Jérôme Le Rousseau

Modeling and imaging with the scalar generalized‐screen algorithms in isotropic media

The phase‐screen and the split‐step Fourier methods, which allow modeling and migration in laterally heterogeneous media, are generalized here so as to increase their accuracies for media with large and rapid lateral variations. The medium is defined in terms of a background medium and a perturbation. Such a contrast formulation induces a series expansion of the vertical slowness in which we recognize the first term as the split‐step Fourier approximation and the addition of higher‐order terms of the expansion increases the accuracy. Employing this expansion in the one‐way scalar propagator yields the scalar one‐way generalized‐screen propagator. We also introduce a generalized‐screen representation of the reflection operator. The interaction between the upgoing and downgoing fields is taken into account by a Bremmer series. These results are then cast into numerical algorithms. We analyze the accuracy of the generalized‐screen method in complex structures using synthetic models that exhibit significant m...

Scalar generalized-screen algorithms in transversely isotropic media with a vertical symmetry axis

The scalar generalized‐screen method in isotropic media is extended here to transversely isotropic media with a vertical symmetry axis (VTI). Although wave propagation in a transversely isotropic medium is essentially elastic, we use an equivalent “acoustic” system of equations for the qP‐waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of “mild” anisotropy. The enhanced accuracy of the generalized‐screen method as compared to the split‐step Fourier methods allows the extension to VTI media. The generalized‐screen expansion of the one‐way propagator follows closely the method used in the isotropic case. The medium is defined in terms of a background and a perturbation. The generalized‐screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation. We cast the theory into numerical algorithms, and assess the accuracy of the generalized‐screen method in a partic...

Stability of Discontinuous Diffusion Coefficients and Initial Conditions in an Inverse Problem for the Heat Equation

We consider the heat equation with a discontinuous diffusion coefficient and give uniqueness and stability results for both the diffusion coefficient and the initial condition from a measurement of the solution on an arbitrary part of the boundary and at some arbitrary positive time. The key ingredient is the derivation of a Carleman-type estimate. The diffusion coefficient is assumed to be discontinuous across an interface with a monotonicity condition.

Depth migration in heterogeneous, transversely isotropic media with the phase‐shift‐plus‐interpolation method

The 2D and 3D depth-migration algorithm, Phase-Shift Plus Interpolation (PSPI), has been modified to take anisotropy into account. The model used is a transversely isotropic (TI) medium with symmetry axis that can be either vertical or tilted. I first review the PSPI algorithm for isotropic media. For the TI model, the Christoffel equation gives velocity as a function of the propagation angle. Building tables linking the values of the different components of wavenumber allows us to circumvent direct inversion of the relationship between the components of the wavenumber and the propagation angle. With given values of and one can directly compute the value of and thus perform the phase-shift. Since the phase-shift occurs in the wavenumber-frequency domain, the modification of the code for TI media is quite straightforward and intuitive. The modification from a medium with vertical symmetry axis to layered media with varying orientation of symmetry axis is also painless. An example shows that, as with the conventional PSPI algorithm, lateral velocity variations can be taken into account; in addition, the algorithm honors lateral variations in the anisotropic coefficients as well.

Uniform controllability properties for space/time-discretized parabolic equations

This article is concerned with the analysis of semi-discrete-in-space and fully-discrete approximations of the null controllability (and controllability to the trajectories) for parabolic equations. We propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters. For time discretization we use θ-schemes with

Poincaré recurrence for observations

{\theta \in [\frac{1}2,1]}

Fourier-Integral-Operator Approximation of Solutions to First-Order Hyperbolic Pseudodifferential Equations I: Convergence in Sobolev Spaces

. For the proofs of controllability we rely on the strategy introduced by Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) for the null-controllability of the heat equation, which is based on a spectral inequality. We obtain relaxed uniform observability estimates in both the semi-discrete and fully-discrete frameworks, and associated uniform controllability properties. For the practical computation of the control functions we follow J.-L. Lions’ Hilbert Uniqueness Method strategy, exploiting the relaxed uniform observability estimate. Algorithms for the computation of the controls are proposed and analysed in the semi-discrete and fully-discrete cases. Additionally, we prove an error bound between the fully discrete and the semi-discrete control functions. This bound is however not uniform with respect to the space discretization. The theoretical results are illustrated through numerical experimentations.

Carleman estimates for anisotropic elliptic operators with jumps at an interface

In arbitrary dimension, we consider the semidiscrete elliptic operator