313
Radar Polarimetry for Rain Estimation
Several researchers (e.g., Ulbrich 1983; Chandrasekar and Bringi 1987; Haddad et
al. 1997) have shown that the retrieved three parameters of the gamma model (N 0 , μ,
and Λ) are not mutually independent. Through disdrometer observations, Zhang et
al. (2001) and Brandes et al. (2004a) found that µ is highly related to Λ. Zhang et al.
(2001) proposed the C-G DSD model with an empirical μ – Λ relation. Reducing the
observation error effect through the method of DSD sorting and averaging based on
two parameters (SATP), Cao et al. (2008) refined the μ – Λ relation with disdrometer
observations in central Oklahoma as
μ = –0.0201Λ 2 + 0.902Λ – 1.718.
(13.14)
Zhang et al. (2003) and Cao and Zhang (2009) further showed that the constraint
relation was physically meaningful. The C-G model, reducing the gamma model to
two parameters, facilitates DSD retrieval from dual-polarization or dual-frequency
radar measurements. Meanwhile, it represents natural DSDs better than other oneor two-parameter models (Brandes et al. 2004b; Zhang et al. 2006; Cao et al. 2008).
13.3.2.2 DSD Retrieval
In a DSD model, rain properties can be retrieved through polarimetric radar measurements, presumably with several different approaches. Previous studies mainly
applied a direct approach to retrieve the DSD (Zhang et al. 2001; Gorgucci et al.
2002, 2008; Brandes et al. 2004a,b; Meneghini and Liao 2007; Anagnostou et
al. 2008). Error in radar measurements had seldom been considered in the direct
approach. Recently, a Bayesian approach and a variational approach have been introduced by Cao et al. (2009, 2010). These two methods aim at the optimal use of radar
measurements to improve the DSD retrieval by reducing the effect of measurement
error. The rest of this subsection briefly describes the direct approach but places
emphasis on the Bayesian and variational approaches for DSD retrieval.
13.3.2.2.1 Direct Approach
In a direct approach, the unknown DSD parameters are solved directly and deterministically from radar measurements. This approach implies that radar measurements
could represent the truth of rain properties. If an exponential model is applied, two
measurements, such as radar reflectivity and differential reflectivity, are required to
solve two DSD parameters (using Equations 13.1 and 13.2). Any polarimetric mea3.1 and 13.2). Any polarimetric mea.1 and 13.2). Any polarimetric mea3.2). Any polarimetric mea.2). Any polarimetric measurement can be used to do the retrieval. Considering that measurement error may
propagate into the retrieval result, direct retrieval mostly applies to Z h and Z dr , which
are believed to be more reliable than other measurements. To solve Equations 13.1
and 13.2, backscattering amplitudes of raindrops are needed. In general, these values
are computed theoretically using the T-matrix method based on assumptions related
to raindrop shape, canting angle, frequency, and temperature (Zhang et al. 2001). The
uncertainty of the direct approach would come partially from these assumptions.
Figure 13.2d shows an example of applying the direct DSD retrieval, in which
the Z h and Z dr measurements as well as a C-G DSD model are used. It is shown that
the DSD retrieval has a very similar result to the R(Z h ,Z dr ) estimator, except for the
north/southwest region (>100 km), where the radar echoes come partially from the
Radar Polarimetry for Rain Estimation
Several researchers (e.g., Ulbrich 1983; Chandrasekar and Bringi 1987; Haddad et
al. 1997) have shown that the retrieved three parameters of the gamma model (N 0 , μ,
and Λ) are not mutually independent. Through disdrometer observations, Zhang et
al. (2001) and Brandes et al. (2004a) found that µ is highly related to Λ. Zhang et al.
(2001) proposed the C-G DSD model with an empirical μ – Λ relation. Reducing the
observation error effect through the method of DSD sorting and averaging based on
two parameters (SATP), Cao et al. (2008) refined the μ – Λ relation with disdrometer
observations in central Oklahoma as
μ = –0.0201Λ 2 + 0.902Λ – 1.718.
(13.14)
Zhang et al. (2003) and Cao and Zhang (2009) further showed that the constraint
relation was physically meaningful. The C-G model, reducing the gamma model to
two parameters, facilitates DSD retrieval from dual-polarization or dual-frequency
radar measurements. Meanwhile, it represents natural DSDs better than other oneor two-parameter models (Brandes et al. 2004b; Zhang et al. 2006; Cao et al. 2008).
13.3.2.2 DSD Retrieval
In a DSD model, rain properties can be retrieved through polarimetric radar measurements, presumably with several different approaches. Previous studies mainly
applied a direct approach to retrieve the DSD (Zhang et al. 2001; Gorgucci et al.
2002, 2008; Brandes et al. 2004a,b; Meneghini and Liao 2007; Anagnostou et
al. 2008). Error in radar measurements had seldom been considered in the direct
approach. Recently, a Bayesian approach and a variational approach have been introduced by Cao et al. (2009, 2010). These two methods aim at the optimal use of radar
measurements to improve the DSD retrieval by reducing the effect of measurement
error. The rest of this subsection briefly describes the direct approach but places
emphasis on the Bayesian and variational approaches for DSD retrieval.
13.3.2.2.1 Direct Approach
In a direct approach, the unknown DSD parameters are solved directly and deterministically from radar measurements. This approach implies that radar measurements
could represent the truth of rain properties. If an exponential model is applied, two
measurements, such as radar reflectivity and differential reflectivity, are required to
solve two DSD parameters (using Equations 13.1 and 13.2). Any polarimetric mea3.1 and 13.2). Any polarimetric mea.1 and 13.2). Any polarimetric mea3.2). Any polarimetric mea.2). Any polarimetric measurement can be used to do the retrieval. Considering that measurement error may
propagate into the retrieval result, direct retrieval mostly applies to Z h and Z dr , which
are believed to be more reliable than other measurements. To solve Equations 13.1
and 13.2, backscattering amplitudes of raindrops are needed. In general, these values
are computed theoretically using the T-matrix method based on assumptions related
to raindrop shape, canting angle, frequency, and temperature (Zhang et al. 2001). The
uncertainty of the direct approach would come partially from these assumptions.
Figure 13.2d shows an example of applying the direct DSD retrieval, in which
the Z h and Z dr measurements as well as a C-G DSD model are used. It is shown that
the DSD retrieval has a very similar result to the R(Z h ,Z dr ) estimator, except for the
north/southwest region (>100 km), where the radar echoes come partially from the