We employ full waveform inversion (FWI) where the reconstruction is based upon iterative minimization techniques. We apply a multilevel scheme to stabilize our iterative reconstruction. We illustrate this idea using both Continuous Galerkin finite element method on unstructured tetrahedral meshes with surface and body waves and finite difference approximation on the regular meshes with body waves only. INTRODUCTION The reconstruction of subsurface parameters using iterative minimization has been initiated by Lailly (1983), Tarantola (1984, 1987) and Mora (1987). This work has been originally realized for time domain wave equations, time-harmonic formulation of the seismic inverse problem was later introduced by Pratt and Worthington (1990).
In this paper, we propose stable low-order Absorbing Boundary Conditions (ABC) for elastic TTI modeling. Their derivation is justified in elliptic TTI media but it turns out that they are directly usable to non-elliptic TTI configurations. Numerical experiments are performed by using a new elastic tensor source formula which generates P-waves only in an elliptic TTI medium. Numerical results have been performed in 3D to illustrate the performance of the ABCs.
Seismic Imaging is still progressing by taking advantage of advanced computational techniques constantly renewed. Nowadays simulations consider more realistic representations of the subsurface, typically moving from Acoustics to Elastodynamics and from Isotropy to Tilted Transverse Isotropy (TTI).
Literature is rich in references about “pseudo-acoustic TTI” RTM. First attempts chose to simplify the elastic TTI approximation, initially depicted in Alkhalifah (1998), leading to several TTI formulations in e.g. Du et al. (2007); Fletcher et al. (2009); Zhang et al. (2011); Duveneck and Bakker (2011), while others investigated equation decoupling, see for instance Zhan et al. (2012). All these references target acoustic TTI RTM only except in Yan and Sava (2011) dealing with elastic TTI RTM. In any case, nothing is mentioned about boundary conditions which are supposed to be non-reflecting for keeping the numerical solution from pollution generated by the boundaries of the computational domain. We propose here to address this issue which is critical when considering elastic TTI modeling.
Isotropic codes are usually based on Perfectly Matched Layers (PML) surrounding the domain of interest, see for instance Collino and Tsogka (2001). Unfortunately, it has been demonstrated in B´ecache et al. (2003) that PMLs are unstable in TTI media. Moreover, the numerical cost of the additional layer is prohibitive in 3D, especially in a RTM framework which is already computationally intensive. Besides, PML also impacts on parallel efficiency, since they requires heterogeneous computations on a large set of data. Hence, the design of stable Absorbing Boundary Condition (ABC) for elastic TTI is an effective alternative that should be considered for RTM. In a previous work, see Barucq et al. (2014), we have proposed a new elastic TTI ABC in 2D and we focus here on the extension to 3D.
We analyze the correlation focusing objective functional introduced by van Leeuwen and Mulder to avoid the cycle-skipping problem in full waveform inversion. While some encouraging numerical experiments were reported in the transmission setting, we explain why the method cannot be expected to work for general reflection data. We characterize the form that the adjoint source needs to take for model velocity updates to generate a time delay or a time advance. We show that the adjoint source of correlation focusing takes this desired form in the case of a single primary reflection, but not otherwise. Ultimately, failure owes to the specific form of the normalization present in the correlation focusing objective.
We have developed an integrated method to obtain high-resolution subsurface elastic parameters using combined wave equation tomography (WET) and full waveform inversion (FWI). Both refraction and reflection data are used. During parameterization, long wavelength and short wavelength structures are separated and mapped into velocity and density to account for kinematics and dynamics, respectively. Full wavefield modeling is used to compute synthetic data that include all reflection and refraction arrivals. To better constrain the reflection amplitude, the near offset data are first inverted using FWI where all the model perturbations are mapped into density. The short wavelength density structure is then converted into vertical travel time domain where it is independent of long wavelength velocity model. As long wavelength structure (velocity) is updated, short wavelength structure is converted back into depth domain for wavefield computation. Finally FWI is applied all the data to retrieve short wavelength structures with resolution up to a quarter wavelength. The method is applied to two synthetic examples; our results shows that one can recover detailed velocity information starting from a model far from the true model.
The full waveform inversion (FWI) of land data are becoming increasingly necessary in hydrocarbon exploration. However, strong surface waves and the existence of complex topography make it difficult to recover the subsurface structure using refraction tomography. To obtain subsurface velocity models of complex topography, we propose the Laplace-Fourier domain FWI with a finite element method (FEM). The mesh was designed to avoid several problems that could affect the inverted results and to minimize the computational cost. Laplace-Fourier domain inversion can recover subsurface P- and S-wave velocities with starting simple initial velocities generated without any prior information.
3D Laplace-domain waveform inversion can recover a large velocity model for successive waveform inversion in the frequency domain. However, the grid interval in 3D Laplace-domain modeling and inversion cannot be sufficiently small because of the heavy computational cost. Therefore, we cannot assess whether or not the modeled wavefield is reliable if our model has an abruptly undulated sea bottom surface. The irregular finite element method can provide a solution; however, it increases the number of bands of the impedance matrix. Instead, we applied the Gaussian quadrature integration method in order to reflect two properties on one element at the irregular sea bottom. In order to verify this modeling algorithm, we compared our modeled wavefield with the analytic solutions for an unbounded homogeneous model, an unbounded two-layer model and an obliquely-inclined two-layer model. The results of the verification tests show that our modeling algorithm better describes a wavefield with an irregular sea bottom in the 3D Laplace domain than the conventional modeling algorithm.