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