Full Waveform Inversion Using the Adjoint Method for Earthquake Kinematics Inversion

Abstract:

Extracting the information contained in seismograms for better description of the Earth structure and evolution is often based on only selected attributes of these signals. Exploiting the entire seismogram, Full Wave Inversion based on an adjoint estimation of the gradient and Hessian operators, has been recognized as a high-resolution imaging technique. Most of earthquake kinematics inversion are still based on the estimation of the Frechet derivatives for the gradient operator computation in linearized optimization. One may wonder the benefit of the adjoint formulation which avoids the estimation of these derivatives for the gradient estimation. Recently, Somala et al. (submitted) have detailed the adjoint method for earthquake kinematics inversion starting from the second-order wave equation in 3D media. They have used a conjugate gradient method for the optimization procedure. We explore a similar adjoint formulation based on the first-order wave equations while using different optimization schemes. Indeed, for earthquake kinematics inversion, the model space is the slip-rate spatio-temporal history over the fault. Seismograms obtained from a dislocation rupture simulation are linearly linked to this slip-rate distribution. Therefore, we introduce a simple systematic procedure based on Lagrangian formulation of the adjoint method in the linear problem of earthquake kinematics. We have developed both the gradient estimation using the adjoint formulation and the Hessian influence using the second-order adjoint formulation (Metivier et al, 2013, 2014). Since the earthquake kinematics is a linear problem, the minimization problem is quadratic, henceforth, only one solution of the Newton equations is needed with the Hessian impact. Moreover, the formal uncertainty estimation over slip-rate distribution could be deduced from this Hessian analysis. On simple synthetic examples for antiplane kinematic rupture configuration in 2D medium, we illustrate the properties of sensitivity kernels for different receivers configurations, the performances of different optimization algorithms (see open software at http://www-ljk.imag.fr/membres/Ludovic.Metivier). We may conclude about the acquisition configuration needed for successful imaging reconstruction of seismic kinematic rupture.