The adjoint-state method applied to the kinematic source inversion: from a benchmark to a real earthquake

Jean Virieux
Université de Grenoble I (Joseph Fourier)

Summary: An earthquake is identified as one of the possible excitation sources for mechanical waves propagating inside the Earth. The earthquake source can be described as a dislocation with a sudden shear slip on a defined fault surfaces within the Earth (Beroza and Kanamori, 2007). We take advantage of the linear relationship between the slip-rate history on the fault and the complete wavefield recorded at receivers, thanks to the representation theorem for point dislocations (Aki and Richards, 2002, pp. 39). At a given receiver, the particle velocity field is a convolution integral over the source geometry between the time-dependent slip-rate function and the traction vector associated with the Green’s functions between each point of the fault surface and the receivers. The traction vectors in every fault point depend on properties of the medium but it is independent of the slip, making the forward problem a linear problem which can be quite fast as soon as we have precomputed stress states for each receiver.
Since Haskell (1964, 1966) promoted the idea of describing the earthquake source as a finite sequence of breaking sub-events away from the point source approximation, many strategies have been applied for the reconstruction of the earthquake slip as an inverse problem. Ide (2007) provides a complete historical review on the evolution of such strategies, called kinematic source inversion. In general, these inverse methods can be divided into two main categories. On the one hand, we recognize those that are based on a linear inverse formulation linking the parameters describing the spatio-temporal slip-rate distribution and the synthetic seismograms (Hartzell and Heaton, 1983; Wald and Heaton, 1994; Wald et al, 1996; Wald and Graves, 2001). A linear inverse formulation implies a large number of parameters to invert, as we are interested in reconstructing at
each time and space samples of the slip-rate function. On the other hand, methods reformulate the problem as a non-linear inversion based on a model reduction strategy of the slip-rate functions. These reduced parametrizations describe the spatio-temporal slip-rate distribution in terms of few source attributes, such as rupture velocity, rise time and slip amplitude (e.g. Campillo and Cotton 1995; Liu and Archuleta, 2004). The second formulation is quite appealing as reduced parameters are directly linked to prior models of slip ruptures we expect. Unfortunately, the null space is still quite important and regularization is required such as frequency hierarchical strategy (Campillo and Cotton 1995). Moreover, the impact of propagation errors (Piatanesi et al, 2004; Razafindrakoto and Mai, 2014; Hallo and Gallovic, 2016) is more difficult to assess when considering non-linear relation and one may hope that the linear formulation will allow a more constrained uncertainty quantification of slip with a robust behavior with respect to propagation errors, once the regularization has been adequately setup in this framework of a linear relationship. In order to do so, we formulate the problem of integral-constrained optimization as a local Newton equation the gradient of which can be estimated efficiently through an adjoint formulation (Plessix, 2006). We shall investigate the influence of the Hessian operator and we shall take benefit of causality related to time evolution for reducing drastically the size of the null space. What are ingredients of regularization we should consider when considering the space-time slip history as our model? We shall first illustrate this investigation on a benchmark synthetic case, named Source Inversion Validation One (SIV1) proposed by Mai et al. (2007), and then by applying the inverse method to the Kumamoto earthquake of 16 April 2016 of magnitude 7.0. We shall conclude on potentialities of this kinematic formulation of earthquake characterization.
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