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## Abstract

A fast, approximate method is described for the calculation of the intensity of multiply scattered lidar returns from clouds. At each range gate it characterizes the outgoing photon distribution by its spatial variance, the variance of photon direction, and the covariance of photon direction and position. The result is that for an *N*-point profile the calculation is *O*(*N*) efficient yet it implicitly includes all orders of scattering, in contrast with the *O*(*N ^{m}*/

*m*!) efficiency of models that explicitly consider each scattering order separately for truncation at

*m*-order scattering. It is also shown how the shape of the scattering phase function near 180° may be taken into account for both liquid water droplets and ice particles. The model considers only multiple scattering due to small-angle forward-scattering events, which is suitable for most ground-based and airborne lidars because of their small footprint on the cloud. For spaceborne lidar, it must be used in combination with the wide-angle multiple scattering model described in Part II of this two-part paper.

## Abstract

A fast, approximate method is described for the calculation of the intensity of multiply scattered lidar returns from clouds. At each range gate it characterizes the outgoing photon distribution by its spatial variance, the variance of photon direction, and the covariance of photon direction and position. The result is that for an *N*-point profile the calculation is *O*(*N*) efficient yet it implicitly includes all orders of scattering, in contrast with the *O*(*N ^{m}*/

*m*!) efficiency of models that explicitly consider each scattering order separately for truncation at

*m*-order scattering. It is also shown how the shape of the scattering phase function near 180° may be taken into account for both liquid water droplets and ice particles. The model considers only multiple scattering due to small-angle forward-scattering events, which is suitable for most ground-based and airborne lidars because of their small footprint on the cloud. For spaceborne lidar, it must be used in combination with the wide-angle multiple scattering model described in Part II of this two-part paper.

## Abstract

Polarization radar offers the promise of much more accurate rainfall-rate *R* estimates than are possible from radar reflectivity factor *Z* alone, not only by better characterization of the drop size distribution, but also by more reliable correction for attenuation and the identification of hail. However, practical attempts to implement retrieval algorithms have been hampered by the difficulty in coping with the inherent noise in the polarization parameters. In this paper, a variational retrieval scheme is described that overcomes these problems by employing a forward model for differential reflectivity *Z*_{dr} and differential phase shift *ϕ*_{dp} and iteratively refining the coefficient *a* in the relationship *Z* = *aR ^{b}* such that the difference between the forward model and the measurements is minimized in a least squares sense. Two methods are used to ensure that

*a*varies smoothly in both range and azimuth. In range,

*a*is represented by a set of cubic-spline basis functions; in azimuth, the retrieval at one ray is used as a constraint on the next. The result of this smoothing is that the retrieval is tolerant of random errors in

*Z*

_{dr}of up to 1 dB and in

*ϕ*

_{dp}of up to 5°. Correction for attenuation is achieved simply and effectively by including its effects in the forward model. If hail is present then the forward model is unable to match the observations of

*Z*

_{dr}and

*ϕ*

_{dp}simultaneously. This enables a first pass of the retrieval to be used to identify the radar pixels that contain hail, followed by a second pass in which the fraction of the

*Z*in those gates that is due to hail is retrieved, this time with the scheme being able to forward-model both

*Z*

_{dr}and

*ϕ*

_{dp}accurately. The scheme is tested on S-band radar data from southern England in cases of rain, spherical hail, oblate hail, and mixtures of rain and hail. It is found to be robust and stable, even in the presence of differential phase shift on backscatter.

## Abstract

Polarization radar offers the promise of much more accurate rainfall-rate *R* estimates than are possible from radar reflectivity factor *Z* alone, not only by better characterization of the drop size distribution, but also by more reliable correction for attenuation and the identification of hail. However, practical attempts to implement retrieval algorithms have been hampered by the difficulty in coping with the inherent noise in the polarization parameters. In this paper, a variational retrieval scheme is described that overcomes these problems by employing a forward model for differential reflectivity *Z*_{dr} and differential phase shift *ϕ*_{dp} and iteratively refining the coefficient *a* in the relationship *Z* = *aR ^{b}* such that the difference between the forward model and the measurements is minimized in a least squares sense. Two methods are used to ensure that

*a*varies smoothly in both range and azimuth. In range,

*a*is represented by a set of cubic-spline basis functions; in azimuth, the retrieval at one ray is used as a constraint on the next. The result of this smoothing is that the retrieval is tolerant of random errors in

*Z*

_{dr}of up to 1 dB and in

*ϕ*

_{dp}of up to 5°. Correction for attenuation is achieved simply and effectively by including its effects in the forward model. If hail is present then the forward model is unable to match the observations of

*Z*

_{dr}and

*ϕ*

_{dp}simultaneously. This enables a first pass of the retrieval to be used to identify the radar pixels that contain hail, followed by a second pass in which the fraction of the

*Z*in those gates that is due to hail is retrieved, this time with the scheme being able to forward-model both

*Z*

_{dr}and

*ϕ*

_{dp}accurately. The scheme is tested on S-band radar data from southern England in cases of rain, spherical hail, oblate hail, and mixtures of rain and hail. It is found to be robust and stable, even in the presence of differential phase shift on backscatter.

## Abstract

The correlated-*k*-distribution (CKD) method is widely used in the radiative transfer schemes of atmospheric models; it involves dividing the spectrum into a number of bands and then reordering the gaseous absorption coefficients within each one. The fluxes and heating rates for each band may then be computed by discretizing the reordered spectrum into *O*(10) quadrature points per major gas and performing a pseudomonochromatic radiation calculation for each point. In this paper it is first argued that for clear-sky longwave calculations, sufficient accuracy for most applications can be achieved without the need for bands: reordering may be performed on the entire longwave spectrum. The resulting full-spectrum correlated-*k* (FSCK) method requires significantly fewer pseudomonochromatic calculations than standard CKD to achieve a given accuracy. The concept is first demonstrated by comparing with line-by-line calculations for an atmosphere containing only water vapor, in which it is shown that the accuracy of heating rate calculations improves approximately in proportion to the square of the number of quadrature points. For more than around 20 points, the root-mean-square error flattens out at around 0.015 K day^{−1} due to the imperfect rank correlation of absorption spectra at different pressures in the profile. The spectral overlap of *m* different gases is treated by considering an *m*-dimensional hypercube where each axis corresponds to the reordered spectrum of one of the gases. This hypercube is then divided up into a number of volumes, each approximated by a single quadrature point, such that the total number of quadrature points is slightly fewer than the sum of the number that would be required to treat each of the gases separately. The gaseous absorptions for each quadrature point are optimized such that they minimize a cost function expressing the deviation of the heating rates and fluxes calculated by the FSCK method from line-by-line calculations for a number of training profiles. This approach is validated for atmospheres containing water vapor, carbon dioxide, and ozone, in which it is found that in the troposphere and most of the stratosphere, heating rate errors of less than 0.2 K day^{−1} can be achieved using a total of 23 quadrature points, decreasing to less than 0.1 K day^{−1} for 32 quadrature points. It would be relatively straightforward to extend the method to include other gases.

## Abstract

The correlated-*k*-distribution (CKD) method is widely used in the radiative transfer schemes of atmospheric models; it involves dividing the spectrum into a number of bands and then reordering the gaseous absorption coefficients within each one. The fluxes and heating rates for each band may then be computed by discretizing the reordered spectrum into *O*(10) quadrature points per major gas and performing a pseudomonochromatic radiation calculation for each point. In this paper it is first argued that for clear-sky longwave calculations, sufficient accuracy for most applications can be achieved without the need for bands: reordering may be performed on the entire longwave spectrum. The resulting full-spectrum correlated-*k* (FSCK) method requires significantly fewer pseudomonochromatic calculations than standard CKD to achieve a given accuracy. The concept is first demonstrated by comparing with line-by-line calculations for an atmosphere containing only water vapor, in which it is shown that the accuracy of heating rate calculations improves approximately in proportion to the square of the number of quadrature points. For more than around 20 points, the root-mean-square error flattens out at around 0.015 K day^{−1} due to the imperfect rank correlation of absorption spectra at different pressures in the profile. The spectral overlap of *m* different gases is treated by considering an *m*-dimensional hypercube where each axis corresponds to the reordered spectrum of one of the gases. This hypercube is then divided up into a number of volumes, each approximated by a single quadrature point, such that the total number of quadrature points is slightly fewer than the sum of the number that would be required to treat each of the gases separately. The gaseous absorptions for each quadrature point are optimized such that they minimize a cost function expressing the deviation of the heating rates and fluxes calculated by the FSCK method from line-by-line calculations for a number of training profiles. This approach is validated for atmospheres containing water vapor, carbon dioxide, and ozone, in which it is found that in the troposphere and most of the stratosphere, heating rate errors of less than 0.2 K day^{−1} can be achieved using a total of 23 quadrature points, decreasing to less than 0.1 K day^{−1} for 32 quadrature points. It would be relatively straightforward to extend the method to include other gases.

## Abstract

With the rapid growth in air travel, there is concern over the radiative impact of contrails and aircraft-induced cirrus on climate. Previous radiation calculations on contrails have almost all used the independent column approximation, which neglects the transport of photons through the sides of the contrail, but in this study the 3D effects are quantified using the Spherical Harmonic Discrete Ordinate Method (SHDOM). The authors have investigated the dependence of shortwave and longwave radiative forcing on contrail aspect ratio, optical depth, solar zenith angle, solar azimuth angle relative to contrail orientation, particle size, particle habit, surface albedo, and surface temperature. It is found that inclusion of 3D transport results in an increase in the positive longwave radiative forcing of the contrail and either an increase or a decrease in the magnitude of the negative shortwave radiative forcing depending on the orientation of the contrail with respect to the sun. Although these two effects are individually quite modest (of order 10%), the fact that the total shortwave and longwave forcings largely cancel during the day means that the relative change in the net radiative forcing due to the 3D effect is substantial; in some cases this results in a doubling of the net forcing of the contrail, in other cases changing its sign. On a more general note, the relatively simple geometry of contrail cirrus provides an ideal test case for explaining the various mechanisms by which 3D photon transport can change the radiative effect of clouds, which can be rather difficult to visualize for more complex cloud scenarios.

## Abstract

With the rapid growth in air travel, there is concern over the radiative impact of contrails and aircraft-induced cirrus on climate. Previous radiation calculations on contrails have almost all used the independent column approximation, which neglects the transport of photons through the sides of the contrail, but in this study the 3D effects are quantified using the Spherical Harmonic Discrete Ordinate Method (SHDOM). The authors have investigated the dependence of shortwave and longwave radiative forcing on contrail aspect ratio, optical depth, solar zenith angle, solar azimuth angle relative to contrail orientation, particle size, particle habit, surface albedo, and surface temperature. It is found that inclusion of 3D transport results in an increase in the positive longwave radiative forcing of the contrail and either an increase or a decrease in the magnitude of the negative shortwave radiative forcing depending on the orientation of the contrail with respect to the sun. Although these two effects are individually quite modest (of order 10%), the fact that the total shortwave and longwave forcings largely cancel during the day means that the relative change in the net radiative forcing due to the 3D effect is substantial; in some cases this results in a doubling of the net forcing of the contrail, in other cases changing its sign. On a more general note, the relatively simple geometry of contrail cirrus provides an ideal test case for explaining the various mechanisms by which 3D photon transport can change the radiative effect of clouds, which can be rather difficult to visualize for more complex cloud scenarios.

## Abstract

Spaceborne lidar returns from liquid water clouds contain significant contributions from photons that have experienced many wide-angle multiple-scattering events, resulting in returns appearing to originate from far beyond the end of the cloud. A similar effect occurs for spaceborne millimeter-wave radar observations of deep convective clouds. An efficient method is described for calculating the time-dependent returns from such a medium by splitting the photons into those that have taken a near-direct path out to and back from a single backscattering event (in the case of lidar, accounting for small-angle forward scatterings on the way, as described in Part I of this paper) and those that have experienced wide-angle multiple-scattering events. This paper describes the modeling of the latter using the time-dependent two-stream approximation, which reduces the problem to solving a pair of coupled partial differential equations for the energy of the photons traveling toward and away from the instrument. To determine what fraction of this energy is detected by the receiver, the lateral variance of photon position is modeled by the Ornstein–Fürth formula, in which both the ballistic and diffusive limits of photon behavior are treated; this is considerably more accurate than simple diffusion theory. By assuming that the lateral distribution can be described by a Gaussian, the fraction of photons within the receiver field of view may be calculated. The method performs well in comparison to Monte Carlo calculations (for both radar and lidar) but is much more efficient. This opens the way for multiple scattering to be accounted for in radar and lidar retrieval schemes.

## Abstract

Spaceborne lidar returns from liquid water clouds contain significant contributions from photons that have experienced many wide-angle multiple-scattering events, resulting in returns appearing to originate from far beyond the end of the cloud. A similar effect occurs for spaceborne millimeter-wave radar observations of deep convective clouds. An efficient method is described for calculating the time-dependent returns from such a medium by splitting the photons into those that have taken a near-direct path out to and back from a single backscattering event (in the case of lidar, accounting for small-angle forward scatterings on the way, as described in Part I of this paper) and those that have experienced wide-angle multiple-scattering events. This paper describes the modeling of the latter using the time-dependent two-stream approximation, which reduces the problem to solving a pair of coupled partial differential equations for the energy of the photons traveling toward and away from the instrument. To determine what fraction of this energy is detected by the receiver, the lateral variance of photon position is modeled by the Ornstein–Fürth formula, in which both the ballistic and diffusive limits of photon behavior are treated; this is considerably more accurate than simple diffusion theory. By assuming that the lateral distribution can be described by a Gaussian, the fraction of photons within the receiver field of view may be calculated. The method performs well in comparison to Monte Carlo calculations (for both radar and lidar) but is much more efficient. This opens the way for multiple scattering to be accounted for in radar and lidar retrieval schemes.

## Abstract

Spaceborne millimeter-wave radar has been identified as a possible instrument to make global measurements in ice clouds, which have an important but poorly understood role in the earth’s radiation budget. In this paper, the authors explore the potential of a dual-frequency spaceborne radar to estimate crystal size in cirrus clouds and, hence, determine ice water content and the shortwave extinction coefficient more accurately than would be possible using a single radar. Calculations show that gaseous attenuation is not a serious problem for a nadir-pointing radar measuring down to cirrus altitudes at frequencies between 35 and 215 GHz, provided the frequencies are chosen to lie in the window regions of the atmospheric absorption spectrum. This enables one to exploit the significant benefits of using frequencies too high to be operated from the ground. Radar reflectivity at 35, 79, 94, 140, and 215 GHz has been calculated from aircraft ice particle size spectra obtained during the European Cloud Radiation Experiment (EUCREX) and the Central Equatorial Pacific Experiment (CEPEX), and it is shown that overall the most promising dual-wavelength combination for measuring crystal size and ice water content is 79 and 215 GHz. For a minimum radar sensitivity of −30 dB*Z,* this combination can measure ice water content and median volume diameter with errors of between 10% and 30% when the reflectivity is greater than −15 dB*Z* (equivalent to an ice water content of around 0.015 g m^{−3}). If only a single wavelength radar were affordable, then, for estimating ice water content, 215 GHz would be the preferred choice. Since the two radars would be likely to use the same antenna, the authors also consider the effect of cloud inhomogeneities to introduce a random error into the reflectivity ratio because of the different beamwidths at each frequency. It is found, using data from the cloud radars at Chilbolton, England, that this is more than 0.2 dB for frequency pairings that include 35 GHz but for all other combinations is less than 0.1 dB, which is comparable to the other errors in the system and much smaller than the typical values being measured. Nonspherical crystals are shown to have a significant effect on the size measured by a nadir-pointing dual-wavelength radar, but the authors present evidence that this can be largely eliminated by viewing at 45° from nadir.

## Abstract

Spaceborne millimeter-wave radar has been identified as a possible instrument to make global measurements in ice clouds, which have an important but poorly understood role in the earth’s radiation budget. In this paper, the authors explore the potential of a dual-frequency spaceborne radar to estimate crystal size in cirrus clouds and, hence, determine ice water content and the shortwave extinction coefficient more accurately than would be possible using a single radar. Calculations show that gaseous attenuation is not a serious problem for a nadir-pointing radar measuring down to cirrus altitudes at frequencies between 35 and 215 GHz, provided the frequencies are chosen to lie in the window regions of the atmospheric absorption spectrum. This enables one to exploit the significant benefits of using frequencies too high to be operated from the ground. Radar reflectivity at 35, 79, 94, 140, and 215 GHz has been calculated from aircraft ice particle size spectra obtained during the European Cloud Radiation Experiment (EUCREX) and the Central Equatorial Pacific Experiment (CEPEX), and it is shown that overall the most promising dual-wavelength combination for measuring crystal size and ice water content is 79 and 215 GHz. For a minimum radar sensitivity of −30 dB*Z,* this combination can measure ice water content and median volume diameter with errors of between 10% and 30% when the reflectivity is greater than −15 dB*Z* (equivalent to an ice water content of around 0.015 g m^{−3}). If only a single wavelength radar were affordable, then, for estimating ice water content, 215 GHz would be the preferred choice. Since the two radars would be likely to use the same antenna, the authors also consider the effect of cloud inhomogeneities to introduce a random error into the reflectivity ratio because of the different beamwidths at each frequency. It is found, using data from the cloud radars at Chilbolton, England, that this is more than 0.2 dB for frequency pairings that include 35 GHz but for all other combinations is less than 0.1 dB, which is comparable to the other errors in the system and much smaller than the typical values being measured. Nonspherical crystals are shown to have a significant effect on the size measured by a nadir-pointing dual-wavelength radar, but the authors present evidence that this can be largely eliminated by viewing at 45° from nadir.

## Abstract

Cloud variability on scales smaller than the gridbox size of numerical forecast and climate models is believed to be important in determining the radiative effects of clouds, and increasingly attempts are being made to parameterize these fluctuations in the radiation schemes of current models. In order to calculate the radiative effects of an inhomogeneous cloud, a model needs to know not only the degree of variability within a gridbox but also the degree to which the inhomogeneities in vertically adjacent levels are overlapped. In this paper these two parameters are derived for ice clouds from an 18-month midlatitude 94-GHz cloud radar dataset and parameterized in terms of horizontal gridbox size (*d*), the vertical shear of the horizontal wind (*s*), and the vertical position in the cloud. The vertical decorrelation length Δ*z*
_{0} (i.e., the depth over which the correlation coefficient of either ice water content or optical extinction coefficient in separate vertical levels falls to *e*
^{−1}) is found to be well represented in the mean by log_{10}Δ*z*
_{0} = 0.3 log_{10}
*d* − 0.031*s* − 0.315, where Δ*z*
_{0} and *d* are in kilometers and *s* is in meters per second per kilometer. As expected, higher shear results in more rapid decorrelation, although the rms deviation from this expression is around a factor of 2.5. It is found that the probability distribution of ice water content within a gridbox is usually well represented by a lognormal or gamma distribution. The fractional variance in ice water content (*f*
_{IWC}) may be expressed to within a factor of 2 by log_{10}
*f*
_{IWC} = 0.3 log_{10}
*d* − 0.04*s* − 0.93, valid for *d* < 60 km, above which *f*
_{IWC} is constant with increasing *d.* The expression for the fractional variance of visible extinction coefficient is the same except with the −0.93 term replaced by −0.96. The *s* dependence indicates a tendency for increased shear to result in *decreased* cloud variability. This can be explained by the presence of ice fallstreaks in a sheared flow: a parcel of air in the middle of a cloud is alternately fed from above by ice-rich and ice-poor air, resulting in a homogenization of the layer at a rate dependent on the shear. A more complicated formula is derived to express the dependence of *f*
_{IWC} on the vertical position within the cloud; it is found that fractional variance tends to be largest at cloud top and decreases into the interior before increasing again in the lowest third of the cloud. Thicker clouds tend to have lower fractional variance. No significant dependence on temperature or absolute altitude was found for either *f*
_{IWC} or Δ*z*
_{0}.

## Abstract

Cloud variability on scales smaller than the gridbox size of numerical forecast and climate models is believed to be important in determining the radiative effects of clouds, and increasingly attempts are being made to parameterize these fluctuations in the radiation schemes of current models. In order to calculate the radiative effects of an inhomogeneous cloud, a model needs to know not only the degree of variability within a gridbox but also the degree to which the inhomogeneities in vertically adjacent levels are overlapped. In this paper these two parameters are derived for ice clouds from an 18-month midlatitude 94-GHz cloud radar dataset and parameterized in terms of horizontal gridbox size (*d*), the vertical shear of the horizontal wind (*s*), and the vertical position in the cloud. The vertical decorrelation length Δ*z*
_{0} (i.e., the depth over which the correlation coefficient of either ice water content or optical extinction coefficient in separate vertical levels falls to *e*
^{−1}) is found to be well represented in the mean by log_{10}Δ*z*
_{0} = 0.3 log_{10}
*d* − 0.031*s* − 0.315, where Δ*z*
_{0} and *d* are in kilometers and *s* is in meters per second per kilometer. As expected, higher shear results in more rapid decorrelation, although the rms deviation from this expression is around a factor of 2.5. It is found that the probability distribution of ice water content within a gridbox is usually well represented by a lognormal or gamma distribution. The fractional variance in ice water content (*f*
_{IWC}) may be expressed to within a factor of 2 by log_{10}
*f*
_{IWC} = 0.3 log_{10}
*d* − 0.04*s* − 0.93, valid for *d* < 60 km, above which *f*
_{IWC} is constant with increasing *d.* The expression for the fractional variance of visible extinction coefficient is the same except with the −0.93 term replaced by −0.96. The *s* dependence indicates a tendency for increased shear to result in *decreased* cloud variability. This can be explained by the presence of ice fallstreaks in a sheared flow: a parcel of air in the middle of a cloud is alternately fed from above by ice-rich and ice-poor air, resulting in a homogenization of the layer at a rate dependent on the shear. A more complicated formula is derived to express the dependence of *f*
_{IWC} on the vertical position within the cloud; it is found that fractional variance tends to be largest at cloud top and decreases into the interior before increasing again in the lowest third of the cloud. Thicker clouds tend to have lower fractional variance. No significant dependence on temperature or absolute altitude was found for either *f*
_{IWC} or Δ*z*
_{0}.

## Abstract

This paper presents a new method for representing the important effects of horizontal radiation transport through cloud sides in two-stream radiation schemes. Ordinarily, the radiative transfer equations are discretized separately for the clear and cloudy regions within each model level, but here terms are introduced that represent the exchange of radiation laterally between regions and the resulting coupled equations are solved for each layer. This approach may be taken with both the direct incoming shortwave radiation, which is governed by Beer’s law, and the diffuse shortwave and longwave radiation, governed by the two-stream equations. The rate of lateral exchange is determined by the area of cloud “edge.” The validity of the method is demonstrated by comparing with rigorous 3D radiative transfer calculations in the literature for two cloud types in which the 3D effect is strong, specifically cumulus and aircraft contrails. The 3D effect on shortwave cloud radiative forcing varies between around −25% and around +100%, depending on solar zenith angle. Even with an otherwise very simplistic representation of the cloud, the new scheme exhibits good agreement with the rigorous calculations in the shortwave, opening the way for efficient yet accurate representation of this important effect in climate models.

## Abstract

This paper presents a new method for representing the important effects of horizontal radiation transport through cloud sides in two-stream radiation schemes. Ordinarily, the radiative transfer equations are discretized separately for the clear and cloudy regions within each model level, but here terms are introduced that represent the exchange of radiation laterally between regions and the resulting coupled equations are solved for each layer. This approach may be taken with both the direct incoming shortwave radiation, which is governed by Beer’s law, and the diffuse shortwave and longwave radiation, governed by the two-stream equations. The rate of lateral exchange is determined by the area of cloud “edge.” The validity of the method is demonstrated by comparing with rigorous 3D radiative transfer calculations in the literature for two cloud types in which the 3D effect is strong, specifically cumulus and aircraft contrails. The 3D effect on shortwave cloud radiative forcing varies between around −25% and around +100%, depending on solar zenith angle. Even with an otherwise very simplistic representation of the cloud, the new scheme exhibits good agreement with the rigorous calculations in the shortwave, opening the way for efficient yet accurate representation of this important effect in climate models.

## Abstract

In this paper an equation is derived for the mean backscatter cross section of an ensemble of snowflakes at centimeter and millimeter wavelengths. It uses the Rayleigh–Gans approximation, which has previously been found to be applicable at these wavelengths due to the low density of snow aggregates. Although the internal structure of an individual snowflake is random and unpredictable, the authors find from simulations of the aggregation process that their structure is “self-similar” and can be described by a power law. This enables an analytic expression to be derived for the backscatter cross section of an ensemble of particles as a function of their maximum dimension in the direction of propagation of the radiation, the volume of ice they contain, a variable describing their mean shape, and two variables describing the shape of the power spectrum. The exponent of the power law is found to be −

## Abstract

In this paper an equation is derived for the mean backscatter cross section of an ensemble of snowflakes at centimeter and millimeter wavelengths. It uses the Rayleigh–Gans approximation, which has previously been found to be applicable at these wavelengths due to the low density of snow aggregates. Although the internal structure of an individual snowflake is random and unpredictable, the authors find from simulations of the aggregation process that their structure is “self-similar” and can be described by a power law. This enables an analytic expression to be derived for the backscatter cross section of an ensemble of particles as a function of their maximum dimension in the direction of propagation of the radiation, the volume of ice they contain, a variable describing their mean shape, and two variables describing the shape of the power spectrum. The exponent of the power law is found to be −