vignettes/Equations.Rmd
Equations.RmdHere I present a description of main equations used in the functions
available in the package climatrends.
Growing degree-days (gdd) is an heuristic tool in phenology that
measures heat accumulation and is used to predict plant and animal
development rates1. Growing
degree-days are calculated by taking the integral of warmth above a base
temperature (\(T_{0}\)). The function
GDD() applies by default the following equation.
Equation [1]
\[GDD = \frac{T_{max} + T_{min}}{2} - T_{0}\]
where \(T_{max}\) is the maximum temperature in the given day, \(T_{min}\) is the minimum temperature in the given day and \(T_{0}\) is the minimum temperature for growth (as per the physiology of the focal organism or ecosystem averages).
Additionally, the function GDD() offers three modified
equations designed for cold environments and for tropical environments.
For cold environments, where \(T_{min}\) may be lower than \(T_{0}\), there are two modified equations
that adjust either \(T_{mean}\)
(variant a) or \(T_{min}\) (variant b).
The variant a changes \(T_{mean}\) to
\(T_{0}\) if \(T_{mean} < T_{0}\) and is expressed as
follow.
Equation [2]
\[ GDD = max \left(\frac{T_{max} + T_{min}}{2} - T_{0}, \; 0 \right)\]
The variant b, is calculated using Equation 1, but adjusts \(T_{min}\) or \(T_{max}\) to \(T_{0}\) if \(T < T_{0}\), the equation is adjusted as follows.
Equation [3]
\[ T < T_{0} \; \rightarrow \; T = T_{0} \]
where \(T\) may refer to \(T_{min}\) and/or \(T_{max}\) when the condition of being below \(T_{0}\) applies.
For tropical areas, where the temperature may surpass a maximum threshold (\(T_{0_{max}}\)), resulting in limited development, the minimum temperature is adjusted using Equation 3 and the maximum temperature is adjusted to a maximum base temperature as follow.
Equation [4]
\[ T_{max} > T_{0_{max}} \; \rightarrow \; T_{max} = T_{0_{max}} \]
where \(T_{0_{max}}\) is the maximum
base temperature for growth, defined in GDD() using the
argument tbase_max.
These modified equations are defined as ‘a’, ‘b’ and ‘c’,
respectively, and can be selected using the argument
equation.
By default, the function returns the degree-days that is accumulated
over the time series using Equation 1. Additionally, the function may
return the daily values of degree-days or the number of days that a
given organism required to reach a certain number of accumulated
degree-days. These values are defined by ‘acc’, ‘daily’ or ‘ndays’ and
can be adjusted using the argument return.as. The required
accumulated gdd is defined with argument degree.days. For
example, the Korean pine (Pinus koraiensis) requires 105 \(^\circ C\) accumulated gdd to onset the
photosynthesis2. In that case,
GDD() will calculate the growing degree-days (\(gdd\)) and sum up the values until it
reaches 105 \(^\circ C\) and return the
number of days required in the given season (\(GDD_{r}\)), as follows.
Equation [5]
\[\parallel GDD_{r} \parallel \: = \; ggd_1 \;+ \; ... \; + \; gdd_n\]
where \(GDD_{r}\) is the length of the vector with accumulated degree-days from day 1 to \(n\).
Late-spring frost is a freezing event occurring after a substantial accumulation of warmth. Frost damage is a known issue in temperate and boreal regions, it is associated with the formation of extracellular ice crystals that cause damage in the membranes3. Freezing occurring after an advanced phenological stage during spring may harm some plant species, resulting in lost of productivity in crop systems4 and important ecological impacts5.
The function late_frost() supports the computation of
late-spring frost events. The function counts for the number of freezing
days with minimum temperature below a certain threshold (argument
tfrost). And returns the number of days spanned by frost
events (temperature below tfrost), latency (event with no
freezing temperature but also no accumulation of growing degree-days)
and warming (when growing degree-days are accumulated enabling the
development of the target organism). Additionally the function returns
the first day of the events. The function calculates the growing
degree-days applying the variant b (Eq. 3), which can be adjusted using
the argument equation passed to GDD() as
explained in the later section. The main inputs are a vector with
maximum and minimum temperatures to compute the degree-days, a vector of
dates (argument date), and, if needed, the
tbase and tfrost, set by default to 4 and -2
\(^\circ C\).
Two functions in climatrends are mainly designed to
capture the effects of climate on the development and stress of crop
species, crop_sensitive() computes indices that aim to
capture the changes in temperature extremes during key phenological
stages (e.g. anthesis), and ETo() computes the reference
evapotranspiration.
The crop ecology indices available in climatrends
are described in Table 3. These indices were previously used in crop
models to project the impacts of climate change on crop yield4,6. Each index has a default
temperature threshold(s) which can be adjusted by using the arguments
*.threshold. Where the * means the index. For
example, to change the defaults for hts_max (high temperature stress), a
vector with the temperature thresholds is passed through the argument
hts_max.thresholds.
Table 3: Crop sensitive indices computed by climatrends.
| Index | Definition | Default thresholds |
|---|---|---|
| hts_mean | High temperature stress using tmean | 32, 35, 38 °C |
| hts_max | High temperature stress using tmax | 36, 39, 42 °C |
| hse | Heat stress event | 31 °C |
| hse_ms | Heat stress event for at least two consecutive days | 31 °C |
| cdi_mean | Crop duration index | 22, 23, 24 °C |
| cdi_max | Crop duration index max temperature | 27, 28, 29 °C |
| lethal | Lethal temperatures | 43, 46, 49 °C |
The reference evapotranspiration measures the influence of the
climate on a given plant’s water need7. The function ETo()
applies the Blaney-Criddle method, a general theoretical method used
when only air-temperature is available locally. It should be noted that
this method is not very accurate and aims to provide the order of
magnitude of evapotranspitation. The reference evapotranspiration is
calculated using the following equation.
Equation [6]
\[ETo = p \times \left(0.46 \times \frac{T_{max} + T_{min}}{2} + 8 \right) \times K_c\]
Where \(p\) is the mean daily percentage of annual daytime hours, \(T_{max}\) is the maximum temperature, \(T_{min}\) is the minimum temperature, and \(K_c\) is the factor for organism water need.
The percentage of daytime hours (\(p\)) is calculated internally by the
‘data.frame’ and ‘sf’ methods in ETo() using the given
latitude (taken from the inputted object) and date (taken
from the inputted day.one). It matches the latitude and
date with a table of daylight percentage derived from Brouwer and
Heibloem7. The table can be
verified using climatrends:::daylight.