An optimal transport approach to linearized inversion of receiver functions

Abstract

Receiver function analysis is widely used to make quantitative inferences about the structure below a seismic station. As these observables are mainly sensitive to traveltimes of phases converted and reflected at seismic discontinuities, the resulting inverse problem is highly non-linear, the solution non-unique, and there are strong trade-offs between the depth of discontinuities and absolute velocities. To overcome this difficulty, we propose to measure the misfit between the predicted and observed data with an optimal transport distance instead of the conventional least-squares distance, a strategy that has shown its assets in the context of full waveform inversion. This approach views a seismogram as a distribution of 'mass'. The optimal transport distance between two waveforms is the minimal cost of transporting one waveform onto the other. We test the optimal transport approach on the inversion of a radial P-wave receiver function. We also show how it can be applied to measure the cross-convolution distance between the radial and vertical components, thus avoiding the need for deconvolution associated with the calculation of the receiver function. The resulting misfit function is minimized with a local optimization algorithm to constrain the receiver-side structure. The benefits of this methodology are studied in simple synthetic tests and with real data. In particular, we show that with its increased sensibility to time-shifts, the optimal transport distance reduces the number of local minima in the misfit function, which, in the case of a linearized inversion, significantly reduces the dependency to the starting model and results in a better convergence towards the solution model. A joint inversion of the P-wave receiver function and surface wave dispersion curves is performed at the Hyderabad station in India.

Geophysical inversion and optimal transport

“…This is encouraging, and suggests that optimization of the Wasserstein based misfit function may be more successful than of the standard L 2 2 norm in terms of avoiding entrapment in local minima. This is also consistent with the observations of Hedjazian et al (2019) in their comparison of Wasserstein misfits with L 2 2 for seismic receiver functions.…”

“…The disadvantage of the LP formulation is that it becomes computationally prohibitive as the number of discrete elements in f and g increases, not least because the number of unknowns to be solved for in the transport matrix, π ij , is equal to the product of the number of elements in f and g. The computational cost can be reduced somewhat with primal-dual optimization methods. For p = 1 this is equivalent to minimization of the Kantorovich-Rubenstein norm between density functions (Lellmann et al, 2014), which has also been extensively applied in seismic full waveform inversion of reflection data (Métivier et al, 2016b,c,a,d) and more recently to receiver function inversion (Hedjazian et al, 2019) as well as body and surface waves in land seismics (He et al, 2019). Another method for 2D discrete cases with p = 2 involves using a regularized Linear Programming formulation through addition of an entropy term.…”

“…Although methods and details vary between studies, these contributions have now established Optimal Transport as a powerful new approach for the construction of novel seismic waveform misfit functions in reflection seismology. More recently, the application of Optimal Transport has been explored for other geophysical problems including seismic receiver function inversion (Hedjazian et al, 2019), gravity inversion (Huang et al, 2019), and simulation of broad-band ground motions (Okazaki et al, 2021). A number of useful surveys of the mathematical and computational aspects of Optimal Transport have been published, including those of Ambrosio (2003), Santambrogio (2015), Kolouri et al (2017), Levy & Schwindt (2018) and Peyré & Cuturi (2019).…”

Emerging Directions in Geophysical Inversion

“…Pioneered in geophysics by Engquist & Froese (2014), this has subsequently been employed for numerous studies, including the work of Métivier et al (2016a,b,c,d) and others (e.g. Huang et al, 2019;He et al, 2019;Hedjazian et al, 2019). However, numerous challenges remain to be fully-overcome.…”

Wasserstein Metric-Driven Bayesian Inversion With Applications to Signal Processing

“…We further note that in FWI there is also considerable interest in the approach by Métivier and co-authors, [23,24,21,22,17]. In their approach the quadratic Wasserstein distance is replaced by a relaxed 1-Wasserstein distance obtained from the Monge-Kantorovich problem via a maximization in the space of bounded 1-Lipschitz functions.…”