where \(\mu(u_{\alpha})\) is the Lagrange multiplier,\(\overline{C}\left(u_{i},u_{j}\right)\) and\(\overline{C}\left(u_{i},u_{\alpha}\right)\) are block-to-block covariance terms. The most important step for the implementation of ATAK is to obtain the point support covariance for deriving the covariance terms. A deconvolution procedure can be used to achieve the point support covariance Goovaerts (2008). In our study, 25 neighboring pixels were employed to predict the target area of ATAK.

2.3 Support vector area-to-area regression kriging

The proposed SVATARK is based on SVR for trend prediction and ATAK for residual prediction. Let \(Z\left(S_{i}\right)\) and\(X_{k}\left(S_{i}\right)\) be the target and k ancillary random variables at coarse pixel \(S_{i}\). The nonlinear regression model between \(Z\left(S_{i}\right)\) and\(X_{k}\left(S_{i}\right)\) can be obtained using Equation (2), denoted by \(f_{\text{SVR}}\left(\bullet\right)\). Assuming that the statistical relationship among variables is scale-invariant, the trend component of the fine spatial resolution can be estimated by using the coarse regression function: