# 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.


---

# Agent Instructions: Querying This Documentation

If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://swatplus.gitbook.io/io-docs/theoretical-documentation/section-1-climate/chapter-1-3-weather-generator/1-3.4-solar-radiation-and-temperature/1-3.4.1-daily-residuals.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.