Quantification of Uncertainty

QUANTIFICATION OF UNCERTAINTY

The nSIGHTS code uses non-linear regression to optimize the values of the fitting parameters and thereby obtain the best match to the measured pressure data and/or flow rates.  The type of test(s) performed and the chosen conceptual model determine the type and number of fitting parameters.  There are a variety of methods in the nSIGHTS code to analyze and quantify uncertainty in the fitting-parameter estimates.   One of these methods is a process referred to as perturbation analysis.

To begin, preliminary analyses are performed in which a reasonable fit is obtained to the chosen constraint.  The resulting values of the fitting parameters are the baseline solution set – a single value for each fitting parameter that provides a satisfactory fit to the data (satisfactory being a judgment call on the part of the analyst).  Perturbation analysis begins by assigning a plus/minus range corresponding to the parameter space one wishes to investigate to each of the baseline fitting-parameter values.  Starting at the baseline value, the fitting parameters are randomly perturbed to fall somewhere within their assigned ranges and are then optimized from these random starting points.  Figure 1 shows an example with 500 initial starting values (blue) as well as the final optimized values (red) derived by perturbing the baseline values of KSS, and n and matching a given constraint.

An example of perturbation analysis

Figure 1. An example of the perturbation analysis process showing the initial perturbed values along with the final optimized values.

The objective of perturbation analysis is to adequately sample the parameter space and locate all of the minima within the parameter space.  Figure 2 shows optimized values of K and SS derived from perturbation analysis where distinct minima are evident in different areas of the parameter space.  By definition, the solution that provides the best quantitative fit to the data, measured in terms of the smallest sum of squared errors (SSE), is the global minimum (assumed true solution), and the other minima are referred to as local minima.  Local minima are effectively localized depressions in the parameter-space topography that trap the inverse regression algorithm during its attempt to find the global minimum – the smallest SSE.

Multiple Minima in Parameter Space

Figure 2. Perturbation analysis results showing the optimized values of hydraulic conductivity and specific storage distributed among multiple minima.

Figure 3 shows an objective-function surface where the SSE value has been calculated at each of the mesh intersections.  The corresponding perturbation-analysis results have been plotted on top of this surface to demonstrate that these perturbation results fall within the various minima located across the parameter-space topography.

Objective Function Surface

Figure 3. Objective-function surface and perturbation results.

Perturbation results can be plotted as perturbation-derived fitting-parameter values versus the corresponding SSE.  These plots show two-dimensional “slices” or projections of higher-dimensional parameter-space surfaces, where the dimension of the parameter space corresponds with the number of fitting parameters.  These two-dimensional projections will exhibit a wide variety of shapes that depend on the characteristics of the higher-dimensional space.  Figure 4 shows a well-defined minimum with little uncertainty in the estimate of K.

Kf Parameter Space -- well defined minimum

Figure 4. Hydraulic conductivity parameter space showing a well-defined minimum.

In contrast, the minimum can be quite wide, as seen in Figure 5.  This indicates that the estimate of hydraulic conductivity is not being well constrained, which can result for a variety of reasons.  Note that if only a single optimization is performed, the algorithm could stop anywhere in this extensive minimum and the true uncertainty in the fitting parameter estimate would remain a dark little secret. Given that the SSE (quality of the fit) doesn’t vary across the minimum, all of the matches to the data would be visually identical.

Kf Parameter Space -- large uncertainty

Figure 5. Hydraulic conductivity parameter space showing large uncertainty.

Multiple minima within a parameter space are common and the characteristics of the parameter spaces can be relatively complex as seen in Figures 6 and 7.  This makes it all too easy for an optimization algorithm to stop at a location within the parameter space that doesn’t correspond to the global minimum. Perturbation analysis gives the analyst a clear picture of the possible solutions and the corresponding uncertainty.

Kf Parameter Space Multiple Minima

Figure 6. Multiple minima within the hydraulic conductivity parameter space.

Ks Parameter Space Multiple Minima

Figure 7. Multiple minima within the skin hydraulic conductivity parameter space.