# Towards Hyper-Dimensional Variography Using the Product-Sum Covariance Model
## 2 Theory
### Modeling of the Hyper-Dimensional Variogram Based on the Product-Sum Model
However, the inferred covariance parameters could be slightly different and dependent on the modeling sequence if tolerances are large, though this can be overcome by modeling at-once, as shown in Section 2.2.2. If we start by modeling first the \(C_{h,t}(h_{h}J_{t})\) in Equation (8), the initial values for \(a_{1}\)\(-\)\(a_{6}\) would be different, yet their products (shown in bold) in Equation (10) would be the same, as they ultimately come from data, which again points to the symmetrical properties of the product sum model with respect to its basic covariance components. The resulting covariance is guaranteed to yield a valid model, as the product of constants \(a_{1}\)\(a_{4}\) that multiplies the first term in Equation (10) is always \(>\)0, while all other products of constants are \(\geq\)0. The four underlined covariance product terms of the final model represent a Hadamard product [38] of two or more positive definite matrices. According to [PERSON] product theorem, a Hadamard product of two positive definite matrices necessarily gives a positive definite matrix [39]. Thus, the resulting model is guaranteed to be valid. In the original sequential modeling procedure starting from one-dimensional variograms, \(\gamma_{s,t}\) (\(h_{s}\),0) and \(\gamma_{s,t}\) (\(0\), \(h_{t}\)) were modeled by pre-determining a spatial and temporal tolerance within which a pair of data can still be considered collocated or coincident. However, in practice the required tolerance might need to be very large to yield enough data points, which would lead to inaccuracies in the modeling of both one-dimensional and higher-dimensional variograms, given that lower dimensional variograms are building blocks of higher dimensional variogram (see Section 2.2.2 for further discussion). Furthermore, correctly representing variability along all axes of anisotropy leads to the reduction in the estimated nugget, if nugget-model variograms are used. For example, in a typical spatial-only variography of satellite retrievals, the variography step is preceded by temporal binning, i.e., data from multiple time periods are aggregated and treated as coincident (e.g., [5; 6]). The consequence is that ignored variability along the temporal axis ends up being superposed to the spatial variability as its unexplained portion--nugget. Similarly, the variability within tolerances imposed by the original sequential product-sum modeling procedure [22] gets superposed to the true variability along the actual modeling axis affecting the unexplained portion of variability. By treating not-exactly-collocated and not-exactly-coincident observations as collocated and coincident, modeler allows some of the spatial variability (within tolerance) to affect the separate temporal variogram, and some of the temporal variability to affect the separate spatial variogram, respectively. In this way, separate spatial and temporal variogram models represent an approximation of the separate variogram models that would be obtained using strictly collocated and strictly coincident observations. This issue was not discussed in the original paper by [PERSON] et al. (2012), although it affects modeling results for the sequential approach (as the nugget has to be modeled in the first step), but not in modeling the \"all at once\" approach (Section 2.2.2). Thus, we change the approach, and model all parameters at once.