## Specific Differential Phase (K_{DP}) Primer

__Specific Differential Phase (K___{DP}): The
difference between propagation constants for horizontally- and
vertically- polarized radar pulses over a given range.

*where:*

_{h} is the phase of the horizontally-polarized pulse
at a given point in the propagation path,

_{v} is the phase of the vertically-polarized pulse
at the same point in the propagation path,

*and:*

r_{1} and r_{2} refer to measurements at range 1 and
range 2 from the radar [km], where r_{1} < r_{2}

To understand the above equations, consider two consecutive radar
pulses that travel the same propagation path. The first pulse is
horizontally polarized, the second is vertically polarized. Along the
propagation path is a uniform field of falling raindrops. As
discussed earlier, falling raindrops are oblate, so the electric field
will encounter more water content in the horizontal direction than in
the vertical.
The horizontally polarized pulse will, therefore, be affected by more
water than the vertically polarized
pulse. Since electromagnetic waves travel more slowly through water
than through air, the horizontally polarized wave will travel more
slowly through the field of raindrops than will the vertically
polarized pulse. This is a two way process -- the backscattered
radiation, horizontally polarized, will travel more slowly back to
the radar than the vertically-polarized backscatter.

_{DP}, or *differential phase*, is
simply the difference in phase
between the horizontally- and vertically- polarized pulses at a given
range along the propagation path. Naturally, differential phase will
increase with range from the radar, so we can take the range
derivative to determine where along the propagation path phase
changes are ocurring. This derivative is called the *specific
differential phase*, or K_{DP}. Note the "2" in the
denominator appears because there is a phase shift on both the
outbound trip and the return trip.

For meteorological echoes, K_{DP} typically ranges from
-1°km^{-1}
to 6°km^{-1}

Values of K_{DP} greater than zero indicate that
_{DP} has increased over the range of
interest (r_{2}-r_{1}). Since
_{h} and _{v}
are always 0, that means the phase of
the horizontally-polarized pulse has gotten larger more rapidly than
the phase of the vertically-polarized pulse. In other words, the
horizontally-polarized pulse has slowed down more than the
vertically-polarized pulse over the given range. This means there is
more hydrometeor content in the horizontal plane, e.g., oblate
hydrometeors.
Likewise, values of K_{DP} below zero indicate vertically oriented
hydrometeors are present in the range of interest.
Values of K_{DP} near zero indicate nearly isotropic
(spherical) hydrometeors are present.
It is important to note that *K*_{DP} is insensitive to isotropic
(spherical) scatterers.. For example, when encountering tumbling
hailstones, both the horizontally- and
vertically- polarized radar pulses will slow down. Because these
hydrometeors are nearly spherical, however, both pulses should change
phase at approximately the same rate, so _{DP} and K_{DP} should not change. For
this reason, K_{DP} is very helpful in rainfall accumulation
estimation, because the amount of rain in a rain-hail mixture can be
directly estimated.

Another advantage of using K_{DP} for rainfall accumulation
estimation is the fact that K_{DP} is immune to the reduction
in reflectivity factor caused by partial beam blockage. The
differential phase will shift at the same rate no matter the
reflectivity factor, as long as some signal can make it to the
scatterers and back.

Back to the main page.