390
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
independent variables are needed to quantify the variability of PSDs within radar
observing volumes. The radar reflectivity observations are sensitive to both these
parameters. Therefore, one cannot accurately describe the PSD within an observing
volume from a single-parameter radar observation. To mitigate this indeterminacy,
it is customary in retrievals from spaceborne single-frequency radar observations to
assume that one PSD parameter can be derived from independent considerations and
to explicitly retrieve the other parameter (Ferreira et al. 2001; Grecu and Anagnostou
2002). Specifically, the PSDs are assumed to follow a normalized gamma distribution, that is, the number of particles of a given size D is
N D N f
D D
D D
w
( )
( )(
) exp[ ( .
)(
)]
=
−
+
µ
µ
µ
/
/
0
0
3 67
,
(16.1)
where
f ( )
.
( .
)
(
)
µ
µ
µ
µ
=
+
+
+
6
3 7
3 67
4
4
4
Γ
.
(16.2)
N w (cm –1 m –3 ) is a normalized concentration parameter equivalent to that of an exponential PSD with the same hydrometeor water content and particle median volume
diameter D 0 (cm), and parameter μ is a shape factor. Given the normalized gamma
distribution, D 0 is explicitly retrieved for each bin within a vertical profile of radar
reflectivity, while only one parameter is used to describe the vertical variability of
N w . This parameter is inferred from the condition that physical variables such as path
integrated attenuation (PIA) and brightness temperatures simulated from the radar
retrievals are in agreement with the actual observations of these variables. It should
be mentioned that, although there are no actual PIA observations, PIA estimates can
be derived from the analysis of the radar surface return (Meneghini et al. 2000). The
shape factor, μ, has a smaller impact on the reflectivity rain relationships (Ferreira
et al. 2001) and is usually assumed known.
From the practical perspective, it is useful to determine how sensitive the brightness temperatures simulated from radar retrievals are to the normalized concentration parameter N w . The simulated PIAs are known to be highly sensitive to N w
(Ferreira et al. 2001; Grecu and Anagnostou 2002), but the determination of N w
based on PIA estimates from the surface reference technique (SRT) of Meneghini
et al. (2000) may be subject to large uncertainties, because the SRT PIAs estimates
are subject to large uncertainties (especially over land). It is therefore beneficial to
incorporate brightness temperature information into the determination of N w .
To investigate the sensitivity of simulated brightness temperatures to N w , TRMM
observations can be used. Shown in the top panel of Figure 16.1 is a vertical cross
section through the field of reflectivity observations collected by the TRMM PR over
Oklahoma on November 30, 2006. The associated water content field derived by the
application of the radar-only retrieval algorithm of Grecu and Anagnostou (2002)
is shown in the bottom panel of Figure 16.1. The retrieval algorithm is based on
an analytical attenuation correction procedure developed by Hitschfeld and Bordan
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
independent variables are needed to quantify the variability of PSDs within radar
observing volumes. The radar reflectivity observations are sensitive to both these
parameters. Therefore, one cannot accurately describe the PSD within an observing
volume from a single-parameter radar observation. To mitigate this indeterminacy,
it is customary in retrievals from spaceborne single-frequency radar observations to
assume that one PSD parameter can be derived from independent considerations and
to explicitly retrieve the other parameter (Ferreira et al. 2001; Grecu and Anagnostou
2002). Specifically, the PSDs are assumed to follow a normalized gamma distribution, that is, the number of particles of a given size D is
N D N f
D D
D D
w
( )
( )(
) exp[ ( .
)(
)]
=
−
+
µ
µ
µ
/
/
0
0
3 67
,
(16.1)
where
f ( )
.
( .
)
(
)
µ
µ
µ
µ
=
+
+
+
6
3 7
3 67
4
4
4
Γ
.
(16.2)
N w (cm –1 m –3 ) is a normalized concentration parameter equivalent to that of an exponential PSD with the same hydrometeor water content and particle median volume
diameter D 0 (cm), and parameter μ is a shape factor. Given the normalized gamma
distribution, D 0 is explicitly retrieved for each bin within a vertical profile of radar
reflectivity, while only one parameter is used to describe the vertical variability of
N w . This parameter is inferred from the condition that physical variables such as path
integrated attenuation (PIA) and brightness temperatures simulated from the radar
retrievals are in agreement with the actual observations of these variables. It should
be mentioned that, although there are no actual PIA observations, PIA estimates can
be derived from the analysis of the radar surface return (Meneghini et al. 2000). The
shape factor, μ, has a smaller impact on the reflectivity rain relationships (Ferreira
et al. 2001) and is usually assumed known.
From the practical perspective, it is useful to determine how sensitive the brightness temperatures simulated from radar retrievals are to the normalized concentration parameter N w . The simulated PIAs are known to be highly sensitive to N w
(Ferreira et al. 2001; Grecu and Anagnostou 2002), but the determination of N w
based on PIA estimates from the surface reference technique (SRT) of Meneghini
et al. (2000) may be subject to large uncertainties, because the SRT PIAs estimates
are subject to large uncertainties (especially over land). It is therefore beneficial to
incorporate brightness temperature information into the determination of N w .
To investigate the sensitivity of simulated brightness temperatures to N w , TRMM
observations can be used. Shown in the top panel of Figure 16.1 is a vertical cross
section through the field of reflectivity observations collected by the TRMM PR over
Oklahoma on November 30, 2006. The associated water content field derived by the
application of the radar-only retrieval algorithm of Grecu and Anagnostou (2002)
is shown in the bottom panel of Figure 16.1. The retrieval algorithm is based on
an analytical attenuation correction procedure developed by Hitschfeld and Bordan