Dr. Michael L. Larsen – Research Overview

Z-R Relationships in Radar Meteorology

 

Basic Background

 

RADAR was an acronym coined in the early 1940’s that stands for RAdio Detection And Ranging.  Over time, the term became common enough that the RADAR capitalization was removed and we merely write radar.

 

A radar system is composed of a transmitter and a receiver, both of which can be at the same location but do not have to be.  The basic principle is that the transmitter sends out a sequence of pulses of electromagnetic radiation that interacts with some matter somewhere, sending scattered radiation back to the receiver.  In principle, one may be able to detect the shape, size, orientation, and distance to the object by taking care to interpret the returned signal carefully.  (Careful interpretation perhaps including time to observation of the scattered signal, total power received, polarization of the received scattered radiation, and other assorted properties).

 

The targets in military applications often are things like airplanes or buildings.  In meteorological applications, the target is usually a “parcel” of air containing stuff like cloud droplets and raindrops.  The basic operational principle remains the same, but in meteorological applications the targets can be quite complicated.

 

One thing that people want to do with the information gained from radar returns is use the strength of the returned signal (power) to interpret if and how strongly it is precipitating at a particular spot.  (Think of the radar maps on the nightly weather report).  The power returned from a radar is usually written “Z”, and the rate of rainfall is usually written “R”, hence this is called the “Z-R” problem.  Given some value of Z, what is R?

 

 

So what’s the Problem?

 

There is no fundamental law of nature that relates Z to R.  Z does not exist as a “conserved quantity” and Z only has any meaning at all because radars exist.  Z is related to the shape of the raindrops, their number, their orientation, their relative positions, and their sizes.  Even the water’s purity plays a part in influencing the value of Z.  If you assume the drops are more or less randomly distributed in space and that they are pure water spheres, you can approximate the volume-averaged value of Z in a known way (sum the droplets’ diameters Di to the 6th power, then multiply by a fairly complicated constant).  R, the rainrate, is related to the fall speed of the raindrops, their diameters, their shapes, the direction and magnitude of the wind fluctuations, and the raindrop number densities.  One can loosely argue that if the drops are more or less randomly distributed in space and fall maintaining a spherical shape, the volume-averaged value of R can be approximately written in a similar way (sum the droplets’ diameters Di to about the 11/3rds power and multiply by a different complicated constant).

 

These relationships for Z and R are both merely approximations, however.  Radars aren’t perfect and have instrumental biases.  Lack of perfect spatial randomness may yield coherent scattering effects.  The R ~ D11/3 approximation is questionable at best, and many of the assumptions we made above just are not valid.  (There are other issues related to how the two quantities are measured as well – Z is measured for a large parcel measured by a radar, while R is usually measured on the ground with a simple raingauge.  These are not comparable volumes).  Nevertheless, the meteorological community has used the above approximations to write Z-R relationships in the form:

 

Z = aRb

 

Where a and b are constants chosen so that the relationship holds.  Perhaps the most famous Z-R relation (by Marshall and Palmer; one of the most famous because it was one of the first) is Z=200R1.6.  In the published literature, there are at least hundreds and probably thousands of published combinations for the constants “a” and “b”, which depend on the radar used, the climatology of the region, the duration of the experiment, and a whole host of other variables.

 

Even when using a Z-R relation specifically tuned to the radar and climate region of interest, however, the error in using Z-R relations to estimate rain-rate from the power returned to the radar can be very substantial.  (As in 50% error or larger).  This could use serious improvement.

 

 

 

 

What is Dr. Larsen doing to try and Solve the Problem?

 

Most studies in Radar meteorology essentially take a bunch of measurements of Z and R and then draw a scatterplot with a least squares fit on log-log scales to find the “best” Z-R relationship given these measurements.  As an empirical approach, you could do far worse.  However, I think you can also do far better.

 

I am working on revisiting the problem using novel techniques to try and get R from Z.  I’m entertaining using different sampling techniques that aren’t as susceptible to sampling bias.  Another path of inquiry utilizes less-biased fitting methods.  Finally, there’s no reason that the form of the best relationship must be a power law.  These, and other methods not mentioned here, are being explored and tested at UNK in an effort to improve the way of using meteorological radar data.

 

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