[11] From (3) and (5), five unknowns (, σ_{air}, *N*_{o}, 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

[12] Assuming that the hydrometeor size distribution can be expressed by a modified gamma distribution, the total reflectivity factor *z* (in units of mm^{6} m^{−3}) is estimated from the zeroth moment of the profiler Doppler velocity spectrum and from the sixth power of the DSD

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, Γ is the complete gamma operator, and the last equality is derived after the integration limits are allowed to be *D*_{min} 0 and *D*_{max} ∞. The reflectivity factor measured by calibrated profilers is a function of the three DSD parameters, *N*_{o}, μ, and Λ.

[13] The observed reflectivity-weighted mean Doppler velocity, *V*_{Doppler} (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 *v*_{fall 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.)

[14] The reflectivity-weighted mean fall speed of the DSD, *V*_{fall 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* [1994] showed that by letting the integration limits become *D*_{min} 0 and *D*_{max} ∞, the mean Doppler velocity simplifies to

when using the fall-speed-to-diameter relationship derived by *Atlas et al.* [1973],

with α_{1} = 9.65 ms^{−1}, α_{2} = 10.3 ms^{−1}, and α_{3} = 0.6 ms^{−1}. The variable *v*_{fall speed}(*D*) has units of ms^{−1}, and the equivalent spherical diameter *D* has units of millimeters. The factor (ρ_{o}/ρ)^{m} represents the adjustment in terminal fall speed due to decreasing atmospheric density with altitude expressed by ρ, relative to the surface density ρ_{o}. A value of *m* = 0.4 is used in this study.

[15] The relations in (7) and (8) imply that for a given fall-speed-to-diameter relationship and, every value of *V*_{Doppler} 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.

[16] The reflectivity-weighted Doppler velocity variance, σ_{z}^{2} (in units of m^{2}s^{−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 *D*_{min} 0 and *D*_{max} ∞, 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.

[17] 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 *W*_{Doppler} = 2σ_{z} (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), *N*_{o} can be estimated using the measured reflectivity and (6). Note that the transformation (*V*_{Doppler}, *W*_{Doppler}) (μ, Λ) is independent of absolute calibration of the profiler.

#### 3.2. Sensitivity to Nonzero Air Motion and Nonzero Spectral Broadening

[18] 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 (9):

To remove the *N*_{o} dependence in the rain rate calculation, (13) is normalized by (6) to produce the ratio *R*/*z* (mm h^{−1} /(mm^{6} m^{−3})) [*Ulbrich*, 1992].

[19] Figure 2 shows the integrated moment SAM model estimates of μ and Λ converted into estimates of *D*_{m} and *R*/*z* as functions of *V*_{Doppler} and *W*_{Doppler}. The values of μ are shown as contours in Figure 2 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}.

[20] 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 *V*_{Doppler} 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 *D*_{m} 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 *D*_{m} and *R*/*z* deviate by 10%.

[21] 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 *W*_{Doppler} dimension in Figure 2. Note that the gradients in Figure 2 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 2 were inserted into (3), and the spectral broadening was increased until the estimated *D*_{m} 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 *D*_{m} and *R*/*z* deviate by 10%.