## 3 NDF Analysis and Space Sampling Cues  
In order to obtain information on how to deploy the sensors, here we study the singular value decomposition of the described single-frequency scattering model. Let us rewrite Equation (1) in the following way  
\[ \begin{split}&\bar{E}_{S}(u,v)=(2\pi)^{2}\zeta\exp\left[jkz_{p} \sqrt{1-(u^{2}+v^{2})}\right]\\ &\int_{-X_{in}}^{X_{in}}\int_{-\eta_{in}}^{\eta_{in}}\exp\left(j2 \pi u\chi\right)\exp\left(j2\pi v\eta\right)J_{eq}(\chi,\eta)d\chi d\eta\end{split}  \tag{4}\]  
where \(u=\sin\theta\cos\phi\) and \(v=\sin\theta\sin\phi\) are the observation variables and \(\chi_{in}=x_{in}/\lambda\) and \(\eta_{in}=y_{in}/\lambda\) are the dimensions of the investigation domain normalized to the wavelength \(\lambda\). Still denote the normalized investigation domain as \(D_{T}\). Note that here \(\bar{E}_{S}\) can be one of the two components of the scattered field in Equation (1). Equation (4) defines a Fourier-like linear non-symmetric operator  
\[\mathcal{A}:J\in L^{2}(D_{T})\rightarrow\bar{E}_{S}\in L^{2}(\Omega) \tag{5}\]  
with \(L^{2}(D_{T})\) and \(L^{2}(\Omega)\) the sets of square integrable complex valued functions supported over \(D_{T}\) and \(\Omega\), \(\Omega\) being the observation domain in the \((u,v)\) plane. Such an operator is compact [31]. Moreover, as the relevant kernel function is of exponential type [32], its singular values are expected to have an exponential fast decay beyond a certain index. Accordingly, this index can be identified as the NDF of the problem. The NDF actually depends also on \(\Omega\) which in turn depends on the measurement aperture. Here, according to the previous choice of the observation variables, it seems natural to consider the measurements taken over a portion of the measurement plane, \(\Sigma\), located at the height \(h\) from the air/soil interface and enclosed within the circle of radius \(r_{O}\), that is \(\Sigma=\{(x,y,z):z=h,x^{2}+y^{2}\leq r_{O}^{2}\}\) or equivalently, \(\Sigma=\{(r,\theta,\phi):r=h/\cos\theta,0\leq\theta\leq\tan^{-1}\left(r_{O}/h \right),0\leq\phi<2\pi\}\). Accordingly, \(\Omega_{C}=\{(u,v):u^{2}+v^{2}\leq\sin^{2}\left[\tan^{-1}\left(r_{O}/h\right) \right]\}\), where the subscript \(C\) refers to the circular shape of the domain. Unfortunately, in this case, even though \(\mathcal{A}\) has a simple expression, its singular system is not known in closed form. To further elaborate on this question, we note that if \(\Omega\) were rectangular, say \(\Omega_{R}=[-U,U]\times[-U,U]\), and considering that the exponential factor outside the integral in (4) is a unitary transformation, the singular values of \(\mathcal{A}\) would be actually known to be linked to the prolate spheroidal wave-functions [33]. Accordingly, the \(\sigma_{n}(\Omega_{R})\)s would exhibit an abrupt decay roughly in correspondence of the index  
\[n=[(2U)^{2}4\chi_{in}\eta_{in}] \tag{6}\]  
Moreover, it can be shown that if \(\tilde{\Omega}\supset\Omega\) then \(\sigma_{n}(\Omega)\leq\sigma_{n}(\tilde{\Omega})\)[34]. Accordingly, by considering two rectangular domains such that \(\Omega_{Rout}\supset\Omega_{C}\supset\Omega_{Rin}\), it is easy to find upper and lower bounds for the NDF of \(\mathcal{A}\) as  
\[8\sin^{2}\theta_{max}\chi_{in}\eta_{in}\leq N_{T}\leq 16\sin^{2}\theta_{ max}\chi_{in}\eta_{in} \tag{7}\]  
where \(\theta_{max}=\tan^{-1}\left(r_{O}/h\right)\). In Figure 2, the behavior of the lower bound of \(N_{T}\) is reported as a function of the normalized dimension of the investigation domain, assumed square, for different values of the ratio \(r_{O}/h\). As can be seen, the number of required data quickly becomes unfeasible as the size of the investigated area increases. The properties of prolate spheroidal wave-functions [33] suggest sampling data in the \(u-v\) plane over a regular grid at a step of  
\[\Delta u =1/(2\chi_{in}) \tag{8}\] \[\Delta v =1/(2\eta_{in})\]  
along \(u\) and \(v\), respectively [33].