The raindrop size distribution is a fundamental quantity used to describe the characteristics of rain. Vertically pointing Doppler radar profilers are well suited to retrieve the raindrop size distributions because of their operating frequency and data collection methodology. Doppler radar profilers operating at UHF are sensitive to both Bragg scattering from the radio refractive index of turbulence and Rayleigh scattering from distributed targets. During light precipitation, both scattering processes are resolved in the Doppler velocity spectra. During moderate to heavy precipitation the ambient air motion is not resolved in the Doppler velocity spectra. The sans air motion (SAM) model is introduced in this study and uses only the Rayleigh scattering portion of the Doppler velocity spectrum to estimate the ambient vertical air motion, the spectral broadening, and the raindrop size distribution. The SAM model was applied to 915 MHz profiler observations in central Florida. There was good agreement between the SAM-model-retrieved rain rate and mass-weighted mean diameter at an altitude of 300 m with simultaneous surface disdrometer observations. The SAM model was applied to the profile of Doppler velocity spectra to yield estimates of rain rate, mass weighted mean diameter, and ambient vertical air motion from 300 m to just under the melting level at 4 km.
 Raindrop size distributions (DSDs) describe the number and size of raindrops in precipitation. The vertical distribution and time evolution of the DSD provides information about the dynamical processes of precipitating clouds. Vertically pointing Doppler radar profilers operating at very high frequency (VHF) and ultrahigh frequency (UHF) provide information on the vertical structure of hydrometeors and the ambient air motions in the precipitating clouds that advect overhead.
 During the past two decades, studies using vertically pointing profilers have successfully retrieved the raindrop size distribution from precipitating clouds. Profilers observe ambient air motion characteristics due to the energy backscattering from changes in the radio refractive index (Bragg scattering) and observe the motion of the hydrometeors due to the energy backscattering off of the distributed particles (Rayleigh scattering). Using the high-transmitted-power 46.5 MHz middle and upper atmosphere (MU) profiler located near Kyoto, Japan, Wakasugi et al. [1986, 1987] resolved both the Bragg and Rayleigh scattering components in a single Doppler spectrum. The observed mean ambient air motion and spectral broadening information from the Bragg scattering component represented the updrafts/downdrafts and turbulence in the radar pulse volume. Raindrop size distribution retrieval models utilizing the Bragg and Rayleigh scattering components in a single Doppler spectrum are called single-Doppler-spectrum (SDS) models.
 Two profilers operating at two different frequencies can extend the sensitivity to the Bragg and Rayleigh scattering processes, not possible with the limited dynamic range of single-frequency profilers. Reliable air motion characteristics have been estimated from a 50 MHz (VHF) profiler at Darwin, Australia, and used as parameterizations to the DSD retrieval using the Doppler velocity spectrum from a collocated 920 MHz (UHF) profiler [Rajopadhyaya et al., 1998, 1999; Cifelli and Rutledge, 1994, 1998; May and Rajopadhyaya, 1996; Cifelli et al., 2000]. This type of DSD retrieval model utilizes a parameterized air motion (PAM) model.
 The Rayleigh scattering portion of the Doppler velocity spectrum represents the hydrometeor size distribution shifted by the ambient air motion and broadened by the turbulence and wind shear in the radar pulse volume. The sans air motion (SAM) model described in this work estimates the ambient air motion, the spectral broadening, and the hydrometeor size distribution from only the Rayleigh scattering portion of the Doppler velocity spectrum. The SAM model can be used when the Bragg scattering component cannot be resolved in the Doppler velocity spectrum (SDS method) and when the ambient air motion and spectral broadening cannot be parameterized (PAM model). Hauser and Amayenc [1981, 1983] developed the initial analytical description for the SAM model using an exponential form of the raindrop size distribution and ignoring the spectral broadening of the Doppler velocity spectrum. Sangren et al.  suggested that the modeling efforts could be improved by including the spectral broadening as a fitted parameter.
 In this study, the SAM model is developed using two methods. The first method uses the integrated moments of reflectivity, mean Doppler velocity, and spectral width calculated from the observed Doppler spectrum. The ambient air motion and spectral broadening are not estimated but are assumed to be zero. The integrated moment version of the SAM model is useful for near-surface observations and stratiform rain regimes where the ambient air motions approach zero. The second method uses the observed Doppler velocity spectrum to determine a best fit model spectrum. The model spectrum estimates the ambient air motion, the spectral broadening, and the raindrop size distribution described by the three parameters of a modified gamma distribution.
 First, this paper presents the general mathematical framework to retrieve the raindrop size distribution from vertical incident profiler observations. Section 3 describes the mathematics for the integrated moment SAM model method assuming zero-mean ambient air motion and negligible spectral broadening. Section 4 develops the spectral SAM model method and presents the accuracy of the DSD retrieval using noisy simulated spectra. Section 5 presents the SAM-model-retrieved raindrop size distributions from UHF (915 MHz) profiler observations made in central Florida and compares the profiler retrievals with simultaneous surface disdrometer observations. The discussion and conclusions are presented in section 6.
2. Mathematical Description of Vertical Profiler Doppler Velocity Spectra
 Doppler radar profilers operating at UHF detect both Bragg scattering from the radio refractive index of turbulence and Rayleigh scattering from distributed targets [Gage et al., 1999] and can be expressed mathematically (following Wakasugi et al. [1986, 1987]) as
where the first term on the right side represents the Bragg scattering component, the second term represents the Rayleigh scattering component, and the last term represents the uniformly distributed random background noise floor. The variables v and represent the independent Doppler velocity at each spectral point and the mean ambient air motion in the radar pulse volume. The Bragg scattering component has been modeled as a Gaussian-shaped probability distribution of turbulent velocities [Tennekes and Lumley, 1973; Gossard, 1994; Currier et al., 1992; Rogers et al., 1993; Rajopadhyaya et al., 1993, 1998, 1999]
where Pair represents the magnitude of the Bragg scattering corresponding to the refractive index irregularities and σair2 represents the variance of the spectral broadening.
 The Rayleigh scattering component observed by a vertically incident profiler results from the convolution of the normalized atmospheric turbulent probability density function with the hydrometeor reflectivity spectral density [Gossard, 1988]. Rajopadhyaya et al. [1998, 1999] expressed this as
where the hydrometeor spectrum in stationary air is represented by
and N(D), D, and dD/dv represent the number concentration of the hydrometeor distribution, the raindrop diameter, and the coordinate transformation from terminal fall speed to diameter space, respectively.
 In this study, the modified gamma functional form describes the raindrop size distribution [Ulbrich, 1983]
where No, μ, and Λ represent the scale, the shape, and the slope parameters. It is assumed that these parameters describe the composite DSD over the range of diameters resolved by the profiler Doppler radar operating at UHF and over the pulse volume and dwell time required to acquire these observations.
3. Sans Air Motion (SAM) Model: Integrated Moment Method
 From (3) and (5), five unknowns (, σair, No, μ, and Λ) describe the Doppler velocity spectrum. If the atmospheric conditions are such that the ambient air motion and spectral broadening are negligible (e.g., near the surface and during stratiform rain), then the three moments of the Doppler velocity spectrum uniquely confine the three parameters of the DSD. Assuming zero vertical air motion and zero spectral broadening enables the development of the integrated moment method sans air motion (SAM) model in a closed mathematical form.
3.1. Zero Ambient Air Motion and Zero Spectral Broadening
 Assuming that the hydrometeor size distribution can be expressed by a modified gamma distribution, the total reflectivity factor z (in units of mm6 m−3) is estimated from the zeroth moment of the profiler Doppler velocity spectrum and from the sixth power of the DSD
where vmin and vmax are the observed integration limits in the velocity domain, Dmin and Dmax are the assumed integration limits in the diameter domain, Γ is the complete gamma operator, and the last equality is derived after the integration limits are allowed to be Dmin → 0 and Dmax → ∞. The reflectivity factor measured by calibrated profilers is a function of the three DSD parameters, No, μ, and Λ.
 The observed reflectivity-weighted mean Doppler velocity, VDoppler (in units of ms−1), is estimated from the first moment of the Doppler velocity spectrum and is expressed in the Doppler velocity and raindrop diameter domains using
where vfall speed(D) is the fall-speed-to-diameter relationship and is the mean ambient air motion (positive indicates upward air motion in this equation). The mean air motion causes a shift in the Doppler spectrum consistent with the chosen sign convention. (Particles with positive definite diameters have velocities defined as positive downward due to the gravitational force of Earth. Fall-speed-to-diameter relationships follow this convention. Meteorological convention defines upward motion as positive. Thus there is a conflict between these two reference frames. All attempts are made in this study to reduce the ambiguity and confusion between these two valid and conflicting conventions by clearly identifying the variables and the polarity of motion in the figures.)
 The reflectivity-weighted mean fall speed of the DSD, Vfall speed, is a function of the shape and slope parameters (μ and Λ) of the gamma distribution and the terminal fall-speed-to-diameter relationship. Ulbrich and Chilson  showed that by letting the integration limits become Dmin → 0 and Dmax → ∞, the mean Doppler velocity simplifies to
with α1 = 9.65 ms−1, α2 = 10.3 ms−1, and α3 = 0.6 ms−1. The variable vfall speed(D) has units of ms−1, and the equivalent spherical diameter D has units of millimeters. The factor (ρo/ρ)m represents the adjustment in terminal fall speed due to decreasing atmospheric density with altitude expressed by ρ, relative to the surface density ρo. A value of m = 0.4 is used in this study.
 The relations in (7) and (8) imply that for a given fall-speed-to-diameter relationship and, every value of VDoppler is associated with a family of (μ, Λ) pairs. This association is independent of the amount of symmetrical spectral broadening in the observed Doppler spectrum.
 The reflectivity-weighted Doppler velocity variance, σz2 (in units of m2s−2), is estimated from the second moment of the Doppler velocity spectrum. Ignoring the convolution effects by assuming that the spectral broadening is zero, then the variance can be expressed in the Doppler velocity and raindrop domains using
Using the fall-speed-to-diameter relationship expressed in (9) and extending the integration limits to Dmin → 0 and Dmax → ∞, the Doppler velocity variance simplifies to
Similar to the expression of the mean Doppler velocity, each Doppler velocity variance is associated with a family of (μ, Λ) pairs.
Figure 1 illustrates how the mean Doppler velocity and spectral width are estimated from the DSD shape and slope parameters using (8) and (11). (By convention in the profiler community, the Doppler velocity variance is reported as the spectral width and is defined as WDoppler = 2σz (in units of ms−1)). Numerically inverting (8) and (11) enables μ and Λ to be estimated from observed values of mean Doppler velocity and spectral width. Figure 1b graphically illustrates this inversion. Once μ and Λ are estimated from the observed mean Doppler velocity and spectral width (either numerically or using Figure 1b), No can be estimated using the measured reflectivity and (6). Note that the transformation (VDoppler, WDoppler) ↔ (μ, Λ) is independent of absolute calibration of the profiler.
3.2. Sensitivity to Nonzero Air Motion and Nonzero Spectral Broadening
 The integrated moment SAM model computes the three parameters of the gamma distribution from the three moments of the Doppler spectrum. From these three parameters the mass-weighted mean diameter Dm and rain rate R can be calculated for each retrieval using the fall-speed-to-diameter relationship of (9):
To remove the No dependence in the rain rate calculation, (13) is normalized by (6) to produce the ratio R/z (mm h−1 /(mm6 m−3)) [Ulbrich, 1992].
Figure 2 shows the integrated moment SAM model estimates of μ and Λ converted into estimates of Dm and R/z as functions of VDoppler and WDoppler. The values of μ are shown as contours in Figure 2 to improve panel-to-panel comparisons. In general, the mean volume diameter Dm increases with increasing VDoppler and decreasing WDoppler. Conversely, the ratio R/z decreases with increasing VDoppler and decreasing WDoppler.
 Estimates shown in Figure 2 assume zero air motion. Nonzero air motions will be manifested as errors in the observed mean Doppler velocity in (7) and will result in a horizontal shift in Figure 2. In the regions of large gradients in the VDoppler dimension in Figure 2, small changes in air motion will cause a large variation. In order to quantify the sensitivity to nonzero air motions, the mean Doppler velocity was artificially increased and decreased until Dm and R/z changed by 10%. In general, the air motion can deviate from zero by ±0.50 and ±0.15 ms−1 before estimates of Dm and R/z deviate by 10%.
 Estimates shown in Figure 2 also assume zero spectral broadening. Spectral broadening causes an increase in the observed spectral width and a shift along the WDoppler dimension in Figure 2. Note that the gradients in Figure 2 are steeper along the VDoppler axis than along the WDoppler axis. This indicates that the DSD integrated quantities are more sensitive to variations in air motion than they are to spectral broadening. In order to quantify the sensitivity to spectral broadening, the DSD parameters retrieved for each pixel in Figure 2 were inserted into (3), and the spectral broadening was increased until the estimated Dm and R/z change by 10%. In general, the spectral broadening can increase to approximately 0.6 and 0.75 ms−1 before the estimates of Dm and R/z deviate by 10%.
4. Sans Air Motion (SAM) Model: Spectra Method
 The spectral method sans air motion (SAM) model estimates the parameters, σair2, No, μ, and Λ by fitting a model spectrum to the observed Rayleigh scattering portion of the Doppler velocity spectrum. At first glance, this five-parameter estimation appears to be an ill-posed problem with multiple solutions. By constraining the observed reflectivity and mean Doppler velocity, the solution set reduces to a three-parameter estimation.
4.1. Spectral Method Constraints
 In this study, two constraints are imposed to reduce the number of free parameters from five to three. The first constraint is the conservation of reflectivity. Both the model spectrum and observed spectrum must have the same total reflectivity as defined in (6). The second constraint is the conservation of mean Doppler velocity. Both the model spectrum and observed spectrum must have the same VDoppler as defined in (7). Using these two constraints, model spectra can be constructed and compared to the observed spectrum in a squared difference sense following Sato et al. 
where Sobs(vi) and Smodel(vi) are the reflectivity spectral density at each velocity above the noise floor for the observed and modeled spectra, respectively. The two constraints narrow the set of possible solutions with the minimum χ2 being determined using standard minimization techniques.
4.2. Sensitivity to Measurement Uncertainties in the Doppler Spectrum
 The SAM model is a conceptual and mathematical framework outlining a procedure to estimate the hydrometeor size distribution from the Rayleigh scattering portion of the Doppler velocity spectrum. The actual computer code used to implement this model can take on many variations with the specific details determining the model's efficiency and efficacy. The numerical code used in this study is the result of continuously improving code being developed at the National Oceanic and Atmospheric Administration Aeromomy Laboratory, and the results presented in this study reflect the status of that code at a single point in time.
 The measurement uncertainty or noise on the observed spectrum will cause errors in the best fit model spectrum relative to the correct (or no noise spectrum) solution. Simulated noisy spectra were constructed and processed with the spectral method SAM model to determine approximate error bounds on the best fit solutions. The DSD parameters for the simulations were determined from 1644 min of Joss-Waldvogel disdrometer (Distromet, Inc., model RD-69) observations collected in August and September 1998 in central Florida. The modified gamma DSD parameters were estimated from each minute DSD using the method of truncated moments described by Ulbrich and Atlas . The median μ parameter estimated from each Dm (±0.05 mm) interval defined the shape of the simulated DSD. For each pair of μ and Dm, 300 simulated Doppler velocity spectra with random noise were constructed. The random noise at each spectral point was selected from a population of random points with the same standard deviation as the measurement uncertainty at each spectral point, which is approximately
where NFFT is the number of independent velocity spectra averaged to form the recorded reflectivity spectral density. The spectral method SAM model was applied to each simulated spectrum, and the mean and standard deviations of the results are shown in Figure 3.
Figure 3a shows the modified gamma shape parameter μ as a function of Dm. The median value from the Joss-Waldvogel disdrometer observations is shown with the solid line, and the mean and 1 standard deviation estimated from the SAM model are shown with circles and vertical lines. As with the other variables shown in Figure 3, the largest standard deviations in the retrievals are for Dm less than approximately 1.5 mm. The decrease in μ with increasing Dm is a consistent feature with this Joss-Waldvogel disdrometer data set. For Dm values greater than 1.5 mm it appears that this SAM model overestimates Dm by approximately 0.05 to 0.10 mm and has an upward bias of approximately 0.10 to 0.25 ms−1 (Figures 3b and (3c). The simulations also indicate that the rain rate to reflectivity ratio is underestimated by about 10% for Dm values greater than 1.5 mm (Figure 3d).
5. Data Observations
 A 915 MHz vertically pointing profiler was deployed in central Florida for August and September 1998 in support of the Tropical Rainfall Measuring Mission (TRMM) Ground Validation Program. Table 1 lists the profiler system characteristics. A Joss-Waldvogel disdrometer located next to the profiler recorded over 280 mm of rain during this two-month campaign. The surface disdrometer records the number and size of raindrops hitting the 50 cm2 sensor head, enabling the direct calculation of reflectivity, rain rate, and Dm [Williams et al., 2000]. Surface disdrometer reflectivity estimates were used to calibrate the 915 MHz profiler as described by Gage et al. .
Table 1. 915 MHz Profiler Parameters Used During Texas-Florida Underflights Experiment B (TEFLUN-B)
3 m shrouded dish
100 or 250 m
Maximum height sampled
Maximum radial velocity
Number of coherent integrations
Number of averaged fast Fourier transforms
Number of spectral points
Number of radar pulses processed
full Doppler spectra
 On 17 September 1998 a convective precipitating system passed over the profiler site. Figure 4 shows the first three moments of the Doppler spectra for the 100 m pulse length mode in time-altitude cross sections for this 6 hour event. During hour 1900 UTC the reflectivities exceeding 50 dBZ and net upward mean Doppler velocities indicate an intense convective rain regime. During 2130 to 2345 UTC a well-defined bright band near 4.5 km and change in mean Doppler velocity through the bright-band altitude indicate a stratiform rain regime.
 The Doppler velocity spectra provide more detail of the convective and stratiform rain regimes than the three integrated moments. Figures 5a and 5b show the vertical profile of stratiform rain observed at 2209:00 UTC, with Figure 5a showing the reflectivity spectral density at each spectral point using log units and Figure 5b showing the total reflectivity at each altitude. The mean downward Doppler velocities of 1 to 2 ms−1 above the bright band correspond to the fall speeds of ice particles, and the downward velocities of 6 to 9 ms−1 below the bright band correspond to the fall speeds of raindrops. Profiler-based precipitation classification algorithms use this clear change in velocity near the melting level to identify stratiform rain [Williams et al., 1995]. Below the melting layer and continuing down to the surface, the mode of Doppler velocity spectra shifts to slower Doppler velocities as the raindrops approach the surface. The increase in atmospheric density with decreasing altitude is the primary factor contributing to this deceleration.
Figures 5c and 5d show the vertical profile of convective rain observed at 1919:00 UTC. Near 1 km the reflectivity spectral density has downward velocities exceeding 12 ms−1. The raindrops do not have terminal velocities exceeding 12 ms−1, but rather, the raindrops are in a downdraft with severe turbulence. Near 2.5 to 3.0 km the mean Doppler velocity is approximately 1 ms−1 upward, and the total reflectivity is greater than 45 dBZ. These raindrops are in an updraft large enough to cause the mean Doppler velocity to be upward. The ambient vertical air motions retrieved from the spectral method SAM model are shown in Figure 5c with black asterisks. The retrieved air motions are consistent with the physical interpretation of the observed Doppler velocity spectra with downward air motions between 0.5 and 1.5 km and upward motions above 1.5 km.
5.1. Integrated Moment Method SAM Model: Surface Comparisons
Figure 6 shows the 1 min resolution surface disdrometer and 915 MHz profiler integrated moment method SAM model retrieved reflectivity, rain rate, and mass-weighted mean diameter at an altitude of 300 m for the 17 September 1998 rain event. Even though one instrument observes precipitation at the surface and the other observes 300 m aloft, there is excellent agreement between the two measurements. Notable disagreements between the disdrometer and profiler retrievals occur during the convective rain regimes. For example, from about 2050 to 2130 UTC, the profiler Dm(R) is less (greater) than the disdrometer estimates. As will be seen in section 5.2, this time interval corresponds to upward ambient air motions retrieved at 300 m using the spectral method SAM model. Not accounting for the updraft in the profiler retrievals underestimates Dm and overestimates R.
5.2. Spectral Method SAM Model: Time-Altitude Retrievals
Figure 7 shows the spectral-method-SAM-model-retrieved parameters for the 17 September rain event. To aid in the visual presentation, the independent retrievals have been smoothed using the eight nearest neighbors in time and altitude. To ensure that there are enough spectral points used in the fitting process, only the retrievals with reflectivities greater than 30 dBZ are shown in Figure 7. The time-altitude patterns of the retrieved parameters are consistent with conceptual models of convective and stratiform rain regimes. For example, during the convective rain near 1900 to 1940 UTC, the high rain rates between 2 and 4 km are associated with small Dm, with larger Dm values near the surface. During this convective rain event the vertical air motion (Figure 7e) indicates a critical level near 2 km, with upward motions above and downward motion below this critical level. The upward vertical air motion lifts the smaller raindrops to higher levels while the larger raindrops fall out because of their larger terminal fall speeds [Atlas and Ulbrich, 2000].
 During the stratiform rain regime from 2215 to 2230 UTC the bright band intensifies, and the rain rate increases (Figures 7a and 7b). The mean mass-weighted diameter also increases during this fall streak event. While rain rate and reflectivity are functions of all three parameters of the DSD, the mean diameter is only a function of the shape of the DSD. Therefore the increase in Dm during the fall streak is due to a change in DSD shape and not an artifact of the increase in reflectivity. The fall streak is associated with a population of particles that have larger mean diameters than the neighboring regions of the stratiform rain.
6. Discussion and Conclusions
 The sans air motion (SAM) model estimates the raindrop size distribution from only the Rayleigh scattering portion of the Doppler velocity spectrum obtained from a vertically pointing profiler. Two different methods of the SAM model are introduced in this study. The first method assumes zero air motion and zero spectral broadening and estimates the three parameters of a modified gamma raindrop size distribution (DSD) from the first three integrated moments of the observed Doppler velocity spectrum (reflectivity, mean Doppler velocity, and spectral width). Sensitivity tests indicate that 10% errors in mass-weighted mean diameter and rain rate occur for air motions exceeding ±0.50 and ±0.15 ms−1, respectively. This integrated moment method should be applied to observations near the surface and during stratiform rain, where the air motions approach zero.
 The second SAM model method uses the observed Doppler velocity spectrum to determine the best model spectrum in a least squared difference sense. The spectral method estimates the ambient air motion, the spectral broadening, and the modified gamma raindrop size distribution. In order to reduce the minimization problem from five to three free parameters, all possible model spectra are constrained to have the same integrated reflectivity and mean Doppler velocity as the observed spectrum. Using simulated noisy Doppler velocity spectra, the spectral method SAM model code appears to overestimate Dm by approximately 0.05 to 0.10 mm and has an upward bias of approximately 0.10 to 0.25 ms−1 which results in underestimating the rain rate to reflectivity ratio by about 10%. Statistical or probabilistic solutions are being developed which estimate the mode and range of each retrieved parameter to account for the uncertainties inherent in the observed spectra.
 The integrated moment method SAM model retrievals at 300 m were compared to surface disdrometer observations. There was very good agreement. The best agreement occurred during stratiform rain when the air motions are smallest and the rain is more homogeneous with less spatial and temporal variability than during convective rain. The spectral method SAM model retrievals up to 4 km are consistent with conceptual models of convective and stratiform rain but have yet to be validated with collaborating observations. Several precipitation events in central Florida with simultaneous profiler and polarimetric scanning radar observations are being analyzed; the results will be presented in a future study.
 This study followed previous profiler DSD retrieval work based on the assumption that the raindrop size distribution can be described mathematically by the modified gamma function. Recent studies on DSD distributions observed by surface and airborne probes indicate that the DSD may be better represented by a normalized distribution [Sempere-Torres et al., 1994; Testud et al., 2001]. The SAM model projects the assumed shape of the DSD onto the observed Doppler velocity spectrum. As better representations of the DSD are developed, future SAM model implementations can incorporate these representations.
 The SAM model will not be applicable in all precipitation studies. The SAM model assumes that the precipitation process is stationary (not evolving with time) and homogenous throughout the radar pulse volume and during the radar dwell time. The model also assumes that the DSD shape is defined by a particular functional form and that the spectral broadening is described by a uniformly weighted convolution. These assumptions limit the SAM model to represent the bulk properties of the precipitation (e.g., Dm and rain rate). The SAM-model-retrieved bulk properties can be utilized to help validate cloud resolving models and spaceborne remote sensing retrieval algorithms including the TRMM precipitation products. Studying the detailed evolution and microphysical processes of precipitation will require the air motion and spectral broadening to be observed and the DSDs estimated using the single Doppler spectra (SDS) or the parameterized air motion (PAM) models.
 The largest errors of the SAM model will occur during convective rain when the variations in the pulse volume are greatest. One example can been seen in Figure 7 between 1910 and 1925 UTC and 2.5 and 4 km. Even though the spectral broadening is enhanced during this period (Figure 7f), underestimating the spectral broadening results in overestimating the DSD width (decreasing μ). The increased DSD width results in underestimating Dm and overestimating the rain rate. The retrieved rain rate during this period exceeds 500 mm h−1, with updrafts exceeding 4 ms−1. The rain rate appears to be excessive but is during a very active and turbulent event. Additional constraints can be added to the SAM model to improve the retrievals during this period so that they fit into conceptual precipitation models.
 In this study, the SAM model was applied to profiler observations below the melting level. The SAM model can be applied to observations above the melting level by specifying the shape of the particle size distribution and solving for the best fall-speed-to-effective-diameter relationship. Thus, in the future, the SAM model can be applied to observations above and below the melting level to resolve the complete profile of ambient vertical air motion, spectral broadening, and particle size distribution.
 Aeronomy Laboratory research for the TRMM field campaigns has been supported in part by funding from NASA Headquarters through the NASA TRMM Project Office. The profiler analysis is supported in part by NASA grant NAG5-9753.