Hervé Chauris

Improving the gradient of the image-domain objective function using quantitative migration for a more robust migration velocity analysis

Migration velocity analysis aims at determining the background velocity model. Classical artefacts, such as migration smiles, are observed on subsurface offset common image gathers, due to spatial and frequency data limitations. We analyse their impact on the differential semblance functional and on its gradient with respect to the model. In particular, the differential semblance functional is not necessarily minimum at the expected value. Tapers are classically applied on common image gathers to partly reduce these artefacts. Here, we first observe that the migrated image can be defined as the first gradient of an objective function formulated in the data-domain. For an automatic and more robust formulation, we introduce a weight in the original data-domain objective function. The weight is determined such that the Hessian resembles a Dirac function. In that way, we extend quantitative migration to the subsurface-offset domain. This is an automatic way to compensate for illumination. We analyse the modified scheme on a very simple 2D case and on a more complex velocity model to show how migration velocity analysis becomes more robust.

Velocity Estimation with the Normalized Integration Method

In the context of velocity estimation, classical full waveform inversion suffers from many local minima. We propose an alternative method referred as the Normalized Integration Method, where the objective function measures the misfit between the integral of the square of the signal. By integrating a positive function, we only compare functions increasing with time. It appears that the objective function has a more convex shape. We first indicate how to efficiently compute the gradient of the misfit function. We then compare the new approach to the classical full waveform inversion through an application on a simple 2D synthetic data set. This example shows that the new approach can be useful for the determination of the long wavelengths of the velocity model.

Finite-difference strategy for elastic wave modelling on curved staggered grids

Waveform modelling is essential for seismic imaging and inversion. Because including more physical characteristics can potentially yield more accurate Earth models, we analyse strategies for elastic seismic wave propagation modelling including topography. We focus on using finite differences on modified staggered grids. Computational grids can be curved to fit the topography using distribution functions. With the chain rule, the elasto-dynamic formulation is adapted to be solved directly on curved staggered grids. The chain-rule approach is computationally less expensive than the tensorial approach for finite differences below the 6th order, but more expensive than the classical approach for flat topography (i.e. rectangular staggered grids). Free-surface conditions are evaluated and implemented according to the stress image method. Non-reflective boundary conditions are simulated via a Convolutional Perfect Matching Layer. This implementation does not generate spurious diffractions when the free-surface topography is not horizontal, as long as the topography is smoothly curved. Optimal results are obtained when the angle between grid lines at the free surface is orthogonal. The chain-rule implementation shows high accuracy when compared to the analytical solution in the case of the Lamb’s problem, Garvin’s problem and elastic interface.

Testing the Behaviour of Differential Semblance for Velocity Optimization

Background velocity estimation is a critical step for the depth imaging process. Because of constant increase of acquired 3D data volumes, this velocit estimation, cast as an inverse problem, should be automated (no picking).

Seismic iterative migration velocity analysis: two strategies to update the velocity model

The objective of seismic imaging is to recover properties of the Earth from surface measurements recorded during active seismic surveys. Migration Velocity Analysis techniques aim at determining a background velocity model (smooth part of the pressure wave velocity model) using the redundancy of seismic data and consist of solving a nested optimisation problem. In the inner loop, an extended reflectivity model (detailed part of the model) is determined from recorded primary reflections through a data-fitting procedure depending on a given background model. In the outer loop, a coherency criterion defined on the extended reflectivity assesses the quality of the background model. The inner problem is usually solved with a single iteration of gradient optimisation, leading to artefacts in the velocity updates. We study the benefits of further iterating on the reflectivity in the inner loop, which also allows the introduction of multiple reflections in the procedure. We propose two strategies for the computation of the gradient of the outer objective function. In the first case, we compute the exact numerical gradient by taking care of the background dependency of all inner iterations. In the second case, we derive an approximate gradient by assuming the optimal reflectivity has been obtained. Both methods are compared on their computational merits and through simple numerical examples on 2D synthetic data sets. The examples illustrate that regularisation of the inner problem is essential to obtain coherent velocity updates. The second approach displays a smaller sensitivity to regularisation and is simpler to implement.

Inversion Velocity Analysis - The Importance of Regularisation

Inversion Velocity Analysis has been recently proposed as an alternative to Migration Velocity Analysis. Under the Born approximation, it consists of first determining the optimal reflectivity model such that the synthetic data set nicely fits with the observed data set. Then, a standard velocity analysis is applied to the inverted reflectivity. The main differences with respect to the classical approach is the use of iterative migration versus standard migration. We propose here an alternative way to compute the gradient of the objective function and demonstrate the importance of the regularisation term introduced to determine the optimal reflectivity model.

Differential Waveform Inversion - A Way to Cope with Multiples?

In the context of velocity estimation, we investigate the differential waveform inversion method. The approach is formulated in the data domain and consists of two main steps. First, for a given shot, we derive the optimal reflectivity section. From the result, we then compute the next shot gather. The new formulation measures the differences between the predicted shot and the observed shot at the nextposition. The main interest of the Differential Waveform Inversion strategy resides in the possibility to take into account surface-related multiples. The key point is the iterative migration.

Alternative Objective Function for Inversion of Surface Waves in 2D Media

The inversion of surface wave properties contributes to the creation of a near-surface model. In seismic exploration, the proper knowledge of the near surface can improve model building in depth. Most surface wave inversion approaches are based on 1D layered models. We propose here to estimate 2D model parameters by using a full waveform inversion approach with an alternative objective function formulated in the frequency-wavenumber domain. In the novel objective function, oscillations are reduced thanks to the exploitation of the dispersive behavior of surface waves that map into localized propagation modes in the frequency-wavenumber domain. Moreover, spatial windowing is used to allow local comparison of modelled and observed data. For the objective function minimization, a gradient-based approach will be used. We implement the adjoint-state method for an efficient gradient computation. We use simple velocity models to show the reliability of our ormulation to localize anomalies, by comparing the gradients computed with the classical full waveform inversion and the novel approach.

Investigating the Differential Waveform Inversion

In the context of velocity estimation, we propose the Differential Waveform Inversion approach defined in the data domain. For reflected data, it first consists of migrating a single shot. From the updated velocity model, we then compute the data at the next shot position and compare it with the observed data. The minimization of the misfit appears to be equivalent to the Differential Semblance Optimization approach formulated in the depth migrated domain, at least for reflected data.

A necessary and sufficient condition for the blind extraction of the sparsest source in convolutive mixtures

This paper addresses sparse component analysis, a powerful framework for blind source separation and extraction that is built upon the assumption that the sources of interest are sparse in a known domain. We propose and discuss a necessary and sufficient condition under which the ℓ0 pseudo-norm can be used as a contrast function in the blind source extraction problem in both instantaneous and convolutive mixing models, when the number of observations is at least equal to the number of sources. The obtained conditions allow us to relax the sparsity constraint of the sources to its maximum limit, with possibly overlapping sources. In particular, the W-disjoint orthogonality assumption of the sources can be discarded. Moreover, no assumption is done on the mixing system except invertibility. A differential evolution algorithm based on a smooth approximation of the ℓ0 pseudo-norm is used to illustrate the benefits brought by our contribution.