316
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
Z H and Z DR could be considered to have independent observation errors. Most previous Bayesian studies have only considered the observation error and assumed ρ = 0.
If the C-G DSD model were applied in the forward operator of Z H and Z DR , model
error would be introduced, and the model errors of Z H and Z DR tend to be correlated.
Although ρ should vary with different Z H −Z DR pairs, the effect of ρ on the retrieval is
not essential. For the sake of simplicity, Cao et al. (2010) assumed a constant ρ = 0.5,
which denotes a moderate correlation between the errors of two variables.
Theoretically, parameter σ 2 stands for the variance of model and measurement
errors. Given that Z H is more reliable than other radar parameters, and its measurement error is generally accepted as 1–2 dB, σ Z H can be assumed to be constant
2 dB. Parameter σ Z DR is assumed to be a function of Z H and Z DR . It is shown that
disdrometer-based Z H and Z DR for rain data fall mostly in a bounded region (Cao et
al. 2008). Dashed lines shown in Figure 13.4 give the upper and lower boundary of
such a region. Within the bounded region, σ Z DR is assumed to be a constant 0.3 dB.
If the observed Z H and Z DR fall outside this region, σ Z DR is believed to be larger than
the one inside the region. This assumption is reasonable, because normal Z DR should
have a small observation error, while abnormal Z DR could be attributed with a large
observation error. The σ Z DR value is given by a function as
σ Z
Z
Z
Z
Z
DR
DR
DR
up
DP
low
DR
=
×
−
(
) +
×
−
(
) +
0 3
0 3
0 3
0 3
0
.
. ,
. ,
.
. . ,
3
above the upper boundary
within the region
bel low the lower boundary
(13.19)
5
Mean: Z DR = 10 (–2.69×10
–4 Z
2
H +0.049 Z H –1.4287)
Upper bound = 2 × Z DR
Lower bound = 0.5 × Z DR –0.2
4
3
2
1
0 0
10
20
30
40
50
60
Z H , dBZ
Z
DR , dB
4.5
3.5
2.5
1.5
0.5
FIGURE 13.4 Sketch figure of Z DR versus Z H from 2DVD measurements. The solid line
denotes the mean curve. (Data from Equation 13.15 of Cao, Q. et al., Journal of Applied
Meteorology and Climatology, 47, 2238, 2008. With permission.) The upper bound and lower
bound are given according to the mean curve. (Data from Cao, Q. et al., Journal of Applied
Meteorology and Climatology, 49, 973, 2010. With permission.)
Multiscale Hydrologic Remote Sensing: Perspectives and Applications
Z H and Z DR could be considered to have independent observation errors. Most previous Bayesian studies have only considered the observation error and assumed ρ = 0.
If the C-G DSD model were applied in the forward operator of Z H and Z DR , model
error would be introduced, and the model errors of Z H and Z DR tend to be correlated.
Although ρ should vary with different Z H −Z DR pairs, the effect of ρ on the retrieval is
not essential. For the sake of simplicity, Cao et al. (2010) assumed a constant ρ = 0.5,
which denotes a moderate correlation between the errors of two variables.
Theoretically, parameter σ 2 stands for the variance of model and measurement
errors. Given that Z H is more reliable than other radar parameters, and its measurement error is generally accepted as 1–2 dB, σ Z H can be assumed to be constant
2 dB. Parameter σ Z DR is assumed to be a function of Z H and Z DR . It is shown that
disdrometer-based Z H and Z DR for rain data fall mostly in a bounded region (Cao et
al. 2008). Dashed lines shown in Figure 13.4 give the upper and lower boundary of
such a region. Within the bounded region, σ Z DR is assumed to be a constant 0.3 dB.
If the observed Z H and Z DR fall outside this region, σ Z DR is believed to be larger than
the one inside the region. This assumption is reasonable, because normal Z DR should
have a small observation error, while abnormal Z DR could be attributed with a large
observation error. The σ Z DR value is given by a function as
σ Z
Z
Z
Z
Z
DR
DR
DR
up
DP
low
DR
=
×
−
(
) +
×
−
(
) +
0 3
0 3
0 3
0 3
0
.
. ,
. ,
.
. . ,
3
above the upper boundary
within the region
bel low the lower boundary
(13.19)
5
Mean: Z DR = 10 (–2.69×10
–4 Z
2
H +0.049 Z H –1.4287)
Upper bound = 2 × Z DR
Lower bound = 0.5 × Z DR –0.2
4
3
2
1
0 0
10
20
30
40
50
60
Z H , dBZ
Z
DR , dB
4.5
3.5
2.5
1.5
0.5
FIGURE 13.4 Sketch figure of Z DR versus Z H from 2DVD measurements. The solid line
denotes the mean curve. (Data from Equation 13.15 of Cao, Q. et al., Journal of Applied
Meteorology and Climatology, 47, 2238, 2008. With permission.) The upper bound and lower
bound are given according to the mean curve. (Data from Cao, Q. et al., Journal of Applied
Meteorology and Climatology, 49, 973, 2010. With permission.)