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Keywords:

  • rain radar;
  • rain ringwaves;
  • surface scattering

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. 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 marching-on procedures in which Zi at each gate i is used to estimate the corresponding rain rate Ri, and in turn, the attenuation ki necessary to correct the apparent Zi+1 in the next gate i + 1 for the two-way path attenuation. It is known that such procedures, like the classical H-B 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 two-way 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 rain-induced surface variance on the drop size distribution (section 3). A selection of DSDs is presented (section 4) and the similarities between the rain-induced 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. Supporting Information

[7] The impact of rain on a water surface significantly modifies its roughness and, in turn, its reflectivity. It is well known that rain simultaneously damps the sea surface waves and produces additional roughness. These effects introduce biases on wind speed estimates from scatterometers [Guymer et al., 1981; Black et al., 1985; Stiles and Yueh, 2002] and modify SAR images [Atlas, 1994; Melsheimer et al., 2001; Alpers and Lin, 2006].

[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 gravity-capillary waves [Worthington, 1963; Le Méhauté et al., 1987; Le Méhauté, 1988]. Hallett and Christensen [1984] and Rein [1993] analyzed these features through high-speed photography, while Hansen [1986] combined such observations with high-resolution 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 two-layer surface wave model of a turbulent layer overlaying a second nonturbulent one.

[10] Wavetank experiments with high-speed 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 rain-roughened 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.

image

Figure 1. (a) Wind only and rain only radial sea surface elevation spectra S(K) for increasing wind friction velocities u* and rain rates R. u* ranges from 20 to 120 cm/s by steps of 10 cm/s and R ranges from 20 to 100 mm/h by steps of 20 mm/h. (b) Rain only component.

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[15] The elevation variance can be written as:

  • equation image

[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 σ2 of the ringwaves and the drop reflectivity Z.

3. Dependence of Rain-Induced Surface Variance on DSD

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. 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 m2v2. 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 m2) of a ringwave generated by a drop after an impact at time t = 0:

  • equation image

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 = mv2/2 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 E0/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 E0 proportional to m2v2, and in turn, a E0/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:

  • equation image

where D is the drop size (mm) and the proportionality constants C and C1 are with dimensions kg−2s2 and m−6s2, respectively.

3.2. Step 2: Full Monodisperse Rain

[20] We now consider rain with constant drop size D. If Nmono is the density of falling drops in air (in m−3), and v(D) is their velocity, the number no of drops that hit a unitary surface per unit of time (in m−2s−1) can be obtained as:

  • equation image

[21] Alternatively, if the rain rate R (in mm h−1) is known, we also have:

  • equation image

where χ is a constant equal to 530, expressed in mm2 m−2 h s−1. The normalized surface energy generated by an individual drop may be written as:

  • equation image

where, in first approximation, fo(t) = et/τ for t > 0 and fo = 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:

  • equation image

[22] Using the expressions above for the number no of drops impacting a unit surface per unit of time, we obtain two possible expressions for the elevation variance:

  • equation image

and

  • equation image

[23] This shows that once σ2 is known along with the characteristics of falling drops (R, D, v) the product C1τ 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:

  • equation image

where fo 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 m3 with a size in the interval [DdD/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 E0 = C1D6v2(D), and that this energy lasts for a given time τ. The number of drops, with a size in the interval [DdD/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:

  • equation image

which shows a D6v3(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 C1τ and to approximate this quantity by the following exponential function:

  • equation image

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 σ2 may be written as:

  • equation image

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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. 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:

  • equation image

[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 σ2 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 ITU-R). 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 σ2, 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:

  • equation image

[33] The above exponential distribution introduced by Marshall and Palmer [1948] has only two parameters N0 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:

  • equation image

The Gamma DSD has three parameters N0, 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:

  • equation image

The parameter NT 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:

  • equation image
  • equation image
  • equation image

In equation (19), μa and μb represent a logarithmic average of diameters, while σa and σb 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 NT and σn2 are noticed in either type of rainfall. However, above 100 mm/h, σn2 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:

  • equation image

[38] Table 1 summarizes a large set of models gathered for the various world climatic regions. Many of them are quoted from Montanari's [1997] compilation. A few others correspond to Li et al. [1994], Timothy et al. [2002], and Maciel and Assis [1990] not included in the Montanari compilation.

Table 1. Raindrop Size Distributions Used in This Studya
 ZType DSD CommentsReferenceb
  • a

    Notes: Z, ITU-R rain climatic zones; MP, Marshall Palmer; gΓ(q), generalized Gamma function of order q; Γ(n), Gamma function with n parameters; LN, lognormal function; W, Weibull distribution; IDS, impact distrometer; ODS, optical distrometer.

  • b

    References marked with an asterisk are recited from Montanari's [1997] compilation.

1KMPNo = 8000Λ = 4.1 R−0.21 Dyed filter paperMarshall and Palmer [1948]
2KMPNo = 1400Λ = 3.0 R−0.21 IDS and radar, thunderJoss et al. [1968]*
3KMPNo = 30000Λ = 5.7R−0.21 IDS and radar, drizzleJoss et al. [1968]*
4KgΓ(0)No = 4230 R0.37Λ = 3.8 R−0.14 Thunderstorm, Doppler radarSekhon and Srivastava [1971]*
5NgΓ(0)No = 605 R0.37Λ = 3.52 R−0.23 Radar attenuationMoupfouma and Tiffon [1982]*
6MgΓ(0)No = 17300 R−0.16Λ = 5.11 R−0.253 Multiple frequency radar, R < 70 mm/hIhara et al. [1984]*
7EgΓ(0)No = 9420 R−0.15Λ = 3.8 R−0.21 IDSWickerts [1982]*
8FgΓ(2)No = 1800000 R1.2Λ = 10 R−0.3 Radar attenuation, Mie, spherical dropsGibbins [1992]*
9PgΓ(2)No = 4.405 1010R1.78Λ = 145.16R0.165 Radar attenuation, Mie, spherical dropsLi et al. [1994]
10NgΓ(2)No = 756600 R1.265Λ = 16.95R0.037 Radar attenuation, Mie, spherical dropsLi et al. [1994]
11PgΓ(3)No = 31444.4 R0.244Λ = 5.753R0.032 ODS and radarKozu et al. [2001]
12NΓ(6)NT = 155R0.39b = 0.14R0.34c = 5.3R−0.15ODS, gradient method R < 150 mm/hMontanari [1997]
13KΓ(6)NT = 135R0.53b = 0.13R0.31c = 6.0R−0.15ODS, gradient method R < 150 mm/hMontanari [1997]
14EΓ(8)NT = 75.2R0.59b = 1.09R0.13/cc = 1.3R0.15−1.2R0.16300 m link, Mie, momentum methodGloaguen and Lavergnat [1995]*
15KLNNT = 157R0.35μ = −0.33 + 0.166ln(R)σn2 = 0.197 − 0.0181ln(R)ODS, momentum method R < 150 mm/hMontanari [1997]
16PLNNT = 136R0.50μ = −0.309 + 0.134ln(R)σn2 = 0.176 − 0.0147ln(R)ODS, momentum method R < 150 mm/hMontanari [1997]
17LLNNT = 108R0.363μ = −0.195 + 0.199ln(R)σn2 = 0.137 − 0.013ln(R)IDS and radarAjayi and Olsen [1985]*
18NLNNT = 0.859R1.535μ = −0.0231 + 0.116ln(R)σn2 = 0.805 − 0.150ln(R)IDS and radar, 70 < R < 120 mm/hMaciel and Assis [1990]
19PLNNT = 47R0.618μ = −0.4102 + 0.203ln(R)σn2 = 0.416 − 0.0598ln(R)IDS and radarTharek and Din [1990]*
20FLNNT = 40R0.64μ = −0.133 + 0.127ln(R)σn2 = 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]*
21FLNNT = 46R0.55μ = −0.451 + 0.264ln(R)σn2 = 0.409 − 0.076ln(R)IDS and radar, thunderstorm 5 < R < 50 mm/hBarclay et al. [1978]*
22FLNNT = 8.8Rμ = 0.567σn2 = 0.099IDS and radar, thunderstorm 5 < R < 200 mm/hBarclay et al. [1978]*
23PLNNT = 276.18R0.381μ = −0.429 + 0.146ln(R)σn2 = 0.156 − 0.0091ln(R)IDSOng and Shan [1997]*
24PLNNT = 203R0.241μ = −0.313 + 0.227ln(R)σn2 = 0.108 − 0.0086ln(R)IDS and radar, stratiform rain R < 20 mm/hTimothy et al. [2002]
25PLNNT = 78.3R0.558μ = −0.312 + 0.118ln(R)σn2 = 0.118 + 0.0086ln(R)IDS and radar, convective rain R < 250 mm/hTimothy et al. [2002]
26 WN0 = 1000b = 0.26R0.44c = 0.95R0.14Drizzle, shower, radar attenuationSekin 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 σ2 vs Z relationship.

image

Figure 2. Drop size distributions under study for R = 50 mm/h. Lines and colors refer to the first column in Table 1 as follows: (left) models 1–14, with continuous lines from 1 to 7, and dotted lines from 8 to 14; (right) models 15–26, with continuous lines from 15 to 21, and dotted lines from 22 to 26.

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5. Selection Criteria on DSDs

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. 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 σ2 and Z. Figure 3 shows that the behaviors of σ2 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.

image

Figure 3. (a) Reflectivity at 13.8 GHz and (b) surface elevation variance obtained for the first 18 DSDs from Table 1 versus rain rate.

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[41] A first criterion on DSDs is based on rain rate consistency. In all DSDs, R appears as an input parameter, say Rin. However R can also be estimated by simply integrating the DSD through the well known relationship [Meneghini and Kozu, 1990]:

  • equation image

[42] In an ideal case Restim and Rin 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 Restim and Rin may be used as a first selection criterion on the DSDs. Here we have chosen a three-step “acceptance” criterion based on calculations at Rin = 100 mm/h with relative errors on Rin not exceeding 10% (class I), 50% (class II), 100% (class III) or more (DSD rejected).

image

Figure 4. Estimated rain rate Restim versus Rin for the DSD types from Table 1.

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[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 σt of individual drops, then the specific attenuation k for all the DSDs under study by:

  • equation image

[44] Figure 5 shows the behavior of k versus Rin 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].

image

Figure 5. Specific attenuation k at 35 GHz versus rain rate Rin for the DSD types from Table 1.

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[45] A similar Mie type calculation has been made on spherical raindrops in order to evaluate their backscattering cross section σb. It has then been integrated to get the rain reflectivity versus rain rate Rin too. An example of this calculated reflectivity, denoted hereafter by Ze, 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 “M-K high” and “M-K 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.

image

Figure 6. Rain radar reflectivities at 35 GHz versus rain rate for the DSD types from Table 1. M-K high and M-K low lines indicate upper and lower limits from Meneghini and Kozu [1990, Figure 4.9] for several ZR relationships.

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[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 σ2(Ze) 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 GHz
ClassDSDTypeError (%)kmin (dB)kmax (dB)Ze1mm/h · 103 (mm6/m3)Ze200mm/h · 105 (mm6/m3)
I1MP50.2540.813.7810.40
I4gΓ(0)10.2047.134.674.94
I6gΓ(0)10.1738.921.3910.79
I17LN00.2443.173.3012.54
I18LN40.0360.300.7712.92
I19LN30.2347.051.779.89
I21LN50.1956.467.357.17
I22LN10.2549.850.801.4 · 10−6
I24LN50.2145.346.0 · 10−83.0 · 10−8
II2MP280.2222.082.6 · 10−33.7 · 10−3
II3MP110.1740.652.600.04
II5gΓ(0)280.0456.233.1312.57
II7gΓ(0)130.4428.873.0119.45
II15LN380.3227.1425.415.81
II16LN340.2631.956.888.27
II23LN360.2729.875.289.51
II25LN490.0924.694.7412.72
III12Γ(6)940.1962.040.4818.71
III20LN530.0826.495.7614.06
III26W700.2464.991.168.36
rejected8gΓ(2)2.8 · 1060.165.1 · 1064.8320.17
rejected9gΓ(2)1001.9 · 10−40.017.6115.23
rejected10gΓ(2)991.8 · 10−30.404.388.70
rejected11gΓ(3)990.210.192.8913.30
rejected13Γ(6)1180.1984.871.347.01
rejected14Γ(8)2841.8343.144.5213.20

6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. 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 σ2(R) and Ze(R) (not converted in dBZ) have been computed for 0 < R < 200 mm/h. This allowed us to obtain (σ2, Ze) pairs for each DSD, at different frequencies, rain rates and classes. Then, we performed first- and second-order polynomial identifications between σ2 and Ze in the logarithmic domain:

  • equation image
  • equation image

where the first-order (p1a and p1b) and second-order polynomial coefficients (p2a, p2b and p2c) are determined in a least squares sense.

[50] Figures 7, 8, and 9 show the scatterplots and the first-order and second-order regression curves at 3, 13.8, and 35 GHz for the three classes specified in Table 2.

image

Figure 7. The σ2Ze first-order (blue lines) and second-order (red lines) logarithmic polynomial fitting at 3 GHz for the three classes of selected DSDs (dotted lines) from Table 2, class I, II or III, respectively.

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image

Figure 8. The σ2Ze first-order (blue lines) and second-order (red lines) logarithmic polynomial fitting at 13.8 GHz for the three classes of selected DSDs (dotted lines) from Table 2, class I, II or III, respectively.

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image

Figure 9. The σ2Ze first-order (blue lines) and second-order (red lines) logarithmic polynomial fitting at 35 GHz for the three classes of selected DSDs (dotted lines) from Table 2, class I, II or III, respectively.

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[51] Table 3 gives the numerical values of the first- and second-order 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 second-order polynomials are more accurate than the first-order ones to derive a law for the σ2(Ze) relationship, even though the gain in accuracy is significant for small values of σ2 and Ze only. Third-order polynomials were also tested, but no significant gain was found in comparison with the already obtained regressions.

Table 3. The σ2Ze Polynomial Fitting Coefficients for the Classes of Selected DSDs
Frequency, GHzStepFirst OrderSecond Order
p1ap1bp2ap2bp2c
 I0.8875−3.8676−0.13252.0951−6.5522
3II0.9691−4.2575−0.02371.1880−4.7507
 III0.9862−4.3387−0.00851.0655−4.5203
 I0.8725−3.8026−0.12912.0571−6.4541
5.3II0.8808−3.8586−0.06861.5342−5.3791
 III0.8966−3.9341−0.05461.4245−5.1770
 I0.8217−3.6955−0.09921.7540−5.8244
10II0.8746−3.9511−0.03071.1686−4.6378
 III0.8837−3.9957−0.01691.0483−4.3859
 I0.8423−3.8257−0.10141.7995−6.0240
13.8II0.9319−4.2590−0.00741.0018−4.4210
 III0.9468−4.33070.00870.8632−4.1341
 I0.9095−3.9936−0.16812.4514−7.4530
24II1.0581−4.6860−0.03521.3764−5.3902
 III1.1188−4.96930.02710.8711−4.4157
 I0.9893−4.0161−0.25053.1322−8.5158
35II1.1093−4.5313−0.19172.7200−7.8524
 III1.2287−5.0421−0.08731.9641−6.5614
 I0.8902−1.8699−0.41033.0267−4.5890
94II1.0189−2.2667−0.65224.4120−6.5918
 III1.2017−2.7149−0.75605.1211−7.6935
Table 4. Error Estimate Mean Values for the Fits at Different Frequencies and Confidence Intervals
Frequency, GHzStep50%90%95%
FirstSecondFirstSecondFirstSecond
3I0.07540.06070.18400.14800.21930.1764
 II0.08230.08180.20070.19950.23920.2377
 III0.08140.08140.19860.19850.23660.2365
5.3I0.05830.03370.14220.08230.16950.0981
 II0.06120.04920.14920.12010.17780.1431
 III0.06660.05880.16250.14340.19360.1709
10I0.05640.03920.13750.09560.16390.1140
 II0.05760.05540.14060.13520.16760.1611
 III0.05940.05860.14490.14300.17270.1704
13.8I0.07210.06060.17590.14790.20970.1762
 II0.08420.08420.20540.20550.24480.2448
 III0.08350.08350.20370.20360.24280.2426
24I0.12060.11010.29410.26850.35060.3201
 II0.17010.17020.41500.41510.49450.4947
 III0.17010.17020.41490.41510.49440.4946
35I0.15770.15040.38490.36700.45870.4374
 II0.24190.23990.59010.58520.70320.6974
 III0.25250.25370.61590.61880.73400.7374
94I0.26110.25860.63700.63090.75920.7519
 II0.33720.32830.82250.80090.98020.9544
 III0.35890.34810.87550.84911.04331.0119

[52] These fits have also been analyzed versus frequency. Figure 10 shows a comparison of all second-order 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 D6 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, σ2 and Ze 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 second-order term appears nonnegligible, while extending the DSD class size from I to III, the dependence becomes more linear in the log-log 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 second-order terms, along with an increase of the error interval.

image

Figure 10. The σ2Ze second-order logarithmic polynomial fitting for the three classes of selected DSDs and the different frequencies. Coefficients are from Table 3.

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[53] For the sake of completeness, a comparison with non-Mie 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 second-order coefficients for class III are given in Table 5. The first-order coefficients close to unity and second-order coefficients relatively small illustrate the quasi-proportionality between σ2 and Ze that can be also observed on the corresponding Figure 11 in which 95% confidence intervals have been indicated also.

image

Figure 11. Second-order polynomial fitting for σ2Ze along with 95% confidence intervals over different frequency bands. Class III DSD.

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Table 5. Second-Order Polynomial Coefficients of σ2Ze Over Different Frequency Bands, and Rayleigh Casea
Frequency BandAverage Coefficients
p2ap2bp2c
  • a

    Class III DSD.

Rayleigh−0.01841.1334−4.6493
3—13.8 GHz−0.01781.1004−4.5543
3—24.0 GHz−0.01281.0749−4.5389
3—35.0 GHz−0.01651.1274−4.6696

7. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. Supporting Information

[54] The dependence of σ2 on D6 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 rain-induced surface variance σ2 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 Ze just above the surface through the σ2(Ze) relationship proposed in this study. The conditions for this example are: DSD class I, fully developed sea state conditions with a wind friction velocity equation image = 20 cm/s (i.e., U10 ≃ 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 Ze just above the surface can provide an estimate of σ2 and, in turn, an evaluation of Δσ° with respect to σ° in the nonrainy case.

image

Figure 12. Sea surface backscattering coefficient change Δσ° at 13.8 GHz and 35 GHz, for nadir incidence and rain rates ranging from 0 to 100 mm/h, related to Ze just above the surface through the σ2(Ze) relationship proposed in this study and numerical models for σ°. Class I DSD, wind friction velocity equation image = 20 cm/s (wind U10equation image 6 m/s), wind sea surface spectrum and electromagnetic scattering models are from Elfouhaily et al. [1997] and Lemaire et al. [1999]; additive surface spectrum due to rain is from Craeye [1998].

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[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. Well-documented 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.

8. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. Supporting Information

[57] A relationship between Z obtained from rain radars and ringwaves height variance σ2 has been established. A possible procedure to derive Δσ° at nadir from Z data has been further suggested once a electromagnetic scattering model has been chosen. Further refinements are possible and could include nonspherical drop shapes, a large database of DSD, as well as iterative retrieval algorithms for both rain rate and surface variance.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Sea Surface Perturbation by Rain
  5. 3. Dependence of Rain-Induced Surface Variance on DSD
  6. 4. Dependence of Rain Radar Reflectivity on DSD
  7. 5. Selection Criteria on DSDs
  8. 6. Rain-Induced Surface Variance Versus Rain Radar Reflectivity
  9. 7. Discussion
  10. 8. Conclusion
  11. Acknowledgments
  12. References
  13. Supporting Information
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rds5585-sup-0001-t01.txtplain text document3KTab-delimited Table 1.
rds5585-sup-0002-t02.txtplain text document1KTab-delimited Table 2.
rds5585-sup-0003-t03.txtplain text document1KTab-delimited Table 3.
rds5585-sup-0004-t04.txtplain text document1KTab-delimited Table 4.
rds5585-sup-0005-t05.txtplain text document0KTab-delimited Table 5.

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