McMaster University
From the SelectedWorks of Barry A. Palynchuk PhD
Spring April 30, 2015
Changes in Heavy Rainstorm Characteristics with
Time and Temperature
Barry A. Palynchuk, PhD, McMaster University
Yiping Guo, PhD, McMaster University
Available at: http://works.bepress.com/barry_palynchuk/12/
CHANGES IN HEAVY RAINSTORM CHARACTERISTICS WITH TIME AND TEMPERATURE.
Barry Palynchuk1 *.
Yiping Guo2 .
1. AECOM Canada Ltd, Montreal, 85 Ste-Catherine St. West,*corresponding author. Tel. 514 390 2620,
e-mail [email protected].
2. McMaster University, Hamilton, ON.
ABSTRACT: The effects of climate change upon extreme rainfall is evaluated, based upon the identification
of individual storms, and the changes in their statistical parameters and distributions. Those changes will be
measured based upon historical time spans, and climatic temperature associated with the events. A brief
review and comparison with other research is provided.
Keywords: storm event; depth; duration; intensity; marginal distribution; temperature.
1. INTRODUCTION
There has been much speculation with regards to the effect of climate change upon changing patterns of
rainfall. One of the major assertions has been that rainfall intensity will increase, or that severe events
will become more frequent (IPCC, 2007, 2012). This prediction has arisen, in part as a result of modelling
studies, as well as from analysis of rainfall statistics. Our approach is analysis of rainstorms as measured
individual events so that the potential climate change impacts may be estimated. Extreme rainstorms are
events that exceed norms, and that do not occur very frequently. They cause hillside and streambank
erosion, with peak stream and sewer flows that will tax the capacity of channels and conduits.
1.1. Objectives of this paper.
In order to examine possible changes in rainstorm characteristics with time, or with changes in temperature, storm events will be identified from hourly archived data using the Inter-Event Time Definition (IETD)
(Eagleson, 1972) to identify individual, infrequent, rainstorm events. This technique separates records of
rainfall based upon a minimum time interval between hourly archived rainfall records. The individual rain
storm events may then be characterized by their total depth, duration, and peak intensity. Because of the
comparative granularity of event definition produced by this technique, these random variables may then be
analyzed in terms of their probability distributions and associated statistical parameters. From that analysis,
the effects of climate scale temperature upon the statistics of storm variables may be evaluated together
with potential changes over time.
1
2
2. STORM EVENT DEFINITION AND ANALYSIS
2.1. Threshold analysis under the assumption of stationarity.
Individual storms are defined by means of a minimum period of time between recorded rainfall. Adams
et al. (1986) examined the basis for selection of the IETD, and applied the concept for the purpose of
characterizing the marginal distributions of rain storm depth, duration, and inter-event time. This definition
of rainstorm events has been used by other researchers (Guo and Adams, 1998a, 1998b; Goel et al., 2000)
to characterize the marginal distribution or rainstorm random variables in order to develop derived probability
distributions of hydrologic outputs such as peak discharge, or runoff depth. More recently, the technique
has been used in combination with threshold exceedance analysis of storm depth. This has been applied to
include measures of peak storm intensity (Palynchuk and Guo, 2011), together with marginal distributions of
rainstorm variables linked with copulas. Similar techniques have been used to model the internal structure
of wet and dry periods within storm events (Hyase-Agyei and Melching, 2012).
Table 1. Station Descriptions, Threshold Analysis of Rainstorm Events March-November,
≤ 24 hours
Name/
Description
Sta. ID
Latitude, degrees N
Longitude, degrees W
Years, hourly rainfall
Total no. rainstorms, m
Springfield
Peoria
IL8719
IL6711
◦
39.848
40.668◦
◦
89.664
89.684◦
1949-2006 1949-2006
4567
4615
O'Hare
IL1549
41.995◦
87.934◦
1962-2006
3777
Pearson
6158733
43.677◦
79.631◦
1960-2003
3172
In this work, hourly-archived rainfall data was analyzed for airport stations at Peoria, Springfield, Chicago(O'Hare),
in the State of Illinois, as well as Pearson, the international airport serving Toronto. Peoria and Springfield
were selected as being relatively rural meteorological stations with long continuous records. Chicago and
Toronto were selected for relatively long records in urbanized areas. Basic information on each station is
shown in Table 1.
The entire record for each station was subject to threshold analysis, in order to develop the dataset of
extreme events. The techniques are summarized as follows for convenience:
• Hourly-archived rainfall data is separated into individual events. The start of a storm event is separated by at least the IETD of 6 hours from the end of the last rainfall record.
• Limit storms to durations less than or equal to 24 hours.
• Select a high storm depth threshold uv , so that there are about 3 to 5 events per year on average
exceeding this depth.
• Evaluate statistical parameters of storm depth (V, v), duration (T, t), and peak intensity (Ip , ip )
• Fit Generalized Pareto Distribution Type I (GPD I) to storm depth, and a bounded distribution to
storm duration:
(1)
P r{V > v} = Ju exp[−(v − uv )/σv ]
3
Where uv is the selected storm depth threshold, and Ju is a natural estimator of the probability of
exceedance of uv . The parameter Ju is the estimator of the probability of exceedence. It is simply:
(2)
Ju = n/m
where n and m are the total number of events exceeding the threshold of storm depth, uv and the
total number of rainstorm events, respectively (Coles, 2001).
(3)
P r{T ≤ t|T ≤ tmax } = t/tmax
where tmax is the selected maximum duration of storms under consideration, in this case, 24 hours.
A uniform distribution is shown in this case, but other bounded equations in the Beta family may be
fitted as appropriate.
• Normalize peak hourly intensity, calculating the Intensity peak factor, per the following equation:
(4)
Ipf = [(Ip /v)t − 1]/(t − 1)
then calculate moments of this reduced variate, to fit to a bounded distribution, usually from the Beta
family:
R ipf
(5)
Fipf =
0
q−1
xp−1 (1 − x)
β (p, q)
dx
where Ipf , ipf is the dimensionless index of peak-hour storm intensity. p and q are Beta distribution
parameters.
• Statistical parameters were estimated by standard techniques, as was Goodness of fit.
• The marginal distributions of V , T , and Ipf may be combined into joint probabilities in order to
assess the return periods of extreme storm events. Generally, it has been found that storm depth V
is independent of storm duration T as well as intensity peak factor Ipf , while T and Ipf are correlated,
in previous analysis carried out at two Toronto, Canada stations. That correlation is addressed with
a Copula relationship.
Table 2. Threshold statistics
Parameter
Storm depth threshold, uv , mm
Total no. of rainstorms, n, v > uv
Average storm depth, v̄, mm
Average duration, t̄, hr
¯
Average intensity peak factor, ipf
Springfield Peoria O'Hare Pearson
IL8719
IL6711 IL1549 6158733
39
37
37
25
184
238
172
144
58.298
51.543 52.881 34.962
13.201
12.036 12.034 11.927
0.294
0.299
0.296
0.339
4
This brief review is to simply establish that standard techniques are applied to ensure that there are probability distributions that model well the marginal distributions of the random variables that describe rainstorm
events. Table 2 provides a summary of the parameters estimated for each of the four meteorological stations
for the available data.
2.2. Methods.
The following sections summarize the differences between mean values, correlations, and differences between distributions. Comparisons are between storm events divided into time spans within the overall period
of record, and temperatures associated with storm events. P-values, the risk of falsely rejecting the null hypothesis, at a threshold for significance of p < 0.10 will be used to assess differences between: 1) means
of storm variables, 2)χ2 tests for differences between distributions, 3) product moment correlation between
variables, and 4) differences of percentages of numbers of events between temperature categories. The
tests are all in routine use. Parameter and test values are not shown, but rather the conclusions arising from
the tests are shown, for the sake of brevity.
2.3. Storm-event analysis - time spans.
2.3.1. Storm variables.
The full set of extreme events were broken into two subsets; events occurring prior to 1980, and those
occurring in 1980 through to the end of the available data. This division of data provided relatively large
sample sizes for early and later events. The same statistical parameters were estimated as was done for
the full dataset. In order to avoid confusion, the data spans will be referred to as pre-1980, and post-1980,
even though the correct description would be pre-1980, and post-1979.
No significant shift in mean values of storm variables describing depth, duration, or intensity peak factor (V ,
T , Ipf ) were found. There may be a trend of increasing mean storm depth at the O'Hare station, but the
trend is not significant.
2.3.2. Threshold-event frequency.
There is a significant decrease in the frequency of occurrence of extreme events exceeding the storm depth
threshold (Ju ) at Peoria. At O'Hare, there is an apparent trend of increasing frequency of extreme events,
but the change is not significant.
2.3.3. Empirical distributions of storm variables.
Storm variables were clustered into groups by 3-hour ranges of storm duration values, i.e., 21 to 24 hours,
18 to 20 hours, etcetera. The frequency of occurrence of all of the events were compared between events
occurring pre- and post-1980.
The results of one of the time spans were used to estimate the expected frequency in the other, then the
differences were compared using the χ2 test, to provide a statistical test of the significance of the differences.
There has been only one significant shift in the empirical distribution of storm depth V , and that is for O'Hare.
One of the primary contributors to that significant difference is a series of storms, 2 in 1987-88, and 2 in
2001-02 each of which exceeded 100mm. The largest event was 246mm occurring in 1987.
5
d) Pearson
Number of Events
c) O'Hare
30
30
25
25
20
20
15
15
10
10
5
5
0
0
1
to
3
4
to
6
7
to
9
10
to
12
13
to
15
16
to
18
19
to
21
22
to
24
Expected (from
1963-1979 dist)
1
Number of Events
a) Springfield
15
15
10
10
5
5
0
3
4
to
6
4
to
6
7
to
9
10
to
12
13
to
15
16
to
18
19
to
21
22
to
24
Expected (from
1960-1979 dist)
Observed
(1980-2003)
b) Peoria
20
to
3
Observed,
1980-2006
20
1
to
7
to
9
10
to
12
13
to
15
16
to
Storm Duration Range, hours
18
19
to
21
22
to
0
24
Expected (from
1949-1979 dist)
1
to
3
4
to
6
7
to
Observed,
1980-2006
9
10
to
12
13
to
15
16
to
18
19
to
21
22
to
24
Expected
(1949-1979
dist)
Observed
(1980-2005)
Storm Duration Range, hours
Figure 1. Storm duration frequency by 3-hour duration increments, pre- and post-1980
There has been significant change in the empirical probability distribution of storm durations at Springfield
and O'Hare. There is some indication of a change in distribution of storm durations between periods prior
to 1980, and the period since that time at Pearson, but it is not statistically significant. Both Springfield and
O'Hare show similar patterns of a reduction in the frequency of storms in the 7 to 9 hour duration range, with
increases in shorter duration ranges, as shown in Figure 1.
2.3.4. Summary of test results.
Table 3. Summary of major time-span changes and trends
Section
Subsection
2.3 Storm-event analysis
Test, Change
- time spans
2.3.1 ∆ means
v̄;t̄;īpf
2.3.2Threshold-event freq. ∆Ju
2.3.3 Empirical Dist
- pre and post 1980
Empirical Dist V
∆fV ;χ2
Empirical Dist T
∆fT ;χ2
Empirical Dist Ipf
∆fIpf ;χ2
Springfield Peoria
IL8719
IL6711
O'Hare
IL1549
Pearson
6158733
NC
NC
NC
⇓
NC
NC
NC
NC
NC
Change
NC
NC
NC
Change
Change NC
Change NC
Change Change
The results of sections 2.3.1, 2.3.2, and 2.3.3 are presented in Table 3. Note that Double-shafted arrows
are used to indicate the direction of change for significant changes, single shaft arrows show the direction
of changes that are not statistically significant. "NC" stands for no change, ∆ stands for changes between
time spans. A significant change between the empirical distributions of intensity peak factor is apparent for
6
all stations, except Springfield. However there is no clear pattern of changes in intensity peak factor among
the empirical distributions for stations at Peoria, O'Hare, and Pearson.
2.3.5. Correlation between storm variables.
One of the key steps in developing a joint probability distribution between storm variables is the determination
of the degree of dependence between them. Pearson's r, the product-moment correlation coefficient, was
determined for correlations of V and T , V and Ipf , and Ipf with T .
Table 4. Summary of major time-span correlation changes and trends
Section
Springfield Peoria O'Hare Pearson
Subsection
Test, Change IL8719
IL6711 IL1549 6158733
2.3.5 Storm-event analysis - time spans
Correl. storm var.
∆r; Ipf - T
NC
NC
NC
NC
Correl. storm var.
∆r; V - Ipf
↑ (−)
↑ (−)
↑ (−)
↓
Correl. storm var.
∆r; V - T
⇑ (−)
↑
NC
NC
Correlation between Ipf and T is strongly negative and significant at all stations under both time spans examined. Correlation between V and Ipf is low, and not significant in most cases. The results for correlations
between V and T between the two time spans are more mixed. For 3 of the 4 stations, Peoria, O'Hare, and
Pearson, correlation increases, but changes are not significant, or only marginally so. For Springfield, there
is a significant change from a positive, to a negative correlation between early and later time spans. Results
are summarized in Table 4.
2.4. Storm-event analysis - temperature.
2.4.1. Storm variable - temperature correlation.
The mean monthly temperature (M M T ) for the month of occurrence of any given extreme rainstorm was
obtained, and storm variables were assessed against this climatological measure, using the product-moment
correlation statistic, r.
All stations show similar patterns of significant negative correlation between increasing temperature and
storm duration, and significant positive correlation between intensity peak factor (Ipf ) and increasing mean
monthly temperature, regardless of time span. Peoria, O'Hare, and Pearson stations show a decrease in
correlation between storm depth with increasing mean monthly temperature. One station only, Springfield,
shows an increased correlation between V and mean monthly temperature.
2.4.2. Temperature range analysis and time-spans.
The events were grouped into temperature ranges separately for each station. Mean values of V , T , and
Ipf in each range were calculated. Product-moment correlation tests showed strong positive correlation
between mean monthly temperature and Intensity peak factor, with strong negative correlation between
storm duration and temperature.
7
d) Pearson
0.0587
0.0587
25%
0.10
0.20
20%
0.2721
15%
0.30
0.3348
0.1448
0.1
0.2049
0.2
20%
15%
0.3
10%
0.0003
g
de
g
to
23
de
g
21
de
1980-2003
21
g
g
19
1960-79
p-value
p-value
b) Peoria
0
30%
0.1
25%
de
de
de
g
1980-2006
0.0910
0.1832
0.0289
35%
0.0091
0.0536
0.1062
30%
0.1167
0.1233
0.00
0.10
25%
0.2
20%
15%
0.5
9
1963-79
2
0.0047
35%
0%
g
to
to
de
19
.5
22
5
7.
to
to
g
de
17
19
.5
22
17
to
g
de
to
14
19
0.3
0.3567
10%
0.20
0.2514
20%
0.2991
15%
0.30
p-value
a) Springfield
to
g
de
9
9
14
de
g
4
to
de
to
6
9
4
to
g
to
0
de
to
0.50
6
0.4808
0.4835
14
0%
0.4
5%
g
0.40
14
10%
5%
Percentage of events
0.1230
25%
0.2045
0.0
0.0514
30%
0.1137
0.00002
35%
0
Frequency of events
30%
0.00
p-value
c) O'Hare
35%
10%
0.4574
5%
0.4
0.4399
0.50
de
.5
27
1949-79
23
.5
to
to
22
Temperature range
0%
g
g
de
g
.5
23
22
de
to
19
14
to
19
de
g
g
de
g
14
de
9
to
9
to
6
to
6
de
g
0.5
0
0.40
5%
0%
1980-2006
p-value
0
to
5
de
g
5
to
9
de
g
9
to
14
de
g
14
to
17
de
g
17
to
19
de
g
19
5
to
.
22
de
g
.5
22
to
24
de
g
24
to
27
de
g
1949-1979
1980-2005
Temperature range
p-value
Figure 2. Storm frequency versus temperature ranges. p-value of difference in proportions
shown on reverse scale, significance assessed as p < 0.10
Using these same temperature ranges, the proportions of storms within each were examined for the two
time spans under investigation. Fig. 2 shows bar charts for each of the stations comparing the frequency
of storms for each time span. The p-values of the differences between frequencies for a given temperature
range are shown in a reverse scale so that a highly significant difference between proportions of events
between the two time spans appears near the top of the chart. In Table 5, %n is the proportion of total
threshold exceedence events in the highest temperature range of mean monthly temperature evaluated for
each station, the measure used to evaluate this change.
For all stations, the frequency of occurrence of storms is significantly higher in the highest temperature range
for each station post-1980. There is no consistent pattern in change in the proportions of storms occurring
in lower temperature ranges.
Table 5. Summary of major time-span and temperature changes and trends
Section
Springfield
Subsection
Test, Change
IL8719
2.4 Storm-event analysis
- temp; change over time
2.4.1 V - MMT correl
∆r, V - MMT
⇑
2.4.1 T - MMT correl
∆r, T - MMT
NC
2.4.1 Ipf - MMT correl
∆r, Ipf - MMT
NC
2.4.2 Temp. range vs. time ∆%n, top MMT
⇑
¯ T (NS)
2.4.3 Avg. of MMT
∆M M
↑
Peoria O'Hare
IL6711 IL1549
Pearson
6158733
⇓
NC
NC
⇑
↓
⇓
NC
NC
⇑
↑
⇓
NC
NC
⇑
↑
The mean monthly temperature associated with each threshold-excess event has been used as a means of
evaluating the relationship between storm variables and temperature. The significance of changes in correlation between the two time spans was evaluated. Table 5 provides the summary of changes in correlations
8
between storm variables and M M T . As well, the average of this mean monthly temperature was calculated
for the two time spans. There was no significant change in the mean of M M T pre and post-1980.
3. INTERPRETATION AND CONCLUSIONS
3.1. Section 2.3 Storm-event analysis - time spans, Table 3:
Means of storm variables did not show any significant change between the time span prior to 1980, and that
following. Average probability of exceedance of threshold events was not significantly different between the
two time spans, with the exception of Peoria, where a significant decrease in the frequency of occurrence
of threshold events appears. Empirical distributions - O'Hare storm depth indicates a significant difference
between earlier and later time spans, driven by 4 storms of depth exceeding 100 mm occurring in the later
time span. Two of the four stations, Springfield and O'Hare, have significantly different distributions of storm
duration between the time span pre-1980, and post-1980 inspite of the lack of change in mean values. The
intensity peak factor empirical distribution is significantly different for 3 of the 4 stations between the two
time spans.
Table 4 - Strong correlations between storm duration and intensity peak factor were unchanged between
time spans. Correlations, or lack of correlation between storm depth and intensity peak factor did not change
between time spans. Correlations between storm depth and duration increased negatively for Springfield
and positively for Peoria.
3.2. Section 2.4. Storm-event analysis - temperature correlation:
Table 5 - Increasing temperature leads to shorter storm durations, and higher relative peak intensity within
storms, this is consistent with high resolution modelling carried out in the UK (Kendon et al, 2014). Most
stations showed decreasing rate of increase of storm depth with temperature. This may be a result of
increasing CO2 leading to a decrease in the intensity of the hydrological cycle, because of a reduction in
the rate of upward radiation of latent heat flux through the troposphere released by precipitation (Allen and
Ingram, 2002). This also confirms the work of Groisman (2010).
Table 6. Sensitivity of rainstorm variables V, T, Ipf to change in MMT for Springfield, (IL8719).
Variable
- V
T
Ipf
Slope of regression
- 0.689 mm/◦ C -0.430 hours/◦ C 0.009/◦ C
Percentage change from mean value, Table 3. - 1.2%
-3.2%
3.3%
Analysis of the statistics forming the basis of product-moment correlation coefficients provides some indication of sensitivity of storm variables to changes in mean monthly temperature. Using the Springfield station
as an example, then the least squares regression coefficient arising from correlations of storm variables
with M M T for the time span post-1980 provide a measure of the average change in storm variables per
degree-change in M M T . Changes in mean storm depth, duration, and intensity peak factor per degree rise
in M M T are shown in Table 6, along with the percentage change. It is clear from this, that even selecting
the station showing the greatest correlation of storm depth with M M T , sensitivity of storm depth is less than
half of that for storm duration and Ipf .
Fig. 2 - All stations have an increase in the frequency of extreme events in the highest temperature ranges.
This is supported by the summary of Karl and Trenberth (2003).
9
Table 5 - No significant increase in average of mean monthly temperatures associated with extreme events;
the increased numbers of events at high temperatures is offset by an increase in the number of events at
lower temperatures; warming late-winter and spring temperatures with increasing rainfall may be an explanation.
Some of the anomolous effects, particularly in rural Illinois, may be due to non-GHG climate change effects.
Groisman (2010) and Changnon et al (2003) have both provided evidence that changes in crop practices
may have a greater impact upon changes in rainfall than GHG-driven warming. This may account for the
contrary changes at Springfield and Peoria.
The definition of individual storms has provided the means to assess the impact of climate change upon
extreme rainfall. Conventional Intensity-duration-frequency (IDF) cannot provide the necessary level of
detail to permit the same assessment, since less information is available. Results are mixed; Hogg and
Hogg (2011) could not determine any clear trends in the Toronto area, while Adamowski et al (2009) project
significant trends in increasing intensity in regions to the east of Toronto.
In this paper, we have been able to show that storm durations decrease and peak intensities increase with
rising mean temperatures. Storm depth is not tightly correlated to rising temperatures, but more extreme
storm events are occurring at higher temperatures. Rising mean temperatures will lead to shorter storms,
with greater peak intensity, but total storm depth will not change to the same degree.
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