fast spatial interpolation using sparse gaussian processes - Monash ...

INTRODUCTION TO THE SPATIAL INTERTICLES THE DA

ARTICLES

FAST SPATIAL INTERPOLATION USING SPARSE GAUSSIAN PROCESSES

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The estimation of the natural ambient radioactivity in this entry to the Spatial Interpolation Comparison 2004 (SIC2004) uses Gaussian processes (GP’s) to predict the underlying dispersal process. GP’s enable us to predict easily levels of radioactivity at previously unseen locations and in addition they allow us to assess the uncertainty in the predicted value. To speed up computation time, which is cubic in the number of examples, a sequential, sparse implementation of the Gaussian process inference (SSGP) was used together with a Gaussian observational noise assumption. The examination of the available data led to a covariance function which is a mixture of exponential and squared-exponential functions. The mixture was chosen so that it incorporates both the local ambiguity in the data and also at the same time it captures the larger-scale variation of the observations. A further characteristic of the competition was the availability of 10 days of “prior observations”. The individual sets comprised of data that was recorded at the same time but at different locations across Germany. Consequently in the modelling stage we assumed that the underlying process governing the observations was invariant across the 10 days of “prior observations”, but that each day, i.e. each dataset, should be treated independently. These prior observations were used to infer the parameters of the covariance function, which are called hyper-parameters. Conforming to the Bayesian inference method, the exact values of the hyper-parameters were not fixed; rather we assigned a probability distribution to these parameters, giving a two level inference scheme. The inferred values of the hyper-parameters were used to set up a prior GP, and the same two level inference was employed when evaluating the method on the released SIC2004 dataset. We show that the sparse approximation with the SSGP method can be used instead of the conventional GP’s without significant loss in accuracy leading to a greatly reduced calculation time. We also compare the SSGP performance with

APPLIED GIS, VOLUME 1, NUMBER 2, 2005 MONASH UNIVERSITY EPRESS

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standard machine learning techniques: the SSGP results compare favourably to the other benchmark techniques.

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FAST SPATIAL INTERPOLATION ARTICLES

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15-3

N

P ( D | w) = ∏ P ( yn | f ( xn , w) )

(1)

n =1

ïÜÉêÉ=

P ( yn | f ( xn , w) ) = áë= íÜÉ= ÇáëíêáÄìíáçå= çÑ= íÜÉ= åçáëÉK= tÉ= ÑìêíÜÉê= ~ëëìãÉ= íÜ~í= íÜÉ=

çÄëÉêî~íáçå= åçáëÉ= áë= d~ìëëá~åI= ïáíÜ= òÉêç= ãÉ~å= ~åÇ= î~êá~åÅÉ= ëáÖã~= ëèì~êÉÇI= ÜÉåÅÉ= ïÉ= Ü~îÉW= 2⎞ ⎛ 1 P ( yn | f ( xn , w), σ 2 ) = exp ⎜ − 2 ( yn − f ( xn , w) ) ⎟ . ⎝ 2σ ⎠

(2)

qÜÉ=Åçãéìí~íáçå=çÑ=íÜÉ=éçëíÉêáçê=éêçÅÉëë=áë=ÇçåÉ=~ÅÅçêÇáåÖ=íç=_~óÉëÛ=êìäÉ=E_Éêå~êÇç= ~åÇ=pãáíÜ=NVVQFI=íÜìë=íÜÉ=~JéçëíÉêáçêá=éêçÄ~Äáäáíó=çÑ=íÜÉ=é~ê~ãÉíÉêë=ï=ÄÉÅçãÉëW=

p p (w | D) =

P ( D | w) p0 ( w)

∫ P( D | w) p0 (w) dw

ïÜÉêÉ=íÜÉ=ÇÉåçãáå~íçê=Ó=~äëç=Å~ääÉÇ=íÜÉ=ÉîáÇÉåÅÉ=

(3)

p( D) = ∫ P( D | w) p0 ( w) dw =Ó=áë=íÜÉ=

åçêã~äáëáåÖ=Ñ~Åíçê=íç=íÜÉ=éçëíÉêáçê=ÇáëíêáÄìíáçåK=qç=ëçäîÉ=Éèì~íáçå=EPF=~é~êí=Ñêçã=âåçïJ áåÖ=íÜÉ=éêáçê=

p0 ( w) I=ïÉ=Ü~îÉ=íç=âåçï=íÜÉ=ÑìåÅíáçå=Ñ~ãáäó f ( x, w) I=áKÉK=íÜÉ=é~ê~ãÉíêáÅ=

Ñçêã= çÑ= íÜÉ= ~ééêçñáã~íçêK= qÜáë= áë= ëáãáä~ê= íç= ÑáñáåÖ= íÜÉ= ÇÉÖêÉÉ= çÑ= íÜÉ= éçäóåçãá~äI= íÜÉ= åìãÄÉê= çÑ= ÅçÉÑÑáÅáÉåíë= áå= íÜÉ= cçìêáÉê= Éñé~åëáçåI= çê= íÜÉ= ~êÅÜáíÉÅíìêÉ= çÑ= íÜÉ= åÉìê~ä= åÉíJ ïçêâK=qÜáë=ÅÜçáÅÉ=áë=~=ê~íÜÉê=ÇáÑÑáÅìäí=çåÉ=íç=ã~âÉK= táíÜáå=íÜÉ=d~ìëëá~å=éêçÅÉëë=EdmF=Ñê~ãÉïçêâ=íÜÉ=éêÉîáçìë=íïç=ëíÉéë=~êÉ=ìåáÑáÉÇW=ïÉ= ëéÉÅáÑó= éêáçêë= çîÉê= íÜÉ= ÑìåÅíáçåë= íÜÉãëÉäîÉëK= qÜáë= äÉ~Çë= íç= ~= åçåJé~ê~ãÉíêáÅ= ëÉíìé= ~åÇ= ÖáîÉë= ãçêÉ= ÑêÉÉÇçã= áå= ã~âáåÖ= éêÉÇáÅíáçåëK= qÜÉ= ~Çî~åí~ÖÉ= áë= íÜ~í= çÑíÉå= Ó= áÑ= íÜÉ= éêáçê= áë= éêçéÉêäó=ÅÜçëÉå=Ó=íÜÉ=ÅçãéäÉñáíó=çÑ=íÜÉ=êÉëìäíáåÖ=dm=áë=ÇêáîÉå=~ìíçã~íáÅ~ääó=Äó=íÜÉ=Ç~í~= ïáíÜçìí=åÉÉÇ=Ñçê=ìëÉê=áåíÉêîÉåíáçåK=qÉÅÜåáÅ~ääóI=dmÛë=Å~å=ÄÉ=îáÉïÉÇ=~ë=~ëëáÖåáåÖ=~=àçáåí= d~ìëëá~å= ÇáëíêáÄìíáçå= Ñçê= ÉîÉêó= ÑáåáíÉ= ÅçääÉÅíáçå= çÑ= ÑìåÅíáçå= î~äìÉë= Em~éçìäáë= ~åÇ= máää~á= OMMOFK= fÑ= ïÉ= ~ëëìãÉ= áåéìí= äçÅ~íáçåë X

f X = [ f ( x1 ),K, f ( xN ) ]

T

=áëW=

p0 ( f X ) =

ïÜÉêÉ=

K N = { K 0 ( xi , x j )}

= { x1 , K , xN } I= íÜÉå= íÜÉ= àçáåí= éêçÄ~Äáäáíó= çÑ=

1

(

N

i , j =1

2π

)

N 2

KN

1 2

⎡ 1 ⎤ exp ⎢ − f XT K N−1 f X ⎥ 2 ⎣ ⎦

(4)

=áë=íÜÉ=âÉêåÉä=ÑìåÅíáçåI=ïÜáÅÜ=~Åíë=~ë=íÜÉ=ã~áå=é~ê~ãÉíÉê=

íç=íÜÉ=d~ìëëá~å=éêçÅÉëë=pÅÜçÉäâçéÑ=~åÇ=pãçä~=OMMOK= qç=çÄí~áå=íÜÉ=éêÉÇáÅíáîÉ=éêçÄ~Äáäáíó=~í=~=íÉëí=äçÅ~íáçå= x* I=çåÉ=Ñáêëí=ÅçåëáÇÉêë=íÜÉ=àçáåí= ÇáëíêáÄìíáçå=çÑ=íÜÉ=ÑìåÅíáçå=î~äìÉë=~í=íÜÉ=áåéìíë=~åÇ=íÜÉ=íÉëí=éçáåíI=ÄìáäÇë=íÜÉ=àçáåí=éçëíÉJ êáçê=çÑ=íÜÉëÉ=î~äìÉë=ìëáåÖ=_~óÉëÛ=êìäÉI==~åÇ==Ñáå~ääó=áåíÉÖê~íÉë=çìí=íÜÉ=ÑìåÅíáçå=î~äìÉë=~í=íÜÉ==

15-4

FAST SPATIAL INTERPOLATION ARTICLES

íê~áåáåÖ=äçÅ~íáçåë=Ó=ïÜáÅÜ=~êÉ=ê~åÇçã=î~êá~ÄäÉë=Ó=íç=~êêáîÉ=~í=íÜÉ=éêçÄ~Äáäáíó=çÑ=íÜÉ=ÑìåÅJ íáçå= î~äìÉ= ~í= ÇáëíêáÄìíáçå=çÑ=

x* K= páåÅÉ= ÄçíÜ= íÜÉ= éêáçê= ~åÇ= íÜÉ= äáâÉäáÜççÇ= áë= d~ìëëá~åI= íÜÉ= éêÉÇáÅíáîÉ= f* = f ( x* ) =áë=~=d~ìëëá~å=ïáíÜ=ãÉ~å=~åÇ=î~êá~åÅÉW= N

µ* = ∑ K 0 ( x* , xi ) α i i =1

(5)

N

σ = K 0 ( x* , x* ) − ∑ K 0 ( x* , xi ) cij K 0 ( x j , x* ) 2 *

i , j =1

(

ïáíÜ= a i =~åÇ= cij =ÄÉáåÖ=ëÅ~ä~ê=ÅçÉÑÑáÅáÉåíë=~åÇ=íÜÉó=~êÉ=ÅçãéìíÉÇ=~ë= α = σ 02 I N + K N

C = (σ 02 I N + K N )

)

−1

y I=

−1

ïÜÉêÉ= ïÉ= ìëÉ= ìåÇÉêäáåáåÖ= íç= ÇÉÑáåÉ= ~= îÉÅíçê= ~åÇ= Å~éáí~ä= äÉííÉêë= Ñçê= = ã~íêáñ=èì~åíáíáÉëK=tÉ=ãÉåíáçå=íÜ~í=íÜÉ=~ÄçîÉ=Éèì~íáçåë=~êÉ=ëáãéäó=~=ÇáÑÑÉêÉåí=ÉñéêÉëëáçå= çÑ=íÜÉ=ÄÉëí=äáåÉ~ê=éêÉÇáÅíçê=Ñçê=íÜÉ=ãÉ~å=~åÇ=î~êá~åÅÉ=ïáíÜáå=íÜÉ=hêáÖáåÖ=Ñê~ãÉïçêâI=~ë= ìëÉÇ=Äó=ÖÉçëí~íáëíáÅá~åë=E`êÉëëáÉ=NVVP=ééK=NMRJNOMFK= ^ë=áí=ï~ë=ãÉåíáçåÉÇI=íÜÉ=ÅÜçáÅÉ=çÑ=íÜÉ=âÉêåÉä=ÑìåÅíáçå=êÉéä~ÅÉë=íÜÉ=ÅÜçáÅÉ=çÑ=íÜÉ=ÑìåÅJ íáçå= Ñ~ãáäóK= ^= Åçããçåäó= ìëÉÇ= âÉêåÉä= áë= íÜÉ= ëèì~êÉÇJÉñéçåÉåíá~ä= âÉêåÉä= ïÜáÅÜ= Ü~ë= íÜÉ= ÑçääçïáåÖ=ÑçêãW= 2⎞ ⎛ 1 K 0 ( x, x ') = A exp ⎜ − 2 x − x ' ⎟ ⎝ 2θ ⎠

ïÜÉêÉ= ÜóéÉêJé~ê~ãÉíÉêë=

(6)

( A, θ ) = ÅÜ~ê~ÅíÉêáëÉ= íÜÉ= êÉëìäíáåÖ= éêçÅÉëë= ~åÇ= íÜÉó= Ü~îÉ= íç= ÄÉ=

ÅÜçëÉå=íç=éêçîáÇÉ=“ÄÉëí=ÑáíÒ=íç=íÜÉ=Ç~í~K=qÜÉ=~ãéäáíìÇÉI=áKÉK=íÜÉ=íóéáÅ~ä=ê~åÖÉ=çÑ=î~êá~íáçå=

áå=íÜÉ=çìíéìí=î~äìÉë=áë=ëéÉÅáÑáÉÇ=Äó A =~åÇ= θ =ëéÉÅáÑáÉë=íÜÉ=äÉåÖíÜJëÅ~äÉ=çÑ=íÜÉ=éêçÅÉëë=çê= íÜÉ= Çáëí~åÅÉ= íç= ïÜáÅÜ= íïç= çÄëÉêî~íáçåë= ïçìäÇ= áåÑäìÉåÅÉ= É~ÅÜ= çíÜÉêK= få= íÜáë= ~êíáÅäÉ= ïÉ== Éãéäçó= íÜÉ= ã~ñáãìã= äáâÉäáÜççÇ= EjiffF= Ñê~ãÉïçêâ= íç= ëÉí= íÜÉ= âÉêåÉä= é~ê~ãÉíÉêë= Ej~ÅJ h~ó= NVVOX= j~Åh~ó= NVVVFK= qÜÉ= éêÉëÉåí~íáçå= çÑ= íÜÉ= ~ìíçã~íÉÇ= ÅÜçáÅÉ= çÑ= íÜÉ= âÉêåÉä= é~ê~ãÉíÉêë=áë=åçí=íÜÉ=~áã=çÑ=íÜáë=ëìÄãáëëáçåX=ÜÉêÉ=ïÉ=çåäó=ÖáîÉ=~å=áåíìáíáîÉ=éáÅíìêÉ=Ó=çåÉ= Å~å=Åçåëìäí=EÉKÖKF=táääá~ãë=~åÇ=o~ëãìëëÉå=ENVVSFI=dáÄÄë=~åÇ=j~Åh~ó=ENVVTF=Ñçê=ÇÉí~áäëK= få= íÜÉ= jiff= Ñê~ãÉïçêâ= íÜÉ= EäçÖJF= çÑ= íÜÉ= ÉîáÇÉåÅÉ= áå= Éèì~íáçå= EPF= áë= ìëÉÇ= íç= ÑáåÇ= íÜÉ= çéíáã~ä=âÉêåÉä=é~ê~ãÉíÉêëK=qç=ëÉÉ=íÜáëI=äÉí=ìë=ÅçåëáÇÉê= θ I=íÜÉ=äÉåÖíÜJëÅ~äÉ=çÑ=íÜÉ=âÉêåÉä= ÑìåÅíáçåK= táíÜ= ~= ÑáñÉÇ= äÉåÖíÜJëÅ~äÉ= íÜÉ= Åçãéìí~íáçå= çÑ= íÜÉ= ÉîáÇÉåÅÉ= áë= ~= Åçåîçäìíáçå= çÑ= d~ìëëá~åëI= íÜìë= ~å~äóíáÅ~ääó= Åçãéìí~ÄäÉK= e~îáåÖ= ~å= ~å~äóíáÅ= ÉñéêÉëëáçå= Ñçê= íÜÉ= EäçÖJF== ÉîáÇÉåÅÉ= P ( D | θ ) I=ïÉ=ã~ñáãáëÉ=áí=ïáíÜ=êÉëéÉÅí=íç= θ K=få=jiff=ïÉ=ÅçåëáÇÉê=íÜÉ=ÉîáÇÉåÅÉ=

íÉêã= ~ë= íÜÉ= “äáâÉäáÜççÇÒ= íç= íÜÉ= ÜóéÉêJé~ê~ãÉíÉêë= ~åÇ= ïÉ= éÉêÑçêã= ~= ã~ñáãìã= ~J éçëíÉêáçêá= é~ê~ãÉíÉê= çéíáãáë~íáçå= Äó= ~ÇÇáåÖ= ~= éêáçê= ~åÇ= ÇáÑÑÉêÉåíá~íáåÖ= íÜÉ= êÉëìäíáåÖ== ÉñéêÉëëáçåK=

FAST SPATIAL INTERPOLATION ARTICLES

15-5

Figure 1 The sparse GP approximation to the posterior process: with conventional GP the approximated posterior mean function is a linear sum of RBF functions at all training points (x signs), with SSGP the sum is relative to the BV set (triangles) and the sum of the 6 RBF functions leads to an accurate approximation of the function – the dashed line on the top is the approximation provided by the posterior mean.

rëáåÖ= ~= ÑáñÉÇ= ëÉí= çÑ= ÜóéÉêJé~ê~ãÉíÉêë= ïÉ= ÅçãéìíÉ= íÜÉ= éçëíÉêáçê= éêçÅÉëëK= låÅÉ= ÅçãJ éìíÉÇI= ïÉ= êÉ~Çáäó= Ü~îÉ= Éëíáã~íÉë= Ñçê= ÄçíÜ= íÜÉ= ÑìåÅíáçå= áíëÉäÑ= ~åÇ= íÜÉ= ìåÅÉêí~áåíó= áå= áíë= î~äìÉ=~í=~=ÖáîÉå=éçáåíK=qÜÉ=éêÉÇáÅíáîÉ=ãÉ~å=áë=ìëÉÇ=áå=~=ã~ñáãìã=~JéçëíÉêáçêá=ëÉííáåÖ=íç= Éëíáã~íÉ=íÜÉ=ìåâåçïå=ÑìåÅíáçåI=áKÉK=í~âáåÖ=íÜÉ=ãÉ~å=çÑ=íÜÉ=éçëíÉêáçê=ÇáëíêáÄìíáçå=~ë= íÜÉ= ~ééêçñáã~íáçå=íç=íÜÉ=ÑìåÅíáçå=EdáÄÄë=~åÇ=j~Åh~ó=NVVTFK=tÉ=ÜáÖÜäáÖÜí=íÜÉ=ã~áå=Çê~ïJ Ä~Åâ= çÑ= íÜÉ= dm= áåÑÉêÉåÅÉW= ~äíÜçìÖÜ= ~å~äóíáÅ~ääó= ~ééÉ~äáåÖI= áí= Å~ååçí= ÄÉ= ìëÉÇ= çå= ä~êÖÉ= Ç~í~ëÉíë=ÇìÉ=íç=íÜÉ=ÜáÖÜ=Åçãéìí~íáçå~ä=Åçëí=çÑ=íÜÉ=ã~íêáñ=áåîÉêëáçåK= få=êÉÅÉåí=óÉ~êë=íÜÉêÉ=Ü~ë=ÄÉÉå=~å=áåÅêÉ~ëÉÇ=çÑ=áåíÉêÉëí=áå=íÉÅÜåáèìÉë=Ñçê=~ééêçñáã~íáçå= íÜ~í=í~âÉ=~Çî~åí~ÖÉ=çÑ=íÜÉ=ÅçåÅÉéí=çÑ=ëé~êëáíó=Etáääá~ãë=~åÇ=pÉÉÖÉê=OMMNX=`ë~íµ=~åÇ=léJ éÉê= OMMOFK= få= íÜÉ= ëÉèìÉåíá~ä= ëé~êëÉ= ~ééêçñáã~íáçå= íç= d~ìëëá~å= éêçÅÉëëÉë= ïÉ= éêçîáÇÉ= ~= éêáåÅáéäÉÇ=~ééêç~ÅÜ=íç=~ééêçñáã~íÉ=íÜÉ=dm=éçëíÉêáçêK=qÜÉ=~ééêçñáã~íáçå=áë=ëìÅÜ=íÜ~í=áå= íÜÉ=ÉåÇI=Ñçê=éêÉÇáÅíáåÖI=ïÉ=Çç=åçí=êÉèìáêÉ=íÜÉ=ïÜçäÉ=ëÉíI=ê~íÜÉê=ïÉ=çåäó=êÉèìáêÉ=~=“ëé~êëÉÒ= ëìÄëÉí=íÜÉêÉçÑK= få=ëé~êëÉ=~ééêçñáã~íáçå=íç=íÜÉ=éçëíÉêáçê=ïÉ=~áã=~í=“ãáåáãáëáåÖÒ=íÜÉ=Åçëí=çÑ=ëíçêáåÖ= íÜÉ=dm=áå=ÅçãéìíÉê=ãÉãçêó=~åÇ=áãéäáÅáíäó=ÇÉÅêÉ~ëáåÖ=íÜÉ=Åçãéìí~íáçå=íáãÉK=qç=~ÅÜáÉîÉ= áíI=ïÉ=~ÇÇ=Éñ~ãéäÉë=íç=íÜÉ=éçëíÉêáçê=dm=çåÉ=Äó=çåÉ=ìëáåÖ=_~óÉëÛ=êìäÉ=áå=~=ëÉèìÉåíá~ä=ã~åJ åÉêK=^í=É~ÅÜ=ëíÉé=ïÉ=ìëÉ=íÜÉ=éçëíÉêáçê=Ñêçã=íÜÉ=éêÉÅÉÇáåÖ=áíÉê~íáçå=~ë=íÜÉ=ÅìêêÉåí=éêáçêK= qÜÉ=ÇÉÅêÉ~ëÉ=áå=Åçãéìí~íáçå=íáãÉ=áë=~ÅÜáÉîÉÇ=ïÜÉå=ïÉ=êÉãçîÉ=~å=áåéìí=äçÅ~íáçå=Ñêçã=íÜÉ= éçëíÉêáçê=Ó=íÜÉ=êÉãçî~ä=ÄÉáåÖ=ÇçåÉ=ëìÅÜ=íÜ~í=íÜÉ=ÇáÑÑÉêÉåÅÉ=ÄÉíïÉÉå=íÜÉ=ÅìêêÉåí=éçëíÉêáçê= ~åÇ=áíë=íêáããÉÇ=~ééêçñáã~íáçå=áë=ãáåáã~äK=tÉ=í~âÉ=áåíç=ÅçåëáÇÉê~íáçå=íÜÉ=éêçÄ~ÄáäáëíáÅ= å~íìêÉ=çÑ=íÜÉ=_~óÉëá~å=áåÑÉêÉåÅÉI=áKÉK=íÜ~í=ïÉ=Ü~îÉ=~ÅÅÉëë=íç=íÜÉ=éçëíÉêáçêë=ÇáëíêáÄìíáçå=çÑ= íÜÉ= ä~íÉåíëI= ÅçåëÉèìÉåíäó= ïÉ= ãáåáãáëÉ= áå= íÜÉ= hìääÄ~ÅâJiÉáÄäÉê= EhiF= ÇáîÉêÖÉåÅÉ= ÄÉíïÉÉå= íÜÉ=íïç=ÇáëíêáÄìíáçåë=E`ë~íµ=OMMOX=`çîÉê=~åÇ=qÜçã~ë=NVVNFK=qÜáë=ÇáîÉêÖÉåÅÉ=ãÉ~ëìêÉë= íÜÉ=ëáãáä~êáíó=çÑ=íïç=éêçÄ~Äáäáíó=ÇáëíêáÄìíáçåë=~åÇ=Å~å=ÄÉ=íÜçìÖÜí=çÑ=~ë=àçáåíäó=ã~íÅÜáåÖ= íÜÉ=Ñáêëí=íïç=ãçãÉåíë=çÑ=íÜÉ=d~ìëëá~åëK=

15-6

FAST SPATIAL INTERPOLATION ARTICLES

lÑíÉå=çåÉ=Å~å=êÉãçîÉI=ïÜáäëí=éÉêÑçêãáåÖ=íÜÉ=ëÉèìÉåíá~ä=ìéÇ~íÉI=~=ä~êÖÉ=éêçéçêíáçå=çÑ= íÜÉ=áåéìí=äçÅ~íáçåë=~åÇ=~í=íÜÉ=ÉåÇ=çÑ=íÜáë=ãçÇáÑáÉÇ=äÉ~êåáåÖ=íÜÉêÉ=áë=~=ëã~ää=ëìÄëÉí=äÉÑí=íç= êÉéêÉëÉåí=íÜÉ=~ééêçñáã~íÉÇ=dmI=ïÉ=Å~ää=íÜáë=ëìÄëÉí=íÜÉ=ëÉí=çÑ=“Ä~ëáë=îÉÅíçêëÒI=çê=_s=ëÉí= ~åÇ=ïÉ=Ü~îÉ=Å~ääÉÇ=íÜÉ=éêçÅÉÇìêÉ=ppdm=Ó=ëÉèìÉåíá~ä=ëé~êëÉ=dm=~äÖçêáíÜãK=qÜÉ=êÉëìäíáåÖ= ~ééêçñáã~íáçå= ìëÉë= çåäó= íÜÉ= _s= ëÉí= áå= ÅçãéìíáåÖ= ÄçíÜ= íÜÉ= éçëíÉêáçê= ãÉ~å= ~åÇ= íÜÉ= EÅçFî~êá~åÅÉK=tÉ=éêçîáÇÉ=~å=áääìëíê~íáçå=áå=cáÖìêÉ=N=ïÜÉêÉ=ïÉ=Å~å=ëÉÉ=íÜÉ=ëé~êëÉ=dm=ÉëíáJ ã~íÉ=çÑ=íÜÉ=ãÉ~å=ÑìåÅíáçå=íçÖÉíÜÉê=ïáíÜ=íÜÉ=íê~áåáåÖ=ëÉí=Eñ=ëáÖåëF=~åÇ=íÜÉ=äçÅ~íáçå=çÑ=íÜÉ= _sÛë=Eíêá~åÖäÉëFK= qÜÉ=ëáÖåáÑáÅ~åí=ÇáÑÑÉêÉåÅÉ=íÜìë=ÄÉíïÉÉå=ÅçåîÉåíáçå~ä=d~ìëëá~å=éêçÅÉëëÉë=~åÇ=íÜÉ=ppdm= ~äÖçêáíÜã= áë= íÜÉ= åÉÉÇ= Ñçê= ä~êÖÉ= ã~íêáñ= áåîÉêëáçå= çéÉê~íáçåëW= íÜÉ= ppdm= ~äÖçêáíÜã= ëÅ~äÉë= ÅìÄáÅ~ääó= çåäó= ïáíÜ= íÜÉ= ëáòÉ= çÑ= íÜÉ= ëã~ääÉê= _s= ëÉíK= páåÅÉ= çåÉ= çÑ= íÜÉ= ã~áå= çÄàÉÅíáîÉë= çÑ= pf`OMMQ=áë=Ñ~ëí=áåíÉêéçä~íáçåI=ïÉ=íÜáåâ=íÜ~í=ppdm=áë=~å=~ííê~ÅíáîÉ=Éëíáã~íáçå=ãÉíÜçÇK= tÉ=åÉñí=ëìãã~êáëÉ=íÜÉ=ã~áå=ëíÉéë=çÑ=íÜÉ=ppdm=~äÖçêáíÜãI=Ñçê=ãçêÉ=áåÑçêã~íáçå=çåÉ= Å~å= Åçåëìäí= E`ë~íµ= OMMOFK= qÜÉ= ~äÖçêáíÜã= Åçåíêçäë= íÜÉ= “åçîÉäíóÒ= ÅçêêÉëéçåÇáåÖ= íç= É~ÅÜ= åÉï=áåéìí=äçÅ~íáçå=Äó=ÅçãéìíáåÖ=íÜÉ=äçëë=ÄÉíïÉÉå=íÜÉ=dm=~ÑíÉê=~ÇÇáåÖ=íÜÉ=çÄëÉêî~íáçå=íç= íÜÉ=dm=Ó=îá~=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=Ó=~åÇ=~=ëÉÅçåÇ=dm=çÄí~áåÉÇ=Ñêçã=íÜÉ=éêÉîáçìë=éçëíÉJ êáçê= ïÜÉå= çåäó= íÜÉ= äçÅ~íáçå= ÅçêêÉëéçåÇáåÖ= íç= íÜÉ= åÉï= áåéìí= Ü~ë= ÄÉÉå= êÉãçîÉÇI= íÜÉ= “Çáëí~åÅÉÒ= ãÉ~ëìêÉ= áë= íÜÉ= hìääÄ~ÅâJiÉáÄäÉê= ÇáîÉêÖÉåÅÉ= E`çîÉê= ~åÇ= qÜçã~ë= NVVNF= ÄÉJ íïÉÉå= íÜÉ= d~ìëëá~å= ÇáëíêáÄìíáçåë= ÇÉêáîÉÇ= Ñêçã= íÜÉ= êÉëéÉÅíáîÉ= dmÛë= í~âÉå= ~í= íÜÉ= áåéìí= äçÅ~íáçåëK= tÉ= ïáää= Å~ää= íÜáë= èì~åíáíó= íÜÉ= “ëÅçêÉÒ= ~ëëçÅá~íÉÇ= íç= ~= ëéÉÅáÑáÅ= áåéìí= ~åÇ= íÜáë= ëÅçêÉ= Å~å= ÄÉ= ÅçãéìíÉÇ= áêêÉëéÉÅíáîÉ= çÑ= íÜÉ= çêÇÉê= çÑ= áåéìí= ~ÇÇáíáçåK= få= íÜÉ= ~äÖçêáíÜã= çåÉ= Ü~ë=íç=ëÉí=~=íÜêÉëÜçäÇ=î~äìÉ= ε

( e.g. = 10−3 ) =~åÇ=~í=íÜÉ=ÉåÇ=çÑ=ÉîÉêó=ëíÉé=íÜÉ=ëÅçêÉë=~ëëçJ

Åá~íÉÇ= íç= íÜÉ= áåéìíë= áå= íÜÉ= _s= ëÉí= ~êÉ= êÉÅçãéìíÉÇK= fÑ= ~åó= çÑ= íÜÉ= ëÅçêÉë= Ñ~ääë= ÄÉäçï= íÜÉ= íÜêÉëÜçäÇ=î~äìÉI=íÜÉå=íÜÉ=êÉëéÉÅíáîÉ=_s=áë=êÉãçîÉÇ=Ñêçã=íÜÉ=_s=ëÉí=EëáåÅÉ=íÜÉ=~äÖçêáíÜã=áë= ëÉèìÉåíá~äI=áÑ=Äó=ãáëí~âÉ=~=_s=áë=êÉãçîÉÇ=Ñêçã=~å=áãéçêí~åí=êÉÖáçå=áå=íÜÉ=áåéìí=ëé~ÅÉI=~í= ~=ä~íÉê=çÅÅ~ëáçå=_sÛë=ïáää=~ééÉ~ê=ïÜáÅÜ=ïáää=Ñáää=áå=íÜÉ=êÉëéÉÅíáîÉ= êÉÖáçåFK=tÜÉå=~å=ÉëíáJ ã~íÉ= çÑ= íÜÉ= éçëíÉêáçê= dm= áë= ~î~áä~ÄäÉI= çåÉ= éêçÅÉÉÇë= íç= íÜÉ= Éëíáã~íáçå= çÑ= íÜÉ= ÜóéÉêJ é~ê~ãÉíÉêë=íç=íÜÉ=ãçÇÉä=Äó=çéíáãáëáåÖ=íÜÉ=äçÖJÉîáÇÉåÅÉ=çÑ=íÜÉ=Ç~í~K=qÜÉêÉ=áë=åç=~å~äóíáÅ= ëçäìíáçå=~åÇ=ïÉ=Éãéäçó=~=ëÅ~äÉÇ=ÅçåàìÖ~íÉ=Öê~ÇáÉåí=ãÉíÜçÇ=íç=ÑáåÇ=íÜÉ=ÅçêêÉÅí=é~ê~ãÉJ íÉêëK=tÜÉå=íÜÉ=é~ê~ãÉíÉêë=~êÉ=ÑçìåÇI=íÜÉ=ppdm=~äÖçêáíÜã=áë=êÉêìå=ìëáåÖ=íÜÉ=åÉï=ÜóéÉêJ é~ê~ãÉíÉê= î~äìÉëK= tÉ= çÄëÉêîÉÇ= Üçï= íÜÉ= ÜóéÉêJé~ê~ãÉíÉêë= î~êáÉÇ= ~ë= ~= ÑìåÅíáçå= çÑ= íÜÉ= åìãÄÉê= çÑ= êÉJÉëíáã~íáçå= ÅóÅäÉëK= qÜáë= áë= íç= ëÉÉ= Üçï= èìáÅâäó= íÜÉ= ÜóéÉêJé~ê~ãÉíÉêë= ÅçåJ îÉêÖÉÇK= tÉ= ìëÉÇ= íÜÉëÉ= éäçíë= íç= ÇÉÅáÇÉ= íÜÉ= åìãÄÉê= çÑ= íáãÉë= ïÉ= ëÜçìäÇ= ìëÉ= íÜÉ= ppdm= ~äÖçêáíÜãK=tÉ=ÇÉÅáÇÉÇ=íç=ìëÉ=S=ÅóÅäÉëI=É~ÅÜ=ÅóÅäÉ=íÜêçìÖÜ=íÜÉ=~äÖçêáíÜã=í~âáåÖ=~ééêçñáJ ã~íÉäó=PM=ëÉÅçåÇëK= 2.1 USE OF PRIOR INFORMATION

qÜÉ= éêáçê= áåÑçêã~íáçå= ÖáîÉå= íç= íÜÉ= é~êíáÅáé~åíë= çÑ= íÜÉ= “pé~íá~ä= fåíÉêéçä~íáçå= `çãé~êáJ ëçåÒ=ÉñÉêÅáëÉ=ï~ë=NM=Ç~í~ëÉíë=ë~ãéäÉÇ=~í=ëéÉÅáÑáÉÇ=äçÅ~íáçåë=~åÇ=~í=NM=ÇáëíáåÅí=íáãÉëI=~í= É~ÅÜ=íáãÉ=íÜÉêÉ=ïÉêÉ=OMM=Ç~í~=éçáåíë ( xi , yi ) =ë~ãéäÉÇI=íÜìë=íÜÉ=Ñìää=éêáçê=Ç~í~ëÉí= D Ü~ë= ÄÉÉå=ÖáîÉå=ëÉé~ê~íÉÇ=áåíç=NM=é~êíë=~ë

D = {D1 , K, D10 } K=páåÅÉ=íÜÉêÉ=ï~ë=åç=ÑìêíÜÉê=áåJ

Ñçêã~íáçå=íÜ~í=ïçìäÇ=ÅçååÉÅí=íÜÉ=áåÇáîáÇì~ä=ëÉíëI=ïÉ=~ëëìãÉÇ=íÜ~í=íÜÉó=~êÉ=áåÇÉéÉåÇÉåí= Äìí=íÜÉ=ë~ãéäÉë=~Åêçëë=É~ÅÜ=áåÇáîáÇì~ä=ëÉí=~êÉ=ÅçååÉÅíÉÇ=áå=íÜ~í=íÜÉó=ïÉêÉ=çÄëÉêîÉÇ=~í=íÜÉ= ë~ãÉ=íáãÉK=

FAST SPATIAL INTERPOLATION ARTICLES

15-7

_ÉÑçêÉ=ïÉ=éÉêÑçêãÉÇ=~åó=Å~äÅìä~íáçåëI=ïÉ=åçêã~äáëÉÇ=íÜÉ=áåéìíë=~åÇ=çìíéìíë=íç=òÉêç= ãÉ~å= ~åÇ= ìåáí= î~êá~åÅÉK= eçïÉîÉêI= áå= çêÇÉê= íç= ã~áåí~áå= íÜÉ= ÇáêÉÅíáçå~ä= ëíêìÅíìêÉ= çÑ= íÜÉ= Ç~í~I=ïÉ=Å~äÅìä~íÉÇ=íÜÉ=î~êá~åÅÉ=ÖäçÄ~ääóK=tÜÉå=ëÜçïáåÖ=íÜÉ=êÉëìäíëI=íÜÉ=åçêã~äáëÉÇ=Ç~í~= áë=íê~åëÑçêãÉÇ=~åÇ=êÉëÅ~äÉÇ=Ä~Åâ=íç=íÜÉ=çêáÖáå~ä=ÅççêÇáå~íÉëK=

Figure 2 Variogram of the prior data, plotted for 20 lags and the different fits to the empirical one. It can be seen that the variogram corresponding to the RBF provides bad estimates at the origin and the mixture kernel has a more accurate fit to the experimental variogram.

qç=Éñ~ãáåÉ=íÜÉ=ëíêìÅíìêÉ=çÑ=íÜÉ=Ç~í~I=ïÉ=éäçííÉÇ=íÜÉ=î~êáçÖê~ã=Ñçê=É~ÅÜ=çÑ=íÜÉ=éêáçê= Ç~í~ëÉíëI= ÇÉéáÅíáåÖ= íÜÉ= î~êá~íáçå= ÄÉíïÉÉå= çÄëÉêî~íáçåë= ~í= áåÅêÉ~ëáåÖ= ä~Ö= ëÉé~ê~íáçå= ÇáëJ í~åÅÉë=çå=cáÖìêÉ=OW=íÜÉ=ÅäçëÉê=éçáåíë=áå=íÜÉ=áåéìí=ëé~ÅÉ=~êÉ=EuJ~ñáëFI=íÜÉ=ãçêÉ=äáâÉäó=íÜÉ= î~äìÉë= çÑ= íÜÉ= çÄëÉêîÉÇ= ê~Çá~íáçå= ~êÉ= íÜÉ= ë~ãÉ= EëÉãá= î~êá~åÅÉ= çå= íÜÉ= vJ~ñáëFI= íÜáë= ÑìåÅJ íáçå~ä= êÉä~íáçå= áë= ÇÉéáÅíÉÇ= áå= íÜÉ= î~êáçÖê~ãK= tÉ= Ñáêëí= Å~äÅìä~íÉÇ= íÜÉ= ÉñéÉêáãÉåí~ä= î~êáçÖê~ã=Ñêçã=íÜÉ=Ç~í~=EÅêçëëÉëFK=tÉ=Éëíáã~íÉÇ=íÜÉ=î~êáçÖê~ã=Ñçê=É~ÅÜ=çÑ=íÜÉ=Ç~óë=áåÇáJ îáÇì~ääóK= iççâáåÖ= ~í= íÜÉ= ÉãéáêáÅ~ä= î~êáçÖê~ã= ïÉ= ÑáííÉÇ= ëÉîÉê~ä= î~êáçÖê~ã= ãçÇÉäëI= ~= ëèì~êÉÇ= ÉñéçåÉåíá~äI= çê= o_cI= Ñêçã= Éèì~íáçå= ESF= ~åÇ= ~= ëáãéäÉ= ÉñéçåÉåíá~ä= Ü~îáåÖ= íÜÉ= ë~ãÉ=Ñçêã=~ë=íÜÉ=o_c=ÉñÅÉéí=íÜ~í=íÜÉ=áåëíÉ~Ç=çÑ=íÜÉ=ëèì~êÉÇ=Çáëí~åÅÉ=áí=ìëÉë=íÜÉ=~ÄëçäìíÉ= î~äìÉ=áå=íÜÉ=ÉñéçåÉåíá~äK=qÜÉëÉ=ãçÇÉäë=ÑáííÉÇ=íÜÉ=Ç~í~=ïÉää=áå=ÅÉêí~áå=êÉÖáçåëI=~ë=ïÉ=Å~å= ëÉÉK=tÉ=ìëÉÇ=íÜÉ=î~êáçÖê~ã=~ë=~=ãÉ~åë=íç=~ëëáëí=áå=íÜÉ=ãçëí=~ééêçéêá~íÉ=ÅÜçáÅÉ=çÑ=âÉêåÉä= ÑìåÅíáçå=Ñçê=íÜÉ=ppdm=~äÖçêáíÜãK=tÉ=ÇÉÅáÇÉÇ=íÜ~í=~=ãáñíìêÉ=çÑ=âÉêåÉä=ÑìåÅíáçåë=áë=~ééêçJ éêá~íÉ= íç= ãçÇÉä= íÜÉ= ê~Çá~íáçå= Ç~í~K= få= ~ÇÇáíáçå= íç= íÜáëI= áåëéÉÅíáåÖ= íÜÉ= äçÅ~íáçåë= çÑ= íÜÉ= éêáçê=Ç~í~I=áí=ÅçìäÇ=ÄÉ=ëÉÉå=íÜ~í=íÜÉêÉ=ïÉêÉ=áåëìÑÑáÅáÉåí=Ç~í~=éçáåíë=~í=ëÜçêí=áåíÉêî~äë=Ñçê= çìê=ãçÇÉä=íç=äÉ~êå=íÜÉ=ëíêìÅíìêÉ=~í=ëÜçêí=ê~åÖÉë=~ÅÅìê~íÉäóK=_~ëÉÇ=çå=ëíìÇáÉë=~Äçìí=Üçï= ~ãÄáÉåí= ê~Çá~íáçå= ÇáëéÉêëÉë= íÜêçìÖÜ= íÜÉ= ~íãçëéÜÉêÉ= Ee~ää= NVUQX= rf`= jÉäÄçêåÉ= OMMQFI= ïÉ=ÇÉÅáÇÉÇ=~í=íÜÉ=ëÜçêí=ê~åÖÉ=äÉîÉä=íÜ~í=~=ëèì~êÉÇJÉñéçåÉåíá~ä=ãçÇÉä=ïçìäÇ=ÄÉ=~ééêçéêáJ ~íÉK= cçê= äçåÖÉê= ê~åÖÉë= ~å= ÉñéçåÉåíá~ä= ãçÇÉä= ïçìäÇ= Ñáí= íÜÉ= Ç~í~= ÄÉëíK= tÉ= ÉãéäçóÉÇ= ~= äáåÉ~ê=ÅçãÄáå~íáçå=çÑ=íÜÉëÉ=íïç=âÉêåÉä=ÑìåÅíáçåë=É~ÅÜ=ïáíÜ=ÇáÑÑÉêÉåí=ê~åÖÉ=~åÇ=~ãéäáíìÇÉW=

⎡ 1 2 ( x − x ') 2 ⎤ ⎡ 1 2 x −x '⎤ K mix ( x, x ') = π A1 exp ⎢ − ∑ i 2 i ⎥ + (1 − π ) A2 exp ⎢ − ∑ i 2 i ⎥ (7) ⎢⎣ 2 i =1 θ1, j ⎥⎦ ⎢⎣ 2 i =1 θ 2, j ⎥⎦

15-8

FAST SPATIAL INTERPOLATION ARTICLES

π = íÜÉ= ãáñáåÖ= ê~íáç= çÑ= íÜÉ= ëèì~êÉÇ= ÉñéçåÉåíá~ä= âÉêåÉä= ïáíÜ= é~ê~ãÉíÉêë= ( A1 ,θ1,1 ,θ1,2 ) = ~åÇ= íÜ~í= çÑ= íÜÉ= ÉñéçåÉåíá~ä= çåÉ= ïáíÜ= é~ê~ãÉíÉêë= ( A2 ,θ2,1 ,θ2,2 ) K= ^äJ

ïáíÜ=

íÜçìÖÜ=ïÉ=~êÉ=åçí=ìëáåÖ=íÜÉ=î~êáçÖê~ã=íç=Éëíáã~íÉ=íÜÉ=áåáíá~ä=é~ê~ãÉíÉêë=Ñçê=íÜÉ=âÉêåÉäë= ïÉ=ìëÉ=áå=çìê=ãçÇÉäI=ÅÉêí~áå=çÄëÉêî~ÄäÉ=ÑÉ~íìêÉë=çÑ=íÜÉ=î~êáçÖê~ã=êÉä~íÉ=íç=íÜÉ=é~ê~ãÉíÉêë= çÑ=íÜÉ=âÉêåÉä=ÑìåÅíáçåëK=qÜÉ=é~ê~ãÉíÉê=^=êÉä~íÉë=íç=íÜÉ=~ãéäáíìÇÉ=çÑ=íÜÉ=âÉêåÉä=ÑìåÅíáçåI= ïÜáÅÜ=Éèì~íÉë=íç=íÜÉ=éä~íÉ~ì=íÜ~í=íÜÉ=î~êáçÖê~ã=êÉ~ÅÜÉëI=Å~ääÉÇ=íÜÉ=ëáääK=qÜÉ=é~ê~ãÉíÉêë=θ= êÉä~íÉ=íç=íÜÉ=äÉåÖíÜJëÅ~äÉ=ïÜáÅÜ=ÅçêêÉëéçåÇë=íç=íÜÉ=ê~åÖÉ=ÑÉ~íìêÉ=çÑ=íÜÉ=î~êáçÖê~ãW=íÜáë=áë= íÜÉ=éçáåí=ïÜÉêÉ=~å=áåÅêÉ~ëÉ=áå=ëÉé~ê~íáçå=Çáëí~åÅÉ=ÄÉíïÉÉå=é~áêë=åç=äçåÖÉê=Å~ìëÉë=~å=áåJ ÅêÉ~ëÉ= áå= íÜÉ= ~îÉê~ÖÉ= ëèì~êÉÇ= ÇáÑÑÉêÉåÅÉ= ÄÉíïÉÉå= íÜÉ= é~áêëK= lÄëÉêîÉ= íÜ~í= åçï= ïÉ= Ü~îÉ= ãçêÉ=é~ê~ãÉíÉêë=êÉä~íáåÖ=íç=íÜÉ=âÉêåÉä=ÑìåÅíáçåW=

( A ,θ 1

1,1

, θ1,2 , A2 , θ 2,1 ,θ 2,2 , π ) =~åÇ=çåÉ=

Ü~ë= íç= ëÉí= íÜÉëÉ= é~ê~ãÉíÉêë= ÄÉÑçêÉ= íÜÉ= çÄàÉÅíáîÉ= Ç~í~ëÉíë= ~êÉ= ëÉÉåI= áKÉK= Ä~ëÉÇ= çå= íÜÉ= NM= î~äáÇ~íáçå= ëÉíë= ~äçåÉK= tÉ= Ü~îÉ= ìëÉÇ= åçåJáëçíêçéáÅ= Åçî~êá~åÅÉ= ÑìåÅíáçåë= ëáåÅÉ= ïÉ= ÄÉäáÉîÉ= íÜ~í= íÜÉ= î~êá~íáçå= áå= íÜÉ= Ç~í~= áë= åçí= ìåáÑçêã= áå= ~ää= ÇáêÉÅíáçåëK= qÜÉ= áåÅêÉ~ëÉ= áå= Åçãéìí~J íáçå~ä= ÅçãéäÉñáíó= Äó= ìëáåÖ= åçåJáëçíêçéáÅ= Åçî~êá~åÅÉ= ÑìåÅíáçåë= áë= íÜ~í= ïÉ= åçï= Ü~îÉ= ~å= ~ÇÇáíáçå~ä=é~ê~ãÉíÉê=íç=çéíáãáëÉK=qÜÉ=çéíáãáë~íáçå=çÑ=íÜÉëÉ=é~ê~ãÉíÉêë=áë=ÇÉëÅêáÄÉÇ=áå=íÜÉ= åÉñí=ëÉÅíáçåK= 2.2 TUNING THE ALGORITHMS

qç=Éëíáã~íÉ=íÜÉ=ÜóéÉêJé~ê~ãÉíÉêë=íç=ìëÉ=ïáíÜ=íÜÉ=ÅçãéÉíáíáçå=Ç~í~I=ïÉ=íê~áåÉÇ=~=dm=Ñçê= É~ÅÜ=çÑ=íÜÉ=NM=Ç~í~ëÉíëX=íÜÉ=ppdm=~äÖçêáíÜã=ï~ë=ìëÉÇ=Ñçê=íê~áåáåÖK=páåÅÉ=íÜÉ=OMM=íê~áåáåÖ= éçáåíë= Ñçê= É~ÅÜ= Ç~í~ëÉí= ïÉêÉ= áåÇÉéÉåÇÉåí= ~åÇ= áÇÉåíáÅ~ääó= ÇáëíêáÄìíÉÇI= íÜÉ= éçëíÉêáçêë= çÄJ í~áåÉÇ= ~êÉ= ~äëç= áåÇÉéÉåÇÉåí= ÅçåÇáíáçåÉÇ= çå= íÜÉ= éêáçêëK= qÜÉ= çéíáãáë~íáçå= çÑ= ÜóéÉêJ é~ê~ãÉíÉêë= ìëÉÇ= ~ää= çÑ= íÜÉ= éçëíÉêáçê= dmÛë= áå= íÜÉ= Åçãéìí~íáçå= çÑ= íÜÉ= Öê~ÇáÉåíëK= tÉ= ~ëJ ëìãÉÇ=íÜ~í=íÜÉ=“íêìÉ=Öê~ÇáÉåíÒ=áë=~=äáåÉ~ê=ÅçãÄáå~íáçå=çÑ=áåÇáîáÇì~ä=Öê~ÇáÉåíë=~åÇ=ëáåÅÉ= íÜÉêÉ=ï~ë=åç=áåÑçêã~íáçå=~Äçìí=íÜÉ=áåÇáîáÇì~ä=Ç~í~ëÉíëI=ïÉ=~ëëìãÉÇ=~å=Éèì~ä=ïÉáÖÜí=íç= ~ää=çÑ=íÜÉãI=íÜìë=íÜÉ=Öê~ÇáÉåí=ï~ëW=

∂ log P ( D) 1 10 ∂ log P( Dk ) = ∑ (8) ∂s ∂s 10 k =1 ïÜÉêÉ=ë=áë=~=é~ê~ãÉíÉê=çÑ=íÜÉ=ãáñíìêÉ=âÉêåÉä=Ñêçã=Éèì~íáçå=ETF=ïÜáÅÜ=ïÉ=ï~åí=íç=çéíáJ ãáëÉK= få= íÜÉ= ÉñéÉêáãÉåíë= ïÉ= ìëÉÇ= íÜÉ= ëÅ~äÉÇ= ÅçåàìÖ~íÉ= Öê~ÇáÉåí= ~äÖçêáíÜã= Ñêçã= íÜÉ= kbqi^_=é~Åâ~ÖÉ=k~ÄåÉó=OMMOK=få=éÉêÑçêãáåÖ=íÜÉ=Öê~ÇáÉåí=çéíáãáë~íáçå=ïÉ=~ÇÇÉÇ=éêáJ çêë= íç= íÜÉ= é~ê~ãÉíÉêë= çÑ= íÜÉ= ãáñíìêÉ= âÉêåÉäK= qÜÉ= éêáçêë= ïÉêÉ= Ñ~ÅíçêáëáåÖ= ~åÇ= ÉÑÑÉÅíáîÉäó= ëÉêîÉÇ=~ë=éÉå~äíáÉëI=ëáãáä~ê=íç=íÜÉ=éÉå~äáëÉÇ=êÉÖêÉëëáçå=Ñê~ãÉïçêâI=éêÉîÉåíáåÖ=íÜÉ=âÉêåÉä= é~ê~ãÉíÉêë= Ñêçã= ÖçáåÖ= íç= “~Äåçêã~äÒ= î~äìÉëI= ÉKÖK= íÜ~í= ïçìäÇ= ã~âÉ= íÜÉ= éçëíÉêáçê= ãÉ~å= Åçää~éëáåÖ=íç=~=ëìã=çÑ=ÇÉäí~=ÑìåÅíáçåëK=páåÅÉ=~ää=é~ê~ãÉíÉêë=Ü~îÉ=íç=ÄÉ=éçëáíáîÉI=ïÉ=ÅÜçëÉ= íç= éìí= ~= éêáçê= çå= íÜÉáê= äçÖJî~äìÉ= ~åÇ= ïÉ= ÅÜççëÉ= íç= Ü~îÉ= ïáÇÉ= d~ìëëá~å= ÇáëíêáÄìíáçåëI= ãÉ~åáåÖ=íÜ~í=íÜÉ=ÜóéÉêJéêáçêë=çîÉê=íÜÉ=äçÖJî~äìÉë=çÑ=dm=ïÉêÉ=d~ìëëá~åë=ïáíÜ=~=ä~êÖÉ=î~êáJ ~åÅÉK=

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15-9

Figure 3 Illustration of combining the observations. The dots on the same horizontal planes show the observations made at the same time. An SSGP fit has been performed for each set of observations on separate planes and the hyperparameter selection is based on optimising a weighted sum of gradients at each individual level.

qç=Éî~äì~íÉ=íÜÉ=ÅçãÄáåÉÇ=ppdm=éÉêÑçêã~åÅÉI=ïÉ=éÉêÑçêãÉÇ=äÉ~îÉJçåÉJçìí=Åêçëë=î~äáJ Ç~íáçå=çå=íÜÉ=NM=Ç~í~ëÉíëW=åáåÉ=Ç~óë=çÑ=Ç~í~=ïÉêÉ=ìëÉÇ=íç=Éëíáã~íÉ=íÜÉ=dm=é~ê~ãÉíÉêë=~åÇ= íÜÉ=çåÉ=Ç~ó=äÉÑí=çìí=ìëÉÇ=íç=ãÉ~ëìêÉ=íÜÉ=Éêêçê=êÉä~íÉÇ=íç=íÜÉ=é~ê~ãÉíÉêë=ïÉ=ÑçìåÇK=cçê=íÜÉ= äÉÑí=çìí=Ç~ó=ïÉ=ìëÉÇ=íÜÉ=ÜóéÉêJé~ê~ãÉíÉêë=êÉëìäíáåÖ=Ñêçã=íÜÉ=åáåÉ=Ç~óë=~ë=íÜÉ=ãÉ~å=î~äìÉ= Ñçê=íÜÉ=ÜóéÉêJéêáçêë=çÑ=íÜÉ=dm=~åÇ=ÇÉÅêÉ~ëÉÇ=íÜÉáê=ÅçêêÉëéçåÇáåÖ=î~êá~åÅÉëI=~ÅâåçïäÉÇÖJ áåÖ=íÜ~í=ëçãÉ=áåÑçêã~íáçå=Ü~ë=~äêÉ~Çó=ÄÉÉå=~ÅèìáêÉÇ=~Äçìí=íÜÉ=é~ê~ãÉíÉêë=íÜÉãëÉäîÉë=Äìí= ïÉ=~êÉ=ëíáää=ìåÅÉêí~áå=~Äçìí=íÜÉáê=~Åíì~ä=î~äìÉëK=qÜáë=ëÉíìé=Éå~ÄäÉë=íÜÉ=êÉJÉëíáã~íáçå=çÑ=íÜÉ= ÜóéÉêJé~ê~ãÉíÉêë=íç=ìëÉ=ïáíÜ=íÜÉ=ÅçãéÉíáíáçå=Ç~í~K= tÉ= ãÉåíáçå= ~å= áãéçêí~åí= ~Çî~åí~ÖÉ= çÑ= íÜÉ= _~óÉëá~å= ãÉíÜçÇçäçÖó= ÉñéäçáíÉÇ= áå= íÜáë= ÅçåíêáÄìíáçåI= å~ãÉäó= íÜÉ= Ñ~Åí= íÜ~í= íÜÉêÉ= áë= åç= åÉÉÇ= íç= Ü~îÉ= ~= ëÉé~ê~íÉ= î~äáÇ~íáçå= ëÉí= íç= ~ëëÉëë= íÜÉ= éÉêÑçêã~åÅÉ= çÑ= íÜÉ= ãÉíÜçÇI= ~åÇ= ~ää= Ç~í~= Å~å= ÄÉ= ìëÉÇ= Ñçê= ~ÅÅìê~íÉ= ÜóéÉêJ é~ê~ãÉíÉê= Éëíáã~íáçå= ìëáåÖ= íÜÉ= ã~êÖáå~ä= Ç~í~= äáâÉäáÜççÇ= Ej~Åh~ó= NVVOFK= qÜáë= ëÅÉå~êáç= ï~ë= ìëÉÇ= Ñçê= íÜÉ= pf`OMMQ= “pé~íá~ä= fåíÉêéçä~íáçå= `çãé~êáëçåKÒ= fí= Ü~ë= ÄÉÉå= íÜÉ= ÇçåÉ= áå= íÜÉ= ÑçääçïáåÖ= ã~ååÉêW= ïÉ= ìëÉÇ= íÜÉ= NM= “éêáçêÒ= Ç~í~ëÉíë= íç= íê~áå= íÜÉ= íÉå= dmÛë= ~åÇ= íÜÉå= ÅçãÄáåÉÇ=íÜÉ=áåÇáîáÇì~ä=Öê~ÇáÉåíë=~ë=áå=Éèì~íáçå=EUF=ïÜáÅÜ=ïÉêÉ=ÑÉÇ=áåíç=~=ÅçåàìÖ~íÉ=Öê~J ÇáÉåí= çéíáãáë~íáçå= ~äÖçêáíÜã= ~åÇ= åÉï= ÜóéÉêJé~ê~ãÉíÉêë= ïÉêÉ= çÄí~áåÉÇK= qÜÉ= éêÉîáçìë= ëíÉéë=ïÉêÉ=êÉéÉ~íÉÇ=~=åìãÄÉê=çÑ=íáãÉë=EPMFI=~åÇ=íÜÉ=é~ê~ãÉíÉêë=ÑçìåÇ=íÜáë=ï~ó=Ü~îÉ=ÄÉÉå= áåëÉêíÉÇ=~ë=ãÉ~åë=áå=íÜÉ=ÜóéÉêJéêáçêë=Ñçê=íÜÉ=dm=ïÉ=ìëÉÇ=Ñçê=íÜÉ=ÅçãéÉíáíáçå=Ç~í~ëÉíë=N= ~åÇ=OK=e~îáåÖ=åç=áåÑçêã~íáçå=~Äçìí=íÜÉ=àçâÉê=Ç~í~ëÉíI=ïÉ=ÇÉÅáÇÉÇ=íÜ~í=íÜÉ=ãçëí=~ééêçJ éêá~íÉ= ~ééêç~ÅÜ= Ñçê= íÜáë= Ç~í~ëÉí= ïçìäÇ= ÄÉ= íç= äççëÉå= çìê= éêáçê= ÄÉäáÉÑë= ~Äçìí= íÜÉ= ÜóéÉêJ é~ê~ãÉíÉêë=ïÉ=ïÉêÉ=ìëáåÖ=áå=çìê=ãçÇÉä=ëáåÅÉ=ïÉ=Ü~Ç=~ÄëçäìíÉäó=åç=áÇÉ~=~Äçìí=ïÜ~í=íÜáë= ÑáäÉ=ïçìäÇ=Åçåí~áåK=

3. RESULTS qÜáë=ëÉÅíáçå=Åçåí~áåë=íÜÉ=êÉëìäíë=çÑ=çìê=ÉñéÉêáãÉåíë=~åÇ=Åçãé~êÉë=~=î~êáÉíó=çÑ=ÄÉåÅÜã~êâ= ãÉíÜçÇë= íç= ëÉÉ= íÜÉ= ÉÑÑÉÅíáîÉåÉëë= çÑ= íÜÉ= ppdm= ~äÖçêáíÜãK= rëáåÖ= j~íä~Ä= ïáíÜ= íÜÉ= ppdm= íççäÄçñ=E`ë~íµ=OMMPF=ãçÇáÑáÉÇ=íç=ãáåáãáëÉ=íÜÉ=ÅçãÄáåÉÇ=Öê~ÇáÉåíI=ïÉ=éêÉÇáÅíÉÇ=íÜÉ=î~äJ ìÉë= ~í= íÜÉ= ìåâåçïå= äçÅ~íáçåë= ìëáåÖ= íÜÉ= ãÉíÜçÇçäçÖó= ÇÉëÅêáÄÉÇ= ~ÄçîÉK= kÉíä~Ä= Ek~ÄåÉó= OMMOFI=~=j~íä~Ä=íççäÄçñ=ï~ë=ìëÉÇ=íç=éêçîáÇÉ=ÄÉåÅÜã~êâ=êÉëìäíë=ìëáåÖ=~=î~êáÉíó=çÑ=çíÜÉê= ã~ÅÜáåÉ= äÉ~êåáåÖ= íÉÅÜåáèìÉë= EjìäíáJä~íÉê= mÉêÅÉéíêçå= EjimF= ~åÇ= o~Çá~ä= _~ëáë= cìåÅíáçåë= Eo_cF=E_áëÜçé=NVVRFFK=qÜÉ=íê~áåáåÖ=çÑ=íÜÉ=jim=ï~ë=~ÅÜáÉîÉÇ=ìëáåÖ=É~êäó=ëíçééáåÖ=íç=ÅçåJ íêçä=íÜÉ=ÉÑÑÉÅíáîÉ=ÅçãéäÉñáíó=çÑ=íÜÉ=åÉíïçêâK=pí~åÇ~êÇ=d~ìëëá~å=éêçÅÉëëÉë=ïÉêÉ=~äëç=ìëÉÇ=

15-10

FAST SPATIAL INTERPOLATION ARTICLES

ëç= íÜ~í= íÜÉ= ÅÜçáÅÉ= íç= ìëÉ= ~= ãáñíìêÉ= çÑ= âÉêåÉä= ÑìåÅíáçåë= ÅçìäÇ= ÄÉ= î~äáÇ~íÉÇK= fí= ëÜçìäÇ= ÄÉ= åçíÉÇ=íÜ~í=íÜÉ=ëÅ~äÉë=çå=áåÇáîáÇì~ä=ìåÅÉêí~áåíó=ã~éë=~êÉ=ÇáÑÑÉêÉåíI=ëç=íÜ~í=íÜÉ=ìåÅÉêí~áåíó= Å~å=ÄÉ=ãçêÉ=É~ëáäó=Ö~ìÖÉÇI=ÜçïÉîÉê=Ñçê=~ää=çÑ=íÜÉ=fëçäáåÉ=äÉîÉä=ã~éëI=íÜÉ=Åçåíçìê=ëÅ~äÉ=áë= Åçåëí~åíK= 3.1. OVERALL RESULTS N = 808

Min

Max

Mean

Median

Std. dev.

Observed

57.00

180.00

98.01

98.80

20.02

SSGP

68.82

125.41

96.75

98.96

14.41

GP (mixture)

67.07

127.20

96.58

98.65

14.98

GP (sqexp)

69.26

123.77

96.87

99.19

14.32

RBF

68.22

129.55

96.85

98.66

14.58

MLP

66.05

129.13

96.80

98.07

14.65

Table 1 Comparison of the estimated and measured values (nSv/h) 1st Dataset.

N = 808

Min

Max

mean

Median

Std. dev.

Observed

57.00

1528.20

105.42

98.95

83.71

SSGP

74.25

634.49

100.78

95.68

39.22

GP (mixture)

87.09

150.81

106.13

102.46

15.51

GP (sqexp)

80.80

161.73

108.51

101.28

22.53

RBF

82.22

160.73

108.22

101.77

21.91

MLP

-129.18

760.02

102.41

94.71

80.03

Table 2 Comparison of the estimated and measured values (nSv/h) 2nd Dataset.

MAE

ME

Pearson’s r

RMSE

SSGP

9.10

-1.27

0.788

12.46

GP (mixture)

9.08

-1.44

0.787

12.47

GP (sqexp)

9.47

-1.15

0.776

12.75

RBF

9.49

-1.19

0.776

12.71

MLP

9.48

-1.22

0.775

12.73

Table 3 Comparison of the errors for 1st Dataset.

MAE

ME

Pearson’s r

RMSE

SSGP

18.55

-4.64

0.856

54.22

GP (mixture)

21.77

0.72

0.350

79.57

GP (sqexp)

22.53

3.09

0.331

79.16

RBF

22.73

3.21

0.334

79.31

MLP

48.41

-3.01

0.384

90.89

Table 4 Comparison of the errors for 2nd Dataset.

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15-11

ESTIMATIONS FOR SSGP METHOD 190 600000

600000

180 170 500000

500000

160 150 400000

400000

140 130 300000

300000

120 110 200000

200000

100 90 100000

100000

80 70 0

0

60 0

100000

0

200000

100000

200000

Figure 4 Isoline levels (nSv/h) for the 1st set (left) and the 2nd set (right) using SSGP.

UNCERTAINTY IN SSGP METHOD 690

165 600000

600000

680

160 155 500000

670 500000

660

150 145

400000

650

400000

640

140 135

300000

630

300000

620

130 125

200000

610

200000

600

120 115

100000

590

100000

580

110 0

105 0

100000

200000

100

570

0

560 0

100000

200000

Figure 5 Isoline levels showing the uncertainty in the estimations obtained for the 1st set (left) and the 2nd set (right) using SSGP. Notice that the grey-levels are different in the two subfigures.

15-12

FAST SPATIAL INTERPOLATION ARTICLES

ESTIMATIONS FOR GP (MIXTURE) METHOD

600000

190

600000

180 170

500000

500000

160 150 400000

400000

140 130 300000

300000

120 110 200000

200000

100 90 100000

100000

80 70 0

0

60 0

100000

0

200000

100000

200000

Figure 6 Isoline levels (nSv/h) for the 1st set (let) and the 2nd set (right) Using GP.

UNCERTAINTY IN GP (MIXTURE) METHOD

188

165 600000

600000

186

160

184

155 500000

500000

182

150 145

400000

180

400000

178

140 135

300000

176 300000

174

130 125

200000

172

200000

170

120 115

100000

168

100000

166

110 105

0

164

0

162

100 0

100000

200000

0

100000

200000

Figure 7 Isoline levels showing the uncertainty associated to the estimations obtained st nd for the 1 set (left) and the 2 set (right) using GP. Notice that the greylevels are different in the two subfigures.

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15-13

3.2. DETECTING ANOMALIES AND OUTLIERS

Figure 8 3D map showing extreme values found in the 2nd set (vertical scale in nSv/h)

3.3. CALCULATION TIME Time taken 1st Dataset

3m28s

2nd Dataset

4m27s Table 5

Calculation Time.

3.4. LEARNED HYPER-PARAMETERS FOR SSGP 1st Dataset

2nd Dataset

SqExp x range

0.02

0.03

SqExp y range

0.24

0.24

SqExp amplitude

0.05

0.05

Exp x range

1.89

0.10

Exp y range

1.11

0.00

Exp amplitude

0.90

60.12

Noise

0.30

41.59

Table 6 SSGP Learnt Hyper-parameters

4. DISCUSSION qÜÉ=êÉëìäíë=Ñçê=íÜÉ=åçêã~ä=Ç~í~ëÉí=áå=q~ÄäÉ=N=ëÜçï=íÜ~í=íÜÉ=ê~åÖÉ=ÄÉíïÉÉå=íÜÉ=ãáåáãìã= ~åÇ= ã~ñáãìã= î~äìÉë= íç= ÄÉ= ãìÅÜ= ëÜçêíÉê= áå= ~ää= çÑ= çìê= ãÉíÜçÇë= íÜ~å= íÜÉ= çÄëÉêîÉÇ= Ç~í~= ~åÇ=íÜÉ=Éëíáã~íÉë=~êÉ=äÉëë=î~êáÉÇK=^ää=íÜÉ=ã~ñáãìã=Éëíáã~íÉë=~êÉ=çÑ=~=ëáãáä~ê=ã~ÖåáíìÇÉ= ïÜáÅÜ=ïçìäÇ=ëìÖÖÉëí=~=äáãáíÉÇ=~ãçìåí=çÑ=áåÑçêã~íáçå=áå=íÜÉ=íê~áåáåÖ=Ç~í~ëÉíK=`çãé~êáåÖ= íÜÉ=Éêêçêë=Eq~ÄäÉ=PF=ÄÉíïÉÉå=íÜÉ=ppdm=~åÇ=íÜÉ=dm=EãáñíìêÉF=ãÉíÜçÇI=áí=Å~å=ÄÉ=ëÉÉå=íÜ~í= íÜÉëÉ=ãÉíÜçÇë=Ü~îÉ=~=ëáãáä~ê=éÉêÑçêã~åÅÉK=fí=áë=ÅäÉ~ê=Ñêçã=íÜáë=íÜ~í=äáííäÉ=áåÑçêã~íáçå=Ü~ë=

15-14

FAST SPATIAL INTERPOLATION ARTICLES

ÄÉÉå=äçëí=Äó=áãéçëáåÖ=ëé~êëáíó=áå=êÉéêÉëÉåíáåÖ=íÜÉ=dmK=pÉäÉÅíáåÖ=íÜÉ=åìãÄÉê=çÑ=Ä~ëáë=îÉÅJ íçêë= íç= êÉéêÉëÉåí= íÜÉ= dm= Äó= áë= ~åçíÜÉê= í~ëâ= çåÉ= ãìëí= í~âÉ= áåíç= ÅçåëáÇÉê~íáçåK= cçê= íÜáë= ÉñÉêÅáëÉ=ïÉ=ÅÜççëÉ=íç=ìëÉ=UM=_sÛëK=rëáåÖ=íÜÉ=éêáçê=Ç~í~ëÉíëI=~= ãáåáã~ä=åìãÄÉê=çÑ=_sÛë= ï~ë=ÅÜçëÉå=íÜ~í=ïçìäÇ=çéíáã~ääó=êÉéêÉëÉåí=íÜÉ=dm=ïáíÜ=ãáåáã~ä=ÉêêçêK= tÜÉå=Åçãé~êáåÖ=íÜÉ=íïç=dm=ãÉíÜçÇë=íÜ~í=ïÉêÉ=íÉëíÉÇ=áí=Å~å=ÄÉ=ëÉÉå=íÜ~í=íÜÉ=âÉêåÉä= ÑìåÅíáçå=ãáñíìêÉ=ãÉíÜçÇ=éêÉÇáÅíë=íÜÉ=íêìÉ=î~äìÉë=ãçêÉ=~ÅÅìê~íÉäó=íÜ~å=íÜÉ=ëèì~êÉÇ=ÉñéçJ åÉåíá~äI= ïÜáÅÜ= ëìééçêíë= çìê= ìåÇÉêäóáåÖ= ~ëëìãéíáçåë= ~Äçìí= íÜÉ= ëíêìÅíìêÉ= çÑ= íÜÉ= Ç~í~K= e~îáåÖ=éêáçê=âåçïäÉÇÖÉ=~Äçìí=íÜÉ=Åçî~êá~åÅÉ=ëíêìÅíìêÉ=çÑ=~ãÄáÉåí=Ä~ÅâÖêçìåÇ=ê~Çá~íáçå= ÅäÉ~êäó=áë=~Çî~åí~ÖÉçìë=áå=éêÉÇáÅíáçåK=qÜÉ=dm=EëèÉñéF=ãÉíÜçÇ=ï~ë=ìëÉÇ=~ë=~=ÄÉåÅÜã~êâ= ~åÇ=ÇçÉë=åçí=í~âÉ=áåíç=~ÅÅçìåí=~åó=éêáçê=áåÑçêã~íáçå=~Äçìí=íÜÉ=ÇáëéÉêë~ä=éêçÅÉëë=ÉñÅÉéí= ÜóéÉêJé~ê~ãÉíÉêë=íÜ~í=Ñáí=~=ëèì~êÉÇ=ÉñéçåÉåíá~ä=ãçÇÉä=íç=íÜÉ=Ç~í~K=qÜáë=ãÉíÜçÇ=Ü~ë=ëáãáJ ä~ê=éÉêÑçêã~åÅÉ=íç=íÜÉ=jim=~åÇ=o_c=ãÉíÜçÇë=ïÜáÅÜ=~äëç=Çç=åçí=í~âÉ=áåíç=~ÅÅçìåí=~åó= ~ëëìãéíáçåë=~Äçìí=íÜÉ=ÇáëéÉêë~ä=éêçÅÉëëK= qÜÉ= å~íìêÉ= çÑ= íÜÉ= ëÉÅçåÇ= Ç~í~ëÉí= ï~ë= ìåâåçïå= éêáçê= íç= íÜÉ= áåíÉêéçä~íáçå= ÉñÉêÅáëÉK= åÇ qÜÉ= O = Ç~í~ëÉí= Éëíáã~íÉë= Eq~ÄäÉë= O= C= QF= ~êÉ= éççêI= Äìí= íÜáë= é~êíáÅìä~ê= ~äÖçêáíÜã= áë= ~= ëãççíÜÉê=~åÇ=áë=åçí=ÇÉëáÖåÉÇ=íç=ÅçéÉ=ïáíÜ=ÉñíêÉãÉ=î~äìÉë=ëáåÅÉ=áíÛë=Ä~ëÉÇ=çå=íÜÉ=~ëëìãéJ íáçå= íÜ~í= çÄëÉêî~íáçåë= ÅäçëÉê= íçÖÉíÜÉê= ~êÉ= ãçêÉ= äáâÉäó= íç= ÄÉ= ëáãáä~ê= íÜ~å= íÜçëÉ= ÑìêíÜÉê= ~é~êíK=qÜÉ=jim=ãÉíÜçÇ=ëÜçïÉÇ=íÜÉ=ïçêëí=éêÉÇáÅíáçå=ÉêêçêI=ÖáîáåÖ=åÉÖ~íáîÉ=Éëíáã~íÉë=~í= ëçãÉ=çÑ=äçÅ~íáçåëX=íÜáë=áë=åçí=ëìêéêáëáåÖ= ëáåÅÉ=áí=áë=~= ã~ÅÜáåÉ=äÉ~êåáåÖ=ãÉíÜçÇ=íÜ~í= ÇçÉë= åçí=í~âÉ=áåíç=~ÅÅçìåí=~åó=éêáçê=âåçïäÉÇÖÉ=~Äçìí=íÜÉ=ìåÇÉêäóáåÖ=éêçÅÉëë=ïÉ=~êÉ=éêÉÇáÅíJ áåÖK=qÜÉ=j^b=Éêêçê=Ñçê=íÜÉëÉ=íïç=ÄÉåÅÜã~êâë=áë=~=äçí=ÜáÖÜÉê=íÜ~å=~åó=çÑ=íÜÉ=dm=ãçÇÉäëX= ÜçïÉîÉê= íÜÉ= éêÉÇáÅíáçåë= ïáíÜ= íÜÉ= jim= ~êÉ= ëäáÖÜíäó= ãçêÉ= ÅçêêÉä~íÉÇ= ïáíÜ= íÜÉ= çÄëÉêîÉÇ= Ç~í~=íÜ~å=íÜÉ=íïç=ëí~åÇ~êÇ=dm=ãçÇÉäëK=qÜÉ=ppdm=ãÉíÜçÇ=ëÜçïë=íÜÉ=ëã~ääÉëí=j^b=~åÇ= íÜÉ=éêÉÇáÅíáçåë=~êÉ=ÅçêêÉä~íÉÇK=qÜÉ=ppdm=ëíáää=ÇáÇ=åçí=ã~å~ÖÉ=íç=ÅçãÉ=ÅäçëÉ=íç=éêÉÇáÅíáåÖ= íÜÉ=ã~ñáãìã=~ë=ëÉÉå=áå=íÜÉ=çÄëÉêîÉÇ=Ç~í~I=Äìí=íÜáë=áë=íç=ÄÉ=ÉñéÉÅíÉÇ=ëáåÅÉ=dmDë=~êÉ=éççê= ~í=éêÉÇáÅíáåÖ=ÉñíêÉãÉ=î~äìÉëK= iççâáåÖ=~í=íÜÉ=j^b=~åÇ=ojpb=Éêêçê=Ñçê=íÜÉ=íïç=Ç~í~ëÉíëI=áí=Å~å=ÄÉ=ëÉÉå=áå=íÜÉ=Ñáêëí= Ç~í~ëÉí=íÜ~í=íÜÉ=ojpb=áë=ëäáÖÜíäó=ä~êÖÉê=íÜ~å=íÜÉ=j^bK=eçïÉîÉê=Ñçê=íÜÉ=ëÉÅçåÇ=Ç~í~ëÉíI= ojpb=áë=ãìÅÜ=ÜáÖÜÉê=íÜ~å=íÜÉ=j^bI=íÜáë=áë=ÄÉÅ~ìëÉ=íÜÉ=ojpb=áë=ãçêÉ=ëÉåëáíáîÉ=íç=ÜáÖÜ= êÉëáÇì~äëK=qÜÉ=Éëíáã~íÉë=~êÉ=~ää=ëäáÖÜíäó=åÉÖ~íáîÉäó=Äá~ëÉÇ=Ñçê=íÜÉ=Ñáêëí=Ç~í~ëÉíK=cçê=íÜÉ=ëÉÅJ çåÇ= Ç~í~ëÉí= áí= Å~å= ÄÉ= ëÉÉå= íÜÉ= Äá~ë= áå= Éëíáã~íÉë= Ü~ë= Öêçïå= ãìÅÜ= ä~êÖÉê= ÇÉéÉåÇ= çå= íÜÉ= ~äÖçêáíÜã=ìëÉÇK= qÜÉ=Åçåíçìê=éäçíë=EcáÖìêÉ=QF=ëÜçï=íÜÉ=Éëíáã~íÉë=çîÉê=íÜÉ=ïÜçäÉ=~êÉ~=çå=íÜÉ=íïç=Ç~í~J ëÉíëK= qÜÉ= ëÉÅçåÇ= Ç~í~ëÉí= éäçí= ÅäÉ~êäó= ëÜçïë= Äáò~êêÉ= ÄÉÜ~îáçìê= áå= çåÉ= ÇáêÉÅíáçå= ïáíÜ= íÜÉ= éêÉÇáÅíáçå= çÑ= íÜÉ= ÇáëéÉêë~ä= çÑ= íÜÉ= Åçåí~ãáå~åí= áåíêçÇìÅÉÇK= qÜÉ= ÜóéÉêJé~ê~ãÉíÉêë= äÉ~êåí= Äó= íÜÉ= ppdm= ~ë= áí= ÅóÅäÉë= íÜêçìÖÜ= íÜÉ= åÉï= Ç~í~= ~êÉ= ëÜçïå= áå=q~ÄäÉ= SK= qÜÉ= äÉ~êåí= ÜóéÉêJ é~ê~ãÉíÉêë= êÉÑÉê= íç= íÜÉ= åçêã~äáëÉÇ= Ç~í~I= ëáåÅÉ= ÇìêáåÖ= íÜÉ= ~äÖçêáíÜã= íÜÉ= Ç~í~= ìëÉÇ= Ü~ë= ÄÉÉå= åçêã~äáëÉÇK= iççâáåÖ= ~í= íÜÉ= äÉ~êåí= ÜóéÉêJé~ê~ãÉíÉêë= áí= Å~å= ÄÉ= ëÉÉå= íÜ~í= íÜÉ= âÉêåÉä= ÑìåÅíáçå=ê~åÖÉ=áå=íÜÉ=ó=ÇáêÉÅíáçå=Ü~ë=Åçää~éëÉÇ=íç=òÉêçK=qÜáë=~ÅÅçìåíë=Ñçê=íÜÉ=ÉääáéíáÅ=ëÜ~éÉ= çÑ=íÜÉ=çìíäáÉêë=~åÇ=íÜÉ=ÑÉ~íìêÉë=áå=íÜÉ=éäçíK=fí=áë=ìåâåçïå=ïÜó=íÜÉ=ÜóéÉêJé~ê~ãÉíÉê=çéíáJ ãáë~íáçå= Ü~ë= êÉëìäíÉÇ= áå= íÜÉëÉ= é~ê~ãÉíÉêëK= få= cáÖìêÉ= U= íÜÉ= å~êêçï= ÇáëéÉêë~ä= áë= ÅäÉ~êäó= îáëáÄäÉK= qÜÉ=äÉ~êåí=ÜóéÉêJé~ê~ãÉíÉêë=ëÜçï=~å=áåÅêÉ~ëÉÇ=åçáëÉ=äÉîÉäK=qÜÉ=åçáëÉ=ÅçêêÉëéçåÇë=íç= íÜÉ=åìÖÖÉí=ÑÉ~íìêÉ=çÑ=íÜÉ=î~êáçÖê~ãK=qÜáë=äÉ~êåí=åçáëÉ=é~ê~ãÉíÉê=ïáää=Å~ìëÉ=íÜÉ=áåíÉêéçä~J íáçå= íç= åçí= çåäó= ÄÉÅçãÉ= ãçêÉ= ìåÅÉêí~áåI= Äìí= ~äëç= Å~ìëÉ= íÜÉêÉ= íç= ÄÉ= ~= ëã~ääÉê= ê~åÖÉ= çÑ= î~êá~íáçå= ÄÉíïÉÉå= íÜÉ= éêÉÇáÅíÉÇ= éçáåíëK= iççâáåÖ= ~í= íÜÉ= ìåÅÉêí~áåíó= ã~éë= EcáÖìêÉ= RF= áí= áë=

FAST SPATIAL INTERPOLATION ARTICLES

15-15

ÅäÉ~ê= íÜ~í= éêÉÇáÅíáçåë= ïáíÜ= íÜÉ= ëÉÅçåÇ= Ç~í~ëÉí= ~êÉ= ÉñíêÉãÉäó= ìåÅÉêí~áåK= qÜÉ= ê~åÖÉ= ~åÇ= é~ííÉêå= çÑ= ìåÅÉêí~áåíó= Ñçê= íÜÉ= Ñáêëí= Ç~í~= ëÉí= ~êÉ= ëáãáä~ê= íç= íÜÉ= ìåÅÉêí~áåíó= áå= íÜÉ= êÉëìäíë= Å~äÅìä~íÉÇ=áå=íÜÉ=dm=ãçÇÉä=EcáÖìêÉ=TFK=qÜÉ=ìåÅÉêí~áåíó=ãÉ~ëìêÉÇ=áå=íÜÉ=dm=Ñçê=íÜÉ=ëÉÅçåÇ= Ç~í~ëÉí=ëÜçïë=~=ëáãáä~êI=Äìí=äÉëë=ÇÉÑáåÉÇ=é~ííÉêå=íç=íÜ~í=ëÜçïå=áå=íÜÉ=Ñáêëí=Ç~í~ëÉíK=cáÖìêÉ= S= ëÜçïë= íÜÉ= éêÉÇáÅíáçå= çÑ= íÜÉ= ÇáëéÉêëÉÇ= Åçåí~ãáå~åí= ìëáåÖ= íÜÉ= dmK= qÜÉ= é~ííÉêå= çÑ= íÜÉ= ÇáëéÉêë~ä=áë=äÉëë=ÉääáéíáÅ=~åÇ=ÅäçëÉê=íç=íÜÉ=é~ííÉêå=çÑ=ÇáëéÉêë~ä=çåÉ=ïçìäÇ=ÉñéÉÅíK= qÜÉ=Å~äÅìä~íáçå=íáãÉ=áë=ëÜçïå=áå=q~ÄäÉ=RK=cçê=íÜÉ=éìêéçëÉë=çÑ=íÜáë=ÅçåíÉëíI=íÜÉ=~äÖçJ êáíÜã= ï~ë= ëÉí= íç= ÅóÅäÉ= íÜêçìÖÜ= íÜÉ= ppdm= ~äÖçêáíÜã= ëÉîÉå= íáãÉë= íç= ~ÅÜáÉîÉ= áãéêçîÉÇ= Éëíáã~íÉëK=páåÅÉ=ãìÅÜ=çÑ=íÜÉ=~äÖçêáíÜã=~åÇ=ÜóéÉêJé~ê~ãÉíÉêë=ïÉêÉ=íìåÉÇ=ìëáåÖ=íÜÉ=éêáçê= Ç~í~=ÄÉÑçêÉ=íÜÉ=~Åíì~ä=ÅçåíÉëí=Ç~í~=ï~ë=ã~ÇÉ=~î~áä~ÄäÉI=éêÉÇáÅíáåÖ=~í=ìåâåçïå=äçÅ~íáçåë= ÅçìäÇ=ÄÉ=ÇçåÉ=~ÑíÉê=çåÉ=çê=íïç=ÅóÅäÉë=íÜêçìÖÜ=íÜÉ=~äÖçêáíÜã=ïáíÜçìí=~=ÖêÉ~í=äçëë=áå=~ÅÅìJ ê~Åó=~åÇ=ÜÉåÅÉ=ÑìêíÜÉê=êÉÇìÅáåÖ=Åçãéìí~íáçå=íáãÉK=^ää=ÉñéÉêáãÉåíë=ïÉêÉ=Å~êêáÉÇ=çìí=çå=~= O= dÜò= m`= ìëáåÖ= j~íä~Ä= SKR= ~åÇ= íÜÉ= ppdm= ~åÇ= kÉíä~Ä= j~íä~Ä= íççäÄçñÉë= E`ë~íµ= OMMPX= k~ÄåÉó=OMMOFK= 5. CONCLUSIONS

qÜÉ= ëé~êëÉ= ÉñíÉåëáçå= íç= íÜÉ= d~ìëëá~å= éêçÅÉëëÉë= éêçîáÇÉë= ~= Ñ~ëíI= ëí~íáëíáÅ~ääó= éêáåÅáéäÉÇ= ~ééêç~ÅÜ=íç=ëé~íá~ä=áåíÉêéçä~íáçåK=^ë=ïáíÜ=ã~åó=ãÉíÜçÇë=çÑ=ëé~íá~ä=áåíÉêéçä~íáçåI=çìíäáÉêë= ~êÉ=é~êíáÅìä~êäó=ÇáÑÑáÅìäí=íç=ãçÇÉäK=qÜÉêÉ=Ü~îÉ=ÄÉÉå=î~êáçìë=~ééêç~ÅÜÉë=~í=ãçÇÉääáåÖ=ãçêÉ= ~ÅÅìê~íÉäó=çìíäáÉêë=ëìÅÜ=~ë=ìëáåÖ=ÅçãÄáå~íáçåë=çÑ=éêÉÇáÅíáçå=ãçÇÉäëI=çåÉ=ëìÅÜ=ãçÇÉä=ìëÉë= ~= äáåÉ~ê= ÅçãÄáå~íáçå= çÑ= ~= åìãÄÉê= çÑ= ãÉíÜçÇë= ïÜÉêÉ= Äó= íÜÉ= ä~êÖÉ= ëÅ~äÉ= ÇáëÅçåíáåìçìë= î~êá~íáçå= áë= ãçÇÉääÉÇ= ìëáåÖ= Ñ~ëí= ï~îÉäÉí= áåíÉêéçä~íáçå= ~åÇ= íÜÉ= êÉëáÇì~äë= ~êÉ= áåíÉêéçä~íÉÇ= ìëáåÖ= ëí~åÇ~êÇ= âêáÖáåÖ= íÉÅÜåáèìÉë= EaÉãó~åçî= Éí= ~äK= OMMNFK= qÜáë= Éåíêó= ï~ë= íç= ëÜçï= íÜÉ= ~Çî~åí~ÖÉë= çÑ= íÜÉ= ppdm= ãçÇÉäK= qÜÉ= ppdm= ÅçìäÇ= ÄÉ= ÅçìéäÉÇ= ïáíÜ= ÉñáëíáåÖ= ~äÖçêáíÜãë= íç= éêÉÇáÅí=ãçêÉ=ÉÑÑÉÅíáîÉäó=ïÜÉêÉ=ÉñíêÉãÉ=î~äìÉë=~êÉ=ãçêÉ=äáâÉäóK= låÅÉ= íÜÉ= å~íìêÉ= çÑ= íÜÉ= Åçî~êá~åÅÉ= ëíêìÅíìêÉ= áë= ìåÇÉêëíççÇI= íÜáë= ãÉíÜçÇ= áë= ÉåíáêÉäó= ~ìíçã~íáÅK= qÜÉêÉ= ~êÉ= ~= ä~êÖÉ= åìãÄÉê= çÑ= íìåÉ~ÄäÉ= é~ê~ãÉíÉêë= íÜ~í= çåÉ= Å~å= ëÉí= íç= ãçêÉ= ~ÅÅìê~íÉäó=éêÉÇáÅí=ìåâåçïå=Ç~í~K=få=ÉãÉêÖÉåÅó=ÅçåÇáíáçåë=íÜÉ=ppdm=~äÖçêáíÜã=ÅçìäÇ=ÄÉ= ìëÉÇ=íç=éêÉÇáÅíI=ïáíÜ=~ÅÅìê~Åó=ÅçããÉåëìê~íÉ=ïáíÜ=~å=çêÇáå~êó=dm=~äÖçêáíÜã=~åÇ=ïáíÜáå= ~= ëáÖåáÑáÅ~åíäó= ëÜçêíÉê= éÉêáçÇ= çÑ= íáãÉK= páåÅÉ= íÜÉ= Åçãéìí~íáçå~ä= ÅçãéäÉñáíó= çåäó= ëÅ~äÉë= ïáíÜ=íÜÉ=åìãÄÉê=çÑ=Ä~ëáë=îÉÅíçêëI=éêÉÇáÅ~íáçå=ïáíÜ=çêÇáå~êáäó=éêçÜáÄáíáîÉäó=ä~êÖÉ=Ç~í~ëÉíë= Å~å=ÄÉÅçãÉ=ã~å~ÖÉ~ÄäÉK=cçê=ëáíì~íáçåë=ïÜÉêÉ=íÜÉêÉ=~êÉ=ÉñíêÉãÉ=î~äìÉëI=íáÖÜíÉê=î~êá~åÅÉë= çå= íÜÉ= éêáçê= ÄÉäáÉÑë= ~Äçìí= íÜÉ= ÜóéÉêJé~ê~ãÉíÉêë= çê= íìêåáåÖ= çÑÑ= ÜóéÉêJé~ê~ãÉíÉê= êÉJ Éëíáã~íáçå=ã~ó=~ääçï=~=ãçêÉ=~ÅÅìê~íÉ=Éëíáã~íÉ=çÑ=íÜÉ=Ç~í~I=ÜçïÉîÉê=éêÉÇáÅíáçå=áå=äçÅ~J íáçåë=ïÜÉêÉ=íÜÉêÉ=~êÉ=ÉñíêÉãÉ=î~äìÉë=ïáää=ëíáää=éÉêÑçêã=éççêäóK= ^åçíÜÉê=~Çî~åí~ÖÉ=çÑ=~=ëé~êëÉ=ãçÇÉä=~ë=éêÉîáçìëäó=ãÉåíáçåÉÇ=áë=íÜÉ=Åçãé~Åí=ï~ó=áå= ïÜáÅÜ= íÜÉ= dm= áë= êÉéêÉëÉåíÉÇ= Äó= ~= ëìÄëÉí= çÑ= Ä~ëáë= îÉÅíçêëK= få= ÉÑÑÉÅíI= íÜÉëÉ= Ä~ëáë= îÉÅíçêë= ÅçãéêÉëë= íÜÉ= Ç~í~= ëç= íÜ~í= íê~åëãáëëáçå= ïçìäÇ= êÉèìáêÉ= ëáÖåáÑáÅ~åíäó= äÉëë= Ä~åÇïáÇíÜK= qÜÉ= êÉ~ä=~Çî~åí~ÖÉ=çÑ=íÜáë=ÅçãéêÉëëáçå=Å~å=ÄÉ=~ééêÉÅá~íÉÇ=ãçêÉ=áÑ=çåÉ=ÅçåëáÇÉêë=íÜÉ=íê~åëãáëJ ëáçå=çÑ=ëìÅÜ=Ç~í~=Ñêçã=~=ë~íÉääáíÉK= cìêíÜÉê=ïçêâ=íÜ~í=áë=ÅìêêÉåíäó=ÄÉáåÖ=ÅçãéäÉíÉÇ=áë=~å=áãéäÉãÉåí~íáçå=çÑ=íÜÉ=ppdm=~äJ ÖçêáíÜã=Ñçê=é~ê~ääÉä=éêçÅÉëëçê=ÅçãéìíÉê= ëóëíÉãëK=cìêíÜÉê=Åçãéìí~íáçå~ä=ëéÉÉÇ=áåÅêÉ~ëÉë= ïáää= ã~âÉ= ppdmÛë= ~å= ÉñíêÉãÉäó= ~ííê~ÅíáîÉ= ëçäìíáçå= áå= ÉãÉêÖÉåÅó= ëáíì~íáçåë= ïÜÉêÉ= èìáÅâ= éêÉÇáÅíáçåë=~êÉ=åÉÉÇÉÇ=~åÇ=íÜÉ=Ç~í~ëÉí=áë=ÉñíêÉãÉäó=ä~êÖÉK=cçê=ëÅÉå~êáçë=ïÜÉêÉÄó=íÜÉêÉ=áë=~= å~íìê~ä=Åçåí~ãáå~åí=áåíêçÇìÅÉÇ=áåíç=íÜÉ=~êÉ~I=~ë=áå=íÜÉ=àçâÉê=ëÉíI=çåÉ=Å~ååçí=êÉ~ääó=éêÉJ ÇáÅí= ïáíÜ= ~åó= ÖêÉ~í= ÇÉÖêÉÉ= çÑ= ÅÉêí~áåíó= Üçï= íÜÉ= Ç~í~= ïáää= ÄÉ= ÇáëíêáÄìíÉÇI= Äìí= ~å=

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FAST SPATIAL INTERPOLATION ARTICLES

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REFERENCES _Éêå~êÇçI=gKjI=pãáíÜ=^Kc=ENVVQFI=_~óÉëá~å=qÜÉçêóK=gçÜå=táäÉó=C=pçåëK= _áëÜçé=`Kj=ENVVRFI=kÉìê~ä=kÉíïçêâë=Ñçê=m~ííÉêå=oÉÅçÖåáíáçåK=kÉï=vçêâI=kKvK=lñÑçêÇ=råáîÉêëáíó= mêÉëëK= `çêåÑçêÇ=aI=`ë~íµ=iI=lééÉê=j=EOMMQFI=ÚpÉèìÉåíá~äI=_~óÉëá~å=dÉçëí~íáëíáÅëW=^=mêáåÅáéäÉÇ=jÉíÜçÇ=Ñçê= i~êÖÉ=a~í~=pÉíëÛI=qÉÅÜåáÅ~ä=oÉéçêíI=^ëíçå=råáîÉêëáíóK= `çîÉê=qKjI=qÜçã~ë=gK^=ENVVNFI=bäÉãÉåíë=çÑ=fåÑçêã~íáçå=qÜÉçêóI=gçÜå=táäÉó=C=pçåëK= `êÉëëáÉ=kK^=ENVVPFI=pí~íáëíáÅë=Ñçê=pé~íá~ä=a~í~K=gçÜå=táäÉó=~åÇ=ëçåëK= `ë~íµ=iI=lééÉê=j=EOMMOFI=pé~êëÉ=låJiáåÉ=d~ìëëá~å=mêçÅÉëëÉëK=kÉìê~ä=`çãéìí~íáçå=îçäK=NQI=ééK= SQNJSSVK=ÜííéWLLïïïKåÅêÖK~ëíçåK~ÅKìâLm~éÉêë= `ë~íµ=iK=EOMMOFI=Úd~ìëëá~å=mêçÅÉëëÉë=J=fíÉê~íáîÉ=pé~êëÉ=^ééêçñáã~íáçåëÛK=mÜa=íÜÉëáëI=^ëíçå== råáîÉêëáíóK= `ë~íµ=i=EOMMPFI=ppdm=qççäÄçñK=ÜííéWLLïïïKåÅêÖK~ëíçåK~ÅKìâLmêçàÉÅíëLppdm= aÉãó~åçî=sI=pçäí~åá=pI=h~åÉîëâá=jI=`~åì=pI=j~áÖå~å=jI=p~îÉäáÉî~=bI=qáãçåáå=sI=máë~êÉåâç=s= EOMMNFI=Út~îÉäÉí=~å~äóëáë=êÉëáÇì~ä=âêáÖáåÖ=îëK=åÉìê~ä=åÉíïçêâ=êÉëáÇì~ä=âêáÖáåÖÛK=píçÅÜ~ëíáÅ= båîáêçåãÉåí~ä=oÉëÉ~êÅÜ=~åÇ=oáëâ=^ëëÉëëãÉåí=NRK= dáÄÄë=jI=j~Åh~ó=aKgK`=ENVVTFI=ÚbÑÑáÅáÉåí=áãéäÉãÉåí~íáçå=çÑ=d~ìëëá~å=éêçÅÉëëÉëÛI=qÉÅÜåáÅ~ä== oÉéçêíI=`~îÉåÇáëÜ=i~Äçê~íçêóI=`~ãÄêáÇÖÉ=råáîÉêëáíóK= e~ää=bKg=ENVUQFI=o~Çá~íáçå=~åÇ=iáÑÉ=OåÇ=bÇK=mÉêÖ~ãçå=mêÉëëI=kÉï=vçêâK= fë~~âë=bKeI=pêáî~ëí~î~=oKj=ENVUVFI=^å=fåíêçÇìÅíáçåë=íç=^ééäáÉÇ=dÉçëí~íáëíáÅëI=lñÑçêÇ=råáîÉêëáíó= mêÉëëK= j~Åh~ó=aKgK`=ENVVOFI=_~óÉëá~å=áåíÉêéçä~íáçåK=kÉìê~ä=`çãéìí~íáçå=îçäK=QI=ééK=QNRJQQTI=qÜÉ=jfq= mêÉëëK= j~Åh~ó=aKgK`=ENVVVFI=`çãé~êáëçå=çÑ=^ééêçñáã~íÉ=jÉíÜçÇë=Ñçê=e~åÇäáåÖ=eóéÉêJé~ê~ãÉíÉêëK= kÉìê~ä=`çãéìí~íáçå=îçäK=NNI=ééK=NMPRJNMSUI=qÜÉ=jfq=mêÉëëK= k~ÄåÉó=fKq=EOMMOFI=kbqi^_W=^äÖçêáíÜãë=Ñçê=m~ííÉêå=oÉÅçÖåáíáçåI=péêáåÖÉê=sÉêä~Ö= ÜííéWLLïïïKåÅêÖK~ëíçåK~ÅKìâLåÉíä~ÄLI=^ëíçå=råáîÉêëáíóK= m~éçìäáë=^I=máää~á=pKr=EOMMOFI=mêçÄ~ÄáäáíóI=o~åÇçã=s~êá~ÄäÉëI=~åÇ=píçÅÜ~ëíáÅ=mêçÅÉëëÉëI=jÅdê~ïJ eáääK= oçìëëÉÉìï=mKgI=iÉêçó=^Kj=ENVUTF=oçÄìëí=oÉÖêÉëëáçå=~åÇ=lìíäáÉê=aÉíÉÅíáçåI=gçÜå=táäÉó=C=pçåëK== pÅÜçÉäâçéÑ=_I=pãçä~=^Kg=EOMMOFI=iÉ~êåáåÖ=ïáíÜ=hÉêåÉäëI=qÜÉ=jfq=mêÉëëK= rf`I=jÉäÄçìêåÉI=^ìëíê~äá~K=kìÅäÉ~ê=fëëìÉë=_êáÉÑáåÖ=m~éÉê=NTI=j~êÅÜ=OMMQK= ÜííéWLLïïïKìáÅKÅçãK~ìLåáéNTKÜíãä= táääá~ãë=`KhKfI=o~ëãìëëÉå=`Kb=ENVVSFI=Úd~ìëëá~å=éêçÅÉëëÉë=Ñçê=êÉÖêÉëëáçåÛK=få=mêçÅÉÉÇáåÖë=çÑ=íÜÉ= kÉìê~ä=fåÑçêã~íáçå=mêçÅÉëëáåÖ=póëíÉãëI=îçäK=UI=qÜÉ=jfq=mêÉëëK=ïïïKåáéëKÅÅ= táääá~ãë=`KhKfI=pÉÉÖÉê=j=EOMMNFI=ÚrëáåÖ=íÜÉ=kóëíêçã=ãÉíÜçÇ=íç=ëéÉÉÇ=ìé=hÉêåÉä=j~ÅÜáåÉëÛK=få== mêçÅÉÉÇáåÖë=çÑ=íÜÉ=kÉìê~ä=fåÑçêã~íáçå=mêçÅÉëëáåÖ=póëíÉãëI=îçäK=NPI=qÜÉ=jfq=mêÉëëK= ïïïKåáéëKÅÅ= = Cite this article as: Ingram, Ben; Csató, Lehel; Evans, David. ‘Fast spatial interpolation using sparse Gaussian processes’. Applied GIS, Vol 1, No 2, 2005. pp. 15-01 to 15-17. DOI: 10.2104/ag050015 Copyright © 2005 Ben Ingram, Lehel Csató and David Evans

FAST SPATIAL INTERPOLATION ARTICLES

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