Relating European temperature extremes to trends in mean

DISS. ETH NO. 20714
Relating European temperature extremes to trends in mean
temperature
A dissertation submitted to
ETH Z URICH
for the degree of
D OCTOR OF S CIENCES
presented by
A NDREAS L USTENBERGER
MSc in Atmospheric and Climate Science, ETH Zurich
born on 9 June 1982
citizen of Lucerne (LU)
accepted on the recommendation of
Prof. Dr. Reto Knutti, examiner
Dr. Erich M. Fischer, co-examiner
Dr. Mark Liniger, co-examiner
2012
Abstract
Extreme weather events have the potential to impact ecosystems, society and infrastructure.
Global and regional climate models are used to estimate changes in the frequency, intensity
and duration of extreme events. However, climate model simulations are costly and timeconsuming. Therefore, there are time periods or emission scenarios for which no simulations
are available. The aim of this thesis is to evaluate the skill of simple statistical models used to
fill in gaps in the time series of the frequency and intensity of temperature extremes for which
no data is available.
The main hypothesis of this thesis is that changes in the frequency and intensity of temperature
extremes are related to trends in European mean temperature. Thus, the objective of the first
part of this thesis is to investigate whether expected changes in the frequency of cold and warm
extremes are linearly related to trends in the annual and seasonal mean temperature. To explore
the potential relationships, linear regression models and two different methods to estimate the
slope parameter are used. The objective of the second part is to approximate expected changes
in the intensity of warm temperature extremes by shifting and scaling seasonal temperature
distributions.
The results show that the frequency of warm extremes is linearly related to changes in the annual mean temperature. The skill of the estimated frequencies depends on the slope parameter
estimation method, the trends in the mean temperature and the degree of linearity. The linearity
is determined by the shape of the parent temperature distribution. In addition, linear trends in
the frequency are higher for temperature distributions with small variance. The magnitude of
the trends and the interannual variability affect the robustness of the estimated slope parameter.
However, the uncertainty due to interannual variability is smaller than the uncertainty induced
by the different climate models. The skill of estimated frequency for temperature extremes
based on absolute temperature thresholds is improved when seasonal means are used. The direction and velocity of the shifted temperature distribution are the reasons that the estimated
frequency of cold extremes performs always worse than the estimated frequency of warm extremes, especially for cold extreme events lasting several consecutive days.
Linear regression models are not appropriate for discrete count data or when the exceedance
probability converges to 0% or 100%. The dependency of these issues on the used climate
model and the location is the reason that alternative methods such as, for instance, logistic regression models do not lead to improved estimates over all of Europe.
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Concerning the second objective, the results show that changes in the intensity of warm extremes can be characterized reasonably well by shifting the full temperature distribution towards higher temperatures for one quarter to one half of Europe across all examined climate
models. In winter, cold extremes warm faster than warm extremes. Otherwise, summer warm
extremes warm faster than summer cold extremes. Therefore, the expected temperature variability is reduced in winter and enhanced in summer. If the variability change is asymmetric,
the additional scaling of the temperature distributions leads to large estimations biases because
the use of central moments is not appropriate. In these cases, the shifted and scaled intensities
of warm extremes perform better when the corresponding half of the temperature distribution is
used to estimate the variance. Over western, northern and parts of central Europe, there are significant changes in the shape of the upper tail of the projected temperature distributions. This
implies that the shape of the parent temperature distribution changes. To account for these complex changes in higher moments, detailed knowledge about future temperature distributions is
necessary and, therefore, contradicts the initial assumption of no available data for future time
periods.
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Zusammenfassung
Extreme Wetterereignisse haben ein Auswirkungspotenzial auf Ökosysteme, Gesellschaft
und Infrastruktur. Globale und regionale Klimamodelle werden verwendet um zukünftige
Änderungen in der Häufigkeit, Intensität und Dauer von Extremereignissen zu schätzen. Simulationen mit Klimamodellen sind jedoch zeitaufwendig und kostspielig. Es gibt daher Zeitperioden oder Emissionsszenarien ohne verfügbare Simulationen. Das Ziel dieser Studie ist
die Eignung von einfachen statistischen Modellen zu beurteilen unvollständige Zeitreihen der
Häufigkeit und Intensität von extremen Temperaturereignissen zu vervollständigen für die keine Daten verfügbar sind.
Die zugrundeliegende Hypothese dieser Studie ist, dass Änderungen in der Häufigkeit
und Intensität von Temperaturextreme linear zusammenhängen mit den Trends in den europäischen mittleren Temperaturen. Der erste Teil dieser Studie beschäftigt sich daher damit zu untersuchen, ob es einen linearen Zusammenhang gibt zwischen den zu erwartenden
Häufigkeitsänderungen und den Trends in den mittleren jährlichen und saisonalen Temperaturen. Lineare Regressionsmodelle und zwei verschiedene Methoden zur Schätzung des Steigungsparameters werden verwendet, um mögliche Zusammenhänge zu studieren. Das Ziel des
zweiten Teils der Studie ist die zu erwartende Intensitätsänderungen von warmen Temperaturextreme annährend zu beschreiben durch skalieren und verschieben von saisonalen Temperaturverteilungen.
Die Resultate zeigen, dass es einen linearen Zusammenhang gibt zwischen der Häufigkeit
von warmen Temperaturextreme und den Änderungen in den mittleren jährlichen Temperaturen. Die Schätzung der Häufigkeiten hängt dabei ab von der Methode zur Schätzung des
Steigungsparameters, den Trends in der mittleren Temperatur und der Linearität. Die Linearität selber ist festgelegt durch die Form der zugrundeliegenden Temperaturverteilung. Ausserdem sind die linearen Trends in der Häufigkeit grösser für Temperaturverteilungen mit kleiner Varianz. Die Grössenordnung des Trends und die interannuelle Variabilität bestimmen
wie robust der geschätzte Steigungsparameter ist. Die Unsicherheit aufgrund der interannuellen Variabilität ist jedoch kleiner als die Unsicherheit durch die verschiedenen Klimamodelle. Häufigkeitsschätzungen für extreme Temperaturereignisse, die auf absoluten Temperaturgrenzwerten basieren, sind besser, wenn saisonale Mittelwerte verwendet werden. Die Richtung und Geschwindigkeit der verschobenen Temperaturverteilung sind Gründe dafür, dass die
Häufigkeitsschätzungen von kalten Extremereignissen immer schlechter sind als für warme
Extremereignisse, vor allem für kalte Extremereignisse, die über mehrere aufeinanderfolgende
Tage bestehen.
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Lineare Regressionsmodelle sind nicht geeignet für diskrete Daten wie Zähldaten oder wenn
die Überschreitungswahrscheinlichkeit gegen 0% oder 100% konvergiert. Die Abhängigkeit
dieser Einschränkungen vom verwendeten Klimamodell und Ort ist der Grund dafür, dass alternative Methoden, wie z.B. das logistische Regressionsmodell, nicht überall in Europa zu
verbesserten Schätzungen führen.
Die Resultate des zweiten Teils zeigen, dass für ein Viertel bis die Hälfte des europäischen
Festlands Änderungen in der Intensität von warmen Temperaturextreme erstaunlich gut beschrieben werden können durch Verschieben der gesamten Temperaturverteilung in Richtung
höhere Temperaturen. Für Wintertemperaturen ist die Erwärmung der kalten Temperaturextreme grösser als für warmen Temperaturextreme. Andererseits ist für die Sommertemperaturen die Erwärmung grösser für warme Temperaturextreme als für kalte Temperaturextreme.
Daher ist die zukünftige Temperaturvariabilität im Winter geringer und im Sommer grösser.
Wenn diese Änderung in der Variabilität asymmetrisch ist, dann führt eine zusätzliche Skalierung der Temperaturverteilungen zu grossen Schätzungsfehlern, weil die Verwendung der
zentralen Momente dafür nicht geeignet ist. Bessere Schätzungen für die verschobenen und
skalierten Intensitäten der warmen Temperaturextreme erhält man, wenn für die Schätzung der
Varianz die entsprechende Hälfte der Temperaturverteilung verwendet wird (d.h. die untere
Hälfte für kalte Extreme und die obere Hälfte für warme Extreme). Über Westeuropa, Nordeuropa und Teilen von Zentraleuropa hat es, zusätzlich zu den Änderungen in der Variabilität,
signifikante Veränderungen im Verteilungsschwanz der simulierten warmen Temperaturextreme. Dies bedeutet, dass sich über diesen Gebieten die Form der Temperaturverteilung ändert.
Um diese komplexen Änderungen in den höheren Momenten zu berücksichtigen, benötigt man
detailliertes Wissen über die zukünftigen Temperaturverteilungen und dies widerspricht der
ursprünglichen Annahme, dass für zukünftige Zeitperioden keine Daten verfügbar sind.
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Contents
Abstract
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Zusammenfassung
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1 Introduction
1.1 European heat wave event 2003: A taste of the future climate? . .
1.2 The climate system: A short overview about a complex system . .
1.3 The probabilistic nature of extreme events . . . . . . . . . . . . .
1.4 Data availability: Prerequisite for new insights . . . . . . . . . . .
1.5 Statistical modelling of extreme events . . . . . . . . . . . . . . .
1.5.1 Extreme value theory . . . . . . . . . . . . . . . . . . . .
1.5.2 Extreme indices . . . . . . . . . . . . . . . . . . . . . . .
1.6 Are temperature extremes related to European mean temperature?
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2 The potential of pattern scaling for projecting temperature-related extreme indices
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Extreme Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 ENSEMBLES regional climate models . . . . . . . . . . . . . . . . .
2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Estimation of the response pattern . . . . . . . . . . . . . . . . . . . .
2.3.2 Evaluation of the estimation skills . . . . . . . . . . . . . . . . . . . .
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Sensitivity of European extreme daily temperature return
trends in mean and variance
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Data and Methods . . . . . . . . . . . . . . . . . . . . . .
3.2.1 ENSEMBLES regional climate models . . . . . .
3.2.2 Return level estimation . . . . . . . . . . . . . . .
3.2.3 Adjustment models . . . . . . . . . . . . . . . . .
3.3 Results and Discussion . . . . . . . . . . . . . . . . . . .
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levels to projected
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3.4
3.5
3.3.1 Performance for the summer season .
3.3.2 Performance for the winter season . .
3.3.3 Limitations of the adjustment models
Conclusions . . . . . . . . . . . . . . . . . .
Auxiliary material . . . . . . . . . . . . . . .
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4 Conclusions and outlook
4.1 Trends in central moments and temperature extremes:
4.2 Skill and limitations of the fill in approaches . . . . .
4.2.1 Linear regression model . . . . . . . . . . .
4.2.2 Adjustment models . . . . . . . . . . . . . .
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Are they related?
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References
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Acknowledgments
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Chapter 1
Introduction
1.1 European heat wave event 2003: A taste of the future
climate?
The French weekly newspaper ”Le Nouvel Observateur” asked on August 2003:
”What if the 2003 heatwave is just the beginning?”
This is a legitimate question given the global temperature increase observed over the past
decades and the various socio-economic and ecological impacts caused by the summer heat
wave in 2003. The study of Della-Marta et al. (2007) shows that the summer heat wave length
has doubled over western Europe since 1880. The centre of the heat wave event 2003 was
mainly located over France, Switzerland and Germany and caused record-breaking maximum
temperatures in June and August. Such prolonged time periods with high temperatures affect
the human physiology (Kalkstein and Smoyer, 1993; Changnon et al., 1996; Semenza et al.,
1996; Smoyer, 1998; Basu, 2002; Tan et al., 2007; Fischer and Schär, 2010) and increase the
mortality, especially for elderly people (Koppe et al., 2004; Kovats and Kristie, 2006; D’Ippoliti
et al., 2010). In France, for instance, the heat wave was responsible for nearly 15,000 excess
deaths during the time period from August 1st to 20th (Fouillet et al., 2006). In Switzerland,
there was an increase in daily mortality of about 7% for the time period from June to August,
mainly in suburban areas and cities such as, for instance, Basel, Geneva and Lausanne (Grize
et al., 2005). The higher mortality in urban areas is due to the amplification of heat stress
(Fischer et al., 2012a). The mortality risk is also increased due to the lack of mobility, thermal
insulation and sleeping on the top floor (Vandentorren et al., 2006). Despite of the heat stress,
people suffered additionally from high amount of photochemically produced ozone and air pollution (Fiala et al., 2003; Pellegrini et al., 2007; Solberg et al., 2008). However, not only the
human physiology was affected but also the ecosystems and, therefore, the agricultural sector.
The estimated economical loss in agriculture was valued at approximately e 13 billion (Létard
et al., 2004). The efficiency of the European ecosystems to produce organic matter from atmospheric carbon dioxide (gross primary productivity) was reduced by nearly 30%. As a result,
the ecosystems act as net carbon sources to the atmosphere (Ciais et al., 2005). Numerous
forest fires in Greece, France, Spain, Italy and Portugal were responsible for the air pollution
and caused loss of forest areas and wildlife (Garcı́a-Herrera et al., 2010). Further ecological
1
2
C HAPTER 1: I NTRODUCTION
impacts were large losses of Alpine glacier mass (Fink et al., 2004), shortage of pollen season
(Gehrig, 2006), water temperature records in lakes (BUWAL, 2004; Jankowski et al., 2006) and
reduced river discharge (BUWAL, 2004). The latter led to problems in power plant cooling and
hydro-electric power restrictions (Beniston and Diaz, 2004).
The unprecedented heat wave event in summer 2003 and the broad range of impacts raised
the question in the scientific community about how to put this event into the context of the
current and future climate. In this regard, Figure 1.1 shows the distribution of seasonal JuneJuly-August mean temperature over northwestern Switzerland for the time period from 1864
to 2000.
Figure 1.1: Distribution of seasonal June-July-August mean temperature for the time period from 1864
to 2000 and northwestern Switzerland (Schär et al., 2004).
Figure 1.2: European land-surface summer temperature anomalies for the time period July 20th to
August 20th for the year 2003 relative to the mean of the years 2000, 2001, 2002 and 2004. The radiative
temperature is derived from the TERRA MODIS (MODerate resolution Imaging Spectroradiometer)
satellite (Image by Reto Stöckli, Robert Simmon and David Herring, NASA Earth Observatory, based
on data from the MODIS land team).
1.2 T HE
CLIMATE SYSTEM :
A SHORT
OVERVIEW ABOUT A COMPLEX SYSTEM
3
The temperature anomaly of summer 2003 is about 5.1◦ C relative to the 1864-2000 mean corresponding to more than five times the standard deviation as depicted in Figure 1.1. In Basel,
the observed deviation of the summer mean maximum temperature from the 1961-1990 seasonal mean was up to 6◦ C standing out as record-breaking event since 1901 (Beniston and
Diaz, 2004). The large summer temperature anomalies in 2003 are impressively shown in Figure 1.2 for the time period from July 20th to August 20th . The figure visualizes the land-surface
temperature anomalies over Europe derived from the TERRA MODIS (MODerate resolution
Imaging Spectroradiometer) satellite with spatial resolution of 1km. The mean is computed for
all cloud free data of the years 2000, 2001, 2002 and 2004 and then subtracted from the measured radiative temperature for the year 2003. The satellite image depicts the spatial structure
of the heat wave and shows that the highest anomalies are found over France. The radiative
temperature was up to nearly 10◦ C hotter than for the other years. In historical context and
under consideration of the uncertainties in palaeoclimatological data, the summer 2003 was
the hottest since the 16th century at continental scale (Chuine et al., 2004; Luterbacher et al.,
2004; Xoplaki, 2005; Menzel, 2005; Coumou and Rahmstorf, 2012). Otherwise, the temperature anomalies in summer 2003 could be an analogue for the future climate caused by the
anthropogenic rise in the Earth’s temperature (Beniston and Stephenson, 2004; Beniston and
Diaz, 2004; Beniston, 2004; Meehl and Tebaldi, 2004; Schär et al., 2004; Stott et al., 2004).
Despite of large uncertainties in the statistics of extreme events, there is a need to investigate long-term trends, predictability and the underlying physical processes. The motivation and
main objective of this thesis is to investigate potential relationships between changes in the frequency and intensity of extreme events and the warming trends in European mean temperature
using the state-of-the-art regional climate models from the European multi-model project ENSEMBLES (Van Der Linden and Mitchell, 2009). Under consideration of gaps in temperature
time series, these relationships are used to project the frequency and intensity of warm extreme
events and, therefore, to fill in the gaps. The skill of the projections and the limitations of the
fill in methods are evaluated and discussed. The findings of this project are interesting for the
impact community, i.e. the insurance companies, engineers and public authorities. The methods could be used to fill the gap between data availability and the need for climatological data
as input for impact models. The purpose of the impact models is to assess the future evolution
of ecological and socio-economic impacts in order to plan appropriate adaptation strategies.
1.2 The climate system: A short overview about a complex
system
The climate system is driven by incoming solar short-wave radiation. Figure 1.3 shows
schematically what happens to the incoming solar radiation within the atmosphere. In the
figure are shown the annual and global mean estimates for the different energy fluxes (Kiehl
and Trenberth, 1997).
The left part of Figure 1.3 sketches that a fraction of the incoming short-wave radiation is
diffused within the atmosphere to all directions due to reflection, refraction and diffraction on
air molecules and aerosols, i.e. fog, rain droplets and clouds. As a result, part of the incoming
solar radiation is scattered back to the space and another part is absorbed by the Earth’s atmo-
4
C HAPTER 1: I NTRODUCTION
Figure 1.3: Estimated global and annual mean short-wave and long-wave energy fluxes (Kiehl and
Trenberth, 1997).
sphere. The remaining fraction of diffused radiation together with the not-diffused direct solar
radiation reaches the Earth’s surface. The surface properties determine the amount of shortwave radiation which will be absorbed. The not absorbed radiation is again reflected. The ratio
between reflected and absorbed short-wave radiation is termed albedo. Snow covered areas and
clouds, for instance, have high albedo values while the albedo of soil is lower and depends on
its composition. The absorbed short-wave radiation is converted to and emitted as long-wave
radiation as shown on the right part of Figure 1.3. The emitted long-wave radiation is partially
absorbed by the greenhouse gases (CO2 , CH4 , water vapour) and aerosols and re-emitted in
all directions. The back radiation shown in Figure 1.3 is the long-wave radiation re-emitted
towards the Earth’s surface. To avoid continuous warming or cooling of the Earth over a long
time period, the absorbed solar short-wave radiation has to be balanced by the long-wave radiation released by the Earth’s atmosphere and surface. This energy balance determines at least
the Earth’s climate and, particularly, the air and surface temperature. The greenhouse gases in
the atmosphere and the resulting long-wave back radiation are the reason for the global mean
temperature of about 14◦ C (Jones et al., 1999).
Because of the Earth’s shape and declination, the amount of incoming solar radiation per
surface area depends on the latitude and season. Thus, the Earth system strives to balance the
differences in the received energy between tropics and high latitudes via oceanic and atmospheric circulation. For the latter, the main driver is the release of energy through condensation
of water vapour depicted in Figure 1.3 as latent heat flux. The circulation patterns and the
amount of received radiation determine the weather patterns and temperature and, therefore,
the climate at a certain place and season. But even for the same latitude and season, the climate
varies between distant locations. The reason is schematically shown in Figure 1.4. Figure 1.4 is
a scheme of components of the climate system, physical processes as well as interactions. The
site-specific processes and interactions affect the occurrence, intensity and duration of weather
events. There are, for instance, milder winter temperatures on the coasts than for continental climate due to the thermal inertia of the oceans. Another example is the onset of rainfall
1.2 T HE
CLIMATE SYSTEM :
A SHORT
OVERVIEW ABOUT A COMPLEX SYSTEM
5
Figure 1.4: Scheme of physical processes, interactions and major components of the climate system
(IPCC, 2007).
triggered by the orography. In case of heat waves, the land-climate interactions can amplify
anomalous warm conditions.
Changes in the Earth’s climate system are triggered by changes in the energy fluxes (Figure
1.3), i.e. in the incoming solar short-wave radiation, the Earth’s albedo and the amount of longwave radiation emitted back to the Earth’s surface. The incoming solar shortwave radiation is
determined by the Sun’s intensity and the Earth’s orbit parameter. Both change in time scales
of 104 to 106 years. Reasons for changes in the Earth’s albedo are changes in the cloud cover,
vegetation, land-use and the amount of aerosols in the atmosphere. The fraction of long-wave
back radiation is related to the greenhouse gas concentration, i.e. mainly to the amount of
carbon dioxide and water vapour. During the last few decades, the evidence increased that human affect the radiative fluxes by changing the composition of the atmosphere and the albedo
through land-use (IPCC, 2001, 2007). As a consequence, the climate system responds directly
or indirectly through various positive and negative feedback processes such as, for instance,
the snow-albedo feedback. This feedback implies that for cold temperatures the snow cover
remains and increases the albedo and, as a consequence, the amount of short-wave radiation
reflected back to the atmosphere and space. Due to the decreased amount of available energy
at the surface, the temperature decreases further. Conversely, if the temperature increases and
the snow cover vanish, the darker soil can absorb more short-wave radiation leading to further
warming. The snow-albedo feedback is an example of a positive feedback and influences the
temperature variability for very cold winters (Yiou et al., 2009; Cohen et al., 2012). Generally, positive feedback processes intensify the initial process while negative feedbacks damp it.
Nevertheless, the climate system responds at least to the anthropogenically induced changes in
the radiation fluxes with an observed increase in the surface mean temperature on various spa-
6
C HAPTER 1: I NTRODUCTION
tiotemporal scales (Stott, 2000; Folland et al., 2001; IPCC, 2001; Hansen et al., 2006; IPCC,
2007; Zorita et al., 2008; Easterling and Wehner, 2009; Hansen et al., 2010; Ceppi et al., 2012).
Attribution studies are used to establish the most likely causes for the recent observed
changes in particular climate elements. For the attribution studies, climate models are run with
combined anthropogenic and natural forcings as well as natural forcings only (Stott, 2003; Stott
et al., 2010). The respond of the climate system is then compared with the observed time series
of climate elements such as, for instance, the temperature. Likelihood measures are derived
to determine the most probable explanation for the observed changes (Min et al., 2004). The
attribution procedure is constrained by the ability of the climate models to represent reliably
the effects of external forcings and natural variability, especially on small spatial scales (Stott
et al., 2010). The attribution of changes in extreme events to specific causes is still challenging
due to gaps in the understanding and the difficulty to represent reliably extremes in climate
models (Stott et al., 2010). Nonetheless, there are studies available trying to link particular
extreme weather events and changes in extremes with changes in the climate system (e.g. Stott
et al., 2004; Meehl et al., 2007; Min et al., 2011; Pall et al., 2011; Dole et al., 2011). But, it is
easier to evaluate how the frequency of extreme events changes for a particular warming rate
when the underlying distribution is known.
To detect significant changes in the climate state, i.e. in the mean, variance and extremes,
the definition of a climatological standard normal as reference climate is crucial. According
to the technical regulations of the World Meteorological Organisation (WMO), normals are
period averages for time periods longer than three consecutive ten-year periods (WMO, 2007).
The discussion about normals and their optimal length started already in the 19th century. During the early 20th century, the suggestions for the optimal length of the reference period ranged
from 11 years (solar cycle) up to 50 years to capture more of the inter-decadal variability. In
1935, the International Meteorological Committee suggested during the Warsaw conference to
take 30 years as time period (WMO, 2007; Hulme et al., 2009). This suggestion is more a compromise because the knowledge about the climate system and its variability was poor at that
time. Hence, there is no robust physical or statistical argument for the recommended normal
length. At the moment, the suggested and widely used normal period ranges from 1961 to 1990
and is employed as baseline for computing anomalies and changes in the climate system.
1.3 The probabilistic nature of extreme events
The frequency, intensity and persistence of weather events at a given location is inferred by
measuring several climate elements such as, for instance, temperature, wind, precipitation or
water vapour over a long time period, i.e. over several decades. Then, the measurements are
statistically characterized using estimators such as, for instance, the mean and variance. These
two estimators describe the expected weather and its variability for a certain time period and
location. Probability (PDF) and cumulative (CDF) density functions are used to summarize
and visualize the whole set of measurements. This implies that PDFs summarize all sitespecific processes and feedbacks shown in Figure 1.4, i.e. they represent at least the climate.
In this context, climate is a statistical quantity and is not directly measurable. An example
1.3 T HE
PROBABILISTIC NATURE OF EXTREME EVENTS
7
of a PDF used to describe statistically temperature is shown in Figure 1.5. In this example,
the temperature data are assumed to be normally distributed. To assume normally distributed
temperature data is a reasonable and often applied approximation. However, this assumption is
critical in the tails which are not well constrained because of the rare nature of extreme events
(Coumou and Rahmstorf, 2012). Nevertheless, the normal distribution family is often chosen
as appropriate statistical model to fit the data.
Figure 1.5: Fitted probability density function (PDF) of temperature. Blue and red areas represent cold
and warm extremes, respectively.
Figure 1.5 illustrates that there are events with small likelihood in the tails of the fitted PDF.
Because extreme events are rare events, this means that the tails of the PDF, coloured as blue
and red areas, represent cold and warm extreme events. In addition, Figure 1.5 implies that
the statistical characterization of extreme events depends on the underlying PDF. The resulting
question is where the tails begin and, therefore, how to define the term ”small likelihood”. A
widely used approach is to set the boundaries of the tails where the probability to exceed a
particular absolute or percentile threshold is less than 10% (IPCC, 2012), i.e. the 90th and 10th
percentile. The statistical definition of the tails and, therefore, of extreme events is not related
to their potential socio-economic or ecological impacts.
Instead of using probabilities, extreme events are often characterized in terms of return periods (Coles, 2001). This quantity describes how long the average time period between two
extreme events of equal or larger intensity is. Thus, 20-year return period events occur with
the probability of 5% per year and are expected on average every 20 years. Because return
periods are related to probabilities, the quantiles of them for a given PDF are the return levels.
These values are more useful for the insurance industry or engineers because they are used as
critical threshold values to design infrastructures such as, for instance, dams and dikes and for
8
C HAPTER 1: I NTRODUCTION
human health. In this context, the exceedance of the threshold has an impact potential. Hence,
return levels as absolute temperature thresholds are another possibility to define extreme events
(IPCC, 2012).
To date, the analysis of extremes is limited to particular climate element, i.e. temperature,
precipitation or wind. However, recent literature suggests that extreme events can be understood as compound events (IPCC, 2012). This means that the occurrence of an extreme event
depends conditionally on the occurrence of one or more other events. These other events do
not necessarily have to be extreme. As an example, droughts are extreme concerning their persistence but the absolute temperatures for each day might not be extreme. Further, the drought
itself can trigger the occurrence of other extreme events. For example, the preceding drought in
spring 2003 and the resulting soil-moisture deficit in summer 2003 led to a positive feedback,
i.e. to an amplification of the warm temperature anomaly in summer 2003 (Seneviratne et al.,
2006; Fischer et al., 2007; Fischer and Schär, 2009, 2010).
1.4 Data availability: Prerequisite for new insights
The data quantity and quality constrains the potential to detect trends in the frequency, intensity and persistence of extreme events as well as the understanding of underlying physical
processes. The latter is the precondition for attribution studies (Stott et al., 2010; IPCC, 2012)
and the development of reliable climate change scenarios. The detection of significant trends
in extreme events is limited by the rareness, i.e. high return periods. As a consequence, the
tails of the temperature distributions are not well constrained (Rahmstorf and Coumou, 2011).
This implies that the full range of possible extreme events is not captured for a given stationary climate and any derived estimates are associated with large sampling uncertainties. As
an example, the study of Frei and Schär (2001) investigates the sensitivity of the detection
probability to the rarity of heavy precipitation in the Alpine region. They estimated that the
probability to detect a frequency change by a factor of 1.5 in a 100-year record of seasonal
counts is about 20% for events with 100 days return period.
Another issue is the data quality, i.e. the completeness and homogeneity. Inhomogeneous
time series exhibit abrupt or gradually changes, i.e. so-called change points, caused by changes
in the station location, instrumentation, observing practices or environment. The correction of
these non-climatic changes is crucial because they can introduce artificial trends or systematic
biases. There are statistical methods available to detect and remove change points for annual,
seasonal and monthly data (Alexandersson and Moberg, 1997; Peterson et al., 1998; Vincent,
1998; Menne and Williams, 2005; Gérard-Marchant et al., 2008). For daily data, the development of procedures is still a topic of ongoing research and only a few methods are currently
available (e.g. Della-Marta and Wanner, 2006).
In general, the data quality and quantity depends on the measurement technique, station
density, the length of the time period as well as the definition of global standards and periodic
calibration. The international coordination of atmospheric observations was improved after the
International Geophysical Year (IGY) in 1957/1958. Near-global observations with satellites
started in the 1970s. Large-scale data for the Earth’s surface are available since the middle
1.4 DATA
AVAILABILITY:
P REREQUISITE
FOR NEW INSIGHTS
9
of the 19th century. For example, the land temperature time series from the Climate Research
Unit (CRU) goes back to 1856 (Brohan et al., 2006). Nowadays, there are monthly, seasonal
and annual temperature and precipitation data available for most parts of the world albeit with
different spatial and temporal coverage. However, extreme events such as, for instance, gusts
of wind (Della-Marta et al., 2010) or heavy precipitation events have generally much smaller
spatiotemporal scales. For instance, heavy precipitation events have temporal scales in the
order of hours. Therefore, data on daily or sub-daily time scales are needed to sample these
events and to detect trends. However, such data have only been available since the second half
of the 20th . These few decades with available data are not long enough to capture large-scale,
i.e. continental to hemispheric scale, and low-frequency climate variability and extremes. In
addition, there are still large areas around the Earth with very sparse and unreliable or even no
data such as, for instance, South America or Africa. The data availability is even more limited
for climate elements other than temperature or precipitation such as for the soil-moisture measurements (Seneviratne et al., 2010). This makes it more difficult to investigate extreme events
as compound events. Further, the inhomogeneous station density and the insufficient temporal
resolution imply that not all extreme events are captured and, therefore, the derived extreme
statistics might be biased. Thus, the lack of knowledge about the processes and feedbacks triggering extreme events together with the issues associated with data quality and quantity are at
least responsible for the difficulties to conduct attribution studies with extremes (Stott et al.,
2010; Schiermeier, 2011).
Extending the data availability with palaeoclimatological reconstructions or numerical climate models might help to complete the gaps in the PDF. However, the palaoclimatological
reconstructions have often spatiotemporal resolutions much coarser than for instrumental measurements and, additionally, larger uncertainties. For the climate model simulations, the reliability of the derived statistics depends generally on the appropriate representation of the
climate system, i.e. the physical processes. Because of the limited horizontal resolution of
climate models, it is not possible to simulate directly processes such as, for instance, convection. These sub-grid processes are parameterized, i.e. they are related empirically to the
resolved large-scale features. To simulate large-scale circulation, among other things, General
Circulation Models (GCM) are used with horizontal resolution in the order of 100-300 km.
To investigate regional or local-scale changes in extremes with GCMs, statistical or dynamical downscaling techniques are applied (Wilby and Wigley, 1997; Frei, 2006; Haylock et al.,
2006; Schmidli et al., 2007; Maraun et al., 2010). The dynamical downscaling procedures use
the GCM outputs as boundary conditions driving regional climate models (RCM) with higher
spatial resolutions up to 25km. Examples for large dynamical downscaling experiments are the
EU-funded FP5 PRUDENCE (Christensen et al., 2002) and FP6 ENSEMBLES multi-model
projects (Van Der Linden and Mitchell, 2009). For the latter, 6 GCMs drives 9 RCMs with
spatial resolutions of 25km and 50km. The skill of the dynamically downscaling experiments
depend once again on the reasonable representation of large-scale features in the GCM (Déqué
et al., 2007). However, studies such as Fischer et al. (2012b) and Déqué et al. (2007) have
shown that the contribution of the RCMs to the overall uncertainty has increased for the newest
generation of regional climate models. In contrast to dynamical downscaling with RCMs, statistical downscaling methods use statistical links between large-scale variables, i.e. grid points
in climate models, and observed values, i.e. point values. The statistical downscaling approach
10
C HAPTER 1: I NTRODUCTION
provides projected extremes on local scale, i.e. with smaller spatial extent relative to the area
averages (Klein-Tank et al., 2009). However, it is not possible to downscale extreme events
with magnitudes larger than observed.
As mentioned before, the regional climate models from the European multi-model project
ENSEMBLES (Van Der Linden and Mitchell, 2009) are used in this thesis. From the available
GCM-RCM pairs, a subset of 12 GCM-RCM chains is selected. The chosen set of GCM-RCM
pairs has the same rotated pole grid and, therefore, there is no further interpolation necessary.
It would be more difficult to compare the interpolated with the non-interpolated values because
of the additional uncertainties and the smoothing of extremes (Haylock et al., 2008).
1.5 Statistical modelling of extreme events
The theoretical background to characterize the distributional properties of extreme events is
formulated in the extreme value theory. The properties are determined by the tails of the underlying parent distribution. Therefore, as discussed in the chapter before, large samples of high
quality data are important to constrain the upper and lower tails.
1.5.1 Extreme value theory
The French mathematician Maurice Fréchet (1878-1973) suggested an asymptotic distribution
for samples of maxima in a paper published in 1927 (Fréchet, 1927). This distribution type has
a heavy upper tail and bounded lower tail and is known as Fréchet distribution. The Fréchet
distribution is nowadays one of the three possible extreme value distributions and applied to
model, for instance, market-returns. Two further distributions were developed to address problems in hydrology, engineering and materials science. The Swedish engineer and scientist
Waloddi Weibull (1887-1979) described the Weibull distribution in 1951. The Weibull distribution was proposed in the context of strength of material and fatigue analysis and was applied
to model failure times. The German mathematician Emil Julius Gumbel (1891-1966) applied
the extreme value theory, i.e. the Gumbel distribution, to a broad range of applications and,
therefore, various types of parent distributions (Gumbel, 1954).
The reason for the applicability of extreme value theory to different research fields is implicitly stated in the extreme value theorem (Fisher and Tippett, 1928; Gnedenko, 1943; Coles,
2001). The theorem asserts that the limit distribution of the largest values can only be one of
the three aforementioned extreme value distributions, independently of the underlying parent
distribution. Therefore, samples of the largest values, i.e. max (X 1 , · · · , X n ), converges for
n → ∞ as following:
max (X 1 , · · · , X n ) − bn
lim P
≤ z → G (z) .
n→∞
an
an > 0 and bn is a sequence of normalizing parameters and G (z) represents one of the three
possible extreme value distribution types. The parent distribution of (X 1 , · · · , X n ) determines
how fast the largest values converge to the limit distribution. In earlier studies, the first step
was to assume an extreme value distribution for a specific application and to estimate then
1.5 S TATISTICAL
11
MODELLING OF EXTREME EVENTS
the unknown distribution parameters. The disadvantage of this proceeding is that the a priori
assumption of an extreme value distribution needs a selection technique. All the following
inferences are based on the assumption that the extreme value distribution suggested by the
selection procedure is appropriate. As a consequence, any uncertainties in the inferences arising
from the distribution selection are not taken into account (Coles, 2001). This issue of a priori
distribution selection is circumvented through combining the three distribution families into
one distribution family (Jenkinson, 1955; Coles, 2001). The so-called generalized extreme
value distribution (GEV) family is defined as following:
− 1ξ !
z−µ
G (z) = exp − 1 + ξ ·
σ
for
1+ξ·
z−µ
σ
>0.
The location µ, shape ξ and scale σ parameters satisfy the conditions −∞ < µ < ∞,
−∞ < ξ < ∞ and σ > 0. The advantage of using the generalized extreme value distriˆ determine which of the three extreme
bution family is that the estimated parameters (µ̂, σ̂, ξ)
value types is appropriate for a particular application. Thus, the Gumbel family is chosen for
the limit ξ → 0 while ξ > 0 and ξ < 0 corresponds to Fréchet and Weibull family, respectively.
The extreme value distribution families are applied to model the largest values, i.e. minima or maxima. However, there is a further extreme value distribution family used to model
exceedances, i.e. (X − u|X > u), where u is a large threshold. The limit distribution was
introduced by the Italian economist Vilfredo Pareto (1848-1923) and is known as generalized
Pareto distribution (GPD) (Jenkinson, 1955; Pickands, 1975; Davison and Smith, 1990; Coles,
2001). The two-parameter generalized Pareto distribution family is defined as follows:
(X − u|X > u) · ξ
H (X − u|X > u) = 1 − 1 +
σ GP
− 1ξ
.
ξ is the shape parameter and σ GP the scale parameter. The shape and scale parameters of the
GPD and GEV distribution families are related to each other. The shape parameter ξ is the same
for both distribution families. The scale parameters of GPD and GEV are linked as following:
σ GP = σ + ξ (u − µ) .
Vilfredo Pareto applied this power-law type distribution to model income and wealth (Pareto,
1897). The tail behaviour of the Pareto distribution is, similar as for the GEV distribution, determined by the shape parameter ξ. For ξ < 0 the distribution of exceedances (X − u|X > u)
σ
has an upper bound at u − GP
. In contrast to this, there is no upper limit for ξ > 0. For
ξ
the limit case ξ → 0, the distribution converges to the exponential distribution with parameter
1
. Similar to the GEV distribution, the appropriate type of extreme value distribution is deσGP
termined by the estimated parameters.
Depending on the extreme value distribution family, i.e. GPD or GEV, there are two often
applied procedures for sampling extremes from datasets. When modelling largest values (maximum or minimum), the block maxima approach is used (Coles, 2001). For this approach, the
12
C HAPTER 1: I NTRODUCTION
whole sample is split into blocks of equal length. Then, the maxima or minima are drawn from
each block. For the resulting sample of largest values, the GEV distribution parameters and
the confidence intervals are estimated. To estimate the distribution parameters, the maximum
likelihood approach (e.g. Coles, 2001) or the L-moments method (Hosking, 1990; Coles, 2001)
are applied. To estimate the confidence interval, the Delta-method (Coles, 2001) or parametric or nonparametric resampling procedures are suggested (Davison and Hinkley, 1997). The
difficulty of the block maxima approach is to find an appropriate block size to minimize the
sampling uncertainty and to avoid biases caused by too small block lengths. Another issue is
that not all available information of the tails is used. This is the advantage of the peak-overthreshold approach using all data beyond the selected threshold u. The peak-over-threshold
method is used to derive the exceedances. For the parameter and confidence interval estimations, the same methods can be applied as for the block maxima approach. The difficulty of the
peak-over-threshold method is to determine an appropriate threshold u0 to minimize the sampling uncertainty and biases. Threshold selection methods base on the idea that for high enough
thresholds u0 the GPD parameter estimates converges to a nearly constant value and the mean
excess E (X − u|X > u) is a linear function of u (Coles, 2001). This is tested visually with
appropriate plots and, therefore, provides no objective criteria for the threshold selection. As
a consequence, the approaches are not applicable to large gridded data sets. For this purpose,
goodness-of-fit procedures (Choulakian and Stephens, 2001) and test statistics (e.g. Kyselý
et al., 2008) are used in this thesis. Generally, the GPD distribution is preferred in this thesis
because the maximum possible amount of information is used to model extreme values. This
is especially important in the context of extremes because the amount of information is already
reduced due to their rarity. For comparison, the GEV distribution is also fitted to the data in
order to test the sensitivity of the temperature extreme estimates to the chosen extreme value
distribution family. The distribution parameters of both distribution families are estimated with
the maximum-likelihood method. This approach is preferred because it provides confidence
intervals for the estimated distribution parameters (Coles, 2001). The quantiles of the fitted
extreme value distributions corresponds to the return levels. This value is exceeded on average
once every T years while T is the return period. When the return period T is given in years,
an equivalent proposition is that the return level is exceeded in one year with the probability
1
. The return periods in this thesis are estimated according to Coles (2001). Their confidence
T
intervals estimates are derived from the Delta-method (Coles, 2001).
Extreme value theory provides a statistical framework to model very rare events. However,
as discussed in the chapter before, the inferences are sensitive to the quantity and quality of the
data. To circumvent part of the difficulties arising from the rarity of extreme events, the use
of extreme indices (Frich et al., 2002; Klein-Tank et al., 2009) has established during the last
decade. For this reason, the extreme indices are also used in this thesis.
1.5.2 Extreme indices
Extreme weather events have often time scales in the order of daily or sub-daily. However,
climate scientists are mainly focused on datasets with monthly time scales up to the 1990s.
Consequently, there were no international datasets of long-term daily terrestrial data available
at that time (Peterson and Manton, 2008). This was also stated in the IPCC second assess-
1.5 S TATISTICAL
MODELLING OF EXTREME EVENTS
13
ment report in 1995 (IPCC, 1995) where is written that ”there are inadequate data to determine
whether consistent global changes in climate variability or weather extremes have occurred
over the 20th century”. Analysis of extremes were available for a few countries such as, for instance, the United States, Canada or Australia. However, it was difficult to compare the results
among these countries because they used different measures of extremes (Peterson and Manton, 2008). In addition, because the time scales of extremes are smaller than for the available
datasets, information about extremes is smoothed and partly filtered out, respectively. These
issues motivated not only to develop daily datasets but also to intensify international cooperation. International cooperation is important to generate and to analyze datasets from Asia,
Africa and South America where gaps in the data exists. For this purpose, the meeting in
November 1999 of the joint Expert Team on Climate Change Detection and Indices (ETCCDI)
in Geneva agreed to set the focus on the analysis and development of extreme indices derived
from daily meteorological data and to fill in gaps in the world map of available data by organizing regional workshops. An example for such a successful workshop is the Asia-Pacific
network (ANP) meeting (e.g. Manton et al., 2001; Zhang et al., 2011). The general idea of the
workshops is that participants bring their own data and they are then guided to apply quality
control and inhomogeneity detection procedures. At least, they compute with software provided by the ETCCDI the extreme indices (Zhang et al., 2011).
The current 27 ETCCDI extreme indices (see: http://cccma.seos.uvic.ca/
ETCCDI/list_27_indices.html) represent partly modified versions of the European
Climate Assessment and Data (ECA&D) indices (Klein-Tank and Können, 2003). The extreme
indices differ in their definition of the underlying distribution and in the selected threshold, i.e.
how far the threshold is located within the tail of the parent distribution. Relative to the rare
extreme events, the extreme indices are generally more robust, more frequent and simple to
compute. From a statistical point of view they represent moderate extreme precipitation and
temperature events, i.e. they occur a few times per year. Further advantages of the indices
are that they have a higher signal-to-noise ratio and represent broad aspects of the local and
regional climate, i.e. temperature, precipitation, humidity, cloudiness, wind and so on. The use
of these standardized indices allows now to compare the results among different studies and
regions, respectively. They are based on absolute (e.g. 0 ◦ C) or percentile thresholds (e.g. 90th
percentile). The former are not suitable to compare the results over very large areas because
they do not describe the same part of the underlying distribution. In contrast to this, extreme
indices with percentile thresholds are related to the local climate and take the mean annual
cycle into account. Because they represent in each area of interest the same part of the underlying distribution, their rarity is fixed. Therefore, the number of days exceeding the threshold
is evenly distributed in space. This allows at least to compare the results between different
regions (Klein-Tank et al., 2009). The percentile thresholds are computed using data of the
reference period, i.e. the time period 1961-1990. But, the study of Zhang et al. (2005) shows
that the sampling uncertainty in the estimated percentile causes discontinuities at the beginning
and the end of the selected reference period. They suppose a bootstrap procedure to estimate
the percentile thresholds and, therefore, to remove the inhomogeneities.
During the last decade, extreme indices are used to monitor changes in the frequency, intensity and persistence of moderate extreme events (e.g. Frich et al., 2002; Klein-Tank and
Können, 2003; Alexander et al., 2006; Russo and Sterl, 2011; Moberg et al., 2006), to evaluate
14
C HAPTER 1: I NTRODUCTION
climate models, to explore expected changes (e.g. Russo and Sterl, 2011; Tebaldi et al., 2006)
as well as for attribution studies (e.g. Kiktev et al., 2003).
Based on studies such as Katz and Brown (1992), Klein-Tank and Können (2003), Simolo
et al. (2011) and Ballester et al. (2009a), there is evidence that the exceedance frequencies,
measured by some extreme indices, are related to changes in the parent distribution. Therefore,
the general question is how extreme temperature events are related to the mean temperature.
1.6 Are temperature extremes related to European mean
temperature?
There are physical and statistical arguments that the observed increase in global mean temperature might affect the frequency, intensity and duration of warm and cold extreme events (Katz
and Brown, 1992; Klein-Tank and Können, 2003; Della-Marta et al., 2007; Ballester et al.,
2009b,a; Simolo et al., 2010, 2011; de Vries et al., 2012) as well as record-breaking events
(Wergen and Krug, 2010; Rahmstorf and Coumou, 2011). This is schematically illustrated in
Figure 1.6a showing two temperature distributions, i.e. a reference distribution and a future
distribution shifted by one standard deviation. For the reference period, the initial exceedance
probability is 10% for the warm and cold extremes, respectively. The red area in Figure 1.6a
illustrates the expected increase in the exceedance frequency of warm extremes while the blue
area represents the decrease for cold extremes. Figure 1.6b shows additionally the exceedance
probability as a function of change in the mean ∆µ. For comparison, the evolution of the exceedance probability in Figure 1.6b is also shown for a PDF with variance two times as large
as for the PDF in Figure 1.6a (dashed lines).
(a)
(b)
Figure 1.6: (a) Reference and future temperature distribution shifted by one standard deviation. The
red and blue areas depict the trend-induced changes in the exceedance probabilities of warm and cold
extremes, respectively. (b) Changes in the exceedance probabilities as a function of change in the mean
∆µ. The blue solid and dashed lines represent cold extremes while the red ones correspond to warm
extremes. The variance for the distribution corresponding to the dashed lines is twice as large as for the
distribution shown in (a).
1.6 A RE
TEMPERATURE EXTREMES RELATED TO
E UROPEAN
MEAN TEMPERATURE ?
15
As mentioned before, the trend in the mean temperature is approximately depicted as a shift of
the whole temperature distribution towards higher temperatures (Frich et al., 2002; Klein-Tank
and Können, 2003; Scherrer et al., 2005; Simolo et al., 2010; Hansen et al., 2010; Simolo et al.,
2011). Figure 1.6 illustrates that the shift induces changes in the exceedance probabilities. The
expected change in exceedance probability is larger for warm extremes than for cold extremes
as depicted in Figure 1.6b. The rate of change for the frequency of cold and warm extremes is
determined by the shape of the distribution, shift direction and velocity as well as the variance.
Figure 1.6b demonstrates also that for a particular trend in the mean, the resulting changes
in the exceedance probabilities of warm and cold extremes are higher when the temperature
distribution has a small variance. In other words, the rate of change decreases with increasing
variance.
The link between the occurrence probability and changes in the mean are already investigated in the 1980s in studies such as, for instance, Mearns et al. (1984) and Wigley (1988). A
few years later the study of Katz and Brown (1992) investigated the sensitivity of exceedance
probabilities to changes in the mean and variance. The study shows that the sensitivity to
changes in the variance is higher than for the mean. This is crucial because studies show that
each part of a temperature distribution warms differently leading at least to changes in the temperature variability (Klein-Tank and Können, 2003; Yiou et al., 2009). This is evident when
trend magnitudes in the warm and cold extremes are compared with each other (Klein-Tank and
Können, 2003; Alexander et al., 2006; Moberg et al., 2006; Russo and Sterl, 2011; de Vries
et al., 2012). Consequently, there is a general tendency of decreasing winter temperature variability and increasing summer temperature variability caused by higher trend magnitudes in the
cold extremes for winter and, respectively, in the warm extremes for summer (Fischer et al.,
2012b).
The adjustments of the location (µ) and scale (σ) parameters of the reference temperature
distribution in order to fit the future distribution is not uncontroversial (Katz, 2010). It is not
clear if the shift of the whole temperature distribution is consistent with the changes in the
physical processes. Nevertheless, several studies show that there is a link between changes in
the mean temperature and frequency. Therefore, the main objective of this thesis is to investigate the relationship between mean temperature and frequency of temperature extremes in
regional climate models representing the recent state-of-the-art of physical understanding and
regional climate modelling. Absolute and percentile threshold based extreme indices are used
as robust measures for the frequency of moderate cold and warm extreme events. But to focus
only on the frequency of occurrence ignores any changes in the severity. Therefore, a further
objective is to assess the relationship between the mean climate and the return levels of warm
extremes. The relationships are statistically modelled and the applicability of these models to
project the frequency and intensity of temperature extremes will be tested. The findings are
interesting for the impact community because they can use these simple statistical models to
complete datasets with gaps in the time series.
Chapter 2
The potential of pattern scaling for
projecting temperature-related extreme
indices
17
19
The potential of pattern scaling for projecting
temperature-related extreme indices
Andreas Lustenberger, Reto Knutti and Erich M. Fischer
Institute for Atmospheric and Climate Science, ETH Zurich, Switzerland
(Published in International Journal of Climatology, 2013, doi:10.1002/joc.3659)
Abstract
Pattern scaling can be used to linearly relate changes in extreme indices to changes in the annual or seasonal mean temperature. This study demonstrates the skills and limitations of two
often used pattern scaling approaches in filling-in gaps in the time series of six temperaturerelated extreme indices. The extreme indices over Europe are derived from daily temperature
output of 12 regional climate models of the multi-model project ENSEMBLES. The response
pattern is estimated using one of the two future time periods (2021-2050 or 2070-2099) and the
reference period (1961-1990). The simulated values from the remaining future time period are
used for evaluating the skills. Both pattern scaling approaches perform reasonably well particularly for percentile-based and over most of the regions also for fixed temperature indices.
Uncertainties due to internal variability can be large if the time period used for estimating the
response pattern is close to the reference period. Limitations of pattern scaling due to violations
of the linearity assumption are related to the shape of the temperature distribution. As a result,
differences in the skills among the extreme indices can be related to the magnitude and shift
direction of the whole temperature distribution. Therefore, skills for estimated extreme indices
derived from the upper tail of the underlying temperature distribution are generally high. Over
some areas, linear regression models used in this study are not appropriate statistical models
because of the bounded and discrete nature of the data. Alternative pattern scaling methods
such as, for instance, the logistic regression model lead to improvements over particular areas
but not over the whole integration area.
20
C HAPTER 2: L INEAR
SCALING OF TEMPERATURE EXTREMES
2.1 Introduction
Changes in the frequency or intensity of extreme events as a result of a changing climate would
have major impacts on society and the environment. In this context, not only climate scientists but also different impact communities and end-users such as, for instance, the reinsurance
companies are highly interested in trends in the frequency, intensity and duration of extreme
events and the understanding of the driving mechanisms. From a statistical point of view, their
long return periods are a major challenge in the assessment of extreme events. Consequently,
there are in general not enough observed or simulated extreme events to reliably estimate their
statistical properties. Additionally, climate models still have difficulty simulating the typically
highly non-linear actions across scales and small-scale nature of extreme events. To avoid
some of these problems we here use a set of extreme indices with return periods in the order of
weeks. Such moderate definitions of extremes include events that do not correspond to the extremeness of events that are generally perceived as weather extremes often associated with high
impacts. However, the moderate definitions used here and thus the short return periods ensure a
sufficiently large sample size for robust trend analysis. According to the extreme value theory,
these events are statistically not extreme because they do not follow the extreme value distributions. Extreme indices are used for assessing and monitoring changes in extremes on global
scale (Alexander et al., 2006) or in regional trend studies (Moberg et al., 2006; Toreti and Desiato, 2008). We here use the definitions from the European Climate Assessment and Dataset
(ECA&D) initiative (http://eca.knmi.nl/indicesextremes/index.php).
Simulations performed with regional (RCM) or global climate models (GCM) are an important tool in climate science to assess and understand changes in the climate system and their
driving processes. However, the associated costs limit the availability of simulated data for
some emission scenarios or time periods. This is not only an issue for climate scientists but
also for the impact community using these simulations as input for their impact models. A
widely-used approach to fill in these gaps is pattern scaling, a method that provides climate
change scenarios for time periods or emission scenarios for which no simulations are available.
The underlying idea is that changes in a regional variable are linearly related to large-scale
or global mean temperature changes. A forcing and time-independent response pattern determines the variable of interest from the selected set of predictors. A fast and easy way to
estimate the response pattern is the time-slice method (Mitchell, 2003). The response pattern
is then given by the differences in the estimated means of the variable of interest scaled by
the mean temperature change between two distinct time periods or emission scenarios. However, any deviation from the underlying Gaussian assumption or, more general, from symmetry
concerning the underlying probability density functions (pdf) leads to biases in the response
pattern. A more robust method is to fit a linear regression using the least-squares method because this approach takes all available data points into account, minimizing the mean-squared
error (Mitchell, 2003). Mitchell and Hulme (1999) and Huntingford and Cox (2000) applied
the least-squares method to a sequence of decadal means. Ruosteenoja et al. (2007) used the
least-squares approach in order to fit several simulations of four different emission scenarios
for a specific GCM on the global mean temperature change simulated by a simple climate
model. Kennett and Buonomo (2006) scaled the time-varying mean and standard deviation of
the regional climate model with the smoothed global mean surface air temperature from the
2.2 DATA
21
driving GCM. In the framework of the ENSEMBLES project, Kendon et al. (2010) described
how pattern scaling can be used to increase the number of GCM-RCM chains.
The main idea of regression is to estimate the conditional mean of a variable of interest.
This implies that a trend-induced change in the location parameter of the dependent variable
has to be related to the trend-induced change in the mean of the explanatory variable (Simolo
et al., 2010). But especially extremes and extreme indices are further sensitive to changes in
the scale or shape parameter of the underlying probability density function (Katz and Brown,
1992). In this context, the objective of this study is to evaluate the applicability of the timeslice and least-squares method to a set of temperature-related extreme indices using the IPCC
SRES A1B emission scenario runs from a subset of GCM-RCM pairs in the framework of the
ENSEMBLES project (Van Der Linden and Mitchell, 2009). The pattern scaling methods are
used in this study to fill in gaps in the time series of extreme indices. Since there is mainly
one emission scenario available in the ENSEMBLES project, the skills of the scaling methods
are therefore tested using different time periods rather than different emission scenarios. The
examination of the underlying assumptions such as, for instance, the linearity assumption is
crucial.
The structure of this study is as follows: Section 2 introduces the regional climate models
and the set of temperature-related extreme indices. In section 3, the least-squares and time-slice
method and their associated assumptions are discussed. The results are presented in section 4
and discussed in section 5. The resulting conclusions are the topic of section 6.
2.2 Data
2.2.1 Extreme Indices
In this study, six temperature-related extreme indices are used, each of them derived from daily
2-m temperature data. The extreme indices are defined as follows:
• Number of frost days (FD): If TN is the daily minimum temperature, FD is the number
of days per year when TN < 0◦ C.
• Number of summer days (SU): SU counts the number of days per year when the daily
maximum temperature TX > 25◦ C.
• Percentage of warm days (TX90P) and cold days (TX10P) per year: For each calendar
day of the reference period 1961-1990, the 90th and 10th percentile of the daily maximum
temperature is computed using a five day time window centred at the calendar day.
• Percentage of warm nights (TN90P) and cold nights (TN10P) per year: Same as above
but for daily minimum temperature.
• Cold-spell days index (CWFI): If TG is the daily mean temperature and TG10P the calendar day 10th percentile of a five day window centred at the calendar day, CWFI counts
the number of days per year when TG < TG10P for at least 6 consecutive days.
22
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• Warm-spell days index (HWFI): If TG is the daily mean temperature and TG90P the
calendar 90th percentile of a five day window centred at the calendar day, HWFI counts
the number of days per year when TG > TG90P for at least 6 consecutive days.
All six extreme indices are computed with the Climate Data Operators (CDO). CDO is
an open source tool developed by the Max-Planck-Institute (https://code.zmaw.de/
projects/cdo/) and provides a set of operators for analyzing climate model outputs. The
extreme indices implemented in CDO are European Climate Assessment & Dataset (ECA&D)
indices. However, six of the extreme indices (FD, SU, TX10P, TX90P, TN10P, TN90P) used
in this study follow the recommended definitions by the Expert Team on Climate Change Detection and Indices (ETCCDI). HWFI and CWFI have definitions which are very similar to the
warm-spell duration (WSDI) and cold-spell duration (CSDI) index suggested by ETCCDI. The
main difference is that the daily mean temperature is used to compute CWFI and HWFI. In
contrast to this, WSDI and CSDI employ daily maximum and minimum temperature, respectively.
The suggested extreme indices are day-count indices based on fixed (FD, SU) or percentile
(TX90P, TX10P, TN90P, TN10P, HWFI, CWFI) thresholds. The fixed extreme indices make a
spatial comparison difficult because they do not sample the same part of the underlying probability distribution of temperature at different sites. The motivation of using extreme indices
based on fixed thresholds is that they are often associated with observed impacts. For spatial
comparison the percentile-based extreme indices are more suitable because they have a constant rarity in context of the local climate.
SU and FD do not take the annual cycle into account in their definitions. This means that
in regions with a pronounced annual cycle one can expect that the seasons affect the internal
variability of these two indices.
2.2.2 ENSEMBLES regional climate models
The daily maximum, minimum and mean 2-m temperatures are simulated by different one-way
nested GCM-RCM chains. The GCM-RCM pairs are defined in the framework of the ENSEMBLES project (Van Der Linden and Mitchell, 2009). Due to limited computational resources,
the ENSEMBLES community decided to use mainly the SRES A1B emission scenario. In this
study, a set of 12 GCM-RCM chains is used with a common grid and transient runs covering
the time period from 1951 to 2099. The horizontal resolution of the RCMs is about 25 km.
Only grid points exhibiting a land area fraction of more than 50% are considered in this study.
Table 2.1 gives an overview on the 12 GCM-RCM pairs, i.e. the institutions, the name of the
RCM and the name of the driving GCM.
2.3 Methods
2.3.1 Estimation of the response pattern
There are two ways to estimate the response pattern, the time-slice approach (Mitchell, 2003)
and the least-squares approach (Mitchell, 2003; Huntingford and Cox, 2000). The annual mean
2.3 M ETHODS
23
Institution
RCM model
Driving GCM
Irish National Meteorological
Service (C4I)
RCA3
HadCM3Q16
Danish Meteorological Institute (DMI)
DMI-HIRHAM
ECHAM5
Danish Meteorological Institute (DMI)
DMI-HIRHAM
ARPEGE
Swiss Federal Institute of
Technology (ETH)
CCLM
HadCM3Q0
Royal Netherlands Meteorological
Institute (KNMI)
RACMO
ECHAM5
Met Office Hadley Centre (HC)
HadRM3Q0
HadCM3Q0
Met Office Hadley Centre (HC)
HadRM3Q3
HadCM3Q3
(low sensitivity)
Met Office Hadley Centre (HC)
HadRM3Q16
HadCM3Q16
(high sensitivity)
Swedish Meteorological and
Hydrological Institute (SMHI)
RCA
ECHAM5
Swedish Meteorological and
Hydrological Institute (SMHI)
RCA
BCM
Swedish Meteorological and
Hydrological Institute (SMHI)
RCA
HadCM3Q3
Max-Planck-Institute for
Meteorology (MPI)
REMO
ECHAM5
Table 2.1: The ENSEMBLES regional climate models (RCM) with the same rotated pole grid and available simulations for the time period from 1951 to 2099.
24
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of the daily mean, daily maximum and daily minimum 2-m temperatures are used as explanatory variables. Additionally, in case of the FD and SU index, the seasonal mean summer (JJA)
and mean winter (DJF) temperatures are used. In cases where the seasonal mean temperature
is employed, the extreme indices are computed per season. The anomalies for each variable are
computed by subtracting the mean of the reference period (1961-1990) from the mid-century
(2021-2050) and late-century (2070-2099) period mean. For the least-squares approach, a linear regression model is used with an unknown regression coefficient β1 and the intercept β0
which is assumed to be zero. The errors ǫi have to satisfy the Gauss-Markov conditions (Plackett, 1950). The basic idea of the least-squares method is to minimize the residual sum of squares
with respect to the regression coefficient β1 . Therefore, β1 can be estimated in the following
way:
Pn
(xi − x) · (yi − y)
β̂1 = i=1Pn
2
i=1 (xi − x)
The subscript i denotes all years of the reference and the scenario period. For each grid point
the significance of the assumed linear trend is tested with a simple t-test of the coefficient β1
with the significance level α = 5%. If the Gauss-Markov conditions are not violated then T
is distributed like Student’s t with n-2 degrees of freedom. The advantage of the least-squares
approach relative to the time-slice approach is that all available data are used. Additionally,
due to minimizing a quantity measuring the errors we can be sure that, if all assumptions are
fulfilled, this approach fits the data better than the time-slice method.
Another possibility to estimate the slope coefficient β1 is the time-slice method. This approach estimates β1 in the following way:
β̂1 =
y Scenario − y Ref erence
xScenario − xRef erence
β1 is only estimated when there is a significant trend-induced change of the mean at the 5%
significance level. In this context, a simple t-test is applied with the null hypothesis H0 that the
means are equal.
The scaling factor β1 is estimated for each grid point and for each of the 12 GCM-RCM
pairs using both pattern scaling approaches. In addition, β1 is only estimated for a particular
grid point when the associated significance test rejects the null hypothesis.
2.3.2 Evaluation of the estimation skills
The response pattern is derived using one of the two future time periods and the reference period. The estimated response pattern is used to (1) interpolate the extreme indices for the midcentury period 2021-2050 based on information from the control period and the late-century
2070-2099, or to (2) extrapolate them for the late-century 2070-2099 based on information
from the control period and the mid-century, respectively. The simulated values from the remaining time period are used for evaluating the skill of the extreme index estimates. The
motivation of this study is to stay on the level of the RCMs, i.e. we assume that we have
only information from the 12 GCM-RCM chains presented in subsection 2.2. An alternative
procedure would be to use the global mean temperature change as predictor or regional information derived from statistical downscaling. However, because the information of the RCM
2.4 R ESULTS
25
are more closely related to regional climate simulated by the GCM-RCM chains, the choice of
RCM-based information delivers better insight into the limitation of pattern scaling applied to
extreme indices.
As a measure of skill, two different statistical quantities are used here. First, the root mean
squared error (RMSE). It is important to keep in mind that the RMSE is sensitive to outliers.
Another statistical quantity is the Pearson’s correlation coefficient ρ. The correlation coefficient measures the degree of correspondence between the estimated and simulated values. The
squared value ρ2 expresses the fraction of the variance in the simulated values explained by
the estimated linear regression. However, ρ2 does not take into account any bias. Therefore,
a high value of ρ2 can still have a large systematic error. For this purpose, the RMSE will be
used additionally in order to quantify potential biases. Similar as the RMSE, ρ2 is sensitive to
outliers.
2.4 Results
Figure 2.1 shows in the top panel the mean TX90P values simulated with the HadCM3Q16HadRM3Q16 model and in the bottom panel the values fitted with the least-squares approach
for the time slice 2070-2099 using the yearly mean daily maximum temperature as explanatory
variable.
Over central and northern Europe, there is an underestimation of the fitted mean TX90P
values. In contrast to this, the TX90P values over southern Europe are in relatively good agreement with the simulated ones. The figure representing the fitted values implies some spatial
dependency of the skills which will be discussed later. Figure 2.2 shows the ρ2 and RMSE
values for the estimated TX90P ((a), (b)) and TX10P ((c), (d)) indices over the integration area
of each GCM-RCM pair. Changes in the yearly mean daily maximum temperature are used as
explanatory variable. The range in ρ2 and RMSE across grid points is depicted with the empirical 25th and 75th percentile. Figure 2.2a and 2.2c show the skill values for the time period
2021-2050 (interpolation) whereas Figure 2.2b and 2.2d those for 2070-2099 (extrapolation).
For both extreme indices, the least-squares approach performs better than the time-slice
approach. The multi-model median RMSE values in case of the least-squares approach are
for both time periods of interest about 23%-57% lower than for the time-slice approach. In
addition, the interpolation for the time period 2021-2050 yields substantially better results than
the extrapolation for 2070-2099. The multi-model interquartile range, as a measure for the
inter-model differences, is up to 50% lower for the least-squares approach and the time period
2021-2050 relative to the time-slice approach. When estimating the extreme indices for the
time period 2070-2099, the inter-model differences in the RMSE values become larger. Especially for the TX10P index, the RCMs driven with the high climate sensitivity GCM can be
characterized as outliers in terms of the RMSE values.
Concerning the ρ2 values, Figure 2.2 shows that for both pattern scaling approaches the
correlation between the simulated and fitted TX90P values are between 1.5 to three times
higher than for the TX10P index. Additionally, the multi-model interquartile range is in almost all cases smaller for the TX90P index. This behaviour implies that, given increasing
mean temperatures, estimating extreme indices associated with the upper tail of the underly-
26
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(a)
60° N
45° N
30° N
30° W
15° W
0°
15° E
30° E
45° E
30° W
15° W
0°
15° E
30° E
45° E
(b)
60° N
45° N
30° N
30
40
50
% / year
60
70
80
Figure 2.1: Mean TX90P for the time slice 2070-2099 (a) simulated with the HadCM3Q16HadRM3Q16 model and (b) fitted with the least-squares approach using the annual mean daily maximum temperature as predictor.
ing temperature distribution (e.g. TX90P) seems to work generally better than for those in the
lower tail. This behaviour is due to the fact that the TX10P index approaches 0% per year
as a result of the warming. Therefore, the relationship between the extreme index and mean
temperature tends to flatten and thus violate the assumption of linearity. The same behaviour
is observed for the TN10P and TN90P index (not shown) derived from the daily minimum
temperature distribution. However, there are very small differences in the skill values between
the TN10P and TX10P extreme index and the TN90P and TX90P index, respectively. In case
of the fixed threshold based extreme indices (FD and SU; not shown), the multi-model median
RMSE values are still lower for the least-squares approach than for the time-slice approach
for the interpolation and extrapolation. Seasonal mean temperature as predictor for FD and
SU leads to multi-model median RMSE values up to 50% smaller than for the corresponding
annual based values. For both pattern scaling approaches, this decrease is for the interpolated
estimations larger than for extrapolated values.
2.4 R ESULTS
27
(a) TX90P: 2021-2050
(b) TX90P: 2070-2099
10
RMSE (% per year)
RMSE (% per year)
10
8
6
4
2
0
0
0.2
0.4
2
ρ
0.6
0.8
4
2
0.2
0.4
ρ2
0.6
0.8
1
(d) TX10P: 2070-2099
10
10
RMSE (% per year)
RMSE (% per year)
6
0
0
1
(c) TX10P: 2021-2050
8
8
6
4
2
C4I−RCA3
DMI−HIRHAM−ARPEGE
8
DMI−HIRHAM−ECHAM5
ETH−CLM
KNMI−RACMO
6
HC−HadRM3Q0
HC−HadRM3Q3
4
HC−HadRM3Q16
MPI−REMO
SMHI−RCA−BCM
2
SMHI−RCA−ECHAM5
SMHI−RCA−HadCM3Q3
0
0
0.2
0.4
ρ2
0.6
0.8
1
0
0
0.2
0.4
2
ρ
0.6
0.8
1
Figure 2.2: Skill of the pattern scaling approach for all 12 GCM-RCM chains expressed as squared
correlation ρ2 and RMSE values. Skills are shown for TX90P in (a) 2021-2050, and (b) 2070-2090, and
for TX10P in (c) 2021-2050, and (d) 2070-2099. The indices are estimated with the time-slice (dashed
line) and least-squares approach (solid line). The annual mean daily maximum temperature is used as
explanatory variable. The range of values over the integration area is represented with the empirical
25th and 75th percentile.
For the multi-model ρ2 values, an improvement up to 28% is evident mainly for the leastsquares approach applied on FD. In general, better skills are found over northern Europe for
FD and over the Mediterranean in case of SU. This is due to the fact that over northern Europe
trends in annual mean temperatures are dominated by trends in the seasonal DJF mean temperature while over southern Europe trends in the JJA mean temperature have a larger influence.
For yearly FD, the ρ2 values (multi-model median and interquartile range) are comparable
with the values of the TX10P and TN10P values. In case of SU, the multi-model median values of this particular skill measurement are lower and the multi-model interquartile range is in
almost all cases higher than for the values of the TX90P and TN90P index. In addition, the
higher interquartile range values for each GCM-RCM pair in case of FD and SU imply a higher
spatial dependency of the ρ2 values relative to the quantile-based extreme indices.
The lowest correlation values (not shown) are found between CWFI and the annual mean
temperature. The median value over the integration area of each GCM-RCM chain is between
-0.5 and -0.56 with a maximum interquartile range of 0.11. The small inter-model differences
give a hint that for this particular extreme index the linearity assumption is violated. In contrast
to the CWFI, the skill values for the HWFI indicate larger systematic errors in the estimations
but better correspondence with multi-model median ρ2 values in the order of 0.35 to 0.75 (not
shown).
To investigate the uncertainty induced by internal variability on the estimations of the slope
28
C HAPTER 2: L INEAR
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Estimated Slope (%/K)
parameter β1 , a 30-year moving window is used together with the fixed reference period 19611990. The moving time window is shifted by one year at each step and moves through the time
period from 1991 to 2099. Figure 2.3 shows the estimated β1 values for the TX90P index based
on the time-slice method (blue line) and the least-squares method (red line) for a single grid
point over northern Scandinavia from the ECHAM5-RCA model. The TX90P index is chosen
because this extreme index does not exhibit any significant violation of the linearity assumption
over the whole integration area. The green lines represent the range of estimated β1 values for
that particular grid point for all 12 GCM-RCM chains.
12
11
10
9
8
7
6
5
4
3
2001−2030 2021−2050 2041−2070 2061−2090
Time Window
Figure 2.3: Grid point from the ECHAM5-RCA model with β1 slopes (TX90P per Kelvin warming of
annual mean daily maximum temperature) estimated with the least-squares (red) and time-slice (blue)
approach using the reference period (1961-1990) and a 30-year moving time window. The horizontal
axis shows the time period covered by the moving 30-year time window. The green lines represent the
range of estimated values for all 12 RCMs.
The blue and red curves indicate that the uncertainty in the slope parameter estimates is largest
when the moving window is close to the reference period. This implies that the warming
signal is partly masked by the internal variability (noise) for the coming decades,. However,
the signal-to-noise ratio increases with time and, therefore, the estimated slope parameter converges to approximately 9 %/K. In addition, the uncertainty in the estimates decreases when the
moving window is shifted towards the end of the 21st century, i.e. both curves tend to flatten.
For the least-squares approach, for instance, the uncertainty decreases by approximately 75%
when the moving time window reaches the end of the 21st century. However, the uncertainty
induced by internal variability and the differences in the estimated slope parameters between
both pattern scaling approaches are at any time much smaller than the inter-model differences
depicted as green lines. Even the largest uncertainty of the estimated β1 due to internal variability, found when the time window is close to the reference period, is smaller than 10% of
the model range. However, the magnitude of decrease in the uncertainty of the estimated β1
exhibits some spatial dependency. The largest decrease in the uncertainty occurs over Scandinavia for three quarters of the GCM-RCM chains in case of the least-squares method and
for all RCMs for the time-slice method. For both pattern scaling methods and all GCM-RCM
2.5 D ISCUSSION
29
chains, significant decreases in the uncertainty induced by internal variability take place over
the Mediterranean.
The aforementioned results are mainly observed for warm extreme indices associated with
temperature thresholds in the upper tail of the underlying temperature distribution. For extreme
indices with temperature thresholds in the lower tail of the temperature distribution (cold extreme indices) other issues, such as nonlinearities and the convergence of the extreme index
values towards zero, have a greater influence on the estimations.
2.5 Discussion
In the last section there is evidence for a violation of the underlying linearity assumption for
some of the investigated extreme indices and especially in case of the CWFI. The degree of
linearity and the trend magnitude of the extreme index are related to trends in the explanatory
variable and the probability of exceeding a particular temperature threshold. Figure 2.4 illustrates this link in case of the TX10P and TX90P index over the Mediterranean simulated from
1951 to 2099 with the HadCM3Q3-HadRM3Q3.
The trend in the yearly mean maximum temperature for the time period from 1951 to 2099,
shown in Figure 2.4a, can be characterized by a second order polynomial regression model
(blue solid line). The fitted curve shows that under the assumption of constant central moments
higher than 1st order the trend implies a non-uniform shift of the whole temperature distribution
towards higher temperatures. The shift itself induces changes in the probabilities of exceeding
a particular temperature threshold and, therefore, in the extreme index. The magnitude and
direction of this induced trend depends on the shift direction and the shape of the temperature
distribution. Figure 2.4c shows how the probability of exceeding the 90th percentile changes
as a function of changing location parameter ∆µ in case of a standard normal distribution with
three different values for the scale parameter σ. Figure 2.4c implies that a shift towards higher
temperatures induces first a nonlinear increase in the percentage of days per year exceeding
the 90th percentile of the reference period but then the trend becomes nearly linear after some
point in time. This is exactly what is seen in Figure 2.4b showing the time series of the TX90P
and TX10P index. In addition, because the induced nonlinear trend in the time series of the
TX90P index resembles the nonlinear trend in the time series of the yearly mean temperature
and both time series are in phase, the resulting trend between the extreme index and the explanatory variable is nearly linear. But Figure 2.4c illustrates that the linear trend magnitude
can be very different for temperature distributions with the same constant shape but different
scale parameter. Further, the figure implies that if changes in the scale parameter occur during
the shift of the whole temperature distribution the resulting trend becomes nonlinear.
In contrast to this, changes in the exceedance probabilities at the lower tail of the temperature distribution, illustrated with the TX10P index, exhibit a decrease in the exceedance
probability. However, the magnitude of the decreasing trend is much smaller than for the increasing trend in the TX90P index. This explains why the skill values for warm extreme indices
(temperature threshold in the upper tail of the temperature distribution) are often much better
than for cold extreme indices (thresholds in the lower tail). Furthermore, this explains why
the estimated response patterns for the warm indices (i.e. TX90P, TN90P, HWFI and SU) are
30
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Annual mean daily maximum temperature
(a)
296
Temperature (K)
295
294
293
292
291
290
289
1951
2000
(b)
2050
2099
2050
2099
Time (Year)
Percentage of days per year
50
TX90P
TX90P fit
TX10P
TX10P fit
40
30
20
10
1951
2000
(c)
Time (Year)
1
σ/σ
Exceedance Probability
Reference
= 1.5
σ/σReference = 1.0
0.8
σ/σReference = 0.5
0.6
0.4
0.2
0
−3
−2
−1
0
1
∆µ (σ)
2
3
4
5
Figure 2.4: a) Time series of the annual mean daily maximum temperature (red circles) simulated
over the Mediterranean from 1951 to 2099 with the HadCM3Q3-HadRM3Q3 under the SRES A1B
emission scenario. The blue solid line represents the second-order polynomial fit. b) Time series of the
TX90P index (red dots) and TX10P (green dots) and the second-order polynomial fits (blue line). c)
Probabilities of exceeding the 90th percentile of a standard normal distribution as a function of changes
in the location parameter for three different values of the scale parameter.
2.5 D ISCUSSION
31
less affected by the internal variability. As a consequence of this trend magnitude and linearity
dependence on the shift direction of the whole temperature distribution, interpolated estimates
for extreme indices (i.e. 2021-2050 based on 1961-1990 and 2070-2099) have higher skill than
the extrapolated values. This implies that under the assumption of linearity there is a dependency of the estimated response pattern on the chosen time periods. The moving time window
applied on the TX90P index shows that if the second time slice is placed in the second half of
the 21st century the uncertainty of the estimated β1 decreases and the estimation converges to
a nearly constant value. This effect is especially pronounced over northern Europe, northeast
Europe and over the Mediterranean for the TX90P and TN90P index. This corresponds to the
regions where the largest warming in the annual mean surface air temperature is expected at
the end of the 21st century (Van Der Linden and Mitchell, 2009). For the TN10P index, TX10P
index and CWFI, the violation of the linearity assumption is the more dominant issue.
A further issue is that the extreme indices used in this study are so called count data, i.e.
they count the number of days or events per time period. Therefore, the range of possible values for the predictands is bounded. In addition, the data is discrete, i.e. for a specific range
of yearly mean temperature it is possible to get the same number of days or events per time
period. In such cases, simple linear regression models are not appropriate statistical models
for dealing with this kind of data. The bounded nature of the data results from the typical
property of a cumulative distribution function being bounded between zero and one, as evident
in Figure 2.4c. The question is whether it is possible to modify the pattern scaling method
taking the aforementioned issues into account. There are two potential solutions. First, there
is an ad-hoc approach setting the estimated values to the boundary value when they reach the
limit of reasonable values (i.e. no days or all days exceed the temperature threshold). Another
possibility is to use a logistic regression model instead of the simple linear regression model
(not shown). In general, there is no best alternative regression model or modification of the
established simple linear regression model leading to an overall improvement over the whole
integration area. The improvement depends on the most dominant issue in each of the simulated trends, i.e. if the nonlinearity or the bounded and/or discrete nature of the data is the
main issue. How pronounced these issues are depends on the extreme index and the simulated
trends. The magnitude of the trends determines how fast an extreme index reaches a particular
boundary value. This is especially pronounced in case of CWFI and HWFI because they have
a higher sensitivity on trends in the underlying temperature distribution. As a result, the trend
magnitude for both is much higher than for the other extreme indices. All these dependencies
explain why the improvements vary between different regions.
For FD and SU, the results suggest that an improvement of the results is possible using seasonal means instead of annual means. This is not surprising because these two indices depend
on how pronounced the annual cycle is. The trends in the annual mean temperature over northern Europe are dominated by trends in the DJF mean temperature and over southern Europe by
trends in the JJA mean temperature. For this purpose, the largest increase in the skills for FD
when using seasonal means are over northern Europe and for SU over the Mediterranean.
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2.6 Conclusions
This study demonstrates the potential and limitations of pattern scaling approaches to fill-in
gaps in projections of temperature-related extreme indices based on local mean temperature
changes. For all extreme indices considered the least-squares approach performs better in interpolating and extrapolating changes in extreme indices from local mean temperature changes
than the time-slice approach. The skill for both methods is mainly limited by the degree of linearity of the underlying relationship between mean temperature and index. The linearity of this
relationship is determined by the shape of the underlying temperature distribution. In addition,
changes in the scale parameter during the shift of the temperature distribution can affect the
linearity. However, the shape of the temperature distribution is the main reason why in case of
increasing temperatures violations of the linearity assumption occur mainly for cold extremes
such as, for instance, the TN10P index or CWFI. As a result, the interpolated and extrapolated
estimations have lower skills for cold extreme indices (i.e. TX10P, TN10P, CWFI and FD) than
for warm extreme indices (i.e. TX90P, TN90P, HWFI and SU). Given an underlying warming
trend, these higher skills are observed for both pattern scaling approaches.
Trends in CWFI are highly sensitive to annual mean temperatures due to the duration minimum of 6 days. Therefore, this extreme index converges faster towards zero than, for instance, the TN10P index. The same problem will occur with ongoing warming for extreme
indices characterizing the upper tail of the temperature distribution (e.g. TX90P, HWFI). For
the HadCM3Q16-RCA3 model which projects a strong warming, this violation is already evident for the TX90P index over parts of southern Europe at the end of the 21st century.
The results of this study further illustrate that for both pattern scaling methods the skills
for the interpolation are better than for the extrapolation. This is to a large extent due to the
internal variability that obscures the long-term trend in the near-future period used to derive
the extrapolation. Testing the uncertainty induced by internal variability on estimations of β1
reveals that estimates of the slope parameter β1 are more robust for the second half of the 21st
century than for time periods close to the reference period. Setting the second time slice at the
end of the 21st century, reductions in the uncertainty of β1 up to 75% are observed especially
over Scandinavia and the Mediterranean for both pattern scaling approaches. Nevertheless, it
is important to note that the model uncertainty exceeds that from internal variability by far.
The results show that simulated trends in the time series of extreme indices can be related
to the direction and magnitude of the change in mean temperatures. Based on a particular
GCM-RCM pair we further demonstrate that the skill of the methods is dependent on regions
and seasons considered. Over northern Europe, for instance, the largest projected changes in
seasonal mean surface air temperature in case of the SRES A1B scenario occur for the season
December-February (Van Der Linden and Mitchell, 2009). On the other hand, for the season
June-August the largest projected changes are expected over southern Europe. This implies
that annual trends over some regions are dominated by particular seasonal trends. As a consequence of this, over such regions it is possible to improve the skills for the estimated FD and
SU values when seasonal means are used instead of annual means. Applying, for instance, the
least-squares approach on FD using seasonal DJF mean temperature changes lead to RMSE
values up to 50% smaller than for annual values.
Another important limitation of the pattern scaling approaches is that count data is bounded
2.6 C ONCLUSIONS
33
and/or discrete leading to non-physical or unreasonable estimations such as, for instance, negative number of days per year. All these issues lead to the conclusion that the traditional pattern
scaling approaches are only useful when the user knows where over the area of interest the
aforementioned issues can occur and how pronounced they are. Alternative pattern scaling
techniques such as, for instance, the logistic regression model or ad-hoc procedures discarding
unreasonable values can improve the estimations. But there is no overall solution possible due
to the spatial dependency of the discussed issues and the dependency on the GCM-RCM pair.
Chapter 3
Sensitivity of European extreme daily
temperature return levels to projected
trends in mean and variance
35
37
Sensitivity of European extreme daily temperature return levels
to projected trends in mean and variance
Andreas Lustenberger, Reto Knutti and Erich M. Fischer
Institute for Atmospheric and Climate Science, ETH Zurich, Switzerland
(Submitted to Journal of Geophysical Research)
Abstract
Both the mean and variance in European daily temperature are projected to change in the future.
Such a change affects also the return period of extreme events. The aim of this study is to investigate the relationship between changes in statistical moments of the daily temperature distribution and return levels of warm extremes. For this purpose, the distributions of daily maximum
temperature during summer and daily minimum temperature during winter for the control period 1961-1990 are scaled and shifted by the expected changes in the mean and variance. The
estimated return levels of warm summer day and winter night extremes are then compared with
the projections for the scenario period 2070-2099, actually simulated by regional climate models. This study shows that changes in return levels of warm summer and winter extremes can be
approximated as a simple shift of the full temperature distribution for one quarter to one half of
Europe across all examined climate models. In other areas, changes in the variance have to be
considered since models project reduced temperature variance in winter and enhanced variance
in summer. When for a particular temperature distribution the lowermost und uppermost percentiles change at different rates, the variance changes asymmetrically. Therefore, scaling the
control temperature distribution symmetrically, i.e. with central moments, does not always lead
to an improvement of the estimates compared with the location-adjusted return level estimates.
Nevertheless, the study shows that if the variance change associated with the upper half of the
temperature distribution is known then the location-scale-adjusted return levels perform better
than the location-adjusted return levels over most areas.
38
C HAPTER 3: S ENSITIVITY
OF TEMPERATURE EXTREMES TO WARMING
3.1 Introduction
Probability density functions (PDFs) describe quantities of the climate system such as, for
instance, the surface temperature distribution over typically several decades. Extreme events
are often defined as events exceeding a certain percentile of the PDF (Alexander et al., 2006;
Frich et al., 2002; Klein-Tank and Können, 2003). An appropriate statistical description of
frequency, intensity and duration of extreme events in the past, present and future climate is
important to detect significant trends. Such trends are of public and scientific interest due to the
large socio-economic and ecological impacts of extremes (Fouillet et al., 2006; Garcı́a-Herrera
et al., 2010; Barriopedro et al., 2011; Coumou and Rahmstorf, 2012). The observed trends in
temperature extremes are often characterized as a first approximation as a simple shift of the
full temperature distribution towards higher temperatures (Simolo et al., 2011, 2010; Ballester
et al., 2009a; Kharin et al., 2007, 2013). This shift in the mean accounts for much but not all of
the changes in the frequency of temperature extremes, and existing work argues that changes in
higher order statistical moments need to be taken into account (e.g. Schär et al., 2004; Scherrer
et al., 2005; Clark et al., 2006; Vidale et al., 2007; Fischer and Schär, 2009, 2010) to understand
changes in the tails of the parent temperature distribution.
Mearns et al. (1984) discussed the nonlinear relationships between changes in the mean
temperature and the probability of high temperature events. The study by Wigley (1988) shows
that return periods of warm extremes exhibit a nonlinear decrease for increasing trends in the
mean. Katz and Brown (1992) extended this work and demonstrated that extremes are even
more sensitive to changes in the variance (scale parameter) than in the mean. Klein-Tank and
Können (2003) stated that observed trends in cold and warm temperature extremes over Europe
from 1946 to 1990 are not in contradiction with the assumption of a simple shift of the temperature distribution. They show on their study that an asymmetry between the trends in the
cold and warm extremes becomes evident when they applied the trend analysis to sub-periods.
These observed asymmetric trends suggest an additional change in the variance. In this context,
Ferro et al. (2005) investigated how changes in distinct parts of the distribution are related to
changes in the moments without any assumption about the distribution family.
Several studies discuss the link between changes in the frequency of exceeding a particular temperature threshold and the moments (e.g. Mearns et al., 1984; Wigley, 1988; Katz and
Brown, 1992; Frich et al., 2002; Klein-Tank and Können, 2003; Ferro et al., 2005; Moberg
and Jones, 2005; Alexander et al., 2006; Moberg et al., 2006; Beniston et al., 2007; Toreti and
Desiato, 2008; Ballester et al., 2009b,a; Simolo et al., 2010, 2011). However, these studies use
moderate definitions of extremes (exceedance of 90th or 95th percentile) that do not account for
any severity variations (Coelho et al., 2008; Katz, 2010).
Therefore, the aim of this project is to investigate the potential link between changes in the
return levels of warm extremes and changes in the mean and higher order central moments of
the underlying temperature distribution using regional climate model data from the European
multi-model experiment ENSEMBLES (Van Der Linden and Mitchell, 2009).
3.2 DATA
AND
M ETHODS
39
3.2 Data and Methods
3.2.1 ENSEMBLES regional climate models
The simulations are forced with the SRES A1B emission scenario. A set of 12 model chains
(see table 3.1), in which a global climate model (GCM) is driving a regional climate model
(RCM), is used. These GCM-RCM chains have a common grid and the transient runs cover the
time period from 1951 to 2099. The horizontal resolution of the GCM-RCM chains is about
25 km. Only grid points with a land area fraction of more than 50% are considered in this study.
Institution
RCM model
Driving GCM
Irish National Meteorological
Service (C4I)
RCA3
HadCM3Q16
Danish Meteorological Institute (DMI)
DMI-HIRHAM
ECHAM5
Danish Meteorological Institute (DMI)
DMI-HIRHAM
ARPEGE
Swiss Federal Institute of
Technology (ETH)
CCLM
HadCM3Q0
Royal Netherlands Meteorological
Institute (KNMI)
RACMO
ECHAM5
Met Office Hadley Centre (HC)
HadRM3Q0
HadCM3Q0
Met Office Hadley Centre (HC)
HadRM3Q3
HadCM3Q3
(low sensitivity)
Met Office Hadley Centre (HC)
HadRM3Q16
HadCM3Q16
(high sensitivity)
Swedish Meteorological and
Hydrological Institute (SMHI)
RCA
ECHAM5
Swedish Meteorological and
Hydrological Institute (SMHI)
RCA
BCM
Swedish Meteorological and
Hydrological Institute (SMHI)
RCA
HadCM3Q3
Max-Planck-Institute for
Meteorology (MPI)
REMO
ECHAM5
Table 3.1: The ENSEMBLES regional climate models (RCM) with the same rotated pole grid and available simulations for the time period from 1951 to 2099.
40
C HAPTER 3: S ENSITIVITY
OF TEMPERATURE EXTREMES TO WARMING
3.2.2 Return level estimation
Return levels of warm extremes in European climate are estimated from daily maximum temperature (TMAX) for the summer season June-July-August (JJA) and daily minimum temperature (TMIN) for the winter season December-January-February (DJF). The reason for exploring warm winter night instead of warm summer night extremes is that periods of warm summer
nights are correlated to high maximum temperatures in the daytime (Klein-Tank, 2010). Warm
winter nights due to warm winter spells have impacts on the avalanche risk, seasonal snow
cover, growing period and the income of the alpine ski resorts (Beniston, 2005). To get identically distributed samples, the data is linearly detrended at each grid point for the control
(1961-1990) and scenario (2070-2099) period. The removal of trends is also important for reducing the inflation of the estimated scale parameter (e.g. Scherrer et al., 2005) caused by the
presence of a trend. A further issue is that daily data is serially correlated due to the synoptic time scales of weather events (e.g. blockings or low pressure systems). This correlation
affects the estimated parameters of the extreme value distribution. To reduce this effect, we
declustered the samples by taking every 5th day from the detrended samples. The 5-day time
lag is selected because this value represents the mean persistence of synoptic systems, i.e. low
and high pressure systems, in the extratropics (Austin and Lentz, 1999). The 5-day time lag
reduces substantially the risk to sample several times from the same event and ensures a reasonable sample size. To investigate the sensitivity of the results to the declustering procedure,
several subsamples are generated beginning each time with another day.
An extremely warm temperature event E is defined as E = (Ti > u) with Ti as the temperature for the corresponding day i = 1, 2, · · · , n drawn from a sample of size n. The exceedances
yi = Ti − u for a high enough temperature threshold u have a two-parameter limiting distribution of the form
− ξ1
ξ·y
,
(3.1)
y ∼ GP D(y) = 1 − 1 +
σGP
where y > 0 and 1 + ξ · y/σGP > 0 (Coles, 2001). ξ is the shape parameter and σGP > 0 the
scale parameter of the Generalized Pareto distribution (GPD). If ξ < 0, then the distribution
has a bounded upper tail. Although the block maxima approach has been widely used (e.g.
Kharin and Zwiers, 2005; Nikulin et al., 2011) for modelling extreme events, the peak-overthreshold (POT) approach is preferable (e.g. Brown et al., 2008; Kyselý et al., 2008) because
more information from the sample is used. But to investigate the sensitivity of the approach in
this study, the analysis has been repeated with the block maxima approach (not shown). The
return level estimates are very similar to those derived from GPD and, therefore, the results for
the POT approach will be discussed only.
The main difficulty of the POT approach is to find an appropriate temperature threshold u.
The threshold should be as small as possible, so that the variance of the estimated parameter
is small but not too small in order to avoid any biases due to the sampling of events which are
statistically not extreme.
Often thresholds are only selected visually and are not chosen based on an objective criterion. This makes it difficult to apply these methods to large gridded data sets. The threshold
choice here is based on the Anderson-Darling statistic (Choulakian and Stephens, 2001) and the
dispersion index statistic (e.g. Kyselý et al., 2008). The null hypothesis H0 of the Anderson-
3.2 DATA
AND
M ETHODS
41
Darling statistic is that the sample yi = Ti − u comes from a GPD with unknown shape and
scale parameter. Similar to the approach of Davison and Smith (1990), the procedure starts
at the 70th percentile of the corresponding temperature distribution and the percentile is raised
successively until H0 is not rejected at the significance level α = 0.1. If the chosen percentile
threshold u is appropriate, the number of exceedances per time period follows a Poisson distribution. Thus, the dispersion index statistic DI = s2 N /mN should have a value of 1. mN and sN
are the estimated mean and standard deviation of the number of exceedances per time period
over N years. The confidence intervals of DI are derived from a χ-square distribution with N-1
degrees of freedom. Based on the results of both tests, the lowest possible percentile threshold,
common to all GCM-RCM chains, time periods and all grid points over the integration area
was assigned so that the sampling uncertainty of (ξ, σGP ) is minimized and the proportion of
points exceeding the percentile threshold is equal for each grid point.
For the POT approach, the N-year return levels zN and their confidence intervals are estimated according to Coles (2001) for each grid point of a particular GCM-RCM chain, each
season and for the return periods N = 2, 5, 10, 15 and 20 years. The parameters (σGP , ξ) and
their 95% confidence intervals are estimated with the maximum likelihood approach. The confidence intervals are used to detect significant changes in the parameters between the control
and scenario period. Over most of the integration area, the temperature PDFs have a bounded
upper tail (not shown), i.e. the shape parameter ξ is negative but can be close to zero. The latter
indicates a more moderate upper tail.
The last step is to evaluate the sensitivity of the N-year return level estimates to the methodological approaches, i.e. the temperature threshold selection and declustering procedure, using
the root mean squared error (RMSE). A threshold is considered robust, i.e. independent of
the methodological approaches, if the RMSE of the return level estimates between the chosen
and another appropriate threshold has a minimum value and, additionally, this value remains
approximately constant. To minimize the uncertainty arising from the sample size, the smallest
temperature threshold that was suggested by both significance tests and proved to be robust was
selected. Based on the test statistics and sensitivity analysis we find the 85th percentile to be an
appropriate temperature threshold for all grid points, GCM-RCM pairs and each season. Next
we examine the sensitivity to the declustering procedure by testing the differences between the
return level estimates derived from different sub-samples. The estimates are more sensitive for
winter TMIN than for summer TMAX presumably due to the higher variance in winter (e.g.
Yiou et al., 2009; Hurrell et al., 2003). However, the sensitivity is negligible compared to the
trends and estimation uncertainties of the return levels.
3.2.3 Adjustment models
We test two models, one that accounts for the change in the temperature mean (location parameter) only and a second that additionally accounts for a change in the temperature variance
(scale parameter). The underlying assumption of this study is that we have data for the control
period (CTL) and we know the changes in the location (µ) and scale parameter (σ) of the par-
42
C HAPTER 3: S ENSITIVITY
OF TEMPERATURE EXTREMES TO WARMING
ent daily minimum and maximum temperature distribution TiCT L . Similar to the approach of
Ferro et al. (2005), the first hypothesis, i.e.
HL : TiSCN = µSCN + TiCT L − µCT L ,
(3.2)
assumes that the temperature PDF of the control period TiCT L is shifted by the change in the
mean temperature ∆µ. The subscript i = 1, 2, · · · , M, where M is the sample size, denotes
the daily temperature values. This implies that the Pareto distribution and, therefore, the parameters (ξ, σGP ) remain constant, and, hence, the return levels exhibit the same trend-induced
change as the location parameter µ of the parent temperature distribution. This assumption is
a simplification because changes in extremes are affected by various physical processes on different spatio-temporal scales (Katz, 2010) and, consequently, the response of each process to
changes in the mean temperature might be different. When the location-adjusted return levels
are outside of the 95% confidence intervals of the projected return levels, then the locationscale-adjusted model will be applied. This model adjusts the location and scale of the daily
minimum and maximum temperature PDF, i.e. the hypothesis
CT L
Ti
− µCT L
SCN
(3.3)
= µSCN + σSCN ·
HLS : Ti
σCT L
is tested. The location-scale adjustment model assumes that the temperature PDF of the control
period is additionally scaled by the expected change in the variance σ.
To evaluate the potential of the location and location-scale adjustment models, the rootmean squared error (RMSE) and the percentage error are used. The percentage error is defined
as
T̂NSCN − T̃NSCN
,
(3.4)
T̃NSCN − T̃NCT L
where T̂NSCN and T̃NSCN are the adjusted and projected N-year return levels for the scenario
period. T̃NSCN − T̃NCT L are the projected N-year return level changes between scenario and
control period. The motivation is to show especially the results for the location adjustment
model because it is the simplest model and there is high confidence in projected patterns of
mean changes (the results for the location-scale adjustment model are shown in auxiliary material). In addition, other applications such as, for instance, pattern scaling (Mitchell and Hulme,
1999; Huntingford and Cox, 2000; Mitchell, 2003) use projected changes in the mean temperature to provide climate change scenarios for time periods or emission scenarios for which no
simulations are available.
3.3 Results and Discussion
3.3.1 Performance for the summer season
In order to explore how well the two approaches work, we compare the adjusted changes (here
used to refer to changes inferred with a statistical adjustment model) against projected changes
(here used to refer to future simulations with a physical climate model) in 2 to 20-year return levels of warm temperature extremes. As an example, Figure 3.1 shows the 20-year JJA
3.3 R ESULTS
AND
D ISCUSSION
43
return levels for the control period (a), the expected changes for the scenario period (b), the
location-adjusted (c) and the location-scale-adjusted method (d) in case of the HadCM3Q0CCLM model, i.e. the GCM-RCM pair with the lowest RMSE values between the locationadjusted and projected 2 to 20-year return levels. The stippled areas in Figure 3.1b represent
significant changes in the projected 20-year JJA return levels.
(a) CTL
(b) SCN-CTL
50° N
°
40 N
°
0
°
10 E
°
20 E
°
7
6
°
60 N
5
4
3
°
50 N
2
°
1
40 N
0°
30 E
(c) HL -CTL
°C
JJA
60 N
8
70° N
HadCM3Q0−CCLM
°
46
44
42
40
38
36
34
32
30
28
26
°C
70° N
10° E
20° E
30° E
0
(d) HLS -CTL
70° N
8
70° N
60° N
50° N
40° N
°
0
°
10 E
°
20 E
°
30 E
6
60° N
5
4
3
50° N
2
40° N
1
°
0
°
10 E
°
20 E
°
30 E
Figure 3.1: 20-year return level estimates for the daily JJA maximum temperature in case of the
HadCM3Q0-CCLM model. (a) 20-year JJA return level estimates for the control period. (b) Projected 20-year JJA return level changes in 2070-2099 relative to 1961-1990. Stippled areas represent
significant changes. (c, d) Adjusted 20-year JJA return level changes in 2070-2099 for location-adjusted
HL (c) and location-scale-adjusted HLS (d) temperature distributions.
The HadCM3Q0-CCLM model projects the largest increase in the 20-year JJA return level over
south-eastern Europe, parts of eastern and central Europe, western France and northern Iberian
Peninsula (Figure 3.1b). This is consistent with the other GCM-RCM pairs projecting the
largest warming in the 2 to 20-year JJA return levels over most of southern and south-eastern
Europe as well as for parts of western, eastern and central Europe. However, no significant
0
°C
HadCM3Q0−CCLM
HadCM3Q0−CCLM
7
44
C HAPTER 3: S ENSITIVITY
OF TEMPERATURE EXTREMES TO WARMING
changes are projected for 5 of 12 GCM-RCM pairs over Scandinavia and parts of the British
Isles. Two GCM-RCM chains project no significant changes over central Europe. This agrees
with the projected changes in the 90th to 99th percentiles of the parent temperature distributions
which are not substantial over these areas. Figure 3.1c shows that the location-adjusted 20year JJA return levels agree well with the projections over south-eastern, western and eastern
Europe. Further, the figure shows that the largest differences between projected and adjusted
20-year JJA return levels are found over Scandinavia, i.e. over the area with no significant
changes in the projected return levels. This is also evident for other GCM-RCM pairs when
no significant warming in the return levels is projected (see Figure 3.2). Figure 3.2 shows
the performance of the location-adjusted method for all GCM-RCM pairs measured as percentage error, i.e. the differences between location-adjusted and projected 20-year JJA return
level estimates (error) relative to the projected increase (signal). The stippled areas represent
location-adjusted estimates which are within the 95% confidence intervals of the actually projected return levels, i.e. where the method works well.
All GCM-RCM pairs tend to underestimate the projections over most of Europe when the
location-adjusted method is used. The fraction of grid points with 2 to 20-year JJA return level
estimates within the projected 95% confidence intervals is about 30-50% for all GCM-RCM
chains. This means that for these grid points the assumption of constant scale σGP and shape
ξ parameter of the Generalized Pareto distribution (GPD) is valid for the location-adjusted
approach. For the remaining grid points, changes in the variance or higher moments have
to be considered. In this context, 20-35% of the grid points exhibit significant changes in
σGP caused by changes in the variance of the parent summer TMAX distribution. Thus, the
location-adjusted model fails mainly over areas where a significant increase in summer temperature variance is projected (Yiou et al., 2009; Fischer and Schär, 2009, 2010; Nikulin et al.,
2011; Fischer et al., 2012b). For the HadCM3Q0-CCLM model, for instance, the variance
increase ranges from 10% to 40% over areas along the Mediterranean and parts of western,
central and eastern Europe.
Taking into account changes in variance, the 2nd central moment, lead to improved return
level estimates (Figure 3.1d) in terms of the RMSE values but they can be still outside of the
confidence intervals. For all climate models, the location-scale-adjusted 2 to 20-year JJA return
level estimates fall within the confidence interval of the corresponding model projections for
40-75% of the European area. However, there are areas over Europe for almost all GCM-RCM
pairs where the location-scale-adjusted approach performs worse than the location-adjusted
approach (see Figure S1 of the auxiliary material). This raises the question whether there are
substantial changes in the shape of the parent temperature distribution affecting the shape of
GPD. This is supported by the fact that the number of grid points with significant changes in
the shape parameter ξ of GPD ranges from 20% to 30% for all GCM-RCM pairs.
3.3.2 Performance for the winter season
The issues discussed for summer warm extremes are also found for adjusted winter warm
extremes. Figure 3.3 depicts the same as Figure 3.1 but now for the 20-year return levels of
winter warm extremes in TMIN and for the HadCM3Q0-CCLM model. This GCM-RCM chain
D ISCUSSION
60° N
50° N
°
40 N
°
0
°
10 E
°
20 E
°
40 N
°
0
°
10 E
°
20 E
°
50 N
°
°
°
10 E
°
20 E
°
60 N
°
50 N
°
°
30 E
0
°
°
10 E
°
20 E
°
50 N
°
40 N
°
0
°
10 E
°
20 E
°
50 N
°
°
°
10 E
°
20 E
50° N
°
°
°
10 E
°
20 E
°
30 E
°
30 E
100
80
60
40
20
0
−20
−40
−60
−80
−100
°
°
50 N
°
10 E
°
20 E
°
30 E
°
10 E
°
20 E
°
30 E
0
°
70° N
°
60 N
50° N
°
60 N
50° N
°
40 N
0
20 E
°
°
°
40 N
°
60 N
30 E
HadCM3Q3−RCA
ECHAM5−RCA
°
10 E
°
40 N
°
70° N
60 N
°
°
50 N
40 N
40 N
0
70° N
°
°
°
°
30 E
70 N
60 N
30 E
°
0
ECHAM5−REMO
HadCM3Q16−HadRM3Q16
°
20 E
°
°
70 N
60 N
°
60 N
30 E
°
70 N
10 E
°
40 N
0
°
100
80
60
40
20
0
−20
−40
−60
−80
−100
70 N
°
40 N
°
40 N
0
HadCM3Q0−HadRM3Q0
°
50° N
°
°
60 N
60° N
30 E
70 N
ECHAM5−RACMO
HadCM3Q0−CCLM
50° N
°
°
HadCM3Q3−HadRM3Q3
60° N
30 E
70 N
BCM−RCA
70° N
ECHAM5−DMI−HIRHAM
70° N
ARPEGE−DMI−HIRHAM
HadCM3Q16−RCA3
70° N
45
%
AND
%
3.3 R ESULTS
40 N
°
0
°
10 E
°
20 E
°
30 E
0
°
Figure 3.2: Percentage errors defined as differences between location-adjusted (HL ) and projected 20year return level estimates (error) for the scenario period and daily JJA maximum temperature related
to the expected changes between control and scenario period (signal). The location-adjusted estimates
over the stippled areas are within the 95% confidence intervals of the projected return levels.
46
C HAPTER 3: S ENSITIVITY
OF TEMPERATURE EXTREMES TO WARMING
has again the lowest spatially averaged RMSE values for the location-adjusted 2 to 20-year DJF
return levels. As in Figure 3.2, Figure 3.4 shows the percentage error of the location-adjusted
method for all GCM-RCM pairs.
(a) CTL
(b) SCN-CTL
12
8
°
70 N
10
8
DJF
60 N
4
2
50° N
0
−2
°
40 N
°C
6
°
HadCM3Q0−CCLM
7
6
°
60 N
5
4
3
50° N
2
°
1
40 N
°
0
°
10 E
°
20 E
°
−4
0°
30 E
(c) HL -CTL
°C
°
70 N
10° E
20° E
30° E
0
(d) HLS -CTL
70° N
8
70° N
60° N
50° N
°
40 N
6
60° N
5
4
3
50° N
2
°
1
40 N
0°
10° E
20° E
30° E
0°
10° E
20° E
30° E
Figure 3.3: 20-year return level estimates for the daily DJF minimum temperature in case of the
HadCM3Q0-CCLM model. (a) Adjusted 20-year DJF return level estimates for the control period.
(b) Projected 20-year DJF return level changes in 2070-2099 relative to 1961-1990. Stippled areas represent significant changes. (c, d) Adjusted 20-year DJF return level changes in 2070-2099 for locationadjusted HL (c) and location-scale-adjusted HLS (d) temperature distributions.
For the HadCM3Q0-CCLM model, the largest projected increases in the 20-year DJF return
levels are found over parts of the Mediterranean, central Europe, south-eastern Europe as well
as parts of eastern Europe and Scandinavia (Figure 3.3b). This agrees with the regions found
in the other climate models. Figure 3.4 illustrates for the HadCM3Q0-CCLM model that the
location-adjusted 20-year DJF return levels are within the projected 95% confidence intervals
over parts of the Mediterranean, southern France, eastern Iberian Peninsula, northern parts of
the British Isles and parts of Scandinavia. For all GCM-RCM chains, the location-adjusted 2 to
0
°C
HadCM3Q0−CCLM
HadCM3Q0−CCLM
7
D ISCUSSION
60° N
50° N
°
40 N
°
0
°
10 E
°
20 E
°
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47
%
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3.3 R ESULTS
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Figure 3.4: Percentage errors defined as differences between location-adjusted (HL ) and projected 20year return level estimates (error) for the scenario period and daily DJF minimum temperature related
to the expected changes between control and scenario period (signal). The location-adjusted estimates
over the stippled areas are within the 95% confidence intervals of the projected return levels.
48
C HAPTER 3: S ENSITIVITY
OF TEMPERATURE EXTREMES TO WARMING
20-year DJF return levels are within the confidence intervals for 25-35% of Europe, mainly over
parts of western, southern and south-eastern Europe. However, the location-adjusted changes
are overestimated over most of Europe (see Figure 3.4) and underestimated over parts of southern and south-eastern Europe. Again, most of the climate models have areas with no significant
changes in the projected 2 to 20-year DJF return levels, mainly located over western Europe
and parts of central Europe (see Figure 3.3b and 3.4). But their location is less consistent across
the different climate models compared with the projected JJA return level changes.
The general projected decrease in winter minimum temperature variance (Kjellström et al.,
2007; Fischer et al., 2012b) causes significant changes in the scale parameter σGP for 30-55%
of the grid points. The largest winter variance decrease is found for 10 of the 12 GCM-RCM
pairs over northern and eastern Europe and, therefore, the location-adjusted DJF return levels
exceed the confidence intervals there at almost all grid points. Similar as for the JJA return
level estimates, there are areas where the location-scale adjustment model fails although the
changes in the scale parameter of the parent distribution are significant (see Figure S2 of the
auxiliary material). In addition, there are significant changes in the shape parameter ξ for 3050% of the European area. Areas with largest changes in the shape parameter ξ exhibit also the
largest biases in the location-adjusted and location-scale-adjusted return levels (dark-green and
dark-red areas in Figure 3.2 and 3.4).
3.3.3 Limitations of the adjustment models
Figure 3.2 and 3.4 imply that the applicability of the location adjustment model is limited to
specific regions where the GCM-RCM chains do not project significant changes in the temperature variance. For the other regions, changes in the variance or even higher moments play
an important role. For all climate models, the location-scale-adjusted 2 to 20-year DJF return
level estimates fall within the confidence interval of the corresponding model projections for
25-65% of Europe. The associated reduction in the spatially averaged RMSE values over all
return periods ranges from 15% to 60%. In addition, there is a reduction in the spatial variance
of the RMSE values of about 20-70% implying lower dependency of the results to the location.
To understand the role of changes in the shape of the parent distribution and to answer why
the location-scale adjustment model does not improve the return level estimates over all of Europe, we explored the changes in the percentiles and return levels for the location-scale-adjusted
and projected temperature PDFs. The estimated changes are given relative to the projected differences in the 50th percentile between control and scenario period. To discuss some particular
aspects, two illustrative examples are chosen and shown in Figure 3.5. The estimated changes
are depicted for summer TMAX in the RCA-ECHAM5 model (Figure 3.5a) and for winter
TMIN in the HadCM3Q0-HadRM3Q0 model (Figure 3.5b) over the PRUDENCE region Middle Europe (Christensen and Christensen, 2007).
Figure 3.5a illustrates that summer warm extremes warm more than the mean and summer
cold extremes less than the mean resulting in an increasing variance. In addition, Figure 3.5a
implies that the increase in the variance is nearly symmetric over most of the area. For all
GCM-RCM pairs, the increasing importance of soil moisture-temperature coupling in a future
3.3 R ESULTS
AND
D ISCUSSION
49
climate (Fischer and Schär, 2009; Fischer et al., 2012b) might partly explain this particular
PDF evolution,
(a)
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Figure 3.5: Projected and location-adjusted changes in percentiles and 5, 10 and 20-year return levels relative to the projected changes in the 50th percentile between scenario (SCN) and control (CTL)
period over the PRUDENCE region Middle Europe. The ratios are shown for the daily JJA maximum
temperature and the ECHAM5-RCA model (a) as well as for the daily DJF minimum temperature and
the HadCM3Q0-HadRM3Q0 model (b).
50
C HAPTER 3: S ENSITIVITY
OF TEMPERATURE EXTREMES TO WARMING
especially over western, central and parts of eastern Europe. This is consistent with the other
climate models showing this variance increase in a zonal area extending from the Mediterranean to the North Sea. The increased variance, as shown in Figure 3.5a, implies that the
location adjustment model underestimates the 20-year JJA return levels over Middle Europe
(not shown) and, additionally, they are not within the 95% confidence intervals of the projected
return levels. After considering changes in the summer temperature variance, the adjusted return levels are within the confidence intervals.
For winter TMIN, the cold tail warms more and the warm tail less than the mean leading
to a decreased variance (de Vries et al., 2012; Fischer et al., 2012b). This behaviour is evident
in all GCM-RCM chains over northern and eastern Europe. One factor driving these differing trends in the winter minimum temperature might be the shortening of the snow season
(Lawrence and Slater, 2010) and changes in the large-scale atmospheric variability (Hurrell
et al., 2003). Figure 3.5b shows that the warming in the cold tails is larger than for the mean.
However, the warming in the warm tails is similar to the warming in the mean. As a result,
the variance reduction is dominated by the projected changes in the cold tails, i.e. the variance
decreases asymmetrically. Therefore, the location-scale-adjusted 2 to 20-year return levels for
winter TMIN are underestimated and not within the confidence intervals of the projected return
levels after considering the asymmetric variance decrease.
The location-scale adjustment method assumes symmetric changes in the variance. However, Figure 3.5b shows that this assumption is not always justified. The asymmetric variability
change, also mentioned in the study of Klein-Tank and Können (2003), is responsible that the
location-scale-adjusted return levels can become worse than the location-adjusted ones. The
resulting under- and overestimation are amplified in the tails due to their higher sensitivity to
changes in the parent temperature distribution as discussed by Katz and Brown (1992). This
leads to very large differences between adjusted and projected return levels corresponding to
the dark-green and dark-red areas in Figure 3.2 and 3.4.
Looking at the mathematical definition of return levels, it is evident that changes in the location parameter of the parent distribution adjust the temperature threshold u. Trend-induced
changes in higher order moments affect the scale and shape parameter, i.e. σGP and ξ, of the
Pareto distribution. We have seen that the information about changed variance is only useful
when the change is symmetric. For all other cases, we need to know how much of the asymmetric scale change can be attributed to the upper tail of the parent temperature distribution.
As an example for this asymmetric scale change, Figure 3.6 shows the Pareto distributions of
the projected, location-adjusted and location-scale-adjusted winter TMIN distributions for the
HadCM3Q3-RCA model and a single grid point located in southern Europe.
Figure 3.6 shows that the location-adjusted 2 to 20-year DJF return levels are within the
projected confidence intervals. However, the slightly different scale parameter σGP for the
location-adjusted winter TMIN distribution leads to increasing disagreement for small return
periods and an overestimation for high return periods. Because the change in the scale parameter of the parent distribution is dominated by the larger trends in the lower half of the
distribution (not shown), the variance decrease is overestimated in the upper half when central
moments are used. Consequently, the location-scale-adjusted return levels are all outside of
the projected confidence intervals. Thus, we computed first the differences between the mean
and all temperature values in the upper half of the PDF. These differences are then used to
3.4 C ONCLUSIONS
51
Probability Density Function
0.8
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Location adjusted
Location−scale adjusted
Location−scale adjusted (modified)
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Figure 3.6: Projected (black dashed line), location (L) and location-scale adjusted (LS) Pareto probability density functions of winter minimum temperature for a single grid point over the Mediterranean
and the HadCM3Q3-RCA model. For estimating the variability change, on the one hand the whole
sample was used, on the other hand only the differences between the mean of the whole sample and all
temperature values in the upper half of the PDF (LS (modified)). The black dots are the projected 2 to
20-year DJF return levels with confidence intervals.
estimate the change in the variance in the upper half labeled in the figure as “modified”. Using
this estimation for the variance change as input for the location-scale adjustment model, the
adjusted Pareto PDF fits almost perfectly the projected PDF. With this approach, the fraction
of grid points with adjusted 2 to 20-year return levels within the projected confidence intervals can be raised, relative to the location-scale adjustment model with variance estimated with
the whole sample, by additional 20% for summer TMAX and by 25% for winter TMIN. But
for this approach, we would need the information about the temperature PDF for the scenario
period contradicting the initial assumption of no knowledge about future temperature PDFs.
3.4 Conclusions
We show that estimated changes in return levels of warm temperature extremes in summer
and winter based on a simple shift of the full daily temperature distribution (location-adjusted)
are within the projected confidence intervals for one quarter to one half of Europe across all
examined climate models. The location-adjusted estimate performs better for warm extremes
in summer than in winter. The shifted return level estimates tend to underestimate the projected changes in warm extremes in summer, and to overestimate the projected changes in
warm extremes in winter over most of Europe. The primary reason is the significant projected
changes in daily temperature variance. In summer, changes in warm extremes are larger than
for cold extremes leading to enhanced variance (see also Schär et al., 2004; Vidale et al., 2007;
52
C HAPTER 3: S ENSITIVITY
OF TEMPERATURE EXTREMES TO WARMING
Kjellström et al., 2007; Fischer and Schär, 2009, 2010; Fischer et al., 2012b). In winter, cold
extremes warm more than warm extremes leading to reduced variance (Kjellström et al., 2007;
de Vries et al., 2012; Fischer et al., 2012b).
If the change in the variance is also accounted for (location-scale-adjusted), the spatially
averaged mean error is generally reduced in both winter and summer (not shown). However,
there are still areas where large estimation biases remain for the location-scale-adjusted estimates. We even find cases where return levels adjusted with location and scale are worse than
those adjusted by location only. This is due to asymmetric changes in the variance, which
causes large estimation biases and implies that the use of central moments is not appropriate.
Therefore, the location-scale-adjusted return levels perform poorly in those regions. The performance of the location-scale adjustment method can be further improved if the change in the
variance is estimated separately for the corresponding half (i.e. lower half for cold extremes
and upper half for warm extremes) of the temperature distribution. With this approach, the
return level estimates are within the projected confidence intervals for 60-80% of Europe in
summer and for 40-70% in winter. But this implies detailed knowledge about the future summer maximum and winter minimum temperature distributions. The variance change and the
degree of asymmetry depend on the regional climate model, i.e. changes in the large-scale
atmospheric circulation and feedback mechanisms including the representation of snow-cover,
soil-moisture, clouds and the surface energy budget.
This study further illustrates that there are significant changes in the shape of the upper
tail of the summer maximum and winter minimum temperature distributions and, therefore, in
the shape parameter of the Pareto distribution over northern, western and central Europe. This
implies changes in the shape or higher-order moments of the parent temperature distribution.
Further work is necessary to understand the physical processes behind these changes. Over
these areas, the use of the location and location-scale adjustment models leads to large estimation biases. This is especially the case for extremes with low probabilities due to their higher
sensitivity to changes in the moments of the parent distribution.
In general, information on future temperature distribution is often not available to account
for asymmetric changes in the variance. One promising option would be to identify relationships between mean and variance changes associated with the upper and lower half of the
temperature distribution, similar to the approach proposed by Yiou et al. (2009).
The evaluation of the performance of both adjustment models in this study was limited to
one emission scenario. However, simulations for several emission scenarios would help investigating the robustness of potential relationships between distribution parameters and return
levels of temperature extremes.
3.5 AUXILIARY
53
MATERIAL
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3.5 Auxiliary material
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Figure S1: Percentage errors defined as differences between location-scale-adjusted (HLS ) and projected 20-year return level estimates (error) for the scenario period and daily JJA maximum temperature related to the expected changes between control and scenario period (signal). The location-scaleadjusted estimates over the stippled areas are within the 95% confidence intervals of the projected return
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OF TEMPERATURE EXTREMES TO WARMING
%
C HAPTER 3: S ENSITIVITY
%
54
40 N
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Figure S2: Percentage errors defined as differences between location-scale-adjusted (HLS ) and projected 20-year return level estimates (error) for the scenario period and daily DJF minimum temperature related to the expected changes between control and scenario period (signal). The location-scaleadjusted estimates over the stippled areas are within the 95% confidence intervals of the projected return
levels.
Chapter 4
Conclusions and outlook
The aim of this thesis was to relate projected changes in the frequency and intensity of European temperature extremes to trends in the mean temperature. The objective of the first part
of the thesis was to investigate whether changes in the frequency of European warm and cold
extremes scale linearly with trends in the annual mean temperature. In the second part, presentday summer maximum and winter minimum temperature distributions were shifted and scaled
by the expected changes in the mean and variance in order to approximate the projected changes
in the intensity of warm extremes. The skill of both approaches to fill in gaps in the time series
of the frequency and intensity was evaluated. Daily data of the recent state-of-the-art regional
climate models from the European multi-model project ENSEMBLES were used in this thesis.
4.1 Trends in central moments and temperature extremes:
Are they related?
A simple shift of the full temperature distribution towards higher temperatures is consistent
with the increasing frequency of warm extremes and the decreasing frequency of cold extremes.
For long-term trends in the intensity of warm extremes, the assumption is reasonable for one
quarter to one half of Europe across all regional climate models. Further, the assumption works
better for the intensities of warm summer extremes than for warm winter extremes. Shift direction and shift velocity affect the long-term trends in the frequency and intensity of temperature
extremes. The variance of the present-day temperature distribution has an additional effect on
the trends in the frequency. The linearity in the frequency trends is determined by the shape of
the parent temperature distribution which is approximately described as normally distributed.
Therefore, given the projected linear trends in the annual mean temperature, the projected increasing frequency in warm extremes scales linearly with the annual mean temperature over
all of Europe. Due to the shape of the cold tail in the temperature distributions, this relationship is nonlinear for the decreasing frequency in cold extremes and, additionally, the trends are
smaller than for warm extremes. For temperature extremes lasting several consecutive days, the
nonlinearity is more pronounced, especially for cold extremes, and the trends are larger. For
absolute threshold-based extremes, the frequency changes are dominated by trends in seasonal
mean temperature.
55
56
C HAPTER 4: C ONCLUSIONS
AND OUTLOOK
The differences in percentiles between present-day and future seasonal temperature distributions illustrate that summer warm extremes warm faster than the summer cold extremes. In
winter, cold extremes warm faster than warm extremes. Therefore, the temperature variability
is expected to be reduced in winter and enhanced in summer (see also Schär et al., 2004; Vidale et al., 2007; Kjellström et al., 2007; Fischer and Schär, 2009, 2010; Fischer et al., 2012b;
de Vries et al., 2012). When the rate of change is different between warm and cold extremes,
the change in the temperature variability is asymmetric. Thus, central moments are not appropriate to account for asymmetric changes. The degree of asymmetry and the variability change
depend on the regional climate model, i.e. the representation of large-scale atmospheric circulation and feedback mechanism.
Significant changes in the temperature variability cause changes in the scale parameter of
the extreme value distributions and, therefore, affect trends in the intensity of warm extremes
over substantial parts of Europe. In addition, different trends in each part of the seasonal temperature distribution change the shape of the parent temperature distribution and, therefore, the
shape of the warm tails. These changes in the warm tails are significant over northern, western
and parts of central Europe.
4.2 Skill and limitations of the fill in approaches
4.2.1 Linear regression model
The linear approach, used to reproduce the frequency of temperature extremes, is mainly limited by violations in the linearity assumption. The shape of the parent temperature distribution
and nonlinear trends in the annual mean temperature affect the degree of linearity in the frequency trends. In this context, the shape and the shift direction of the temperature distribution
are the reason why this approach works reasonably well for warm extremes. The estimates
perform even better for temperature distributions with small variance. Warm extremes lasting
for several consecutive days exhibit a higher sensitivity to changes in the annual mean temperature. Therefore, the trends for these multi-day warm extremes are larger and partly nonlinear.
As a result, the estimates have large systematic errors. In contrast to this, the skill of the linear fill in approach is limited for cold extremes due to the occurrence of nonlinear trends, the
convergence to 0% and the smaller trends compared with the warm extremes. The degree of
nonlinearity depends on the representation of the cold tail by the corresponding climate model.
For multi-day cold extreme events, these issues are even more pronounced and, therefore, the
approach performs poorly. In case of absolute-threshold based warm and cold extremes, the
skill of the linearly scaled frequencies is improved when seasonal mean temperatures are used
instead of annual mean temperatures.
Because of the internal variability, the skill of the estimated frequencies depends on the
available data used to estimate the trends, i.e. the slope parameter. The trend estimates are
less robust for data from scenario periods close to the reference period, and more robust for
data from scenario periods at the end of the 21st century. The latter is especially the case for
warm extremes because of their larger trends. In addition, methods such as the least-squares
approach use all available data to estimate the trends and, therefore, lead to more robust trend
4.2 S KILL
AND LIMITATIONS OF THE FILL IN APPROACHES
57
estimates. As a consequence, the frequencies interpolated to the first half of the 21st century
perform better than those extrapolated to the second half of the 21st . However, the uncertainty
in the trend estimates due to internal variability is smaller than the uncertainty due to the different climate models.
Logistic regression models are appropriate to deal with discrete count data and to capture
nonlinear trends in the frequency. However, the dependency of these issues on the used climate
model and the location is the reason that the estimates are not improved over all of Europe.
4.2.2 Adjustment models
Intensity estimates, derived from shifted seasonal temperature distributions, are overestimated
in winter and underestimated in summer due to the projected changes in temperature variance.
For 40-75% of Europe in summer and for 25-65% in winter, the intensity estimates perform
reasonably well when the shifted temperature distributions are additionally scaled with the expected changes in the variance. For areas with asymmetric variance changes, the symmetric
scaling of shifted present-day temperature distributions leads to large biases in the intensity
estimates of warm extremes. The approach performs better when the variance is estimated
with temperature data from the upper half of the seasonal temperature distributions. With this
method of variance estimation, the shifted and scaled intensities perform reasonably well for
60-80% of Europe in summer and for 40-70% in winter. Both adjustment models are not
appropriate over areas with significant changes in the shape of the tails of the temperature distributions.
Another promising approach to capture the asymmetry would be to explore the relationships between mean and variance changes associated with the upper and lower half of the
parent temperature distribution, similar as in Yiou et al. (2009). With this approach, changes
in the intensity could be estimated using the expected change in the mean as only information.
However, even when asymmetric variance changes are considered, there are still areas with
large estimation biases caused by significant changes in the shape of the warm tails. In this
context, quantile regression models are appropriate to study how trends in different percentiles
are related to changes in the moments of the temperature distribution and how the shape of the
temperature distribution evolves in time. This is crucial to understand how much of the changes
in the shape of the tails are related to changes in the shape of the temperature distribution. At
least, the aim is to develop a statistical model which could be used to derive directly the parameters of the extreme value distributions from the estimated changes in the parameters of the
temperature distributions, similar as in the study of Nogaj et al. (2007).
For skewed temperature distributions, the sensitivity of extremes to changes in the moments
of the parent temperature distribution might also change. To investigate this issue, the approach
in Katz and Brown (1992) could be applied to skew-normal distributions which have a shape
parameter to account for non-zero skewness (Azzalini, 2005).
In this thesis, the skill and limitations of the two fill in approaches are discussed for one
emission scenario. However, more emission scenarios are necessary to test the robustness of
these methods and to evaluate the uncertainty arising from different emission scenarios. The
motivation is to develop statistical models used to fill in gaps in emission scenarios for which
58
C HAPTER 4: C ONCLUSIONS
AND OUTLOOK
no data are available. However, not only more emission scenarios are necessary but also more
ensemble members for each climate model. In the ENSEMBLES project, there is one run
available for each climate model. But more ensemble members are crucial to quantify the
uncertainties in the estimated extremes arising from natural variability and to evaluate more reliably the benefits of regional climate models to represent, among other things, the small-scale
nature of present-day extreme events.
Both fill in approaches do not work reliably for extreme precipitation events because
changes in precipitation extremes do not scale with temperature changes. A possibility would
be to use the moments of the precipitation distribution as explanatory variable, similar as for
the adjustment method. The next step would then be to investigate how changes in the tails are
related to changes in the moments.
Another possibility is to define a set of explanatory variables which affects the intensity
and frequency of precipitation extremes. This approach is motivated by the idea that extreme
events can be viewed as compound events (IPCC, 2012). This implies that the occurrence of
an extreme precipitation event depends on the occurrence of other events which could be extreme or not. To explore the potential relationships between the precipitation extremes and the
selected explanatory variables, artificial neural networks are appropriate mathematical models.
The advantage of artificial neural networks is that they are able to capture non-linear and complex relationships.
Studies such as, for instance, Boberg and Christensen (2012) have shown that there are
temperature-dependent biases over certain areas due to the insufficient spatial resolution of the
climate models. In this context, the motivation is to develop a temperature-dependent bias correction procedure for temperature extremes. After applying the fill in approaches, the correction
procedure is used to derive frequency and intensity estimates which are physically consistent.
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Acknowledgments
I would like to express my sincere gratitude to my supervisor Prof. Dr. Reto Knutti for his
patient guidance, the constructive criticism, his readiness to help at any time and his motivating
encouragement. I am very grateful to make my first experiences as a scientist in his group.
I would like to offer my special thanks to Dr. Erich Fischer for his support, constructive criticism, proof-reading, numerous hints and the very interesting and intellectually stimulating
discussions.
I would like to thank all the members of the climate physics group for the helpfulness and the
good time at the IAC.
I would also like to thank Dr. Mark Liniger and Prof. Dr. Christoph Schär for their feedbacks
and hints in the two PhD interviews and to be a part of my PhD Defense.
A special thank goes to my office mate Nathalie Schaller for the good time in the last years and
her readiness to help.
I would like to sincerely thank my parents and brothers for their encouragement and willingness
to help wherever requested.
My biggest and sincere thanks go to my wife Jenny and my son Noah. They enrich my life in
every possible way.
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