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)

MRF prior where and n j =#{i:i~j}

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 s 1,..,s n. Write  (s) has covariance function

More generally Consider k independent stationary spatial fields W i (s) and a random vector Z. Write and create a nonstationary process by Its covariance (with  =Cov(Z)) is

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 motion C(s,t)=min(s,t)  k (s)=2 1/2 sin((k-1/2)  t)/((k-1/2)  ) Conversely,

Discrete case Eigenexpansion of covariance matrix Empirically SVD of sample covariance Example: squared exponential k=1520

Tempering Stationary case: write with covariance To generalize this to a nonstationary case, use spatial powers of the k : Large  corresponds to smoother field

A simulated example

Estimating  (s) Regression spline Knots u i picked using clustering techniques Multivariate normal prior on the  ’s.

Piazza Road revisited

Tempering More fins structure More smoothness

Covariances A BCD

Karhunen-Loeve expansion revisited and where   are iid N(0, i ) Idea: use wavelet basis instead of eigenfunctions, allow for dependent  i

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. 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)

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 x3 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