# Two-Dimensional Linear Inversion of GPR Data with a Shifting Zoom along the Observation Line
## 1 Introduction  
The classical way to achieve reliable and well-focused ground-penetrating radar (GPR) images is based on the migration procedure widely used and implemented in most available codes [1; 2]. This data processing method originates from seismology [3; 4; 5], and was later also adapted for the solution of inverse problems in Maxwell's equations [6; 7]. Another class of reconstruction approaches, also deployed in real world applications, exploits linear models of electromagnetic scattering [6]. Either way, the adoption of linear models entails few limitations with regards to the possibility of dealing with mutual interactions between targets [8] and reconstructing the electromagnetic properties of a target in a quantitative way [9]. On the other hand, the Born Approximation method can be extended to a quadratic approximation, which permits mitigation of the effect of non-linearity, in terms of false solution occurrences [10]. The reliability of the inverse scattering approach based on quadratic approximation has been also validated with data collected in the laboratory [11]. Relative to non-linear methods, linear models allow reliable algorithms, based on the minimization of a cost function, which are not affected by the false solutions issue [10; 11; 12]. The stability of the inversion approach is ensured by the exploitation of regularization schemes. Linear electromagnetic scattering inversion algorithms have been developed and analyzed in relation to their effectiveness with GPR problems and other electromagnetic geo-sounding techniques [13; 14; 15; 16; 17]. In addition, linearinverse scattering models have applications in other, non-conventional configurations, such as the \"differential ones\" [18; 19] and for radar on airborne vehicles [20] and UAV [21]. Relative to migration algorithms, a linear inverse scattering approach is able to account, in a rigorous way, for different issues, such as the presence of targets in the near field of the GPR system, truncation of the measurement domain and possible losses in the soil (often ignored in a migration algorithm, or rather counteracted with the application of some gain variable vs. depth). On the other hand, the linear inverse scattering approach requires filling and performing the inversion (in a generalized and regularized sense) of the matrix--which is the discretized version of the linear integral scattering operator--and this can be computationally demanding. The most popular approaches for regularizing a linear inversion are the Truncated Singular Value Decomposition (TSVD) [22; 23; 24; 25] and the Tikhonov method [26]. In this paper, we adopt a TSVD scheme, which permits, relative to migration, a more rigorous analysis of the performance of the inversion approach, in terms of its resolution limits. However, as previously mentioned, it has the main drawback of being much more demanding in terms of computational cost. Consequently, in real GPR surveys (with scanning lines of several tens or even hundreds of wavelengths long), it is not possible to compute SVD for the relatively large matrices involved. This, in general, prevents the possibility of \"inverting\" an entire radar image in a single step. The first strategy introduced to counteract this issue was to subdivide the overall scan profile into adjacent sub-profiles, sufficiently small enough to allow for the inversion of the matrix, and then, to join the inversion results achieved for each sub-profile side-by-side [27; 28]. However, this approach poses several problems (described in detail in Section 4) with regards to the targets located in the proximity of the lateral edges of two adjacent sub-profiles, or those that cross these lateral edges. A strategy aimed at mitigating these issues was presented in [29], where a double sequence of investigation domains was exploited with a shift of half an investigation domain between two successive sub-profiles.