## 1. Introduction

The knowledge of the cloud properties has been recently identified as a mandatory step to reach if the operational weather and climate change forecasts are to be improved (Stephens et al. 2002; Stephens 2005). In the framework of the space missions devoted to monitoring the microphysical, radiative, and dynamic properties of clouds at a global scale using cloud radar and lidar combinations [CloudSat/Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO), part of the Afternoon Train^{*}, Stephens et al. (2002], there is a need for ground-based and airborne validation of the radar–lidar measurements and products from these space missions. The synergy between radar and lidar instruments (ground based, airborne, and spaceborne) is such that in moderately thick clouds, the liquid/ice water content (IWC) and effective radius of droplets/crystals can be accurately retrieved from radar and lidar measurements (Tinel et al. 2005; Wang and Sassen 2002; Donovan and van Lammeren 2001; Okamoto et al. 2000). The domain of application of the radar–lidar synergy is however limited to a given range of clouds (optical thickness less than 3, roughly). As an example, prefrontal and mixed-phase clouds, which are very common in midlatitude regions, are generally not fully traversed by the lidar. In the present paper we therefore propose an original method complementary to the radar–lidar algorithm, which makes use of the two measurements of a Doppler cloud radar (35 or 95 GHz), namely, the radar reflectivity and the Doppler velocity, in order to recover the effective radius and terminal fall velocity of crystals, the ice water content, and the visible extinction, and therefore the visible optical depth. Previously, Matrosov et al. (2002) proposed a method combining Doppler velocity and reflectivity assuming the Rayleigh scattering. Such a method is not really applicable to a radar at 95 GHz when the Rayleigh approximation is no longer valid. In the present paper, we describe a new Doppler radar method available for radars from 3 to 95 GHz (but these could be extended to higher frequencies if needed). Furthermore, it is noteworthy that in the longer term this method might be applied to the Doppler radar of Earth Cloud Aerosol and Radiation Explorer (EarthCARE; European Space Agency 2004).

This radar retrieval method is described in section 3. It relies on a set of relationships between the cloud properties and the radar measurements, scaled by the intercept parameter of the normalized particle size distribution (Delanoë et al. 2005), the principle of which is described in section 2. These relationships are derived from the different moments of the normalized particle size distribution. The performance of the method is then evaluated in section 4 using a large microphysics database and comparisons with a radar–lidar method. We also conducted an analysis of the sensitivity of the retrieval to the Doppler velocity averaging period. Conclusions are given in section 5.

## 2. General background

### a. The particle size distribution

*N**

_{0}) significantly reduced this variability (Delanoë et al. 2005; the main results of this paper are summarized in the appendix). In the present study, the PSD is made independent from the ice water content and the mean volume-weighted diameter [see Delanoë et al. (2005) for further details]. A general expression of the PSD can be written as follows:where

*N*(

*D*

_{eq}) is the PSD,

*F*(

*D*

_{eq}/

*D*) is the normalized PSD,

_{m}*D*

_{eq}is the “equivalent melted” diameter (which is the diameter the ice particle would have if it was a spherical water particle of the same mass), and

*N**

_{0}(m

^{−4}) is the intercept parameter of the PSD proportional to IWC/

*D*

_{m}^{4}. The volume-weighted diameter

*D*is the ratio of the fourth to the third moment of the PSD.

_{m}The relationship between the maximum diameter of an ice crystal and its equivalent melted diameter involves an assumption on the ice crystal density, which is a critical point for all-radar and radar–lidar methods (Delanoë et al. 2005). The way this assumption is dealt with in the present radar method has been discussed in Delanoë et al. (2005).

In the same paper, Delanoë et al. (2005) have investigated the stability of the normalized PSD shape in ice clouds, using a very extensive airborne in situ microphysics dataset, including different types of ice clouds, and both midlatitude [1998 Cloud Lidar and Radar Experiment (CLARE98), 1999 Cloud by Ground-Based and Airborne Radar and Lidar (CARL99), European Clouds and Radiation Experiment (EUCREX), Fronts and Atlantic Storm Track Experiment (FASTEX), Atmospheric Radiation Measurement (ARM) Intensive Observation Period (IOP)] and tropical [the Central Equatorial Pacific Experiment (CEPEX) and the Cirrus Regional Study of Tropical Anvils and Cirrus Layers Florida-Area Cirrus Experiment (CRYSTAL-FACE)] campaigns. They found that as in the case of the raindrop size distribution the normalized PSD was fairly identical from one cloud to another, and therefore proposed to use a single analytical formulation (the so-called modified gamma shape) to describe scaled ice cloud PSDs. Delanoë et al. (2005) have shown that the mean relative error on each cloud and observable parameter is clearly minimized when using the modified gamma shape, leading to biases of less than 3% for all clouds and instrumental parameters.

This stability in the shape of the normalized PSD of ice clouds is an important result, because it also implies that a moment *X* of the normalized PSD (i.e., IWC/*N**_{0}, *α*/*N**_{0}, *Z*/*N**_{0}) can be related to any moment *Y* by a power-law relationship *Y* = *aX ^{b}*. When assuming a modified gamma shape, the equivalent

*n*th moment of the normalized PSD can be analytically determined.

### b. Cloud parameters and radar observables

*Z*(assuming Rayleigh scattering) and ice water content are proportional to the sixth and third equivalent moments of the PSD, respectively. Contrary to the case of Rayleigh scattering, the reflectivity coefficient is however not directly proportional to a moment of the PSD in the case of Mie scattering,where

*D*is the maximum (unmelted) diameter,

*|K*

_{w}|^{2}= 0.93 at 3 GHz, and

*σ*

_{bsc}is the Mie backscattering coefficient depending on the maximal diameter;

*λ*is the wavelength and

*ρ*, the density–diameter relationship, can be expressed aswhere

*D*is in centimeters and

*ρ*is in grams per cubic centimeter.

Here, *a _{ρ}* can be expressed as a function of

*b*and the limit diameter

_{ρ}*D*

_{l}:

*a*

_{ρ}= 0.917/(

*D*

_{1}× 10

^{4})

^{bρ}, where

*D*

_{l}(

*μ*m) is the “limit diameter.” The rationale for this limit diameter is that for small diameters the ice density exceeds that of the solid ice, which cannot happen in the real world. When the solid ice density is reached the corresponding diameter is the limit diameter, and below this limit diameter the ice density is set at that of solid ice (

*ρ*= 0.917 g cm

_{i}^{−3}).

*N**

_{0}formalism

*Z*is rewritten aswhere

_{e}*F*(

*D*/

*D*) is the shape of the normalized PSD.

_{m}*υ*(

*D*) is the terminal fall velocity for an ice particle with the diameter

*D*. The

*υ*(

*D*) relationship is related to the ice density and cross-sectional area [cf. Eqs. (6) and (7)] assumptions (Khvorostyanov and Curry 2002; Mitchell et Heymsfield 2005). The terminal fall velocity for each diameter is derived from the formulation of Mitchell and Heymsfield (2005),where

*g*is the gravitational acceleration,

*ρ*is the air density,

_{a}*ν*is the kinematic viscosity, and

*a*(

_{d}*D*) and

*b*(

_{d}*D*) are the coefficients of the relationship between

*X*[Best (also called Davies) number] and the Reynolds number, respectively. These coefficients are calculated using Mitchell and Heymsfield (2005),where

*D*is in centimeters and

*A*is in inverse centimeters squared.

*N**

_{0}and

*D*and the density of liquid water

_{m}*ρ*,In the geometric optics approximation, the visible extinction coefficient

_{w}*α*is proportional to the

*σ*th moment of the physical PSD, where

*σ*is the exponent of the area–diameter relationship,The effective radius

*r*can be defined as the ratio of the third to the second equivalent moment of the PSD. This translates into a direct analytical relationship between

_{e}*r*and

_{e}*α*,where

*ρ*is the density of solid ice (0.917 g cm

_{i}^{−3}), IWC is in grams per cubic centimeter, and

*α*is in inverse meters.

All of these parameters depend on the density–diameter and area–diameter relationships, so the choice of these parameterizations is crucial. For instance, we obtain a bias of about 50% (standard deviation 5%) between IWC computed in using the Brown and Francis (1995) aggregate density–diameter relationships (*ρ* = 0.07*D*^{−1.1}, with *D* in millimeters) and the relationship for aggregates proposed by Mitchell (1996). For *α*, the bias can reach 36% (standard deviation 9%).

In conclusion, if we can estimate the ice density *N**_{0} and *D _{m}*, then we can access an extensive documentation of the ice cloud properties, including ice water content, effective radius, visible extinction (and cloud optical depth), and number concentration. The method proposed in this paper, denoted as radar only (RadOn) herein, consists in estimating these quantities from two radar measurements (radar reflectivity and Doppler velocity). This method is described in the next section.

## 3. Principle of the method

The different steps of the method (summarized on the flowchart of Fig. 1) are described in detail in the following sections.

### a. Terminal fall velocity retrieval from Doppler velocity

*V*and the vertical air motion

_{t}*w*. To estimate

*V*, a statistical approach (hereinafter referred to as the

_{t}*V*–

_{t}*Z*approach) has been recently proposed in the case of frontal cyclones and nonprecipitating ice clouds (Protat et al. 2003). It consists of developing statistical relationships between terminal fall velocity and radar reflectivity, which can be expressed aswhere

*Z*is in millimeters to the sixth power per cubic meter and

*a*and

*b*are the coefficients of

*V*–

_{t}*Z*relationship obtained by linear regression.

Within nonprecipitating clouds (i.e., clouds that do not produce precipitation at the ground), the vertical air motions are generally small, even at small scales of motion, as opposed to the case of convective systems. In any case, however, the vertical air motions are not negligible with respect to the terminal fall speed. For a long time span, however (from one to a few hours), the mean vertical air motion should vanish with respect to the mean terminal fall speed, which is much less fluctuating. A statistical power-law relationship between the terminal fall speed and radar reflectivity may therefore be derived from this statistical approach. Figure 2 shows a scatterplot between Doppler velocity and reflectivity from which a *V _{t}*–

*Z*relationship is derived for a given cloud. Following this approach the scatter around the fitted curve is attributed to the vertical wind component only. Figure 3c shows the corresponding time–height section of

*V*when this relationship is applied to the time–height cross section of reflectivity.

_{t}The implicit assumption of this approach is that the cloud microphysical characteristics do not change within the cloud (the nature of the *V _{t}*–

*Z*relationship, e.g.). It is clear that this approach is not perfect; in particular, it is expected that the

*V*–

_{t}*Z*relationship will probably change in the vertical, especially for thick ice clouds, for which aggregation will produce a diminution of crystal densities and number.

An alternative approach has been used in the radar method of Matrosov et al. (2002) and has been implemented in RadOn, which consists of estimating terminal fall velocities from 20-min averages of the Doppler velocities. In RadOn this approach has been slightly refined by using 20-min running means with a 10-s resolution, which allows for the small-scale variability in the retrievals that is due to possible changes in the cloud microphysical characteristics to be partially kept (referred to as the “running mean” approach in the following). This approach has the great advantage of limiting the assumption of steady microphysics to a 20-min duration horizontally, and to avoid any assumption in the vertical, which seems sensible. The major drawback though is that the vertical air motion will be filtered out in a much less accurate way (occurrences of positive values of fall speed can even be found in the upper part of the clouds, which cannot be treated). The impact of these two approaches on the cloud microphysics retrieval will be analyzed in detail in section 3e, after the method is described.

### b. Density and particle habit retrieval from V_{t}–Z relationship

For different density–diameter relationships and particle habits, we have computed theoretical relationships between the reflectivity-weighted terminal fall velocity and the equivalent reflectivity at 3, 35, and 95 GHz, using a very extensive airborne in situ microphysical dataset, including different types of ice clouds, and both midlatitude (CLARE98, CARL99, EUCREX, FASTEX, ARM IOP) and tropical (CEPEX, CRYSTAL-FACE) datasets. This database is described in detail in Delanoë et al. (2005).

In this study, we assume five different area–diameter relationships spanning the most common particle types observed in stratiform clouds. These five area–diameter relationships have been extracted from Mitchell (1996) (solid spheres, hexagonal plates, hexagonal columns, unspherical aggregates, assemblages of planar polycrystals in cirrus clouds). Then for each habit, each *D*_{l} in the range from 10 to 200 *μ*m, and each *b _{ρ}* in the range from −1.4 to −0.5), we compute the corresponding

*V*–

_{t}*Z*relationship using the whole microphysical in situ database.

By comparing these theoretical power-law *V _{t}*–

*Z*relationships with the radar relationship obtained in section 3a, it is therefore possible to determine indirectly the most representative density–diameter and area–diameter relationships for a given cloud. The relationship that produces the smallest difference in the least squares sense with the radar relationship is finally selected. This indirect retrieval is one of the unique features of the method. In other radar or radar–lidar methods, these density–diameter and area–diameter relationships are fixed and not adapted from a cloud to another (see, e.g., Hogan et al. 2005; Matrosov et al. 2002; Tinel et al. 2005; Wang and Sassen 2002; Donovan and van Lammeren 2001; Okamoto et al. 2000).

### c. D_{m} retrieval from the vertical velocity V_{t}

Once the ice density and area relationships are estimated from the radar and theoretical *V _{t}*–

*Z*relationships, then all of the relationships of section 2b are computed using this ice density. The remaining unknowns used to access the ice cloud properties are

*N**

_{0}and

*D*. To estimate

_{m}*D*, we have developed relationships between

_{m}*V*and

_{t}*D*parameterized by the retrieved ice density using the extensive microphysics dataset. The procedure to produce these

_{m}*V*–D

_{t}_{m}relationships is exactly the same as that described for the

*V*–

_{t}*Z*relationships at the end of the previous section.

*D*if we assume a normalized PSD shape and a density relationship;

_{m}*V*can then be written as a function of

_{t}*D*as follows:where (

_{m}*g*,

*l*) are related to the retrieved ice density and particle habit.

Then, at this step, only *N**_{0} remains to be retrieved in order to access the cloud parameters.

### d. N*_{0} from Z_{e} and D_{m}

*N**

_{0}, there is an analytical relationship between

*N**

_{0}and

*D*and

_{m}*Z*when an analytical shape is assumed for the normalized PSD,where

_{e}*Z*is in millimeters to the sixth power per cubic meter and

_{e}*D*is in meters, and

_{m}*I*(

*D*) is an integral function that depends on the ice particle density and the mean volume-weighted diameter,Using the retrieved

_{m}*N**

_{0}and

*D*, and

_{m}*F*(

*D*/

*D*) the normalized PSD, the cloud properties can be finally retrieved from Eq. (8) for IWC, Eq. (9) for

_{m}*α*, and Eq. (10) for

*r*.

_{e}### e. Illustration of ice cloud retrieval using RadOn

The method described in section 3 has been applied to continuous Doppler cloud radar measurements at 95 GHz collected in the frame of the European Cloudnet project over the Site Instrumental de Recherche par Télédétection Atmosphérique (SIRTA; in Palaiseau, France). The illustrative case shown here is a thick midlatitude prefrontal ice cloud. A backscatter lidar was also operating at that time, but the optical depth of the ice cloud was such that only few hundred meters of the cloud were penetrated by the lidar until complete extinction. This case is therefore a good illustration of the how the radar method and the radar–lidar method complement each other, as the radar method in this particular case allows the upper part of the ice cloud that cannot be reached by the lidar to be explored.

As discussed in section 3a, we have implemented two methods in RadOn to estimate terminal fall speed from the radar measurements, the so-called *V _{t}*–

*Z*and running mean approaches, which have different advantages and drawbacks.

This is illustrated in Fig. 4, for which both approaches have been applied on the 14 April 2003 case of Figs. 1 –3. The retrievals using the *V _{t}*–

*Z*approach shows that overall, effective radii are in the correct range for such thick ice clouds (from 20 to 50

*μ*m), but there is no independent estimate of effective radius available for validation. This figure highlights the pros and cons of both methods—on one hand, there are structures in the Doppler velocity field that are not perfectly correlated with the reflectivity as it is shown by Figs. 3a,b, which suggests that there are local changes in the

*V*–

_{t}*Z*relationship. In this case the running mean approach seems more relevant. However, in the upper part of the cloud large artifacts are observed on IWC and

*α*, which are not linked to any structure in the reflectivity or Doppler velocity field. These artifacts are responsible for the large discrepancies observed on the time series of optical depth (Fig. 4, lower right panel).

The reason for this instability of the retrieval is explained in Fig. 5, which displays a scatterplot of *N**_{0} versus *D _{m}*. As shown by this figure, a small error in

*D*for the small

_{m}*D*range (and therefore the small

_{m}*V*range) has a dramatic impact on the estimate of

_{t}*N**

_{0}, which can vary over several orders of magnitude here. It is therefore believed that the small errors resulting from a less accurate filtering of the vertical air velocities when averaging over 20 min are responsible for the large artifacts observed in the upper part of the cloud. As a result, the current version of RadOn makes use of the

*V*–

_{t}*Z*approach. Furthermore, as will be shown by the comparisons with the radar–lidar methods in section 4, this version seems to provide fairly accurate and roughly unbiased retrievals of cloud microphysics. Moreover, in the case of the running mean approach, some positive values of fall speed occur, for which the algorithm is obviously not able to retrieve clouds parameters.

In the next section, a further attempt to evaluate the RadOn method is developed, by quantifying the errors statistics using an extensive aircraft in situ database described earlier, and by comparing the outputs of the RadOn and radar–lidar retrievals in the common sampling area.

## 4. Performance of the method

In this section, rms errors of the RadOn method are characterized, and the different sources of errors are estimated separately in order to evaluate which source of error is most significant. In section 4a(1), we use the microphysics database to assess the global errors, the variability of these errors within the IWC and *α* ranges, and the variability of the errors when different density–diameter and area–diameter relationships are assumed. We then assess the sensitivity of the method to a radar calibration uncertainty, to a residual of vertical air motions in the terminal fall speed, and to both at the same time. We finally come up with an error statistic that is assumed to be reasonably representative of the true errors (end of section 4a). Then, we have estimated and additional sensitivity to the averaging time used for the filtering of the Doppler velocities in order to access terminal fall speed (section 4b). Last, we carry out an intercomparison with a radar–lidar method using 2 yr of continuous radar–lidar observations over three European ground-based sites (section 4c).

### a. Error analysis using the microphysics database

#### 1) Global rms errors on the cloud and radiative parameters without including the *V*_{t} and density errors

_{t}

In this section , we first develop an error analysis of the method, which does not include the density and *V _{t}* retrieval errors, and we compare these errors with those obtained in the most recent radar methods using the same error analysis. For this purpose, we compute the reflectivity-weighted velocity and the radar reflectivity at 95 GHz from the extensive microphysics database, assuming the Brown and Francis (1995) density–diameter relationship (

*ρ*= 0.07

*D*

^{−1.1}, with

*D*in millimeters). These two input parameters (

*V*and

_{t}*Z*) are introduced in RadOn and the outputs are compared with the “true” parameters computed directly from the microphysics database (Fig. 6), using the same density–diameter relationship as that in RadOn. By doing this, we estimate all sources of errors except the errors associated with the density–diameter relationship and possible errors on

*V*and

_{t}*Z*. These other sources of errors are estimated in section 4a(2). In RadOn,

*D*is first retrieved from

_{m}*V*and subsequently

_{t}*N**

_{0}directly from

*Z*and

_{e}*D*, assuming a density–diameter relationship and an analytical shape for the normalized PSD. We first estimate the errors on these two parameters of the normalized PSD (

_{m}*N**

_{0}and

*D*). Figure 7 shows the

_{m}*N**

_{0}retrieved using the analytical shape (A8) as a function of the true

*N**

_{0}computed directly from the true PSDs for the entire database. The obtained mean relative error and standard deviation on

*N**

_{0}, which includes the error resulting from the assumption on the normalized PSD shape, are of about −2.5% and 15%, which by construction translates into roughly the same errors on the retrieved cloud parameters (Delanoë et al. 2005). This error is mostly due to the assumption on the normalized PSD shape. Figure 8 shows the

*D*retrieved from

_{m}*V*using Eq. (12) as a function of the true

_{t}*D*computed directly from the database. As is previously observed for

_{m}*N**

_{0}, the error is mostly due to the assumption on the normalized PSD shape, a small bias of about −1.5% with less than 11% as standard deviation.

We now turn to the description of the errors on the retrieved cloud properties themselves. The results of this error analysis are summarized in Table 1. All retrieved parameters are in good agreement with the in situ calculations; the bias for the IWC is very small (around 0.4%) with a moderate standard deviation [less than 18%, which is much less than the errors of 50%–100% generally acknowledged for IWC–*Z*/IWC–*Z*–*T* relationships with the same sources of errors included; Protat et al. (2007, hereinafter PR07)]. The visible extinction is also in agreement with the in situ calculation, with a bias of about −3.6% and a standard deviation of 19%. The retrieved effective radius is slightly overestimated by the model (5%), but the standard deviation is less than 11%, which is probably due to the fact that *r _{e}* depends only on

*D*and not on

_{m}*N**

_{0}[Eq. (10)]. This framework using the microphysics database also offers the opportunity to compare RadOn with other radar-only methods recently published in the literature. We have selected three methods. Two of these, Hogan et al. (2006, hereinafter HO06) and PR07, are statistical relationship methods relating the ice water content to the radar reflectivity and ambient temperature through empirical relationships. IWC–

*Z*–

*T*HO06 relationships are derived from EUCREX and CEPEX datasets, while IWC–

*Z*–

*T*PR07 relationships have been derived from a larger database (CLARE98, CARL99, EUCREX, FASTEX, ARM IOP, CEPEX, and CRYSTAL-FACE). However, both use the same methodology rigorously. The third one is the Doppler radar method initiated by Matrosov et al. (2002), previously mentioned in the introduction. This latter method uses a 35-GHz radar with a fixed density–diameter relationship, and derives a

*D*

_{0}diameter from the terminal fall velocity

*V*and IWC, and

_{t}*α*from

*D*

_{0}and

*Z*. Results are summarized in Table 1.

For IWC, the standard deviation of RadOn in the Matrosov et al. (2002) Doppler radar methods is better than that of the statistical IWC–*Z*–*T* relationships (less than 20% against more than 58%). However, the bias produced by the Matrosov method is higher than IWC–*Z*–*T* PR07 and RadOn, most likely because of the radar frequency used in this test [Matrosov et al. (2002) is devoted to a 35-GHz radar because it has been built using the Rayleigh approximation] and the fact that the density–diameter relationship is fixed in the Matrosov method while it is adapted for each cloud situation in RadOn. So when particles become larger, the error strongly increases because of the effects of Mie scattering.

For the *α* retrieval, both Doppler radar methods are very close in terms of standard deviation (15% for Matrosov and 19% for RadOn), although the bias of RadOn is much less (−47% and −3.6%, respectively). This larger bias likely results from the Mie effect, as discussed previously.

#### 2) Error analysis as a function of *A*(*D*) and *ρ*(*D*) for a radar at 95 GHz

The purpose of this section is to evaluate whether the errors assuming the Brown and Francis (1995) density–diameter relationship are similar when other particle habits and density–diameter relationships are considered. Figures 9, 10 and 11 show biases and standard deviations of the relative difference between cloud parameters retrieved from RadOn and those derived from the in situ measurements. Biases and standard deviations are given as a function of five different area–diameter relationships. For each relationship, we change the exponent *b _{ρ}* of the density–diameter relationships. For this illustration, we fixed the upper limit of the particle diameter to 100

*μ*m (

*D*

_{l}). Considering other limit diameters yields similar results (not shown).

As shown in Fig. 9 the bias on the IWC is very small (less than 0.4%) and the standard deviation is around 18.1%, assuming *b _{ρ}* = −1.1. If we consider extreme values of the exponent of

*ρ*(

*D*) (from −1.4 to −0.5), the errors are not very different. The overestimation of IWC is less than 5.5% with a standard deviation less than 20% for

*b*= −1.4. IWC is underestimated (around 8%) with the same standard deviation (21%) for

_{ρ}*b*= −0.5. The error does not depend on the area–diameter relationship when

_{ρ}*b*varies between −0.5 and −1.1.

_{ρ}Figure 12 shows the bias and standard deviation as a function of *b _{ρ}* for IWC. The bias is positive when

*b*< −1.1 and negative above −1.1. The standard deviation increases slowly with

_{ρ}*b*, from 15% to 21%. Whatever the density–diameter relationship, as is shown by Fig. 10, we underestimate

_{ρ}*α*from −2% to −10%. Contrary to the case of IWC, the standard deviation of the relative difference in

*α*is strongly dependent on the area–diameter relationship. It increases strongly as

*b*decreases and the particle type departs strongly from the sphere (Fig. 13). The largest error (bias = −10% and std = 37.8%) is obtained when the area–diameter relationship is

_{ρ}*A*(

*D*) = 0.05

*D*

^{1.4}, which corresponds to the hexagonal columns for the particle larger than 300

*μ*m (Mitchell 1996). However, in the literature this area–diameter relationship is encountered only when

*b*is less than −1 (Mitchell 1996). When the area–diameter relationship is closer to the spherical particle (or plates) the bias is around 4% and the standard deviation is 28%. Globally, biases and standard deviations increase with

_{ρ}*b*.

_{ρ}The effective radius is proportional to the ratio IWC/*α* (Francis et al. 1994). As shown in Fig. 11, *r _{e}* is slightly overestimated (from 4% to 16%). As for

*α*, biases and standard deviations are larger for the combination

*A*(

*D*) = 0.05

*D*

^{1.4}and

*b*= −0.5. However, if we remove this unlikely configuration (discussed previously), the bias does not exceed 10% and the standard deviation is less than 16%. It is noteworthy that the same study conducted with a radar frequency of 35 GHz yielded similar results (not shown).

_{ρ}#### 3) Error variability as a function of IWC and *α*

In this section, we estimate the relative root-mean-square difference (rmsd) between cloud parameter (IWC, *α*) computed directly from the measured PSDs and the RadOn retrieval of these parameters as a function of the logarithm of the evaluated parameter. The rmsd characterizes both the bias and the standard deviation of the relative difference. In this section, we focus solely on the ice water content and the visible extinction, because the effective radius is deduced from the ratio IWC/*α* and its rmsd is less than 15% (not shown), whatever the area–diameter relationship considered. This study has been conducted using several combinations of *A*(*D*) and *ρ*(*D*) relationships, as previously, but we only show the result for the Brown and Francis (1995) relationship.

Figure 14a shows the relative rmsd on IWC as a function of log(IWC) and confirms the previous results; the error does not depend on the choice of *A*(*D*). The rmsd is nearly constant (about 20%) through the log(IWC) range. However, as shown in Fig. 14b, *α* is a little bit more dependent on *A*(*D*). The rmsd decreases slightly when log(*α*) increases in the range from 10^{−6} to 10^{−3} and is 10% greater for the hexagonal columns than for the other *A*(*D*) (as seen in the previous section).

#### 4) RadOn sensitivity to a calibration error and to *V*_{t} errors

_{t}

Unfortunately, the radar measurements are not perfect and an error on the *V _{t}* retrieval from the

*V*–

_{t}*Z*relationship can reach from ±5 to ±10 cm s

^{−1}(Protat et al. 2003), which roughly corresponds to the synoptic environmental uplift in a cloud. Potential effects of a change in the microphysical characteristics or an increase of the

*V*error resulting from the time lag used to filter out the small-scale vertical air motions can be treated as sources that increase the

_{t}*V*variability for a given

_{t}*Z*. This is why we have also estimated the errors arising from an addition of a Gaussian noise of a given standard deviation to

*V*. In addition, although cloud radars can be fairly accurately calibrated, it is difficult to achieve accuracies better than 1 dB, which needs to be accounted for in our error calculations. It is also important to note that these two errors indirectly produce errors in the retrieval of the density–diameter relationship. As a result, this last sensitivity test can be viewed as a representative estimate of the total true errors of RadOn.

_{t}In this section we first estimate the sensitivity of RadOn to a calibration error on radar reflectivity (+1 and then +2 dB*Z*) and to a random noise on terminal fall velocity (±5, ±10 cm s^{−1}), in addition to the other sources of errors estimated previously. As is shown in Table 2, a +1 (+2) dB*Z* calibration error translates into a bias on IWC and *α* of about 18% and 25% (35% and 40%), which is larger than previously estimated in Table 1. However, the standard deviation of the errors on the clouds parameters is less sensitive (e.g., +1 dB*Z* leads to *σ*_{IWC} = 13.4, as compared with 18% in Table 1). Moreover, there is only the effect on the PSD shape and *D _{m}* retrieval on the effective radius.

When we apply a Gaussian noise with a ±5 cm s^{−1} standard deviation on *V _{t}*, the biases on IWC and

*α*do not increase much (less than around 5%) and the standard deviation is around 32%. If we increase the standard deviation of the Gaussian noise up to ±10 cm s

^{−1}, the bias reaches 22.5% and the standard deviation is 70%–80%. Again, the effective radius is not sensitive to the

*V*–

_{t}*Z*random error, the bias is only due to the PSD assumption, and the

*D*retrieval error and standard deviation is less than 12%.

_{m}As discussed previously it is expected that there should be a residual vertical air motion after filtering due to the synoptic uplift. Therefore, we now investigate the error of the method if the terminal fall velocity is shifted with a +10 cm s^{−1} vertical air motion contribution. The determination of the density–diameter and area–diameter relationships is also affected by the error on *V _{t}*. Figure 15 represents the relative rms difference between the true IWC and

*α*[from the in situ measurements when we apply the Brown and Francis (1995) relationship] and IWC and

*α*obtained by adding up to +10 cm s

^{−1}to

*V*. The errors are shown as a function of the true IWC and

_{t}*α*. In this case the error in IWC does not exceed 30% and remains less than 20% for the IWC range of 2 × 10

^{−3}to 0.1 g cm

^{−3}. The increase of the error owing to this bias is unexpectedly small, and even smaller than without the bias in the small IWC bins. The effect of this bias is to increase the error for the large IWCs, but surprisingly to reduce the errors for the small IWCs. This is most likely the result of a different adjustment of the density–diameter and area–diameter relationships in the retrieval. Regarding

*α*, the effect of a 10 cm s

^{−1}bias on

*V*is obviously larger than that for IWC, with errors of around 40%–50%, corresponding to an increase by around 30% over the whole a range with respect to the case of perfect

_{t}*V*s.

_{t}When a 1-dB*Z* calibration error is introduced in *Z*, the error statistics on IWC is not modified with respect to the case of a perfect calibration for IWC less than 10^{−1.5} g m^{−3} (the error is less than 20%), while it increases up to 30% for IWC larger than 10^{−1.5} g cm^{−3} (a 10% increase with respect to the case of a perfect calibration). The increase of the error on *α* is around 10% over the whole range, yielding relative errors on *α* of around 30%. It is noteworthy that the effect on *α* of a calibration error is much less than the effect of a bias on *V _{t}*.

To estimate an error statistic that is as realistic as possible for RadOn, we finally added both 10 cm s^{−1} to *V _{t}* and 1 dB

*Z*to

*Z*. This translates into an error on IWC of less than 30% for IWC less than 0.06 g m

^{−3}, which increases up to 40% for the larger IWCs. However, IWC is not really sensitive to a calibration error or a

*V*bias if we treat them separately, except for the large IWCs. The error on the extinction is around 55%–60% over the whole range of extinctions. Errors resulting from

_{t}*V*and calibration are additive in the case of extinction, which is not the case for IWC, except for large IWCs.

_{t}### b. Sensitivity of the retrieval to the Doppler velocity averaging period

Using the 14 April 2003 case (previously described in section 3e), we have investigated the impact of the time lag used to filter out the vertical air motions in the Doppler velocities on the retrieval. To do so, we have derived *V _{t}*–

*Z*relationships every 1, 2, and 3 h, and then we have compared the results with the

*V*–

_{t}*Z*derived using the whole time period (approximately 17 h for this particular case). The results have been summarized in Table 3 for

*V*,

_{t}*α*, and IWC. As suggested by Table 3, the effect of the averaging period on

*V*and

_{t}*α*is essentially to increase the standard deviation of the error from 8% for a 3-h averaging up to 16% for a 1-h averaging for

*V*, and from 20% for a 3-h averaging up to 56% for a 1-h averaging for

_{t}*α*. Regarding IWC, the averaging tends to produce a bias (which was not the case for

*V*and

_{t}*α*) increasing from −9% for 3 h to −22% for a 1-h averaging. Conversely, the increase in standard deviation (11%, less than in Table 1, up to 24% for 1-h averaging) is smaller than that observed for

*V*and

_{t}*α*.

As it appears from Table 3—that using a 1-h averaging period tends to degrade the quality of the retrieval—we suggest that the averaging period should not be less than 2 h.

### c. Comparison with radar–lidar method

To evaluate the performance of RadOn statistically, it is compared in the present section with the radar–lidar retrieval method of Tinel et al. (2005) in the cloud areas sampled by both the radar and the lidar. This is done using the whole Cloudnet radar–lidar database (Illingworth et al. 2007) collected during 2 yr from three European instrumented sites (Chilbolton, United Kingdom, Cabauw, Netherlands, and Palaiseau, France).

From Figs. 16 –17, it appears clearly that the RadOn and radar–lidar methods are statistically consistent, with a roughly unbiased estimate of IWC and *α* over the whole IWC/*α* variability range.

The standard deviation is also not very large (22%), as shown by the fact that most points are close to the 1:1 line (gray to white dots). It is however observed that for intermediate to large IWCs (>10^{−2} g m^{−3}) a bias appears, with IWCs retrieved with RadOn being larger than those retrieved with the radar–lidar method. This is due to the Mie effect (Mie 1908), which is not accounted for in the radar–lidar method, while it is in RadOn; thus, this is more likely to be a problem of the radar–lidar method.

It is particularly impressive to see in Fig. 17 how the estimate of extinction by the radar is good when compared with the much more direct estimate by the radar–lidar method. This is despite the fact that extinction is only very indirectly derived in RadOn from the *A*(*D*) relationship selected in the step described in section 3b, whereas it is directly proportional to the backscatter measured by the lidar. This good agreement indirectly validates the retrieval of density–diameter and area–diameter relationships. Furthermore, as mentioned by Hogan et al. (2005), the visible extinction retrieved from the radar–lidar algorithm is not sensitive to the *A*(*D*) and *ρ*(*D*) assumptions.

## 5. Conclusions

A new method for retrieval of ice cloud properties from ground-based Doppler radar observations (called RadOn) has been described in the present paper. From the Doppler velocity and radar reflectivities we retrieve IWC, *α*, *r _{e}*, and the optical thickness using the normalized PSD concept [

*N*(D) =

*N**

_{0}

*F*(

*D*/

*D*)]. The terminal fall velocity is derived from the Doppler velocity using a

_{m}*V*–

_{t}*Z*relationship. From this relationship, we estimate the most representative density–diameter and area–diameter relationships, by comparing the radar

*V*–

_{t}*Z*with in situ

*V*–

_{t}*Z*relationships. The relationship that produces the smallest difference in the least squares sense with the radar relationship is selected. Once the density– and area–diameter relationships are fixed, we derive

*D*from

_{m}*V*and

_{t}*N**

_{0}from

*D*and

_{m}*Z*.

This method has then been evaluated using an extensive microphysics in situ database. We have first carried out an error analysis assuming a perfect measurement of the terminal fall velocity and the radar reflectivity and no error in the density–diameter relationship retrieval. All retrieved parameters are in good agreement with the in situ calculations; the bias for the IWC is very small (around 0.4%), with a moderate standard deviation (less than 18%). The visible extinction is also in agreement with the in situ calculation, with a bias of about −3.6% and a standard deviation very close to 19%. The retrieved effective radius is overestimated by the model (5%), but the standard deviation is less than 11%. Moreover, this analysis has been conducted with several density–diameter and area–diameter relationships, yielding similar results.

We then carried out a new error evaluation, taking into account a radar calibration error and a residual in the terminal fall velocity. The impact of a radar calibration error, when we assume no error on the density retrieval, is to produce a bias of 20%–25% in the cloud parameters (for 1-dB*Z* error), while a random *V _{t}* error tends to increase the standard deviation from 18% to 32% (for a ±5 cm s

^{−1}random noise). Because the two effects are very similar on IWC and

*α*, the effective radius is not really affected. When we take into account both density retrieval errors—the potential radar calibration error and a 10 cm s

^{−1}bias in

*V*—the final relative rms error in IWC and

_{t}*α*are 30%–40% and 55%–60%, respectively.

We also compared the RadOn retrievals with coincident radar–lidar retrievals and showed that the estimate of extinction by radar is good when compared with the much more direct estimate by the radar–lidar method. This is despite the fact that extinction is only very indirectly derived in RadOn from the *A*(*D*) relationship selected, whereas it is directly proportional to the backscatter measured by the lidar. This good agreement indirectly validates the retrieval of density–diameter and area–diameter relationships. We also presently participate to an intercomparison exercise using measured in situ IWC profiles and PSDs, from which radar reflectivities and Doppler fall speeds are simulated. This should be compared with the present error estimates when available.

This method is presently being systematically applied to the cloud radar measurements collected over the three instrumented sites of the European Cloudnet project to evaluate the representation of ice clouds in numerical weather prediction models and to build up a cloud climatology.

## Acknowledgments

This work was carried out with the support of EU Cloudnet Contract EVK2-CT-2000-00064, and the funding from the Centre National d’Etudes Spatiales (CNES) and Institut National des Sciences de l’Univers (INSU) in the frame of the RALI project. The authors are also grateful to Drs. R. J. Hogan and E. J. O’Connor for providing the categorization and classification Cloudnet files that made the algorithm application to the Cloudnet database easier.

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## APPENDIX

### Recall of the Principle of the Normalization of the PSD and Main Results

In this appendix, we summarize the important results of the normalized particle size distribution study of Delanoë et al. (2005), and we encourage the reader to refer to the original paper for more details.

The convenient formulation of the “equivalent melted diameter” has been chosen for use instead of the physical diameter, which corresponds to the diameter the ice particle would have if it was a spherical water particle of the same mass. This formulation implies that a density–diameter relationship must be assumed. By definition, the particle size distribution *N*(*D*_{eq}) is the number of particles per unit volume and per interval of diameter (m^{−4}), where *D*_{eq} is the melted equivalent diameter (m). This formalism, known as the “normalized PSD,” consists of scaling the diameter and *N*(*D*_{eq}) axes in such a way that the PSDs are independent of the ice water content (IWC) and the mean volume-weighted diameter (*D _{m}*).

^{−3}) is proportional to the third moment of the PSD and

*D*(m) is proportional to the ratio of the fourth to the third moment of the PSD:A general expression of the PSD can be written aswhere

_{m}*N**

_{0}is the scaling parameter for the concentration axis,

*D*is the scaling parameter for the diameter axis, and

_{m}*F*denotes the normalized PSD.

*F*(

*X*) satisfies the following equation:Then, considering (A2), we can also writeIt follows from (A4) and (A5) that to make the normalized PSD independent of IWC and

*D*, the third moment of the PSD must be constant. This constant has been chosen in such a way that the

_{m}*N**

_{0}parameter is equal to the intercept parameter

*N*

_{0}of the exponential Marshall and Palmer (1948) PSD, which yieldsThe

*N**

_{0}parameter is a function of IWC and

*D*, which can be written asWe have investigated the statistical properties of the normalized particle size distribution in ice clouds. To do so, an extensive database of airborne in situ microphysical measurements has been constructed and analyzed. Qualitatively, it is first obtained that there is a remarkable stability in the shape of the normalized PSD for the normalized diameters

_{m}*D*

_{eq}/

*D*smaller than 2, and a larger variability for larger diameters. A global analysis has therefore been conducted in order to assess the errors introduced on radar- and lidar-related parameters (reflectivity, specific attenuation, visible extinction) and cloud parameters (ice water content, effective radius, terminal fall velocity) derived by the use of a single analytical PSD shape for all the PSDs in a large in situ database instead of the “true” shape of each normalized PSD of the database. Different analytical shapes have been evaluated in this way. It has been obtained that the so-called modified gamma shape could be used as an accurate approximation of the normalized PSD for any normalized ice particle size distribution, and for any instrumental or cloud parameter to be derived from the normalized PSD, because it has the unique advantage of well fitting the particular “S shaped” structure of the ice cloud PSDs, where most data points are located. This normalized modified gamma shape [Γ

_{m}*(*

_{m}*α*,

*β*)] can be expressed aswhere

*α*and

*β*can be variationally adjusted to the measured PSDs. A value of

*α*= −1 and

*β*= 3 produced the smallest errors in Delanoë et al. (2005), with a weak bias and a small standard deviation of −0.66% and 4.75%, respectively. These results are still valid when we change the density–diameter relationship.

Comparison error analysis between RadOn and the other methods.

Sensitivity of RadOn retrievals to calibration and *V _{t}*–

*Z*errors.

Sensitivity of the retrieval to the Doppler velocity averaging period.

^{}

* The National Center for Atmospheric Research is sponsored by the National Science Foundation

^{1}

The Afternoon Train or “A Train” is the nickname given to a group of satellites that fly close together and pass over the equator in the early afternoon.