How do you determine the axial ratio from a RHCP wave to a LHCP wave
from an antenna? In essence, I need an axial ratio transformation
from
RHCP to LHCP.

If you are working with an antenna that radiates RHCP it is totally
insensitive to LHCP.
If you have an antenna that radiates "RHCP" (or LHCP) the axial
ratio
is
the ratio of the Minor axis to the Major axis of the E field. So, it
varies from zero to one as the radiated field varies from Linear to
Circular.
You mentioned insensetivity above, are there any other polarities
insensitive to others ? I note that weather antennas are now migrating
to
dual diversity antennas for horizontal and vertical, is it possible to
totally isolate those too.?
I may make mistakes by using non scientific terms. When I read my
post,
I realize that nothing is actually defined by my above statement. I
would
submit that a LHCP antenna will receive no RHCP signal. I'm sure you
know
that any linearly polarized antenna is insensitive to linearly polarized
signals that are orthogonal to the line of polarity of that antenna.

Weather experts have found that using both horizontal and vertical
polarizations coming from the same antenna tho in separate pulses that
with oncomming clouds and weather systems that the received picture is
more defined where size of droplets can be determined as well as other
things.

Lots of questions, lots of possible answers.. here goes:
1) While a theoretical antenna of one polarization sense is
insensitive to the opposite sense, in reality, antennas are perfect,
particularly "off boresight". that's what axial ratio tells you.. how
"perfect" is the CP (1 is perfect, 0 is a linear pol, non circular
antenna). Think of it as a continuum from a straight line (linear) to
a circle. Elliptical pol is the general case, and axial ratio is the
ratio between the major and minor axes of the ellipse (to fully
specify the antenna, you'd also need to specify the angle of one of
the axes and the "handed ness")

2) The radiation from an antenna, in a particular direction, can be
represented by a linear combination of two orthogonal polarizations
(orthogonal here has a special meaning... two linear pols separated by
90 degrees is but one case). The two popular systems are CP and linear
(that is you say that the field is (a+bj) horizontal + (c+dj)
vertical. You can do the same for RHCP and LHCP. (and, actually, for
any two linear pols that are different). This is one way to "make" a
cp antenna (combine two linear antennas with the right relative
amplitudes and phases).

3) bear in mind that an antenna may be perfectly CP on boresight
(axial ratio=1) and not CP in other directions. In fact, it can be
proved that no antenna can have the same polarization everywhere (see
the "hairy ball theorem")

Much more info can be found on the web.. wikipedia has an entry on
polarization,antenna books talk about it (Orfanidis' book is online at
Rutgers) Kraus has a nice discussion (particularly appropriate since
axial mode helical antennas are particularly good at being CP)


Now to weather radar...
spherical particles (raindrops) are perfect "depolarizers"... send any
pol to the droplet, and what comes back is equally distributed between
both pols. (spheres are used as cal targets for this reason)
ice crystals, on the other hand, show significant polarization
effects.
Since the polarization effects are also "size of particle" dependent
(relative to wavelength) you can also figure out things like particle
size distribution. that way you can tell the difference between a few
big raindrops and lots of little raindrops.

Most other physical phenomena show some polarization effects, which is
heavily used in radar remote sensing.
You transmit radar pulses of two different polarizations, and receive
the returns with a dual pol receiver, so you essentially get 4
different complex numbers (HH, HV, VV, VH, for linear pol) for the
backscatter.
An example might be looking at forests.. in the summer, when they have
their leaves on, they pretty much randomize the polarization, but in
the winter, the Hpol backscatter is very different from the Vpol
backscatter. Plowed soil has a very different polarization signature
from flat earth. Conifers look different from broadleaf plants.