In this work, a new forward polarimetric radar operator for the COSMO numerical weather prediction (NWP) model is proposed. This operator is able to simulate measurements of radar reflectivity at horizontal polarization, differential reflectivity as well as specific differential phase shift and Doppler variables for ground based or spaceborne radar scans from atmospheric conditions simulated by COSMO. The operator includes a new Doppler scheme, which allows estimation of the full Doppler spectrum, as well a melting scheme which allows representing the very specific polarimetric signature of melting hydrometeors. In addition, the operator is adapted to both the operational one-moment microphysical scheme of COSMO and its more advanced two-moment scheme. The parameters of the relationships between the microphysical and scattering properties of the various hydrometeors are derived either from the literature or, in the case of graupel and aggregates, from observations collected in Switzerland. The operator is evaluated by comparing the simulated fields of radar observables with observations from the Swiss operational radar network, from a high resolution X-band research radar and from the dual-frequency precipitation radar of the Global Precipitation Measurement satellite (GPM-DPR). This evaluation shows that the operator is able to simulate an accurate Doppler spectrum and accurate radial velocities as well as realistic distributions of polarimetric variables in the liquid phase. In the solid phase, the simulated reflectivities agree relatively well with radar observations, but the simulated differential reflectivity and specific differential phase shift upon propagation tend to be underestimated. This radar operator makes it possible to compare directly radar observations from various sources with COSMO simulations and as such is a valuable tool to evaluate and test the microphysical parameterizations of the model.

Weather radars deliver areal measurements of precipitation at a high temporal
and spatial resolution. Most recent operational weather radar systems have
dual-polarization and Doppler capabilities (called polarimetric below), which provide not only information about the intensity of
precipitation, but also about the type of precipitation (e.g., phase,
homogeneity and shape of hydrometeors). Additionally, the Doppler capability
of weather radars allows monitoring the radial velocity of hydrometeors. In
view of their capacities, weather radars offer great opportunities for
validation of and assimilation in numerical weather prediction (NWP) models.
This is unfortunately far from being a trivial task since radar observables
that are derived from the backscattered power and phase from precipitation
cannot be simply put into relation with the state of the atmosphere as
simulated by the model. There is thus the need for a conversion tool, able to
simulate synthetic radar observations from simulated model variables: a
so-called

Over the past few years, several forward radar operators have been developed.
One of the first efforts was made by

Finally,

Most available radar operators are primarily designed to simulate operational
PPI (plane position indicator) scans from operational weather radars at S, C,
or X bands. However, in research, other types of radar data are available
which can also be relevant in the evaluation of a NWP model, especially for
the simulated vertical structure of precipitation. Some examples of radar
data used for research include satellite swaths at higher frequencies, such
as measurements of the GPM-DPR satellite at Ku and Ka band

The purpose of this work is to design a state of the art forward polarimetric radar operator for the COSMO NWP model taking into account the physical aspects of beam propagation and scattering as accurately as possible, while ensuring a reasonable computation time on a standard desktop computer. The radar operator also needs to be versatile and able to simulate a variety of radar variables at many frequencies and for different microphysical schemes, in order to be used in the future as a model evaluation tool with operational and research weather radar data. As such, this radar operator includes a number of innovative features: (1) the ability to simulate the full Doppler spectrum at a very low computational cost, (2) the ability to simulate observations from both ground and spaceborne radars, (3) a probabilistic parameterization of the properties of solid hydrometeors derived from a large dataset of observations in Switzerland, (4) the inclusion of cloud hydrometeors (which contribution becomes important at higher frequencies). Besides, the radar operator has been thoroughly evaluated using a large selection of radar data at different frequencies and corresponding to various synoptic conditions.

The article is structured as follows: in Sect. 2, a description of the COSMO NWP model as well as the radar data used for the evaluation of the operator is given. In Sect. 3, the different steps of the polarimetric radar operator are extensively described and its assumptions are discussed in details. Section 4 focuses on the qualitative and quantitative evaluation of the simulated radar observables using real radar observations from both operational and research ground weather radars, as well as GPM satellite data. Finally, Sect. 5 summarizes the main results and opens perspectives for possible applications of the operator.

The COSMO Model is a mesoscale limited area model initially developed as the
Lokal-Modell (LM) at the Deutscher Wetterdienst (DWD). It is now operated and
developed by several weather services in Europe (Switzerland, Italy, Germany,
Poland, Romania, and Russia). Besides its operational applications, it is also
used for scientific purposes in weather dynamics, microphysics and prediction
and for regional climate simulations. The COSMO Model is a non-hydrostatic
model based on the fully compressible primitive equations integrated using a
split-explicit third-order Runge–Kutta scheme

In COSMO, grid-scale clouds and precipitation are parameterized operationally
with a one-moment scheme similar to

In COSMO, with the exception of ice crystals and rain in the
two-moment scheme, mass–diameter relations as well as velocity–diameter
relations are assumed to be power-laws. For rain in the two-moment scheme, a
slightly more refined formula by

In the one-moment scheme, which is used operationally, the only free
parameter of the PSDs is

In the two-moment scheme, both

Table

Parameters of the hydrometeor PSDs and power-laws for the two microphysical schemes (separated by a slash sign).

Non-precipitating quantities (cloud droplets and cloud ice) do not have a spectral representation in the one-moment scheme of COSMO, but are instead treated as bulk, with the total number of particles being a function of the air temperature.

In the operational setup, for the parameterization of atmospheric turbulence,
the COSMO model uses a prognostic turbulent kinetic energy (TKE) closure at
level 2.5 similar to

For the evaluation of polarimetric variables, the final product from the
Swiss operational radar network was used. The Swiss network consists of five
polarimetric C-band radars, performing PPI scans at 20 different elevation
angles

For the evaluation of simulated Doppler variables (mean radial velocity and
Doppler spectrum at vertical incidence), observations from a mobile X-band
radar (MXPol) deployed in Payerne in Western Switzerland in Spring 2014 were
used. The radar was deployed in the context of the PARADISO measurement
campaign

An overview of the specifications of all radars used in this study is given
in Table

Besides ground radar data, measurements from the dual-frequency precipitation
radar (DPR,

Specifications of the ground radars used in the evaluation of the radar operator.

Location of the Swiss operational radars. The three radars used in
the context of this study are surrounded by black circles which indicate the
maximum range of radar data (100 km) used for the evaluation of the radar
operator (Sect.

In order to compare the COSMO drop size distribution
parameterizations with real observations, data from three Parsivel-1 optical
disdrometers were used. These instruments were deployed at a short distance
from each other, near the Payerne MeteoSwiss station. Like the X-band radar
presented above, these instruments were deployed in the context of the
PARADISO measurement campaign. The measured drop size distributions were
corrected with measurements from a 2-dimensional video disdrometer (2DVD)
using the method of

A list and short description of all five events used for
the evaluation of the radar operator with data from the operational C-band
radars (Sect.

For the comparison of simulated GPM swaths with real observations, the 100 overpasses with the largest precipitation fluxes recorded between March 2014 and the end of 2016 were selected. Overall, this selection is a balanced mix between widespread low-intensity precipitation and local strong convective storms.

List of all events used for the comparison of simulated radar
observables with real ground radar observations. The last column indicates
the context of the comparison. A indicates the comparison with the
operational C-band radars (Sect.

The radar operator simulates observations of

Forward operator workflow.

Microwaves in the atmosphere propagate along curved lines at speeds

The Equivalent Earth Model is a simple yet often used model, in which the
atmospheric refractive index

In case of non-standard temperature profiles, such as a temperature
inversion, the profile of

The choice of the refraction model (Earth equivalent or atmospheric
refraction) is left to the user of the radar operator, noting that the
computation cost for the latter is slightly larger. The whole evaluation of
the radar operator presented in Sect.

Once the distance at the ground

Once the coordinates of all radar gates have been defined, the model
variables must be interpolated to the location of the radar gates. This is
done with trilinear interpolation (linear interpolation in three dimensions). The advantage of
interpolating model variables before estimating radar observables, instead of
doing the opposite, is twofold. At first, it is much more computationally
efficient, because computing radar observables requires numerical integration
over a particle size distribution at every bin, which is costly. Secondly,
computing radar observables after linear interpolation
allows preservation of the mathematical relation between them. Indeed, radar
variables are far from being independent. For example, in the liquid phase

Technical details about the trilinear interpolation procedure are given in Appendix

In the one-moment scheme, for a given hydrometeor

As in the COSMO microphysical parameterization (see

For the two-moment scheme, the method is similar, except that both mass and
number concentrations are needed to retrieve

Equation (

except for the ice crystals in the one-moment scheme, where COSMO does not consider any spectral representation.

in TableIn our radar operator, cloud droplets are neglected because the radar
operator is designed for common precipitation radar frequencies (2.7 up to 35 GHz), for which the contribution of cloud droplets is very small

Unfortunately, in the one-moment scheme of COSMO, only one single moment is
known, which corresponds to

Taking advantage of these results, it is possible to retrieve a PSD for ice
crystals in the radar operator by (1) using the COSMO temperature to retrieve
an estimate for

Beam broadening increases the sampling volume with increasing range
and is caused by the fact that the normalized power density pattern of the
antenna (shown in red/blue tones) is not completely concentrated on the beam
axis. The blue dots correspond to the integration points used in the
quadrature scheme (in this case with

Part of the transmitted power is directed away from the axis of the antenna
main beam, which will increase the size of the radar sampling volume with
range, an effect known as beam-broadening. Depending on the antenna beamwidth
this effect can be quite significant and needs to be accounted for by
integrating the radar observables at every gate over the antenna power
density pattern. Equation (

In our operator, similarly to

Another often used simplification is to neglect side lobes in the power
density pattern and to approximate

This integration can be accurately approximated with a Gauss–Hermite
quadrature

Another advantage of using a quadrature scheme is that is makes it easy to
consider partial beam-blocking (grayed out area in
Fig.

The choice of this simple Gaussian quadrature was validated by comparison
with an exhaustive integration scheme during three precipitation events (two
stratiform and one convective). The exhaustive integration consists in the
decomposition of a real antenna pattern (obtained from lab measurements) into
a regular grid of 200

Bias and RMSE in terms of

The mathematical formulation of the radar observables involves the

The scattering matrix

The FSA subscript indicates the forward scattering alignment convention, in
which the positive

The direction of the far-field scattered wave is given by the
spherical angles

In the FSA convention, the scattering matrix is also called the

All radar observables for a simultaneous transmitting radar can be defined in
terms of a backscattering covariance matrix

and

The radar backscattering cross sections

All polarimetric variables at the radar gate polar coordinates

Estimation of

The

In order to make the overall computation time reasonable, the scattering
properties for the individual hydrometeors are pre-computed for various
common radar frequencies and stored in three-dimensional lookup tables:

For liquid precipitation (raindrops), the aspect-ratio model of

For solid precipitation, estimation of these parameters is a much more arduous task, since solid particles have a very wide variability in shape. Few aspect-ratio models have been reported in the literature and even less is known about the orientations of solid hydrometeors.

In terms of aspect-ratio,

In terms of orientation distributions, both

Given the large uncertainty associated with the geometry of solid
hydrometeors, a parameterization of aspect-ratios and orientations for
graupel and aggregates was derived using observations from a
multi-angle snowflake camera (MASC). A detailed description of the MASC can
be found in

The inverse of aspect ratio,

Note that using the properties of the inverse distribution, the distribution of aspect-ratios can easily be obtained from the distributions of their inverses:

Figure

Fitted probability density functions for the inverse of the
aspect-ratio

Figure

Note that even if this increase in the polarimetric signature of aggregates
and graupel seems particularly drastic, comparisons with real radar
measurements indicate that the operator is still underestimating the
polarimetric variables in snow (Sect.

Polarimetric variables at X-band (9.41 GHz) as a function of the
mass concentration for snow and graupel when using canting angle and
aspect-ratio parameterizations from the literature

A similar analysis could not be performed for hail, as no MASC observations
of hail were available. Hence, the canting angle distribution is assumed to
be Gaussian with zero mean and a standard deviation of 40

For ice-crystals, the aspect-ratio model is taken from

In the following, the term (complex)

For the permittivity of rain

The permittivity of composite materials, such as snow, which consists of a
mixture of air and ice, can be estimated with a so-called Effective Medium
Approximation (EMA). A well known EMA is the Maxwell–Garnett approximation

Note that other EMAs exist, such as the

Dry solid hydrometeors consist of inclusions of ice in a matrix of air. In
this case

The densities

The matrices

Since the computation of the

The received power at the radar antenna decreases with the square of the
range, which leads to a decrease of signal-to-noise ratio (SNR) with the
distance. To take into account this effect, all simulated radar variables at
range

Stratiform rain situations are generally associated with
the presence of a melting layer (ML), characterized by a strong signature in
polarimetric radar variables

The fraction of wet hydrometeor mass is obtained by converting the total mass of rain and dry hydrometeors within the melting layer into melting aggregates and melting graupel.

The wet fraction within melting hydrometeors can be estimated by the fraction of mass coming from rainwater over the total mass. This results in equal wet fraction for wet snow and wet graupel:

For the mass of wet hydrometeors, the quadratic relation proposed by

For the terminal velocity

This relationship is also used by

For the canting angle distributions, a linear shift of

For a given diameter, the distribution of aspect-ratio for melting
hydrometeors is the renormalized sum of the gamma distribution of dry
aspect-ratios obtained from the MASC observations (Eq.

In Eq. (

The necessary volume fractions of all components

In the first step, Eq. (

Once the mass concentrations and the wet fractions are known, it is possible
to retrieve a particle size distribution for melting hydrometeors. Two
different retrieval methods have been implemented and compared: a

This approach is based on

The functional form

Note that in

In our model, this PSD is further adjusted by multiplying it with a mass
conservation factor

This approach is more empirical and simply assumes that, during melting, the PSD of melting hydrometeors will gradually shift from the PSD of their dry counterpart to the DSD of rain, with increasing wet fraction.

As in the

These two methods were compared by simulating all RHI scans of the PARADISO
campaign (label B in Table

Figure

As a conclusion, as it allows for a more realistic simulation of the melting
layer and agrees better with radar observation, the empirical

Average vertical observed and simulated (with the

Due to the sharp transition it causes in the simulated polarimetric variables, the melting layer effect causes major difficulties when integrating radar variables over the antenna power density. Indeed, the Gauss–Hermite quadrature scheme is appropriate only for continuous functions and will work well with a small number of quadrature points only for a relatively smooth function. Using a small number of quadrature points in the case of a melting layer was found to create unrealistic artifacts with the presence of several shifted melting layers of decreasing intensities. Globally increasing the number of quadrature points by a significant amount is not a viable solution since the computation time will increase linearly. Instead, the best compromise was found by increasing the number of quadrature points only at the edges of the melting layer, where the transitions are the strongest. In practice this is done by using ten times more quadrature points (oversampling factor of 10) in the vertical than normally, but taking into account only the 10 % of quadrature points with the highest weights for the computation of radar variables, except near the melting layer edges where all points are used.

Unfortunately, some trades-off are required to run such a simple oversampling
scheme. Because the number of quadrature points is not constant at every
radar gate (as not all sub-beams cover the whole radar beam trajectory), the
order of attenuation computation and integration have to be reversed, i.e.
attenuation computation is done only at the very end, once all radar
variables (including

As illustrated in Fig.

Estimating

It can be shown

where

In this section we propose a simple scheme able to compute the Doppler
spectrum at any incidence at a very small computational cost (less than 10 %
of the total cost). Unlike

Using the specified hydrometeor terminal velocity relations, it is possible to not only compute the average radial velocity, but also the Doppler spectrum: the power weighted distribution of scatterer radial velocities within the radar resolution volume.

This is done by first computing the resolved velocity classes of the Doppler
spectrum

For every hydrometeor

Once this is done, the corresponding diameters

Trigononometric expression of the radial velocity as the
power-weighted sum of the projection into the beam axis of the 3-dimensional
wind field (

Any statistical moment can then be computed from this spectrum. The average radial velocity, for example is simply the first moment of the Doppler spectrum:

The standard deviation of the Doppler spectrum, often referred to as the
spectral width, is a function of both radar system parameters and
meteorological parameters that describe the distribution of hydrometeor
density and velocity within the sampling volume

In the forward radar operator,

where

For

A constant value of 1.6 is used in the radar operator.

andThis makes it possible to estimate both

In reality, attenuation will cause a decrease in observed radar
reflectivities at all velocity bins within the spectrum. To take into account
this effect, the path integrated attenuation in linear units at a given radar
gate (

Observed computation times for three types of scans and two
computers. The desktop has an 8 core i7-4770S CPU with 3.1 GHz
(30.5 GFlops s

The radar operator was adapted to be able to simulate swaths from spaceborne
radar systems, such as the GPM dual-frequency radar

Though being mostly written in Python, the forward radar operator was
optimized for speed as all computations are parallelized and its most time
consuming routines are implemented in C. In addition, the scattering
properties of individual hydrometeors are pre-computed and stored in lookup
tables. Table

The computation times are usually reasonable even on a standard desktop computer, except when simulating the melting layer effect on a PPI scan at low elevation. However, it can be seen that the forward radar operator scales very well with increasing number of computation power and nodes, since the computation time decreases more or less linearly with increasing computer performance.

In this section, a comparison of simulated radar fields with radar observations is performed. It is important to realize that discrepancies between measured and simulated radar variables can be caused both by of the following reasons.

The inherent inexactitude of the model which manifests itself by differences in magnitude as well as temporal and spatial shifts in the simulated state of the atmosphere, compared with the real state of the atmosphere.

Limitations of the forward radar operator, e.g., imperfect assumptions on hydrometeor shapes, density and permittivity, inaccuracies due to numerical integration, non-consideration of multiple scattering effects.

When validating the radar operator, only the second factor is of interest but as the discrepancies are often dominated by the first factor, validation becomes a difficult task.

Hence, for evaluation purposes, it is important to run the model in its best
configuration, in order to limit as much as possible its inaccuracy. This
is why the model was run in analysis mode, with a 12 h spin-up time,
using analysis runs of the coarser COSMO-7 (7 km resolution) as input and
boundary condition. Note that even though COSMO has recently become
operational at a resolution of 1 km over Switzerland, the simulations
performed in this work were still done at a 2 km resolution. Note that the
present evaluation was done with the standard one-moment scheme, for sake of
simplicity, but Appendix

Evaluation of the radar operator was first done by visual inspection on a time step basis and was followed by a more quantitative evaluation over the course of the whole precipitation events.

Figures

Example of simulated and observed (with the Swiss La Dôle C-band
radar) PPI at 1

Same as Fig.

Figure

Example of RHI showing the observed and simulated melting layer
during the PARADISO campaign in Spring 2014 (Table

Figure

Example of comparisons at several altitude levels between GPM radar
observations at Ka band

Evaluation of the simulated average radial velocities was performed by comparison of simulated velocities with observations from the MXPol X-band radar deployed in Payerne in Western Switzerland in Spring 2014 in the context of the PARADISO measurement campaign.

A total of 720 RHI scans (from 0 to 180

Distributions of simulated (blue) and observed (red) radial velocities at X-band during six days of precipitation in Western Switzerland.

Scatter-plot of the measured and simulated radial velocities (for
all events). The red line shows the 1

During the PARADISO campaign, MXPol was also retrieving the Doppler spectrum
at vertical incidence, which allows comparing simulated spectra with real
measurements. Figure

Simulated and measured daily averaged Doppler spectrum at X-band at
vertical incidence during 6 days of precipitation in Western Switzerland.
The dashed line represents the radial velocity calculated from the spectrum
(Eq.

Evaluation of polarimetric variables (

In order to test the quality of the simulated polarimetric variables, five
events corresponding to different synoptic situations with widespread
precipitation over Switzerland were selected (Table

The duration of all events ranges between 12 and 24 h with a resolution
in time of 5 min (which corresponds to the temporal resolution of the
available radar data). A total of 1017 PPI scans were simulated at
1

The

The comparison between simulated and observed radar variables was performed
separately in the liquid and solid phases. Indeed, the uncertainty in the
liquid phase is expected to be lower than in the ice phase because the
scattering properties of raindrops are more reliable than in snowfall. The
simulated model temperatures were taken as a criterion to separate the
phases; the liquid phase corresponds to

Figure

Figure

In the solid phase, the radar operator tends to underestimate

Observed (red) and simulated (green) distributions of polarimetric
variables (

Observed (red) and simulated (green)

In order to further investigate these surprisingly large discrepancies in the distributions of polarimetric variables between the different COSMO rain DSD parameterizations, a comparison with ground measurements from three Parsivel disdrometer was performed. The disdrometers measurements were integrated over a time interval of 5 min to yield volumic DSDs. The same events used for the Doppler evaluation were used: six events over Payerne in Switzerland dominated by stratiform rainfall. The COSMO DSDs were obtained at the lowest model level, on the grid cell comprising all three Parsivels.

Figure

As conclusion, changing the shape parameter in the COSMO microphysics is a
delicate task, as without re-tuning other parameters in the model, it might
lead,

Average measured (blue bins) and parameterized rain DSDs at the ground in Payerne over six stratiform precipitation events. The dashed black line corresponds to the best fit of a gamma DSD on the measurements

In order to evaluate the simulation of GPM swaths, the distributions of
simulated and observed reflectivities at both Ku and Ka band were compared
for 100 GPM overpasses over Switzerland, corresponding to the overpasses with
the largest precipitation fluxes (c.f. Sect.

Figure

Observed (red) and simulated (blue = one-moment, green =
two-moment) reflectivities at Ku band

In order to evaluate the addition of ice crystals to
the forward operator, a two-fold analysis was performed. First, the simulated
polarimetric variables obtained with and without considering ice crystals
were compared with real observations by MXPol during three pure snowfall
events in the Swiss Alps in Davos (Table

Observed and simulated (with and without ice crystals) distributions of polarimetric variables during three pure snowfall events for the one-moment microphysical scheme.

Due to their smaller sizes, the effect of ice crystals on

QQ-plots of the quantiles of simulated

As a conclusion, adding ice crystals improves the quality of the simulated

In this work we propose a new polarimetric radar forward operator for the COSMO NWP model which is able to simulate measurements of reflectivity at horizontal polarization, differential reflectivity and specific differential phase shift on propagation for ground based or spaceborne (e.g., GPM) radar scans, while taking into account most physical effects affecting the propagation of the radar beam (atmospheric refractivity, beam-broadening, partial beam-blocking and attenuation). Integration over the antenna pattern is done with a simple Gauss–Hermite quadrature scheme. This scheme was compared with more advanced schemes that also take into account antenna side lobes, but was shown to offer on average the best trade-off, due to its better representation of the main lobe and lower computational cost. The operator was extended with a new Doppler scheme, which allows to efficiently estimate the full Doppler spectrum, by taking into account all factors affecting the spectral width (antenna rotation, turbulence, wind shear and attenuation), as well as a melting layer scheme able to reproduce the very specific polarimetric signature of melting hydrometeors, even though the COSMO model does not explicitly simulate them. Finally, the operator was adapted both to the operational one-moment microphysical scheme of COSMO and to its more advanced two-moment scheme. Performance tests showed that the operator is sufficiently fast and efficient to be run on a simple desktop computer.

The scattering properties of individual hydrometeors are pre-computed with
the

The operator was evaluated by a comparison of the simulated fields of radar observables with observations from the operational Swiss radar network, from a high resolution X-band research radar and from GPM swaths. Visual comparisons between simulated and measured polarimetric variables showed that the operator is indeed able to simulate realistic looking fields of radar observables both in terms of spatial structure and intensity and to simulate a realistic melting layer both in terms of thickness and polarimetric signature. Comparisons of the radial velocities measured by the X-band radar and simulated by the radar operator, in the vicinity of the Payerne radiosounding site showed an excellent agreement with a high determination coefficient. The operator was also able to simulate realistic Doppler spectra at vertical incidence, with realistic fall velocities and reflectivities below and above the melting layer, as well as within the melting layer, thanks to the melting scheme. A comparison of the distributions of polarimetric variables as well as the relations between these variables with measurements from the Swiss operational C-band radar network was performed. In the liquid phase, the radar operator is generally able to simulate realistic distributions of polarimetric variables and realistic relations between them. A comparison with measurements from Parsivel disdrometers revealed that the agreement between simulated and observed polarimetric variables depends strongly on the shape parameter used in the drop size distribution of raindrops.

However, in the solid phase the polarimetric variables tend to be
underestimated when using the

Ultimately, this operator provides a convenient way to relate outputs of a NWP model (state of the atmosphere, precipitation) to polarimetric radar measurements. The evaluation of the operator has shown that this tool is a promising way to test the validity of some of the hypothesis of the microphysical parameterization of COSMO. Future work will focus on a detailed sensitivity analysis of the main parameters and assumptions of the radar operator, taking again a large dataset of radar observations as reference. In the liquid phase, the analysis should focus on the geometry of raindrops as well as the parameterization of the DSD. In the ice phase, the potential benefit of using more sophisticated methods to estimate the scattering properties of solid hydrometeors will be investigated.

The radar operator code is available at

Interpolation is computationally faster if the radar gate coordinates are
first converted from the World Geodetic System 1984 (WGS) latitude and longitude coordinates
to the local pole-rotated model coordinates, where the model variables are
defined on a regular grid. To this end, the spherical WGS coordinates of the
radar gate (

Finally, the Cartesian coordinates

For every radar gate, the eight neighbor model nodes can efficiently be
identified by direct mapping of the (

Location of the eight neighbors of a radar gate

In the two-moment scheme all prescribed PSDs are initially defined as a function of particle mass.

However, in the context of this radar operator, it is much more convenient to
work with diameter-based PSDs. This conversion can be done by using the
prescribed mass–diameter relations which are part of the microphysical
scheme:

with

By equating

Note that besides these differences in PSD retrieval, the two-moment scheme also yields slightly different hydrometeor scattering properties, since the mass–diameter relations differ from the one-moment scheme.

Equation (

The phase shift upon backscattering

Despite being
simulated, this quantity was not used in the context of this thesis as it
cumulative and thus cannot be related in an easy way to other radar
observables. Besides, it is often very noisy on real radar data. In fact its
derivative

The final volume-integrated polarimetric estimates

DW designed and implemented the forward radar operator, performed all experiments detailed in this work, and wrote the manuscript. AB contributed to the design and discussion of the work, as well as to the writing of the manuscript.

The authors declare that they have no conflict of interest.

The authors would like to thank MeteoSwiss for providing the data from the Swiss operational radar network. The authors are also thankful to Jacopo Grazioli for the processing of the raw MXPol radar data and to Timothy Hughes Raupach for the processing of Parsivel data. Edited by: Marcos Portabella Reviewed by: two anonymous referees