Figure 4a shows our convention for the plaquette-square occupation fractions appearing in the program. The nine square centers used in the 2D program take on integer coordinate values\(\left\{x_{i}\right\}=\left\{0,1,2,0,1,2,0,1,2\right\}\),\(\left\{y_{i}\right\}=\left\{0,0,0,1,1,1,2,2,2\right\}\), covariance coefficients \(\left\{x_{i}y_{i}\right\}=\)\(\left\{0,0,0,0,1,2,0,2,4\right\}\), and second moment coefficients\(\left\{x_{i}^{2}+y_{i}^{2}\right\}=\)\(\left\{0,1,4,1,2,5,4,5,8\right\}\). Solutions are reported using the convention:
\(\text{\ \ \ LP}_{2D}\left[{u\ }_{x}(t),{u\ }_{y}(t)\right]\ =\left\{minVAR,\left\{p\left[1\right],p\left[2\right],p\left[3\right],p\left[4\right],p\left[5\right],p\left[6\right],p\left[7\right],\left[8\right],p\left[9\right]\right\}\right\}\)(2a)
where minVAR is the sought for minimum variance and p[1] thru p[9] are the minVAR grid occupation numbers normalized to unity.
Figure 4b illustrates a specific solution. The lower right corner cells of the plaquette, (cells 2,3,5 and 6) contain minVAR’s Eulerian representation of the selected point, coordinates of which are indicated by the red marker, along a hypothetical wind trajectory (red curve). Grid occupation numbers are illustrated using the grayscale of Fig. 1 with numerical values given in Eq. 2b. Cell 3, with p[3] = 0.014, appears empty in grayscale as its value is below the shading threshold. The trajectory is representative of one that might be input to the advection routine from a separate computational fluid dynamics code or meteorological model. Using coordinates of the selected point as an example,\({x=u\ }_{x}(t)\ =\ 1.07292,\ {\ y=u\ }_{y}(t)\ =\ 0.808117,\)the solution from Table 2 is:
\begin{equation} \text{LP}_{2D}\left[x,\ y\right]=\text{LP}_{2D}\left[1.07292,\ 0.808117\right]\ =\nonumber \\ \end{equation}
\(\ \ \ \ \ \ \ \ \ \{0.222667,\ \{0,\ 0.177891,\ 0.0139921,\ \ 0,\ 0.749189,\ 0.0589279,\ 0,\ 0,\ 0\}\}\). (2b)
The \(p[i]\)’s listed in 2b, satisfy the four equality and nine inequality constraints of Table 2, as do many other solutions. What makes this solution unique is that it also minimizes the cost function to give the minimum possible variance representation of the test point’s location on the grid. The centroid, or center of mass of the spread, matches the arguments of the LP, its minVar variance (0.222663) is consistent with the \(p[i]\) values listed in 2b. While a single point by itself has no intrinsic variance, its Eulerian representation does. Section 4 generalizes this idea from single points to clouds of points having physical variance added to but separable from minVAR.