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**File Type**

Shi, Jia (Rice University) | De Hoop, Maarten (Rice University) | Faucher, Florian (INRIA) | Calandra, Henri (Total)

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).

Artificial Intelligence, body wave, boundary value problem, compression, frequency, full waveform inversion, full-waveform inversion, international exposition, inverse problem, inversion, iteration, mesh, multilevel algorithm, reconstruction, representation, Reservoir Characterization, seg seg international, stability, Upstream Oil & Gas

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

Technology: Information Technology > Artificial Intelligence > Representation & Reasoning > Mathematical & Statistical Methods (0.35)

**Summary**

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.

**Introduction**

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.

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

**SUMMARY**

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.

**Summary**

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.

Yoo, Jewoo (_) | Lee, Jaejoon (_) | Shin, Changsoo (_) | Calandra, Henri (_)

**Summary**

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.

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (0.87)

We propose to use wave-equation tomography (WET) method to build long-wavelength velocity structure for full waveform inversion (FWI). In WET, full wavefield modeling is performed and cross-correlation time delay between the arrivals from synthetic and real waveforms is used as objective function. Adjoint method is used to calculate the gradient in each iteration efficiently. Since WET and FWI share similar inversion structure, we use a hybrid misfit function to combine the two methods as an integrated workflow that is able to estimate high-resolution structure from poor starting model. To stabilize WET and make it converge to global minimum, we precondition the time delay measures with maximum cross-correlation coefficients and perform adaptive scale smoothing to the gradients. By exploring the band-limited feature of seismic wavefield, WET can provide better resolution than ray-based travel time tomography, which is under high frequency approximation. To illustrate the advantage of wave-equation tomography, we show in a 2D synthetic test that WET provides subsurface information that is critical for successful FWI. We also test 2D Marmousi model and satisfactory inversion results are achieved without much manual manipulating.

adjoint, Artificial Intelligence, frequency, full waveform inversion, FWI, gradient, inversion, iteration, optimization problem, Reservoir Characterization, resolution, subsurface, time delay, tomography, Upstream Oil & Gas, wave equation tomography, wave-equation tomography, wavefield, waveform, Waveform Inversion

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

Technology: Information Technology > Artificial Intelligence > Representation & Reasoning > Optimization (0.49)

When waveform inversion is performed in the Laplace-Fourier domain, wave propagation should be described through Laplace-Fourier domain modeling. However, because the modeling operator matrix organized by a complex-valued angular frequency is not satisfied with the positive definite, direct matrix solvers or iterative matrix solvers supporting nonsymmetrical linear systems should be used. In this study, 3D 2

Technology:

- Information Technology > Hardware (1.00)
- Information Technology > Graphics (0.89)

We present a method for approximately inverting the Hessian of full waveform inversion as a dip-dependent and scaledependent amplitude correction. The terms in the expansion of this correction are determined by least-squares fitting from a handful of applications of the Hessian to

algorithm, annual meeting, application, approximation, Artificial Intelligence, Curvelet transform, Demanet, freedom, Hessian, inverse, inverse Hessian, inversion, matrix, migrated image, migration, model space, operator, preconditioner, pseudodifferential, Reservoir Characterization, Upstream Oil & Gas, wave-equation hessian

Although wave equation based migration techniques, such as Reverse Time Migration (RTM), have been popular for years, the acoustic approximation is still applied frequently. Even when considering anisotropic behavior, modifications to the acoustic wave equation are invoked to facilitate changes in wavespeed along different directions. In a classical marine survey, seismic waves are generated in the water layer and observations are recorded in the form of pressure fluctuations. In that case, using a purely acoustic wave equation is a reasonable assumption, since conversions between compressional and shear motions along the ocean bottom are weak. However, with the availability of Ocean Bottom Cables (OBCs) or in land surveys, shear waves do play an important role. On one hand, incorporating elastic information in imaging may enhance the coherence of arrivals and thus provide better images. On the other hand, recorded shear signals might contaminate the image if falsely interpreted as reflected compressional waves. Following these considerations, we carry out a 3D elastic experiment investigating proper imaging conditions for elastic migration.

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.

analytic solution, Gaussian, Gaussian quadrature, Gaussian quadrature integration, gaussian quadrature integration method, information technology software, integration, inversion, IT software, Laplace domain, laplace-domain modeling, matrix, method, numerical solution, obliquely-inclined interface, quadrature, Reservoir Characterization, reservoir description and dynamics, sea bottom, seismic processing and interpretation, Software Engineering, unbounded two-layer model, Upstream Oil & Gas

Industry:

- Information Technology > Software (1.00)
- Energy > Oil & Gas > Upstream (1.00)

Thank you!