METHODS FOR ANALYZING
EXTREME EVENTS UNDER
CLIMATE CHANGE
RICHARD W. KATZ
BARBARA G. BROWN
ENVIRONMENTAL AND SOCIETAL
IMPACTS GROUP
NATIONAL CENTER FOR
ATMOSPHERIC RESEARCH
AUGUST 1992
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CONTENTS
Section Page
ABSTRACT 2
CONTENTS 3
1. INTRODUCTION 4
2. SUMMARY OF ACCOMPLISHMENTS 6
2.1. General theory ......... 6
2.2. Extreme temperature example ....... 8
2.3. Regional analysis/Spatial analogue ....... 10
2.4. Extreme precipitation events . . . . . . .12
2.5. Heat island effect . . . . . .14
3. EXTENSIONS AND IMPLICATIONS 16
REFERENCES 18
TABLE 20
FIGURES 21
APPENDIX 1 30
Papers Based on Research Supported by Cooperative Agreement
APPENDIX 2 31
"Extreme Events in a Changing Climate: Variability is More Important than Averages"
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SECTION 1
INTRODUCTION
This report summarizes work completed through a cooperative agreement between the
National Center for Atmospheric Research (NCAR) and the Environmental Protection Agency
(EPA) to study methods for analyzing extreme climate events. The overall goal of this project
is to develop statistical models for extreme climate events that will be useful for the construc-
tion and application of scenarios of future climate. The results of this study will provide
methods for determining how the likelihood of extreme climate events may change as other
more general climate parameters (e.g., the mean or variance) change.
Research in the first year focused on the general problem of expressing climate change in
terms of the likelihood of extreme events. A statistical "paradigm" for climate change was
formulated, and theoretical properties of the relative sensitivity of extreme events were
derived. During the second year, these theoretical results were extended to treat more realistic
situations for climate variables (e.g., autocorrelation, finite samples). The interpretation of the
theory in terms of a "spatial analogue" for climate change was also begun. Research in the
third year concentrated on refinements that naturally arise in actual applications or case
studies of extreme climate events. For instance, the accuracy of approximations based on
extreme value theory was investigated for extreme maximum and minimum temperature
events, and a specialized treatment for extreme precipitation events was devised. Finally, as
another analogue for climate change, the so-called "heat island effect" was considered.
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These accomplishments are summarized in greater detail in Section 2. Their possible
extensions and their implications for the generation of scenarios of future climate are
discussed in Section 3. A list of papers produced under this cooperative agreement is given in
Appendix 1, and a reprint of an article that appeared in Climatic Change is included as
Appendix 2.
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SECTION 2
SUMMARY OF ACCOMPLISHMENTS
2.1. General theory
Appreciation of the need for a statistical paradigm for climate change arises when
considering how the relative frequency of extreme events might change as more conventional
statistics, such as the mean or standard deviation, change. A climate variable X is assumed to
have a probability distribution with a location parameter p and a scale parameter c. If this
distribution were the normal, then p would be the mean and c the standard deviation. Climate
change is envisioned to involve a combination of two different statistical operations: (i) the
distribution is shifted, producing a change in location (p); and (ii) the distribution is rescaled,
producing a change in scale (a).
Figure 1 illustrates this concept for one hypothetical choice of distribution. The two forms
of climate change are included: (i) a change in the location parameter p to a new, in this case
larger, value p*; and (ii) a change in the scale parameter a to a new, in this case larger, value
g\ Especially noteworthy is how much these distributions differ in the tails, the shape of
which determines the probability of extreme events. Katz (1991) treated this statistical
paradigm for climate change in more detail.
Some standard statistical theory for extremes can be applied to reveal some broad
generalizations that can be made about the relative sensitivity of extreme events to the
location and scale parameters. Attention is focused on two specific types of extreme events:
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(i) the exceedance of a threshold [event £, = > c}, where the constant c denotes a
threshold]; and
(ii) the maximum of a sequence of length n exceeding a threshold [event E2 =
{max(X„X2 X„)>c}].
The sensitivity of an extreme event to the location parameter p or the scale parameter a is
defined to be the corresponding partial derivative of the probability of the event; that is,
dP(E)fdp or dP(E)/da. Because extreme events vary in their likelihood, it is reasonable to deal
with the relative sensitivity, [dP(E)/d\i]/P(E) or [dP(E)/da]/P(E), comparing the sensitivity of
an event to its probability.
Katz and Brown (1992a) show that the relative sensitivity of an extreme event (either £,
or EJ to the scale parameter c becomes proportionately greater than its relative sensitivity to
the location parameter p as the event becomes more extreme (i.e., the larger the threshold c).
Moreover, in many instances, the relative sensitivity of an extreme event to both p and a in-
creases as the event becomes more extreme. These theoretical properties are illustrated in the
next subsection for an extreme temperature example.
Some of the theoretical results for extreme event El (i.e., the maximum of a sequence
exceeding a threshold) are based on large sample approximations that do not directly take into
account certain prominent statistical features, like autocorrelation, of climate time series.
Nevertheless, simulation studies show that these results are actually quite robust (Katz and
Brown, 1992b). These results are also presented in the next subsection.
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22. Extreme temperature example
Extreme high temperature events of a form known to be deleterious to the corn crop in
the midwestem U.S. are considered (Meams el al., 1984). The July time series of daily
maximum temperature at Des Moines, Iowa is utilized (mean p = 30 °C, standard deviation a
= 3.9 °C). Figure 2 shows plots of the relative sensitivity of extreme event £, to p and c as
the threshold c increases (i.e., as the event becomes more extreme). In this application, the
event £, corresponds to the temperature exceeding a threshold on a given day in July. These
curves are based on the assumption of a normal distribution for daily maximum temperature.
The relative sensitivity of £, to p increases at an approximately linear rate, whereas the
relative sensitivity to a increases at an approximately quadratic rate, for large threshold c.
Figure 3 shows plots of the relative sensitivity of extreme event £2 to p and a as the
threshold c increases. In this application, the event £2 corresponds to the temperature ever
exceeding the threshold within the entire month of July (i.e., n = 31). These curves are based
on the assumption of a Type I extreme value distribution for the maximum of a sequence
[i.e., a distribution function of the form G(x) = exp(-e")] (Katz and Brown, 1992a). The
relative sensitivity of £2 to p is approximately constant, whereas the relative sensitivity to a
increases at an approximately linear rate, for large c.
To convert these results into more concrete terms, Table 1 gives the probability of event
£j for a threshold of c = 38 °C, when p and a are changed by ± 0.5 °C. Relative to the
current probability of 0.020, P(£j) changes by roughly twice as much for a change in o as for
the corresponding change in p. Table 1 also includes the probability of event £2 for the same
threshold and changes in p and a. Again, the relative changes in P(£j) are roughly twice as
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large when a is varied as when p is varied. Unlike event £, which always remains rare for all
of the values of p and a considered, event E2 becomes quite likely when a is increased and
somewhat rare when a is decreased.
As mentioned previously, the Type I extreme value distribution serves only as an
approximation in determining P(Ej). By means of a simulation study, the exact relative
sensitivity of the maximum of a finite sequence of a normally distributed, autocorrelated time
series can be determined. It is more convenient to actually perform the simulation in terms of
the hazard rate of the exact distribution of the maximum (i.e., the hazard rate H for a
distribution function F is H(x) = F'(x)/[ 1 - F(r)]). Katz and Brown (1992a) established that
this hazard rate curve has the same shape as the relative sensitivity of the extreme event E2 to
the mean p.
Figure 4 shows the simulated hazard rate for the exact distribution of the maximum of a
sequence of length n = 30 with a first-order autocorrelation coefficient of (j) = 0.5 (for the Des
Moines application, n = 31 and <|> » 0.58). For comparison sake, the hazard rate for the Type
I extreme value distribution (i.e., equivalent to the dashed curve in Figure 3) and the hazard
rate for the exact distribution of the maximum of a sequence of n = 30 normally distributed,
independent observations (i.e., = 0) are also included in Figure 4. It is evident that these
curves are quite similar, with any discrepancies for the exact curve being in the direction of
even more sensitivity than either the asymptotic theory or the exact theory under indepen-
dence would predict. Katz and Brown (1992b) treated this issue in more detail.
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2.3. Regional analysis!Spatial analogue
One approach to the interpretation of the assumptions on which the theoretical results
presented in Section 2.1 are based involves the so-called "spatial analogue" for climate
change. Actual differences in climate across space are substituted for hypothetical changes
over future time horizons. This concept is similar to the "regional analysis" approach that is
employed in hydrology, for instance, to estimate flood probabilities.
Time series of daily maximum temperature for July at 30 sites in the U.S. Midwest and
of daily minimum temperature for January at 28 sites in the U.S. Southeast were subjected to
such a regional analysis. Figure 5 gives a plot of the relative frequency of the maximum
temperature on a given day in July exceeding a threshold of c = 35 °C (i.e., event £,) versus
the standardized threshold of (c - p)/a for each of the 30 stations in the Midwest. Here the
location and scale parameters, p and a, were estimated using the sample means and standard
deviations of the July daily maximum temperatures for the individual stations. The points fall
remarkably close to a smooth decreasing curve, in agreement with our statistical paradigm for
climate change (Section 2.1). Similar results were obtained for the analogous case of January
minimum temperature in the Southeast (Brown and Katz, 1991).
As an additional check, Figure 6 gives a plot of the relative frequency of the temperature
ever exceeding c = 35 °C during the entire month of July (i.e., event EJ versus the same stan-
dardized threshold. The plot has a greater degree of scatter than Figure 5, in part because
these relative frequencies are based on a much smaller sample (i.e., only one observation for
each July instead of 31 for event £,). Nevertheless, the indication of an underlying relation-
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ship is present. Again, similar results were obtained for minimum temperature (Brown and
Katz, 1991).
These checks have served so far to help interpret our statistical paradigm for climate
change. Additional conditions, however, were imposed in examining the relative sensitivity of
extreme events. In particular, the relative sensitivity of extreme event E2 (i.e., the maximum
of a sequence exceeding a threshold) was derived through the Type I extreme value approxi-
mation (Figure 3). Further regional analysis of the same daily time series of July maximum
temperature was performed to investigate whether this approximation is appropriate. When the
two parameters of the Type I extreme value distribution are derived indirectly by assuming a
normal distribution for daily maximum temperature, the approximation is quite inaccurate
(Brown and Katz, 1992). Although this behavior indicates that the conventional assumptions
about the statistical properties of time series of daily maximum temperature do not all hold, it
does not necessarily conflict with our statistical paradigm for climate change.
In this regard, Figure 7 shows the results when the parameters are estimated directly from
the monthly maxima. The event E2 of the maximum exceeding a threshold of c « 37.8 °C is
considered for each of the 30 midwestem sites. There is reasonably good agreement between
the theoretical probabilities based on the Type I extreme value distribution and the observed
relative frequencies. For minimum temperatures in the Southeast, the Type I approximation is
not as accurate, with the Type ID extreme value distribution (i.e., a distribution function of
the form G(x) = exp[-(-^)°], a > 0, x < 0) being a considerable improvement Although no
simple physical explanation for this disagreement exists, such behavior for daily minimum
temperature extremes has been noted previously (e.g., Farag6 and Katz, 1990). Because the
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Type HI distribution requires an additional shape parameter, our statistical paradigm for
climate change might need to be made more complex to encompass this situation. Brown and
Katz (1992) provided further details on these issues.
2.4. Extreme precipitation events
Katz and Garrido (1992) employed a somewhat more specialized approach to address the
issue of how the frequency of extreme precipitation events might change with an overall
change in climate. Such an approach is required, because the forms of distribution generally
fit to precipitation totals do not satisfy the location and scale model on which our statistical
paradigm for climate change is based (Section 2.1). Nevertheless, results that are qualitatively
similar to those stated in Section 2.1 and illustrated in Section 2.2 can still be obtained in this
case.
The climate variable, Y say, represents the precipitation totaled over a month or season.
The extreme event of interest is the total precipitation exceeding a threshold c, say E = {Y >
c}, referred to as the "right-hand tail event." Analogous to the approach followed in Section
2.1, the relative sensitivity of event E is defined with respect to the median (rather than the
location parameter or mean) and to the scale parameter of the distribution of total precipita-
tion y. The median is adopted as a measure of central tendency, because the distribution of
total precipitation has a substantial degree of positive skewness.
A technique based on a power transformation to normality is employed to account for this
skewness. Here X = T, for some s, 0 < s < 1, is assumed to have a normal distribution with
mean and variance ox2 [written N(px, a/)] (Katz and Garrido, 1992). Let mY denote the
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median of the distribution of total precipitation Y [i.e., mY satisfies F^mY) = 1 - Fy(mY) = 1/2,
where Fy denotes the distribution function of Y]. This median mY is related to the mean of
the normal distribution for the transformed variable X by mY - p/, where r = 1/5. Analogous
to the relative sensitivity to the location parameter previously treated (Section 2.1), the
relative sensitivity of the right-hand tail event £ to the median mY is [dP(E)/dmr]/P(E).
The scale parameter, X say, of the distribution of total precipitation Y is introduced by
considering a new random variable Yx = XY, 0 < X < oo, representing a multiplicative effect.
The relative sensitivity of the right-hand tail event £ to the scale parameter is defined as
{[3/>(£x)/3X]//,(£x)} where £x = {}\ > c) is the analogous extreme event for the rescaled
variable Yx. The partial derivative is evaluated at X = 1, because this value corresponds to an
instantaneous change from the current climate. In effect, an additional parameter X has been
included in the power transform distribution of total precipitation Y (Katz and Garrido, 1992).
Figure 8 shows the relative sensitivity of the right-hand tail event £ to the median and to
the scale parameter for summer total precipitation at Segovia, Spain. Both relative sensitivities
increase as the event becomes more extreme (i.e., as the threshold c increases). Moreover, the
relative sensitivity to the scale parameter is greater than that to the median for virtually all
values of the threshold c, and it increases at a much faster rate as c increases. Similar results
are obtained for the relative sensitivity of the left-hand tail event (i.e., {y < c}) to the median
and to the scale parameter (Katz and Garrido, 1992).
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25. Heat island effect
Another analogue for climate change is the "urban heat island," in which real temporal
climate changes associated with human activity have been produced inadvertently for local
environments. As metropolitan areas develop, a warming can occur that is comparable in
magnitude to that anticipated for the enhanced greenhouse effect (i.e., 2-3 °C according to
Changnon, 1992). This heat island has been detected in cities across the globe, ranging from
the tropics to high latitudes and, to a lesser extent, in relatively small communities.
Research on the heat island effect has dwelt on average temperatures, with little mention
of any changes in variability or in the frequency of extreme events. One notable exception is
the recent work by Balling et al. (1990). They examined the trend in the occurrence of
extreme maximum and minimum temperatures at Phoenix, Arizona, an area that has experi-
enced a marked heat island effect in recent decades. Among other things, the inadequacy of a
statistical model for climate change in which simply the mean (or location parameter) is
allowed to change (i.e., the variance or scale parameter is held constant) was established.
Although changes in the mean are apparently sufficient to explain the trend in occurrence of
extreme minimum temperatures, such a model overestimates the frequency of extreme
maximum temperatures.
A reanalysis of the same Phoenix temperature data has been performed. The goal is to
establish whether our statistical paradigm for climate change (Section 2.1), allowing for a
change in variance or scale, could satisfactorily explain the observed trend in the occurrence
of extreme maximum temperatures. Figure 9 shows the trend in summer (July-August)
standard deviation of minimum and maximum daily temperatures. For minimum temperature,
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the slight apparent increase in standard deviation makes only a minor contribution to the
decrease in the frequency of extreme low temperature events. On the other hand, the more
substantial decrease in the standard deviation for maximum temperature does make a major
contribution to the frequency of extreme high temperature events. In particular, it explains
why the change in the mean alone results in too high a frequency of extreme high temper-
atures. Tarleton and Katz (1993) included further details on this example.
The urban heat island, thus, provides a real-world application in which changes in
variability need to be taken into account to anticipate changes in the frequency of extreme
events. Of course, as pointed out by Balling et al. (1990), the heat island effect is not
necessarily analogous to the enhanced greenhouse effect Further, being situated in a desert
region, the heat island effect for Phoenix is not necessarily typical of that for other urban
areas.
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SECTION 3
EXTENSIONS AND IMPLICATIONS
A myriad of ways exist in which this study of extreme events and climate change could
be extended. For instance, all of our work has concentrated on time series of a single climate
variable treated in isolation. It would be natural to consider simultaneously the extremes of
two or more variables, either different variables (such as temperature and precipitation) for
the same site or the same variable at several locations (e.g., fields of temperature or precipita-
tion). Other more specialized approaches could also be taken. For example, extreme precipita-
tion events could be more systematically studied by developing an underlying stochastic
model (i.e., on a daily or hourly time scale) for the precipitation process. It would be more
physically meaningful to change these basic model parameters (e.g., the frequency or intensity
of "storms"), using probabilistic methods to induce the effects on any extreme events of
interest
As it stands, this study has significant implications for scenarios of future climate.
Stochastic weather generators that are convenient to employ for simulating climate variables,
such as daily maximum and minimum temperature and precipitation amounts, do exist (e.g.,
Richardson, 1981). Such models were originally intended to be used to simulate time series
for the present climate. However, these models have not been extensively validated with
respect to their ability to reproduce the frequency of extreme events. In particular, our results
concerning the lack of fit of the Type I extreme value distribution to extreme minimum
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temperature events in the U.S. Southeast (Section 2.3) would appear to cast doubt upon the
use of Richardson's model when extremes are of importance. Specifically, this model
essentially represents both time series of daily minimum and maximum temperature as first-
order autoregressive processes having normal distributions. Under these assumptions, the Type
I extreme value distribution is known to be a good approximation for the maximum or
minimum of a sequence (i.e., as generated by Richardson's model). In other words, the Type
ID extreme value distribution that sometimes arises in practice as a better fit to extreme
minimum temperature events could not be reproduced by Richardson's model.
Recently, it is becoming increasingly popular to utilize these same stochastic weather
generators to simulate time series for a changed climate (Wilks, 1992). But it would be
potentially misleading to apply such models by changing only those parameters (e.g., mean
values) for which some information about future changes is currently available. For instance,
Richardson's model requires the specification of the means and variances of the normal
distributions for daily maximum and minimum temperature, as well as their contemporaneous
cross correlation coefficient and individual first-order autocorrelation coefficients (technically,
these parameters are conditional on whether or not precipitation occurs). As our results
convincingly establish, attention needs to be devoted to how these other model parameters
(especially variances) might change as well. Without this information, stochastic weather
generators may produce scenarios of future climate whose frequency of extremes turns out to
be far off the mark.
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REFERENCES
Balling, R.C., Jr., J.A. Skindlov and D.H. Phillips, 1990: The impact of increasing summer
mean temperatures on extreme maximum and minimum temperatures in Phoenix, Arizona.
Journal of Climate, 3, 1491-1494.
Brown, B.G., and R.W. Katz, 1991: Characteristics of extreme temperature events in the U.S.
Midwest and Southeast: Implications for the effects of climate change. Preprints, AMS
Seventh Conference on Applied Climatology, Salt Lake City, UT, pp. J30-J36.
Brown, B.G., and R.W. Katz, 1992: Regional analysis of temperature extremes: Implications
for climate change. To be submitted to Journal of Climate.
Changnon, S.A., 1992: Inadvertent weather modification in urban areas: Lessons for global
climate change. Bulletin of the American Meteorological Society, 73, 619-627.
Farag6, T., and R.W. Katz, 1990: Extremes and design values in climatology. Report No.
WCAP-14, WMO/TD-No. 386, World Meteorological Organization, Geneva, 43 pp.
Katz, R.W., 1991: Towards a statistical paradigm for climate change. Preprints, AMS Seventh
Conference on Applied Climatology, Salt Lake City, UT, pp. 4-9.
Katz, R.W., and B.G. Brown, 1992a: Extreme events in a changing climate: Variability is
more important than averages. Climatic Change, 21, 289-302.
Katz, R.W., and B.G. Brown, 1992b: Sensitivity of extreme events to climate change: The
case of autocorrelated time series. To be submitted to Environmetrics.
Katz, R.W., and J. Ganido, 1992: Sensitivity of extreme precipitation events to climate
change. Submitted to Water Resources Research.
Mearns, L.O., R.W. Katz and S.H. Schneider, 1984: Extreme high-temperature events:
Changes in their probabilities with changes in mean temperature. Journal of Climate and
Applied Meteorology, 23, 1601-1613.
Richardson, C.W., 1981: Stochastic simulation of daily precipitation, temperature, and solar
radiation. Water Resources Research, 17, 182-190.
Tarleton, L.F., and R.W. Katz, 1993: Effects of urban heat island on temperature variability
and extremes. Preprints, AMS Eighth Conference on Applied Climatology, Anaheim, CA (in
press).
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Wilks, D.S., 1992: Adapting stochastic weather generation algorithms for climate change
studies. Climatic Change, 22, 67-84.
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TABLE 1
Probability of extreme events, £, and E2, with threshold of c = 38 °C associated with changes
in mean p and standard deviation a of July daily maximum temperatures at Des Moines, Iowa
(current climate of p = 30 °C and a = 3.9 °C) [Source: Katz and Brown, 1992a],
Change in p
(°C)
Change in a
(°C)
P(EX)
(Relative Change)
PiEJ
(Relative Change)
0
0
0.020
0.492
+0.5
0
0.027 (+34.7%)
0.612 (+24.5%)
0
+0.5
0.034 (+70.8%)
0.713 (+44.9%)
-0.5
0
0.015 (-27.7%)
0.384 (-22.0%)
0
-0.5
0.009 (-54.0%)
0.264 (-46.2%)
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Figure 1. Hypothetical distribution of a climate variable with location parameter p and scale
parameter a (solid line); location parameter p* and scale parameter a (dashed line); and
location parameter p and scale parameter a* (long and short dashed line) [Source: Katz and
Brown, 1992a].
r~r
m /**
CLIMATE VARIABLE X
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Figure 2. Relative sensitivity of extreme event Elt temperature exceeding threshold on given
day in July, to mean (dashed line) and standard deviation (solid line) [Source: Katz and
Brown, 1992a].
>
I-
(/>
z
LU
cn
LlI
5
-j
UJ
cc
TEMPERATURE THRESHOLD (°C)
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Figure 3. Relative sensitivity of extreme event E2, temperature ever exceeding threshold
during entire month of July, to mean (dashed line) and standard deviation (solid line) [Source:
Katz and Brown, 1992a].
30 32
TEMPERATURE THRESHOLD TO
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Figure 4. Simulated hazard rate for the exact distribution of the maximum (dotted curve) from
a time series of length n = 30 with a first-order autocorrelation coefficient <}) = 0.5. Curves
showing theoretical hazard rates for the exact distribution of the maximum under indepen-
dence (dashed curve) and for the Type I extreme value distribution (solid curve) are included
for comparison [Source: Katz and Brown, 1992b].
> I > l i I > I
3.0 -
w
^ 20
LN
g "
J I I L
¦ ' ' I I I ¦ I ¦ I L
1 .0
3.0
MAXIMUM
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Figure 5. Relative frequency of extreme event temperature exceeding c = 35 °C on given day in July, versus standardized
threshold (c - p)/a for 30 stations in the U.S. Midwest [Source: Katz and Brown, 1992a].
0 . 5
0 . 0 H
0 . 0
0.5 1.0 1.5
STANDARD I ZED THRESHOLD
2 . 0
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Figure 6. Relative frequency of extreme event E2, temperature ever exceeding c = 35 °C during entire month of July, versus stan-
dardized threshold (c - p)/a for 30 stations in the U.S. Midwest [Source: Katz and Brown, 1992a].
O 0 . 8
0 . 0
0.5 1.0 1.5
STANDARD I ZED THRESHOLD
2 . 0
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Figure 7. Observed relative frequencies of extreme event E2, temperature in July ever
exceeding c = 37.8 °C, versus probabilities estimated using Type I extreme value distribution
with location and scale parameters fit directly to monthly maxima (for 30 stations in the U.S.
Midwest) [Source: Brown and Katz, 1992].
1.0
1 1 1 1 1 1 1 1—T
z
Id
D
(3
Ul
tr
>
tr
ui
W
OD
o
.4 .6
THEORETICAL PROBABILITY
1 .0
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28
Figure 8. Relative sensitivity of right-hand tail event E (i.e., summer total precipitation exceeding threshold c) to the median (solid
line) and to the scale (dashed line) as a function of c at Segovia, Spain [Source: Katz and Garrido, 1992],
>H
Fh
M
CO
w
CO
w
>
H
E-«
<
J
8
6 -
4 -
g 2-
0
1 ' 1 r
40 80 120
THRESHOLD (MM)
160
200
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Figure 9. Time series of standard deviation of daily maximum (solid line) and minimum (dashed line) temperature for July-August
at Phoenix, Arizona, for the period 1948-1990 [Source: Tarleton and Katz, 1993].
U
U
W
Q
O
H
Eh
<
M
>
w
Q
§
<
Q
E-«
CO
3 -
2 -
1940
1950
1960 1970
YEAR
1980
1990
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APPENDIX 1
Papers Based on Research Supported by Cooperative Agreement
Journal Articles
Katz, R.W., and B.G. Brown, 1992: Extreme events in a changing climate: Variability is more
important than averages. Climatic Change, 21, 289-302.
Katz, R.W., and J. Garrido, 1992: Sensitivity of extreme precipitation events to climate
change. Submitted to Water Resources Research.
Katz, R.W., and B.G. Brown, 1992: Sensitivity of extreme events to climate change: The case
of autocorrelated time series. To be submitted to Environmetrics.
Brown, B.G., and R.W. Katz, 1992: Regional analysis of temperature extremes: Implications
for climate change. To be submitted to Journal of Climate.
Conference Proceedings
Katz, R.W., and B.G. Brown, 1989: Climate change for extreme events: An application of the
theory of extreme values. Preprints, AMS Eleventh Conference on Probability and Statistics
in Atmospheric Sciences, Monterey, CA, pp. 10-15.
Brown, B.G., and R.W. Katz, 1991: Characteristics of extreme temperature events in the U.S.
Midwest and Southeast: Implications for the effects of climate change. Preprints, AMS
Seventh Conference on Applied Climatology, Salt Lake City, UT, pp. J30-J36.
Katz, R.W., 1991: Towards a statistical paradigm for climate change. Preprints, AMS Seventh
Conference on Applied Climatology, Salt Lake City, UT, pp. 4-9.
Brown, B.G., and R.W. Katz, 1992: Estimating the sensitivity of extreme events to climate
change: The effects of autocorrelation and choice of extreme value distribution. Preprints,
Fifth International Meeting on Statistical Climatology, Toronto, Ontario, pp. 297-300.
Tarleton, L.F., and R.W. Katz, 1993: Effects of urban heat island on temperature variability
and extremes. Preprints, AMS Eighth Conference on Applied Climatology, Anaheim, CA (in
press).
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APPENDIX 2
Reprint of "Extreme Events in a Changing Climate: Variability is More Important than
Averages" [R.W. Katz and B.G. Brown, 1992: Climatic Change, V. 21, pp. 289-302]
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EXTREME EVENTS IN A CHANGING CLIMATE:
VARIABILITY IS MORE IMPORTANT THAN AVERAGES
RICHARD W KATZ and BARBARA G BROWN
Environmental and Societal Impacts Group, National Center for Atmospheric Research *, Boulder,
CO 80307, USA
Abstract. Extreme events act as a catalyst for concern about whether the climate
is changing Statistical theory for extremes is used lo demonstrate that the
frequency of such events is relatively more dependent on any changes in the varia-
bility (more generally, the scale parameter) than in the mean (more generally, the
location parameter) of climate Moreover, this sensitivity is relatively greater the
more extreme the event These results provide additional support for the conclu-
sions that experiments using climate models need to be designed to detect
changes in climate variability, and that policy analysis should not rely on scenarios
of future climate involving only changes in means
I. Introduction
Recent hot spells and droughts, as well as evidence of a gradual warming trend in
global mean temperature (Hansen and Lebedeff, 1987, 1988), have led to a
heightened awareness of possible greenhouse gas-induced climate change.
Although it is natural that society tends to notice the extremes and variability of
weather, climate change experiments, based on the use of general circulation
models (GCMs) of the atmosphere, have to date dwelt on potential changes in
average climate (Schlesinger and Mitchell, 1987; Schneider, 1989) Likewise,
efforts to establish the statistical significance of apparent changes in the observed
climate have only been successful in detecting changes in average conditions (e g,
Solowand Broadus, 1989)
Assessments of the economic impacts of global climate change also focus on
averages, rather than on variability or extremes (Adams et al., 1990). But the
primary impacts of climate on society result from extreme events, a reflection of the
fact that climate is inherently variable (Parry and Carter, 1985). A hot spell during
the summer of 1983 in the midwestern U.S. resulted in a substantial decrease in
corn yields (Mcarns et al, 1984). Freezes during the winters of 1983 and 1985
killed a significant fraction of the citrus trees in the state of Florida (Miller and
Glantz. 1988) Droughts are a frequent cause of adverse societal impacts, for
instance, the recurrent episodes of famine in Africa (Glantz, 1987).
In spite of the need to examine how the frequency of extreme events might
change as the mean climate changes (Wtglcy, 1985; Mitchell el al, 1990), attempts
* The National Center for Atmospheric Research is sponsored by the National Science Foundation
Climatic Change 21. 289-302. 1992
© 1992 Kluwer Academic Publishers Printed in the Netherlands
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290
Richard W Kati and Barbara O. Brown
to quantify (he nature of such relationships have been rare (Meams et al, 1984;
Wigley, 1988). Here some standard statistical theory for extremes is applied to
reveal some amazingly broad generalizations that can be made about the relative
sensitivity of extreme events to the mean and variability (more generally, the
location and scale parameters) of climate. The extreme events considered in-
clude the exccedance of a threshold by a climate variable, (i) on a particular
occasion; or (ii) on one or more occasions within a certain time period (i.c., by the
maximum value of a sequence of observations). In essence, our results indicate
that (i) extreme events arc relatively more sensitive to the variability of climate than
to its average, and (ii) this sensitivity is relatively greater the more extreme the
event.
2. Statistical Model for Gimate Change
lust as the need for a paradigm to monitor global climate change has been recog-
nized (Wood, 1990), a prototype model is needed to define climate change in
statistical terms. A given climate variable X has some probability distribution (with
distribution function Fx[x) — P\X < je|), possessing a location parameter ft and a
scale parameter a\ that is, the distribution of the standardized variable
a
is assumed to not depend on cither ft or a. In the special case of Fx being the
normal distribution, the location parameter ft is simply the mean and (he scale
parameter o is simply the standard deviation. For nonnormal distributions, it is
more meaningful from a statistical perspective to deal with location and scale
parameters, rather than with the mean and standard deviation.
The relation (I) implies that the distribution function Fx can be expressed in
terms of the distribution function Fz of the standardized variable Z as
Fx{x)-Fz\{x-vVo\. (2)
Differentiating (2), the probability density function of X, F'x(x), can be expressed
as
F'x{x)-(\/o)Fj\(x-»)/o\. (3)
By use of (2) and (3), properties of location and scale parameter distributions can
be obtained directly from those for the simpler standardized variable (I).
Climate change is envisioned to involve a combination of two different statistical
operations: (l) the distribution function Fx is shifted, producing a change in loca-
tion ft, and (n) F, is rcscalcd, producing a change in scatc o Figure 1 illustrates this
concept for one arbitrary choice of Fx that happens to be positively skewed For
Climatic Change July 1992
Extreme Events m a Changing Climate
291
this particular distribution, the location parameter ft is the mode, rather than the
mean, whereas the scale parameter c),.where the constant c
denotes a threshold!,
(ii) the maximum of a sequence exceeding a threshold [event E2 — |max(A",, X2,
-<*„)> cfI
z
o
h-
U
z
z>
(1.
>-
i-
tn
z
lli
~
>
I-
m
<
CD
O
-------
292
Richard W. Katt and Barbara G. Brown
X
Ui
_J
3
UJ
(-
<*
Z
o
0 2 4 6 B 10 12
TIME t
Fig. 2. Hypothetical climate time series (solid line) of length n- II. along with threshold value c
(dashed line) used to define extreme events
Figure 2 shows a hypothetical climate time scries of length n - 11. Since X* < c, the
event E, did not occur at the particular time f 6. On the other hand, since the
threshold was exceeded on at least one occasion (actually by both X2 and A\), the
event E2 did occur.
In the notation previously introduced, the probability of (he extreme event £, is
given by
P(E,)-l~Fx(c). (4)
If the sequence {A"|, X2, ...,Xn) consists of independent and identically distributed
random variables, then the probability of the extreme event £2 is given by
P(£3)-1-|Mc))". (5)
Rather than rely on this exact expression (S), the theory of extreme values (Lead-
better et at, 1983) can be invoked to produce an approximation that holds even
when the sequence is autocorrelatcd, as is typical of climate variables. Specifically,
P{E2)~ 1 -C|a„(c-&„)], (6)
for large sequence length n. Here G is one of three possible extreme value distribu-
tions:
(i) Type/(or Gumbel)
C(*)-exp(-c»), (7)
Climatic Change July 1992
Extreme Events in the Changing Climate 293
(ii) Type II (or Frechct)
G(jc)-cxp(-jc~°), a, *>0; (8)
(lii) Type III (or Wcibull)
G(x) — cxp(-(—*)°|, a > 0, x<0. (9)
In (6), fl„ > 0 and bn are normalizing constants that depend on the original distri-
bution Fx.
For the most part, we will assume that the approximation (6) holds for G being
the Type I extreme value distribution (7). This assumption is valid if Fx is any one
of the distributions commonly fit to climate variables (c g., exponential, gamma,
lognormal, normal, squared normal, Weibull) (Essenwangcr, 1976). Nevertheless,
the Type II or III extreme value distributions have sometimes been found to pro-
vide a better fit in practice to the empincal distribution of the maximum of climate
sequences (eg., Farago and Katz, 1990). Tiago dc Oliveira (1986) and Buishand
(1989) review the application of the theory of extreme values to climatology.
3. Sensitivity of Extreme Events
We arc interested in how the probability P(E) of an extreme event E would change
as the location or scale parameters, /< or a, change. The sensitivity of an extreme
event to the location or to the scale is defined to be the corresponding partial deri-
vative of the probability of the event, that is, dP(E)/dp or dP(E)/da. Since
extreme events vary in their likelihood, it is reasonable to deal with the relative sen-
sitivity.
dP{E)
d/t
/ P{E)
dP(E)
do
/ P(E),
(10)
comparing the sensitivity of an event to its probability.
3 1. Extreme Event E,
Using (2)-(4), it follows that the two sensitivities can be expressed as
dP<>E<) r t . nn
— />(e), (II)
dn
2aS)-(£^)l5(„ «,2,
Comparing (II) and (12),
Climatic Change July 1992
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294
Richard W Kalz and Barbara G Brown
dP(E,) I c — fi\ dP(Et) (n)
da \ a I dfi
Equation (13) implies that the (relative) sensitivity of an extreme event to the scale
becomes proportionately greater than its (relative) sensitivity lo the location as the
event becomes more extreme (i e, the larger the threshold c). Since the (relative)
sensitivity to the scale is always greater than the (relative) sensitivity lo the location,
provided (c - ft)/a> 1, this condition could even be taken as an objective criterion
for what constitutes an 'extreme' event.
Another important issue concerns whether the relative sensitivity of an extreme
event lo the location (or to the scale) increases or decreases as the event becomes
more extreme. One way to attack this problem involves recognizing lhat the relative
sensitivity of event £, to the location is mathematically equivalent to the so-called
'hazard rate' or 'failure rate' for distribution function Fx\namely,
In fact, it follows directly from (11) that
^^ ]//>(£,)-tf(c) (15)
dfi
The hazard rate is a fundamental measure in engineering studies of system reliabili-
ty (Hillier and Lieberman, 1986), representing the 'rale' of occurrence of the event
\X = x) given that X > x.
The question of whether the relative sensitivity of £, to ft increases as the
threshold c increases is equivalent to asking whether Fx has a hazard rate H(x)
that is an increasing function for large x, a question that is commonly addressed in
the statistical literature (Johnson and Kotz, 1970). For instance, since the normal
distribution has a hazard rate that is an increasing function |in particular, ll(x)
increases at an approximately linear rate for large x|, the relative sensitivity of C, to
(i increases as c increases in this case. However, other forms of distribution Fx have
hazard rates that arc not strictly increasing functions, implying that the relative sen-
sitivity of £| to p need not always increase For instance, the exponential distribu-
tion has a constant hazard rate, and the Weibull distribution has either a strictly
increasing or strictly decreasing hazard rate, depending on the value of its shape
parameter.
Equation (13) implies that, if the relative sensitivity of extreme event £, to the
location either increases, remains constant, or decreases at slower than a linear rate
as the event becomes more extreme, then the relative sensitivity to the scale must
increase For instance, if Fx were the normal distribution, then the relative sensitivi-
ty of E, to a would increase at an approximately quadratic rate for large c. If rx
were the exponential distribution, then the relative sensitivity lo a would increase
at a linear rate.
Climatic Change July 1992
Extreme Events in a Changing Climate
295
3.2. Extreme Event E2
If we make use of the large-sample approximation (6), then the same relation (13)
between the sensitivity to the location and scale parameters, ft and a, also holds for
extreme event £2. no matter which of the three types of extreme value distributions
(7)-(9) arises. Moreover, analogous to (15), the relative sensitivity of C2 f ,s 'hc
same as the hazard rate, not for Fx, but for the appropriate choice of extreme value
distribution G. For the Type I extreme value distribution (7), the hazard rate is an
increasing function (with H(x) being approximately constant for large Jt|. Conse-
quently, in many cases the relative sensitivity of £2 to p should increase as the
threshold c increases. The Type II extreme value distribution (8) has a hazard rate
H(x) that is decreasing for large x. So it is possible in some circumstances that the
relative sensitivity of E2 to n would decrease as c increases.
Equation (13) can again be employed to infer the relative sensitivity of £2 to the
scale parameter a from that of £2 to p. If the Type I extreme value approximation
(7) were employed, then the relative sensitivity of £2 to a would increase at an
approximately linear rate as c increases. If the Type II extreme value approximation
(8) were employed, then the relative sensitivity of £2 to a would gradually level off
lo a constant Katz and Drown (1989) presented a more technical discussion of the
relative sensitivity of extreme event £2.
3.3 Other Extreme Events
Although only two forms of extreme event, namely £, and £2, have been explic-
itly treated here, the theoretical results concerning relative scnsiUvity apply much
more generally. For instance, a perfectly analogous theory exists for the event of
falling below a relatively small threshold, say X
-------
296
Richard W Kalz and Barbara G Brown
cally, during the month of July). The July time series of daily maximum temperature
at Des Moines, Iowa is utilized (data previously analyzed by Mcarns etal, 1984). It
is assumed that Fx is the normal distribution, so that the location parameter fi is
the mean and the scale parameter a is the standard deviation. The 31 years of his-
torical data indicate that n is about 30 *C and a is about 3.9 *C. Thresholds of
c - 35 and 38 *C are of special interest.
Figure 3a shows plots of the relative sensitivity of extreme event Et, the tem-
perature exceeding a threshold c on a given day in July, to n and a as c increases
(ie., as the event becomes more extreme). As specified by (13), the two curves
intersect when the threshold is one standard deviation above the mean. The separa-
tion between the two curves becomes greater as c increases, as anticipated, with the
relative sensitivity to n increasing at an approximately linear rate in contrast with
the approximately quadratic rate for the relative sensitivity to a. Strictly speaking,
the relative sensitivity curves shown in Figure 3a apply only to infinitesimal changes
in either n or a. To convert these results into more concrete terms, Tabic I gives the
probability of event Ex for a threshold of c - 38 "C when ft and a are changed by
±0.5 *C. Relative to the current probability of 0.020, P(E,) changes by roughly
twice as much for a change in a as for the corresponding change in fi.
Figure 3b shows plots of the relative sensitivity of extreme event C2, the tem-
perature ever exceeding c within the entire month of July (i.c, n — 31), to ft and a
as a function of c. These relative sensitivities are based on the use of the Type I
extreme value approximation for the maximum (7), with the normalizing constants,
a„ and bm, for the case of Fx being the normal distribution (see Lcadbcttcr et al,
1983, p. 14) Again, the two curves intersect at the point specified in (13) More
TABLE I Probability of extreme events, E, and
El% with threshold of c~38*C associated with
changes in mean and standard deviation a of July
daily maximum temperatures at Des Moines, Iowa
Change
in
Probability of extreme
a
event (relative
change)
CC)
CC)
P(E,)
nc,)
0*
o-
0 020
0 492
-HIS
0
0 027
0612
(+34 7%)
(+24 5%)
0
+05
0 034
0713
(+70 8%)
(+44 9%)
-05
0
0015
0 384
(-277%)
(-22 0%)
0
-05
0009
0 264
(-54 0%)
(-46 2%)
¦ Current climate of ft — 30 "C and o - 3 9 'C
Climatic Change Jufy 1992
Extreme Events m a Changing Climate
297
SO 32 34 36 38 40 42 44
TEMPERATURE THRESHOLD (°C)
Ui
in
uj
>
30 32
TEMPERATURE THRESH0L0 (°C)
Tig 3 Relative sensitivity of extreme event to mean (dashed line) and standard deviation (solid line)
versus threshold c (a) event temperature exceeding threshold on given day in July, (b) event E2,
temperature ever exceeding threshold dunng entire month of July
Climatic Change July 1992
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298
Richaid W Katz and Barbara G. Brown
importantly, the two relative sensitivities rapidly separate as the event becomes
more extreme. As the theory has established, the relative sensitivity to fi increases,
but gradually levels off to a constant, whereas the relative sensitivity to a increases
at an approximately linear rate for large c. Tabic I also includes the probability of
event £2 for the same threshold and changes in n and a. Again, the relative changes
in P(£2) are roughly twice as large when a is varied sis when n is varied. Unlike
event £, which always remains rare, event £2 becomes either quite likely or
somewhat rare depending on these seemingly small changes in a.
Since the relative sensitivity curves shown in Figure 3b are based on an approxi-
mation derived in extreme value theory, the question arises as to the realism of the
application to time series of daily maximum temperature. Using a First-order
autorcgrcssivc process to represent daily maximum temperature (the same
stochastic model assumed by Mearns er a/., 1984), both the effects of taking the
maximum over a relatively small sample (i.e., n — 31 days) and the positive day-to-
day autocorrelation of daily temperature were investigated in a simulation study. At
least qualitatively, our theoretical results turn out to be quite robust. Moreover, any
discrepancies that exist appear to be in the direction of even greater actual relative
sensitivity than the theory predicts (Katz and Brown, 1992)
As a check on the plausibility of our assumptions about climate change (Section
2), the relative frequency of occurrence of these two extreme events was recorded
at 30 sites within the midwcstcrn U.S that possess relatively long records of daily
maximum temperature (i c., 40-90 years). Figure 4 gives plots of these relative
frequencies versus the standardized threshold (c-fi)/a for c-35'C. Because
daily maximum temperature is known to have an approximately normal distribu-
tion (e g., Mearns ei a!., 1984), the location and scale parameters // and a were esti-
mated by the sample mean and standard deviation of the July time series at the cor-
responding site. If a change in the future climate were analogous to a spatial reloca-
tion, then our model for climate change would imply that the scatter plot for event
£, simply represents the right-hand tail of the distribution of daily maximum tem-
perature 1 — Fx Since the points (Figure 4a) fall remarkably close to a smooth
decreasing curve, these results serve as motivation for our statistical concept of
climate change. The plot for event £2 (Figure 4b) has a greater degree of scatter, in
part because these relative frequencies were obtained from a much smaller sample
(i c., only one observation for each July instead of 31 for event £,) Nevertheless,
the indication of an underlying relationship is present Brown and Katz (1991)
presented a more extensive validation of this climate change model.
5. Implications and Extensions
These theoretical results concerning the relative sensitivity of extreme events to the
average and variability (or, more generally, location and scale parameters) of
climate are quite compelling. Of course, our statistical model for climate change
may be an oversimplification of the actual circumstances of future climate change
Climatic Change July 1992
Extreme Events in a Changing Climate
299
(a) 0.0^
0.5 1.0 15
STANDARDIZED THRESHOLD
2 0
t 0
>•
0
. 9
o
z
UJ
z>
o
0
. e
Ul
q:
u.
UJ
0 .
, 7
>
h-
<
Ul
0 .
6
K
O) 0 . 5
0.0 05 1.0 15 20
STANDARDIZED THRESH0L0
l:ig 4 Relative frequency of extreme cvcnl versus standardized threshold (r - n)/o for 10 stations in
the US Midwest (a) event temperature exceeding c - 15 *C on given day in July, (b) event /J,
temperature ever exceeding r — 35 #C during entire month of July
Climatic Change July 19V2
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300
Richard W Katz and Barbara G. Brown
Moreover, since the theory only deals with sensitivities (i.e., partial derivatives), it
does not eliminate the possibility that the magnitude of the change in the mean
could be enough larger than that for the variance to still have a dominant effect on
the actual change in the frequency of extreme events.
Still, the implications for climate change experiments that rely on GCMs arc
clear. More emphasis needs to be placed on the validation of such models in terms
of their ability to reproduce climate variability, not just the mean climate. Experi-
ments specifically designed to detect changes in the overall variability of climate
also remain to be performed (Katz, 1988a; Rind et al., 1989, Mearns et al., 1990).
Discussion of the generation of scenarios of future climate is common (Lamb,
1987; Katz, 1988b), but so far, the enhanced greenhouse effect in terms of changes
in the overall variability of climate is not well known.
There are also some lessons for policy analysis that attempts to deal with the
societal impacts of climate change. Assessments that rely on scenarios of future
climate involving only changes in mean values or that infer changes in the frequen-
cy of extreme events from only changes in means (i.e., holding the variability of
climate constant) are suspect. Although it is true that climate modelers are current-
ly unable to provide policy analysts with much information on how cither the varia-
bility of climate or the frequency of extreme events would change, these issues need
to be addressed before impact assessments for greenhouse gas-induced climate
change can be expected to gain much credibility.
The characterization of the sensitivity of extreme events to climate change could
be extended to situations more representative of climate variables. Besides con-
sidering small samples and autocorrelation (as in work in progress mentioned in
Section 4), results could be obtained that allow for diurnal or seasonal cycles.
Further, extreme events involving climate variables with nonnormal distributions
(e g., 'drought' or 'flood' events defined in terms of total precipitation) would con-
stitute interesting case studies. Although some of the distributions treated do not
fall within the framework of location and scale parameter families, quite analogous
results have been derived (Garrido and Katz, 1992). Waggoner (1989) has at-
tempted to study this issue for precipitation, but without making any explicit use of
the theory of extreme values. Finally, more complex forms of extreme events such
as ones that take into account the duration of an excursion above a threshold (eg.,
runs of consecutive hot days as in Mearns et al, 1984) or below (c.g, cold spells as
in LcBoutillier and Waylen, 1988) could be investigated.
The issue of how to verify the applicability of this theory to actual climate
change remains problematic. Nevertheless, several observational studies of climate
could be performed that should at least produce some complementary results. For
instance, the spatial analogue introduced in Section 4 would provide one technique
for examining relationships between the probability of extreme events and statistics
such as means and variances (see Brown and Katz, 1991). Another approach
would involve the study of how the frequency of extreme events has changed for
sites which are known to have already experienced a change in climate over time -
Climatic Change July 1992
Extrtme Events in a Changing Climate
301
one such instance is the so-called 'heat island' effect associated with the growth of
cities (Balling eta!., 1990).
Acknowledgements
This work was funded in part by the U.S. Environmental Protection Agency
through Cooperative Agreement CR-8915732-01-0 with the National Center for
Atmospheric Research (NCAR) These results have not been subject to the agen-
cy's peer and policy review and therefore do not necessarily reflect the views of the
agency, and no official endorsement should be inferred. Interest in this topic was
stimulated by a collaborative research project between NCAR and the Hungarian
Meteorological Service, jointly funded by the National Science Foundation (U.S -
Eastern Europe Cooperative Science Program) and the Hungarian Academy of
Sciences. Wc thank Mary W. Downton for computer programming assistance and
Sharon K. LcDuc for comments on this work.
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