## 2 Sampling Representation over the Upper Hemisphere
As will be shown later, the OSI algorithm also requires knowledge of those samples falling in the lower hemisphere; otherwise, they have to be set to zero, thus determining a severe truncation error in the reconstruction of the NF data required by [PERSON]'s NF/FF transformation. Again, the application of the image principle offers the possibility of overcoming this issue. As a matter of fact, these missing samples, which should have been acquired by the probe and rotated probe, can be synthesised from the corresponding ones collected on the upper hemisphere by applying the following \"mirroring rule\":  
\[\left\{ \begin{array}{l}V_{p}(r,\pi-\vartheta,\varphi)=-V_{p}(r,\vartheta, \varphi)\\ V_{r}(r,\pi-\vartheta,\varphi)=V_{r}(r,\vartheta,\varphi)\end{array} \right. \tag{7}\]  
which has been properly developed for the first-order probe adopted here. It must be stressed that, to make possible the mirroring of the collected voltage samples according to (7), the sample arrangement must be symmetrically distributed with respect to the plane \(z=0\) and must not contain the equator as sampling parallel due to the presence in that plane of the infinite PEC ground plane. The NF samples directly acquired on the upper hemisphere and those mirrored via (7) are efficiently interpolated to obtain the NF data needed by [PERSON]'s NF/FF transformation. To this end, the following two-dimensional OSI algorithm [33; 34] is conveniently employed. It exploits two properly matched OSI expansions, one allowing the interpolation along the meridians and the other along the parallels. Both expansions use few samples nearest to the given output point. To reconstruct the voltage at a point \(Q\) of [PERSON]'s spherical grid, the algorithm works as follows: It considers the meridian passing through \(Q(\vartheta,\varphi)\) and uses the following OSI expansion along the parallels [33]  
\[V(\tau_{k},\varphi)=\sum_{j=j_{0}-\vartheta+1}^{j_{0}+q}V\Big{(}\tau_{k}, \varphi_{j,k}\Big{)}\Omega_{N_{j}}\Big{(}\varphi-\varphi_{j,k},\overline{ \varphi}\Big{)}D_{N_{j}^{\prime\prime}}\left(\varphi-\varphi_{j,k}\right) \tag{8}\]  
to reconstruct the \(2p\) intermediate samples at the intersection points (\(\vartheta(\tau)\), \(\varphi\)) of the meridian at \(\varphi\) with the involved sampling parallels. In (8), \(V(\tau_{k}=k\Delta\tau,\varphi_{j,k}=j\Delta\varphi_{k})\) are the involved \(2q\) non-redundant voltage samples directly measured or synthesised on each of \(2p\) parallels over the spherical observation surface with steps  
\[\Delta\tau=\frac{\pi}{K^{\prime\prime}};\;\Delta\varphi_{k}=\frac{2\pi}{2N_{j }^{\prime\prime}+1} \tag{9}\]  
where \(K^{\prime\prime}+1\) is the number of the parallels (poles included), symmetrically distributed with respect to the plane \(z=0\), with \(K^{\prime\prime}=\lfloor\chi K^{\prime}\rfloor+1\) and necessarily odd to avoid the equator as a sampling parallel, \(2N_{j}^{\prime\prime}+1\) is the number of the samples on each of them,  
\[K^{\prime}=\lfloor\chi^{\prime}W_{\tau}\rfloor+1;\;N_{j}^{\prime\prime}= \left\lfloor\chi N_{j}^{\prime}\right\rfloor+1 \tag{10}\]  
\[N_{j}^{\prime}=\lfloor\chi^{*}W_{\varphi}(\tau_{k})\rfloor+1;\;\chi^{*}=1+( \chi^{\prime}-1)[\sin\vartheta(\tau_{k})]^{-2/3} \tag{11}\]  
and \(\lfloor\cdot\rfloor\) is the floor function. Moreover, \(j_{0}=\lfloor\varphi/\Delta\varphi_{k}\rfloor\) is the index of the sample nearest to each point intersecting the meridian at \(\varphi\), \(\overline{\varphi}=q\Delta\varphi_{k}\), \(N_{j}=N_{j}^{\prime\prime}-N_{j}^{\prime}\), \(\chi\) an oversampling factor needed to control the truncation error [33], and  
\[ \begin{array}{l}D_{N_{j}^{\prime\prime}}\left(\varphi\right)=\frac{\sin \left[(2N_{j}^{\prime\prime}+1)\varphi/2\right]}{(2N_{j}^{\prime\prime}+1)\sin \left(\varphi/2\right)};\\ \Omega_{N_{j}^{\prime}}(\varphi,\overline{\varphi})=\frac{T_{N_{j}}[2\cos^{2 }(\varphi/2)/\cos^{2}(\overline{\varphi}/2)-1]}{T_{N_{j}}[2/\cos^{2}(\overline {\varphi}/2)-1]}\end{array}  \tag{12}\]\(T_{N_{j}}(\cdot)\) being the Tschebyscheff polynomial of degree \(N_{j}\).