A methodology for estimating and downscaling the probability associated with
the duration of heatwaves is presented and applied as a case study for Indian
wheat crops. These probability estimates make use of empirical-statistical
downscaling and statistical modelling of probability of occurrence and streak
length statistics, and we present projections based on large multi-model
ensembles of global climate models from the Coupled Model Intercomparison
Project Phase 5 and three different emissions scenarios: Representative Concentration Pathways (RCPs) 2.6, 4.5, and
8.5. Our objective was to estimate the probabilities for heatwaves with more
than 5 consecutive days with daily maximum temperature above
35

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

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

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.

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

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.

It is generally accepted that optimal temperatures for wheat are in the range
17–23

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

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

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

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

Several studies

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

The probability of long-lasting heatwaves (with

Our main working hypothesis

One obstacle to such analyses was the poor data availability and quality over
India, which restricted our ability to extract representative numbers for the
hot events and connect these to climate model projections. We made use of
additional information concerning mean temperatures and spell length
statistics to support the analysis, which included using “high-quality”
European data from the European Climate Assessment and Dataset (ECA&D;

Comparison between winter

Given a dependency between the mean temperature and the mean spell duration
(

We present two types of probability estimates here:
(1)

The probability of at least one event in a season (probability type 1) was
estimated based on a statistical model assuming the Poisson distribution
conditioned by the seasonal mean maximum temperature. Rather than using a GLM
calibrated on

To estimate the probability of a 5-day or longer heatwave (probability
type 2), the projections of seasonal mean maximum temperature were used
together with a GLM calibrated on daily maximum temperature data to infer
changes in the mean hot spell duration length (

We used ESD to make future projections
for the February–April mean daily maximum temperature for a set of locations
in India (see the Supplement for map) with multi-model ensembles as
in

We used the mathematical framework described in the previous section to
analyse the probability of events and their duration. To obtain projections
of the probability of one or more heatwaves (

To produce maps of probabilities, the results were gridded using the same
kriging method as in

In summary, this downscaling study brings in several novel aspects, including utilising large multi-model ensembles of GCM simulations, downscaling essential statistical characteristics of heatwave durations, and producing outlooks for the probability of future heatwaves lasting more than 5 days. These results were based on PCA of the local temperatures, which enhances the signal and can make the results more robust for a situation where the data are both scarce and considered to be of questionable quality.

The daily maximum temperature

More details about the data, processing, and analysis are provided in the
Appendix and the Supplement, which provide results from an R Markdown script,
available from figshare

A comparison between interannual and geographical variations in the
mean duration

An evaluation of the downscaled results for the February–April mean maximum
temperature

Estimated probability (expressed in %) for an episode with
temperatures exceeding 35

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

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 (

Quantile–quantile plot between the cold

The second assumption was that the spell duration statistics had a geometric
distribution (

Estimated probability (expressed in %) for duration greater than 5
consecutive days with temperatures exceeding 35

The projected probability of a hot event (

Projected probability of

Figure

Projected probability of

A number of studies suggest a more pronounced change in climatic extremes
compared to changes in the mean

The analysis of the mean number of heatwaves lasting more than 5 days

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

The question of the degree of validity of the relationship

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

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.

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.

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

We presented a case study for Indian wheat crops to test a methodology for
estimating probabilities of long-lasting heatwaves, based on statistical
modelling of streak lengths, their dependency on the seasonal mean of daily
maximum temperature, and empirical-statistical downscaling of multi-model
ensembles. Wheat crops appear to be subject to increased risks of heat stress
in 2100 due to more frequent heatwaves with daily maximum temperature
exceeding 35

Code for reproducing this experiment is provided in the
Supplement as an R Markdown script (pdf and Rmd files). The data are freely
available from figshare:

The geometric distribution

The probability based on the geometric distribution refers to a single
heatwave event, and the probability of a long-lasting heatwave is higher with
an increasing number of heatwaves.

The estimation of probabilities was based on

We used a strategy described in

The analysis was carried out in the R computing environment

The supplement related to this article is available online at:

REB designed and carried out the analysis, whereas the co-authors contributed to writing the paper. JS is also the project leader of CixPAG.

The authors declare that they have no conflict of interest.

This work was funded by the Norwegian Research Council through the CixPAG project (grant number: 244551) and the Norwegian Meteorological Institute. Christian Wilhelm Mohr provided coordinates for the IGP region. Edited by: Sarah Perkins-Kirkpatrick Reviewed by: David Keellings and Yun Li