Abstract
In this paper, we study 3-D multiparameter full waveform inversion (FWI) in viscoelastic media based on the generalized Maxwell/Zener body including arbitrary number of attenuation mechanisms. We present a frequency-domain energy analysis to establish the stability condition of a full anisotropic viscoelastic system, according to zero-valued boundary condition and the elastic–viscoelastic correspondence principle: the real-valued stiffness matrix becomes a complex-valued one in Fourier domain when seismic attenuation is taken into account. We develop a least-squares optimization approach to linearly relate the quality factor with the anelastic coefficients by estimating a set of constants which are independent of the spatial coordinates, which supplies an explicit incorporation of the parameter Q in the general viscoelastic wave equation. By introducing the Lagrangian multipliers into the matrix expression of the wave equation with implicit time integration, we build a systematic formulation of multiparameter FWI for full anisotropic viscoelastic wave equation, while the equivalent form of the state and adjoint equation with explicit time integration is available to be resolved efficiently. In particular, this formulation lays the foundation for the inversion of the parameter Q in the time domain with full anisotropic viscoelastic properties. In the 3-D isotropic viscoelastic settings, the anelastic coefficients and the quality factors using bulk and shear moduli parametrization can be related to the counterparts using P and S velocity. Gradients with respect to any other parameter of interest can be found by chain rule. Pioneering numerical validations as well as the real applications of this most generic framework will be carried out to disclose the potential of viscoelastic FWI when adequate high-performance computing resources and the field data are available.
Highlights
Full waveform inversion (FWI) is an attractive tool to obtain high-resolution subsurface parameters in complex geological structures, by iteratively minimizing the waveform misfit between the synthetic data and the observed seismograms (Tarantola 1987; Virieux & Operto 2009)
It is important to emphasize that Q (x, ω) is a frequency-dependent quality factor coming from GMB/generalized Zener body (GZB) model associated with eq (38), while Q(x) is a frequency-independent target value for attenuation
We formulate in a consistent way the 3-D multiparameter FWI in viscoelastic media based on GMB using arbitrary number of attenuation mechanisms
Summary
Full waveform inversion (FWI) is an attractive tool to obtain high-resolution subsurface parameters in complex geological structures, by iteratively minimizing the waveform misfit between the synthetic data and the observed seismograms (Tarantola 1987; Virieux & Operto 2009). Since the pioneer work of Romanowicz (1995) at global scale, estimation of the attenuation factor has focused the attention of seismologists (see Romanowicz & Mitchell 2007, for a review) Both the theory and the practice on multiparameter inversion using an arbitrary number of SLS mechanisms allow the inversion of the parameter Q in the time domain in full anisotropic viscoelastic medium: Fichtner & van Driel (2014) have provided simple expressions for frequency (in)-dependent Q models with numerical demonstration from regional- and global-scale time-domain wave propagation, complementing and simplifying nicely expressions provided by Tromp et al (2005). This new formulation builds a consistent framework with potential applications in both seismology and exploration geophysics
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