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Model Predictive Uncertainty

Model Predictive Uncertainty. Sensitivity analysis …. GM Seam Inflows. Permian Inflows. Tertiary Sands Inflow. Hydraulic property heterogeneity. correlation length. Hydraulic property correlation decreases with distance. correlation length. C(K 1 , K 2 ). distance.

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Model Predictive Uncertainty

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  1. Model Predictive Uncertainty

  2. Sensitivity analysis …..

  3. GM Seam Inflows

  4. Permian Inflows

  5. Tertiary Sands Inflow

  6. Hydraulic property heterogeneity correlation length

  7. Hydraulic property correlation decreases with distance correlation length C(K1 , K2 ) distance

  8. Hydraulic property correlation decreases with distance correlation length variogram C(K1 , K2 ) distance

  9. Hydraulic property measurement points

  10. Hydraulic property correlation decreases with distance correlation length variogram C(K1 , K2 ) distance

  11. Hydraulic property realisation

  12. Hydraulic property realisation

  13. constrained by point measurements

  14. constrained by point measurements

  15. Particularly useful in pathline analysis ...

  16. ...

  17. Sensitivity analysis [calibrated model] …..

  18. Estimated parameter values p2 Objective function minimum p1

  19. Estimated parameter values Objective function minimum p2 Maximum probability for p1 and p2 p1

  20. Estimated parameter values Allowed parameter values p2 Maximum probability for p1 and p2 p1

  21. Estimated parameter values Allowed parameter values p2 Maximum probability for p1 and p2 p1

  22. Estimated parameter values – nonlinear case Allowed parameter values p2 Maximum probability for p1 and p2 p1

  23. Field or laboratory measurements and model output:- Model output value calibration dataset prediction q2 q1 q3 etc distance or time

  24. Field or laboratory measurements and model output:- Model output value Lower predictive limit calibration dataset q2 q1 q3 etc distance or time

  25. Field or laboratory measurements and model output:- Model output value Upper predictive limit calibration dataset q2 q1 q3 etc distance or time

  26. Field or laboratory measurements and model output:- Model output value Confidence interval for prediction calibration dataset q2 q1 q3 etc distance or time

  27. Estimated parameter values – nonlinear case Allowed parameter values p2 Maximum probability for p1 and p2 p1

  28. Estimated parameter values – nonlinear case knowledge constraints p2 Allowed parameter values p1

  29. A certain model prediction p2 Increasing value p1

  30. Defining a confidence interval p2 The critical points p1

  31. Residuals Model output value calibration dataset prediction q2 q1 q3 etc distance or time

  32. The variance of the residuals is:- 2 =  / (m - n) m = number of observations n = number of parameters

  33. Field or laboratory measurements and model output:- Model output value Confidence interval for prediction calibration dataset q2 q1 q3 etc distance or time

  34. Field or laboratory measurements and model output:- Model output Predictive uncertainty interval value calibration dataset q2 q1 q3 etc distance or time

  35. Software for predictive uncertainty analysis • UCODE • assumes model linearity • only works with a few parameters • PEST • full nonlinear predictive analysis • unlimited number of parameters

  36. For linear models…

  37. Estimated parameter values:- p2 Extreme values of p1 and p2 p1

  38. A simple lumped parameter model

  39. A simple lumped parameter model par1 par2 par5 par6 par3 par4

  40. The covariance matrix of the estimated parameter set is given by C(p) = 2 (Mt QM)-1 For a nonlinear model replace M by J, the Jacobian matrix. C(p) = 2 (Jt QJ)-1

  41. Let M (ie. green M) represent the action of the model in predictive mode and o the model outputs in predictive mode. Then C(o) = MC(p)Mt For a nonlinear model:- C(o) = JC(p)Jt Notice that predictions can be correlated.

  42. probability value of prediction #1

  43. Maximum probability Bivariate probability density function.

  44. For linear and nonlinear models…

  45. PEST’s predictive analyzer p2 Initial parameter estimates The critical point p1

  46. PEST’s predictive analyzer p2 The critical point Initial parameter estimates p1

  47. Major problem with this approach • assumes that there is an objective function minimum • assumes that this defines a unique set of parameters • thus it assumes that parameters are “lumped” and that there aren’t many of them

  48. objective function contours

  49. A Confined Aquifer head Fixed Inflow T1 T3 T2 Fixed head

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