A differential semblance algorithm for the inverse problem of reflection seismology

AbstractThis paper presents an analysis of stability and convergence for a special case of differential semblance optimization (DSO). This approach to model estimation for reflection seismology is a variant of the output least squares inversion of seismograms, enjoying analytical and numerical properties superior to those of more straightforward versions. We study a specialization of DSO appropriate to the inversion of convolutional-approximation planewave seismograms over layered constant-density acoustic media. We prove that the differential semblance variational principle is locally convex in suitable model classes for a range of data noise. Moreover, the structure of the convexity estimates suggest a family of quasi-Newton algorithms. We describe an implementation of one of these algorithms, and present some numerical results

On the linear growth mechanism driving the stationary accretion shock instability

During stellar core collapse, which eventually leads to a supernovae explosion, the stalled shock is unstable due to the standing accretion shock instability (SASI). This instability induces large-scale non spherical oscillations of the shock, which have crucial consequences on the dynamics and the geometry of the explosion. While the existence of this instability has been firmly established, its physical origin remains somewhat uncertain. Two mechanisms have indeed been proposed to explain its linear growth. The first is an advective-acoustic cycle, where the instability results from the interplay between advected perturbations (entropy and vorticity) and an acoustic wave. The second mechanism is purely acoustic and assumes that the shock is able to amplify trapped acoustic waves. Several arguments favouring the advective-acoustic cycle have already been proposed, however none was entirely conclusive for realistic flow parameters. In this article we give two new arguments which unambiguously show that the instability is not purely acoustic, and should be attributed to the advective-acoustic cycle. First, we extract a radial propagation timescale by comparing the frequencies of several unstable harmonics that differ only by their radial structure. The extracted time matches the advective-acoustic time but strongly disagrees with a purely acoustic interpretation. Second, we present a method to compute purely acoustic modes, by artificially removing advected perturbations below the shock. All these purely acoustic modes are found to be stable, showing that the advected wave is essential to the instability mechanism.Comment: 17 pages, 10 figures, accepted for publication in MNRA

A Transfer Function Approach to Structural Vibrations Induced by Thermoacoustic Sources

To decrease NOx emissions from a combustion system, lean premixed combustion in combination with an annular combustor is used. One of the disadvantages is an increase in sound pressure levels in the combustion system, resulting in an increased excitation of the surrounding structure, the liner. This causes fatigue, which limits the life time of the combustor. To model the interaction between flame, acoustics and structure, a transfer function approach is used. In this approach, the components are represented by the frequency dependent linear transfer between their inputs and outputs. For the flame a low pass filter with convective time delay is used as transfer function between velocity perturbations at the burner outlet and the flame as acoustic volume source. The acoustic transfer from volume source to velocity perturbation at the burner outlet is obtained from a harmonic finite element analysis, in which a temperature field from CFD calculations is used. The calculated response is subsequently curve-fitted using a pole-zero model to allow for fast calculations. The finite element model includes the two way coupling between structural vibrations and acoustics, which allows extraction of the vibration levels. The different transfers are finally coupled in one model. Results show frequencies of high acoustic response which are susceptible to thermoacoustic instability. Damping mechanisms and the phase relation between the different components determine stable or unstable behavior and the amplitude of the resulting perturbations. Furthermore there are also frequencies of high structural response. Especially when the two coincide, the risk of structural damage is high, whereas when they move away from each other, the risk decreases

Stable Realization of a Delay System Modeling a Convergent Acoustic Cone

This paper deals with the physical modeling and the digital time simulation of acoustic pipes. We will study the simplified case of a single convergent cone. It is modeled by a linear system made of delays and a transfer function which represents the wave reflection at the entry of the cone. According to [1], the input/output relation of this system is causal and stable whereas the reflection function is unstable. In the continuous time-domain, a first state space representation of this delay system is done. Then, we use a change of state to separate the unobservable subspace and its orthogonal complement, which is observable. Whereas the unobservable part is unstable, it is proved that the observable part is stable, using the D-Subdivision method. Thus, isolating this latter observable subspace, to build the minimal realization, defines a stable system. Finally, a discrete-time version of this system is derived and is proved to be stable using the Jury criterion. The main contribution of this work is neither the minimal realization of the system nor the proofs of stability, but it is rather the solving of an old problem of acoustics which has heen achieved using standard tools of automatic control

Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: explicit time-stepping and efficient mass matrix inversion

We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces result in less stringent CFL restrictions than equivalent piecewise continuous or discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based on L2 n-widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods