconsidered
Two (or more) tomograms that share a borehole appear inconsistent at the borehole	Electrical anisotropy	Use an inversion package with a forward model that allows for electrical conductivity anisotropy
	Outlier data are present in at least one dataset	Check the dataset for outliers and edit Try the L1 norm for data misfit
	Parsing data into individual tomograms when they should be considered together	Invert all data at once, rather than breaking it into pieces
Tomograms show vertical streaking, or high or low electrical conductivity patches only at boreholes	Resolution may vary greatly from the sides to the middle of the tomogram	Create synthetic or hypothetical forward models of the experiment to evaluate resolution and likely artifacts Examine plots of the inversion’s sensitivity or resolution matrix Explicitly model the boreholes

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