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

1:3.4.3

where the superscript denotes the inverse of the matrix and the superscript T denotes the transpose of the matrix. and are defined as

1:3.4.4

1:3.4.5

is the correlation coefficient between variables and on the same day where and may be set to 1 (maximum temperature), 2 (minimum temperature) or 3 (solar radiation) and is the correlation coefficient between variable and with variable lagged one day with respect to variable . 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 and matrices become

1:3.4.6

1:3.4.7

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

1:3.4.8

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.