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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

Melting snow, graupel, and hail are often modeled as uniform mixtures of air–ice–water or ice–water. Two-layered models have also been proposed in which the particle consists of a dry snow or ice core surrounded by water or a wet snow mixture. For both types of particle models, the mixtures are characterized by effective dielectric constants. This information, along with particle shape, size, and orientation, provides the necessary data for calculating the scattering characteristics of the particles. The most commonly used formulas for the effective dielectric constant, ɛ_{eff}, are those of Maxwell Garnett and Bruggeman. To understand the applicability and limitations of these formulas, an expression for ɛ_{eff} is derived that depends on the mean internal electric fields within each component of the mixture. Using a conjugate gradient numerical method, the calculations are carried out for ice–water mixtures. Parameterization of the results in terms of the fractional water volume and the electromagnetic wavelength provides an expression for ɛ_{eff} for wavelengths between 3 and 28 mm. To circumvent the laborious task of parameterizing ɛ_{eff} with wavelength for air–ice–water mixtures, several approximate formulations are proposed. Tests of the accuracy of the formulas are made by calculating the mean and variance from different particle realizations and by comparison to a previous method. Tests of the applicability of the formulas for ɛ_{eff} are made by changing the shape, size, and orientations of the inclusions. While the formulas are adequate over a certain range of inclusion sizes and for a change in shape from cubic to spherical, they are not applicable to highly eccentric, aligned inclusions such as rods or plates.

## Abstract

Melting snow, graupel, and hail are often modeled as uniform mixtures of air–ice–water or ice–water. Two-layered models have also been proposed in which the particle consists of a dry snow or ice core surrounded by water or a wet snow mixture. For both types of particle models, the mixtures are characterized by effective dielectric constants. This information, along with particle shape, size, and orientation, provides the necessary data for calculating the scattering characteristics of the particles. The most commonly used formulas for the effective dielectric constant, ɛ_{eff}, are those of Maxwell Garnett and Bruggeman. To understand the applicability and limitations of these formulas, an expression for ɛ_{eff} is derived that depends on the mean internal electric fields within each component of the mixture. Using a conjugate gradient numerical method, the calculations are carried out for ice–water mixtures. Parameterization of the results in terms of the fractional water volume and the electromagnetic wavelength provides an expression for ɛ_{eff} for wavelengths between 3 and 28 mm. To circumvent the laborious task of parameterizing ɛ_{eff} with wavelength for air–ice–water mixtures, several approximate formulations are proposed. Tests of the accuracy of the formulas are made by calculating the mean and variance from different particle realizations and by comparison to a previous method. Tests of the applicability of the formulas for ɛ_{eff} are made by changing the shape, size, and orientations of the inclusions. While the formulas are adequate over a certain range of inclusion sizes and for a change in shape from cubic to spherical, they are not applicable to highly eccentric, aligned inclusions such as rods or plates.

## Abstract

Airborne dual-wavelength and dual-polarization radar data are analyzed for measurements taken in stratiform rain in the western Pacific during September 1990. The focus of the paper is on the vertical profiles of the linear depolarization ratio, LDR ( 10 GHz); the reflectivity factor, dBZ ( 10 GHz); and the dual-frequency ratio, DFR (10, 34.45 GHz). Statistical characterizations of the maxima of these quantities and the relative locations at which they occur suggest that the eccentricity of the melting particles is fairly large and that the shape and size of the particles are correlated. To try to explain these features, two types of simulation are presented. In the first, a set of measured drop size distributions is used in the context of a standard model of the melting layer. Variations in snow density, as well as shape, size, and orientation distributions are used to study the relationship between these parameters and the radar measurements. To reduce the amount of ambiguity in the estimation, a second type of simulation is described in which the size distribution of the snow is estimated. Comparisons between the simulated and measured profiles indicate that radar measurements can be used to derive certain characteristics of the particle size and shape distributions in the melting layer.

## Abstract

Airborne dual-wavelength and dual-polarization radar data are analyzed for measurements taken in stratiform rain in the western Pacific during September 1990. The focus of the paper is on the vertical profiles of the linear depolarization ratio, LDR ( 10 GHz); the reflectivity factor, dBZ ( 10 GHz); and the dual-frequency ratio, DFR (10, 34.45 GHz). Statistical characterizations of the maxima of these quantities and the relative locations at which they occur suggest that the eccentricity of the melting particles is fairly large and that the shape and size of the particles are correlated. To try to explain these features, two types of simulation are presented. In the first, a set of measured drop size distributions is used in the context of a standard model of the melting layer. Variations in snow density, as well as shape, size, and orientation distributions are used to study the relationship between these parameters and the radar measurements. To reduce the amount of ambiguity in the estimation, a second type of simulation is described in which the size distribution of the snow is estimated. Comparisons between the simulated and measured profiles indicate that radar measurements can be used to derive certain characteristics of the particle size and shape distributions in the melting layer.

## Abstract

Radars have played an important role in the observation of precipitation and will continue to do so in the future. With the recent introduction of space-based radar for measuring precipitation on the Tropical Rainfall Measurement Mission (TRMM) satellite, weather radar applications now range from local to global scales. The radar basis for characterizing precipitation lies in the scattering and propagation properties of electromagnetic waves through precipitation, and is summarized in this paper. The methodologies for converting the backscattering and propagation measurements such as radar reflectivity, differential reflectivity, differential propagation phase, and attenuation to precipitation estimates are provided for both ground-based and space-based radars. Quantitative precipitation estimation has been a challenging problem for over four decades. This challenge has inspired extensive progress in the area of precipitation microphysics, remote sensing techniques, and in situ observations. Another major advance in quantitative precipitation estimation is the understanding of the critical role player by practical engineering considerations. Techniques for developing precipitation algorithms from space and ground observations as well as strategies for validating the estimates are also presented. Following a summary of the various challenges, the discussion focuses on those areas with potential for significant future progress for the estimation of both local and global precipitation.

## Abstract

Radars have played an important role in the observation of precipitation and will continue to do so in the future. With the recent introduction of space-based radar for measuring precipitation on the Tropical Rainfall Measurement Mission (TRMM) satellite, weather radar applications now range from local to global scales. The radar basis for characterizing precipitation lies in the scattering and propagation properties of electromagnetic waves through precipitation, and is summarized in this paper. The methodologies for converting the backscattering and propagation measurements such as radar reflectivity, differential reflectivity, differential propagation phase, and attenuation to precipitation estimates are provided for both ground-based and space-based radars. Quantitative precipitation estimation has been a challenging problem for over four decades. This challenge has inspired extensive progress in the area of precipitation microphysics, remote sensing techniques, and in situ observations. Another major advance in quantitative precipitation estimation is the understanding of the critical role player by practical engineering considerations. Techniques for developing precipitation algorithms from space and ground observations as well as strategies for validating the estimates are also presented. Following a summary of the various challenges, the discussion focuses on those areas with potential for significant future progress for the estimation of both local and global precipitation.

## 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

The High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) dual-frequency conically scanning airborne radar provides estimates of the range-profiled mean Doppler and backscattered power from the precipitation and surface. A velocity–azimuth display analysis yields near-surface estimates of the mean horizontal wind vector *υ*
_{
h
} in cases in which precipitation is present throughout the scan. From the surface return, the normalized radar cross section (NRCS) is obtained, which, by a method previously described, can be corrected for path attenuation. Comparisons between *υ*
_{
h
} and the attenuation-corrected NRCS are used to derive transfer functions that provide estimates of the wind vector from the NRCS data under both rain and rain-free conditions. A reasonably robust transfer function is found by using the mean NRCS (⟨NRCS⟩) over the scan along with a filtering of the data based on a Fourier series analysis of *υ*
_{
h
} and the NRCS. The approach gives good correlation coefficients between *υ*
_{
h
} and ⟨NRCS⟩ at Ku band at incidence angles of 30° and 40°. The correlation degrades if the Ka-band data are used rather than the Ku band.

## Abstract

The High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) dual-frequency conically scanning airborne radar provides estimates of the range-profiled mean Doppler and backscattered power from the precipitation and surface. A velocity–azimuth display analysis yields near-surface estimates of the mean horizontal wind vector *υ*
_{
h
} in cases in which precipitation is present throughout the scan. From the surface return, the normalized radar cross section (NRCS) is obtained, which, by a method previously described, can be corrected for path attenuation. Comparisons between *υ*
_{
h
} and the attenuation-corrected NRCS are used to derive transfer functions that provide estimates of the wind vector from the NRCS data under both rain and rain-free conditions. A reasonably robust transfer function is found by using the mean NRCS (⟨NRCS⟩) over the scan along with a filtering of the data based on a Fourier series analysis of *υ*
_{
h
} and the NRCS. The approach gives good correlation coefficients between *υ*
_{
h
} and ⟨NRCS⟩ at Ku band at incidence angles of 30° and 40°. The correlation degrades if the Ka-band data are used rather than the Ku band.

#
^{}
Attenuation Correction over Ocean for the HIWRAP Dual-Frequency Airborne Scatterometer

## Abstract

An important objective in scatterometry is the estimation of near-surface wind speed and direction in the presence of rain. We investigate an attenuation correction method using data from the High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) dual-frequency scatterometer, which operates at Ku and Ka band with dual conical scans at incidence angles of 30° and 40°. The method relies on the fact that the differential normalized surface cross section, *δσ*
^{0} = *σ*
^{0}(Ka) − *σ*
^{0}(Ku), is relatively insensitive to wind speed and direction and that this quantity is closely related to the magnitude of the differential path attenuation, *δA* = *A*(Ka) − *A*(Ku), arising from precipitation, cloud, and atmospheric gases. As the method relies only on the difference between quantities measured in the presence and absence of rain, the estimates are independent of radar calibration error. As a test of the method’s accuracy, we make use of the fact that the radar rain reflectivities just above the surface, as seen along different incidence angles, are approximately the same. This yields constraint equations in the form of differences between pairs of path attenuations along different lines of sight to the surface. A second validation method uses the dual-frequency radar returns from the rain just above the surface where it can be shown that the difference between the Ku- and Ka-band-measured radar reflectivity factors provide an estimate of differential path attenuation. Comparisons between the path attenuations derived from the normalized surface cross section and those from these surface-independent methods generally show good agreement.

## Abstract

An important objective in scatterometry is the estimation of near-surface wind speed and direction in the presence of rain. We investigate an attenuation correction method using data from the High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) dual-frequency scatterometer, which operates at Ku and Ka band with dual conical scans at incidence angles of 30° and 40°. The method relies on the fact that the differential normalized surface cross section, *δσ*
^{0} = *σ*
^{0}(Ka) − *σ*
^{0}(Ku), is relatively insensitive to wind speed and direction and that this quantity is closely related to the magnitude of the differential path attenuation, *δA* = *A*(Ka) − *A*(Ku), arising from precipitation, cloud, and atmospheric gases. As the method relies only on the difference between quantities measured in the presence and absence of rain, the estimates are independent of radar calibration error. As a test of the method’s accuracy, we make use of the fact that the radar rain reflectivities just above the surface, as seen along different incidence angles, are approximately the same. This yields constraint equations in the form of differences between pairs of path attenuations along different lines of sight to the surface. A second validation method uses the dual-frequency radar returns from the rain just above the surface where it can be shown that the difference between the Ku- and Ka-band-measured radar reflectivity factors provide an estimate of differential path attenuation. Comparisons between the path attenuations derived from the normalized surface cross section and those from these surface-independent methods generally show good agreement.

## 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.