Introduction
Weather statistics and society
People have learnt to cope with
climate variations and severe weather over historical times and have adapted
to various weather-related risks. In this respect, climate can be regarded as
the statistical description of various weather variables
, giving a picture of “typical” types of
weather and what to expect. This statistical description includes the mean,
variance, autocorrelation, periodicity, and duration of
various climatological events. Weather-related risks are a product of
probability and consequence, where the probability is provided by the
statistical distribution or a probability density function (pdf). The
statistical character of weather is influenced by physical processes, and
variations and changes to the climate can be linked to a number of physical
conditions. Some of the most severe types of past weather-related events
affecting society have included harvest failures due to cold summers or
prolonged droughts
. For the
case of droughts, one important statistic is their duration, even though high
temperature and winds are contributing factors in terms of water stress.
Likewise, the duration of events matters for livelihoods when there are
periods with temperature below, above, or within a range of thresholds. For
example, local statistical temperature characteristics control the prospects
for various aspects of society, such as wheat crops in India, typical skiing
conditions in Norway, or heatwave risks in continental Europe.
It is often tricky to estimate durations defined by a variable crossing
threshold values, especially if it is based on models which are subject to
biases and systematic errors
. It is also impossible
to provide a detailed forecast into the far future, but statistical
properties, such as the parameters describing the shape of a pdf, are more
predictable than single events. Some statistical parameters tend to respond
more systematically to changes in physical conditions, while others are
insensitive. One trivial illustration is that the mean temperature exhibits a
clear dependency on conditions such as the seasonal cycle, latitude, and
altitude, whereas its autocorrelation is not very sensitive to such factors
. The mean seasonal temperature lends itself to
climate change projections; provided that the daily temperature anomalies
follow a normal distribution, it is also expected to affect the statistics of
hot spell duration (here we use “hot spell” and “heatwave” as synonyms).
One strategy for estimating durations of episodes, therefore, is to make use
of statistical models to estimate statistical characteristics.
used an empirical distribution function to
analyse the dry spell lengths over western Africa and found a relationship
that may be used for predictions of the average frequency of dry and wet
spells based on the mean annual rainfall. analysed
the duration of dry spells over the Iberian Peninsula, assuming the spell
duration statistics could be approximated by a Weibull distribution, and
found decreasing trends in the length of wet intervals. A similar strategy
was used in a study to estimate the number of rain-on-snow events over
Svalbard , although the statistic was a count of
occurrences rather than the duration of intervals. The statistics of counts
and duration (e.g. a streak of dry days) follow different types of
distributions, where the former is expected to behave more like a Poisson
process (Poisson distribution) and the latter is expected to follow the
geometric distribution .
pioneered the use of statistical theory for
heatwaves and proposed a statistical framework to model the frequency,
duration, and intensity of heatwaves. Making use of the expected
characteristics of stochastic processes, they used a Poisson distribution to
describe the frequency (number) of events, the geometric distribution to
estimate the number of consecutive days (duration), and a generalised Pareto
distribution to quantify their intensity. They applied the statistical
framework to analyse trends in heatwave statistics in three temperature
records from Phoenix (Arizona, USA), Fort Collins (Colorado, USA), and Paris
(France). analysed the variability of
heatwaves over Florida, both in space and time, and reported both that there
is considerable spatial variability in heatwave characteristics and that
heatwaves have become increasingly frequent and intense throughout Florida.
They made use of extreme-value analysis to quantify the heatwave intensity,
the Poisson distribution to describe the number of heatwaves, and the
geometric distribution to estimate their duration.
used the statistical framework proposed by
and bias-corrected temperatures from a
30-member ensemble of global climate models for the projection of heatwave
statistics in China. Global climate models, however, are not designed to
represent local climate characteristics accurately, and it is therefore
common to downscale the model output in order to get a description that is
representative of the regional and local features
.
However, there have not been many studies on changes in the probability of
future heatwaves based on the downscaling of large multi-model ensembles in
general, and particularly not in India, where good-quality
open-access data are scarce.
Furthermore, we are not aware of any previous attempts to downscale the
duration statistics by means of empirical-statistical downscaling (ESD).
While the statistics for frequency or duration is more straightforward, as
their respective distributions rely on single-parameter distributions related
to the mean number or duration, extreme-value distributions are trickier
since they involve several parameters with a less clear connection to
large-scale conditions.
Here we apply the methodology for downscaling duration statistics to examine
critical temperatures for growing wheat in India, which vary between the
different phenological stages. The mean duration of hot spells with
temperature above a critical threshold has an important effect on
agriculture, especially if the statistics of duration follow a geometric
distribution for which the mean is directly connected to the parameter that
sets the shape of the pdf. The probability of lasting hot spells with a
duration exceeding a given threshold in the current climate can be inferred
from statistical properties found in the observations. An important question
is how global warming will lead to more long-lasting hot spells with a
detrimental effect on the wheat crops. A novel aspect of the strategy
presented in this paper is the downscaling of probabilities directly, rather
than downscaling a physical variable and then using it to estimate the
parameters for the pdf.
Consequences of temperature on agriculture
Wheat is one of the major crops in India, and the largest wheat growing
regions are in the Indo-Gangetic Plain (IGP) – particularly in the
north-western states Uttar Pradesh, Punjab, Haryana, and Rajasthan
– and in the central state Madhya Pradesh
(Directorate of Economics and Statistics, 2017) in addition to Bihar in the
north-east. In these states, wheat is grown over the winter season, sown
between mid-November (north-west) and mid-December (central), and harvested
in late March to mid-April. While this period is typical for the variety
known as winter wheat, the main variety that is grown during this period is
spring wheat. Wheat goes through three distinct growing and maturation
phases, from the vegetative phase from germination and seedling
development (1); through a reproductive phase with branching, elongation, and
heading (2); to a flowering, grain setting and filling, and maturation
phase (3).
Wheat is differentially temperature sensitive across its various development
stages and through different mechanisms, and effects on growth or yield are
gradual and variety specific. summarised from
many studies non-lethal temperatures for wheat in the range of 18 to
47 ∘C, but this covers a broad range of world cultivars across
several growing stages. Wheat and any other plants grow and develop within
thermal limits called cardinal temperatures. These limits characterise a
Gaussian curve with the extreme points and a narrow range of temperatures
where the morphophysiological (relating to, or concerned with, biological
interrelationships between form and function) events are maximal, and hence
termed the optimum temperature range “top”. After this range, the minimum
basal temperatures “Tmin” and maximum “Tmax” are found, after
which growth and morphophysiological activity are paralysed by deficiency and
excess of energy, respectively.
It is generally accepted that optimal temperatures for wheat are in the range
17–23 ∘C over the entire growing season, with a Tmin of
0 ∘C and Tmax of 37 ∘C, beyond which growth stops
.
In India, there are many varieties of wheat grown across the states,
differing in their sensitivity to temperatures and other parameters, and
there are also breeding programmes for heat tolerance
. Across its various
growth stages, the national recommendations for wheat growth (Directorate of
wheat development, 2015) state a daily average between 20 and 25 ∘C
as optimal temperature. Critical minimum temperatures are around
3.5–5.5 ∘C, and the maximum around 35 ∘C. Temperatures
above the optimum (25 ∘C) lead to decreased grain yields, and
temperatures above 30 ∘C at maturity (around mid-March) lead to
forced maturity and yield loss. Warming is already affecting wheat yields
across the world, and for each degree increase in global mean temperature,
there is a reduction in global wheat grain production of about 6 %
.
For some wheat varieties, the first and second growing phases benefit from
cold exposure known as vernalisation, which improves yield by shortening the
duration to flowering, and thus leave more time to grain formation and
filling before high temperatures set in .
Vernalisation is not critical to yield per se, and the duration and
temperature requirements (chill-degree days) differ for different winter
wheat types .
For all Indian wheat varieties, the main challenge is the high temperatures
in the final growing phase, late in the season from February to April
. The most temperature-susceptible reproductive
stages are the period priors to flowering and during flowering and
fertilisation . Extremely high temperatures
drastically affect wheat during the reproductive phase, particularly during
pollination, but there is no evidence of the temperature effect on the leaf
area and the production of vegetative biomass. The harmful effect on the
reproduction and grain filling under high temperatures conditions intensifies
with dry events during the spring or summer , which is
the period where the phases of reproduction and grain filling occur
preferentially .
There does not seem to be a consensus between studies on the exact critical
temperature limits, and the effects of increasing temperature on yield appear
to be gradual. Signs of thermal shock proteins have been found in several
wheat varieties in the vegetative and reproductive phase, suggesting that they were able
to extend their tolerance limits to high temperatures through genetic
breeding . Three days of 30 ∘C showed a
reduction of grain set by almost 70 % , and
temperature regimes of 36 and 31 ∘C (day and night, respectively) for 2 days resulted in
55 %–85 % grain sterility .
suggested 30 ∘C as an upper limit
(daily maximum temperature Tmax) around the flowering period as short
periods (4 days) above this limit impact yield.
similarly found that temperatures above
30 ∘C slow grain filling, damaging the plant. Other studies
e.g. have suggested higher critical
temperatures: an exposure to daily Tmax above 36 ∘C and
Tmin 31 ∘C during the period immediately before flowering
(January) may result in sterility and reduced yield. Simulated yield studies
show possible reductions of about 10 %–15 % by the end of the
century if 40 ∘C is exceeded for only 1 day
.
Several studies
have found that wheat is becoming more sensitive to increasing minimum
temperatures and that a continuous exposure to a daily minimum temperature
(Tmin) exceeding 12 ∘C for 6 days and Tmax exceeding
34∘C for 7 days past flowering (February) constrains yields
.
In summary, the period February–April is most critical, with all
temperatures above optimal decreasing wheat yield. Studies on the more
sensitive varieties suggest a daily maximum in February of 30 ∘C as
a limit above which yield is reduced. However, to simplify the analysis, the
threshold for maximum temperature before limiting wheat crop yields was set
to 35 ∘C for 5 consecutive days based on published research
. Based on this information, our
objective was to estimate the likelihood for long-lasting future heatwaves
with detrimental consequences for Indian wheat production. We explored a new
methodology within downscaling, making use of large multi-model ensembles to
get an ad hoc representation of uncertainties associated with
interannual-to-decadal variability and model differences.
Results
An evaluation of the downscaled results for the February–April mean maximum
temperature Tmax‾ suggested high skill for the leading PCA in
terms of the cross-validation, with correlations in the range 0.79–0.87
(Supplement). When the downscaled results for the PCAs were used to recover
the format of the original temperature records, an evaluation of the RCP4.5
ensemble indicated good skill for Tmax‾ over the wheat
growing IGP region, but low skills in the south (Supplement). The skill of
downscaling was low for the stations in southern India, as both the trends in
the downscaled results and the range of interannual variability were lower
than seen in the observations over the common overlapping period
(1970–2015). The differences in skill can be explained from the leading PCA,
which had strongest weights for the locations with high skill and weakest
weights where the skill was low.
Estimated probability (expressed in %) for an episode with
temperatures exceeding 35 ∘C over more than 5 consecutive days in
February–April. The observed frequency was based on the individual
observational record and length of time series, and it is not exactly
equivalent to the estimated probability for 2010. The location names in bold
font mark stations within the IGP region.
Observed
Predicted
RCP4.5
RCP2.6
RCP8.5
frequency
2010
2050
2100
2050
2100
2050
2100
PBO ANANT
60
83
83
84
83
83
84
85
MACHILIPA
63
70
72
73
71
71
73
76
NELLORE
79
77
78
78
78
78
78
79
GAUHATI
9
48
51
53
50
49
53
59
DIBRUGARH
2
20
26
28
24
22
27
36
PATNA
84
67
71
74
70
69
74
81
AHMADABAD
63
78
81
83
80
80
83
88
VERAVAL
12
61
65
68
64
64
67
74
BHUJ-RUDR
67
77
79
80
78
78
80
84
SURAT
100
77
79
80
78
78
80
83
HISSAR
60
62
68
72
66
66
71
81
GADAG
74
77
78
78
78
78
78
80
KOZHIKODE
16
70
72
72
71
71
72
75
THIRUVANA
2
70
70
71
70
70
71
72
JAGDALPUR
77
76
77
78
77
77
78
81
PENDRA RO
70
68
73
77
72
72
77
86
GWALIOR
49
68
73
76
72
72
75
84
INDORE
65
75
78
79
77
76
79
84
JABALPUR
53
71
74
77
73
73
76
82
BHOPAL
53
72
76
78
75
75
78
84
BOMBAY
21
67
70
71
69
68
71
76
NAGPUR SO
65
79
81
83
81
81
83
87
POONA
88
78
79
80
79
79
80
83
SHOLAPUR
60
84
85
86
84
84
85
88
BHUBANE
95
77
80
82
79
79
81
86
BIKANER
74
69
75
78
73
72
76
85
JAIPUR
58
66
72
75
70
70
74
84
JODHPUR
58
73
77
80
76
76
79
87
CUDDALO
51
66
67
68
67
67
68
70
MADRAS
91
72
72
73
72
72
73
74
TIRUCHCHI
79
78
78
78
78
78
78
78
AGARTALA
42
62
65
67
64
64
67
73
NEW DELHI
67
57
65
69
62
62
68
80
LUCKNOW
65
68
73
76
72
71
75
84
CALCUTTA
88
70
73
75
72
72
74
81
An evaluation of the OLR used to estimate the mean number of heatwaves for
the different sites suggested a statistically significant dependency on the
seasonal mean daily maximum temperature at the 1 % level, with an R2
of 0.2 (Supplement). There was a great deal of scatter about the fitted line,
which suggests that there may be other important factors or that the data
have variable quality.
In order to trust the results and analysis presented for the duration of the
heatwaves, we also needed to test the underlying assumptions about the
statistical nature of the data (H1 and H2). The
first assumption was that the temperature is approximately normally
distributed and that there is a systematic dependency between the mean
duration of hot episodes and the mean temperature (H1). We
tested this dependency by looking at the best available data (ECA&D data
from European stations), assuming that the way LH‾
depends on Tmax‾ is a universal property for daily
temperatures on Earth that is close to the dependency found for data with a
normal distribution. Figure shows one set of test results for the
relationship between the mean seasonal temperature and mean duration of cold
spells in winter and warm spells (with Tmax>20 ∘C) in summer
over Europe. The observational data (red symbols) are shown together with
results from an analysis repeated with synthetic normally distributed data
(grey). The results of this test confirmed the systematic dependency of the
mean spell duration on the seasonal mean temperature. The results from a
similar test on data from India were consistent with these results, albeit
with a smaller statistical sample and a substantial scatter
(Fig. ). The fitted curve could account for 10 % of the
variance according to an analysis of variance, and the results were
statistical significant at the 1 % level.
Quantile–quantile plot between the cold (a) and
warm (b) spell length and the fitted geometric distribution for the
selected ECA&D stations.
![]()
The second assumption was that the spell duration statistics had a geometric
distribution (H2). Figure shows a comparison between
the spell duration statistics based on the European ECA&D data and the
geometric distribution as a quantile–quantile plot. The results suggested
that the assumption of a geometric distribution was reasonable for
short-to-moderate duration but not for durations longer than a single season
(90 days). For the case of summer, the duration statistics exhibited a high
bias for durations greater than 30 days. The tests of the underlying
assumptions suggested that they were reasonable for both warm and cold
seasons at least in Europe. A comparison between histograms of heatwave
durations in India and fitted geometric distributions based on
LH‾ suggested a reasonable match (Supplement). The
evaluation of hypotheses H1 and H2 provided support
for making projections of the probabilities based on the ESD of
Tmax‾ from large multi-model ensembles. A summary of the
results of the probability projections are found in Tables and
. The estimated probability of at least one heatwave in a
February–April season predicted for the present day (2010) was in a
reasonable agreement with the observed frequency of events for most stations,
but there were some exceptions where the modelled estimates were
substantially higher than the observed frequency (first two columns in
Table ). However, none of these exceptions affected the stations
in the IGP region (bold font in Table .) The sites with a mismatch
between the observed frequency and estimated probability will be discussed
further later on.
Estimated probability (expressed in %) for duration greater than 5
consecutive days with temperatures exceeding 35 ∘C during
February–April. The observed frequency was based on the number of
February–April heatwaves lasting more than 5 days divided by the total
number of heatwaves in February–April. The location names in bold font mark
stations within the IGP region.
Observed
Predicted
RCP4.5
RCP2.6
RCP8.5
frequency
2010
2050
2100
2050
2100
2050
2100
PBO ANANT
32
20
20
21
20
20
21
21
MACHILIPA
21
15
16
16
16
16
16
17
NELLORE
34
18
18
18
18
18
18
18
GAUHATI
5
11
12
12
11
11
12
13
DIBRUGARH
14
8
8
9
8
8
9
9
PATNA
42
15
16
17
16
15
17
19
AHMADABAD
30
18
19
20
19
19
20
23
VERAVAL
3
13
14
15
14
14
15
17
BHUJ-RUDR
30
18
19
19
18
18
19
21
SURAT
36
18
18
19
18
18
19
20
HISSAR
32
14
15
16
15
14
16
19
GADAG
45
18
18
18
18
18
18
19
KOZHIKODE
13
16
16
16
16
16
16
17
THIRUVANA
1
15
16
16
16
16
16
16
JAGDALPUR
45
17
18
18
18
18
18
19
PENDRA RO
32
15
17
18
16
16
18
21
GWALIOR
23
15
17
17
16
16
17
21
INDORE
27
17
18
18
18
17
19
21
JABALPUR
27
16
17
17
16
16
17
20
BHOPAL
29
16
17
18
17
17
18
21
BOMBAY
4
15
15
16
15
15
16
17
NAGPUR SO
37
18
19
20
19
19
20
22
POONA
40
18
19
19
18
18
19
20
SHOLAPUR
35
20
21
21
21
21
21
23
BHUBANE
38
18
19
20
19
19
19
22
BIKANER
38
15
17
18
16
16
18
21
JAIPUR
38
14
16
17
16
15
17
21
JODHPUR
38
16
18
19
17
17
19
22
CUDDALO
22
14
15
15
15
15
15
16
MADRAS
30
16
16
16
16
16
16
17
TIRUCHCHI
40
18
18
18
18
18
18
18
AGARTALA
22
14
14
15
14
14
15
16
NEW DELHI
33
12
14
15
14
14
15
19
LUCKNOW
41
15
16
17
16
16
17
21
CALCUTTA
34
15
16
17
16
16
17
19
The projected probability of a hot event (Tmax>35 ∘C) turning
into a heatwave (LH>5 days) for 2010 was crudely compared with
the observed number of heatwaves divided by the total number of events with
Tmax>35 ∘C (first two columns in Table ). For most
stations in the IGP region (shown in bold font), the observed frequencies
were higher than the probabilities predicted for the present.
Table also contains many southern states that do not produce any
wheat but were included to enlarge the sample size to get an improved
estimate of the heatwave duration statistics regardless of their effects on
the wheat crops. The results presented in Table also suggest that
the estimated Pr(LH>5) was in the range of
8 %–20 % for 2010 and will increase to 9 %–23 % in 2100
assuming the high-emission scenario RCP8.5.
Projected probability of (a) one or more events with daily
maximum temperature above 35 ∘C lasting longer than 5 days during
the February–April season for Patna and (b) the probability that
the heatwave lasts more than 5 days, given temperature above
35 ∘C. These curves represent one of the stations presented in
Tables and . The fitted trend curves were fourth-order
polynomials for different emission scenarios ,
where green represents RCP2.6, blue RCP4.5, and red RCP8.5.
![]()
Figure shows for one selected location (Patna; row six in the
table) (a) the probability of one or more heatwaves in a season and (b) the
probability that a hot event lasts more than 5 days, based on the ensemble
median of the downscaled projections for three different emission scenarios.
Figure presents the projected probabilities for 2100 assuming
emission scenario RCP4.5 (the fourth columns in Tables and
) for all stations in India. According to the results presented in
Fig. a, continuing global warming will imply an increased
probability of long-lasting heatwaves in Patna, and Fig. b
indicates that the likelihood for future 5-day heatwaves will depend on
the future emissions, where the probability may increase by almost as much as
a third from present-day values for the high-emission scenario: the
probability of a 5-day or longer heatwave is approximately 15 % at the
present time, but it is expected to increase to 19 % in 2100 in a
continued high-emission scenario (RCP 8.5). For the intermediate-emission
scenario RCP4.5, the results suggest an increase from 15 % to 17 %
probability and an increase which is about half of that associated with
RCP8.5. Hence, lower-emission scenarios give smaller increases. The maps
presented in Fig. suggest greater probabilities for heatwaves in
the central parts of India. The variable skill of downscaling at different
locations implies that the results are less accurate for some parts of India,
namely the far eastern and southern parts.
Projected probability of (a) one or more events with daily
maximum temperature above 35 ∘C lasting longer than 5 days during
the February–April season in 2100 assuming the RCP4.5 emission scenario and
(b) the probability that the heat lasts longer than 5 days given
a hot day in 2100 assuming the RCP4.5 emission scenario. The map was
generated by gridding estimates shown in the fourth column in
Table .
![]()
Discussion
A number of studies suggest a more pronounced change in climatic extremes
compared to changes in the mean
.
The shape of the pdf for temperature may change with a shift in the mean
μ, and the relationship between the mean and the shape of the pdf was
tested on the actual temperature data used herein. A scatter plot between
seasonal mean and seasonal standard deviation σ showed that it tends
to decrease with increasing mean values (Supplement). Hence, since the mean
often is not a good predictor for extreme values, we used the mean
temperature to estimate the mean of another pdf; in this case, the seasonal
mean daily maximum temperature was used to estimate the mean number of events
and mean duration of heatwaves: n5d‾=f(Tmax‾) and LH‾=g(Tmax‾).
The analysis of the mean number of heatwaves lasting more than 5 days
n5d‾ and the mean duration of heatwaves had some
caveats, and an assessment of the conformity of n5d to the
normal distribution suggested divergence towards the tail of the
distribution. One plausible reason for the deviation was that the mean
was taken from small samples of Poisson-distributed data, whereas the mean was
expected to converge to the normal distribution with large sample size. The
divergence from the normal distribution may also have been a result of
variable data quality. Nevertheless, using the mean duration and the mean
number of events could justify using OLMs instead of GLMs since aggregated variables are expected to
be closer to being normally distributed than the underlying data.
A more traditional approach is to downscale the temperature day by day, for
instance through the means of regional climate models (RCMs), and then apply
extreme-value theory to the model results. RCMs will not give a direct
answer, as they have biases and suffer from other shortcomings. Hence,
RCM-based studies also come with a set of uncertainties. However, there is a
great benefit in having more than one approach as different strategies for
estimating the results have different strengths and weaknesses independent of
each other.
According to both Tables and , the observed frequency
of heatwaves was substantially lower than the estimated corresponding
probability for seven sites in the far north-eastern parts of India or near
India's western coast (Gauhati, Dibrugarh, Veraval, Kozhikode,
Thiruvananthapuram, Bombay, Agartala), but for the 12 sites in interior parts
of India where wheat is grown (the IGP region) and along India's east coast,
they indicated a good match with a 25 % difference or smaller. All of
these temperature records were deteriorated by missing data; to produce
usable spell duration statistics, it was necessary to fill in short gaps of
missing data by the means of linear interpolation. As the proportion of
missing values was in the range of 7 %–34 %, the observed number of
events in Tables and needs to be interpreted with
caution. The sites with large mismatch were missing more than 15 % of
values in their data record (Gauhati: 15.5 %; Dibrugarh: 34 %;
Veraval: 19.5 %; Kozhikode: 23.5 %; Thiruvananthapuram: 24.6 %;
Bombay: 18.4 %; Agartala: 22.6 %). A more detailed diagnostics of the
data quality and the discrepancy between observed frequencies of heatwaves
longer than 5 days and estimated likelihoods is provided in the Supplement,
which suggests that the poor matches coincided with stations that carried low
weights in the leading PCA. Some discrepancies between the downscaled
probability and the observed frequency must also be expected since the former
was based on a Bayesian-type analysis whereas the latter was based on
observed counts. The bias in the estimated probability of a hot spell lasting
more than 5 days compared to estimated frequency for the observations for the
IGP region suggested that the estimates of probability type 2 may be less
skillful than those of type 1. One reason may be that quality of the Indian
data was low, which may be the reason for the differences in the scatter
plots between LH‾ and Tmax‾ in India
and Europe (Figs. 1b and 2 and Supplement). The hot spells were also not
quite geometrically distributed (Fig. 3), which also could introduce an
additional bias.
The question of the degree of validity of the relationship
LH‾=g(Tmax‾) depends on the data
quality and volume. While there was a weak link in India, there was a clear
link over Europe. Furthermore, tests applied to ideal synthetic data
indicated a connection between the two, and similar noisy scatter at the
upper (lower for cold spells) tail of the ideal synthetic stochastic data (x∼N(μ,σ2)) in Fig. 1 suggested that estimates for more extreme
cases were subject to increased sampling fluctuations. The noisy picture
given by the scatter plots may also suggest that there were other
unaccounted-for factors which influence the mean duration or the mean number
of heatwaves. Another question is its validity in the future, as the
connection may change if the shape of the pdf for Tmax changes under
global warming. The agreement between the link established for the European
data and the ideal data (Fig. 1) suggests a universal trait as long as the
daily temperature is approximately normally distributed, but a bias is likely
to be present if the standard deviation diminishes (Supplement).
We wanted to demonstrate how this downscaling methodology makes the best use
of the sketchy data, as the estimates themselves are based on more robust
statistical parameters such as the mean duration LH‾
and the PCA of the mean temperature Tmax‾. These quantities
may be considered to be fairly resistant to errors as long as there are not
too many of them, and that they are both random and unbiased. Furthermore, the PCA is
resistant to errors in single temperature series as long as they represent a
small number of the stations and are uncorrelated with errors at other sites.
However, the interpolation of gaps with missing data introduced new
uncertainties, and the presence of missing data also made it tricky to get
accurate estimates for the heatwave durations. Missing data and errors
introduced through interpolation represent one possible explanation for the
poor match between the observed frequency of heatwaves in Table
and estimated likelihood for 5-day heatwaves at some of the sites. Moreover,
the sites with the largest mismatch were not in regions where wheat crops are
important, but we included them here to maximise the signal in the PCA and to
enhance the chance of getting a good estimate of the dependency between large
and small scales needed for empirical-statistical downscaling.
The analysis presented here was based on a novel methodology where the
probability associated with heatwave duration was calculated from downscaled
seasonal mean temperature estimates rather than inferring it from downscaled
daily data. There has been some similar work, but none that have involved
downscaling of large multi-model ensembles to make projections for heatwaves
over India. did not include downscaling and used a
Weibull distribution to describe the spell duration statistics rather than
the geometric distribution. We chose the latter since it is based on the
number of successive probabilities (hot days; see the Appendix). The analysis
presented by was more similar to our
projections of heatwave statistics over India, but they used bias-corrected
GCM results for China rather than downscaling over India. We, on the other
hand, combined statistical modelling of heatwave statistics with the
empirical-statistical downscaling of February–April mean daily maximum
temperature involving several multi-model ensembles.
The probabilities presented here were subject to a number of uncertainties:
(a) the unknown nature of future emissions, (b) shortcomings in the global
climate models, (c) limitations of the empirical-statistical downscaling
method, (d) uncertainties associated with the connection between the mean
daily maximum temperature and the duration statistics, and (e) errors in the
observations. By including three different emission scenarios (RCPs 2.6, 4.5,
and 8.5), the analysis provided some indication of the sensitivity of the
probabilities to the nature of the emissions. Both Fig. and
Tables – indicate that future emissions mattered for
the likelihood of longer lasting heatwaves, which have negative effects on
wheat crops. Figures – present an evaluation of the
connection between the mean daily maximum temperature and the heatwave
duration statistics and reveal that it is not “perfect”, particularly for
very long lasting heatwaves (>30 days). This connection nevertheless
provides a reasonable estimate, and the comparison between synthetic normally
distributed random data with similar autocorrelation suggested that this
connection is robust. However, the connection would be sensitive to a change
in the autocorrelation, although the autocorrelation appears to be
insensitive to variations in physical conditions
.
It is impossible to predict the course of natural variability, and even a
single climate model may produce different projections with widely different
outcomes on local and regional scales .
Probabilities account for such variability, and the analysis presented here
made use of the median of the simulated temperature from large multi-model
ensembles and a Bayesian-inspired approach to account for both natural
variability and model differences. Such ensembles cannot be considered to be
unbiased statistical samples as different models
have similar biases since they share many components. The model differences,
however, have been found to be less pronounced than the year-to-year
variations and can for all intents and purposes
be used as an imperfect description of the statistical spread when
better information is lacking. The estimation of future probabilities also makes the
question of statistical significance less relevant, since statistical
significance refers to the probability that a change in a random variable is
due to chance, assuming that the variable has a stochastic nature. In this
case, the estimation of a change in probabilities is on the same level as the
estimation of the probability levels commonly used in statistical
significance tests.