We consider the velocity estimation problem for the scalar wave equation using the array response matrix of sensors. In the active configuration, the sensors probe the unknown medium to be imaged with a pulse and measure the backscattered waves which gives directly the array response matrix. In the passive configuration, the sensors are passive receivers that record the signals transmitted by unknown, ambient noise or opportunistic sources and the array response matrix can be obtained by cross correlating the recorded signals. Under such circumstances, conventional Full Waveform Inversion (FWI) is carried out by nonlinear least-squares fitting of the array response matrix. It turns out that the FWI misfit function is high-dimensional and non-convex with many local minima. A novel approach to FWI based on a data driven reduced order model (ROM) of the wave equation operator is introduced and it is shown that the minimization of ROM misfit function performs much better.

We consider random wave coupling along a flat boundary in dimension three, where the coupling is between surface and body modes and is induced by scattering by a randomly heterogeneous medium. In an appropriate, anisotropic scaling regime we obtain a system of radiative transfer equations which are satisfied by the mean Wigner transform of the mode amplitudes. Interestingly, seismograms recently acquired with SEIS on Mars (InSight mission) show a behavior that fits the hypotheses of our analysis about the properties of its crust. We provide a rigorous probabilistic framework for describing solutions to the mentioned system using that it has the form of a Kolmogorov equation for some Markov process. We then prove statistical stability of the smoothed Wigner transform under the Gaussian approximation. We conclude with analyzing the nonlinear inverse problem for the radiative transfer equations and establish the unique recovery of phase and group velocities as well as power spectral information for the medium fluctuations from the observed smoothed Wigner transform.

Backprojection has proven useful in imaging large earthquake rupture processes. The method is generally robust and requires relatively simple assumptions about the fault geometry or the Earth velocity model. It can be applied in both the time and frequency domain. Backprojection images are often obtained from records filtered in a narrow frequency band, limiting its ability to uncover the whole rupture process. Here, we develop and apply a novel frequency-difference backprojection (FDBP) technique to image large earthquakes, which imitates frequencies below the bandwidth of the signal. The new approach originates from frequency-difference beamforming, which was initially designed to locate acoustic sources. Our method stacks the phase-difference of frequency pairs, given by the autoproduct, and is less affected by scattering and -time errors from 3-D Earth structures. It can potentially locate sources more accurately, albeit with lower resolution. We validated the FDBP algorithm with synthetic tests and benchmarked it against conventional backprojection. We successfully applied the method to the 2015 M7.8 Gorkha earthquake, and tested two stacking approaches - Band Width Averaged Autoproduct and its counterpart (BWAP and non-BWAP). The FDBP method shows promise in resolving complex earthquake rupture processes in tectonically complex regions.

In helioseismology one studies cross-correlations of line-of-sight velocities at the solar surface in order to invert for parameters in the solar interior. In the frequency domain the cross-correlation data takes the form $C(\pmb{r_1}, \pmb{r_2}, \omega)=\psi(\pmb{r_1}, \omega)^* \psi(\pmb{r_2}, \omega)$ with freqeuency $\omega$ and two points $\pmb{r_1}, \pmb{r_2}$ on the solar surface. This data set is of immense size and unfeasible to store, such that it is in need of an apriori averaging in space in frequency. Helioseismic holography is a physically motivated averaging scheme, which is based on backpropagation of surface fluctuations [1]. In this talk we show that the traditional holograms can be understood as the first step of an iterative inversion procedure [2]. This way we can extend traditional helioseismic holography to a full quantitative regularization method, which has two main advantages compared to traditional helioseismic inversions: By changing the order of backpropagation and local correlation we can use the whole cross-correlation data implicitly by avoiding the computation explicitly. Furthermore the iterative setup allows us to tackle nonlinear problems, which are only rarely studied in helioseismology. We validate iterative helioseismic holography on synthetics of large-scale axisymmetric flows like solar differential rotation and meridional flows. Finally we show some interesting future applications of iterative helioseismic holography which can not be studied with traditional helioseismology so far.

[1] C. Lindsey, D. Braun. Helioseismic Holography, Astrophysical Journal 485(2), p.895-903, 1997. doi:10.1086/304445

[2] T. Hohage, H. Raumer, C. Spehr. Uniqueness of an inverse source problem in experimental aeroacoustics, Inverse Problems 36(7), 2020. doi:10.1088/1361-6420/ab8484