# 1:3.4.1 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.