1. Introduction
Traditional radar methods of measuring precipitation are based on relations between water equivalent reflectivity factor (hereafter “reflectivity”), Ze, and precipitation rate, R. Even for a simpler case of rainfall, there are significant variations in Ze–R relations due to variability of the drop size distributions. Estimating snowfall parameters from radar measurements is more challenging. Variability in snowflake shapes, densities, and fall velocities results in greater uncertainties in Ze–R relations for snowfall. Notwithstanding these uncertainties, radar gives an opportunity to obtain near-real-time estimates of snowfall rate and accumulation, providing information that is not readily available with other remote sensors.
As for rainfall, snowfall radar estimates are typically obtained with longer wavelength (centimeter) radars such as those that operate at S band (10–11-cm wavelength, λ), C band (λ ∼ 5–6 cm), or X band (λ ∼ 3 cm). Snowfall rate as inferred from radar measurements is usually expressed in melted liquid equivalent, S, rather than in solid snow rate. Two different approaches are used to derive Ze–S relations. In one approach both Ze and S are calculated from theoretical or experimental snowflake size distributions (SSD) and they are related in a statistical manner (e.g., Sekhon and Srivastava 1970; Matrosov 1992). In the second approach, radar measurements of reflectivity are related to the snowfall rate as observed by nearby snow gauges (e.g., Super and Holroyd 1998; Hunter and Holroyd 2002). Rasmussen et al. (2003) provides a review of some Ze–S relations suggested for use with longer-wavelength radars. For such radars, snowflakes are typically much smaller than radar wavelength and scattering does not deviate significantly from the Rayleigh law.
At millimeter radar wavelengths such as those at Ka band (λ ∼ 8 mm) and, especially, W band (λ ∼ 3 mm), scattering by typical snowflakes is generally non-Rayleigh. Radars operating at these wavelengths have been primarily intended for applications in cloud remote sensing and they have typically not been used for snowfall measurements. It has been shown, however, that the use of such radars in pair with centimeter-wavelength radars can potentially improve the accuracy of snowfall parameter retrievals (Matrosov 1998; Liao et al. 2005; Wang et al. 2005) because the dual wavelength reflectivity ratio provides independent information on characteristic size of snowflakes. Knowing size information, one can better infer snowfall rate from single reflectivity measurements.
Currently, a number of vertically pointing millimeter-wavelength radars operate on a continuing basis in many locations including those that regularly or sometimes get snowfalls. Examples of such radars are vertically pointing Ka-band (34.6 GHz) millimeter-wavelength cloud radars (MMCR) operated by the U.S. Department of Energy’s Atmospheric Radiation Measurement Program (DOE ARM) in Oklahoma and the North Slope of Alaska (NSA). Another MMCR-type radar has been recently deployed in the Canadian Arctic. The W-band (94 GHz) nadir-looking CloudSat radar will provide global information on reflectivity profiles of hydrometeors (Stephens et al. 2002). Although the main objective of all these radars is cloud remote sensing, their measurements of snowfall can also be interpreted quantitatively, thus providing valuable information on snowfall accumulation and the vertical structure of snowfall rate, which can vary significantly (Mitchell 1988).
Since for most of the radars mentioned above there are no collocated centimeter-wavelength radars for application of dual-wavelength ratio techniques, Ze–S relations specifically tailored for millimeter wavelengths are needed for quantitative interpretation of snowfall measurements. This article presents results of modeling Ze–S relations for the common millimeter-wavelength frequencies of 34.6 and 94 GHz based on available information on snowflake-size distributions and densities. The first of the aforementioned approaches for deriving Ze–S relations is employed here because there are no reliable datasets available that combine collocated millimeter-wavelength radar and snow gauge measurements. This modeling is concerned with so called “dry” snow, the term usually used for describing snow with negligible amounts of liquid water/riming. “Dry” snowfalls usually contribute most significantly to snow accumulation at temperatures less than about −8°C.
2. Snowflake model
a. Snowflake shape
The simplest geometrical shape that allows describing nonspherical hydrometeors is that of spheroid. In their classical observational in situ study, Magono and Nakamura (1965) used tracing outlines of snowflakes that revealed a generally spheroidal shape. The oblate spheroidal model is used widely in the radar community to model raindrops and it was also shown to satisfactorily describe and explain depolarization observations of snowflakes at Ka band (Matrosov et al. 1996, 2001). Note also that, at radar frequencies, the small details in a particle structure usually do not significantly affect the backscatter properties that depend largely on the overall shape (Dungey and Bohren 1993), which, in the case of spheroid, is determined by the spheroid aspect ratio, r. This is especially true for smaller snowflake size parameters x = πDλ−1 (Schneider and Stephens 1995). The applicability of the spheroid model for larger snowflakes with small values of r (e.g., dendrites that are larger than several millimeters) at W band is not so obvious. This model, however, satisfactorily explains experimental 94-GHz aggregate snow reflectivities (Matrosov et al. 2005a).
Based on the arguments given above, and also due to the fact that very large dendrites are not very common, the oblate spheroidal model was adopted here for snowflake modeling. Another reason to use this model is the lack of detailed shape information on aggregates, which makes the input information for the computational models that are able to account for fine shape details (e.g., the discrete dipole approximation; Lemke and Quante 1999) not very well defined. In the spheroid model, low aspect ratio values of about 0.1–0.3 are suitable for modeling pristine dendrite snow crystals, while values of r ≈ 0.5–0.7 are usually observed for larger aggregate crystals (Magono and Nakamura 1965; Korolev and Isaac 2003). The value around r ≈ 0.6 also best explains experimental dual-wavelength ratios from the airborne radars operating at 9.6 and 94 GHz (Matrosov et al. 2005a).
b. Snowflake orientation
Magono and Nakamura (1965) noted that, due to aerodynamic forcing, snowflakes typically fall with their major dimensions oriented horizontally. Polarimetric radar depolarization measurements at different polarimetric bases and at different elevation angles also reveal that, in the absence of strong horizontal winds (and wind shear), single shape snowflakes are oriented preferably horizontally with a standard deviation (SD) of about 9° (Matrosov et al. 2005b). Thus it was assumed during modeling that symmetry axes of oblate spheroids that approximate snowflakes are oriented with the 0° mean and SD = 9° with respect to vertical. While there is no polarization dependence of radar measurables for vertically pointing radar for such snowflake orientations, snowflake shape and, to a lesser extent, the value of SD influence the magnitude of Ze at millimeter-wavelength frequencies.
c. Snowflake mass
Equation (3c) reflects the fact that the density of very large snowflakes does not exhibit any significant dependence on size.
d. Snowflake fall velocity
Many theoretical (e.g., Mitchell and Heymsfield 2005) and experimental (e.g., Magono and Nakamura 1965) studies indicate that terminal fall velocities of larger snowflakes practically do not depend upon the size of the flake. Taking this into consideration, it was assumed that (6) describes aggregate snowflake fall velocities in the range 0.01 cm < D < 1 cm and the constant value of υt was then assumed for D ≥ 1 cm.
3. Backscatter cross sections of individual snowflakes
Figure 1 shows backscatter cross sections, σb, of horizontally oriented aggregate snowflakes with aspect ratios that are in the range 0.6 ≤ r ≤ 1 when viewed at vertical incidence. The mass of snowflakes was modeled according to (3) and was preserved as r was changing. This means that effective bulk densities are different for snowflakes with the same size but different aspect ratios. Though Mie-type backscatter oscillations are present for all considered aspect ratios, as can be seen from Fig. 1, backscatter cross sections of nonspherical snowflakes already begin to deviate noticeably from the spherical model already at major dimensions of about D = 0.15 cm for ν = 34.6 GHz, and D = 0.05 cm for ν = 94 GHz. This indicates that, for at least vertical-viewing geometry and millimeter radar wavelengths, the assumption that the backscatter cross section of an arbitrary snowflake would be the same as that of a sphere of ice having the same mass is not generally valid. This assumption, which is often used when calculating radar properties of snowflakes is, however, valid for lower radar frequencies in the Rayleigh scattering regime.
While, as mentioned above, r ≈ 0.6 is expected to adequately describe natural snowflake aggregates, it is also instructive to examine backscatter cross sections for particles with smaller aspect ratios. Figure 2 presents results for 0.2 ≤ r ≤ 0.5. Note that the T-matrix calculation routine often becomes unstable for particles with aspect ratios smaller than about 0.2, so no results are presented for r < 0.2. It can be seen from Fig. 2 that the dependence of backscatter cross sections σb on r becomes progressively less pronounced as r decreases. This is true especially for smaller particles.
Figure 3 shows backscatter cross sections for single pristine dendritic crystals (P1d) as a function of their size. Though the calculations presented in this figure were made for r = 0.2 they are expected also to be representative for smaller values of r because of the lesser sensitivity of σb to r at smaller values of aspect ratios and also because pristine crystals usually are smaller than aggregates and they are not expected to exceed a few millimeters in size. Note also that because the effective bulk density of snowflakes changes with their size, the results presented in Figs. 1 –3 for one frequency cannot be obtained exactly from the results for the other frequency using simple frequency scaling.
4. Reflectivity of ensembles of snowflakes
a. Snowflake size distributions
One of the most complete experimental studies of snowflake size spectra in terms of snowflake real sizes was presented by Braham (1990) who reported data that contained 49 experimental SSDs. Figure 4 shows these SSDs approximated by the exponential function with parameters No and Λ from Braham (1990). The majority of snowflakes in Braham’s study appeared to be aggregate snowflakes with some amount of single spatial dendrites. Heavier snowfalls are characterized by smaller values of the parameter Λ. Later experimental studies (e.g., Harimaya et al. 2000) showed experimental SSD features similar to those reported by Braham (1990).
b. Ze–S relations at millimeter wavelengths
Figure 5 shows the scatterplots of radar reflectivity versus liquid equivalent snowfall rate for frequencies 34.6 and 94 GHz calculated for the experimental SSDs with parameters from Fig. 4. Aggregate snowflake aspect ratio was assumed to be 0.6 and the mass and terminal fall velocities were calculated using (3) and (6). The best power-law fits drawn with Ze as an independent variable for the aggregate snowflake assumption are Ze = 56S1.20 (34.6 GHz) and Ze = 10S0.8 (94 GHz), where Ze in mm6 m−3and S in mm h−1. Note that first the S–Ze relations were developed considering Ze as an independent variable and then these relations were converted to the conventional Ze–S form. The data scatter around the best-fit lines is rather modest. Due to stronger non-Rayleigh scattering effects, reflectivities at 94 GHz are significantly smaller than those at 34.6 GHz for the same values of snowfall rate S. The millimeter wavelength Ze–S relations are quite different from those obtained by various researches for longer wavelength radars (Rasmussen et al. 2003).
While it is expected that aggregate snowflakes are the dominant hydrometeor type in appreciable snowfalls, for comparison reasons, calculations were also performed using the dendritic crystal model and the same experimental SSDs. The mass–size and fall velocity–size relations in this case are given by (5) and (7). The results for which dendrite reflectivities were greater than 0 dBZ are also shown in Fig. 5. For 34.6 GHz the aggregate best fit also rather well describes the dendrite data. It is not quite so for the 94-GHz data, although deviations of the dendrite data from the best power-law fit for aggregates are still not very significant. Reflectivities that are less than about 0 dBZ are more characteristic of nonprecipitating clouds and were not considered here.
c. Variability of Ze–S relations due to assumptions
The main assumptions affecting radar reflectivity values, Ze, include those about the particle mass and shape. While in the Rayleigh scattering regime, backscatter cross sections of individual particles are proportional to the square of particle masses, the resonance effects diminish the backscatter–mass dependence. The shape effects, which are not very pronounced for small particles, however, become progressively more important (especially for vertical incidence at horizontally oriented scatterers) as particles grow larger and transition to the sizes that are outside the Rayleigh scattering regime. The assumption about particle mass also affects the liquid equivalent snowfall rates, S. The values of S are also affected by the assumption about the terminal fall velocity–size relation.
The aggregate particle mass–size relation given by (3) and the terminal fall velocity–size relation given by (6) and an assumption about the r = 0.6 aspect ratio were used to produce the data presented in Fig. 5. Table 1 shows how the best fit power-law Ze–S relations would vary if one of the main assumptions is changed. To assess this variability the particle mass and fall velocity were changed by 20% compared to values from Eqs. (3) and (6), respectively, and the particle aspect ratio was changed from 0.6 to 0.8. A more random orientation of snowflakes will be manifested similarly as an increase in r. These changes in the main assumptions may represent the reasonable uncertainties of corresponding assumptions. Indeed, the fall velocity approach of Mitchell and Heymsfield (2005), for example, provides fall velocities of 1 cm aggregates (which is a typical aggregated snowflake size) of about 1 m s−1. The Khvorostyanov and Curry (2002) approach provides 1.4 m s−1 for the same particle size. This is about ±20% from their mean value of 1.2 m s−1. Two different mass–size relations for aggregates from Mitchell (1996) provide similar percentage differences if applied to larger aggregate sizes.
It can be seen from Table 1 that there is an appreciable sensitivity of the Ze–S relations to main assumptions. Assuming the independence of the assumption uncertainties, one can expect variability of at least a factor of 2 in the coefficients of the Ze–S relations. Variability in the exponents of these relations is more modest, though it also contributes to the overall uncertainty in the correspondence between Ze and S. It should be noted also that, compared to the low observational angles, the vertical radar beam observations (which is the most common configuration for Ka- and W-band radars) and the preferable orientation of snowflakes with their major dimension in the horizontal plane result in larger variability of backscatter due to changes in the particle aspect ratio.
5. Attenuation of radar signals in snowfall
Figure 6 shows scatterplots of the attenuation coefficients at −5°C versus liquid-equivalent snowfall rates at Ka- and W-band radar frequencies calculated for the assumed snowflake aspect ratio r = 0.6 and the same experimental SSDs that were used in reflectivity modeling. Complex refractive indices of solid ice corresponding to a temperature of −5°C are about 1.78 −i0.0024 (34.6 GHz) and 1.78 −i0.0043 (94 GHz).
The best-fit power law approximations for the data in Fig. 6 are a = 0.011S1.1 and a = 0.12S1.1 for 34.6 and 94 GHz, respectively. It can be seen that the dry snow attenuation at Ka band is fairly small and can be neglected for most practical cases. The attenuation at W band, however, is about one order of magnitude higher than that at Ka band and, probably, needs to be accounted for when measuring snowfall with higher precipitation rates in thick snow layers. The total attenuation in such cases can be appreciable and reach or even exceed 1 dB, which is on the order of a typical absolute calibration uncertainty of radars. For wet and/or rimed snow, attenuation can be significantly larger.
It should be noted also that the attenuation in the atmospheric gases (i.e., water vapor and oxygen) generally should be accounted for when interpreting measurements at W band at longer ranges (Stephens et al. 2002). Gases attenuation at Ka band at vertical incidence that typically amounts to fractions of 1 dB often can be neglected especially in dry cold atmospheres.
6. Summary and conclusions
Millimeter-wavelength radars operating at Ka and W bands are being increasingly used in the atmospheric remote sensing. These radars are typically employed either in a vertical-pointing mode (for ground based radars) or in a nadir-pointing mode (for airborne and spaceborne radars). A number of ground-based millimeter wavelength cloud profiling radars have been deployed in the colder climates and spaceborne radars are usually designed to operate at polar orbits, thus providing frequent coverage of high latitudes where snowfall is common. Although these radars’ primary use is remote sensing of nonprecipitating clouds, they can also potentially be used for snowfall measurements. Adding quantitative snowfall information derived from radar measurements will be useful in many climate and cloud physics applications.
In this study, modeling of reflectivity–liquid-equivalent snowfall rate was performed for the common frequencies employed at Ka band (∼35 GHz) and W band (∼94 GHz). “Dry” snowfall, which is characterized by negligible amounts of liquid water in snow and snowflake riming, was studied. Nonspherical aggregate and single crystal dendritic snowflake models were used for calculations. These models included realistic mass–size and terminal fall velocity–size relations and assumed some flutter around the preferred orientation of snowflakes with their maximum dimensions in the horizontal plane. Experimental size distributions describing real (not melted) snowflake sizes were employed in this study.
Most common aggregate snowflake sizes are already outside the Rayleigh scattering regime at Ka- and especially at W-band frequencies. For vertical/nadir-pointing geometry, backscatter properties of snowflakes noticeably depend on snowflake overall shape (i.e., aspect ratio). The typical radar reflectivity–snowfall rate relations for aggregate snowflakes calculated for a set of basic assumptions (see Table 1) are Ze = 561.20 (34.6 GHz) and Ze = 10S0.8 (94 GHz). These relations are generally valid also for snowfall consisting of single dendrite crystals (P1d), though snowfall rates and thus accumulations of such snowfalls are usually significantly less compared to the snowfalls consisting of aggregates. Though these relations were developed for “dry” snowfall, they are expected to be applicable for snowfall with some light degree of riming. Dry snow typically forms at colder temperatures (usually less than about −12°C) and is characterized by mean vertical Doppler velocities that are less than about 1.2–1.4 m s−1. A significant increase in snowflake riming and/or wetness is likely to cause changes in Ze–S relations.
The strong non-Rayleigh behavior of backscatter at millimeter wavelengths in Ka and W bands results in reflectivity being proportional to the lower moment of the snowflake size distribution. As a result the exponents in the Ze–S relations are closer to unity compared to the longer wavelength radars. The proportionality of Ze and S to similar moments of the size distribution can, in part, explain small data scatter in Fig. 5. For precipitation radar frequencies (3–10 GHz), both coefficients and exponents in Ze–S relations are larger. Typically for such frequencies, coefficients vary from 50 to 400 while the exponents vary between 1.5 and 2 (e.g., Rasmussen et al. 2003).
Uncertainties in the coefficients of Ze–S relations caused by variability in the snowflake shape, and in the parameters of mass–size and terminal fall velocity–size relations are rather substantial. These coefficient uncertainties can easily be as high as a factor of ∼2; thus, the errors in the snowfall rate and accumulation estimations for radar reflectivity measurements could be appreciable. These errors can potentially be reduced if the coefficient tuning is performed based on collocated radar and snow gauge measurements. Such a tuning approach is often used in rainfall measurements when parameters of standard Ze–R (R here is rainfall rate) relations are adjusted based on the rain gauge measurements that are deployed in the radar coverage area. There are, however, no high quality collocated datasets for millimeter-wavelength radars and snow gauge data available at the moment, so this direction for refinement of Ze–S relations should be left for further studies.
Acknowledgments
This research was supported by the Office of Science (Biological and Environmental Research), U.S. Department of Energy, Grant DE-FG02-05ER63954, and the CloudSat project.
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Backscatter cross sections of horizontally oriented aggregate snowflakes with aspect ratios 0.6, 0.8, and 1.0 at vertical incidence and frequencies (a) 34.6 and (b) 94 GHz as a function of snowflake major dimension.
Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1
As in Fig. 1 but for aspect ratios 0.2, 0.3, and 0.5.
Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1
Backscatter cross sections of horizontally oriented fordendritic crystals (P1d) as a function of crystal major dimension.
Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1
Scatterplot of the intercept and slope parameters in experimental snowflake size distributions.
Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1
Scatterplots of radar reflectivity at vertical incidence and snowfall rate at frequencies (a) 34.6 and (b) 94 GHz for experimental snowflake size distributions.
Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1
Scatterplots of the attenuation coefficient at vertical incidence and snowfall rate for experimental snowflake size distributions.
Citation: Journal of the Atmospheric Sciences 64, 5; 10.1175/JAS3904.1
Variability of Ze–S (Ze in mm6 m−3, S in mm h−1) relations due to main assumptions about snowflake mass m, terminal fall velocity υt, and aspect ratio r.
