1. 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)

2. 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)

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

5. 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

10. Nott & Dunsmuir, 2002, Biometrika Consider a stationary process W(s), correlation R, observed at sites s 1,..,s n. Write  (s) has covariance function

11. 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

12. 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.

14. 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,

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

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

17. A simulated example

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

19. Piazza Road revisited

20. Tempering More fins structure More smoothness

21. Covariances A BCD

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

23. Spatial wavelet basis Separates out differences of averages at different scales Scaled and translated basic wavelet functions

24. 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

25. W-transform

26. Covariance expansion For covariance matrix  write Useful if D close to diagonal. Enforce by thresholding off-diagonal elements (set all zero on finest scales)

27. 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.

28. ROM covariance

29. Some open questions Multivariate Kronecker structure Nonstationarity Covariates causing nonstationarity (or deterministic models) Comparison of models of nonstationarity Mean structure