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: