Understanding the stationarity properties of rainfall is critical when using stochastic weather generators. Rainfall stationarity means that the statistics being accounted for remain constant over a given period, which is required for both inferring model parameters and simulating synthetic rainfall. Despite its critical importance, the stationarity of precipitation statistics is often regarded as a subjective choice whose examination is left to the judgement of the modeller. It is therefore desirable to establish quantitative and objective criteria for defining stationary rain periods. To this end, we propose a methodology that automatically identifies rain types with homogeneous statistics. It is based on an unsupervised classification of the space–time–intensity structure of weather radar images. The transitions between rain types are interpreted as non-stationarities.

Our method is particularly suited to deal with non-stationarity in the context of sub-daily stochastic rainfall models. Results of a synthetic case study show that the proposed approach is able to reliably identify synthetically generated rain types. The application of rain typing to real data indicates that non-stationarity can be significant within meteorological seasons, and even within a single storm. This highlights the need for a careful examination of the temporal stationarity of precipitation statistics when modelling rainfall at high resolution.

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

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

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

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 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.

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.

Figure

Examples of rain fields over Vaud Alps, Switzerland.

A visual inspection of the rain fields displayed in Fig.

a distribution of intensities that is often skewed

well-defined spatial patterns

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.

It is worth noting that not only the space–time statistics of rainfall change
between rain events, as shown in Fig.

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.

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.

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.

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

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

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

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

TI.1: eastward component of the displacement vector, i.e.

TI.2: northward component of the displacement vector, i.e.

TI.3: correlation coefficient between the two radar images

Computation of indices characterizing rainfall space–time statistics
for a single radar image. This procedure is repeated for each image with

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

To define rain types, we adopt an approach based on a Gaussian mixture model (GMM)

The inference of the model parameters (i.e

Once fitted, the GMM can be used to derive a probabilistic classification of
any vector

A classification of the entire image dataset can thus be obtained by
assigning to each image

As mentioned above, the classification procedure can only be applied to radar
images with a significant proportion of rainy pixels (

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.

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.

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

The multivariate Gaussian latent random field (

In this model, the advection of rainstorms is assumed to be constant and
linear along a vector

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:

For generating the synthetic images, we use the stochastic rainfall model
described in Sect.

Parameters of the stochastic rainfall model used for the generation of synthetic images.

Identification of non-stationarities in rain statistics for a
synthetic case study.

In Fig.

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.

Observation dataset used for validation.

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 (

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.

Reproduction of the cumulative rain height (averaged over the whole
network) for the three tested hypotheses.

Reproduction of rainfall statistics for the three tested hypotheses.

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. (

Contrary to H1, hypothesis H2 leads to a slight overestimation of the
simulated rain height, in particular for the first 30 days
(Fig.

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.

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

The classification framework presented in Sect.

The automatic selection of the number of Gaussian mixtures used in the GMM
model leads to 11 rain types for this year. Figure

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.

The distributions of the space–time indices displayed in
Fig.

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

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.

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.

Correspondence between rain types derived from two distinct calibration datasets.

Figure

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.

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

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 (

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.

The succession of rain types identified in
Sect.

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.

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

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.

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

The source code used for this study is freely available
on the following repository:

LB, MV and GM designed the study. LB implemented the MATLAB utilities and performed the numerical experiments. LB wrote the paper with input and corrections from MV and GM. Edited by: Carlo De Michele Reviewed by: Nadav Peleg and one anonymous referee

The authors declare that they have no conflict of interest.