## Abstract

The power spectrum of the phase difference fluctuations between two IR beams propagating side by side is strongly influence by rain along the path. According to our measurements over a 1.37-km path in convective rainfall averaging 18 mm/h, the power spectrum of phase difference fluctuations in rain extends over the 300-Hz to 2-kHz band, and it is as much as 18 dB above the no-rain spectrum at ~600 Hz during periods of heavier rain (>30 mm/h). The shape of the spectrum is different at different stages of the storm probably because of changing drop size distributions. A simple dimensionless parameter derived from the power spectrum of phase difference fluctuations is 0.186 ± 20% for a rain rate of 41 (mm/h) km ± 25%, but this value may depend on the chopped transmitter waveform. The simple optical design and data analysis procedure allow power spectra to be obtained from a single signal channel. We conclude that measurement of phase difference fluctuations in 10.6-*μ*m propagation should allow the path-averaged rainfall rate to be inferred over path lengths of at least 1.5 km if a properly calibrated system is used.

© 1985 Optical Society of America

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### Equations (13)

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(1)
$$\begin{array}{cc}{E}_{j}={E}_{0}\phantom{\rule{0.2em}{0ex}}\text{exp}\left\{i\left[\omega t+{\varphi}_{j}\left(t\right)\right]\right\},& j=\mathrm{1,2},\end{array}$$
(2)
$$\begin{array}{ll}I\left(t\right)\hfill & =EE*=\text{dc}\phantom{\rule{0.2em}{0ex}}\text{terms}+{{E}_{0}}^{2}\left\{\text{exp}\left[i\Delta \varphi \left(t\right)\right]+\text{exp}\left[-i\Delta \varphi \left(t\right)\right]\right\}\hfill \\ \hfill & =\text{dc}\phantom{\rule{0.2em}{0ex}}\text{terms}+{{E}_{0}}^{2}\phantom{\rule{0.2em}{0ex}}\text{cos}\Delta \varphi \left(t\right),\hfill \end{array}$$
(3)
$$\Delta \varphi \left(t\right)={\varphi}_{1}\left(t\right)-{\varphi}_{2}\left(t\right).$$
(4)
$$\Delta \varphi \left(t\right)=\Delta {\varphi}_{0}+\delta \left(t\right),\text{with}\phantom{\rule{0.2em}{0ex}}\delta \left(t\right)\ll \pi ,$$
(5)
$$\begin{array}{ll}\text{cos}\left[\Delta {\varphi}_{0}+\delta \left(t\right)\right]\hfill & =\text{cos}\Delta {\varphi}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}\delta \left(t\right)-\text{sin}\Delta {\varphi}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}\delta \left(t\right)\hfill \\ \hfill & \begin{array}{l}=\text{cos}\Delta {\varphi}_{0}\left[1-\delta {\left(t\right)}^{2}/2!+\dots \right]\\ \phantom{\rule{0.4em}{0ex}}-\text{sin}\Delta {\varphi}_{0}\left[\delta \left(t\right)-\delta {\left(t\right)}^{3}/3!,\dots \right]\end{array}\hfill \\ \hfill & \simeq -\delta \left(t\right)\phantom{\rule{0.2em}{0ex}}\text{sin}\Delta {\varphi}_{0},\text{ac}\phantom{\rule{0.2em}{0ex}}\text{only}.\hfill \end{array}$$
(6)
$$F\left[a\left(t\right)b\left(t\right)\right]=\left(1/2\pi \right)A\left(f\right)*B\left(f\right).$$
(7)
$$\overline{P\left(\Delta \varphi ,f\right)}\simeq \overline{P\left(\delta ,f\right)}\simeq \overline{2P\left(I/{{E}_{0}}^{2},f\right)}\propto \overline{P\left(i,f\right)}$$
(8)
$$\begin{array}{cc}{E}_{j}={a}_{j}\left(t\right){E}_{0}\phantom{\rule{0.2em}{0ex}}\text{exp}\left\{i\left[\omega t+{\varphi}_{j}\left(t\right)\right]\right\},& j=\mathrm{1,2}.\end{array}$$
(9)
$$\Delta \varphi \left(t\right)={\text{cos}}^{-1}\left[I\left(t\right)/{a}_{1}\left(t\right){a}_{2}\left(t\right){{E}_{0}}^{2}\right],$$
(10)
$${E}_{r}={E}_{0}+{E}_{0}{a}_{jk}\left({b}_{k}\right)\text{exp}\left[i{\varphi}_{jk}\left(t\right)\right],\begin{array}{l}j=\mathrm{1,2},\hfill \\ k=1\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}n,\hfill \end{array}$$
(11)
$${\varphi}_{jk}\left(t\right)=f\left[\left|{d}_{k}\left(t\right)\right|,{x}_{k},\lambda \right]$$
(12)
$$d\lesssim {\left(\lambda x\right)}^{1/2},$$
(13)
$${P}^{\prime}=\left(1/BW\right){\displaystyle {\int}_{200\phantom{\rule{0.2em}{0ex}}\text{Hz}}^{2.5\phantom{\rule{0.2em}{0ex}}\text{kHz}}\left[P\left(f\right)/{P}_{r}\right]}\phantom{\rule{0.2em}{0ex}}df,$$