Copula function and its application to estimate sand and bulk density of soil

Background and objectives: Spatial and temporal variations of soil characteristics occur in large and small scales. Study of these variations is very time-consuming and costly especially in large scales. In order to the fast and reliable determination of soil properties, various interpolation techniques have been developed and applied. The most widely used interpolation techniques in various sciences is the Kriging types. The copula function is one of the new interpolation techniques that are widely used in sciences such as hydrology. Thus, the aim of this study was to evaluate the spatial variations of some soil physical properties using copula function and to compare with geostatistics techniques.

Materials and methods: Sampling by regular networking was done in an area of 484 ha located in 10 km from the west of Baft city, Kerman province and finally, 121 surface soil samples were collected. After air drying, the apparent bulk density was determined using the Hunk, then the soil samples were passed through a 2 mm sieve to determine the percentage of sand. To interpolate, four functions of the Archimedean copula including the Clayton, Frank, Gumbel and Joe functions, and geostatistics techniques including simple, ordinary, universal and disjunctive Kriging and the Inverse Distance Weighting (IDW) method were used. The results were analyzed using Root Mean Square Error (RMSE), determination coefficient (R2), Mean Absolute Error (MAE), and Mean Bias Error (MBE).

Results: Based on the descriptive statistics, soil bulk density and soil sand followed a normal and skewed distribution, respectively. In order to fit the copula function, the distribution functions of the studied variables were firstly determined. The results showed that the sand and bulk density followed the Frechet (3P) and Wakeby distribution functions, respectively. Also, based on the Pearson correlation coefficient, the correlation between pairs of points was determined in distances less than 2000 m and distances more than 2000 m were known as an independent distance. The estimation efficiency based on the determination coefficient (R2) showed that value of determination coefficient for copula function for the sand variable, 6% and for bulk density 8%, more than conventional geostatistics techniques were obtained. Also, the estimation error of copula function was minimum that indicate good performance of copula function to estimate the spatial variation of soil physical properties.

Conclusion: The results of study showed that copula function, especially the median copula, have the better performance for estimation the studied soil properties. One of the most important reasons for this superiority is the ability to fit the marginal distribution function on the data in copula, while it is not possible in geostatistics techniques. Other reasons include the ability to express the correlation between the data at different intervals and the lack of sensitivity to outlier data in copula relative to conventional geostatistics techniques. Due to the skewness nature of soil data, as well as the need for more accurate analysis and interpretation of actual soil data, copula functions can be widely used to estimate of soil properties.