LaplaceDomain CrosstalkFree SourceEncoded
Elastic Full Waveform Inversion Using TimeDomain Solvers
Geophysics, 2024

Zhaolun Liu
Saudi Aramco 
Jürgen Hoffmann
DNO ASA 
Etienne Bachmann
Princeton University 
Congyue Cui
Princeton University 
Frederik J. Simons
Princeton University 
Jeroen Tromp
Princeton University
Objectives
To reduce the computational time of elastic fullwaveform inversion (FWI) using timedomain solvers, we develop a new version of source encoding that (1)~only needs two numerical simulations per iteration, namely, one sourceencoded forward simulation and one sourceencoded adjoint simulation, (2)~will not generate any crosstalk noise, (3)~poses no restriction on acquisition geometry, e.g., a fixed receiver spread for random polarity encodings or a dense source array for plane wave encodings, and (4)~can easily implement time windowing.
Methods
We present a new version of Laplacedomain crosstalkfree source encoding for elastic fullwaveform inversion (FWI) using timedomain solvers. As shown in Fig.1, our method of sourceencoding for FWI assigns to each source at random for each iteration a unique complex frequency, introducing a damping factor to take care of attenuating late arrivals. The misfit criterion is the sum of the squared errors in the ``shifted'' Laplace coefficients between observed and synthetic data. Additionally, we derive a phaseonly misfit. The sourcetime function takes the form of a weighted cosine or sine with exponentially increasing amplitude. We simultaneously activate all the sources and carry out one sourceencoded forward simulation followed by one sourceencoded adjoint simulation, using a timedomain solver. The encoded forward and adjoint wavefields are run until they reach steady state. The gradient is calculated through zerolag crosscorrelation between the steadystate encoded forward and adjoint wavefields over a time period proportional to the inverse of the encoded frequency spacing. Owing to the orthogonality between the trigonometric terms of the encoding operator, no crosstalk is introduced during gradient calculation, and there are no limitations on acquisition geometry. By tuning the damping factor, we can timewindow the data even when only one or a few sparse frequencies are being sampled. Timewindowing allows for the selection of specific arrivals during the various stages of the inversion.
Results
We demonstrate the effectiveness of our multibin WD algorithm by applying it to invert shingling events (see Fig. 1c) from a complex layered model (see Fig. 1a). The nearsurface model inverted by multibin WD is close to the true model (see Fig. 1g), and the travel time, waveform, and dispersion curves of the synthetic data calculated from the inverted model by multibin WD match well with those from the observed model (see Figs. 1h1i). In comparison, waveequation traveltime inversion (WT, see Figs. 1d1f) can only update the model (see Figs. 1d) to match the travel time of the data (see Fig. 1e), but the waveform and dispersion curves of the synthetic data from WT are still discrepant with the observed ones. Next, we apply multibin WD to invert surface waves for Swave wave speed (VS) (see Fig. 2). The seismic data set is collected in Wyoming, US for the critical zone study (see Fig. 2a). Eight bins are used, and the corresponding dispersion curves are calculated (see Fig. 2b) and used to invert for the VS model. The initial and inverted VS model are shown in Figs. 2c and 2d. The waveform comparison before and after inversion is shown in Figs. 2e and 2f, respectively. We can see that the waveform matches very well after multibin inversion, even though our misfit is based on the dispersion curves. These results demonstrate the effectiveness of the multibin WD method.