# Towards Hyper-Dimensional Variography Using the Product-Sum Covariance Model
## 3 Application  
As an example of how our method can address spatial scaling problems, we explored sensitivity of gross primary productivity (GPP) to spatial variability in photosynthetically-active radiation (PAR). Clouds influence the quantity and quality of light for GPP [40; 41]. However, gridded meteorological datasets are often too coarse (~100 km) to resolve mesoscale (~10 km) weather systems (e.g., thunderstorms) that affect PAR. Furthermore, modeled GPP calculated from coarse-resolution PAR can be biased due to nonlinearity in the GPP-PAR relationship. We explored the impact of spatial variability in PAR on GPP, using our kriging method to estimate shortwave radiation from Oklahoma Mesonet stations (PAR was approximated as half the downwelling shortwave radiation; [42]. We modeled GPP as a function of PAR (black curve in Figure 1), using a two-leaf canopy model applied to winter wheat in Oklahoma [43].  
The spatio-temporal modeling of PAR was done under a moving window setup [44] in such a way that spatial covariance for every 5-min time slice was modeled separately, allowing sampling of the data within +/-30 preceding and following 5-min time periods. Thus, problems resulting from the non-stationarity of the PAR mean due to the diurnal cycle were avoided. We assumed that the V(\({}^{3}\)) variogram offers an adequate representation of the variability, and treated time, horizontal and vertical distances as orthogonal components in variogram space, and thus modeled separately. The vertical and temporal directions were modeled using Gaussian, and horizontal using an exponential model. After modeling the covariance, we kriged the PAR to a regular ~10 \(\times\) 10 km grid mimicking the kriging approach in [PERSON] et al. [6], and obtained the average PAR for May 2006 shown in Figure 1.  
We estimated the impact of spatial variability in PAR on GPP, using GPP calculated from spatially-averaged PAR as a basis for comparison. Due to nutrient or other limitations, GPP can become light-saturated, such that further increases in PAR yield diminishing returns for GPP at high PAR values. This saturation effect is characterized by the nonlinear shape of the GPP-PAR curve (Figure 1, black line). Forcing a GPP model with spatially-averaged PAR could lead to overestimated GPP, since the full spatial distribution of PAR would include values in the light-saturated part of the curve while the spatial average would fall closer to the unsaturated part. In that case, higher values of PAR increase average PAR but do not substantially increase average GPP.  
Indeed, we found that GPP calculated from the spatially-averaged PAR was up to 1.5 umol m\({}^{-2}\) s\({}^{-1}\) higher (Figure 1, red circle) than the average of GPP calculated from the spatially-variable PAR (Figure 1, blue line). The difference between these two estimates is 24% of the observed standard deviation in daytime half-hourly GPP at a site near Lamont, Oklahoma [43], and indicates the importance of spatial variability in climate drivers of GPP. This likely represents an upper bound on such effects at this site, as the GPP-PAR curve used here has a high degree of nonlinearity to illustrate the utility of our method for a general class of problems. Related problems for which this method is applicable include modeling spatiotemporal variability in soil moisture and leaf area index, and the nonlinear influence of these variables on evapotranspiration and surface energy partitioning [45].