HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus GmbHGöttingen, Germany10.5194/hess-18-5093-2014Characterising the space–time structure of rainfall in the Sahel with a view to estimating IDAF curvesPanthouG.https://orcid.org/0000-0002-6906-3654VischelT.theo.vischel@ujf-grenoble.frhttps://orcid.org/0000-0003-4230-4953LebelT.QuantinG.MoliniéG.https://orcid.org/0000-0002-5480-320XLTHE – UMR5564, Univ. Grenoble, IRD, CNRS, Grenoble, FranceT. Vischel (theo.vischel@ujf-grenoble.fr)11December201418125093510730June201423July20143November20143November2014This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://www.hydrol-earth-syst-sci.net/18/5093/2014/hess-18-5093-2014.htmlThe full text article is available as a PDF file from https://www.hydrol-earth-syst-sci.net/18/5093/2014/hess-18-5093-2014.pdf
Intensity–duration–area–frequency (IDAF) curves are increasingly demanded for characterising
the severity of storms and for designing hydraulic structures. Their computation requires
inferring areal rainfall distributions over the range of space scales and timescales that are the most
relevant for hydrological studies at catchment scale. In this study, IDAF curves are computed for
the first time in West Africa, based on the data provided by the AMMA-CATCH Niger network,
composed of 30 recording rain gauges having operated since 1990 over a 16 000 km2 area
in south-western Niger. The IDAF curves are obtained by separately considering the time (intensity–duration–frequency, IDF) and
space (areal reduction factor, ARF) components of the extreme rainfall distribution. Annual
maximum intensities are extracted for resolutions between 1 and 24 h in time and from point
(rain gauge) to 2500 km2 in space. The IDF model used is based on the concept of scale
invariance (simple scaling) which allows the normalisation of the different temporal resolutions
of maxima series to which a global generalised extreme value (GEV) is fitted. This parsimonious framework allows one to use the
concept of dynamic scaling to describe the ARF. The results show that coupling a simple scaling in
space and time with a dynamical scaling that relates to space and time allows one to satisfactorily model the
effect of space–time aggregation on the distribution of extreme rainfall.
Introduction
Torrential rain and floods have long been a major issue for hydrologists. For
one, defining and computing their probability of occurrence is a scientific
challenge per se, largely because it is a scale-dependent exercise.
Second, and equally important, is the fact that they cause heavy
environmental, societal, and economical damages – including human casualties –
thus being a major concern for populations and decision makers.
The request of providing both an objective assessment of the probability of
occurrence of high-impact rainfall and a tool for civil engineering structure
design has found an answer through the calculation of
intensity–duration–frequency (IDF) curves. These curves, generally computed
from rain-gauge data, are intended to characterise the evolution of extreme
rainfall distributions at a point when the duration of rainfall accumulation
changes. However, rainfall at point location is not of the greatest interest when
it comes to the hydrological and socio-economic impacts of extreme events,
since it is essentially the convolution of the rainfall intensities over
a catchment that characterises the severity of storms and creates the real
threat.
This is why intensity–duration–area–frequency (IDAF) curves were conceived
as a spatial extension of the IDF curves. Generally established by combining
IDF curves and areal reduction factors (ARFs), they provide an estimation of
extreme areal rainfall quantiles over a range of timescales and spatial scales.
Theoretical studies on IDF and ARFs have been an active research topic over
the past 20 years or so among
others.
IDF practical studies are also numerous but focused on regions where long
series of sub-daily rainfall are available
e.g..
On the other hand, when ARFs are computed from rain-gauge networks
, it requires a fair density of
rain gauges in order to obtain accurate estimates of areal rainfall. The computation of IDAF curves
must therefore deal with two major data requirements: (i) a high-density network of rain gauges
and (ii) an array of long sub-daily rainfall series. In addition to these
requirements, scientists face the challenge of producing coherent ARF and IDF models, if they
wish for
their IDAF model to be statistically consistent. This explains why there are so few studies dealing
with the implementation of an IDAF model over a given region
e.g..
In fact, in some regions of the world there are virtually no IDF, ARF, and
IDAF models that have ever been conceived because of data limitations. This
is especially the case in many tropical regions, such as West Africa; in fact, one
reason being that a harsh environment and resource scarcity have made
it
very challenging to operate recording rain-gauge networks. The few IDF
studies available in the region
are essentially empirical with no theoretical background that allows for the
upgrade of
their results in order to produce IDAF curves. ARF studies are still fewer,
the most noticeable being an attempt by and
at computing ARF values for a return
period of 10 years, with no explicit inference of the areal rainfall
distributions. All in all, there has never been any IDAF model derived for
West Africa or sub-regions of West Africa. Yet, flood management – for which
IDAF curves are a very useful tool – is now a major concern for West African
countries. As a matter of fact and despite that West Africa is known for
having experienced a major lasting drought over 1970–2000, numerous severe
floods and exceptional inundations have struck the region over the last 2
decades .
Moreover, flood damages in the region have been constantly increasing
since 1950 .
Study area. The background maps displays the elevation (metres).
While operational networks of the West African National Weather Services do
not allow the establishment of IDAF curves in a consistent way – because
they do not provide any long-term sub-daily rainfall series – there are other
sources of data that can be used for that purpose. Among them are the 5 min
rainfall series of the long-term AMMA-CATCH observing system covering
a 16 000 km2 area in south-western Niger from 1990 to present
(Fig. ). In this study we will make use of 30 series
providing continuous 5 min rainfall records from 1990 to 2012.
This unique data set enables us to characterise the relationship between extreme rainfall
distributions computed at various spatio-temporal scales and to propose IDAF curves for this
characteristic Sahelian region.
IDAF curves in a generalised extreme value distribution and scale invariance framework
IDAF curves provide an estimate of areal rain rates – averaged over
a given duration D and a given area A – for a given frequency of
occurrence (currently expressed in term of return period Tr).
In practice, IDAF curves are generally obtained by aggregating a temporal
component and a spatial component represented respectively by the
IDF computed at a point (A=0) and
by the ARF computed for a range of durations. In
this framework, the most general formulation of an IDAF equation is as
follows:
IDAF(D,A,Tr)=IDF(D,Tr)×ARF(D,A,Tr).
Assessing IDAF curves requires (i) inferring appropriate statistical
distributions of rainfall to estimate the return periods and (ii) describing
the statistical links between the distributions obtained at different
space scales and timescales.
Several recent studies have confirmed that the generalised extreme value
(GEV) distribution provides a suitable
framework to describe the distribution of extreme rainfall at a point
e.g..
Also, many authors have shown that rainfall displays scale invariance
properties
,
both in space and time. The temporal scaling properties give access to
a direct analytical formulation of IDF curves
,
while the spatial scaling properties allow one to upscale IDF curves into IDAF
curves
,
thus providing an integrated space–time characterisation of extreme rainfall
distributions. Under certain assumptions, namely the GEV distribution of
point annual rainfall maxima and simple scaling in both time and space, an
analytical formulation of the various components of Eq. ()
may be obtained, as will be detailed below.
GEV distribution
Let us define I(D,A) as a random variable representing the annual maxima of
rainfall accumulated over a given duration D and area A, and i(D,A)
a sample of I(D,A). In the general framework of the block maxima sampling
scheme , working on annual maxima generally
ensures that the block size is large enough for the maxima distribution to
follow a GEV distribution , written as
G(i;μ,σ,ξ)=exp-1+ξi-μσ-1ξfor1+ξi-μσ>0,
where i is a generic notation for any value associated with a realisation of I(D,A), μ being
the location parameter, σ>0 the scale parameter, and ξ the shape parameter of the GEV
distribution. The shape parameter describes the behaviour of the distribution tail: a positive
(negative) shape corresponds to a heavy-tailed (bounded) distribution. When ξ is
equal to 0, the GEV reduces to the Gumbel distribution (light-tailed distribution):
G(i;μ,σ)=exp-exp-i-μσ.
Simple scaling in time and analytical formulation of IDF curves
The simple-scaling framework has been extensively used for deriving IDF
curves
.
An analytical formulation of the ARF was also given by
in a multi-scaling framework. However, simple
scaling is much more tractable than the multi-scaling approach and is more
robust in terms of parameter inference; thus, this is the approach chosen
here.
The annual maximum point rainfall random variable {I(D,0)} follows a simple-scaling relation for
a given duration D (with respect to a reference duration Dref) if
I(D,0)=dλη×I(Dref,0),
where λ is a scale ratio (λ=D/Dref), η is a scale exponent, and
=d denotes an equality in distribution. Note that, for every duration D for which
Eq. () holds, the normalised random variable {I(D,0)/Dη} has the same
statistical distribution as the normalised reference distribution
{I(Dref,0)/Drefη}; this property will be used in the optimisation
procedure in Sect. . Equation () implies
E[I(D,0)]=λη×E[I(Dref,0)]
and, more generally, a scaling of all the moments that can be written as
E[Iq(D,0)]=λk(q)×EIq(Dref,0)
or
ln{E[Iq(D,0)]}=k(q)ln(λ)+lnEIq(Dref,0).
The notion of simple scaling is related to how k(q) evolves with q. When this evolution is linear
k(q)=ηq
and simple scaling holds (as opposed to multi-scaling if this relation in non-linear).
Checking whether the simple-scaling hypothesis is admissible over a given range of durations is thus
equivalent to verifying through the data set whether the two following conditions are fulfilled
:
Eq. (): log–log linearity between the statistical moments of any given order q
Eq. (): linearity between k(q) and q.
Figure a illustrates these two conditions.
Spatial scaling, dynamical scaling, and ARF model
In its most general sense, the ARF is the ratio between
areal rainfall and point rainfall, either for a given observed rain event or
in a statistical sense. Here we are interested in deriving a statistical ARF
that can be used for obtaining an analytical formulation of IDAF curves
(which implies that the ARF does not depend on the return period considered);
thus, this ARF denotes the ratio between the areal distribution and the point
distribution of the annual rainfall maxima:
I(D,A)=dARF(D,A)×I(D,0).
Note that Eq. () implies the following relationship:
ARF(D,A)=E[I(D,A)]E[I(D,0)].
In this study, the ARF model proposed by is used. This model is
based on two assumptions (which will have to be verified; see Sect. ):
The studied rainfall variable is characterised by a simple-scaling relationship both in time
and space.
Time and spatial scales are linked by a so-called dynamic-scaling property written asDDref=AArefz,where z is the dynamic-scaling exponent.
When these assumptions are verified, show that the ARF can be written as
ARF(D,A)=1+ωAaDbη/b=1+ωAzDbη/b,
where η is the scaling exponent characterising the temporal simple scaling, a and b are two positive constant-scaling exponents linked by the relation z=a/b, and ω is a factor of normalisation.
This ARF formulation implies that iso-ARFs are lines in the plane {ln(A),ln(D)} as shown in Fig. b
.
GEV simple-scaling IDAF model
By assuming that the maximum annual rainfall is GEV-distributed and that the scaling relations in
time and space (Sects. and ) are verified, then
the IDAF model is
I(D,A)=dI(Dref,0)×λη×ARF(D,A).
Visual model checks: (a) simple scaling; (b) ARF.
As shown in the Appendix , the compatibility of the simple-scaling and GEV frameworks
is defined by the following equations:
I(D,A)∼GEVμ(D,A),σ(D,A),ξ(D,A)μ(D,A)=μref×λη×ARF(D,A)σ(D,A)=σref×λη×ARF(D,A)ξ(D,A)=ξref,
where μref, σref , and ξref correspond to the GEV
parameters computed for the arbitrary reference duration Dref at a
point, and λ=D/Dref.
Data and implementation
Rainfall observing systems usually do not provide direct measurements at all
the space scales and timescales required for an IDAF study; thus, it is necessary to
derive from the raw data set an elaborated data set that allows one to
verify the various assumptions found in the theoretical framework defined in
Sect. .
Accordingly, this section describes both the rainfall samples initially available on our Sahelian
region of south-western Niger and the process used to obtain the final data set from which the IDAF
curves were computed. This process consists of two major steps:
space–time aggregation of the 5 min point rain rates in order to obtain the average
rain rates for various space (A) and time (D) resolutions
extraction of extreme rainfall samples for each of the above resolutions.
The rainfall data set: AMMA-CATCH Niger records
The AMMA-CATCH Niger observing system was set up at the end of the 1980s as
part of the long-term monitoring component of the HAPEX-Sahel experiment
. Since then, it has continuously operated a large
array of meteorological and hydrological instruments, providing a unique set
of high-resolution hydrometeorological data, covering
a 16 000 km2 area in south-western Niger. For the purpose of this
research, a subset of thirty 5 min rainfall series was selected
(Fig. ), covering the entire 1990–2012 period. At each
station, all years with more than 25 % of missing data have been removed
in order to limit any sampling effect due to missing data in a particular
year. After this quality control, all stations remain with at least 20 years
of valid data, constituting our raw data sample.
To estimate areal rainfall intensities, this study makes use of the dynamical
kriging interpolation method proposed by .
Rain fields are produced over the domain of study at a time resolution of
5 min and a spatial resolution of 1 km2. Dynamical kriging
takes advantage of the time structure of 5 min rain fields to complement the
purely spatial information provided by the gauge network. Instead of using a
3-D variogram (the inference of the space–time cross-covariance being
notoriously non-robust), the method relies on the construction of Lagrangian
rain fields which display a stable spatial structure represented by a nested
exponential variogram. Dynamical kriging is an exact interpolator in the
sense that the measured point values are replicated exactly; this
interpolator is then used to produce 5 min rainfall grids, with a grid mesh
of 1 km2.
Space–time rainfall aggregation
The starting elements of the space–time aggregation process are the
discretized fields of rain accumulated over a time increment Δt=5min and averaged over a square pixel of side length Δxy=1km. In the following, these rain fields are denoted as r*(x*,y*,t*), where t* is the ending time of the 5 min time step, and {x*,y*} is the centre of the 1 km2 pixel.
Spatial aggregation of 5 min rain fields
Let A be a surface over which the rainfall intensity is averaged. In this
study, A is a square of side Nx×Δxykm=Ny×Δxykm (corresponding to Nx×Ny pixels of 1 km2). From the 5 min rain fields we can compute series of space
averaged 5 min rain field accumulations, rA*, as
rA*(x*,y*,t*)=1A∑m=0Nx-1∑n=0Ny-1r*x*+m-Nx-12Δxy,y*+n-Ny-12Δxy,t*.
From these spatially averaged rain fields, spatial rainfall series have been
extracted at rain-gauge locations. For each rain-gauge location (located at
{x,y}), the nearest spatial rainfall series, rA* (located at
{x*,y*}), is extracted. Figure a illustrates this
approach the black circle of the right panel represents a rain-gauge located
at {x,y}. In total, 12 scales of spatial aggregations have been retained
to build the rainfall series: 1 km2 (the pixel on which the
station is located is selected) then 4, 9, 16, 25, 49, 100, 225, 400, 900,
1600, and 2500 km2.
Space and time aggregation procedures. (a) Illustration of
the procedure leading to select (right case) or to reject (left case) a gauge
for becoming a centre for spatial aggregation; (b) time aggregation
at a point: comparing a hyetograph of 5 min rainfall to a hyetograph of
1 h rainfall.
To limit border effects, the spatial aggregation is performed only in areas
where the spatial distribution of stations is more or less isotropic. Each of
the 30 measurement stations are considered individually; a circle centred on
the station is plotted and divided in eight cardinal sectors (each sector has
an angle of 45∘). Only rain gauges having at least one other rain
gauge present in at least seven of the eight sectors are retained for spatial
aggregation (see Fig. a); the distance of the other
gauges from the centre station is not taken into account for the selection,
only the presence or absence of a rain gauge in the sector. Only 13
gauges (out of 30) satisfy this criterion (Fig. ). They
are referred to in the following as a central rain gauge (CR) because their
localisation is used as a central point, from which the 12 areas of
aggregation are delimited. In total there are thus 156 areal series (12 areas
of aggregation centred on 13 different locations).
Time aggregation of 5 min point series and 5 min spatial series
A time aggregation procedure is applied to the 30 point 5 min rainfall series and to the 156
spatial rainfall series.
Let D be a given duration of Nt 5 min time steps (D=Nt×Δt). The time aggregation is done by using a moving time window of length D
over which the 5 min rainfall intensity is averaged (this moving window
procedure is carried out in order to make sure that we will be able to
extract the maximum for each duration considered). The time
aggregation can be written as
rD,0*(x,y,t*)=1D∑p=0Nt-1r0*(x,y,t*-p×Δt)
in the case of 5 min point series located at {x,y} (A=0), and
rD,A*(x*,y*,t*)=1D∑p=0Nt-1rA*(x*,y*,t*-p×Δt)
for a given surface A in the case of 5 min spatial rainfall series located at {x*,y*}.
Thus, Nt=12 for D=1h, Nt=24 for D=2h and so
forth. This procedure is illustrated in Fig. b. The 11 different time
resolutions considered in this study range from 1 to 24 h (1,
2, 3, 4, 6, 8, 10, 12, 15, 18, and 24 h) and are all obtained from the original 5 min series.
Extraction of extreme rainfall: annual block maxima
The use of GEV distribution to model the extreme rainfall series requires
using the block maxima procedure to extract rainfall extremes. It consists of
defining annual blocks of observations separately for each of the 11
different time resolutions considered and to take the maxima within each
block. A sample of 23 (1990–2012) annual maximum rainfall values
{i(D,A)} is thus obtained for each spatial aggregation and duration.
In summary
There are 13 reference locations.
Around each of the 13 reference locations, 12 areas of increasing size 1, 4, 9, 16, 25, 49,
100, 225, 400, 900, 1600, and 2500 km2 are defined.
For each of these 156 (13×12) areas, 11 time series of 23 (1990–2012) annual maximum
values are constructed, corresponding to 11 different durations of rainfall accumulation 1, 2, 3,
4, 6, 8, 10, 12, 15, 18, 24 h.
Example of IDAF model inference at the Niamey Aéroport rain gauge:
(a) checking of the temporal simple-scaling conditions (left: linear
relationship between the logarithm of the statistical moments of order q
and the durations D, right: linear relationship between k(q) and q) and
estimation of the temporal simple-scaling exponent; (b) left:
empirical cumulative distribution of annual maxima, right: global fitting of
the GEV parameters; (c) comparison between empirical and modelled
ARF.
Inferences of the individual components of the model
The proposed model has seven parameters: the temporal scale exponent
(η), the three ARF parameters (a, b, ω), and the three GEV
parameters (μref, σref, ξref).
After having tested different optimisation procedures most notably
a global maximum likelihood estimation and the two-step method proposed
by, a three-step method was finally retained,
since it gave the best results in the evaluation of the IDAF model (see
Sect. ). These three steps are explained in the following
paragraphs (Sects. to ).
For each step, an illustration based on the result obtained for the Niamey
Aéroport station is given in Fig. .
Temporal simple scaling: estimation of η
The temporal scaling of the IDAF model is described by the η parameter.
The inference of η is achieved in two steps. The first one consists of
computing k(q) for different moments q through a linear regression
between the logarithm of the statistical moments of order q (E[Iq]) and
the durations D (see Fig. a, left panel); next, η is
obtained by a linear regression between k(q) and q (see
Fig. a, right panel). At the Niamey Aéroport station,
the value obtained for η is equal to -0.91.
GEV parameters: μref, σref, ξref
The GEV parameters μref, σref,
ξref are estimated on the point samples, using the property
that all normalised samples {i(D,0)/Dη} must come from the same
distribution if simple scaling holds. All normalised samples are pooled in
one single sample on which the GEV parameters are estimated this
methodology corresponds to the second step of the two-step method proposed
by. Figure b illustrates
this process at the Niamey Aéroport rain gauge: initial samples are
displayed on the left panel normalised samples are plotted on the right
panel. The fitted GEV and the estimated GEV parameters are also given in this
figure.
In comparison with fitting the GEV parameters separately to each sample
constituted for each duration, this method aims at limiting sampling effects
by fitting the GEV parameters on a single sample gathering all rainfall
durations. The maximum likelihood and the L-moments methods were tested for
estimating the GEV parameters. The estimation provided by these methods gave
similar results, probably due to the large sample size. The results of the
L-moments method are presented here, while this method is generally considered
better than the Maximum Likelihood Estimation (MLE) for the estimation of high quantiles when the length of
the series is short .
Spatial scaling
The estimation of a, b, and ω was carried out by minimising the mean
square difference between the empirical ARF (Eq. ) and the
model ARF (Eq. ), as originally proposed by
. Other scores (mean and max. absolute
error, bias, etc.) and variables (difference between the observed and model
mean areal rainfall) have been tried but gave poorer results in validation
(Sect. ). Figure c shows the comparison
between the empirical ARF and the model ARF at the Niamey Aéroport
station and the parameters obtained for the theoretical ARF model.
Regional model
The point parameter inference (μref, σref,
ξref and η) has been performed on each of the 30 point
rainfall series, which thus provide 30 IDF models. The complete IDAF model
has been fitted to each of the 13 rain gauges CR. The obtained IDF and IDAF
parameters display neither any coherent spatial pattern nor any trend
over the domain, as may be seen in Fig. . Sampling
effects due to the small area and the short length of the series may explain
this point, since a trend has been observed on a larger domain at the
regional scale for daily rainfall .
Assuming a spatial homogeneity of rainfall distribution (no spatial pattern),
annual maxima series have been pooled together to obtain regional samples.
The regional samples were used to fit the IDAF model over the domain in order
to limit sampling effects. The point regional sample pools together the 30
rainfall samples directly provided by the 30 rain gauges i(D,0); the 12
areal regional samples obtained for each of the 12 spatial resolutions
{i(D,A);A=1,…,2500km2} result from pooling together
the 13 individual series (CRs) computed as explained in
Sect. .
Obtained parameters for the global IDAF model (a) and corresponding
GEV parameter values for the different durations D for the point scale
A=0 (b).
Map of the obtained IDF parameters (η, μref,
σref, and ξref fitted on the 30 rain gauge
samples) and IDAF parameters (η, μref,
σref, ξref, a, b, and ω fitted on
the 13 CR samples). The grey box shows box plots of the parameter
values obtained at the different rain gauges.
Table presents the parameters obtained for the global
IDAF model. The obtained GEV parameters are μref=40.6mmh-1, σref=10.8mmh-1, and ξref=0.1. When upscaled to the daily duration
μ(24 h) = 2.29 mmh-1 (55.0 mmday-1) and σ(24 h) = 0.61 mmh-1
(14.6 mmday-1). It is worth noting that these latter values are coherent
with those obtained for a much larger area in this region by
, working on the data of
126 daily rain gauges covering the period 1950–1990. Note also that the
temporal scale exponent (η) is large (0.9), which means that the
intensity strongly decreases as the duration increases. This is not
surprising given the strong convective nature of rainfall in this region.
Similar values of the temporal scaling exponents are obtained in regions
where strong convective systems occur
,
while lower values are obtained in regions where extreme rainfall is
generated by different kinds of meteorological systems for example in
many mid-latitude regions; see
e.g.. The
dynamic-scaling exponent is roughly equal to 1 which means that increasing
the surface by a given factor leads to a similar ARF change than increasing
the duration by the same factor (keeping in mind that this rule applies only
to the range of time–space resolutions explored here).
Checking of the temporal simple-scaling conditions for the regional
samples defined by the 30 available rain gauges for point resolution (top),
and the 13 CRs (see Sect. ) for
resolutions at 100 km2 (middle) and 2500 km2 (bottom).
Comparison between empirical ARF (obtained with the regional
samples: 30 rain gauges for point resolution and 13 CRs for other
spatial resolutions) and the modelled ARF IDAF model.
IDAF model evaluation
The evaluation of the IDAF model is carried out in two successive stages.
First each component used to build the final model (temporal simple scaling,
ARF model, and GEV distribution) is checked individually; next, the global
goodness of fit (GOF) is tested using the Anderson–Darling (AD) and the
Kolmogorov–Smirnov (KS) tests.
In Fig. two series of graphs are plotted in order to
verify whether the simple-scaling hypothesis holds for the time dimension. On
the left are the plots of ln(E[Iq]) vs. ln(D) designed to check the
log–log linearity between these two variables (Eq. ); on
the right are the plots of q vs. k(q) aimed at checking the linearity
between these two variables (Eq. ). At all three spatial
scales, there is a clear linearity of the plots, meaning that the two
conditions for accepting the temporal simple-scaling hypothesis are
fulfilled. Note that the graphs shown are those obtained on the regional
samples for three different spatial scales only (point scale, 100 and
2500 km2), but the quality of the fitting is similar for all the
other spatial scales.
Simple scaling in space and dynamical scaling (e.g. the relationship between
time and spatial scaling) are checked in Fig. . This figure
compares the empirical ARFs (Eq. ) computed on the regional
samples and the ARFs obtained with the model (Eq. ) for all
the space scales and timescales pooled together. With a determination coefficient
(r2) of 0.98, and a very small root mean square error, it appears that the model
restitutes very well the empirical ARF at all space scales and timescales, except
at the hourly time step and for the three largest surfaces (900, 1600, and
2500 km2), for which the model significantly underestimates the
observed reduction factor. At such space scales and timescales the finite size of the
convective systems generating the rain fields creates a significant external
intermittency seeon the distinction between internal and external
intermittency. It thus seems that the simple-scaling
framework holds only as long as the influence of the external intermittency
is negligible or weak. Consequently, it is likely that the overestimation of
the ARF by the simple scaling-based model would be
observed for larger space scales and timescales than the ones the AMMA-CATCH data set
allows one to explore.
Figure illustrates that the global model is also able to
reproduce very correctly the mean areal rainfall intensity over the whole
time–space domain explored here, except again for the hourly time step and
the largest surfaces.
As the IDAF model is primarily designed to estimate high quantiles, its
ability to represent the mean is not a sufficient skill. It is thus of
primary importance to evaluate its ability to also represent correctly high
return levels and extreme quantiles. This was realised by visually inspecting
return level plots and by using GOF statistical tests
computed in a cross-validation mode (all the stations are used to calibrate
the model except one which is used to validate the model prediction). These
tests are used to quantitatively assess how well the theoretical GEV
distribution based on the IDAF model fits the empirical Cumulative Distribution Functions (CDF) of the observed
annual maxima for each spatio-temporal resolution. Each test provides
a statistic and its corresponding p value. The p value is used as an
acceptation/rejection criterion by fixing a threshold of non-exceedance (here
1, 5, and 10 %).
The return level plots displayed in Fig. for two reference
locations and three time steps allow a visual inspection of the capacity of the
global IDAF to fit the empirical samples. The p values of the two GOF tests
are given in the inset caption. As could be expected, there is a significant
dispersion of the results obtained on individual samples. The difficulty of
reproducing correctly the empirical distribution when combining the smallest
time steps with the largest areas is confirmed. While similar graphs were
plotted for the other 11 reference locations, it is obviously difficult to
obtain a relevant global evaluation from the visual examination of such
plots.
Comparison between empirical mean areal rainfall intensity (obtained
with the regional samples: 30 rain gauges for point resolution and 13 CRs
for other spatial resolutions) and global IDAF model for different
spatio-temporal aggregations.
Empirical return level plot obtained at two rain gauges in
comparison with the global IDAF model for different durations (1 h,
6 h, and 24 h from top to bottom) and different spatial
aggregations (from point to 2500 km2).
Anderson–Darling GOF test (cross-validation): percentage of rejected series for 1 % (a) and
10 % (b) significance level for the global IDAF model. Note that there are 30 series
for the point scale (0 km2) and 13 for the other spatial aggregation (CR).
Figure aims at tackling this limitation by representing this
information in a more synthetic way. In this figure the percentage of
individual series for which the IDAF model is rejected by the
AD GOF test is mapped for each duration and spatial
aggregation for two levels of significance (1 and 10 %). Here it is worth
remembering that we have 30 individual series for the point scale, and 13
different individual series for each of the 12 spatial scales, meaning that,
for a given time step, the percentage of rejections/acceptations are computed
from a total of 186 (30+12×13) test values. Here again, the limits
of the model for small time steps and large areas are clearly visible; one
can also notice a larger number of rejections for small areas and the highest
durations (duration higher than 12 h and area smaller than
25 km2). Apart from that, the number of rejections of the null
hypothesis remains low. The KS test (not shown) displays similar results with
fewer rejections of the null hypothesis. It thus appears fair to
conclude that, over the range of space scales and timescales covered by the AMMA-CATCH
network, a simple scaling approach allows for computing realistic ARFs, the limit of validity being reached for areas roughly
larger than 1000 km2 at the hourly time step.
Discussion and conclusions
Up to now the rarity of rainfall measurements at high space–time resolution
in tropical Africa has not allowed comprehensive studies on
the scaling properties of rain fields in that region to be carry out. From 1990 the
recording rain-gauge network of the AMMA-CATCH observing system has sampled
rainfall in a typical Sahelian region of West Africa at a time resolution of
5 min and a space resolution of 20 km, over an area slightly larger
than 1∘×1∘. This data set was used here for
characterising the space–time structure of extreme rainfall distribution,
the first time such an attempt has been made in this region where rainfall is
notoriously highly variable.
Simple scaling was shown to hold for both the time and the space dimensions
over a space–time domain ranging from 1 to 24 h and from the point scale to
2500 km2; it was further shown that dynamical scaling relates the
timescales to the space scales, leading to propose a global IDAF model valid
over this space–time domain, under the assumption that extreme rainfall
values are GEV distributed.
Different optimisation procedures were explored in order to infer the seven
parameters of this global IDAF model. A three-step procedure was finally
retained, the global IDAF model being fitted to a global sample built from
all the different samples available for a given space scales and timescale. This model
has been evaluated through different graphical methods and scores. These
scores show that the ARFs yielded by the IDAF model fit
significantly well (in a statistical sense) with the observed ARFs over our space–time domain, except for the part of the domain
combining the smallest timescales with the largest space scales. This
limitation is likely related to the larger influence of the external
intermittency of the rain fields at such space scales and timescales.
Despite the growing accuracy of rainfall remote-sensing devices, this study
demonstrates that dense rain-gauge networks operating in a consistent way
over long periods of time are still keys to the statistical modelling of
extreme rainfall. In the numerous regions where rainfall is undersampled by
operational networks and where satellite monitoring is not accurate enough to
provide meaningful values of high rainfall at small space scales and timescales,
dense networks covering a limited area may provide the information necessary
for complementing the operational networks and satellite monitoring. In West
Africa, south of the Niger site, AMMA-CATCH has been operating another site
of similar size in a Sudanian climate since 1997 (Ouémé Catchment,
Benin), providing ground for a similar study in a more humid tropical
climate.
As mentioned in the discussion of Sect. 5, there is however a limitation of
these two research networks, linked to their spatial coverage. Extending the
area sampled by these networks to something of the order of 2∘×2∘ would indeed allow for studying more finely the effect of the
limited size of the convective systems onto the statistical properties of the
associated rain fields. However, this means enlarging the area by a factor of 4,
making it much more costly and difficult from a logistical point of view to
survey properly. For the years to come, AMMA-CATCH remains committed to
operating both the Niger and the Benin sites for documenting possible
evolutions of the rainfall regimes at fine space scales and timescales in the
context of global change as well as for verifying whether the scaling
relationships proposed here still hold for quantiles at higher time periods.
As a matter of fact, one strong hypothesis of the model proposed here is that
the ARF is independent of the return period. This hypothesis seems verified
for return periods smaller than the length of our time series, but it is not
possible to infer whether this really holds for higher return periods.
Therefore, developing an IDAF model able to account for a possible evolution
of the ARF with the return period level is a path that has to be explored,
copulas being a candidate for such a development
e.g..
It is also envisioned to test other IDAF model formulations based on alternative approaches for
modelling the scale relationships, among which the method proposed by
seems of particular interest.
A brief explanation of the transition between the simple-scaling framework used to described the
space–time scaling of maximum annual rainfall (Eq. ) and the GEV model used to model
the statistical distribution of these maxima (Eqs. to ) is given
here.
The random variables I(Dref,0) and I(D,A) are modelled by a GEV model:
ProbI(Dref,0)≤i(Dref,0)=exp-1+ξ(Dref,0)i(Dref,0)-μ(Dref,0)σ(Dref,0)-1ξ(Dref,0)
and
Prob(I(D,A)≤i(D,A)=exp-1+ξ(D,A)i(D,A)-μ(D,A)σ(D,A)-1ξ(D,A).
Letting c=λη×ARF(D,A), then Eq. () becomes
I(D,A)=dI(Dref,0)×c
and if Eq. () holds, then
ProbI(D,A)≤i(D,A)=ProbI(Dref,0)×c≤i(Dref,0)×c=ProbI(Dref,0)≤i(Dref,0)
and
ProbI(D,A)≤i(D,A)=exp-1+ξ(Dref,0)i(Dref,0)-μ(Dref,0)σ(Dref,0)-1ξ(Dref,0).
By replacing I(Dref,0) by I(D,A)/c we obtain
ProbI(D,A)≤i(D,A)=exp-1+ξ(Dref,0)I(D,A)c-μ(Dref,0)σ(Dref,0)-1ξ(Dref,0)
or
ProbI(D,A)≤i(D,A)=exp-1+ξ(Dref,0)I(D,A)-μ(Dref,0)×cσ(Dref,0)×c-1ξ(Dref,0).
The equality between Eq. () and Eq. () gives
μ(D,A)=μref×cσ(D,A)=σref×cξ(D,A)=ξref.
Equations ()–() correspond to Eqs. ()–() in the main text.
Acknowledgements
This research was partly funded by the LABEX OSUG@2020 (ANR10 LABX56), partly by the French
project ESCAPE (ANR10-CEPL-005), and partly by IRD and INSU through the support to the AMMA-CATCH
observing system. Gérémy Panthou's Ph.D. grant is funded by SOFRECO.
Edited by: P. Molnar
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