1) The document proposes an approach called locally stationary geostatistics that relaxes the assumption of strict stationarity by allowing statistics like the mean and variogram to vary locally based on location. 2) Key aspects include defining location-dependent weights to transform sample data, estimate local distributions and statistics, and perform simulation and estimation techniques like kriging under the assumption of local stationarity. 3) Benefits include models that are richer in local features, improved accuracy, and narrower uncertainty, while drawbacks include increased computational and data demands.

1. Locally Stationary Geostatistics
David F. Machuca-Mory and Clayton V. Deutsch
Sampling Data Calibration of the distance weighting function parameters Selection of anchor points Recalculate and store Inference of local distributions
n
Problem Statement: locations distance weights F (u; zk ;= Prob{Z (u) ≤ zk |=
o) o} ∑ ω (uα ; o) ⋅ I (uα ; zk ) ∈ [0,1]
-Smooth adaptation Minimize the number
α =1
to local features ∀uα ∈ D, k = K
1,...,
of reference points
•Standard geostatistical simulation and -Avoid overfitting for the inference of
estimation techniques are constrained by the location-dependent
statistics without
assumption of strict stationarity. Program : LDWgen Introducing excessive
error.
• This assumption may be to rigid modelling the Program : LDWgen
patterns produced by different geological Program : Histpltsim
processes.
Proposed Approach: The Assumption of
Hermite models of the local normal Local normal scores transformation Location-dependent measures of Local variogram model fitting
Local Stationarity scores transformations • It is performed semiautomatically using minimum least squares
spatial continuity criterion.
•Under this assumption the distributions and z = ϕ Z ( y; o ) 
Q
∑ φq (o)H q [ y]
• Location-dependent variogram:
• Geological knowledge can be incorporated for
their statistics are specific of each location o: q =0 = −1
(
= F G ( y j ); o ϕ Z ( y j ; o)
zj ) γ (h; o)
=
1 N (h )
∑ ω ′(uα , uα + h; o) ⋅ [ y (uα ) − y (uα + h) ]
2
guiding the fitting of anisotropy parameters.
n j = 1,..., n 2 α =1
1 • A locally changing variogram shape is allowed:
φq (o)  ∑ (z j −1 − z j ) ⋅ H q −1 ( y j ) ⋅ g ( y j )
{
Prob {Z (uα ) < z1 ,..., Z (u n ) <= Prob Z (uα + h) < z1 ,..., Z (u n + h) < z K ; o j
z K ; oi } } j =2 q • Location-dependent covariance:
N (h )
   3 h′ b ( o )  
γˆ (h; o) c(o). 1 − exp  − 
=   0 < b(o) ≤ 2
∀ uα , uα + h ∈ D, and only if i =j Program : herco_loc Program : nscore_loc C (h; o)
= ∑ ω ′(uα , uα + h; o) ⋅ y(uα ) ⋅ y(uα + h) − m−h (o) ⋅ m+h (o)    az′ (o)   
 
α =1  Program : varfit_loc
•These are obtained by weighting the sample • Location-dependent correlogram:
C (h; o)
values inversely proportional to their distance to =ρ (h; o)
2 2
σ −h (o) ⋅ σ +h (o)
∈ [−1, +1] Program : gamvlocal
the prediction point o. N (h )
∑ ω ′(uα , uα + h; o) ⋅ [ z (uα ) − m-h (o) ]
Interpolate and Store
N (h ) 2 2
σ −h (o)
= ,
m-h (o)
= ∑ ω ′(uα , uα + h; o) ⋅ z (uα ) ,
•The same set of weights modify all the required Interpolate and store the local Interpolate and store local
α =1
α =1
N (h )
∑ ω ′(uα , uα + h; o) ⋅ [ z (uα + h) − m+h (o)]
N (h ) 2 2
categorical proportions
σ + h (o)
=
m+h (o)
= ∑ ω ′(uα , uα + h; o) ⋅ z (uα + h)
Hermite coefficients variogram parameters
α =1
statistics.
α =1
•A Gaussian kernel can be used as weighting φ0 (o) =
= E[ Z (u); o] m(o)
function: Q
 ( d (u ; o) )2 
2
σ Z (o)  ∑ φq2 (o)
q =1
α
ε + exp  − 
 2s 2 
ωGK (uα ; o) =  
n  ( d (u ; o) )2 
nε + ∑ exp  − α

α =1  2s 2 
 
•2-point weights are obtained by averaging 1-
Locally Stationary MultiGaussian Kriging Locally Stationary Sequential Gaussian Simulation Locally Stationary Sequential Indicator Simulation
point weights: Point Kriging 1.Read all required local parameters
n (o )
ω (uα , uα + h; o)
= ω (uα ; o) ⋅ ω (uα + h; o) 2.Follow random path
∑ λβ( LSSK ) (o) ρ (u β − uα ; o) = (o − uα ; o)
ρ α = n(o)
1,..., 1.Read all required local parameters
β =1
2.Follow random path 3.Transform surrounding data locally
Benefits: =1 − ∑ λα
2  n (o ) ( LSSK )
σ LSSK (o)
C (0; o) 

(o) ρ (o − uα ; o)  4.Build local covariances matrix
3.Transform surrounding data locally
 α =1 
•Resulting models are richer in local features n (o )  n (o ) ( LSSK )  4.Build local covariances matrix 5.Perform locally stationary simple
= ∑ λα
*
Z LSSK (o) ( LSSK )
(o)[ Z (uα )] + 1 − ∑ λα (o)  m(o) kriging
and look geologically more realistic. = 1= 1 α  α  5.Perform locally stationary simple
kriging 6.Draw a random value
•This may result in improved accuracy Block Kriging
6.Draw a random value 7.Backtransform the simulated value
Q
and add it to the data set
•The uncertainty of posterior distributions is z* (v(o=
p )) ϕv ( y* (v(o)); o)
p = ∑ φq (o) ⋅r q (o) ⋅ H q [ y*p (v(o))] 7.Backtransform simulated value
q =0 and add it to the data set 8. Continue from 2
generally narrower Q
= ∑ φq (o) ⋅r q (o) ⋅ H q [YLSSK (v(o)) + σ LSSK (v(o)) ⋅ t p ]
* 8. Continue from 2
• Spatial connectivity is improved. q =0
Program : kt3d_lMG Program : ultisgsim Program : sisim_loc
•A better performance is observed in transfer
functions.
Model performance Model Check Model Check
Drawbacks:
•Increased demand of computer and
professional resources.
•Local statistics are unreliable if data is scarce.