The waveform inversion problem is inherently ill-posed. Traditionally, regularization schemes are used to address this issue. For waveform inversion, where the model is expected to have many details reflecting the physical properties of the Earth, regularization and data fitting can work in opposite directions: the former smoothing and the latter adding details to the model. We propose constraining estimated velocity fields by reparameterizing the model. This technique, also called model-space preconditioning, is based on directional Laplacian filters: It preserves most of the details of the velocity model while smoothing the solution along known geological dips. Preconditioning also yields faster convergence at early iterations. The Laplacian filters have the property to smooth or kill local planar events according to a local dip field. By construction, these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models. We illustrate with 2D synthetic and field data examples how preconditioning with nonstationary directional Laplacian filters outperforms traditional waveform inversion when sparse data are inverted and when sharp velocity contrasts are present. Adding geological information with preconditioning could benefit full-waveform inversion of real data whenever irregular geometry, coherent noise and lack of low frequencies are present.

Reverse time migration (RTM) backscattered events are produced by the cross-correlation between waves reflected from sharp interfaces (e.g. the top of salt bodies). Commonly, these events are seen as a drawback for the RTM method because they obstruct the image of the geologic structure. Many strategies have been developed to filter out the artifacts from the conventional image. However, these events contain information that can be used to analyze kinematic synchronization between source and receiver wavefields reconstructed in the subsurface. Numeric and theoretical analysis indicate the sensitivity of the backscattered energy to velocity accuracy: an accurate velocity model maximizes the backscattered artifacts. The analysis of RTM extended images can be used as a quality control tool and as input to velocity analysis designed to constrain salt models and sediment velocity.The analysis in this thesis suggest that we can use backscattering events along with reflection data to define a joint optimization problem for velocity model building. The gradient required for model optimization suffers from cross-talk, similar to the more conventional RTM images. In order to avoid the cross-talk, I use a directional filter based on Poynting vectors which preserves the components of the wavefield traveling in the same direction.Using backscattered waves for constraining the velocity in the sediment section requires defining the top of salt in advance, which implies a dynamic workflow for model building in salt environments where both sediment velocity and salt interface change iteratively during inversion.

CitationLi V, Wang H, Tsvankin I, Díaz E, Alkhalifah T (2017) Inversion gradients for acoustic VTI wavefield tomography and, ultimately, provide a higher resolution. Here, we implement forward and adjoint wavefield 9 extrapolation for VTI (transversely isotropic with a vertical symmetry axis) media using a gener-10 alized pseudospectral operator based on a separable approximation for the P-wave dispersion rela-11 tion. This operator is employed to derive the gradients of the differential semblance optimization 12(DSO) and modified image-power objective functions. We also obtain the gradient expressions for

A B S T R A C TFor some acquisition geometries, the cost of Full Waveform Inversion (FWI) can be considerably reduced by inverting simultaneously encoded shots. Encoded-shot strategies have the undesirable effect of leaving crosstalk noise in the final result. For FWI, changing the coding sequence periodically mitigates this effect. Another alternative is to use preconditioning, whereby the gradient is smoothed at every iteration along predefined directions. Preconditioning steers the solution towards accurate models while attenuating crosstalk artefacts. It also increases convergence speed and robustness to noise present in the data.

One of the main challenges for full-waveform inversion (FWI) is taking into account both anisotropy and elasticity. Here, we perform elastic FWI for a synthetic 2D VTI (transversely isotropic with a vertical symmetry axis) model based on the geologic section at Valhall field in the North Sea. Multicomponent surface data are generated by a finite-difference code. We apply FWI in the time domain using a multiscale approach with three frequency bands. An approximate inverse Hessian matrix, computed using the L-BFGS-B algorithm, is employed to scale the gradients of the objective function and improve the convergence. In the absence of significant diving-wave energy in the deeper part of the section, the model is updated primarily with reflection data. An oblique displacement source, which excites sufficiently intensive shear waves in the conventional offset range, helps provide more accurate updates in the Shear-wave vertical velocity, especially in the shallow layers. We test three model parameterizations, which exhibit different radiation patterns and, therefore, create different parameter trade-offs. Whereas most examples are for a constant-density model, we also generate a density field using Gardner's relationship and invert for the density along with the velocity parameters. The parameterizations that combine velocities and anisotropy coefficients generally yield superior results to the one that includes only velocities, provided that a reasonably accurate initial model is available.

No abstract

Reverse time migration (RTM) backscattered events are produced by the cross-correlation between waves reflected from sharp interfaces (e.g. the top of salt bodies). Commonly, these events are seen as a drawback for the RTM method because they obstruct the image of the geologic structure. Many strategies have been developed to filter out the artifacts from the conventional image. However, these events contain information that can be used to analyze kinematic synchronization between source and receiver wavefields reconstructed in the subsurface. Numeric and theoretical analysis indicate the sensitivity of the backscattered energy to velocity accuracy: an accurate velocity model maximizes the backscattered artifacts. The analysis of RTM extended images can be used as a quality control tool and as input to velocity analysis designed to constrain salt models and sediment velocity.The analysis in this thesis suggest that we can use backscattering events along with reflection data to define a joint optimization problem for velocity model building. The gradient required for model optimization suffers from cross-talk, similar to the more conventional RTM images. In order to avoid the cross-talk, I use a directional filter based on Poynting vectors which preserves the components of the wavefield traveling in the same direction.Using backscattered waves for constraining the velocity in the sediment section requires defining the top of salt in advance, which implies a dynamic workflow for model building in salt environments where both sediment velocity and salt interface change iteratively during inversion.

W aveform inversion (FWI) requires a good starting model and/or data at low frequency (<1 Hz) for convergence. However, this is not a necessary condition, but an artifact of the objective function defined using differences of observed and simulated data. Image-domain tomographic methods using the same wavefields and wave equations can converge to a reasonable solution from poor starting models and without long-offset and/or low-frequency data. Cascading imagedomain and data-domain wavefield tomography eliminates the need for extremely low-frequency in the acquired data.