## 1. Introduction

Mountain barriers can significantly amplify or weaken preexisting atmospheric disturbances through dynamical processes, such as upslope ascent, leeside descent, and the formation of gravity waves. These processes also modify the amount and spatial distribution of precipitation, leading to enhanced intensities upstream and reduced quantities on the lee side (Roe 2005). If orographically induced ascent dominates synoptic-scale lifting, precipitation could be intensified significantly by up to one order of magnitude (Kunz and Kottmeier 2006b). This may trigger hydrometeor-related hazards, such as floods, landslides, or avalanches. Therefore, accurate knowledge of precipitation statistics over complex terrain is a prerequisite for water and risk management. Besides, advancing our knowledge of the characteristics of orographic precipitation is a condition sine qua non for further improvement of precipitation forecasts.

Interpolation techniques of various degrees of complexity have been employed widely to estimate gridded or area-integrated precipitation from irregularly spaced rain gauge data. State-of-the-art techniques relate precipitation totals to orographic characteristics and atmospheric parameters (e.g., Basist et al. 1994; Daly et al. 1994; Kilsby et al. 1998; Drogue et al. 2002). Other geostatistical methods, such as kriging, consider spatial covariance structures of precipitation (e.g., Faulkner and Prudhomme 1998; Goovaerts 2000), sometimes complemented by terrain and/or climate variables (e.g., Hevesi et al. 1992; Guan et al. 2005). A weakness of these methods results from the lack of representativeness of station data with respect to the surroundings, especially over complex terrain. Furthermore, purely statistical approaches are unable to reflect the dynamical and thermodynamical processes decisive for real precipitation events.

Numerical weather forecast models can basically provide reliable precipitation fields and, for this reason, are frequently used to study the processes of orographic precipitation. Jiang (2003), for example, investigated the interaction between flow stagnation and precipitation by using different numerical models and setups. Colle (2004) studied the sensitivity of orographic precipitation to changing ambient and terrain properties with the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5). In a recent study, Kirshbaum and Smith (2008) examined the influence of temperature and moist stability on orographic precipitation for a set of idealized, convection-resolving simulations with the Weather Research and Forecasting Model (WRF). These numerical models, however, often fail to elucidate and separate the mechanisms and processes decisive for spatial precipitation structures. Sensitivity studies, on the other hand, are limited by high computational costs, in particular, for 3D setups.

*P*is obtained by vertical integration of the condensation rate

*dq*/

_{s}*dt*from the lifting condensation level (LCL) to the cloud top (CT):

*ρ*and

_{w}*ρ*are the densities of liquid water and moist air, respectively, and

*q*is the saturation mixing ratio. The quasi-diagnostic approach according to (1) was applied successfully in several studies of orographic precipitation for single events (Sinclair 1994; Neiman et al. 2002; Barstad and Smith 2005; Kunz and Kottmeier 2006b) and in climatological studies (Thompson et al. 1997; Crochet et al. 2007).

_{s}In this study, two linear orographic precipitation models are applied to both idealized situations and real case studies. Based on the same basic physics, different solution and parameterization schemes are employed in the models. By using linear theory (Smith 1980, 1989), several flow effects evolving over complex terrain, such as the formation of tilting gravity waves or flow passing around an obstacle, are considered. Precipitation on the ground is estimated from condensation rates associated with orographic lifting and additional parameterization schemes for hydrometeor drifting. The main purposes of this paper are to investigate the principal characteristics of orographic precipitation with respect to hydrometeor time scales, background precipitation, and atmospheric conditions; and to test the model skills. Sensitivities of areal precipitation to changing model and atmospheric parameters are elaborated for a set of idealized cases. Spatial precipitation patterns of several stratiform events between 1971 and 2000 for a typical European low mountain region extending from eastern France to southwestern Germany are investigated. The models are calibrated by appropriate estimates of background precipitation and hydrometeor time scales for a sample of orographic precipitation events from a 10-yr calibration period (1971–1980). For a sample of events from a 20-yr application period (1981–2000), the extent to which the models are able to reproduce observed precipitation qualitatively as well as quantitatively without any adjustment is investigated.

The paper is organized as follows. The models are described in section 2. Section 3 presents the different datasets and methods in a brief overview. Sensitivities of orographic precipitation for idealized conditions and a real event are discussed in section 4. Precipitation fields in a climatological context are shown and evaluated in section 5. Findings are summarized and discussed in section 6.

## 2. The models

Results of two different linear precipitation models are evaluated and compared to identify the most decisive mechanisms for quantitative and qualitative precipitation patterns.

### a. The model of PSB

*τ*and fallout

_{c}*τ*written as

_{f}*q*and

_{c}*q*are the cloud water density and hydrometeor density, respectively;

_{f}**v**is the undisturbed incoming horizontal wind vector; and

*S*is the source term defined as the mass flux of precipitation generated by vertical lifting. The conversion rate of cloud water to hydrometeors (

*q*/

_{c}*τ*) is proportional to the cloud water density integrated over a vertical column. The precipitation rate in (3) (

_{c}*P*=

*q*/

_{f}*τ*) is proportional to the hydrometeor column density. If

_{f}*S*is positive because of ascent, then the sink in (2) acts as a source in (3) and, finally, it leads to precipitation. For a narrow ridge or long

*τ*–

_{c}*τ*, hydrometeors generated on the windward side can be advected into the lee where they evaporate in descent regions (negative

_{f}*S*). Because the conversion rate (

*q*/

_{c}*τ*) in (2) is in balance with

_{c}*S*assuming constant advection, negative

*S*due to descent gives rise to evaporation with the same amount as the precipitation rate. This may result in an exaggeration of evaporation downstream of the mountains, as shown by Barstad et al. (2007) for idealized conditions. Because negative precipitation as a net result is unphysical, it is truncated away.

*P̂*(

*k*,

*l*), and the terrain,

*ĥ*(

*k*,

*l*), can be recovered by an inverse Fourier transform to obtain the precipitation field on the ground,

*σ*=

*uk*+

*υl*is the intrinsic frequency with the horizontal wavenumbers

*k*and

*l*and the undisturbed horizontal wind components

*u*and

*υ*;

*H*is the water vapor scale height;

_{w}*N*is the saturated Brunt–Väisälä frequency (Lalas and Einaudi 1974); and

_{m}*P*

_{∞}is the background precipitation due to large-scale lifting. The factor

*C*=

_{w}*ρq*Γ

_{υ}*/*

_{m}*γ*is the uplift sensitivity with the water vapor mixing ratio

*q*, the moist adiabatic laps rate Γ

_{υ}*, and the actual laps rate*

_{m}*γ*, computed from vertical profiles of radiosoundings. To obtain an analytical expression for the transfer function (4), the two characteristic time scales

*τ*and

_{c}*τ*are set as constant over a vertical column and throughout the region.

_{f}The first bracket in the denominator of (4) describes how the source term is modified by airflow dynamics. Here, the vertical wavenumber *m*(*k*, *l*), which depends on wind speed and stratification, controls the depth and tilt of the forced ascent (or descent). The other two brackets describe the shift and—in case of descent—the reduction of precipitation during the advection time scales *τ _{c}* and

*τ*. Note that these two parameters are mathematically analogous and, thus, can be equated.

_{f}### b. The model of PKU

Based on the same basic equations, the model after Kunz and Kottmeier (2006a), PKU, is resolved vertically into 20 layers. It uses simple parameterization schemes for microphysics, and precipitation is computed step-by-step—both in the spatial and frequency domains.

*U*, and saturated Brunt–Väisälä frequency

*N*(Lalas and Einaudi 1974). To apply the lower boundary condition at

_{m}*z*=

*h*instead of

*z*= 0, where

*h*(

*x*,

*y*) is the terrain, the basic equations are formulated for an isosteric (

*υ*= 1/

*ρ*) vertical coordinate according to Smith (1988). The set of equations are rewritten in terms of an undisturbed height coordinate

*z*and the vertical displacement

_{u}*η*which is expressed by the double Fourier integral

*N*

_{m}^{2}≫

*σ*

^{2}),

*m*is given by the dispersion relation

*U*is the undisturbed wind speed perpendicular to the model domain (270°). To account for all wind direction

*β*, the model domain is rotated accordingly. Mass flux of precipitation

*S*is derived from the equation of state, the Clausius–Clapeyron equation, and the first law of thermodynamics assuming adiabatic vertical motions (Haltiner and Williams 1980),

*R*and

_{d}*R*are the gas constants for dry air and water vapor, respectively;

_{υ}*c*is the specific heat of dry air at constant pressure;

_{p}*L*is the latent heat of condensation; and

_{υ}*T*is the ambient temperature. In this formulation,

*S*is directly proportional to the vertical wind speed

*w*. Hence, lifting is associated with a positive mass flux of precipitation, whereas descent causes a negative mass flux, referred to as evaporation.

After condensation, hydrometeors drift some distance within the ambient flow before reaching the ground. The condensed cloud water, *S* [*x*_{1}, *y*_{1}, *z*(*k*)], is assumed to drift on a certain density surface *k* until the hydrometeors are of sufficient size to fall to the ground. This cloud conversion time scale *τ _{c}* is parameterized in terms of height, yielding low time scales (∼100 s) near the ground and higher ones on the order of 1000 s aloft. This value may be treated as a tuning parameter in the model. Once formed, precipitation starts to fall from the position [

*x*

_{2},

*y*

_{2},

*z*(

*k*)] to the surface. The fallout time scale

*τ*is parameterized by the mean terminal velocity assumed to be constant for ice particles with

_{f}*w*= 1.5 m s

_{i}^{−1}and to depend on rain intensity and air density for rain drops (

*w*≈ 4–8 m s

_{r}^{−1}). If hydrometeors encounter a descent region with unsaturated air, then evaporation acts to reduce

*S*. The evaporation rate is assumed to be proportional to the negative flux

*S** (

*x*,

*y*,

*z*). The algorithm is similar to that presented in Barros and Lettenmaier (1994). Background precipitation due to synoptic-scale lifting

*P*

_{∞}, is simply added to the source field

*S*[

*x*,

*y*,

*z*(

*k*)].

The loss of water in a vertical column above the LCL due to upstream precipitation is considered by reducing *q _{s}* in the vertical profile. This yields a reduced

*S*′ (

*x*,

*y*,

*z*) for repeated uplifts downstream of another hill. The water balance thus is fulfilled approximately in the model domain.

## 3. Datasets and methods

The study area is located in southwestern Germany and eastern France (Fig. 1). The terrain exhibits a specific complexity with some rolling terrain in the north, the flat and broad Rhine valley, and a chain of three low mountain ranges: Vosges (France), Black Forest, and Swabian Jura (both in Germany). Mean annual precipitation varies between 600 mm in the valleys and 2000 mm around the highest peaks. According to a combined radar and ombrometer study by Hannesen (1998), the ratio between stratiform and convective precipitation type is around 0.5 over the Rhine Valley and the mountains of the Black Forest and 0.35 over the Swabian Jura. Approximately two-thirds of the stratiform precipitation totals over the Black Forest can be ascribed to orographic lifting. On the 2.5-km model grid, the highest mountain, Feldberg in the southern Black Forest, has an elevation of 1348 m, whereas the lowest point in the northern Rhine Valley is at 90 m.

To allow disturbances to decay before reaching the downstream boundaries, the model domain is enlarged to a 256 × 256 grid and smoothed on all lateral sides for the FFT. All subsequent calculations are conducted on a 100 × 100 grid.

### a. Datasets

#### 1) Radiosoundings

*ρ*

_{d}as the density of dry air, is reduced only slightly, yielding a negative bias of −23.9 kg m

^{−1}s

^{−1}(≈−7.5%) on the days considered (Fig. 2a). Almost the same is true for the precipitable water (pw), which is reduced by −1.7 kg m

^{−2}(≈−10.4%) on average (Fig. 2b). This reduction may be partly attributed to moisture loss due to precipitation fallout between the two locations. But also the higher elevation of the station of Stuttgart (315 m) compared to Nancy (212 m) leads to lower integrated moisture content of approximately 1 kg m

^{−2}on average. Owing to the good agreement of atmospheric parameters at least on days considered in this study, it is reasonable using the Stuttgart instead of the Nancy sounding data.

*U*,

*β*, and stability

*N*. To emphasize the lower-level atmospheric conditions, the flow parameters Λ̃ are obtained by vertical integration of the profiles from the surface up to

_{m}*z*= 5000 m, applying “water vapor weighting”:

_{t}*N*, the averaging routine is applied to

_{m}*N*

_{m}^{2}. In the few slightly unstable cases,

*Ñ*was set to a near-neutral value of 0.0003 s

_{m}^{−1}.

In PSB, the uplift sensitivity factor *C _{w}* [see Eq. (4)] is computed by averaging the saturation water vapor density

*ρq*over the lowest 500 m. The water vapor scale height

_{υ}*H*as the height where the integrated water vapor density dropped to

_{w}*e*

^{−1}, is determined directly from the sounding.

#### 2) Climate stations

A network of 216 climate stations (Fig. 1) of DWD equipped with rain gauges is used to select appropriate heavy precipitation events and to evaluate simulated precipitation fields. All stations were in continuous operation during the entire period from 1971 to 2000. The entire dataset was thoroughly controlled, homogenized, and corrected for errors by the Potsdam Institute for Climate Impact Research (PIK). The measuring interval of the 24-h rain totals starts at 0730 LT, that is, 0530 UTC during summertime and 0630 UTC during wintertime. Unfortunately, precipitation observations are available only for Germany.

#### 3) Reanalysis data

To cover also the regions outside of Germany, high-resolution reanalysis data were used in addition for qualitative evaluation of the results of the linear models. These model runs were computed at the Institute for Meteorology and Climate Research (IMK) with the climate version of the Consortium for Small-scale Modeling (COSMO; COSMO-CLM) model of DWD (Rockel et al. 2008). The model was initialized by the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analyses (ERA-40; Simmons and Gibson 2000). The resolution of the final dataset, labeled CCLM–ERA-40, is approximately 7 km × 7 km.

### b. Evaluation of model results

^{−1}) must be converted into totals (in mm) to be compared with observations. This is done by multiplying the intensities with the time lag between two consecutive soundings, that is, 12 h. For comparison purposes, the model fields are interpolated to the station locations by an inverse Cressman scheme (Cressman 1959),

*P*is the model precipitation at the four grid points around an observation site. The weights

_{i}*W*, are calculated as follows:

_{i}*D*is the horizontal distance between an observation site and the model grid point.

_{i}Various methods have been developed in the past to verify model results or to estimate model skills. A good overview of standard verification methods is provided by Wilks (1995) or by Nurmi (2003). In this study, four different widely used skill scores are employed. Both the Heidke skill score (HSS) and the equitable threat score (ETS; see the appendix for definitions) are based on categorical verification with a contingency table (Table A1). The data are entered into the 2 × 2 decision matrix, depending on whether a certain threshold was exceeded in the simulations and observations. To prevent variable sample sizes from entering the statistics, thresholds are defined by the median and the 16th and 84th percentiles of the distribution function of rain totals observed. In case of a normal distribution, the latter two values represent twice the standard deviation *σ*.

*N*is the number of observation/simulation pairs and

*O*are the totals observed. While the bias measures a systematic deviation between simulations and observations, the rmse indicates the magnitude of the difference, with a strong effect of large deviations.

_{i}### c. Selection and characteristics of past events

The period of 1971 to 2000 was subdivided into two time frames: (i) 1971–1980 as a calibration period (CP) for the models and (ii) 1981–2000 as the application period (AP).

Rain totals as a mean of several stations were determined for three different subterritories: the northern Rhine Valley (*R*_{NRV}) with 13 stations, the northern Black Forest (*R*_{NBF}) with 7 stations, and the southern Black Forest (*R*_{SBF}), again with 7 stations. For the total sample, all 24-h precipitation events satisfying the following conditions were selected:

*R*_{NBF}and*R*_{SBF}> 24 mm,*R*_{NBF}and*R*_{SBF}> 2 ×*R*_{NRV}, and*R*_{NBF}=*R*_{SBF}± 30%.

The result after considering the different criteria is a total of 148 events for the entire period. As 24 events had to be rejected from the list because of missing humidity and/or wind profiles at higher levels in the soundings, 40 events remain for the calibration period and 84 events for the application period. A total of 100 events (71 in AP) occurred in the winter half year from October to March and 24 (13 in AP) in the summer from April to September. Atmospheric conditions most decisive for the simulation results—in particular, wind direction and wind speed, static stability, and incoming water vapor flux—are almost similar for AP and CP (cf. Fig. 12). In all cases, the rainfall duration was at least 18 h. The exact duration cannot be determined because data available for the entire 30-yr period have a resolution of only 6 h (synoptic stations of DWD).

## 4. Case studies and model comparison

### a. Idealized perspective

Precipitation simulations under idealized conditions according to the different realizations of both models, PSB and PKU, are discussed below. This will help to understand better the precipitation patterns of real cases. The models are initialized by *U* = 20 m s^{−1} and *N _{m}* = 0.002 s

^{−1}, reflecting mean atmospheric conditions during the events selected (cf. Fig. 12). Considering a near-surface temperature of

*T*= 281.6 K, vertical profiles of temperature and moisture are calculated.

_{s}*H*= 1 km is the peak mountain height,

*a*= 10 km (20 km) is the half-width, and

*x*

_{0}=

*y*

_{0}= 50 km defines the center point set in the middle of the domain of 100 km × 100 km with Δ

*x*= Δ

*y*= 1 km.

If only flow dynamics but neither hydrometeor drifting nor evaporation are taken into account, then the simulations from both models are almost identical (Fig. 3a). In the descent region downstream of the crest, precipitation drops to zero at once. Differences in the simulation results—in particular, the location of the second peak, caused by the gravity wave at a larger distance from the mountain—are due to the hydrostatic balance assumed in the PKU model.

Considering hydrometeor drifting and evaporation on the lee side significantly reduces maximum intensities and broadens the precipitation areas shifted downstream (Fig. 3b). For both mountain widths displayed (*a* = 10 and 20 km), the maximum intensities are found farther upstream in the PKU than in the PSB simulations as a consequence of the different inclusion of hydrometeor drifting. This is obvious, especially for the broader mountain. Upstream of the crest, the model results are similar, especially in the case of a narrow mountain. On the lee side, however, PSB exaggerates evaporation considerably, as already discussed in the model section. On the other side, PKU simulated precipitation also far downstream of the crest, which is not observed for real events.

*a*and

*H*, and the free model parameters,

*P*

_{∞}and

*τ** =

*τ*+

_{c}*τ*, modify precipitation totals. Model results are summarized by the drying ratio (DR), which is defined according to Smith and Barstad (2004) as

_{f}*P*

*F*

_{wυ}is the area-integrated water vapor flux. Referring to this definition,

*DR*quantifies the effect of removing moisture from the atmosphere by precipitation. Since

*P*

*P*

*F*

_{wυ}in (16) is restricted to the centerline of the width Δ

*y*. In the experiments, one parameter was varied while the others were kept constant. The increments in the variations are given by the difference between the minimum and the maximum, divided by 10 (Fig. 4). Simulated

*DR*varies between 1% and 28%, indicating that only a small fraction of the water vapor is converted into precipitation.

As the mountain width *a* increases, so does DR but nonlinearly in both models (Fig. 4a). The increase is determined by the superposition of two effects: decreased vertical lifting, but over a larger area upstream of the crest, and reduced spillover, resulting in less evaporation for broader mountains. The narrower the mountain, the higher leeside evaporation, as can be seen from the increasing differences of DR between the standard PKU run and one neglecting evaporation (PKU_noeva in Fig. 4a). Consequently, if time scales for mountain overflow *a*/*U* (≈500 s in the control runs) are shorter than hydrometeor time scales *τ** only a small fraction of atmospheric moisture can be converted into precipitation. This is the case for very narrow mountains and/or high horizontal wind speeds.

Because of the proportionality of *ĥ* in Fourier space, DR increases linearly with *H* in the PSB simulations (Fig. 4b). This is not the case in PKU because higher mountains cause lifting also at higher levels, where ice particles may form. These are advected in descent regions because of their slow terminal fall velocity parameterized in this model. The differences between the two models disappear for higher temperatures (not shown).

As can be seen in Figs. 4c and 4d, DR is very sensitive to the choice of the free model parameters *τ** and *P*_{∞}. The higher *τ**, the higher is the horizontal drift; thus is also the leeside evaporation, which is much more pronounced in PSB. It should be noted that the characteristic time scales of both models cannot be compared directly because of the different considerations given to hydrometeor formation and fall distances. As background precipitation *P*_{∞} increases, so does DR in a linear way (Fig. 4d). Because of the higher evaporation rate in PSB, the increase in DR is slightly less than in PKU. For atmospheric conditions characteristic of widespread precipitation events, DR is more sensitive to *P*_{∞}, which usually varies between 0.2 and 1.0 mm h^{−1} (cf. Fig. 7 in section 5), than to *τ**, which is anywhere between 2000 and 3500 s.

In general, the results obtained from both models are in good agreement regarding the amplitude and shape of curves. Several details, however, reveal the differences in taking into account microphysics in the models, in particular concerning vertical resolution and evaporation. The consequences of the different model realizations for real events will be studied in detail in the next section.

### b. Case study: 11 December 1997

Simulations of real cases are associated with several uncertainties in the wake of the simplified models, first of all, because of the two parameters *τ* and *P*_{∞} unknown a priori. But also ambient conditions in terms of stability (*N _{m}*), wind (

*U*,

*β*), and moisture (

*q*) are not known precisely because of the low spatial and temporal resolution of the soundings used as model input. The main objective of this section is to explore how sensitive simulation results on a 2.5-km grid are to changing model and atmospheric parameters under real conditions.

_{υ}For this study a widespread precipitation event of 11 December 1997 was chosen. On that day, sustained advection of moist air combined with synoptic-scale ascent due to a midtropospheric shortwave trough caused continuous heavy rainfall particularly over the mountains of southwestern Germany. The whole stratiform precipitation event persisted for about 58 h with a maximum peak on 11 December. Vertical profiles from the Stuttgart sounding reveal a high moisture content (pw = 23.3 kg m^{−2}), low stability (*Ñ _{m}* = 4.3 × 10

^{−3}s

^{−1}), and strong westerly winds (

*Ũ*= 19.1 m s

^{−1}). The very combination of those parameters already is an indication of substantial precipitation enhancement at the western flanks of the Black Forest.

#### 1) Sensitivity of orographic precipitation to changing model parameters

To determine appropriate values of *τ _{c}*,

*τ*, and

_{f}*P*

_{∞}, model simulations with different values of those parameters were carried out. While

*τ*=

*τ*

_{c}≡

*τ*varies between 500 and 2500 s (Δ

_{f}*τ*= 200 s),

*P*

_{∞}is in a range between 0 and 1.0 mm h

^{−1}(Δ

*P*

_{∞}= 0.1 mm h

^{−1}).

The two quantities *τ* and *P*_{∞} try to push the precipitation fields in different directions. While DR decreases with increasing *τ*, it is the other way around for *P*_{∞} (cf. Figs. 4c and 4d). Considering only area-integrated values, the two effects could cancel out as can be seen in the 2D sensitivity plots for PSB simulations (Fig. 5a). The DR is calculated here for a subarea of the total domain with *y* ≥ 30 km and *x* ≥ 100 km, where observations are available. In agreement with the idealized simulations, DR increases almost linearly with *P*_{∞}. Under the assumption of constant *P*_{∞}, DR decreases most strongly for low values of *τ*. In that range, a slight horizontal shift of hydrometeors strongly enhances leeside evaporation, resulting in the high sensitivity of DR. On the other hand, most of the spillover already evaporated for high *τ*. Hydrometeors may also experience further lifting downstream when advected over a longer distance. Thus, variations of DR are only marginal in that range. This finding is important for the discussion below of model tuning and application in a climatological perspective.

As for DR, ERR is more sensitive to variations of *τ*, but only as long as values are low (Fig. 5b). For higher values on the order of 800 s, the opposite is true. The “best model run” of PSB is determined by the local minima of the error matrix, ERR = fct(*τ*, *P*_{∞}), yielding 1000 ≤ *τ* ≤ 1200 s and 0.5 ≤ *P*_{∞} ≤ 0.6 mm h^{−1}. In both models, *P*_{∞} was set uniquely to 0.6 mm h^{−1}, whereas *τ _{c}* and

*τ*were set equal to 1100 s in the PSB model but varied between 100 s near the surface and 1700 s above freezing level in the PKU model. Even if the time scales cannot be compared directly, they are at least in the same range. Both models (Figs. 5c and 5d) are able to reproduce the general structure of the precipitation fields observed (Fig. 5f). Upstream and over the highest mountains of the Black Forest (mean wind direction from the west), the precipitation sums and their spatial distribution show a good agreement to the observations. The comma-shaped region with weak precipitation observed in the northeast, however, is not rendered correctly in both models. The overestimate in the model results may be due to prevailing horizontal differences in atmospheric conditions. Moreover, the second precipitation peak in the south is underestimated by the models. Note that the number of rain gauges in that region and other mountain regions is low. Hence, the true precipitation pattern on this day is not known. Comparison of the results of both models in detail makes it clear that PSB tends to underestimate precipitation in descent regions.

_{f}Omitting the Vosges affects simulated precipitation almost over the whole test area because of changes in the underlying atmospheric water budget, but also in flow dynamics. According to PSB simulations (Fig. 5e), rain sums increase up to 25 mm over the Rhine Valley and between 5 and 10 mm over the western slopes of Black Forest mountains. In contrast to this, totals slightly decrease over the eastern parts of the investigation area. Overall, the influence of the Vosges to simulated precipitation downstream is only moderate, except for the Rhine valley. Furthermore, the spatial distribution of precipitation is not affected.

#### 2) Sensitivity of orographic precipitation to changing atmospheric parameters

Besides the parameters *τ* and *P*_{∞}, model performance strongly depends on the representation of the ambient conditions in terms of (*Ñ _{m}*,

*Ũ*,

*β̃*,

*q̃*). Since these parameters are determined only from two radiosoundings which, in addition, may be affected by the mountains of the northern Black Forest, we will show exemplarily how variations in ambient conditions change model results.

_{υ}Atmospheric parameters as obtained from Eq. (10) are continuously perturbed by a multiplicative factor (var_mult in Fig. 6) increasing linearly from 0.5 to 2.0 in increments of 0.05. Wind direction, *β̃*, was varied ±30°. The best results for PSB in terms of the lowest rmse are obtained with the original values. This confirms the reliability of the initializing approach with water vapor–weighted bulks from radiosoundings. In contrast, the lowest rmse is given for slightly perturbed atmospheric parameters in the PKU model. Enhancement of *q _{υ}* by 25% yields a 13% decrease of rmse. The differences between the two models are a consequence of the different ways of considering atmospheric moisture, which is resolved vertically only in PKU. In general, the skill of both models to reproduce realistically rain patterns is quite similar.

*U*than to stability

*N*. According to linear theory of unsaturated stratified flow (Smith 1979), the two parameters should scale in opposite directions. This, however, holds true for the flow regime but not for precipitation, where the horizontal wind speed controls not only

_{m}*F*

_{wυ}but also the horizontal drift of hydrometeors and subsequent evaporation. Hence, a similarity theory with simplified analytical expressions, as for linear theory, cannot be established for orographic precipitation.

According to several other tests, the results exemplarily shown here are also valid for most of the events selected for the sample (not shown). Thus, deriving model parameters from the soundings by the methods described is appropriate for the purposes in this study.

## 5. Precipitation climatology

In this section model skills to reproduce reliable precipitation fields for various real events will be estimated. During the CP (1971–80), the best combination of the model parameters (*τ*, *P*_{∞}) is assessed by comparing modeled and observed precipitation fields. These fixed values determined during CP are used for simulating the events of the AP (1981–2000). During AP, the models are neither tuned in any way nor are the output fields averaged or smoothed.

### a. Calibration period: Evaluation of model parameters

To find the best combination of the parameters (*τ*, *P*_{∞}) representative in the mean, the same optimization procedure as described above was carried out for all 40 events of the CP. The parameter values yielding the lowest ERR [see Eq. (17)] on average will be used for simulating the events selected for the AP.

The five realizations with the lowest ERR are added to the sample to enlarge sample size and not to rely on a single combination of parameter values only, whereas a different combination may yield nearly identical results. In the total sample of 40 × 5 = 200 cases, a count was made to see how many optimized model runs are based on the same (*P*_{∞}, *τ*) values. As Fig. 7a shows for PSB, almost all runs have a *τ* in the range between 1200 and 2000 s and a *P*_{∞} between 0.2 and 0.6 mm h^{−1}. Even though the spread is large, a clear majority of optimal runs lie within two areas: *τ* = [1400, 1800] s and *P*_{∞} = 0.5 mm h^{−1}, or *τ* = [1700, 2000] s and *P*_{∞} = 0.4 mm h^{−1}.

In addition, we calculated a mean *P*_{∞}, *τ*) matrix was obtained for 1600 s ≤ *τ* ≤ 2000 s and *P*_{∞} = 0.4 mm h^{−1}. Following this estimate, parameters are defined by *P*_{∞} = 0.4 mm h^{−1} and *τ* = 1600 s. These values will be used in simulating the events selected during the AP. While direct measures of *τ* are generally not obtainable, *P*_{∞} can be estimated from rain gauge measurement in the northern Rhine Valley, which is not severely affected by descending air and thus leeside drying. Averaged over all 13 appropriate stations (white circles in Fig. 1) and all events between 1971 and 2000, mean precipitation is *R*_{NRV} = 0.48 ± 0.22 mm h^{−1} (median: 0.42 mm h^{−1}), which is close to the value obtained by the optimization procedure.

### b. Application period: Precipitation statistics

The ability of the models to reproduce realistically spatial distributions of precipitation without any model tuning is discussed below. All 84 events of the AP selected were simulated by using the fixed parameters as determined above for the CP. As the PKU model is resolved vertically and has a parameterization scheme for *τ _{f}* , only the fixed value of

*τ*is used for the highest level of the model.

_{c}Both models reproduce approximately the mean rainfall observed (Fig. 8). Major differences in the totals between valleys and mountains document the high impact of orographically induced lifting, yielding mean totals in excess of 40 mm over the mountains. In contrast, background precipitation is low on average and well estimated by a predefined intensity of *P*_{∞} = 0.4 mm h^{−1} (≈10 mm). Maximum totals over and upstream of the highest mountains of the Black Forest are approximately the same in the observations and simulations, though more stretched out horizontally in the simulations. Note, however, that the number of rain gauges in these areas is low; therefore, direct comparison to modeled fields is impossible.

Comparing the results of the two models shows PSB to produce not enough rain over broad and deep valleys, such as the southern Rhine valley. Downstream of the very steep Vosges, precipitation is almost zero because of the strong descent and, thus, the exaggeration of evaporation as already shown for idealized conditions (cf. Fig. 3b). In contrast, downstream of Black Forest mountains, where the terrain drops gently, evaporation is considerably reduced, resulting in a slight overestimation of precipitation (see also Fig. 11). As shown by Barstad et al. (2007), leeside evaporation may be partly unphysical in the PSB model, as the vertical integration will allow drying aloft to cancel the moisture spillover nearer to the ground. This is clearly not happening in the real world. Besides, recall that the same time delays were used for forward and backward processes (i.e., raindrop formation and evaporation). To conclude, leeside precipitation may be strongly biased in the PSB model.

In PKU, the parameterization scheme for evaporation, including an upper bound that is reached asymptotically, prevents the overestimation of leeside drying. Hence, precipitation totals in descent regions are modeled more realistically. This is also obvious from the scatter diagram with totals observed versus simulated (Fig. 9). As can be seen by the clustering of observed totals between 10 and 20 mm, model performance is controlled by rain totals. Both models tend to overestimate the totals in general, particularly for lower amounts. However, low rain sums observed are significantly underestimated by PSB but not by PKU. These stations are restricted to the Rhine Valley, where evaporation is overestimated in PSB as discussed above. Aside from these discrepancies, the high correlation coefficients of *r* = 0.87 and *r* = 0.89 for PSB and PKU, respectively, confirm that both models reproduce precipitation distributions in good quality.

As observation data are available for Germany only, we used CCLM–ERA40 reanalysis data as another reference (Fig. 8b). Compared to the linear models, the skill of CCLM to reproduce reliable precipitation distributions is not better, despite the highly sophisticated model physics. In particular the precipitation peaks over the Black Forest are substantially underestimated compared to observation. A good agreement is reached for the other parts of southwest Germany in general, where orographic effects are not dominant. The most striking feature is the extensive precipitation peak over the southern Vosges, which is not captured in both linear models. In the PSB model, precipitation in this area is underestimated by a factor of approximately 1.5. This underestimate may be ascribed, to some extent, to the initialization of the models by the Stuttgart soundings, which display ambient conditions after moisture removal from the atmosphere at the first two mountain chains. As PKU computes microphysics only on the inner grid, precipitation is generally too low near the western and northern boundaries. Assuming a mean wind speed of 18 m s^{−1} (see Fig. 12) and a time delay of *τ _{c}* +

*τ*= 3200 s, this yields a drift distance of 58 km. Within that range and, of course, depending on the location relative to the upstream boundary, precipitation is inevitably too low.

_{f}To estimate quantitatively the reliability of the simulation results not only for the mean, we calculated rmse, bias, HSS, and ETS separately for each event. Results are shown as a box plot summarizing the distribution function of the respective error measure (Fig. 10). The higher the threshold represented by the percentiles, the higher the skill of the models in terms of HSS and ETS. This means that the models reproduce the highest precipitation totals over the mountains at best. Moreover, the PSB model exhibits better performance than the PKU model according to all error measures except for the bias. The negative bias of the PSB runs is caused by, in particular, the overestimated evaporation in descent regions. In contrast, the more sophisticated representation of evaporation in PKU causes higher totals in the valleys and, hence, almost unbiased results on average. By way of comparison, Fig. 10 also shows (for PSB only) error and skill measurements when applying the optimization routine for the variables *P*_{∞}–*τ*. In all cases, capability increases and the simulations are almost unbiased, especially because of the adjustment of *P*_{∞}.

The question arises whether specific areas can be identified where model performance is low in a systematic way. To provide an indication, relative bias and rmse, normalized by the totals observed, are quantified as a function of the location from the same model runs (Fig. 11). As discussed above for the mean precipitation fields (Fig. 8), PSB simulations tend to strongly underestimate precipitation especially in the Rhine Valley, yielding a relative rmse and bias of up to 100% and −50%, respectively. In contrast, the relative rmse is lowest for both models over the crests of the Black Forest, a region with the highest orographically induced totals. Downstream of the mountains, the rmse increases again because of an overestimate of the totals. Since the terrain drops only slowly from the crests of the Black Forest to the Stuttgart region, causing a very moderate descent, spillover is reduced only slightly. As indicated by a positive bias of up to 40%, the PKU model overestimates precipitation totals in the southern parts of the Rhine Valley. In contrast to PSB, PKU simulations are almost unbiased downstream of the Black Forest.

### c. Atmospheric conditions during orographic precipitation events

The ingredients most relevant for high orography-induced rain totals will be studied next. For this purpose, the atmospheric conditions prevailing during the total sample of events were analyzed by combining radiosonde and model data.

All events are related to specific atmospheric conditions (Fig. 12). In most cases, the atmosphere showed a slightly stable stratification, as was indicated by a positive *Ñ _{m}* of (4.65 ± 2.32) × 10

^{−3}s

^{−1}(mean ± standard deviation). Almost all events have flow directions from the west and high horizontal wind speeds (

*Ũ*= 18.2 ±4.5ms

^{−1}). Because of the asymmetric shape of the Black Forest mountains, with the steepest slopes on the west side, orographically induced vertical velocities and, hence, orographic precipitation are highest for westerly flows. This holds particularly true for low LCL of, on average, 858 ± 200 m. In many cases, the mountain peaks therefore are assumed to be covered by clouds.

The high wind speed along with large amounts of precipitable water (pw = 16.3 ± 4.7 kg m^{−2}) results in high values of *F*_{wυ}. In reality, only a fraction of *F*_{wυ} is converted into precipitation as defined by DR. For the events considered, DR was in a range between 8% and 23% (10th and 90th percentiles). Even if *F*_{wυ} and the totals observed over the northern Black Forest are not directly correlated, Fig. 13 shows that *F*_{wυ} can be regarded as an upper bound for precipitation. In other words, high totals over the mountains require strong moisture influxes in general. This relation does not work the other way around, as low precipitation totals also coincide with high *F*_{wυ}. To include model data in the analysis, *P*_{∞} and *τ* were allowed to vary for the total sample of events between 1971 and 2000. The optimization approach is used to identify and select the best model runs of PSB for statistics. In Fig. 13, different regimes can be identified: low synoptic-scale ascent (*P*_{∞} ≤ 0.2 mm h^{−1}) corresponding to low precipitation totals, medium ascent (0.2 < *P*_{∞} ≤ 0.7 mm h^{−1}) with large variability of totals, and significant synoptic-scale ascent (*P*_{∞} > 0.7 mm h^{−1}) coinciding with both high orographic precipitation and high incoming water vapor flux. This indicates that heavy precipitation over mountains is restricted to situations with strong forcings in terms of synoptic-scale lifting and a combination of strong pressure gradients determining *U* and a high moisture content.

The other free parameters in the PSB model, *τ _{f}* and

*τ*, are in a range between 1100 and 2200 s (10th and 90th percentiles) with a median of 1400 s. As discussed in the previous section, model simulations are not very sensitive to the exact value of

_{c}*τ*in the range estimated here (cf. Fig. 5a). Therefore, no systematic relation is found between the

*τ*s and other model values or ambient parameters, such as the temperature determining the hydrometeor phases and thus their terminal velocity. Interestingly, the observations also do not provide any indication of the location of precipitation peaks being controlled by the ambient temperature (not shown here).

## 6. Summary and conclusions

Simulations of orographic precipitation with two different linear precipitation models are presented. Both models are based on 3D airflow dynamics from linear theory for saturated conditions and use a simple parameterization scheme for microphysics. Model runs for idealized conditions and a real case study reveal the principal characteristics of orographic precipitation over low mountain ranges and the sensitivity of the results to changing model and atmospheric parameters. Precipitation fields over the complex terrain of southwestern Germany and eastern France were calculated on a 2.5-km grid. A representative sample of widespread events of a 10-yr calibration period (1971–1980) was considered to adjust the free model parameters used to simulate a sample of events of a 20-yr application period (1981–2000). Both models are shown to be able to derive reliable precipitation amounts and distributions in good agreement with observations. The highest precipitation totals were seen to occur on days with strong synoptic-scale lifting and high horizontal water vapor influx.

From the results of both models, the following major issues can be inferred, which are relevant in estimating orographic precipitation: (i) The basic mechanisms of orographic precipitation enhancement are linear and can be described by simplified equations as long as precipitation characteristics are stratiform. The good agreement between the results of linear models and observations suggests that the simplified approaches consider the essential dynamics and physics most decisive in the generation and spatial distribution of orographic precipitation. The marginal differences of the linear models over the mountains suggest that resolution in the vertical is not necessary. (ii) The magnitude and spatial distribution of precipitation is the result of the interaction of characteristic time scale *τ* for the advection of the hydrometeors and background precipitation *P*_{∞} associated with large-scale lifting. While total precipitation *R* or drying ratio DR increases with *P*_{∞}, the reverse is true for *τ* because of the evaporation of hydrometeors in descent regions. (iii) While the horizontal incoming water vapor flux *F*_{wυ} provides an upper limit of conversion of moisture into precipitation, the synoptic-scale ascent determines the overall precipitation enhancement. Hence, heavy precipitation over mountains is restricted to situations with strong forcings in terms of synoptic-scale lifting, horizontal wind speed, and moisture content. It may also be concluded from the findings that low mountain ranges can amplify initial atmospheric disturbances but not initiate them.

During the calibration period, the best model results were obtained with *P _{∞}* = 0.4 mm h

^{−1}and

*τ*=

_{f}*τ*= 1600 s. Alternatively,

_{c}*P*can be determined from station data at locations not or only marginally affected by orographic precipitation or leeside drying. With

_{∞}*P*

_{∞}averaged over appropriate station data in the Rhine Valley, mean background precipitation of 0.48 ± 0.22 mm h

^{−1}are obtained for all events. This is close to the value obtained by the optimization procedure and thus confirms the applicability of this approach. Surprising is the almost consistency of the

*τ*values, which are in a range between 1500 and 2000 s for the events considered. Depending on a specific region, these parameters may exhibit a larger variability. Besides, these estimates are relatively high compared to other studies. For the Olympic Mountains in Washington state, the eastern Mediterranean coastal ranges, and the southern Alpine region, Smith and Barstad (2004) and Barstad and Smith (2005) estimate cloud delays by 1000 s using the same PSB model. For the Oregon climate, best model runs were performed with

*τ*=

_{f}*τ*= 1500 s, consistent with our findings (Smith et al. 2005). Estimates derived from DR, using stable isotope data from stream water samples as a reference and PSB simulations, are generally lower, for example, 500 s for western Oregon (Smith et al. 2005) and 800 s for the Olympic range (Smith and Evans 2007). However, background precipitation, which would increase predicted DR and thus

_{c}*τ*as well, was neglected. The

*τ*values in this study are estimated on the basis of two different models by applying separately and independently an optimization procedure. Hence, our confidence in obtaining the best parameter values for the test area is high.

A potential weakness, particularly in the PSB model, is the underrepresentation of leeside precipitation due to the vertical integration. Drying of the air aloft may cancel moisture spillover near the ground, leading to a strong underestimation of leeside precipitation (Barstad et al. 2007). By considering several layers, the PKU model avoids this unphysical effect. Furthermore, recall that the models are linearized about a state of background precipitation. Downstream of steep mountains and at Fr ≈ 1, linear theory breaks down. Thus, it cannot be expected that the models yield reliable rain sums over that regions. Finally, another weakness arises from the initializing method with single radiosoundings, which are available at 12-h intervals only. If atmospheric conditions change on a shorter time scale or length scale, an initialization routine with input from a coarser grid model is expected to provide more realistic results.

The methodology used to derive climatological precipitation fields as presented in this paper is suitable for a broad range of applications involving the generation of orographic precipitation. It can be used to interpolate rain gauge measurements over complex terrain to obtain gridded precipitation for hydrological purposes. Simulation results can be employed to complement radar data in regions unseen by radar. Finally, initializing the models with reanalysis data, such as ERA-40 or National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) available globally, could help to derive detailed precipitation statistics for regions poorly monitored until now.

## Acknowledgments

Special thanks are given to Ron B. Smith for very interesting and helpful discussions of orographic precipitation and modeling during a research visit of the author at Yale University and afterward. The author thanks F.-W. Gerstengarbe of PIK for providing validated and homogenized data of the climate station network, DWD for providing radiosonde data, H.-J. Panitz of IMK for providing the CCLM–ERA-40 datasets, Idar Barstad, and Daniel J. Kirshbaum for fruitful discussions. Three helpful reviews are much appreciated.

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## APPENDIX

### Skill Scores and Error Measures

*a*–

*d*, the HSS is given by

(a) The *F*_{wv} and (b) pw derived from radiosonde observations at the stations of Stuttgart and Nancy at 0000 and 1200 UTC for the precipitation events between 1990 and 2000 used in this study. Indicated are 1:1 and regression lines and correlation coefficient after Pearson.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

(a) The *F*_{wv} and (b) pw derived from radiosonde observations at the stations of Stuttgart and Nancy at 0000 and 1200 UTC for the precipitation events between 1990 and 2000 used in this study. Indicated are 1:1 and regression lines and correlation coefficient after Pearson.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

(a) The *F*_{wv} and (b) pw derived from radiosonde observations at the stations of Stuttgart and Nancy at 0000 and 1200 UTC for the precipitation events between 1990 and 2000 used in this study. Indicated are 1:1 and regression lines and correlation coefficient after Pearson.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Idealized simulations of orographic precipitation from PSB and PKU as cross sections through the centerline of the mountain considering only (a) dynamics and (b) in addition hydrometeor drift and leeside evaporation for *a* = 10 km (solid lines) and *a* = 20 km (dashed lines) with *τ** = *τ _{c}* +

*τ*= 1000 s.

_{f}Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Idealized simulations of orographic precipitation from PSB and PKU as cross sections through the centerline of the mountain considering only (a) dynamics and (b) in addition hydrometeor drift and leeside evaporation for *a* = 10 km (solid lines) and *a* = 20 km (dashed lines) with *τ** = *τ _{c}* +

*τ*= 1000 s.

_{f}Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Idealized simulations of orographic precipitation from PSB and PKU as cross sections through the centerline of the mountain considering only (a) dynamics and (b) in addition hydrometeor drift and leeside evaporation for *a* = 10 km (solid lines) and *a* = 20 km (dashed lines) with *τ** = *τ _{c}* +

*τ*= 1000 s.

_{f}Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Idealized simulations of the DR for changing (a) *a*, (b) *H*, (c) *τ** = *τ _{c}* +

*τ*, and (d)

_{f}*P*

_{∞}. In (a), the PKU results without leeside evaporation are also shown (PKU_noeva). Parameters of the control run:

*a*= 10 km,

*H*= 1 km,

*P*

_{∞}= 0 mm h

^{−1},

*τ** = 1000 s,

*U*= 20 m s

^{−1},

*N*= 0.002 s

_{m}^{−1}, and

*T*= 282 K.

_{s}Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Idealized simulations of the DR for changing (a) *a*, (b) *H*, (c) *τ** = *τ _{c}* +

*τ*, and (d)

_{f}*P*

_{∞}. In (a), the PKU results without leeside evaporation are also shown (PKU_noeva). Parameters of the control run:

*a*= 10 km,

*H*= 1 km,

*P*

_{∞}= 0 mm h

^{−1},

*τ** = 1000 s,

*U*= 20 m s

^{−1},

*N*= 0.002 s

_{m}^{−1}, and

*T*= 282 K.

_{s}Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Idealized simulations of the DR for changing (a) *a*, (b) *H*, (c) *τ** = *τ _{c}* +

*τ*, and (d)

_{f}*P*

_{∞}. In (a), the PKU results without leeside evaporation are also shown (PKU_noeva). Parameters of the control run:

*a*= 10 km,

*H*= 1 km,

*P*

_{∞}= 0 mm h

^{−1},

*τ** = 1000 s,

*U*= 20 m s

^{−1},

*N*= 0.002 s

_{m}^{−1}, and

*T*= 282 K.

_{s}Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

The (a) DR and (b) ERR as a function of *τ* = *τ _{f}* =

*τ*and

_{c}*P*

_{∞}from PSB simulations for 11 Dec 1997. Precipitation fields (24-h totals) simulated with (c) the PSB model for

*τ*= 1100 s and

*P*

_{∞}= 0.6 mm h

^{−1}, (d) with the PKU model for

*τ*= 1700 s and

_{c}*P*

_{∞}= 0.6 mm h

^{−1}, (e) as difference without and with Vosges in PSB and according to (f) observations.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

The (a) DR and (b) ERR as a function of *τ* = *τ _{f}* =

*τ*and

_{c}*P*

_{∞}from PSB simulations for 11 Dec 1997. Precipitation fields (24-h totals) simulated with (c) the PSB model for

*τ*= 1100 s and

*P*

_{∞}= 0.6 mm h

^{−1}, (d) with the PKU model for

*τ*= 1700 s and

_{c}*P*

_{∞}= 0.6 mm h

^{−1}, (e) as difference without and with Vosges in PSB and according to (f) observations.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

The (a) DR and (b) ERR as a function of *τ* = *τ _{f}* =

*τ*and

_{c}*P*

_{∞}from PSB simulations for 11 Dec 1997. Precipitation fields (24-h totals) simulated with (c) the PSB model for

*τ*= 1100 s and

*P*

_{∞}= 0.6 mm h

^{−1}, (d) with the PKU model for

*τ*= 1700 s and

_{c}*P*

_{∞}= 0.6 mm h

^{−1}, (e) as difference without and with Vosges in PSB and according to (f) observations.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

The RMSE for variable atmospheric parameters according to simulations with (a) PSB and (b) PKU (see text for further explanations). The black (*U*, *N _{m}*,

*q*) and gray (

_{υ}*β*) vertical bars indicate the values of the control run.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

The RMSE for variable atmospheric parameters according to simulations with (a) PSB and (b) PKU (see text for further explanations). The black (*U*, *N _{m}*,

*q*) and gray (

_{υ}*β*) vertical bars indicate the values of the control run.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

The RMSE for variable atmospheric parameters according to simulations with (a) PSB and (b) PKU (see text for further explanations). The black (*U*, *N _{m}*,

*q*) and gray (

_{υ}*β*) vertical bars indicate the values of the control run.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

(a) Number of PSB model runs with the lowest ERR as a function of the (*P*_{∞}, *τ*) matrix for the CP and (b) *P*_{∞}, *τ*) matrix from the average of the same events.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

(a) Number of PSB model runs with the lowest ERR as a function of the (*P*_{∞}, *τ*) matrix for the CP and (b) *P*_{∞}, *τ*) matrix from the average of the same events.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

(a) Number of PSB model runs with the lowest ERR as a function of the (*P*_{∞}, *τ*) matrix for the CP and (b) *P*_{∞}, *τ*) matrix from the average of the same events.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Mean rain sums of all events selected within the AP according to (a) observations, (b) CCLM–ERA-40 reanalysis data, and model simulations with (c) PSB and (d) PKU.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Mean rain sums of all events selected within the AP according to (a) observations, (b) CCLM–ERA-40 reanalysis data, and model simulations with (c) PSB and (d) PKU.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Mean rain sums of all events selected within the AP according to (a) observations, (b) CCLM–ERA-40 reanalysis data, and model simulations with (c) PSB and (d) PKU.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Scatter diagram between observed and simulated rain totals (24 h) for all events selected within the AP from PSB and PKU models, including regression lines and Pearson correlation coefficients.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Scatter diagram between observed and simulated rain totals (24 h) for all events selected within the AP from PSB and PKU models, including regression lines and Pearson correlation coefficients.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Scatter diagram between observed and simulated rain totals (24 h) for all events selected within the AP from PSB and PKU models, including regression lines and Pearson correlation coefficients.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Box plot of HSS and ETS for different thresholds according to the mean and 84th percentile of the distribution function (denoted by the subscript) and rmse and bias quantified from observed and simulated rain totals from PSB and PKU models of all events selected within the AP. The cross-hatched boxes (PSB_{var}) are the results of PSB runs with changing variables (*P*_{∞}, *τ*) applying the optimization routine.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Box plot of HSS and ETS for different thresholds according to the mean and 84th percentile of the distribution function (denoted by the subscript) and rmse and bias quantified from observed and simulated rain totals from PSB and PKU models of all events selected within the AP. The cross-hatched boxes (PSB_{var}) are the results of PSB runs with changing variables (*P*_{∞}, *τ*) applying the optimization routine.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Box plot of HSS and ETS for different thresholds according to the mean and 84th percentile of the distribution function (denoted by the subscript) and rmse and bias quantified from observed and simulated rain totals from PSB and PKU models of all events selected within the AP. The cross-hatched boxes (PSB_{var}) are the results of PSB runs with changing variables (*P*_{∞}, *τ*) applying the optimization routine.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Error measures (a),(c) rmse and (b),(d) bias between observations and simulation results of the (a),(b) PSB and (c),(d) PKU models derived for all rain events selected within the AP. Both the rmse and bias are normalized by the respective totals observed.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Error measures (a),(c) rmse and (b),(d) bias between observations and simulation results of the (a),(b) PSB and (c),(d) PKU models derived for all rain events selected within the AP. Both the rmse and bias are normalized by the respective totals observed.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Error measures (a),(c) rmse and (b),(d) bias between observations and simulation results of the (a),(b) PSB and (c),(d) PKU models derived for all rain events selected within the AP. Both the rmse and bias are normalized by the respective totals observed.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Atmospheric parameters of the 24-h precipitation events selected for CP and AP derived from radiosounding observations in Stuttgart at 0000 and 1200 UTC with mean, quartile distance, minimum, and maximum values.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Atmospheric parameters of the 24-h precipitation events selected for CP and AP derived from radiosounding observations in Stuttgart at 0000 and 1200 UTC with mean, quartile distance, minimum, and maximum values.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Atmospheric parameters of the 24-h precipitation events selected for CP and AP derived from radiosounding observations in Stuttgart at 0000 and 1200 UTC with mean, quartile distance, minimum, and maximum values.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Observed 24-h rain totals vs incoming water vapor flux and *P*_{∞}, indicated by the color of the markers.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Observed 24-h rain totals vs incoming water vapor flux and *P*_{∞}, indicated by the color of the markers.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Observed 24-h rain totals vs incoming water vapor flux and *P*_{∞}, indicated by the color of the markers.

Citation: Journal of Hydrometeorology 12, 1; 10.1175/2010JHM1231.1

Contingency table for dichotomous categorical verification.