Introduction
Identification of spatial patterns of various soil properties such as soil fertility status or salinity/sodicity problem is a prerequisite to designing any management strategy for fertilization programme or reclamation of salt affected soils. Traditionally, salt affected soils especially sodic soils are reclaimed using a single uniform rate of gypsum irrespective of the presence of spatial patterns of gypsum requirement in a field. This practice causes underapplication or overapplication of gypsum in different areas of the field when spatial patterns of gypsum requirement are present. Considerable research work has been carried out by many scientists to identify the magnitude and extent of spatial variability of soil fertility status and crop productivity (Trangmar et al., 1987; Bhatti et al., 1991; Wollenhaupt et al., 1994) and variablerate fertilizer management (Mulla et al., 1992; Bhatti and Mulla, 1995). Some research work has also been carried out on spatial variability of salinity/sodicity (Wagenet and Jurinak, 1978; Hajrasuliha et al., 1980; Bhatti and Bakhsh, 1995).
Materials and Methods
Spatial patterns of gypsum requirement of a salt affected field were studied before and after gypsum application. For this purpose a salt affected field was selected in Shandau village of Dera Ismail Khan district. The field was divided into six parallel transacts 154 m long and 5 m wide. Each transect was sampled at 15 m intervals from 025 cm depth. Gypsum requirement of all the samples was determined. Each transect was divided into two strips. In one strip gypsum was applied at a single uniform rate based on the mean value of gypsum requirement. In the adjacent strip, variable rates of gypsum were applied to match the gypsum requirement of different locations. Wheat crop was planted in the whole field after gypsum application. All the recommended cultural practices were followed during the growth period of crop. Wheat crop was harvested at 15 m interval from an area of 1 m^{2} from both the strips on all the six transacts. Wheat yields were converted to per hectare basis. Surface soil samples were collected after wheat harvest from both the strips in each transect at 15 m intervals. All soil samples thus collected were analyzed for gypsum requirement. Three dimensional plots of gypsum requirement before and after treatment and grain yields were made to visualize spatial patterns. Descriptive statistics (mean and coefficient of variation) for all the data on gypsum requirement and wheat yields were computed. Differences between gypsum requirements before and after treatment were compared using ttest of significance. Similarly, ttest was also applied to compare the differences in gypsum requirement and wheat yields between the two management strategies. Geostatistical technique of auto correlation analysis was used to identify the magnitude and extent of spatial dependence of observations on gypsum requirement. Auto correlation analysis provides a quantitative estimate of the degree to which sample points are correlated with one another by virtue of separation distance. Because samples taken from locations closer to one another are typically more closely related than are samples taken from locations farther apart. Auto correlation statistics of semivariance was calculated, which when graphed yields a sernivariogram. Semivariance is defined as:
where,
G(h) = semivariance for lag distance h
z_{i }= measured sample value at point I
z_{i+h}, = measured sample value at point h+I and
N(h) = total number of sample pairs for the lag interval h
The resulting graph of semivariance vs. different lag distances yields a semivariance in a data set. Different models are used to fit the semivariograrn of the data. The model is considered to be the best fit when the residual sums of squares (RSS) term is not further minimized by significant changes in model parameters.

Fig. 1:  Spatial patterns in gypsum requirement before gypsum application 

Fig. 2:  Spatial patterns in gypsum requirement in variable strategy after gypsum application 
RSS is calculated for the regression of actual vs. model estimated sernivariance at each lag distance. Different models used include linear, linear to till, spherical. exponential and gaussian model. Each model Is defined in terms of nugget variance (C_{o}), sill (structural variance C, +nugget variance C_{0}) and range (a) parameters. Spherical and exponential models were found to be the best fit for the present study.

Fig. 3:  Spatial patterns in gypsum requirement in uniform strategy after gypsum application 
Spherical model is a modified quadratic function in which at some distance 'a', pairs of points no longer be auto correlated and the semivariogram asymptotes. The formula used for this model is:
G(h) 
= 
C_{0}+C_{1} (1.5(h/a)0.5(h/a)^{3} I for h<a 
G(h) 
= 
C_{0}+C_{1} for h>a 
Where h 
= 
lag interval 
C_{0} 
= 
nugget variance >0 
C_{1} 
= 
tructural variance >0 and a 
a 
= 
range 
Exponential model is similar to the spherical in that it approaches the sill gradually, but different from the spherical in the rate at which the sill is approached and in the fact that the model and the sill never actually converge. The formula used for this model is:
G(h) 
= 
C_{o}+C_{1} 11exp(h/aH 
Where h 
= 
lag interval 
C_{0} 
= 
nugget variance >0 
C_{1} 
= 
structural variance >0 and 
a 
= 
range parameter (not range) 
Range in the exponential model is usually assumed to be the point at which the model includes 95% of the sill (C_{0}+C_{1}), this can be estimated as 3* a.
Results and Discussion
Descriptive statistics of gypsum requirements before and after gypsum application (Table 1) showed that the mean value of OR reduced significantly both in uniform (t = 11.82, p = 0.0000) and variablerate strategy (t = 12.78, p = 0.0000). However, the GR was significantly lower in variablerate strategy than in the uniformrate strategy after gypsum application. It = 5.24, p = 0.0000). Coefficient of variation for GR increased in both the strategies, Grain yield of wheat (Table 1) was lower in variablerate strategy than in the uniformrate strategy but the difference between the two strategies was nonsignificant (t = 0.39, p = 0.6975).
Table 2:  Parameters of semmariogrem models for GR 

Table 3:  Parameters of sernivariograrn models for grain yields 

Moreover, the coefficient of variation for grain yield was higher in uniformrate strategy then in the variablerate strategy. Regarding the spatial patterns of GR, three dimensional plots of GA before and after treatment with gypsum showed that there were strong spatial patterns in GR before gypsum application (Fig. 1). After gypsum application, the spatial patterns were still present in both the management strategies (Fig. 2 and 3) and were more obvious in variablerate strategy. Sernivariogram analysis of GR before gypsum application (Table 2) showed that there were strong spatial patterns. A summary of the best fitting semivariogram model parameters (Table 2) showed that the exponential model was the best fit for GR with a nugget of 14.55 and a range of influence of about 295 m. The r^{2}value for this model was 0.49. Exponential model was also the best fit for GR after gypsum application in uniformrate strategy but the value was very small, showing poor spatial pattern. While spherical model was the best fit for the semivariogram of Gil in variablerate strategy with a nugget of 0.445 and a range of influence of 48.5 m. The r^{2} value for this model was 0.72. Semivariogram analysis of grain yields in both the management strategies (Table 3) showed that there were strong spatial patterns in grain yields.
Exponential model was the best fit for grain yield semivariogram in uniformrate strategy with a nugget of 2.64×10^{5} and range of influence of 39 m. Spherical model was the best for the semivariogram of grain yield in variablerate strategy with a nugget of 3.45×10^{5} and a range of about 100 m.
Application of gypsum using a single uniform rate as well as variable rates reduced the GR significantly. Strong spatial patterns were also observed after treatment with gypsum in variablerate strategy. These were due to the treatment with different rates. Similarly, strong spatial patterns of grain yields were observed in both the management strategies. Such fields should be divided into different management units and be managed differently to increase the efficiency of added inputs.