Tikrit Journal of Pure Science

T he problem tackled in this paper is the estimation of variogram function Indicator of spatial stochastic process for the Levels of groundwater, by the method of weighted Least squares. This methods is well known in regression analysis in estimating the coefficient of ression model. After defining the indicator variable the parameters of Indicator variogram estimated based on mean squares error. The final formula of weighted least squares estimator can be not be solved exactly, then through the use of iterative Newten - Raphson algorithm and for some iterations the convergence of solution is obtained with certain termination criterion or number of repeats (that used in this paper)


Introduction
Many statisticians prefer to name the estimation of the variogram function Indicator at certain locations by the prediction process to distinguish it from the word estimate for the parameters in a particular probability distribution, and that the word estimation has been widely used by geostatistics researchers in the study of the spatial random process [1].In developed countries, estimation of the variocram function Indicator has become a priority in protecting and cleaning the environment.Estimating and reconciling variogram are critical stages in estimating or spatial prediction, when a spatial prediction is performed to obtain a prediction in locations to be predicted in a spatial phenomenon that may be minerals, groundwater, plants, or contamination of the environment or a satisfactory environment, the estimation of the variogram function must be conducted in a careful manner from a lesser estimation error or the prediction results will be in accurate and unreliable.The experimental (approximate) variogram function that measures spatial continuity with in this work is used to make spatial predictions.In order to calculate these spatial predictions, we have an effective theoretical model (variogram) to fit the experimental variogram function we obtain from the diagram.For this purpose, we should know model variogram theoretical indicator (real) as a synthesis written in small groups of approved (authorized functions) I.e, dependable meaning, for example (joke effect, Exponential model, spherical model, Gaussian model ( see table )1) shown in the appendix (A) with positive equations and each basic structure of these functions depends on a limited number of parameters eg.jokes effect, Sill, Isotropic, nonuniform properties, and as soon as we know the basic formulations and all parameters will be possible to determine the appropriate theoretical model compared variogram Indicator by drawing and estimating model parameters theory by ordinary least squares )OLS ( and generalized least squares )GLS ( [2].By moving spatial statistics as in time series [3]., Shows that the composite model differs in different directions across the data perspective .The weighted least squares method would be best to obtain an optimal estimator for the Variogram function Indicator [4.5.6].This research included two parts, where the part one contains the theoretical aspect of the research in terms of formulating the question of appreciation Formulation of estimation problem by weighted least squares .The second part contains the applied side of the research, the results were encouraging as the Gaussian and spherical models were used on these data The aim of this research is to estimate the variogram function Indicator by the least squares method estimated weighted Indicator variogram data on water tanker factories in groundwater in Bashiqa / Nineveh Governorate/ Northern Iraq.Finally, all the algorithms are programmed into the search using the Matlab system.

Section Ι Estimation by Generalized least Squares (GLS) 1-1-Indicator Variable:
The pointer variable is often referred to only as the pointer in the spatial statistics literature and is essentially a binary variable and takes values 1 and 0 Just .these variables typically represent the presence or absence of a particular attribute (or feature) in the region consideration.For example, in the mining phenomenon we know the variable is equal 1 when there is a metal and 0 In the absence of the metal .

1-2-Variogram Function Indicator:
After the pointer variable is declared, a binary random process is obtained where this random (or accidental) process is spatial{ Z ( x ); x D } Since assuming that this process fulfills the basic stability Hypothesis of intrinsic stationary in which: a-Mathematical expectation exists and does not depend on the location x that is : b-For all distances h , the increase [Z ( x + h ) -Z ( x )] it has a specific variation and is not dependent on the location x that is : As that 2(h) is the variogram function Indicator of the distance h.Z(x) is the spatial variable of the site x with tow dimensions x=(u,v).We assume that the binary random Stochastic process B[Z(x)≤ Z C ] is stable and isotropic, so that 2(h) depends on the distance h only.Now, the classic estimator of variogram function indicator is defined as: [8.9.10] these sources adopted the primary source [11], and surely N ( h ) represents the number of pairs of views that are separated by distance h.Now can define a function variogram pointer is a function representing the degree of heterogeneity of the continuity of the phenomenon (metal ore wells or groundwater or air gases) under study to analyze the spatial heterogeneity within the structure of the specific occurrence of the phenomenon of the region, the increasing dimension h between observations as the heterogeneity becomes large until the height of the function stabilizes (h) at a certain offset like h = a and this offset a it called long, and after a long note will fade covariance function in variogram as it stabilizes the value of fixed equal contrast views  2 .This explains outside the study area, I.e, there is no effect of the phenomenon studied after the range, or there may be an impact in very small quantities.Given the importance of the variogram function (h) scientists have been able to identified various models that can be expressed in Table (1) shown in the appendix (A).Now you can give be the following formula of the Semi-Variogram function by shifting (h) .
It is the square of the differences between spatial observations that are shifted apart h , as that N(h) represents the number of views pairs Z ( x i ), Z ( x i + h ), [12.13].Called the equation ( 4) Semivariogram function Indicator because there is a half at the right end of the equation.

1-3-drafting issue Formulation of the Problem:
The appropriate methods of the experimental variogram function.I.e, containing the unknown parameters proposed so far ignore the visual diagram represented  ( h) Since h = h 1 , h 2 ,…., h n and then find a theoretical variogram that is close to it fit the theoretical model with the experimental model and this method is a useful tool because the parameters of the experimental model is not estimated with the least possible error [14,15] The formula ( 5 ) represents the estimated least normal squares to  and of course we will get after taking the derivative of equation ( 5) for  and equal to zero to get (the least squares estimator), although equation ( 5) does not have a geometric appearance because it does not consider the distribution of the variability function of the estimator is important in addition to the variability of the spatial variable Z ( x ) Therefore, the appropriateness of the generalized least squares estimator assumed that the variogram estimator obtained at distances ) h 1 , h 2 ,….h n ( That n they are fixed and represent the number of views that are apart from each other h .The Ordinary Least Squares )OLS) Is the estimation of the parameter θ and suppose that it is written so that: Then to get the estimator we use the Generalized Least Squares method (GLS) of reducing the amount there is an intermediate stage between  ~ and  is the least-squares method which states that the estimated  * : We get it by reducing the amount )) ( 2 ( ^i h r Va  To obtain the final formula for the weighted least squares estimator, the proofs listed below must be proved :

) [18] :
Let it be { Z ( x )} stable new random spatial process of second order though i = 1,2, ..., n Z( x i ) views at sites x 1 , x 2 ,…, x n then: Since h = ( x i -x j ( represents distance between locations and  (.) Semi -Variogram function.

1-4-Generalized least Squares (GLS):
Returning to the classical estimator defined by equation ) 3 ( and suppose that { Z ( x )} is Gausian (normal) and as it is known that any conciliation of a linear variable Gausian is also Gausian if : It represents a variable where the Chisquare distribution is distributed with equal freedom 1 the correlation between the two variables can be written ( Z ( y + h 2 ) -Z ( y )) 2 , ( Z ( x + h 1 ) -Z ( x )) 2 and offset h 1 and h 2 in the following form: Applying the two versions of the theorem (2) and theorem ( 3) [19.20] .It should be noted that the border contains C(0) all will be reduced as a result of simplification as we get an equation that contains only limits in it (h) Variogram function .
 . Represents distance between location points, Note that equation ( 9) leads to equation (10) , That is proved by theorem (1), that is : From the standard definition of least generalized squares in the quadratic form: Because of that V diagonal matrix means that heterogeneity is equal to zero so the correlation Corr ( Z ( x i + h ) -Z ( x i )), ( Z ( y i + h ) -Z ( y i ) = 0 Then we can get [2], [3] : results in : After simplification, the following equation is produced: Formula( 14) was obtained from the assumptions of the researcher (Cressie) [17], and adopted (Chiles and Delfiner) [18].The approximate estimate of the least weighted squares can be obtained from the reduction G to me  it is minimized G by taking the derivative, we cannot get a complete equation that can be analyzed to get  * .In this case , we use the algorithm Newton Raphson to obtain the estimated  * .This will be explained in the practical side.

Section ΙΙ Calculating the quality of the Variogram Function using the Generalized least Squares (GLS) Introduction
The practical aspect includes the Newton Raphson algorithmto obtion the estimator and estimating the variogram function indicator for real data taken from the site ( x i ) and values Z ( x i ) Since i = 1, 2,… n and the total of data from inside Iraq water wells in the area Ba'shiqah /province of Nineveh / Iraq [21] .They are the aquatic conductivity plants in 45 Exploration well in the area, has been taking a subset of these the data (30) wells to get a regular network Regular grid, and Table No(   for the spherical and exponential models.9-End.

2-3-Estimate Generalized least Squares:
Now we will use Newton Ravson ' s algorithm to get the estimator  * in the equation (14).In order to apply the Generalized Least Squares we must derive the equation ( 14) derived partial first and second, that is: We have obtained the required results by repeating the above method.  in the algorithm is as follows: Since  ) k+ 1 ( appreciation  when repeating appreciation  when repeating k , q k the first derivative vector relative to the parameter  for G in equation( 14) when repeating k and G k The second derivative matrix of the equation G for master  and G known as the Hessian matrix .
If we assume that : So By Levenberg -Marquardt parameter [16.22] .We stopped the implementation of the program at 100 iterations to get the best approximation between the real and approximate values curve and the lowest mean error square between them and the exponential and spherical models .

2-4-Conclusions
Because we did not get the complete solution for estimating the least squares, Newton Ravson ' s algorithm was used to obtain the final solution for estimation.And we listed the four directions

Appendix (A)
and now we can know the indicator b (x) ,From the constant variable Z (x) , Simply equal to 1, if :which means , Less from the b(x) indicator or equal to a Specified Threshold Z C and (0) otherwise b(x) = x is location within the study area/R is Euclid's space / P=2 tow dimensions or P=3 three dimensions / D a field or area under study .By defining the indicator we have divided the measurement of the spatial variable Z (x) Into two sections the first The Z (X) ≤ Z C And the other section Z (x)> Z C Respectively.This division is known as Indicator variable.If the variable z(x) Represents a watch for a random process Z (x), The indicator b (x) It is considered as a view of the random indices function B[Z(x)≤ Z C ] this function is a new binary random stochastic process [7].
. Which should be a measure of convergence between the experimental model and the theoretical model by the total difference between the boxes variogram demo and variogram indicator theory, which must be less than what can, and of course a function variogram demo containing unknown parameters represent a vector  as:  =(  1 ,  2 , ... ..,  n ) will therefore encode function variogram index theoretical form (h : ), That is : The smallest thing possible,as that S( ) it represents the sum of the residuum boxes or error between variogram indicator theoretical and variogram indicator, when the: It symbolizes the vector of random variables ) , i = 1, 2,... n .with a contrast matrix where : weighted matrix .Diagonal matrix with zeros in all inputs except contrast , ( , i = 1,2,….n on the main diameter . 2 ) , and she was Corr ( x 1 , x 2 ) = ρ Then Corr ( x 1 2 , x 2 2 ) = ρ 2 1-3-2-Theorem ) 2) [17] : For each pair of new spatial random variables [Z ( x + h ), Z ( x )] Covariance exists and depends on the separation distance h, The stability of covariance leads to the stability of variance and the variogram indicator and a correlation can be obtained between covariance C ( h ) The function of the indicator variogram and contrast .As in the following formula : and get their expense matrix V after adapting the theoretical pharyocram model, which may be spherical, exponential, or Gausian, we set the parameters  Place the values that we get from the drawing variogram index to get appropriate (Spherical Model) in terms of the model  Which (h ; ) and from V can get V(( .

3 Fig. 1 :Fig. 2 :
Fig. 1: The four angles used to calculate the semivariogram function represent the indicator

3 - 4 - 5 - 6 - 7 - 8 -
Calculate the first derivative of equation (14.)Calculate the second derivative of equation (14.)Using Newton Ravson ' s algorithm by taking the values θ 01 , θ , 02 θ 03 and stopping condition is 100 iterations.Calculate the spherical model and the exponential model of the new values (obtained from the Newton-Ravson algorithm from step(5).Comparing old and new results (ie improved values) using the least squares method with the graph.Calculate the lowest error ratio between the real and estimated values using equation (15) This was a special program that used a system Matlab The growth exponential primary values obtained from the drawing variogram function, which were taken from the drawing function semi variogram indicator of Figure( 2 ) in the total of the data for 1 ,  2 ,  3  01 = 0.6  02 = 1.2  03 = 9.3The spherical model (Spherical Variogram) Can be written as follows:The exponential model (Exponential Variogram) It is as follows:


θ 1 , θ 2 , θ 3 The values of λ are as follows: at 100 iterations and a value  = 300 best in exponential model: In the spherical model, the best was achieved when repeating 100 and value  = But we did not just get the above results, we calculated the quality of the reconciliation of variogram by the estimator of least squares weighed by the mean error square (MSE ( Mean Squares error which is calculated from the following law: Of a special program specially prepared for this purpose by the system MATLAB was obtained less error ratio between the real values and the estimated values for the data in the study site using the value of the spherical model MSE = 7.9911 Figure (3) illustrates this.Using the exponential model, the lowest error rate was MSE = 1.9551 Figure (4) illustrates this.That is, the results given by the spherical model are better than exponential patterns .

Fig. 3 :Fig. 4 :
Fig. 3: is the best estimate of the spherical model

Table 1 : Models of Variogram functions
Both a,  0 ,  unknown parameters called covariance components or semi indicator variogram functions

Table 3 : Results of the Parameters of the Parameters of Data at the Study Location
h n ( h )  (  , h ) h n ( h )  (  , h ) h n ( h )  (  , h ) h n ( h )  (  , h )