Page 1

JME

Journal of Mining & Environment,

Vol.2, No.2, 2011, 25-40.

On the reduction of the ordinary kriging smoothing effect

H. Rezaee1, O. Asghari1*, J.K. Yamamoto2

1- School of Mining, University of Tehran, Tehran

2- Department of Environmental and Sedimentary Geology, Institute of Geosciences, University of São Paulo, São Paulo, Brazil

Received 16 Nov 2011; received in revised form 30 Mar 2012; accepted 5 Apr 2012

*Corresponding author: o.asghari@ut.ac.ir (O. Asghari).

Abstract

This study proposes a simple but novel and applicable approach to solve the problem of smoothing effect of

ordinary kriging estimates. This approach is based on transformation equation in which Z scores are derived

from ordinary kriging estimates and then rescaled by the standard deviation of sample data with addition of

the mean value of original samples to the results. It bears great potential to reproduce the histogram and

semivariogram of the primary data. Actually, raw data are transformed into normal scores in order to avoid

the asymmetry of ordinary kriging estimates. Thus ordinary kriging estimates are first rescaled using the

transformation equation and then back-transformed into the original scale of measurement. To test the

proposed procedure, stratified random samples have been drawn from an exhaustive data set. Corrected

ordinary kriging estimates follow the semivariogram model and back-transformed values reproduce the

sample histogram, while preserving local accuracy.

Keywords: ordinary kriging; smoothing effect; normal score transform; histogram reproduction;

semivariogram model reproduction; local accuracy

1. Introduction

Ordinary Kriging (OK), one of the most reliable

local estimation methods, suffers from a main

problem which has been known as “Smoothing

Effect”. OK estimates do not reproduce the

sample histogram because of reduced variance as

a consequence of the smoothing effect. In the OK

estimation process low values are overestimated

and high values underestimated making the

estimated histogram narrower than the sample

histogram. Considering

representative of the population from which was

drawn, it is important that estimates follow the

sample histogram in order to make inferences

about the population. In the same way it is also

important that estimates reproduce the spatial

correlation model as

semivariogram. Actually the histogram and the

semivariogram model characterize the population

the sample as

described by the

or spatial phenomenon under study. Therefore, the

challenge in geostatistics is to get estimated maps

reproducing both the sample histogram and the

semivariogram model. However, the reproduction

of histogram and semivariogram model must be

done without loss of local accuracy.

The solution for smoothing in kriging calls for

some post-processing OK estimates in order to

correct the smoothing effect and keeping the local

accuracy that characterizes the OK estimation

process.

The cornerstone of all research done to correct the

smoothing effect is to apply some post processing

to the Kriged values, instead of applying some

changes to the Kriging equations themselves [1, 2,

3]. Guertin [1] proposed a nonlinear correction

function based upon an analytical representation

of the bivariate distribution of the true grade

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Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

26

against the estimated one. Olea and Pawlowsky

[2] have proposed a procedure called compensated

kriging in which through a numerical comparison

they used cross-validation to detect and model

such as the smoothing effects while the next

inversion of the model produced a new estimator.

Based on Olea and Pawlowsky [2] an intermediate

state of Kriging and simulation was derived for

the corrected estimates. A spectral approach

proposed by Yao [4, 5] to conditional simulation

capitalized on the speed of the Fast Fourier

Transform (FFT). The strengths and weaknesses

of a kriging approach vs. a simulation approach

were recalled by Journel et al. [3]. However,

according to these authors, semivariogram

reproduction is achieved at a loss of local

accuracy, confirming that global accuracy and

local accuracy are conflicting objectives.

Yamamoto [6] proposed a four-step procedure for

correcting the smoothing effect of ordinary

kriging estimates that was shown to be effective

for the reproduction

semivariogram without loss of local accuracy,

sharing both local and global accuracies.

Yamamoto [7] applied the post-processing

algorithm for lognormal kriging estimates in order

to avoid biased back-transformed estimates as

given by conventional procedure proposed by

Journel [8]. Yamamoto [9] compared the post-

processing algorithm for correcting the smoothing

effect of ordinary kriging estimates [6] with

sequential Gaussian simulation realizations and

showed the superiority of Corrected Ordinary

kriging estimates to any individual simulation

realization.

In this study a novel approach is proposed that

yields global accuracy without loss of local

accuracy. To apply the suggested procedure on the

kriged estimates, a great deal of skills is needed

which makes it hard at least for practitioners. The

method presented in this study is easy to use while

it reproduces sample

semivariogram model.

of histogram and

histogram and the

2. The

correcting kriging smoothing

The post-processing algorithm proposed by

Yamamoto [6] and updated by Yamamoto [7] is

based on four-step procedure. In the first step

smoothing errors are derived from the cross-

validation procedure. After this step, we have for

every data point the estimated (

actual (

Z

) values and also the interpolation

standard deviation (

post- processing algorithm for

oOKxZ*

) and

ox

o S ) [10]. These values are

combined to derive a new random variable named

number of interpolation standard deviations:

o OK

o

S

where

o

Z*

is the ordinary kriging estimate

from cross-validation,

Z

and

In the second step,

o

NS is estimated at nodes of a

regular grid or unsampled locations resulting in

o

NS*

. In the next step, we run ordinary

kriging to estimate the variable x

the same regular grid as defined for

doing these steps for every grid node we have:

o

Z*

,

NS*

for the post-processing in the fourth step as

follows:

oOKo OK

NSxZxZ

On the right side the term

correcting amount that is added to or subtracted

from the ordinary kriging estimate depending on

the signal for

o

NS*

. But sometimes this term

exceeds in such a way that the corrected estimate

falls outside the range of neighbor data points,

thus this term is replaced by a new variable

usually less than

NS*

correcting amount must be rescaled to reproduce

the sample variance using a constant factor that

multiplies all correcting amounts. It is important

to note that this factor multiplies only the

correcting amount and not the ordinary kriging

estimate (

o

Z*

). This procedure guarantees

the local accuracy of corrected estimates. Details

of this procedure can be found in Yamamoto

[6, 7].

o

o

xZxZ

NS

*

OKx

ox

is the actual value

o S is the interpolation standard deviation.

ox

Z

at nodes of

NS . After

o

OKx

o S and

oox

, that are combined

x

o

*

oo

S

.

****

x

ooo

S

.

NS

is the

ox

oox

. Moreover, the

OKx

3. The new approach for correcting the

smoothing effect

A completely new approach is proposed in this

paper and it is based on a well-known

transformation equation in statistics [11] that

allows calculation of raw score when the Z score

is known:

XSZX

.

where X is the raw score, and X and

mean and standard deviation of the raw score.

Replacing the terms in (1) to get corrected

estimates we have:

X

(1)

X

S are the

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Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

27

x

Var

x

x

x

ZE

ZVar

Z

xZExZ

Z

o OK

oOKoOK

oOK

.

*

**

**

(2)

where

Z

o OKxZ*

o

Var

Z

x

is the sample mean.

This equation gives corrected estimates presenting

the sample standard deviation

the sample mean

ZE

requirements for histogram reproduction.

Before going further it is important to note that

Equation 2 is based on Z score and therefore the

distribution of

o

Z*

plays an important role in

this process. Z score transformation retains the

shape of

o

Z*

, that is if

lognormal distribution,

lognormal distribution and consequently Z scores

will present the same shape as the former

distributions. Application of Equation 2 for

skewed distributions can result in extreme values

for corrected estimates. Thus, it is important to

work with distributions as symmetric as possible.

Evidently, it calls for a data transformation such

as normal score transform as described by

Deutsch and Journel [12]:

1

n

where G(y) is the standard normal cdf and

is the rank for ith

z

associated with a set of n

data values

nixz

i

,,

1

For illustration purposes Figure 1 shows

distribution of Z scores for ordinary kriging

estimates from raw data and normal score data.

The range of Z scores for raw data is from -1.19 to

12.18 whereas for normal score data is from -3.14

to 3.40. Besides, the shape of raw data Z scores is

positively skewed such as the sample histogram

while the shape of normal data Z scores is almost

symmetric because some

introduced during the estimation process. Since Z

scores are measured in a number of standard

deviations, it is not convenient with scores less

than -3 or greater than 3. Usually Z scores enlarge

the range of normal score data from (-2.33 to

is the ordinary kriging estimate,

o

x

is the sample standard deviation and

OK

oOK OK

Z

xZEx

*

**

is the Z score and

xVar

ZE

x

Z Var

and

x

that are minimum

OKx

OKx

x

will follow a

Z

follows a

oOKxZ*

1

xr

Gxy

i

i

ixr

ix

.

asymmetry was

2.33) to (-3.14 to 3.40) and this may represent a

problem when back-transforming

estimates to the original scale of measurement.

However, the worst scenario comes from Z scores

of raw data in which they range from -1.19 to

12.18. A large number of corrected estimates will

likely fall outside the range of original data. This

subject will be examined in the Section 5 Results

and Discussion.

Further work takes place in the Gaussian domain.

Statistics for sample data are then calculated:

i

n

1

ExYExY Var

After computing and modeling the experimental

semivariogram for xY

we run ordinary Kriging

for estimation at unsampled locations:

i

1

corrected

n

ixYxYE

1

x

2

2

Y

n

iio OK

xYxY

*

(3)

Once again we can compute the mean and

variance for OK estimates

o

YVar

. Because the smoothing effect

o

YVar

will be less than

is important to correct for the smoothing effect

before back-transforming

o OKxYE

*

and

OKx

OKx

*

*

x

Y Var

and it

oOKxY*

into the

o

Z*

original scale of measurement

proposed by Yamamoto [7] for lognormal kriging

estimates.

Equation 2 becomes:

xYE

After this correction

Y

reproduce the sample histogram

semivariogram model for x

work with transformed values, it is important to

check if the semivariogram model for x

reproduced, because it guarantees that back-

transformation will be carried out with the same

spatial correlation.

Now

o

Y*

and

Y

into the original scale using the inverse operation:

oOKoOK

xYFxZ

o OKoOK

xYFxZ

OKx

as

x

x

YVar

xYVar

xYExY

Y

o OK

o OKo OK

o OK

.

*

**

**

(4)

o OKx

**

is supposed to

xY

and the

. Since we do not

Y

Y

is

OKx

o OKx

**

are back-transformed

*1*

(5)

**1**

(6)

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Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

28

Back-transformed values after correcting the

smoothing effect

o

Z

sample histogram of

Z

paper we want to check Equation 4 for

reproducing the semivariogram model and

Equation 6 for reproducing the sample histogram.

OKx

**

should be close to the

x

. Therefore, in this

4. Materials and methods

In this study we depart from an exhaustive data

set composed of 50x50 values on a regular grid.

This exhaustive data set follows a lognormal

distribution and it was derived from the public

domain file true.dat [12]. The secondary variable

of this data set was transformed into normal

values using the normal score transform [12].

After that normal scores were transformed into a

new variable using the exponential function:

exZ

This new variable presents a perfect lognormal

distribution (Figure 2).

xY

Figure 1. Histograms for Z scores calculated from both raw data and normal score data.

0

0.09

10

20

30

40

50

60

60

20

(%)

(%)

(%)

(%)

(%)

(%)

3.627.16 10.7014.2417.78

Zlog10

Zlog10

0

-2.33

5

10

15

-1.40-0.47 0.471.402.33

Y(x)

0

0.08

10

20

30

40

50

3.627.1610.70 14.2417.78

0

-2.33

5

10

15

20

25

30

-1.40-0.470.471.402.33

Y*OK

0

-1.19

10

20

30

40

50

1.494.16 6.84 9.5112.18

Zscore(Z*OK(x))

0

-3.14

5

10

15

20

-1.84-0.530.78 2.093.40

Zscore(Y*OK(x))

ORDINARY KRIGING

ZSCORES

ORDINARY KRIGING

ZSCORES

NORMAL

SCORE

TRANSFORM

TRANSFORMED DATARAW DATA

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Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

29

Figure 2. Spatial distribution of a lognormal variable (A); Cumulative frequency curve and histogram of the lognormal

variable in exhaustive data set

Now, from this exhaustive data set we can draw

samples for this study. In order to avoid clustering

of sampled data points, we have used the stratified

random sampling technique. Samples with 25, 49

and 100 data points have been drawn from the

exhaustive data set. Since samples should give a

good representation of the exhaustive data set, we

compared sample distributions with exhaustive

distribution (Figure 3).

It is clear that small size samples do not represent

well the parent population or the exhaustive data

set. On P-P plots we can measure an average

distance to the reference line [6] that gives us an

idea about how close points are to the reference

line (Table 1).

Table 1: Average distances measured on P-P plots for

stratified random samples.

Sample size

25

1 4.59

2 4.21

3 3.48

4 2.71

5 3.66

6 3.11

Samples

50

2.33

2.75

2.41

1.88

2.44

1.79

100

1.21

1.63

1.91

1.47

1.33

1.97

Looking at Figure 3 and Table 1, one readily

concludes that samples with 100 data points are

more representative of the exhaustive data set.

Even when sampling 100 points the maximum

value was not reproduced, but this happens when

we are working with lognormal distribution. In

this study we will consider these samples as

representative of the exhaustive data set and

consequently they can be used to make statistical

inferences about the exhaustive data set as well as

the spatial distribution shown by the parent

population (Figure 2). Parameters for the

exhaustive data set and statistics for samples are

presented in Table 2.

As we can see in Table 2, samples follow either

lognormal distribution with CV > 1.2 (samples 1-

2-3) or positively-skewed distribution with CV <

1.2 (samples 4-5-6). Since sample distributions

are not symmetric, we have to transform raw data

into normal scores that follow a bell-shaped

distribution presenting a mean zero and a variance

equal to one. Thus, all data have been transformed

into normal scores using the normal score

transform as described in Deutsch and Journel

[12, p. 138 and 209-211]. Experimental

semivariograms have

modeled for normal scores data (Figure 4).

Thus, all calculations are made in the normal

score domain in which we get the OK estimates

after Equation 3. The grid on which further

calculations are done is the same as primary

exhaustive data set. It should be noted that the

estimation is performed only on the nodes which

belong to the convex hull [13]. Even in the

Gaussian domain, OK estimates will present some

smoothing that must be corrected using Equation

4. Corrected estimates can be back-transformed

into original scale of measurement according to

Equation 6.

Moreover, this procedure can be compared with

the former procedure for correcting the smoothing

in kriging as published by Yamamoto [6,7].

Actually, this paper will consider the same

procedure for back-transforming

kriging estimates [7]. OK estimates in the normal

scores domain (Equation 3) will be corrected for

been calculated and

lognormal

0

10

20

30

40

50

01020304050

A

28.586

20.779

15.102

10.968

7.970

5.791

4.208

3.058

2.222

1.614

1.173

0.852

0.619

0.450

0.327

0.238

0.173

0.125

0.091

0.066

0.048

0.035

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.010.101.010100

Zlog10

CUMULATIVE %

B

0

15

30

45

60

75

90

0.0414.3128.59

%

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Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

30

smoothing

transformed using the inverse operation as

Equation 5 or 6. For comparison purposes back-

transformed values from corrected estimates

according to the former algorithm [6,7] will also

be considered in this paper.

(Equation 4) and then back-

5. Results and discussion

Figure 5 presents the image maps of back-

transformed values based on Equation 5 which are

not corrected after kriging. These are given for

comparison purposes. Results for corrected

estimates from raw data will not be displayed but

discussed accordingly.

Figure 3. Cumulative frequencies for samples (thin black lines) compared with exhaustive distribution (thick red line) for

samples with: 25 points (A), 49 points (B) and 100 points (C)

Table 2. Parameters and statistics for exhaustive data and samples.

Parameter/

Statistics

N

Mean

Std. Dev.

CV

Maximum

Upper Q.

Median

Lower Q.

Minimum

Exhaustive

Samples

1 2 3 4 5 6

2500

1.640

2.057

1.254

28.596

1.961

0.999

0.509

0.035

100

1.688

2.172

1.286

17.784

1.954

1.046

0.542

0.085

100

1.658

2.300

1.387

19.067

2.080

1.084

0.555

0.109

100

1.747

2.614

1.496

20.817

1.954

0.861

0.514

0.088

100

1.671

1.939

1.160

12.333

1.918

0.940

0.592

0.056

100

1.732

1.879

1.085

9.002

1.971

0.966

0.580

0.068

100

1.583

1.764

1.114

13.330

1.750

1.104

0.604

0.087

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.010.101.010100

VARIABLE

CUMULATIVE %

A

P-P PLOT

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.010.101.010100

VARIABLE

CUMULATIVE %

B

P-P PLOT

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.010.101.010100

VARIABLE

CUMULATIVE %

C

P-P PLOT

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Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

31

Figure 4. Experimental and modeled semivariograms for normal score data.

Figure 6 illustrates the image maps of back-

transformed estimates (into original scale) for

both approaches used in this study

It is very clear that corrected estimates produce

enhanced maps in which low areas are lower and

high areas are even higher. Moreover both

approaches give similar and highly correlated

results. Statistics calculated for back-transformed

values after Equation 5 are in Table 3 and after

Yamamoto [6-7] are presented in Table 4 and

statistics for back-transformed values based on

this paper’s approach are listed in Table 5.

Comparing these statistics on Table 3 with sample

statistics (Table 2), we realize back-transformed

values after Equation 5 are biased. Actually, these

values are mean biased and present reduced

standard deviations. Besides, statistics of upper

tails are strongly biased (maximum values and

upper quartiles).

Both methods give similar results. Statistics for

back-transformed values (Tables 4 and 5) are very

close to the sample statistics (Table 2). Therefore,

unbiased back-transform is only possible when

estimates in the transform domain are corrected

previously to the inverse operation as proposed by

Yamamoto [7] for lognormal kriging estimates. It

0.00

0.23

0.46

0.69

0.92

1.15

0510152025

DISTANCE

SEMIVARIOGRAM

A

0.00

0.21

0.43

0.64

0.86

1.07

0510152025

DISTANCE

SEMIVARIOGRAM

B

0.00

0.21

0.41

0.62

0.83

1.04

0510152025

DISTANCE

SEMIVARIOGRAM

C

0.00

0.21

0.42

0.63

0.84

1.05

0510152025

DISTANCE

SEMIVARIOGRAM

D

0.00

0.23

0.46

0.69

0.92

1.15

0510152025

DISTANCE

SEMIVARIOGRAM

E

0.00

0.22

0.44

0.66

0.87

1.09

0510152025

DISTANCE

SEMIVARIOGRAM

F

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Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

32

is important to note that any correction based on

the kriging variance or its square root will result

in biased back-transformed values. Figure 7

presents cumulative curves and P-P plots

comparing the sample distribution with back-

transformed values.

Figure 5. Image maps of back-transformed values after equation (4). Letters A-F correspond to samples 1-6. Legend: cross =

sample data location

0

10

20

30

40

50

010203040 50

A)

17.784

9.879

7.384

5.042

4.486

3.226

2.860

2.272

1.842

1.431

1.132

0.955

0.686

0.566

0.529

0.403

0.355

0.295

0.226

0.154

0.087

0.085

0

10

20

30

40

50

01020304050

B)

19.067

12.287

7.197

4.697

3.970

2.808

2.464

2.181

1.792

1.451

1.278

1.008

0.653

0.594

0.507

0.432

0.312

0.192

0.170

0.145

0.111

0.109

0

10

20

30

40

50

01020304050

C)

20.817

12.692

8.467

5.829

5.060

4.140

3.093

2.116

1.769

1.174

0.938

0.851

0.701

0.546

0.452

0.289

0.214

0.201

0.178

0.140

0.117

0.088

0

10

20

30

40

50

01020304050

D)

12.333

9.417

7.741

5.979

4.797

3.910

2.674

2.008

1.879

1.602

1.031

0.873

0.776

0.628

0.489

0.412

0.322

0.256

0.192

0.150

0.102

0.056

0

10

20

30

40

50

01020304050

E)

9.002

8.286

7.282

6.072

5.272

4.659

3.471

2.113

1.685

1.556

1.134

0.880

0.767

0.631

0.494

0.357

0.240

0.186

0.153

0.124

0.105

0.068

0

10

20

30

40

50

01020304050

F)

13.330

7.830

6.139

5.227

4.163

3.384

2.687

2.041

1.568

1.434

1.168

1.003

0.730

0.661

0.539

0.393

0.350

0.318

0.258

0.177

0.136

0.087

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Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

33

0

10

20

30

40

50

010 20304050

A)

17.784

9.879

7.384

5.042

4.486

3.226

2.860

2.272

1.842

1.431

1.132

0.955

0.686

0.566

0.529

0.403

0.355

0.295

0.226

0.154

0.087

0.085

0

10

20

30

40

50

0 1020 304050

B)

17.784

9.879

7.384

5.042

4.486

3.226

2.860

2.272

1.842

1.431

1.132

0.955

0.686

0.566

0.529

0.403

0.355

0.295

0.226

0.154

0.087

0.085

0

10

20

30

40

50

01020304050

C)

19.067

12.287

7.197

4.697

3.970

2.808

2.464

2.181

1.792

1.451

1.278

1.008

0.653

0.594

0.507

0.432

0.312

0.192

0.170

0.145

0.111

0.109

0

10

20

30

40

50

01020304050

D)

19.067

12.287

7.197

4.697

3.970

2.808

2.464

2.181

1.792

1.451

1.278

1.008

0.653

0.594

0.507

0.432

0.312

0.192

0.170

0.145

0.111

0.109

0

10

20

30

40

50

01020304050

E)

20.817

12.692

8.467

5.829

5.060

4.140

3.093

2.116

1.769

1.174

0.938

0.851

0.701

0.546

0.452

0.289

0.214

0.201

0.178

0.140

0.117

0.088

0

10

20

30

40

50

01020304050

F)

20.817

12.692

8.467

5.829

5.060

4.140

3.093

2.116

1.769

1.174

0.938

0.851

0.701

0.546

0.452

0.289

0.214

0.201

0.178

0.140

0.117

0.088

0

10

20

30

40

50

01020304050

G)

12.333

9.417

7.741

5.979

4.797

3.910

2.674

2.008

1.879

1.602

1.031

0.873

0.776

0.628

0.489

0.412

0.322

0.256

0.192

0.150

0.102

0.056

0

10

20

30

40

50

01020304050

H)

12.333

9.417

7.741

5.979

4.797

3.910

2.674

2.008

1.879

1.602

1.031

0.873

0.776

0.628

0.489

0.412

0.322

0.256

0.192

0.150

0.102

0.056

Page 10

Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

34

Figure 6. Image maps of back-transformed values to the original scale based on Yamamoto (2005 and 2007) are on the left

(A-C-E-G-I-K – corresponding to samples 1 to 6) and on the right (B-D-F-H-J-L – corresponding to samples 1 to 6) are image

maps back-transformed based on the new approach (this paper). Legend: cross = sample data location.

Table 4. Statistics for back-transformed values after Yamamoto (2005 and 2007).

Statistics

Samples

1 2 3 4 5 6

N 2332

1.680

2.103

1.252

17.784

1.943

1.064

0.547

0.086

2233

1.713

2.471

1.442

19.067

2.079

1.055

0.556

0.109

2285

1.713

2.439

1.424

19.611

2.039

0.904

0.517

0.088

2272

1.678

1.940

1.156

12.333

1.917

0.941

0.589

0.075

2253

1.720

1.930

1.122

9.002

1.972

0.954

0.575

0.083

2264

1.588

1.764

1.111

13.330

1.920

1.061

0.607

0.090

Mean

Std. Dev.

Coeff. Var.

Maximum

Upper Q.

Median

Lower Q.

Minimum

Table 5. Statistics for back-transformed values according to this paper’s approach.

Statistics

Samples

1 2 3 4 5 6

N 2332

1.653

2.020

1.222

17.784

2.068

1.049

0.547

0.085

2233

1.615

2.115

1.310

19.067

2.117

1.027

0.572

0.109

2285

1.738

2.404

1.383

20.817

2.046

0.849

0.526

0.088

2272

1.674

1.875

1.120

12.333

1.977

0.909

0.590

0.056

2253

1.755

1.935

1.103

9.002

1.972

0.949

0.591

0.068

2264

1.589

1.750

1.102

13.330

1.975

1.065

0.587

0.087

Mean

Std. Dev.

Coeff. Var.

Maximum

Upper Q.

Median

Lower Q.

Minimum

0

10

20

30

40

50

0102030 4050

I)

9.002

8.286

7.282

6.072

5.272

4.659

3.471

2.113

1.685

1.556

1.134

0.880

0.767

0.631

0.494

0.357

0.240

0.186

0.153

0.124

0.105

0.068

0

10

20

30

40

50

010 20 304050

J0

9.002

8.286

7.282

6.072

5.272

4.659

3.471

2.113

1.685

1.556

1.134

0.880

0.767

0.631

0.494

0.357

0.240

0.186

0.153

0.124

0.105

0.068

0

10

20

30

40

50

0 1020304050

K)

13.330

7.830

6.139

5.227

4.163

3.384

2.687

2.041

1.568

1.434

1.168

1.003

0.730

0.661

0.539

0.393

0.350

0.318

0.258

0.177

0.136

0.087

0

10

20

30

40

50

01020304050

L)

13.330

7.830

6.139

5.227

4.163

3.384

2.687

2.041

1.568

1.434

1.168

1.003

0.730

0.661

0.539

0.393

0.350

0.318

0.258

0.177

0.136

0.087

Page 11

Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

35

Figure 7. Cumulative curves and P-P plots comparing sample distribution with cumulative distributions of back-transformed

estimates. Legend: red cross = sample data; green circle = back-transformed estimates after Yamamoto (2005 and 2007); blue

square = back-transformed estimates according to this paper; black diamond = back-transformed after equation (5). Letters

A-F correspond to samples 1-6.

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.01 0.101.0 10 100

VARIABLE

CUMULATIVE %

A

P-P PLOT

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.11.0 10100

VARIABLE

CUMULATIVE %

B

P-P PLOT

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.010.101.010100

VARIABLE

CUMULATIVE %

C

P-P PLOT

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.010.101.010100

VARIABLE

CUMULATIVE %

D

P-P PLOT

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.010.101.010

VARIABLE

CUMULATIVE %

E

P-P PLOT

0.01

0.05

0.10

0.50

1.00

5.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

95.00

99.00

99.50

99.90

99.95

99.99

0.010.101.010100

VARIABLE

CUMULATIVE %

F

P-P PLOT

Page 12

Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

36

Examining

approaches

consequently are valid approaches for correcting

the smoothing effect of ordinary kriging

estimates. Average distances on P-P plots confirm

that both approaches give similar results (Table

5). However, back-transformed values after

Equation 5 show different distributions to their

respective sample distributions. Consequently,

these values cannot be used to make statistical

inference about the parent population.

Now we can check for semivariogram model

reproduction after these approaches (Figure 8).

Looking at this figure we conclude the former

algorithm seems to reproduce the semivariogram

model better. The new proposal also reproduces

Figure

give

7, we

close

conclude

results

both

and very

the semivariogram model, but experimental

semivariograms show more continuity than

sample semivariograms.

Since we have the exhaustive data set, we can

compare actual values with back-transformed

estimates (Figure 9). This comparison gives an

idea of the local precision as measured by the

correlation coefficient in a scattergram.

Once again correlation coefficients are very close

to each other proving both methods give a good

correlation with actual values. Moreover, it also

confirms that samples are representative of the

exhaustive data set.

Now results provided by both approaches can also

be compared on scattergrams and the correlation

coefficient can be computed (Figure 10).

Figure 8. Experimental semivariograms computed from back-transformed estimates: open circle for values after Yamamoto

(2005 and 2007); full circle for values after this paper; star = sample semivariogram; thick line = semivariogram model.

Letters A-F correspond to samples 1-6.

0.00

0.21

0.42

0.63

0.84

1.05

0510152025

DISTANCE

SEMIVARIOGRAM

A

0.00

0.20

0.40

0.59

0.79

0.99

0510152025

DISTANCE

SEMIVARIOGRAM

B

0.00

0.19

0.38

0.58

0.77

0.96

0510152025

DISTANCE

SEMIVARIOGRAM

C

0.00

0.20

0.40

0.60

0.80

1.00

0510152025

DISTANCE

SEMIVARIOGRAM

D

0.00

0.21

0.42

0.63

0.84

1.05

0510152025

DISTANCE

SEMIVARIOGRAM

E

0.00

0.20

0.41

0.61

0.81

1.02

0510152025

DISTANCE

SEMIVARIOGRAM

F

Page 13

Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

37

0.010.10 1.010100

0.01

0.10

1.0

10

100

Z**OK

Zlog10

CORRELATION COEF.=0.902

A)

0.01 0.101.0 10100

0.01

0.10

1.0

10

100

Z***OK

Zlog10

CORRELATION COEF.=0.872

B)

0.010.101.0 10100

0.01

0.10

1.0

10

100

Z**OK

Zlog10

CORRELATION COEF.=0.911

C)

0.010.101.010100

0.01

0.10

1.0

10

100

Z***OK

Zlog10

CORRELATION COEF.=0.882

D)

0.01 0.101.010100

0.01

0.10

1.0

10

100

Z**OK

Zlog10

CORRELATION COEF.=0.920

E)

0.010.101.010100

0.01

0.10

1.0

10

100

Z***OK

Zlog10

CORRELATION COEF.=0.878

F)

Page 14

Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

38

Figure 9. Scattergrams comparing actual values with back-transformed values to the original scale based on Yamamoto

(2005 and 2007) are on the left (A-C-E-G-I-K – corresponding to samples 1 to 6) and on the right (B-D-F-H-J-L –

corresponding to samples 1 to 6) are scattergrams comparing actual values with back-transformed based on the new

approach (this paper).

0.01 0.101.010100

0.01

0.10

1.0

10

100

Z**OK

Zlog10

CORRELATION COEF.=0.915

G)

0.010.10 1.0 10 100

0.01

0.10

1.0

10

100

Z***OK

Zlog10

CORRELATION COEF.=0.892

H)

0.010.10 1.010100

0.01

0.10

1.0

10

100

Z**OK

Zlog10

CORRELATION COEF.=0.912

I)

0.010.101.0 10100

0.01

0.10

1.0

10

100

Z***OK

Zlog10

CORRELATION COEF.=0.878

J)

0.010.101.010100

0.01

0.10

1.0

10

100

Z**OK

Zlog10

CORRELATION COEF.=0.903

K)

0.010.10 1.010 100

0.01

0.10

1.0

10

100

Z***OK

Zlog10

CORRELATION COEF.=0.879

L)

Page 15

Rezaee et al./ Journal of Mining & Environment, Vol.2, No.2, 2011

39

Figure 10. Scattergrams comparing back-transformed estimates after Yamamoto (2005 and 2007) with back-transformed

estimates according to this paper. Letters A-F correspond to samples 1-6.

High correlation coefficients shown in Figure 10

confirm once again both methods give equivalent

results. Evidently, back-transformed values are

just close to each other but not equal because they

are based on completely different approaches.

Finally we can check the effectiveness of the

proposed method regarding corrected scores after

the transformation equation. It can be tested by

counting the number of times that corrected scores

fall outside the permissible range (Tables 6 and 7,

0.010.101.010100

0.01

0.10

1.0

10

100

Z***OK

Z**OK

CORRELATION COEF.=0.961

A)

0.11.010100

0.1

1.0

10

100

Z***OK

Z**OK

CORRELATION COEF.=0.958

B)

0.010.101.010100

0.01

0.10

1.0

10

100

Z***OK

Z**OK

CORRELATION COEF.=0.955

C)

0.010.101.0 10100

0.01

0.10

1.0

10

100

Z***OK

Z**OK

CORRELATION COEF.=0.945

D)

0.010.101.0 10

0.01

0.10

1.0

10

Z***OK

Z**OK

CORRELATION COEF.=0.961

E)

0.010.101.010100

0.01

0.10

1.0

10

100

Z***OK

Z**OK

CORRELATION COEF.=0.954

F)