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IX. Transient Model Nonlinear Regression and Statistical Analysis.

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1 IX. Transient Model Nonlinear Regression and Statistical Analysis

2 Nonlinear Regression When all K and S parameters are log-transformed, the regression for the transient problem will converge, and optimal estimates of the nine model parameters will be obtained. EXERCISE 9.7: Estimate parameters for the transient system by nonlinear regression.

3 Evaluate Model Fit Now, we will perform the same analysis of the regression results for the transient problem that was performed for the steady-state problem. EXERCISE 9.8: Evaluate measures of model fit Statistical measures of overall model fit, S, s 2, and s, are shown in Figure 9.13, p. 246.

4 Evaluate Model Fit EXERCISE 9.9: Use Graphs for Analyzing Model Fit and Evaluate Related Statistics EXERCISE 9.9a: Evaluate graphs of weighted residuals and weighted and unweighted simulated and observed values. See Figure 9.14, p. 247 of Hill and Tiedeman and statistic R in Figure 9.13. Which graphs are most useful to understanding model fit? Is R helpful?

5 Weighed Residuals vs. Simulated Values Figure 9.14a of Hill and Tiedeman (page 247)

6 Weighted Observed Values vs. Weighted Simulated Values Figure 9.14b of Hill and Tiedeman (page 247)

7 Evaluate Model Fit EXERCISE 9.9b. Evaluate graphs of weighted residuals against independent variables and the runs statistic. The runs statistic is given in Figure 9.16, p. 249. EXERCISE 9.9c: Assess independence and normality of the weighted residuals. The normal probability graph and the R N 2 statistic are shown in Figure 9.17, p. 250.

8 Normal Probability Graph Figure 9.17 of Hill and Tiedeman (page 250)

9 Evaluate Parameter Estimates EXERCISE 9.10: Evaluate Estimated Parameters EXERCISE 9.10a. Composite scaled sensitivities. EXERCISE 9.10b: Parameter estimates and confidence intervals. EXERCISE 9.10c: Reasonable parameter ranges. EXERCISE 9.10d: Parameter correlation coefficients.

10 Composite Scaled Sens. Figure 9.18 of Hill and Tiedeman Final Composite Scaled Sensitivities (page 251) Figure 9.11 of Hill and Tiedeman Initial Composite Scaled Sensitivities (page 243)

11 Confidence Intervals Figure 9.19 of Hill and Tiedeman: Confidence Intervals for Transient Regression (page 252) Figure 7.7 of Hill and Tiedeman: Confidence Intervals for Steady State Regression (page 153)

12 Final Parameter Correlation Coefficients Q_1&2SS_1HK_1K_RBVK_CBSS_2HK_2RCH_1RCH_2 Q_1&21.00-0.75-0.99-0.089-0.50-0.056-0.95-0.17-0.91 SS_1 1.000.74-0.190.82-0.600.700.120.68 HK_1 1.000.00030.510.0570.910.180.90 K_RB 1.00-0.380.420.280.0050.095 VK_CB 1.00-0.700.430.0900.44 SS_2 symmetric 1.000.0780.0210.065 HK_2 1.000.140.88 RCH_1 1.00-0.23 RCH_2 1.00 Table 9.7 of Hill and Tiedeman (page 253)

13 Model Linearity EXERCISE 9.11: Test for linearity. See Figure 9.20, p. 253. The modified Beale’s measure is 84. The model is effectively linear if this measure is less than 0.04, and the model is nonlinear if this measure is greater than 0.44.

14 IX. Transient Predictions

15 Update: Ground-Water Management Issues Results from the recalibrated model can now be used to update the advective transport predictions. Many of landfill developer’s concerns have been addressed: Model has been calibrated with head and flow data collected under same stress conditions that will exist during operation of the landfill, and under which the advective transport will be predicted. Uncertainty of most flow model parameters has been reduced, compared to their uncertainty in steady-state model. Advective travel will be analyzed under steady-state pumping conditions, because these are the conditions under which the landfill will operate.

16 Predicting Advective Transport Figure 9.21 of Hill and Tiedeman (page 255) Exercise 9.12a: Plot predicted path

17 Predicting Advective Transport Landfill

18 Figure 9.22 of Hill and Tiedeman (page 256) Parameters Important to Advective Paths EXERCISE 9.12b: Evaluate the model’s ability to simulate predictions using composite and prediction scaled sensitivities, and parameter correlation coefficients.

19 Q_1&2SS_1HK_1K_RBVK_CBSS_2HK_2RCH_1RCH_2 Q_1&21.00-0.75 -0.99 -0.089-0.50-0.056 -0.95 -0.17-0.91 SS_1 1.000.74-0.190.82-0.600.700.120.68 HK_1 1.000.00030.510.0570.910.180.90 K_RB 1.00-0.380.420.280.0050.095 VK_CB 1.00-0.700.430.0900.44 SS_2 symmetric 1.000.0780.0210.065 HK_2 1.000.140.88 RCH_1 1.00-0.23 RCH_2 1.00 Q_1&2SS_1HK_1K_RBVK_CBSS_2HK_2RCH_1RCH_2 Q_1&21.00-0.65 -0.99 -0.066-0.40-0.035 -0.92 -0.37-0.84 SS_1 1.000.63-0.260.80-0.710.580.220.53 HK_1 1.00-0.0500.420.0360.840.380.82 K_RB 1.00-0.430.420.320.0160.076 VK_CB 1.00-0.750.300.150.32 SS_2 symmetric 1.000.0630.0280.047 HK_2 1.000.310.79 RCH_1 1.00-0.17 RCH_2 1.00 Table 9.7 of Hill and Tiedeman: without predictions Table 9.8 of Hill and Tiedeman: with predictions

20 Prediction Uncertainty: Linear Simultaneous Confidence Intervals Fig 8.15b, p. 210 From calibration with transient data From calibration with steady-state data Fig 9.23a, p. 258 EXERCISE 9.12c: Evaluate prediction uncertainty using inferential statistics.

21 Fig 8.15d, p. 210Fig 9.23d, p. 258 From calibration with transient data From calibration with steady-state data Prediction Uncertainty: Nonlinear Simultaneous Confidence Intervals

22 Finally: Should the landfill be approved? Why or why not?

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