INTRODUCTION

The Geographic Information System (GIS) is an effective and worthwhile tool in the estimation of the spatial distribution of environmental variables (Dogan, 2007a; Ordu and Demir, 2009; Rabah et al., 2011). The spatial interpolation methods are commonly used for estimating temperature or precipitation when climate stations are not dense (Maciej, 1998; Dogan, 2007b). Rainfall regime is the most important climatic parameter strongly influencing agricultural activities, natural and water resources in hydrological studies (Tzimopoulos et al., 2008). It can considerably vary with time and space; even within a few kilometers distance and very short time scales (Ayanlade, 2008). Hence, having rainfall/precipitation maps is of essential needs for earth science researchers, environment planners and watershed managers (Ninyerola et al., 2007).

Interpolation can be undertaken utilizing simple mathematical models (e.g., inverse distance weighting, trend surface analysis and splines and Thiessen polygons), or more complex models (e.g., geo-statistical methods, such as kriging) (Negreiros et al., 2011). There are many studies showing the application of interpolation methods for estimating temperature or precipitation. The methods include distance weighting (Eischeid et al., 1995; Lennon and Turner, 1995; Ashraf et al., 1997; Dodson and Marks, 1997), interpolating polynomials (Stewart and Cadou, 1981; Tabios and Salas, 1985; Eischeid et al., 1995), kriging (Phillips et al., 1992; Hammond and Yarie, 1996; Holdaway, 1996; Ashraf et al., 1997) and splines (Hulme et al., 1995; Lennon and Turner, 1995). Among the methods, the geo-statistical interpolation method has found grounds in climatology because it is based on the spatial variability of variables of interest and can quantify the estimation uncertainty (Martinez-Cob, 1996; Holawe and Dutter, 1999; Sun et al., 2008).

The study of precipitation on a global scale has found interests and there have been attempts to map worldwide precipitation. In this respect, there are many studies using various methods ranging from more classic approaches such as the Times Survey Atlas of the World (Bartholomew, 1922) to more recent initiatives that utilize GIS techniques such as the Digital Atlas of the World Water Balance.

To validate a predictive model and choose appropriate algorithm Cross-Validation (CV), which is a statistical method, has been widely used (Arlot and Celisse, 2010). Past applications of the CV approach have given a range of results which have not always been consistent (Falivene et al., 2010). For instance, the comparison between two widely used algorithms such as kriging and IDWA have resulted in inconsistent conclusions; some authors have concluded that kriging yields better interpolations (Weber and Englund, 1994; Goovaerts, 2000; Teegavarapu and Chandramouli, 2005), some have not found any significant difference in the results (Dirks et al., 1998; Moyeed and Papritz, 2002) and others have advocated that the IDWA methods yields better interpolations (Weber and Englund, 1992; Lu and Wong, 2008).

This research aimed to implement and compare the accuracy of different interpolation methods including Inverse Distance Weighted Averaging (IDWA), regularized and tension spline, spherical, circular, exponential and Gaussian kriging. The accuracy assessment was implemented using Cross Validation (CV) on the interpolated annual precipitation of the Fars Province, Iran.

MATERIALS AND METHODS

Study area and data: The study area is located at the south and south west of Iran (50°36'-55° 35' E and 27°03'-31°40' N), Fars Province, covering an area of about 133000 km² (Fig. 1). Agriculture and animal husbandry are the main economic activities of people. The region has various climates, including arid (in south and south-east), temperate (in the center) and mountainous (in north and north-west). The mean annual temperature is 18°C and annual average rainfall is about 285 mm, of which 55% is received in the winter, 27% in the fall, 16% in the spring and 2% in the summer (Http: llirimo.ir). Twenty years historical rainfall data (from 1990-91 to 2009-10) published by the I.R. of Iran Meteorological Organization (IRIMO) were used to develop different spatial interpolation models. Rainfall data were observed from 92 locations at synoptic and rain gauge meteorological stations. In order to reduce the influence of outliers, 6 meteorological stations were added to database of the 92 stations, two stations (Bushehr and Bandar Abbas stations), located near the shores of Persian Gulf, were excluded. This provided adjusted and homogenized climatic data as these two stations were located close to large lakes.

Geospatial interpolation: ArcGIS and GS+ softwares were used for the visualization of model outputs and the geostatistical analysis, respectively (Negreiros et al., 2010).

Fig. 1:	Location of the study region and meteorological stations

Different interpolation methods including Inverse Distance Weighted Averaging (IDWA), regularized and tension spline as the determinist methods and spherical, circular, exponential and Gaussian kriging as the stochastic methods were used for interpolating yearly rainfall. A brief description of the methods used is provided below:

Inverse distance weighted averaging (IDWA): Inverse distance weighted averaging estimates the variable of interest by assigning more weight to closer points (Burrough and McDonnell, 1998; Goula Bi et al., 2011):

(1)

where, d_i is the distance between x₀ and x_i, p is a power parameter and n represents the number of sampled points used for the estimation.

The main variable affecting the accuracy of IDWA is the value of the power parameter. Weights diminish as the distance increases, especially when the value of the power parameter increases (Isaaks and Srivastava, 1989; Robinson and Metternicht, 2006). The powers (1 to 5) were examined to select those models with lower error (i.e., IDWA (1), IDWA (2)… IDWA (5).

Spline: The spline method can be thought of as fitting a rubber-sheeted surface through the known points using a mathematical function. Splines consist of polynomials, which describe pieces of a line or surface and they are fitted together so that they join smoothly (Anderson, 2002):

(2)

where, N is the number of sampled points used for the estimation, λ is Coefficient of linear equations, r_j is distance from the Sample of point and T is determined by the user.

Kriging: Kriging weights are derived from a statistical model of spatial correlation expressed as semivariograms that characterize the spatial dependency and structure in the data. The experimental variogram measures the average degree of dissimilarity between un-sampled values and a nearby data value and consequently can depict autocorrelation at various distances. The value of the experimental variogram for a separation distance of h (referred to as the lag) is half of the average squared difference between the value at z(x_i) and the value at z(x_i + h) (Robinson and Metternicht, 2006):

(3)

where, N (h) is the number of data pairs within a given class of distance and direction.

If the values at z(x_i) and z(x_i + h) are auto-correlated, the result of Eq. 3 relative to an uncorrelated pair of points will be small, Using an analysis of the experimental variogram, a suitable model (i.e., spherical, circular, exponential and Gaussian) is fitted. This is made using weighted least squares and relevant parameters (e.g., range, nugget and sill) are then used in the kriging procedure. The examination of the annual precipitation data indicated that they were not significantly different from a normal distribution.

Models validation: The common validation method in climatological studies has variously been termed as cross-validation (Nalder and Wein, 1998). Cross validation (leaving-one-out method) is based on removing one data point at a time and performing the interpolation for the location of the removed point using the remaining samples. At the final step of cross validation, the difference (residual) between observed and predicted values of that point are calculated. The leaving-one-out approach is repeated until every sample has been, in turn, removed (Davis, 1987) and estimates are calculated for each point. The overall performance of each interpolation method is calculated using Mean Bias Error (MBE), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), (Solaimani, 2009). Moreover, correlation coefficient (r), a quantitative index of association between observed data (O) and predicted values (P) for N number of cases are calculated. It is described by Pearson’s product-moment correlation coefficient (Willmott, 1982):

(4)

(5)

(6)

RESULTS AND DISCUSSION

A summary of errors obtained from the application of interpolation models using cross validation tests together with correlation coefficient (r) is presented in Table 1. It is expected that an optimal interpolation algorithm should be basically associated with minimum cross-validation error (Falivene et al., 2010). Similarly, the application of interpolation models using inconsistent and un-homogenized climatic data may produce considerable errors in predictions. Hence, two stations of Bushehr and Bandar Abbas due to their vicinity to Persian Gulf and inconsistency and un-homogeneity that might give to the data base were removed from the considered stations. The inclusion of these stations produced extremely high errors whereas their removal reduced MAE and RMSE in all the models Table 1.

The results showed that exponential kriging produced the most accurate interpolated precipitation data (RMSE = 83.5, MAE = 60.0, MBE = 0-3.7 and r = 0.902). This method was therefore chosen as the most reliable model for interpolating precipitation and used for the final map production (Fig. 2). On the contrary, the worst performance has been observed where the IDWA (2) and (3), Gaussian, kriging and regularized spline methods were used (Table 1).

Table 1:	Difference measures of precipitation interpolation model performance
DM: Difference measure, SIM: Spatial interpolation model, MAE: Mean absolute error, RMSE: Root mean square error, r: Correlation coefficient, KC, KE, KG and KS, circular, exponential, Gaussian, spherical kriging, respectively; IDWA (2) and (3): Inverse distance weighted averaging with power 2 and 3, SR and ST: Regularized and tension spline

Fig. 2:	Interpolated precipitation map using exponential kriging

Fig. 3:	Comparison between observed and estimated precipitation using exponential kriging (KE)

Previous studies confirm our findings in this research. For instance, Weber and Englund (1994) and Goovaerts (2000) showed that the kriging method produced more accurate results compared to IDWA. They justified this as the kriging method could take the relative positions of sampling points and their distance from the interpolated point into account.

The results also showed that mean error was low for all kriging models, except for Gaussian, but the lowest value of mean error was resulted from the exponential kriging (KE) application. Moreover, the highest value of r (r = 0.902) was obtained using the KE compared to all other models. On the basis of MBE, the tension spline and Gaussian kriging produced lower errors. This might stem from the fact that MBE would not provide enough diagnostic to justify its inclusion, over other measures, in an array of model evaluation measures (Willmott, 1982).

The lowest MAE and RMSE were obtained using the exponential, circular and spherical kriging methods. Willmott (1982) argued that RMSE and MAE could be among the best overall measures of the model performance as both measures summarize the mean difference in the units of observed and predicted values. In a comparison of methods for estimating annual precipitation, the exponential and Gaussian kriging methods created the highest (r = 0.902) and the lowest (r = 0.842) coefficient of correlation, respectively (Table 1). Figure 3 shows the comparison between observed and estimated annual precipitation. In overall, the Fars Province can be divided into the following rainfall-altitudes relationships parts:

CONCLUSION

This research aimed to implement and compare the accuracy of different interpolation methods using cross validation errors for interpolating total annual precipitation of the Fars Province, Iran. The results showed that the application of the exponential kriging method produced less errors between observation and prediction (RMSE = 83.5 and MAE = 60.0 mm) and it was chosen to predict long term average precipitation. The tension spline, circular and spherical kriging methods were also reasonably performed well and could be used as the second order of priority. In contrast, the Gaussian kriging, IDWA and regularized spline methods were associated with an unacceptable level of errors and could not be justifiably used for interpolating long term average precipitation in the study area.

ACKNOWLEDGMENT

We wish to thank the I.R. of Iran Meteorological Organization (IRIMO) for providing us the precipitation data of meteorological stations in Fars Province.