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# Building initial velocity model for FWI by slope tomography

B. Tavakoli and S. Sambolian have developed at Geoazur high-resolution slope tomography as a tool to build accurate initial velocity model for FWI ( Tavakoli et al., 2017; Tavakoli et al., 2019; Sambolian et al., 2019). Unlike the seminal formulation of Billette and Lambaré (1998) based upon ray tracing and Fréchet derivatives, our implementation relies on eikonal solver and the adjoint-state method . Slope tomography exploits the local coherency of seismic arrival in finely-sampled collection of seismograms to perform a dense semi-automatic picking of kinematic attributes including traveltimes and slopes (the horizontal component of the slowness vector at source and receiver positions) (Figure 1). A locally-coherent event, parametrized by the two-way traveltime and the source & receiver slopes, is tied in depth to a scatterer (Figure 1, right panel).

Figure 1: (Left) Picking slopes and traveltime of a locally-coherent event in common-shot and common-receiver gathers. (Right) A locally-coherent event in the data space is tied to a scattering position in depth. This position is found at the intersection between the isochrone defined by the two-way traveltime and the ray leaving the receiver with the picked slope (Figure 2) . Parsimonious slope tomography then update wavespeeds by minimizing the slopes at sources ( Sambolian et al., 2019).

In reflection mode, the so-called parsimonious slope tomography (PAST) use two attributes (one slope and the two-way traveltime) to position in depth the reflection/diffraction point by a kinematic migration process (Figure 2), while the reminder slope is used to update wavespeeds ( Sambolian et al., 2019). This kinematic migration process relies on two focusing equations as initially proposed by Chauris et al. (2002) in the framework of migration-based velocity analysis.

$$t(s;x) + t(r;x) = T^*\tag{1}\label{1}$$ and $$p_r = p_r^*\tag{2}\label{2} ,$$ where $$s$$ and $$r$$ stands for the source and receiver position, respectively, $$x$$ stands for the sought scatterer position, and $$^*$$ labels observable attributes. Simply, put the localization process consists of finding the intersection between the isochrone (the isotime surface) and the ray leaving $$r$$ with the slope $$p_r^*$$.

We formulate slope tomography as a nonlinear constrained problem. We solve this constrained problem with the so-called adjoint state method implemented with a Lagrange-multiplier implementation ( Plessix, 2006; Chavent, 2008; Kern, 2016). The focusing equations are used as state equation (or constraints) in PAST and the coordinates of the scatterers (the unknowns of these state equations) are state variables. The picked traveltimes and receiver slopes are the right-hand sides of the focusing equations while the data misfit function reduces to the least-square norm of the slope residuals at source positions.

Figure 2: Sketch illustration of kinematic migration of two-way traveltimes and slope at receiver for positioning of the scatterer in depth.

In contrast to Sambolian et al., 2019, Tavakoli et al., 2017 process the positions of the scatterers as optimization variables in adjoint-state slope tomography . This extends the search space through a relaxation of the focusing equations of Chauris (the migration and the demigration velocities are not anymore the same) but makes slope tomography more ill-posed.

An application of PAST on towed-streamer data from Carnarvon Basin, north-west of Australia are illustrated in Figures 3-5. See Sambolian et al., 2019 for more details.

Figure 3: A case study of PASTT on towed-streamed data from Carnarvon Basin, north-west of Australia (courtesy of CGG). (a-b) initial (a) and final (b) velocity model of PASTT. (c-d) Same as (a-b) with superimposed positions of the scatterer positions (from Sambolian et al., 2019).

Figure 4: Depth migrated image built using the PAST velocity model (Figure 19b) as background model (from Sambolian et al., 2019).

Figure 5: Common-image gathers in the depth-offset domain associated with the migrated image shown in Figure 11 (from Sambolian et al., 2019).

We recently show how to extend slope tomography to first-arrival traveltime tomography ( Tavakoli et al., 2018; Sambolian et al., 2019). We show an application on sparse OBN data collected offshore Japan for the regional exploration of the eastern Nankai subduction zone. Then, we develop joint first-arrival and reflection slope tomography from coincident towed-streamer and OBS data ( Sambolian er al., 2019). Future work involves the adaptation of reflection slope tomography to sparse multi-component OBN data, where estimation of the slope at OBS positions will be an issue due to the sparsity of the acquisition.

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