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.
Introduction
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 forward radar operator.
Over the past few years, several forward radar operators have been developed.
One of the first efforts was made by who designed a
polarimetric operator for the COSMO model, able to simulate horizontal
reflectivity ZH, differential reflectivity ZDR,
and linear depolarization ratio (LDR) observations. The operator relies on the
T-matrix method
to estimate scattering properties of individual
hydrometeors. Assumptions about shape, density, and canting angles, which
cannot be obtained from the NWP model were obtained from a sensitivity study.
A limitation of this operator is that it does not perform any integration
over the antenna power density pattern and thus neglects the beam broadening
effect which can be quite significant at longer distances from the radar
.
developed a three-dimensional stochastic radar
simulator able to simulate raw time series of weather radar data. Doppler
characteristics are retrieved by moving discrete scatterers with the
three-dimensional model wind field, which allows producing sample-to-sample
time series data, instead of theoretical moments as with conventional radar
simulators. Thanks to this, the radar simulator is able to generate the full
Doppler spectrum; however, this is at the expense of a high computation cost and
without taking attenuation into account.
developed a polarimetric radar operator able to simulate
ZH, ZDR as well as the specific differential phase on
propagation Kdp and adapted it for two different microphysical
schemes: one single-moment scheme and one two-moment scheme. The authors also
proposed a method to simulate the effect of the melting layer with a weather
model that does not explicitly simulate wet hydrometeors. They used this
operator to simulate realistic polarimetric radar signatures of a supercell
storm from simulations obtained with the Advanced Regional Prediction System
(ARPS; ). However, the validation of the operator was
limited to idealized cases at S-band only.
developed an advanced forward radar operator for a
research cloud model with spectral microphysics able to simulate
ZH, ZDR, LDR, and Kdp. Scattering
amplitudes of smaller particles are estimated with the Rayleigh approximation
whereas the T-matrix method is used for larger hydrometeors.
However, note
that this cloud model is computationally expensive and is not used for
operational weather prediction.
elaborated a polarimetric forward radar operator for
the French non-hydrostatic mesoscale research NWP model Meso-NH
based on the forward conventional radar operator of
which simulates all operational polarimetric radar
observables: ZH, ZDR, the differential phase shift
upon propagation ϕDP, the co-polar correlation coefficient
ρhv and Kdp. The operator uses the T-matrix method
for rain, snow, and graupel particles and Mie scattering for pristine ice
particles. Beam-broadening is taken into account by approximating the
integration over the antenna normalized power density pattern with a
Gauss–Hermite quadrature scheme.
Finally, developed a forward radar operator for the
COSMO model. The operator is designed for operational purposes (assimilation
and validation) with an emphasis on performance and modularity. It simulates
Doppler velocity with fall speed and reflectivity weighting as well as
attenuated horizontal reflectivity, with different levels of approximation
that can be specified. Note that the operator is currently not able to
simulate polarimetric variables.
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
as well as power weighted distributions of scatterer
radial velocities (Doppler spectra), commonly recorded by many research
radars.
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.
Description of the dataCOSMO model
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 . The
spatial discretization is based on a fifth-order upstream advection scheme on
an Arakawa C-grid with Lorenz vertical staggering. Height-based Gal-Chen
coordinates are used in the vertical . The model
uses a rotated coordinate system where the pole is displaced to ensure
approximatively horizontal resolution over the model domain. Sub-grid scale
processes are taken into account with parameterizations.
In COSMO, grid-scale clouds and precipitation are parameterized operationally
with a one-moment scheme similar to and
, with five hydrometeor categories: rain, snow, graupel,
ice crystals, and cloud droplets. Snow is assumed to be in the form of rimed
aggregates of ice-crystals that have become large enough to have an
appreciable fall velocity. Cloud ice is assumed to be in the form of small
hexagonal plates. In the version of COSMO that is being used (5.04), ice
crystals have a bulk non-diameter dependent terminal velocity, that depends
on their mass concentration. The particle size distributions (PSD) are
assumed to be exponential for all hydrometeors, except for rain where a gamma
PSD is assumed. A more advanced two-moment scheme with a sixth hydrometeor
category, hail, was developed for COSMO by and
extended by and . As this scheme
significantly increases the overall computation time it is currently not used
operationally.
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 is used. For ice
crystals, the two-moment scheme, in contrast with the one-moment scheme uses
a spectral (diameter-dependent) representation of ice crystal terminal
velocities. For both microphysical schemes, all PSDs can be expressed as
particular cases of generalized gamma PSDs.
N(D)=N0Dμexp-Λ⋅Dνm-3mm-1,
where N0 is the intercept parameter in units of
mm-1-μ m-3, μ is the dimensionless shape parameter,
Λ is the slope parameter in units of mm-ν and ν
is the dimensionless family parameter.
In the one-moment scheme, which is used operationally, the only free
parameter of the PSDs is Λ which can be obtained from the prognostic
mass concentrations. N0 is either assumed to be constant during the
simulation, or in the case of snow, to be temperature dependent. μ is
equal to zero (exponential PSDs) for all hydrometeors, except for rain where
it is set to 0.5 by default and ν is always equal to one.
In the two-moment scheme, both Λ and N0 are prognostic parameters,
and can be obtained from the prognostic moment of order zero (number
concentration) and from the mass concentration. μ and ν are defined
a priori.
Table gives the values of the PSD parameters μ,
N0, and ν as well as the mass–diameter power-law
parameters a and b and the terminal velocity–diameter
power-law parameters α and β for all hydrometeor types and the
two microphysical schemes.
Parameters of the hydrometeor PSDs and power-laws for the two microphysical schemes (separated by a slash sign). ∅
indicates that the hydrometeor is not simulated in this scheme, a dash
indicates that this parameter is not used in this parameterization, and
“free” indicates a prognostic parameter. As in Eq. (),
N0 is expressed in units of mm-1-μm-3. Note that the value
of μ for rain can be specified in the COSMO user set-up, 0.5 being the
default value. The parameters a and b correspond to the power-law: m(D)=aDb, with m is in kg and D in mm. The parameters α and β
correspond to the power-law: vt(D)=αDβ, with
vt being the terminal fall velocity in m s-1, and D is
the diameter in mm.
1for snow, a relation of N0 with the temperature is used .
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 . The main difference is the use
of variables that are conserved under moist adiabatic processes: total cloud
water and liquid water potential temperature. Additionally, a so-called
“circulation term” is included which describes the transfer of nonturbulent
subgrid kinetic energy from larger-scale circulation toward TKE. The reader
is referred to and the model documentation
for a more in-depth description of the various COSMO
sub-grid parameterizations.
Radar data
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 . The final quality-checked measurements are
corrected for ground clutter, calibrated and aggregated at a resolution of
500 m. In this work, ZH was used as provided, ZDR was
corrected with a daily radar-dependent calibration constant provided by
MeteoSwiss, and Kdp was estimated from ΨDP using
the Kalman filter ensemble method of . Note that
two of the operational radars were installed only recently (2014 and
2016) and were thus not used in this study (see Fig. ).
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 . The PARADISO dataset provides a great
opportunity to evaluate the simulated radial velocities, as Payerne is the
location from which the radiosoundings, which are assimilated every 3 h in the model, are launched.
An overview of the specifications of all radars used in this study is given
in Table . The location of the Swiss operational radars
used in the evaluation of the radar operator (Sect. ) and
their maximum considered range (100 km) are shown in
Fig. .
Besides ground radar data, measurements from the dual-frequency precipitation
radar (DPR, ), on-board the core satellite of the Global
Precipitation Measurement mission (GPM, ) were used to
validate the simulation of spaceborne radar swaths. The GPM-DPR radar
operates at both Ku (13.6 GHz) and Ka (35.6 GHz) bands. At Ku-band, the
satellite swath covers approximately 245 km in width, with a horizontal
resolution approximatively 5 km and a 250 m vertical (radial) resolution. At
Ka-band, the satellite swath is more narrow, covering only 125 km in width.
Specifications of the ground radars used in the evaluation of the radar
operator.
MXPolSwiss radar networkLocationPayerne: 46.813∘ N,Albis: 47.284∘ N, 8.512∘ E, 891 m a.s.l.6.943∘ E, 495 m a.s.l.La Dôle: 46.425∘ N, 6.099∘ E, 1680 m a.s.l.Monte Lema: 46.040∘ N, 8.833∘ E, 1604 m a.s.l.Frequency f9.41 GHz (X-band)5.6 GHz (C-band)Pulse width τ0.5 µs0.577 µsPRF1666 Hz500 to 1500 Hz (depends on elevation)FFT length128-3 dB beamwidth1.45∘1∘Sensitivity (SNR = 10 dB)11 dBZ at 10 km0 dBZ at 10 km
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. ). Note that as they were installed only
quite recently, no data from the Weissfluhgipfel and Plaine Morte radars were
used in this study.
Parsivel data
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 . For more details regarding
these instruments, see . All disdrometers were
located within the same COSMO grid cell, so the measured DSDs were simply
averaged before comparing them with the COSMO parameterizations.
Precipitation events
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. ) and all six events from the PARADISO
campaign used for the evaluation of the radar operator with data from MXPol
(Sect. ) and from Parsivel data
(Sect. ) is given in Table .
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. ), B indicates the
comparison with the X-band radar (Sect. ), and the
Parsivel data (Sect. ) in Payerne and C indicates the
evaluation of ice crystals with the X-band radar in the Swiss Alps in Davos
(Sect. ).
EventDescriptionUsed for1 February 2013Heavy snowfall event with strong westerly geostrophic winds.A22 March 2014Stationary front with widespread stratiform liquid precipitation over Switzerland.B8 April 2014After the crossing of a cold front, presence of mostly liquidwidespread stratiform precipitation over Switzerland.A/B1 May 2014Occlusion over Switzerland with mild temperatures andwidespread stratiform precipitationB7 May 2014Wake of a cold front with scattered stratiform precipitationB11 May 2014Wake of a cold front with strong scattered stratiform and occasionally convective precipitationB14 May 2014Occlusion over Switzerland with mild temperatures andwidespread stratiform precipitationB8 November 2014The first two weeks of November 2014 were characterized by very heavy rainfall over the Southern Alps with strong Foehn winds, due to the presence of a very strong low pressure system over the Mediterranean (Xandra).A9 January 2015Crossing of a warm front over Switzerland with widespread stratiform precipitation and snowfall over the Swiss Alps.C26 January 2015Snowfall event over the Swiss Alps with very similar characteristics to the 9 January 2015 eventC23 February 2015Crossing of a cold front over Switzerland with some widespread and medium-intensity snowfallC13 August 2015Strong summer convection triggered by the presence of very warm and wet subtropical air over Switzerland.A7 June 2016Presence of warm and moist air over Western Europe with a succession of thunderstorms.ADescription of the polarimetric radar operator
The radar operator simulates observations of ZH, ZDR,
Kdp, average Doppler (radial) velocity, and of the full Doppler
spectrum based on COSMO simulations and user-specified radar characteristics,
such as its position, its frequency, the 3 dB antenna beamwidth
Δ3dB, the pulse duration τ, and the pulse repetition
frequency (PRF). Figure summarizes the main steps of this
procedure, which will be more extensively detailed in the further section.
Forward operator workflow.
Propagation of the radar beam
Microwaves in the atmosphere propagate along curved lines at speeds v<c
as the permittivity of the atmosphere ϵ is larger than ϵ0,
the permittivity of vacuum. In the case of large atmospheric permittivity
gradients the beam can even be refracted back to the surface, which can cause
distant ground objects to appear on the radar scan. Obviously in order to
simulate the propagation of the radar beam, the effect of atmospheric
refraction needs to be taken into account. In the radar operator, computing
the distance at the ground s, and the height above ground h for every
radial distance r (see Fig. ), can be done in two
ways.
Equivalent Earth Model
The Equivalent Earth Model is a simple yet often used model, in which the
atmospheric refractive index n=ϵ is assumed to be a
horizontally homogeneous linear function of height dndh=const.
This approximation is simple and often used in practice, as it does not
require any knowledge about the current state of the atmosphere, and is quite
accurate as long as the assumed vertical profile of n is valid in the first
kilometers of the atmosphere.
Atmospheric refraction model
In case of non-standard temperature profiles, such as a temperature
inversion, the profile of n can vary significantly from the one assumed by
the Equivalent Earth Model, which can lead to strong underestimation of the
beam refraction. Fortunately proposed a more generic
and accurate model that is based on the vertical profile of atmospheric
refractivity derived from the model data. This vertical profile can be
approximated from the temperature T, the partial pressure of water vapor
Pw, and the total pressure P. The height at a given
range can then be estimated by solving a second order ordinary differential
equation derived from Snell's law for spherically stratified layers. Again,
this model assumes horizontal homogeneity of the atmospheric refractivity.
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. was performed
with the more advanced model of .
Interpolation of model variables
Once the distance at the ground s and the height above ground h are
obtained from the refraction model, it is easy to retrieve the latitude, longitude, and height
coordinates (ψWGS, λWGS, h) of the
corresponding radar gate, knowing the beam elevation θ0 and azimuth
ϕ0 angles, as well as the position of the radar.
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
ZH is closely co-fluctuating with ZDR, in the form of
a power-law that tends to stagnate at large reflectivities. Some tests were
performed on random Gaussian fields of rain mass concentration. The results
indicate that when computing the radar observables first and then
interpolating them, this theoretical relation becomes more and more linear
when the interpolation resolution increases, which is
quite unrealistic. On the contrary, when computing the radar variables after
interpolating the rain concentration field, the theoretical relationship is
always preserved, regardless of the interpolation technique
that is being used.
Technical details about the trilinear interpolation procedure are given in Appendix .
Retrieval of particle size distributions
In the one-moment scheme, for a given hydrometeor j, the COSMO specific
mass concentration QM(j) in kg m-3 is proportional to a
specific moment of the particle size distributions (PSD), since the COSMO
parameterizations assumes simple power-laws for the mass–diameter relations:
m(j)(D)=a(j)Db(j). Because all COSMO PSDs belong to the class
of generalized gamma PSDs, QM can be expressed as
follows:
As in the COSMO microphysical parameterization (see ), the
PSDs are assumed to be only weakly truncated and the integration bounds
[Dmin(j), Dmax(j)] are replaced by [0, ∞),
in order to get an analytical solution and avoid the cost of numerical root
finding. Note that this truncation hypothesis is done only for the retrieval
of Λ and not when computing the radar observables
(Sect. and Appendix ). For the
one-moment scheme, by integrating the Eq. (), one gets the
following expression for the free parameter Λ(j).
For the two-moment scheme, the method is similar, except that both mass and
number concentrations are needed to retrieve Λ and N0. The
corresponding mathematical formulation is given in
Appendix .
Equation () allows retrieving the PSD parameters
for all hydrometeors
except for the ice crystals in the one-moment
scheme, where COSMO does not consider any spectral representation.
in
Table at every radar gate using the model variable
QM(j), and, for the two-moment scheme, the prognostic number
concentration QN(j) (M0) as well. Knowing the PSDs
(N(j)(D)) makes it possible to perform the integration of polarimetric
variables over ensemble of hydrometeors as will be described in the next
steps of the operator.
In 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
. However, at higher frequencies and in weak precipitation,
the contribution of ice crystals can be significant, especially for
ZDR, as these crystals can be quite oblate
. Therefore, ice crystals are considered explicitly,
even though they do not have a spectral representation in the one-moment
scheme of COSMO. Instead, a realistic PSD is retrieved with the double-moment
normalization method of . This formulation of the PSD
requires to know two moments of the PSD as well as an appropriate normalized
PSD function. proposes best-fit relations between the
moments of ice crystals PSDs as well as fits of generating functions for
different pair of moments. Precisely, assuming moments 2 (M2)
and 3 (M3) of the size distributions are known,
suggest parameterizing the PSD in the following way:
Nice(D)=M24⋅M3-3ϕ23(x),withx=DM2M3,
with
ϕ23(x)=490.6exp(-20.78x)+17.46x0.6357exp(-3.290x).
Unfortunately, in the one-moment scheme of COSMO, only one single moment is
known, which corresponds to M3, since the value of the b
parameter in the mass–diameter power-law for ice crystals is equal to 3 (see
Table ). Fortunately, also
provide best-fit relations relating M2 to other moments of the
PSD. According to these relationships, M3 can be estimated from
M2 with
M3≈a(3,Tc)M2b(3,Tc),
where a(3,Tc) and b(3,Tc) are polynomial functions of the in-cloud
temperature (in ∘C) and the moment order (3 in this case).
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 a(3,Tc) and b(3,Tc), (2) inverting
Eq. () to get an estimate of M2, and
(3) use Eqs. () and () to estimate the
PSD of ice crystals.
Integration over the antenna pattern
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 J,K=3 for illustration purposes) and
their size depends on their corresponding weights. The effect of atmospheric
refraction on the propagation of the radar beam is also illustrated: r is
the radial distance (radar range), s is the ground distance, and h the
distance above ground of a given radar gate, which need to be estimated
accurately.
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 () formulates the antenna integration
for an arbitrary radar observable y and a normalized power density pattern
of the antenna represented by f2, as in .
In our operator, similarly to and
, we set W(r0-r)=1 if r∈r0-cτ4,r0+cτ4 and W(r0-r)=0 otherwise.
Indeed since the model resolution (1–2 km) is about one order of magnitude
larger than the typical gate length of a modern radar (80–250 m), effects
related to the finite receiver bandwidth can be neglected. Integration over
r can still be done a posteriori by using a higher radial
resolution and aggregating the simulated radar observables afterwards.
Another often used simplification is to neglect side lobes in the power
density pattern and to approximate f2 by a circularly symmetric Gaussian.
These simplifications reduce the integration to Eq. ().
This integration can be accurately approximated with a Gauss–Hermite
quadrature :
Iy(ro,θo,ϕo)≈∑j=1Jwj′cosθ0+zj′∑k=1Kwk′y(r0,θ0+zj′,ϕ0+zk′),
where wj′=σwj, wk′=σwk and zj′=σzj, zk′=σzk with σ=Δ3dB22log2,
where Δ3dB is the 3 dB beamwidth of the antenna in
degrees. wj and zj are respectively the weights and the roots of the
Hermite polynomial of order K (for elevational integration) and wk and
zk are the weights and roots of the Hermite polynomial of order K (for
azimuthal integration). For the integration in the radar operator, default
values of J=5 and K=7 are used according to . The
quadrature points thus correspond to separate sub-beams with different
azimuth and elevation angles that are resolved independently. A schematic
example of this quadrature scheme is shown in Fig.
for J,K=3.
Another advantage of using a quadrature scheme is that is makes it easy to
consider partial beam-blocking (grayed out area in
Fig. ). Note that in our operator, the blocked
sub-beams are simply lost (i.e., are not considered in the integration) and no
modeling of ground echoes is performed. However, as was done in the
evaluation of the operator (Sect. ), these beams can
easily be identified and removed when comparing simulated radar observables
with real measurements.
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 × 200 sub-beams. Such an integration is
obviously extremely computationally expensive and can not be considered as a
reasonable choice of quadrature in practice. Four other quadrature schemes
were tested, (1) a sparse Gauss–Hermite quadrature scheme
, (2) a custom hybrid Gauss–Hermite and Legendre quadrature
scheme based on the decomposition of the real antenna diagram in radial
direction with a sum of Gaussians, (3) a Gauss–Legendre quadrature scheme
weighted by the real antenna pattern, and (4) a recursive Gauss–Lobatto scheme
based on the real antenna pattern. All schemes were
tested in terms of bias and root-mean-square-error (RMSE) in horizontal
reflectivity ZH and differential reflectivity ZDR
as a function of beam elevation (from 0 to 90∘), taking the
exhaustive integration scheme as a reference.
Figure shows an example for one of the two
stratiform events. It was observed that the simple Gauss–Hermite scheme was
the one which performed the best on average (lowest bias and RMSE for both
ZH and ZDR), with schemes (1) and (3) performing
almost systematically worse. Schemes (2) and (4) tend to perform slightly
better at low elevation angles in particular situations where strong vertical
gradients are present, generated for instance by a melting layer or by strong
convection. This is due to the fact that in these situations, the
contribution of the side lobes can become quite important, for example when
the main beam is located in the solid precipitation above the melting layer
but the first side lobe shoots through the melting layer or the rain
underneath. However, considering that these schemes are more computationally
expensive and tend to perform worse at elevations > 3∘, it was
decided to keep the simple Gauss–Hermite scheme, which seems to offer the
best trade-off. However, as an improvement to the operator, it could be
possible to use an adaptive scheme that depends on the specific state of the
atmosphere and the beam elevation.
Bias and RMSE in terms of ZH during one day of
stratiform of precipitation (around 120 RHI scans), for the five possible
quadrature schemes. The exhaustive quadrature scheme is used as a reference.
The other two events show similar results.
Derivation of polarimetric variables
The mathematical formulation of the radar observables involves the
scattering matrixS, which relates the scattered electric
field Es to the incident electric field Ei for a given scattering angle.
EhsEvs=e-ik0rrSFSAEhiEvi,
where k0 is the wave number of free space (k0=2π/λ).
The scattering matrix SFSA is a 2×2 matrix of
complex numbers in units of m-1e.g.,.
SFSA=shhshvsvhsvvFSA
The FSA subscript indicates the forward scattering alignment convention, in
which the positive z-axis is in the same direction as the travel of
the wave (for both the incident and scattered wave). A sketch illustrating
the reference unit vectors for the scattered wave in the FSA convention is
given in Fig. .
The direction of the far-field scattered wave is given by the
spherical angles θs and ϕs, or by the unit
vector ψ^s. In the FSA convention, the horizontal
and vertical unit vectors are defined as h^s=ϕ^s and v^s=ϕ^s. The unit vectors for the spherical coordinate system
form the triplet
(ψ^s,θ^s,ϕ^s),
which in the FSA convention becomes
(ψ^s,v^s,h^s),
with ψ^s=v^s×h^s. This figure was adapted from Bringi and
Chandrasekar (2001).
In the FSA convention, the scattering matrix is also called the
Jones matrix . In the following the
coefficients of the backscattering matrix (scattering towards the radar) will
be denoted by sb, and the coefficients of the forward scattering
matrix (scattering away from the radar) by sf.
All radar observables for a simultaneous transmitting radar can be defined in
terms of a backscattering covariance matrix Cb and a forward
scattering vector Sf. For a given hydrometeor of type (j) and
diameter D.
Sf,(j)(D)=shhf,(j)svvf,(j)∈C2×1,
where the superscripts b and f indicate backward, respectively forward
scattering directions and s are elements of the scattering matrix
SFSA (Eq. ) that relates the
scattered electric field to the incident electric field for a given particle
of diameter D.
The radar backscattering cross sections σb are easily
obtained from Cb:
All polarimetric variables at the radar gate polar coordinates
(ro,θo,ϕo) are function of Cb and Sf and
can be obtained by first integrating these scattering properties over the
particle size distributions, summing them over all hydrometeor types and
finally integrating them over the antenna power density. The exhaustive
mathematical formulation of all simulated radar observables is given in
Appendix . Additionally, real radar observations of
ZH and ZDR are affected by attenuation, which
needs to be accounted for to simulate realistic radar measurements. The
specific differential phase shift on propagation Kdp also needs
to be modified in order to account for the specific phase shift on
backscattering (see Appendix ).
Scattering properties of individual hydrometeors
Estimation of Cb,(j) and Sf,(j) for individual hydrometeors is
performed with the transition-matrix (T-matrix) method. The T-matrix method
is an efficient and exact generalization of Mie scattering by randomly
oriented nonspherical particles . Since the
shape of raindrops is widely accepted to be well approximated by spheroids
e.g.,, the
T-matrix method provides a well suited method for the computation of the
scattering properties of rain. This method was also used for the solid
hydrometeors (snow, graupel, hail and ice crystals), at the expense of some
adjustments, that will be described later on.
The T-matrix method requires knowledge about the permittivity, the shape as
well as the orientation of particles. Since particles are assumed to be
spheroids, the aspect-ratio ar, defined in the context of this work as the
ratio between the smallest dimension and the largest dimension of a particle,
is sufficient to characterize their shapes. The orientation o is defined as
the angle formed between the horizontal and the major axis (canting angle
∈ [-90, 90]) and can be characterized with the Euler angle β
(pitch).
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:
diameter, elevation and temperature for dry
hydrometeors and diameter, elevation and wet fraction for wet hydrometeors (Sect. ). On run time,
these scattering properties are then simply queried from the lookup tables,
for a given elevation angle and temperature and wet fraction.
Aspect-ratios and orientationsRain
For liquid precipitation (raindrops), the aspect-ratio model of
is used and the drop orientation us assumed to be
normally distributed with a zero mean and a standard deviation of 7∘
according to .
Snow and graupel
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, report values ranging
between 0.6 and 0.8 for dry aggregates and between 0.6 and 0.9 for graupel
while reports a median aspect-ratio of 0.6 for
aggregates and a strong mode in graupel aspect-ratios around 0.9.
In terms of orientation distributions, both and
consider a Gaussian distribution with zero mean and
a standard deviation of 40∘ for aggregates and graupel in their
simulations.
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 . MASC observations recorded during one
year in the Eastern Swiss Alps were classified with the method of
, giving a total of around 30 000 particles for both
hydrometeor types. The particles were grouped into 50 diameter classes and
inside every class a probability distribution was fitted for the aspect-ratio
and the orientations. For sake of numerical stability, the fit was done on
the inverse of the aspect-ratio (large dimension over small dimension). In
accordance with the microphysical parameterization of the model, the
considered reference for the diameter of solid hydrometeors is their maximum
dimension.
The inverse of aspect ratio, 1/ar, is assumed to follow a gamma
distribution, whereas the canting angle o is assumed to be normally
distributed with zero mean, and the parameters of these distributions depend
on the considered diameter bin ⌊D⌋.
o:go(o,D)=N0,σo(D),1ar:g1/ar(1/ar,D)=(1ar-1)Λar(D)-1exp-1ar-1M(D)M(D)Λar(D)Γ(Λar(D))b,
where Λar and M are the shape and scale
parameters of the gamma aspect-ratio probability density function and
σo is the standard deviation of the Gaussian canting angle
distribution. These parameters depend on the diameter D. Technically
Λ, M and σo have been fitted separately for each
single diameter bin of MASC, then their dependence on D has been fitted by
power-laws for each parameter, which also allows further integration over the
canting angle and aspect-ratio distributions for all particle sizes. Note
also that the gamma distribution is rescaled with a constant shift of 1, to
account for the fact that the smallest possible inverse of aspect-ratio is 1
and not 0.
Note that using the properties of the inverse distribution, the distribution
of aspect-ratios can easily be obtained from the distributions of their
inverses:
gar(ar,D)=1ar2g1/ar(1/ar,D).
Figure shows the fitted densities for every
diameter and every value of inverse aspect-ratio and canting angle. Overlaid
are the empirical quantiles (dashed lines) and the quantiles of the fitted
distributions (solid lines). Generally the match is quite good. The fitted
models are able to take into account the increase in aspect-ratio spread and
decrease in canting angle spread with particle size, which are the two
dominant trends that can be identified in the observations.
Fitted probability density functions for the inverse of the
aspect-ratio (a) and the canting angle (b). The power-laws relating
the particle density function parameters to the diameter are displayed in the
grey boxes on the top-left. Note that the fit was performed on the inverse of
the aspect-ratio (major axis over minor axis).
Figure shows the effect of using this MASC-based
parameterization instead of the values from the literature
on the resulting polarimetric variables. Whereas
only a small increase is observed for the horizontal reflectivity
ZH, the difference is quite important for ZDR and
Kdp, especially for graupel. The MASC parameterization tends to
produce a stronger polarimetric signature. It is interesting to notice that
ZDR tends to decrease with the mass concentration, which is
rather counter-intuitive as ZDR is thought to be independent of
concentration effects. This can be explained by the fact that, in COSMO, the
density of snowflakes decreases with their size (they become less compact)
and therefore the permittivity computed with the mixture model decreases as
well. When the concentration increases, the proportion of larger (and more
oblate) snowflakes increases but given their smaller permittivity, the
overall trend is a slight decrease in ZDR. This trend hence
reflects an assumption in COSMO, not necessarily the reality.
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
(solid line) and when using the parameterization based on MASC data (dashed
line).
Hail
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∘, while
the aspect-ratio model is taken from .
arhail=1-0.02D,ifD<10mm0.8,ifD≥10mm
Ice crystals
For ice-crystals, the aspect-ratio model is taken from
for hexagonal columns, while the canting angle distribution is assumed to be
Gaussian with zero mean and a standard deviation of 5∘, which
corresponds to the upper range of the canting angle standard deviations
observed by in cirrus and midlevel clouds.
Permittivities
In the following, the term (complex) permittivity will be used for
the relative dielectric constant of a given material. It is defined as
follows:
ϵ=ϵ′+iϵ′′,
where ϵ′ is the real part, related to the phase velocity of the
propagated wave, and ϵ′′ is the imaginary part, related to the
absorption of the incident wave.
Rain
For the permittivity of rain ϵr, the well known model of
for the permittivity of water at microwave
frequencies is used. Note that recently, a new model for water permittivity
has been proposed by , which appears to provide a
better agreement with field observations at high frequencies. However, for
common precipitation radar frequencies (<30 GHz) and temperatures (>-20∘) both models agree very well.
Snow, graupel, hail and ice crystals
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
, in which the effective medium consists of a matrix
medium with permittivity ϵmat and inclusions with
permittivity ϵinc:
ϵeff=ϵmat1+2fvolincϵinc-ϵmatϵinc+2ϵmat1-fvolincϵinc-ϵmatϵinc+2ϵmat,
where ϵeff is the effective permittivity of the composite
material, and fvolinc is the volume fraction of the
inclusions.
Note that other EMAs exist, such as the and
approximations. If none of the components is a strong
dielectric, all these EMAs approximately agree to first order
. The interested reader is referred to ,
for an intercomparison of these EMA in the context of simulated reflectivity
fields.
Dry solid hydrometeors consist of inclusions of ice in a matrix of air. In
this case ϵmat≈1, which leads to a simplified
form of the mixing formula e.g.,.
ϵ(j)=1+2fvoliceϵice-1ϵice+21-fvoliceϵice-1ϵice+2,
where fvolice is the volume fractions of ice within the
given hydrometeor (snow, graupel or hail) and ϵice is the
complex permittivity of ice, which can be estimated with
's formula.
The densities ρ(j) can be easily obtained from the COSMO
mass–diameter relations ρ(j)=a(j)D(b)π/6D3
and the density of ice is assumed to be constant ρi=916 kg m-3.
Integration of scattering properties
The matrices Cb,(j)(D) (Eq. )
and Sf,(j)(D) (Eq. ) are obtained by integration
over distributions of canting angles and, for snow and graupel,
aspect-ratios. For Cb,(j) this gives the following for snow and graupel:
Cb,(j)(D)=12π∫02π∫-π/2π/2∫01cb,(j)(D,ar,α,o)cos(o)go(o,D)gar(ar,D)dαdodar,
and for rain and hail, where ar is constant for a given diameter:
Cb,(j)(D)=12π∫02π∫-π/2π/2cb,(j)(D,α,o)cos(o)go(o,D)dαdo,
where cb,(j)(D,α,o) are the scattering properties for a fixed
diameter, canting angle o, and yaw Euler angle (azimuthal orientation)
α. go(o) and gar are the probabilities of o and ar for a
given diameter D as obtained from Eqs. () and
(). Note that the final scattering properties are averaged over
all azimuthal angles α, which are all considered to be equiprobable.
The cos(o) in the equation is the surface element which arises
from the fact that the integration over α and o is a surface
integration in spherical coordinates. The procedure for Sf is
exactly the same.
Since the computation of the T-matrix for a large number of canting angles
and aspect-ratios can be quite expensive, two different quadrature schemes
were used, one Gauss–Hermite scheme for the integration over the Gaussian
distributions of canting angles, and one recursive Gauss–Lobatto scheme
for the integration over aspect-ratios.
Taking into account the radar sensitivity
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 rg are censored if
ZH(rg)<S+G+SNRthr+20⋅log10rgr0,
where G is the overall radar gain in dBm, S is the radar antenna
sensitivity in dBm, ZH is the horizontal reflectivity factor in
dBZ, and SNRthr corresponds to the desired signal-to-noise
threshold in dB (typically 8 dB in the following). r0 is a distance used
to normalize the argument of the logarithm. If all units are consistent then
r0=1.
Simulation of the melting layer effect
Stratiform rain situations are generally associated with
the presence of a melting layer (ML), characterized by a strong signature in
polarimetric radar variables e.g.,. In order
to simulate realistic radar observables, this effect needs to be taken into
account by the radar operator. Unfortunately COSMO does not operationally
simulate wet hydrometeors, even though a non-operational parameterization was
developed by . proposed a method to
retrieve the mass concentration of wet snow aggregates by considering
co-existence of rain and dry hydrometeors as an indicator of melting. A
certain fraction of rain and dry snow is then converted to wet snow, which
shows intermediate properties between rain and dry snow, depending on the
fraction of water within (wet fraction). As a first try to simulate the
melting layer we have implemented the method of and adapted
it to also consider wet graupel. However, two issues with this method have
been observed. First of all the co-existence of liquid water and wet
hydrometeors causes a secondary mode in the Doppler spectrum within the
melting layer, due to the different terminal velocities, a mode that was
never observed in the corresponding radar measurements. Secondly, the
splitting of the total mass into separate hydrometeor classes (rain and wet
hydrometeors) causes an unrealistic decrease in reflectivity just underneath
the melting layer. It was thus decided to use an alternative parameterization
in which only wet aggregates and wet graupel exist within the melting layer.
At the bottom of the melting layer, where the wet fraction is usually almost
equal to unity, these particle behave almost like rain and at the top of the
melting layer, where the wet fraction is usually very small, these particles
behave like their dry counterparts. Note that contrary to
, which explicitly consider separate prognostic variables
for the meltwater on snowflakes, our scheme is purely diagnostic and is meant
to be used in post-processing, when the COSMO model has been run without a
parameterization for melting snow.
Mass concentrations of wet hydrometeors
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.
Qms=Qs+QrQsQs+Qg,Qmg=Qg+QrQgQs+Qg,
where the superscripts s, g and r indicate dry snow, dry graupel and
rain, and ms and mg indicate wet snow and graupel. Note that the mass of
rainwater is added to the mass of wet hydrometeors proportionally to their
relative fractions.
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:
fwetms=fwetmg=QrQs+Qg+Qr.
Diameter dependent propertiesMass
For the mass of wet hydrometeors, the quadratic relation proposed by
is used:
mm(D)=fwetm2mr(D)+1-fwetm2md(D),
where the superscript d indicates the corresponding dry hydrometeor and the
superscript m indicates the melting hydrometeor. The considered diameter
D is the actual maximum dimension of a melting particle, and not the melted
diameter.
Terminal velocity
For the terminal velocity vtm of melting hydrometeors, the equation is
computed from the terminal velocities of rain and dry hydrometeors, using a
best-fit obtained from wind tunnel observations by .
vtm(D)=ϕvtr(Dr)+(1-ϕ)vtd(D),
where ϕ=0.246fwetm+(1-0.246)fwetm7. Dr is the equivalent melted diameter of the
particle. Dr is related to D by
Dr(D)=ρm(D)ρwater1/3D.
This relationship is also used by and .
Canting angle distributions
For the canting angle distributions, a linear shift of σcant
(the standard deviation of the Gaussian distribution of canting angle) with fwetm is considered:
σcantm(D)=fwetmσcantr(D)+(1-fwetm)σcantd(D).
Aspect-ratio
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. )
and the aspect-ratio distribution of rain, linearly weighted by the melting
fraction fwetm. Since for rain the aspect-ratio is considered
constant for a given diameter, the distribution would be a Dirac. Instead, in
order to perform the weighted sum, the distributions of aspect-ratios in rain
are represented by a very narrow Gaussian distribution
(σa-rr = 0.001) centered around the corresponding
aspect-ratio.
Permittivity
In Eq. (), we have previously introduced the
general two-component Maxwell-Garnett EMA. However, melting hydrometeors are
a mixture of three components: water, ice, and air. To compute their
permittivity, the general two-component formulation is used recursively,
first to derive the permittivity of dry snow (as was done previously for dry
snow, graupel, hail and ice crystals), and then the permittivity of the dry
snow and water mixture.
The necessary volume fractions of all components fvol can again
be estimated with the mass–diameter model:
fvolwater=fwetmρmρwater,fvolice=ρm-fvolwaterρwaterρice,fvolair=1-fvolwater-fvolice,
where ρm=mm(D)π/6D3 is the density of the melting
hydrometeor.
In the first step, Eq. () is used with
fvolinc=fvolicefvolice+fvolair, ϵmat≈1,
ϵinc=ϵice, to yield
ϵd, the permittivity of the dry part of the melting
hydrometeor. However, for the second step, the estimated permittivity of the
melting hydrometeor will depend on whether water is treated as the matrix and
snow as the inclusions or the opposite, giving two different possible
outcomes. To overcome this issue, a formulation proposed by
is used, where the final permittivity is a weighted
sum of both permittivities and where the weights are function of the wet
fraction. This method is also used by . Precisely,
Eq. () is used first with
fvolinc=fvolwater and
ϵmat=ϵd, ϵinc=ϵwater, to yield ϵm,(1), and at
second with fvolinc=fvolair+fvolice and
ϵmat=ϵwater,
ϵinc=ϵd, to yield
ϵm,(2). The final ϵm is a weighted
sum of ϵm,(1) and ϵm,(2):
ϵm=12(1+τ)ϵm,(1)+(1-τ)ϵm,(2),
where parameter τ is a function of fwetm:
τ=Erf21-fwetmfwetm-1iffwetm>0.01.
Particle size distribution for melting hydrometeors
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
flux-based approach and a more empirical weighted PSD
approach.
Flux-based approach
This approach is based on and assumes a one-to-one
correspondence between rain and dry solid hydrometeors, i.e., one
snowflake/graupel leads to one raindrop during the melting process (no
shedding/aggregation). This implies that one can match the flux of melting
hydrometeors with the equivalent flux of rainwater:
Nr(Dr)vtr(Dr)dDr=Nm(D)vtm(D)dD⇒Nm(D)=Nr(D)vtr(D)vtm(D)dDrdD,
where vt is the hydrometeor terminal velocity.
The functional form dDrdD can be estimated from
Eqs. () and (), by taking into account the
fact that the mass–diameter relation of the dry hydrometeor equivalent is a
power-law: md(D)=adDbd.
dDrdD=13(fwet2)+CDbd-3-2/3C(bd-3)Dbd-3+(fwet2)+CDbd-31/3,
with C=ad6(1-fwet2)2πρwater.
Note that in , the functional form dDrdD
was neglected.
In our model, this PSD is further adjusted by multiplying it with a mass
conservation factor κ to ensure that the integral of the PSD weighted
by the particle mass matches the mass concentrations of wet hydrometeors
Qm. Hence Nm,corr(D)=κNm(D) with
κ=Qm∫DminDmaxmm(D)Nm(D)dD,
where mm(D) is the mass of a melting particle of diameter D
(Eq. ).
Weighted PSD approach
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.
Nm(D)=fwetNr(Dr)dDrdD+(1-fwet)Nd(D)
As in the flux-based approach, this PSD is then corrected to ensure
conservation of the simulated mass concentration by Nm,corr(D)=κNm(D), with κ as in Eq. ().
These two methods were compared by simulating all RHI scans of the PARADISO
campaign (label B in Table ), and comparing them with
radar observations recorded by MXPol at X-band. These events correspond
mostly to stratiform precipitation with an omnipresent melting layer.
Figure shows the vertical of profile of ZH and ZDR
averaged over all scans at which time a melting layer was detected on the
radar observations, using the method of . In
the computation of this vertical profile, for every scan only the 10 first
kilometers from the radar have been considered for ZH, and kilometers 7 to
10 have been considered for ZDR, which is ill-defined at high elevation.
To remove biases in the simulated precipitation intensities, the values of
ZH and ZDR have been normalized by subtracting from every scan the
average value in the liquid phase below the melting layer. Moreover, to
remove biases in the height of the isotherm 0∘, the reference height
is the height relative to the peak of the detected bright-band peak (maximum
of ZH). It can be seen clearly, that the weighted PSD approach
produces a much more realistic bright-band peak in ZH, when compared with
the radar observations. Moreover, the transition to the solid phase is also
more realistic, even though the simulated reflectivities in dry snow seem too
small, which is a different problem. In terms of ZDR, the simulations tend
to produce a peak that is too narrow, and no approach seems significantly
better than the other. Besides agreeing better with the radar observations in
terms of bright-band peak, the weighted PSD approach has another
major advantage: it allows for a seamless transition between the PSD of
melting hydrometeors and the PSD of dry solid hydrometeors above the melting
layer. Contrarily, in the flux-based approach, there is no
continuity for fwet=0, as the modeled wet PSD does not converge
perfectly towards the PSD of dry hydrometeors. This results in very abrupt
transitions in polarimetric variables above the melting layer (several dBZ
over one or two radar gates), and unrealistic increases in reflectivity
when very weak concentrations of rain are present above the isotherm
0∘.
As a conclusion, as it allows for a more realistic simulation of the melting
layer and agrees better with radar observation, the empirical
weighted PSD approach was retained in the radar operator.
Average vertical observed and simulated (with the
flux-based and weighted approaches) profiles of
ZH and ZDR. The x-axis corresponds to the
average shift with respect to the average values in the liquid phase below
the ML. The y-axis corresponds to the distance with respect to the
peak of the bright-band.
Integration scheme
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 kh and kv) have been integrated over the antenna
diagram. This is a somewhat strong simplification but it is the only way to
perform a local oversampling, which is the only computationally feasible way
to simulate the melting layer effect with volumetric integration. The effect
of this approximation was investigated for the strong convective event of the
13 August 2015 (with J=5, K=7 and an oversampling factor of 10). The
results indicate an overestimation of the final ZH by an average of 0.58 dBZ, with respect to the normal integration scheme. This bias is caused by
the underestimation of the attenuation effect. However, for ZDR, the bias
is negligible (0.03 dB), which is likely due to the fact that this
simplification affects ZH and Zv to a similar extent.
Retrieval of Doppler velocitiesAverage radial velocity
As illustrated in Fig. , the average radial velocity
vrad is the power-weighted sum of the projections of U
(eastward wind component), V (northward wind component), W (vertical wind
component), and vt, the hydrometeor terminal velocity, onto the axis of
the radar beam defined by elevation θ0 and azimuth ϕ0.
Estimating vrad requires knowing the terminal velocity of
precipitating hydrometeors. In this work, we use the power-law relations
prescribed by COSMO's microphysical parameterizations with parameters as
given in Table .
It can be shown e.g., that, in the hypothesis of
radial homogeneity inside a radar resolution volume, the average radial
velocity at a given radar gate characterized by coordinates r (range),
ϕ (azimuth) and θ (elevation) is given by Eq. (),
where σhb,(j)(D) is the backscattering radar
cross-section at horizontal polarizations for an hydrometeor of type j and
diameter D and I is the quadrature antenna integration operator defined
in Eq. ().
where
vrad(j)(D,r,ϕ,θ)=U(r,ϕ,θ)sinϕ+V(r,ϕ,θ)cosϕcosθ+W(r,ϕ,θ)-vt(j)(D)sinθ,
Doppler spectrum
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 , this approach is not
based on sampling and is thus deterministic, but the computational cost is
much smaller.
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 vr,bins[i], for every bin i, based on the specified
radar FFT window length NFFT and Nyquist velocity
vNyq.
vrad,bins[i]=(i-NFFT2)vnyqNFFT∀i=-N2,…,N2,
where vNyq is the Nyquist velocity, in m s-1, given by
vNyq=100PRF⋅λ2,
where λ is the radar wavelength in cm.
For every hydrometeor j and every velocity bin i, given the
three-dimensional wind components (U, V, W), one can estimate the
hydrometeor terminal velocity vt that would be needed to yield the radial
velocity vrad,bins[i]:
Once this is done, the corresponding diameters D(j)[i] can be retrieved
by inverting the diameter-velocity power-laws (see
Table ). Finally, for a given radar gate defined by
coordinates (ro, ϕo, θo) the Doppler spectrum S in linear
Ze units (mm6 m-3), for a given velocity bin i is
Trigononometric expression of the radial velocity as the
power-weighted sum of the projection into the beam axis of the 3-dimensional
wind field (U,V,W) and the hydrometeor terminal velocity vt.
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 . Assuming
independence of the spectral broadening mechanisms, the square of the
velocity spectrum width σv2 (i.e., standard deviation of the
spectrum) can be considered as the sum of all contributions
.
σv2=σs2+σα2+σd2+σo2+σt2,
where σs2 is due to the wind shear, σα2 to the
rotation of the radar antenna, σd2 to variations in hydrometeor
terminal velocities, σo2 to changes in orientations or vibration of
hydrometeors and σt2 to turbulence.
In the forward radar operator, σs2 is already taken into
account by the integration scheme, σd2 by the use of the
diameter-velocity relations, and σo2 by the integration of
the scattering properties over distributions of canting angles. Thus, the
spectrum computed in Sect. () needs to be corrected only
for turbulence and antenna motion. gives the following
estimation for σα.
σα=ωλcosθo2πΔ3dBlog2,
where ω is the angular velocity (in rad s-1). Note that
σα is equal to zero at vertical incidence, which is the most
common configuration for Doppler spectrum retrievals.
For σt, gives the following estimation, originally
derived by , which is based on the hypothesis of isotropic
and homogeneous turbulence, with all contributions to turbulence coming from
the inertial subrange.
σt=ϵtro1.35B3/20.721/3if σr≪rσθϵtσr1.35B3/21115+415(r2σθ2σr-2-3/21/3else,
where B is a constant between 1.53 and 1.68
A constant value of
1.6 is used in the radar operator.
and ϵt is the eddy dissipation
rate (EDR) expressed in units of m2 s-3. ϵt is the rate at
which turbulent kinetic energy is converted into thermal internal energy. It
is a model variable, simulated by the turbulence parameterization and can be
obtained as any other variable used in the radar operator, by interpolation to the radar gates. Finally
σr and σθ depend on the radar specifications: σr=0.35cτ/2 (τ is the pulse duration in s) and
σθ=Δ3dB/4log(2).
This makes it possible to estimate both σo and σt using the
specified radar system parameters and simulated turbulence variables. If one
assumes the spectral broadening caused by the antenna motion and turbulence
to be Gaussian with zero mean e.g.,, the
corrected spectrum can be obtained by convolution with the corresponding
Gaussian kernel.
Scorr[i]=∑j=0NFFTS[j]G(vrad,bins[i]-vrad,bins[j])∑j=0NFFTG(vrad,bins[i]-vrad,bins[j]),
where G is the Gaussian kernel defined by
G(x)=1σt+α2πexp-x22σt+α2,
where σt+α=σt+σα
Attenuation computation in the Doppler spectrum
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 (kh in Eq. ) is distributed uniformly
throughout the spectrum.
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-1) and 32 GB of RAM, the server has a 12 core i7-3930K
with 3.20 GHz (59 GFlops s-1) and 32 GB of RAM.
The radar operator was adapted to be able to simulate swaths from spaceborne
radar systems, such as the GPM dual-frequency radar
at both Ku and Ka bands. The main modifications to the standard routine
concern the beam tracing, which is estimated from the GPM data (in HDF5
format) by using the WGS84 coordinates at the ground and the radar position
in Earth-centered–Earth-fixed coordinates to retrieve the coordinates of
every radar gate. Currently, the atmospheric refraction is neglected which
leads to an average positioning error of 55 m, the error being minimal at the
center of the swath (where the incidence angle is nearly vertical) and
maximal at the edges of the swath. The integration scheme remains unchanged
and a fixed beamwidth of 0.5∘ is used according to GPM
specifications. An important advantage of simulating satellite radar
measurements over simply comparing the precipitation intensities at the
ground, is that it allows a three-dimensional evaluation of the model data.
Computation time
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 gives some indication of the computation
times encountered for different types of simulated scans. The RHI scan
consists of 150 different elevations in the main direction of the
precipitation system, with a maximal range of 20 km and a radial resolution
of 75 m. The melting layer is simulated with the quadrature oversampling
scheme described in Sect. . The RHI scan was also
computed with the full Doppler scheme (Sect. ). The PPI
scan consists of 360 different azimuth angles at 1∘ elevation at
C-band, with a maximal range of 150 km and a radial resolution of 500 m. All
scans were performed in a stratiform rain situation (8 April 2014 for ground
radars and 4 April 2014 for GPM), with a wide precipitation coverage. The
advanced refraction scheme by was used for all scans
except the GPM swath. To integrate over the antenna density pattern 3
quadrature points in the horizontal and 5 in the vertical were used for all
scans (with an oversampling factor of 10 at the ML edges).
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.
Evaluation of the operator
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 gives some additional
indications and results for the two-moment scheme.
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.
Qualitative comparisonsPPI scans at C-band
Figures and show two examples of
simulated and observed PPI scans from the La Dôle radar in western
Switzerland at 1∘ elevation during one mostly convective event (13 August 2015) and one mostly stratiform event (8 April 2014). The displayed
radar reflectivities are raw uncorrected ones, and the attenuation effect is
taken into account for simulated reflectivities. It can be seen that in both
cases, the model is able to locate the center of the precipitation event
quite accurately but tends to overestimate its extent, especially in the
convective case. Generally, the simulated ZH, ZDR and
Kdp are of the same order of magnitude as the observed ones, with
the exception of the stratiform case, where the simulated Kdp is
underestimated on the edges of the precipitating system. The simulated radial
velocities seem very realistic and agree well with observations both in terms
of amplitude and spatial structure.
Example of simulated and observed (with the Swiss La Dôle C-band
radar) PPI at 1∘ elevation during the 13 August 2015 convective event
(Table ). Panels (a, c, e, g) correspond to the
simulated radar observables and (b, d, f, h) to the observed ones. The
displayed variables are, from top to bottom, the horizontal reflectivity
factor (in dBZ) (a, b), the differential reflectivity (in dB) (c, d), the specific
differential phase shift upon propagation (in ∘km-1) (e, f), and the
radial velocity (in m s-1) (g, h).
Same as Fig. but for the stratiform event on
the 8 April 2014 (Table ).
RHI with melting layer at X-band
Figure shows one example of simulated and observed RHI
scan in a stratiform situation (22 March 2014) with a clearly visible melting
layer at low altitude. It can be seen that the forward radar operator is
indeed able to simulate a realistic polarimetric signature within the melting
layer with a clearly visible bright-band in ZH, an increase in ZDR
followed by a sharp decrease in the solid phase above and higher values of
Kdp. The extent of the melting layer seems also to be quite accurate when
compared with radar measurements. Note that, in this case, the model slightly
overestimates the signature in ZDR and ZH below the melting layer, but
this is related to the fact that COSMO tends to overestimate the rain
intensity during this particular event. In terms of radial velocities, again
the model simulates some very realistic patterns that agree well with the
observations, with two shear transitions at around 1 and 3.5 km altitude
followed by a strong increase in velocities at higher altitudes.
Example of RHI showing the observed and simulated melting layer
during the PARADISO campaign in Spring 2014 (Table ).
Panel (a) corresponds to the simulated radar observables, panel (b) to the observed values at X-band. Note that there is an area with
velocity folding (blue area in the middle of a larger red area) around 5 km
altitude and 10–15 km horizontal distance on the radar RHI scan.
GPM swath
Figure shows an example of simulated and measured GPM swath
at Ku band at different altitudes. Again the forward radar operator produces
a realistic horizontal and vertical structure as well as plausible values of
reflectivities, given the fact that in this case the simulated average rain
rate is lower than the GPM estimated average rain rate (0.38 mm s-1
vs. 0.46 mm s-1).
Example of comparisons at several altitude levels between GPM radar
observations at Ka band (a) and the corresponding radar operator
simulation from the COSMO model (b) for one GPM overpass.
Doppler variables
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∘ elevation) were simulated
over six days of mostly stratiform precipitation (c.f.
Table ). Figure shows a comparison
of the distributions of radial velocities between the simulation and the
radar observations. Note that in the scope of this work, the
term density indicates the frequency density, in analogy with a
probability density function. It represents the proportion of samples within
every bin divided by the width of the bin, such that the integral of the
empirical distribution is equal to one. It is thus in units of x-1,
where x is the unit of the considered variable (in this particular case x= m s-1). The scatter-plot in Fig. shows the
excellent overall agreement when considering all events and scans.
Simulations observations match very well, both in terms of distributions and
in terms of one-to-one relations, which shows that the radar operator is
indeed able to simulate accurate radial velocities. Since wind observations
from the radiosoundings performed in Payerne are assimilated into the model,
one can expect it to perform well in this regard. These results indeed
confirm these expectations.
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 : 1 relation. The coefficient of
determination (R2) is 0.9.
During the PARADISO campaign, MXPol was also retrieving the Doppler spectrum
at vertical incidence, which allows comparing simulated spectra with real
measurements. Figure shows the daily averaged
simulated and measured Doppler spectra during the same six days of
precipitation. Generally, the simulated spectrum is able to reproduce the
transition from high velocities near the ground (in liquid precipitation) to
smaller velocities in altitude (solid precipitation). The height of this
transition, which corresponds roughly to the isotherm 0∘, as well as
the simulated velocities above and below the isotherm 0∘ agree quite
well with the observations. Thanks to the melting layer scheme, the operator
is able to produce a quite realistic transition between solid and liquid
phase. Indeed, when the melting scheme is disabled, the simulated Doppler
spectra show a very abrupt and unrealistic transition in velocities. In terms
of reflectivity, the bright-band effect is clearly visible on the simulated
spectra and its magnitude relative to the reflectivities below and above the
melting layer agrees well with observations. However, in absolute terms, some
events show a good agreement (22 March 2014, 7 May 2014), while in others,
the simulated reflectivities tend to be overestimated over the whole spectrum (8 April, 14 and 1 May 2014). However, we think that these
discrepancies are mostly caused by the larger precipitation intensities
simulated by the model during these days. Precipitation measurements with a
rain gauge collocated with the radar tend to confirm this hypothesis. For the
two events with the strongest discrepancies (1 and 14 May), the gauge
measured in total 1.9 and 1.2 mm of precipitation, whereas the model
simulated 16.9 and 2.1 mm of precipitation in the closest grid cell.
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. ).
Polarimetric variables
Evaluation of polarimetric variables (ZH,
ZDR and Kdp) is difficult, because their agreement
with radar observations depends heavily on the temporal and spatial accuracy
of simulated precipitation fields. However, when averaging over a
sufficiently large number of samples, the radar operator should at least be
able to simulate realistic distributions of polarimetric variables, as well
as realistic relations between these polarimetric variable.
, for example, validated their operator, inter alia,
by comparing simulated and observed membership functions between the
polarimetric functions.
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 simulated polarimetric variables were compared with observations from
three operational C-band radars (La Dôle, Albis, and Monte Lema).
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∘ elevation with a maximum range of 100 km (in order to limit the
effect of beam-broadening). Both observed and simulated radar data were
censored with an SNRthr value of 8 dB (Eq. ).
The shape parameter of the gamma DSD used in COSMO for rain has a
strong influence on the outcome of the radar operator. Indeed, the skewness
of the gamma distribution is inversely proportional to μrain,
so DSDs with small values of μrain will have longer right
tails. This is of particular importance when simulating polarimetric
variables that are related to statistical moments of a high order, such as
ZDR. Two values of μrain have been tested,
μrain=0.5, which is the default value in the model and
μrain=2, which corresponds to the upper range of recommended
values in the model. Note that the COSMO model has been run twice, once with
μrain=0.5 and once with μrain=2.
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 T>5∘ and the solid phase
to T<-5∘ as in . Areas with temperatures
in between have been ignored in order to limit the contribution of wet snow,
which is not directly simulated by COSMO. It was observed that increasing the
temperature margin between liquid and solid phases did not change
significantly the main results and conclusions. However, decreasing it would
affect quite significantly the observed radar signatures due to the inclusion
of measurements from the melting layer, which have a much stronger
polarimetric signature than dry snow.
Figure shows the corresponding histograms of
observed and simulated polarimetric variables and precipitation intensities
at the ground in the liquid phase, for μrain=2. The
histograms for μrain=0.5 (not displayed) show only minor
differences. The simulated distributions agree well with the observed ones in
terms of broad features, which confirms the fact that the operator is able to
simulate realistic radar observables at least in liquid phase. One can
observe that the radar operator is not able to simulate negative
ZDR, which can be explained by the assumptions about the drop
shapes and orientations, which make it almost impossible for a drop to have a
vertical dimension larger than its horizontal dimension. In addition, the
radar operator seems to produce slightly smaller values of ZH
than observed, but this can be attributed to the fact that COSMO tends to
simulate smaller precipitation intensities than the ones estimated from the
radar reflectivities (bottom-right of Fig. ).
Indeed, the discrepancies in ZH agree well with the discrepancies
in precipitation intensities.
Figure shows the observed (from MeteoSwiss radars)
and the simulated ZH-ZDR and ZH-Kdp relations averaged over all radars and all events in the
liquid and solid phases. It appears that the radar operator is able to
simulate realistic relations between polarimetric variables at least in the
liquid phase. In terms of ZDR, a value of μrain=2
seems more appropriate than a value of 0.5, which tends to overestimate the
differential reflectivity for a given horizontal reflectivity. For
Kdp the trend is reversed. A possible explanation is that
ZDR is independent of the mass concentration and highly dependent
on the length of the DSD tail, i.e., small differences in the numbers of large
and oblate drops can cause large differences in differential reflectivity.
However, Kdp depends on both the mass concentration and the tail
of the DSD, and is quite sensitive to the mode of the DSD. However, one must
also keep in mind that the “observed” Kdp values are in fact
estimated from noisy Ψdp measurements and as such are likely
to be underestimated . This dependency of
simulated polarimetric variables on small changes in the DSD shape
illustrates quite well the difficulty parameterizing the DSDs to match both
the lower order moments used in weather prediction (number and mass
concentration) and the higher order moments, to which the radar observables
are related.
In the solid phase, the radar operator tends to underestimate ZDR
and Kdp, which is a trend also observed by
. This is likely due to the combination of the
imperfect parameterization of snow PSD in the model, the crude assumptions
about the permittivity of snow and graupel (mixture model derived from the
COSMO density parameterizations), and the estimation of the scattering
properties (T-matrix is likely not correct for ice-phase hydrometeors).
Observed (red) and simulated (green) distributions of polarimetric
variables (ZH,ZDR and Kdp) as well
as the precipitation intensities on the ground (in log scale) for the
one-moment scheme with μrain=2 in the liquid phase.
Observed (red) and simulated (green) ZH-ZDR and ZH-Kdp relationships for the
COSMO one-moment scheme in liquid and solid phases. These membership
functions are computed by dividing all simulated values in bins of
reflectivity of 1 dBZ of width, and computing the quantiles of the dependent
variable (on the y-axis) within every bin.
Comparison of the COSMO rain DSDs with ground measurements
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 shows a comparison of the average
measured rain DSD and the COSMO parameterized DSDs over the six days of
precipitation. It is obvious that the COSMOS DSDs with μrain=0.5 tends to produce too many small drops when compared with the Parsivel
data. However, one must keep in mind that due to the instrument's limitations,
the Parsivel, as most disdrometers, has difficulty measuring very small
drops and might underestimate their numbers .
However, one can still observe with certitude that the mode of the COSMO
parameterized DSDs is located too much on the left, especially for
μrain=0.5. When fitting a gamma DSD on the measured data, the
optimal value of μrain is around 3.4, which indicates that the
match with the real radar observations could possibly be even better by
increasing even more the value of μrain. However, one must keep
in mind the numerous difficulties in the comparison of these DSDs. First of
all, the sampling volumes are vastly different around 80 millions m3 for the COSMO grid cell, around 10 000 m3 for the three
Parsivels integrated over a time interval of 5 min and averaged over 520
of these time intervals. Secondly, the shape of the DSDs depend strongly on
the simulated precipitation intensity which is not always agreeing with
observations (rain gauges). Regarding the first point, giving the large
homogeneity of the studied precipitation events (widespread stratiform rain),
the representativity issue comparison still has some relevance. Concerning
the second point, since precipitation intensity is a moment of the DSD, one
can expect a better agreement of Parsivel observations with more realistic
COSMO microphysics, especially for larger particles.
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, in fine, to a degradation of the surface precipitation. Using
it solely off-line in the context of the forward radar operator
might be a better choice, as it can help to reduce the bias in simulated
polarimetric variables.
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
GPM swaths
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 shows the overall distributions of
reflectivity at both frequency bands as well as the distributions of
estimated GPM precipitation intensities and COSMO simulated intensities at
the ground. Note that all reflectivities below 14 dBZ have been discarded as
this corresponds roughly to the radar sensitivities at Ka and Ku band
. Although the distributions are very consistent,
some minor discrepancies are present, mostly for low reflectivities (at Ka
band only) and high reflectivities which appear more frequently in the
simulations than in the measurements from the GPM-DPR. Again, this is
consistent with the differences in simulated precipitation intensities (in
panel c). COSMO tends to produce a larger number of precipitation
intensities ≥30 mm h-1 as well as a larger number of
precipitation intensities below 0.15 mm h-1 which corresponds roughly
to 14 dBZ. Note that similar observations in terms of underestimation of
surface rainfall intensities by GPM with respect to the Swiss operational
rain gauge and radar precipitation products have been reported by
. Overall, the simulated distributions of reflectivity
at both frequency bands are realistic and agree quite well with the
observations for both microphysical schemes. Note that when neglecting ice
crystals the match is much poorer (see Sect. ).
Observed (red) and simulated (blue = one-moment, green =
two-moment) reflectivities at Ku band (a) and Ka band (b), as well as the
precipitation intensities (in log-scale) at the ground (c) estimated by GPM
and simulated by COSMO.
Effect of ice crystals
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 ). Since no
liquid precipitation or melting layer was present during these events, the
attenuation effect is expected to be negligible. Note that the analysis
focused on the one-moment scheme but the effect on the two-moment scheme is
expected to be quite similar. Figure shows a
comparison of the distributions of polarimetric variables in the solid phase
averaged over all three events for the one-moment microphysical scheme. On
ZH, the effect of adding ice crystals is characterized by an
additional mode around 8 dBZ, which is not present on radar observations.
This mode is caused by the large homogeneity in the simulated ice crystals,
which, according to the microphysical parameterization, are all assumed to be
hexagonal plates. In reality, ice crystals can have a large variability of
shapes e.g.,, and their backscattering
coefficients can be quite different , which would result
in a much more spread out reflectivity signature of ice crystals. On
ZDR, one can see that, when neglecting ice crystals, one
completely removes the right tail of the distribution (values above 0.2 dBZ)
that is clearly visible on the observed values. When considering ice
crystals, which have a quite strong signature in differential reflectivity,
this right tail gets accurately reproduced and matches well with the
observations. However, even when adding ice crystals, the radar operator is
not able to reproduce the negative ZDR values that are quite
frequent in the observations. On Kdp, a similar effect can be
observed, though not as clear. Still, the addition of ice crystals creates an
additional mode in the distribution of simulated values which slightly better
matches with the observed one (longer tail and good agreement of the
additional mode with the mode of the observed distribution). Just as with
ZDR, the radar operator is not really able to simulate negative
values of Kdp, which are also frequent in the observations. However, these
discrepancies could also be due in part to uncertainties in the radar
observations, coming from possible miscalibration (for ZDR) and
inaccuracies in the retrieved Kdp values. Still, overall at
X-band, the addition of ice crystals leads to a much better representation of
ZDR in solid precipitation, a slightly better representation of
Kdp and no significant improvement in ZH.
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 ZH
should increase with the frequency. To investigate this effect, a second
comparison was performed on the simulation of GPM swaths, with and without
ice crystals. The resulting distributions of ZH at Ku and Ka band
were compared with means of QQ-plots of observed versus simulated quantiles.
Figure shows these QQ-plots at Ka band for both the
one-moment and the two-moment scheme. The red line is the 1 : 1 which implies
a perfect match with the observed quantiles. The results at Ku band are not
displayed as they are visually very similar to the results at Ka band. For
the one-moment scheme, a much better agreement with observations is observed
for small quantiles (up to 20 dBZ) when adding ice crystals. Without ice
crystals, small quantiles tend to be underestimated. Large simulated
quantiles tend to be overestimated when compared with GPM observations. For
very large quantiles, this overestimation is slightly stronger when adding
ice crystals but this might be a sampling effect as large quantiles are very
sensitive to outliers. For the two-moment scheme, adding ice crystals does
not seem to significantly improve the agreement with observed quantiles.
QQ-plots of the quantiles of simulated ZH values
versus the quantiles of observed GPM ZH values at Ka band. The
red line corresponds to the 1 : 1 line indicating a perfect match with observed
quantiles.
As a conclusion, adding ice crystals improves the quality of the simulated
ZDR and Kdp in pure solid precipitation at X-band and
when simulating horizontal reflectivities at K band.
Conclusions
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 T-matrix method and stored into lookup tables for various frequencies.
The permittivities for the complex hydrometeors (snowflakes, hail and
graupel) are obtained with a mixture model by using the mass–diameter
relations of COSMO to estimate their densities. The other required parameters
for the T-matrix method (canting angle distributions and aspect-ratios) are
obtained from the literature (for rain, hail and ice crystals) and from
measurements performed in the Swiss Alps with a multi-angle snowflake camera
(MASC), for snow and graupel. A large number of MASC pictures were used to
estimate realistic parameterizations of the distributions of aspect-ratio and
canting angle of graupel and aggregates, leading to a good agreement with
measured quantiles. Integration of the hydrometeors scattering properties
over these distributions was shown to increase the polarimetric signature of
solid hydrometeors, which tends to be often underestimated in radar
operators.
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 T-matrix method to simulate hydrometeor
scattering properties, even with the local MASC parameterization. Finally the
effect of considering ice crystals in the simulation or not was investigated
and it was observed that at X-band the agreement with observed differential
reflectivity and differential phase shift improves significantly, whereas at
GPM frequencies, the simulated distributions of reflectivity are more
realistic, especially for smaller reflectivities.
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.
Code availability
The radar operator code is available at
https://github.com/wolfidan/cosmo_pol (Wolfensberger and Berne, 2018).
Trilinear interpolation
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 (ψWGS= lon, λWGS= lat) are
first projected to Earth-centered–Earth-fixed (ECEF) coordinates (x,y,z)
and then rotated to the pole-rotated system using two rotation matrices, one
for the longitudinal rotation of the pole ΔλWGS,
and one for the latitudinal rotation of the pole
ΔψWGS, to yield (xm,ym,zm).
Finally, the Cartesian coordinates (xm,ym,zm) in the model pole-rotated
system, are projected back to spherical coordinates to yield
(ψm,λm), the spherical coordinates of radar gates in the model
pole-rotated system.
For every radar gate, the eight neighbor model nodes can efficiently be
identified by direct mapping of the (ψm, λm) coordinates (which
as stated are on a regular grid) and by binary search through all vertical
model levels. Once the neighbors have been identified
(Fig. ), interpolation is done by first
linearly interpolating all neighbors with identical (ψm, λm) to
the height z of the radar gate: (Au,Al)→A⋆, (Bu,Bl)→B⋆, (Cu,Cl)→A⋆, (Du,Dl)→D⋆. The resulting points
(A⋆, B⋆,C⋆, D⋆) are then bilinearly
interpolated to the horizontal location of the radar gate.
Location of the eight neighbors of a radar gate R. The position of the radar gate is shown by a red star.
Specificities of the two-moment scheme
In the two-moment scheme all prescribed PSDs
are initially defined as a function of particle mass.
Nm(x)=N0,mxμmexp(-Λmxνm),
where the subscript m denotes that the quantity is mass-based and Nm(x)
is in units of kg-1 m-3.
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: D(x)=amxbm⇒x=Dam1bm and
by considering that Nm(D)=Nd(x)⋅dDdx=am(bm-1)xbm-1Nd(x), where the subscript
d denotes that the quantity is
diameter-based and Nd(x) is in units of mm-1 m-3. Replacing this
in Eq. () yields
By equating M0 with the number concentration QN and adMbd with the mass concentration QM, where ad=am-1/bm and bd=1/bm, one is able to retrieve the N0,d and
Λd from the prognostic parameters of the PSDs.
N0,d=νdQNΓμd+1νdΛdμd+1νdandΛd=1adΓμd+1νdΓμd+bd+1νdx‾-νd/bd,
where x‾=QM/QN is the average particle mass.
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.
Polarimetric equations
Equation () give the basic
polarimetric equations integrated over ensembles of hydrometeors for every
radar gate defined by a given set of spherical coordinates xg=(rg,θg,ϕg), where rg is the range, θg is the elevation
angle θg and ϕg is the azimuth angle. The backscattering
covariance matrix Cb, forward scattering vector Sf,
and backscattering cross-sections σb for a given hydrometeor
(j), are defined as in Eqs. (), () and
(). λ is the wavelength in cm.
Zh(xg)=λ4π5|Kw|2∑j=0H∫Dmin(j)Dmax(j)N(j)(D,xg)⋅σhb,(j)(D,xg)dDmm6m-3Zv(xg)=λ4π5|Kw|2∑j=0H∫Dmin(j)Dmax(j)N(j)(D,xg)⋅σvb,(j)(D,xg)dDmm6m-3Kdp(xg)=0.18πλ∑j=0H∫Dmin(j)Dmax(j)N(j)(D,xg)⋅ℜS1f,(j)(D,xg)-S2f,(j)(D,xg)dD∘km-1δhv(xg)=180πλarg∑j=0H∫Dmin(j)Dmax(j)N(j)(D,xg)C2,1b,(j)(D,xg)dD∘kh(xg)=λ∑j=0H∫Dmin(j)Dmax(j)N(j)(D,xg)ℑS1f,(j)(D,xg)dDkm-1kv(xg)=λ∑j=0H∫Dmin(j)Dmax(j)N(j)(D,xg)ℑS2f,(j)(D,xg)dDkm-1
where Zh and Zv are the linear reflectivity factors at horizontal and
vertical polarizations, Kdp, is the specific differential phase
shift upon propagation, δhv is the total differential phase
shift upon backscattering, and kh and kv are the attenuation
coefficients in linear scale.
The phase shift upon backscattering δhv is not taken into
account in Kdp, because the radar Kdp retrieval
method that is being used is able to remove the
contribution of δhv. However, besides Kdp, the
total phase shift Ψdp is also simulated
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 Kdp, estimated from radar observations with robust
differentiation techniques, is much more useful and widely used.
, which
combines the phase shift due to backscattering and propagation. Additionally,
the effect of two-way attenuation is taken into account for ZH
and Zv. This yields the following polarimetric products at every
radar gate and for every sub-beam (Eq. ).
The final volume-integrated polarimetric estimates
ZHatt‾,
ZDRatt‾, Kdp‾, and
Ψdp‾ are obtained by integrating the necessary
quantities over all sub-beams with the quadrature antenna integration
operator I defined in Eq. (). The linear reflectivity
factors are also converted to logarithmic scale.
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.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
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
ReferencesAndsager, K., Beard, K. V., and Laird, N. F.: Laboratory measurements of axis
ratios for large rain drops, J. Atmos. Sci., 56, 2673–2683,
10.1175/1520-0469(1999)056<2673:LMOARF>2.0.CO;2, 1999.Auer, A. H. and Veal, D. L.: The dimensions of ice crystals in natural clouds,
J. Atmos. Sci., 27, 919–926,
10.1175/1520-0469(1970)027<0919:TDOICI>2.0.CO;2, 1970.Augros, C., Caumont, O., Ducrocq, V., Gaussiat, N., and Tabary, P.:
Comparisons between S-, C- and X-band polarimetric radar observations and
convective-scale simulations of the HyMeX first special observing period, Q. J. Roy. Meteor. Soc., 142, 347–362, 10.1002/qj.2572, 2016.Babb, D. M., Verlinde, J., and Rust, B. W.: The Removal of Turbulent Broadening
in Radar Doppler Spectra Using Linear Inversion with Double-Sided
Constraints, J. Atmos. Ocean. Tech., 17, 1583–1595,
10.1175/1520-0426(2000)017<1583:TROTBI>2.0.CO;2, 2000.Bailey, M. P. and Hallett, J.: A comprehensive habit diagram for atmospheric
ice crystals: conformation from the laboratory, AIRS II, and other field
studies, J. Atmos. Sci., 66, 2888–2899, 10.1175/2009JAS2883.1, 2009.Baldauf, M., Seifert, A., Förstner, J., Majewski, D., Raschendorfer, M.,
and Reinhardt, T.: Operational convective-scale numerical weather prediction
with the COSMO model: description and sensitivities, Mon. Weather Rev., 139,
3887–3905, 10.1175/MWR-D-10-05013.1, 2011.Battaglia, A., Sturniolo, O., and Prodi, F.: Analysis of polarization radar
returns from ice clouds, Atmos. Res., 59–60, 231–250,
10.1016/S0169-8095(01)00118-1, 2001.Beard, K. V. and Chuang, C.: A new model for the equilibrium shape of
raindrops, J. Atmos. Sci., 44, 1509–1524, 10.1175/1520-0469(1987)044<1509:ANMFTE>2.0.CO;2, 1987.Blahak, U.: Towards a better representation of high density ice particles in a
state-of-the-art two-moment bulk microphysical scheme, in: 15th International
Conf. on Clouds and Precipitation, Cancun, Mexico,
available at: https://www.researchgate.net/publication/228387376_Towards_a_better_representation_of_high_density_ice_particles_in_a_state-of-the-art_two-moment_bulk_microphysical_scheme (last access: 14 March 2018),
2008.Blahak, U.: RADAR MIE LM and RADAR MIELIB Calculation of Radar Reflectivity
from Model Output, in: Tech. Report No. 28, Consortium for Small Scale
Modelling (COSMO),
available at: http://www.cosmo-model.org/content/model/documentation/techReports/docs/techReport28.pdf (last access: 14 March 2018),
2016.
Bohren, C. F. and Huffman, D. R.: Absorption and scattering of light by
small particles, Wiley, 1983.Bringi, V. N. and Chandrasekar, V.: Polarimetric doppler weather radar,
Cambridge University Press, Cambridge, UK, 10.1017/CBO9780511541094, 2001.Bruggemann, D. A. G.: Berechnung verschiedener physikalischer Konstanten von
heterogenen Substanzen, I. Dielektrizitätskonstanten und Leitfähigkeiten
der Mischkörper aus isotropen Substanzen, Ann. Phys., 24, 636–679, 10.1002/andp.19354160705, 1935.Caumont, O., Ducrocq, V., Delrieu, G., Gosset, M., Pinty, J.-P., Parent du
Châtelet, J., Andrieu, H., Lemaître, Y., and Scialom, G.: A radar
simulator for high-resolution nonhydrostatic models, J. Atmos. Ocean. Tech., 23, 1049–1067, 10.1175/JTECH1905.1, 2006.Cheong, B. L., Palmer, R. D., and Xue, M.: A Time Series Weather Radar
Simulator Based on High-Resolution Atmospheric Models, J. Atmos. Ocean. Tech., 25, 230–243, 10.1175/2007JTECHA923.1, 2008.Doms, G., Förstner, J., Heise, E., Herzog, H.-J., Mironov, D., Raschendorfer, M., Reinhardt, T., Ritter, B., Schrodin, R.,
Schulz, J.-P., and Vogel, G.: A description of the nonhydrostatic regional COSMO model, Part
II: Physical Parameterization,
available at: http://www.cosmo-model.org/content/model/documentation/core/cosmoPhysParamtr.pdf (last access: 26 January 2018),
2011.
Doviak, R. and Zrnić, D.: Doppler radar and weather observations, second
edition, Dover Publications, Mineola, N. Y., 2006.
Fabry, F.: Radar Meteorology, Principles and Practice, Cambridge University
Press, Cambridge, UK, 2015.Fabry, F. and Zawadzki, I.: Long Term Observations of the Melting Layer of
Precipitation and Their Interpretation, J. Atmos. Sci., 52, 838–851,
10.1175/1520-0469(1995)052<0838:LTROOT>2.0.CO;2, 1995.Field, P. R., Hogan, R. J., Brown, P. R. A., Illingworth, A. J., Choularton,
T. W., and Cotton, R. J.: Parametrization of ice-particle size distributions
for mid-latitude stratiform cloud, Q. J. Roy. Meteor. Soc., 131, 1997–2017,
10.1256/qj.04.134, 2005.
Figueras i Ventura, J., Schneebeli, M., Leuenberger, A., Gabella, M., Grazioli,
J., Raupach, T. H., Wolfensberger, D., Graf, P., Wernli, H., Berne, A., and
Germann, U.: The PARADISO campaign, description and first results, in: 37th
Conference on Radar Meteorology, Norman, USA, 2015.Frick, C. and Wernli, H.: A Case Study of High-Impact Wet Snowfall in Northwest
Germany (25–27 November 2005): Observations, Dynamics, and Forecast
Performance, Weather Forecast., 27, 1217–1234,
10.1175/WAF-D-11-00084.1, 2012.Furukawa, K., Nio, T., Konishi, T., Masaki, T., Kubota, R., Oki, T., and
Iguchi, T.: Current status of the dual-frequency precipitation radar on the
global precipitation measurement core spacecraft and the new version of GPM
standard products, in: SPIE 10000, Sensors, Systems, and Next-Generation
Satellites XX, 1000003 (19 October 2016), 10000, 10000–10000–6, 10.1117/12.2240907,
2016.Gal-Chen, T. and Somerville, R. C. J.: On the use of a coordinate
transformation for the solution of the Navier-Stokes equations, J.
Comput. Phys., 17, 209–228, 10.1016/0021-9991(75)90037-6, 1975.Gander, W. and Gautschi, W.: Adaptive Quadrature – Revisited, BIT, 40, 84–101, 10.1023/A:1022318402393, 2000.Garrett, T. J., Fallgatter, C., Shkurko, K., and Howlett, D.: Fall speed
measurement and high-resolution multi-angle photography of hydrometeors in
free fall, Atmos. Meas. Tech., 5, 2625–2633,
10.5194/amt-5-2625-2012, 2012.Garrett, T. J., Yuter, S. E., Fallgatter, C., Shkurko, K., Rhodes, S. R., and
Endries, J. L.: Orientations and aspect ratios of falling snow, Geophys. Res.
Lett., 42, 4617–4622, 10.1002/2015GL064040, 2015.Gautschi, W.: Orthogonal Polynomials, Quadrature, and Approximation:
Computational Methods and Software (in Matlab), 1–77, Springer Berlin
Heidelberg, Berlin, Heidelberg, 10.1007/978-3-540-36716-1_1, 2006.Germann, U., Galli, G., Boscacci, M., and Bolliger, M.: Radar precipitation
measurement in a mountainous region, Q. J. Roy. Meteor. Soc., 132,
1669–1692, 10.1256/qj.05.190, 2006.Grazioli, J., Schneebeli, M., and Berne, A.: Accuracy of Phase-Based Algorithms
for the Estimation of the Specific Differential Phase Shift Using Simulated
Polarimetric Weather Radar Data, IEEE Geosci. Remote S., 11,
763–767, 10.1109/LGRS.2013.2278620, 2014.Hufford, G. A.: A model for the complex permittivity of ice at frequencies
below 1 THz, Int. J. Infrared Milli., 12, 677–682,
10.1007/BF01008898, 1991.Iguchi, T., Hanado, H., Takahashi, N., Kobayashi, S., and Satoh, S.: The
dual-frequency precipitation radar for the GPM core satellite, in: IGARSS
2003, 2003 IEEE International Geoscience and Remote Sensing Symposium,
Proceedings (IEEE Cat. No.03CH37477), Toulouse, France, 3, 1698–1700,
10.1109/IGARSS.2003.1294221, 2003.Jones, R. C.: A New Calculus for the Treatment of Optical SystemsI. Description
and Discussion of the Calculus, J. Opt. Soc. Am., 31, 488–493,
10.1364/JOSA.31.000488, 1941.Jung, Y., Ming, X., and Zhang, G.: Assimilation of Simulated Polarimetric Radar
Data for a Convective Storm Using the Ensemble Kalman Filter, Part I:
Observation Operators for Reflectivity and Polarimetric Variables, Mon.
Weather Rev., 136, 2228–2245, 10.1175/2007MWR2083.1, 2008.
Kang, E.: Radar System Analysis, Design, and Simulation, Artech House radar
library, Artech House, Norwooed, Massachusetts, USA, 2008.Labitt, M.: Coordinated radar and aircraft Observations of turbulence, Proj.
Rep. ATC 108 1468, Massachussetts Institute of Technology, Lincoln Lab.,
Cambridge, available at: https://www.ll.mit.edu/mission/aviation/publications/publication-files/atc-reports/Labitt_1981_ATC-108_WW-15318.pdf (last access: 12 December 2017),
1981.Lafore, J. P., Stein, J., Asencio, N., Bougeault, P., Ducrocq, V., Duron, J.,
Fischer, C., Héreil, P., Mascart, P., Masson, V., Pinty, J. P.,
Redelsperger, J. L., Richard, E., and Vilà-Guerau de Arellano, J.: The
Meso-NH Atmospheric Simulation System. Part I: adiabatic formulation and
control simulations, Ann. Geophys., 16, 90–109,
10.1007/s00585-997-0090-6, 1998.Lee, G., Zawadzki, I., Szyrmer, W., Sempere-Torres, D., and Uijlenhoet, R.: A
general approach to double-moment normalization of drop size distributions,
J. Appl. Meteorol., 43, 264–281,
10.1175/1520-0450(2004)043<0264:AGATDN>2.0.CO;2, 2004.Liebe, H., Hufford, G., and Manabe, T.: A Model for the Complex Permittivity of
Water at Frequencies Below 1 THZ, Int. J. Infrared Milli., 12, 659–675,
10.1007/BF01008897, 1991.Lin, Y.-L., Farley, R. D., and Orville, H. D.: Bulk Parameterization of the
Snow Field in a Cloud Model, J. Clim. Appl. Meteorol., 22, 1065–1092,
10.1175/1520-0450(1983)022<1065:BPOTSF>2.0.CO;2, 1983.Liu, G.: A Database Of Microwave Single-Scattering Properties For Nonspherical
Ice Particles, B. Am. Meteorol. Soc., 89, 1563–1570,
10.1175/2008BAMS2486.1, 2008.Magono, C. and Lee, C. W.: Meteorological classification of natural snow
crystals, J. Fac. Sci., Hokkaido Univ., Series VII, 2, 321–335,
10.5331/seppyo.24.33, 1966.Matrosov, S. Y.: Assessment of radar signal attenuation caused by the melting
hydrometeor layer, IEEE T. Geosci. Remote Sens., 46, 1039–1047,
10.1109/TGRS.2008.915757, 2008.Mellor, G. L. and Yamada, T.: Developement of a turbulence closure model for
geophysical fluid problems, Rev. Geophys. Space Ge., 20, 851–875,
10.1029/RG020i004p00851, 1982.Meneghini, R. and Liao, L.: Comparisons of Cross Sections for Melting
Hydrometeors as Derived from Dielectric Mixing Formulas and a Numerical
Method, J. Appl. Meteorol., 35, 1658–1670,
10.1175/1520-0450(1996)035<1658:COCSFM>2.0.CO;2, 1996.
Mishchenko, M., Travis, L., and Lacis, A.: Scattering, Absorption, and Emission
of Light by Small Particles, Cambridge University Press, Cambridge, UK, 2002.Mishchenko, M. I., Travis, L. D., and Mackowski, D. W.: T-matrix computations
of light scattering by nonspherical particles: A review, J. Quant. Spectrosc. Ra., 55, 535–575,
10.1016/0022-4073(96)00002-7, 1996.Mitra, S. K., Wohl, O., Ahr, M., and Pruppacher, H. R.: A Wind Tunnel and
Theoretical Study of the Melting Behavior of Atmospheric Ice Particles. IV:
Experiment and Theory for Snow Flakes, J. Atmos. Sci., 47, 584–591,
10.1175/1520-0469(1990)047<0584:AWTATS>2.0.CO;2, 1990.Noel, V. and Sassen, K.: Study of Planar Ice Crystal Orientations in Ice Clouds
from Scanning Polarization Lidar Observations, J. Appl. Meteorol., 44,
653–664, 10.1175/JAM2223.1, 2005.Noppel, H., Blahak, U., Seifert, A., and Beheng, K. D.: Simulations of
a hailstorm and the impact of CCN using an advanced two-moment cloud
microphysical scheme, Atmos. Res., 96, 286–301,
10.1016/j.atmosres.2009.09.008, 2010.Oguchi, T.: Electromagnetic wave propagation and scattering in rain and other
hydrometeors, P IEEE, 71, 1029–1078,
10.1109/PROC.1983.12724, 1983.Pfeifer, M., Craig, G. C., Hagen, M., and Keil, C.: A polarimetric radar
forward operator for model evaluation, J. Appl. Meteorol. Clim., 47,
3202–3220, 10.1175/2008JAMC1793.1, 2008.Praz, C., Roulet, Y.-A., and Berne, A.: Solid hydrometeor classification and
riming degree estimation from pictures collected with a Multi-Angle Snowflake
Camera, Atmos. Meas. Tech., 10, 1335–1357,
10.5194/amt-10-1335-2017, 2017.Raupach, T. H. and Berne, A.: Correction of raindrop size distributions
measured by Parsivel disdrometers, using a two-dimensional video disdrometer
as a reference, Atmos. Meas. Tech., 8, 343–365,
10.5194/amt-8-343-2015, 2015.Rogers, R. R., Baumgardner, D., Ethier, S. A., Carter, D. A., and Ecklund,
W. L.: Comparison of Raindrop Size Distributions Measured by Radar Wind
Profiler and by Airplane, J. Appl. Meteorol., 32, 694–699,
10.1175/1520-0450(1993)032<0694:CORSDM>2.0.CO;2, 1993.Rutledge, S. A. and Hobbs, P.: The Mesoscale and Microscale Structure and
Organization of Clouds and Precipitation in Midlatitude Cyclones. VIII: A
Model for the “Seeder-Feeder” Process in Warm-Frontal Rainbands, J.
Atmos. Sci., 40, 1185–1206,
10.1175/1520-0469(1983)040<1185:TMAMSA>2.0.CO;2, 1983.Ryzhkov, A. V.: The Impact of Beam Broadening on the Quality of Radar
Polarimetric Data, J. Atmos. Ocean. Tech., 24, 729–744,
10.1175/JTECH2003.1, 2007.Ryzhkov, A. V., Pinsky, M., Pokrovsky, A., and Khain, A.: Polarimetric Radar
Observation Operator for a Cloud Model with Spectral Microphysics, J. Appl.
Meteorol. Clim., 50, 873–894, 10.1175/2010JAMC2363.1, 2011.Schneebeli, M., Sakuragi, J., Biscaro, T., Angelis, C. F., Carvalho da Costa,
I., Morales, C., Baldini, L., and Machado, L. A. T.: Polarimetric X-band
weather radar measurements in the tropics: radome and rain attenuation
correction, Atmos. Meas. Tech., 5, 2183–2199,
10.5194/amt-5-2183-2012, 2012.Seifert, A. and Beheng, K. D.: A two-moment cloud microphysics parameterization
for mixed-phase clouds. Part 1: Model description, Meteorol. Atmos. Phys.,
92, 45–56, 10.1007/s00703-005-0112-4, 2006.
Smolyak, S. A.: Quadrature and interpolation formulas for tensor products of
certain class of functions, Dokl. Akad. Nauk SSSR+, 148, 1042–1053, transl.:
Soviet Math. Dokl., 4, 240–243, 1963.Speirs, P., Gabella, M., and Berne, A.: A comparison between the GPM
dual-frequency precipitation radar and ground-based radar precipitation rate
estimates in the Swiss Alps and Plateau, J. Hydrometeorol., 18, 1247–1269,
10.1175/JHM-D-16-0085.1, 2017.
Straka, J. M., Zrnic, D. S., and Ryzhkov, A. V.: Bulk hydrometeor
classification and quantification using polarimetric radar data: synthesis of
relations, J. Appl. Meteorol., 39, 1341–1372,
10.1175/1520-0450(2000)039<1341:BHCAQU>2.0.CO;2, 2000.Szyrmer, W. and Zawadzki, I.: Modeling of the Melting Layer. Part I: Dynamics
and Microphysics, J. Atmos. Sci., 56, 3573–3592,
10.1175/1520-0469(1999)056<3573:MOTMLP>2.0.CO;2, 1999.Thurai, M., Huang, G., Bringi, V., Randeu, W., and Schönhuber, M.: Drop
shapes, model comparisons, and calculations of polarimetric radar parameters
in rain, J. Atmos. Ocean. Tech., 24, 1019–1032, 10.1175/JTECH2051.1, 2007.Thurai, M., Gatlin, P., Bringi, V. N., Petersen, W., Kennedy, P.,
Notaroš, B.,
and Carey, L.: Toward Completing the Raindrop Size Spectrum: Case Studies
Involving 2D-Video Disdrometer, Droplet Spectrometer, and Polarimetric Radar
Measurements, J. Appl. Meteorol. Clim., 56, 877–896,
10.1175/JAMC-D-16-0304.1, 2017.Toyoshima, K., Masunaga, H., and Furuzawa, F. A.: Early Evaluation of Ku- and
Ka-Band Sensitivities for the Global Precipitation Measurement (GPM)
Dual-Frequency Precipitation Radar (DPR), vol. 11 of Scientific Online
Letters on the Atmosphere (SOLA), Meteorological Society of Japan,
14–17, 10.2151/sola.2015-004, 2015.Turner, D. D., Kneifel, S., and Cadeddu, M. P.: An Improved Liquid Water
Absorption Model at Microwave Frequencies for Supercooled Liquid Water
Clouds, J. Atmos. Ocean. Tech., 33, 33–44,
10.1175/JTECH-D-15-0074.1, 2016.Wicker, L. J. and Skamarock, W. C.: Time-splitting methods for elastic models
using forward time schemes, Mon. Weather Rev., 130, 2088–2097,
10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2, 2002.Wolfensberger, D., Scipion, D., and Berne, A.: Detection and characterization
of the melting layer based on polarimetric radar scans, Q. J. Roy. Meteor.
Soc., 142, 108–124, 10.1002/qj.2672, 2016.Wolfensberger, D. and Berne, A.: A forward polarimetric radar operator for
the COSMO NWP model, 10.5281/zenodo.1298618, 2018.Xue, M., Droegemeier, K. K., and Wong, V.: The Advanced Regional
Prediction System (ARPS); A multi-scale nonhydrostatic atmospheric
simulation and prediction model. Part I: model dynamics and verification,
Meteorol. Atmos. Phys., 75, 161–193, 10.1007/s007030070003, 2000.Zeng, Y., Blahak, U., Neuper, M., and Jerger, D.: Radar Beam Tracing Methods
Based on Atmospheric Refractive Index, J. Atmos. Ocean. Tech., 31,
2650–2670, 10.1175/JTECH-D-13-00152.1, 2014.Zeng, Y., Blahak, U., and Jerger, D.: An efficient modular volume-scanning
radar forward operator for NWP models: description and coupling to the COSMO
model, Q. J. Roy. Meteor. Soc., 142, 3234–3256, 10.1002/qj.2904, 2016.