JAG
Kriging and thin plate splines for mapping
variables
Eric P J Boer’*,
Kirsten
M de Beursl
and A Dewi
Volume
l
3 - Issue 2 - 2001
climate
Hartkampz
1 Wageningen University and Research Center, Mathematical and Statistical Models Group, Dreyenlaan 4, 6703 HA Wageningen,
The Netherlands (e-mail: E.P.J.BoerQato.dlo.nI;
K.deBeursQesrinl.com)
2 Natural Resources Group, International Maize and Wheat Improvement
06600 Mexico D.F., Mexico (e-mail: [email protected])
Center (CIMMYT),
KEYWORDS:
dictions
elevation,
mum temperature,
interpolation
Mexico,
techniques,
maxi-
precipitation
growth
119981, for example,
state
of kriging
and three
forms
discussed in this paper to predict
monthly
mean
improve
using elevation
splines performed
results
plate
maximum
in Jalisco
the prediction
trivariate
of thin
monthly
precipitation
Results show that techniques
mation
because
the modeling
of
(CIMMYT)
thin
maximum
countries.
temper-
mean precipi-
is more
trouble-
Crop
to
McDonnell
[ 1998,
of measurement
a regular
grid
of
p. 1581 show
a table
tics of ten classes of interpolation
published
recently
is made
techniques.
tional
tional
Burrough
&
can provide
question
In this paper,
information
several
classified
ways
et al,
network
of
will
the
variables
simulation
is often
how
be generated
with
from
measurements
radiation,
etc.-
an
Climate
a network
essential
interpolation.
Jalisco
State
of
in this
paper.
A
(DEM) can be used as additional
to increase the prediction
accuracy
of the cli-
of this study is to find an optimal
elevation
way
data of the area into the interpo-
to increase
the prediction
maps.
In total,
forms
of thin
plate splines will
MATERIALS
input
for
maps (surfaces)
four
forms
accuracy
of kriging
of
and
be presented.
temperature,
Accuracy
METHODS
Two data sets are considered
years,
crop
of long-term
mean
The second
from
1940-I
-
for a square
the
solar
state
months
of pre-
because
coefficient)
author
146
of
April,
in this paper.
(2 19 years,
precipitation
values
The first data
from
1940-1990)
at 193 measurement
set consists of 136 long-term
990)
data. These data were
can
stations
AND
set consists
ered,
*Corresponding
over a sparse
DATA SETS
addi-
used classes
of measurement
of precipitation,
through
The
long-term
the
Model
in
models.
and
measured
climate
monthly
models.
to crop
addi-
of including
into two widely
provide
input
precipitation
stations
pro-
be used as a case study
Elevation
stations.
growth
climate
are used to
of these
et al, 19981. Climate
temperature,
techniques
three
be discussed.
Climate
Center
and sustainabili-
interpolation
- thin plate splines and krig-
techniques
Improvement
models
Hartkamp
mean
maximum
Mexico,
lation
20001, a com-
of those
of pre-
and limitations
essential
monthly
of including
In papers
Dirks
[eg,
The main purpose
can be used to increase the prediction
of interpolation
ing - will
are
characteris-
1998;
Goovaerts,
several
An important
information
accuracy.
1998;
between
with
in
mate maps [de Beurs, 19981.
points
techniques.
et a/,
[Goodale
1998; Pardo-lguzquiza,
parison
points.
variability
systems in developing
simulation
systems
information
applied
Systems (GIS). Data collect-
ed on a sparse Tablenetwork
interpolated
are commonly
growth
duction
Digital
INTRODUCTION
techniques
is
et a/
interpolation
productivity
maize and wheat
the opportunities
maps
in the data and their non-Gaussian
Information
the greater
and Wheat
evaluate
monthly
interpolation
Maize
aims to improve
long-term
in Geographical
sites
Rosenthal
results is most likely due to
coarse grid for spatial
ty of smallholder
plate
character.
Statistical
that
simulation
The International
infor-
From these
and trivariate
of precipitation
interpolated
cipitation.
are
Mexico.
as additional
best. The results of monthly
at
simulation.
temperature
considerably.
regression-kriging
some due to higher variability
splines
State
ature are much clearer than the results of monthly
tation,
crop growth
the relatively
Four forms
techniques,
for
Postal 6-641,
conditions
crop
their
and
of weather
important
ABSTRACT
Lisboa 27, Apartado
monthly
extracted
maximum
from
(2 19
temperature
ERIC [IMTA,
19961
area (600 km x 600 km, called D), covering
Jalisco,
May,
Mexico.
August
for these
between
and
months
elevation
In this
study,
September
the correlation
and
only
the
are consid(Pearson
precipitation
is
JAG Volume 3 - Issue 2 - 2001
Geostatistics and thin plate splines
l
greater than 0.5. Figure 1 shows the measurement
stochastic models while the method of thin plate splines
is a deterministic interpolation technique with a local stochastic component. It is well known that under certain
conditions these two interpolation techniques are equivalent to one another [Kent & Mardia, 19941. In this
paper, however, at least the function for modeling the
spatial correlation is chosen differently. The modeling of
the trend and the neighborhood used for prediction can
differ too.
sta-
tions and a DEM of the area.
Figure 2 shows scatterplots of long-term monthly maximum temperature (T,,,)
and long-term monthly mean
precipitation (Pmea, ) against elevation for August. The
correlation between T,,,, and elevation is -0.7 for April
and May and -0.9 for August and September. For Pmean
these values are 0.6 (April and May), -0.5 (August) and 0.6 (September). The scatterplot of P,,,,
shows statistically less attractive features.
Let the actual meteorological measurements be denoted
as z(st), Z(Q),..., z(sn), where si = (xi,y$ is a point in 0,
Xi and yi are the coordinates of point si and IZ is equal to
the number of measurement points. The elevation at a
point s in the area D will be denoted as q(s). A mea-
INTERPOLATION TECHNIQUES
Interpolation techniques can be divided into deterministic and stochastic models. Kriging technique is based on
A
L
0
200
600
400
Kllometera
FIGUREI: The meteorological stations [IMTA, 19961 and a DEM [USGS,
19971 of Jalisco State, situated northwest of Mexico City
..
..
..
.
..
.
+
=-a
. .
:m. ‘L
L
.
.
..
.
.
c
. .
.
.
.
.
.
.
.
.
.
n
.
c
. .
=
.I
,.
=
.
n m.
.I
9
. .
+.,
.
:
.
.
1
. .
.
.
. .
.A#
+
9,’
l..._:
.
.
.
‘=..
..
$‘.
1004
500
1500
2000
0
2504
FIGURE
2: SCatterplOtS
for
500
loo0
Tmax and fmean against elevation
for August.
147
.=
# .’ .
.
1500
ElWatiOll
ElWH!!Il
m
$-$. ‘m 8
.
0
..
.+
+
200fJ
2500
JAG
Volume
3 - Issue 2 - 2001
g and the parameters
fit and pz can
Geostatistics and thin plate splines
surement
considered
as realization
able will be denoted
be considered
by a lower
as a realization
of a stochastic
case. Therefore,
of stochastic
(ie, outcome
of a set of stochastic
some
spatial
locations
other
is specified
and whose
by some probabilistic
by minimizing:
Q(S)
that
dependence
The function
be estimated
q(s) can
variable
variables
value.
vari-
l
$
have
[z(Si) - S(Q) - Pl’J(Si) - P2q2(%)]2 + M!(S).
(6)
on each
giving
mechanism).
the
same solution
structure
as for
bivariate
thin
plate splines.
Bivariate
thin plate
Wahba
[I9901
spline
described
the theory
In the case of bivariate
thin
ments z(Si) are modeled
as:
f is an unknown
and
e(Si) are random
that
e(Si) are realizations
random
splines,
plate splines.
Trivariate
the measure-
Hutchinson
i = 1, . ..) 7%
deterministic
errors.
smooth
Commonly,
plate
(1)
j(si)
f can be estimated
Namely,
=
f(si,
qi)
J~(fl
is enlarged
the
minimization
for bivariate
there
bivariate
on the location
plate
function
(7)
Si. The function
[see Wahba,
is solved
splines.
of
thin
, ....n
i=l
E(Si7 qi),
problem
way
bivariate
&si,qj):
by several terms
thin
into
the
function
+
is another
elevation
replacing
qi is the elevation
where
that
the covariable
z(Si7 qi)
by minimizing
shows
in (1) by a t nvariate
it is assumed
of zero mean and uncorrelated
spline
[I9981
splines.
function
errors.
The function
thin plate
incorporating
Z(Si) = f(Si) + E(R),
where
of thin
plate
19901 and
in the same way as
The solution
can be writ-
ten as:
2
[4Si)
-
f(Si)12
+
XJz(f)
P(s)= &aj4j(S)
i=l
where
J~fl
is a measure
of smoothness
by means of the following
where
integral:
7 { (g)2+2(g)2+
($,‘)dx&
and (si,qJ
(3)
(8)
i=l
$1 = 1, $2 = x, $3 = y and $4
@j are polynomials,
= q; and Y =
Jz(f) = 7
cbiQ(hi)
+
j=l
of J1 calculated
h. The Euclidean
is calculated
hi between
distance
(s,q)
by
-m--m
and h is the so-called
lates the trade-off
to the
data
smoothing
smoothing
between
and
the
parameter
the closeness
smoothness
parameter
which
hi = J(Z
regu-
of the function
of the
function.
h can be estimated
Scaling
The
becomes
scale of variation
by generalized
lowed
(2) is solved
with?as
f(s) =
kaj4j(s)
+2 W(h)
j=l
Equation
+(Y
Ordinary
where
distance
(4) can be calculated
s and
Wackernagel,
Uj and bi in
the following
hi between
The coefficients
-yiJ2).
by solving
a linear
system
thin plate
a partial
tional
thin
plate
plate
linear
spline
where
+
can be enlarged
by incorporating
AdSi)
the function
of known
are modeled
the basis of scatterplots
S(R)
model
combination
The measurements
z(Si) =
model
to calculate
(GCV) on different
with
scal-
the lowest
+
functions
in the following
lated
+
4%)~
to be estimated,
function
and
PI and
1, . ..) 72
g(s) being an unknown
82 are parameters
with
else1993;
are modeled
in
are considered
F in point
errors.
1,2,...,n
i =
E(Si)r
may
of zero
mean
p(s)
of
contain
E{F(s)} to model
The trend
constant
(9)
as realizations
Si, which
p(s) =
E(Si) are realizations
correlation
can be quantified
(5)
a
possible
and uncorre-
is assumed
to
be
p.
between
where
smooth
between
unknown
148
we
two
the measurement
by means of the semivariance
y(s, h) = +r[F(s)
f(S) = g(s) + 81 q(s) + 82 q*(s) is the
function
+
function
random
The spatial
i =
Cressie,
way, on
in Figure (2):
P2Q2(Si)
f(Q)
equal to an unknown
intoJ:
are well explained
1989;
19951. The measurements
function
trends;
an
kriging
Srivastava,
way:
deterministic
to
addi-
(q) into (1). This is done by adding
information
unknown
spline
&
in this case,flsJ
random
spline
thin
of ordinary
z(Si) =
n.
The bivariate
[I9981
and select the scaling
[Isaaks
where,
Partial
We fol-
directions
of Hutchinson
kriging
The principles
$1 = 1, $2 = x and $3 = y; and
Y = hz In(h) of the Euclidean
Si (h, = J(x--x~)~
(4)
i=l
Qj are polynomials,
of order
and the
units
value of the GCV.
ear combination
where
in different
cross validation
ings of elevation
the lin-
19981, because
[Hutchinson,
can vary in different
the suggestion
the generalized
problem
important
X, y and q can be expressed
cross validation.
The minimization
- Xi)2 + (y - yi)2 + (q - q$!.
assume
points.
that
- F(s + h)]
h is the
Assume
that
Euclidean
the trend
points
function:
(IO)
distance
is constant
Geostatistics
and thin plate splines
JAG
cross-semivariance
and y(s,h) is independent of S. A parametric function is
used to model the semivariance for different values of h.
In this research, the spherical model - c Sph(a) - is used.
c{;(:)-;(k)“},
h>a
c,
where c is the scale parameter of the semivariance function and a is a parameter which determines the so-called
range of spatial dependence. The random errors (and/or
the spatial nugget random function) have a variance CO.
For the stochastic variable 2 the following semivariance
function is used: CO Nug(0) + c Sph(a).
is1
where the weights wi are derived from the kriging equations by means of the semivariance function; n is the
number of measurement points within a radius from
point SO(in this study we have taken a radius of 240 km).
The parameters of the semivariance function and the
nugget effect can be estimated by the empirical semivariance function. An unbiased estimator for the semivariance function is half the average squared difference
between paired data values.
&
$y[%(Si,
- %(Si+ q2
(15)
The interpolated value at an arbitrary point SOin D is the
realization of the (locally) best linear unbiased predictor
of F(so) and can be written as weighted sum of the measurements:
f(scl)=
2 WliZ(Si)+ 2 “ZjP(Sj)
i-1
Odeh et al [I9951 compared, among other techniques,
three forms of regression-kriging (comparable with kriging with external drift). The idea of regression-kriging, in
this paper, is that we characterize the trend component
p(s) of the model for the random function F(S) as an
unknown linear combination of known functions (regression model). In ordinary kriging the trend component is
modeled as constant; in the usual form of universal kriging the trend component is modeled as a polynomial of
a certain degree. In our application the trend is modeled
as:
(13)
where n(h) is equal to the number of data pairs of measurement points separated by the Euclidean distance h.
0 +&q(S)
+
(17)
h?(S)
The interpolated value at location se can be calculated by
a linear combination
of the regression model and a
weighted sum (ordinary kriging) of regression residuals
cokriging
Cokriging makes use of different variables, modeled as
realizations of stochastic variables. In this study, elevation - f&) - of the area D is used as covariable to predict values of T,,
and P,,,,.
The spatial dependence is
characterized
by two semivariance functions yzz(s,h),
,-&,h) and the cross-semivariance function:
Z*(Si)
=
%(Si)
-
8
-
bljlq(Si)
-
bq2(Si).
This results in:
P(%) = s + Wq(So) + &12(SO) + e
Y&,
(16)
j=l
Regression-kriging
a=1
Ordinary
function
where ml is the number of measurements of Z(S) taken
within a radius (of 240 km) from SO(out of the modeling
data set), m2 are the number of meteorological stations
within a radius of 240 km from SO (out of the modeling
and validation set). The weights wli and ~2~ can be
determined using the two semivariance functions and
the cross-semivariance function.
The interpolated value at an arbitrary point SOin D is the
realization of the (locally) best linear unbiased predictor
of F(se) and can be written as weighted sum of the measurements.
“I(h) =
3 - Issue 2 - 2001
where n(h) is the number of data pairs where both variables are measured at an Euclidean distance h.
(11)
(
Volume
%,@) = ~~~~~(Si)-~(.i+h)][q(sl)-q(si+h)]
2n(h) i=l
Olhla
y(h) =
l
Wz*(%)
(18)
i=l
h) = ;E {[Z(S) - Z(S + h)][Q(s) - Q(s + h)] (14)
The difficulty of this form of regression-kriging,
and of
universal kriging in general, is that the parameters of the
regression model and the parameters of the semivariance
function of the spatial correlated regression residuals
should
be
estimated
simultaneously
[Laslett
&
McBratney, 19901. Under the assumption of normality,
the parameters can be estimated by restricted maximum
likelihood (REML), which is one of the techniques to estimate the parameters of the regression model and the
parameters of the semivariance function simultaneously
[Gotway & Hartford, 19961.
To ensure that the variance of any possible linear combination of the two stochastic variables is positive, a socalled linear model of coregionalization
is applied. This
model implies that each semivariance and cross-semivariante function must be modeled by the same linear combination of semivariance functions [Isaaks & Srivastava,
19891. Furthermore,
the matrix of coregionalization
should be positive semi-definite. A nested semivariance
function is used with a nugget and two spherical semivariance functions with different ranges. The cross-semivariance function
can be estimated by the empirical
149
Geostatistics and thin plate splines
Trivariate
JAG
The semivariance functions
regression-kriging
Finally, trivariate regression-kriging
will be introduced.
Trivariate regression-kriging is a form of regression-kriging, where trivariate ordinary kriging is applied on the
regression residuals. The trend is chosen equally as in
trivariate thin plate splines. The interpolated value at a
location so can be calculated by:
The weights wi are determined by the semivariance function, which is a function
of the Euclidean distance
between two points (si, qi) and (s, q). The units are of
different order and scaling becomes important, the same
scaling is used as for trivariate thin plate splines. In this
case, REML is not applied because of limitations of the
software used. The residual semivariance function is now
estimated from the OLS regression residuals.
of interpolation
Only the
tions are
We used
increased
the point
as for all
show the
validation
techniques
To compare the interpolation
techniques, the original
data set is divided into a modeling data set and a validation set of 25 measurement points. The 25 points are not
chosen randomly, but are selected by the authors, so
that the area is still reasonably covered by measurement
points. Five validation sets are chosen from each data
set. Each validation set contains different measurement
points from the original data sets. Predictions on the
locations of the validation points - f(si) - and the measured values at these locations - Z(Si) - are compared by
the two criteria: the Mean Square Error (MSE) and the
Maximal Prediction Error (MPE).
@(si)
- z(si)]’
(21)
a=1
where n, (= 25) is the number of validation
for ordinary
3 - Issue 2 - 2001
kriging are esti-
recorded elevations of the meteorological staused for interpolation with ordinary cokriging.
a DEM of the area but prediction accuracy
substantially as just the recorded elevation at
to be predicted (validation point) was available,
other interpolation techniques. Tables 1 and 2
results of all 7 interpolation techniques for 5
sets for T,,, and Pm,,, respectively.
The results of Tables 1 and 2 demonstrate the benefit of
using the covariable elevation. Especially for Tmax the
differences of interpolation with elevation and without
elevation are convincing. This is due to the high correlation between elevation and Tma,. Comparing regressionkriging, cokriging, trivariate regression-kriging, trivariate
thin plate splines and partial thin plate spline for Tmax
shows an advantage for the two interpolation techniques
which
made use of three-dimensional
coordinates
(trivariate). The differences between the results of the
interpolation
techniques are less clear for Pm,,,. Only
for validation sets 1 and 2 did the trivariate interpolation
techniques perform more accurately concerning the MSE.
(20)
MSE = $
Volume
mated by weighted least squares with GSTAT [Pebesma &
Wesseling, 19981. For cokriging the semivariance functions, by means of the linear model of coregionalization,
are estimated by COREG [Bogaert et al, 19951. The residual semivariance function for trivariate regression-kriging
is estimated in a relatively simple way. First the trend is
estimated by ordinary least squares (OLS), followed by
the estimation of the spatial variability of the regression
residuals. For regression-kriging, where the semivariance
function depends only on x and y, the parameters of the
regression model and the parameters of the semivariance
function
are estimated simultaneously
by the REML
option of PROC MIXED in SAS [Littell et al, 19961. Figures
3 and 4 show some examples of fitted semivariance functions (with models and parameter values) for T,,,,, and
P mean for cokriging and trivariate regression-kriging.
(19)
Comparison
l
points.
RESULTS
The automatic calculation procedure of thin plate splines
allows a straightforward
analysis of these techniques.
There is no need for any prior estimation of the spatial
dependence
of
measurement
points.
ANUSPLIN
[Hutchinson, 19971 is used to perform the analyses. For
trivariate thin plate splines it is useful to optimize the elevation scale [Hutchinson,
19981. Therefore the square
root generalized cross validation for trivariate thin plate
splines is determined at different scales of elevation
kilometer).
hectometer
and
(meter,
decameter,
Decameter is the optimal scaling for Tmax and kilometer
for Pm,,,. This way of scaling is found to be sufficient,
because no major differences between the GCV of two
successive scales are found. Trivariate regression-kriging
is applied with the same scaling as trivariate thin plate
splines.
The prediction results of Pm,,, for validation set 1 for
August and September are very poor for bivariate thin
plate splines and partial thin plate splines. This is mainly
caused by two validation points in the South-East of the
area, which have a large prediction error. Probably, this
is caused by a local trend at adjacent measurement stations.
DISCUSSION
In this paper 7 interpolation techniques are discussed, 5
including and 2 excluding elevation as additional information. The two techniques excluding elevation perform,
especially for Tmax considerably less accurately than the
techniques including elevation. The MSE and MPE values
150
Geostatistics
and thin plate splines
JAG
l
Volume
3 - issue 2 - 2001
400000
350000
3Ocim
g
25mOc
1
200000
I
/
/
15OcoO
100X300
1
50000
2.52 Nug(0) + 5.1 Sph(l.2)
0.5
di
0 -_
Aance
- 586
-13.6 Nug(0)
+ 2.9 Spq$
Sph(l.4a rii x elevation
Sph(3)
_899
-259
;/./I
159
/
0 L
0
2
/
/
+204
/A7
.-
1.5
/263
-
73.6 Nug(0) + 67264 Sph(l.2)
0.5
elevation
+ 382461 Sph(3) I
disltance
1.5
2
2
-200 -
8
-400
5
‘1
-600
z
-600
1Qo8
\j +516
\.\
t
g
\\*526
-1000
\
2.46 Nug(0) + 3.15 Sph(B%$/
0
0
20
40
60
L
60
1oc
distance
PURE
3: Estimated semivariance functions for cokriging and trivariate regression-kriging for Tmm, April, validation set I. Upper left:
upper right: elevation; Lower left: cross semivariance function between T,,,, and elevation; Lower right: residual semivariance
fut??ion TmM for trivariate regression-kriging.
400000
*915
20
7w
+H835
___--
+ 91 3m_--e
35OMJO
+ 969
300000
15
k?
p: 250000
E
s
/
k? 10
E
I
5
5
,/* 377
-
2ooOOo
/
+ 662
/
50000
0
3.53 Nug(0) + 12.3 Sph(0.6) + 3.59 S$#
b
0.5
diitance
A<3
1.5
elevation
24.65 Nug(0) + 47660 Sph(0.6) + 401131 Sph(3) 2
/’
0
2
2
0
0.5
di:tance
:F
1800
1.5
2
* 622
* 936
1600
+ 1&?3wq
*1526
,’
1400
$
5
=
2
c
,/’
1200
1g36
*930
J
i
,/
/1324
/
1000
t
1:
600
2
0
600
400
i
,/+
-I
200
,/
1034
*66
+754
/j266
april x elevation
9.33 Nug(0) + 766.2 Sph(0.6) + 1200 Sph(3) L
0
0
0.5
disknce
1.5
5.16 Nug(0) + 4.86 Sph(1.Ti’ 4) 2
:
0
2
0.5
dikmce
1.5
2
FIGURE 4: Estimated semivariance functions for cokriging and trivariate regression-kriging for Pmean, April, validation set 1. Upper
left: P,,,.
Upper right: elevation; Lower left: cross semivariance function between Pmean and elevatron; Lower right: residual semivariance function Pmean for trivariate regression-kriging.
151
Geostatistics and thin plate splines
JAG
l
Volume 3 - Issue 2 - 2001
TABLE 1: Results of the Mean Square Error (MSE) and the Maximum Prediction Error
(MPE) for 5 validation sets (VI-v5) of a long-term monthly maximum temperature (I”,,).
The values with the lowest MSE and MPE of the 7 interpolation techniques are written in
italics
Interuolation techniaue
vl
ordinsry kriging
reunion-~ig~g
cokriging
trivariate regression-kriging
bivsriate thin plate splines
tfivariate thii plate splines
partial thin plate spliiea
v2
ordinary kriging
regression-kriging
cokriging
trivariate regression-kriging
bivariate thin plate splines
trivariate thin plate splines
partial thii plate splines
ordinary kriging
v3
regression-kriging
cokriging
trivariate regression-kriging
bivariate thii plate splines
trivariate thin plate splines
partial thin plate splines
v4
ordinary kriging
reunion-~i~ng
cokriging
trivariate regression-kriging
bivariate thii plate splines
trivariate thin plate splines
partial thin plate splinea
ordiiarykriging
v5
regression-kriging
cokriging
trivariate region-~i~ng
bivariate thin plate spliies
trivariate thin plate splines
partial thin plate splines
MPE
MSE
wr
10.3
5.0
may aw
9.8
4.9
5.5
;:I
10.5
4.8
4.8
7.2
4.5
4.3
3.5
8.4
3.8
4.1
5.9
5.0
9.9
4.5
5.2
5.9
4.2
3.7
3.5
7.3
4.0
3.8
5.6
4.3
;.‘5
7.5
5.8
7.1
7.4
3.9
5.1
9.4
7.5
3.8
3.9
6.8
3.8
4.6
2.0
7.4
2.6
2.6
;.“o
7.3
5.3
6.2
7.2
4.2
5.0
3.7
7.6
3.7
4.1
6.4
2.4
4.3
2.0
7.1
3.6
2.3
4.0
are lower when elevation is used as additional information for prediction,
especially when the correlation
between the two variables is high.
8.2
4.0
3.9
2.9
8.8
3.1
3.4
3.9
2.0
1.9
2.1
4.9
1.8
2.2
5.0
1.7
sw
5.0
2.1
2.3
1.5
5.8
1.6
2.0
5.7
1.2
2.0
1.3
8.3
1.8
8.4
3.4
4.1
2.9
8.3
3.0
3.7
3.9
2.0
1.8
2.0
5.0
1.8
2.1
5.2
1.8
2.1
1.9
5.2
2.3
2.0
5.4
1.9
2.6
1.8
6.1
1.8
2.2
5.4
1.1
1.7
1.2
7.5
1.7
1.1
1.0
::s”
5.0
2.1
1.7
w
may aug sep
6.1
5.0
5.5
4.6
6.1
5.8
4.9
5.6
4.9
5.7
4.5
4.4
5.1
6.9
5.9
5.5
4.9
6.6
5.4
5.3
5.7
5.3
5.9
5.2
6.6
5.3
6.4
6.4
3.7
5.6
3.9
6.5
4.3
3.7
6.8
4.4
5.5
3.5
6.7
5.3
3.5
5.2
6.7
5.7
5.4
4.7
6.4
5.5
5.0
5.1
5.2
6.0
5.3
5.9
6.1
6.2
6.6
4.0
5.9
4.0
6.6
4.2
4.1
7.0
3.8
5.0
3.6
7.0
5.6
3.4
5.2
4.2
4.3
3.8
5.8
4.3
3.9
5.3
3.3
3.0
3.3
5.6
3.5
3.7
5.4
3.7
4.4
3.8
5.2
4.3
9.5
6.4
4.1
5.1
2.9
6.4
2.7
3.9
5.7
2.8
3.3
2.2
6.8
2.8
2.8
5.4
4.2
4.5
8.7
5.2
4.1
4.4
5.6
3.7
2.9
3.2
6.0
3.4
3.7
5.3
3.9
4.5
4.0
5.1
4.4
4.5
6.8
4.2
5.6
3.5
6.8
$.I
4.4
5.5
2.4
2.8
2.1
6.2
2.6
2.7
elevation is lower, which causes smaller differences
between including and excluding elevation. In general,
precipitation data are clearly non-Gaussian. Although a
transformation can be considered, this has been reported
to have disadvantages for local estimation [Roth, 19981.
From the techniques which include elevation, trivariate
thin plate splines and trivariate regression-kriging
seem
to perform best. Cokriging, is the most time-consuming
interpolation
technique
to implement
in this study.
Therefore, in this case study, cokriging is not preferable.
The main reason for cokriging having relatively poor prediction results is the fact that a linear relation is assumed
between climate variable and elevation. The other techniques (including elevation) used in this paper, do not
assume such relation because regression models with
more regressors and trivariate techniques are used.
Beek et al [I9921 stress the importance of interpolation
techniques for crop growth simulation. In this paper more
advanced forms of kriging and thin plate splines are
applied. Especially, the trivariate forms of kriging and thin
plate splines performed well. The main advantage of thin
plate splines over kriging is the operational simplicity of
this technique, which can be very important from a practical point of view. The kriging procedure requires more
effort and experience. For Tmax the predictions results of
trivariate regression-kriging
are slightly more accurate
compared to the results of trivariate thin plate splines, but
the differences are small. Within the geostatistical framework trivariate regression-kriging,
as described in this
paper, seems to be an attractive option.
The results of Tmax are much clearer to interpret than the
results of P,,,,.
There is much more variability in the
P Mean predictions resulting from a higher variability in the
data. The correlation between the climate variable and
152
Geostatistics
and thin
plate
TABLE
JAG
splines
2: Results of the Mean
Square
Error (MSE) and the Maximum
Prediction
l
Volume 3 - Issue 2 - 2001
Error (MPE)
for 5 validation sets (~14) of a long-term monthly mean precipitation (P,,,). The values
with the lowest MSE and MPE of the 7 interpolation techniques are written in italics.
Interpolation technique
MPE
MSE
apr
26.2
26.7
26.4
26.4
26.2
26.4
26.9
may
22.4
22.7.
23.4
22.6
22.9
22.9
aug
84.8
89.2
77.3
62.2
172.5
58.9
183.6
=p
80.6
78.7
59.5
66.5
195.3
63.6
169.5
14.9
10.9
13.6
14.0
12.8
10.5
10.9
44.9
43.9
42.5
44.8
46.0
44.7
43.8
256.2
252.4
249.3
260.6
250.3
255.0
248.7
208.4
203.5
203.1
194’.4
201.4
6.8
4.2
17.6
17.5
19.7
17.8
19.6
17.8
17.1
93.4
74.9
82.6
95.2
88.1
90.0
87.9
114.6
104~
115.1
138.8
138.2
145.9
140.7
25.4
18.7
18.5
17.3
25.7
16.4
165.6
162.0
174.9
200.1
190.0
217.3
138.0
129.i
148.9
195.1
157.0
196.6
18.4
17.2
24.4
27.7
23.0
21.3
25.5
24.5
182.3
161.1
135.2
153.8
149.0
83.3
136.6
88.2
139.4
164.2
135.1
148.0
132.5
132.1
131.8
129.1
apr
36.7
36.1
35.3
83.Q
44.1
33.7
36.9
may
62.4
64.4
58.1
50.6
78.5
50.8
67.4
aug
1251.9
1182.1
1132.8
728.4
3562.6
753.6
3456.6
?23.6
1111.2
820.0
520.1
3418.2
525.9
3101.4
17.5
13.7
19.3
17.6
13.8
13.3
7.8
3.5
5.8
6.7
6.4
3.0
3.4
158.5
168.0
172.3
170.7
170.2
168.7
166.8
55.4
51.9
48.3
49.3
55.2
48.5
52.5
4715.1
4712.9
4687.3
4479.5
4802.2
4492.2
5319.8
1804.2
1683.5
1652.7
1787.4
1762.8
1665.3
1720.1
4261.5
4175.3
4062.4
9554.d
3834.3
3623.4
4037.6
1206.7
971.0
1012.4
1252.6
1264.2
1230.0
1397.2
60.3
44.9
59.6
47.0
71.6
45.1
42.4
2477.1
2467.8
2537.4
3252.8
3069.9
3696.9
2858.4
3150.!
3321.0
3517.8
4038.0
4029.3
4137.1
3767.8
14.3
11.2
bivariate thii plate spliies
trivariate thii plate splines
partial thin plate spliies
17.4
9.5
13.6
13.7
15.9
9.5
9.1
ordiiary kriging
v5
regression-kriging
cokriging
trivariate regression-kriging
bivariate thii plate splines
trivariate thin plate spliies
partial thii Dlate sdines
8.6
6.8
12.2
8.4
7.6
6.8
6.8
61.5
68.7
91.1
58.5
84.3
64.2
67.7
2193.9
1891.4
2276.2
2170.4
1607.1
2419.2
1724.2
2873.6
2291.8
2538.7
2723.6
1984.!
2923.4
2050.3
8.6
8.3
10.0
8.3
7.0
8.7
7.9
ordii
kriig
vl
regression-kriging
cokriig
trivariate regression-kriging
bivariate thiu plate splinea
trivariate thin plate spliies
partial thii plate splines
ordii
kriig
v2
regression-kriging
cokriging
trivariate regression-kriing
bivariate thin plate spliies
trivariate thin plate splines
partial thii plate spliies
ordinary kriging
V3
regreasion-kriging
cokriig
trivariate regression-kriging
bivariate thin plate spliiea
trivariate thin plate spliiea
partial thii plate spliies
ordinary kriging
v4
regression-krigiug
cokriging
trivariate regession-kriging
la.0
10.0
12.2
14.0
11.1
11.4
191.a
198.6
Goodale, C.L., J.D. Aber & S.V. Ollinger, 1998. Mapping monthly
precipitation, temperature,
and solar radiation for Ireland with
polynomial regression and a DEM. Climate Research 10: 35-49.
ACKNOWLEDGEMENTS
The authors thank A.C. van Eijnsbergen and A. Stein for
fruitful discussions at various stages of this work.
Goovaerts, P., 2000. Geostatistical approaches for incorporating elevation into the spatial interpolation
of rainfall. Journal of
Hydrology 228: 113-129.
Gotway, C.A. & A.H. Hartford,
1996. Geostatistical methods for
incorporating
auxiliary information
in the prediction of spatial
variables. Journal of Agricultural, Biological, and Environmental
Statistics 1: 17-39.
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de interpolation
por
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la
temperatura
maxima
mensual
y la precipitation
media mensual
en el Estado de Jalisco en Mexico.
Se muestra
que las tecnicas
que utilizan datos altimetricos
coma information
adicional
mejoran considerablemente
la prediccidn
de resultados.
Entre estas
tecnicas,
el kriging de regresion
trivariada
y 10s filtros (thin plate
splines) trivariados
dieron 10s mejores
resultados.
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de temperatura
maxima
mensual
son m6s nitidos que 10s resultados de precipitacidn
media mensual,
porque
la modelizacion
de la precipitation
es m&s dificultosa
debido
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154