locally stationary geostatistics
TRANSCRIPT
Locally Stationary GeostatisticsDavid F. Machuca-Mory and Clayton V. Deutsch
Location-dependent measures of spatial continuity• Location-dependent variogram:
• Location-dependent covariance:
• Location-dependent correlogram:
Local variogram model fitting• It is performed semiautomatically using minimum least squares criterion.
• Geological knowledge can be incorporated for guiding the fitting of anisotropy parameters.
• A locally changing variogram shape is allowed:
Locally Stationary MultiGaussian KrigingPoint Kriging
Block Kriging
Locally Stationary Sequential Gaussian Simulation
1.Read all required local parameters
2.Follow random path
3.Transform surrounding data locally
4.Build local covariances matrix
5.Perform locally stationary simplekriging
6.Draw a random value
7.Backtransform simulated valueand add it to the data set
8. Continue from 2
Sampling Data Selection of anchor points locations
Minimize the numberof reference points for the inference oflocation-dependent statistics without Introducing excessiveerror.
Recalculate and store distance weights
Interpolate and Store categorical proportions
Interpolate and store local variogram parameters
Calibration of the distance weighting function parameters
-Smooth adaptation to local features-Avoid overfitting
Model performance Model Check Model Check
Locally Stationary Sequential Indicator Simulation1.Read all required local parameters
2.Follow random path
3.Transform surrounding data locally
4.Build local covariances matrix
5.Perform locally stationary simplekriging
6.Draw a random value
7.Backtransform the simulated valueand add it to the data set
8. Continue from 2
Inference of local distributions
1( ; ; ) Prob{ ( ) | } ( ; ) ( ; ) [0,1]
, 1,...,
u o u o u o u
u
n
k k kF z Z z I z
D k K
α αα
α
ω=
= ≤ = ⋅ ∈
∀ ∈ =
∑
Local normal scores transformation
( )1 ( ); ( ; )
1,...,
o o
j j Z jz F G y y
j n
ϕ−= =
=
Hermite models of the local normal scores transformations
0( ; ) ( ) [ ]
Q
Z q qq
z y H yϕ φ=
= ∑o o
1 12
1( ) ( ) ( ) ( )n
q j j q j jj
z z H y g yq
φ − −=
− ⋅ ⋅∑o
[ ]( )
2
1
1( ; ) ( , ; ) ( ) ( )2
hh o u u h o u u h
Ny yα α α α
αγ ω
=
′= + ⋅ − +∑
( )
1( ; ) ( , ; ) ( ) ( ) ( ) ( )
h
h +hh o u u h o u u h o oN
C y y m mα α α αα
ω −=
′= + ⋅ ⋅ + − ⋅∑
2 2
( ; )( ; ) [ 1, 1]( ) ( )h +h
h oh oo o
Cρσ σ−
= ∈ − +⋅
( )
1( )
1
( ) ( , ; ) ( ) ,
( ) ( , ; ) ( )
h
-h
h
+h
o u u h o u
o u u h o u h
N
N
m z
m z
α α αα
α α αα
ω
ω
=
=
′= + ⋅
′= + ⋅ +
∑
∑
[ ]
[ ]
( )22
1( )
22
1
( ) ( , ; ) ( ) ( ) ,
( ) ( , ; ) ( ) ( )
h
h -h
h
h +h
o u u h o u o
o u u h o u h o
N
N
z m
z m
α α αα
α α αα
σ ω
σ ω
−=
+=
′= + ⋅ −
′= + ⋅ + −
∑
∑
( )3
ˆ( ; ) ( ). 1 exp 0 ( ) 2( )
o
z
hh o o o
o
b
c ba
γ′
′ = − − < ≤
Interpolate and store the local Hermite coefficients
0 ( ) [ ( ); ] ( )o u o oE Z mφ = =
2 2
1( ) ( )o o
Q
Z qq
σ φ=∑
( )( )
1( ) ( ; ) ( ; ) 1,..., ( )
nLSSK nβ α αβ
βλ ρ ρ α
=− = − =∑
oo u u o o u o o
( )2 ( )
1( ) (0; ) 1 ( ) ( ; )
nLSSK
LSSK C α αα
σ λ ρ=
= − −
∑
oo o o o u o
( ) ( )* ( ) ( )
1 1( ) ( )[ ( )] 1 ( ) ( )
n nLSSK LSSK
LSSKZ Z mα α αα α
λ λ= =
= + −
∑ ∑
o oo o u o o
* * *
0
*
0
( ( )) ( ( ( )); ) ( ) ( ) [ ( ( ))]
( ) ( ) [ ( ( )) ( ( )) ]
o o o o o o
o o o o
p v p q q pq
q q LSSK LSSK pq
z v y v r H y v
r H Y v v t
ϕ φ
φ σ
=
=
= = ⋅ ⋅
= ⋅ ⋅ + ⋅
∑
∑
Problem Statement:•Standard geostatistical simulation andestimation techniques are constrained by theassumption of strict stationarity.• This assumption may be to rigid modelling thepatterns produced by different geologicalprocesses.
Proposed Approach: The Assumption ofLocal Stationarity•Under this assumption the distributions andtheir statistics are specific of each location o:
•These are obtained by weighting the sample values inversely proportional to their distance to the prediction point o.•The same set of weights modify all the required statistics.•A Gaussian kernel can be used as weighting function:
•2-point weights are obtained by averaging 1-point weights:
Benefits:•Resulting models are richer in local features and look geologically more realistic.•This may result in improved accuracy•The uncertainty of posterior distributions is generally narrower• Spatial connectivity is improved.•A better performance is observed in transfer functions.
Drawbacks:• Increased demand of computer and professional resources.•Local statistics are unreliable if data is scarce.
{ } { }1 1Prob ( ) ,..., ( ) ; Prob ( ) ,..., ( ) ;
, and only if
u u o u h u h o
u u h =n K i n K jZ z Z z Z z Z z
D i jα α
α α
< < = + < + <
∀ + ∈ ,
( )
( )
2
2
2
21
( ; )exp
2( ; )
( ; )exp
2
u o
u ou o
GKn
ds
dn
s
α
αα
α
ε
ω
ε=
+ − = + −
∑
( , ; ) ( ; ) ( ; )α α α αω ω ω+ = ⋅ +u u h o u o u h o
Program : LDWgen
Program : LDWgen
Program : Histpltsim
Program : nscore_loc
Program : gamvlocal
Program : herco_loc
Program : kt3d_lMG
Program : varfit_loc
Program : sisim_locProgram : ultisgsim