1. Bringing Inverse Modeling to the Scientific Community Hydrologic Data and the Method of Anchored Distributions (MAD) Matthew Over 1, Daniel P. Ames 2, & Yoram Rubin 1 1. Department of Civil and Environmental Engineering, University of California, Berkeley 2. Department of Geosciences and the Department of Civil Engineering, Idaho State University

2. Presentation Goals Introduce MAD ▫Bayesian Principles ▫Inversion of model parameters ▫Anchors and their purpose Incorporating MAD with HydroDesktop ▫Plug-in that utilizes central database ▫Analyzing HydroModeler output ▫Using map tools for visualization

3. The Bayesian Approach Variables : : Observed data of any scale or type : Geo-statistical modeling parameters

4. Bayesian Philosophy Applied to MAD Adding anchor parameters,, expanding data types,, and using proportionality:

5. Bayesian Philosophy Applied to MAD Adding anchor parameters,, expanding data types,, and using proportionality: Prior

6. Bayesian Philosophy Applied to MAD Adding anchor parameters,, expanding data types,, and using proportionality: PriorLikelihood

7. Bayesian Philosophy Applied to MAD Adding anchor parameters,, expanding data types,, and using proportionality: PosteriorPriorLikelihood

8. Goal of inversion MAD aims to determine probability distributions of modeling parameters that are conditioned on the available data. The joint distribution of parameters can be used to generate appropriate spatial fields of the target variable, e.g. conductivity or permeability MAD does NOT aim to directly ‘invert’ the target variable for each grid cell of the domain.

9. Let’s discuss the MAD procedure 1. Generation of prior distribution of modeling parameters. Based on engineering judgment, existing reports, etc. 2. Calculation of likelihood function. Based on comparison of physical observations and simulated results. 3. Determination of the posterior distribution. The model parameters are conditioned on the available field data.

10. Example Field Study Analysis Goal: Characterizing hydraulic conductivity in the domain – defines target variable Assumed geo-statistical model: exponential covariance – defines Available data: 1)Conductivity in boreholes - defines 2)Pumping test drawdown in boreholes - defines

11. Step 1: Define Prior Distribution The field study yields ranges for several of the geo-statistical parameters with respect to log(K): Mean: -2 to 2 [dimensionless] Variance: 1 to 1.5 [dimensionless] Length Scale: 8 to 12 meters How can we improve the estimates of these parameters?

12. Step 1: Define Prior Distribution Step 1: Defined prior distribution – marginal distributions depicted

13. Step 2: Likelihood Calculation Sample the prior distribution of the geo- statistical parameters Each sample has an infinite number of random fields that obey the geo-statistical model Generate an ensemble of random fields – for each prior distribution sample. Simulate the pumping test (indirect measurement physical process) and compare to observations in the boreholes

14. Step 2: Likelihood calculation 1 prior sample (Limit behavior requires larger sample size than example) (Mean 1.57, Variance 1.00, Scale 8.04)

15. Step 2: Likelihood calculation For a given sample, generate an ensemble of random log(K) fields. Limit behavior requires large ensemble

16. Step 2: Likelihood calculation For each random field in the ensemble, simulate the physical process defined by observed Estimate the density of simulated drawdown in observation wells, compare to field data at steady state

17. Step 2: Likelihood calculation For each random field in the ensemble, simulate the physical process defined by observed Estimate the density of simulated drawdown in observation wells, compare to field data at steady state

18. Step 2: Likelihood calculation For each random field in the ensemble, simulate the physical process defined by observed Estimate the density of simulated drawdown in observation wells, compare to field data at steady state

19. Step 3: Posterior Distributions Combining likelihood with the prior distribution yields the conditional posterior distribution of geo-statistical model parameters Model is conditioned on the data Model can be updated sequentially as new data is available

20. Step 3: Compute posterior distributions Weighted by the likelihood, new marginal distributions of geo-statistical parameters reflect field data

21. Haven’t you forgotten something? Where are the anchors? An anchor is a theoretical device located away from the borehole that captures information relevant to the target variable The random field is no longer free at anchor locations, but is conditioned to the anchor value sampled from the prior distribution Process repeats exactly as before, but the prior distribution has more dimensions

22. How Anchors Work – An Example Consider drawdown in a borehole: ▫Not simply a function of the hydraulic conductivity in the borehole ▫Can be function of the hydraulic conductivity in the vicinity of the borehole, entire aquifer, etc. Information about the hydraulic conductivity away from a borehole is transmitted via the drawdown This indirect relationship allows inference about non-local hydraulic conductivity from local borehole data

23. Assembling the MAD analytical tool in the existing HIS framework MAD exists as a 3 ‘block’ process which will be developed as a plug-in for HydroDesktop The MAD plug-in is being designed to communicate with the existing HIS database The MAD plug-in is open source The development of a MAD plug-in management webpage where users can communicate with developers

24. MAD plug-in framework inside HydroDesktop: Proposed Architecture HydroDesktop HydroModeler HIS Database MAD Block 3 Post-Processing MAD Block 2 Likelihood MAD Block 2 Likelihood MAD Block 1 Pre-Processing MAD Block 1 Pre-Processing

25. MAD plug-in framework inside HydroDesktop: Proposed Architecture HydroDesktop HydroModeler HIS Database MAD Block 3 MAD Block 3 MAD Block 2 MAD Block 2 MAD Block 1 MAD Block 1 MAD Block 3 Post-Processing MAD Block 2 Likelihood MAD Block 2 Likelihood MAD Block 1 Pre-Processing MAD Block 1 Pre-Processing

26. MAD plug-in framework inside HydroDesktop: Proposed Architecture HydroDesktop HydroModeler HIS Database MAD Block 3 MAD Block 3 MAD Block 2 MAD Block 2 MAD Block 1 MAD Block 1 MAD Block 3 Post-Processing MAD Block 2 Likelihood MAD Block 2 Likelihood MAD Block 1 Pre-Processing MAD Block 1 Pre-Processing

27. MAD plug-in framework inside HydroDesktop: Proposed Architecture HydroDesktop HydroModeler HIS Database MAD Block 3 MAD Block 3 MAD Block 2 MAD Block 2 MAD Block 1 MAD Block 1 MAD Block 3 Post-Processing MAD Block 2 Likelihood MAD Block 2 Likelihood MAD Block 1 Pre-Processing MAD Block 1 Pre-Processing

28. MAD plug-in framework inside HydroDesktop: Proposed Architecture HydroDesktop HydroModeler HIS Database MAD Block 3 MAD Block 3 MAD Block 2 MAD Block 2 MAD Block 1 MAD Block 1 MAD Block 3 Post-Processing MAD Block 2 Likelihood MAD Block 2 Likelihood MAD Block 1 Pre-Processing MAD Block 1 Pre-Processing

29. MAD plug-in framework inside HydroDesktop: Proposed Architecture HydroDesktop HydroModeler HIS Database MAD Block 3 MAD Block 3 MAD Block 2 MAD Block 2 MAD Block 1 MAD Block 1 MAD Block 3 Post-Processing MAD Block 2 Likelihood MAD Block 2 Likelihood MAD Block 1 Pre-Processing MAD Block 1 Pre-Processing

30. MAD plug-in framework inside HydroDesktop: Proposed Architecture HydroDesktop HydroModeler HIS Database MAD Block 3 MAD Block 3 MAD Block 2 MAD Block 2 MAD Block 1 MAD Block 1 MAD Block 3 Post-Processing MAD Block 2 Likelihood MAD Block 2 Likelihood MAD Block 1 Pre-Processing MAD Block 1 Pre-Processing

31. In Development Currently a GUI for the MAD plug-in is being designed Features: ▫Wizard that guides user through process ▫Map window for placing boreholes and anchors ▫Layers that contain pertinent analysis results/info Initial plans are to provide a 2-D analysis tool that enforces any pertinent probability convergence requirements without user choice.

32. MAD Plug-in Plans Additional options in the wizard and GUI ▫Density estimation techniques ▫Prior generation and sampling methods ▫Advanced user scripting ▫Convergence options/optimization criterion Possible development of OpenMI output for easy exchange of information with HydroModeler

33. In conclusion… The MAD plug-in is an open-access, open-source software in development The MAD plug-in will mimic the 3 module structure of the inversion method and interface with existing HydroDesktop functionality The MAD plug-in will be placed under CUAHSI custody at the end of the current NSF grant