Daily Residuals

Residuals for maximum temperature, minimum temperature and solar radiation are required for calculation of daily values. The residuals must be serially correlated and cross-correlated with the correlations being constant at all locations. The equation used to calculate residuals is:

$\chi_i(j)=A{\chi_{i-1}}(j)+B{\varepsilon_i}(j)$ 1:3.4.1

where $\chi_i(j)$ is a 3 × 1 matrix for day $i$ whose elements are residuals of maximum temperature ($j=1$), minimum temperature ($j=2$) and solar radiation ($j=3$), $\chi_{i-1}(j)$) is a 3 × 1 matrix of the previous day’s residuals, $\varepsilon_i$ is a 3 × 1 matrix of independent random components, and $A$ and $B$ are 3 × 3 matrices whose elements are defined such that the new sequences have the desired serial-correlation and cross-correlation coefficients. The $A$ and $B$ matrices are given by

$A=M_1*M_0^{-1}$ 1:3.4.2

$B*B^T=M_0-M_1*M_0^{-1}*M_1^T$ 1:3.4.3

where the superscript $-1$ denotes the inverse of the matrix and the superscript T denotes the transpose of the matrix. $M_0$ and $M_1$ are defined as

$M_0=\left[\begin{array}{ccc} 1 & \rho_0(1,2) & \rho_0(1,3) \\ \rho_0(1,2) & 1 & \rho_0(2,3) \\ \rho_0(1,3) & \rho_0(2,3) & 1 \end {array} \right ]$ 1:3.4.4

$M_1=\left[\begin{array}{ccc} \rho_1(1,1) & \rho_1(1,2) & \rho_0(1,3) \\ \rho_1(2,1) & \rho_1(2,2) & \rho_1(2,3) \\ \rho_1(3,1) & \rho_1(3,2) & \rho_1(3,3) \end {array} \right ]$ 1:3.4.5

$\rho_0(j,k)$ is the correlation coefficient between variables $j$ and $k$ on the same day where $j$ and $k$ may be set to 1 (maximum temperature), 2 (minimum temperature) or 3 (solar radiation) and $\rho_1(j,k)$ is the correlation coefficient between variable $j$ and $k$ with variable $k$ lagged one day with respect to variable $j$. Correlation coefficients were determined for 31 locations in the United States using 20 years of temperature and solar radiation data (Richardson, 1982). Using the average values of these coefficients, the $M_0$ and $M_1$ matrices become

$M_0=\left[\begin{array}{ccc} 1.000 & 0.633 & 0.186 \\ 0.633 & 1.000 & -0.193 \\ 0.186 & -0.193 & 1.000 \end {array} \right ]$ 1:3.4.6

$M_1=\left[\begin{array}{ccc} 0.621 & 0.445 & 0.087 \\ 0.563 & 0.674 & -0.100 \\ 0.015 & -0.091 & 0.251 \end {array} \right ]$ 1:3.4.7

Using equations 1:3.4.2 and 1:3.4.3, the A and B matrices become

$A=\left[\begin{array}{ccc} 0.567 & 0.086 & -0.002 \\ 0.253 & 0.504 & -0.050 \\ -0.006 & -0.039 & 0.244 \end {array} \right ]$ 1:3.4.8

$B=\left[\begin{array}{ccc} 0.781 & 0 & 0 \\ 0.328 & 0.637 & 0 \\ 0.238 & -0.341 & 0.873 \end {array} \right ]$ 1:3.4.9

The A and B matrices defined in equations 1:3.4.8 and 1:3.4.9 are used in conjunction with equation 1:3.4.1 to generate daily sequences of residuals of maximum temperature, minimum temperature and solar radiation.