RESSTE workshop - Avignon - November 2018 Can the SPDE approach replace traditional Geostatistics for industrial applications? N. Desassis, R. Carrizo Vergara, M. Pereira D. Renard, T. Romary, X. Freulon MINES-ParisTech - G´ eosciences RESSTE - Avignon 1 / 39
Context The Geostatistics team of Mines-ParisTech Production of methodology for the society Production of softwares (RGeostats, Geovariances) Mineral ressources oriented RESSTE - Avignon 2 / 39
Constraints imposed by the industry Research of innovative solutions to increase productivity Quite conservative (changes allowed in a stable workflow) RESSTE - Avignon 3 / 39
Computational ressources RESSTE - Avignon 4 / 39
Computational ressources Generally more limited Uranium deposit - Arlit - Niger RESSTE - Avignon 5 / 39
Workflow 1) Modeling RESSTE - Avignon 6 / 39
Workflow 1) Modeling RESSTE - Avignon 6 / 39
Workflow 1) Modeling - Multivariate case RESSTE - Avignon 7 / 39
Workflow 1) Modeling - Multivariate case RESSTE - Avignon 7 / 39
Workflow 2) Conditional simulations Let � � Z D Z = Z T where Z D is the vector of data and Z T the vector of targets RESSTE - Avignon 8 / 39
Workflow 2) Conditional simulations Let � � Z D Z = Z T where Z D is the vector of data and Z T the vector of targets Covariance matrix � Σ DD � Σ DT Cov ( Z ) = Σ = Σ TD Σ TT RESSTE - Avignon 8 / 39
Workflow 2) Conditional simulations Let � � Z D Z = Z T where Z D is the vector of data and Z T the vector of targets Covariance matrix � Σ DD � Σ DT Cov ( Z ) = Σ = Σ TD Σ TT Conditional expectation (kriging) T = E [ Z T | Z D ] = Σ TD Σ − 1 Z ⋆ DD Z D RESSTE - Avignon 8 / 39
Workflow 2) Conditional simulations Let � � Z D Z = Z T where Z D is the vector of data and Z T the vector of targets Covariance matrix � Σ DD � Σ DT Cov ( Z ) = Σ = Σ TD Σ TT Conditional expectation (kriging) T = E [ Z T | Z D ] = Σ TD Σ − 1 Z ⋆ DD Z D Conditional variance (covariance matrix of the errors) T − Z T ) = Σ TT − Σ TD Σ − 1 Var [ Z T | Z D ] = Cov ( Z ⋆ DD Σ DT RESSTE - Avignon 8 / 39
Handling large data sets and large grid Kriging with large data sets is performed by using moving neighborhoods Conditional simulations are performed by using non conditional simulations and kriging of the residuals RESSTE - Avignon 9 / 39
Principle Let Z ( x ) = Z SK ( x ) + Z ( x ) − Z SK ( x ) where Z SK ( x ) = � n j =1 λ j ( x ) Z ( x j ) simple kriging Z ( x ) − Z SK ( x ) kriging residuals Z SK and Z − Z SK are two independent Gaussian random functions RESSTE - Avignon 10 / 39
Principle 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle 0 −600 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle 0 −600 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle 0 −600 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● −600 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● −600 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● −600 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● −600 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● −600 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Principle 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● −600 ● 0 200 400 600 800 1000 RESSTE - Avignon 11 / 39
Context Selective exploitation Punctual grade Z ( x ) , x ∈ D with mean m and covariance function C RESSTE - Avignon 12 / 39
Context Selective exploitation Punctual grade Z ( x ) , x ∈ D with mean m and covariance function C Selective Mining Unit (SMU): v RESSTE - Avignon 12 / 39
Context Selective exploitation Punctual grade Z ( x ) , x ∈ D with mean m and covariance function C Selective Mining Unit (SMU): v RESSTE - Avignon 12 / 39
Context Selective exploitation Punctual grade Z ( x ) , x ∈ D with mean m and covariance function C Selective Mining Unit (SMU): v Regularized grade on SMUs Z ( v ) = 1 � Z ( x ) dx | v | v RESSTE - Avignon 12 / 39
Context Selective exploitation Punctual grade Z ( x ) , x ∈ D with mean m and covariance function C Selective Mining Unit (SMU): v Regularized grade on SMUs Z ( v ) = 1 � Z ( x ) dx | v | v RESSTE - Avignon 12 / 39
Context Selective exploitation Punctual grade Z ( x ) , x ∈ D with mean m and covariance function C Selective Mining Unit (SMU): v Regularized grade on SMUs Z ( v ) = 1 � Z ( x ) dx | v | v From exploration data Z ( x 1 ) , . . . , Z ( x n ) RESSTE - Avignon 12 / 39
Context Selective exploitation Punctual grade Z ( x ) , x ∈ D with mean m and covariance function C Selective Mining Unit (SMU): v Regularized grade on SMUs Z ( v ) = 1 � Z ( x ) dx | v | v From exploration data Z ( x 1 ) , . . . , Z ( x n ) RESSTE - Avignon 12 / 39
Support effet What can we say about Z ( v ) ? RESSTE - Avignon 13 / 39
Support effet What can we say about Z ( v ) ? Same mean m RESSTE - Avignon 13 / 39
Support effet What can we say about Z ( v ) ? Same mean m Block covariance function C v ( h ) = Cov ( Z ( v ) , Z ( v + h )) 1 � � = C ( x − y ) dxdy | v | 2 v v + h RESSTE - Avignon 13 / 39
Support effet What can we say about Z ( v ) ? Same mean m Block covariance function C v ( h ) = Cov ( Z ( v ) , Z ( v + h )) 1 � � = C ( x − y ) dxdy | v | 2 v v + h P ( Z ( v ) ≥ z ) for any cutoff z ? RESSTE - Avignon 13 / 39
Support effet What can we say about Z ( v ) ? Same mean m Block covariance function C v ( h ) = Cov ( Z ( v ) , Z ( v + h )) 1 � � = C ( x − y ) dxdy | v | 2 v v + h P ( Z ( v ) ≥ z ) for any cutoff z ? RESSTE - Avignon 13 / 39
Support effet What can we say about Z ( v ) ? Same mean m Block covariance function C v ( h ) = Cov ( Z ( v ) , Z ( v + h )) 1 � � = C ( x − y ) dxdy | v | 2 v v + h P ( Z ( v ) ≥ z ) for any cutoff z ? RESSTE - Avignon 13 / 39
Support effet What can we say about Z ( v ) ? Same mean m Block covariance function C v ( h ) = Cov ( Z ( v ) , Z ( v + h )) 1 � � = C ( x − y ) dxdy | v | 2 v v + h P ( Z ( v ) ≥ z ) for any cutoff z ? Block simulations are required to generate several scenarios RESSTE - Avignon 13 / 39
Direct block simulations The number of SMU can be large (e.g 1 million) Conditional simulations by using discretization of the blocks can be time consuming Solution : use of a change of support model to describe the multivariate distribution of the points and the blocks and perform conditional simulations of the regularized variable without discretization Several hours for 100 simulations with around 100 000 observations RESSTE - Avignon 14 / 39
Handling covariance non-stationarities RESSTE - Avignon 15 / 39
Handling covariance non-stationarities Current solutions Deform the space Cut the domain into several sub-domains in which stationarity is acceptable RESSTE - Avignon 15 / 39
More complex environments RESSTE - Avignon 16 / 39
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