1. Chapter 6 Calibration and Application Process

2. 6.1 The Calibration Process
Calibration of a flow model refers to a demonstration that the model is capable of producing field-measured heads and flows which are the calibration values.
Calibration is accomplished by finding a set of parameters, boundary conditions, and stresses that produce simulated head and fluxes that match filed-measured values within a preestablished range of error.

3. Finding this set of values amounts to solving what is known as the inverse problem.
In an inverse problem the objective is to determine values of the parameters and hydrologic stresses from information about head, whereas in the forward problem system parameters such as hydraulic conductivity, specific storage, and hydrologic stresses such as recharge rate are specified and the model calculates heads.

5. (1) manual trial-and-error adjustment of parameters;
There are basically two ways of finding model parameters to achieve calibration, i.e., of solving the inverse problem:
(1) manual trial-and-error adjustment of parameters;
(2) automated parameter estimation.

6. (1) Trial-and-error calibration: Trial-and-error calibration may produce nonunique solutions when different combinations of parameters yield essentially the same head distribution.
The trial-and-error process is influenced by the modeler’s expertise and biases.

8. (2) Automated calibration: Automated inverse modeling is performed using specially developed codes that use either a direct or indirect approach to solve the inverse problem.
The indirect approach is similar to performing trial-and-error calibrations in that the forward problem is solved repeatedly.

9. Automated Calibration：
using specially developed codes that use either a direct or indirect approach to solve the inverse problem.
In a direct solution, the unknown parameters are treated as dependent variables in the governing equation and heads are treated as independent variables.
Problem: unstable of solution

10. indirect solution: An inverse code automatically checks the head solution and adjusts parameters in a systematic way in order to minimize an objective function.
Methods used to minimize objective functions are often based on Gauss-Newton or gradient search methods.
to control instability by proper zonation of aquifer parameters

11. To date, automated inverse models have had limited application
To date, automated inverse models have had limited application. They are criticized because of problems with nonuniqueness and instability

12. 6.2 Evaluating the Calibration
The results of the calibration should be evaluated both qualitatively and quantitatively.

13. (1) Traditional measures of calibration:
Comparison between contour maps of measured and simulated heads provides a visual, qualitative measure of the similarity between patterns, thereby giving some idea of the spatial distribution of error in the calibration.

15. A scatterplot of measured against simulated heads is another way of showing the calibrated fit.

17. A listing of measured and simulated heads together with their differences is a common way of reporting calibration results.

18. (2) Distribution of error:
The distribution of head residuals may be shown in discrete form (Fig. 8.13a) or using contours (Fig. 8.13b).

19. (3) Sensitivity analysis:
The purpose of a sensitivity analysis is to quantify the uncertainty in the calibration model caused by uncertainty in the estimates of aquifer parameters, stresses, and boundary conditions.
During a sensitivity analysis, calibrated values of hydraulic conductivity, storage parameters, recharge, and boundary conditions are systematically changed within the previously established plausible range. The magnitude of change in heads from the calibrated solution is a measure of the sensitivity of the solution to that particular parameter.

20. Sensitivity analysis is typically performed by changing one parameter value at a time.

21. (4) Model verification:
Owing to uncertainties in the calibration, the set of parameter values used in the calibrated model may not accurately represent field values. Model verification will help establish greater confidence in the calibration.
Verification is accomplished when the verification targets are matched without changing the calibrated parameter values.
A calibrated but unverified model can still be used to make predictions as long as careful sensitivity analyses of both the calibrated model and the predictive model are performed and evaluated.

22. 6.3 Prediction
In a predictive simulation, the parameters determined during calibration and verification are used to predict the response of the system to future events.
The confidence to be placed in model predictions depends largely on the results of the calibration, sensitivity analyses, and verification tests.
Faust et al. (1981) suggest that a predictive simulation not be extended into the future more than twice the period for which calibration data are available.

23. Two major pitfalls are involved in making predictions: uncertainty in the calibrated model and uncertainty about future hydrologic stresses. Each of these requires a different type of sensitivity analysis.