Analyzing Nonlinearity in Inverse Scattering Problems (ISPs): A Breakdown of Scatterer Strength

The degree of nonlinearity in inverse scattering problems (ISPs) depends significantly on the strength of the scatterers. For weak scatterers, where the norm of the product of the contrast function and the scattering operator is much less than 1, the higher-order Taylor terms in the scattering equation can be ignored, simplifying the problem into a linear one. However, real-world measurements are often noisy, making direct inversion of the linear operator unreliable. To address this, regularization techniques like Tikhonov regularization are employed to balance the accuracy of data fitting and the stability of the solution. When dealing with moderate scatterers, where the norm of the product of the contrast function and the scattering operator is less than 1, multiple scattering effects become pronounced. This necessitates the inclusion of low-degree polynomials in the estimation of scattered fields, transforming the problem into a nonlinear optimization task. Consequently, nonlinear optimization methods are required to find a solution. Lastly, strong scatterers, characterized by a norm of the product of the contrast function and the scattering operator greater than 1, exhibit intensive multiple scattering effects. This leads to a complex relationship between scattered fields and the contrast function, often requiring higher-degree polynomials or even nonpolynomial representations. Even in the absence of noise, obtaining a solution for strong scatterers can be highly challenging. In summary, the analysis of nonlinearity in ISPs highlights the diverse complexities arising from varying scatterer strengths, emphasizing the need for tailored solution approaches depending on the specific problem.