estimation of the parameters of isotropic semivariogram model
TRANSCRIPT
Applied Mathematical Sciences, Vol. 9, 2015, no. 103, 5123 - 5137
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.54293
Estimation of the Parameters of Isotropic
Semivariogram Model through Bootstrap
K. N. Sari
Department of Mathematics and Natural Sciences
Bandung Institute of Technology, Indonesia
U. S. Pasaribu
Department of Mathematics and Natural Sciences
Bandung Institute of Technology, Indonesia
O. Neswan
Department of Mathematics and Natural Sciences
Bandung Institute of Technology, Indonesia
A. K. Permadi
Department of Petroleum, Bandung Institute of Technology
Indonesia
Copyright © 2015 K. N. Sari et al. This article is distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited.
Abstract
In isotropic semivariograms, ordinary least squares can estimate nugget effect and
sill by partitioning its range. By conducting simulation, a semivariogram model
with previously given parameters will be estimated through bootstrap method. Least
square-bootstrap (LS-Bootstrap) will be applied to estimate the parameters of the
model after resampling the errors of the model. The selection of the resulting
semivariogram model from bootstrap method will be affected by the number of
distance lags, the precision level of the range partitions, the number of bootstrap
iterations, and the given reference model. The exponential and Gaussian models are
sufficiently good in the estimation for the models with the same references. Meanwhile, the estimation yielded from spherical model is quite far from the reference
5124 K. N. Sari et al.
exponential and Gaussian models, with the mean square error value reaching 713.
The estimation with bootstrap method which is the same as the reference model will
be faster to converge with the maximum iteration of 50. Besides, bootstrap method
enables to obtain the point estimates and interval estimates of the nugget effect, sill,
and range parameters.
Keywords: Bootstrap, confidence interval, isotropy, least square, semivariogram
1. Introduction
There are some well-known resampling methods such as randomization test,
cross-validation, bootstrap, and jackknife. These methods are often applied in
regression, time series, and principle component analysis. Randomization test has
been used to test linear independence between random variables in wavelength
selection in near-infrared spectral analysis [1]. Cross-validation has been applied to
estimate unbiased prediction error [2]. Both of bootstrap and jackknife have been
to estimate residuals of a predictor of the random variables and apply in model
autoregressive (AR)-sieve bootstrap model in time series [3]. Both also have been
used to give stability index of principle component analysis result [4].
Resampling methods above are also widely used in geostatistics especially
bootstrap and jackknife. Those methods often applied in kriging. Kriging is a
method of calculating estimates of regionalized variable at a point, over an area, or
within a volume, and uses as a criterion the minimization of an estimation variance
[5]. Until now, kriging used jackknife to obtain accurate estimate for variable values
observed by removing one by one observation values in a certain location, and then
estimating the parameters with the rule that the square error has to be as small as
possible. While, Den Hertog et al. [2006, 6] developed bootstrapped kriging to
estimate the predictor variance as a function for an unobservable location. In
application, Waterman [7] used weighted jackknife-ordinary kriging in estimation
of gold and copper ore deposits at Grasberg, Papua. That method was better than
jackknife ordinary kriging method because it was used for data that had outliers and
not symmetrical.
In kriging, one method used to estimate the value at an unobservable location
is determining semivariogram model. Semivariogram is a measure of variance from
difference between two spatial locations that separated by certain distance.
Semivariogram is divided based on the presence or absence of the influence of the
angle between pair of locations respectively called anisotropic and isotropic
semivariogram. That model has 3 parameters such as nugget effect, sill, and range.
Nugget effect is initial semivariance when autocorrelation is highest; or just the
uncertainty where distance (d) is close to 0, sill is the value that semivariance flat,
and range is lag distance where the sill is reached. There are 7 semivariogram
models such as: nugget effect, linear, spherical, exponential, power functions,
Estimation of the parameters of isotropic semivariogram model… 5125
Gaussian, and hole effect. Mostly, the model commonly used are exponential,
Gaussian, and spherical. The three parameters and semivariogram models are
illustrated in Figure 1.
Figure 1. Plot of semivariogram models (γ(d)) and 3 parameters model (nugget
effect, sill, and range)
The main problem in semivariogram modeling is to estimate parameters model.
Till now, those parameters are estimated by applying numerical methods because
the nonlinearity of the semivariogram function involved. Several methods have
been proposed to estimate parameters of the semivariogram models, such as: least
squares [8], generalized least squares [8], maximum likelihood [9], restricted
maximum likelihood [9], and weighted least squares. Zimmerman and Zimmerman
[10] found that weighted least squares is sometimes the best procedure and never
does badly, whereas the others are subject to erratic behavior in some requirements
has made it the primary choice among semivariogram model estimation methods.
Some background for weighted least squares method is the ordinary least squares
and generalized least squares.
In that estimation, the number of lag distance as sample points only a few
frequently, so it needs the nonparametric techiques such as jackknife and bootstrap
to add sample point. Along with the development of computation field, the
nonparametric technique are developed rapidly. Moreover, using of resampling
methods in the estimation semivariogram model is still limited. Therefore, this
paper will applied bootstrap method and the least square method to estimate the
vector of parameters isotropic semivariogram model. It’s expected that the
parameter estimation results obtained from applying least square-bootstrap (LS-
Bootstrap) method will be close to the given semivariogram model within a certain
mean square error.
5126 K. N. Sari et al.
2. Estimation of The Parameters of Isotropic Semivariogram
Model
2.1 Notation
The following notations will be used to formulate mathematical symbols that
needed to study the estimation of parameters semivariogram model:
d - Distance between pair of locations
0θ , 0θ̂ - Nugget effect and its estimator
1θ , 1θ̂ - Sill and its estimator
2θ , 2θ̂ - Range and its estimator
0 1 2, , 'θ θ θθ - Parameters vector
0 1 2ˆ ˆ ˆˆ , , 'θ θ θθ - Estimated model-parameters vector
0 1 2ˆ , ,B θ θ θθ
- Estimation of parameters vector by bootstrap
1ˆ( )
ˆ( )
ˆ( )n
γ d
γ d
γ d
- Empirical semivariogram
1( , )
( , )
( , )n
γ d
γ d
γ d
θ
θ
θ
- Semivariogram model
2
ˆ( ) ( , )d
γ d γ d θ
- Sum of the squared residuals
1
( )n
i
i
d
- Sum of the experimental semivariogram where n is
the number of distance lag
ˆ
1
1 i
nd
i
e
- Sum of the exponential function
ˆ
1
( ) 1 i
nd
i
i
d e
- Sum of the multiplication between the experimental
semivariogram and the exponential function
2ˆ
1
1 i
nd
i
e
- Sum of the squared the exponential function
2
ˆ
1
1 i
nd
i
e
- Sum of the Gaussian function
2
ˆ
1
( ) 1 i
nd
i
i
d e
- Sum of the multiplication between the experimental
semivariogram and the Gaussian function
2
2
ˆ
1
1 i
nd
i
e
- Sum of the squared the Gaussian function
Estimation of the parameters of isotropic semivariogram model… 5127
3
1
ˆ ˆ1.5 0.5n
i i
i
d d
- Sum of the spherical function
3
1
ˆ ˆ( ) 1.5 0.5n
i i i
i
d d d
- Sum of the multiplication between the experimental
semivariogram and the spherical function
2
3
1
ˆ ˆ1.5 0.5n
i i
i
d d
- Sum of the squared the spherical function
p - Number of lag distance
B - Number of bootstrap iteration
2.2 Estimation of parameters
This research applies ordinary least squares to find the estimator of θ by
minimizing 2
ˆ( ) ( , )d
γ d γ d θ . In ordinary least squares, all the empirical
semivariogram are assumed to be uncorrelated with each other and to have the same
variance. The objective function to be minimized in ordinary least squares is can be
written in matrix notation as ˆ ˆ( ) ( , ) ' ( ) ( , ) 'γ d γ d γ d γ d ε ε θ θ .
Through ordinary least squares method, obtained θ̂ . Range will be determined
first by partitioning it using systematic random sampling by taking minimum
distance interval until maximum of lag distance. Then, nugget effect and sill will
be estimated using ordinary least squares method, as in the estimation of the
parameters in simple linear regression model. In this paper, parameter estimations
are conducted for 3 semivariograms, i.e. exponential (exp), Gaussian (gauss), and
spherical (sph). These models are respectively formulated as follow:
exp 0 1
2
ˆ ˆˆ ( ) 1 expˆ
dγ d θ θ
θ
,
2
gauss 0 1
2
ˆ ˆˆ ( ) 1 expˆ
dγ d θ θ
θ
, and
3
sph 0 1
2 2
ˆ ˆˆ ( ) 1.5 0.5ˆ ˆ
d dγ d θ θ
θ θ
.
2.3 Parameters estimation for exponential model
Let 2
1ˆ
ˆ
,
1
( )n
i
i
d a
, ˆ
1
1
1 i
nd
i
e b
, ˆ
1
1
( ) 1 i
nd
i
i
d e c
and
2ˆ
1
1
1 i
nd
i
e d
then the estimators are 1
0 1ˆ ˆ ba
n n
and
11
1 2
1
1
ˆ
abc
n
bd
n
.
5128 K. N. Sari et al.
To obtain the estimation of nugget effect, the sill will be estimated first. Meanwhile,
the range will always appear in the two parameters. By considering exponential
model form, the range can be written as
1
0 1
ˆlnˆ
ˆ ˆ ˆln ( )i id d
.
2.3 Parameters estimation for gauss model
Let 2
1ˆ
ˆ
,
1
( )n
i
i
d a
, 2
ˆ
2
1
1 i
nd
i
e b
,
2ˆ
2
1
( ) 1 i
nd
i
i
d e c
and
2
2
ˆ
2
1
1 i
nd
i
e d
, then the estimators are 2
0 1ˆ ˆ ba
n n
and
22
1 2
2
2
ˆ
abc
n
bd
n
.
If range is also estimated by applying ordinary least square, the formula will be
obtained as follows:
2 2 2
ˆ ˆ ˆ2 2 2
0 1
1 1 1
ˆ ˆ( ) i i i
n n nd d d
i i i i
i i i
d d e d e d e
2 2ˆ2
1
1
ˆ 0i
nd
i
i
d e
To estimate range, the solution for 2
ˆ1 id
iu e
should be computed, also the
solution of 2 2 2 2 2
0 1 1
1 1 1 1
ˆ ˆ ˆ( ) 0n n n n
i i i i i i i i i
i i i i
d d u d u d u d u
.
2.4 Parameters estimation for spherical model
Let 2
1ˆ
ˆ
,
1
( )n
i
i
d a
, 3
3
1
ˆ ˆ1.5 0.5n
i i
i
d d b
, 3
3
1
ˆ ˆ( ) 1.5 0.5n
i i i
i
d d d c
and 2
3
3
1
ˆ ˆ1.5 0.5n
i i
i
d d d
, then the estimators are 3
0 1ˆ ˆ ba
n n and
33
1 2
3
3
ˆ
abc
n
bd
n
.
If the range is also estimated using ordinary least square, then the formula in the
form of polynomial will be obtained: 2 3 5ˆ ˆ ˆ ˆ 0 p q r s t where
2 3 3 4
1 0 1
1 1 1 1
ˆ ˆ ˆ1.5 , ( ) , 2 ,n n n n
i i i i i
i i i i
p d q d d d r d
6
1 0
1 1 1
ˆ ˆ0.5 , ( )n n n
i i i i
i i i
s h t d d d
.
Each of the parameters estimation above contains another parameter such as
range and sill. The range always appears in nugget effect and sill formula,
meanwhile the sill always appears in nugget effect formula. With this reason, the
process of estimating parameter can only be executed numerically, by computing
the range first as written above. Along with the rapid development in computation
methods, the parameter estimation methods are also developing, i.e the ones using
nonparametric methods.
Estimation of the parameters of isotropic semivariogram model… 5129
2.5 Bootstrap method
Bootstrap method, initially introduced by Efron in 1979. This is a procedure
for resampling data with replacement, expecting the samples can represent the
population. Bootstrap enables to do statistical inference without making any
assumptions of strong distributions for any estimator’s sampling distribution or
without any available analytical formulation. The data used doesn’t need to be
assumed strictly in its distribution, it is small-sized and original, not resulting from
any simulation [11].
In the estimation of the semivariogram parameters, if real data is unavailable,
then it is a must to do simulation to generate normally distributed errors with 0 as
the mean and 1 as the variance which will be added to semivariogram model to
obtain the semivariogram values considered as experimental semivariogram.
Isotropic semivariogram model can take one of the models: exponential, Gaussian,
and spherical. Then bootstrap method will be applied to resample the errors of
semivariogram model to obtain the estimations of model parameters as many as the
bootstrap iterations. Semivariogram model resulting from bootstrap will be
obtained by taking average of those three parameter estimations of the model.
Bootstrap procedure for estimation will be described in Chapter 3.
3. Results and Discussions
3.1 LS-Bootstrap Procedure
Firstly, least square is used to estimate effect nugget and sill by partition its
range, the following steps are conducted:
i. Suppose one of the reference semivariogram model: (1) exp, (2) gauss, or (3)
sph with respectively parameters vector 0 1 2, ,θ θ θref ref ref
θ .
ii. Select p for forming semivariogram function.
iii. Select d1, d2, ... , dp with d1 < d2 < ... < dp. di for i = 1,2,...,p is choosen
systematically with divided distance interval 2 times range with p dan d1 will
be fixed randomly.
iv. Compute the expected semivariogram ˆ( ), 1,2,...,iγ d i p with di on iii.
v. Take iε is normal distribution with mean 0 and variance 1.
vi. Add , 1,2,...,iε i p to the ˆ( )iγ d .
vii. Estimate parameter vectors, i.e. 0 1 2ˆ ˆ ˆˆ , ,θ θ θθ by least square for respectively
the semivariogram model: (1) exp, (2) Gauss, or (3) sph as estimation model.
viii. Make 3 semivariogram model *ˆ ( )iγ d (exp, Gauss, and sph) with input the θ̂
on vii.
ix. Compute the mean square error (MSE), formulated as 2
*
1
1ˆ ˆ( ) ( )
n
i i
i
γ d γ dp
.
x. Repeat step i.
5130 K. N. Sari et al.
From the 1st procedure, the least square estimator estimator θ̂ is close to θ , in other
word called unbiased estimator.
Secondly, the bootstrap is used to resample the error of semivariogram model
from the first procedure above. The following steps are conducted:
i. Suppose one of the reference semivariogram models: (1) exp, (2) gauss, or
(3) sph with respectively parameters vector 0 1 2, ,θ θ θref ref ref
θ .
ii. Compute the errors *ˆ ˆ( ) ( ), 1,2,...,i i iε γ d γ d i p .
iii. Define p for each semivariogram model.
iv. Take iε as the bootstrap sample with p as the size and B as the number of
bootstrap repetitions.
v. Add , 1,2,...,iε i p to the ˆ( )iγ d .
vi. Estimate B parameter vectors, i.e. 0 1 2ˆ ˆ ˆˆ , ,j j j jθ θ θθ , 1,2,...,j B by least square
for respectively the semivariogram model: (1) exp, (2) gauss, or (3) sph as
estimation model.
vii. Compute the average of the estimated parameter vectors 0 1 2ˆ , ,B θ θ θθ where
0 0
1
1 ˆ ,B
i
i
θ θB
1 1
1
1 ˆB
i
i
θ θB
and 2 2
1
1 ˆB
i
i
θ θB
.
viii. Make 3 semivariogram models *ˆ ( )iγ d (exp, Gauss, and sph) with input the
parameter estimation on vii.
ix. Compute the MSE, formulated as 2
*
1
1ˆ ˆ( ) ( )
n
i i
i
γ d γ dp
.
x. Repeat step i.
The best estimated model can be determined by finding the one with the least MSE.
Besides, by using bootstrap, the (1-α)% confidence interval can be estimated for the
parameter vector, where α is the significance level.
3.2 Simulation of the number of sample point and Isotropic Semivariogram
Model
The data used will be the permeability of a reservoir at Jatibarang field,
Indonesia. Jatibarang reservoir is one of the famous reservoirs because of its special
characteristics. There are volcanic stones with fractures and low sulfur content. The
volcanic layer is the largest oil producer among the Jatibarang reservoirs. This
reservoir is located in the north of West Java and the oil field area has an elongated
position +10 kilometers north-south and +16 kilometers west-east. Since 1969,
there have been +200 opened and in 1998 the production reached a cumulative
production of nearly 13 million m3 [12].
Out of the existing 132 wells, 12 wells have been selected through
systematic random sampling to be analyzed with spatial analysis. The data consists
of the coordinates of the selected 12 wells and k-fracture values. k-fracture values
show the oil permeability near a well in milli-Darcy (mD) field unit. By applying
Surfer software, obtained 3 semivariogram models with the parameter estimates as
Estimation of the parameters of isotropic semivariogram model… 5131
follows: expˆ ( ) 171.2 245 1 exp
1.900
dγ d
,
2
gaussˆ ( ) 125.4 185 1 exp
0.552
dγ d
and 3
sphˆ ( ) 29.32 268.1 1.5 0.5
0.881 0.881
d dγ d
.
The selection of p was simulated to form three isotropic semivariogram
models. p was taken systematically but the first distance interval will be fixed
randomly, later will be called as sample point. p is selected 6 sample points with
consideration of many lag distance may be formed for a Gaussian model that has a
curve inflection. Furthermore, by doing 1st procedure, the result of estimation of
the parameter vector is presented in Table 1 and Figure 1.
Table 1 The parameter vector and MSE for three semivariogram models (exp, gauss, and sph).
The reference model The parameter vector and MSE
Exp Gauss Sph
Exp
(171.2, 245, 1.900)
(155.39, 264.83, 1.98)
81.6
(209.75, 183.97, 1.98)
115.5
(189.26, 157.50, 1.98)
519.4
Gauss
(125.4, 185, 0.552)
(69.87, 265.13, 0.53)
174.5
(116.56, 197.79, 0.53)
36.4
(111.41, 148.72, 0.53)
1,156.5
Sph
(29.32, 268.1, 0.881)
(77.95, 292.26, 0.95)
802.2
(138.89, 196.62, 0.95)
1,170.6
(103.42, 188.86, 0.95)
303.7
.(a) (b)
(c)
Figure 2 Semivariogram model plots for the three reference semivariogram (a) the exponential
reference model, (b) the Gaussian reference model, and (c) the spherical reference model.
5132 K. N. Sari et al.
Figure 3a MSE plots for the three semivariogram models with p sample points and B number of
bootstrap iterations (25, 50, 75, and 100) for the exponential model as the reference.
Figure 3b MSE plots for the three semivariogram models with p sample points and B number of
bootstrap iterations (25, 50, 75, and 100) for the Gaussian model as the reference.
Estimation of the parameters of isotropic semivariogram model… 5133
Figure 3c MSE plots for the three semivariogram models with p sample points and B number of
bootstrap iterations (25, 50, 75, and 100) for the spherical model as the reference.
From Figure 2a, exponential model is the best model that estimated its
reference model with point estimator for each model parameter is closed and the
MSE value is the smallest reaching 81.6. The spherical model has the highest MSE
to estimate the exponential as the reference model. From Figure 2b, Gaussian model
is the best estimator for Gaussian model as the reference model with MSE reaching
36.4. That estimation is followed by exponential model that estimate better than
spherical model because exponential model is base form from Gaussian model.
While from Figure 2c, the spherical model can estimate better its reference model
than exponential and Gaussian model. The conclusion, estimation of parameter
vector using ordinary least square with certain reference model could be estimated
by same model with MSE value less than 100 for exponential and Gaussian model,
and MSE value less than 500 for spherical model.
Figure 3 show that exponential model always close to Gaussian model. From
Figure 3a, if the reference model is exponential, it is clear that exponential model
could estimate better than Gaussian model, while spherical model gave a far model
estimates with MSE reaching 713 in the 50th and 75th bootstrap iteration. By
selecting 4 sample points and 50 bootstrap iterations, exponential model can
estimate the reference model very well with the MSE is 56. Then, for each bootstrap
iteration, some repetitions are applied to see the convergence of MSE values. If less
than 10 number of sample points is selected, for the estimations using exponential,
Gaussian, and spherical models, the MSE will converge respectively at the 50th,
50th, and 75th iteration. It can be concluded that if the exponential model as the
5134 K. N. Sari et al.
reference is estimated using Gaussian and spherical models for small-sized samples
(not more than 8), then the MSE values for each models are, respectively, 70 – 348
and 290 – 662. While for big-sized samples (more than 8), the MSE values for each
models are 134 – 230 and 577 – 713. So, for those three models, it is sufficient to
take 4-6 sample points.
If the reference model is Gaussian in Figure 3b, it is clear that Gaussian model
will be the best estimates, followed by exponential model which has slightly
different model form. By selecting 6 sample points and 50 bootstrap iterations,
Gaussian model can estimate the reference model very well with the MSE is 39. If
less than 10 number of sample points is selected, for the estimations using
exponential, Gaussian, and spherical models, the MSE will converge at the 50th
iteration for all models. It can be concluded that if the Gaussian model as the
reference is estimated using exponential and spherical models for small-sized
samples (not more than 8), then the MSE values for each models are, respectively,
118 – 279 and 681 – 1,357. While for big-sized samples (more than 8), the MSE
values for each models are 156 – 287 and 940 – 1,339. So, for those three models,
it is sufficient to take 4-6 sample points.
If the model is estimated by the spherical in Figure 3c, then the MSE will
converge at the 25th iteration. If the spherical model as the reference, by selecting 4
sample points and 25 bootstrap iteration, the spherical model can estimate the
reference model very well with the MSE of equal to 90. If less than 10 number of
sample points is selected, for the estimations using exponential, Gaussian, and
spherical models, the MSE will converge respectively at the 75th, 50th, and 25th
iteration. It can be concluded that if the spherical model as the reference is estimated
using exponential and Gaussian models for small-sized samples (not more than 8),
then the MSE values for each models are, respectively, 526 – 1,395 and 615 – 1,753.
While for big-sized samples (more than 8), the MSE values for each models are 797
– 1,834 and 1,210 – 1,651. So, for those three models, it is sufficient to take 4-6
sample points.
From the result, the parameter vector can be estimated by selecting the number
of sample points and bootstrap iterations. For the exponential, Gaussian, and
spherical model as the reference, number of sample point and number of iteration
bootstrap are selected respectively 4 and 50, 6 and 50, 4 and 50. The results of that
estimation are presented in Table 2 and Figure 4.
Table 2 Estimations of parameter vector for 3 semivariogram models. The reference
semivariogram model and the number of bootstrap iteration (B) are given.
The reference model Point Estimator, Confidence interval of parameter vector, and MSE value
Exp Gauss Sph
Exp
(171.2, 245, 1.900)
(148.61, 277.81, 1.95)
29.99, 183.82
252.10, 326.68
0.59, 3.30
83.0
(245.39, 118.40, 1.95)
173.26, 265.43
102.50, 157.22
0.59, 3.30
273.4
(142.57, 161.86, 1.95)
0, 314.67
3.13, 360.39
0.59, 3.30
402.9
Estimation of the parameters of isotropic semivariogram model… 5135
Table 2. (Continued): Estimations of parameter vector for 3 semivariogram models. The reference
semivariogram model and the number of bootstrap iteration (B) are given.
Gauss
(125.4, 185, 0.552)
(36.29, 317.48, 0.56)
0, 80.78
281.28, 399.29
0.11, 0.99
349
(147.65, 148.57, 0.56)
63.45, 173.15
127.89, 200.45
0.11, 0.99
166
(59.86, 158.27, 0.56)
0, 233.55
0.61, 230.29
0.11, 0.99
1,032
Sph
(29.32, 268.1, 0.881)
(21.94, 332.88, 0.81)
0, 61.91
293.95, 396.36
0.18, 1.44
1,521
(143.36, 205.04, 0.81)
0, 177.42
170.57, 302.10
0.18, 1.44
1,474
(78.90, 138.34, 0.81)
0, 237.77
1.09, 381.17
0.18, 1.44
215
. . (a) (b)
(c)
Figure 4 Plots of reference semivariogram model and three semivariogram models (exponential,
Gaussian, and spherical) from bootstrap result for (a) the exponential model as the reference with p
= 4, B = 50, (b) the Gaussian model as the reference with p = 6, B = 50 and (c) the spherical
model as the reference with p = 4, B = 50.
4. Conclusion
By applying bootstrap method, the estimations of the semivariogram
parameters such as nugget effect and sill with partitioned range in a certain
precision will give point and interval estimations with a certain confidence level.
The number of the range partitions affects the MSE value between the experimental semivariogram and its reference model. The selection of the reference semivariogram
5136 K. N. Sari et al.
model will also affect the appropriate semivariogram model. With the exponential
as reference model, the estimation of exponential and Gaussian model will be the
best estimation of the reference model. Meanwhile, the spherical model will be the
poorest estimation of the model with the MSE reaching 713. The model estimations
using bootstrap method will converge faster to the reference semivariogram model
with the maximum number of iterations of 50. The computation time spherical
model as the reference model, the estimation was obtained within the fastest
computation time with the number of bootstrap iterations of 25. Meanwhile, the
spherical model is the slowest computation time for estimating the exponential and
Gaussian model with the number of bootstrap iterations of 75.
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Received: April 17, 2015; Published: August 3, 2015