Where t1 and t2 are the
averages of the time series of climate variables in the first and second
time slice respectively, and St1 is the standard
deviation of the interannual variability from the first time slice
(t1) for climate variable k .
Changes in probability of local climate extremes
Extreme value theory provides a statistical framework for making
informed assessments regarding the likelihood of exceptionally rare or
extreme events. It aims to predict the probability of such events, by
measuring the deviations from the central tendency in the frequency
distribution of the events. Building upon the foundations of extreme
value theory, specifically the Generalized Extreme Value (GEV)
distribution, we can analyze a sample of extreme events and derive the
parameters that best characterize the underlying probability
distribution of these extremes. The GEV has three parameters: shape,
scale, and location (Gaines & Denny, 1993; Katz et al., 2005). The
shape parameter delineates three possible distributions: i) light-tailed
(Gumbel), heavy-tailed (Frerchet), and iii) bounded (Weubull) (see Katz
et al., 2005). The location parameter specifies where the distribution
is “centred”, while the scale parameter quantifies its “spread”.
The climetrics R package leverages the statistical method
presented by Katz et al., (2005) to assess extremes. In this approach,
the probability of extreme events is defined based on climate variables
and is calculated as percentiles (e.g., the 95th and
5th percentiles of the distributions of the
temperature and precipitation, respectively), for each grid cell within
the baseline period (t1). Subsequently, the percentiles
of the future distributions in the second time period
(t2), corresponding to the extreme values within the
baseline, are computed. These percentiles representing the probability
of exceeding historical extreme values.
To quantify the probability of extreme events for any two variables at
each grid cell, the algorithm combines the two probabilities by summing
them and then subtracting the product of the two probabilities to
prevent double-counting. The probability of the second time period
(t2 ) is then subtracted from the probability of
the first time period (t1 ). Positive values
indicate an increased probability of extremes in the second time period
while negative values suggest a decrease. These calculations provide
insights into specific aspects of extreme climate events (e.g., hot and
dry spells). The same methodology can be applied to incorporate other
factors like wind speed.
The formula for calculating the probability of exceeding a threshold (x)
in given by:
\begin{equation}
\mathbf{\text{P\ }}\left(\mathbf{A\ \cap B}\right)\mathbf{=P(A)\times P(B)}\nonumber \\
\end{equation}
P (A ∩ B) is the probability of the joint occurrence of both events
(A&B), which calculates by multiplying the individual probabilities of
each event: P(A) × P(B).
Changes in areas of analogous climates:
The algorithm measures different aspects of risk arising from climate
change by quantifying changes in the spatial distribution of future
climate conditions relative to those in the recent past. To this end, it
begins by using as input a map of climate zones for each time period.
Such climate zones can be provided by users. Else, the climetricsR package generates them using the updated version of Köppen-Geiger
climate classification (Peel et al., 2007) for both the baseline
(t1) and the second time period
(t2) using temperature (minimum, mean, and
maximum) and precipitation. The Köppen-Geiger climate classification
relies on annual temperature and precipitation, which are subjected to a
sufficiently large time or ensemble averaging.
The climetrics R package employs the method developed by
Ohlemüller et al., (2006) to quantify the percentage of changes in the
areas of similar classes between the baseline
(t1) and the second time period
(t2). In this context, positive values indicate
expansions or gain in these areas, whereas negative values indicate
signify contraction or losses. This analysis provides valuable insights
into the shifting patterns of analogous climate regions over time.
\begin{equation}
\Delta C_{\text{ij}}\ =\ \sqrt{\left(C_{t2j}-C_{t1i}\right)^{2}}\nonumber \\
\end{equation}
Ct1i = Climate Zones int1Ct2j = Climate Zones int2ΔCij = Changes in areas of Analogues climates
Novel climates:
Novel climates are characterized by environmental conditions that lack
analogues in the recorded past (Saxon et al., 2005; Ackerly et al.,
2010). Within the climetrics R package, the method proposed by
Williams et al., (2007) is employed to quantify dissimilarities between
the baseline (t1) and either future orpast time
slices (t2) using the Standardized Euclidean
Distance (SED) as described in the “Standardized Local Anomalies”
section.
The algorithm measures the SED for each grid cell between the first
(t1) and second time periods
(t2). To identify a novel climate at each cell,
it assesses the climate realization in the second time slice
(t2) for the grid cell against climate
realizations from the baseline (t1) across all
grid cells, ultimately retaining the minimum SED
(SEDmin).
SEDmin represents the upper limit of local climate
change indices. Since the pool of potential climatic analogues is
global, a high value of SEDmin indicates that the
climate conditions second time period (t2) have
no close analogues anywhere in the baseline (t1).
In essence, The larger the SEDmin score, the greater
dissimilarity of the future climate relative to the global pool of
potential climate analogues. This metric effectively highlights the
presence of novel and unprecedented climate conditions.
Change in distance to analogous climates:
The climetrics R package implemented the method developed by
Ohlemüller et al., (2006) to quantify similarities between climate
zones.
For a grid cell i in the baseline (t1),
the algorithm calculates the geographic distance to all other cellsj that belong to the same climate classification as grid celli. This computation employs the great-circle distance (Zar,
1989), also known as orthodromic distance, which measures the shortest
distance between two points on the surface of a sphere, measured along
the surface of the sphere.
Then for each cell i, the algorithm computes the median of these
great-circle distances below the 10th percentile of
the distribution of all such distances for both the baseline
(t1) and the second time periods
(t2). Subsequently, it maps the changes in these
distances over time.
To illustrate the changes in distances between the baseline
(t1) and the second time period
(t2), with regard to a specific dimension of
climate change, the algorithm calculates the difference in distances
between these two time points (Δkm = kmt2 –
kmt1). A negative value indicates a temporal
decrease in distance over time, signifying a closer similarity between
the climate zones. Conversely, a positive value indicates an increase in
distance, suggesting greater dissimilarity between the climate zones
over time.
Climate change velocity
Climate change velocity is a
measure of the local rate of movement or displacement of climatic
conditions across Earth’s surface (Loarie et al., 2009; Sandel et al.,
2011). The climetrics R package employs three distinct algorithms
to assess climate change velocity. The first algorithm (“dVe”)
represents a distanced-based velocity measurement developed by Sandel et
al., (2011). It calculates the changing velocity in terms of
geographical distance per unit of time (km/year). The algorithm,
firstly, calculates the temporal gradient by measuring the local
difference between the baseline (t1) and the
second time period (t2). Secondly, it calculates
spatial gradients by taking the slope of the specific climate parameter
for the baseline (t1) using a 3x3 grid-cell
neighbourhood.
\begin{equation}
dVe\ (km\ /year)=\frac{\text{Temporal}}{\text{Spatial}}=\frac{\left(t1-t2\ \right)\text{yea}r^{-1}}{\left(Slope\ 3*3\right)km^{-1}}\nonumber \\
\end{equation}
t1 = Climate variable for the baseline
t2 = Climate variable for the second time period
(future/past)
The second algorithm for quantifying velocity (“ve”) inclimetrics , is implemented based on the method and code developed
by Hamman et al., (2015). This algorithm accounts for the fact that no
two grid cells poses identical climate values. Instead, it relies on a
user-defined threshold to identify a climate match for the climate
surface of the baseline in the climate surface representing the second
period (t2 ). Then, it calculates the geographic
distance of all matching t2 climate cells to the
baseline cell and finds the shortest geographic distance. The distance
is then divided by the number of years between t1and t2 , providing the measure of velocity (see
Hamann et al., 2015).
The third algorithm of climate change velocity (“gVe”) adopts a
gradient-based methodology inspired by the work of Burrows et al.
(2011). This approach focuses on quantifying velocity by dividing the
long-term trend of climate variables by the spatial gradient along the
same direction. For an angle, θ, with 0° as North and 180° as South, the
velocity of climate change along those angles, Vθ, is given by:
\begin{equation}
V\theta=\frac{\text{Temporal\ trend}}{S_{\text{NS}}\ cos\theta\ \ +S_{\text{EW}}\text{\ \ sinθ}}\nonumber \\
\end{equation}
where SNS is the North-South spatial gradient andSEW is the East-West spatial gradient. When θ is
perpendicular to the angle of the velocity of climate change, the
velocity of climate change in that direction is infinite, since the
denominator in equation 8 becomes zero.