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- Author or Editor: R. Meneghini x

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## Abstract

One of the impediments to the interpretation of radar signatures from the melting layer is the uncertainty over the dielectric mixing formula for ice-water mixtures. In the commonly used Maxwell Garnett mixing formula, the dielectric constant for ice inclusions in a water matrix differs from that for water inclusions in an ice matrix for the same fraction of meltwater. While the choice of materials for the matrix and inclusion is clear for either small or large fractions of meltwater, it is not obvious how these are to be chosen in the intermediate ranges of melting. In this paper, cross sections derived from the various mixing formulas are compared to a conjugate gradient-fast Fourier transform numerical method. In the numerical method the particle is divided into equi-volume subcells in which the composition of the particle is controlled by assigning a probability of water to each subcell. For a uniform distribution of water and ice, where the probability of water in a subcell is independent of its location within the particle, the numerical results for fractional water contents of less than about 0.7 indicate that the scattering coefficients are closest to those predicted by the Maxwell Garnett mixing formula if an ice matrix with water inclusions is assumed. However, if the meltwater is highly concentrated near the boundary of the particle or if the fractional volume of water is greater than about 0.8, the Maxwell Garnett formula is in fair agreement with the numerical results, if the roles of ice and water are interchanged. A discussion of the relevance of these results to the modeling of melting snow aggregates and the interpretation of radar signatures of the bright band is given in the final section of the paper.

## Abstract

One of the impediments to the interpretation of radar signatures from the melting layer is the uncertainty over the dielectric mixing formula for ice-water mixtures. In the commonly used Maxwell Garnett mixing formula, the dielectric constant for ice inclusions in a water matrix differs from that for water inclusions in an ice matrix for the same fraction of meltwater. While the choice of materials for the matrix and inclusion is clear for either small or large fractions of meltwater, it is not obvious how these are to be chosen in the intermediate ranges of melting. In this paper, cross sections derived from the various mixing formulas are compared to a conjugate gradient-fast Fourier transform numerical method. In the numerical method the particle is divided into equi-volume subcells in which the composition of the particle is controlled by assigning a probability of water to each subcell. For a uniform distribution of water and ice, where the probability of water in a subcell is independent of its location within the particle, the numerical results for fractional water contents of less than about 0.7 indicate that the scattering coefficients are closest to those predicted by the Maxwell Garnett mixing formula if an ice matrix with water inclusions is assumed. However, if the meltwater is highly concentrated near the boundary of the particle or if the fractional volume of water is greater than about 0.8, the Maxwell Garnett formula is in fair agreement with the numerical results, if the roles of ice and water are interchanged. A discussion of the relevance of these results to the modeling of melting snow aggregates and the interpretation of radar signatures of the bright band is given in the final section of the paper.

## Abstract

Estimates of rain rate derived from a spaceborne weather radar will be most reliable over an intermediate range of values. At light or heavy rain rates, where the signal-to-noise ratios are degraded either by small values of the backscattered power or by large attenuation, the accuracy will be poor. In forming an area average of the rain rate, an alternative to the averaging of the high-resolution estimates, irrespective of their individual accuracies, is a multiple threshold approach. The method is based on the fact that the Fractional area above a particular rain-rate threshold *R _{j}
* is related to the cumulative distribution of rain rates evaluated at

*R*. Varying the threshold over the effective dynamic range of the radar yields the cumulative distribution function over this range. To obtain the distribution at all rain rates, a lognormal or gamma test function is selected such that the mean-square error between the test function and the measured values is minimized. Once the unknown parameters are determined, the first-order statistics of the areawide rain-rate distribution can be found. Tests of the method with data from the SPANDAR radar provide comparisons between it and the single threshold and the direct averaging approaches.

_{j}## Abstract

Estimates of rain rate derived from a spaceborne weather radar will be most reliable over an intermediate range of values. At light or heavy rain rates, where the signal-to-noise ratios are degraded either by small values of the backscattered power or by large attenuation, the accuracy will be poor. In forming an area average of the rain rate, an alternative to the averaging of the high-resolution estimates, irrespective of their individual accuracies, is a multiple threshold approach. The method is based on the fact that the Fractional area above a particular rain-rate threshold *R _{j}
* is related to the cumulative distribution of rain rates evaluated at

*R*. Varying the threshold over the effective dynamic range of the radar yields the cumulative distribution function over this range. To obtain the distribution at all rain rates, a lognormal or gamma test function is selected such that the mean-square error between the test function and the measured values is minimized. Once the unknown parameters are determined, the first-order statistics of the areawide rain-rate distribution can be found. Tests of the method with data from the SPANDAR radar provide comparisons between it and the single threshold and the direct averaging approaches.

_{j}## Abstract

The radar return powers from a three-frequency radar, with center frequency at 22.235 GHz and upper and lower frequencies chosen with equal water vapor absorption coefficients, can be used to estimate water vapor density and parameters of the precipitation. A linear combination of differential measurements between the center and lower frequencies on one hand and the upper and lower frequencies on the other provide an estimate of differential water vapor absorption. The coupling between the precipitation and water vapor estimates is generally weak but increases with bandwidth and the amount of non-Rayleigh scattering of the hydrometeors. The coupling leads to biases in the estimates of water vapor absorption that depend primarily on the phase state and the median mass diameter of the hydrometeors. For a down-looking radar, path-averaged estimates of water vapor absorption are possible under rain-free as well as raining conditions by using the surface returns at the three frequencies. Simulations of the water vapor attenuation retrieval show that the largest source of error typically arises from the variance in the measured radar return powers. Although the error can be mitigated by a combination of a high pulse repetition frequency, pulse compression, and averaging in range and time, the radar receiver must be stable over the averaging period. For fractional bandwidths of 20% or less, the potential exists for simultaneous measurements at the three frequencies with a single antenna and transceiver, thereby significantly reducing the cost and mass of the system.

## Abstract

The radar return powers from a three-frequency radar, with center frequency at 22.235 GHz and upper and lower frequencies chosen with equal water vapor absorption coefficients, can be used to estimate water vapor density and parameters of the precipitation. A linear combination of differential measurements between the center and lower frequencies on one hand and the upper and lower frequencies on the other provide an estimate of differential water vapor absorption. The coupling between the precipitation and water vapor estimates is generally weak but increases with bandwidth and the amount of non-Rayleigh scattering of the hydrometeors. The coupling leads to biases in the estimates of water vapor absorption that depend primarily on the phase state and the median mass diameter of the hydrometeors. For a down-looking radar, path-averaged estimates of water vapor absorption are possible under rain-free as well as raining conditions by using the surface returns at the three frequencies. Simulations of the water vapor attenuation retrieval show that the largest source of error typically arises from the variance in the measured radar return powers. Although the error can be mitigated by a combination of a high pulse repetition frequency, pulse compression, and averaging in range and time, the radar receiver must be stable over the averaging period. For fractional bandwidths of 20% or less, the potential exists for simultaneous measurements at the three frequencies with a single antenna and transceiver, thereby significantly reducing the cost and mass of the system.

## Abstract

This study compares precipitation rate profiles derived from a single frequency radar and radiometer with such profiles derived from a dual-frequency radar.

Measurements obtained during the 1985–86 CRL/NASA rain measuring experiment from airborne X- and Ka-band radars and an X-band passive microwave radiometer were used to derive rainfall rate profiles over the Atlantic Ocean. The rainfall retrieval employs the classical Hitschfeld-Bordan radar equation constrained by a measurement of the path integrated extinction derived from passive radiometry.

The path-integrated extinction obtained from the radiometric measurements was compared with that obtained from coincident dual-frequency radar reflection measurements from the ocean surface. The mean rainfall rate derived from the path-integrated extinction retrieved from the measured microwave radiances agreed within 25% with the mean rainfall rate obtained from the reflected radar signals.

An analysis of the errors in the retrieval algorithm showed that errors in the path-integrated extinction significantly affect the retrieved rainfall profiles near the surface. A least squares linear extrapolation of the profile in the lowest kilometer was used to revise the boundary condition in the retrieval. The profiles were solved iteratively until the rainfall rate at the surface was within the range of scatter about the linear profile at higher altitudes.

An optimization analysis was applied to the derivation of rainfall rate profiles retrieved from a dual-frequency radar data. The results of the retrieval were compared to those obtained from the radar-radiometer retrievers.

The availability of only an X-band radiometer limited the retrieval of rainfall rate profiles to maritime cases. It appears that it will be possible to measure rainfall under most conditions when radiometers operating at several higher frequencies become available on future airborne radar experiments.

## Abstract

This study compares precipitation rate profiles derived from a single frequency radar and radiometer with such profiles derived from a dual-frequency radar.

Measurements obtained during the 1985–86 CRL/NASA rain measuring experiment from airborne X- and Ka-band radars and an X-band passive microwave radiometer were used to derive rainfall rate profiles over the Atlantic Ocean. The rainfall retrieval employs the classical Hitschfeld-Bordan radar equation constrained by a measurement of the path integrated extinction derived from passive radiometry.

The path-integrated extinction obtained from the radiometric measurements was compared with that obtained from coincident dual-frequency radar reflection measurements from the ocean surface. The mean rainfall rate derived from the path-integrated extinction retrieved from the measured microwave radiances agreed within 25% with the mean rainfall rate obtained from the reflected radar signals.

An analysis of the errors in the retrieval algorithm showed that errors in the path-integrated extinction significantly affect the retrieved rainfall profiles near the surface. A least squares linear extrapolation of the profile in the lowest kilometer was used to revise the boundary condition in the retrieval. The profiles were solved iteratively until the rainfall rate at the surface was within the range of scatter about the linear profile at higher altitudes.

An optimization analysis was applied to the derivation of rainfall rate profiles retrieved from a dual-frequency radar data. The results of the retrieval were compared to those obtained from the radar-radiometer retrievers.

The availability of only an X-band radiometer limited the retrieval of rainfall rate profiles to maritime cases. It appears that it will be possible to measure rainfall under most conditions when radiometers operating at several higher frequencies become available on future airborne radar experiments.

## Abstract

Data from the Tropical Rainfall Measuring Mission (TRMM) precipitation radar represent the first global rain-rate dataset acquired by a spaceborne weather radar. Because the radar operates at an attenuating wavelength, one of the principal issues concerns the accuracy of the attenuation correction algorithms. One way to test these algorithms is by means of a statistical method in which the probability distribution of rain rates at the high end is inferred by measurements at the low to intermediate range and by the assumption that the rain rates are lognormally distributed. Investigation of this method and the area–time integral methods using a global dataset provides an indication of how well methods of this kind can be expected to perform over different space–timescales and climatological regions using the sparsely sampled TRMM radar data. Identification of statistical relationships among the rain parameters and an understanding of the rain-rate distribution as a function of time and space may help to test the validity of the high-resolution rain-rate estimates.

## Abstract

Data from the Tropical Rainfall Measuring Mission (TRMM) precipitation radar represent the first global rain-rate dataset acquired by a spaceborne weather radar. Because the radar operates at an attenuating wavelength, one of the principal issues concerns the accuracy of the attenuation correction algorithms. One way to test these algorithms is by means of a statistical method in which the probability distribution of rain rates at the high end is inferred by measurements at the low to intermediate range and by the assumption that the rain rates are lognormally distributed. Investigation of this method and the area–time integral methods using a global dataset provides an indication of how well methods of this kind can be expected to perform over different space–timescales and climatological regions using the sparsely sampled TRMM radar data. Identification of statistical relationships among the rain parameters and an understanding of the rain-rate distribution as a function of time and space may help to test the validity of the high-resolution rain-rate estimates.

## Abstract

Simulations of observations from potential spaceborne radars are made based on storm structure generated from the three-dimensional (3D) Goddard cumulus ensemble model simulation of an intense overland convective system. Five frequencies of 3, 10, 14, 35, and 95 GHz are discussed, but the Tropical Rainfall Measuring Mission precipitation radar sensor frequency ( 14 GHz) is the focus of this study. Radar reflectivities and their attenuation in various atmospheric conditions are studied in this simulation. With the attenuation from cloud and precipitation in the estimation of reflectivity factor (dBZ), the reflectivities in the lower atmosphere in the convective coresare significantly reduced. With spatial resolution of 4 km X 4 km, attenuation at 14 GHz may cause as large as a 20-dBZ difference between the simulated measurements of the peak (Z_{mp}) and near-surface reflectivity (Z_{mp}) in the most intense convective region. The Z_{mp} occurs at various altitudes depending on the hydrometeor concentrations and their vertical distribution. Despite the significant attenuation in the intense cores, the presence of the rain maximum is easily detected by using information of Z_{mp}. In the stratiform region, the attenuation is quite limited (usually less than 5 dBZ), and the reduction of reflectivity is mostly related to the actual vertical structure of cloud distribution. Since Z_{mp} suffers severe attenuation and tends to underestimate surface rainfall intensity in convective regions, Z_{mp} can be more representative for rainfall retrieval in the lower atmosphere in these regions. In the stratiform region where attenuation is negligible, however, Z_{mp} tends to overestimate surface rainfall and Z_{mp} is more appropriate for rainfall retrieval. A hybrid technique using a weight between the two rain intensities is testedand found potentially useful for future applications. The estimated surface rain-rate map based on this hybrid approach captures many of the details of the cloud model rain field but still slightly underestimates the rain-rate maximum.

## Abstract

Simulations of observations from potential spaceborne radars are made based on storm structure generated from the three-dimensional (3D) Goddard cumulus ensemble model simulation of an intense overland convective system. Five frequencies of 3, 10, 14, 35, and 95 GHz are discussed, but the Tropical Rainfall Measuring Mission precipitation radar sensor frequency ( 14 GHz) is the focus of this study. Radar reflectivities and their attenuation in various atmospheric conditions are studied in this simulation. With the attenuation from cloud and precipitation in the estimation of reflectivity factor (dBZ), the reflectivities in the lower atmosphere in the convective coresare significantly reduced. With spatial resolution of 4 km X 4 km, attenuation at 14 GHz may cause as large as a 20-dBZ difference between the simulated measurements of the peak (Z_{mp}) and near-surface reflectivity (Z_{mp}) in the most intense convective region. The Z_{mp} occurs at various altitudes depending on the hydrometeor concentrations and their vertical distribution. Despite the significant attenuation in the intense cores, the presence of the rain maximum is easily detected by using information of Z_{mp}. In the stratiform region, the attenuation is quite limited (usually less than 5 dBZ), and the reduction of reflectivity is mostly related to the actual vertical structure of cloud distribution. Since Z_{mp} suffers severe attenuation and tends to underestimate surface rainfall intensity in convective regions, Z_{mp} can be more representative for rainfall retrieval in the lower atmosphere in these regions. In the stratiform region where attenuation is negligible, however, Z_{mp} tends to overestimate surface rainfall and Z_{mp} is more appropriate for rainfall retrieval. A hybrid technique using a weight between the two rain intensities is testedand found potentially useful for future applications. The estimated surface rain-rate map based on this hybrid approach captures many of the details of the cloud model rain field but still slightly underestimates the rain-rate maximum.

## Abstract

This paper represents the first attempt to use Tropical Rainfall Measuring Mission (TRMM) rainfall information to estimate the four-dimensional latent heating structure over the global Tropics for one month (February 1998). The mean latent heating profiles over six oceanic regions [Tropical Ocean and Global Atmosphere (TOGA) Coupled Ocean–Atmosphere Response Experiment (COARE) Intensive Flux Array (IFA), central Pacific, South Pacific Convergence Zone (SPCZ), east Pacific, Indian Ocean, and Atlantic Ocean] and three continental regions (South America, central Africa, and Australia) are estimated and studied. The heating profiles obtained from the results of diagnostic budget studies over a broad range of geographic locations are used to provide comparisons and indirect validation for the heating algorithm–estimated heating profiles. Three different latent heating algorithms, the Goddard Space Flight Center convective–stratiform heating (CSH), the Goddard profiling (GPROF) heating, and the hydrometeor heating (HH) algorithms are used and their results are intercompared. The horizontal distribution or patterns of latent heat release from the three different heating retrieval methods are very similar. They all can identify the areas of major convective activity [i.e., a well-defined Intertropical Convergence Zone (ITCZ) in the Pacific, a distinct SPCZ] in the global Tropics. The magnitudes of their estimated latent heating release are also in good agreement with each other and with those determined from diagnostic budget studies. However, the major difference among these three heating retrieval algorithms is the altitude of the maximum heating level. The CSH algorithm–estimated heating profiles only show one maximum heating level, and the level varies among convective activity from various geographic locations. These features are in good agreement with diagnostic budget studies. A broader maximum of heating, often with two embedded peaks, is generally derived from applications of the GPROF heating and HH algorithms, and the response of the heating profiles to convective activity is less pronounced. Also, GPROF and HH generally yield heating profiles with a maximum at somewhat lower altitudes than CSH. The impact of different TRMM Microwave Imager (TMI) and precipitation radar (PR) rainfall information on latent heating structures was also examined. The rainfall estimated from the PR is smaller than that estimated from the TMI in the Pacific (TOGA COARE IFA, central Pacific, SPCZ, and east Pacific) and Indian Oceans, causing weaker latent heat release in the CSH algorithm–estimated heating. In addition, the larger stratiform amounts derived from the PR over South America and Australia consequently lead to higher maximum heating levels. Sensitivity tests addressing the appropriate selection of latent heating profiles from the CSH lookup table were performed.

## Abstract

This paper represents the first attempt to use Tropical Rainfall Measuring Mission (TRMM) rainfall information to estimate the four-dimensional latent heating structure over the global Tropics for one month (February 1998). The mean latent heating profiles over six oceanic regions [Tropical Ocean and Global Atmosphere (TOGA) Coupled Ocean–Atmosphere Response Experiment (COARE) Intensive Flux Array (IFA), central Pacific, South Pacific Convergence Zone (SPCZ), east Pacific, Indian Ocean, and Atlantic Ocean] and three continental regions (South America, central Africa, and Australia) are estimated and studied. The heating profiles obtained from the results of diagnostic budget studies over a broad range of geographic locations are used to provide comparisons and indirect validation for the heating algorithm–estimated heating profiles. Three different latent heating algorithms, the Goddard Space Flight Center convective–stratiform heating (CSH), the Goddard profiling (GPROF) heating, and the hydrometeor heating (HH) algorithms are used and their results are intercompared. The horizontal distribution or patterns of latent heat release from the three different heating retrieval methods are very similar. They all can identify the areas of major convective activity [i.e., a well-defined Intertropical Convergence Zone (ITCZ) in the Pacific, a distinct SPCZ] in the global Tropics. The magnitudes of their estimated latent heating release are also in good agreement with each other and with those determined from diagnostic budget studies. However, the major difference among these three heating retrieval algorithms is the altitude of the maximum heating level. The CSH algorithm–estimated heating profiles only show one maximum heating level, and the level varies among convective activity from various geographic locations. These features are in good agreement with diagnostic budget studies. A broader maximum of heating, often with two embedded peaks, is generally derived from applications of the GPROF heating and HH algorithms, and the response of the heating profiles to convective activity is less pronounced. Also, GPROF and HH generally yield heating profiles with a maximum at somewhat lower altitudes than CSH. The impact of different TRMM Microwave Imager (TMI) and precipitation radar (PR) rainfall information on latent heating structures was also examined. The rainfall estimated from the PR is smaller than that estimated from the TMI in the Pacific (TOGA COARE IFA, central Pacific, SPCZ, and east Pacific) and Indian Oceans, causing weaker latent heat release in the CSH algorithm–estimated heating. In addition, the larger stratiform amounts derived from the PR over South America and Australia consequently lead to higher maximum heating levels. Sensitivity tests addressing the appropriate selection of latent heating profiles from the CSH lookup table were performed.

## Abstract

Information about the vertical microphysical cloud structure is useful in many modeling and predictive practices. Radiometers and radars are used to observe hydrometeor properties. This paper describes an iterative retrieval algorithm that combines the use of airborne active and wideband (10–340 GHz) passive observations to estimate the vertical content and particle size distributions of liquid and frozen hydrometeors. Airborne radar and radiometer observations from the third Convection and Moisture Experiment (CAMEX-3) were used in the retrieval algorithm as constraints. Nadir profiles were estimated for 1 min each of flight time (approximately 12.5 km along track) for anvil, convective, and quasi-stratiform clouds associated with Hurricane Bonnie (August 1998). The physically based retrieval algorithm relies on high frequencies (≥150 GHz) to provide details on the frozen hydrometeors. Neglecting the high frequencies yielded acceptable estimates of the liquid profiles, but the ice profiles were poorly retrieved. The wideband observations were found to more than double the estimated frozen hydrometeor content as compared with retrievals using only 90 GHz and below. The convective and quasi-stratiform iterative retrievals quickly reached convergence. The complex structure of the frozen hydrometeors required the most iterations for convergence for the anvil cloud type. Nonunique profiles, within physical and theoretical bounds, were retrieved for thin anvil ice clouds. A qualitative validation using coincident in situ CAMEX-3 observations shows that the retrieved particle size distributions are well corroborated with independent measurements.

## Abstract

Information about the vertical microphysical cloud structure is useful in many modeling and predictive practices. Radiometers and radars are used to observe hydrometeor properties. This paper describes an iterative retrieval algorithm that combines the use of airborne active and wideband (10–340 GHz) passive observations to estimate the vertical content and particle size distributions of liquid and frozen hydrometeors. Airborne radar and radiometer observations from the third Convection and Moisture Experiment (CAMEX-3) were used in the retrieval algorithm as constraints. Nadir profiles were estimated for 1 min each of flight time (approximately 12.5 km along track) for anvil, convective, and quasi-stratiform clouds associated with Hurricane Bonnie (August 1998). The physically based retrieval algorithm relies on high frequencies (≥150 GHz) to provide details on the frozen hydrometeors. Neglecting the high frequencies yielded acceptable estimates of the liquid profiles, but the ice profiles were poorly retrieved. The wideband observations were found to more than double the estimated frozen hydrometeor content as compared with retrievals using only 90 GHz and below. The convective and quasi-stratiform iterative retrievals quickly reached convergence. The complex structure of the frozen hydrometeors required the most iterations for convergence for the anvil cloud type. Nonunique profiles, within physical and theoretical bounds, were retrieved for thin anvil ice clouds. A qualitative validation using coincident in situ CAMEX-3 observations shows that the retrieved particle size distributions are well corroborated with independent measurements.

## Abstract

In this study, two different particle models describing the structure and electromagnetic properties of snow are developed and evaluated for potential use in satellite combined radar–radiometer precipitation estimation algorithms. In the first model, snow particles are assumed to be homogeneous ice–air spheres with single-scattering properties derived from Mie theory. In the second model, snow particles are created by simulating the self-collection of pristine ice crystals into aggregate particles of different sizes, using different numbers and habits of the collected component crystals. Single-scattering properties of the resulting nonspherical snow particles are determined using the discrete dipole approximation. The size-distribution-integrated scattering properties of the spherical and nonspherical snow particles are incorporated into a dual-wavelength radar profiling algorithm that is applied to 14- and 34-GHz observations of stratiform precipitation from the ER-2 aircraftborne High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) radar. The retrieved ice precipitation profiles are then input to a forward radiative transfer calculation in an attempt to simulate coincident radiance observations from the Conical Scanning Millimeter-Wave Imaging Radiometer (CoSMIR). Much greater consistency between the simulated and observed CoSMIR radiances is obtained using estimated profiles that are based upon the nonspherical crystal/aggregate snow particle model. Despite this greater consistency, there remain some discrepancies between the higher moments of the HIWRAP-retrieved precipitation size distributions and in situ distributions derived from microphysics probe observations obtained from Citation aircraft underflights of the ER-2. These discrepancies can only be eliminated if a subset of lower-density crystal/aggregate snow particles is assumed in the radar algorithm and in the interpretation of the in situ data.

## Abstract

In this study, two different particle models describing the structure and electromagnetic properties of snow are developed and evaluated for potential use in satellite combined radar–radiometer precipitation estimation algorithms. In the first model, snow particles are assumed to be homogeneous ice–air spheres with single-scattering properties derived from Mie theory. In the second model, snow particles are created by simulating the self-collection of pristine ice crystals into aggregate particles of different sizes, using different numbers and habits of the collected component crystals. Single-scattering properties of the resulting nonspherical snow particles are determined using the discrete dipole approximation. The size-distribution-integrated scattering properties of the spherical and nonspherical snow particles are incorporated into a dual-wavelength radar profiling algorithm that is applied to 14- and 34-GHz observations of stratiform precipitation from the ER-2 aircraftborne High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) radar. The retrieved ice precipitation profiles are then input to a forward radiative transfer calculation in an attempt to simulate coincident radiance observations from the Conical Scanning Millimeter-Wave Imaging Radiometer (CoSMIR). Much greater consistency between the simulated and observed CoSMIR radiances is obtained using estimated profiles that are based upon the nonspherical crystal/aggregate snow particle model. Despite this greater consistency, there remain some discrepancies between the higher moments of the HIWRAP-retrieved precipitation size distributions and in situ distributions derived from microphysics probe observations obtained from Citation aircraft underflights of the ER-2. These discrepancies can only be eliminated if a subset of lower-density crystal/aggregate snow particles is assumed in the radar algorithm and in the interpretation of the in situ data.

## Abstract

Rainfall retrieval algorithms often assume a gamma-shaped raindrop size distribution (DSD) with three mathematical parameters *N*
_{
w
}, *D*
_{
m
}, and *μ*. If only two independent measurements are available, as with the dual-frequency precipitation radar on the Global Precipitation Measurement (GPM) mission core satellite, then retrieval algorithms are underconstrained and require assumptions about DSD parameters. To reduce the number of free parameters, algorithms can assume that *μ* is either a constant or a function of *D*
_{
m
}. Previous studies have suggested *μ*–Λ constraints [where Λ = (4 + *μ*)/*D*
_{
m
}], but controversies exist over whether *μ*–Λ constraints result from physical processes or mathematical artifacts due to high correlations between gamma DSD parameters. This study avoids mathematical artifacts by developing joint probability distribution functions (joint PDFs) of statistically independent DSD attributes derived from the raindrop mass spectrum. These joint PDFs are then mapped into gamma-shaped DSD parameter joint PDFs that can be used in probabilistic rainfall retrieval algorithms as proposed for the GPM satellite program. Surface disdrometer data show a high correlation coefficient between the mass spectrum mean diameter *D*
_{
m
} and mass spectrum standard deviation *σ*
_{
m
}. To remove correlations between DSD attributes, a normalized mass spectrum standard deviation *D*
_{
m
}, with *D*
_{
m
} and *μ* are created from *D*
_{
m
} and *μ*.

## Abstract

Rainfall retrieval algorithms often assume a gamma-shaped raindrop size distribution (DSD) with three mathematical parameters *N*
_{
w
}, *D*
_{
m
}, and *μ*. If only two independent measurements are available, as with the dual-frequency precipitation radar on the Global Precipitation Measurement (GPM) mission core satellite, then retrieval algorithms are underconstrained and require assumptions about DSD parameters. To reduce the number of free parameters, algorithms can assume that *μ* is either a constant or a function of *D*
_{
m
}. Previous studies have suggested *μ*–Λ constraints [where Λ = (4 + *μ*)/*D*
_{
m
}], but controversies exist over whether *μ*–Λ constraints result from physical processes or mathematical artifacts due to high correlations between gamma DSD parameters. This study avoids mathematical artifacts by developing joint probability distribution functions (joint PDFs) of statistically independent DSD attributes derived from the raindrop mass spectrum. These joint PDFs are then mapped into gamma-shaped DSD parameter joint PDFs that can be used in probabilistic rainfall retrieval algorithms as proposed for the GPM satellite program. Surface disdrometer data show a high correlation coefficient between the mass spectrum mean diameter *D*
_{
m
} and mass spectrum standard deviation *σ*
_{
m
}. To remove correlations between DSD attributes, a normalized mass spectrum standard deviation *D*
_{
m
}, with *D*
_{
m
} and *μ* are created from *D*
_{
m
} and *μ*.