3.4.3. Scattering from Slow-Moving Vehicles

Another example of the benefits of bistatic scattering at HF is the improved skywave detectability of slow land-based vehicles. Measurements of the spatial spectra of two types of terrain are presented in Figure 17a,b. These are based on very high-resolution digital elevation maps, covering thousands of square kilometers, with postings at 5 m spacing.

**Figure 17.** Power spectra of terrain elevation for (**a**) a hilly region, and (**b**) a relatively flat region.

From Figure 17a, corresponding to a hilly region, the surface elevation η follows a one-dimensional power law with *<sup>S</sup>*(κ) <sup>∝</sup> <sup>κ</sup>−<sup>3</sup> while, for the relatively flat region treated in Figure 17b, the power law behaves as *<sup>S</sup>*(κ) <sup>∝</sup> <sup>κ</sup>−2.4. It is not unreasonable to regard these exponents as rough limits on typical terrestrial surfaces when the vegetation density is light. Now, the local HF bistatic scattering coefficient for natural land surfaces can be modelled with first-order small perturbation theory, taking account of the dependence of the scattering coefficient on the local angle of incidence, weighted by the medium-scale slope probability density function. To proceed, we facet the resolution cell into n patches that are planar in the mean, but with small scale roughness,

$$\begin{split} \sigma\_{\text{cell}}(\theta\_{\text{scat}},\phi\_{\text{scat}},\theta\_{\text{inc}},\phi\_{\text{inc}}) &= \int \sigma(\theta\_{\text{scat}}',\phi\_{\text{scat}}',\theta\_{\text{inc}}',\phi\_{\text{inc}}';\overrightarrow{r}) \mathrm{P}(\theta\_{\text{inc}}',\phi\_{\text{inc}}') d\theta\_{\text{inc}}' d\phi\_{\text{inc}}' d\overleftarrow{r} \\ &= \sum\_{m=1}^{n} \, \_{m} \sigma(\theta\_{\text{scat}}'(m),\phi\_{\text{scat}}'(m),\theta\_{\text{inc}}'(m),\phi\_{\text{inc}}'(m)) \mu(\Delta\_{m}) \end{split} \tag{13}$$

where the patches over each of which a local normal vector is defined are obtained by faceting the surface using a local flatness criterion.

The general expression for the first-order bistatic scattering coefficient takes the form

$$\begin{aligned} \sigma\_{pq}^{(1)}(\theta\_{\text{scat}}, \varphi\_{\text{scat}}, \theta\_{\text{inc}}, \varphi\_{\text{inc}}) &= 2^6 \pi k\_0^4 P\_{pq} \sum\_{m=1,2} S(k\_0 \sin \theta\_{\text{scat}} \cos(\varphi\_{\text{scat}} - \varphi\_{\text{inc}}) \\ &- k\_0 \sin \theta\_{\text{inc}} \quad k\_0 \sin \theta\_{\text{scat}} \sin(\varphi\_{\text{scat}} - \varphi\_{\text{inc}})) \end{aligned} \tag{14}$$

where *q* and *p* index the incident and scattered polarisation states and the function *Ppq* accounts for the polarisation-dependence. To apply this to the tilted facets, we need only to transform to local coordinates via the appropriate rotation matrix, apply the scattering formula, then perform the inverse transformation.

There are two important considerations here that relate to the merits of bistatic scattering geometry. First, substituting the measured power laws of the land surfaces in the scattering formula, and writing the expression for the case of θ*scat* = θ*inc* = <sup>π</sup> <sup>2</sup> for clarity, we find

$$\sigma\_{pq}^{(1)}\left(\frac{\pi}{2}, q\_{\rm scatt}, \frac{\pi}{2}, q\_{\rm inc}\right) \propto \left. k\_0^4 P\_{pq} \right| k\_0 \cos(q\_{\rm scat} - q\_{\rm inc}) - \left. k\_0 \right|^{-a} \equiv k\_0^{4-a}. \left. P\_{pq} \left| \cos(q\_{\rm scat} - q\_{\rm inc}) - 1 \right|^{-a} \tag{15}$$

where α [2.4, 3,]. There are two features of interest here. The first is the slow fall-off of the power spectrum, which implies that the scattering coefficient will increase with radar frequency, so clutter can be reduced by operating at lower frequencies. The other feature is the variation of the scattering coefficient with bistatic angle. The case of the VV element of the polarization scattering matrix is plotted in Figure 18 over a typical range of bistatic angles. It is evident that by employing a moderately bistatic scattering geometry, the land clutter can be reduced by a factor of 2–3. As spectral leakage is a key limiting factor in slow land target detection, this is a significant gain in detectability.

**Figure 18.** The variation of the VV scattering coefficient as a function of bistatic angle.
