Kriging. 16 August 2004

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1 T T Kriging Allan Aasberg Nielsen Infmatics and Mathematical Modelling Technical niversit of Denmark, Building 321 DK-2800 Lngb, Denmark aa aa@immdtudk 16 August 2004 Kriging (after the South African mining engineer and Profess Danie Krige) is a term used f a famil of methods f minimum err variance estimation Consider a linear ( rather affine) estimate at a location based on measurements "!#$ %&' ("! at locations ) + -, )0/1 ) ) + 2,43! (1) here is a constant and ) is the eight applied to ) We consider ) as particular realisations of random variables 5 ), 6 $ 5 5 "!#$ 5 5 7! We think of 5 as consisting of a deterministic mean and a stochastic residual 5 8:9;,=< ith a constant covariance >?A@;B > C@8 )GFH JI is a displacement vect measuring distance and direction beteen locations ) and KI F the linear (affine) estimat e get 5 + 2,=3! 6 (2) The actual err LF is unknon but f the expectation of the estimation err e get EM%5 F 5 N EM&5 FO F 3! 6 N :9 FP F 3!?Q (3) here 9R +9; S is the expectation value of 5 and Q is a vect of expectation values Q 9; W 9; We ant our estimat to be unbiased, ie e request EM%5 LF 5 N +\ XZ 91 9 XZ (4) 9R LFO- (F 3!1Q ]\K (5) The estimation variance is

2 T T T ^R_ ` arm&5 F 5 %N arm%5 %N2, arm a,43! 6 N F4b CovM%5 2,43! 6 N (6) ^ _a,=3!dc 3 F4b 3! CovM&6 5 N here c is the variance/covariance matrix (also knon as the dispersion matrix) of 6, c DM&6 N ith elements > )eif > g@h)iib > )df= KI that are the covariances beteen observations at locations R) and KI CovM&6 5 N is a column vect of covariances beteen observations at locations R) and ^ _ quadratic function in 3 ` is a Simple Kriging (SK) In simple kriging e assume that 9; W is knon (often assumed constant) If e insert from (5) into (3) e get The kriging eights () are found b minimising ^ _ ` 5 FP9 3! 6 F Q (7) ^ _ ` 3 b c 3 F4b CovM&6 5 %N k (8) hich gives the simple kriging sstem c 3 CovM%6 5 N (9) > ' lll > m > ( lll > - X L X > > ' X (10) The minimum squared estimation err is ^R_ no ^R_,43! c 3 F4b CovM%6 5 N ; ^R_ F 3! CovM&6 5 N (11) In SK the mean is knon In practice one must estimate 9; pri to the estimation design an estimation algithm that requires no pri mean Ordinar Kriging (OK) In dinar kriging e assume that the mean is constant ( p9 ) f 5 and the the estimation of 5 points that enter into EM&5 LF 5 N q9r r mslf 3!at hfo- +\ (12) f an 9 t is a vect of ones This is possible onl if ( +\ and 3u! t s The eights ) are found b minimising ^ ` _ ith 3v! t ps Introduce ith Lagrange multiplier Fxb 2

3 T T T ^ _ `, bz? 3!8t FHsS (13) 3 b c 3 F4b CovM&6 5 N2, b t k (14) b 3! t F{s \ (15) hich gives the dinar kriging sstem c 3}, t CovM&6 5 N (16) tj! 3 s (17) T > lll > ~ s > (lll > - s s lll s \ XZ T x XZ T > > A s XZ (18) The minimum squared estimation err is ^R_ Ro ^R_,=3! c 3 F b CovM&6 5 N h ^R_ F 3! CovM%6 5 N F? (19) OK implies a re-estimation of 9? f each ne estimation suppt niversal Kriging (K) In universal kriging e assume that the mean can be ritten as a linear combination of knon functions (ideall determined b the phsics of the problem dealt ith) 9; W;ƒ /R? A ;]!dˆ (20) If no knoledge exists about the fm of ˆ e can use lo der (tpicall second der) polnomials and consider them as local Tal expansions B convention xps Q 9; R 9; EM%5 F 5 %N +! ˆ dfp F 3! Q (21) XZ! ˆ W! ˆ XZ ˆ! W~ ˆ! m XZ Š] (22) ith 3

4 T T T T ŠŒ ˆ! ˆ! XZ Š! $ ˆ R lll ˆ g (23) EM&5 -F 5 N +! ˆ SGFO- (F 3! Š #]! ˆ SGF4Š! 3 GFP- q\ (24) f an This is possible onl if ( +\ and Š! 3 ˆ The eights L) are found b minimising ^ ` _ ith Š! 3 ˆ Introduce ith Ž, s Lagrange multipliers Fxbz here 4p ÿ ' 7! ^R_ `, b! CŠ! 3 F ˆ Sm (25) 3 b c 3 F b CovM&6 5 N2, b ŠD k (26) bk CŠ! 3 F ˆ m k (27) hich gives the universal kriging sstem c 3}, ŠD CovM&6 5 N (28) Š! 3 ˆ (29) > ' lll > ~ r lll W > ( lll > - z lll r W lll \ lll \ R lll \ lll \ XZ ( L K XZ > > A z K S S XZ (30) The minimum squared estimation err is ^ _ o ^ _,43! c 3 F4b CovM%6 5 &N ; ^ _ F 3! CovM&6 5 %N F4! ˆ (31) When appling K c and CovM%6 5 &N must be estimated after removal of the trend that e are estimating along ith the stochastic part This is a problem that limits the use of K (one could argue that this characteristic doesn t go too ell ith the adective universal ) We can use K to appl an external drift f example b setting Ž s!gˆ 8ƒ /R A a, & (32) An external drift is easil alloed f befe estimating c and CovM%6 5 N 4

5 > Factial Kriging (FK) In factial kriging e assume that the random variable can be ritten not as a sum of a deterministic mean and a stochastic err but rather as a sum of, s independent stochastic terms facts ) 5 8,,+lll, (33) Since the facts are independent e get f the covariance function f 5 > C@;; > g@h, > C@h,+lll, >a g@h (34) here > ) g@h is the covariance function f ) The facts are dered so that lo indices crespond to sht range phenomena and high indices crespond to phenomena ith increasingl longer range f example stemming from, s nested structures identified in the model of > C@; We can filter out the covariance contributions of an number of consecutive sht range noise facts (ith indices ranging from 0 to F{s ith \ ) leaving the long range signal facts onl The long range signal component of 5 is 5; %š œ / œa ) Its estimat is 5G š 3u! 6 We ant the expectation EM%5h %š F 5h %š N to be zero As in the OK case this leads to 3u! t ps The estimation variance in this case is ^ _ ` arm&5 %š F 5 %š N arm&5 %š N2,43!dc 3 F b 3! CovM%6 5 %š N (35) The eights () are found b minimising ^ _ ` ith 3v! t ps Introduce ith Lagrange multiplier Fxb ^ _ `, bz? 3! t FHsS (36) 3 b c 3 F b CovM&6 5 %š N2, bz t k (37) bk 3!;t F{s \ (38) hich gives the factial kriging sstem (f the long range part) c 3}, t CovM%6 5h %š N (39) t! 3 s (40) T > ' lll > ~ s > (lll > - s s lll s \ XZ T x XZ T > %š > A š s XZ (41) The elements in the RHS vect are > A) %š œ / œ œ g@g A)Ch@G A)1: (FP ) 5

6 œa ž 5 n š œ /R ) is the sht range noise part of 5 Its estimat is 5 n š 3Ÿ! 6 We ant the expectation EM%5 n š F 5 n š N to be zero Since e ant the noise expectation EM%5 n š N to be zero EM 5 n š N is zero also This leads to 3u! t q\ The estimation variance in this case is ^ _ ` arm&5 n š F 5 n š N arm&5 n š N2,43! c 3 F b 3! CovM&6 5 n š N (42) The eights () are found b minimising ^ _ ` ith 3v! t ]\ Introduce ith Lagrange multiplier Fxb ^ _ `, bz 3!8t (43) 3 b c 3 F4b CovM%6 5 n š N2, b t k (44) b 3! t \ (45) hich gives the factial kriging sstem (f the sht range part) c 3$, t CovM&6 5 n š N (46) t! 3 \ (47) T > ' lll > m s > (ƒlll > - s s lll s \ The elements in the RHS vect are > A) n š XZ T œ ž œ /R > XZ T > n š > A n š \ œ C@h ')gh@g ')d: LFP ) XZ (48) We see that compared to the OK sstem, in the FK sstem the LHS is unchanged and that the covariances on the RHS are replaced ith relevant long sht range components, and that f the long range part the eights sum to one as in OK and f the sht range part the sum to zero References AG Journel and ChJ Huibregts (1978) Mining Geostatistics Academic Press EH Isaaks and RM Srivastava (1989) Applied Geostatistics Oxfd niversit Press C Deutsch and AG Journel (1998) GSLIB: Geostatistical Softare Librar and ser s Guide, second edition Oxfd niversit Press T ao, T Mukeri, A Journel and G Mavko (1999) Scale matching ith factial kriging f improved posit estimation from seismic data Mathematical Geolog 31(1),

OFTEN we need to be able to integrate point attribute information

OFTEN we need to be able to integrate point attribute information ALLAN A NIELSEN: GEOSTATISTICS AND ANALYSIS OF SPATIAL DATA 1 Geostatistics and Analysis of Spatial Data Allan A Nielsen Abstract This note deals with geostatistical measures for spatial correlation, namely