## 7 Conclusions  
The truncated singular value decomposition has always been used as a regularization tool in order to obtain a unique solution (the minimum norm one) to a rank-deficient inverse problem. This is exactly what has been done in imaging radiometry by aperture synthesis for a long time. The number of singular values kept in the inversion is here equal to the number of Fourier frequencies in the experimental frequency coverage of the interferometric array of antennas. This comfortable situation has never been questioned unless recent studies have evidenced a potential issue when the shortest spacing between the antennas of an interferometric array is chosen is such a way that some viewing directions of the synthesized field cannot be taken into account in the inversion process because they have neither a physical meaning nor a mathematical one.  
This issue has been illustrated with the aid of numerical simulations conducted within the frame of the SMOS-HR project. However, it is also explained why this problem will affect any interferometric array as soon as the shortest spacing between the elementary antennas becomes smaller than a geometrical limit, which only depends on the geometry of the underlying sampling grids. It has been shown that, in such a situation, the inversion of the complex visibilities becomes unstable in the presence of noise, and this instability is characterized by the undesirable presence of a phantom in the retrieved brightness temperature maps. The origin of this phantom has been explained, and a solution to cure the interferometric array from that issue has been proposed and assessed.  
From the linear algebra point of view, this study has illustrated that the truncated singular value decomposition can also be used as a conditioning tool in order for the unique solution of a rank-deficient inverse problem to be a stable one with regard to the propagation of noise or errors during the inversion process. Provided that the number of additional singular values to discard prior to the inversion is acceptable, the radiometric sensitivity and the angular resolution do not suffer from this supplementary truncation. The smaller the shortest spacing below that geometrical limit, the stronger the propagation of noise and the larger the number of additional singular values to be discarded for reducing the presence of that phantom in the retrieved brightness temperature maps.  
**Author Contributions:** Conceptualization, E.A. and L.Y.; methodology, E.A. and L.Y.; software, E.A.; validation, E.A. and L.Y.; formal analysis, E.A. and L.Y.; investigation, E.A. and L.Y.; resources, E.A.; data curation, E.A.; writing--original draft preparation, E.A.; writing--review and editing, E.A. and L.Y.; visualization, E.A.; supervision, E.A.; project administration, E.A. and L.Y. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported by the french national research council (CNRS) within the frame of the SMOS-HR (Soil Moisture and Ocean Salinity at High-Resolution) project led by the french space agency (CNES) currently undergoing phase A studies.  
**Data Availability Statement:** Data are available upon request by email to the corresponding author.  
**Acknowledgments:** The authors are very grateful to [PERSON] (CESBIO) and [PERSON] (CNES) for their constant support and encouragement during this work. Data processing support was provided by CESBIO and CNES. The authors are indebted to the associate editor and to the four anonymous reviewers for their helpful reviews which greatly improved the manuscript.  
**Conflicts of Interest:** The authors declare no conflict of interest.