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.