groundwater permeability easy to solve the forward problem: flow of groundwater given permeability...
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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
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