Bayesian and Geostatistical Approaches to Inverse Problems

1
Bayesian and Geostatistical Approaches to Inverse
Problems

• Peter K. Kitanidis
• Civil and Environmental Engineering
• Stanford University

2
Outline

• Important points
• Current Work

3
Inverse Problem

• Estimate functions from sparse and noisy
  observations
• The unknowns are sensitive to data gaps or flaws
  (Problem is ill-posed in the sense of Hadamard)
• Data are insufficient to zero in on a unique
  solution
• Usually, it is the small-scale variability that
  cannot be resolved.

4
(No Transcript)
5
Cheney, M. (1997), Inverse boundary-value
problems, American Scientist, 85 448-455.
6
Bayesian Inference Applied to Inverse Modeling
Likelihood of unknown parameter given data
Posterior distribution of unknown parameter
Prior distribution of unknown parameter
y measurements s unknown
7
Bayesian Inference Applied to Inverse Modeling
Information from observations
Combined information (data and structure)
Information about structure
y measurements s unknown
8
How do you get the structure?

• We often use an empirical Bayes in which the
  structure pdf is parameterized and inferred from
  the data the approach is rigorous and robust.
• Alternative interpretation We use
  cross-validation.
• In specific applications, we may use geological
  or other information to describe structure.

9
Computational cost

• Reduce cost by dealing with special cases, or
• Bite the bullet and use computer intensive
  numerical methods (MCMC, etc.)

10
The importance of properly weighing observations
A source identification problem
Identify the pumping rate at an extraction well
from head observations, in a neighboring
monitoring well
11
Over-weighting Observations
Five slides from Kitanidis, P. K. (2007), On
stochastic inverse modeling, in Subsurface
Hydrology Data Integration for Properties and
Processes edited by D. W. Hyndman, F. D.
Day-Lewis and K. Singha, pp. 19-30, AGU,
Washington, D. C.
12
Under-weighting Observations
13
(No Transcript)
14
(No Transcript)
15
Optimal Weighting
16
The cost of computations

• Moores law Cost of computations is halved
  every 1.5 years. Thus, between 1975 and 2006
  2(31/1.5)1.7E6.
• 5,000 of computer usage for a project in 1975.
• 19755,000 -- adjust for inflation -gt
  200620,000.
• 20,000/1.7E6 corresponds to 1 cent worth of
  computational power in 2006.

17
From the BOISE HYDROGEOPHYSICAL RESEARCH SITE
(BHRS)
18
METHODMarkov Chain Monte Carlo

• Based on Michalak and Kitanidis (2003 and 2004)
• Use EM method on marginal distributions to find
  optimal parameters for structure and epistemic
  error.
• Employ a Gibbs sampler to build a set of
  conditional realizations of posterior pdf. (A
  large enough set of conditional realizations has
  the same statistical properties as the actual
  posterior distribution.)

19
A problem of forensic environmental engineering
PCE data at location PPC13. Measurement data and
fitted concentrations resulting from the
estimated boundary conditions.
Michalak, A.M., and P.K. Kitanidis (2003) A
Method for Enforcing Parameter Nonnegativity in
Bayesian Inverse Problems with an Application to
Contaminant Source Identification, Water
Resources Research, 39(2), 1033,
doi10.1029/2002WR001480.
20
Location PPC13. Estimated time variation of
boundary concentration at the interface between
the aquifer and aquitard. The end time
represents the sampling date (June 6, 1996).
21
PCE data at location PPC13 with non-negativity
constraint. Measurement data and fitted
concentrations resulting from the estimated
boundary conditions.
22
Location PPC13 with non-negativity constraint.
Estimated time variation of boundary
concentration at the interface between the
aquifer and aquitard. The end time represents
the sampling date (June 6, 1996).
23
TRACER RESPONSESynthetic Case
Without Error
With Error
Fienen, M. N., J. Luo, and P. Kitanidis (2006), A
Bayesian Geostatistical Transfer Function
Approach to Tracer Test Analysis, Water Resour.
Res., 42, W07426, 10.1029/2005WR004576.
24
Current Work

• Large variance and highly nonlinear problems
  (Convergence of Gauss-Newton, usefulness of
  Fisher matrix, etc.)
• Tomographic inverse problems (development of
  protocols, processing of large data sets.)

25
Current Work (cont.)

• Identification of zone boundaries.
• Solution methods for very large data sets.
• Making tools available to users.

26
Identification of zone boundariesExample

• Linear tomography
• Zones small-scale variability
• measurement error (2)

Four slides from the work of Michael Cardiff
27
Example Problem Performance
28
We are developing

• Stochastic analysis of zone uncertainty
• Merging of structural (level set) and
  geostatistical inverse problem concepts
• Use of level sets for joint inversion

29
Toolbox for

• COMSOL Multiphysics is a commercial general
  purpose PDE solver.
• We are adding inverse-model capabilities,
  including adjoint-state sensitivity analysis and
  stochastics.
• See Cardiff, M, and P. K. Kitanidis, Efficient
  solution of nonlinear underdetermined inverse
  problems with a generalized PDE solver,
  Computers and Geosciences, in review, 2007.

30
Our approach is

• Stochastic (aka probabilistic or statistical)
  We assign a probability to every possible
  solution.

• Bayesian Because the Bayesian approach provides
  a general framework.

• Geostatistical We have adopted the best ideas
  from the geostatistical school.

• Practical Our methods are evolving, with
  particular emphasis on practicality, robustness,
  and computational efficiency.

31
For More Info

• See publications on the WWW
• http//www.stanford.edu/group/peterk/publications.
  htm