4.5 Quantification of Inversion Quality

Several approaches are commonly used to gain insight into the reliability of tomograms. For small inverse problems, it is possible to calculate the model resolution matrix (e.g., Menke, 1984) and present the diagonals, rows, and columns of these matrices as cross-sectional images. Conceptually, the model resolution matrix is the lens or filter through which the inversion sees the study region. For a linear inverse problem, the parameter estimates are expressed by Equation 14.

[latex]\displaystyle m=[J^{\mathrm{T}}C{_{D}}^{-1}J+\varepsilon D^{\mathrm{T}}D]^{-1}J^{\mathrm{T}}C{_{D}}^{-1}d_{obs}\approx [J^{\mathrm{T}}C{_{D}}^{-1}J+\varepsilon D^{\mathrm{T}}D]^{-1}J^{\mathrm{T}}C{_{D}}^{-1}Jm_{true}[/latex] (14)

In this case, the model resolution matrix R is defined as shown in Equation 15.

[latex]\displaystyle R=[J^{\mathrm{T}}C{_{D}}^{-1}J+\varepsilon D^{\mathrm{T}}D]^{-1}J^{\mathrm{T}}C{_{D}}^{-1}J[/latex] (15)

Consequently, the parameter estimates are the product of the true parameter values and the resolution matrix as shown in Equation 16.

[latex]\displaystyle m=Rm_{true}[/latex] (16)

For linear problems, where J is independent of mtrue, R can be calculated prior to data collection. Given an estimate of measurement errors, the model resolution matrix can be calculated using Equation 15 and used as a tool to assess and refine hypothetical survey designs and regularization criteria. In interpreting inversion results, R is useful for identifying likely inversion artifacts (Day-Lewis et al., 2005). The model resolution matrix quantifies the spatial averaging inherent to tomography; hence, it gives insight into which regions of a tomogram are well resolved versus poorly resolved. This information is valuable if tomograms are to be converted to quantitative estimates of porosity, concentration, or other hydrogeologic parameters. Calculation of resolution matrices, however, remains computationally prohibitive for many problems, particularly those involving 3-D inversion. Hence, few commercially available software packages support calculation of R, and it is instead more common to look at an inverse problem’s cumulative squared sensitivity vector (S) as shown in Equation 17.

[latex]\displaystyle S=\mathrm{diag}(J^{T}J)[/latex] (17)

Here, J is the sensitivity matrix defined in Equation 10a and diag( ) indicates the diagonal elements of a matrix. The sensitivity matrix can be used to gain semi-quantitative insight into how resolution varies spatially over a tomogram. Pixels with high values of sensitivity are relatively well informed by the measured data, whereas pixels with low values of sensitivity are poorly informed. It is important to note that, in contrast to R, S does not account for the effects of regularization criteria (as contained in D) or measurement error (as contained in CD). Rather, S is based only on the survey geometry and measurement sensitivity. An example sensitivity map is provided in the case study in Section 5.2 and qualitatively in Figure 2. Another question is whether inversion results are consistent with our conceptual models of the site—this is a different definition of inversion quality. A good review exploring this idea is presented by Linde (2014).

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Electrical Imaging for Hydrogeology Copyright © 2022 by Kamini Singha, Timothy C. Johnson, Frederick D. Day Lewis and Lee D. Slater. All Rights Reserved.