122
C.G. Collier
An equation as complex as desired, may be derived between PI and functions of
S(TBB) and its derivative with respect to time:
PI = Ao + A· S(TBB) + A'~ S(TBB);
(6.8)
dt
forTBB
These approaches have now been underpinned by a more logical mathematical
approach. Doneaud et al. (1981, 1984) derived the volumetric rainfall, V, from
V = f fRdadt =Rc f f dadt = RcLA/lti
(6.9)
TA
TA
i
where R is the instantaneous local rain rate
da and dt are incremental elements of area and time respectively
Rc is the average rain rate.
The integrals are taken over the entire area A for duration T. The double integral
is the area-time integral (A TI).
Following this work Chiu and Kedem (1990) developed a logistic regression model
to estimate the fractional rainy area, which estimates the conditional probability that
rain rate over an area exceeds a fIXed threshold given the values of related covariates.
Tests showed that this approach is superior to multiple regression. However, variabilities of meteorological parameters must be accounted for if this technique is to be
applied to estimating rain rate from space.
Atlas et al. (1960) developed a unified theory for the estimation of both the total
rainfall from an individual convective storm over its lifetime, and the area wide
instantaneous rain rate from a multiplicity of such storms, by use of measurements of
the areal coverage of the storms within a threshold rain intensity isopleth or the
equivalent threshold radar reflectivity. Equation 6.6 was generalised to,
V = [A{t). T]. S{t)
(6.10)
where the
ATI
"C
S("C)
A("C).T
threshold
Rc( "C)/ and may be defined in terms of the probability
density
function
(pdf)
j RP(R) dR/j PeR) dR
o
0
the fraction of the total volumetric rain rate
Divide Eq. 6.10 by total area observed Ao then V/Ao is the average area wide rain
rate and A("C)/Ao is the fractional area, F("C), covered by rain within the threshold,
"C.
= F("C) Rc("C)/
(6.11)
C.G. Collier
An equation as complex as desired, may be derived between PI and functions of
S(TBB) and its derivative with respect to time:
PI = Ao + A· S(TBB) + A'~ S(TBB);
(6.8)
dt
forTBB
These approaches have now been underpinned by a more logical mathematical
approach. Doneaud et al. (1981, 1984) derived the volumetric rainfall, V, from
V = f fRdadt =Rc f f dadt = RcLA/lti
(6.9)
TA
TA
i
where R is the instantaneous local rain rate
da and dt are incremental elements of area and time respectively
Rc is the average rain rate.
The integrals are taken over the entire area A for duration T. The double integral
is the area-time integral (A TI).
Following this work Chiu and Kedem (1990) developed a logistic regression model
to estimate the fractional rainy area, which estimates the conditional probability that
rain rate over an area exceeds a fIXed threshold given the values of related covariates.
Tests showed that this approach is superior to multiple regression. However, variabilities of meteorological parameters must be accounted for if this technique is to be
applied to estimating rain rate from space.
Atlas et al. (1960) developed a unified theory for the estimation of both the total
rainfall from an individual convective storm over its lifetime, and the area wide
instantaneous rain rate from a multiplicity of such storms, by use of measurements of
the areal coverage of the storms within a threshold rain intensity isopleth or the
equivalent threshold radar reflectivity. Equation 6.6 was generalised to,
V = [A{t). T]. S{t)
(6.10)
where the
ATI
"C
S("C)
A("C).T
threshold
Rc( "C)/
density
function
(pdf)
j RP(R) dR/j PeR) dR
o
0
the fraction of the total volumetric rain rate
Divide Eq. 6.10 by total area observed Ao then V/Ao is the average area wide rain
rate
"C.
(6.11)