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When is matching percentiles useful ?

Power-law regressions have proved very popular to find a relation (some would say any relation) between the radar reflectivity Z of rain and the rain rate R itself. To do this,

  1. one collects raindrop samples during several time intervals,
  2. one calculates the corresponging rain volume R,
  3. one calculates the corresponding radar reflectivity factor Z,
Jet Propulsion Laboratory
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spacer 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
  1. collect rain rate samples from events falling within the given class
  2. collect radar reflectivity measurements from (not necessarily the same) events falling within the given class
  3. matching percentiles between the two sample distributions
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?"



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 events;
  • 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:

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