An adaptive phase space method with application to reflection traveltime tomography

Abstract

In this work, an adaptive strategy for the phase space method for traveltime tomography (Chung et al 2007 Inverse Problems 23 309-29) is developed. The method first uses those geodesics/rays that produce smaller mismatch with the measurements and continues on in the spirit of layer stripping without defining the layers explicitly. The adaptive approach improves stability, efficiency and accuracy. We then extend our method to reflection traveltime tomography by incorporating broken geodesics/rays for which a jump condition has to be imposed at the broken point for the geodesic flow. In particular, we show that our method can distinguish non-broken and broken geodesics in the measurement and utilize them accordingly in reflection traveltime tomography. We demonstrate that our method can recover the convex hull (with respect to the underlying metric) of unknown obstacles as well as the metric outside the convex hull.

A Hybrid Adaptive Phase Space Method for Reflection Traveltime Tomography

“…The underlying medium outside the convex hull of the scatterer is reconstructed by a optimization based iterative method. The newly developed method is more stable than the previous adaptive phase method proposed in [34] due to the introduction of an auxiliary fidelity function to guide the layer stripping process and a direct detection of all non-broken rays. To image the boundary of the scatterer, we use a direct imaging method that can locate points on the boundary of the scatterer by selecting those broken-once rays that hit the scatterer almost normally and tracing back those rays to half traveltime in the reconstructed medium.…”

“…In [34][35][36], the authors have developed a phase-space approach for transmission and reflection traveltime tomography for acoustic and elastic media by using the Stefanov-Ulhmann identity formulated in [45]. The method is advantageous over traditional methods in inverse kinematic problems [3,[46][47][48][49], because it uses multiple arrival times systematically and has the potential to handle anisotropic metrics as well, while these traditional methods can only recover isotropic metrics by utilizing first arrival times.…”

“…However, for the challenging case where both the medium and the scatterers are unknown, adaptive phase space method developed in [34] only uses non-broken ray to recover the medium outside the convex hull of the scatterers. In this work, we combine the adaptive phase space method, which is an optimization based iterative method, with a direct imaging method using selected broken-once ray data.…”

An inverse problem from condensed matter physics

“…Along the lines of the reduced ODE's (3) and (5), a set of simple ODE's with anisotropic traps were proposed and studied in which vortices interact with the background potential and with each other, see [32]. Note also the earlier work on vortices in BEC in isotropic potentials, [31,33,34].…”

A Penalization-Regularization-Operator Splitting Method for Eikonal Based Traveltime Tomography

“…Consequently, the available first-arrival traveltime information may not suffice to determine the sound speed so that the resulting sound speed model may have limited resolution when the to-be-imaged structure is very complicated, and this can be observed in our numerical results for the Marmousi model. To improve resolution of traveltime tomography, we have to take into account multiarrivals as demonstrated in [15,2,3]. It is an ongoing effort to develop a penalization-regularization approach for multiarrival traveltime tomography.…”