In this study we propose and demonstrate a data-driven approach in an “information-theoretic” framework to quantitatively estimate precipitation. In this context, predictive relations are expressed by empirical discrete probability distributions directly derived from data instead of fitting and applying deterministic functions, as is standard operational practice. Applying a probabilistic relation has the benefit of providing joint statements about rain rate and the related estimation uncertainty. The information-theoretic framework furthermore allows for the integration of any kind of data considered useful and explicitly considers the uncertain nature of quantitative precipitation estimation (QPE). With this framework we investigate the information gains and losses associated with various data and practices typically applied in QPE. To this end, we conduct six experiments using 4 years of data from six laser optical disdrometers, two micro rain radars (MRRs), regular rain gauges, weather radar reflectivity and other operationally available meteorological data from existing stations. Each experiment addresses a typical question related to QPE. First, we measure the information about ground rainfall contained in various operationally available predictors. Here weather radar proves to be the single most important source of information, which can be further improved when distinguishing radar reflectivity–ground rainfall relationships (

Quantitative precipitation estimation at high temporal and spatial resolution and in high quality are important prerequisites for many hydrometeorological design and management purposes. Besides rain gauges that have their own limitations

While the advantage of weather radar is that it provides 3-D observations at a high spatial and temporal resolution and with large coverage, unfortunately its use relies on some assumptions, which are sometimes justified and sometimes not. It is further hampered by considerable error and uncertainty arising from measuring the radar reflectivity factor

In this paper we will focus on the first two aspects, namely the

In this context, it is the aim of this paper to suggest and apply a framework which would use relationships between data expressed as empirical discrete probability distributions (dpd's), and would measure the strength of relations and remaining uncertainties with measures from information theory. Comparable approaches have been suggested by

In particular, we investigate the effect of applying a purely data-based, probabilistic

The remainder of the paper is structured as follows: in the next section, we briefly present the experiments carried out in the paper; in Sect.

We conduct a total of six experiments. In Experiment 1, we investigate the information on ground rainfall contained in various predictors, such as weather radar observations alone or in combination with additional, operationally available hydrometeorological predictors. In Experiment 2, we investigate the effect of limited data on the uncertainty of ground rainfall estimation for various data-based models. In Experiment 3, we examine the degree to which the empirical

Since its beginnings in communication theory and the seminal paper of Claude Shannon

Please note that while the concepts of IT are universal and apply to both continuous and discrete data and distributions, we will, for the sake of clarity and brevity, restrict ourselves to the latter case and work with discrete (binned) probability distributions throughout all experiments.

The most fundamental quantity of IT, information

Information can be described as the property of a signal that effects a change in our state of belief about some hypothesis (

Compared to working with probabilities, using its log-transforms – information – has the welcome effect that counting the total information provided by a sequence of events is additive, which is computationally more convenient than the multiplicative treatment of probabilities.

Information entropy

While information is a function of the occurrence probability of a particular outcome only, entropy is influenced by and is a measure of the shape of the entire pdf. Therefore, a pdf with only a single possible state of probability

Like the variance of a distribution, entropy is a measure of spread, but there are some important differences: while variance takes the values of the data into account and is expressed in (squared) units of the underlying data, entropy takes the probabilities of the data into account and is measured in bit. Variance is influenced by the relative position of the data on the measure scale and dominated by values far from the mean; entropy is influenced by the distribution of probability mass and is dominated by large probabilities. Some welcome properties of entropy are that it is applicable to data that cannot be placed on a measure scale (categorical data), and that it allows comparison of distributions from different data due to its generalized expression in bit.

So far, we know that entropy is as a measure of the expected information of a single distribution. We could also refer to this as a measure of self-information, or information we have about individual data items when the data distribution is known. If we do not only have a data set

If

Entropy and conditional entropy measure information contained in the shape of distributions and the underlying data. Their calculation depends on prior knowledge of a pdf of these data and implies the assumption that this pdf is invariant and representative. In practice, this condition is not always fulfilled, either because we construct the pdf from limited data or because the system generating the data is not invariant. In these cases we work with approximations, or models, of the pdf, which means we estimate the information attached to a signal based on imperfect premises, which we in turn pay for with increased uncertainty. Cross entropy

If both the entropy of a distribution and the cross entropy between the true distribution and an approximation (a model) thereof are known, we can separate these two components of total uncertainty: uncertainty due to the shape of the true distribution, and uncertainty because we do not know it exactly. The latter is measured by the Kullback–Leibler divergence

Suppose we have a data set of many repeated, joint observations of several variates (data tuples), e.g., a time series of joint observations of

With range and binning chosen, we can map the data set into a multivariate, discrete frequency distribution as shown in Fig.

Illustration of a bivariate conditional probability distribution as a simple data-based predictive model.

So far we have used entropy to measure uncertainty as if we have had to guess all values of a data tuple randomly drawn from an a priori known distribution. However, often we do not only know the distribution a priori, but we also have knowledge of the values of parts of the data tuples (e.g., we know

In short, we can consider a dpd constructed from a data set as a “minimalistic predictive model”, which, given some values for the predictor variates, provides a probabilistic prediction of the target variate. It is minimalistic in the sense that it involves only a small number of assumptions (representativeness of the dpd) and user decisions (choice of the binning scheme). Compared with more common modeling approaches, where relations among data are expressed by deterministic equations, the advantage of a dpd-based model is that it yields probabilistic predictions in the form of conditional dpd's, which include a statement about the target value and the uncertainty associated with that statement. Also, as dpd-based modeling does not involve strong regularizations, it reduces the risk of introducing incorrect information. When building a standard model, choosing a particular deterministic equation to represent a data relation, which is a common form of regularization, involves the risk of introducing bias, and it ignores predictive uncertainty. In this context,

Things are different, however, if we apply the model to a new situation, i.e., if we construct the model from one data set and use it with predictor data of another under the assumption that the predictor–target relation expressed by the model also holds for the new situation. This is clearly only the case if the learning and the application situation are identical; however, in most cases the situations will differ and we will pay for this inconsistency with additional uncertainty. We can measure this uncertainty using the Kullback–Leibler divergence (Eq.

We will apply this approach to construct and analyze predictive models throughout the paper, and use it to learn about the information content of various predictors about our target, ground rainfall

Expressing predictions by probability distributions, and expressing uncertainty as “information missing” as described in the previous sections has the advantage that we can build default models providing lower and upper bounds of uncertainty, which we can then use as benchmarks to compare other models against. The smallest possible uncertainty occurs if the predictive distribution of the target is a Dirac, i.e., the entire probability mass is concentrated in a single bin. In such a case, irrespective of the number of bins covering the value range, the entropy of the distribution is zero:

Given that we successfully avoid infinite Kullback–Leibler divergence, the worst thing that can happen is that our predictions are completely uninformative, i.e., we provide a uniform distribution across the entire value range of the target. In this case, the entropy of the distribution is equal to the logarithm of the number of bins:

Finally, for the special cases where we know that the model we use is perfect (typically because we apply it to the same situation it was constructed from), we know that Kullback–Leibler divergence is zero, and total uncertainty equals conditional entropy. In this case, we can state another upper bound for uncertainty: if at worst the predictors we use are completely uninformative, conditional entropy will be equal to the unconditional entropy of the target distribution (see Eq.

We will use these lower and upper bounds for total uncertainty throughout the experiments to put the performance of the tested models into perspective.

In experiments 2 and 3 we investigate the effect of limited sample size, i.e., the information loss (or uncertainty increase) if we do not construct a model from a full data set, but from subsets drawn thereof. This corresponds to the real-world situation of building models from available, limited data. For the sake of demonstration, we assume in the experiments that a long and representative reference data set is available for evaluation. While this is clearly not the case in real-world situations, we can get answers from such experiments for practically relevant questions such as “what is a representative sample, i.e., how many observations are required until a model built from the data does not change further with the addition of observations?” or “should we build a model from locally available but limited data, or should we use a model learned elsewhere, but from a large data set?”.

Throughout the experiments, we apply simple random sampling without replacement to take samples from data sets. In order to reduce effects of chance, we repeat each sampling 500 times, calculate the results for each sample and then take the average.

This study uses data from various sources collected during a 4-year period (1 October 2012–30 September 2016) within the CAOS (Catchments As Organised Systems) research project. For more detail on project goals and partners see

The position of the Attert catchment in Luxembourg and Belgium with superimposed orography (in m a.s.l.) and the locations of the MRR's, disdrometers, rain gauges, the synoptic station (ASTA) and radar site (Neuheilenbach), and the rawinsonde launching (sounding) station (Idar-Oberstein – WMO-ID 10618) as well as the scale and orientation of the small-scale map. © OpenStreetMap contributors 2019. Distributed under a Creative Commons BY-SA License.

Summary of the raw data used in the experiments: description, summary statistics and binning.

The project was conducted in the Attert Basin, which is located in the central western part of the Grand Duchy of Luxembourg and partially in eastern Belgium (see Fig.

The study area is situated in the temperate oceanic climate zone (Cfb according to the Köppen–Geiger classification;

In our study, we investigated both the effects of location and season on drop-size distributions and ultimately on QPE.

We used 10 min reflectivity data from a single polarization C-band Doppler radar located in Neuheilenbach (see Fig.

We also used drop size spectra measured at 1500 and 100 m above ground by two vertical pointing K-band METEK micro rain radars (MRR)

We also deployed six second-generation OTT particle size and velocity

Next to the rain rate retrieved from the disdrometer data, we also used additional observations from standard tipping-bucket rain gauges at the Useldange, Roodt and Reichlange sites (Fig.

In addition to direct observations of precipitation and drop size spectra, we collected a number of operationally available data to test their value as additional predictors for QPE. We selected standard meteorological in situ observations such as 2 m temperature, relative humidity, zonal and meridional wind speeds, season indicators such as month and tenner-day of the year as well as synoptic indices such as convective available potential energy (CAPE) and classified circulation pattern. The latter two are operationally provided by the German Weather Service. All predictors are listed in Table

The in situ observations were taken from ASTA station Useldange (Fig.

With respect to classified circulation patterns, the German Weather Service provides a daily objective classification of the prevailing circulation pattern over Europe into one of 40 classes

Using all of these data in the methods as described in Sect.

Rainfall is an intermittent process, and quite expectedly most of our 4-year series contained zeroes for rainfall. If we had used this complete data set for analysis, the results would have been dominated by these dry cases; however, these cases were not what we were interested in. Therefore, we applied a data filter to select only the hydro-meteorologically relevant cases with measurements from all available stations and at least two rain gauges showing rainfall

In Fig.

Discrete probability distributions of the most important variables used in the experiments (filtered data set). The description of variables and binning can be found in Table

In this experiment we explore the information content regarding ground rainfall

We used ground rainfall observations from the eight rain gauges in the test domain as target data, filtered the raw data with the “minimum precipitation” filter and created models using various predictor combinations. We measured the usefulness of the predictors with entropy and conditional entropy (Eqs.

Entropy of the target (RR0) and the benchmark distribution (RR0 uniform), conditional entropies of the target given one or several predictors, non-normalized and normalized. Underlying data are from the filtered data set. Predictors are ordered by descending conditional entropy.

There are two upper benchmarks we can use to compare the different QPE models against: unconditional entropy if we know nothing but the binning of the target and use a uniform (maximum entropy) distribution for prediction (case 1 in Table

The next important source of information is radar reflectivities: if we use it as a single predictor (case 3), uncertainty is reduced by 0.28 bit or 15.3 % compared with RR0. Note that this approach directly applies the reflectivity data provided by the radar, no side information was added nor existing information in the data destroyed by applying an additional

Using each of the other predictors separately did not reduce uncertainty much (not shown); therefore, we only show results for the cases where they were applied as two-predictor models in combination with radar reflectivity (cases 4 to 11). If we compare the relative conditional entropies of their predictions to those of the radar-only model, we can see that neither the ground meteorological observations nor CAPE contained much additional information (cases 4 to 8). Instead, the three most informative models (cases 9 to 11) either distinguish the relation between

Based on these results, we built and evaluated three-predictor models only with combinations of these relatively informative predictors (cases 12 and 13). The information gain from using three predictors in combination is considerable, and it reduced uncertainty to 68.4 % (case 12) and 62.6 % (case 13) compared with the “target-distribution-only case” (case 2). The benefit of applying a season-dependent and circulation-pattern-dependent relation between

Entropy of the unconditional target distribution (filtered data set, black line), entropy of the benchmark uniform distribution (filtered data set, red line), cross entropies between conditional distributions of the target given one, two and three predictors of the filtered data set and samples thereof (green, purple, yellow and dashed yellow lines, respectively). The “

An obvious conclusion from these findings would be to build better models by simply adding more predictors, which according to the information inequality equation (Eq.

The data base and filter used here are identical to the previous experiment, so a set of 11 984 joint observations of the target (rain rate at the ground) and predictors (radar reflectivity, circulation pattern, tenner-day and month of the year) were available. From these data we built and tested a total of four predictive models, as described in Experiment 1: a one-predictor model applying radar reflectivity only, two two-predictor models applying radar reflectivity and tenner-day or month of the year, and a three-predictor model using radar reflectivity, month of the year and circulation pattern. The difference to Experiment 1 is that we now do not only apply the entire data set but also randomly drawn samples thereof to build the model (see Sect.

The results are shown in Fig.

On the right margin of Fig.

We can interpret the lines in Fig.

The learning behavior of the models, which differs with the number of predictors used, is a manifestation of the “curse of dimensionality”, and visual examination of learning curves of different models as plotted in
Fig.

In this experiment we investigate the information content in spatial position by learning and applying site-specific relations between radar reflectivity and ground rain rate (in the previous experiments we applied them in a spatially pooled manner). We used data filtered with the “minimum precipitation” filter again, so a set of 11 984 joint observations of ground rain rate and radar reflectivity were available. We included spatial information by simply using the ID number of each station (“Statnum” in Table

Entropy of the unconditional, target distribution (filtered data set, black line, same as in Fig.

The red and black horizontal lines are the same as in Fig.

Overall, the information gain of using site-specific

In this experiment we evaluate the effect of functional compression by measuring the information loss when using a deterministic function to express the relation between radar reflectivity and ground rain rate instead of the empirical relation derived from the data. As before, we used all joint observations of radar reflectivity and ground rainfall passing the “minimum precipitation” filter. Each data pair is shown as a blue dot in Fig.

Relation between target RR0 and predictor dBZ1500Rad. Empirical relation as given by the filtered data set (blue dots). Deterministic power-law relation according to

Let us suppose we would not have been in the comfortable situation of having joint observations of target and predictor to construct a data-based model, or suppose it would take too much storage or computational resources to either store or apply such a model. In these cases, it could be reasonable to approximate the “scattered” relation as contained in the data either using a deterministic function gained from other data or a deterministic function fitted to the data. In fact, this is standard practice. Expressing a data relation by a function drastically reduces storage space, is easy to apply and preserves the overall relation among the data. However, what we lose is information about the strength of that relation as expressed by the scatter of the data. Instead, when applying a deterministic function we claim that the predictive uncertainty of the target is zero.

Our aim here is to quantify the information loss associated with such deterministic functional compression. Applying a deterministic model is in principle no different than using a model learned from a subset of the data, as we did in Experiment 2: we use an imperfect model, which results in additional uncertainty which we can measure via Kullback–Leibler divergence between the true data relation and the model we apply (see Eq.

For demonstration purposes we applied two typical deterministic

As described above, we additionally used our information-based approach and calculated the conditional Kullback–Leibler divergence between the predictive distributions given by the empirical (perfect) and the deterministic (imperfect) models for all available data. From Eq. (

For our data, conditional Kullback–Leibler divergence for the Marshall–Palmer model was 3.43 bit, and it was 2.69 bit for the optimized model. Added to the conditional entropy of the empirical

In this experiment we explore how the information content about ground rainfall in reflectivity observations is related to the measurement position along a vertical profile above the rain gauge, and how it is related to the measurement device. To this end, we used data from two sites, Petit-Nobressart and Useldange (see Fig.

In contrast to the previous experiments, we were restricted to two (instead of eight) sites with MRR's installed. This made the application of the standard “minimum precipitation” filter (see Sect.

As in Experiment 1, we used entropy and conditional entropy to measure the information content of the available predictors and added the entropy of both a uniform and the observed distribution of ground rainfall as benchmarks. The results are shown in Table

Entropy of the target (RR0) and the benchmark distribution (RR0 uniform), conditional entropies of the target given various predictors along the radar path, non-normalized and normalized, for Petit-Nobressart and Useldange, respectively. Underlying data are from the filtered data set. Predictors are ordered by decreasing distance to the target.

Weather radar data, even if they are measured at distance and at height contain considerable information about ground rainfall: using them as predictors reduced uncertainty by 15.7 % (100 % to 84.3 %) compared with the benchmark entropy of the target distribution (Table

So why go to the extra trouble of operating an MRR? The advantages of this instrument are evident when moving down to elevations inaccessible by weather radar. Changes in drop size distribution along the pathway of rainfall from cloud to ground can be considerable, and the closer to the ground the observation is taken the stronger its relation to ground rainfall: using MRR data from the lowermost bin (

Further information gains can be achieved when measuring reflectivity directly at the ground: using the disdrometer measurements further reduced uncertainty by 7.5 % (73.8 % to 66.3 %, Table

In this final experiment we compare two methods of QPE, radar-based and rain-gauge-based, and additionally explore the benefits of jointly tapping both sources of information. As we were not dependent on the availability of MRR data as in the previous experiment, we could again make use of the full eight-site set of observations filtered with the “minimum precipitation” filter.

We used the same data-based approach of constructing empirical dpd's as a predictive model as in all previous experiments; the only difference between the three tested QPE models was the type of predictor used: for the radar-based QPE, we used weather radar observations at 1500 m above ground in a one-predictor model to predict ground rainfall at the same site. This is the same approach that we applied in Experiment 1. For the rain-gauge-based QPE, we built a one-predictor model based on a straightforward approach comparable to leave-one-out cross validation: each of the eight available stations (see Fig.

For the joint QPE model we extended the approach used for rain gauge interpolation to a two-predictor model: again each of the eight available stations was used once as a target station; however, in this case, not only were observations from each of the respective remaining stations used as predictors, but radar observations measured at height above the target station were also applied. This is a typical approach when merging radar and rain gauge data for QPE: we use data from rain gauges observed at a horizontal distance from the target, and radar data observed at a vertical distance from the target.

As we built and compared models with different numbers of predictors in this experiment, and as the models for each particular target station were built from a subset of the data only, it could be worthwhile exploring the additional uncertainty due to the effect of sample size here as in experiments 2 and 3. However, for this particular experiment we found it more useful for the reader (and us) to discuss results as a function of the distance between stations, as it provides a link to the large body of literature on spatial rainfall structure analysis and station-based rainfall interpolation. The results are shown in Fig.

Entropy of the unconditional, target distribution (filtered data set, black line, same as in Fig.

Just as in Figs.

However, for the QPE model based on rain gauge observations (light blue line), the interpolation distance does play a role: as is to be expected, the smaller the distance between the target and the predictor station, the higher the information content of the predictor and the smaller the conditional entropy, i.e., the blue line rises from left to right. For short distances between 2 and 4 km, conditional entropy is 1.34 bit, which is lower than for the radar-only QPE. If we take the unconditional entropy of the target again as a reference as in Experiment 1 (Table

The above results indicate that each of the two QPE methods has its particular strengths. In other words they add nonredundant information; therefore, we can expect some benefits when joining them in a two-predictor model. This is indeed the case if we take a look at the related conditional entropies plotted as the pink line in Fig.

In Experiment 1 we also built and tested two-predictor models. The best among them, using radar data and tenner-days of the year for predictors (Table

Reliable QPE is an important prerequisite for many hydrometeorological design and management purposes. In this context we pursued two aims with this paper: the first was to suggest and demonstrate a probabilistic framework based on concepts of information theory, in which predictive relations are expressed by empirical discrete probability distributions directly derived from data. The framework allows for the integration of any kind of data deemed useful and explicitly acknowledges the uncertain nature of QPE. The second aim was to investigate the information gains and losses associated with various data and practices typically applied in QPE. For this purpose we conducted a total of six experiments using a comprehensive set of data comprising 4 years of hourly aggregated observations from weather radar, vertical radar (MRR), disdrometers, rain gauges and a range of operationally available hydrometeorological observables, such as large-scale circulation patterns, ground meteorological variables and season indicators.

In Experiment 1, we measured the information on ground rainfall contained in various operationally available predictors with entropy and conditional entropy. Weather radar proved to be the single-most important source of information, which could be further improved by distinguishing

In Experiment 2, we tested the robustness of QPE models developed in the previous experiment by measuring the additional uncertainty due to limited learning data with Kullback–Leibler divergence and cross entropy. The main lesson learned here was that this added uncertainty is strongly dependent on the number of predictors used in the model, and that for unfavorable constellations (multiple-predictor models learning from small samples) this effect quickly dominates total uncertainty: this is the well-known “curse of dimensionality”. For the data set used in this study, we found a two-predictor model using a month-specific relation between radar reflectivity and ground rainfall to provide the best trade-off between performance and robustness.

In Experiment 3, we examined the degree to which the empirical

In Experiment 4, we evaluated the effect of functional compression, i.e., the information loss when replacing the data-inherent probabilistic relationship between reflectivity and rain rate by a deterministic functional approximation. We measured the additional uncertainty with Kullback–Leibler divergence, which adds to the uncertainty caused by the incomplete information of the predictor regarding the target measured by conditional entropy. We used two standard power-law

In Experiment 5, we investigated information gains along the radar path using weather radar, MRR and disdrometer data from two sites. The main insights here were that the information content of radar reflectivity measured at height (1500 m above ground) does not differ much between weather radar and MRR, but considerable additional information is gained by using observations from lower elevations (100 m above ground), thereby avoiding information losses due to changes in drop size spectra along the pathway of precipitation from cloud to ground. This emphasizes both the importance of VPR corrections for accurate QPE and of the required MRR observations. Despite these information gains, considerable uncertainty remained even when using as predictor reflectivity observations taken at the same spot and by the same device (disdrometer) as the target variable (rain rate) itself. This indicates a somewhat ambiguous relationship between radar reflectivity and rain rate as measured by the disdrometer, which could potentially be sharpened by taking the precipitation type into account.

In the last experiment, Experiment 6, we built QPE models based on radar data only, rain gauge data only and a combination thereof and evaluated their information content as a function of the distance between the target and predictor rain gauge. Comparing the first two revealed that a separation distance of

We would like to emphasize that the results from these experiments are partly contingent on the choice of the data filter: the “minimum precipitation” (at least two rain gauges with rainfall

Quantitative statements about the information content of particular predictors or the relative performance of competing QPE models may differ according to these user choices. However, we would like to point out that all of the different experiments in this study could be formulated, conducted and evaluated in a single framework and relied on a single property – information. Therefore, it is not so much the particular results we want to emphasize here, but rather the probabilistic, data- and information-based framework we applied. By its probabilistic concept, it explicitly acknowledges the uncertain nature of QPE, and by expressing probabilities in terms of information, it facilitates both interpretation and computation. In this framework, predictive relationships are directly derived from data and are expressed as discrete probability distributions. The advantage of this is that this avoids the introduction of unwanted side information as much as possible, e.g., by parametric choices, and it avoids the deletion of existing information, e.g., by data transformation or lossy compression. Altogether, this facilitates tracking sources and sinks of information. However, these advantages come at a price: learning robust data-based relations requires a considerable amount of available target and predictor data, and applying them for predictions is computationally more expensive than using deterministic functions.

The reflectivity and rain rate data measured by the six disdrometers and two MRRs are published by the GFZ data service repository

UE and MN directly contributed to the conception and design of the work, collected and processed the various data sets, conducted a thorough data quality control, undertook the data analysis and interpretation, and wrote the paper.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Linking landscape organisation and hydrological functioning: from hypotheses and observations to concepts, models and understanding (HESS/ESSD inter-journal SI)”. It is not associated with a conference.

We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) and the Open Access Publishing Fund of Karlsruhe Institute of Technology (KIT). This research contributes to the “Catchments As Organized Systems” (CAOS) research group funded by the DFG. We gratefully acknowledge the fruitful cooperation and joint operation of the meteorological observation network within CAOS, namely Hans-Stefan Bauer and Volker Wulfmeyer. We also gratefully acknowledge the providers of meteorological data used in this study: the German Weather Service (DWD) for radar and classified circulation pattern data, the Administration des services techniques de l'agriculture (ASTA) for rain gauge and climate data and the Hydrometry Service Luxembourg for rain gauge data. Furthermore, we especially thank Jean-François Iffly from the Luxembourg Institute of Science and Technology (LIST) for his extensive commitment to finding the right locations to place the instruments and his outstanding help with maintaining them. We further thank Kirsten Elger from GFZ Helmholtz Centre Potsdam for her patient help regarding data publication in the GFZ repository; our colleagues Ralf Loritz, Sibylle Haßler, Conrad Jackisch and Erwin Zehe for fruitful discussions on data analysis; and Jan Handwerker from the KIT Institute of Meteorology and Climate Research for help concerning radar meteorology issues. Last but not least, we thank the rest of the CAOS team for the warm-hearted project meetings and discussions.

The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Alexander Gelfan and reviewed by two anonymous referees.