Doppler radar analysis techniques - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 428

restriction, but they cannot fly too close to storms for safety, and do not sense
motions close to the ground well, owing to ground clutter contamination. In addi-
tion, it takes a relatively long time ( 5min) for an aircraft to fly by a storm.
Unless two aircraft are available to do simultaneous scans, one aircraft typically
passes by a storm and the radar antenna scans alternately in the fore and aft
directions at an angle to the aircraft ( Figure A.3 ). Since overlapping beams are
not valid for the exact time, the analysis of the wind field from Doppler radar
wind measurements is called ''pseudo dual-Doppler analysis.
For a network of just two Doppler radars, since we have two equations (in
terms of two measured quantities V r 1 and V r 2 ) in four unknown variables

(u W t ), we need two additional, independent equations. Typically, an equa-

tion of continuity is used in conjunction with a kinematic lower-boundary
condition. The equation of continuity that is usually used is the incompressible
form (relates spatial derivatives of u, v ,andw with respect to each other—the
Boussinesq form) or one modified to include vertical variations in density (the
anelastic form); these equations are discussed in Chapter 2.
To complicate matters further, the radar measures the Doppler velocity of
scatterers such as raindrops and hailstones, which have terminal fall speeds rela-
tive to the air. It is customary to approximate fall speeds based on a formula that
relates reflectivity to fall speed. Such a method is expected to work better at longer
wavelengths such as S-band (10 cm) than at shorter wavelengths such as C-band
(5 cm) or X-band (3 cm), because it is more likely at shorter wavelength that the
hydrometeors detected will be in the Mie range, rather than in the Rayleigh
range. For example, at X-band, large hailstones are well into the Mie range (for
hailstones 2-3 cm or larger in diameter), and resonance effects lead to a non-
monotonic relationship between the radar reflectivity factor and hail size ( Figure
A.4 ). In the future, estimates of fall speed may be refined using polarimetric
measurements and fuzzy logic.
For a network of three Doppler radars, there are three equations related to
network geometry, in four unknowns. For a network of four Doppler radars,
there are four equations related to geometry, in four unknowns, so in principle
one could solve for u, v , w, and W t exactly without resort to any other equations
such as continuity or vorticity. In practice, however, the more radar members
there are in a network, the less likely it is that a storm will be positioned opti-
mally. Moreover, no matter how many radars there are, none of them will resolve
much of vertical velocity, which is nearly perpendicular to all beams except for
mid to high levels when the convective storm is very close to the radars (i.e.,
within 10-15 km).
To solve for u, v , and w in practice, in a dual-Doppler network, one typically
iteratively finds an exact solution to (A.1), (A.2), and the continuity equation, or
uses variational analysis to find the best ''fit'' to the observations, subject to
boundary conditions (at the surface and possibly aloft) and ''weak'' constraints
such as continuity or a vorticity equation or both. To find an exact solution, one
can make a first guess, for example, by neglecting w and making use of a reflectiv-
ity-based relationship for terminal fall velocity W t , and solving only (A.1) and

; v; w ;