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4 - STATISTICAL RETRIEVAL OF SURFACE RAINFALL RATE
It has been observed that upwelling T B ’s at frequencies greater than about 19 GHz are almost insensible to the
structure of the lower rain layers and that the variability in the upper part of the cloud causes a large dispersion
of the observed T B 's against the rainfall rate (Smith et al., 1992). These considerations suggested that one
should assess a method to identify the gross vertical structure of the precipitating cloud from radiometric
measurements and then evaluate the rainfall rate from the retrieved vertical structure by means of a proper
fallout model (Mugnai et al., 1993). As compared to other methods inferring the complete structure of the
cloud, this algorithm searches the cloud profile within a cloud-radiation database and exhibits fairly good
performances in terms of processing time. The time consuming task of solving the radiative transfer equation
is accomplished only once, when generating the cloud-radiation dataset.
4.1. Maximum likelihood estimation algorithm
If the set of hydrometeor vertical profiles is identified by a vector g, consisting of equivalent water contents of
each hydrometeor category, and the multi-frequency T B measurement is expressed by vector t, the surface rain
rate is found by searching the most probable cloud structure within the cloud-radiation dataset, i.e. by
maximizing the following discriminant function with respect to g:
d(t,g) = ln[p(tlg)] + ln[p(g)]-ln[p(t)] (2)
where p(tlg) is the conditional probability density function (pdf) of t conditioned to g, and p(g) and p(t) are the
pdfs of g and t, respectively. Eq. (1) has been derived by means of the Bayes theorem that allows one to
express the posterior pdf p(glt) as a function of the prior pdf p(tlg). Assuming a Gaussian multidimensional
joint-distribution of vector g, the covariance matrix C g and the mean vector m are sufficient to describe the
statistics of g, and its probability density function is given by:
P(e) = ^?'^de.(C,'i eXP ['i <e - m)TC ‘" (e ' m)1 ] ®
where det(Cg) and Cg- 1 indicate respectively, the determinant and the inverse matrix of Cg. If t' is the Tb
vector associated to g by the radiative transfer equation, and the errors (t-t') are assumed uncorrelated at the
different frequency bands with the same variance a t 2 , by substituting Eq.(3) into Eq.(2) and considering that
p(t) is a common term, it results:
d(t ,g ) = -^ T ( t- t ’) T ( t -t')-y ln [ d e t ( Cg )]-i(g-m) T C g - 1 ( g -m) (4)
where ’T" represents the transpose operator. Once the vector g has been selected, the surface r ainf all rate is
computed from the retrieved hydrometeor profiles by means of the mentioned fallout model.
4.2. Rain rate retrieval over land
For the precipitation event over Liguria, some rainfall measurements from rain-gauges were available and they
were used for comparison with satellite rain estimates. Note, however, that comparison of satellite rain rate
retrievals with rain-gauge measurements is highly problematic because of the different spatial and temporal
integrations. The SSM/I, in particular, provides estimations over areas of about 200-220 Km 2 , while rain-
gauge measurements are representative only of the instrument site; moreover the location of the available rain-
gauges was mountainous, and thus characterized by strong and rapid precipitation variations.
Figure 5 shows the rainfall rate map as obtained by applying the ma ximum likelihood
algorithm to the SSM/I image, zoomed for the flush flood event over Liguria. The contour map of rain rate
shows a high variability of rainfall rate within the precipitation area with values that can reach even 40 mm/h.
Figure 6 shows the comparison of SSM/I derived rain rates with those measured by the rain-gauges present
within the SSM/I pixels. The comparison shows a discreet agreement in the average, even though
discrepane«
rain-gauge ’
Figure 5.
Figure 6.
