# Tomographic Inversion Methods for Retrieving the
## 3 The External Reconstruction Tomographic Algorithm (ERTA)  
In this section, we summarize the basic mathematics of the so-called exterior reconstruction problem and show some outcomes of our implementation of the ERTA. Let us denote the water vapor function to be reconstructed as \(f:\mathbb{R}^{2}\rightarrow\mathbb{R}\), expressed in polar coordinates. The measured IWV field coincides with its Radon Transform, indicated in the following as \(g\) and defined as  
\[g(\rho,\phi)=\int_{L(\rho,\phi)}f(r,\theta)ds, \tag{3}\]  
where \(L\) is a line identified by the couple (\(\rho\), \(\phi\)), with \(\rho\) the distance of \(L\) from the origin and \(\phi\) the angle formed by the \(\theta=0\) axis and the normal to the line. The inversion technique known as _Exterior Radon Transform_ has been introduced for the first time by [PERSON], whose important contribution was the demonstration that the values of the function \(f\)\(\forall\)\(r\geq r_{o}\) can be uniquely retrieved from the knowledge of its Radon Transform \(g\)\(\forall\)\(\rho\geq\)\(r_{o}\)[21]. Successively, the technique has been expanded in other works by [PERSON] and [PERSON] [24; 25; 26; 27]. In Figure 2, the geometry of the ERTA acquisition system is shown (in the figure, the radial distance \(r\) is normalized with respect to the minimum possible radial distance \(r_{o}\), i.e., the Earth radius in our case). Hence, the boundaries of the normalized interval \([1,K]\) correspond to \(r_{o}\) and to the maximum altitude of the annular region, respectively. The functions \(f(r,\theta)\) and \(g(\rho,\phi)\) are periodical in the angle variables \(\theta\) and \(\phi\), respectively, so that they can be expanded as Fourier series, that is  
\[f(r,\theta)=\sum_{n=-\infty}^{+\infty}f_{n}(r)e^{in\theta} \tag{4}\]  
\[g(\rho,\phi)=\sum_{n=-\infty}^{+\infty}g_{n}(\rho)e^{in\phi} \tag{5}\]  
The first important theoretical result is that each term of the two series can be further decomposed by means of bases of functions, whose elements \(f_{n,l}(r,\theta)\) and \(g_{n,l}(\rho,\phi)\), \(l\ \in\ \mathbb{N}\), are described by Jacobi polynomials and whose major characteristic is that the Radon transform maps such functions onto each other [24]. More specifically, consider the following definitions:  
\[f_{n,l}(r,\theta)=\left\{ \begin{array}{ll}r^{-2}Q_{l}\Big{(}-\frac{1}{2}, \frac{1}{2};r^{-2}\Big{)}e^{in\theta}&for\ even\ n\\ r^{-3}Q_{l}\Big{(}-\frac{1}{2},\frac{1}{2};r^{-2}\Big{)}e^{in\theta}&for\ odd\ n \end{array} \right. \tag{6}\]  
\[g_{n,l}(\rho,\phi)=\rho^{-1-n}Q_{l}\Big{(}n,0;\rho^{-2}\Big{)}e^{in\phi} \tag{7}\]  
Figure 2: Geometry of the ERTA acquisition system. where \(n\) is an integer and  
\[Q_{l}(\alpha,\beta;x)=\sqrt{\frac{2^{1+\alpha+\beta}}{h_{l}(\beta,\alpha)}}P_{l}( \beta,\alpha;2x-1) \tag{8}\]  
with \(P_{l}(\beta,\alpha;2x-1)\) Jacobi polynomials and  
\[h_{l}(\alpha,\beta)=\frac{2^{\alpha+\beta+1}}{2l+\alpha+\beta+1}\frac{\Gamma( l+\alpha+1)\Gamma(l+\beta+1)}{\Gamma(l+1)\Gamma(l+\alpha+\beta+1)} \tag{9}\]  
where \(\Gamma(\kappa)\) is the Euler Gamma function. If \(R(\cdot)\) denotes the Radon operator, then it can be demonstrated that  
\[R(f_{n,l})(\rho,\phi)=0\text{ for }l<\left\lfloor{\left\lfloor{n}\right\rfloor /2}\right\rfloor \tag{10}\]  
\[R(f_{n,l})(\rho,\phi)=c_{n,l^{\prime}}g_{n,l^{\prime}}(\rho,\phi)\text{ \ for }l \geq\left\lfloor{\left\lfloor{n}\right\rfloor/2}\right\rfloor \tag{11}\]  
with \(l^{\prime}=l-\left\lfloor{\left\lfloor{n}\right\rfloor/2}\right\rfloor\), while \(\left\lfloor{\cdot}\right\rfloor\) denotes the integer part and  
\[c_{n,l}=\frac{\sqrt{2\pi}}{\sqrt{\left\lvert{n}\right\rfloor+2l+1}} \tag{12}\]  
Such results demonstrate not only the previously mentioned mapping property of elements of the set \(\left\{{f_{n,l}(r,\theta)}\right\}\) onto elements of the set \(\left\{{g_{n,l}(\rho,\phi)}\right\}\), but also a second important result of this tomographic approach, which is the existence of nonnull functions \(f_{n,l}(r,\theta)\) whose Radon transform is null.