Introduction
Stochastic rainfall models are statistical models that aim at simulating
realistic random rains. For this purpose, they generate rainfall simulations
which reproduce, in a distributional sense, a set of key rainfall statistics
derived from an observation dataset . The practical
interest of stochastic rainfall models is notably to complement numerical
weather models for the simulation of rainfall heterogeneity at fine scales,
and to quantify the uncertainty associated with rainfall reconstructions.
Indeed, numerical weather models face challenges for reproducing rainfall
heterogeneity in space and time, in particular at fine scales
. Some of the main applications of stochastic
rainfall models are therefore the fast generation of synthetic rainfall
inputs for local impact studies related for instance to hydrology
or agronomy , and the downscaling of aggregated precipitation products such as
rain observations or numerical model
outputs . In all cases, the target is the
transposition of observed rain statistics into synthetic rain simulations.
Recently, considerable attention has been paid to increasing the resolution
of stochastic rainfall models so that they can mimic rainfall at sub-daily
timescales. Currently, several high-resolution stochastic rainfall models
are able to deal with precipitation data at typical resolutions of 1 min to
1 h in time and of 100×100 m2 to 1×1 km2 in space
(see for example ). At such scales, not only the
marginal distribution of observed rain intensity matters but the space–time
dependencies within rain fields are also important features of the rain
process . In particular, the impact of the
advection and diffusion of spatial rainfall patterns (e.g. rain cells or rain
bands) have to be modelled . As a
consequence, most sub-daily stochastic rainfall models consider rainfall as a
random space–time process.
An underlying hypothesis in stochastic rainfall modelling is that of
stationarity: the statistics of rainfall are supposed to be constant over a
given (space–time) modelling domain. This enables (1) the inference of
rainfall statistics from an observation dataset, and (2) the reproduction of
these statistics in simulations. The definition of stationary domains can be
regarded as a modelling choice, often subjective and left to the judgment of
modellers . It consists of defining pools of data that are
considered similar enough (in a statistical sense) to perform model
inference. In the case of stochastic rainfall modelling, the identification
of stationary datasets or sub-datasets relies on some phenomenological
guesses about rainfall, which serve as fuzzy guidelines to delineate
stationary domains. Depending on the application and modelling choices, the
parametrization of sub-daily stochastic rainfall models is considered as
changing at scales ranging from seasons to single rainstorms .
One possible approach to delineate pools of homogeneous rain observations in
a more quantitative way is to classify them prior to modelling. A set of
predefined criteria is used to build a metric of similarity between the
observations, and a classification algorithm is applied to the resulting
similarity measures in order to define clusters of closely related rain
observations. The result of such a classification procedure, often referred
to as rain typing, is the identification of a limited number of rain types which gather rain observations that share similar properties. Until recently,
rain typing mainly focused on classifications based on rain intensity only,
with the aim to assess the physical processes responsible for rain generation
(e.g. distinguish convective and stratiform rains) . In the last years, the emergence of metrics
characterizing rainfall spatial or space–time behaviour paved the way for new rain typing
methods based on the arrangement of rain fields in space and in time
.
In this context, the present paper focuses on the temporal non-stationarity
of rainfall space–time statistics in view of sub-daily stochastic rainfall
modelling. We intentionally restrict our investigation to temporal
non-stationarities and consider stationarity in space (i.e. constant
statistics over the whole area of interest) as a prerequisite modelling
assumption. The goal is therefore to identify periods of time during which
rainfall space–time statistics remain as constant as possible over a given
area. The proposed framework relies on the classification of radar images
based on their space–time features. The resulting classes are then used to
define rain types that group rain fields with similar statistical signatures.
Finally, the transition between rain types is interpreted as a break in the
temporal stationarity of rainfall statistics.
The remainder of this paper is structured as follows:
Sect. gives a general overview of rainfall space–time
patterns visible in radar images. Section describes a rain
typing method based on the previously identified patterns, and explains how
the resulting rain types can be used to identify stationary periods. Then,
Sect. assesses the performance of this method for both
synthetic and real case studies. Next, Sect. discusses the
dependence of the proposed rain typing method to the stochastic model in use,
as well as the implications of the observed patterns of non-stationarity for
sub-daily stochastic rainfall modelling. Finally, Sect.
gives general conclusions.
Overview of rainfall space–time patterns observed in radar images
Prior to the design of a quantitative method to identify
non-stationarities in rainfall statistics, the current section seeks to
illustrate with some typical examples the diversity of space–time patterns
that can be observed in rain fields, and to give an overview of their
temporal evolutions.
We illustrate this study with data collected over the Vaud Alps, Switzerland
(Fig. a). The area of interest encompasses a network of
high-resolution rain gauges used later for validation and covers an area of
60×60 km2. For reasons of data availability, we focus in this paper
mostly on summer rains observed from 1 July to 31 August 2017. During
this period, only periods corresponding to rain events are considered. A rain
event is defined as a rainy period isolated by at least 30 min of dry
conditions before and after, and resulting in at least 2 mm of cumulated rain
height (in average over the area of interest). The dataset is comprised of 17
rain events causing around 250 mm of cumulative rain height.
Figure b, c display different rain fields observed by the
Swiss weather radar network operated by the MeteoSwiss weather agency
. Weather radars are remote sensing devices providing
comprehensive images of rain fields at the regional scale (coverage up to
about 200 km from the radar device), with a high resolution (in the present
case, 10 min in time and 1 km in space). The resulting rain rate estimates
are known to be biased , but at the same time radar images
are currently the most reliable and exhaustive source of information about
the spatial organization of rain fields and their temporal evolution : this is why radar images are
used in the current section to illustrate rainfall space–time behaviour, and
in the following to extract rainfall statistics.
Examples of rain fields over Vaud Alps, Switzerland.
(a) Situation map. The area of interest is delineated by the blue
square. The red star denotes the location of the rain gauge network used for
validation (Sect. ). (b) Examples of rainstorms
with different space–time behaviours. (c) One example of a rain field
with a temporally changing behaviour. In (b) and (c) the red
arrows displayed in the last panels denote the advection of the rainstorm in
20 min.
A visual inspection of the rain fields displayed in Fig.
allows understanding of the benefit of jointly characterizing the spatial and
the temporal behaviour of high-resolution rain fields.
At the scale of
interest, all rain fields shown in Fig. b, c have in common
strong space–time interactions structured by the following:
a distribution of intensities that is often skewed , with a variable amount of zero values
due to within-storm rain intermittency ;
well-defined spatial patterns , which can be linked to the processes responsible
for rainfall production such as rain cells, rain bands or rainstorms;
a temporal behaviour shaped by the advection and diffusion of spatial patterns over time .
Despite common space–time attributes, the three rain events of
Fig. b look very different. For example, the Type A event is
characterized by a strong spatial intermittency (i.e. dry and wet locations
coexist in the same radar image) combined with well-defined areas generating
intense rainfall, while the Type C event shows a lower fraction of dry
locations, is less spatially structured and generates low rain rates. Hence,
it should be possible to find statistical metrics able to distinguish these
three rain types based on their spatio-temporal characteristics.
It is worth noting that not only the space–time statistics of rainfall change
between rain events, as shown in Fig. b, but also these
statistics can change within a single rain event, as illustrated in
Fig. c. In this case, a widespread and spatially continuous
rain field (first two images) is replaced by disconnected rain aggregates (last
two images). This change in the space–time features is very rapid and takes
place in less than 30 min. Such abrupt changes in the space–time behaviour
within a rain event are relatively common in our dataset, as discussed later.
Starting from this example, this paper investigates how to detect
non-stationarities in rainfall space–time statistics using radar images as
primary information. We adopt a Eulerian approach and investigate the
temporal variability of rainfall statistics over a given area of interest, as
perceived by an Earth-fixed observer.
Assessing rain statistics stationarity from radar images
Extracting space–time information from radar images
To assess the stationarity of rainfall space–time statistics, we propose to
start by extracting information on the rainfall space–time behaviour from
radar images. To this end, 10 statistical metrics are derived for every radar
image (Fig. ), which are split in three categories that
reflect the three main characteristics of rain fields identified in
Sect. :
Intensity indices (IIs). These relate to the probability density function (histogram) of the rain intensities measured in a given radar image. The following indices are used:
II.1: fraction of the image covered by rainy pixels (informs the intra-storm rain
intermittency).
II.2: mean rain intensity computed over all rainy pixels.
II.3: an 80 % quantile
of rain intensities characterizing heavy rain pixels.
Spatial indices (SIs). These characterize the spatial arrangement of patterns within rain fields. They are selected among the indices proposed by . They are computed based on binary images representing rain masks (Fig. ). Such binary images are obtained by thresholding radar images at a rain intensity of
0.1 mm h-1, and by assigning a 0 value to the pixels under the threshold and a 1 value otherwise. Then, connected components, hereafter referred to as rain aggregates, are identified in every binary image. Their morphological properties are used to derive the following indices:
SI.1: fraction of rainy area covered by the largest rain aggregate in the image. This is a first indication of how the rain field is split into aggregates. Let Np be the total number of rainy pixels in the binary image and Nm be the number of pixels of the largest aggregate; then SI.1=NmNp.
SI.2: connectivity index. It is equal to 1 if the rain field is fully connected (one single rain aggregate) and tends to zero if the rain field is split into many disconnected aggregates. Let Nc be the number of rain aggregates in the binary image, then SI.2=1-Nc-1Np+Nc.
SI.3: perimeter index, characterizing the sinuosity of the contours of rainy areas. It is equal to 1 if all rain aggregates are squares and tends to 0 if the rain aggregates are very sinuous. Let p be the total perimeter of rain aggregates, i.e. the sum of the perimeters of all rain aggregates, then
SI.3=4×NppifNp=Np,SI.3=2×2×Np+1pifNp≠Np.
SI.4: area index, characterizing the spread of the rain aggregates. It is equal to 1 if the radar image contains one single aggregate, and tends to zero if the rainy pixels are only in the corners of the image. Let Aconvex be the area of the convex hull
encompassing all the rain aggregates, then: SI.4=NpAconvex.
Temporal indices (TIs). These characterize the temporal evolution of the rain fields. They assess the advection of rainstorms over the ground as well as the temporal deformation of spatial rain patterns.
Let It and It+1 be two subsequent images. In addition, let ri,j be the cross-correlation between It+1 and It translated
by a vector D=i.E+j.N of coordinates i and j along the eastward and northward directions respectively.
Finally, let rmax be the maximum correlation and Dmax=imax.E+jmax.N the corresponding displacement vector. Then the TIs are defined
by the following:
TI.1: eastward component of the displacement vector, i.e. imax. This index corresponds to the advection of the rainstorm along the west–east direction between times t and t+1.
TI.2: northward component of the displacement vector, i.e. jmax. This index corresponds to the advection of the rainstorm along the south–north direction between times t and t+1.
TI.3: correlation coefficient between the two radar images It and It+1 after removing advection effects, i.e. rmax. This index equals one if the spatial rain patterns remain identical between two subsequent radar images (up to a translation), and tends to zero if the images are completely different.
Computation of indices characterizing rainfall space–time statistics
for a single radar image. This procedure is repeated for each image with
>10 % rainy pixels. Note that the temporally subsequent image
is required to compute the time indices (TIs).
Classification of radar images based on rainfall space–time statistics
The 10 indices defined above are used to classify the radar images in order
to obtain a limited number of rain types. To ensure the reliability of these
indices, only images with a significant proportion of rainy pixels are used
for classification. Indeed, if the number of rainy pixels is low, the SIs are not meaningful and the TIs cannot be computed
because the image correlation procedure fails. We therefore only classify the
images with more than 10 % rainy pixels, the remaining rainy images (rain
fraction <10 %) being typed afterwards.
To define rain types, we adopt an approach based on a Gaussian mixture model (GMM)
classifier. This classifier has been
selected because it allows for an automatic selection of the number of
classes, and because it does not require any a priori information about the
joint distribution of the rain indices. The idea of the GMM is to approximate
the joint distribution of the 10 statistical indices. This approximation
being a combination of Gaussian functions, those can be considered as
representing several discrete categories. In the GMM, the joint distribution
p^(x) of the space–time indices forming a vector x∈R10 is approximated by a weighted sum of K multivariate
Gaussian distributions N(x|μk,Σk),k=1,…,K,
with respective mean vector μk and covariance matrix Σk:
p^(x)=∑k=1Kπk×N(x|μk,Σk).
The inference of the model parameters (i.e
πk,μk,Σk,k=1,…,K) is performed with an
expectation–maximization (EM) algorithm . A full
covariance model is used for the covariance matrices Σk
in order to take into account a possible correlation between the indices. The
number K of Gaussian mixtures used in the GMM model is selected by
minimization of the BIC (Bayesian information criterion) derived from EM fits computed for different
numbers K , with the goal of selecting the GMM model
resulting in the best fit while maintaining a parsimonious parametrization.
Here the MATLAB Statistics and Machine Learning Toolbox has been used to fit
the GMM model, with the function “fitgmdist”.
Once fitted, the GMM can be used to derive a probabilistic classification of
any vector x of indices. The probability that a vector belongs to the
population Gj whose distribution is the jth mixture component N(x|μj,Σj) is given by the following:
p^(x∈Gj)=πj×N(x|μj,Σj)∑k=1Kπk×N(x|μk,Σk).
A classification of the entire image dataset can thus be obtained by
assigning to each image I the class that corresponds to the most probable
mixture component (in practice, the MATLAB “cluster” function is used):
G(I)=maxj(p^(x∈Gj)).
As mentioned above, the classification procedure can only be applied to radar
images with a significant proportion of rainy pixels (>10 %). In addition, it
can be desirable to avoid high-frequency successions of rain types, for
instance if the targeted stochastic rainfall model requires long-lasting
pools of data to perform parameter inference. To do this, we impose a
temporal persistence threshold for the rain types. To this end, all images
that lead to temporal clusters of classes that do not reach a certain
duration are set to unclassified. Here we use a 60 min duration threshold.
After cleaning the classification, all the time steps that are unclassified
(either because less than 10 % of the pixels are rainy or because the image
belongs to a <60 min cluster) receive the type of the classified image that
is temporally the closest (i.e. nearest-neighbour interpolation along the time
axis). The complete rain typing framework is summarized in
Fig. .
Rain typing framework.
Since the rain typing method presented above aims at defining groups of rain
fields sharing similar statistical signatures, the transitions between rain
types can be interpreted as non-stationarities. Similarly, periods with a
constant rain type are interpreted as stationary periods.
Validation and application
In this section, we validate our rain typing approach in the context of
stochastic rainfall modelling. The validation study comprises four steps:
first, the proposed approach is tested in a synthetic case in order to
determine whether we are able to identify known non-stationarities. Then,
real data are used to compare our rain typing strategy with two alternative
hypotheses of rainfall stationarity: (1) rainfall statistics are stationary
at a seasonal scale and (2) rainfall statistics are stationary at a rainstorm scale. Next, the rain typing method is applied to radar data covering
the full year 2017, in order to assess its ability to handle various rainfall
situations. Finally, the sensitivity of the classification to the calibration
dataset is assessed by comparing rain types computed for the summer of 2017
based on two calibration periods: (1) the summer of 2017 and (2) the full
year 2017. Prior to the validation itself, the next subsection describes the
stochastic rainfall model used for validation.
Stochastic rainfall model
The validation of the rain typing approach uses a stochastic rainfall model
designed for local-scale (area of a few square kilometres) and high-resolution
(up to 1 min) data. This model involves 11 parameters and aims at modelling
both the marginal distribution of observed rain intensities and the
space–time dependencies that exist within rain fields. It is briefly
introduced hereafter; for more details the reader is referred to
. In this model, the marginal distribution of rain rates is
accounted for by considering that rain measurements (R) originate from the
censoring and power transform (involving parameters a0,a1,a2) of a
standardized multivariate Gaussian random field (Z) tainted by an additive
measurement noise (ϵ∼N(0,σϵ))
(Eq. ):
R=Z+ϵ-a0a11a2ifZ+ϵ>a0,R=0ifZ+ϵ≤a0,
The multivariate Gaussian latent random field (Z) is characterized by an
asymmetric Gneiting space–time covariance function
ρ which accounts for both the advection and the diffusion of spatial
rain patterns (Eq. ). For two rain observations separated by a
spatial lag h and a temporal lag u, the covariance is given by the following:
ρ(h,u)=1(u/d)2δ+1exp-(||h+V.u||)/c)2γ(u/d)2δ+1βγ.
In this model, the advection of rainstorms is assumed to be constant and
linear along a vector V defined by its amplitude VS and direction
Vθ. The regularity parameters γ (for space) and δ (for
time) control the slopes at the origin of the covariance function and thereby
regulate the small-scale variability of the rain fields, and ultimately their
smoothness. The scale parameters c (for space) and d (for time), in units
of distance and time respectively, control the decorrelation distances of
rain patterns. Finally, the separability parameter β controls the
space–time interactions. When β=0, the covariance function is
space–time separable.
Detection of rainfall non-stationarity in a controlled setting
The ability of the rain typing method to detect possible non-stationarities
is tested by applying it to synthetic time series of radar images. These
images are generated using the stochastic rainfall model presented above,
with model parameters changing abruptly. This produces (temporal)
non-stationary synthetic rain fields. The rain typing method is then applied
to the simulated radar-like images (resolution: 1×1 km2 in
space, 10 min in time; footprint: 60×60 km2) in order to assess if it is able to
retrieve the prescribed patterns of temporal non-stationarity.
For generating the synthetic images, we use the stochastic rainfall model
described in Sect. with model parameters corresponding
to three typical rain behaviours identified by visual inspection (Table 1).
Figure a shows an example of simulated rain field for each
rain type.
Parameters of the stochastic rainfall model used for the generation of synthetic images.
a0
a1
a2
σϵ
γ
c
δ
d
β
VS
Vθ
Type 1
-0.73
0.78
0.41
0.0
0.76
4642
0.89
590
0.97
5.6
18
Type 2
0.0
0.64
0.41
0.0
0.49
6840
0.86
892
0.91
1.2
-11
Type 3
-0.83
1.16
0.45
0.0
0.38
13 995
0.81
1702
0.95
0.9
-39
Identification of non-stationarities in rain statistics for a
synthetic case study. (a) Examples of synthetic rain fields
simulated for each rain type. (b) Segmentation of the time axis into
periods with stationary rain statistics. First row: reference. Rows 2–4:
segmentation obtained by rain typing applied to synthetic radar-like images.
Dotted black lines represent dry periods.
In Fig. b, the series titled “reference” shows the rain
types prescribed to the stochastic rainfall model for the generation of the
synthetic radar images. Based on stochastic simulations, three sets
(realizations) of synthetic images are generated. Each realization is
classified to determine whether it is possible to identify the reference rain
types. Results show that the proposed method can consistently detect the
prescribed rain types and their temporal evolution, with an agreement between
the reference and the estimated rain types of 97 %, 96.3 % and 92.6 % for
realizations 1 to 3. It also properly estimates the number of rain types
prescribed in the reference. The only noticeable difference between the
reference and the simulations is the emergence of a very infrequent fourth
rain type (accounting for 0 %, 1 % and 1.6 % of the estimated rain types for
realizations 1 to 3) at the beginning or at the end of some rain events (in
green in Fig. b). This is because at these periods, the rain
does not cover the whole area of interest, and in certain situations it can
produce rain fields with different space–time statistics, which induces this
artificial fourth rain type. Except for this fourth rain type, results show
that in this synthetic experiment, the proposed method performs well in
detecting non-stationarities of rainfall space–time statistics and, in turn,
periods during which these statistics remain stationary.
Impact of rainfall non-stationarity on stochastic modelling of an actual dataset
To further validate our rain typing method, we apply it to a real dataset
acquired in the Vaud Alps, Switzerland, during the summer of 2017. In such a
real case study, the true succession of rain types is obviously unknown. To
assess the performance of the proposed rain typing method, we compare it with
two other hypotheses of stationarity that can be found in the literature. We
therefore consider three cases, illustrated in Fig. :
Hypothesis H1. The time axis is broken down into rain types interpreted as stationary
time periods (the approach proposed in this paper). Applying the rain typing
method presented in Sect. to the period of interest leads
to six rain types.
Hypothesis H2. The statistical signature of rainfall is constant over
meteorological seasons . This
leads to one stationary pool of rain events for the period of interest.
Hypothesis H3. Rainfall statistics are constant within a single rainstorm
but change between storms . Here 17 rain
events are identified following the definition adopted in
Sect. .
Observation dataset used for validation. (a) Measurement
network. Red dots denote rain gauge locations. (b) Picture of a rain
gauge with the Vallon de Nant catchment in the background.
(c) Observed rain rate averaged over the network.
(d) Segmentation of the time axis into stationary periods for the
three tested hypotheses. For each hypothesis, segments with the same colour
denote periods for which rainfall is expected to have similar space–time
statistics.
To compare these three hypotheses, we apply the same stochastic model as
above to rain data collected by a dense network of eight high-resolution rain
gauges set up in a small (3×6 km2, Fig. a) alpine
catchment called “Vallon de Nant”, situated within the area of interest
presented in Sect. . Hence, in the following, radar images
will be used only to carry out the rain typing presented in
Sect. in order to define the hypothesis H1. The remainder of the validation, i.e. stochastic model calibration and simulation under the
three tested hypotheses, will be carried out on rain rate time series
acquired by rain gauges, and not on radar images. The goal is indeed to keep
the following validation as independent as possible from the radar images
used for rain typing. By doing so, we seek to prove that the proposed method
captures the stationarity of the rainfall process itself, and not only the
stationarity of radar images.
Once the periods of stationarity have been built for each of the three
hypotheses, the stochastic model is calibrated for each stationary period.
This means that for each hypothesis, a set of model parameters is inferred
from observations for each postulated stationary dataset. Then, synthetic
rain fields are generated by unconditional simulation under the three
hypotheses of stationarity, and in each case 50 realizations (i.e. 50
simulated synthetic rain histories) are compared to actual measurements. To
assess the realism of the different scenarios, Fig. shows
the simulated cumulative rain heights. Next, Fig. shows
quantile–quantile (q-q) plots for four statistics selected to assess the
marginal distribution of rain rates and its space–time arrangement: number of
rain gauges measuring zero rain at each time step, rain intensity, standard
deviation (in time) of rain intensities separated by a time lag of 5 min, and
standard deviation (in space) of rain intensities at each time step.
Reproduction of the cumulative rain height (averaged over the whole
network) for the three tested hypotheses. (a) Rain is stationary by rain
type (H1), (b) rain is stationary during the entire summer period (H2),
(c) rain is stationary by rain event (H3). The red line corresponds to the
observations and the black lines correspond to simulations.
Reproduction of rainfall statistics for the three tested hypotheses.
(a–d) Rain is stationary by rain type (H1), (e–h) rain is stationary during
the entire summer period (H2), (i–l) rain is stationary by rain event (H3).
(a, e, i) Quantile–quantile (q-q) plot of simulated vs. observed rain
intermittency, (b, f, j) q-q plot of simulated vs. observed rain rate, (c, g, k) q-q plot of simulated vs.
observed temporal variability of rainfall (at lag
5 min), (d, h, l) q-q plot of simulated vs. observed spatial variability of
rainfall. The quantiles used in the q-q plots are centiles. Each centile is
denoted by a black cross.
Results show that H1 tends to slightly underestimate the cumulative rain due
to an underestimation of very high intensities. This underestimation of heavy
rainfall is common to all the three cases and probably originates from the
stochastic model itself, which is not designed to handle extreme rainfall
due to the simple transform function selected in Eq. (). This could
be improved by adopting a transform function accounting for extreme rainfall
(see for example ) but at the price of a more complex
parametrization, which is not regarded as essential here because the observed
rain rates are mostly low to moderate, and only the 99th centile is affected
by rain rate underestimation. Apart from this underestimation of high rain
rates, hypothesis H1 allows the other metrics to be correctly reproduced.
Contrary to H1, hypothesis H2 leads to a slight overestimation of the
simulated rain height, in particular for the first 30 days
(Fig. ). This is due to the overestimation of moderate rain
rates that compensates for the underestimation of extremely high values. This
bias in the simulated marginal distribution is due to the lack of flexibility
of H2 that imposes a single underlying stochastic model for the entire summer
period. This does not allow enough flexibility to capture the diversity of
structures emerging from high-resolution data. This is also visible for the
simulated variability in space and time, which tends to be overestimated for
the low centiles but underestimated for the high centiles.
Under hypothesis H3, simulation results are close to those of H1, with a
greater propensity of underestimating heavy rainfall. In addition, the
standard deviation in time is not perfectly reproduced for the middle
quantiles. This slightly lower performance of H3 compared to H1 can be
attributed either to the non-inclusion of intra-storm non-stationarities in
this hypothesis, leading to a poorer reproduction of the true rainfall
dynamics, or to a poor inference of model parameters in case of short rain
events caused by the low amount of observations available for such very short
stationary periods.
To sum up, the proposed method consisting of typing rain fields according to
their space–time statistical signature derived from radar images (H1) leads
to more realistic rainfall simulations than the other approaches, H2 and H3.
Seasonality of rain type occurrence
The two previous sections have shown that the
proposed rain typing method is able to reliably identify rain types when
applied to summer rains. To complement the previous findings, the current
section investigates the ability of our rain typing framework to classify
rainfall from other seasons. To this end, rain typing is performed for the
same area of interest as above (Fig a), but this time for a
complete year of radar observations. Here no comparison data are available
since the rain gauge network used for validation in
Sect. had to be removed after the summer due to
harsh local winter conditions. For this reason, this section concentrates on a
qualitative analysis of the timing of rain type occurrences throughout the
year, with interpretation of some resultant rain types.
The classification framework presented in Sect. is
applied to radar data covering the entire year 2017. It should be noted that
rainfall and snowfall are processed without distinction, and thus the
resulting classification produces precipitation types rather than rain types
strictly speaking. However, to be coherent with the rest of the paper, we
will continue to refer to rain types despite possible mixes between rain and
snow during the winter months.
The automatic selection of the number of Gaussian mixtures used in the GMM
model leads to 11 rain types for this year. Figure shows
the monthly occurrence of these rain types, and Fig. shows
the related marginal distributions of the 10 indices used to characterize
rainfall space–time behaviour.
Monthly occurrence of rain types during the year 2017.
Marginal distributions of the indices used for rain type
classification. Each graph represents a different index. Colours denote rain
types.
The monthly occurrence of rain types shows a clear seasonality for most
rain types (Fig. ), with some types occurring mostly during
winter months (types 1, 6 and 7), others during summer months (types 2, 3 and
5 ) and one during spring and fall (type 4). By contrast, rain types 8 to 11
seem relatively unseasonal. The seasonality observed in rain type occurrence
is in good agreement with the local climatology of rainfall with more
snowfall and stratiform rainfall expected during winter, more sleet showers
and rain showers expected in spring and autumn, and more thunderstorm induced
convective rainfall expected during summer. Of course there is no one-to-one
match between the rain types identified by our classification and the
physical types listed above, for two reasons: first, the rain typing method
is purely statistical and we cannot expect a direct match with the physical
processes responsible of rainfall generation. All the attempts at linking the
rain types with physical processes made in this section should therefore be
regarded as qualitative interpretations instead of well-defined
relationships. The second reason for the imperfect match is that one rain
generation process (e.g. a thunderstorm generating convective rainfall) can
lead to several rain types (e.g. in this case types 2, 3 and 5).
The distributions of the space–time indices displayed in
Fig. allow refinement of the interpretation of the rain types.
As an illustration, one rain type typical of each season will be described
hereafter in light of the distribution of indices resulting from the
classification:
Rain type 6 is typical of winter months, and probably corresponds to stratiform rainfall.
The rain fields classified in this type are featured by a large fraction of
the area of interest covered by rainfall, which leads to a small number of
large rain aggregates, and in turn to high connectivity and area indices.
Such rainstorms are moving eastward and are well correlated in time, which
is typical of stratiform rains over Switzerland. Finally the resulting rain
intensities are low.
Rain type 4 is typical of spring and autumn months, and probably corresponds to sleet or rain showers.
The corresponding rain fields are very scattered (low fraction of rain
coverage and low area index) and are poorly connected. In addition, it is
interesting to note that such rain fields have a low temporal correlation,
which reflects a strong variability in time, as is the case for fast-changing
mid-season events.
Rain type 5 is typical of summer months, and probably corresponds to heavy convective rainfall produced by thunderstorms.
Indeed, the rain fields classified in this type generate localized (moderate
fraction of rain coverage and contribution of main aggregate) but heavy (high
mean and Q80 intensity) rainfall. Also, such storms mostly originate from
the south-west (eastward and northward advections >0), which is typical for
large thunderstorms over the Swiss Alps.
It is important to note that even if different rain types dominate at
different months, several rain types occur at each month. In other words, at
a given location, there is no single typical summer rain or winter rain, but
rather a variable collection of rain types that occur each month. It follows
for instance that the rain type 6 is typical of winter months, but it can
also occur during summer, although at a lower frequency than the typical
summer rain types 2, 3 and 5.
Sensitivity of the rain typing approach to the size of the calibration dataset
To complete the assessment of our rain typing
strategy, we study the sensitivity of the classification method to the
dataset used to calibrate the GMM model. To this end, the radar images
corresponding to the periods covered by the 17 rain events of interest
occurring during summer 2017 are typed based on two different GMM models:
A first GMM model (model A) is calibrated using the radar images of summer 2017 (July and August). This corresponds
to the GMM model used in Sect. , and this
classification results in 6 rain types that all occur during the period of
interest.
A second GMM model (model B) is calibrated using the radar images of the full year 2017. This corresponds to the GMM
model used in Sect. . This model results in 11 rain
types, 9 of which occur during the period of interest (rain types 7 and 11
are absent).
Correspondence between rain types derived from two distinct calibration datasets.
Figure shows how the results of rain typing compare when
carried out based on both GMM models A and B. Since the number of rain types
differs between the models, there is no one-to-one correspondence. This is
because the calibration of model B on a larger dataset allows for a more
detailed segmentation of the indices into rain types. Indeed, model A focuses
only on the summer, and if some typical winter rain types occasionally occur
during this period (which is the case as shown in
Sect. ), these rain fields are too infrequent to
generate their own cluster. They are then assigned to the closest Gaussian
mixture. In contrast, the same rain fields have proportionally more relatives
in model B, and they can therefore be grouped in a separate rain type. This
distinct segmentation of the indices space shows that one cannot
unequivocally associate a type derived from model B to a type derived from
model A. A rain type in model B can straddle the border between two rain
types in model A.
Although imperfect, the links between the two classifications are strong. To
assess these links, we first make coincide the two classifications by
assigning to each rain type of model B its most common counterpart in model
A. In this case, the rain types 1, 2, 3, 4, 5, 6, 8, 9, 10 from model B are
paired with respectively rain types 1, 3, 6, 4, 3, 6, 6, 4, 1 in model A. By
doing so, we obtain a 73.6 % match between the two classifications. Then, we
compare the timing of the intra-event rain type transitions for the two raw
classifications (i.e. without the previous pairing). Due to the unequivocal
links between the two outputs, model B leads to a more fragmented
classification (26 transitions instead of 16 for model A). Despite this, most
of the transitions in model A (9 among 16) have a counterpart in model B
(i.e. a transition appears in model B during the same hour than the
transition in model A). This tends to confirm that the intra-event
non-stationarities identified in Sect. are well
defined, and that their detection is robust.
Discussion
Model dependence in rain typing
The aim of this work is to develop a rain typing
strategy able to identify stationary periods for further stochastic rainfall
modelling at a sub-daily resolution, with an emphasis on very high resolution
models (up to 1 min resolution). Although the main ideas developed throughout
this paper can be applied to many different stochastic rainfall models, the
detailed settings of the rain typing strategy must be tuned from case to case
in order to be compatible with the targeted rainfall model. Indeed, because
the final aim is to identify time periods over which stationary statistics
can be inferred, the rain typing method necessarily depends on the properties
to be modelled and the nature of the model. Therefore, we discuss hereafter
how the stochastic rainfall model used in this study has influenced the
settings of the rain typing method.
As primary data source, raw radar images have been preferred to combined rain
gauge–radar products because their higher temporal
resolution (10 min instead of 60 min) is more in agreement with the
resolution of the stochastic model used (i.e. 1 min). Radar images acquired
at higher temporal resolution (e.g. 5 min) are acknowledged to potentially
improve the classification, but such images were unfortunately not available
for this study. In counterpart of the high temporal resolution of raw radar
products, the observed rain intensities can be impaired by local biases, in
particular in mountainous regions such as the one considered in the present
application. However, we think that our method is not significantly affected
by such biases because radar shadow effects due to topography are constant in
time. Therefore, they do not affect the temporal variability of the rain
indices used for classification.
Once the radar product has been selected, another important choice is the
size of the area of interest from which the information will be extracted. Since the study area is very small (3×6 km2), we could in theory
have taken a window of the same size to analyse radar images. However, a
significantly larger window (60×60 km2) has been chosen because the
space indices used for classification rely on the count and on the spatial
arrangement of rain aggregates, which requires several aggregates within the
window of interest for reliable computations. This implies a window slightly
larger than the expected scale of rain aggregates, which ranges from 10 to
30 km in the present case . On the other hand, the window
extent is restricted by the wish to limit the variety of rain behaviours
existing in that window, thereby preserving as much as possible the
stationarity of rainfall in space.
Finally and more importantly, the indices selected for classification must be
consistent with the statistics embedded in the stochastic rainfall model.
Here 10 indices are needed to characterize the evolution of the main features
of rainfall considered by the stochastic model, namely the marginal
distribution of rain intensity, the spatial arrangement of rain aggregates,
and the advection–diffusion of rainstorms. This large number of indices
reflects the complexity of the stochastic rainfall model in use. In case of a
different model (due to for example a single-site study, a different resolution or
another climatology) the set of indices could be modified.
Consequences of non-stationarity on sub-daily stochastic rainfall modelling
The succession of rain types identified in
Sect. and has two important
implications on how stationarity should be regarded in sub-daily stochastic
rainfall models.
Due to the variability observed in rainfall statistics, we believe that it is
often incorrect to assume rain stationarity over long periods, such as months
or seasons, for characterizing of rainfall statistics. In the perspective
adopted in this paper, the seasonality observed in monthly rain statistics is
attributed to a variation in rain type occurrence rather than to a smooth
change in rainfall behaviour (see Sect. ). As such, it
would be incorrect to assume constant model parameters over long periods of
time because of the risk of mixing distinct rainfall statistics during model
calibration. This results in the emergence of artificial rain types whose
statistics are an average of the statistics of the rain types that actually
occur during the period of interest. In the extreme case, such artificial
rain types may not even correspond to any actual rainfall event. This leads
in turn to improper space–time dependencies in the simulated rain fields.
A second striking observation is that rainfall statistics can change
drastically within a single rain event. As a result, the hypothesis of rain
stationarity along entire rain events can be invalidated in some instances.
At least in our dataset, such non-stationary events seem relatively frequent
(at least 7 non-stationary rain events out of 17 have been identified in our
data). This observation is not new, since it may lead to temporal asymmetry
(or temporal irreversibility) in rain rate time series, which is discussed by
. Our framework offers a way to deal with this phenomenon
through the identification of stationary periods prior to the stochastic
modelling of rain. Then, stochastic modelling is carried out separately for
each rain type, and the results are merged afterwards. This allows the
generation of synthetic rain fields presenting a temporal asymmetry, even if the stochastic
model itself is only capable of generating symmetric rain fields for a given
set of model parameters. The temporal asymmetry is then carried by the
temporal arrangement of rain types within a single rainstorm.
Conclusions
This paper proposes a quantitative method to identify stationary rainfall
periods, that is, periods during which a set of 10 statistics representative
of rainfall space–time behaviour at the local scale remains broadly constant.
It is based on a classification of radar images into groups of rain fields
sharing similar statistics when observed at high resolution. For reasons of
data availability, we focused our investigation on summer rains over the Vaud
Alps, Switzerland. However, most of the results obtained in this context are
expected to be transferable to other seasons, as illustrated in
Sect. , as well as to other mid-latitude areas in cases
where extratropical rainstorms significantly influence the precipitation
regime.
The application of the proposed rain typing method in the context of
sub-daily stochastic rainfall modelling shows that our method is able to
(1) identify abrupt changes in rainfall statistics during a controlled synthetic
experiment, and to (2) delineate relevant stationary periods when applied to
an actual dataset. The succession of rain types obtained for our observation
dataset is characterized by the coexistence of several distinct rain types,
with switches between types occurring within single rainstorms. In the
context of sub-daily stochastic rainfall modelling, this observation
highlights the need to delineate stationary periods based on actual
observations rather than subjective assumptions about the rain process.
A possible future work would be to use the proposed method to
simultaneously type precipitation fields observed by radars and simulated by
numerical models. This may provide a new metric to assess the precipitation
component of high-resolution numerical weather or climate models. Indeed, the
proper reproduction of rain types and rain type successions in model outputs
would indicate a correct simulation of the overall space–time behaviour of
rainfall by the model.
It could also be interesting to apply the proposed rain typing method to long-term archives of radar images in order to investigate the temporal behaviour
of rain type occurrence. The resulting information could be used as the starting
point for the design of a statistical model of rain type occurrence, and in
turn a stochastic rain type generator. Coupled with already existing
high-resolution stochastic rainfall models, this would allow the design of
high-resolution stochastic rainfall generators that are able to reproduce local
rainfall statistics over long simulation periods under the assumption of a
steady climate. An alternative to the development of a stationary rain type
generator would be the design of a synoptically conditioned stochastic
rainfall generator by linking the occurrence
of rain types with the state of the atmosphere simulated by one or several
climate models. This could be achieved by assessing the statistical
relationships between rain types derived from radar observations and
meteorological variables derived from climate model reanalyses. If strong
dependences are found, then the future evolution of climate variables
simulated by general circulation models could provide precious
insights into the possible evolution of rain type occurrence in a changing
climate.