METR 6613
Homework #4
Brett Roberts
2011-10-10
Problem 2
Part 1
The scattering amplitudes for a range of equivolume drop diameters were calculated using both the
Rayleigh and T-Matrix methods. The real component (Fig. 1a), imaginary component (Fig. 1b),
magnitude (Fig. 1c), and phase (Fig. 1d) are shown.
The two calculation methods gave relatively similar results for the real component, with the T-Matrix
method giving slightly smaller values for the larger drop sizes. Because the real component was much
larger than the imaginary component for all drop sizes, the magnitude of the scattering amplitude also
showed good agreement between the two methods. However, the imaginary components did not agree
as well, diverging quite significantly for drops near and above 3 mm in diameter. This divergence also
appears in the phase.
Part 2
The following expression was used to calculate the scattering amplitude using the Rayleigh
approximation:
𝐷𝑒 3 1
𝜀𝑟 − 1
𝑓𝑎,𝑏 = 𝑘 2 ( )
2 3 1 + 𝐿𝑎,𝑏 (𝜀𝑟 − 1)
Where k is the radar wavenumber, De is the equivolume drop diameter, εr is the relative dielectric
constant, and L is the L-factor.
Figure 2 shows the ratio |fa|/|fb| using two values for the dielectric constant: one for drops (as used in
the above calculations), and one for dry snow with a fractional volume of 0.1. The ratio increases
exponentially with diameter for drops, but remains near unity for dry snow. Examining the equation
above, it is apparent that the much smaller magnitude of the dielectric constant for dry snow (~1)
compared with raindrops (~80) is responsible for this phenomenon. Specifically, the term L(εr – 1)
becomes very small when εr ~ 1, which renders the L-factor unable to significantly affect the scattering
amplitude. Therefore, the difference between the vertical and horizontal scattering amplitude is small
for the dry snow case, since the L-factor is the only variable on the RHS related to the orientation.
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METR 6613
Homework #4
Brett Roberts
2011-10-10
Figure 1. Scattering amplitude for a range of equivolume drops.
Figure 2. Ratio |fa|/|fb| for drop vs. dry snow dielectric constant.
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