Simultaneous ambient air motion and raindrop size distributions retrieved from UHF vertical incident profiler observations

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1 RADIO SCIENCE, VOL. 7, NO.,,./RS, Simultaneous ambient air motion and raindrop size distributions retrieved from UHF vertical incident profiler observations Christopher R. Williams Cooperative Institute for Research in Environmental Sciences, University of Colorado at Boulder, Boulder, Colorado, USA Tropical Dynamics and Climate Group, NOAA Aeronomy Laboratory, Boulder, Colorado, USA Received December ; revised August ; accepted October ; published April. [] 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 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 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 m to just under the melting level at km. INDEX TERMS: Meteorology and Atmospheric Dynamics: Precipitation (); Meteorology and Atmospheric Dynamics: Remote sensing; Meteorology and Atmospheric Dynamics: Instruments and techniques; KEYWORDS: drop size distribution, DSD, profiler, precipitation. Introduction Copyright by the American Geophysical Union. -//RS - [] 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. MHz middle and upper atmosphere (MU) profiler located near Kyoto, Japan, Wakasugi et al. [, 7] 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 MHz (VHF) profiler at Darwin, Australia, and used as parameterizations to the DSD retrieval using the Doppler velocity spectrum from a collocated

2 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS MHz (UHF) profiler [Rajopadhyaya et al.,, ; Cifelli and Rutledge,, ; May and Rajopadhyaya, ; Cifelli et al., ]. 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 [, ] 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. [7] First, this paper presents the general mathematical framework to retrieve the raindrop size distribution from vertical incident profiler observations. Section describes the mathematics for the integrated moment SAM model method assuming zero-mean ambient air motion and negligible spectral broadening. Section develops the spectral SAM model method and presents the accuracy of the DSD retrieval using noisy simulated spectra. Section presents the SAM-model-retrieved raindrop size distributions from UHF ( 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.. 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., ] and can be expressed mathematically (following Wakasugi et al. [, 7]) as S obs ðþ¼p v air S air ðv wþþs air ðv wþ* S hyd ðþ v þnoise; ðþ 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 w 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, 7; Gossard, ; Currier et al., ; Rogers et al., ; Rajopadhyaya et al.,,, ] " # S Bragg ðþ¼p v air S air ðv wþ ¼ P air v w pffiffiffiffiffi exp ð Þ ; p sair s air ðþ where P air represents the magnitude of the Bragg scattering corresponding to the refractive index irregularities and s air 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, ]. Rajopadhyaya et al. [, ] expressed this as S Rayleigh ðþ¼s v air ðv wþ* " S hyd ðþ v # v w ¼ pffiffiffiffiffi exp ð Þ * S hyd ðþ; v p sair s air ðþ

3 WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS - where the hydrometeor spectrum in stationary air is represented by S hyd ðþ¼n v ðdþd dd dv ; ðþ 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, ] NðDÞ ¼ N o D m exp½ LDŠ; ðþ where N o, m, and L 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.. Sans Air Motion (SAM) Model: Integrated Moment Method [] From () and (), five unknowns (w, s air, N o, m, and L) 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... 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 mm m ) is estimated from the zeroth moment of the profiler Doppler velocity spectrum and from the sixth power of the DSD z ¼ Z v max v min S obs ðþdv v ¼ ¼ N ogð7 þ mþ L 7þm ; Z D max D min N o D þm exp½ LDŠdD () where v min and v max are the observed integration limits in the velocity domain, D min and D max are the assumed integration limits in the diameter domain, G is the complete gamma operator, and the last equality is derived after the integration limits are allowed to be D min! and D max!. The reflectivity factor measured by calibrated profilers is a function of the three DSD parameters, N o, m, and L. [] The observed reflectivity-weighted mean Doppler velocity, V Doppler (in units of ms ), is estimated from the first moment of the Doppler velocity spectrum and is expressed in the Doppler velocity and raindrop diameter domains using V Doppler ¼ ¼ vr max v min v max R v min DR max D min vs obs ðþdv v S obs ðþdv v ¼ V fall speed w v fall speed ðdþd þm exp½ LDŠdD w; DR max D min D þm exp½ LDŠdD ð7þ where v fall speed (D) is the fall-speed-to-diameter relationship and w is the mean ambient air motion (positive w 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, V fall speed, is a function of the shape and slope parameters (m and L) of the gamma distribution and the terminal fall-speed-to-diameter relationship. Ulbrich and Chilson [] showed that by letting the integration limits become D min! and D max!, the mean Doppler velocity simplifies to h V fall speed ¼ a a þ a i ð7þmþ r m o ; ðþ L r

4 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS when using the fall-speed-to-diameter relationship derived by Atlas et al. [7], r m v fall speed ðdþ ¼ ½a a expð a DÞŠ o ; ðþ r with a =. ms, a =. ms, and a =.ms. The variable v fall speed (D) has units of ms, and the equivalent spherical diameter D has units of millimeters. The factor (r o /r) m represents the adjustment in terminal fall speed due to decreasing atmospheric density with altitude expressed by r, relative to the surface density r o. A value of m =. is used in this study. [] The relations in (7) and () imply that for a given fall-speed-to-diameter relationship and w, every value of V Doppler is associated with a family of (m, L) pairs. This association is independent of the amount of symmetrical spectral broadening in the observed Doppler spectrum. [] The reflectivity-weighted Doppler velocity variance, s z (in units of m s ), 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 s z ¼ R v max v min Sobs v V Doppler ðþdv v vr max v min DR max v fall speed ðd D ¼ min S obs ðþdv v DR max D min Þ V fall speed D þm exp½ LDŠdD D þm exp½ LDŠdD : ðþ Using the fall-speed-to-diameter relationship expressed in () and extending the integration limits to D min! and D max!, the Doppler velocity variance simplifies to " s z ¼ a þ a # ð7þmþ þ a 7þm ð Þ L L r m o : ðþ r Similar to the expression of the mean Doppler velocity, each Doppler velocity variance is associated with a family of (m, L) pairs. [7] Figure illustrates how the mean Doppler velocity and spectral width are estimated from the DSD shape and slope parameters using () and (). (By convention in the profiler community, the Doppler velocity variance is reported as the spectral width and is defined as W Doppler =s z (in units of ms )). Numerically inverting () and () enables m and L to be estimated from observed values of mean Doppler velocity and spectral width. Figure b graphically illustrates this inversion. Once m and L are estimated from the observed mean Doppler velocity and spectral width (either numerically or using Figure b), N o can be estimated using the measured reflectivity and (). Note that the transformation (V Doppler, W Doppler ) $ (m, L) is independent of absolute calibration of the profiler... 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 D m and rain rate R can be calculated for each retrieval using the fall-speed-to-diameter relationship of (): D m ¼ R R D NðDÞdD ¼ D NðDÞdD R R D þm exp½ LDŠdD D þm exp½ LDŠdD ðþ ¼ Gð þ mþ Gð þ mþl ; R ¼ p Z vd ð ÞD NðDÞdD ¼ p " No Gð þ mþ a ð Þ a þ a # ðþmþ : ðþ L L þm To remove the N o dependence in the rain rate calculation, () is normalized by () to produce the ratio R/z (mm h / (mm m )) [Ulbrich, ]. [] Figure shows the integrated moment SAM model estimates of m and L converted into estimates of D m and R/z as functions of V Doppler and W Doppler. The values of m are shown as contours in Figure to improve panel-to-panel comparisons. In general, the mean volume diameter D m increases with increasing V Doppler and decreasing W Doppler. Conversely, the ratio R/z decreases with increasing V Doppler and decreasing W Doppler. [] Estimates shown in Figure 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. In the regions of

5 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS - a. Reflectivity Weighted Mean Doppler Velocity (color) Reflectivity Weighted Spectral Width ( σ) (contours) As a Function of µ and λ. ms -. Shape Parameter, µ Slope Parameter, λ, cm Spectral Width, ms b. Gamma Function Slope Parameter, λ, (color) Gamma Function Shape Parameter, µ, (contour) As a Function of Spectral Width and Doppler Velocity cm Doppler Velocity, ms - Figure. (a) Reflectivity-weighted mean Doppler velocity (shading) and reflectivity-weighted spectral width (contour) calculated as a function of shape and slope parameters (m and L) of a modified gamma raindrop size distribution. (b) The slope (color) and shape (contour) parameters of a modified gamma raindrop size distribution estimated from the mean Doppler velocity and spectral width. See color version of this figure at back of this issue.

6 - - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS.. a. Mean Volume Diameter, Dm, (color) Gamma Function Shape Parameter, µ, (contour) As a Function of Spectral Width and Doppler Velocity mm.. -. Spectral Width, ms Doppler Velocity, ms b. Rain Rate to Reflectivity Ratio, R/z, (color) Gamma Function Shape Parameter, µ, (contour) As a Function of Spectral Width and Doppler Velocity - log(r/z) Spectral Width, ms Doppler Velocity, ms - -. Figure. (a) Mass-weighted mean diameter D m (color) and shape parameter m (contour) estimated as a function of mean reflectivity-weighted Doppler velocity and spectral width. (b) Same as Figure a but for rain rate to reflectivity ratio R/z (color). See color version of this figure at back of this issue.

7 WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS - 7 large gradients in the V Doppler dimension in Figure, 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 D m and R/z changed by %. In general, the air motion can deviate from zero by ±. and ±. ms before estimates of D m and R/z deviate by %. [] Estimates shown in Figure also assume zero spectral broadening. Spectral broadening causes an increase in the observed spectral width and a shift along the W Doppler dimension in Figure. Note that the gradients in Figure are steeper along the V Doppler axis than along the W Doppler 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 were inserted into (), and the spectral broadening was increased until the estimated D m and R/z change by %. In general, the spectral broadening can increase to approximately. and.7 ms before the estimates of D m and R/z deviate by %.. Sans Air Motion (SAM) Model: Spectra Method [] The spectral method sans air motion (SAM) model estimates the parameters w, s air, N o, m, and L 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... 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 (). The second constraint is the conservation of mean Doppler velocity. Both the model spectrum and observed spectrum must have the same V Doppler 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. [] c ¼ XN i¼ flog½s obs ðv i ÞŠ log½s model ðv i ÞŠg ; ðþ where S obs (v i ) and S model (v i ) 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 c being determined using standard minimization techniques... 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 min of Joss-Waldvogel disdrometer (Distromet, Inc., model RD-) observations collected in August and September 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 m parameter estimated from each D m (±. mm) interval defined the shape of the simulated DSD. For each pair of m and D m, 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 sðþ¼ v S obsðþ v p ffiffiffiffiffiffiffiffiffiffiffiffi ; ðþ NFFT 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.

8 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS Dm Difference (mm) Air motion (ms - ) µ Percent Difference R/z (%) a. Shape Parameter, µ b. Dm Difference, (model - input) c. Air Motion, positive = upward d. Percent Difference of R/z..... Mean Diameter, Dm (mm) Figure. Spectral method SAM model retrievals from simulated spectra as a function of prescribed D m. Circles are the retrieved mean, and vertical lines are standard deviation. (a) DSD shape parameter, where solid line is the model input Table. MHz Profiler Parameters Used During Texas- Florida Underflights Experiment B (TEFLUN-B) Parameter Value Frequency MHz Wavelength. cm Peak power W Antenna m shrouded dish Beam width deg Pulse length or m Maximum height sampled. km Maximum radial velocity ± ms Number of coherent integrations Number of averaged fast Fourier transforms Number of spectral points Number of radar pulses processed, Dwell time s Recording full Doppler spectra [] Figure a shows the modified gamma shape parameter m as a function of D m. The median value from the Joss-Waldvogel disdrometer observations is shown with the solid line, and the mean and standard deviation estimated from the SAM model are shown with circles and vertical lines. As with the other variables shown in Figure, the largest standard deviations in the retrievals are for D m less than approximately. mm. The decrease in m with increasing D m is a consistent feature with this Joss-Waldvogel disdrometer data set. For D m values greater than. mm it appears that this SAM model overestimates D m by approximately. to. mm and has an upward bias of approximately. to. ms (Figures b and c). The simulations also indicate that the rain rate to reflectivity ratio is underestimated by about % for D m values greater than. mm (Figure d).. Data Observations [7] A MHz vertically pointing profiler was deployed in central Florida for August and September in support of the Tropical Rainfall Measuring Mission (TRMM) Ground Validation Program. Table lists the profiler system characteristics. A Joss-Waldvogel disdrometer located next to the profiler recorded over mm of rain during this two-month campaign. The surface disdrometer records the number and size of raindrops hitting the cm sensor head, enabling the direct calculation of reflectivity, rain rate, and D m [Williams et al., ]. Surface disdrometer reflectivity estimates were used to calibrate the MHz profiler as described by Gage et al. []. [] On 7 September a convective precipitating system passed over the profiler site. Figure shows the first three moments of the Doppler spectra for the m pulse length mode in time-altitude cross sections for this

9 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS TEFLUN-B, Triple-N-Ranch, FL, 7 September (Day #) MHz, Pulse Width = m, ALRC =. a. Reflectivity dbze b. Doppler Velocity - ms c. Spectral Width ms- : : : : : Hour of Day (UT) : : Figure. Time-altitude cross section of (a) reflectivity, (b) mean Doppler velocity (downward is red), and (c) spectral width for MHz profiler observations at Triple-N-Ranch in central Florida on 7 September. TEFLUN-B, Texas-Florida Underflights Experiment B. The Aeronomy Laboratory radar constant (ALRC) determines the reflectivity calibration and was. for this installation. See color version of this figure at back of this issue.

10 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS hour event. During hour UTC the reflectivities exceeding dbz and net upward mean Doppler velocities indicate an intense convective rain regime. During to UTC a well-defined bright band near. 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 a and b show the vertical profile of stratiform rain observed at : UTC, with Figure a showing the reflectivity spectral density at each spectral point using log units and Figure b showing the total reflectivity at each altitude. The mean downward Doppler velocities of to ms above the bright band correspond to the fall speeds of ice particles, and the downward velocities of to ms 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., ]. 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 c and d show the vertical profile of convective rain observed at : UTC. Near km the reflectivity spectral density has downward velocities exceeding ms. The raindrops do not have terminal velocities exceeding ms, but rather, the raindrops are in a downdraft with severe turbulence. Near. to. km the mean Doppler velocity is approximately ms upward, and the total reflectivity is greater than 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 c with black asterisks. The retrieved air motions are consistent with the physical interpretation of the observed Doppler velocity spectra with downward air motions between. and. km and upward motions above. km... Integrated Moment Method SAM Model: Surface Comparisons [] Figure shows the min resolution surface disdrometer and MHz profiler integrated moment method SAM model retrieved reflectivity, rain rate, and mass-weighted mean diameter at an altitude of m for the 7 September rain event. Even though one instrument observes precipitation at the surface and the other observes 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 to UTC, the profiler D m (R) is less (greater) than the disdrometer estimates. As will be seen in section., this time interval corresponds to upward ambient air motions retrieved at m using the spectral method SAM model. Not accounting for the updraft in the profiler retrievals underestimates D m and overestimates R... Spectral Method SAM Model: Time-Altitude Retrievals [] Figure 7 shows the spectral-method-sam-modelretrieved parameters for the 7 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 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 to UTC, the high rain rates between and km are associated with small D m, with larger D m values near the surface. During this convective rain event the vertical air motion (Figure 7e) indicates a critical level near 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, ]. [] During the stratiform rain regime from to 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 D m 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.. Discussion and Conclusions [] The sans air motion (SAM) model estimates the raindrop size distribution from only the Rayleigh scatter-

11 WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS - TEFLUN-B, 7 September, :: UTC MHz, Pulse Width = m, ALRC =. b. Reflectivity dbze/ms a. Reflectivity Spectral Density (dbze/(ms - )) Doppler Velocity (Upward) Doppler Velocity (ms - ) (Downward) (dbze) TEFLUN-B, 7 September, :: UTC MHz, Pulse Width = m, ALRC =. c. Reflectivity Spectral Density (dbze/(ms - )) d. Reflectivity dbze/ms - Doppler Velocity Air Motion (Upward) Doppler Velocity (ms - ) (Downward) (dbze) Figure. Reflectivity spectral density in units of dbz/ms and total reflectivity at each range gate. (a and b) Stratiform rain on 7 September at : UTC. Mean Doppler velocity and total reflectivity are indicated with blue asterisks. (c and d) Convective rain on 7 September at : UTC. Spectral method SAM model air motion estimates are indicated with black asterisks. See color ersion of this figure at back of this issue

12 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS TEFLUN-B, Triple-N-Ranch, FL, 7 September (Day #) JWD Disdrometer and MHz at m, Integrated Moment Method a. Reflectivity Reflectivity (dbze) JWD Surface Disdrometer Profiler at m Rain Rate (mm hr - ). Diameter (mm) b. Rain Rate c. Mean Mass Weighted Diameter, Dm JWD Surface Disdrometer Profiler at m JWD Surface Disdrometer Profiler at m : : : : : Hour of Day (UT) : : Figure. Surface disdrometer observations and MHz profiler integrated moment SAM model retrievals at m above the ground for 7 September. (a) Observed reflectivity, (b) rain rate, and (c) massweighted mean diameter. JWD, Joss-Waldvogel disdrometer. See color version of this figure at back of this issue.

13 WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS - TEFLUN-B, Triple-N-Ranch, FL, 7 September (Day #) MHz, Pulse Width = m, ALRC =., Spectral Method a. Reflectivity b. Rain Rate c. Mean Mass Weighted Diameter, Dm : : : : : : : Hour of Day (UT) dbz e mm mm hr d. Gamma DSD Shape Parameter, µ - e. Vertical Ambient Air Motion, ω, Positive = Upward f. Turbulent Broadening, σ : : : : : : : Hour of Day (UT) ms - ms Figure 7. Spectral method SAM model retrievals for MHz profiler observations from m to km above the ground for 7 September. (a) reflectivity, (b) rain rate, (c) mean mass-weighted diameter, (d) shape parameter, (e) vertical ambient air motion (upward motion is positive), and (f ) spectral broadening. See color version of this figure at back of this issue.

14 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS ing 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 % errors in mass-weighted mean diameter and rain rate occur for air motions exceeding ±. and ±. ms, 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 D m by approximately. to. mm and has an upward bias of approximately. to. ms which results in underestimating the rain rate to reflectivity ratio by about %. 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 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 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. [7] 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., ; Testud et al., ]. 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., D m 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 and UTC and. and km. Even though the spectral broadening is enhanced during this period (Figure 7f ), underestimating the spectral broadening results in overestimating the DSD width (decreasing m). The increased DSD width results in underestimating D m and overestimating the rain rate. The retrieved rain rate during this period exceeds mm h, with updrafts exceeding ms. 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

15 WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS - 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. [] Acknowledgments. 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 NAG-7. References Atlas, D., and C. W. Ulbrich, An observationally based conceptual model of warm oceanic convective rain in the tropics, J. Appl. Meteorol.,,,. Atlas, D., R. S. Srivastava, and R. S. Sekhon, Doppler radar characteristics of precipitation at vertical incidence, Rev. Geophys.,,, 7. Cifelli, R., and S. A. Rutledge, Vertical motion structure in maritime continent mesoscale convective systems: Results from a MHz profiler, J. Atmos. Sci.,,,. Cifelli, R., and S. A. Rutledge, Vertical motion, diabatic heating and rainfall characteristics in N. Australia convective systems, Q. J. R. Meteorol. Soc.,,,. Cifelli, R. C., C. R. Williams, W. L. Ecklund, D. K. Rajopadhyaya, S. K. Avery, K. S. Gage, and P. T. May, Drop-size distribution characteristics in tropical mesoscale convective systems, J. Appl. Meteorol.,, 7 777,. Currier, P. E., S. K. Avery, B. B. Balsley, and K. S. Gage, Use of two wind profilers for precipitation studies, Geophys. Res. Lett.,, 7,. Gage, K. S., C. R. Williams, W. L. Ecklund, and P. E. Johnston, Use of two profilers during MCTEX for unambiguous identification of Bragg scattering and Rayleigh scattering, J. Atmos. Sci.,, 7,. Gage, K. S., C. R. Williams, P. E. Johnston, W. L. Ecklund, R. Cifelli, A. Tokay, and D. A. Carter, Doppler radar profilers as calibration tools for scanning radars, J. Appl. Meteorol.,,,. Gossard, E. E., Measuring drop size distributions in cloud with clear-air sensing Doppler radar, J. Atmos. Oceanic Technol.,,,. Gossard, E. E., Measurement of cloud droplet spectra by Doppler radar, J. Atmos. Oceanic Technol.,, 7 7,. Hauser, D., and P. Amayenc, A new method for deducing hydrometeor-size distributions and vertical air motions from Doppler radar measurements at vertical incidence, J. Appl. Meteorol.,, 7,. Hauser, D., and P. Amayenc, Exponential size distributions of raindrops and vertical air motions deduced from vertically pointing Doppler radar using a new method, J. Clim. Appl. Meteorol.,, 7,. May, P. T., and D. K. Rajopadhyaya, Wind profiler observations of vertical motion and precipitation microphysics of a tropical squall line, Mon. Weather Rev.,,,. Rajopadhyaya, D. K., P. T. May, and R. A. Vincent, A general approach to the retrieval of raindrop size distributions from wind profiler Doppler spectra: Modeling results, J. Atmos. Oceanic Technol.,, 7 77,. Rajopadhyaya, D. K., P. T. May, R. C. Cifelli, S. K. Avery, C. R. Williams, W. L. Ecklund, and K. S. Gage, The effect of vertical air motions on rain rates and median volume diameter determined from combined UHF and VHF wind profiler measurements, J. Atmos. Oceanic Technol.,,,. Rajopadhyaya, D. K., S. K. Avery, P. T. May, and R. C. Cifelli, Comparison of precipitation estimation using single- and dual-frequency wind profilers: Simulations and experimental results, J. Atmos. Oceanic Technol.,, 7,. Rogers, R. R., D. Baumgardner, S. A. Ethier, D. A. Carter, and W. L. Ecklund, Comparison of raindrop size distributions measured by radar wind profiler and by airplane, J. Appl. Meteorol.,,,. Sangren, K. L., P. S. Ray, and G. B. Walker, A comparison of techniques to estimate vertical air motions and raindrop size distributions, J. Atmos. Oceanic Technol.,,,. Sato, T., D. Hiroshi, H. Iwai, I. Kimura, S. Fukao, M. Yamamoto, T. Tsuda, and S. Kato, Computer processing for deriving drop-size distributions and vertical air velocities form VHF Doppler radar spectra, Radio Sci.,, 7,. Sempere-Torres, D., J. M. Porrà, and J.-D. Creutin, A general formulation for raindrop size distribution, J. Appl. Meteorol.,,,. Tennekes, H., and J. L. Lumley, A First Course in Turbulence, pp., MIT Press, Cambridge, Mass., 7. Testud, J., S. Oury, R. A. Black, P. Amayenc, and X. Dou, The concept of Normalized distribution to describe raindrop spectra: a tool for cloud physics and cloud remote sensing, J. Appl. Meteorol.,,,. Ulbrich, C. W., Natural variations in the analytical form of the raindrop size distribution, J. Clim. Appl. Meteorol.,, 7 77,. Ulbrich, C. W., Algorithms for determination of rainfall integral parameters using reflectivity factor and mean Doppler fall speed at vertical incidence, J. Atmos. Oceanic Technol.,,,. Ulbrich, C. W., and D. Atlas, Rainfall microphysics and radar properties: Analysis methods for drop size spectra, J. Appl. Meteorol., 7,,. Ulbrich, C. W., and P. B. Chilson, Effects of variations in precipitation size distribution and fallspeed law parameters on relations between mean Doppler fallspeed and reflectivity factor, J. Atmos. Oceanic Technol.,,,. Wakasugi, K., A. Mizutani, M. Matsuo, S. Fukao, and S. Kato, A direct method for deriving drop-size distributions and vertical air velocities from VHF Doppler radar spectra, J. Atmos. Oceanic Technol.,,,.

16 - WILLIAMS: SIMULTANEOUS AMBIENT AIR MOTION AND RAINDROP SIZE DISTRIBUTIONS Wakasugi, K., A. Mizutani, M. Matsuo, S. Fukao, and S. Kato, Further discussion on deriving dropsize distribution and vertical air velocities directly from VHF Doppler radar spectra, J. Atmos. Oceanic Technol.,, 7 7, 7. Williams, C. R., W. L. Ecklund, and K. S. Gage, Classification of precipitating clouds in the tropics using MHz wind profilers, J. Atmos. Oceanic Technol.,,,. Williams, C. R., A. Kruger, K. S. Gage, A. Tokay, R. Cifelli, W. F. Krajewski, and C. Kummerow, Comparison of simultaneous raindrop size distributions estimated from two surface disdrometers and a UHF profiler, Geophys. Res. Lett., 7, 7 7,. C. R. Williams, CIRES/NOAA Aeronomy Laboratory, Mail Stop R/AL, Broadway, Boulder, CO -7, USA. (cwilliams@al.noaa.gov)

17 RADIO SCIENCE, VOL. 7, NO.,./RS, a. Reflectivity-Weighted Mean Doppler Velocity (shading) Reflectivity-Weighted Spectral Width ( σ) (contour) As a Function of µ and λ ms -.. Shape Parameter, µ Slope Parameter, λ, cm -. 7 Spectral Width, ms b. Gamma Function Slope Parameter, λ, (color) Gamma Function Shape Parameter, µ, (contour) As a Function of Spectral Width and Doppler Velocity cm Doppler Velocity, ms - Figure. (a) Reflectivity-weighted mean Doppler velocity (shading) and reflectivity-weighted spectral width (contour) calculated as a function of shape and slope parameters (m and L) of a modified gamma raindrop size distribution. (b) The slope (color) and shape (contour) parameters of a modified gamma raindrop size distribution estimated from the mean Doppler velocity and spectral width. -

18 - RADIO SCIENCE, VOL. 7, NO.,./RS,.. a. Mean Mass-Weighted Diameter, Dm, (color) Gamma Function Shape Parameter, µ, (contour) As a Function of Spectral Width and Doppler Velocity mm.. -. Spectral Width, ms Doppler Velocity, ms b. Rain Rate to Reflectivity ratio, R/z, (color) Gamma Function Shape Parameter, µ, (contour) As a Function of Spectral Width and Doppler Velocity - log(r/z) Spectral Width, ms Doppler Velocity, ms - -. Figure. (a) Mass-weighted mean diameter D m (color) and shape parameter m (contour) estimated as a function of mean reflectivity-weighted Doppler velocity and spectral width. (b) Same as Figure a but for rain rate to reflectivity ratio R/z (color). -

19 RADIO SCIENCE, VOL. 7, NO.,./RS, TEFLUN-B, Triple-N-Ranch, FL, 7 September (Day #) MHz, Pulse Width = m, ALRC =. a. Reflectivity dbze ms- Upward b. Doppler Velocity Downward c. Spectral Width - ms : : : : : Hour of Day (UT) : : Figure. Time-altitude cross section of (a) reflectivity, (b) mean Doppler velocity (downward is red), and (c) spectral width for MHz profiler observations at Triple-N-Ranch in central Florida on 7 September. TEFLUN-B, Texas-Florida Underflights Experiment B. The Aeronomy Laboratory radar constant (ALRC) determines the reflectivity calibration and was. for this installation. -

20 RADIO SCIENCE, VOL. 7, NO.,./RS, TEFLUN-B, 7 September, :: UTC a. Reflectivity Spectral Density (dbze/(ms- )) b. Reflectivity Doppler Velocity dbze/ms (Upward) Doppler Velocity (ms-) (Downward) dbze - 7 TEFLUN-B, 7 September, :: UTC - c. Reflectivity Spectral Density (dbze/(ms )) d. Reflectivity Doppler Velocity Air Motion (Upward) Doppler Velocity (ms ) (Downward) dbze dbze/ms Figure. Reflectivity spectral density in units of dbz/ms and total reflectivity at each range gate. (a and b) Stratiform rain on 7 September at : UTC. Mean Doppler velocity and total reflectivity are indicated with blue asterisks. (c and d) Convective rain on 7 September at : UTC. Spectral method SAM model air motion estimates are indicated with black asterisks

21 RADIO SCIENCE, VOL. 7, NO.,./RS, TEFLUN-B, Triple-N-Ranch, FL, 7 September (Day #) JWD Disdrometer and MHz at m, Integrated Moment Method a. Reflectivity Reflectivity (dbze) JWD Surface Disdrometer Profiler at m Rain Rate (mm hr - ). Diameter (mm) b. Rain Rate JWD Surface Disdrometer Profiler at m c. Mean Mass-Weighted Diameter, Dm JWD Surface Disdrometer Profiler at m : : : : : Hour of Day (UT) : : Figure. Surface disdrometer observations and MHz profiler integrated moment SAM model retrievals at m above the ground for 7 September. (a) Observed reflectivity, (b) rain rate, and (c) massweighted mean diameter. JWD, Joss-Waldvogel disdrometer. -

22 RADIO SCIENCE, VOL. 7, NO.,./RS, TEFLUN-B, Triple-N-Ranch, FL, 7 September (Day #) MHz, Pulse Width = m, ALRC =., Spectral Method Reflectivity Rain Rate Mean Mass Weighted Diameter, Dm : : : : : : : Hour of Day (UT) dbz e mm hr - mm Gamma DSD Shape Parameter, µ Vertical Ambient Air Motion, ω, Positive = Upward f. Spectral Broadening, σ Downward : : : : : : : Hour of Day (UT) ms - ms c. a. b. e. d. - upward. Figure 7. Spectral method SAM model retrievals for MHz profiler observations from m to km above the ground for 7 September. (a) reflectivity, (b) rain rate, (c) mean mass-weighted diameter, (d) shape parameter, (e) vertical ambient air motion (upward motion is positive), and (f ) spectral broadening. -