Bringing Inverse Modeling to the Scientific CommunityHydrologic Data and the Method of Anchored Distributions (MAD)Matthew Over1, Daniel P. Ames2, & Yoram

Rubin1

1. Department of Civil and Environmental Engineering, University of California, Berkeley2. Department of Geosciences and the Department of Civil Engineering, Idaho State University

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

The Bayesian Approach

Variables:

: Observed data of any scale or type

: Geo-statistical modeling parameters€

P θ z( )∝ P zθ( )P θ( )

Bayesian Philosophy Applied to MAD

Adding anchor parameters, , expanding data types, , and using proportionality:

€

P θ,ϑ za ,zb( )∝ P za ,zb θ,ϑ( )P θ,ϑ( )€

za & zb

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ϑ

Bayesian Philosophy Applied to MAD

Adding anchor parameters, , expanding data types, , and using proportionality:

€

P θ,ϑ za ,zb( )∝ P za ,zb θ,ϑ( )P θ,ϑ( )€

za & zb

€

ϑ

Prior

Bayesian Philosophy Applied to MAD

Adding anchor parameters, , expanding data types, , and using proportionality:

€

P θ,ϑ za ,zb( )∝ P za ,zb θ,ϑ( )P θ,ϑ( )€

za & zb

€

ϑ

PriorLikelihood

Bayesian Philosophy Applied to MAD

Adding anchor parameters, , expanding data types, , and using proportionality:

€

P θ,ϑ za ,zb( )∝ P za ,zb θ,ϑ( )P θ,ϑ( )€

za & zb

€

ϑ

Posterior PriorLikelihood

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.

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.

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 - defines2) Pumping test drawdown in boreholes -

defines

€

θ

€

za

€

zb

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?

Step 1: Define Prior Distribution•Step 1: Defined prior distribution –

marginal distributions depicted

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

Step 2: Likelihood calculation

•1 prior sample (Limit behavior requires larger sample size than example)

•(Mean 1.57, Variance 1.00, Scale 8.04)

Step 2: Likelihood calculation

•For a given sample, generate an ensemble of random log(K) fields.

•Limit behavior requires large ensemble

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

€

zb

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

€

zb

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

€

zb

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

Step 3: Compute posterior distributions•Weighted by the likelihood, new marginal

distributions of geo-statistical parameters reflect field data

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

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

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

MAD plug-in framework inside HydroDesktop: Proposed Architecture

HydroDesktop

HydroModeler

HIS Database

MAD Block 3 Post-Processing

MAD Block 2Likelihood

MAD Block 1Pre-Processing

MAD plug-in framework inside HydroDesktop: Proposed Architecture

HydroDesktop

HydroModeler

HIS Database

MAD Block 3

MAD Block 2

MAD Block 1

MAD Block 3 Post-Processing

MAD Block 2Likelihood

MAD Block 1Pre-Processing

MAD plug-in framework inside HydroDesktop: Proposed Architecture

HydroDesktop

HydroModeler

HIS Database

MAD Block 3

MAD Block 2

MAD Block 1

MAD Block 3 Post-Processing

MAD Block 2Likelihood

MAD Block 1Pre-Processing

MAD plug-in framework inside HydroDesktop: Proposed Architecture

HydroDesktop

HydroModeler

HIS Database

MAD Block 3

MAD Block 2

MAD Block 1

MAD Block 3 Post-Processing

MAD Block 2Likelihood

MAD Block 1Pre-Processing

MAD plug-in framework inside HydroDesktop: Proposed Architecture

HydroDesktop

HydroModeler

HIS Database

MAD Block 3

MAD Block 2

MAD Block 1

MAD Block 3 Post-Processing

MAD Block 2Likelihood

MAD Block 1Pre-Processing

MAD plug-in framework inside HydroDesktop: Proposed Architecture

HydroDesktop

HydroModeler

HIS Database

MAD Block 3

MAD Block 2

MAD Block 1

MAD Block 3 Post-Processing

MAD Block 2Likelihood

MAD Block 1Pre-Processing

MAD plug-in framework inside HydroDesktop: Proposed Architecture

HydroDesktop

HydroModeler

HIS Database

MAD Block 3

MAD Block 2

MAD Block 1

MAD Block 3 Post-Processing

MAD Block 2Likelihood

MAD Block 1Pre-Processing

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.

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

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