1. Quantify prediction uncertainty (Book, p. 174-189) Prediction standard deviations (Book, p. 180): A measure of prediction uncertainty Calculated by translating parameter uncertainty through to the predictions: Activate all parameters when calculating !!! Calculate parameter var-cov matrix with all parameters Calculate prediction sensitivities for all parameters

2. Quantify prediction uncertainty Linear confidence and prediction intervals ( p. 176-177 ) Intervals can be individual or simultaneous Form: confidence interval prediction interval Prediction intervals account for ‘measurement’ error. Use to compare simulated results to field measurements.  is the significance level, c(  ) is the critical value and is different for different types of intervals (Table 8.1, p. 176).

3. Individual vs. Simultaneous Intervals Individual linear intervals Defined as an interval that has a specified probability of containing the true predicted value. Exact for correct, linear models with normally distributed residuals. The more these requirements are violated, the less accurate the intervals become. Simultaneous linear intervals On two or more predictions, each has a specified probability of containing the true value. Always ≥ linear intervals, because of greater difficulty in defining intervals that simultaneously include true values of two or more predictions. Largest intervals are for case where # of predictions = # of parameters Common types: Bonferoni & Sheffé

4. Exercise 8.2a: Calculate linear confidence intervals on predicted advective transport Linear confidence intervals c an be computed in UCODE_2005 using program Linear_Uncertainty.exe. Linear_Uncertainty uses V(b) from the regression run output, along with information from an extra ucode run with the prediction conditions (for computing prediction sensitivities) to calculate prediction standard deviations. Then it calculates the different types of individual and simultaneous intervals using the app ropriate statistics.

5. Calculating linear intervals with UCODE_2005. From Poeter +, 2005, p. 158)

6. Linear Intervals Do Exercise 8.2a (p. 208-209) and the Problem, including answering Question 5: What is the uncertainty in the predictions? Correction to book: p. 208, second line from the bottom, should read “Answer Question 5…”

7. Linear Individual Linear Simultaneous (Scheffe d=NP) Results of Exercise 8.2a Linear Confidence Intervals for Question 5: What is the prediction uncertainty? Figure 8.15a, p. 210 Figure 8.15b, p. 210

8. Results of Exercise 8.2a (continued) Linear Confidence Intervals for Question 5: What is the prediction uncertainty? Figure 8.16, p. 211

9. Nonlinear Intervals Method involves finding the minimum and maximum predicted value on a confidence region for the parameters, which is defined as (book, p. 178) S(b)  S(b’) + (s 2 x crit) + a crit=critical value Developed by Vecchia and Cooley (1987, WRR) Each limit of each interval requires a regression run that is often more difficult than the regression runs used for calibration. Maximum prediction Minimum prediction

10. Calculating nonlinear intervals with UCODE_2005. Modified from Poeter +, 2005, p. 193)

11. Nonlinear Intervals Do exercise 8.2b Computer instructions: the input files are provided for you in initial\ex8\ucode-opr-ppr-runs\ex8.2b directory, as noted in the computer instructions. The nonlinear intervals are in ex8.2b._intconf

12. Nonlinear Individual Nonlinear Simultaneous (Scheffe d=NP) Results of Exercise 8.2b Nonlinear Confidence Intervals for Question 5: What is the prediction uncertainty? Do the Problem on p. 212 Figure 8.15c, p. 210 Figure 8.15d, p. 210

13. Linear Individual Linear Simultaneous (Scheffe d=NP) Figure 8.15a, p. 210 Figure 8.15, p. 210 Nonlinear Individual Nonlinear Simultaneous (Scheffe d=NP)

14. Our Final Analysis and the County Decision Our Analysis Though it looks likely that the particle goes to the well, results are not conclusive. Consider using parameter values for which the particle goes to the river in an advective-dispersive model to analyze concentrations at the well. If concentrations high, results become more conclusive. County decision No additional modeling right now Wait for the new data and use it to recalibrate

15. Monte Carlo Analysis (Book, p. 185-189) Change some aspect of model input, run model, evaluate selected changes in model results. Can change parameter values, definition of hydrogeology, etc. When changing parameter values, can generate new sets from V(b) if model was calibrated by regression. For changing hydrogeology, a common geostatistical approach is ‘simulation’, which uses kriging as part of the method. Can just do forward simulations, or can involve inverse modeling as well. Commonly need to do numerous model runs to obtain enough ‘data’ to make supportable conclusions. This is now often feasible, with the level of computational power in PCs. Results commonly displayed as histograms showing distribution of model output values; can also calculate statistics from the results, such as means and variances. Suggestion: only use sets of generated parameter values that produce a reasonable fit to the calibration data (Beven)

16. Can confidence intervals replace traditional sensitivity analysis? (p. 184-185) Traditional sensitivity analysis quantify uncertainty in the calibrated model caused by uncertainty in the estimated parameter values change hydraulic conductivity, storage, recharge and boundary conditions systematically within previously established plausible range Weaknesses of traditional method Plausible range does not reflect significant information provided through model calibration. Results exaggerate uncertainty. Suggested method to account for parameter correlation exacerbates this exaggeration.

17. Can confidence intervals replace traditional sensitivity analysis? Weaknesses of both methods Only consider uncertainty in the parameter values. Uncertainty in model construction generally neglected entirely Advantages of confidence intervals Account for information provided through the modeling process.