360
Eugene E. Clothiaux, Thomas P. Ackerman, David M. Babb
The measured optical depth Tm,ir is inferred from the infrared window measurements of sky
brightness temperature using (13.85).
As before, the radar reflectivity is defined by (13.90b), where Nt and Dn are now a function of
height, i.e., Nt,j and Dn,j respectively. The reflectivity-weighted mean Doppler velocity due to
all of the particles in a radar resolution volume is
v . _ Jo oo vj(D)D 6 n mg (D)dD
/,) -
J;' D6 nmg (D)dD.
(13.93)
Based on Pruppacher and Klett (1978), Matrosov et al. (1994) assume that the fall speed of
the particles are related to the diameter by
v/(D) = ADB
(13.94)
where A and B are constants that depend upon the crystal shape. Substituting (13.94) into
(13.93) and using (13.77b) yields
V = A [r(v + B + 6)] DB
(13.95)
1,)
r(v + 6)
n,)'
where air density and viscosity effects have been neglected.
At this point 2Ng + 2 unknowns have entered into the set of equations, i.e., Nt,j and Dn,j where
j = 1, ... , Ng , A and B. However, there are only 2Ng + 1 measurements consisting of Zj and
V/,j, where j = 1, ... , Ng , and Tm,ir' Since the variation of B is relatively small, i.e., 0.75 to 1.4,
Matrosov et al. (1994) set B = 1. Picking an initial value of Ao for A, Matrosov et al. (1994)
solve (13.90b) and (13.95) for Nt,j and Dn,j at each range gate. Computing the infrared window
optical depth Tir from (13.92) and comparing the result with the measured optical depth Tm,ir,
they are able to derive a new value for A, i.e.,
( . )B/4
A = Ao Tm,1T
,
TIT
(13.96)
such that when (13.92) is recomputed with this value of A and the existing values of Nj,t and
Dj,t, the measured optical depth Tm,ir results. Using A from (13.96) to solve (13.90b) and
(13.95) for Nt,j and Dn,j, Matrosov et al. (1994) arrive at their final estimates of the particle
concentration and scaling diameter as a function of height within the cloud layer. The ice
water contents IWC and the ice water path follow immediately from (13.78a) and (13.78b),
respectively.
For the retrievals to work, the reflectivity-weighted mean Doppler velocity V/,j must represent
the fall velocities of the particles in the air with no vertical motion. Since there are generally
updrafts and downdrafts within a cloud, the raw radar derived value of VI,; cannot be used
in (13.95). Based on work by Orr and Kropfli (1993), Matrosov et al. (1994) circumvent
this problem by averaging over several hours all of the Doppler velocities Vj,j at each resolution
volume whose corresponding reflectivities fall within a narrow reflectivity bin. If the reflectivity
value is indicative of a specific particle distribution within the radar resolution volume and if
the average vertical motion of the air over the averaging interval is near zero, then the average
velocity generated for each reflectivity bin of each resolution volume corresponds to the vertical
fall speed of the particles in still air at the resolution volume location. Therefore, the reflectivity
observed for each resolution volume is used as an index into the reflectivity bins from which an
estimate of the average, still air particle fall speed is recovered.
Cloud Drop Size Distribtions from the First Three Radar Doppler Moments
Frisch et al. (1995) utilize the first three spectral moments, i.e., reflectivity, Doppler velocity
Eugene E. Clothiaux, Thomas P. Ackerman, David M. Babb
The measured optical depth Tm,ir is inferred from the infrared window measurements of sky
brightness temperature using (13.85).
As before, the radar reflectivity is defined by (13.90b), where Nt and Dn are now a function of
height, i.e., Nt,j and Dn,j respectively. The reflectivity-weighted mean Doppler velocity due to
all of the particles in a radar resolution volume is
v . _ Jo oo vj(D)D 6 n mg (D)dD
/,) -
J;' D6 nmg (D)dD.
(13.93)
Based on Pruppacher and Klett (1978), Matrosov et al. (1994) assume that the fall speed of
the particles are related to the diameter by
v/(D) = ADB
(13.94)
where A and B are constants that depend upon the crystal shape. Substituting (13.94) into
(13.93) and using (13.77b) yields
V = A [r(v + B + 6)] DB
(13.95)
1,)
r(v + 6)
n,)'
where air density and viscosity effects have been neglected.
At this point 2Ng + 2 unknowns have entered into the set of equations, i.e., Nt,j and Dn,j where
j = 1, ... , Ng , A and B. However, there are only 2Ng + 1 measurements consisting of Zj and
V/,j, where j = 1, ... , Ng , and Tm,ir' Since the variation of B is relatively small, i.e., 0.75 to 1.4,
Matrosov et al. (1994) set B = 1. Picking an initial value of Ao for A, Matrosov et al. (1994)
solve (13.90b) and (13.95) for Nt,j and Dn,j at each range gate. Computing the infrared window
optical depth Tir from (13.92) and comparing the result with the measured optical depth Tm,ir,
they are able to derive a new value for A, i.e.,
( . )B/4
A = Ao Tm,1T
,
TIT
(13.96)
such that when (13.92) is recomputed with this value of A and the existing values of Nj,t and
Dj,t, the measured optical depth Tm,ir results. Using A from (13.96) to solve (13.90b) and
(13.95) for Nt,j and Dn,j, Matrosov et al. (1994) arrive at their final estimates of the particle
concentration and scaling diameter as a function of height within the cloud layer. The ice
water contents IWC and the ice water path follow immediately from (13.78a) and (13.78b),
respectively.
For the retrievals to work, the reflectivity-weighted mean Doppler velocity V/,j must represent
the fall velocities of the particles in the air with no vertical motion. Since there are generally
updrafts and downdrafts within a cloud, the raw radar derived value of VI,; cannot be used
in (13.95). Based on work by Orr and Kropfli (1993), Matrosov et al. (1994) circumvent
this problem by averaging over several hours all of the Doppler velocities Vj,j at each resolution
volume whose corresponding reflectivities fall within a narrow reflectivity bin. If the reflectivity
value is indicative of a specific particle distribution within the radar resolution volume and if
the average vertical motion of the air over the averaging interval is near zero, then the average
velocity generated for each reflectivity bin of each resolution volume corresponds to the vertical
fall speed of the particles in still air at the resolution volume location. Therefore, the reflectivity
observed for each resolution volume is used as an index into the reflectivity bins from which an
estimate of the average, still air particle fall speed is recovered.
Cloud Drop Size Distribtions from the First Three Radar Doppler Moments
Frisch et al. (1995) utilize the first three spectral moments, i.e., reflectivity, Doppler velocity