First-arrival traveltime tomography (FATT) has been one of the commanding tomographic approaches for long wavelength velocity model building. Nowadays, FATT models are often used as initial models for more resolving imaging methods such as full-waveform inversion (FWI). In this context, improving the accuracy of FATT model is crucial to mitigate the nonlinearity of FWI. In spite of the widespread use of FATT at all scales (from near surface to global scale), it suffers from ill-posedness in terms of non-uniqueness of the solution due to the deficient information carried out by the sole traveltime attribute. We promote the use of the traveltime perturbation with respect to the source/receiver positions (horizontal component of the slowness vector or slope) as a supplement to the first-arrival traveltime (Tavakoli et al., 2018). For multi-component controlled-source experiments, we can generally take advantage of the fine source sampling to measure slopes by tracking local coherency over few neighboring traces in addition to traveltimes in common-receiver gathers, while slopes at the receiver position may be estimated by polarization analysis. The information of differential nature carried out by slopes yields a higher order sensitivity with respect to the subsurface parameters. We implement our first-arrival slope+traveltime (FASTT) tomography with eikonal solver and the matrix-free adjoint-state method, as previously proposed by Taillandier et al.(2009) for FATT. We illustrate the resolution power of FASTT relatively to FATT for coarse ocean bottom seismometer (OBS) acquisitions using two benchmarks based upon the EAGE/SEG Overthrust model at the exploration scale and a synthetic model of the eastern-Nankai subduction zone at the deep crustal scale (Górszczyk et al., 2018), then assess the quality of the tomographic models as initial models for FWI. We finally present an application to real OBS data collected during the SFJ experiment in the eastern Nankai Trough (Tokai area).

Dense acquisition are more and more available in exploration and earthquake seismology. Tomographic approaches can now consider not only travel times but also the wavefront itself across the seismic network (Zhang and Thurber, 2003; Yuan et al., 2016). For dense controlled-source seismic experiments, double differences of travel times between receivers in a common-shot gather (resp between sources in a common-receiver gather) are estimated, namely the horizontal component of the slowness vector at source and receiver positions designed as slopes. These slopes associated with the two-way traveltimes are interpreted as a reflection/diffraction from a small reflector segment or diffractor are used in tomographic inversion (Lambaré, 2008; Tavakoli F. et al., 2017). Picking of locally-coherent events leads to dense volumetric dataset and hence higher-resolution tomographic results (Guillaume et al., 2008). The reflection setting introduces implicitly another class of unknowns which are scatterer positions. Resulting inverse problem is awkward due to the intrinsic coupling between velocities and scatterer positions. The first choice alternates positions and wavespeeds. The second performs the joint estimation of the two parameter classes. The third one relies on the projection of the scatterer positions subspace onto the wavespeed subspace leading to a reduced-space inversion. This reduced-space formulation can be implemented in the slope tomography using adjoint-state method. Two focusing equations, which depend on two observables among the three available ones (two-way traveltime and one slope in 2D), gives exact solutions of positions which are injected as constraints in the slope tomography (Chauris et al., 2002). These constraints explicitly enforce the positions in the velocity estimation problem, which reduces now to a mono-variate inverse problem by minimization of single-slope residuals, not yet used. 2D synthetic (see figure) and real data case studies show faster convergence toward more accurate minimizer achieved by this variable projection method compared to the alternated and joint strategies. This method, which can be extended to 3D configurations, draws also interesting perspective for the joint hypocenter-velocity inversion problem in earthquake seismology.

Accurate kinematic models are fundamental to enhance our knowledge of the seismic cycle as well as to improve surface ground motion prediction. However, the solution of the ill-posed kinematic inverse problem is non-unique (e.g., Cohee & Beroza, 1994; Wald & Heaton, 1994; Cotton & Campillo, 1995 and Minson et al., 2013) and, according to current acquisition systems surrounding active faults, this problem is highly underdetermined, in spite of its rather simple formulation as a linear inverse problem. Non-linear formulations of the problem, based on model reduction strategies, alleviate the underdetermination of the problem. However, non-linear formulations imply drastic assumptions on the rupture history and they complicate the use of linear algebra tools to assess the uncertainties of results. Regardless of the assumed inverse formulation, the incorporation of physical constrains and prior information into the inverse problem is necessary to provide more robust and plausible solutions. In this work (Sanchez-Reyes et al. 2018), we present a new hierarchical linear time domain kinematic source inversion method able to assimilate data traces through evolutive time windows. This progressive approach, both on the data and model spaces, does require mild assumptions based on prior knowledge or preconditioning strategies on the slip rate local gradient estimations. Contrary to similar approaches (Fan et al., 2014), this strategy benefits from the sparsity and causality of the seismic rupture while still ensuring the positivity of the solution. While standard regularization terms are used for stabilizing the inversion, strategies based on parameter reduction leading to a non-linear relationship between the source history and the observed seismograms are avoided. Rise time, rupture velocity and other attributes can be extracted later on from the slip-rate inversion we perform. . Satisfactory results are obtained on synthetic benchmarks proposed by the Source Inversion Validation project (Mai et al. 2016) and for the 2016 M$_w$7.0 Kumamoto mainshock. Our specific formulation combined with simple prior information, as well as numerical results obtained so far, yields interesting perspectives for a quasi-real-time implementation and to ease the uncertainty quantification of such ill-conditioned problem.