HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-6461-2017Does nonstationarity in rainfall require nonstationary intensity–duration–frequency curves?GanguliPoulomipoulomi.ganguli@alumnimail.iitkgp.ac.ingangulip@mcmaster.cahttps://orcid.org/0000-0002-2372-1121CoulibalyPaulinDepartment of Civil Engineering, McMaster Water Resources and Hydrologic
Modelling Group, McMaster University, 1280 Main Street West, Hamilton, ON L8S 4L7, Canadanow at: GFZ German Research Centre for Geosciences, Sect. 5.4 Hydrology, 14473 Potsdam, GermanyPoulomi Ganguli (poulomi.ganguli@alumnimail.iitkgp.ac.in,
gangulip@mcmaster.ca)18December20172112646164837June201710November201731October201712June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/6461/2017/hess-21-6461-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/6461/2017/hess-21-6461-2017.pdf
In Canada, risk of flooding due to heavy rainfall has risen in recent
decades; the most notable recent examples include the July 2013 storm in the
Greater Toronto region and the May 2017 flood of the Toronto Islands. We
investigate nonstationarity and trends in the short-duration precipitation
extremes in selected urbanized locations in Southern Ontario, Canada, and
evaluate the potential of nonstationary intensity–duration–frequency (IDF)
curves, which form an input to civil infrastructural design. Despite apparent
signals of nonstationarity in precipitation extremes in all locations, the
stationary vs. nonstationary models do not exhibit any significant
differences in the design storm intensity, especially for short recurrence
intervals (up to 10 years). The signatures of nonstationarity in rainfall
extremes do not necessarily imply the use of nonstationary IDFs for design
considerations. When comparing the proposed IDFs with current design
standards, for return periods (10 years or less) typical for urban drainage
design, current design standards require an update of up to 7 %, whereas
for longer recurrence intervals (50–100 years), ideal for critical civil
infrastructural design, updates ranging between ∼2 and 44 % are
suggested. We further emphasize that the above findings need re-evaluation in
the light of climate change projections since the intensity and frequency of
extreme precipitation are expected to intensify due to global warming.
Introduction
Short-duration extreme rainfall events can have devastating consequences,
damage to crops and infrastructures, leading to severe societal and economic
losses in Canada (CCF, 2013; TRCA, 2013). In a warming climate, extreme
precipitation events are expected to intensify due to moistening of the
atmosphere (Donat et al., 2016; Fischer and Knutti, 2016; Pendergrass et al.,
2015; Prein et al., 2016; Pfahl et al., 2017). Using observational records, a
review of the literature suggests a dependency between mean and extreme
precipitation on temperature (O'Gorman, 2015). The increased water-holding
capacity of warmer air, as governed by the Clausius–Clapeyron (C–C)
relation (Lenderink and van Meijgaard, 2008; O'Gorman and Schneider, 2009;
Wasko and Sharma, 2015, 2017), intensifies heavy rainfall at a rate of
approximately 7–8 % ∘C-1 of warming. On a local
scale, for sub-hourly and up to 6-hourly extreme precipitation, increases at
or above the C–C rate have been found in the Netherlands (Lenderink and van
Meijgaard, 2008; Lenderink et al., 2017), Switzerland (Ban et al., 2014),
Germany (Berg et al., 2013), the UK (Blenkinsop et al., 2015), the
Mediterranean (Drobinski et al., 2016), most of Australia (Wasko and Sharma,
2015, 2017; Schroeer and Kirchengast, 2017), North America (Shaw et al.,
2011), and China (Miao et al., 2016), while in India (Ali and Mishra, 2017)
and northern Australia (Hardwick Jones et al., 2010) negative rates have been
reported. The extent of urbanization also contributes to extreme regional
precipitation through the urban heat island effect and aerosol concentration
(Dixon and Mote, 2003; Mölders and Olson, 2004; Guo et al., 2006; Mohsen
and Gough, 2012; Wang et al., 2015). One of the first attempts to derive
nonstationary IDF through the Bayesian inference (BI) approach for extreme
value analysis was by Cheng and AghaKouchak (2014), where the authors
introduced a linear trend in the parameters of the selected distribution.
Agilan and Umamahesh (2017) used six physical processes, namely, time,
urbanization, local temperature changes, the annual global temperature
anomaly (as an indicator of global warming), El Niño–Southern
Oscillation (ENSO), and the Indian Ocean Dipole (IOD) as covariates for the
nonstationary extreme precipitation analysis in the city of Hyderabad, India.
Their analysis indicated that the local processes,
urbanization, and local temperature changes are
the best covariates for short-duration rainfall, whereas global processes,
such as global warming, the ENSO cycle, and IOD, are the best covariates for
the long-duration rainfall. In their study, time was never qualified as the
best covariate for modelling local-scale extreme rainfall intensity. Singh
et al. (2016) performed nonstationary frequency analysis of Indian Summer
Monsoon Rainfall extreme (ISMR; defined as cumulative rainfall over
continental India from 1 June to 30 September) and found evidence of
significant nonstationarity in ISMR extremes in urbanizing or developing
urban areas (transitioning from rural to urban), as compared to completely
urbanized or rural areas. However, their analysis was performed at a spatial
resolution of 1∘ using gridded daily precipitation data obtained from
the Indian Meteorological Department (IMD). Ali and Mishra (2017) showed that
a strong (higher than the C–C rate) positive relationship exists between
short-duration rainfall extremes, dew point, and tropospheric temperature
(T850; or the temperature in the upper troposphere at 850 hPa) over
23 urban locations in India. The latter two were subsequently used as
covariates for nonstationary design storm estimates. The results indicated an
increase in rainfall maxima at a majority of locations assuming nonstationary
conditions over stationary atmospheric conditions. In contrast, in another
studies, over Melbourne and Victoria, in Australia, Yilmaz et al. (2014,
2017) found superiority of stationary models over nonstationary models. To
develop nonstationary models, the authors (Yilmaz et al., 2014, 2017)
considered both the time dependency and dependency on large-scale climate
oscillations affecting Australian rainfall. Using temperature as a covariate
for nonstationarity, Wasko and Sharma (2017) investigated the sensitivity of
extreme daily precipitation and streamflow to changes in daily temperature.
Their results suggested a little evidence of an increase in
streamflow, with an increase in heavy rainfall events at higher temperature.
However, most of these previous studies have analysed changes in expected
point estimates of nonstationary vs. stationary design storm intensity
(hereafter referred to as DSI), but have not reported the statistical
significance of the difference between the two methods of estimates. To our
best knowledge, no thorough comparison of stationary vs. nonstationary
methods for deriving IDF statistics has been conducted in Southern Ontario,
Canada. For densely populated Southern Ontario, observations and multiple
climate models suggest increasing trends in regional surface temperature and
extreme precipitation in recent decades (Stone et al., 2000; Paixao et al.,
2011; Mailhot et al., 2012; De Carolis, 2012; Burn and Taleghani, 2013;
Shephard et al., 2014; Deng et al., 2016). A recent study shows an increase
in local surface temperature of 3.06±0.18∘C century-1
in the Greater Toronto Area (GTA) since the 1960s (Berkeley Earth, 2017). In
July 2013, a single storm event has resulted in 126 mm of rainfall in
the GTA, causing total insured losses of around USD 940 million, and is
claimed to be the third-most expensive weather-related event in Canada (CDD,
2015; TRCA, 2013).
Extreme rainfall statistics are often mathematically expressed using the
concept of exceedance probability or T-year return period (i.e. T=1/(1-Fp(P)), where Fp(P) is the cumulative probability of the underlying
distribution), and graphically as a decision relevant metric in the form of
intensity–duration–frequency (IDF) curves (or relations) (ASCE, 2006; CSA,
2010; EC, 2012). These curves are based on a comprehensive statistical
analysis of historical rainfall records and are widely used for the design
and operation of storm-water and sewerage systems, and other engineered
hydraulic structures (Coulibaly and Shi, 2005; Durrans and Brown, 2001; Lima
et al., 2016; Madsen et al., 2009; Rana et al., 2013; Sandink et al., 2016;
Yilmaz et al., 2014a). At a given return period and storm duration, the
average DSI is determined from the IDF relationship. The IDF curves are based
on fitting a theoretical probability distribution to short-duration
(sub-hourly, hourly, and daily) annual maximum precipitation (AMP). The
approach can be implemented both locally (at site) or regionally (Svensson
and Jones, 2010; regional frequency analysis (RFA) or pooled). The RFA is
used when available record lengths are short or at locations where no
observed data are available (Castellarin et al., 2012; Komi et al., 2016).
However, various RFA estimation methods have certain drawbacks; for instance,
the index flood method is sensitive to the homogeneity assumption and
formation of regions; in a Bayesian method of regionalization, the prior
distributions of parameters are often not precise enough and do not add
precision to the estimates. Komi et al. (2016) summarize limitations and
advantages of some of the widely used RFA techniques. In the present study,
the available records across all sites range between 47 and 66 years, which
are more than the climatology (often over time periods of 30 years) of
a region. Therefore, we employ at-site frequency analysis herein. This also
allows a consistent comparison with the Environment Canada (EC) IDFs that
have been used in practice in the study area. For Canada, information for
preparation of IDFs and nation-wide IDF curves is archived at the EC
Engineering Climate Datasets (EC, 2014;
http://climate.weather.gc.ca/prods_servs/engineering_e.html), which is
produced based on short-duration available rainfall records from the
tipping-bucket rain gauges (TBRGs). Nevertheless, the methodology to derive
existing IDF curves has certain drawbacks, for example that the current IDF
curves in Canada are based on the assumption of stationarity, which implies
that statistical properties of hydroclimatic time series will remain the same
over the period of time. However, the impact of urbanization and
human-induced climate change (IPCC SREX, 2012; Villarini et al., 2009a; Milly
et al., 2009; Kunkel, 2003) raises the question of whether the stationarity
assumption to derive IDF curves is still reliable for urban infrastructural
planning (Sarhadi and Soulis, 2017; Cheng and AghaKouchak, 2014; Jakob, 2013;
Yilmaz et al., 2014a; Yilmaz and Perera, 2013).
The nonstationary behaviour of rainfall extremes is already being reflected
in the increase in the frequency or magnitude of such events, resulting in
a shift in its distribution (Figure SPM 0.3 in the IPCC SREX, 2012; Fig. S1
in the IPCC AR5 working Group Report, Stocker et al., 2013). For instance,
seasonal and annual extreme precipitation in the northern–central and
eastern US in 2013 (Knutson et al., 2014), extreme rainfall in the Golden Bay
region in New Zealand (Dean et al., 2013), an increase in the summer
precipitation rate in northern Europe (Yiou and Cattiaux, 2013), and
successive winter storm events in southern England in 2013/14 leading to
severe winter floods (Schaller et al., 2016) are primarily attributable to
intrinsic natural variability and partly to anthropogenic influences. The
asymmetric changes in the distribution of extremes owing to climate change
have subsequently been validated for winter temperature extremes over the
Northern Hemisphere (Kodra and Ganguly, 2014) and regional short-duration
precipitation extremes in India and Australia (Mondal and Mujumdar, 2015;
Westra and Sisson, 2011). Two of the recent studies (Deng et al., 2016;
Mailhot et al., 2012) analysed a large ensemble of CMIP3 global climate model
(GCM) runs and a sub-set of regional climate models that are part of the
North American Regional Climate Change Assessment Program (NARCCAP) in terms
of impact-relevant metrics over Canada. Both studies confirmed a relative
increase in intensity and magnitude of rainfall extremes, especially over
Southern Ontario. This issue has come to our attention in the Guideline for
Canadian water resources practitioners (CSA, 2010), which urges the need for
updated IDF calculations:
[...] climate change will likely result in an increase in the intensity and
frequency of extreme precipitation events in most regions in the future. As
a result, IDF values will optimally need to be updated more frequently than
in the past [...].
Furthermore, so far very few studies have reported the difference between the
updated vs. EC-generated IDFs, taking into account nonstationarity in design
consideration. Simonovic and Peck (2009) compared updated vs. EC IDFs for the
city of London, Ontario, and reported that EC IDF curves show a difference of
the order of around 20 %. However, their analysis was based on the
stationarity assumption of precipitation extremes. Similarly, Coulibaly
et al. (2015) have compared EC IDFs with stationary GEV-based IDF curves
across Southern Ontario; however, no nonstationary methods were investigated.
Motivated by these research gaps, here we address several important questions
pertaining to short-duration precipitation extremes over Southern Ontario, to
improve pro-active management of storm-induced urban flooding. First, is
there any signature of statistically significant nonstationary trends
(gradual or monotonic changes), change points, or regime shifts (occurrence
of any abrupt changes in the mean/variance of the distribution) in
short-duration AMP in densely and moderately populated urbanized locations
across Southern Ontario? Second, does nonstationarity in the time series
necessitate the use of nonstationary IDFs, barring economic consideration and
the mathematical complexity involved in the design? Third, how can we use
this knowledge to assess the credibility of existing EC-generated IDFs
against the backdrop of a changing climate? We do not attempt to provide
a methodological comparison of EC-generated IDFs vs. the current approach,
but will focus on differences in estimated DSI values between the updated and
EC IDFs. Further, to this end, we test the hypothesis that signatures of
nonstationarity in rainfall extremes do not necessitate the use of
nonstationary IDFs for design considerations. In general, urban drainage
areas have substantial proportions of impervious or semi-impervious land
cover, which significantly reduce response time to extreme precipitation and
increase the peak flow, resulting in storm-induced floods (Miller et al.,
2014). Hence, it is the short-duration precipitation extremes which control
the design of urban infrastructure (Mishra et al., 2012). Therefore, we focus
our analysis on AMP intensity. We select Southern Ontario as a test bed
because of the majority of stations with more than 30 years of available
rainfall records (Adamowski and Bougadis, 2003; Deng et al., 2016; Shephard
et al., 2014). Recent studies have indicated that the region is more
vulnerable to climate change than any other part of Canada (Deng et al.,
2016; Mailhot et al., 2012). Furthermore, Southern Ontario is one of the
prominent economic hubs, with the largest population concentration in Canada
(Bourne and Simmons, 2003; Kerr, 1965; Partridge et al., 2007). In this
context, we explore a robust statistical framework to evaluate possible
nonstationary trends, analyse the frequency of urban precipitation extremes,
and assess the risk of severe rain-induced urban flooding in Southern Ontario
(Table 1).
Selected station locations, population distribution, and hourly and
daily data availability.
1 CMA and CA denote census metropolitan
area and census agglomeration respectively. Statistics Canada defines a CMA
as having a population density of at least 100 000, where the urban core of
that area has at least 50 000 people, whereas CA must have an urban core
population density of at least 10 000. A population centre (or urban area)
is an area with at least a population of 1000 and a density of 400 or more
people per square kilometre. All population information is collected from the
Statistics Canada (https://www12.statcan.gc.ca/) website.
2 Missing values are infilled using observations from the nearest
Environment Canada station, ID 6144475 (latitude 44∘ and longitude
-81.5∘), located at 111.5 km geodesic distance. Annual
maxima values of missing years or durations are obtained by disaggregating
daily data to hourly and sub-hourly time steps.
(a) Selected urbanized sites in Southern Ontario. Southern
Ontario (41–44∘ N, 84–76∘ W) is the southernmost region
of Canada and is situated on a southwest–northeast transect, bounded by
lakes Huron, Erie, and Ontario. The nine locations on the map are (from the
southwestern to northeastern corner): Windsor Airport, London International
Airport, Stratford Wastewater Treatment Plant (WWTP), Fergus Shand Dam,
Hamilton Airport, Toronto International Airport, Oshawa Water Pollution
Control Plant (WPCP), Trenton Airport, and Kingston Pumping Station.
Topography map indicates the maximum slope of 670 m above mean sea
level. (b) The population map shows six of the the sites: Windsor
Airport, London International Airport, Hamilton Airport, Toronto
International Airport, Oshawa WPCP, and Kingston P. Station are located
either in or the vicinity of densely populated urbanized area. The remaining
three sites are located in the moderately populated area. The short-duration
AMP records in all locations vary between the minimum of 46 years and the
maximum of 66 years.
Schematics of the process flow (blue – input steps, orange –
process steps, and green – decision steps). All three tests –
Mann–Kendall, Pettitt, and Mann–Whitney – check for shifts in the mean.
While Mann–Kendall checks for monotonic
trends, the other two tests, Pettitt and Mann–Whitney, check for a change
point or regime shift in the time series.
Spatial distribution of trends, change points, and nonstationarities
in rainfall extremes of several durations in nine
urbanized locations, Southern Ontario (a–g). The up and down triangles in white indicate (statistically insignificant)
upward
and downward shifts; the up and down triangles in cyan and orange indicate shifts with change points only; the up and down triangles
in dark blue and red show the presence of (statistically significant) trends,
including change point(s). A “×” symbol in the
triangle indicates nonstationarity detected through Priestley and Subbarao test statistics. All tests are performed at 10 %
significance levels, i.e. p value <0.10.
DSI estimates of the median (horizontal line within the box plot)
and 95 % credible intervals for 100-year return periods
of stationary vs. nonstationary models across nine sites (a–i). The boxplots indicate the uncertainty in estimated DSI
using Bayesian inference.
Percentage changes (top panel) and Z statistics (bottom panel) of
at-site T-year event estimates for T=2-year to T=100-year return
periods (a–d) with durations between 15 min and 24 h in nine
urbanized locations, Southern Ontario. The Z statistic represents the
statistical significance of differences in DSI obtained from the best
selected nonstationary vs. stationary model. The Z statistic is
statistically significant when |Z|>1.64 at the 10 % significance level.
The shades in blue and red denote decrease and increase in Z statistics,
with the strength of the shading representing the magnitude of the test
statistics. The durations with significant autocorrelations are excluded from the analysis.
Central tendency (median, b) and the bounds (95 %
credible interval, a and c) of the updated nonstationary
vs. EC-generated T=2- and 10-year event estimates for DSI at selected
return periods with durations between 15 min and 24 h. The DSI and
associated 95 % confidence limits of EC-generated IDF are obtained from
the national archive of Engineering
Climate Datasets (http://climate.weather.gc.ca/). The shades in blue and red denote decrease and increase in DSI. The strength
of the shading represents the magnitude of the ratio between updated and
EC-generated DSI.
Central tendency (median, b) and the bounds (95 %
credible interval, a and c) of the updated nonstationary
vs. EC-generated T=50- and 100-year event estimates for DSI at selected
return periods with durations between 15 min and 24 h. The DSI and
associated 95 % confidence limits of EC-generated IDFs are obtained from
the national archive of Engineering
Climate Datasets (http://climate.weather.gc.ca/). The shades in blue and red denote decrease and increase in DSI. The strength
of shading represents the magnitude of the ratio between updated and EC-generated DSI.
Estimated nonstationary vs. EC-generated IDFs for T=2-, 5-, 10-,
25-, 50- and 100-year return periods for the selected urbanized locations in
Southern Ontario, Canada. The updated and EC IDFs are shown using solid and
dotted lines respectively.
Study area and dataStudy area
Southern Ontario is situated on a southwest–northeast transect, in the
southernmost Canadian region, and is separated from the United States by
lakes Erie, Huron, and Ontario (Fig. 1). The study area includes nine densely
and moderately populated urbanized and anthropogenically altered locations of
the Windsor–Kingston corridor (Fig. 1; Table 1). The last column in Table 1
shows a list of missing years and AMP values for each duration at each
station. The digital elevation model (DEM) of the study area was derived from
the Shuttle Radar Topography Mission (SRTM) 90 m Digital Elevation
Database v4.1 (Jarvis et al., 2008), which indicates a shallow slope with
a maximum altitude of 670 m above mean sea level (MSL). The proximity
to Great Lakes and topographic effect, especially in areas to the lee of Lake
Erie, Lake Ontario, and the Georgian Bay significantly modifies the climate
in the region (Baldwin et al., 2011). Convective showers and thunderstorms
primarily modulate the summer rainfall, but fall rainfall is dominated by
reduced convective activity and increased lake effect precipitation (Lapen
and Hayhoe, 2003). Further, the topographic features and associated westerly
winds in the Niagara Escarpment and the Oak Ridge Moraine play a significant
role in modulating rainfall in the Toronto region. On the other hand, the
Windsor metropolitan area, the southernmost urbanized location in the region,
has a humid continental climate, which results in warm summer temperatures
(30 ∘C or higher), with the greatest precipitation in the spring and
summer seasons, and lowest in the autumn and winter (Sanderson and Gorski,
1978). Moreover, because of the part of Windsor–Detroit international
transborder agglomeration, the extreme summer precipitation in the city of
Windsor is primarily influenced by convection and urban heat island effect
(Sanderson and Gorski, 1978; De Carolis, 2012).
Hydrometeorological data
We identified the station locations (Fig. 1b) based on the quality of
long-range rainfall records (e.g. 30 years or more) and 2011 Census
information archived at the Statistics Canada (SC, 2016) website
(https://www12.statcan.gc.ca). The geographic areas of these locations
are extracted from 2011 Census digital boundary shape files
(https://www12.statcan.gc.ca/census-recensement/2011/geo/bound-limit/bound-limit-2011-eng.cfm).
The Toronto metropolitan area is the most populous (over 5 million in
population) and known to be one of the fastest growing population bases in
Canada (http://torontosvitalsigns.ca/main-sections/demographics/),
while Fergus is the least populated (population of around 19 000) (Table 1)
city. The other cities have population ranges between ∼500000
(Hamilton) and 30 000 (Stratford) (Table 1). We obtained AMP observations at
particular durations (15 and 30 min, 1, 2, 6, 12, and 24 h) with a few data
gaps from Engineering Climate Datasets archived and maintained by the EC (EC,
2014; http://climate.weather.gc.ca/prods_servs/engineering_e.html). The
rainfall records collected from TBRG are thoroughly quality controlled
(Shephard et al., 2014). These records have been previously analysed for the
assessment of national extreme rainfall trends (Burn and Taleghani, 2013;
Shephard et al., 2014). We consider seven storm durations ranging from 15 and
30 min (the typical time of concentration for small urban catchments) to 1,
2, 6, 12, and 24 h (the standard time of concentration for larger
watersheds) following a previous study (Bougadis and Adamowski, 2006). Except
for a few stations (for example, Toronto International Airport and Trenton
Airport), for most of the sites, the AMP observation is available either
until the year 2007 or before (Table 1). Also, we found missing values in the
AMP time series in all sites. We obtained daily and hourly rainfall records
from Environment and Climate Change Canada (ECCC, 2017).
Methods
Figure 2 shows schematics of the overall analysis. In the subsequent
subsection, we will discuss each of these steps in detail.
Infilling missing AMP record
We infilled missing values and updated the AMP records by successively
disaggregating daily rainfall values to hourly and sub-hourly time steps
using a multiplicative random cascade (MRC)-based disaggregation tool. The
cascade-based disaggregation model for continuous rainfall time series was
suggested by (Olsson, 1995, 1998). The technique was later successfully
implemented by (Güntner et al., 2001; Jebari et al., 2012; Rana et al.,
2013) for temporal disaggregation of point rainfall and the development of
IDF curves from short-duration rainfall extremes. Due to freezing weather
conditions during winter, most of the TBRGs' are inoperative from early
November to late April of the following year. Therefore, when short-duration
rainfall records were not available, the AMP values over moving windows of
n durations (n varies from 15 to 30 min, and between 1, 2, 6, 12, and
24 h) are extracted from May to October (warm season) disaggregated rainfall
volumes for remaining years. There are several reasons for selecting warm
periods: first, extreme rainfall events mostly occur in the study area during
the warm season (Cheng et al., 2010); second, the focus of our analysis is an
investigation of extreme rainfall related flood risks and development of IDF
curves using extreme rainfall statistics. We adjusted the occasional
overestimation of extreme values at a higher order cascade step by
a statistical post-processing method. We employed the quantile matching (QM)
approach (Li et al., 2010), which claims to outperform other simple bias
correction methods and which corrects not only the mean, but also the
variance of the distribution of interest (Gudmundsson et al., 2012;
Teutschbein and Seibert, 2012). QM is based on equidistant cumulative
probability distribution matching of observed and disaggregated AMP time
series using a three-parameter generalized extreme value (GEV) distribution.
Although like other statistical post-processing techniques, QM relies on the
stationarity assumption of the time series, in our case, we applied QM to
entire time series of both observed and disaggregated AMP, which comes from
the same station location (or similar spatial resolution) and a similar
period. Therefore, we avoid potential consequences of inflation by quantile
mapping (Maraun, 2013) in our analysis. We discuss the implementations of
MRC, adjustment of extremes, and associated model fits in more detail in the
Supplement (Sect. S1).
Detection of nonstationarity
A series of statistical tests are employed to detect the presence of
nonstationary trends and abrupt shifts in the short-duration AMP before
frequency analysis. The multiple tests allow a more rigorous and
comprehensive assessment of the overall trend in the time series since
certain tests are complementary to each other (Sadri et al., 2016; Yilmaz
et al., 2014, 2017). Figure 2 shows the schematics of the overall analysis.
Most of the trend and change-point detection algorithms assume observations
are mutually independent. The presence of autocorrelations in the time series
overestimates or underestimates the statistical significance of trend and
change-point detection algorithms (Serinaldi and Kilsby, 2016; von Storch and
Navarra, 1999). We employed a Ljung–Box test with 20 lags to the
short-duration AMP time series at each site to check whether they show
statistically significant autocorrelation (at the 5 and 10 % significance
levels). For the time series with no serial autocorrelation, we test for
trending behaviour and nonstationarity. It is also important to note that the
presence of nonstationarity may not be evaluated merely on the basis of
trends or abrupt shifts in the time series, even if the increasing or
decreasing trends are statistically significant (Yilmaz et al., 2014). First,
we check for the presence of nonstationarity in the time series by employing
a unit root-based augmented Dickey–Fuller (ADF; Dickey and Fuller, 1981)
test. However, the test may have a low power against
stationarity near
unit root processes (Dritsakis, 2004; Chowdhury and Mavrotas, 2006).
Therefore, as a complement to the unit root test, a KPSS test (Kwiatkowski
et al., 1992) is employed to validate the results of the ADF test. Since both
the ADF and KPSS tests assume linear regression or normality of the
distribution, alternatively, a log transformation can convert a possible
exponential trend present in the data into a linear trend. Therefore,
following previous studies (Gimeno et al., 1999; Van Gelder et al., 2006),
the AMP time series is log-transformed before applying stationarity tests.
However, Yilmaz et al. (2014) did not observe the presence of any significant
nonstationarity in extreme rainfall time series in the city of Melbourne,
even after employing ADF and KPSS tests. Therefore, as an alternative, we
also employed a frequency-based Priestley and Subbarao test (“PSR” test;
Priestley and Rao, 1969), which is able to better capture the nonlinear
dynamical nature of a hydrological system than the former two tests (Ali and
Mishra, 2017; Hamed and Rao, 1998). Next, we detected the presence of smooth
and abrupt changes in the time series. The continuous or monotonic trends in
short-duration rainfall extremes are identified using non-parametric
Mann–Kendall trend statistics with correction for ties (Hamed and Rao, 1998;
Reddy and Ganguli, 2013) at the 5 and 10 % significance levels. In
general, the abrupt change (or change point) in the time series occurs at
a single point in the record and bifurcates the time series into two halves,
either with different means or variances, or both dissimilar means and
variance together at each part. The change point in the location (or mean) is
identified using non-parametric Pettit (Pettitt, 1979) and Mann–Whitney
tests (Ross et al., 2011). As indicated by previous studies (Xie et al.,
2014; Yue and Wang, 2002), the rank-based nonparametric Mann–Whitney test is
not really distribution-free, and the power of the test is often affected by
the properties of sampled data. In practice, when a real change point is
unknown, often a Mann–Whitney test, in general, does not work well, and the
Pettitt method can yield a plausible change-point location along with its
statistical significance. However, the significance of the Pettitt test can
be obtained using an approximated limiting distribution (Xie et al., 2014;
Sect. S2). The shift in scale (or variance) is detected using a
non-parametric Mood test (Ross et al., 2011; see Fig. 2 for details). We
applied nonparametric tests due to their robustness to non-normality, which
usually appears in the hydroclimatic time series. Further, in order to reduce
the number of underlying assumptions required for testing a hypothesis, such
as the presence of a specific kind of trend or change point in the data,
nonparametric tests are employed. For the time series with significant
autocorrelation, we employed a trend-free pre-whitening procedure (TPFW;
Sect. S2) as described in Yue et al. (2002, 2003) and later modified by
Petrow and Merz (2009). Then, we applied trend and change-point detection
algorithms to the pre-whitened AMP extremes. In order to test the issue of
multiple comparisons associated with statistical analysis, we analysed
p values of five statistical tests, i.e. a Ljung–Box test, a KPSS test, a
Mann–Kendall trend test, a Priestley–Subbarao test, and a Pettitt test
using the false discovery rate (FDR) method (not shown here) as suggested by
Benjamini and Hochberg (1995). However, we excluded ADF, Mann–Whitney, and
Mood tests from the analysis, since unlike other tests, the higher p value
in ADF statistics indicates the presence of nonstationarity in the time
series. On the other hand, the latter two tests do not offer any p values.
Extreme value analysis of sub-daily and daily precipitation extremes
Nationwide EC IDF curves were developed using a particular family of
distribution functions from extreme value theory (i.e. a Gumbel distribution
or extreme value type I, hereafter referred to as EVI). However, the EV1
distribution has certain limitations, such as that it is a non-heavy-tailed
distribution and is characterized by constant skewness and kurtosis
coefficients (Markose and Alentorn, 2005; Pinheiro and Ferrari, 2016).
However, the short-duration AMP intensities often exhibit fat-tailed
behaviour, indicating large skewness and kurtosis. In fact, a few studies in
the past have shown that EV1 fits poorly to the historical rainfall extremes
(Burn and Taleghani, 2013; Coulibaly et al., 2015). Therefore, in the present
study, we perform frequency analysis of extreme precipitation using a GEV
distribution. The choice of the GEV distribution was based on previous
studies where various distribution functions were compared in the study area
(Coulibaly et al., 2015; Switzman et al., 2017). The GEV distribution is
a combination of the Gumbel, Fréchet, and Weibull distributions and is
fitted to block or AM time series (Cheng and AghaKouchak, 2014; Katz et al.,
2002; Katz and Brown, 1992). The GEV distribution is characterized by three
parameters, the location, the scale, and the shape of the distribution, which
describes the centre of the distribution, the deviation
around the mean, and the shape or the tail of the distribution (Katz et al.,
2002; Katz and Brown, 1992). The cumulative distribution function of the
stationary (time-invariant) GEV model is given by Coles et al. (2001) and
Gilleland and Katz (2016):
G(z)=exp-1+ζz-μσ+-1/ζ if ζ≠0exp-exp-z-μσ+ if ζ→0
where y+=maxy,0 and
z∈μ-σ/ζ,+∞ when ζ>0,z∈-∞,μ-σ/ζ when ζ<0, and z∈-∞,+∞ when ζ=0.μ is a location parameter, σ is a scale parameter and ζ is
a shape parameter determining the heaviness of the tail. The shape parameter
ζ determines the higher moments of the density function and also the
skew in the probability mass. The “+” sign indicates the positive part of
the argument. Equation () encompasses three types of DFs based on
the sign of the shape parameter, ζ: (i) the Fréchet, with a finite
lower bound of μ-σ/ζ and an unbounded, heavy
upper tail, (ζ>0), (ii) the Weibull, unbounded below and with a finite
upper bound of μ-σ/ζ, (ζ<0), and (iii)
the Gumbel, unbounded below and above with a light upper tail ζ=0,
formally obtained by taking the limit as ζ→0. The Gumbel
distribution is described by an unbounded light-tailed distribution and the
tail decreases rapidly following an exponential decay. The Fréchet
distribution is a heavy-tailed distribution, and the tail drops relatively
slowly following a polynomial decay (Towler et al., 2010). On the other hand,
the Weibull distribution is a bounded distribution. Here we compare the
performance of both the stationary and nonstationary forms of the GEV
distribution. For stationary models, we estimate parameters using BI coupled
with a differential evaluation Markov chain (DE-MC) Monte Carlo (MC)
simulation as suggested by Cheng et al. (2014) and Cheng and AghaKouchak
(2014). For nonstationary models, the shape parameter is assumed constant
throughout. Here it should be noted that modelling temporal changes in
ζ requires long-term observations, which are often not available in
practice (Cheng et al., 2014). Hence, following previous studies (Cannon,
2010; Cheng et al., 2014; El Adlouni et al., 2007; Gu et al., 2017), we
incorporate time-varying covariates into GEV location (GEVt-I), and into
both location and scale parameters (GEVt-II) respectively, to describe
trends as a linear function of time (in years):
μt=μ1t+μ0σt=σ1t+σ0
Since the scale parameter must be positive throughout, it is often modelled
using a log link function (Gilleland and Katz, 2016)
lnσt=σ1t+σ0⇒σt=expσ1t+σ0
where t is the time (in years), and λ=μ1,μ0,σ1,σ0,ζ are the
parameters.
Then we estimate parameters of the nonstationary GEV distribution by
integrating BI combined with DE-MC simulation. For AMP intensity, we derive
the time-variant parameter(s) from the 50th (the median or the mid-point of
the distribution) percentile(s) of the DE-MC sampled parameter(s). We obtain
the associated 95 % credible intervals (the bounds) from the 2.5th and
97.5th percentiles of the simulated posterior samples (see Sect. S3 for
details). We perform the calculations following Cheng and AghaKouchak (2014)
using a MATLAB-based software package, Nonstationary Extreme Value Analysis
(NEVA, version 2.0). The Bayes factor followed by the Akaike information
criterion (AIC) with a small sample correction (AICc) are
used to identify the best model. AICc claims to avoid
overfitting the data as compared to the traditional AIC (Burnham and
Anderson, 2004; Hurvich and Tsai, 1995). Here we assess model fitness based
on a least square sense of AIC statistics considering the maximum deviation
between empirical (obtained from the rank-based plotting position formula)
and modelled cumulative distributions (CDF; Dawson et al., 2007; Hu, 2007;
Karmakar and Simonovic, 2007). For calculation of AICc
statistics, we consider the median of the DE-MC sampled parameters, which can
be considered an average or expected value of risk in the historical
observation. Besides this, we also assess the performance of models using
probability–probability (PP) plots. The derived model parameters are then
utilized to obtain DSI using the concept of a T-year return period. We
discuss the methods to estimate DSI and T-year return periods using
stationary and nonstationary methods in detail in Sect. S4. To test
(statistically) significant differences in the estimated DSI from the
best-selected stationary vs. nonstationary models, we calculate standardized
z statistics for selected return periods (Madsen et al., 2009; Mikkelsen
et al., 2005). We applied the two-sided option with 10 % significance
levels to assess the statistical significance of the test statistics (see
Sect. S5 for details). Finally, we compared the DSI obtained from
nonstationary and stationary models with existing EC-generated DSI estimates.
Results
The extreme rainfall statistics show high standard deviation with positive
skew behaviour. The skewness is a measure of the asymmetry in the AMP
distribution. Positive values of skewness indicate that data are skewed to
the right. The skewness of sub-hourly precipitation extremes varies between
0.22 and 4.45, with highest being 30 min AMP record at Hamilton and least
being at Oshawa respectively. Likewise, for hourly extremes, the skewness
ranges between 0.54 and 2.54, with least being 1 h AMP at Oshawa and highest
is 1 h AMP at Hamilton respectively. The majority of stations show positive
excess kurtosis, which indicates the data have a distinct peak near the mean,
which decline rapidly, and have heavy tails. We find the presence of
statistically significant autocorrelations in the AMP time series of Toronto
International Airport, Hamilton Airport, and Fergus Shand Dam (Sect. S2). We
apply nonparametric TPFW to precipitation extremes with a significant
autocorrelation (Tables S4.1, S5.1 and S12). However, two successive TPFWs
fail to correct the effect of autocorrelation in 12 and 24 h duration
extremes in Shand Dam. Hence we exclude those two time series from frequency
analysis (Table S12). The ADF test for nonstationarity is statistically
significant in all durations, as indicated by the higher p values. As
a complement to the ADF test, we also employed KPSS and PSR tests (Fig. 2;
Sect. S2) to check significant nonstationarity. Figure 3 shows the spatial
distribution of trends, change points and nonstationarity in short-duration
rainfall extremes. We find co-occurrences of trends, change points and
nonstationarities in extremes at multiple locations. In general, the three
sites in the extreme northeast, the Oshawa WPCP, Trenton Airport, and
Kingston P. Station show evidence of statistically significant upshifts and
nonstationarities in the time series, whereas the rest of the sites in the
southwest exhibit downshifts and statistically significant nonstationarities
(Fig. 3). For 2 h and beyond durations, London International Airport shows
the presence of statistically significant downward trends with change points.
An increase (decrease) in mean precipitation implies an increase (decrease)
in heavy precipitation and vice versa.
Further, it could also alter the shape of the right-hand tail, changing
overall asymmetry in the distribution (Fig. S1), and hence affecting the
nature of extremes (Stocker et al., 2013). Furthermore, the presence of
opposite signs of trends within a proximity of sites are prominent in all
durations, for example, except for 1 h duration, extremes in all durations
at Toronto International Airport and Oshawa WPCP show downwards and upward
shifts respectively. Our findings confirm the other study (Burn and
Taleghani, 2013), where authors report a lack of spatial structures and
presence of different trends within a close vicinity of stations. Further, we
find statistically significant monotonic increase and abrupt step changes,
both in mean and variance in Oshawa and Trenton respectively (Tables S6 and
S10), whereas London shows a (significant) decrease (Table S9) from 6 h
duration and beyond. A few sites, such as Windsor, Kingston, and Stratford
show (significant) step changes as confirmed by Mann–Whitney and Mood tests
(Tables S7, S8, and S11) respectively. On the other hand, Toronto, Hamilton,
and Fergus Shand Dam (Tables S4, 4.1, S5, 5.1, and S12) do not exhibit any
significant gradual or abrupt changes in the AMP time series. The ADF tests
show the presence of nonstationarity in all durations across the sites. To
further validate the results of the ADF test, KPSS and PSR tests are
employed. The KPSS test detects the presence of nonstationarity at three out
of nine sites for 24 h rainfall extreme at the 5 % significance level,
whereas the results of the PSR test indicate nonstationarity across five
sites in 24 h rainfall extremes. While the KPSS test alone could not detect
the presence of nonstationarity in any of the extreme series in Oshawa and
Stratford respectively, the results of the PSR test could not discern
nonstationarity in any of the short-duration rainfall extremes in Windsor.
Both of these tests taken together detect the presence of nonstationarity in
rainfall extremes across six out of nine sites. We find that even if trends
in individual sites may not seem significant, the magnitude of trends (as
measured by slope per decade, Tables S4–S12 in Sect. S2) is never zero in
any of the sites. Although we present a range of statistical tests to
investigate plausible shifts in the time series, we do acknowledge that
statistical inferences may be affected by the problem of multiple comparisons
resulting in a set of false positive outcomes. On analysing the FDR-based
multiple comparison procedure, we find that except Windsor, in all sites the
presence of trend and nonstationarities in the time series turns out to be
significant with the highest number of statistical tests showing adjusted
p value <0.10 in Trenton followed by Hamilton. This indicates our
analysis is not affected by the issue of multiple comparisons.
Performance of stationary and nonstationary models for Toronto
Pearson International Airport.
* GEVt-0 is a stationary model,
whereas GEVt-I and GEVt-II are nonstationary models
with a time-variant mean, and both a time-variant mean and standard deviation
respectively. The selected best fitted nonstationary model is marked in bold
letters. The Bayes factor, γ<1, indicates that the nonstationary
model fits better than the stationary model. However, in cases where γ>1, to compare with the stationary model, the nonstationary model is selected
following minimum AICc criteria. LB and UB indicate
the lower and upper bounds of DSI at a 100-year return period.
Performance of stationary and nonstationary models for Hamilton
Airport.
* GEVt-0 is a stationary model, whereas
GEVt-I and GEVt-II are nonstationary models with a
time-variant mean, and both a time-variant mean and standard deviation
respectively. The selected best fitted nonstationary model is marked in bold
letters. The Bayes factor, γ<1, indicates that the nonstationary
model fits better than the stationary model. However, in cases where γ>1, to compare with the stationary model, the nonstationary model is selected
following minimum AICc criteria. LB and UB indicate
the lower and upper bounds of DSI at a 100-year return period.
Performance of stationary and nonstationary models for Windsor
Airport.
* GEVt-0 is a stationary model, whereas
GEVt-I and GEVt-II are nonstationary models with a
time-variant mean, and both a time-variant mean and standard deviation
respectively. The selected best fitted nonstationary model is marked in bold
letters. The Bayes factor, γ<1, indicates that the nonstationary
model fits better than the stationary model. However, in cases where γ>1, to compare with the stationary model, the nonstationary model is
selected following minimum AICc criteria. LB and UB
indicate the lower and upper bounds of DSI at a 100-year return period.
Performance of stationary and nonstationary models for London
International Airport.
* GEVt-0 is a stationary model, whereas
GEVt-I and GEVt-II are nonstationary models with a
time-variant mean, and both a time-variant mean and standard deviation
respectively. The selected best fitted nonstationary model is marked in bold
letters. The Bayes factor, γ<1, indicates that the nonstationary
model fits better than the stationary model. However, in cases where γ>1, to compare with the stationary model, the nonstationary model is selected
following minimum AICc criteria. LB and UB indicate
the lower and upper bounds of DSI at the 100-year return period.
A weak trend can also have a significant impact on the results of probability
analysis (Porporato and Ridolfi, 1998). Hence even if precipitation extremes
often exhibit statistically insignificant trends in a few durations, we
assess the performance of both nonstationary and stationary models in all
sites. Tables 2–5 list performances of nonstationary vs. stationary models
for selected airport locations, whereas Tables S13–S17 present results of
the distribution fit for the remaining stations. Barring a few exceptions,
the shape parameters in most of the models range between -0.30 and +0.3,
which is an acceptable range of GEV shape parameters, as shown in an earlier
study (Martins and Stedinger, 2000). Our results corroborate well with recent
research (Papalexiou et al., 2013; Wilson and Toumi, 2005), which showed that
a distribution with fat tails (with a GEV shape parameter, ζ<0) fits
better for the precipitation extremes. The nonstationary models are selected
by employing a Bayes factor and a minimum AICc
criterion. For example, for the 6 h duration storm at Hamilton Airport, the
nonstationary GEVt-I (nonstationary model with time-varying GEV
location) model performed best, as shown by both test metrics. However, in
certain cases, nonstationary models do not pass a Bayes-factor test. In such
cases, we select the best nonstationary model (i.e. between
GEVt-I and GEVt-II) following
AICc test statistics. Overall values of the Bayes
factor indicate that there is no strong evidence to favour or reject any of
the three models. In general, although we find the stationary model cannot be
rejected, it does not imply that there is no change. We may be unable to
detect the apparent signal of nonstationarity due to the strong natural
variability present in the data. However, it should be noted that the
objective of the present analysis is to compare the design storm obtained
from the stationary vs. the best nonstationary model, and not to analyse the
best distribution between them.
As a measure of uncertainty, we also report the 95 % credible interval of
design rainfall quantiles at a 100-year return period as a ratio between the
upper and lower bounds, which ranges between factors of 1.2-to-1 and 3.9-to-1
in all cases. The performance of time-varying GEV models (Fig. S9) closely
follows the spatial pattern as indicated in the trend map (Fig. 3). For
example, Trenton Airport, which showed significant upward trends with change
points and nonstationarity, is better modelled by the nonstationary GEV
distributions for most of the durations. Likewise, we find that in a few
cases, GEVt-II fits best if the time series exhibit (significant)
evidence of nonstationarity as detected by PSR-test statistics, for example,
15 min and 12 h extremes in London and Toronto International Airport
(Tables 2 and 5) respectively. However, in many cases, the performances of
nonstationary models are often comparable and even superseded by their
stationary counterparts (Sect. S3). In fact, the scatter of data points in
the PP plots (Figs. S10–S12) suggests a close resemblance between
stationary and nonstationary models across all durations. Figure 4 shows the
relation between DSI and durations (ranges between 15 min and 24 h) for
100-year return periods estimated by stationary vs. nonstationary GEV
distributions. The interquartile range of the boxplot shows the uncertainty
in estimated rainfall quantiles obtained using BI. However, the spread of the
boxes simulated by the nonstationary model is found to be relatively narrower
as compared to the one simulated by the stationary model for most of the
sites (Fig. 4), indicating less uncertainty in the estimated quantiles. For
shorter return periods, such as 10 and 5 years, the DSI from stationary vs.
nonstationary models shows more or less subtle differences (Figs. S13–S14)
than those for the 100-year period.
Figure 5 displays the differences in DSI obtained using the best performing
nonstationary model relative to the stationary models using percentage
changes and z statistics for different durations and return periods. While
percentage change indicates a magnitude of change, the z statistics show
the statistical significance of the relative difference in estimated DSI
using the two different methods. The percentage differences at the 2- and
10-year return periods are small relative to longer return periods. At the
100-year return period, a maximum positive difference of around 44 % is
observed at the 12 h storm duration in Toronto International Airport
(Table S18.1). The standardized z statistics show positive (negative)
values, indicating an increase (decrease) in DSI values assuming
nonstationarity in return period estimates against their stationary
counterparts. However, a comparison between T-year event estimates from
both models indicates statistically indistinguishable differences in rainfall
intensity. We find that, for all return periods and durations, z statistics
range between -1 and +1 across all nine sites (Sect. S5). Nonetheless,
extreme precipitation intensity shows either positive or negative
(statistically insignificant) changes in signs. The difference between DSI
shows a decrease, at 1 and 2 h storm duration in Toronto, 6 h storm
duration in London, and 15 min and 12 h storm duration at Trenton Airport
for 50- and 100-year return periods (Fig. 5; Sect. S5). In contrast, Toronto,
Windsor, and London International Airport show an increase in DSI values at
15 min duration (Fig. 5; Sect. S5), although the increase is statistically
insignificant. Further, we note that, except for the 6 and 12 h storm
durations, the performance statistics show a comparable and in a few cases
even better performance of the stationary vs. nonstationary GEV models across
most of the sites (Sect. S3). At 2- and 10-year return periods, which are
typical for most urban drainage planning, the differences are close to zero
(Fig. 5, Tables S27 and S28 in Sect. S5) for most of the duration.
Figures 6 and 7 compare the T-year event estimates of updated vs.
EC-generated IDFs for different return periods, taking into account both
stationary and nonstationary conditions. The median and associated lower and
upper bounds of the ratio of regional updated vs. EC-generated T-year
event estimates can be interpreted as analogous to the most likely minimum
and maximum plausible scenarios. While the positive value of the ratio
indicates a required increase in DSI, the negative value indicates a decrease
in the DSI estimate. Considering nonstationarity, at the T=10-year return
period (Fig. 6; Sect. S4), the ratio of updated vs. old estimates of DSI is
of the order of ∼1.01–1.08, which indicates that the required
increases are of the order of 1.4 to 7.2 %. At the T=2-year return
period, except for Oshawa and Windsor, the majority of sites show a decrease
in DSI for most of the storm durations (Fig. 6, middle row). In contrast, an
increase in the estimated ratio is more pronounced at the 50- and 100-year
return periods, which are of the order of ∼1.03–1.80 (Fig. 7,
Sect. S4). While for Toronto International Airport and Hamilton Airport, we
find no increase in the short-duration rainfall extremes of less than 1 h
and the 50-year return period considering the nonstationary condition, the
increase is more pronounced for longer durations and return periods (12 and
24 h, and the 50- and 100-year return periods; Sect. S4). For longer
recurrence intervals, while the heat maps of minimum bounds and the most
likely scenario show a smaller number of stations and durations to reach
a ratio of 1.5 and beyond, the maximum bounds suggest a sharp increase in the
ratio across all durations and locations. Further, for return periods of
50 years and more, the increase in the ratio is more prominent in the upper
bound of the stationary models (Fig. 7, left two columns) as compared to the
nonstationary models. The resulting increase in T-year event estimates is
because of the relatively wider confidence interval of estimated DSIs in
stationary models than that of the nonstationary models (Fig. 4; Sect. S3).
In general, for longer return periods, our analysis reveals that the increase
in the ratio of updated vs. EC-generated rainfall intensity is more prominent
in sites with statistically significant signatures of nonstationarity, upward
trends, and change points. For example, the updated DSIs of Oshawa WPCP,
Windsor, and Trenton Airport shows an increase in the ratio for most of the
durations and return periods as compared to the EC-generated DSI values
(Sect. S4). On the other hand, except for the 100-year return period events,
the hourly precipitation extremes in London International Airport, in
general, show a decrease in the ratio (Tables S23.1–23.2) across all return
periods, which is predominantly due to the presence of significant downward
trends with change points in the time series.
Based on the study results and in anticipation of stakeholders' participation
in adaptive management, we present updated IDFs for four selected urbanized
locations across Southern Ontario (Fig. 8). In order to distinguish between
stationary and nonstationary methods of analysis, we also present updated IDF
assuming stationary condition relative to EC IDF in the same plot (in the top
panel). The comparison of remaining sites is presented in Fig. S15. Thus we
made the first attempt to compare the results of updated vs. EC-generated
IDFs considering both nonstationary and stationary conditions, which are the
part of contemporary design standards and widely used by the stakeholders and
practitioners. Overall, the updated IDFs closely follow the pattern of trends
analogous to EC-generated IDFs, except for the 100-year return period. The
difference is more pronounced considering nonstationary condition, especially
at Toronto International Airport (Fig. 8), Oshawa WPCP, and Stratford WWTP
(Fig. S15). At longer durations and return periods, stations in metropolitan
areas (such as Toronto International Airport, Hamilton Airport, Oshawa WPCP,
and Windsor Airport) show large differences in DSIs, whereas moderately
populated locations such as Kingston P. Station and Fergus Shand Dam show
a relatively small change. Considering the nonstationary condition, the
maximum increase at Fergus Shand Dam is noted as 18.7 % for the 2 h
storm duration and 100-year return period, whereas an increase of around
44.5 % is shown for the 12 h storm duration at Toronto International
Airport. For T=10 years or less, we find a decrease in the range of ∼2–40 % in the T-year event estimates (Sect. S4). Meanwhile, for the
T=10-year return period, we find that the increase is of the order of
∼1.4 to 7.2 % across several stations. Considering nonstationarity,
at T=50 years and more, the required increase ranges between ∼2.8
and 44.5 %. We find that the largest increase is for the 12 h rainfall
extreme in Toronto International Airport (∼32–44.5 %;
Table S18.1), followed by a 2 h extreme at Stratford WWTP (∼27–36 %; Table S25.1). However, considering the stationarity condition
at T=50 years and more, the required increase ranges between ∼1.4
and 26 %. It should be noted that above results are based on an average
risk approach for extreme value analysis considering median of the sampled
parameters in the historical observation and not taking into account the
overall risk envelope (i.e. minimum and maximum bounds). In summary, our
findings confirm that updates of the order of ∼2–44 % are required
based on locations and return periods to mitigate the risk of
precipitation-induced urban flooding irrespective of the choice of methods
used in the IDF estimation (Sect. S4). The results are consistent with
Simonovic and Peck (2009), in which the authors recommend an average update
of about 21 %, with a difference, range between ∼11 and 35 %
for the updated vs. EC-IDF in the London metropolitan area. However, they
assumed stationarity condition to develop at-site IDF. The above results also
highlight the need to update existing EC IDFs, which are generated using
Gumbel probability distributions and do not fit the data well.
Discussions and conclusions
This paper has sought to assess signatures of nonstationarity in densely and
moderately populated urbanized locations across Southern Ontario, which is
one of the major economic hubs in Canada. We update short-duration rainfall
extremes with latest available ground-based observations and present
a comprehensive analysis to evaluate nonstationary vs. the stationary method
of IDF estimation. This analysis yields two principal findings. First,
detectable non-stationarity in rainfall extremes does not necessarily lead to
significant differences in design storm values. Second, comparison of at-site
T-year event estimates of updated vs. EC-generated IDFs shows at T=10 years, the return period commonly used for urban drainage design, current
design standards require updates of up to 7 % to mitigate the risk of
urban flooding. Meanwhile, for a longer recurrence interval (T=50 years
or more), typical for critical civil infrastructural design, comparison of
updated vs. EC-generated IDF curves shows a difference ranges between 2 and
44 % based on locations. These findings pose an important question: does
the presence of nonstationarities in rainfall extremes require the design of
nonstationary IDF curves? We argue that although it is crucial to recognize
nonstationarity in precipitation extremes, the stationary form of IDFs can
still represent the extreme rainfall statistics for the present-day climate
over the Southern Ontario region. Our results are consistent with Yilmaz
et al. (2014) and Yilmaz and Perera (2013), in which authors found despite
the presence of (statistically) significant trends in rainfall extremes;
nonstationary GEV models did not show any additional advantages over the
stationary models. As supported by the previous study (Singh et al., 2016),
we attribute that the little or no changes in extreme rainfall statistics in
the urbanized setting is due to the stabilization of urban development
leading to no substantial variations in the land use pattern. Hence, no
significant changes in synoptic-scale circulations, which in turn affect
space–time pattern in rainfall extremes (Moglen and Schwartz, 2006).
Preliminary investigations based on regional and global climate model
simulations in the study area confirm a considerable uncertainty in the
projection of short-duration and high-intensity extreme rain events
(Coulibaly et al., 2015). While short-duration precipitation extremes are
typically controlled by synoptic-scale moisture convergence (Ruiz-Villanueva
et al., 2012; Westra and Sisson, 2011), the daily extremes are often
modulated by large-scale circulation patterns and local orographic factors
(Carvalho et al., 2002; Gershunov and Barnett, 1998; Trenberth, 1999). The
roles of natural variability and multidecadal modes of sea-surface
temperature (SST) in modulating Canadian extreme rainfall intensity have
already been shown in the past (Gan et al., 2007; Shabbar et al., 1997).
Further, a review of the literature suggests that heavy precipitation does
not necessarily lead to high stream discharge (Ivancic and Shaw, 2015; Do
et al., 2017; Wasko and Sharma, 2017). The analysis of Do et al. (2017)
reveals that the trend in streamflow is more consistent across the
continental scale, and neither the anthropogenic activities such as the
presence of dams nor the vegetation cover have any significant effect on the
results of trend estimates. Interestingly, the consensus among all three
studies (Ivancic and Shaw, 2015; Do et al., 2017; Wasko and Sharma, 2017) is
that the catchment size, which regulates the flow response because of the
antecedent moisture content, is the most important contributing factor in
modulating the nature of a trend in stream discharge. The smaller (especially
urban) catchments may have increased flood peaks; in contrast, larger
(agricultural and rural) catchments may experience decreased runoff due to
lower soil moisture. This can be attributed to the fact that high temperature
leads to drying up of soil more quickly in larger catchments, thus forcing
a large portion of precipitation not to become an overland flow. Finally,
using more than 9000 daily streamflow records globally, Do et al. (2017)
showed more stations with significant decreasing trends in annual maximum
streamflow than that of significant increasing trends, indicating limited
evidence of increasing flood hazards. Their findings are corroborated by
Wasko and Sharma (2017), in which the authors showed that only in the most
extreme cases, for small catchments, does an increase in precipitation at a
higher temperature lead to an increase in streamflow.
The statistical uncertainty in modelling nonstationarity can result from
multiple sources, for instance, extrapolating the effects detected with
observed historical series to the more extreme values that have not yet been
experienced, model choices resulting from selection of covariates in the
nonstationary distributions (Agilan and Umamahesh, 2017), and the treatment
of nonstationarity introduced through either a linear (Ali and Mishra, 2017;
Cheng et al., 2014; Westra et al., 2012) or polynomial (Villarini et al.,
2009b) trend, or a change point (Renard et al., 2013) in the model. The key
question remains how to update design events in a nonstationary climate. This
becomes more challenging after trends and change points are detected in
hydrometric time series. To address climate change adaptation needs under
nonstationarity and uncertainty, some of the concepts discussed in the
literature are design life level (Rootzén and Katz, 2013) to quantify the
probability of exceeding a fixed threshold during the design life of
a project, replacing the commonly used concept of average return period with
reliability (Read and Vogel, 2015) and a risk-based decision-making approach
integrating the concept of expected regret (Rosner et al., 2014).
However, apart from statistical uncertainty, one of the important sources of
uncertainty in future planning periods is the use of climate model output.
Modelling nonstationarity in the future time period is complicated by the
choice of spatial resolution of climate models, lack of understanding of
model physics due to different model choices, and inherent uncertainties in
climate model realizations resulting from different initial conditions, which
is especially apparent over regional scales and decadal planning horizons
(Ganguli et al., 2017; Hawkins and Sutton, 2009; Kumar and Ganguly, 2017;
Meehl et al., 2009).
The increase in design storm during the update process could also indicate
a tendency towards an increase in mean precipitation and (or) a shift in the
distribution, affecting its tail behaviour. However, a few caveats remain:
for example, a critical question could be “is an increase in DSI potentially
linked to more frequent and more intense precipitation extremes or is it an
artifact of the new dataset in the update process?” It is worthwhile noting
that the results shown here are manifestations of the present-day climate
using ground-based hydrometeorological observations, and the specific
insights are nonetheless subject to the quality of available rainfall
records. It remains an open-ended question to what extent we can credibly
develop IDFs in a changing climate (Coulibaly et al., 2015) since there is no
uniformly accepted method of generating IDF information and related
projection uncertainty in the light of climate change. Firstly, highlighting
advantages and limitations of nonstationary vs. stationary methods of
analyses (Koutsoyiannis and Montanari, 2015; Montanari and Koutsoyiannis,
2014; Serinaldi and Kilsby, 2015; Serinaldi et al., 2018) is beyond the scope
of the current study. Secondly, one of the limitations of the present
analysis includes the lack of accounting for the consistency of the IDF
curves in terms of shape enforcement. The lack of shape enforcement in the
IDF curves (Figs. 8 and S15) is the result of fitting separate distributions
to the series of different durations. Finally, although in several instances
we do find evidence of step changes in short-duration rainfall extremes, we
have not introduced any change-point model in
the GEV parameters (Renard et al., 2013).
Future research should include two aspects. First, investigation of physical
drivers (such as temperature, decadal, and multidecadal modes of SST)
influencing short-duration rainfall extremes, and inclusion of these
covariates in nonstationary IDF development. Second, modelling
nonstationarity by introducing a step-change model in GEV location and scale
parameters. Finally, given that the findings reported herein are for the
current period (e.g. historical extreme rainfall time series), we recommend
a careful extrapolation of these findings with regards to future climate
projections, in which the frequency and magnitude of extreme rainfall are
expected to intensify (Mailhot et al., 2012; Deng et al., 2016; Fischer and
Knutti, 2016; Prein et al., 2016, 2017; Pfahl et al., 2017; Switzman
et al., 2017). Further work should consider nonstationary methods for
deriving future IDFs in Southern Ontario.
The annual maximum rainfall data used in this study are
downloaded from the Environment Canada (EC, 2014), Engineering Climate
Datasets archive. The hourly (HLY) and daily (DLY) rainfall data are obtained
from Environment and Climate Change Canada (ECCC, 2017). The 2011 census data
and census digital boundary shape files are obtained from the Statistics Canada
(SC, 2016) website.
The Supplement related to this article is available online at https://doi.org/10.5194/hess-21-6461-2017-supplement.
PG and PC designed the experiment. PG carried out
the experiment. PG prepared the manuscript with contributions from PC.
Acknowledgements
This work was supported by the Natural Science and Engineering Research
Council (NSERC) Canadian FloodNet (grant number: NETGP 451456). The first
author of the paper would like to thank Jonas Olsson of the Swedish
Meteorological and Hydrological Institute (SMHI), Norrköping, Sweden, for
sharing MATLAB-based random cascade disaggregation tools and implementation
details by email. While change-point and nonstationarity analyses were
conducted in statistical software R version 3.30 with add-on packages
“trend”, “fractal”, and “cpm”, the remaining analyses were performed in
a MATLAB platform (MATLAB R2016a). The nonstationary GEV analyses were
performed using the MATLAB-based NEVA toolbox, available at the University of
California, Irvine, website: http://amir.eng.uci.edu/neva.php (as
accessed on May 2016). The work was completed and communicated when the first
author was a postdoctoral research fellow at McMaster Water Resources and
Hydrologic Modeling Lab, McMaster University, Canada. Edited by: Thomas Kjeldsen Reviewed by: two
anonymous referees
References
Adamowski, K. and Bougadis, J.: Detection of trends in annual extreme rainfall, Hydrol. Process., 17, 3547–3560, 2003.
Agilan, V. and Umamahesh, N. V.: What are the best covariates for developing non-stationary rainfall
Intensity–Duration–Frequency relationship?, Adv. Water Resour., 101, 11–22, 2017.Ali, H. and Mishra, V.: Contrasting response of rainfall extremes to increase in surface air and dewpoint temperatures at
urban locations in India, Sci. Rep.-UK, 7, 1228, 10.1038/s41598-017-01306-1, 2017.ASCE: Standard Guidelines for the Design of Urban Stormwater Systems,
Standard Guidelines for Installation of Urban Stormwater Systems, and
Standard Guidelines for the Operation and Maintenance of Urban Stormwater
Systems, ASCE/EWRI 45-05, 46-05, and 47-05, American Society of Civil
Engineers, Reston, VA, available at:
https://ascelibrary.org/doi/book/10.1061/9780784408063 (last access: 9
December 2016), 2006.
Baldwin, D. J. B., Desloges, J. R., and Band, L. E.: Physical geography of
Ontario, in: Ecology of a Managed Terrestrial Landscape: Patterns and
Processes of Forest Landscapes in Ontario, University of British Columbia
Press, Vancouver, 2011.
Ban, N., Schmidli, J., and Schär, C.: Evaluation of the convection-resolving regional climate modeling approach in
decade-long simulations, J. Geophys. Res.-Atmos., 119, 7889–7907, 2014.
Berg, P., Moseley, C., and Haerter, J. O.: Strong increase in convective precipitation in response to higher temperatures,
Nat. Geosci., 6, 181–185, 2013.
Benjamini, Y. and Hochberg, Y.: Controlling the false discovery rate: a practical and powerful approach to multiple
testing, J. Roy. Stat. Soc. B Met., 57, 289–300, 1995.Berkley Earth: Local climate change: 44.20∘ N, 80.50∘ W,
available at: http://berkeleyearth.lbl.gov, last access: December 2017.
Blenkinsop, S., Chan, S. C., Kendon, E. J., Roberts, N. M., and
Fowler, H. J.: Temperature influences on intense UK hourly precipitation and
dependency on large-scale circulation, Environ. Res. Lett., 10, 054021, 2015.
Bougadis, J. and Adamowski, K.: Scaling model of a rainfall
intensity–duration–frequency relationship, Hydrol. Process., 20,
3747–3757, 2006.
Bourne, L. S. and Simmons, J.: New fault lines? Recent trends in the Canadian Urban System and their implications for
planning and public policy, Can. J. Urban Res., 12, 22–47, 2003.
Burn, D. H. and Taleghani, A.: Estimates of changes in design rainfall values
for Canada, Hydrol. Process., 27, 1590–1599, 2013.
Burnham, K. P. and Anderson, D. R.: Multimodel inference understanding AIC
and BIC in model selection, Sociol. Methods Res., 33, 261–304, 2004.Cannon, A. J.: A flexible nonlinear modelling framework for nonstationary
generalized extreme value analysis in hydroclimatology, Hydrol. Process., 24,
673–685, 10.1002/hyp.7506, 2010.
Carvalho, L. M., Jones, C., and Liebmann, B.: Extreme precipitation events in
southeastern South America and large-scale convective patterns in the South
Atlantic convergence zone, J. Climate, 15, 2377–2394, 2002.Castellarin, A., Kohnová, S., Gaál, L., Fleig, A., Salinas, J. L.,
Toumazis, A., Kjeldsen, T. R., and Macdonald, N.: Review of
applied-statistical methods for flood-frequency analysis in Europe, available
at: http://nora.nerc.ac.uk/19286/ (last access: 20 May 2017), 2012.CCF (Canadian Climate Forum): Extreme Weather, 1, 1–4, Ottawa, available at: http://www.climateforum.ca/ (last
access: December 2016), 2013.CDD (Canadian Disaster Database): Public Safety Canada, available at:
https://www.publicsafety.gc.ca/cnt/rsrcs/cndn-dsstr-dtbs/index-en.aspx
(last access: December 2016), 2015.Cheng, L. and AghaKouchak, A.: Nonstationary precipitation
intensity-duration-frequency curves for infrastructure design in a changing
climate, Sci. Rep.-UK, 4, 7093, 10.1038/srep07093, 2014.Cheng, C. S., Li, G., Li, Q., and Auld, H.: A synoptic weather typing approach to simulate daily rainfall and extremes in
Ontario, Canada: potential for climate change projections, J. Appl. Meteorol. Clim., 49, 845–866, 10.1175/2010JAMC2016.1,
2010.
Cheng, L., AghaKouchak, A., Gilleland, E., and Katz, R. W.: Non-stationary extreme value analysis in a changing climate,
Climatic Change, 127, 353–369, 2014.
Chowdhury, A. and Mavrotas, G.: FDI and growth: what causes what?, World Econ., 29, 9–19, 2006.
Coles, S.: An Introduction to Statistical Modeling of Extreme Values,
Springer, 1–183, 2001.Coulibaly, P. and Shi, X.: Identification of the effect of climate change on
future design standards of drainage infrastructure in Ontario, Rep. Prep.
McMaster Univ. Funding Minist. Transp. Ont., 82, available at:
http://www.cspi.ca/sites/default/files/download/Final_MTO_Report_June2005rv.pdf
(last access: 9 December 2016), 2005.Coulibaly, P., Burn, D., Switzman, H., Henderson, J., and Fausto, E.:
A comparison of future IDF curves for Southern Ontario, Technical Report,
McMaster University, Hamilton, available at:
https://climateconnections.ca/wp-content/uploads/2014/01/IDF-Comparison-Report-and-Addendum.pdf
(last access: 9 December 2016), 2015.
CSA (Canadian Standards Association): Technical Guide – Development,
Interpretation and Use of Rainfall Intensity-duration-frequency (IDF)
Information: Guideline for Canadian Water Resources Practitioners, CSA Group,
Ottawa, 2010.Dawson, C. W., Abrahart, R. J., and See, L. M.: HydroTest: A web-based toolbox of evaluation metrics for the standardised
assessment of hydrological forecasts, Environ. Modell. Softw., 22, 1034–1052, 10.1016/j.envsoft.2006.06.008, 2007.
Dean, S. M., Rosier, S., Carey-Smith, T., and Stott, P. A.: The role of
climate change in the two-day extreme rainfall in Golden Bay, New Zealand,
December 2011, B. Am. Meteorol. Soc., 94, S61–S63, 2013.De Carolis, L.: The urban heat island effect in Windsor, ON: an assessment of
vulnerability and mitigation strategies, City Windsor Ont., 2012, 1–52,
available at:
https://www.citywindsor.ca/residents/environment/Environmental-Master-
Plan/Documents/Urban Heat Island Report (2012).pdf, last access: December
2016.
Deng, Z., Qiu, X., Liu, J., Madras, N., Wang, X., and Zhu, H.: Trend in frequency of extreme precipitation events over
Ontario from ensembles of multiple GCMs, Clim. Dynam., 46, 2909–2921, 2016.
Dickey, D. A. and Fuller, W. A.: Likelihood ratio statistics for
autoregressive time series with a unit root, Econom. J. Econom. Soc., 49,
1057–1072, 1981.
Dixon, P. G. and Mote, T. L.: Patterns and causes of Atlanta's urban heat island–initiated
precipitation, J. Appl. Meteorol., 42, 1273–1284, 2003.
Do, H. X., Westra, S., and Leonard, M.: A global-scale investigation of
trends in annual maximum streamflow, J. Hydrol., 552, 28–43, 2017.
Donat, M. G., Lowry, A. L., Alexander, L. V., O'Gorman, P. A., and Maher, N.: More extreme precipitation in the world's
dry and wet regions, Nat. Clim. Change, 6, 508–513, 2016.Dritsakis, N.: Tourism as a Long-Run Economic Growth Factor: An Empirical
Investigation for Greece Using Causality Analysis, Tour. Econ., 10, 305–316,
10.5367/0000000041895094, 2004.Drobinski, P., Da Silva, N., Panthou, G., Bastin, S., Muller, C., Ahrens, B.,
Borga, M., Conte, D., Fosser, G., Giorgi, F., Güttler, I., Kotroni, V.,
Li, L., Morin, E., Önol, B., Quintana-Segui, P., Romera, R., and Torma,
C. Z.: Scaling precipitation extremes with temperature in the Mediterranean:
past climate assessment and projection in anthropogenic scenarios, Clim.
Dynam., 1–21, 10.1007/s00382-016-3083-x , 2016.
Durrans, S. and Brown, P.: Estimation and internet-based dissemination of extreme rainfall information,
Transp. Res. Rec. J. Transp. Res. Board, 1743, 41–48, 2001.
EC (Environment Canada): Documentation on Environment Canada Rainfall
Intensity–DurationFrequency (IDF) Tables and Graphs V2.20, Government of
Canada, 2012.EC (Environment Canada): Documentation on Environment Canada Rainfall
Intensity–DurationFrequency (IDF) Tables and Graphs V2.30, December, 2014,
Government of Canada, available at:
http://climate.weather.gc.ca/prods_servs/engineering_e.html (last
access: December 2016), 2014.ECCC (Environment and Climate Change Canada): Technical Documentation –
Digital Archive of Canadian Climatological Data, Government of Canada, data
available at:
ftp://client_climate@ftp.tor.ec.gc.ca/Pub/Documentation_Technical/Technical_Documentation.pdf
(last access: December 2016) from:
https://www.canada.ca/en/environment-climate-change.html, 2017.El Adlouni, S., Ouarda, T. B. M. J., Zhang, X., Roy, R., and Bobée, B.:
Generalized maximum likelihood estimators for the nonstationary generalized
extreme value model, Water Resour. Res., 43, W03410,
10.1029/2005WR004545, 2007.IPCC (Intergovernmental Panel on Climate Change): Managing the Risks of
Extreme Events and Disasters to Advance Climate Change Adaptation, A Special
Report of Working Groups I and II of the Intergovernmental Panel on Climate
Change, edited by: Field, C. B., Barros, V., Stocker, T. F., Qin, D., Dokken
D. J., Ebi,K. L., Mastrandrea, M. D., Mach, K. J., Plattner, G.-K., Allen, S.
K., Tignor, M., and Midgley, P. M., Cambridge University Press, Cambridge,
United Kingdom and New York, NY, USA, 582 pp., available at:
https://www.ipcc.ch/pdf/special-reports/srex/SREX_Full_Report.pdf,
2012. Fischer, E. M. and Knutti, R.: Observed heavy precipitation increase confirms theory and early models, Nat. Clim. Change,
6, 986–991, 2016.Gan, T. Y., Gobena, A. K., and Wang, Q.: Precipitation of southwestern Canada: wavelet, scaling, multifractal analysis,
and teleconnection to climate anomalies, J. Geophys. Res.-Atmos., 112, D10110, 10.1029/2006JD007157, 2007.Ganguli, P., Kumar, D., and Ganguly, A. R.: US power production at risk from water stress in a changing climate,
Sci. Rep.-UK, 7, 11983, 10.1038/s41598-017-12133-9, 2017. Gershunov, A. and Barnett, T. P.: ENSO influence on intraseasonal extreme rainfall and temperature frequencies in the
contiguous United States: observations and model results, J. Climate, 11, 1575–1586, 1998.Gilleland, E. and Katz, R. W.: Extremes 2.0: an extreme value analysis
package in R, J. Stat. Softw., 72, 1–39, 10.18637/jss.v072.i08, 2016.Gimeno, R., Manchado, B., and Mìnguez, R.: Stationarity tests for financial time series, Physica A, 269, 72–78,
10.1016/S0378-4371(99)00081-3, 1999. Gu, X., Zhang, Q., Singh, V. P., Xiao, M., and Cheng, J.: Nonstationarity-based evaluation of flood risk in the Pearl
River basin: changing patterns, causes and implications, Hydrolog. Sci. J., 62, 246–258, 2017.Gudmundsson, L., Bremnes, J. B., Haugen, J. E., and Engen-Skaugen, T.: Technical Note: Downscaling RCM precipitation to
the station scale using statistical transformations – a comparison of methods, Hydrol. Earth Syst. Sci., 16, 3383–3390,
10.5194/hess-16-3383-2012, 2012.Güntner, A., Olsson, J., Calver, A., and Gannon, B.: Cascade-based disaggregation of continuous rainfall time series:
the influence of climate, Hydrol. Earth Syst. Sci., 5, 145–164, 10.5194/hess-5-145-2001, 2001.Guo, X., Fu, D., and Wang, J.: Mesoscale convective precipitation system
modified by urbanization in Beijing City, Atmos. Res., 82, 112–126,
10.1016/j.atmosres.2005.12.007, 2006.
Hamed, K. H. and Rao, A. R.: A modified Mann–Kendall trend test for autocorrelated data, J. Hydrol., 204, 182–196, 1998.Hardwick Jones, R., Westra, S., and Sharma, A.: Observed relationships between extreme sub-daily precipitation, surface
temperature, and relative humidity, Geophys. Res. Lett., 37, L22805, 10.1029/2010GL045081, 2010. Hawkins, E. and Sutton, R.: The potential to narrow uncertainty in regional climate predictions, B. Am. Meteorol. Soc.,
90, 1095–1107, 2009.Hu, S.: Akaike information criterion, Cent. Res. Sci. Comput., NC State University, Raleigh, NC, available at:
http://www4.ncsu.edu/~shu3/Presentation/AIC_2012.pdf (last access: 12 December 2016), 2007.
Hurvich, C. M. and Tsai, C.-L.: Model selection for extended quasi-likelihood models in small samples,
Biometrics, 51, 1077–1084, 1995. Ivancic, T. J. and Shaw, S. B.: Examining why trends in very heavy precipitation should not be mistaken for trends in very
high river discharge, Climatic Change, 133, 681–693, 2015. Jakob, D.: Nonstationarity in extremes and engineering design, in: Extremes in a Changing Climate,
Springer, 363–417, 2013.Jarvis, A., Reuter, H. I., Nelson, A., and Guevara, E.: Hole-filled SRTM for
the globe Version 4, available at:
http://www.cgiar-csi.org/data/srtm-90m-digital-elevation-database-v4-1
(last access: 13 December 2016), 2008. Jebari, S., Berndtsson, R., Olsson, J., and Bahri, A.: Soil erosion estimation based on rainfall
disaggregation, J. Hydrol., 436, 102–110, 2012.Karmakar, S. and Simonovic, S.: Flood Frequency Analysis Using Copula with Mixed Marginal Distributions, Water
Resour. Res. Rep., available at: http://ir.lib.uwo.ca/wrrr/19 (last access: 15 December 2016), 2007. Katz, R. W. and Brown, B. G.: Extreme events in a changing climate: variability is more important than averages, Climatic
Change, 21, 289–302, 1992. Katz, R. W., Parlange, M. B., and Naveau, P.: Statistics of extremes in hydrology, Adv. Water Resour., 25, 1287–1304,
2002.Kerr, D.: Some aspects of the geography of finance in Canada, Can. Geogr.-Geogr. Can., 9, 175–192,
10.1111/j.1541-0064.1965.tb00825.x, 1965.
Knutson, T. R., Zeng, F., and Wittenberg, A. T.: Seasonal and annual mean precipitation extremes occurring during 2013:
a US focused analysis, B. Am. Meteorol. Soc., 95, S19–S23, 2014.Kodra, E. and Ganguly, A. R.: Asymmetry of projected increases in extreme temperature distributions, Sci. Rep.-UK, 4,
5884, 10.1038/srep05884, 2014.Komi, K., Amisigo, B. A., Diekkrüger, B., and Hountondji, F. C.: Regional flood frequency analysis in the Volta River
Basin, West Africa, Hydrology, 3, 5, 10.3390/hydrology30100005, 2016. Koutsoyiannis, D. and Montanari, A.: Negligent killing of scientific concepts: the stationarity case, Hydrolog. Sci. J.,
60, 1174–1183, 2015.Kumar, D. and Ganguly, A. R.: Intercomparison of model response and internal variability across climate model ensembles,
Clim. Dynam., 1–13, 10.1007/s00382-017-3914-4, 2017.Kunkel, K. E.: North American trends in extreme precipitation, Nat. Hazards, 29, 291–305, 10.1023/A:1023694115864,
2003. Kwiatkowski, D., Phillips, P. C., Schmidt, P., and Shin, Y.: Testing the null hypothesis of stationarity against the
alternative of a unit root: How sure are we that economic time series have a unit root?, J. Econ., 54, 159–178, 1992.Lapen, D. R. and Hayhoe, H. N.: Spatial analysis of seasonal and annual temperature and precipitation normals in Southern
Ontario, Canada, J. Great Lakes Res., 29, 529–544, 10.1016/S0380-1330(03)70457-2, 2003.Lenderink, G. and van Meijgaard, E.: Increase in hourly precipitation extremes beyond expectations from
temperature changes, Nat. Geosci., 1, 511–514, 10.1038/ngeo262, 2008.Lenderink, G., Barbero, R., Loriaux, J. M., and Fowler, H. J.: Super-Clausius–Clapeyron scaling of extreme hourly
convective precipitation and its relation to large-scale atmospheric conditions, J. Climate, 30, 6037–6052,
10.1175/JCLI-D-16-0808.1, 2017.Li, H., Sheffield, J., and Wood, E. F.: Bias correction of monthly precipitation and temperature fields from
Intergovernmental Panel on Climate Change AR4 models using equidistant quantile matching, J. Geophys. Res.-Atmos., 115, D10101,
10.1029/2009JD012882, 2010. Lima, C. H., Kwon, H.-H., and Kim, J.-Y.: A Bayesian beta distribution model for estimating rainfall IDF curves in
a changing climate, J. Hydrol., 540, 744–756, 2016. Madsen, H., Arnbjerg-Nielsen, K., and Mikkelsen, P. S.: Update of regional intensity–duration–frequency curves in
Denmark: tendency towards increased storm intensities, Atmos. Res., 92, 343–349, 2009. Mailhot, A., Beauregard, I., Talbot, G., Caya, D., and Biner, S.: Future changes in intense precipitation over Canada
assessed from multi-model NARCCAP ensemble simulations, Int. J. Climatol., 32, 1151–1163, 2012. Maraun, D.: Bias correction, quantile mapping, and downscaling: revisiting the inflation issue, J. Climate, 26,
2137–2143, 2013. Markose, S. and Alentorn, A.: The Generalized Extreme Value (GEV) distribution, implied tail index and option pricing,
J. Deriv., 18, 35–60, 2005.
Martins, E. S. and Stedinger, J. R.: Generalized maximum-likelihood generalized extreme-value quantile estimators
for hydrologic data, Water Resour. Res., 36, 737–744, 2000.Meehl, G. A., Goddard, L., Murphy, J., Stouffer, R. J., Boer, G., Danabasoglu, G., Dixon, K., Giorgetta, M. A.,
Greene, A. M., Hawkins, E., Hegerl, G., Karoly, D., Keenlyside, N., Kimoto, M., Kirtman, B., Navarra, A., Pulwarty, R., Smith, D.,
Stammer, D., and Stockdale, T.: Decadal prediction, B. Am. Meteorol. Soc., 90, 1467–1485, 10.1175/2009BAMS2778.1, 2009.Miao, C., Sun, Q., Borthwick, A. G. L., and Duan, Q.: Linkage between hourly precipitation events and atmospheric
temperature changes over China during the warm season, Sci. Rep.-UK, 6, srep22543, 10.1038/srep22543, 2016.Mikkelsen, P. S., Madsen, H., Arnbjerg-Nielsen, K., Rosbjerg, D., and Harremoës, P.: Selection of regional historical
rainfall time series as input to urban drainage simulations at ungauged locations, Atmos. Res., 77, 4–17,
10.1016/j.atmosres.2004.10.016, 2005.Miller, J. D., Kim, H., Kjeldsen, T. R., Packman, J., Grebby, S., and Dearden, R.: Assessing the impact of urbanization on
storm runoff in a peri-urban catchment using historical change in impervious cover, J. Hydrol., 515, 59–70,
10.1016/j.jhydrol.2014.04.011, 2014.Milly, P. C. D., Betancourt, J., Falkenmark, M., Hirsch, R. M., Kundzewicz, Z. W., Lettenmaier, D. P., and
Stouffer, R. J.: Stationarity Is Dead: Whither Water Management?, Science, 319, 573–574, 10.1126/science.1151915, 2008.Mishra, V., Dominguez, F., and Lettenmaier, D. P.: Urban precipitation extremes: How reliable are regional climate
models?, Geophys. Res. Lett., 39, L03407, 10.1029/2011GL050658, 2012. Moglen, G. E. and Schwartz, D. E.: Methods for adjusting US geological survey rural regression peak discharges in an urban
setting, US Geological Survey Scientific Investigation Report 2006-5270, 1–55, 2006. Mohsin, T. and Gough, W. A.: Characterization and estimation of Urban Heat Island at Toronto: impact of the choice of
rural sites, Theor. Appl. Climatol., 108, 105–117, 2012. Mölders, N. and Olson, M. A.: Impact of urban effects on precipitation in high latitudes, J. Hydrometeorol., 5,
409–429, 2004.Mondal, A. and Mujumdar, P. P.: Modeling non-stationarity in intensity, duration and frequency of extreme rainfall over
India, J. Hydrol., 521, 217–231, 10.1016/j.jhydrol.2014.11.071, 2015. Montanari, A. and Koutsoyiannis, D.: Modeling and mitigating natural hazards: stationarity is immortal!, Water
Resour. Res., 50, 9748–9756, 2014.O'Gorman, P. A.: Precipitation extremes under climate change, Curr. Clim.
Change Rep., 1, 49–59, 10.1007/s40641-015-0009-3, 2015.
O'Gorman, P. A. and Schneider, T.: The physical basis for increases in
precipitation extremes in simulations of 21st-century climate change, P.
Natl. Acad. Sci. USA, 106, 14773–14777, 2009.
Olsson, J.: Limits and characteristics of the multifractal behaviour of a high-resolution rainfall time series, Nonlinear
Proc. Geoph., 2, 23–29, 1995.Olsson, J.: Evaluation of a scaling cascade model for temporal rain- fall disaggregation, Hydrol. Earth Syst. Sci., 2,
19–30, 10.5194/hess-2-19-1998, 1998.Paixao, E., Auld, H., Mirza, M. M. Q., Klaassen, J., and Shephard, M. W.:
Regionalization of heavy rainfall to improve climatic design values for
infrastructure: case study in Southern Ontario, Canada, Hydrol. Sci. J., 56,
1067–1089, 10.1080/02626667.2011.608069, 2011.Papalexiou, S. M., Koutsoyiannis, D., and Makropoulos, C.: How extreme is
extreme? An assessment of daily rainfall distribution tails, Hydrol. Earth
Syst. Sci., 17, 851–862, 10.5194/hess-17-851-2013, 2013.Partridge, M., Olfert, M. R., and Alasia, A.: Canadian cities as regional
engines of growth: agglomeration and amenities, Can. J. Econ., 40, 39–68,
10.1111/j.1365-2966.2007.00399.x, 2007.
Pendergrass, A. G., Lehner, F., Sanderson, B. M., and Xu, Y.: Does extreme
precipitation intensity depend on the emissions scenario?, Geophys. Res.
Lett., 42, 8767–8774, 2015.
Petrow, T. and Merz, B.: Trends in flood magnitude, frequency and seasonality
in Germany in the period 1951–2002, J. Hydrol., 371, 129–141, 2009.
Pettitt, A. N.: A non-parametric approach to the change-point problem, Appl. Statist., 126–135, 1979.Pfahl, S., O'Gorman, P. A., and Fischer, E. M.: Understanding the regional pattern of projected future changes in extreme
precipitation, Nat. Clim. Change, 7, 423–427, 10.1038/nclimate3287,
2017.Pinheiro, E. C. and Ferrari, S. L. P.: A comparative review of
generalizations of the Gumbel extreme value distribution with an application
to wind speed data, J. Stat. Comput. Sim., 86, 2241–2261,
10.1080/00949655.2015.1107909, 2016.
Porporato, A. and Ridolfi, L.: Influence of weak trends on exceedance
probability, Stoch. Hydrol. Hydraul., 12, 1–14, 1998.Prein, A. F., Rasmussen, R. M., Ikeda, K., Liu, C., Clark, M. P., and
Holland, G. J.: The future intensification of hourly precipitation extremes,
Nat. Clim. Change, 7, 48–52, 10.1038/nclimate3168, 2016.Prein, A. F., Liu, C., Ikeda, K., Trier, S. B., Rasmussen, R. M., Holland, G.
J., and Clark, M. P.: Increased rainfall volume from future convective storms
in the US, Nat. Clim. Change, 7, 880–884, 10.1038/s41558-017-0007-7,
2017.
Priestley, M. B. and Rao, T. S.: A test for non-stationarity of time-series,
J. Roy. Stat. Soc. B Met., 31, 140–149, 1969.Rana, A., Bengtsson, L., Olsson, J., and Jothiprakash, V.: Development of IDF-curves
for tropical india by random cascade modeling, Hydrol. Earth Syst. Sci. Discuss.,
10.5194/hessd-10-4709-2013, 2013.Read, L. K. and Vogel, R. M.: Reliability, return periods, and risk under
nonstationarity, Water Resour. Res., 51, 6381–6398,
10.1002/2015WR017089, 2015.
Reddy, M. J. and Ganguli, P.: Spatio-temporal analysis and derivation of
copula-based intensity–area–frequency curves for droughts in western
Rajasthan (India), Stoch. Env. Res. Risk A., 27, 1975–1989, 2013.
Renard, B., Sun, X., and Lang, M.: Bayesian methods for non-stationary extreme value analysis, in: Extremes in a Changing
Climate, Springer, 39–95, 2013.Rootzén, H. and Katz, R. W.: Design Life Level: quantifying risk in
a changing climate, Water Resour. Res., 49, 5964–5972,
10.1002/wrcr.20425, 2013.Rosner, A., Vogel, R. M., and Kirshen, P. H.: A risk-based approach to flood
management decisions in a nonstationary world, Water Resour. Res., 50,
1928–1942, 10.1002/2013WR014561, 2014.Ross, G. J., Tasoulis, D. K., and Adams, N. M.: Nonparametric monitoring of
data streams for changes in location and scale, Technometrics, 53, 379–389,
10.1198/TECH.2011.10069, 2011.Ruiz-Villanueva, V., Borga, M., Zoccatelli, D., Marchi, L., Gaume, E., and
Ehret, U.: Extreme flood response to short-duration convective rainfall in
South-West Germany, Hydrol. Earth Syst. Sci., 16, 1543–1559,
10.5194/hess-16-1543-2012, 2012.Sadri, S., Kam, J., and Sheffield, J.: Nonstationarity of low flows and their
timing in the eastern United States, Hydrol. Earth Syst. Sci., 20, 633–649,
10.5194/hess-20-633-2016, 2016.
Sanderson, M. and Gorski, R.: The effect of metropolitan Detroit–Windsor on
precipitation, J. Appl. Meteorol., 17, 423–427, 1978.
Sandink, D., Simonovic, S. P., Schardong, A., and Srivastav, R.: A decision
support system for updating and incorporating climate change impacts into
rainfall intensity-duration-frequency curves: Review of the stakeholder
involvement process, Environ. Modell. Softw., 84, 193–209, 2016.SC (Statistics Canada): 2011 Census – Boundary files, Government of Canada,
available at:
http://www12.statcan.gc.ca/census-recensement/2011/geo/bound-limit/bound-limit-2011-eng.cfm,
last access: November 2016.
Schaller, N., Kay, A. L., Lamb, R., Massey, N. R., van Oldenborgh, G. J.,
Otto, F. E. L., Sparrow, S. N., Vautard, R., Yiou, P., Ashpole, I.,
Bowery, A., Crooks, S. M., Haustein, K., Huntingford, C., Ingram, W. J.,
Jones, R. G., Legg, T., Miller, J., Skeggs, J., Wallom, D., Weisheimer, A.,
Wilson, S., Stott, P. A., and Allen, M. R.: Human influence on climate in the
2014 southern England winter floods and their impacts, Nat. Clim. Change, 6,
627–634, 2016.Schroeer, K. and Kirchengast, G.: Sensitivity of extreme precipitation to
temperature: the variability of scaling factors from a regional to local
perspective, Clim. Dynam., 1–14, 10.1007/s00382-017-3857-9, 2017.Sarhadi, A. and Soulis, E. D.: Time-varying extreme rainfall
intensity-duration-frequency curves in a changing climate, Geophys. Res.
Lett., 44, 2454–2463, 10.1002/2016GL072201, 2017.
Serinaldi, F. and Kilsby, C. G.: Stationarity is undead: Uncertainty
dominates the distribution of extremes, Adv. Water Resour., 77, 17–36, 2015.Serinaldi, F. and Kilsby, C. G.: The importance of prewhitening in change
point analysis under persistence, Stoch. Env. Res. Risk A., 30, 763–777,
10.1007/s00477-015-1041-5, 2016.Serinaldi, F., Kilsby, C. G., and Lombardo, F.: Untenable nonstationarity: An
assessment of the fitness for purpose of trend tests in hydrology, Adv. Water
Resour., 111, 132–155, 10.1016/j.advwatres.2017.10.015, 2018.
Shabbar, A., Bonsal, B., and Khandekar, M.: Canadian precipitation patterns
associated with the Southern Oscillation, J. Climate, 10, 3016–3027, 1997.
Shaw, S. B., Royem, A. A., and Riha, S. J.: The relationship between extreme
hourly precipitation and surface temperature in different hydroclimatic
regions of the United States, J. Hydrometeorol., 12, 319–325, 2011.
Shephard, M. W., Mekis, E., Morris, R. J., Feng, Y., Zhang, X., Kilcup, K.,
and Fleetwood, R.: Trends in Canadian short-duration extreme rainfall:
including an intensity–duration–frequency perspective, Atmos. Ocean, 52,
398–417, 2014.Simonovic, S. P. and Peck, A.: Updated rainfall intensity duration frequency
curves for the City of London under the changing climate, Department of Civil
and Environmental Engineering, The University of Western Ontario, available
at: http://ir.lib.uwo.ca/wrrr/29/ (last access: 13 January 2017), 2009.Singh, J., Vittal, H., Karmakar, S., Ghosh, S., and Niyogi, D.: Urbanization
causes nonstationarity in Indian Summer Monsoon Rainfall extremes, Geophys.
Res. Lett., 43, 11269–11277, 10.1002/2016GL071238, 2016.
Stocker, T. F., Qin, D., Plattner, G. K., Tignor, M., Allen, S. K.,
Boschung, J., Nauels, A., Xia, Y., Bex, V., and Midgley, P. M.: Climate
change 2013: the physical science basis. Intergovernmental panel on climate
change, Working Group I Contribution to the IPCC Fifth Assessment Report
(AR5), New York, 2013.
Svensson, C. and Jones, D. A.: Review of rainfall frequency estimation methods, J. Flood Risk Manag., 3, 296–313, 2010.Switzman, H., Razavi, T., Traore, S., Coulibaly, P., Burn, D. H.,
Henderson, J., Fausto, E., and Ness, R.: Variability of future extreme
rainfall statistics: comparison of multiple IDF projections, J. Hydrol. Eng.,
22, 04017046, 10.1061/(ASCE)HE.1943-5584.0001561, 2017.
Teutschbein, C. and Seibert, J.: Bias correction of regional climate model simulations for hydrological climate-change
impact studies: Review and evaluation of different methods, J. Hydrol., 456,
12–29, 2012.
TRCA (Toronto Region Conservation Authority): Resilient City: Preparing for
Extreme Weather Events, City of Toronto, Canada, 2013.Towler, E., Rajagopalan, B., Gilleland, E., Summers, R. S., Yates, D., and
Katz, R. W.: Modeling hydrologic and water quality extremes in a changing
climate: a statistical approach based on extreme value theory, Water Resour.
Res., 46, W11504, 10.1029/2009WR008876, 2010.
Trenberth, K. E.: Atmospheric moisture recycling: role of advection and local
evaporation, J. Climate, 12, 1368–1381, 1999.
Van Gelder, P., Wang, W., and Vrijling, J. K.: Statistical estimation methods
for extreme hydrological events, in Extreme Hydrological Events: New Concepts
for Security, Springer, 2006.Villarini, G., Serinaldi, F., Smith, J. A., and Krajewski, W. F.: On the
stationarity of annual flood peaks in the continental United States during
the 20th century, Water Resour. Res., 45, W08417, 10.1029/2008WR007645,
2009a.Villarini, G., Smith, J. A., Serinaldi, F., Bales, J., Bates, P. D., and
Krajewski, W. F.: Flood frequency analysis for nonstationary annual peak
records in an urban drainage basin, Adv. Water Resour., 32, 1255–1266,
10.1016/j.advwatres.2009.05.003, 2009b.
von Storch, H. and Navarra, A.: Analysis of Climate Variability: Applications
of Statistical Techniques, Springer, 1–303, 1999.
Wasko, C. and Sharma, A.: Steeper temporal distribution of rain intensity at
higher temperatures within Australian storms, Nat. Geosci., 8, 527–529,
2015.
Wasko, C. and Sharma, A.: Continuous rainfall generation for a warmer climate
using observed temperature sensitivities, J. Hydrol., 544, 575–590, 2017.
Wang, X., Huang, G., Liu, J., Li, Z., and Zhao, S.: Ensemble projections of
regional climatic changes over Ontario, Canada, J. Climate, 28, 7327–7346,
2015.
Westra, S. and Sisson, S. A.: Detection of non-stationarity in precipitation
extremes using a max-stable process model, J. Hydrol., 406, 119–128, 2011.Westra, S., Alexander, L. V., and Zwiers, F. W.: Global increasing trends in
annual maximum daily precipitation, J. Climate, 26, 3904–3918,
10.1175/JCLI-D-12-00502.1, 2012.Wilson, P. S. and Toumi, R.: A fundamental probability distribution for heavy
rainfall, Geophys. Res. Lett., 32, L14812, 10.1029/2005GL022465, 2005.
Xie, H., Li, D., and Xiong, L.: Exploring the ability of the Pettitt method
for detecting change point by Monte Carlo simulation, Stoch. Env. Res.
Risk A., 28, 1643–1655, 2014.
Yilmaz, A. G. and Perera, B. J. C.: Extreme rainfall nonstationarity
investigation and intensity–frequency–duration relationship, J. Hydrol.
Eng., 19, 1160–1172, 2013.Yilmaz, A. G., Hossain, I., and Perera, B. J. C.: Effect of climate change and variability on extreme rainfall
intensity–frequency–duration relationships: a case study of Melbourne,
Hydrol. Earth Syst. Sci., 18, 4065–4076, 10.5194/hess-18-4065-2014,
2014.Yilmaz, A. G., Imteaz, M. A., and Perera, B. J. C.: Investigation of
non-stationarity of extreme rainfalls and spatial variability of rainfall
intensity–frequency–duration relationships: a case study of Victoria,
Australia, Int. J. Climatol., 37, 430–442, 10.1002/joc.4716, 2017.
Yiou, P. and Cattiaux, J.: Contribution of atmospheric circulation to wet north European summer precipitation of 2012,
B. Am. Meteorol. Soc., 94, S39, 2013.
Yue, S. and Wang, C. Y.: Power of the Mann–Whitney test for detecting
a shift in median or mean of hydro-meteorological data, Stoch. Env. Res.
Risk A., 16, 307–323, 2002.
Yue, S., Pilon, P., Phinney, B., and Cavadias, G.: The influence of autocorrelation on the ability to detect trend in
hydrological series, Hydrol. Process., 16, 1807–1829, 2002.
Yue, S., Pilon, P., and Phinney, B. O. B.: Canadian streamflow trend
detection: impacts of serial and cross-correlation, Hydrol. Sci. J., 48,
51–63, 2003.