Mean Temperature
Model Overview
The Mean Temperature model is explained in Chilling Accumulation: its Importance and Estimation. The model was developed for use in temperate climates by people who do not have access to hourly temperature measurements but do have have access to the temperature measurements from a max/min thermometer (also known as a high/low thermometer).
The model maps the mean temperature over the coldest month of the year to the total number of chill units using a relationship based on measured data. The mean temperature is calculated by averaging the mean of the maximum temperatures and mean minimum temperatures.
Max/Min Thermometer Emulation
To emulate a max/min thermometer, BiTWeather.chill calculates the daily maximum $x_{max}(d) = \Theta_{max}(d)$ and minimum $x_{min}(d) = \Theta_{min}(d$) temperatures from the weather data's temperature measurements using
\[ x_{max}(d) = \Theta_{max}(d) = \max_{n \in d} \Theta_n(d) \]
\[ x_{min}(d) = \Theta_{min}(d) = \min_{n \in d} \Theta_n(d) \]
where $\Theta_n(d)$ is the temperature in $\degree C$ of measurement $n$ on day $d$. After that, BiTWeather.chill applies the Mean Temperature model.
Model Implementation
First, BiTWeather.chill calculates the mean maximum temperature $x_{max} = \overline{\Theta_{max}}$ and the mean minimum temperature $x_{min} = \overline{\Theta_{min}}$ using
\[ x_{max} = \frac{1}{{d_2 - d_1 + 1}} \cdot \sum_{d=d_1}^{d_2} x_{max}(d) = \overline{\Theta_{max}} = \frac{1}{{d_2 - d_1 + 1}} \cdot \sum_{d=d_1}^{d_2} \Theta_{max}(d) \]
\[ x_{min} = \frac{1}{{d_2 - d_1 + 1}} \cdot \sum_{d=d_1}^{d_2} x_{min}(d) = \overline{\Theta_{min}} = \frac{1}{{d_2 - d_1 + 1}} \cdot \sum_{d=d_1}^{d_2} \Theta_{min}(d) \]
where $d_1$ is the first day of the calculation period and $d_2$ is the last day of the calculation period. After that, it calculates the mean temperature $x = \overline{\Theta}$ using
\[ x = \frac{x_{max} + x_{min}}{2} = \overline{\Theta} = \frac{ \overline{\Theta_{max}} + \overline{\Theta_{min}} }{2} \]
The data in Figure 1 of Chilling Accumulation: its Importance and Estimation shows the mapping from mean temperature to total chill units. This mapping can be closely approximated by the linear equation
\[ y = m \cdot x + b \]
where
\[ x_0 = \frac{5 \degree C}{9 \degree F} \cdot (62 \degree F - 32 \degree F) = \frac{50}{3} \degree C \space,\space y_0 = 200\space \text{chill units} \]
\[ x_1 = \frac{5 \degree C}{9 \degree F} \cdot (44 \degree F - 32 \degree F) = \frac{20}{3} \degree C \space,\space y_1 = 1200\space \text{chill units} \]
\[ m = \frac{y_1 - y_0}{x_1 - x_0} = -100 \space \frac{\text{chill units}}{\degree C} \space,\space b = - m * x_0 + y_0 = \frac{5600}{3} \space \text{chill units} \]
Software Name Mappings
Because of naming restrictions imposed by the Julia language and Julia coding style, I cannot use the parameter, variable and function names from this summary for the parameter, variable and function names in the program. So, I have mapped names in the summary to names in the program while attempting to keep them consistent.
Parameter Name Mappings
Variable Name Mappings
- $\Theta_n \Rightarrow$ temperature
Equation Name Mappings
- $x_{max}(d) \Rightarrow$ x_max_d
- $x_{min}(d) \Rightarrow$ x_min_d
- $x_{max} \Rightarrow$ x_max
- $x_{min} \Rightarrow$ x_min
- $x \Rightarrow$ x
- $y \Rightarrow$ y
Last Reviewed on 06 February 2021