Groundwater permeability

Easy to solve the forward problem: flow of groundwater given permeability of aquifer

Inverse problem: determine permeability from flow (usually of tracers)

With some models enough to look at first arrival of tracer at each well (breakthrough times)

Notation

is permeability

b is breakthrough times

expected breakthrough times

Illconditioned problems: different permeabilities can yield same flow

Use regularization by prior on log()

MRF

Gaussian

Convolution with MRF (discretized)

b̂()L( b) ∝ exp(−

12σ2 (bk −b̂k ())2

k∑ )

MRF prior

where

and nj=#{i:i~j}

π(x λ) ∝ λm/2 exp(− 12 λxTWx)

W =−1 if j~knj if j=k0 otherwise

⎧

⎨⎪

⎩⎪

Kim, Mallock & Holmes, JASA 2005

Analyzing Nonstationary Spatial Data Using

Piecewise Gaussian Processes

Studying oil permeability

Voronoi tesselation (choose M centers from a grid)

Separate power exponential in each regions

Nott & Dunsmuir, 2002, Biometrika

Consider a stationary process W(s), correlation R, observed at sites s1,..,sn.Write

(s) has covariance function

W(s) = R(s −s i )[ ]TR(s i −s j )⎡⎣ ⎤⎦

−1W(s i )[ ]

krigingpredictor1 24 4 4 4 4 4 34 4 4 4 4 4

+ (s)krigingerror

{

=λ(s)T [W(s i )] + (s)

R (s, t) =R(t−s)

− R(s −s i )[ ]TR(s i −s j )⎡⎣ ⎤⎦

−1R(t−s i )[ ]

More generally

Consider k independent stationary spatial fields Wi(s) and a random vector Z. Write

and create a nonstationary process by

Its covariance (with =Cov(Z)) is

μ i (s) = λ i (s)Z

Z(s) = wi (s)μ i (s) + wi (s)12 i (s)∑∑

R(s, t) = wi (s)wi (t)λ i (s)Tλ j (t)i,j∑

+ wi (s)12 wi (t)

12Ri

(s, t)i

∑

Fig. 2. Sydney wind pattern data. Contours of equal estimated correlation with two different fixed sites, shown by open squares: (a) location 33·85°S, 151·22°E, and (b) location 33·74°S, 149·88°E. The sites marked by dots show locations of the 45 monitored sites.

Karhunen-Loéve expansion

There is a unique representation of stochastic processes with uncorrelated coefficients:

where the k(s) solve

and are orthogonal eigenfunctions.Example: temporal Brownian motionC(s,t)=min(s,t)

k(s)=21/2sin((k-1/2)πt)/((k-1/2)π)Conversely,

Z(s) = αkφk (s)∑ Var(αk ) =λk

C(s, t)φk (t)dt=λkφk (s)∫

C(s, t) = λkφk (s)φk (t)∑

Discrete caseEigenexpansion of covariance matrix

Empirically SVD of sample covariance

Example: squared exponential

k=1 5 20

Tempering

Stationary case: write

with covariance

To generalize this to a nonstationary case, use spatial powers of the λk:

Large corresponds to smoother field

Z(s j ) = αkφk∑ (s j ) + ε(s j )

C(si,s j ) = λkφk (s i )φk (s j ) + σ2 1(i=j)∑

C(s, t) = λk(s ) / 2∑ λk

(t) / 2φk (s)φk (t)

A simulated example

(s) = 0.01s

2

⎛

⎝⎜⎞

⎠⎟

3

Estimating (s)

Regression spline

Knots ui picked using clustering techniques

Multivariate normal prior on the ’s.

log(s) =0 + 1s + j+2(s;uj )j=1

r

∑ (s;u) = s − u

2log s − u

Piazza Road revisited

Tempering

More fins structure

More smoothness

Covariances

A B C D

Karhunen-Loeve expansionrevisited

and

where α are iid N(0,λi)Idea: use wavelet basis instead of

eigenfunctions, allow for dependent αi

Cov(Z(s1),Z(s2 )) =C(s1,s2 )

= λ iφi (s1)φi (s2 )i=1

∞

∑

Z(s) = α iφi (s)i=1

∞

∑

Spatial wavelet basis

Separates out differences of averages at different scales

Scaled and translated basic wavelet functions

Estimating nonstationary covariance using wavelets

2-dimensional wavelet basis obtained from two functions and:

First generation scaled translates of all four; subsequent generations scaled translates of the detail functions. Subsequent generations on finer grids.

S(x1,x2 ) =φ(x1)φ(x2 )H(x1,x2 ) =(x1)φ(x2 )V(x1,x2 ) =φ(x1)(x2 )D(x1,x2 ) =(x1)(x2 )

detail functions

W-transform

Covariance expansion

For covariance matrix write

Useful if D close to diagonal.

Enforce by thresholding off-diagonal elements (set all zero on finest scales)

=ΨDΨT; D = Ψ−1Σ(ΨT )−1

Surface ozone model

ROM, daily average ozone 48 x 48 grid of 26 km x 26 km centered on Illinois and Ohio. 79 days summer 1987.

3x3 coarsest level (correlation length is about 300 km)

Decimate leading 12 x 12 block of D by 90%, retain only diagonal elements for remaining levels.

ROM covariance

Some open questions

Multivariate

Kronecker structure

Nonstationarity

Covariates causing nonstationarity (or deterministic models)

Comparison of models of nonstationarity

Mean structure