In the dendritic growth layer (corresponding to temperatures between -20 and -10 °C), supersaturation levels significantly influence the degree of branching in plate-like crystals, resulting in a wide variety of morphologies . categorized these forms as sector, broad-branch, stellar, dendritic, and fern-like structures. This high degree of complexity of the internal structure of dendritic ice crystals cannot be explicitly described in habit prediction schemes such as the one implemented in . The habit prediction schemes therefore describe ice crystals as porous spheroids, whose aspect ratio and density is varied by the habit prediction. A detailed description is given in Sect. . While the description of ice crystals as spheroids gives reasonably well results when comparing the aspect ratio, mass and maximum dimension (Dmax) to laboratory studies , it is well known that the internal structure of ice particles is essential for scattering in the microwave regime e.g.. In order to forward simulate the output of models using habit prediction schemes, a large variety of ice crystal shapes has to be used, in order to match the predicted aspect ratios, densities, masses, and Dmax.

To capture the morphological diversity of dendritic ice crystals, we therefore employed the Reiter-algorithm as implemented in the aggregation model. The Reiter algorithm is a two-dimensional cellular automaton defined on a hexagonal lattice. Each grid cell represents vapour, boundary, or ice and contains a scalar supersaturation value. The evolution of the crystal proceeds in discrete time steps consisting of a diffusion step and an attachment (growth) step. In the diffusion step, supersaturation is redistributed between neighbouring cells using a weighted average controlled by the parameter α, which specifies the relative strength of diffusion. Higher values of α lead to faster smoothing of the supersaturation field, resulting in more open, dendritic structures, while smaller values promote more compact shapes.

Boundary cells, those adjacent to ice, can accumulate mass from the surrounding supersaturation field. The rate of this accumulation is governed by β, which sets how quickly vapour condenses onto the growing crystal edges. Once the accumulated mass of a boundary cell exceeds a threshold determined by γ, the cell is converted into ice. The parameter γ therefore controls the overall growth rate and the sharpness of branching features: smaller γ values allow rapid conversion to ice and more finely branched structures, whereas larger γ values promote slower, smoother growth. When β=0 and γ>0, plates are formed, and regardless of β, at high γ, plates, or broad branched stellar forms are produced. For levels of low γ and β, dendritic growth happens, while for larger values of β, stellar forms appear. Figure 3 in shows an overview of the effect of γ and β on the form of the crystal.

Iterating the diffusion and attachment rules yields crystal patterns that exhibit hexagonal symmetry, dendritic branching, and sector structures consistent with natural snow crystals. Despite its conceptual simplicity, the Reiter algorithm captures the principal morphological characteristics of atmospheric ice particles.

We used the Reiter algorithm to generate a large variety of dendritic shapes. Specifically, we set α=1 and varied β between 0.2 and 0.7 and γ between 0.0001 and 0.006. Dmax ranged from 10 µm to 7 mm. Figure  presents a subset of the resulting ice crystal shapes.

The variety in aspect ratios generated by the Reiter algorithm alone is not enough to cover the entire aspect ratio space predicted by the habit prediction. In extreme cases, a particle can change growth regime from e.g., the columnar growth regime assumed at temperatures colder than -22 °C into the plate-like growth regime at temperatures between -20 and -10 °C. have shown that when such a regime change happens, the particle is likely to decrease its aspect ratio towards unity, effectively generating a “thick” column. To capture these particles, we adapted the aggregation model from to allow the generation of ice crystals with adjustable aspect ratios at fixed Dmax. This was done by effectively stacking dendritic particles of the same shape on top of each other, increasing the aspect ratio towards unity. In total, we generated 1850 dendritic ice crystals.

At smaller sizes or under low supersaturation conditions, solid plates are more likely to form. To represent this regime, we generated 625 plate crystals with varying sizes, masses, and aspect ratios with the aggregation model and our adaptation to allow more flexible aspect ratios. In the temperature range between -10 and -5 °C, ice crystals tend to grow into columnar shapes . Accordingly, we used the aggregation model to generate 125 columnar crystals, again varying size, mass, and aspect ratio. A description of the plates and columns is given in Fig. 1 of .

The resulting mass–size and aspect ratio–size relationships for all 2600 ice crystals are illustrated in Fig. .

For particle shape, we chose a grid resolution dependent on Dmax such that gridres=Dmax⋅2.5×10-2.

Similarly to ice crystals, the microphysical properties of aggregates vary largely depending on the ice crystals which generated the aggregate and stochastic parameters determined during the collision process, such as the angle between the colliding partners, or the contact point e.g.. Therefore, we have randomly selected 450 aggregates of plates, dendrites, mixtures of dendrites and columns (mix2) with a riming equivalent liquid water content (see ) between 0 and 1 kgm-2 from the snow library of snowScatt . The resulting database comprises 175 rimed particles, 106 mixtures of columns and dendrites, 105 dendrite aggregates, and 63 plate aggregates. A subset of their shapes in presented in Fig. , the microphysical properties of all aggregates are shown in Fig. .

After determining the particle shapes, their scattering properties must be computed. The discrete dipole approximation (DDA) is a widely used method for calculating the scattering of arbitrarily shaped particles by solving the volume integral equation form of Maxwell's equations for a discretized representation of each particle. Originally developed by , the method replaces the particle with an array of polarizable dipoles. The interactions among these dipoles and with the incident electromagnetic field yield a system of linear equations, the solutions of which give the dipole polarizations and thus allows to compute the scattered electromagnetic field.

Since its initial development, increases in computational power and the availability of efficient public implementations such as ADDA and DDSCAT have made the DDA a popular choice for modelling particles of arbitrary shape and composition. The accuracy of the DDA is considered to be limited by two sources of error, namely the shape error and the discretization error which are both dependent on the dipole size . These two sources of error refer to the accuracy at which a certain shape can be approximated by a set of cubic voxels and how well those same cubic volumes can represent the spatially varying electromagnetic field. A commonly used criterion for the latter is to guarantee that the parameter |m|kd<0.05, where m is the refractive index, k the wavenumber, and d is the dipole spacing . At W-Band (94 GHz, the largest frequency in this database) this implies a dipole spacing below 14 µm; our largest dipole spacing of 10 µm satisfies this requirement.

The shape error is particularly critical for strongly asymmetric shapes, such as thin plates, whose thickness may span only a single layer of dipoles or heavily-branched dendrites with even finer internal structures. For this reason, we adopt a variable dipole spacing that depends on particle size. Some studies , adopted specific investigation techniques to analyse these two error sources by varying the dipole resolution. Since this resolution is size-dependent in our database a comprehensive application of this approach to all shapes would be particularly effort-taking. Instead, we limited our analysis on two random dendritic particles from our database and reduced the spacing by factors of two and four. The resulting changes in the computed scattering properties were only 0.12 %–0.2 %, depending on polarization and wavelength, indicating that the discretization is sufficient.

We would like to emphasize that uncertainties arising from DDA discretization are likely negligible compared with the much larger uncertainties associated with selecting appropriate particle shapes, as the true in-cloud geometries are unknown. We attempt to mitigate these uncertainties by simulating a wide variety of particles, but the error inherent in choosing any particular shape remains unknown and cannot currently be constrained by in-situ observations.

The full ADDA setup used for the simulations can be inferred from the shared analysis codebase and has been derived from the experience gained in our research group regarding the calculation of the scattering properties of complex shaped frozen and partially melted hydrometeors . In particular provides an overview of the performance of various combinations of ADDA configurations for such scattering targets. In particular we used Lattice Dispersion Relation (LDR) and Filtered Coupled Dipoles (FCD), respectively for polarization formulation and interaction term.

We first define two fundamental reference frames, namely the radar reference frame (RRF), and the particle reference frame (PRF). The RRF (often called laboratory reference frame) can be considered as the most general one, it is fixed with a hypothetical ground-based radar, the z-axis goes along the vertical with respect to the radar location, while the orientation of the horizontal xy-axes is arbitrary. The radar reference frame defines the coordinates of the propagation direction of the radar beam and its polarization planes (Fig. a). The propagation direction of the radar beam is defined by the radar elevation angle el and azimuth az. The vertical polarization plane spans the beam propagation direction and the z-axis, while the horizontal polarization plane is orthogonal to the vertical and also contains the beam propagation vector.

The PRF is fixed with the snow particle, and is assumed to correspond to the particle axes of inertia with the shortest inertia axis aligned with the z-axis of the PRF. The transformation from PRF to RRF is performed by means of 3 Euler rotations (angles α^, β^, and γ^) that follow the commonly used zyz-convention (Fig. b). With respect to the PRF, the propagation direction of any vector is defined by the polar coordinates θ and ϕ.

The polarimetric scattering properties of a complex-shaped snowflake in the RRF would in general depend on all 5 angles (el, az, α^, β^, γ^). Note that a rotation of the angles az of the RRF and α^ of the PRF act on axes that are parallel, which implies that only the difference az-α^ is relevant for the definition of the overall geometry. By assuming az=0 the dependence on the angle az can be dropped without loss of information.

At a given particle orientation, the incident electromagnetic field Ei can be related to the scattered (far-) field Es via the Amplitude scattering matrix S : 1E∥sE⊥s=exp⁡(ikr)rS1S4S3S2E∥iE⊥i, the subscripts ∥ and ⊥ indicate parallel and perpendicular orientations with respect to the scattering plane, i=-1, k=2π/λ is the wavenumber, r is the distance between the scatterer and the observation point, and S1,…,4 are the complex elements of the Amplitude matrix. Please note that the ADDA code outputs a slightly different definition of the Amplitude matrix that includes a -ik factor in the denominator. The Amplitude matrix can be converted to the Mueller matrix Z, which relates the Stokes vector of the incoming and scattered fields : 2IscaQscaUscaVsca=1r2Z11Z12Z13Z14Z21Z22Z23Z24Z31Z32Z33Z34Z41Z42Z43Z44IinQinUinVin, where Z11,…,44 are the real elements of the Mueller matrix. The Stokes vector consists of the elements I, Q, U and V. I describes the total intensity of the incoming/scattered field, Q and U which describe the linear polarization, in case of Q the preference of horizontal versus vertical polarization, and in case of U the preference of ± 45° polarization. V describes the circular polarization. In contrast to the Amplitude matrix, the Mueller matrix contains only information about the intensity of the scattered wave, not the absolute phase. The Amplitude matrix can be converted into the Mueller matrix following well established formulas. Both matrices were calculated with the ADDA version 1.4.0 and are available in the scattering database at C- (5.6 GHz), X- (9.6 GHz), Ka- (35.6 GHz) and W- (94.0 GHz) band as well as for el between 0 and 180° in 5° steps and az between 0 to 337.5° in 22.5° steps for every scattering direction. Such a coarse azimuthal sampling is sufficient because the scattering properties are later averaged over azimuth (see Sect. ). An azimuthal resolution as coarse as 45° yields identical azimuthally averaged scattering properties. The scattering properties were calculated at a temperature of 270 K, using the refractive index model described in , Chap. 5 for ice.

The full set of formulas to calculate the Z11,…,44 elements from the Amplitude matrix can be found in and in the code we shared. For convenience we report here the equations for the four elements that are relevant for the radar quantities we calculate in McRadar 3Z11=12|S1|2+|S4|2+|S3|2+|S2|2,4Z12=12|S1|2-|S4|2+|S3|2-|S2|2,5Z21=-ReS1⋅S4∗+S2⋅S3∗,6Z22=-ImS1⋅S4∗-S2⋅S3∗, where the asterix apex denotes the complex conjugate while Im(var),Re(var) the imaginary and real parts of the variable var, respectively.

After deriving the Amplitude and Mueller scattering matrices using the definition, typical radar parameters can be calculated as follows :

Radar reflectivity factor at horizontal polarization7Zehh=2λ4π4|K|2Z11|180°+Z22|180°+Z12|180°+Z21|180°

Radar polarimetric quantities are sensitive to the shape and orientation of the particles. The steady-state falling orientation of a complex-shaped particle, such as a snowflake, is determined by the complex equilibrium of gravitational and aerodynamic forces. The problem of finding the fall attitude of a snowflake is further complicated by the fact that such a stationary orientation might not even exist (i.e., snowflakes can continuously tumble and spin; ), furthermore, hydrometeors typically fall in an environment that is itself non-stationary and turbulent.

The attitude of a snowflake during its fall can be described by the probability distribution of the three Euler angles that define its orientation with respect to the default fall configuration. For this study, the default orientation is defined as aligning the longest axis of inertia horizontally (i.e. this corresponds to the Euler angle β^=0). The distributions of Euler angles defining the particle orientations are commonly defined by fα^(α^), fβ^(β^), fγ^(γ^), but please beware that it is not always possible to factorize the probability distribution of the particle orientation on the product of the individual distribution of the three Euler angles. The average value of a quantity S(α^,β^,γ^) that depends on the orientation of the particle can be calculated as 11〈S〉α^,β^,γ^=∫02π∫0π∫02πS(α^,β^,γ^)fα^(α^)fβ^(β^)×sin⁡(β^)fγ^(γ^)dα^dβ^dγ^.

A radar observes a vast population of hydrometeors whose orientation in space can be defined by angular distributions. They also sense those particles over a certain amount of time, during which particles change their orientation through, i.e. tumbling and spinning movements. For this reason we are not interested in the scattering properties of a single oriented particle. Rather, we are interested in orientation-averaged scattering properties of the particles population. This can be represented as an ensemble of multiple particles oriented differently but otherwise identical. It is widely accepted that the α^ and γ^ angles follow a uniform distribution . Averaging over the distributions of α^ and γ^ leads to the representation of polarimetric scattering properties that follow the so-called azimuthally random orientation (ARO) assumption . Under these conditions, the polarimetric scattering properties of snowflakes depend on only two angles, namely the radar elevation angle el and the particle canting angle β^ 12SARO(el,β^)=∫02π∫02πSRRF(el,α^,β^,γ^)4π2dα^dγ^.

In the previous section, we explained how it is possible to leverage the assumption that hydrometeors have no preferential orientation in the azimuth to reduce the dimensionality of the polarimetric scattering data structure. However, to compute Eq. () one would still need to perform a sufficient number of scattering calculations on individual sets of elevation angles and particle orientations (el,α^,β^,γ^). This is a considerable computational effort, thus, following previous studies we reduce the dimensionality of the problem by realizing that any scattering problem in the RRF (identified by the five angles el, az, α^, β^, γ^) can be represented by a corresponding scattering in the PRF, provided that one can operate a proper transformation between the two reference frames. In the PRF any scattering problem is identified simply by the two polar coordinates (θ0,ϕ0) that define the propagation direction of the incoming electromagnetic wave in the PRF. Mathematically (el,az,α^,β^,γ^)⇒(θ0,ϕ0).

The transformation matrices ρ and ρ-1 that allow one to convert the Amplitude matrices from the PRF to the RRF are provided by and depend on the polar coordinates of the direction of the incoming and scattered beams as well as the three Euler angles that define the orientation of the scatterer. Here, we drop the dependency on the scattered directions for the sake of simplicity, since, for radar applications, the only two relevant scattering directions are the forward and backward ones. Mathematically, one can write 13SRRF(el,az,α^,β^,γ^)=ρT(el,az,α^,β^,γ^)×SPRF(θ0,ϕ0)ρ(el,az,α^,β^,γ^).

Thus, we perform individual scattering calculations in the PRF with a regular (θ0,ϕ0) grid. The polar angle θ0 ranges from -90 to 90° with increments of 5° while the azimuth angle ϕ0 ranges from 0 to 360° with increments of 22.5°. The set of directions used is represented as red dots in Fig. b. This approach is less efficient than other strategies to sample points over a unitary sphere (such as the icosahedral sampling strategy of ), because it would allow for an increased density of sampling points close to the poles. However, it is fundamentally simple and has the advantage of integrating perfectly over azimuths in the case of non-tumbling (β^=0) particles.

The azimuth averaging of Eq. () is then performed numerically by sampling the database of individual scattering directions using a nearest-neighbour approach, hence (el,az,α^,β^,γ^)⇒(θ0NN,ϕ0NN). The transformation of Eq. () is then applied to the sampled matrix before executing the quadrature rule which is equivalent to perform the following integral 14SARO(el,β^)=∫02π∫02πρT(el,az,α^,β^,γ^)×SPRF(θ0NN,ϕ0NN)×ρ(el,az,α^,β^,γ^)4π2dα^dγ^.

At an elevation angle of 30° and β^ = 0°, Ze, ZDR, and KDP at 9.6 GHz (X-band), 35.6 GHz (Ka-band), and 94 GHz (W-band) for all particles in the database are shown in Fig. . A notable feature is that small ice crystals in the database can exhibit negative ZDR values, which is counter-intuitive since the particles are horizontally aligned with their largest dimension. For particles with an aspect ratio different from unity, one would generally expect a positive ZDR.

The particles showing negative ZDR are dendritic crystals with aspect ratios close to unity (0.7–0.9). Based solely on aspect ratio, their ZDR should be positive. However, the internal mass distribution is asymmetric: the vertical dimension has a higher mass density than the horizontal dimension (see examples in Fig. ). When the aspect ratio is near unity, this density difference becomes significant, resulting in a higher vertical Ze and thus a negative ZDR. This effect is not observed for solid plates with similar size and aspect ratio because their mass distribution is nearly uniform along the vertical and horizontal axes, making aspect ratio the dominant factor for ZDR.

Comparing Ze, ZDR, and KDP at W-Band of the ice crystals in this database with those in (see their Fig. 4 and Fig. C1 in ), we find similar features. Ice crystals with masses exceeding 10-8 kg, corresponding to a maximum dimension of approximately 3 mm (see Fig. ), enter the non-Rayleigh scattering regime, producing a ZDR peak of 12 dB and a flattening of Ze. This behaviour is consistent with , where dendritic crystals reach a maximum W-band ZDR of 9 dB at Dmax = 3 mm. Likewise, KDP in our database aligns with , reaching 0.2 mm°km-1 at Dmax = 8 mm when normalized by wavelength (Fig. ).

Aggregates in our database show some differences from those in . At X-band, aggregates reach a maximum Ze of  27 dBz at a mass of 10-4 kg (corresponding to Dmax = 3.5 cm), whereas in , the maximum Ze for Dmax ≈ 3 cm is only 9.5 dBz. This higher reflectivity in our dataset is primarily due to rimed aggregates. Unrimed aggregates in our database reach only 5 dBz at Dmax ≈ 3.5 cm, comparable to .

The transition of aggregates into the Mie-regime occurs at slightly higher masses in our dataset: around 5 × 10⁻⁸ kg at W-band, with a pronounced flattening at 5 × 10⁻⁷ kg (Dmax = 2–3 mm). In contrast, reports the transition at 5 × 10⁻⁹ kg. As noted in , their aggregates enter the non-Rayleigh scattering regime earlier than those in , likely due to structural differences. Our aggregates consist primarily of plates, dendrites, or a mixture of dendrites and columns, whereas the aggregates are more akin to aggregates of needles.

ZDR values of aggregates are similar between the two databases: at W-band, most aggregates exhibit ZDR between 2–3 dB. Only high-density rimed aggregates in our dataset reach up to 5.5 dB. KDP also remains in a comparable range to .

McRadar is a flexible radar forward simulator designed to interface directly with the level 1b LUTs and forward simulate microphysical model output into radar observables while maintaining consistency in physical properties. Similar to existing forward operators such as the Cloud-resolving model Radar SIMulator CR-SIM , McRadar calculates the Doppler spectrum, specific differential phase shift (KDP), and attenuation based on the microphysical inputs. Instrument-specific noise can be added to the Doppler spectra, and spectral broadening effects due to turbulence, finite beamwidth, and wind shear are optionally included following .

McRadar also allows the user to specify particle canting angle distributions (e.g., wobbling), enabling more realistic simulations. From the Doppler spectra, standard radar moments, including equivalent reflectivity (Ze), mean Doppler velocity, differential reflectivity (ZDR), KDP, and linear depolarization ratio (LDR), are derived.

Allowing the ice particle to grow via deposition, during which the shape of the particle evolves using a habit prediction , results in a multitude of particle shapes, and therefore a large variability of particle size, mass, and aspect ratio for which scattering properties need to be found. To find the scattering properties of a particle that best fits with its microphysical properties (i.e., mass, size, and aspect ratio) to the model-predicted particle, McRadar performs a nearest-neighbour selection or regression on the level 1b LUTs, depending on user preference. This is implemented using the scikit-learn Python library, which supports efficient neighbour search algorithms. By default, McRadar uses neighbours-based regression, where the output is a weighted average of the n nearest neighbours based on inverse distance to the query point. This supervised learning approach follows the methodology described by , and more details can be found in the official documentation: https://scikit-learn.org/1.5/modules/neighbors.html (last access: August 2025).

For monomers, the scattering properties are retrieved using the nearest match in mass, Dmax, and aspect ratio in logarithmic space. For aggregates, the nearest match is found based on mass and Dmax in logarithmic space.

Showing the flexibility of the DDA database in terms of large variability of particle properties is done by forward simulating the trajectories (or growth histories) of ice crystals that are sedimenting towards the ground while growing through deposition. The habit of ice crystals strongly depends on the ambient temperature and super-saturation i.e.. Recently, a habit prediction has been implemented in McSnow . The habit of an ice crystal can be described by its aspect ratio ϕ. In McSnow, ϕ is defined as the ratio of the c to the a-axis ϕ=ca, where the a-axis is the maximum dimension of the basal face, and the c-axis is the maximum dimension of the prism face. The change of ϕ can be calculated with 15dcda=dϕ=αc(T)αa(T)ϕ=Γ(T)ϕ, with the deposition coefficients of axis c (αc) and axis a (αa) and the temperature (T) dependent inherent growth function Γ. Γ has been determined by different studies through comparison with observational data as well as laboratory experiments e.g.. The change of mass due to depositional growth can then be calculated with the well-known mass growth equation i.e.Chap. 7 16dmdt=4πCDvf‾Si-1Fki+Fdi, with the mass of the particle m, the water vapour diffusivity Dv, ambient saturation ratio with respect to ice Si, particle ventilation coefficient f‾. Fki, is linked with the latent heat release due to sublimation and the diffusion of heat away from the ice particle, and Fdi is linked with the diffusion of water vapour towards the ice particle. They are expressed as 17Fki=LsRvT-1;Fdi=RvTDves,i(T), with Ls the latent heat of sublimation, T the ambient temperature, Rv the gas constant of water vapour and es,i(T) the saturation vapour pressure above ice at the temperature T.

The change of ϕ is taken into account through the capacity C (with subscript p for plate-like particles, c for columnar particles): 18Cp=αϵpsin-1(ϵp);Cc=cϵcln⁡(1+ϵc)ϕ with ϵp=1-ϕ2 and ϵc=1-ϕ-2.

An additional variable necessary to determine the habit of an ice crystal is its overall density. This density is, for example, reduced from solid ice density through branching of dendrites at temperatures close to -15 °C or hollowing of columns at temperatures between -5 and -10 °C. In McSnow, the volume enveloping ice crystals is simulated as a spheroid, which does not allow to simulate branching or hollowing directly. Instead, the density of the spheroid is adapted by amplifying the increase of the ice volume V relative to the mass m 19dVdt=1ρdepodmdt.

The deposition density ρdepo depends on the ambient temperature and is proportional to ρiΓ(T) for plate-like particles and ρiΓ(T)-1 for columnar particles with ice density ρi. For further details about the habit prediction, the reader is referred to .

For this application, the modified inherent growth function (IGF2) from has been used. While McSnow is a Lagrangian Monte-Carlo particle model, designed to simulate populations of particles, it is also possible to simulate the evolution of single ice particles due to various microphysical processes, in this case depositional growth and sedimentation. Keeping the ambient super-saturation with respect to ice constant at 5 %, setting the temperature at the ground to 0 °C and assuming a constant lapse rate of 6 Kkm-1, 5 particles were simulated starting at different heights (temperatures). The particles were initialized with a diameter of 10 µm and an aspect ratio of unity. During the simulation, the particles grow by deposition as predicted by the habit prediction while sedimenting through different temperature regimes towards the ground. No interactions between the particles have been assumed.

Depending on the temperature at which a particle was initialized, the physical properties vary significantly (see Fig. ). The particles that started at temperatures colder than -22 °C are growing columnar (with an aspect ratio larger than 1), while the particles initialized at temperatures warmer than -22 °C grow plate-like (with an aspect ratio smaller than 1). The particle that was initialized at -15 °C experiences the strongest depositional growth, reaching a maximum dimension (Dmax) of 3.5 mm and an aspect ratio of 0.0015, consistent with laboratory studies . The density of this particle is significantly reduced, reaching a minimum of 390 kgm-3. This indicates significant branching, as is expected by the predominantly dendritic growth at -15 °C. The smallest mass and Dmax is reached by the particle nucleated at -10 °C. This particle initially grows plate-like, but is sedimenting out of the plate growth region into the columnar growth region at temperatures warmer than -10 °C. Therefore, the particle only slowly increases its mass and Dmax while at the same time increasing its aspect ratio toward unity.

This large variability of ice microphysical properties produced by the McSnow simulation highlights the need for a scattering database consisting of ice particles with a similar variability. For the database in this study, we therefore generated specific ice crystals that can provide this large variability in ice microphysical properties. The corresponding forward simulations using the nearest-neighbour regression with n=10 to access the DDA LUTs, are shown in the second and third row of Fig. .

The most interesting scattering properties are expected from particles nucleated within the dendritic growth zone (between -10 and -20 °C). Those particles experience the strongest depositional growth, and reduce their density through branching. While growth effects such as branching can be seen in a reduction of ZDR, e.g. at -14.2 °C for the particle nucleated at -15 °C, also differences at different frequencies can be observed. While at X-Band, ZDR of the particle nucleated at -15 °C only slightly increases towards warmer temperatures, at W-Band a strong peak at -8 °C is observed, where a ZDR of 14.5 dB is reached. Looking at the single particle scattering properties in Fig. , it is evident that particles with a mass between 10-8 and 10-6 kg are producing large ZDR at W-Band. This mass corresponds to approximately a Dmax of 3 mm, which is close to the wavelength of W-band, causing non-Rayleigh scattering effects. While both ZeH and ZeV are in the non-Rayleigh regime (ZeH and ZeV are not increasing as fast at W-Band as at X-Band), ZeV is affected stronger than ZeH, causing the large increase in ZDR.

In-situ observations sometimes reveal particles which must have grown in multiple growth regimes . For example, capped-columns are observed when an ice crystal initially grows columnar, then sediments into the plate-like growth regime and develops plates on both ends of the column. In this simulation, the particle nucleated at -25 °C could potentially develop as a capped column in nature. In the model, these kinds of particles are, however, not represented. Only the change in aspect ratio and density of the particle can be captured. The transition from the columnar to the plate-like growth regime causes a reduction of the aspect ratio towards unity. While the addition of capped-columns is in principle possible, we have decided not to include these particles in the scattering database, as this introduces the need to include the growth history of the ice particle in the forward simulations. Rather, in the database, we have generated plates, dendrites, and columns with aspect ratios close to unity. In , it has also been shown in a laboratory study, that when a column transitions into the plate-like growth regime, under certain circumstances it can also just reduce its aspect ratio rather than producing a plate at each end, which might justify our approach.

Because forward simulations rely on a finite set of particle geometries, they are inevitably affected by sampling variability, i.e., uncertainty and potentially biases arising from the limited number of shapes used to represent the full natural population. The database includes 1000 particles, a significantly larger number of particles as compared to the database, thus allowing for more variability in particle properties. However, this variability is still not large enough to cover the variability of ice particle properties in McSnow simulations, especially the aspect ratios lack in variability compared to our database (see Fig. ). Compared to our database, the database contains approximately 1/3 particles, thus still potentially introducing a bias by not considering enough particles. The impact is likely two-fold: first, by using a smaller number of particles, the microphysical properties (mass, Dmax, aspect ratio) of the simulated particle might not be matched well enough, and second, considering a smaller number increases the statistical noise introduced by picking a random particle.

To investigate potential biases arising from particle representation, we performed an idealized one-dimensional simulation using McSnow. McSnow is particularly well-suited for this purpose, as it explicitly simulates the microphysical evolution of individual particles. For this case study, we adopt the same atmospheric setup as in Sect. : a surface temperature of 0 °C, a lapse rate of 6 Kkm-1, and a constant relative humidity with respect to ice of 105 %. Ice particles are initialized in a nucleation layer spanning 2–6 km altitude, corresponding to temperatures from -10 to -30 °C. In this layer, we assume a constant nucleation rate of 0.375 particless-1m-3, leading to a maximum concentration of 2000 particlesm-3 at -15 °C.

During the simulation, ice particles may grow by deposition and aggregation. Since we use the temperature-dependent sticking efficiency from , aggregation becomes an important growth mechanism only at temperatures warmer than -20 °C. In the current McSnow setup, aggregates follow predefined mass–size and area–size relationships; here, we assume aggregates composed of side planes, columns, and bullet rosettes as described in .

For our reference radar forward simulation, we use all particles of our database, and n=10 nearest neighbours to estimate the scattering properties of each simulated super-particle. To show the benefit of our database, which uses approximately 3 times more particles as the database, we then randomly downsampled our database to contain only 1/3 of particles. Assuming a homogeneous coverage in the microphysical space, in order to have a similar look-up radius we reduced the number of nearest neighbours considered for the look-up to n=3. We then forward simulated the McSnow output. In total, we performed the random downsampling and subsequent forward simulations 100 times.