HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-5823-2017Simple scaling of extreme precipitation in North AmericaInnocentiSilviasilvia.innocenti@ete.inrs.cahttps://orcid.org/0000-0002-3095-533XMailhotAlainFrigonAnnehttps://orcid.org/0000-0002-3621-214XCentre Eau-Terre-Environnement, INRS, 490 de la Couronne, Québec, G1K 9A9, CanadaConsortium Ouranos, 550 Sherbrooke Ouest, Montrèal, H3A 1B9, CanadaSilvia Innocenti (silvia.innocenti@ete.inrs.ca)24November201721115823584611November201630November20165October201722October2017This 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/21/5823/2017/hess-21-5823-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/5823/2017/hess-21-5823-2017.pdf
Extreme precipitation is highly variable in space and time. It is therefore
important to characterize precipitation intensity distributions on several
temporal and spatial scales. This is a key issue in infrastructure design and
risk analysis, for which intensity–duration–frequency (IDF) curves are the
standard tools used for describing the relationships among extreme rainfall
intensities, their frequencies, and their durations. Simple scaling (SS)
models, characterizing the relationships among extreme probability
distributions at several durations, represent a powerful means for improving
IDF estimates. This study tested SS models for approximately 2700 stations in
North America. Annual maximum series (AMS) over various duration intervals
from 15 min to 7 days were considered. The range of validity, magnitude, and
spatial variability of the estimated scaling exponents were investigated.
Results provide additional guidance for the influence of both local
geographical characteristics, such as topography, and regional climatic
features on precipitation scaling. Generalized extreme-value (GEV)
distributions based on SS models were also examined. Results demonstrate an
improvement of GEV parameter estimates, especially for the shape parameter,
when data from different durations were pooled under the SS hypothesis.
Introduction
Extreme precipitation is highly variable in space and time as various
physical processes are involved in its generation. Characterizing this
spatial and temporal variability is crucial for infrastructure design and to
evaluate and predict the impacts of natural hazards on ecosystems and
communities. Available precipitation records, however, are sparse and cover
short time periods, making a complete and adequate statistical
characterization of extreme precipitation difficult. The resolution of
available data, whether observed at meteorological stations or simulated by
weather and climate models, often mismatches the resolution needed for
applications e.g.,, thus
adding to the difficulty of achieving complete and adequate statistical
characterizations of extreme precipitation.
The need for multi-scale analysis of precipitation has been widely recognized
in the past among
others and much
effort has been put into the development of relationships among extreme
precipitation characteristics on different scales. The conventional approach for characterizing scale
transitions in time involves the construction of
intensity–duration–frequency (IDF) or the equivalent
depth–duration–frequency (DDF) curves
.
These curves are a standard tool for hydraulic design and risk analysis as
they describe the relationships between the frequency of occurrence of
extreme rainfall intensities (depth) Xd and various durations de.g.,. Analysis is usually conducted by separately
estimating the statistical distributions of Xd at the different durations
see for discussions about commonly used probability
distributions. The parameters or the
quantiles of these theoretical distributions are then empirically compared to
describe the variations of extreme rainfall properties across temporal
scales.
Despite its simplicity, this procedure presents several drawbacks. In
particular, it does not guarantee the statistical consistency of
precipitation distributions, independently estimated at the different
durations, and it limits IDF extrapolation on non-observed scales or at ungauged
sites. Uncertainties of estimated quantiles are also presumably larger
because precipitation distribution and IDF curve parameters are fitted
separately.
Scaling models based on the
concept of scale invariance , have been proposed to link
rainfall features on different temporal and spatial scales. Scale invariance
states that the statistical characteristics (e.g., moments or quantiles) of
precipitation intensity observed on two different
scales d and λd can be
related to each other by a power law of the form:
f(Xλd)=λ-Hf(Xd),
where f(.) is a
function of X with invariant shape when rescaling the variable X by a
multiplicative factor λ and for some values of the exponent H∈R. In the simplest case, a constant multiplicative factor
adequately describes the scale change. The corresponding mathematical models
are known as simple scaling (SS) models . SS models
are attractive because of the small number of parameters involved, as opposed
to multiscaling (MS) models which involve more than one
multiplicative factor in Eq. ()
e.g.,.
A single scaling exponent, H, is used to characterize the extreme
rainfall distribution on all scales
over which the scale invariance property holds. As a consequence, a
consistent and efficient estimation of extreme precipitation characteristics
is possible, even on non-sampled temporal scales, and a parsimonious
formulation of IDF curves based on analytical results is available
e.g.,.
Theoretical and physical evidence of the scaling properties of precipitation
intensity over a wide range of durations has been provided by several
studies. MS has been demonstrated to be appropriate for modelling the temporal
scaling features of the precipitation process (i.e., not only the extreme
distribution) and for the extremes in event-based representations of rainfall
(stochastic rainfall modelling) e.g.,and references
therein. These
multifractal features of precipitation last within a finite range of temporal
scales (approximatively between 1 h and 1 week) and concern the temporal
dependence structure of the process. They have been connected to the large
fluctuations of the atmospheric and climate system governing precipitation
which are likely to produce a “cascade of random multiplicative effects”
.
At the same time, many studies confirmed the validity of SS for approximating
the precipitation distribution tails in IDF estimation for examples of
durations ranging from 5 min to 24 h
see.
This type of scaling is substantially different from the temporal scaling
since it only refers to the power law shape of the marginal distribution of
extreme rainfall. Application of the SS models to precipitation records
showed that the scaling exponent estimates may depend on the considered range
of durations e.g., and the climatological and
geographical features of the study regions
e.g.,.
However, the application of the SS framework has been mainly restricted to
specific regions and small observational datasets. A deeper analysis of the
effects of geoclimatic factors on the SS approximation validity and on
estimated scaling exponent is thus needed.
The present study aims to deepen the knowledge of the scale-invariant
properties of extreme rainfall intensity by analyzing SS model estimates
across North America using a large number of station series. The specific
objectives of this study are (a) assess the ability of SS models to
reproduce extreme precipitation distribution, (b) explore the variability of
scaling exponent estimates over a broad set of temporal durations and
identify possible effects of the dominant climate and pluviometric regimes on
SS, and (c) evaluate the possible advantages of the introduction of the SS
hypothesis in parametric models of extreme precipitation.
Note that, although modifications in precipitation distributions are expected
as a result of climate changes
e.g.,, the proposed approach
implicitly relies on the assumption of stationarity for extreme rainfall.
This choice has been motivated by both the limited evidence for changes in
rainfall intensities for North America extremes during recent decades and the
difficulties of assessing distribution changes from short recorded series,
especially for sub-daily extremes and references
therein.
The article is structured as follows. In Sect. the
statistical basis of scaling models is presented, while data and their
preliminary treatments are described in Sect. .
Section presents the distribution-free estimation of SS
models and their validation using available series. Section
focuses on the spatial variability of SS exponents and discusses the
scaling exponent variation from a regional perspective. Finally, the SS
estimation based on the generalized extreme-value (GEV) assumption is
discussed in Sect. , followed by a discussion and
conclusions (Sect. 7). Table S1 of the Supplement lists in alphabetic order
the recurrent acronyms used in the text.
Simple scaling models for precipitation intensity
When the equality in Eq. () holds for the cumulative
distribution function (CDF) of the precipitation intensity X sampled at two
different durations d and λd, the SS can be
expressed as follows :
Xd=distλHXλd,
where H∈R and =dist means that the same
probability distribution applies for Xd and Xλd, up to a
dilatation or contraction of size λH. An important consequence of
the SS assumption is that Xd and λHXλd have the
same distribution. Hence, if Xd and Xλd have finite moments
of order q, E[Xdq] and E[Xλdq], these moments are thus
linked by the following relationship :
E[Xdq]=λHqE[Xλdq].
This last relationship is usually referred to as the wide sense
simple scaling property and signifies that simple scaling
results in a simple translation of the log moments between scales:
lnEXdq=lnEXλdq+Hqlnλ.
Moreover, without loss of generality, λ can always be expressed as
the scale ratio λ=d/d* defined for a reference duration d*
chosen, for simplicity, as d*=1. Therefore, the SS model can be
estimated and validated over a set of durations d1<d2<…<dD by
simply checking the linearity in a log–log plot of the X moments versus the
observed durations dj, j=1,2,…,D (see, for instance,
; ; Fig. 1 of ;
and Fig. 2a of ). If H estimated for the first moment
equals the exponents (slopes) for the other moments, the precipitation
intensity X can be considered scale invariant under SS in the interval of
durations d1 to dD.
More sophisticated methods have also been proposed for detecting and
estimating scale invariance (for instance, dimensional analysis,
; ; ;
; spectral analysis and wavelet estimation
; ; ; and
empirical probability distribution function (PDF) power law detection
; ; ).
However, estimation through the moment scaling analysis is by far the
simplest and most intuitive tool to check the SS hypothesis for a large
dataset. For this reason, the presented analyses are based on this method.
According to the literature, the values of the scaling exponents H
generally range between 0.4 and 0.8 for precipitation intensity considered
on daily and shorter timescales
e.g., (note that
for the rainfall depth the scaling exponent Hdepth=1-H
applies). Values from 0.3 to 0.9 have also been reported for some specific
cases e.g.,for scaling intervals defined within 1 and
24 h.
Higher H values have been generally observed for shorter-duration
intervals, and regions dominated by convective precipitation
e.g.,and references
therein. Nonetheless, some
studies performing spatio-temporal scaling analysis reached a different
conclusion. For instance, , analyzing extreme precipitation
events from radar data for durations between 5 min and 6 h and spatial
scales between 1 and 50 km, indirectly showed that stratiform precipitation
intensity generally displays higher temporal scaling exponents than
convective intensity. For short-duration intervals (typically less than
1 h), previous studies have also reported more spatially homogeneous H
estimates than for long-duration intervals e.g.,and references
therein. This suggests that processes involved in the
generation of local precipitation are comparable across different regions.
More generally, higher H values are associated with larger variations in
moment values as the scale is changed (i.e., a stronger scaling), while H
close to zero means that the Xd distributions for different durations d
more closely match each other.
Simple scaling GEV models
Annual maximum series (AMS) are widely used to select rainfall extremes
from available precipitation series. Various theoretical arguments and experimental
evidences support their use for extreme precipitation inference
e.g.,.
Based on the asymptotic results of the extreme value theory
, the AMS distribution of a random variable X is well
described by the GEV distribution family. If we
represent the AMS by x1,x2,…,xn, the GEV CDF can be
written as follows :
F(x)=exp-1+ξx-μσ-1/ξ,
where ξ≠0, -∞<x≤μ+σ/ξ if ξ<0 (bounded
tail), and 1/μ+σξ≤x<+∞ if ξ>0 (heavy tail).
If ξ=0 (light-tailed shape, Gumbel distribution),
Eq. () reduces to the following:
F(x)=exp-exp-x-μσ,
where -∞<x<+∞. In Eqs. ()
and (), the parameters μ∈R, σ>0
and ξ respectively represent the location, scale, and shape parameters of
the distribution. The shape parameter describes the characteristics of the
distribution tails. Thus, high-order quantile estimation is particularly
affected by the value of ξ.
In applications, the GEV distribution is frequently constrained by the
assumption that ξ=0 (i.e., to the Gumbel distribution), due to the
difficulty of estimating significant values of the shape parameter when the
recorded series are short e.g.,.
However, based on theoretical and empirical evidence, many authors have shown
that this assumption is too restrictive for extreme precipitation, and may
lead to important underestimations of the extreme quantiles
e.g.,.
Instead, approaches aimed at increasing the sample size may be used to
improve the estimation of the GEV distribution shape parameter for
instance, the regional frequency analysis (RFA),. Among these
approaches, SS models constitute an appealing way to pool data from different
samples (durations) and reduce uncertainties in GEV parameters.
For the GEV distribution it is straightforward to verify that, if X=distGEV(μ,σ,ξ), then λX=distGEV(λμ,λσ,ξ) for any
λ∈R. This means that the GEV family described by
Eqs. () and () satisfies
Eq. () and thus complies with statistical scale
invariance for any constant multiplicative transformation of X. Hence, when
the scale invariance is further assumed for the change of observational scale
from duration d to λd (as in Eq. ), the wide sense SS
definition (Eq. ) gives the following:
μd=dHμ*,σd=dHσ*,andξd=ξ*,
where μ*, σ*, and ξ* represent the GEV parameters for a
reference duration d* chosen, for simplicity, as d*=1, so that λ=d.
SS GEV estimation
Taking advantage of the scale-invariant formulation of the GEV distribution,
many authors have proposed simple scaling IDF and DDF models for extreme
precipitation series
e.g.,. In
these cases, the scaling exponent and the GEV parameters are generally
estimated in two separate steps: first, the H value is empirically
determined through a log–log linear regression, as described above; then,
GEV parameters μ*, σ*, and ξ* for the reference duration
d* are estimated on the pooled sample of all available durations. In this
case, classical estimation procedures, such as GEV maximum likelihood (ML)
or probability-weighted moment (PWM)
, can be used.
In a few other cases, a generalized additive model ML (GAM-ML) framework
has also been used to obtain the joint estimate of
H,μ*,σ*, and ξ* through the introduction of the duration as
model covariate e.g.,.
Data and study region
Four station datasets were used for the construction of intensity AMSs at different durations: the daily maxima precipitation
data (DMPD) and the hourly Canadian precipitation data (HCPD) datasets
provided by Environment and Climate Change Canada and the
(in french Ministère du Développement Durable, de l'Environnement et de la Lutte contre les Changements Climatiques) for Canada, and the hourly precipitation data (HPD) and 15 min
precipitation data (15PD) datasets made available by the National Oceanic and
Atmospheric Administration agency
(http://www.ncdc.noaa.gov/data-access/land-based-station-data) for
the United States. The total number of stations was approximately 3400, with
roughly 2200 locations having both DMPD and HCPD series, or both HPD and 15PD
series. The majority of stations are located in the United States and in the
southern and most densely populated areas of Canada. In northern regions the
station network is sparse and the record length does not generally exceed 15
or 20 years. Moreover, for most of DMPD and HCPD stations, the annual
recording period does not cover the winter season and available series
generally include precipitation measured from May to October. For this
reason, the year from which the annual maxima was sampled was
limited to the recording season going from June to September for northern
stations (stations located north of the 52nd parallel) and from June to
September for the southern stations. As a result, 122 days a year were used
for northern stations and 184 days a year for remaining stations.
List of available datasets and their main characteristics.
DatasetRegionNo. ofOperationalTemporalPrevalentcstationsperiodbresolutionresolution (mm)Daily maximum precipitation dataa (DMPC)Canada3701964–20071, 2, 6, 12 h0.1 (82.25 %)Hourly Canadian precipitation data (HCPD)Canada6651967–20031 h0.1 (70 %)Hourly precipitation data (HPD)USA25311948–20131 h0.254 (82.5 %)15-Min precipitation data (15PD)USA20291971–201315 min2.54 (80.42 %)
a Daily maxima depth series over a 24 h window
beginning at 08:00 LT. b Main station network operational period
corresponding to 25th percentile of the first recording year and the 75th
percentile of the last recording year of the stations. c Prevalent
instrument resolution, estimated by the lowest non-zero value for each
series, and corresponding percentage of stations with this resolution.
Data were collected through a variety of instruments (e.g., standard,
tipping-bucket, and Fischer–Porter rain gauges) and precipitation values
were processed and quality-controlled using both automated and manual methods
HPD and 15PD online documentation. Most often,
observations were recorded by tipping-bucket gauges with tip resolution from
0.1 to 2.54 mm ; 15 min series usually present
the coarser instrument resolution, with a minimum non-zero value of 2.54 mm,
observed for about 80.5 % of 15PD stations. The effects of such a coarse
instrument resolution on simple scaling estimates could be important, leading
to empirical Xd CDFs becoming step-wise functions with a low number of
steps. Some preliminary analyses aiming at evaluating these effects on SS
estimates are presented in the Supplement (see Figs. S2 and S3). However, the
15PD dataset is important considering the associated network density and its
fine temporal resolution, and thus it has been retained for our study. The
main characteristics of the available datasets are summarized in
Table .
The scaling AMS datasets were constructed according to the following steps.
Three duration sets were defined: (a) 15 min to 6 h with a
15 min step, (b) 1 to 24 h with a 1 h step, and (c) 6 to 168 h (7 days) with
a 6 h step. These duration sets are hereinafter referred to as
short-duration (SD), intermediate-duration (ID), and long-duration (LD)
datasets, respectively (see Fig. a).
Meteorological stations that were included in each final dataset were selected
according to the following criteria: (1) precipitation series must have at
least 85 % of valid observations for each May-to-October (or June-to-September) period, otherwise the corresponding year was considered as
missing; (2) each station must have at least 15 valid years; (3) for each
station, it was possible to compute AMS for all durations considered in the
scaling dataset (e.g., HCPD and HPD stations were not included in the SD
dataset because only hourly durations were available). Note that, in order to
exclude outliers possibly associated with recording or measurement errors,
extremely large observations were discarded and assimilated to missing data.
In particular, as in some previous studies
e.g.,, an iterative procedure was
applied prior to step (ii) (1) to discard observations larger than 10 times
the second largest value of the series.
A moving window was
applied to 15PD, HCPD, and HPD series to estimate aggregated series at each
duration. For DMPD series, a quality check was also implemented in order to
guarantee that precipitation intensities recorded each day at different
durations were consistent with each other. For instance, each pair of DMPD
rainfall intensity (mm) (xd1,xd2) observed at durations d1<d2
must respect the condition xd2/xd1≥d1/d2 derived from the
definitions of daily maximum rainfall intensity and depth; otherwise all DMPD
values recorded that day were discarded and assimilated to missing data.
For each selected station, annual maxima were extracted for each valid year
and duration. For stations with both DMPD and HCPD series, or 15PD and HPD
series, for each year, the annual maxima extracted from these two series were
compared and the maximum value was retained as the annual maximum for that
year.
Major characteristics of each scaling AMS dataset are reported in
Table .
Final datasets used in scaling analysis and corresponding AMS
characteristics.
Methodology steps: (a) definition of the SD, ID, and LD
scaling datasets; (b) identification of durations and scaling
intervals within each matrix of Figs.
and ; (c) moment scaling analysis (MSA)
regression for the estimation of the slope coefficients Kq;
(d) slope test: regression of Kq on the moment order q and
Student's t test for the null hypothesis H0: h^1=K1;
(e) examples of valid and non-valid SS stations according to the
slope and GOF tests; (f) example of valid SS station proportion
values and normalized RMSE values, r‾‾xd, as
represented, in Figs. and .
SS estimation through moment scaling analysis (MSA)
Moment scaling analysis (MSA) for the SD, ID, and LD datasets was carried out
to empirically validate the use of SS models for modelling AMS empirical
distributions. Assessing the validity of the SS hypothesis for various
duration intervals also aimed at determining the presence of different
scaling regimes for precipitation intensity distributions.
In order to identify possible changes in the SS properties of AMS
distributions, various scaling intervals were defined for the MSA.
In particular, all possible subsets with 6, 12, 18, and 24 contiguous
durations were considered within each dataset. Figures
and show the 136 scaling intervals thereby
defined: 40 scaling intervals for SD and ID, and 56 scaling intervals for LD.
For instance, the top-left matrix of Fig. a presents the
6-duration scaling intervals 15 min–1 h 30 min, 30 min–1 h 45 min,
…, 4 h 45 min–6 h defined for the SD dataset (i.e., the 19 scaling
intervals containing 6 contiguous durations defined with a 15 min
increment). More schematically, Fig. b shows an example of
the first five 6-duration scaling intervals for the ID dataset (i.e., 1–6 h,
2–7 h, …, 5–10 h, containing six contiguous durations defined with
an increment of 1 h). This procedure was defined in order to evaluate the
sensitivity of the SS estimates to changes in the first duration d1 of the
scaling interval and in the interval length (i.e., the number of durations
included in the scaling interval).
For each scaling interval (for simplicity, their index has been omitted),
the validity of the SS hypothesis was verified according to the following steps:
MSA regression: for each q=0.2,0.4,…,2.8,3, the
slopes Kq of the log–log linear relationships between the empirical
q moments Xdq of Xd1,Xd2,…,XdD and the corresponding durations d1,d2,…,dD in the
scaling interval [d1,dD] were estimated by ordinary least squares (OLS)
(see Fig. 1c for a graphic example). Orders q≥3 were not considered
because of the possible biases affecting empirical high-order moment
estimates.
Slope test: to verify the SS assumption that the estimated Kq exponents
vary linearly with the moment order q, i.e., Kq≈Hq, an OLS
regression between the MSA slopes Kq and q was applied (see
Fig. d). For the regression line Kq=h^0+h^1q, a Student's t test was then used to test the null
hypothesis H0: h^1=K1. If H0 was not rejected at
the significance level α=0.05, the SS assumption was considered
appropriate for the scaling interval and the simple scaling exponent H=K1 was retained.
Goodness-of-fit (GOF) test:
for each duration d, the goodness of fit of the Xd distribution under SS
was tested using the Anderson–Darling (AD) and the Kolmogorov–Smirnov (KS)
tests. These tests aim at validating the appropriateness of the scale
invariance property for approximating the Xd CDF by the distribution of
Xd,ss=d-HXd*. To this end, each AMS, xdj=xdj,1,xdj,2,…,xdj,i,…xdj,n,
recorded at duration dj was rescaled at the reference duration d* by
inverting Eq. ():xdj*=djHxdj,1,djHxdj,2,…,djHxdj,i,…djHxdj,n,where n represents the number of observations (years) in xdj. Then,
the pooled sample, xd*, of the D rescaled
AMS, xdj*, was used to
define Xd* under the SS assumption:xd*=xd1*,…,xdj*,…,xdD*.Since, in Eq. (), D represents the number of
durations dj in the scaling interval, n×D rescaled observations
were included in xd*.
As in previous applications e.g.,, the AD and KS tests
were then applied at significance level α=0.05 to compare the
empirical distributions Cunnane plotting formula, of
the SS sample, xd,ss=d-Hxd*, and the non-SS sample,
xd. In fact, despite the low power of KS and AD tests for small sample
tests, they represent the only suitable solution to the problem of comparing
empirical CDFs when the data do not follow a normal distribution. Because
both AD and KS are affected by the presence of ties in the samples (e.g.,
repeated values due to rounding or instrument resolution), a permutation test
approach was used to estimate test p values. According to
this approach, data in xd and xd,ss were pooled and randomly
reassigned to two samples with the same sizes as the SS and non-SS samples.
Then, the test statistic distribution under the null hypothesis of equality
of the Xd,ss and Xd distributions was approximated by
computing its value over a large set of random samples. Finally, the test
p value was obtained as the proportion of random samples presenting a test
statistic value larger than the value observed for the original sample.
Proportion of stations satisfying both the slope and GOF tests
applied at the 0.95 confidence level, for each duration (vertical axis) and
scaling interval (horizontal axis) for the SD, ID, and LD datasets
(panels a, b, and c, respectively). White circles
indicate proportions between 0.25 and 0.90. See Fig. 1b and f for the
identification of durations and scaling intervals within each matrix.
Cross-validation normalized RMSE averaged over all valid SS stations
(r‾‾xd) for each duration (vertical axis) and
scaling interval (horizontal axis) in the SD, ID, and LD datasets
(panels a, b, and c, respectively). White circles
indicate values between 0.15 and 0.3. See Fig. 1b and f for the
identification of durations and scaling intervals within each matrix.
The SS model validity and the mean error resulting from approximating the
Xd distribution by the SS model were then evaluated in a cross-validation
setting. For this analysis, each duration was iteratively excluded from each
scaling interval and the scaling model re-estimated at each station by
repeating steps 1 to 3 (MSA regression, slope test, and GOF tests).
Predictive ability indices, such as the mean absolute error (MAE) and the
root mean squared error (RMSE) between empirical and SS distribution
quantiles, were then estimated for highest quantiles for valid SS stations.
In particular, to focus on return periods of practical interest for IDF
estimation, only quantiles larger than the median were considered (i.e., only
return periods greater than 2 years).
For each station s, the normalized RMSE, ϵ‾xd,s, was estimated:
ϵ‾xd,s=ϵxd,sx‾d,s,
where ϵxd,s and x‾d,s are, respectively, the
RMSE and the mean value of all Xd quantiles of order p>0.5. Then, the
average over all stations of the normalized RMSE,
ϵ‾‾xd, was computed for each scaling
interval and duration:
ϵ‾‾xd=1ns∑nss=1ϵ‾xd,s,
where ns is the number of valid SS stations in the dataset. Note
that ϵ‾‾xd is a measure of error, meaning
that values of ϵ‾xd,s closer to 0 correspond to a
better fit than larger values.
Model estimation and validation
Figure presents the results of steps 1 to 3 of the
methodology for evaluating the SS validity. For all the three scaling
datasets, no particular pattern was observed for slope test results, and at
most 2 % of the stations within each scaling interval displaying a non-linear evolution of the scaling exponent with the moment order. For this
reason, Fig. a–c show, for each scaling interval and
duration, the proportion of valid SS stations without differentiating for
slope or GOF test results. As shown in the example in
Fig. e, for each scaling interval, valid SS stations were
defined as stations that rejected neither the slope test for the scaling
interval nor the GOF tests for each duration included in this scaling
interval.
As expected, the proportion of valid SS stations decreased when the number of
durations within the scaling interval increased and with decreasing d1.
This is particularly evident for short d in SD and ID datasets. More GOF
test rejections were observed for longer scaling intervals (not shown), due
to the higher probability of observing large differences between xd and
xd,ss quantiles when xd,ss had larger sample size and
included data from more distant durations. However, several factors can
impact GOF test results when shorter d1 are considered. First, GOF tests
are particularly sensitive to the presence of very large values in
short-duration samples. Second, when considering durations close to the
temporal resolution of the recorded series (i.e., 15 min in SD and 1 h in
ID and LD), stronger underestimations could affect the measure of
precipitation because intense rainfall events are more likely to be split
between two consecutive time steps. Finally, preliminary analyses (Figs. S2
and S3 in the Supplement) showed that the largest GOF test rejections could
also be connected to the coarse instrument resolution of 15PD series, which,
similar to the temporal resolution effect, induces larger measurement errors
in the shortest duration series. Note that comparable resolution issues were
previously reported by some authors while estimating fractal and
intermittency properties of rainfall processes
e.g., and IDF
e.g.,.
Valid SS station proportions between 0.99 and 1 were always observed for GOF
tests in ID and LD datasets, except for some durations shorter than 3 h (ID
dataset) or 6 h (LD dataset). When considering both GOF and slope tests, with
the exception of some durations ≤ 1 h, the proportion of stations
satisfying SS was higher than 0.9, and the majority of scaling intervals (65,
90, and 98 % of the scaling intervals in SD, ID, and LD, respectively)
included at least 95 % of valid SS stations. For each scaling interval,
only valid SS stations were considered in the rest of the analysis.
These findings were also confirmed by cross-validation experiments. The
proportion of valid SS stations resulting from cross-validation slope and GOF
tests were similar to, even if slightly lower than, proportions displayed in
Fig. (see Fig. S4 of the Supplement).
Figure presents, for each scaling interval and
duration, the station average, ϵ‾xd, of the normalized
RMSE. These graphics show that mean relative errors on intensity quantiles
did not generally exceed 5 % of the precipitation estimates for
6-duration scaling intervals (Fig. , first column).
Greater errors were observed for durations at the border of the scaling
intervals. Not surprisingly, this result underlines that, in a
cross-validation setting, both the MSA estimation of H and the
Xd,ss approximation are less sensitive to the exclusion of an
inner duration of the scaling interval than to the exclusion of d1 or
dD. Conversely, the extrapolation under SS of the Xd distribution is
generally less accurate for durations at the boundaries or outside the
scaling interval used to estimate H. Moreover, as for the valid SS station
proportion, the performances of the model deteriorated with decreasing d1
and with increasing scaling interval length, especially for durations at the
border of the scaling intervals. However, for more that 70 % of 12-, 18-,
and 24-duration scaling intervals, ϵ‾‾xd≤0.1 for each duration included in the scaling interval.
Values of ϵ‾‾xd≥0.25 were observed for 15 min in
12-duration or longer scaling intervals, pointing out the weaknesses of the
model in approximating short-duration extremes when the scaling interval
included durations ≥ 3 h.
Estimated scaling exponents and their variability
In order to evaluate the sensitivity of SS to the considered scaling
interval, the variability of H with d1 has been analyzed. Then, the
spatial distribution of the scaling exponents for each scaling interval was
studied to assess the uncertainty in H estimation and the dependence of SS
exponents on local geoclimatic characteristics.
Column (i): median and relevant quantiles of the scaling exponent
distribution over all valid SS stations for each 6-duration scaling interval.
Column (ii)–(iv): median and relevant quantiles of the distribution of the
scaling exponent deviation ΔH(j) (defined in
Eq. ). The average number of valid SS stations over the
scaling intervals (identified by their first duration, d1) is indicated at
the top of each graph.
Investigating the variability of the scaling exponent with the scaling
interval is particularly important since, if SS is assumed to be valid
between some range of durations, one should expect that H remains almost
unchanged over the various scaling intervals included in this range. For this
reason, the variation ΔH(j) of the scaling exponents computed
for overlapping scaling intervals with the same d1 but different lengths
was analyzed. For each station and d1, ΔH(j) was defined as follows:
ΔH(j)=H(j)-H(6),
where j=12,18, or 24 represents the number of durations considered in
the specified scaling interval, H(j) is the corresponding scaling
exponent, and H(6) is the scaling exponent estimated for the 6-duration
scaling interval with the same d1. If SS is appropriate over a range of
durations, ΔH(j) is expected to be small for scaling intervals
defined within this range.
Figure i–iv show for all relevant scaling intervals
the median, interquantile range (IQR), and quantiles of order 0.1 and 0.9 of
the ΔH(j) distribution over valid SS stations. Adding new
durations to the scaling intervals, the median ΔH(j), as well as
its IQR, increased for all d1. Nonetheless the median scaling exponent
variation was generally smaller than 0.05, except for a relatively small
proportion of stations. Equally important, |ΔH(j)| was generally
centered on 0 and for all d1≥ 1 h more than 50 % of stations had
|ΔH(12)|≤ 0.025 (SD dataset) and |ΔH(18)|≤ 0.03 (ID dataset) (Fig. , columns ii–iii).
For some stations, a dramatic difference could exist in IDF estimations
obtained with the different definitions of the scaling interval. For
instance, for the 24-duration scaling interval “1–24 h” (ID dataset), the
median ΔH(24) was equal to 0.047
(Fig. b, column iv). For the interval
“15 min–6 h” (SD dataset), ΔH(24) was even larger, with a
median scaling exponent variation approximately equal to 0.087 and with
25 % of stations having ΔH(24)≥0.11
(Fig. a, column iv). Finally, changes in H values
were also important when comparing 6- and 12-duration scaling intervals when
d1≤1 h (SD and ID datasets) and in LD dataset
(Fig. , column ii).
The median, Interquantile Range (IQR), and quantiles of order 0.1 and 0.9 of
the H distribution across stations, are presented in
Fig. , column (i), for each 6-duration scaling
interval. The smallest median H values were observed for d1≤30 min
in Fig. a, column (i), and for the longest d1s in
Fig. c, column (i). Scaling intervals beginning at 15
and 30 min also displayed the smallest variability across stations. Although
fewer stations were available for these intervals (only 15PD stations were
used and the number of valid SS stations was smaller), this result is
consistent with previous reports in the literature demonstrating that H
values are spatially more homogeneous for short durations.
A larger dispersion of H values was observed when d1 ranged between
approximately 1 and 5 h, in particular in the SD dataset, for which the
10th–90th percentile difference almost covered the entire range of observed
H values (Fig. , column i). This result could be
partially explained by the use of scaling intervals with equally spaced
durations. This implies that the mean distance between the logarithms of
durations in the scaling interval decreases as d1 increases. Hence, the
OLS estimator of H used in the MSA regression may have larger variance for
longer d1, especially when scaling intervals include few durations. Larger
uncertainty may thus have an impact on the H estimation for the longest
d1 scaling intervals of SD. However, as shown in the following sections, H
spatial distribution may also explain the greater variability of the scaling
exponent for d1 greater than a few hours.
Spatial distribution of the scaling exponent for the first (i.e.,
with minimum d1) 6-, 12-, and 24-duration scaling intervals (first,
second,
and third column, respectively) for SD, ID, and LD datasets
(panels a, b, and c, respectively). These scaling
intervals correspond to the first column of matrices in
Figs. and .
Spatial distribution of the scaling exponent for the last (i.e., with
maximum d1) 6-, 12-, and 18-duration scaling intervals (first, second, and
third column, respectively) for SD, ID, and LD datasets
(panels a, b, and c, respectively). These scaling
intervals correspond to the last column of matrices in
Figs. and .
Climatic regions of (grey borders) and regions
defined for this analysis (regions A1 to F in the legend; colored borders).
Abbreviations for each region are in parenthesis.
The largest median H values were observed for d1 greater than 10 h
(Fig. b, column i) and lower than 2 days
(Fig. c, column i), with approximately half of the
stations having H≥0.8. This means that a stronger scaling (i.e.,
larger H values) is needed to relate extreme precipitation distributions at
approximately 12 h to distributions on daily and longer scales. It may
therefore be expected that the stations characterized by H closer to 1 are
located in geographical areas where differences in precipitation
distributions are important among temporal scales included in these scaling
intervals.
Examples of the spatial distributions of the scaling exponent are given in
Figs. and for the first and
last d1 for each interval length and dataset, respectively. Since only one
24-duration scaling interval was defined for both the SD and ID datasets,
only scaling intervals containing 6, 12, and 24 (Fig. )
or 18 (Fig. ) durations are presented. This avoids the
redundancy of showing the “15 min–6 h” (SD dataset) and
“1–24 h” (ID dataset) scaling intervals twice.
Generally, the scaling exponent displayed a strong spatial coherence and
varied smoothly in space, although a more scattered distribution of H
characterizes maps in Fig. . In this last figure, the
local variability of H may be attributed to the larger estimation
uncertainties affecting longer d1 scaling intervals, as previously
mentioned. Meaningful spatial variability and clear spatial patterns emerged
for d1≥1 h. In fact, for stations located in the interior and
southern areas of the continent, a shift from weaker scaling regimes (smaller
H) to higher H values was observed as d1 increases (e.g., second and
third rows of Fig. ). On the contrary, a smoother
evolution of H over the scaling intervals characterized the northern
coastal areas, especially in north-western regions, and the Rockies, where H>0.75 values were rarely observed even for greater d1 values.
Regional analysis
Regional differences in scaling exponents were investigated. Only the results
for the 6-duration scaling intervals are presented, similar results having
been obtained for longer scaling intervals (see the Supplement, Figs. S6
and S7, for 12- and 18-duration scaling intervals). Stations were pooled into
six climatic regions based on the classification suggested by
(see Fig. ). Stations outside the
domain covered by the Bukovsky regions were attributed to the nearest region.
Regions with less than 10 stations were not considered (regions without
colored borders in Fig. ); regions A1 (W_Tun) and A2
(NW_Pac) were kept separated since only 14 stations were available in region
A1 (W_Tun) for ID and LD datasets.
To provide deeper insights about regional features of precipitation
associated with specific scaling regimes, two variables related to the
precipitation events observed within AMS were also analyzed: the mean number
of events per year, Neve‾, and the mean wet time per
event, Twet‾, contributing to AMS within each scaling
interval. For a given year and station, annual maxima associated with different
durations of a given scaling interval were considered to belong to the same
precipitation event if the time intervals over which they occurred
overlapped. The mean wet time per event contributing to AMS,
Twet‾, was defined as the mean number of hours with
non-zero precipitation within each event. Details on the calculation of
Neve‾, Twet‾, and the corresponding
results are presented in the Supplement (Sect. S2 and Figs. S5 and S6).
Regional variation of the scaling exponents.
Figure shows the distribution of H within each
region. Three types of curves can be identified. First, curves in
Fig. a–c have a characteristic smooth S shape.
Conversely, Fig. d displays a rapid increase of
H for scaling intervals defined in ID and LD datasets until d1=2 days,
preceded and followed by two plateaus: one plateau for the longest d1 with
remarkably high H values, and one for the shortest d1 with small H
values. Finally, an inverse-U-shaped curve can be seen in
Fig. e and f, with globally high H values
already reached at sub-daily durations in dry regions (region E).
For d1≤24 h, Fig. a displays lower values
of H than Fig. e–f, meaning that smaller
variation in AMS moments are observed in A1 and A2 when the scale is changed.
This difference can be partially explained by the weaker impact of convection
processes in generating very short-duration extremes in north-west coastal
regions with respect to southern areas (regions E and F). For northern
regions, in fact, the transition between short and long duration
precipitation regimes may be smoothed out by cold temperatures which moderate
short-duration convective activity, especially for W_Tun (region A1). The
topography characterizing the northern Pacific coast may then explain the
smoothing effect for the curve of region NW_Pac (A2). In this case, in fact,
the precipitation rates on daily and longer scales are enhanced by the
orographic effect acting on synoptic weather systems coming from the Pacific
Ocean .
Similarly, mountainous regions in C (Fig. c)
displayed the smallest variations of H over d1, indicating that
analogous scaling regimes characterize both short- and long-duration scaling
intervals. Again, this may be related to the important orographic effects of
precipitation in these regions that are involved in the generation of
extremes for both sub-daily and multi-daily timescales. The mean number of
events per year in regions A and C was higher than in regions E–F, in
particular for SD scaling intervals, and displayed steeper decreases with
increasing d1 (Fig. S5a and c in the Supplement).
The main differences between regions B and A were the stronger scaling regimes
observed in B, which were mainly due to contributions from stations located
in the south-eastern part of the E_Bor region (not shown). For scaling
intervals in the ID dataset, region B was also characterized by the highest
mean number of events per year, with most of the stations presenting
N‾eve>2 for d1=1 h and d1=2 h and sharp
decreases of N‾eve with increasing d1 (Fig. S5b in the
Supplement). Moreover, a remarkably large range of N‾eve
was observed for 1 h ≤d1≤6 h, suggesting that B may be highly
heterogeneous.
Median and interquantile range (IQR) of the scaling exponent
distribution over valid SS stations within each region of
Fig. for 6-duration scaling intervals for the SD (left
curve), ID (central curve), and LD (right curve) datasets. For each region,
the mean number of valid SS stations over the scaling intervals is indicated
in brackets in the legend. See Fig. for region
definition.
Two distinct scaling regimes can be observed for SW_Pac (region D) at,
respectively, d1≤3 h (SD dataset) and d1≥2 days (ID dataset)
(region D in Fig. d). These plateaus may be
interpreted by recalling that 1-H=Hdepth. On the one hand, the
low and constant H observed for d1≤3 h indicates that the average
precipitation depth increases with duration at the same growth rate for all
these intervals. On the other hand, H approximately equal to 0.9 at daily
and longer durations demonstrates that the average precipitation depth
associated with long-duration annual maxima remained roughly unchanged when
the duration increased from 1.5 to 7 days (λHdepth≈1 in Eq. ). This, along with the fact that the scaling
exponent increased almost monotonically for 1 h ≤d1≤ 24 h (ID
and LD datasets), suggests that extremes at durations shorter than ∼3 h (SD dataset) drive annual maxima precipitation rates on longer scales,
with the rapid and continuous decay in mean intensity caused by the
increasing size of the temporal scale of observation.
For SW_Pac (region D), the relative absence of long-lasting weather systems
able to produce important extremes for long durations was confirmed by the
analysis of N‾eve and T‾wet (see
Figs. S5 and S6 of the Supplement). In fact, the mean number of events per
year was relatively high for short durations (the median
N‾eve is equal to 1.82 for d1=15 min and to 1.4 for
d1=1 h), while it rapidly decreased below 1.1 events per year for
d1≥6 h (ID dataset) and for d1≥18 h (LD dataset). With the
exception of d1=6 h (LD dataset), at least 90 % of SW_Pac stations
had N‾eve≤1.25 for all d1>3 h. In other
regions, median N‾eve were never smaller than 1.1 for the
SD and ID datasets, except for d1≥12 h in region E.
These results suggests that both the distinctive topography of the west coast
and the characteristic large-scale circulation of the south-west areas of the
continent are crucial factors determining the transition between the two
scaling regimes in region D.
Median H values displayed inverse-U shapes for the remaining regions with
very small IQR, despite the high number of valid SS stations: a slow
transition from lower to higher H is observed approximately between 1 and
12 h (region E) or 30 h (region F). The strongest scaling regimes were
observed for 1 h ≤d1≤ 2 days in arid western regions
(Fig. e), while median H values greater than 0.8
were only observed for approximately 6 h ≤d1≤ 2 days in more
humid areas (Fig. f). In both region E and F, very
short-duration extremes are typically driven by convective processes, while a
transition to different precipitation regimes may be expected between 1 h
and a few hours. However, H shows a smoother increase in Fig. 7f with
respect to Fig. 7e. This may indicate that in eastern areas (region F)
sub-daily duration extremes are more likely associated with embedded convective
and stratiform systems, or to mesoscale convective systems, which are less
active in western dry areas of region E . On the contrary,
differences between short- and long-duration extreme precipitation intensity
seem stronger for south-western dry regions
(Fig. e), where less intense summer extremes are
expected compared to eastern areas (see Supplement, Fig. S1). In particular,
H tended to scatter in a range of higher values for approximately 1 h ≤d1≤ 12 h, indicating that precipitation intensity moments
strongly decrease as the duration increases.
In summary, these results suggest a regional effect on precipitation scaling
of both local geographical characteristics, such as topography or coastal
effects, and general circulation patterns. In general, the weakest scaling
regimes were observed for short d1 and along the west coast of the
continent and seem to be connected to scaling intervals and climatic areas
characterized by homogeneous weather processes. Low H values correspond in
fact to small variations in AMS distribution moments. On the contrary,
stronger scaling regimes were observed for longer d1 in the other regions
of the study area. This indicates that important changes occur in AMS moments
across duration and, thus, in extreme precipitation features. According to
these results, it would be important to take into account the climatological
information included in the scaling exponent to improve SS and IDF
estimation. Even more important, these results give useful guidelines for
modeling the spatial distribution of H, which could help for the definition
of IDF relationships at non-sampled locations.
Simple scaling GEV estimation
Results presented in this section are limited to a descriptive analysis of
GEV parameter estimates for 6-duration scaling intervals. Similar results
were generally obtained for 12-,18-, and 24-duration intervals
(see Supplement, Figs. S10–S16). An assessment of the potential improvements
carried out by simple scaling GEV (SS GEV) models with respect to non-SS GEV
models is also presented.
In our study, the PWM procedure was applied to
estimate SS GEV parameters μ*, σ*, and ξ*
(Eq. ) from xd* (Eq. ). For
each duration d, PWM were also used to estimate non-SS parameters μd,
σd, and ξd from each of the non-SS samples xd. Preliminary
comparisons of various estimation methods (PWM, classical ML estimators, and
GAM-ML; see Sect. ), showed that PWM slightly
outperformed the other methods.
Quantiles estimated from the SS and the non-SS GEV were compared with
empirical quantiles. Global performance measures, such as RMSE, were computed
to evaluate the overall fit of the estimated GEV to the empirical Xd
distributions. In particular, mean errors between SS and non-SS quantile
estimates and empirical quantiles were compared using the relative total RMSE
ratio, Rrmse‾, defined as follows:
Rrmse‾=Rss‾-Rnon-ss‾Rnon-ss‾,
where
Rmod‾=∑Dd=d1ϵd,modx‾d
represents the normalized mean square difference between model and empirical
quantiles of order p>0.5 for all the durations included in the scaling
interval. See Eq. () for the definition of
ϵd,mod for each station.
Distribution over valid SS stations of SS GEV parameters (grey and
black lines) for 6-duration scaling intervals and non-SS GEV parameters (red
solid and dashed lines) for reference durations. Location and scale
parameters (first and second column, respectively) are scaled at d*= 1 h
(SD and ID datasets) and d*= 24 h (LD dataset). Distributions for the
shape parameter (third column) are presented for ξ>0 and ξ<0,
excluding cases where ξ=0 (Gumbel distribution).
Spatial distribution over valid SS stations of SS GEV position
(first column), scale (second column), and shape (third column; grey symbols
indicate Gumbel distributions, ξ*=0) parameters scaled at d*=1 h
for the first 6-duration scaling interval (i.e., interval with minimum d1)
of SD (a), ID (b), and LD (c) datasets.
Stacked histograms of the fractions of valid SS stations with ξ<0 (in red), ξ=0 (in grey), and ξ>0 (in blue) resulting from the
Hosking test applied at the 0.95 confidence level for each duration (non-SS
GEV, first column) and each 6-duration scaling interval (SS GEV, second
column) for SD (a), ID (b), and LD (c) datasets.
Distribution of the relative total RMSE ratio,
Rrmse‾, for ξ*<0 (first column), ξ*=0 (second
column), and ξ*>0 (third column) for 6-duration scaling intervals in
SD (a), ID (b), and LD (c) datasets. The average
number of valid SS station over the scaling intervals is indicated in the
right-top corner of each graph.
Estimated SS GEV parameters
Figure presents the
distributions over valid SS stations of the SS GEV parameters rescaled at
d*=1 h (Fig. a
and b) and d*=24 h
(Fig. c).
For the SD dataset, even for scaling intervals which did not include the
reference duration d*, the μ* and σ* distributions appeared
to be similar to the non-SS μd and σd distributions
(Fig. , first row).
Similarly, for 6 h ≤d1≤ 2 days in the LD dataset, the SS
location and scale parameter distributions are in relatively close agreement
with the corresponding non-SS parameter distributions. Conversely, for the ID
dataset, both μ* and σ* distributions are more positively skewed
than the corresponding non-SS distributions. Finally, for d1≥2 days
in the LD dataset, μ* and σ* had distributions shifted toward
lower values than μ24h and σ24h. Moreover,
the relative differences Δμ=(μ*-μd)/μd and
Δσ=(σ*-σd)/σd were estimated for each
station, duration, and scaling interval. Two important results came out of
this analysis (see Figs. S11 and S12 of the Supplement). On the one hand,
median values of Δμ and Δσ were generally smaller
than ±5 and ±10 %, respectively. On the other hand,
Δσ showed large positive values when ξd=0 (i.e., Gumbel
distributions), while small Δσ<0 were estimated when ξd≠0 (not shown for conciseness). These results are interesting since the
estimation of the scale parameter σ of a GEV distribution may be
biased when the shape parameter is spuriously set to zero (ξ=0). Hence,
while non-SS μd values can be considered to be accurate estimates of the
Xd location parameter, small uncertainties should be expected for the
scale parameter only when the ξd value is correctly assessed. In
addition, μ* and σ* displayed a strong spatial coherence. Their
spatial distributions were characterized by an obvious north-west to
south-east gradient (Fig. shows
examples for the scaling intervals 15 min–1.5 h, 1–6 h, and 6–36 h).
Notable differences between SS GEV and non-SS GEV estimates were observed for
the shape parameter
(Fig. , third column,
and Fig. ). Firstly, for cases that have shape parameters
strictly different from zero (third column of
Fig. ), ξ* absolute
values were smaller than non-SS ξd absolute values. Secondly, the
distributions of ξ* across stations were generally more peaked around
their median value than the corresponding non-SS distributions. Finally, for
the non-SS model the majority of stations had shape parameter ξd
non-significantly different from zero, while the fraction of SS GEV shape
parameters ξ*≠0 was always greater than 39 % asymptotic
test for PWM GEV estimators applied at level 0.05;. In
particular, for each duration, non-SS models estimated light-tailed
distributions (i.e., ξd=0) for more than 85 % of the stations,
except that for d=15 min and d=30 min (Fig. ,
first column). Conversely, for all scaling intervals with d1>15 min, SS
GEV shape parameters were significantly different from zero for 40 to
45 % of valid SS stations (Fig. , second column).
Moreover, when using scaling intervals of 12 durations or more, the
proportion of ξ*>0 was always important (greater than 35 % for all
18- and 24-duration scaling intervals; see the Supplement, Fig. S10).
The previous results suggest that pooling data from several durations may
effectively reduce the sampling effects impacting the estimation of ξ,
allowing more evidence of non-zero shape parameters, and, in many cases, of
heavy-tailed (ξ>0) AMS distributions. This conclusion is consistent with
previous reports, namely that 100- to 150-year series are necessary to
unambiguously assess the heavy-tailed character of precipitation
distributions e.g.,. These
studies typically reported values of ξ≈0.15e.g.,, which are close to ξ* values
estimated in the present analysis for cases with ξ*>0.
However, uncertainties on ξ* estimates remain important. Support for
this comes from the spatial distribution of ξ*, which was still highly
heterogeneous, with local variability dominating on small
scales (e.g.,
Fig. , third column).
Improvement with respect to non-SS models
The proportion of series for which the SS model RMSE,
ϵd,ss, was smaller than the non-SS GEV RMSE,
ϵd,non-ss, was analyzed (see the Supplement, Fig. S11).
For cases with non-zero ξ*, more than 60 % of stations had
ϵd,ss<ϵd,non-ss over most scaling
intervals and durations. The 6-duration scaling intervals
“15 min–1 h 30 min” (SD dataset) and “1–6 h” (ID dataset) showed
the largest fractions of stations with increasing errors. On the contrary,
increasing errors (ϵd,ss>ϵd,non-ss)
were observed for all scaling intervals and durations for most stations
(generally more than 70 %) with ξ*=0.
Figure presents the
Rrmse‾ distribution over valid SS stations. When the SS
shape parameters were not significantly different from zero
(Fig. , second column), the relative
increases in total RMSE were usually smaller than 0.1 in SD dataset, and only
scaling intervals with d1<1 h had greater Rrmse‾.
For the ID and LD datasets, the medians of the total relative RMSE ratio
distributions were smaller than 0.05 for d1≥4 h and d1≥24 h, respectively. Furthermore, more than 90 % of stations had
Rrmse‾<0.125 for d1≥6 h (ID dataset) and d1≥30 h (LD dataset). When ξ*≠0, an increase of the mean error
in high-order quantile estimates was observed for d1=15 min (SD dataset)
and d1=1 h (ID dataset) for at least half of the stations
(Fig. , first column; note the different
scale on the y axis). However, for all other d1, negative
Rrmse‾ values were observed for the majority of
stations for all scaling intervals, with a median reduction up to 30 % of
the mean error. Note that also for 12- and 18-duration scaling intervals the
median Rrmse‾ where generally negative for d1>1 h
and ξ*≠0 (Figs. S14 and S15 of the Supplement). Conversely,
Rrmse‾ increased for the majority of stations in all
24-duration scaling intervals with d1<12 h (Fig. S17 of the
Supplement).
Note also that no particular spatial pattern characterized the
Rrmse‾ estimates.
Discussion and conclusion
This study investigated simple scaling properties of extreme precipitation intensity
across Canada and the United States.
The ability of SS models to reproduce extreme precipitation intensity distributions over
a wide range of sub-daily to weekly durations was evaluated.
The final objective was to identify duration intervals and geographical areas
for which the SS model can be used for an efficient production of IDF curves.
The validity of SS models was empirically confirmed for the majority of the
scaling intervals. In particular, based on the comparison of SS distributions
to empirical quantiles, the hypothesis of a scale-invariant shape of the
Xd distribution held for all duration intervals spanning from 1 h to
7 days. Less convincing results were obtained for durations shorter than
1 h, especially for the longest scaling intervals (24-duration intervals).
One possible explanation is that the coarse instrument resolution of the
available 15 min series may strongly impact both the validation tools (for
instance, GOF tests) and SS estimates. These results provide important
operative indications concerning the inner and outer cut-off durations for
AMS scaling and show the importance of a deeper analysis to evaluate the
impact of dataset characteristics (e.g., their temporal and measurement
resolutions, or the series length) on the scale-invariant properties of
extreme precipitation.
The majority of the estimated scaling exponents ranged between 0.35 and 0.95,
showing a smooth evolution over the scaling intervals and a well-defined
spatial structure. Six geographical regions, initially defined according to a
climatological classification of North America into 20 regions, displayed
different features in terms of scaling exponent values. Specifically,
distinct median values of H were observed for the various geographical
regions, each characterized by a different precipitation regime. This is
consistent with results reported in the literature for some specific regions
and smaller observational datasets e.g.,and references
therein. Moreover, while small
and smooth changes of H over the scaling intervals were observed in regions
containing the majority of stations, one region, SW_Pac, displayed two
dramatically distinct scaling regimes separated by a steep transition
occurring between a few hours and 24 h. These results limit the
applicability of SS models in SW_Pac, and were connected to the local
features of intense precipitation events by the analysis of the mean number
of events per year and the mean wet time of these events.
Weak scaling regimes, characterized by relatively small H values (H close
to 0.5), were generally observed for scaling intervals containing very short
durations (e.g, less than 2 h) and for regions on the west coast of the
continent (regions A1, A2, and D; see Fig. ). For
these scaling intervals and regions, we can expect that extreme precipitation
events observed at various durations will have similar statistical
characteristics, being governed by homogeneous weather processes.
The interpretation of high H values (e.g., H>0.8), observed between 1 and
several days, depending on the region, is more complex. These scaling regimes
correspond to mean precipitation depth that varies little with duration. This
suggests an important change in precipitation regimes occurring at some
durations included in the scaling interval. One interesting example was
region SW_Pac (region D) for scaling intervals of durations longer than
1 day. In this case, the analysis of the mean number of events per year
sampled in AMS suggested that very few long-duration extreme events were
produced by large-scale dynamic precipitation systems.
For scaling intervals of durations longer than 4 days, scaling exponents
seemed to converge to approximately 0.7 for all regions, except west-coast
regions (regions A1, A2, and D).
These results suggest that SS represents a reasonable working hypothesis for
the development of more accurate IDF curves. This may have important
implications for infrastructure design and risk assessment for natural
ecosystems, which would benefit from a more accurate estimation of
precipitation return levels. In addition, the spatial distribution of the
scaling exponent and its dependency on climatology should be taken into
account when defining SS duration intervals for practical estimation of IDF.
The accuracy of the SS approximation may in fact depend on the range of
considered temporal scales. Equally critical, estimated H values were found
to gradually evolve with the considered scaling intervals. In this respect,
interesting extensions of the analysis should consider methods for the
quantification of the uncertainty in H estimations as well as the
possibility of modeling the scaling exponent as a function of both the
observational duration and the AMS distribution quantile or moment order, i.e., by the use of a
multiscaling framework for IDFs. Equally important, the events sampled by the
AMS also showed different statistical features within different geographical
regions, and some specific results (e.g., for the SW_Pac region) stimulate
the interest for an analysis of the scaling property of extreme precipitation
by the use of a temporal stochastic scaling approach.
The evaluation of SS model performances under the assumption of GEV
distributions for AMS intensity was then performed. Results indicate that the
proposed SS GEV models may lead to a more reliable statistical inference of
extreme precipitation intensity than that based on the conventional non-SS
approach. In particular, a better assessment of the GEV shape parameter seems
possible when pooling data from several durations under the scaling
hypothesis. The use of the SS approximation may introduce biases in high
quantile estimates when AMS distributions move drastically away from perfect
scale invariance (short durations and/or longest scaling intervals).
Nonetheless, decreases in the SS GEV RMSE with respect to non-SS GEV models
for d1 longer than a few hours and/or scaling intervals shorter than 24
durations indicate that quantile errors in IDF estimates can be generally
reduced.
Caution is advised when interpreting these results due to the fact that high-order empirical quantiles were used as reference estimates of true Xd
quantiles, which could be a misleading assumption, especially when available
AMS are short. Moreover, two important limitations of the presented SS
approach must be stressed. Firstly, a more comprehensive assessment of the
scaling exponent uncertainty and of the influence of dataset characteristics
on the estimation of AMS simple scaling is recommended for a reliable
estimation of simple scaling IDF curves. Secondly, the proposed model relies
on the implicit hypothesis of stationarity of AMS over the observed period
while growing evidence supports the ongoing changes in extreme precipitation
intensity, frequency, duration, and spatial patterns as a result of climate
change e.g.,. In particular,
short-duration extreme rainfall is expected to respond to global warming with
a different sensitivity to temperature than those expected on daily or longer
timescales e.g.,,
which implies a change in the temporal scaling properties of precipitation
over time.
Hence, considering these limitations and our general results, any future
extension of this study should investigate the possibility of introducing
spatial information in scaling models as well as the characterization of
the possible evolution of the scaling
exponent in a warmer climate in order to identify valuable approaches
allowing non-stationarity of SS model parameters.
The 15 min precipitation data (15PD) and hourly
precipitation data (HPD) were freely obtained from NOAA Climate Prediction
Center (CPC)
(http://www.ncdc.noaa.gov/data-access/land-based-station-data)
. Hourly Canadian precipitation data (HCPD) and
maximum daily precipitation data (DMPC) for Canada were acquired from
Environment and Climate Change Canada (ECCC) and from the MDDELCC of Québec
(data available upon request by contacting info-climat@mddelcc.gouv.qc.ca).
The Supplement related to this article is available online at https://doi.org/10.5194/hess-21-5823-2017-supplement.
The authors declare that they have no conflict of
interest.
Acknowledgements
Silvia Innocenti's scholarship was partly provided by the OURANOS consortium.
Financial support for this project was also provided by the Collaborative
Research and Development Grants program from the Natural Sciences and
Engineering Research Council of Canada. We also thank Guillaume Talbot for
preliminary data processing and Dikra Khedhaouiria for useful
discussions. Edited by: Jan
Seibert Reviewed by: Juliette Blanchet and three anonymous
referees
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