Ordinary Kriging (OK)

The OK interpolation method is the most common and simple among the Kriging techniques, which incorporates statistical properties of the measured data (spatial autocorrelation) (Bhunia et al. 2018). Ordinary kriging is also described by the acronym BLUE, as mentioned above. OK assumes that the constant mean is unknown and defined as the mean of samples (local mean). As mentioned above, the first step for the application of Kriging is the estimation of the semi- variogram. The equation used for OK interpolation is given in eqs. 1 and 2:

------------------------------- (1)

----------------------------------------- (2)

where ZOK (xo) is the interpolated value for point (xo), Z(xi) is the known value, λi is the OK weight for the Z(xi) value. In addition, λi values must be evaluated to obtain an unbiased estimation and to minimize the error variance (Pellicone et al. 2018).

Inverse Distance Weightage (IDW)

IDW is a simple kind of deterministic method and mostly used in soil science (Bhunia et al., 2018). This method assumes that the measured values at a closer distance have greater weight than those further away. Nearer points are given more weightage and the weightage reduces with increase in distance. It is an exact interpolation method with the maximum and minimum values occurring in the sample points only; thus, any clustering or presence of outliers can affect the prediction. The infuence of a known value is inversely related to the distance from the unknown data point. Consequently, this method gives greater weights to values closest to the prediction position and the weights reduce as a function of distance. IDW determines cell values using a linearly weighted combination of a set of sample points and is given in eqs. 3.

--------------------------- (3)

where Z(xo) is the estimated unknown value, x is the set of spatial coordinates (x1, x2), and r is the weight related to the distances dij which is the distance between the estimation point of the n data points.

Radial Basis Function (RBF)

The Radial Basis Function (RBF) is an interpolator which is based on a form of artificial neural networks i.e. input layer, hidden layers, and output layer (Johnston et al. 2001; Ali et al. 2021). In RBF, the generated surface requires passing through each measured point while minimizing the total curvature of the surface (Johnston et al. 2001). In addition, RBF can predict values above the maximum and below the minimum. The interpolator basis functions that are covered by the RBF include a thin-plate spline, tension-based spline, completely regularized spline, multi-quadric function and inverse multi-quadric spline

Empirical Bayesian Kriging (EBK)

According to (Krivoruchko et al. 2019) EBK consists of two geostatistical models: the intrinsic random function kriging (IRFK) and the linear mixed model (LMM). In EBK, the stochastic spatial process is represented locally as a stationary or nonstationary random feld and the parameters of the locally defined random feld are allowed to vary across space (Gribov et al. 2020). Empirical Bayesian kriging also differs from other kriging methods by considering the error introduced by estimating the underlying semi-variogram. Classic Kriging techniques obtained semi-variogram is evaluated from known data positions and is considered as the single, true semi-variogram and it is used to make predictions at unknown positions by not taking the uncertainty in the semi-variogram estimation into account, thus underestimating the standard errors of predictions (Pellicone et al. 2018; Schneider et al. 2001). EBK is a geostatistical interpolation method that automates the most difficult aspects of building a valid kriging model through a process of sub setting the study area, coupled with multiple simulations to obtain the best ft (Krivoruchko et al. 2019). This process finally creates a spectrum of semi-variograms and each of these is an estimate of the true semi-variogram for the subset (Pellicone et al. 2018). So here the standard errors of prediction are more accurate than kriging methods (Adhikary et al., 2011). EBK function consists in the following:

------------------------ (5)

where p stands for a parameter, Zp for critical level of the parameter, ip is equal to 1 or zero depending on whether p is lower or higher than zk, respectively (eq. 5), Sj is a kriging weight estimated on the basis of a cross-variogram between ip (x, p) and U(x); and U(x) is given in eqs. 6 and 7.

----------------------------------------- (6)

----------------------------------------------- (7)

where R is rank of Rth order statistics of parameter measured at location x.

In a simplified way unknown sample estimation is done in EBK in the following way (Rajesh et al., 2019):

i) Estimation of semi-variogram from known data, followed by simulation of a new value from the estimated semi-variogram.

ii) ii) Estimation of another new semi-variogram using the newly formed simulated data. The weights of the new semi-variogram model are calculated based on the Bayes’ rule.

During this process, the prediction of new values and their standard errors are calculated at unsampled locations. A semivariogram spectrum is generated by repetition of the above two steps.

Reference

- Adhikary, P.P., Dash, J., Bej, R., Chandrasekaran, H., 2011. Indicator and probability kriging methods for delineating Cu, Fe, and Mn contamination in groundwater of Najafgarh Block, Delhi, India. Environ. Monit. Assess. 176 (1–4), 663–676.
- Ali G, Sajjad M, Kanwal S, Xiao T, Khalid S, Shoaib F, Gul HN (2021) Spatial–temporal characterization of rainfall in Pakistan during the past half-century (1961–2020). Sci Rep 11:. https://doi.org/10.1038/s41598-021-86412-x.
- Bhunia, G.S., Shit, P.K., Maiti, R., 2018. Comparison of GIS-based interpolation methods for spatial distribution of soil organic carbon (SOC). J. Saudi Soc. Agric. Sci. 17 (2), 114–126.
- Pellicone G, Caloiero T, Modica G, Guagliardi I (2018) Application of several spatial interpolation techniques to monthly rainfall data in the Calabria region (southern Italy). Int J Climatol 38:3651–3666. https://doi.org/10.1002/joc.5525
- Johnston K, Ver Hoef JM, Krivoruchko K, Lucas N (2001) Using ArcGIS geostatistical analyst. Analysis 300:300
- Krivoruchko K, Gribov A (2019) Evaluation of empirical Bayesian kriging. Spat Stat 32:. https://doi.org/10.1016/j.spasta.2019. 100368
- Gribov A, Krivoruchko K (2020) Empirical Bayesian Kriging implementation and usage. Sci Total Environ 722:. https://doi.org/10. 1016/j.scitotenv.2020.137290
- Schneider B, Martinoni D (2001) A distributed geoprocessing concept for enhancing terrain analysis for enviromental modeling. Trans GIS 5:165–178. https://doi.org/10.1111/1467-9671.00074
- Rajesh, R., Elango, L., Brindha, K., 2019. Methods for assessing the groundwater quality.In: GIS and Geostatistical Techniques for Groundwater Science, pp. 57–78.

**Contributors :** Akanksha Sikarwar, Rahul Mishra, Seema Bhardwaj, Vimal Shukla, Mayank Vyas, Rahul Prajapati and Yogesh Sikaniya

Indian Institute of Soil Science ,Bhopal (Madhya Pradesh); R.V.S.K.V.V, Gwalior (Madhya Pradesh); CSAUAT,Kanpur (U.P)

Last Modified : 12/12/2023

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