1. Introduction

Knowledge of the raindrop size distribution (DSD) is of fundamental interest to the atmospheric sciences community in relation to modeling studies and radar-rainfall estimation. Modelers seek realistic DSD representation in microphysical parameterizations, where DSD is critical for latent heating, radiative transfer, and quantitative precipitation estimation. Radar meteorologists require knowledge of the DSD to relate radar reflectivity Z to rainfall rate R. Because there is not a one-to-one correspondence between the two quantities, a wide range of Z–R relations have appeared in the literature (Battan 1973). For example, different relations are often prescribed either for different rainfall processes, for example, stratiform versus convective (Tokay and Short 1996), or in different climate regimes, for example, continental versus maritime (Bringi et al. 2003), owing to the different microphysical processes that contribute to DSD evolution.

Most of the earth’s rain falls over the oceans, especially in the tropics. For studies of the global hydrologic cycle, it is therefore critical that rainfall measurements be made here. Since its launch in November 1997, the National Aeronautics and Space Administration (NASA) Tropical Rainfall Measuring Mission (TRMM) satellite estimates rainfall at tropical and subtropical latitudes, employing a 13.8-GHz precipitation radar (PR) and the TRMM Microwave Imager (TMI; Kummerow et al. 1998). However, because the PR operates at a single frequency, it is subject to errors resulting from the regional variability of DSD (Berg et al. 2006). The upcoming Global Precipitation Measurement (GPM) mission will improve upon TRMM by employing a dual-frequency precipitation radar (Iguchi et al. 2003). The dual-frequency radar offers an additional measurement with the potential to reduce the uncertainties in DSD estimation that are inherent with single-frequency radar. In addition, microwave radiometers on the core satellite and constellation satellites will provide a global precipitation measuring platform with high temporal frequency and spatial coverage.

The dual-frequency radar is one of several multiparameter radar technologies developed to estimate DSD more precisely. Other such technologies include polarimetric radar measurements (Bringi et al. 2002), vertically pointed radars (Williams 2002), and a microwave link (Rincon and Lang 2002). Of these, dual-frequency radar is the most suitable for satellite-based observations (Kuo et al. 2004). The choice of frequencies is subject to power and antenna size requirements as well as attenuation (Iguchi et al. 2003). The frequencies should also be sufficiently separated so that reflectivity measurements have different dependencies on the DSD (Meagher and Haddad 2006). In consideration of these requirements, Ku- (13.6 GHz) and Ka-band (35 GHz) frequencies have been designated for the GPM core satellite.

Previous studies involving dual-frequency radar have attempted to improve upon single-frequency Z–R relations through a variety of techniques. Eccles and Mueller (1971) employed 3- and 10-GHz frequencies to estimate liquid water content via differential attenuation, which is more closely related to rain rate than reflectivity. Goldhirsh and Katz (1974) advanced dual-frequency techniques by deriving two parameters of an exponential DSD function directly from the two radar measurements, although their method was constrained by the assumption of constant rain rate over a path. Meneghini et al. (1992) pioneered the use of downward-looking airborne dual-frequency measurements of reflectivity and attenuation to formulate the retrieval of DSD from spaceborne radar measurements. A crucial element of this study was the use of frequencies where non-Rayleigh scattering allowed the retrieval of DSD parameters from attenuation-corrected reflectivities.

In general, the availability of two independent reflectivity measurements allows for the retrieval of only two parameters describing the DSD. This naturally poses questions of how many parameters are sufficient to describe the range of DSDs observed in nature, and how to optimally model the DSD with only two independent parameters. These questions have been the subject of extensive study since DSD measurements have been available. Although there has been a recent study in Bayesian techniques where no a priori distribution is prescribed (Haddad et al. 2006), the three-parameter gamma distribution is mostly employed to retrieve the DSD from reflectivity and/or attenuation measurements. In this study we use a form of the gamma distribution expressed as

where N₀ is the intercept parameter, Λ is the slope parameter, and m is the shape parameter. The exponential distribution is a special form of the gamma distribution for which the shape parameter is equal to zero. In an exponential distribution, the drop concentration per unit diameter N(D) increases with decreasing diameter at a rate that is determined by the slope parameter. Marshall and Palmer (1948) introduced a special form of the exponential size distribution where N₀ is constant and Λ is a function of rain rate. Since their pioneering study, both surface and airborne measurements of the DSD have illustrated that the shape of the spectra deviates from the exponential distribution at small and large drop sizes. There is often some degree of curvature around a peak concentration near 1-mm diameter. The gamma distribution with a positive shape parameter may accurately represent this curvature as shown in Tokay and Short (1996).

When dual-frequency measurements are applied to the three-parameter gamma distribution, the lack of a third radar measurement introduces a challenge, and the number of independent parameters must be reduced. This reduction can be achieved either by assuming a value for one of the parameters, or by specifying relationships among some of the parameters. Indeed, there have been several studies in regard to empirical relations between the gamma parameters. Haddad et al. (1996) argued that the parameters given in Eq. (1) are not independent and introduced a new set of three parameters. Ulbrich (1983) illustrated that the shape and intercept parameters of the gamma distribution can be related. Zhang et al. (2001) and Brandes et al. (2003) introduced a constrained gamma distribution where a relationship between the slope and shape parameters was specified for use in polarimetric radar-rainfall retrievals. Although high correlation coefficients in these studies confirm the existence of such relations between the gamma parameters, the relations do not necessarily hold at all rain intensities and for all precipitation events.

In this study, we seek to develop optimal constraints on the three-parameter gamma distribution that allow for accurate DSD retrieval from dual-frequency reflectivity measurements. The framework for these measurements is described in section 2. In section 3, we describe the disdrometer datasets we employed to develop the constraints. The gamma parameters are fit to these measurements by a process described in section 4, and the results of this process are given in section 5. Relationships between the shape and slope parameter from these fits are derived in section 6 and evaluated in section 7. In the final section, we offer both a summary and concluding remarks.

2. Dual-frequency radar framework

Numerous algorithms (e.g., Meneghini et al. 1992; Mardiana et al. 2004; Iguchi 2005; Liao and Meneghini 2005) have been proposed to retrieve rainfall estimates from dual-frequency radar measurements. These algorithms aim to estimate the DSD at each range gate once the attenuation correction has been applied from previous range gates. In this study, we focus on DSD estimation and assume that the reflectivity measurements have been corrected for attenuation. These measurements of effective reflectivity Z_e are the integral product of backscattering cross section σ_b and the DSD. In general, Z_e is expressed as

where λ is the radar wavelength and |K|² is the dielectric factor, which is related to the complex index of refraction of the target and is sensitive to its temperature and the radar frequency. The backscattering cross section is a function of both the size and temperature of the drop and the radar frequency. The general solution to the backscattering cross section of a homogeneous sphere interacting with an electromagnetic plane wave is given by Mie theory. For drops much smaller than the radar wavelength, where the size parameter (πD/λ) is much less than 1, the backscattering cross section is well approximated by Rayleigh theory. At 13.6 and 35 GHz, however, the size parameter nears or exceeds 1 for drops larger than about 1 mm in diameter. At these drop sizes, the full Mie theory must be used to calculate the backscattering cross section at these two frequencies, which diverges from Rayleigh scattering (Fig. 1). As a result, the effective reflectivity of a DSD containing these drops is frequency dependent, a property that is exploited in the dual-frequency radar retrieval technique described in this study.

The DSD dependence of Z_e can be shown by substituting the gamma distribution [Eq. (1)] into Eq. (2),

The difference between the reflectivity measurements is described by the dual-frequency ratio (DFR). This quantity is independent of the intercept parameter and is defined as

The DFR may be examined in shape parameter–slope parameter space (Fig. 2) where both gamma parameters are bound by the limits suggested by Tokay and Short (1996), who found that the slope parameter ranged between 1 and 20, while the shape parameter remained between −2 and 30. This upper limit on the shape parameter was lowered to 20 because subsequent research has shown that high shape parameters are a nonphysical product of biases in the method of moments (Smith and Kliche 2005). For a given DFR, an infinite number of solutions exists for the shape and slope parameters within these bounds. To reduce this set to a single solution, a constraint, such as a constant shape parameter (Liao and Meneghini 2005) or a relationship between the shape and slope parameters (Brandes et al. 2003), must be imposed, even though DSDs that deviate from these constraints could lead to retrieval errors.

Where the DFR is less than 1, the contours in Fig. 2 are hyperbolic curves. This characteristic represents a well-known ambiguity (Meagher and Haddad 2006) that is an obstacle for dual-frequency radar DSD retrievals. This ambiguity manifests itself in dual solutions for the slope parameter when a constraint of constant shape parameter is given. This is because the DFR of a single drop does not increase monotonically with drop size, but decreases from 1 at the limit of a small-drop diameter to a minimum of 0.53 at 1.8-mm diameter before increasing to values greater than 1 at larger drop diameters (Fig. 1). Thus, a DSD consisting of drops smaller than 1.8 mm may have the same DFR as a DSD including larger drops. This ambiguity can be reduced to some extent because the distribution containing only the small drops will, in all likelihood, have a lower reflectivity than the DSD with both small and large drops (Liao and Meneghini 2005), but then it is necessary to prescribe a threshold reflectivity at which the solutions switch over.

To further interpret Fig. 2, it is useful to draw upon relations between the gamma parameters and physical DSD parameters. The mean mass diameter (D_m) is the ratio of the fourth to third moments of the DSD and is related to the slope parameter of the gamma distribution (Ulbrich and Atlas 1998):

Many dual-frequency DSD retrieval studies (e.g., Meneghini et al. 1997; Kuo et al. 2004; Liao and Meneghini 2005) cite the median volume diameter D₀ instead, which is closely related to D_m:

The relationship between DFR and mean mass diameter is illustrated in Fig. 2. The D_m contours are nearly parallel to the DFR contours except at the apex of the DFR contours, indicating that DFR is a good proxy for D_m in most of the shape and slope parameter space. In general, mean mass diameter decreases with increasing slope parameter and decreasing shape parameter. This means that the smallest mean mass diameter is associated with a nearly exponential distribution with a steep slope. Conversely, the largest mean mass diameter is associated with a downward-curved spectrum where the slope is shallow, allowing the peak concentration to be at relatively large drop sizes.

When the peak concentration is at 1–2-mm diameter, the DFR is at a minimum because the lowest dual-frequency ratio is at these sizes (Fig. 1). The value of this minimum depends on the width of the distribution, which is related to the shape parameter. As shape parameter increases, the distribution becomes narrower and the minimum DFR decreases (Fig. 2). Therefore, the DFR provides a lower bound on the shape parameter, although this information is only useful when the limit is higher than a physically realistic lower bound of −2 (Ulbrich 1983). Likewise, the physically realistic upper bound of 20 on a shape parameter will give no solution for a slope parameter with an extremely low (<0.6) DFR measurement.

In summary, parameters of the gamma distribution are related to drop concentration (intercept parameter), the proportionality of large and small drops (slope parameter), and the variance of drop size (shape parameter). The DFR is a function of the latter two physical quantities, but with only two reflectivity measurements, an additional constraint is needed on the gamma distribution to provide a solution for all three parameters. For global rainfall measurements, it is critical that these assumptions are physically realistic and accommodate the wide variety of DSDs observed in nature. In this study, we employ disdrometer measurements to develop constraints that are optimized for dual-frequency radar DSD retrieval algorithms.

3. Datasets and instrumentation

The DSD measurements used in this study were obtained at various NASA TRMM satellite validation sites and from field campaigns related to TRMM (Table 1). These sites represent a variety of precipitation regimes, allowing examination of the sensitivity of our proposed dual-frequency radar DSD retrieval algorithm to regional variations in drop spectra. The datasets were classified into nine regions based on previous studies of DSD characteristics. The observations from two different islands of the Kwajalein atoll, located in the central tropical Pacific Ocean (CTP), contained similar DSD features. These islands are located near the intertropical convergence zone (Houze et al. 2004), and are thus representative of oceanic precipitation in the deep Tropics. Likewise, datasets from three consecutive summers showed nearly identical characteristics in deep convective rainfall in the Florida Keys (FK; Tokay et al. 2003a), and they were also merged. In contrast, the data from Brazil and Darwin, Australia, showed different characteristics depending on wind regime. The rainfall in the easterly wind regime in Brazil was found to be similar to rain during the break period in Darwin (Tokay et al. 2002); thus, these datasets are combined for our analysis (BE). However, the Brazil westerly regime (BW) and Darwin monsoon period (DM) DSDs were distinct and are treated separately in this study. Data from Wallops Island, Virginia (Tokay et al. 2005), were classified by storm type via examination of archive radar imagery. The resulting datasets represent precipitation from tropical cyclones (WT) and cellular convection (WC). We also include DSD measurements from the western tropical Pacific warm pool during the Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Experiment (TOGA COARE; Tokay and Short 1996) and the South China Sea Monsoon Experiment (SCSMEX) as stand-alone datasets. These final two regions are designated as TC and SCS, respectively.

Prior to deriving DSD parameters and bulk descriptors of rainfall from DSD observations, it is essential to be aware of the limitations of the instrument that was used to obtain them, in this study the Joss–Waldvogel disdrometer (JWD; Joss and Waldvogel (1967). The JWD is an impact-type disdrometer that transforms the vertical momentum of a raindrop impacting the 50 cm² sampling area into an electric pulse of which the amplitude is a function of the drop diameter. Drop diameters ranging from 0.3 to 5.3 mm are detected in 127 bins, which are aggregated by the disdrometer software into 20 bins so that a sufficient number of drops are present in each bin during a 1-min sampling period to represent the DSD at a wide range of rainfall intensities.

Like any other disdrometer, the JWD has shortcomings. These include dead time, which hinders the instrument’s ability to detect more than one drop at a time, and background noise, which limits the detection of very small drops. The small disdrometer sampling area is a compromise between sampling a sufficient number of drops at low rainfall rates while minimizing the dead-time problem at high rainfall rates. The assumption of terminal fall speed is necessary to determine drop concentrations per unit volume from a surface-based measurement, but drop velocities can diverge from their theoretical terminal fall speed in stagnant air, resulting in under- or overestimates of drop diameter (Salles and Creutin 2003). Also, the impact method cannot distinguish the size of drops larger than 5.3 mm in diameter because of the insignificant increment in terminal fall speeds beyond this size.

These errors in the JWD measurements may lead to biases in derived quantities. For example, reflectivity is particularly sensitive to large-drop concentrations and the truncation at 5.3 mm could result in a negative reflectivity bias in heavy rainfall. To overcome these limitations, data from optical disdrometers (e.g., Hauser et al. 1984; Löffler-Mang and Joss 2000; Kruger and Krajewski 2002; Barthazy et al. 2004), which measure the size and velocity of hydrometeors, have been compared with JWD measurements. Tokay et al. (2001) compared JWD-derived reflectivity, rain rate, and gamma parameters with the same quantities derived from video disdrometer (2DVD) measurements, which have the ability to measure drop sizes beyond 5.3 mm. The 2DVD also measured more small drops than the JWD in heavy rainfall as a result of the noise problem, but the impact on calculated rain rate was within 3%, and calculated reflectivities were within 1 dB. The ability of both JWD and 2DVD disdrometers to represent the reflectivity from the much larger sampling volume of radar profilers has been shown by Gage et al. (2000), who found good agreement between the two in stratiform rain.

Another significant issue is the sensitivity of derived gamma parameters to instrument errors. Tokay et al. (2001) showed that the gamma parameters derived from the average coincident spectra from both disdrometers were nearly identical, although subsets with fewer samples indicated a positive bias in the parameters derived from the 2DVD. While instruments such as the 2DVD promise to improve DSD measurements in the future, the current widespread deployment of the JWD in field experiments (Tokay et al. 2003b) makes it well suited to compare long-term records of DSD characteristics in different climate regimes.

The DSD, which is expressed in units of concentration per diameter interval in each bin (m⁻³ mm⁻¹), is derived by normalizing the raw disdrometer drop counts in each bin (C_i),

where N_i is the drop concentration in the ith bin (m⁻³ mm⁻¹), τ is the time period (60 s) over which drops were counted, α is the disdrometer sampling area, υ_t,i is the terminal velocity of the midsize drop in the ith bin (m s⁻¹), following Beard (1976), and ΔD_i is the width of the ith bin (mm).

4. Analysis procedure

The goal of this study is to develop constraints on the three-parameter gamma distribution that results in the optimal retrieval of rain rate, liquid water content, and mean mass diameter from disdrometer-calculated dual-frequency radar measurements. To achieve this goal, we first address the potential sampling problems of the JWD, which can lead to biases in the parameters derived from the observations (Smith et al. 1993; Smith and Kliche 2005). To reduce errors related to undersampling, some form of averaging is usually performed on the disdrometer observations. Lee and Zawadzki (2005) have advocated compositing DSDs over reflectivity intervals, rather than averaging over time intervals or random samples, because reflectivity-based composites result in the most consistent Z–R relations when different regression techniques are used. However, it is possible that different rainfall physics may occur at a given reflectivity interval, and in these cases the average spectra cannot represent the underlying true variability. An analysis of the variability of the spectra in each reflectivity interval is given in appendix A and shows that there is generally more variability at reflectivities below 40 dBZ_13.6, where both convective and stratiform processes are common, and also in the continental regions. This variability should be taken into account when considering the parameters derived from the composite DSDs.

The composites were made for each region in 2-dBZ_13.6 intervals from 10 to 60 dBZ_13.6. This bin width resulted in sample sizes on the order of 100–1000 for most of the composite spectra. However, at the high end of this reflectivity range, fewer samples were available for inclusion in each composite, raising the question of whether or not these composites were truly representative of the region. Thus, a cutoff was established by examining the fitted gamma parameters for consistency from one reflectivity interval to the next. A minimum of 20 samples was found to be necessary under this criterion, corresponding to maximum reflectivities between 42 and 56 dBZ_13.6 for each region.

The ability of the composites to represent regional differences in DSD was evaluated by examining the spectra at each reflectivity interval. The composites from different datasets within the same climate region tended to form distinct clusters when compared with other regions, especially at reflectivities above 35 dBZ_13.6. This demonstrates that the differences between composites from different regions are significant relative to the differences within a region.

To develop constraints on the gamma distribution from these composites, the gamma parameters are fitted to each composite DSD using a method that incorporates the simulated dual-frequency radar measurements. The success of this or any other fitting algorithm depends on the degree to which the data resemble a gamma distribution. If there are significant deviations, such as multiple modes or inflection points, then some information will be lost in the fitted distribution. For applications to dual-frequency radar retrievals, the information contained in the DFR is primarily representative of the shape of the distribution at large drop sizes. This is because the reflectivities are approximately sixth-moment quantities and are thus heavily influenced by the concentration of larger drops. With these radar measurements, we desire to retrieve a DSD that also accurately reproduces integral rainfall quantities, which are related to the third and fourth moments. Therefore, the gamma distributions we derive should reflect the shape of the distribution at larger drop sizes, even if this comes at the expense of accurately representing the concentration of small drops.

The fitting procedure (Fig. 3) begins with the calculation of reflectivity at 13.6 and 35 GHz from a discrete version of Eq. (2), where the numeric integration is performed from the minimum to maximum diameter. We also calculate integral rainfall quantities—rain rate R (mm h⁻¹), liquid water content W (g m⁻³), and mean mass diameter D_m (mm)—for each composite DSD using Eqs. (8)–(10):

where η is the number of bins (20, for our datasets), σ_b is the backscattering cross section (mm²) calculated for each frequency at a temperature of 20°C, D_i is the drop diameter (mm) at the middle of the ith bin, υ_t is the terminal velocity (m s⁻¹) as described in section 3, N_i is the drop concentration (m⁻³ mm⁻¹) defined in Eq. (7), and ρ_w is the density of liquid water (g cm⁻³).

Next, we fit the slope and intercept parameters for a given shape parameter using the simulated dual-frequency reflectivity values. The slope parameter is calculated by numerically solving Eqs. (4) with the given shape parameter. This produces either one, two, or no solutions, depending on the DFR and value of the shape parameter imposed. The intercept parameter N₀ is then calculated by solving Eq. (3) for N₀ and substituting the previously obtained solutions for m and Λ:

where Z_e is either the 13.6- or 35-GHz reflectivity calculated from the disdrometer measurement. These derived gamma parameters correspond to a distribution that exactly reproduces both of the reflectivity measurements.

We derived these distributions for the same range of shape parameter used in Fig. 2, from −2 to 20, and incremented by 0.1. To complete the fitting process, the optimal shape parameter is selected by minimizing the sum of the absolute values of relative error E for rain rate, liquid water content, and mean mass diameter:

where Q_retr and Q_obs are the gamma-retrieved and disdrometer-observed integral rainfall quantities, respectively. The magnitude of this error is lowest for the distribution that provides the closest estimate of that integral rainfall quantity. By summing the error magnitude from different integral rainfall quantities, we seek the gamma parameters that not only reproduce the DFR exactly, but also offer the best approximation of lower-moment integral rainfall quantities.

During the application of this fitting algorithm to the composite DSDs, two regimes were identified regarding the behavior of these errors and the corresponding optimal solution. At reflectivities less than 36–42 dBZ_13.6, depending on region, the DFR is less than 1, and thus two sets of solutions for slope parameter exist for a given shape parameter. This ambiguity is a direct result of the dual roots to Eq. (4). At higher reflectivities, the DFR is greater than 1, and thus only one solution exists. The DFR was sufficiently high in all cases to ensure the existence of one or two solutions.

For the dual-solution cases, the correct solution was determined by seeking the simultaneous minimization of integral rainfall quantity errors. This occurs for the second solution at reflectivities less than 22 dBZ_13.6 (Fig. 4a), and for the first solution at reflectivities greater than 28 dBZ_13.6 (Fig. 4e). The superiority of one solution over the other in these intervals is illustrated in Figs. 4b,f.

The reflectivity at which the optimal solution switched from second to first varied among different regions, but was always between 22 and 28 dBZ_13.6. For some composites in this intermediate range, second-solution error magnitudes for each integral rainfall quantity reached absolute minima at widely separated values of shape parameter (Fig. 4c). In these cases, relative error minima were also observed at the lowest shape parameter. Here, both the first and second solution fit the observed DSD well (Fig. 4d). This is not surprising because these solutions are close to one another in shape parameter–slope parameter space near the minimum shape parameter on a DFR curve (Fig. 2). Thus, this minimum shape parameter was considered optimal in these cases.

The curves in Figs. 4a,c,e also provide an indication of the errors that can be expected if a nonoptimal shape parameter is applied. Everywhere, except for a small range near the transition from the second to first solution for slope parameter, a lower-than-optimal shape parameter is associated with an overestimate of rain rate and liquid water content and an underestimate of mean mass diameter, and vice versa. Near the transition, however, these trends reverse. This can be explained by examining the DFR curves in Fig. 2. In the linear branches of these curves, D_m increases with shape parameter, but at the apex, D_m decreases with increasing shape parameter along the curve. The opposite sign of the D_m error with respect to rain rate and liquid water content errors is a result of the bias in dual-frequency retrievals toward higher moments of the DSD. For a lower-than-optimal value of shape parameter, the retrieved DSD will be flatter than the observed DSD. Thus, in order to reproduce the shape of the DSD at larger drop sizes, concentrations will be overestimated at the small drops, with the reverse being true for higher-than-optimal shape parameters.

5. Regional variability of optimal shape parameter

We obtained optimal shape parameters from the composite DSDs for all reflectivity intervals at each region. The behavior of the optimal shape parameter with respect to reflectivity and climate region reflects physical differences in the DSD among these regions. All regions showed a similar progression from the second to first solution with increasing reflectivity, but the values of the optimal shape parameter vary considerably between regions and between different reflectivity intervals within a region.

The optimal shape parameters at each reflectivity interval for all datasets, shown in Fig. 5 and appendix B, represent the different DSD characteristics in each region. Recall that the optimal shape parameter is more representative of the change in slope for mid- to large-sized drops than the curvature of the distribution immediately around the peak concentration at small drop sizes. Additionally, the slope parameter is calculated directly from the DFR and is also largely a measure of the exponential decay of the DSD at the large drops (D > 3 mm). Therefore, it is not surprising that the fitted gamma DSDs match observed DSDs very well at large-drop bins (Fig. 4). Although this fitting procedure can lead to significant concentration errors for the small-sized drops (D < 1 mm), particularly in DSDs with a wide range of drop sizes, these errors are of minimal importance to the quantities we seek to retrieve.

At the lowest reflectivity range we examined (10–20 dBZ_13.6), many regions exhibit a minimum shape parameter near 15 dBZ_13.6, indicating DSDs that are approaching exponential. The increase of shape parameter on either side of this minimum occurs for different reasons on each side. On the low-reflectivity end the largest drop diameters are approximately 1.5 mm and the shape parameter represents the curvature of the concentration peak that occurs near 0.5 mm. On the high-reflectivity side of this minimum, the optimal shape parameter is more representative of the change in slope of medium-sized (1 < D < 3 mm) drop concentrations than the curvature around the peak.

At 22–35 dBZ_13.6, corresponding to both light and moderate rainfall, the optimal shape parameter is fairly steady at each region, but varies considerably from region to region. The steadiness of the shape parameter within each region, along with the increase of DFR with increasing 13.6-GHz reflectivity, indicates that the mean drop size increases with reflectivity in this range. The variability of the shape parameter between regions is a result of the differences in the shape of the DSD at drop sizes larger than 1.5 mm. The high shape parameters were associated with spectra that curve downward toward larger drop sizes, whereas the slopes of low shape parameter spectra were nearly constant at these drop sizes. Even though we did not explicitly separate convective and stratiform rainfall in our datasets, it is noteworthy that the higher shape parameters come from datasets that are dominated by convective rainfall (WT and WC).

At reflectivities above 35 dBZ_13.6 the results from oceanic and continental regions diverged, with shape parameters increasing for oceanic regions while remaining approximately constant for continental regions. The oceanic datasets reached a maximum shape parameter of 10–12 around the 40–45-dBZ_13.6 intervals before decreasing to 6–9 at 50 dBZ_13.6. This disparity in optimal shape parameter is caused by physical differences between continental and oceanic rainfall in deep convective precipitation and has been observed previously (Bringi et al. 2003). The higher value of the shape parameter in oceanic rainfall indicates a narrower distribution of drop sizes in oceanic rainfall relative to continental rainfall. At a given reflectivity, the oceanic spectrum has a smaller mean drop size than continental rainfall. This leads to significant differences in rain rate at a given reflectivity; for example, at 42 dBZ_13.6, rain rate varied from 9.9 mm h⁻¹ in the Wallops convection dataset to 20.7 mm h⁻¹ in the Darwin monsoon dataset.

6. Relationships between the shape and slope parameters

The variability of the shape parameter from region to region corresponds to significant variation in rain rates at a given reflectivity. This suggests that the use of a fixed shape parameter for dual-frequency rainfall retrievals will result in systematic regional biases. However, if the shape and slope parameters are correlated, then a relationship between the two, hereinafter denoted as an m–Λ relationship, may be employed during these retrievals to account for some of the observed variability. Such a relationship has been developed from disdrometer datasets by Zhang et al. (2001) and subsequently employed in dual-polarimetric radar-rainfall retrievals (Brandes et al. 2003) and radar data assimilation (Zhang et al. 2006). Although this relationship is supported by the work of Seifert (2005) via rain shaft model simulations, the video disdrometer datasets from which it was derived were limited to rainfall from Florida and Colorado, which is primarily continental in origin. Additionally, a minimum rain rate of 5 mm h⁻¹ was imposed. Our data points that adhere to these criteria are in general agreement with the relationship of Zhang et al. (2001). However, the oceanic datasets have higher shape parameters and the light rainfall (R < 5 mm h⁻¹) DSDs have lower shape parameters than this relationship would suggest (Fig. 6a). These deviations are sufficiently significant to preclude a single m–Λ relationship from being used for all retrievals.

When only the m–Λ pairs from a particular dBZ_13.6 interval are considered (Fig. 6b), a strong linear relationship among the points from different regions emerges. For each reflectivity interval i, a least squares linear fit was made of the form

The coefficients a_i and b_i and the correlation coefficient are given for each reflectivity interval in appendix C. Positive values of a_i and b_i at all intervals indicate that at a given reflectivity, a higher shape parameter is associated with a higher slope parameter. This has the first-order effect of damping variations in D_m, but through combining Eqs. (5) and (13), it can be shown that D_m will increase with m if 4a < b. Conversely, D_m decreases with m if 4a > b. The first case is true for all reflectivity intervals below 36 dBZ_13.6, but above this level, the m–Λ relationships give a D_m that decreases with m. This is consistent with the higher rain rates we find in the regions with a high shape parameter at a given reflectivity.

The strength of each m–Λ relationship is given by the squared correlation coefficient, which averaged 0.934 for all reflectivity intervals. This coefficient was near 0.9 at low-reflectivity intervals, but greater than 0.97 at all intervals above 36 dBZ_13.6, indicating strong m–Λ relationships at high rainfall rates. This is significant because the greatest regional variation of shape parameter is found for these high rainfall rates, but an m–Λ relationship at the appropriate reflectivity interval accounts for this variation and could potentially result in more accurate DSD retrievals.

The parameters of the m–Λ relationships also evolve systematically with respect to reflectivity. The coefficients a_i and b_i both decrease with increasing 13.6-GHz reflectivity, which can be seen by the progression of the best-fit lines in Fig. 6b. As reflectivity increases this trend results in a smaller slope parameter for a given shape parameter, which is consistent with the presence of more large drops in high-reflectivity DSDs. This finding is also consistent with the results of Zhang et al. (2003), who determined that although statistical error in fitting methods can create m–Λ relationships, the coefficients represent a true relationship between DSD parameters. The following second-order polynomials fit the coefficients in Eq. (13) as functions of 13.6-GHz reflectivity Z:

The correlation coefficients are 0.68 and 0.93 for Eqs. (14) and (15), respectively.

7. Evaluation of constraints

We compare two constraints on the gamma distribution for dual-frequency radar-rainfall retrieval. The first constraint is a fixed shape parameter and the second is a relationship between shape and slope parameters. The parameters of these constraints are allowed to vary with reflectivity. For the fixed shape parameter constraint, we use a mean shape parameter (see Table B1) to calculate rain-rate error E_i for each reflectivity interval at each site. The average magnitude of these errors is shown in Fig. 7. We also calculate the total error weighted by rain amount,

where N_DSD is the number of composites over which the weighted error is calculated, E_i is given by Eq. (12), and w_i is the contribution of rainfall in the ith reflectivity interval to total rainfall,

where n_i is the number of 1-min observations in the ith reflectivity interval. Likewise, we use m–Λ relationships (see Table C1) to calculate the same error statistics for this constraint. The weighted rain-rate error averaged over all regions for the fixed shape parameter constraint is 5.70%, as compared with 4.43% for the m–Λ relationships. At high reflectivities the strong correlation between shape and slope parameter contributes to lower errors for the m–Λ constraint than for the fixed shape parameter constraint (Fig. 7), which is reflected in the total weighted error.

An alternative to using the tables in the appendices is to apply polynomial relationships derived from them. The second-order polynomial fit for mean shape parameter as a function of dBZ_13.6 (Z) is

The polynomial fits for the coefficients of the m–Λ relationships are given in Eqs. (14) and (15), respectively. These fitted polynomials are used in place of the tables to calculate the rain-rate error statistics. In terms of total weighted error, the polynomial relationships show some degradation of performance relative to that shown in the tables in the appendices, and the Z_13.6–m relationship outperforms the m–Λ relationship.

As a final basis for comparison, a Z_13.6–R relationship was derived from the datasets with a linear least squares fit to illustrate the advantage of dual- over single-frequency rainfall retrievals,

The weighted error from the Z_13.6–R relation is 9.81%, about twice the amount resulting from the dual-frequency methods. Figure 7 indicates comparable performance of the single-wavelength Z_13.6–R relationship to the dual-frequency methods at low reflectivities (10–20 dBZ_13.6), but the dual-frequency methods outperform the Z_13.6–R relationship at medium and high reflectivities (dBZ_13.6 > 20), resulting in the lower weighted errors for these methods.

The decreasing magnitude of the rain-rate error with increasing reflectivity is noted for both constraints. The advantage of the dual-frequency methods over the Z_13.6–R relation is also greatest at high reflectivities. These trends are a consequence of the additional information provided by the DFR, which is strongly related to the shape of the large-drop end of the spectrum. As reflectivity increases, these well-retrieved large drops increasingly contribute to the total rain rate and hence reduce the error magnitude. We also note that liquid water content errors (not shown) are greater in magnitude than rain-rate errors, a result of the stronger dependence of the former quantity on the poorly retrieved small-drop end of the spectrum.

The fixed shape parameter constraint results in regional biases that must be removed for applications such as long-term climate studies. A possible solution is to use different reflectivity–shape parameter relationships in different regions, but this requires definition of these regions, which cannot be done globally with our limited datasets. Furthermore, even in the same region, DSD characteristics may depend on wind regime (Tokay et al. 2002). In these situations, a fixed shape parameter would fail to capture this variability, possibly resulting in seasonal biases.

The m–Λ relationships show the potential to overcome the biases inherent in using a constraint of fixed shape parameter. These relationships are strongest at high reflectivities (dBZ_13.6 > 35), where the regional biases of fixed shape parameter are most problematic. Another advantage of the m–Λ relationships is shown at 20–25 dBZ_13.6. Here, the two solutions for the fixed shape parameter constraint are given by the double intersection of a DFR contour. The m–Λ relationships we derived only intersect each contour once, eliminating this ambiguity. However, there are two reasons why an m–Λ relationship might fail to provide accurate solutions. A low correlation coefficient could give solutions far from the optimal values. This occurs at 25–35 and 10–15 dBZ_13.6, where even a single-frequency Z_13.6–R relation outperforms the m–Λ constraint. Another problem is that even a highly correlated fit could fail if it is nearly parallel to the DFR contours. This is because only a small change in DFR, which might result from an error in attenuation correction, could lead to a large change in the retrieved DSD.

8. Summary and conclusions

The retrieval of a three-parameter gamma distribution representing DSD from dual-frequency radar measurements requires an a priori constraint. In this study, we have employed disdrometer observations from distinct climatic regimes to develop constraints that are representative of the wide regional variation in drop spectra. We derived two constraints—a fixed shape parameter and the shape parameter–slope parameter relationship. The shape parameter–slope parameter relationships provide a slight advantage over the fixed shape parameter terms of overall rain amount–weighted rain-rate error, and both dual-frequency methods are approximately twice as accurate as the Z_13.6–R relationship. However, each constraint has strengths and weaknesses; the m–Λ relationships generally perform better than the fixed shape parameter at high reflectivities, and vice versa at low reflectivities. The regional DSD differences in shape parameter were found to be most significant both in magnitude and in the context of lower internal DSD variability (appendix A) at high reflectivities (>40 dBZ_13.6), which coincides with the strongest m–Λ relationships, so it is at these high reflectivities that our results are most significant and applicable.

The performance of these constraints with polarimetric radars or at different frequencies than those used here should be similar to our results in principle because DFR at different frequencies and the polarimetric differential reflectivity Z_DR are both also weighted toward the sixth moment of the DSD, and thus the retrieved parameters will result in optimal rainfall estimation. However, the slopes of the DFR curves may be different than those shown in Fig. 2, and certain properties of our relationships, such as the elimination of dual solutions near 20 dBZ_13.6, may not be applicable at other frequencies. The angle at which the DFR or Z_DR curves intersect the m–Λ relationships also determines the susceptibility of DSD retrieval to measurement error, which may thus be different for other frequencies/polarizations than in this study.

The constraints that have been developed in this study may be incorporated into DSD profile retrieval algorithms that are being considered for the dual-frequency radar on the GPM core satellite. However, while these constraints appear to be valid for DSDs from a wide variety of climate regimes, they need to be verified against independent DSD observations from other regions. For validation of our constraints above the ground, profiler observations of the vertical structure of DSDs in precipitation systems could be analyzed.

Regarding applications to GPM, it is important to note that dual-frequency retrievals will be limited by the sensitivity of the radar receiver. The 13.6-GHz radar will only be sensitive to reflectivities higher than about 17 dB, whereas at 35 GHz, the minimum sensitivity will be 12 dB, according to recent design specifications (Iguchi et al. 2003). The latter will allow GPM to estimate rainfall that is currently undetectable by the TRMM PR, which may contribute to disagreements between the PR and TMI (Berg et al. 2006), but these retrievals will be done at a single frequency. In heavy rainfall, attenuation will limit the range of 35-GHz measurements, and in extreme cases the same may occur at 13.6 GHz. In these situations, DSD estimates at range gates where both frequencies are available could be used to formulate optimal Z_13.66–R relations at subsequent attenuated range gates.

In this study, we have investigated only one component of satellite-based dual-frequency radar-rainfall retrieval algorithms. The effects of ice and snow, brightband interaction, and attenuation from atmospheric water vapor, cloud water, and hydrometeors are not considered herein. Despite these additional challenges, our results show considerable promise for the ability of dual-frequency radar measurements, augmented by optimal constraints, to accurately measure DSD.