Total backscattering cross section of rain

Hydrometeors are essentially particles of water within the atmosphere, they can take the form of liquid water as in rain, mist and fog or ice as in clouds, hail and snow. Plane electromagnetic waves travelling through air containing precipitation are scattered and absorbed by the particles of ice, snow or water. Water with its larger dielectric constant scatters more strongly than ice (see Table 1.1). In addition, it has a much larger dielectric loss and the attenuation due to thermal dissipation is therefore much greater for water particles than for ice particles [2, 29]. This makes rain more accountable for the degradation of performance of millimeter-wave radars than ice. Hence, this section concentrates only on the effects of rains on the millimeter-wave propagation.

The rain affects the propagation of electromagnetic waves in different ways. If radiowave is propagating through a region containing raindrops, it will be scattered, depolarized, absorbed and delayed in time. All these ef- fects of rain on the wave propagation are related to the operating frequency and polarization of the wave as well as to the rain rate, which influences the form and size distribution of the raindrops. These physical relationships

sented in many publications, mentioned at the beginning of this Chapter. Although, this section will be restricted on the examination of the backscat- tering effect if it can be made applicable for an automatic identification of rain and its consequence on the detection performance of automotive radar sensor systems.

As it is illustrated detailed in section 2.3.1, three quantities with the dimensions of area are derived for a particle in the path of a plane travelling wave. The backscattering cross section σb is defined in such a way that σb

multiplied by the incident intensity would be the total power radiated by an isotropic source, which radiates the same power in the backward direction as the scatterer. The scattering cross section σs is such that σs multiplied

by the incident intensity, is the power scattered by the particle. The total absorption cross section σa is such that σatimes the incident intensity, is the

total power taken from the incident wave.

Based on the relationship between these cross sections and the ratio of precipitation size to the wavelength, the effects of rain on wave propagation can be explicitly examined and sought for their practical applicability (re- fer sections 2.3.1 and 5.1.1). For example, the backscattering properties of precipitation may make possible extensive use of millimeter-wave radar sets for identification purpose of precipitation and could be applied to find out precipitation effect on the detection performance of radars.

The radar signal from precipitation is, however, not steady like that from a point target. The received power at any instant is made up of the resultant of signals from a very large number of precipitation, and depends on their exact arrangement in space, which is continually changing. It means, an instantaneous observation of received power gives very little information about the precipitation, and the signal at any given range has to be averaged until a large number of independent returns have been received. This can be well achieved by FMCW automotive radar, which illuminates the scatterers for relatively longer time (in the order of some milliseconds) and can be also averaged out measurements of rain returns from several antenna elements.

The receiving patterns of LRR2 (Figure 3.2) are confined within narrow solid angles. The characteristics of the precipitation (the density ρ and cross

sections σ) can be thus assumed to be uniform throughout the rain cell volume VRCR in Figure 5.2. Automotive radar dR Rmin V_RCR θ, Φ

Figure 5.2: Plan view showing narrow beam monostatic radar consisting of raindrop scatterers. θ and φ are the half-power beamwidths in the azimuth and elevation, respectively.

In such a case, the ratio of the power received to the power transmitted can be calculated according to the backscattering radar equation (2.107). This can be considerably simplified from the volume integral to the length integral using the expression VRCR = ((πR2minθφ) / (8ln2)) dR [2, § 4-3], and

gives the following general equation Pr Pt = π (4π)38ln2λ 2_G t(ηi)2θφ Z Rmin+dR Rmin ρσb R2e−2γdR . (5.1)

The fact that the target normally fills the beam at any range accounts for the inverse square law for received power versus range, in place of the usual fourth-power law for a point target such as a vehicle. The total backscattering cross section ρ< σb > (see (2.110)) per cubic meter of rain cell volume VRCR

in the antenna beam can be obviously equated from (5.1) as follows ρ< σb > = 114 λ2_G t(ηi)2θφe(−2ρ<σt>Rmin)RR min+dR Rmin 1 R2e(−2ρ<σ t>R)dR Pr Pt . (5.2) All parameters in (5.2), except ρ< σb > and ρ< σt >, are either known or

the raindrops are in the order of the size of the wavelength (Figure 5.1). And then, the calculations of the cross sections can be treated by the Mie- theory [2, § 2-8]. The optical distance γ in (5.1) can also expressed with the attenuation due to rain ρ< σt > (see (2.108)). This is nearly proportional

to the rain intensity I and given by ρ< σt > = KtI. The constant Kt, on

the other hand, depends on the rain intensity I and some typical values are given in [2, § 3-2]. For millimeter and optical frequencies, Kt ≈ 0.4 −

0.8 dB/km/mm/hr at rainfall intensities I < 10 mm/hr, and Kt decreases

to 0.15 − 0.3 dB/km/mm/hr at rainfall intensities I > 10 mm/hr.

So far, the attenuating effect of the precipitation e−2γ_{, through which}

the signal may have to pass, has been considered. It shows that the influence of the rain intensity over the attenuation makes it difficult to determine the backscattering cross section ρ< σb > of the rain using (5.2) by only measuring

the received power.

For precipitation that is found in near distance to the radar sensor, it could be, however, possible to neglect the attenuation term in (5.2). That is, ρ< σb > can be approximated as a function of rain intensity I by measur-

ing the average backscattered power Pr. Experimental investigation on this

relationship will be presented and discussed in detail in the next section.