Abstract
 Top of page
 Abstract
 1. Introduction
 2. Sea Surface Perturbation by Rain
 3. Dependence of RainInduced Surface Variance on DSD
 4. Dependence of Rain Radar Reflectivity on DSD
 5. Selection Criteria on DSDs
 6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 7. Discussion
 8. Conclusion
 Acknowledgments
 References
 Supporting Information
[1] Raindrops impacting the rough sea modify its surface and its backscattering coefficient. This roughness change essentially depends on the rain content in very large drops, which is highly variable from one drop size distribution model to another. However, it has been observed that the radar reflectivity of raindrops has a drop size dependence very similar to that of the ringwaves induced by rain on the surface. From a numerical analysis on various drop size distributions, rain rates, and frequencies from 3 to 35 GHz, a relationship between the sea surface elevation variance of ringwaves resulting from drop impact and the rain radar reflectivity Z is established. It is found to be weakly dependent on the raindrop size distribution model. This link is expected to lead to better estimates of the surface roughness, and in turn, via electromagnetic scattering models, it could improve algorithms for near nadir rain radar retrieval.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Sea Surface Perturbation by Rain
 3. Dependence of RainInduced Surface Variance on DSD
 4. Dependence of Rain Radar Reflectivity on DSD
 5. Selection Criteria on DSDs
 6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 7. Discussion
 8. Conclusion
 Acknowledgments
 References
 Supporting Information
[2] The measurement of rain has always been of great interest for human activities and its significance has increased in view of climatic challenges. Global rain monitoring became possible with microwave ground based remote sensing techniques like weather radars, as well as with the launch of remote sensing satellites. Major achievements have been reached thanks to several satellite missions with payloads containing multifrequency radiometers, and more recently the successful TRMM mission. Future perspectives are ahead with the Global Precipitation Mission (GPM) (R. K. Kakar et al., Global Precipitation Measurement (GPM) progress, paper presented at 2nd International GPM Ground Validation Workshop, National Taiwan University, Taipei, Taiwan, 2005) planned for the near future. The core instrument on TRMM as in GPM is a precipitation radar (PR) that measures the rain reflectivity Z from space at or near nadir.
[3] To determine accurate rain profiles a resolution high enough, consistent with rain cell sizes, is needed. For ground based stations the antenna size and transmitted power are not limiting factors as it is the case for spaceborne payloads. Therefore in ground based radars non attenuating low microwave frequencies can be used while space radars will have to use higher but attenuating frequencies.
[4] The algorithms retrieving vertical rain profiles from apparent Z obtained by rain radars often make use of power law relationships between rain rate R, specific attenuation k and reflectivity Z. These are basically marchingon procedures in which Z_{i} at each gate i is used to estimate the corresponding rain rate R_{i}, and in turn, the attenuation k_{i} necessary to correct the apparent Z_{i+1} in the next gate i + 1 for the twoway path attenuation. It is known that such procedures, like the classical HB algorithm [Hitschfeld and Bordan, 1954], suffer from significant propagation errors and instabilities due to the fact that, among others, the first gates correspond to the 0°C isotherm where mixed rain and melting hydrometeors coexist. This affects the classical rain only relationships and introduces significant uncertainties. Therefore researchers introduced additional constraints in order to stabilize the retrieval algorithms. A possible constraint is the total PIA (Path Integrated Attenuation) that can be exploited in different forward or backward techniques like the SRT (Surface Reference Technique) and companion methods for a single frequency PR [Iguchi et al., 2000; Meneghini et al., 2001], the DSRT (Dual Surface Reference Technique) or the DWT (Dual Wavelength Technique) for a dual frequency PR [Kozu et al., 1991; Meneghini et al., 1992; Nakamura and Iguchi, 2007].
[5] The twoway PIA may be evaluated from various sources, like collocated radiometric measurements or, when the PR flies above the ocean, from the estimate of the attenuation of the ocean surface echo inside rain, provided that a good estimate of the surface reflectivity may be found. One way to obtain this information is to use the surface reflectivity of the ocean just outside the rain cell as a reference [Meneghini et al., 2001; Ferreira et al., 2001; Durden et al., 2003; Meneghini and Liao, 2007]. In this technique, the backscattering coefficients of the surface σ° inside rain and outside rain are assumed to be equal. However, it is well known that the impact of raindrops on the sea surface significantly modifies its backscattering coefficient [Moore et al., 1979]. The purpose of this paper is to contribute to further improvements of rain radar retrieval algorithms by establishing an additional link between the ocean surface echo perturbed by rain and the rain reflectivity above the surface. The idea is based on the observation that those two quantities have similar dependence on drop size distribution (DSD).
[6] This paper is organized as follows: the perturbation of the sea surface by the rain impact is analyzed (section 2) in order to derive the dependence of the raininduced surface variance on the drop size distribution (section 3). A selection of DSDs is presented (section 4) and the similarities between the raininduced surface variance and the rain reflectivity dependencies versus the DSD are highlighted. Numerical simulations of both quantities versus rain rate for various DSDs (section 5) lead after a selection procedure to a polynomial identification linking directly surface variance and rain reflectivity (section 6).
2. Sea Surface Perturbation by Rain
 Top of page
 Abstract
 1. Introduction
 2. Sea Surface Perturbation by Rain
 3. Dependence of RainInduced Surface Variance on DSD
 4. Dependence of Rain Radar Reflectivity on DSD
 5. Selection Criteria on DSDs
 6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 7. Discussion
 8. Conclusion
 Acknowledgments
 References
 Supporting Information
[8] As for the surface roughening, when a single raindrop hits a quiet water surface, it generates a crater with a crown which collapses to form a vertical stalk of water, and then subsides to spawn rings of gravitycapillary waves [Worthington, 1963; Le Méhauté et al., 1987; Le Méhauté, 1988]. Hallett and Christensen [1984] and Rein [1993] analyzed these features through highspeed photography, while Hansen [1986] combined such observations with highresolution radar measurements at 9 GHz and grazing angles for individual splashes. Stalks have been identified as the dominant feature for backscattered power at grazing angles for individual drops of 3 and 4 mm [Wetzel, 1990].
[9] Besides the splash products, rain also damps the sea waves if the rain intensity is high enough. The very first study of this effect seems to be that of Reynolds [1900], who in 1875 mentioned the damping effect of sea waves by rain which is well known to sailors. This damping has been modeled by Manton [1973], who established an “eddy viscosity” of water related to the turbulence induced by the raindrop impacts. Wavetank experiments under very heavy artificial rain (150–600 mm/h at 50% of drop terminal velocity) by Tsimplis and Thorpe [1989] allowed them to estimate the eddy viscosity [Tsimplis, 1992]. Nuysten [1990] extended the modeling of the damping effect to a twolayer surface wave model of a turbulent layer overlaying a second nonturbulent one.
[10] Wavetank experiments with highspeed photography of surface features induced by single drop impacts falling very close to terminal velocity (within 1%) have further been performed, together with scattering measurements. These wavetank measurements have been extended to the surface backscattering in case of monodisperse and polydisperse artificial rains, with drops falling also at terminal velocity. Experiments to analyze the drop size dependence have been conducted as well [Bliven et al., 1993; Sobieski and Bliven, 1995; Bliven et al., 1997; Sobieski et al., 1999]. These have shown that, at scatterometric incidences (30° from nadir), the dominant features of backscattered power due to the impact of drops are not the stalks but the generated ringwaves. These yield an additional component in the surface elevation spectrum, superimposed to the surface spectrum due to instantaneous wind speed and wind history.
[11] Multifrequency (S, C and X bands) and multipolarization scatterometric and Doppler measurements, in laboratory as well as in open field, along with rain data confirmed the previous findings. More specifically extensive measurements by Braun et al. [2002] and Braun and Gade [2006] indicated that the dominant scattering mechanism of a rainroughened water surface observed at all incidence angles at VV polarization is Bragg scattering from ringwaves. For HH polarization, the radar backscattering mechanisms are dependent on incidence angle: at steep incidences Bragg scattering from ringwaves is dominant, while with increasing angle this effect diminishes and scattering from nonpropagating splash products, like stalks, becomes more visible, as delineated from the conclusions of Wetzel [1990].
[12] Ku band measurements during the KWAJEX experiment, as well as on the Cowlitz river by Contreras et al. [2003] and Contreras and Plant [2004], extended the previous results to open field rain conditions. They concluded from Doppler spectra that VV backscatter during rain is mainly due to ringwaves, while HH backscatter is also from ringwaves at moderate incidence angles (20°–60°) but contains substantial contributions from stationary splash products at high incidence angles.
[13] One significant outcome of these studies lies in the fact that for incidence angles not too far from nadir the rain effect appears, first, to be mostly related to the roughening of the surface due to ringwaves, and second, that this effect may reasonably be considered as additive. This approximatively additive characteristic allowed Draper and Long [2004a, 2004b] to retrieve both wind and rain fields from SeaWinds data. On the contrary for higher incidence angles it is probably the combined effect of both ringwaves and stalks may cause significant increases in sea surface radar cross section as can be suggested from the data collected by Weissman and Bourassa [2008].
[14] In this paper, we focus on the change of the sea surface spectrum of the surface displacements in presence of rain and wind. A model for the additive spectral component to the wind spectrum due to the presence of the ringwaves has been derived for monodisperse rain by Bliven et al. [1997] and Craeye et al. [1997]. The wind only radial spectrum S(K), where K is the wave number (cm^{−1}) and this additional component due to rain are shown in Figure 1 for increasing wind friction velocities u_{*} and rain rates R. The wind spectrum shown here is obtained from Lemaire et al. [1999], but could be replaced by other models available in the literature [e.g., Elfouhaily et al., 1997]. Figure 1b shows the details of the rain radial spectrum in the frequency domain for the monodisperse case (2.8 mm diameter drops) whose analytical expression is given by Lemaire et al. [2002] for various drop sizes.
[15] The elevation variance can be written as:
[16] It should be noted that nonlinear combinations of rain and wind spectra have also been studied by Craeye [1998] and recently in the very comprehensive paper by Contreras and Plant [2006]. However, this aspect will be left outside the scope of this paper, which looks for a direct relationship between the elevation variance σ^{2} of the ringwaves and the drop reflectivity Z.
3. Dependence of RainInduced Surface Variance on DSD
 Top of page
 Abstract
 1. Introduction
 2. Sea Surface Perturbation by Rain
 3. Dependence of RainInduced Surface Variance on DSD
 4. Dependence of Rain Radar Reflectivity on DSD
 5. Selection Criteria on DSDs
 6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 7. Discussion
 8. Conclusion
 Acknowledgments
 References
 Supporting Information
[17] Experiments with single drops by Craeye et al. [1999] have shown that the energy transferred by the drop into surface ringwaves is not proportional to the kinetic energy of the drop. Indeed, it was observed that the largest drops have a much larger relative contribution to the surface elevation variance than small ones. In other words, for large drops, a much larger fraction of the incident kinetic energy is transformed into surface waves. More precisely, this study has shown that the contribution of a given drop to the surface elevation variance is nearly proportional to the square of its momentum m^{2}v^{2}. Other experimental data obtained for various monodisperse artificial rain events, with various drops sizes [Lemaire et al., 2002], confirmed the drop size scaling rule based on the squared momentum law. In the following paragraphs, we provide three steps supporting the squared momentum model.
3.1. Step 1: Single Drop
[18] Let us first define the normalized surface energy E(t) (dimensions m^{2}) of a ringwave generated by a drop after an impact at time t = 0:
where r is the horizontal distance from the point of impact, z(r, t) is the water height and the integral over dS is taken over the horizontal surface where the ringwave extends. Owing to the viscosity of water, we may expect an exponential decay of energy versus time. The corresponding lifetime is denoted by τ. This exponential decay is a mere approximation, among others, because the ringwaves need to be fully developed (the point of impact is almost at rest again) for this model to be reasonable. Nevertheless, the reasoning below holds with a more accurate time dependence than the exponential model above.
[19] Let us denote by K = mv^{2}/2 the kinetic energy of the impacting drop. As will be detailed below, the fraction of the drop kinetic energy transferred to surface waves, i.e., the E_{0}/K ratio, is not constant and exhibits a strong dependence on drop size. Actually, it appears that, in good approximation, this fraction increases linearly with the volume (or mass) of the drop. This point can be deduced from Le Méhauté [1988], where the amplitude z of the wave is found to be proportional to m v. This indeed leads to a value of E_{0} proportional to m^{2}v^{2}, and in turn, a E_{0}/K ratio proportional to m. Measurements presented by Craeye et al. [1999] for two different drop sizes support this relationship. This is also quantitatively consistent for the full monodisperse rain case [Lemaire et al., 2002], analyzed hereafter. In the following, we will use the model:
where D is the drop size (mm) and the proportionality constants C and C_{1} are with dimensions kg^{−2}s^{2} and m^{−6}s^{2}, respectively.
3.2. Step 2: Full Monodisperse Rain
[20] We now consider rain with constant drop size D. If N_{mono} is the density of falling drops in air (in m^{−3}), and v(D) is their velocity, the number n_{o} of drops that hit a unitary surface per unit of time (in m^{−2}s^{−1}) can be obtained as:
[21] Alternatively, if the rain rate R (in mm h^{−1}) is known, we also have:
where χ is a constant equal to 530, expressed in mm^{2} m^{−2} h s^{−1}. The normalized surface energy generated by an individual drop may be written as:
where, in first approximation, f_{o}(t) = e^{−t}/τ for t > 0 and f_{o} = 0 for t < 0. Therefore, if we consider the surface as the superposition of individual ringwaves, the average total normalized energy can be obtained by assuming ergodicity of the height random variable. Doing so, we obtain the elevation variance as:
[22] Using the expressions above for the number n_{o} of drops impacting a unit surface per unit of time, we obtain two possible expressions for the elevation variance:
and
[23] This shows that once σ^{2} is known along with the characteristics of falling drops (R, D, v) the product C_{1}τ can be found.
[24] Expression (8) will be extended in the next section to natural rain conditions. The drop size dependence of expression (9) has been found consistent with data presented by Lemaire et al. [2002], where elevation variances obtained with identical rain rates and different drop sizes are compared. This further supports model (3).
[25] It should be noted that the expressions above still hold for a nonexponential decay of the energy brought to the surface by a single drop. In this case, the lifetime must be defined as:
where f_{o} is defined above and has a maximum value equal to 1.
3.3. Step 3: Natural Rain
[26] Natural rain is characterized by its drop size distribution (DSD) N(D), defined in such a way that N(D) dD is the number of drops per m^{3} with a size in the interval [D − dD/2, D + dD/2]. Following the reasoning of the previous paragraph, it can be assumed that each drop brings to the surface an energy proportional to its squared momentum E_{0} = C_{1}D^{6}v^{2}(D), and that this energy lasts for a given time τ. The number of drops, with a size in the interval [D − dD/2, D + dD/2], that hit a unit surface per unit of time is v(D) N(D) dD. The integration of individual ringwave energies over the distribution of drops that hit the surface then yields:
which shows a D^{6}v^{3}(D) dependence for the ringwaves height variance.
[27] From the studies by Craeye [1998] and Lemaire et al. [2002], we found that τ(R) may be obtained from monodisperse rain experimental data. These allowed us to estimate the product C_{1}τ and to approximate this quantity by the following exponential function:
which is a slowly decreasing function of rain rate up to values of R ≃ 100mm/h. For light rain rates, τ(R) appears almost constant, because interactions are negligible, while it tends to become smaller when the rain rate increases, because of an increase of turbulence and dissipation rate. From this weak dependence for monodisperse rains, we also concluded that the lifetime may be assumed to depend on the rain rate R only, which implies that the surface elevation variance σ^{2} may be written as:
and thus depends on the sixth power of D and the third power of terminal velocity v(D). This drop velocity in rain is a well known quantity [Gunn and Kinzer, 1949]. Owing to air friction effects, it presents a rather low dependence on drop diameter D and a saturation effect for large drops with a terminal drop velocity v(D) ≃ 9 m/s.
[28] For the sake of completeness and to highlight the consistency of this model, it should still be noted that the results from Le Méhauté [1988] and Sobieski et al. [1999] have been exploited to reproduce scattering data by Contreras and Plant [2006], based on tuning of the ringwave maximum age and of the radius of the drop impulse. With this procedure, they obtained ringwave lifetimes of about τ = 4 s to match observations from the KWAJEX campaign.
4. Dependence of Rain Radar Reflectivity on DSD
 Top of page
 Abstract
 1. Introduction
 2. Sea Surface Perturbation by Rain
 3. Dependence of RainInduced Surface Variance on DSD
 4. Dependence of Rain Radar Reflectivity on DSD
 5. Selection Criteria on DSDs
 6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 7. Discussion
 8. Conclusion
 Acknowledgments
 References
 Supporting Information
[29] For scattering by particles that are very small compared to the wavelength λ, the Rayleigh approximation holds, and the backscattering cross section is proportional to the sixth power of the drop diameter D. However the conditions for the Rayleigh scattering approximation are not necessarily fulfilled in existing and future satellite configurations. Hence, exact Mie computations are necessary, and the rain radar reflectivity will take a modified form:
[30] The functional similarity between the surface elevation variance and the atmospheric drop reflectivity in equations (13) and (14) is striking. Obviously, both quantities resemble the sixth moment of the DSD, which makes them highly sensitive on the very large drop content, for which the terminal velocity v(D) is almost constant. However, it is well known that different rain events may have the same rain rate, but quite different DSDs. Hence, there will be no unique relationship between rain rate and surface energy, as is also the case between rain rate and reflectivity. Therefore let us derive a direct relationship between the calculated surface energy σ^{2} and the rain reflectivity Z, in order to analyze its dependence on the choice of the drop size distribution, and especially for their large drops content.
[31] The evolution of raindrop sizes is governed by complex processes, depending on many parameters changing from the source region in clouds to the bottom of the atmosphere. Attempts to quantify DSDs are mostly empirical and many contributions have been made by numerous authors for the various climate regions and cloud types around the globe (specified by ITUR). It is not the purpose here to review all the experience gained in that field over decades. Our approach has been to exploit the results of a systematic compilation and analysis of numerous models made by Montanari [1997] relevant for our study. As research is going on, new parameterizations of the DSD regularly appear and will continue to do so. Here, we limit ourself to assess the relationships between σ^{2}, Z and R on a sufficiently wide range of reasonably well accepted DSD models.
[32] Several functional forms can be used to fit an experimental DSD. Having in mind our further classification and selection purposes, let us organize them according to exponential, Gamma, lognormal or Weibull types. Because exponential functions are simple, their use has been attractive and has lead to a long tradition in rain radar applications:
[33] The above exponential distribution introduced by Marshall and Palmer [1948] has only two parameters N_{0} and Λ(R). Some observations, however, indicated that natural rain DSDs contain fewer of both very large and very small drops than predicted by the exponential distribution. For this purpose, a generalized Gamma distribution for representing raindrop spectra has been proposed:
The Gamma DSD has three parameters N_{0}, μ, Λ(R) and allows to describe a broader range of DSDs than the exponential law, which then becomes a special case of a Gamma distribution with μ = 0. An extensive analysis of this distribution and of its impact on the coefficients in the power law relationship between rain specific attenuation k and the rain rate R may be found in the work of Haddad et al. [2004], where new parameterizations are proposed.
[34] This Gamma DSD has been further extended to provide more degrees of freedom to fit measurements:
The parameter N_{T} accounts for the density number of drops, while b and c stand for the mean diameter and the skewness of drops, respectively. These quantities may be expressed as power laws of the rain rate R.
[35] Although not as widely used as the various forms of the Gamma distribution, the lognormal DSD given hereafter has found applications to both rain and cloud measurements. This distribution may be written as:
In equation (19), μ_{a} and μ_{b} represent a logarithmic average of diameters, while σ_{a} and σ_{b} in (20) are their standard deviations. In convective rain events, μ is always positive, varying from 0 to 0.5, whereas in stratiform cases, μ takes both positive and negative values between −1.5 and 1.5. No marked boundaries in N_{T} and σ_{n}^{2} are noticed in either type of rainfall. However, above 100 mm/h, σ_{n}^{2} is almost independent from rain rate [Timothy et al., 2002].
[36] Under heavy rainfall rates, the Gamma and the lognormal laws give somewhat similar results; however, as mentioned by Montanari [1997], the lognormal model fits the measurements better than any other distribution.
[37] The Weibull DSD is practical for rainfall composed mostly of small particles. This function is also claimed to provide good comparative results with measured attenuation and rainfall rate. It is given by:
Table 1. Raindrop Size Distributions Used in This Study^{a}  Z  Type   DSD   Comments  Reference^{b} 


1  K  MP  N_{o} = 8000  Λ = 4.1 R^{−0.21}   Dyed filter paper  Marshall and Palmer [1948] 
2  K  MP  N_{o} = 1400  Λ = 3.0 R^{−0.21}   IDS and radar, thunder  Joss et al. [1968]* 
3  K  MP  N_{o} = 30000  Λ = 5.7R^{−0.21}   IDS and radar, drizzle  Joss et al. [1968]* 
4  K  gΓ(0)  N_{o} = 4230 R^{0.37}  Λ = 3.8 R^{−0.14}   Thunderstorm, Doppler radar  Sekhon and Srivastava [1971]* 
5  N  gΓ(0)  N_{o} = 605 R^{0.37}  Λ = 3.52 R^{−0.23}   Radar attenuation  Moupfouma and Tiffon [1982]* 
6  M  gΓ(0)  N_{o} = 17300 R^{−0.16}  Λ = 5.11 R^{−0.253}   Multiple frequency radar, R < 70 mm/h  Ihara et al. [1984]* 
7  E  gΓ(0)  N_{o} = 9420 R^{−0.15}  Λ = 3.8 R^{−0.21}   IDS  Wickerts [1982]* 
8  F  gΓ(2)  N_{o} = 1800000 R^{1.2}  Λ = 10 R^{−0.3}   Radar attenuation, Mie, spherical drops  Gibbins [1992]* 
9  P  gΓ(2)  N_{o} = 4.405 10^{10}R^{1.78}  Λ = 145.16R^{0.165}   Radar attenuation, Mie, spherical drops  Li et al. [1994] 
10  N  gΓ(2)  N_{o} = 756600 R^{1.265}  Λ = 16.95R^{0.037}   Radar attenuation, Mie, spherical drops  Li et al. [1994] 
11  P  gΓ(3)  N_{o} = 31444.4 R^{0.244}  Λ = 5.753R^{0.032}   ODS and radar  Kozu et al. [2001] 
12  N  Γ(6)  N_{T} = 155R^{0.39}  b = 0.14R^{0.34}  c = 5.3R^{−0.15}  ODS, gradient method R < 150 mm/h  Montanari [1997] 
13  K  Γ(6)  N_{T} = 135R^{0.53}  b = 0.13R^{0.31}  c = 6.0R^{−0.15}  ODS, gradient method R < 150 mm/h  Montanari [1997] 
14  E  Γ(8)  N_{T} = 75.2R^{0.59}  b = 1.09R^{0.13}/c  c = 1.3R^{0.15}−1.2R^{0.16}  300 m link, Mie, momentum method  Gloaguen and Lavergnat [1995]* 
15  K  LN  N_{T} = 157R^{0.35}  μ = −0.33 + 0.166ln(R)  σ_{n}^{2} = 0.197 − 0.0181ln(R)  ODS, momentum method R < 150 mm/h  Montanari [1997] 
16  P  LN  N_{T} = 136R^{0.50}  μ = −0.309 + 0.134ln(R)  σ_{n}^{2} = 0.176 − 0.0147ln(R)  ODS, momentum method R < 150 mm/h  Montanari [1997] 
17  L  LN  N_{T} = 108R^{0.363}  μ = −0.195 + 0.199ln(R)  σ_{n}^{2} = 0.137 − 0.013ln(R)  IDS and radar  Ajayi and Olsen [1985]* 
18  N  LN  N_{T} = 0.859R^{1.535}  μ = −0.0231 + 0.116ln(R)  σ_{n}^{2} = 0.805 − 0.150ln(R)  IDS and radar, 70 < R < 120 mm/h  Maciel and Assis [1990] 
19  P  LN  N_{T} = 47R^{0.618}  μ = −0.4102 + 0.203ln(R)  σ_{n}^{2} = 0.416 − 0.0598ln(R)  IDS and radar  Tharek and Din [1990]* 
20  F  LN  N_{T} = 40R^{0.64}  μ = −0.133 + 0.127ln(R)  σ_{n}^{2} = 0.086 − 0.005ln(R)  IDS and radar, showers 5 < R < 50 mm/h, Data fitted by Ajayi and Olsen [1985]  Barclay et al. [1978]* 
21  F  LN  N_{T} = 46R^{0.55}  μ = −0.451 + 0.264ln(R)  σ_{n}^{2} = 0.409 − 0.076ln(R)  IDS and radar, thunderstorm 5 < R < 50 mm/h  Barclay et al. [1978]* 
22  F  LN  N_{T} = 8.8R  μ = 0.567  σ_{n}^{2} = 0.099  IDS and radar, thunderstorm 5 < R < 200 mm/h  Barclay et al. [1978]* 
23  P  LN  N_{T} = 276.18R^{0.381}  μ = −0.429 + 0.146ln(R)  σ_{n}^{2} = 0.156 − 0.0091ln(R)  IDS  Ong and Shan [1997]* 
24  P  LN  N_{T} = 203R^{0.241}  μ = −0.313 + 0.227ln(R)  σ_{n}^{2} = 0.108 − 0.0086ln(R)  IDS and radar, stratiform rain R < 20 mm/h  Timothy et al. [2002] 
25  P  LN  N_{T} = 78.3R^{0.558}  μ = −0.312 + 0.118ln(R)  σ_{n}^{2} = 0.118 + 0.0086ln(R)  IDS and radar, convective rain R < 250 mm/h  Timothy et al. [2002] 
26   W  N_{0} = 1000  b = 0.26R^{0.44}  c = 0.95R^{0.14}  Drizzle, shower, radar attenuation  Sekin et al. [1987]* 
[39] It should be stressed that Table 1 is obviously not exhaustive, as further interesting work on DSD is going on [Uijlenhoet, 2001; Zhang et al., 2001; Hendrantoro and Zawadzki, 2003]. Figure 2 displays 26 DSDs in two groups: the exponential and the Gamma DSDs (Figure 2, left), and the lognormal and the Weibull ones (Figure 2, right). In the next section these DSD will not all be exploited, but a selection will be made in the context of the searched goal: the design of a σ^{2} vs Z relationship.
5. Selection Criteria on DSDs
 Top of page
 Abstract
 1. Introduction
 2. Sea Surface Perturbation by Rain
 3. Dependence of RainInduced Surface Variance on DSD
 4. Dependence of Rain Radar Reflectivity on DSD
 5. Selection Criteria on DSDs
 6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 7. Discussion
 8. Conclusion
 Acknowledgments
 References
 Supporting Information
[40] Figure 2 highlights the fact that some DSDs (dotted lines) present a very particular behavior that could introduce a large bias when deriving a general relationship between σ^{2} and Z. Figure 3 shows that the behaviors of σ^{2} and Z given by equations (13) and (14) are very similar except for some particular DSDs. In order to limit the mentioned bias, we established three selection criteria on the DSDs.
[41] A first criterion on DSDs is based on rain rate consistency. In all DSDs, R appears as an input parameter, say R_{in}. However R can also be estimated by simply integrating the DSD through the well known relationship [Meneghini and Kozu, 1990]:
[42] In an ideal case R_{estim} and R_{in} should be equal. This is actually not the case as can be seen in Figure 4 that displays the estimated recalculated rain rate for the DSDs classified as in Figure 2. The discrepancies between R_{estim} and R_{in} may be used as a first selection criterion on the DSDs. Here we have chosen a threestep “acceptance” criterion based on calculations at R_{in} = 100 mm/h with relative errors on R_{in} not exceeding 10% (class I), 50% (class II), 100% (class III) or more (DSD rejected).
[43] A second criterion we chose is based on attenuation consistency, since this quantity is of great importance in rain radar retrieval algorithms. Similarly to rain rate consistency, we apply classical Mie calculation assuming spherical shapes in order to evaluate the extinction cross section σ_{t} of individual drops, then the specific attenuation k for all the DSDs under study by:
[44] Figure 5 shows the behavior of k versus R_{in} at 35 GHz for the DSD models highlighted in Figure 2. Similar calculations have been performed for various frequencies between 3 and 94 GHz. Again it appears that some of the specific attenuation curves clearly depart from the general trend, and that some of these can certainly be questioned because they yield very large differences with respect to the typical values that may be found in literature [Meneghini and Kozu, 1990].
[45] A similar Mie type calculation has been made on spherical raindrops in order to evaluate their backscattering cross section σ_{b}. It has then been integrated to get the rain reflectivity versus rain rate R_{in} too. An example of this calculated reflectivity, denoted hereafter by Z_{e}, is shown for 35 GHz in Figure 6. To check their validity, the previous calculations have been compared with other reflectivity models available in the literature as, for instance, those in the work of Meneghini and Kozu [1990]. In Figure 6 in addition to data points calculated in this paper, the upper and lower limits entitled “MK high” and “MK low” that can be found as upper and lower limits in Figure 4.9 of Meneghini and Kozu [1990] are reproduced and indicate the consistency of our calculations.
[46] Combining those criteria we have constructed three sets of acceptable DSDs depending on an increasing relative error on the recalculated rain rate, as mentioned above, and yielding reasonable attenuation and rain radar reflectivity ranges. We have discarded the other ones.
[47] Three successive steps were then considered for the evaluation of the σ^{2}(Z_{e}) relationship searched for: first, with the more strict criteria in the first one (step I), then relaxing them in steps II and III, as will be shown in the next section. Table 2 shows the DSDs from Table 1 reorganized into four classes: the three accepted classes and the rejected one.
Table 2. Selection Results of DSDs Under Study With Parameters Calculated at 35 GHzClass  DSD  Type  Error (%)  k_{min} (dB)  k_{max} (dB)  Z_{e1mm/h} · 10^{3} (mm^{6}/m^{3})  Z_{e200mm/h} · 10^{5} (mm^{6}/m^{3}) 

I  1  MP  5  0.25  40.81  3.78  10.40 
I  4  gΓ(0)  1  0.20  47.13  4.67  4.94 
I  6  gΓ(0)  1  0.17  38.92  1.39  10.79 
I  17  LN  0  0.24  43.17  3.30  12.54 
I  18  LN  4  0.03  60.30  0.77  12.92 
I  19  LN  3  0.23  47.05  1.77  9.89 
I  21  LN  5  0.19  56.46  7.35  7.17 
I  22  LN  1  0.25  49.85  0.80  1.4 · 10^{−6} 
I  24  LN  5  0.21  45.34  6.0 · 10^{−8}  3.0 · 10^{−8} 
II  2  MP  28  0.22  22.08  2.6 · 10^{−3}  3.7 · 10^{−3} 
II  3  MP  11  0.17  40.65  2.60  0.04 
II  5  gΓ(0)  28  0.04  56.23  3.13  12.57 
II  7  gΓ(0)  13  0.44  28.87  3.01  19.45 
II  15  LN  38  0.32  27.14  25.41  5.81 
II  16  LN  34  0.26  31.95  6.88  8.27 
II  23  LN  36  0.27  29.87  5.28  9.51 
II  25  LN  49  0.09  24.69  4.74  12.72 
III  12  Γ(6)  94  0.19  62.04  0.48  18.71 
III  20  LN  53  0.08  26.49  5.76  14.06 
III  26  W  70  0.24  64.99  1.16  8.36 
rejected  8  gΓ(2)  2.8 · 10^{6}  0.16  5.1 · 10^{6}  4.83  20.17 
rejected  9  gΓ(2)  100  1.9 · 10^{−4}  0.01  7.61  15.23 
rejected  10  gΓ(2)  99  1.8 · 10^{−3}  0.40  4.38  8.70 
rejected  11  gΓ(3)  99  0.21  0.19  2.89  13.30 
rejected  13  Γ(6)  118  0.19  84.87  1.34  7.01 
rejected  14  Γ(8)  284  1.83  43.14  4.52  13.20 
6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 Top of page
 Abstract
 1. Introduction
 2. Sea Surface Perturbation by Rain
 3. Dependence of RainInduced Surface Variance on DSD
 4. Dependence of Rain Radar Reflectivity on DSD
 5. Selection Criteria on DSDs
 6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 7. Discussion
 8. Conclusion
 Acknowledgments
 References
 Supporting Information
[48] The similarity of the integrals appearing in the expressions of the elevation variance (13) and of the reflectivity factor (14) has already been pointed out. Hence, those two quantities, even if not proportional to each other, will have similar behaviors. In other terms, the relationship between rain reflectivity and elevation variance is expected to present a rather weak dependence on the actual model chosen for the DSD.
[49] Most of the rain radar retrieval algorithms make use of power law relationships between each pair of the three quantities R, k and Z. This choice will not be criticized here and a similar formulation will be searched. Therefore for the selected DSDs, both σ^{2}(R) and Z_{e}(R) (not converted in dBZ) have been computed for 0 < R < 200 mm/h. This allowed us to obtain (σ^{2}, Z_{e}) pairs for each DSD, at different frequencies, rain rates and classes. Then, we performed first and secondorder polynomial identifications between σ^{2} and Z_{e} in the logarithmic domain:
where the firstorder (p_{1a} and p_{1b}) and secondorder polynomial coefficients (p_{2a}, p_{2b} and p_{2c}) are determined in a least squares sense.
[50] Figures 7, 8, and 9 show the scatterplots and the firstorder and secondorder regression curves at 3, 13.8, and 35 GHz for the three classes specified in Table 2.
[51] Table 3 gives the numerical values of the first and secondorder coefficients of the polynomials, for seven frequencies between 3 and 94 GHz for each DSD class. Table 4 shows the corresponding mean error estimates for confidence intervals of 50%, 90%, and 95%, respectively. From Table 4 we observe that the chosen models approximate the simulated data well and that the secondorder polynomials are more accurate than the firstorder ones to derive a law for the σ^{2}(Z_{e}) relationship, even though the gain in accuracy is significant for small values of σ^{2} and Z_{e} only. Thirdorder polynomials were also tested, but no significant gain was found in comparison with the already obtained regressions.
Table 3. The σ^{2}–Z_{e} Polynomial Fitting Coefficients for the Classes of Selected DSDsFrequency, GHz  Step  First Order  Second Order 

p_{1a}  p_{1b}  p_{2a}  p_{2b}  p_{2c} 

 I  0.8875  −3.8676  −0.1325  2.0951  −6.5522 
3  II  0.9691  −4.2575  −0.0237  1.1880  −4.7507 
 III  0.9862  −4.3387  −0.0085  1.0655  −4.5203 
 I  0.8725  −3.8026  −0.1291  2.0571  −6.4541 
5.3  II  0.8808  −3.8586  −0.0686  1.5342  −5.3791 
 III  0.8966  −3.9341  −0.0546  1.4245  −5.1770 
 I  0.8217  −3.6955  −0.0992  1.7540  −5.8244 
10  II  0.8746  −3.9511  −0.0307  1.1686  −4.6378 
 III  0.8837  −3.9957  −0.0169  1.0483  −4.3859 
 I  0.8423  −3.8257  −0.1014  1.7995  −6.0240 
13.8  II  0.9319  −4.2590  −0.0074  1.0018  −4.4210 
 III  0.9468  −4.3307  0.0087  0.8632  −4.1341 
 I  0.9095  −3.9936  −0.1681  2.4514  −7.4530 
24  II  1.0581  −4.6860  −0.0352  1.3764  −5.3902 
 III  1.1188  −4.9693  0.0271  0.8711  −4.4157 
 I  0.9893  −4.0161  −0.2505  3.1322  −8.5158 
35  II  1.1093  −4.5313  −0.1917  2.7200  −7.8524 
 III  1.2287  −5.0421  −0.0873  1.9641  −6.5614 
 I  0.8902  −1.8699  −0.4103  3.0267  −4.5890 
94  II  1.0189  −2.2667  −0.6522  4.4120  −6.5918 
 III  1.2017  −2.7149  −0.7560  5.1211  −7.6935 
Table 4. Error Estimate Mean Values for the Fits at Different Frequencies and Confidence IntervalsFrequency, GHz  Step  50%  90%  95% 

First  Second  First  Second  First  Second 

3  I  0.0754  0.0607  0.1840  0.1480  0.2193  0.1764 
 II  0.0823  0.0818  0.2007  0.1995  0.2392  0.2377 
 III  0.0814  0.0814  0.1986  0.1985  0.2366  0.2365 
5.3  I  0.0583  0.0337  0.1422  0.0823  0.1695  0.0981 
 II  0.0612  0.0492  0.1492  0.1201  0.1778  0.1431 
 III  0.0666  0.0588  0.1625  0.1434  0.1936  0.1709 
10  I  0.0564  0.0392  0.1375  0.0956  0.1639  0.1140 
 II  0.0576  0.0554  0.1406  0.1352  0.1676  0.1611 
 III  0.0594  0.0586  0.1449  0.1430  0.1727  0.1704 
13.8  I  0.0721  0.0606  0.1759  0.1479  0.2097  0.1762 
 II  0.0842  0.0842  0.2054  0.2055  0.2448  0.2448 
 III  0.0835  0.0835  0.2037  0.2036  0.2428  0.2426 
24  I  0.1206  0.1101  0.2941  0.2685  0.3506  0.3201 
 II  0.1701  0.1702  0.4150  0.4151  0.4945  0.4947 
 III  0.1701  0.1702  0.4149  0.4151  0.4944  0.4946 
35  I  0.1577  0.1504  0.3849  0.3670  0.4587  0.4374 
 II  0.2419  0.2399  0.5901  0.5852  0.7032  0.6974 
 III  0.2525  0.2537  0.6159  0.6188  0.7340  0.7374 
94  I  0.2611  0.2586  0.6370  0.6309  0.7592  0.7519 
 II  0.3372  0.3283  0.8225  0.8009  0.9802  0.9544 
 III  0.3589  0.3481  0.8755  0.8491  1.0433  1.0119 
[52] These fits have also been analyzed versus frequency. Figure 10 shows a comparison of all secondorder polynomials, whose coefficients are given in Table 3. In Figure 10 we observe that the fits are quite close to each other for all frequencies below 14 GHz, and that for the fit at 35 GHz the deviation increases, while for 94 GHz the curves depart strongly from the other ones. This is not surprising because at such a high frequency the scattering of hydrometeors strongly departs from the D^{6} dependence used as an approximate basis for this study. In other words, when in terms of wavelength the largest drops leave the Rayleigh region to the resonance region, σ^{2} and Z_{e} no longer represent similar moments of the DSD and the latter will bring little information on the former. This is certainly what happens for frequencies beyond 40 GHz. It should also be underlined that for class I the secondorder term appears nonnegligible, while extending the DSD class size from I to III, the dependence becomes more linear in the loglog domain. This means that the inclusion of DSDs from less strict classes, in the sense of our selection criteria, simultaneously decreases the significance of the secondorder terms, along with an increase of the error interval.
[53] For the sake of completeness, a comparison with nonMie scattering has been conducted by repeating the complete procedure with the Rayleigh approximation. A global average fit combining the different frequencies in bands up to 35 GHz has also been made. The results of the corresponding secondorder coefficients for class III are given in Table 5. The firstorder coefficients close to unity and secondorder coefficients relatively small illustrate the quasiproportionality between σ^{2} and Z_{e} that can be also observed on the corresponding Figure 11 in which 95% confidence intervals have been indicated also.
Table 5. SecondOrder Polynomial Coefficients of σ^{2}–Z_{e} Over Different Frequency Bands, and Rayleigh Case^{a}Frequency Band  Average Coefficients 

p_{2a}  p_{2b}  p_{2c} 


Rayleigh  −0.0184  1.1334  −4.6493 
3—13.8 GHz  −0.0178  1.1004  −4.5543 
3—24.0 GHz  −0.0128  1.0749  −4.5389 
3—35.0 GHz  −0.0165  1.1274  −4.6696 
7. Discussion
 Top of page
 Abstract
 1. Introduction
 2. Sea Surface Perturbation by Rain
 3. Dependence of RainInduced Surface Variance on DSD
 4. Dependence of Rain Radar Reflectivity on DSD
 5. Selection Criteria on DSDs
 6. RainInduced Surface Variance Versus Rain Radar Reflectivity
 7. Discussion
 8. Conclusion
 Acknowledgments
 References
 Supporting Information
[54] The dependence of σ^{2} on D^{6} highlighted here supports the fact that the surface roughness due to impacting rain increases very fast with drop size. As a result, mostly the large drops affect this surface roughness. This effect is very difficult to estimate when only the rain rate is known, because the large drop content strongly depends on the models chosen for the DSD. It has been shown in this paper that this uncertainty can be reduced when drop reflectivity data Z are available, as it is the case for nadir looking spaceborne rain radar data. Indeed, drop reflectivity and surface elevation variance correspond to similar moments of the drop size distribution. Therefore the raininduced surface variance σ^{2} can been calculated from the Z estimated just above the surface following the proposed scheme.
[55] Further, the rain spectral component added to the wind spectral one can be estimated more accurately than using the rain rate only. The surface backscattering coefficient change Δσ° can then be calculated by using available electromagnetic models and codes, for the remote sensing geometries of interest, and introduced in rain radar algorithms for further improvement of rain rate retrievals. In Figure 12, an example for such a calculation of Δσ° at 13.8 GHz and 35 GHz is given for nadir incidence and for rain rates ranging from 0 to 100 mm/h. Δσ° is related to Z_{e} just above the surface through the σ^{2}(Z_{e}) relationship proposed in this study. The conditions for this example are: DSD class I, fully developed sea state conditions with a wind friction velocity = 20 cm/s (i.e., U_{10} ≃ 6 m/s wind at 10 m height) using the wind sea surface spectrum and electromagnetic scattering models from Elfouhaily et al. [1997] or Lemaire et al. [1999], respectively. The additive surface spectrum due to rain is taken from Craeye [1998]. The arrows on Figure 12 indicate how a measurement of Z_{e} just above the surface can provide an estimate of σ^{2} and, in turn, an evaluation of Δσ° with respect to σ° in the nonrainy case.
[56] In the example shown here, Δσ° is limited up to 1 to 2 dB for the range of the chosen conditions. It would be interesting to validate such a procedure against measured data. As direct Δσ° measurements at nadir noncontaminated by rain attenuation are very sparse, what is one motivation of this paper, an indirect approach to compare the range of Δσ° with measurements could be followed. Welldocumented field radar data at Ku band for various incidence angles versus wind and rain are available in the work of Contreras et al. [2003]. Nadir Δσ° are not available. Analyzing the reported results and the evolution of σ° for decreasing incidence angles from 76 down to 14, we observe that the simulated σ° using this procedure for similar environmental conditions are consistent with the data, even though the sign of Δσ° (increase or decrease versus rain rate) depends on the sea state conditions. Therefore more well documented data including sea state conditions for combined wind and rain are needed to further validate the radar backscattering by the surface at nadir or near nadir.