# Towards Hyper-Dimensional Variography Using the Product-Sum Covariance Model
## 2 Theory
### Modeling of the Hyper-Dimensional Variogram Based on the Product-Sum Model
#### 2.2.2 Modeling \"All at Once\"  
To avoid defining a tolerance, we alter the original procedure by estimating all covariance parameters simultaneously. This simultaneous parameter estimation makes the model more applicable to scattered data and data with variable spatial coverage, as is often the case with airborne observations or satellite data. Defining the tolerance followed naturally from the original sequential modeling procedure proposed in [PERSON] et al. (2012), starting by modeling separate one-dimensional variograms used in subsequent steps to yield a generalized product-sum model. In the original procedure, the first step assumed computing the sample spatial and temporal variograms corresponding to \(\gamma_{s,t}\) (\(h_{s}\),\(0\))and \(\gamma_{s,t}\) (\(0\),\(h_{t}\)):  
\[\hat{\gamma}_{s,t}(r_{s},0) = \frac{1}{2|N(r_{s})|}\sum_{N(r_{s})}[Z(s+h_{s},t)-Z(s,t)]^{2} \tag{11}\] \[\hat{\gamma}_{s,t}(0,r_{t}) = \frac{1}{2|M(r_{t})|}\sum_{M(r_{t})}[Z(s,t+h_{t})-Z(s,t)]^{2} \tag{12}\]  
where \(r_{s}\) and \(r_{t}\) were, respectively, the vector lag with spatial tolerance \(\delta_{s}\) and the lag with temporal tolerance \(\delta_{t}\). \(\left|\,N(r_{s})\,\right|\) and \(\left|\,M(r_{t})\,\right|\) are the cardinalities of the following sets:  
\[N(r_{s}) = \{(s+h_{s},t)\in A,\;(s,t)\in A:\;\left\|r_{s}-h_{s}\right\|<\; \delta_{s}\} \tag{13}\] \[M(r_{t}) = \{(s,t+h_{t})\in A,\;(s,t)\in A:\;\left\|r_{t}-h_{t}\right\|<\; \delta_{t}\} \tag{14}\]  
The need to define the tolerance follows from the fact that, in order to construct separate spatial variogram based on coincident observations (\(\hat{\gamma}_{s,t}(r_{s},0)\)), or separate temporal variogram based on collocated observations (\(\hat{\gamma}_{s,t}(0,r_{t})\)), the observational data set in practice has to be dense, or tolerances have to be relatively large to yield enough data for the construction of variograms. In other words, there are often too few collocated pairs of points that cover a wide range of temporal distances, and coincident points that cover a wide range of spatial distances, which imposes increasing the tolerances. The trade-off is that the variabilities captured by separate variograms become virtually higher, as temporal and spatial variability between pairs of points separated within the tolerance gets superposed on spatial and temporal variability of coincident, and collocated points, respectively. However, if the hyperdimensional variograms are fitted to the raw variograms at once, the need to define tolerances vanish. All variogram parameters are estimated simultaneously, including the ones that correspond to separate spatial and temporal variograms. The fitting procedure exploits all found pairs of points at their exact spatial and temporal distances and fits the optimal parameters at once. The modeling starts by fitting the V(n) variogram corresponding to Equation (3). into an experimental (or alternatively a raw) variogram, and replacing k using Equation (7). Then the V(n-1) variogram component on the right-hand side of the expression is again substituted using Equation (3) and Equation (7), and maximal dimensionality of the basic variograms is again reduced by one. The procedure is recursively repeated until all variogram terms on the right-hand side have dimensionality one. While the expression itself can be very large, the number of parameters to estimate all at once is relatively small and computationally feasible. For example, to model V(3) variogram using common 3-parameter exponential or Gaussian model, it would be required to simultaneously estimate only 9 parameters (nugget, 3 times V(1) sil and range parameters, and two k-parameters). For temporally evolving anisotropic full 3D space, only 12 estimated parameters would be required. Generally, the number of parameters for an \(n\)-dimensional variogram to be estimated is \(3n\).