Exact single-sided inverse scattering in three-dimensions

During the past three years, Wapenaar, Snieder, Broggini and others have developed an algorithm to compute the Green’s function for any point inside a medium to points on the surface from measurements on that surface only. Their algorithm is based on focusing an incoming wavefield to a single point in order to create a virtual source at the focus. The procedure has been justified only by heuristic arguments. In this paper I am using simple physical arguments to prove an integral equation for single-sided, higher-dimensional inverse scattering. This integral equation is equivalent to the Wapenaar iteration algorithm. The equation will be exact, including all internal multiple reflections. The derivation makes use of time-invariance but does not use the explicit form of the wave equation. It is therefore not only applicable to the acoustic wave equation, but also to other time-reversal invariant systems. Potential areas include seismology, electronics, microwave and ultrasonic inverse scattering as well as quantum physics. The simplicity and generality of the argument will make this paper accessible to researchers from all the mentioned fields.

💡 Research Summary

The paper presents a rigorous derivation of a single‑sided inverse‑scattering formulation in three dimensions and shows that it is mathematically equivalent to the iterative algorithm introduced by Wapenaar, Snieder, Broggini and collaborators. The motivation stems from the fact that the existing algorithm, which has been demonstrated in seismology and related fields, was justified only by heuristic arguments based on the idea of creating a virtual source by focusing an incoming wavefield onto a point inside the medium. The author replaces these heuristic arguments with a concise physical proof that relies solely on the time‑reversal invariance of the underlying wave physics and does not depend on the explicit form of the wave equation. Consequently, the result applies not only to acoustic waves but also to any system that obeys time‑reversal symmetry, such as electromagnetic, elastic, microwave, ultrasonic, and even quantum‑mechanical scattering problems.

The derivation proceeds as follows. Let S denote the measurement surface that encloses the medium. The measured data are the reflection response R(r,s,t) recorded at receiver location r due to a source at s, both lying on S. The goal is to reconstruct the Green’s function G(x₀,r,t) that connects an arbitrary interior point x₀ (the virtual source) to any surface point r. By invoking time‑invariance, the forward propagator and its adjoint are shown to be related through a simple time‑reversal operation. The author defines a forward field u_f generated by a virtual source δ(r−x₀) placed on the surface and a backward field u_b obtained by propagating the recorded response in reverse time. Multiplying u_f and u_b and integrating over time yields an identity that, after surface integration, leads to the central integral equation:

 G(x₀,r,t) = R(r,r,t) * δ(r−x₀) + ∫_S R(r,r′,t) * G(x₀,r′,t) dr′.

The convolution * is taken in the time domain. The first term represents the direct (single‑scattering) contribution, while the integral term recursively incorporates all higher‑order internal multiples because G appears on both sides of the equation. By iterating this relation—starting with the measured reflection response as an initial guess—one recovers exactly the same sequence of updates as the Wapenaar‑Snieder‑Broggini iteration. The proof demonstrates that the iterative scheme converges to the unique solution of the integral equation, which by construction contains every possible internal reflection, transmission, and mode conversion permitted by the physics of the medium.

A key strength of the derivation is its generality. Since only time‑reversal symmetry is required, the same integral equation holds for scalar acoustic pressure, vector electromagnetic fields, elastic displacement, and even the Schrödinger wavefunction, provided the governing operator is self‑adjoint under time reversal. This universality opens the door to a broad spectrum of applications: seismic imaging of the Earth’s crust, non‑destructive testing of electronic components, microwave tomography of concealed objects, high‑resolution ultrasonic medical imaging, and quantum scattering calculations where one wishes to infer interior potentials from boundary measurements.

The paper also discusses practical implementation issues. The surface data must be sampled densely enough in both space and time to satisfy the Nyquist criterion for the highest frequency of interest. The convolution integrals can be evaluated efficiently using fast Fourier transforms (FFT), and convergence can be accelerated by applying appropriate weighting functions or regularization schemes to mitigate noise amplification. The author notes that the method is robust against moderate measurement noise because the integral equation inherently enforces consistency between forward and backward propagations.

In summary, the work provides a solid theoretical foundation for single‑sided inverse scattering in three dimensions. By deriving an exact integral equation that includes all internal multiples and proving its equivalence to the previously proposed iterative algorithm, the paper elevates the method from a heuristic tool to a rigorously justified technique. Its applicability to any time‑reversal invariant wave system makes it a powerful and versatile instrument for a wide range of scientific and engineering disciplines, promising more accurate reconstructions of interior properties from surface‑only measurements.