locally stationary geostatistics

1
Locally Stationary Geostatistics David 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 Kriging Point 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 simple kriging 6.Draw a random value 7.Backtransform simulated value and add it to the data set 8. Continue from 2 Sampling Data Selection of anchor points locations Minimize the number of reference points for the inference of location-dependent statistics without Introducing excessive error. 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 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 simple kriging 6.Draw a random value 7.Backtransform the simulated value and 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 k F z Z z I z Dk K α α α α ω = = = = Local normal scores transformation ( ) 1 ( ); ( ;) 1,..., o o j j Z j z F Gy y j n ϕ = = = Hermite models of the local normal scores transformations 0 (;) () [] Q Z q q q z y H y ϕ φ = = o o 1 1 2 1 () ( ) ( ) ( ) n q j j q j j j z z H y gy q φ = o [ ] () 2 1 1 (;) ( , ;) ( ) ( ) 2 h ho u u ho u u h N y y α α α α α γ ω = = + + () 1 (;) ( , ;) ( ) ( ) () () h h +h ho u u ho u u h o o N C y y m m α α α α α ω = = + + 2 2 (;) (;) [ 1, 1] () () h +h ho ho o o C ρ σ σ = ∈− + () 1 () 1 () ( , ;) ( ) , () ( , ;) ( ) h -h h +h o u u ho u o u u ho u h N N m z m z α α α α α α α α ω ω = = = + = + + [ ] [ ] () 2 2 1 () 2 2 1 () ( , ;) ( ) () , () ( , ;) ( ) () h h -h h h +h o u u ho u o o u u ho u h o N N z m z m α α α α α α α α σ ω σ ω = + = = + = + + () 3 ˆ (;) ( ). 1 exp 0 () 2 () o z h ho o o o b c b a γ = < Interpolate and store the local Hermite coefficients 0 () [ ( ); ] () o u o o EZ m φ = = 2 2 1 () () o o Q Z q q σ φ = () ( ) 1 () ( ;) ( ;) 1,..., ( ) n LSSK n β α α β β λ ρ ρ α = = = o o u u o o u o o () 2 ( ) 1 () (0; ) 1 () ( ;) n LSSK LSSK C α α α σ λ ρ = = o o o o o u o () () * ( ) ( ) 1 1 () ( )[ ( )] 1 () () n n LSSK LSSK LSSK Z Z m α α α α α λ λ = = = + o o o o u o o * * * 0 * 0 ( ( )) ( ( ( )); ) () () [ ( ( ))] () () [ ( ( )) ( ( )) ] o o o o o o o o o o Q q p v p q q p q Q q q q LSSK LSSK p q z v y v r H y v r H Y v v t ϕ φ φ σ = = = = = + Problem Statement: Standard geostatistical simulation and estimation techniques are constrained by the assumption of strict stationarity. This assumption may be to rigid modelling the patterns produced by different geological processes. Proposed Approach: The Assumption of Local Stationarity Under this assumption the distributions and their 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 1 Prob ( ) ,..., ( ) ; Prob ( ) ,..., ( ) ; , and only if u u o u h u h o u u h = n K i n K j Z z Z z Z z Z z D ij α α α α < < = + < + < + , ( ) ( ) 2 2 2 2 1 ( ;) exp 2 ( ;) ( ;) exp 2 u o u o u o GK n d s d n s α α α α ε ω ε = + = + ( , ;) ( ;) ( ;) α α α α ω ω ω + = + u u ho u o u ho Program : LDWgen Program : LDWgen Program : Histpltsim Program : nscore_loc Program : gamvlocal Program : herco_loc Program : kt3d_lMG Program : varfit_loc Program : sisim_loc Program : ultisgsim

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Page 1: Locally Stationary Geostatistics

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

Qq

p v p q q pq

Qq

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