Converted wave tomography based on inverse level set and adjoint formulation

Abstract

SUMMARY Shear wave velocity (Vs) is a fundamental property of elastic media whose estimation from PS converted waves is challenging and requires modelling the boundary where P to S conversion occurs. This paper presents a PS tomography where seismic wave conversion/reflection points correspond to reflectors modelled with the level-set function set to zero [ϕ(x, z) = 0]. The proposed method aims for stable Vs inversion in a seismic acquisition setting using multicomponent receivers. Synthetic models simulating true Vs, Vp and the location of the geological reflector are used in the study. The inversion starts by locating a flat reflector, ϕ(x, z) = 0, which defines the zone Ω1 between the surface and the reflector, where the initial Vs and Vp fields are also set. To calculate the traveltimes of incident PT (P wave that propagates in Ω1 from source to the reflector), converted PS and reflected PP waves, for both observed and modelled data (forward problem), the methodology proposed by Rawlinson and Sambridge is adopted. This method uses the arrival times of the P waves, Tpt, from the seismic source at each reflector point as secondary sources generating the times Tps and Tpp. These times are calculated as a solution to the eikonal equation by using the Fast Marching method. The PS and PP residual times are minimized by updating Vs, Vp and ϕ(x, z) = 0 through adjoint variables designed from a formulation using Lagrange Multipliers in a variational context. The performance of the algorithm is evaluated for models with synclinal, sinusoidal and monoclinal reflector geometries using numerical tests considering the inversion of: (1) ϕ, given the true values of Vs and Vp; (2) ϕ and Vs, given the true value of Vp; (3) ϕ and Vp, given the true value of Vs and (4) the three parameters ϕ, Vs and Vp, simultaneously. Good results are obtained by inverting Vs and ϕ, given the true value of Vp. The simultaneous inversion of the three parameters exhibits promising results, despite the illumination problems caused by the different distribution of the PS, PP and PT time gradients due to the geometry of the reflectors and the acquisition setting (sources–receivers in the same plane). The proposed tomography estimates Vs and reflector positions which could help in statics corrections and improve the lithological characterization of near surface.