Performance of Spatial Distribution Quality by Ordinary Fuzzy Kriging for Soil Properties under Uncertainty

Abstract


INTRODUCTION
There are many spatial interpolation methods that are used to find the best performance of a spatial distribution based on the theory of regionalized variables.Kriging techniques are the most important to prediction values in the study area, later on, Georges Matheron, which to the first scientific approach to the kriging method.The main purpose of spatial data analysis is to obtain the best estimate of the values of a particular phenomenon in the study area with minimal variance errors through kriging techniques such as universal kriging [7].Spatial variability of the studied data has an accurate prediction of covariance for any application (such as mining field, level of groundwater, environmental sciences, soil data, pollution, …, etc.).Various statistics and analysis numerical approaches to the best model have been introduced.Multivariate techniques have proven to be effective in the spatial prediction of landslides using a high degree of accuracy [11], [12].Many studies dealt with the fuzzy system starting to know the fuzzy logic, fuzzy numbers, and the membership model [16], [18].Other studies took the prediction using an application of the analytical hierarchy process.[1], [2].And also, the fuzzy inference system interest
This paper describes a method of fuzzy kriging depending on the variogram function.Kriging specifically explains the subject of uncertainty about the empirical variogram.This work demonstrates how to "fuzzy spacing", which achieves variance kriging.Fuzzy kriging interests soil scientists, it assumes that the reader is familiar with the basic ideas of fuzzy theory [13], [15].

Interpolation methods
Geostatistical techniques include kriging to interpolate the value of regionalized variable theory.Geostatistics is applied in different fields such as (depth, soil science, hydrology, groundwater, environmental sciences, …, etc.) [12].

The variogram function
The variogram function is defined in spatial statistics of the stochastic process * ( ) +.For each pair of points in the sample data, the variogram function is defined by Matheron as a measure of the half mean-squared difference between their values at locations or with at the distance (lag h). Nugget effect denoted as (C 0 ) defined as the discontinuity at the origin point (or measurement with error).
 Sill (C+C 0 ) defined the limit of function to infinity, where C is partial sill [17], [18].and the empirical variogram function can be written as:  where ( ) is the pairs of observations [8].

Fuzzy Theory
The published research" Fuzzy sets" is the first research of the professor and head of the Department of Electrical Engineering at California university at Berkeley Lotfi A. Zadeh which was published in 1965.This scientist is considered the first to sudy " The Fuzzy " after identifying it and linking it with Probability theory to get the mathematical logic.Zadeh used the membership of classical binary logic and developed it for a set of mathematical principles to represent the membership degrees of multivalued fuzzy Logic rather than the classical set.The first introduces some concepts of fuzzy set theory after Zadeh [2], [3], [9].The fuzzy set A is defined as a set of pairs of elements and the corresponding membership degrees less than or equal to 1 denoted by *( ( )) + where X is a collection of element numbers ,called the membership function of set A, [10].If we have two fuzzy sets A and B then the intersection of A and B denoted by Let M and N, fuzzy subsets of sets X and Y with ( ) and ( ) then a furry subset of has ⋂ ( ) * ( ) ( )+.

Fuzzy kriging procedure
Parameter values, whether accurate or inaccurate, at certain points where the parameters are to be estimated these values are the inputs of fuzzy kriging.The estimated value for any location represents the outcome of the fuzzy kriging.The experimental variogram is used as the best tool to find the theoretical variogram function.Fuzzy kriging can be calculated by taking fuzziness and fitting a variogram curve [14], [17].

The Hypothetical Fuzzy Variogram
The experimental variogram function is defined Corresponding, the fuzzy variogram for fuzzy ̌ is defined as ̌( ) and for the formula: From equations (3), and (4) we write:

Ordinary Kriging
The kriging technique is used to estimate a value at a point of real spatial data of the study area.The variables satisfice the second-order stationarity.In ordinary kriging, we want to estimate a value of ( ) using the data values from neighboring sample point ( ).The predictor of ordinary kriging linearly with weights can be written as: is the weights, the estimate variance is defined by : By minimizing the estimate variance with condition on the weight, the ordinary kriging system: ) Where is the Lagrange parameter and are the weights also the ordinary kriging system can be defined in the form: The estimate variance is defined as: Where is location of the real data , then ( ) ( ) .[5], [8].

Cross validation
 In order to obtain effective a prediction we used The value of G measures by using sample mean.
where ( ) are the observations of the variables, ( ) the predictor values and are ̅ the sample mean, to evaluate the complete prediction G is equal to 1 , while the prediction is less accurate when G is a negative value, while if G is a positive value that means a more positive prediction and G is zero refer to the sample mean should be used.
 By using the kriging variance, we can define the accuracy of prediction mean square error (MSE) and calculated by:

Study Area
This research adopted the soil data from Mosul city in Iraq.These data contain (100) values of each soil data (Mg, Cl, and NO3).Table (1) show the statistics of soil data for (Mg, Cl, and NO3) including (min, max, median, mode, and standard deviation        Table (3) describes the results of the variogram parameters for Mg, Cl, and NO3 in all thetas, these parameters represent the nugget effect, sill, and range.The parameters of the variogram function are defined on the assumption of uncertainty.Property as: nugget effect (C 0 ), sill (C 0 +C) and partial variance (C). the kriging variance after prediction of points of regionalized variables.
The step of defuzzification gave the approach the set of the fuzzy spacing by computing the fuzzy mean value: where ( ) is the membership of the set fuzzy [5], [18].6), compute in these locations to get the accuracy of prediction process.and using kriging variance according equation (9).Most of the values of kriging variance are very small and also we used G measure according to equation (10) and obtained mean square error (MSE) according equation (11).These results in Table (6) proves the accuracy of the kriging technique and, likely, supplies a good prediction.Also, most of the G values are closer to 1 of ordinary kriging technique.In addition, that the weights are equal to one ∑ (condition of unbiased predictor).

Conclusions
Soil data are applied in this work to describe the spatial distribution of several parameters or properties by variogram function beside the fuzzy variogram.The kriging variance could be fuzzification for crisp information and the variance of kriging is very small.A fuzzy variogram was obtained by applying the process models with soil data.The fuzzy variogram function can be made wider, by changing the value we get the organic function of the regionalized variables The results obtained indicate small differences in the prediction.The weights are close to one, we obtained the closest data contains the largest weights, while the further data contains the smallest weights.The variograms are described by exponential, and spherical models.Fuzzy systems appear from applications in soil data is still a developing field through information about soil differences.In addition, the uncertainty they represent by kriging anisotropy.To conclude, the variogram gave the fuzzy specification to derive a fuzzy set of spacing with kriging variance.Fuzzy kriging spacing gave an ideal method for determining sampling composition.
the variogram function, ( ) regionalized variables containing a point ( ).Cressie defined variogram as the variance ( ) ( ( ) ( )) also, we can write the variogram also function as the expectation .second-order stationarity with mean a covariance ( ) i ) , ( )ii) Covariance function epend on the distance of h The scientists (Chilled, and Wachanagel) give the properties of variogram function as the following: the curve of the variogram function by the vriogram parameters as the following:


Figure (1): parameters of variogram  if we have the set of sample at locations where denoted of ( )  and the empirical variogram function can be written as:  ( )

Figure ( 3 )
Figure (3): results for Cl data of variogram: (a) for all theta, (b) average of each two thetas.

Figure
Figure (3a) illustrates the curves of the variogram function for (Cl data) for all theta between the distance (or lag h) on the x-axis and the variogram on the y-axis.Figure (3b) shows the curves (red) the average of thetas (0 o ,90 o ) and the black curve for two thetas (45 o , 135 o) .

Figure ( 4 )Figure ( 4 )
Figure (4): results of variogram function for NO3 data (a) in all theta, (b) average of variogram Figure (4) above describe the curves of variogram function (a) in all theta of compass, and (b) the curve of average of variogram function rely on the distance. 0

Figure ( 5 ):
Figure (5): Triangular membership function Where , let v 1 is fuzzy set, then the membership of a spherical variogram ( ) in is given as: * ( | ) ( | ) (13) Where ( | ) The ranges used to define triangular membership functions for the nugget effect and partial variance of Mg or ( No3 ) Then , and can be defined as: ( | ) The furzy variogram for soil NO 3 in this field

Figure ( 5 )
Figure (5): curves of soil data between spacing and kriging variance (a) for Mg data, (b) for Cl Data.

Figure ( 5 )
Figure (5) illustrates the curves of soil data (a) for Mg data, (b) for Cl data, and (c) for NO3, showing a plot between the spacing on the x-axis and kriging variance on the y-axis.When we compare the curves of all data we show the similar behavers between the variogram function results with the parameters (nugget effect, sill, and range ( prediction used for six random locations by applying equation (

Table ( 1
): data statistics of soil data (Mg, Cl, and NO 3 )

Table ( 2): results
of the variogram function for Mg

Table ( 3
): results of parameters of variogram function.

Table ( 6
) shows a comparison between soil data of Mg, Cl