SENSITIVITY ANALYSIS FOR INVERSE PROBLEMS SOLVED BY SINGULAR VALUE DECOMPOSITION

Truncated singular value decomposition (SVD) can be used to mitigate inverse modeling convergence problems caused by parameter insensitivity and(or) parameter interdependence (correlation). To make SVD more transparent and informative, we consider the relation of SVD to an alternative method.

In truncated SVD, SVD parameters are defined as linear combinations of process-model parameters; that is, each SVD parameter is calculated by summing terms equal to a coefficient times a process-model parameter. The number of SVD parameters equals the number of process-model parameters. Important SVD parameters have larger singular values. SVD parameters with small singular values can be omitted from the regression to achieve convergence. The rule of thumb is to omit SVD parameters with singular values less than the largest singular value by five or six orders of magnitude. Representation of the process-model parameters within estimated SVD parameters is measured using the identifiability statistic in PEST, here called the SVD parameter loading (SVD-PL) statistic. For each process-model parameter, SVD-PL is calculated by squaring and summing the associated coefficients from all estimated SVD parameters. Summation over all SVD parameters equals 1.0.

The alternative method is based solely on process-model parameters and uses composite scaled sensitivity (CSS) and parameter correlation coefficients (PCC). The rule of thumb is to omit all process-model parameters with CSS values less than the largest CSS value by about two orders of magnitude. Also, omit enough parameters with PCC absolute values larger than about 0.98 such that the set of estimated parameters has smaller PCC absolute values.

Our test case involves the USDA’s Root Zone Water Quality Model (RZWQM2) applied at the Merced River basin, CA. For each of five soil layers, there are three parameters: saturated hydraulic conductivity (Ks), water content at field capacity (WFC), and bulk density (BD). There is also a globally applied N transformation parameter (R45). There are 1,670 observations of aqueous nitrate and bromide concentrations, soil nitrate and organic matter content, and soil moisture content and water tension.

Regression experiments suggest that 15 of the 16 process-model parameters and all 16 SVD parameters could be estimated by regression. CSS values vary from 19.5 to 0.1; the three highest are WFC1>WFC2>BD2. The largest parameter correlation was for WFC1 and BD1 (PCC=-0.94).

Singular values vary from 1,870,000 to 336. SVD-PL suggests that the top four SVD parameters are dominated, in order, by BD2, WFC2, and WFC1, which is similar to the list identified using CSS. Additionally, BD1 is almost as dominant as WFC1, which is consistent with the high PCC between these two parameters.

The test case displays how resolving inverse model convergence problems using CSS and PCC is closely connected to SVD. SVD integrates parameter sensitivity and correlation, while CSS and PCC evaluate them individually. Each approach provides useful insights. Together, the methods provide a powerful set of tools for model calibration.

BD, bulk density; CSS, composite scaled sensitivity; Ks, saturated hydraulic conductivity; PCC, parameter correlation coefficient; RZWQM2, Root Zone Water Quality Model; SVD, singular value decomposition; SVD-PL, singular value decomposition parameter loading statistic; WFC, water content at field capacity.