GeoEnv -July Simulations. D. Renard N. Desassis. Geostatistics & RGeostats

Transcription

1 GeoEnv -July 2014 Simulations D. Renard N. Desassis Geostatistics & RGeostats 1

2 o Why are simulations necessary? Simulations Estimation (Kriging) produces smooth results We need a different method which can: reproduce the variability Give valid (non biased) solution to complex criterion (non linear) Example of Volumetrics problem in the Oil industry: get the volume of a reservoir below an impermeable horizon and above the oil-water contact Estimated Top Oil Volume OWC Geostatistics & RGeostats 2

3 oyeu Island Example Geostatistics & RGeostats 3

4 oyeu Island Example 40 samples on 8 bathymetric profiles No sample ON the island Samples and Kriged results True map Representation profile Geostatistics & RGeostats 4

5 oyeu Island Example 9 simulations conditioned by the bathymetric profiles Geostatistics & RGeostats 5

6 oyeu Island Example 9 simulated profiles conditioned by the bathymetric profiles Geostatistics & RGeostats 6

7 oyeu Island Example Probability map to belong to the island Geostatistics & RGeostats 7

8 oyeu Island Example We can calculate the function of interest per simulation and derive statistics 0. Fréque ence Surface 0. Fréquen nce nce Fréquen Volume 10 Hauteur 40 Estimation Simulations Réalité Surface (km 2 ) Volume (km 3 ) Hauteur (m) Geostatistics & RGeostats 8

9 o Spatial Law Simulations We cannot rely on the first two moments: Three realizations with same histogram, same covariance, same 3-point statistics Geostatistics & RGeostats 9

10 o Spatial Law Simulations We must know the spatial law which characterizes the variable of interest: ( ( ) <,..., ( ) < ) (,..., ) P Z x z Z x z x x 1 1 n n 1 n In general, the spatial law is not tractable Gaussian framework: Definition: { } ( ) Y x Y x1 Y x n ( ) gaussian ( ),..., ( ) gaussian vector Simplification in the (multi-) gaussian case: Knowing the first two moments is sufficient to describe the whole spatial law Most of algorithms based on large number of independent replicates tends to normality: Central Limit Theorem Stability properties Geostatistics & RGeostats 10

11 o Definition Gaussian Anamorphosis The variable Z is a Gaussian transformed variable if: [ Y x ] Z( x) = Φ ( ) φ is a monotonous increasing function (called Gaussian Anamorphosis) Z = φ( Y ) Y = 1 φ ( Z ) Raw Gaussian The Gaussian Anamorphosis is fitted using Hermitepolynomials: it is used to convert simulated results backwards. Geostatistics & RGeostats 11

12 o Conditional Law Simulations Consider the gaussian vector We can write: where Y = Y + Y Y 0 SK 0 n Y = SK λ α Y α α = 1 ( Y Y Y ) 0 1 Then the following vector is bi-gaussian: SK,,..., n ( Y, Y Y ) SK 0 SK Orthogonality property of Simple Kriging: ( ) Cov Y, Y Y = 0 SK 0 SK Then we can write: Y0 = YSK +σ SKG(0,1) Conditional law: ( ) Law Y Y = y,..., Y = y = G( Y, σ ) n n SK SK Geostatistics & RGeostats 12

13 o Basic Method Simulations The conditional law is based on the Simple Kriging of available information Hence the simulation basic algorithm: 2 1 Draw the first simulated value Y S (0) according to G( m, σ ) 2.1 Perform Simple Kriging at next target using the previously simulated samples. We obtain Y S* and σ 2 S 2.2 Draw the simulated value according to 2.3 Return to 2.1 until all targets are processed S S G Y σ * 2 ( S, S ) Obviously the kriging system grows with the rank of the target. This algorithm becomes intractable when the number of targets is large Geostatistics & RGeostats 13

14 o Gibbs Sampler Simulations A similar simulation algorithm: 1 Draw spatial uncorrelated gaussian values at targets according to Perform the following iteration several times: 2.1 Consider one target site at random 2 G( m, σ ) Perform a simple kriging using all other information. We obtain Y S* 2 and σ S * Draw the simulated value at target according to G( Y, σ ) 2.3 Iterate 2.1 until all targets have been processed S S This algorithm (also) becomes intractable when the number of targets is large Geostatistics & RGeostats 14

15 o Turning Bands Simulations Transform the simulation of RF in R d by several independent simulations in R Along one band S, generate the RF Y 1 (s) with a given covariance: S Spread the n bands in R d S 2 S 1 S 3 1 Y x = Y < x S > n d 1 ( ). n i = 1 ( ) i Geostatistics & RGeostats 15

16 o Spherical model Turning Bands C 3 3 r 1 r ( r) = a 2 a 3 D 3 r a C3 D ( r) 3 r r C 1 D ( r ) = r a C1 D ( r ) a a 1-D Simulation Simulation using turning bands (1, 10, 1000 bands) 16

17 o Exponential model Turning Bands ( ) exp r C3 r = a C3 D( r) r r C ( r) = 1 1 a exp a C1 D ( r ) 1-D Simulation Simulation using turning bands (1, 10, 1000 bands) 17