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IX. Transient Model Nonlinear Regression and Statistical Analysis

IX. Transient Model Nonlinear Regression and Statistical Analysis. 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 .

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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, s2, and s, are shown in Figure 9.13, p. 246.

  4. Evaluate Model Fit • EXERCISE 9.9: UseGraphs for Analyzing Model Fitand 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 RN2 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 EstimatedParameters • 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. CompositeScaled Sens. Figure 9.11 of Hill and TiedemanInitial Composite Scaled Sensitivities (page 243) Figure 9.18 of Hill and TiedemanFinal Composite Scaled Sensitivities (page 251)

  11. ConfidenceIntervals 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. Q_1&2 SS_1 HK_1 K_RB VK_CB SS_2 HK_2 RCH_1 RCH_2 Q_1&2 1.00 -0.75 -0.99 -0.089 -0.50 -0.056 -0.95 -0.17 -0.91 SS_1 1.00 0.74 -0.19 0.82 -0.60 0.70 0.12 0.68 HK_1 1.00 0.0003 0.51 0.057 0.91 0.18 0.90 K_RB 1.00 -0.38 0.42 0.28 0.005 0.095 VK_CB 1.00 -0.70 0.43 0.090 0.44 SS_2 symmetric 1.00 0.078 0.021 0.065 HK_2 1.00 0.14 0.88 RCH_1 1.00 -0.23 RCH_2 1.00 Final Parameter Correlation Coefficients 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, andthe 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. PredictingAdvectiveTransport Figure 9.21 of Hill and Tiedeman (page 255) Exercise 9.12a: Plot predicted path

  17. Landfill Predicting Advective Transport

  18. 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. Figure 9.22 of Hill and Tiedeman (page 256)

  19. Q_1&2 Q_1&2 SS_1 SS_1 HK_1 HK_1 K_RB K_RB VK_CB VK_CB SS_2 SS_2 HK_2 HK_2 RCH_1 RCH_1 RCH_2 RCH_2 Table 9.7 of Hill and Tiedeman: without predictions Q_1&2 Q_1&2 1.00 1.00 -0.65 -0.75 -0.99 -0.99 -0.066 -0.089 -0.50 -0.40 -0.035 -0.056 -0.92 -0.95 -0.17 -0.37 -0.91 -0.84 SS_1 SS_1 1.00 1.00 0.63 0.74 -0.26 -0.19 0.80 0.82 -0.60 -0.71 0.70 0.58 0.12 0.22 0.68 0.53 HK_1 HK_1 1.00 1.00 -0.050 0.0003 0.51 0.42 0.057 0.036 0.91 0.84 0.38 0.18 0.90 0.82 K_RB 1.00 -0.43 0.42 0.32 0.016 0.076 K_RB 1.00 -0.38 0.42 0.28 0.005 0.095 VK_CB 1.00 -0.75 0.30 0.15 0.32 VK_CB 1.00 -0.70 0.43 0.090 0.44 SS_2 symmetric 1.00 0.063 0.028 0.047 SS_2 symmetric 1.00 0.078 0.021 0.065 HK_2 1.00 0.31 0.79 HK_2 1.00 0.14 0.88 RCH_1 1.00 -0.17 RCH_1 1.00 -0.23 RCH_2 1.00 RCH_2 1.00 Table 9.8 of Hill and Tiedeman: with predictions

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

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

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

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