and one thus ends up with a "scatter diagram",
log(R) on one axis and log(Z)
on the other, and one fits a line through the "samples".
While this procedure is simple and reassuringly equally popular in many other fields
("if everybody's doing it, it can't be all bad, can it?" hmm ...), it does in
fact have several drawbacks, to wit
- By the time you have collected a sufficiently large number of samples, the
characteristics of the rain may have changed so drastically that your samples
include apples, oranges, mangosteen, etc
- While the line minimizing the r.m.s. distance through the scatter will
produce a relation having the (relatively) smallest r.m.s. error, there is no
guarantee that this error is actually small. In fact, given the amazing
variations between the
different Z-R power-laws that have been derived to date, the error one would make
using any one power law could easily exceed 100%.
. . .
Along came the suggestion to use a priori an objective classification of rain
(using quantitative measures such as horizontal gradients, cloud depth, etc),
then to study each class separately and
Tradionalists (habituees of the time-tested regression) were outraged. "What is
this `matching' nonsense?" they clamored. "How can you not need simultaneous
measurements of the same events?"
- collect rain rate samples from events falling within the given class
- collect radar reflectivity measurements from (not necessarily the same) events
falling within the given class
- matching percentiles between the two sample distributions
It turns out that
- if one wants to assume little or only subjective a priori climatological or
physical information about the rain event, the regression method results in an
approximation to the optimal Z-R relation which
minimizes the variance associated to this (perhaps quite large) class of rain
- if one can objectively classify the rain event a priori according to
quantitative climatological, physical and geometric considerations, the
percentile-matching method produces the optimal Z-R relation associated with the
specific rain regime at hand.
To retrieve a figure-less copy of the preprint (accepted in Quart. J. Roy.
Meteor. Soc., August 1996) explaining the mathematical rationale
behind the regression approach and, more importantly, the PMM approach:
Go back to the top page of the JPL TRMM site