Various fields of application, such as risk assessments of the insurance
industry or the design of flood protection systems, require reliable
precipitation statistics in high spatial resolution, including estimates for
events with high return periods. Observations from point stations, however,
lack of spatial representativeness, especially over complex terrain. Current
numerical weather models are not capable of running simulations over
thousands of years. This paper presents a new method for the stochastic
simulation of widespread precipitation based on a linear theory describing
orographic precipitation and additional functions that consider synoptically
driven rainfall and embedded convection in a simplified way. The model is
initialized by various statistical distribution functions describing
prevailing atmospheric conditions such as wind vector, moisture content, or
stability, estimated from radiosonde observations for a limited sample of
observed heavy rainfall events. The model is applied for the stochastic
simulation of heavy rainfall over the complex terrain of southwestern Germany.
It is shown that the model provides reliable precipitation fields despite its
simplicity. The differences between observed and simulated rainfall
statistics are small, being of the order of only

Severe pluvial flood events resulting from persistent rainfall over large
areas have the potential to cause high economic losses of several billion
euros (EUR) in central Europe. In Germany, the two extreme floods of 2002
and 2013 with estimated return periods of more than 200 years

To account for the former issue, geostatistical interpolation routines, such
as kriging

To account for the limited observation period, several studies have employed
stochastic weather generators to simulate precipitation events at single grid
points

Furthermore, robust estimates of precipitation extremes with high return periods, for example, for an event happening once in 200 years, require a large sample size of several thousands of events. Current numerical weather prediction (NWP) models, though having a high spatial resolution of several kilometers, are not able to simulate thousands of events due to their complexity and the resulting high computing costs.

In this study we present a semi-physical two-dimensional stochastic
precipitation model (SPM2D), which was designed for the stochastic simulation
of a very large number of precipitation fields in high spatial resolution. It
is based on the linear theory for orographic precipitation by

The paper is structured as follows. Sect. 2 introduces the basic concept of the SPM2D. Section 3 briefly describes the data sets used in this study. Section 4 presents the results of the calibration based on a set of 200 representative historical heavy rainfall events and examines sensitivities of the model depending on varying ambient conditions. Section 5 shows some characteristics of the selected events. Results of the stochastic simulations are discussed in Sect. 6, and Sect. 7 lists some conclusions. Further information is given in a short Appendix and Supplement.

The SPM2D model is designed for the simulation of widespread, pluvial
precipitation over complex terrain such as low mountain ranges, a typical
feature of European topography. The investigation area for this study is the
federal state of Baden-Württemberg (BW) in southwestern Germany
(Fig.

Topographic map (in meters above mean sea level; m a.m.s.l.)
of southwestern Germany and surrounding areas with
main river networks and lakes as well as substantial orographic structures.
The national borders (slim solid black contours) and the border of the
federal state of Baden-Württemberg (bold solid black contour) are shown,
as well as the model domain (red box), which extends from 46.6 to
50.8

As described in the following Sect.

After a description of the model components (Sect.

Overall, the model SPM2D quantifies total precipitation

As the SBM does not reliably reproduce the observed spatial variability of
precipitation also over flat terrain for physical reasons

The linear orographic SBM

The SBM is based on the linear theory of three-dimensional (3-D) stratified,
hydrostatic flow over mountains with uniform incoming horizontal wind speed
and stability

One of the key components of the linear model is a pair of linear
steady-state equations for the advection of vertically integrated cloud water
and hydrometeor density during characteristic timescales of cloud water
conversion

A powerful method for the solution of the advection equations for cloud
physics together with 3-D flow is to apply a two-dimensional (2-D) Fourier
transform. Precipitation at the ground is obtained via an inverse FFT given
by the transfer function:

Whereas the nominator of Eq. (

The vertical wavenumber

Combining Eq. (

Under the assumption that the prevailing synoptic conditions during the
12 h of model integration are approximately horizontally homogeneous and
stationary,

Probability of observed 24 h rainfall totals greater than 50 mm,
expressed as the average days per year for Baden-Württemberg; the black
box indicates the area where background precipitation

As further development, two types of modifications are applied to the SBM: adjustments to the existing orographic precipitation using additional calibration parameters and additional precipitation components originating from different physical processes.

The original orographic precipitation equation of the SBM
(Eq.

Different effects of the implemented internal free parameters

The uplift sensitivity factor

An additional parameter,

Finally, the last additional calibration parameter,

Apart from large-scale lifting connected to low-pressure systems or waves in
the flow patterns, precipitation is also substantially enhanced by
synoptic-scale weather fronts. Active fronts may increase precipitation
considerably due to cross-frontal circulations and lifting in the warm sector
of a cyclone

In order to estimate

The frontal enhancement factor is a function of space realized by a
rectangular area

Schematic of a Gaussian-shaped distribution of the frontal
enhancement factor with a maximum of

Embedded convection mainly occurs when lifting is locally enhanced at
mid- and upper-tropospheric levels, leading to a decrease of thermal stability by
the release of latent heat of condensation

Because embedded convection is mainly induced by orographic lifting, we
implemented a multiplicative factor to the precipitation fields related to
both orographic and large-scale lifting, similar to the frontal part:

Schematic of embedded convection implementation by using rectangular
cells (blue). The orientation is defined by the wind direction (arrow); each
cell is assigned to an individual factor

As embedded convection occurs several times a day and at several locations, we used a variable number of convective footprints in the model. The complete convective precipitation field for each time step is spatially smoothed to avoid sharp gradients. For this, we applied a moving average of 10 grid points to preserve the high spatial variability of convection.

Stochastic modeling of precipitation events with the SPM2D is based on appropriate PDFs of all input parameters required by the model. These PDFs are estimated using an adequate set of representative past heavy rainfall events. Because the characteristics of the ambient conditions and thus the precipitation regimes change throughout the year, we seasonally differentiate the estimated PDFs between spring (March–April–May – hereafter MAM), summer (June–July–August – JJA), autumn (September–October–November – SON), and winter (December–January–February – DJF).

In the first step, a sufficient and appropriate subset of relevant historic
events was identified. Here, an event is defined as a period of 1 or more
days with persistent precipitation above a certain threshold of daily totals.
An extension to multi-day events is reasonable for considering time delays in
discharge response or flood waves traveling along river networks

We define the historic event set according to maximum areal precipitation.
For this, we simply accumulate the (equidistant) 24 h rainfall
totals

Flow chart of the individual components of the SPM2D (solid boxes) and the corresponding input variables (PDFs; dashed boxes). Loops are highlighted as ellipsis or bold dashed arrows. The constant model parameters are illustrated as the shaded box.

Event precipitation starts on the first day that exceeds

In the next step, we identified the PDFs most appropriate for statistically
describing each of the seven atmospheric input parameters, the event
duration

To find the PDF that best fits the data, we estimated the appropriate number
of histogram classes according to

The presented SPM2D is an event-based model in the sense that a specified
number

The SPM2D presented in this study is based on two different types of data sets: gridded precipitation data also used to calibrate and verify the model and vertical profiles from radiosondes to initialize the model. Furthermore, the SPM2D is also validated using reanalysis.

Rainfall statistics in our study are based on the regionalized precipitation (REGNIE)
data provided by the German Weather Service (DWD). REGNIE is a
gridded data set of 24 h totals (06:00 to 06:00 UTC) based on several thousand
climate stations more or less evenly distributed across Germany (so-called RR
collective). The REGNIE algorithm interpolates the observations to a regular
grid of 1 km

In our study, we use REGNIE data from 1951 to 2016 to identify the top200
event set (see previous section); to estimate the duration of the events, the
front factor

The seven input parameters (see Sect.

The vertical profiles, provided by the Integrated Global Radiosonde Archive (IGRA)
from the National Climatic Data Center

The stochastic generation of enhanced precipitation streaks associated with
embedded convection, namely their length and width (

The SPM2D simulation results are additionally validated with high-resolution
reanalysis from the non-hydrostatic Consortium for Small-scale Modeling (COSMO)
model in climate mode

The SPM2D is calibrated with the top200 events (training data) by adjusting
the free model parameters

In order to determine appropriate values of the free parameters, a large
number of model simulations were carried out. Whereas one parameter was
successively varied, the others were kept constant. The selected ranges and
increments of the parameters listed in Table

Range of values of the free model parameters used for the calibration of the model.

The model skill was evaluated using the skill score

Applying the method as described above, the highest value for

The sensitivity of the skill score

To demonstrate how variations of atmospheric conditions translate into
precipitation, we conduct a sensitivity study with the SBM

Areal mean precipitation (24 h totals; median of the top200 event
set) as a function of

Comparison of

Mean precipitation shows a high sensitivity to changes in

Qualitatively similar behavior to the model is found for the medians of
RMSE and skill score

The skill score

After the parameter adjustment, the SBM

On that day, a pronounced low-pressure system with its center over Croatia
led to the sustained advection of moist air masses from northerly directions
around 20

Overall, the SBM

One reason for the discrepancy between observed and simulated precipitation
might be the suboptimal location of the Stuttgart sounding used for the model
initialization. The sensitivity study as described in
Sect.

Changes in

After separating the historic event set into the four main seasons, we
estimate, for each of the 10 input parameters, the PDF that best fits the
distribution of the observations (Table

Estimated best-fitting PDFs for event duration (

The input parameters are considered as independent and uncorrelated. To
justify this assumption, we performed a correlation analysis of all possible
combinations of input parameters using the

The histogram of the duration

Histogram of top200 event duration for Baden-Württemberg according to REGNIE (bars), and estimated best-fitting PDFs (dotted lines) for the summer (blue) and the winter (red).

Concerning

An overview of the range of the seven input parameters of the model is shown
as box plots in Fig.

Atmospheric parameters required as input for the SPM2D derived from radio sounding observations at Stuttgart for the top200 events with mean, interquartile distance, minimum, and maximum values; the left whisker of each pair represents the summer, and the right one represents the winter season. The units for each variable are given in the brackets below the variable names.

Horizontal wind speed

Precipitation fields for the median of

Overall, a total number of

Spatial 24 h mean values range between 1.2 and 79.7 mm in the SPM2D, and
1.3 to 97.0 mm in the SBM

Both the median and the 90th-percentile (p90) precipitation fields of the
top200 event set and the SPM2D agree well concerning the spatial distribution
and the precipitation amounts (Figs.

Precipitation fields for the 90th percentile (p90) of

The areal rainfall of the SPM2D median (Fig.

The areal rainfall for the p90 field (Fig.

For other percentiles the differences between REGNIE and the SPM2D are very
small for both the spatial mean and the maximum precipitation at any grid
point in the model domain (Fig. S3a). The differences become considerable
only above the 95th percentile. The SPM2D tends to overestimate lower
precipitation amounts because the minimum values at any grid point are higher
in the model than in the observations and invert for the 99th percentile
only. In contrast, the differences between the SBM

To estimate precipitation distributions for specific return periods, we fit a
Gumbel PDF to the annual maximum series of both REGNIE and the SPM2D. As it is
not possible to directly estimate the time period and a corresponding annual
maximum series for the stochastic event set, we count the number of
stochastic values exceeding the 99th percentile of observations

For a 10-year return period, the SPM2D shows only small differences in REGNIE
of less than

Relative difference of the precipitation amounts for return periods
of

On the level of the major river catchments, the differences are small, too.
For the Neckar catchment, for example (Fig.

Single grid point deviations and the ensuing spatial mean values as described
above are sensitive to local conditions and uncertainties in both REGNIE and
the SPM2D. Hence, we evaluate the model in a similar way by calculating spatial
mean precipitation first and then estimating the corresponding return periods
(see Appendix A3). Again, the difference between the SPM2D and REGNIE is small
for entire BW, with slightly lower values from the simulations
(Fig.

Daily rainfall totals (areal means) as a function of return
period

We have presented a novel method for estimating the statistics of heavy
rainfall based on a stochastic model approach. Total precipitation is
calculated from the linear superposition of four different parts: orographic
precipitation, synoptic background precipitation, frontal precipitation, and
precipitation from convection embedded into mainly stratiform clouds. The
linear theory of orographic precipitation according to

The focus of the presented investigations was on the federal state of Baden-Württemberg in southwestern Germany, with the striking low mountain
ranges of the Black Forest and Swabian Jura. The following main conclusions can be drawn:

The SPM2D has high skill for simulating both historic and stochastic heavy rainfall events. The simulated precipitation fields and magnitudes are reliable despite the simplified approach of the model initialized by a set of atmospheric variables obtained from radio soundings. The differences between the SPM2D and REGNIE are small, with deviations of less than 10 %. Local differences, however, may also result from the regionalization procedure of REGNIE, mainly because of the low density of rain gauges over mountainous terrain.

The comparison of the SPM2D with the underlying linear approach of

The solution of the model equations in Fourier space by an FFT allows for the simulation of a large number of events and to operate the model in stochastic mode. Otherwise, the FFT restricts the model domain to a symmetric equidistant and mesoscale extent.

The extent of the model domain, furthermore, has to be limited to ensure the validity of the assumption of spatially homogeneous distributed atmospheric conditions and synoptic forcing. This allows, for instance, the usage of a single vertical profile.

The presented stochastic approach is easily applicable to other investigation areas. Atmospheric variables for the initialization of the model can be estimated either from radio soundings, such as within this study, or from using reanalysis or data from NWP models. Therefore, it can be applied to any region of the world with similar precipitation characteristics, even if there is only a limited number of ground-based observations available.

As shown in our study, the SPM2D is sensitive to perturbations of ambient
conditions. Therefore, high-quality input data, especially of the atmospheric
parameters, are essential. In contrast, the sensitivity of
precipitation and RMSE to changing input parameters is limited in a range of
around

The input parameters can be considered as independent, as just a few cases revealed higher correlations. The sensitivity of the model for these parameters, however, turned out to be weak. Additionally, the correlation coefficients between the model input parameters vary among the seasons.

To transfer the method to another investigation area and future risk assessments, just a few steps are necessary, the first being a proper sample of historical heavy rainfall events. In the next step, the statistics (PDFs) of the prevailing ambient conditions, background precipitation, and duration of the event set have to be calculated. Finally, the non-stochastic part of the SPM2D has to be calibrated by determining appropriate values for the free model tuning parameters.

The output of the SPM2D is a certain number of independent heavy precipitation events and not a continuous time series. We have presented a method for converting this to a equivalent time period, which is mostly necessary for risk assessments, by counting the number of days in the SPM2D above a defined threshold and normalizing it by the corresponding probability of the observations. Using this total time span it is possible to estimate the return period of every single event and a corresponding new PDF. The time between two events is assumed as dry period.

The presented SPM2D is part of the project FLORIS (FLOod RISk estimation for southwestern Germany), which represents a novel risk assessment methodology for an entire domain and not only for single catchments usually considered in the insurance industry. Within the framework of this project, the SPM2D was applied to other federal states in central Germany. The modeled precipitation fields are used as input data for hydrological and hydraulic simulations from which the flood risk can be estimated, for example, those required for an event happening once in 200 years according to the insurance regulation of Solvency II. However, the results of the SPM2D basically can be used for several different applications such as water management or the design of flood protection measures.

The REGNIE data used in this paper are freely available for
research and can be requested at the DWD (

We used the 20 probability density functions (PDFs) preset in the MATLAB
statistical toolbox

List of the tested and suitable PDFs preset in the MATLAB statistical toolbox (the short acronyms in brackets are for further orientation).

In this study we use the skill score

For the estimation of return periods, the annual maximum series with
length

The supplement related to this article is available online at:

MK designed the research. FE performed the research, the modellling, and the analysis. MK was involved in the interpretation of the results. FE mainly wrote the paper. MK extensively edited the paper and provided substantial comments and constructive suggestions for scientific clarification and further improvements.

The authors declare that they have no conflict of interest.

The authors thank a local insurance company for funding the project. We also would like to thank the German Weather Service (DWD) and the Integrated Global Radiosonde Archive (IGRA) for providing different observational data sets and CGIAR-CSI for the orographic data. Special thanks go to James Daniell, Andreas Kron. and Simon Hoellering from the Karlsruhe Institute of Technology (KIT) for constructive discussions during the project and for valuable suggestions for the model development. We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) and open-access publishing fund of KIT. We are grateful to the constructive and very helpful comments and suggestions of the reviewers that helped to improve the scientific quality of this paper. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Jan Seibert Reviewed by: Nadav Peleg and three anonymous referees