Regularization for Hypothesis Testing and Exploration of Predictive Uncertainty

1. Regularization for Hypothesis Testing and Exploration of Predictive Uncertainty

2. Stability Achieved Through Parsimony • The World: • Variable • Unknown • Little information • The Modeler: • Tired • Stressed • Budget constrained • Would like some personal life…… Hypothetical model

3. Stability Achieved Through Parsimony The Approach: • Small(ish) number of zones • Based on geology • Satisfies conceptual model • Some are fixed or tied • Some have prior information Stable and often quite rapid in terms of the regression since sensitivities obtained quickly K2 K7 K6 K1 K5 K4 K3 K8 K1 = 30 K6 = 40 K1 = 30; weight = 15.0 K6 = 40; weight = 15.0 log(K1) - log(K2) = 0 log(K1) - log(K3) = 0

4. Stability Achieved Through Parsimony The Difficulty: • How much lumping is best? • What if the lumped zones contradict the true hydrogeologic framework? • What if the ‘mean’ in my zone is not what I need to know? • What if there is uncertainty in the knowledge of my parameters – my prior information?

5. Pilot Points Estimate Values at Points: • Parameter values are defined at the pilot points • Can be aquifer parameters, recharge, etc • Location selected on variety of information

6. Pilot Points Interpolate from Points to Grid: • Interpolation using kriging or other methods • Kriging has some advantages – e.g., Gaussian. Extreme values are always at the pilot points, which can be a good or bad thing.

7. Pilot Points Spatially Varying Properties: • Can be used in conjunction with zones • Can have different interpolation and prior information in different zones

8. Pilot Points and Prior Information Prior Information on Pilot Points: • Prior information on value or between pilot points • In the latter - points linked to each other by prior information equations. Used to tend toward a smooth distribution. P2 P3 P1 log(P1) - log(P2) = 0 log(P1) - log(P3) = 0

9. Pilot Points and Regularization Prior Information and Regularization: • Inter-parameter weights can be predetermined – equal, f(distance), etc • Can be based on same geostatistics as kriging • Can be determined through constrained optimization – Tikhonov Regularization P2 P3 P1 log(P1) - log(P2) = 0 Weight = ? log(P1) - log(P3) = 0 Weight = ?

10. Regularized Inversion Pilot Points and Regularization: • Supports the representation of smoothly-varying aquifer properties, which is consistent with many geologic contexts. For example, deltaic deposits. • Enables modeler to set a ‘target objective function’ – how well to fit the data. Typically - greater departures from smooth produces better fit. • Can be placed throughout the domain, focused where data (aquifer tests, water levels) are densest: • Leads to good fits • Suggests this is the only place where heterogeneity reigns? • When used in predictive analysis – can illustrate the mechanism that leads to a very ‘bad’ or very ‘good’ simulated outcome.

11. Regularized Inversion(Doherty,2003, Ground Water) Study Area Model Domain and Observed Heads Model Grid

12. Regularized Inversion Study Area RMS = 3.5 inches RMS = 2 inches

13. Large Numbers of Parameters Can, however, lead to: • Numerical instability • Non-unique results • Long run times • Heightened anxiety, loss of personal life …

14. Eigenvector Analysis(Tonkin and Doherty, 2005, WRR) Eigenvector Analysis • Rather than estimating every parameter individually, estimate combinations of parameters • This is related to what we do when we a-priori ‘tie’ parameters – that is: • We define parameter combinations by analyzing parameters sensitivities and observation weights • But – this is accomplished formally through eigenvector analysis (related to principal component analysis) • We call these combinations of parameters – “super parameters”!