Assessing the Spatial Distribution of Rainfall and the Effect of Altitude in Iran (Hamadan Province)

Kriging

Kriging methods rely on the notion of autocorrelation. There is difference between classical statistics and geostatistics technique. Over classical statistics, observations are assumed to be independent which means there is no correlation among observation samples. In geostatistics, autocorrelation is known as a distance function. The information on spatial locations allows researchers compute distances among observation samples and model autocorrelation as a function of distance. This is related to a basic principle of geography issues: Observation samples closer together tend to be more similar than those that are farther apart.^12,13 The experimental variogram measures the average degree of dissimilarity between unsampled values and a nearby data value and thus can depict autocorrelation. ¹⁴ The value of the experimental variogram for a separation distance of h (referred to as the lag) is half the average squared difference between the value at z(x_i) and the value at z(x_i + h) ¹⁵ :

where N(h) is the data pair number 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 equation (1) will be small, relative to an uncorrelated pair of points. From an analysis of the experimental variogram, a suitable model is fitted, usually by weighted least squares, and the parameters (eg, range, nugget, and sill) are then used in the kriging procedure. In this study, ordinary kriging and simple kriging are used with their models.

Co-kriging

Co-kriging is defined as a multivariate version of kriging that minimizes the variance of estimation error by considering the spatial correlation between the variables of interest and secondary variables. The cross-semivariogram is used to quantify cross-spatial autocovariance between the original variable and the covariate. ¹⁶ The cross-semivariance is computed by equation (2):

where γ u v h is the cross-semivariance between u and v variables, Z u ( x ) is the primary variable, and Z v ( x ) is the secondary variable.

Radial basis functions

Radial basis function (RBF) methods are “a series of exact interpolation techniques; that is, the surface must go through each measured sample value. Each basis function has a different shape and results in a slightly different interpolation surface.” Radial basis function showed that radial basis function neural network gives the same results as a standard geostatistical technique: prediction mapping of the regression function and mapping of the prediction variance. Radial basis function neural network is expressed by equation (3):

where λ is a zero-mean noise and h is the RBF.

Radial basis functions are a class of functions, the key feature of which is that the distance from the center determines their response. Most commonly used RBF is the Gaussian RBF. Gaussian model provides an example of local RBF that gives significant response only in a neighborhood of its center.^17–20

Inverse distance weighting

Inverse distance weighting (IDW) is “a deterministic, nonlinear interpolation technique that uses a weighted average of the attribute (ie, phenomenon) values from nearby sample points to estimate the magnitude of that attribute at nonsampled locations.” The weight a particular point is assigned in the averaging calculation depends on the sampled point’s distance to the nonsampled location. These weights are controlled on the bases of power of 10. Often with an increase in power, the effect of the points that are farther diminishes. Lesser power distributes the weights more uniformly between neighboring points. We should keep in mind that in this method the distance between the points counts, so the points of equal distance have equal weights.^21,22 In this method, the weight factor is calculated by equation (4).

where λ i is the weight of point, D i is the distance between point, i is the unknown point, and α is the power 10 of weight.

Global polynomial interpolation

Global polynomial interpolation (GPI) is “a quick deterministic interpolator that is smooth. There are very few decisions to make regarding model parameters. It is best used for surfaces that change slowly and gradually.” However, there is no assessment of prediction errors and it may be too smooth. Locations at the edge of the data can have a large effect on the surface. There are no assumptions required of the data.

Local polynomial interpolation

Local polynomial interpolation (LPI) is “a moderately quick interpolator that is smooth. It is more flexible than the global polynomial method, but there are more parameter decisions.” The method provides prediction, prediction standard error, and condition number surfaces that are comparable with ordinary kriging with measurement errors.