Discretization of multipole sources in a finite difference setting for wave propagation problems

Abstract

Seismic sources are commonly idealized as point-sources due to their small spatial extent relative to seismic wavelengths. The acoustic isotropic point-radiator is inadequate as a model of seismic wave generation for seismic sources that are known to exhibit directivity. Therefore, accurate modeling of seismic wavefields must include source representations generating anisotropic radiation patterns. Such seismic sources can be modeled as linear combinations of multipole point-sources. In this paper we present a method for discretizing multipole sources in a finite difference setting, an extension of the moment matching conditions developed for the Dirac delta function in other applications. We also provide the necessary analysis and numerical evidence to demonstrate the accuracy of our singular source ap-1 proximations. In particular, we develop a weak convergence theory for the discretization of a family of symmetric hyperbolic systems of first-order partial differential equations, with singular source terms, solved via staggered-grid finite difference methods. Numerical experiments demonstrate a stronger result than what is presented in our convergence theory, namely, optimal convergence rates of numerical solutions are achieved point-wise in space away from the source if an appropriate source discretization is used.

Incorporating source directivity in wave-based virtual acoustics: Time-domain models and fitting to measured data

“…Using source terms involving repeated differentiation of the Dirac delta function, one may construct a multipole expansion for the source 32,33 . For a multipole of order higher than one, however, directivity is in general dependent on frequency-the longitudinal quadrupole being the simplest such example.…”

“…(35) Comparing (35) to (33) shows that the functionŝ a l,m (ω) can be computed from the coefficientsW l,m (R, ω) of the source directivity aŝ…”

“…In recent work, source models have been framed through the use of the wave equation accompanied by an additional term containing the three-dimensional (3D) Dirac delta function under the action of a series of Carte-sian spatial differential operators. In particular, the multipole representation has been employed, both in acoustics applications 32 and in more general settings 33 . A more natural approach is to associate such differential operators in Cartesian coordinates with spherical harmonic directivity patterns.…”

Projection in negative norms and the regularization of rough linear functionals

“…Observe that we have used the notation fn to distinguish between the solution of ( 14) and the solution f n of the semi-infinite mixed system (11), or equivalently, the best-approximation (10).…”

“…To overcome this drawback, we can distinguish adaptive approaches based on a posterior error estimates in the natural W 1,p -setting of these equations (see, e.g., [7,8,5,29]); or approaches based on error estimates in fractional (and Hilbert) Sobolev norms H s (see, e.g., [31]); or approaches based on weighted Muckenhoupt norms (see, e.g., [2,3,4]). On another hand, we can also find methods based on mesh-grading techniques (see, e.g., [6,23]), and methods based on regularization techniques (see, e.g., [48,46,45,11,35,10,34]). Many of the former results also apply for line sources (see, e.g., [23,32]).…”