David F. Aldridge

Equations for finite-difference, time-domain simulation of sound propagation in moving inhomogeneous media and numerical implementation.

Finite-difference, time-domain (FDTD) calculations are typically performed with partial differential equations that are first order in time. Equation sets appropriate for FDTD calculations in a moving inhomogeneous medium (with an emphasis on the atmosphere) are derived and discussed in this paper. Two candidate equation sets, both derived from linearized equations of fluid dynamics, are proposed. The first, which contains three coupled equations for the sound pressure, vector acoustic velocity, and acoustic density, is obtained without any approximations. The second, which contains two coupled equations for the sound pressure and vector acoustic velocity, is derived by ignoring terms proportional to the divergence of the medium velocity and the gradient of the ambient pressure. It is shown that the second set has the same or a wider range of applicability than equations for the sound pressure that have been previously used for analytical and numerical studies of sound propagation in a moving atmosphere. Practical FDTD implementation of the second set of equations is discussed. Results show good agreement with theoretical predictions of the sound pressure due to a point monochromatic source in a uniform, high Mach number flow and with Fast Field Program calculations of sound propagation in a stratified moving atmosphere.

The Berlage wavelet

Symmetric wavelets are commonly used in seismic modeling studies. However, accurate simulation of many physical wave propagation phenomena requires a causal waveform possessing a certain degree of differentiability. The purpose of this note is to quantitatively describe the characteristics of a relatively unfamiliar wavelet, called the Berlage wavelet, that is appropriate for such studies.

Padé approximation in time-domain boundary conditions of porous surfaces

Formulation and implementation of time-domain boundary conditions (TDBCs) at the surface of a reactive porous material are made challenging by the slow decay, complexity, or noncausal nature of many commonly used models of porous materials. In this paper, approaches are described that improve computational efficiency and enforce causality. One approach involves approximating the known TDBC for the modified Zwikker-Kosten impedance model as a summation of decaying exponential functions. A second approach, which can be applied to any impedance model, involves replacing the characteristic admittance with its Padé approximation. Then, approximating fractional derivatives with decaying exponentials, a causal and recursive TDBC is formulated.

Insight into the output of reverse-time migration: what do the amplitudes mean?

Summary With the purpose of attaching meaning to the waveforms imaged by reverse-time migration, we obtain an expression for the output of such an algorithm over a simple subsurface model of a dipping interface. We invoke the cross-correlation imaging condition and make extensive use of the stationary phase approximation to analyze the migrated image. Our result quantifies the meaning of the amplitudes output from reverse-time shot-profile migration and should have relevance for direct migration of passive seismic data and frequency-swept source signals. A numerical example of reverse-time migration supports our theoretical results.

Statistically perturbed geophone array responses

Seismic‐receiver arrays implemented under typical field conditions are subject to a variety of perturbing influences. The array responses that are actually achieved differ, perhaps substantially, from the nominal response associated with ideal conditions (precise positioning, vertical plants, identical geophones, perfect ground coupling, etc.). Variations in receiver array response may degrade the effectiveness of multichannel processing and analysis schemes that rely upon channel‐to‐channel waveform constancy. In effect, array‐response variation is a form of noise added to recorded waveforms and is thus potentially harmful. A rigorous physical treatment of the response of a geophone array to incident plane‐wave elastic radiation forms the point of departure for assessing the importance of response perturbations. The hard‐wired multiple seismometer group, long transmission line, and recording‐system input impedance are considered an electromechanical system. An individual geophone may have arbitrarily spe...

3D finite‐difference simulation of acoustic waves in turbulent moving media

A finite‐difference algorithm appropriate for modeling acoustic waves in a fully heterogeneous moving 3D media has been developed. The model is characterized by: acoustic velocity, density, and the three components of the background media velocity. The approach solves a set of coupled 1st order velocity‐pressure differential equations appropriate for an adiabatic divergence‐free background velocity. The equations are staggered in time and space and the algorithm uses second order temporal and fourth order spatial finite‐differences. Since approximations are not adopted in the solution of the equations all arrivals are modeled with fidelity providing the spatial and temporal grids are chosen appropriately. The algorithm can include either a pressure or velocity free surface on the bottom boundary and absorbing boundaries on other model flanks. Designed to run on large scale parallel computational platforms, the algorithm has been validated for four machine architectures. Comparisons are presented to an ana...

Investigating the point seismic array concept with seismic rotation measurements.

Spatially-distributed arrays of seismometers are often utilized to infer the speed and direction of incident seismic waves. Conventionally, individual seismometers of the array measure one or more orthogonal components of rectilinear particle motion (displacement, velocity, or acceleration). The present work demonstrates that measure of both the particle velocity vector and the particle rotation vector at a single point receiver yields sufficient information to discern the type (compressional or shear), speed, and direction of an incident plane seismic wave. Hence, the approach offers the intriguing possibility of dispensing with spatially-extended received arrays, with their many problematic deployment, maintenance, relocation, and post-acquisition data processing issues. This study outlines straightforward mathematical theory underlying the point seismic array concept, and implements a simple cross-correlation scanning algorithm for determining the azimuth of incident seismic waves from measured acceleration and rotation rate data. The algorithm is successfully applied to synthetic seismic data generated by an advanced finite-difference seismic wave propagation modeling algorithm. Application of the same azimuth scanning approach to data acquired at a site near Yucca Mountain, Nevada yields ambiguous, albeit encouraging, results. Practical issues associated with rotational seismometry are recognized as important, but are not addressed in this investigation.

Numerical simulation of resonances in microtomographic models.

Summary To quantitatively assess material properties in porous media and their associated scale dependence, we simulate acoustic wavefield resonances in numerical models derived from microtomographic data gathered for sintered glass bead packs. The 3D wave propagation simulations utilize a staggered-grid finite-difference (FD) formulation of the first order velocity-pressure system of equations. The FD code is capable of modeling both stress-free and periodic boundaries and this flexibility facilitates the modeling of an infinitely-extending “slab” resonator from the microtomographic-based models. We find that the porous materials act as homogeneous effective media at low frequencies and we observe the transition into the regime of strong scattering from the pore structure, where an effective medium description is no longer valid. Future work will focus on simulating viscous loss, elastic wave propagation, and resonances resulting from a cube sample geometry instead of a “slab” in order to reproduce the experimental setup typically used in resonant ultrasound spectroscopy (RUS).

Seismic wave propagation in 3D randomly‐heterogeneous elastic media

Three-dimensional (3D) seismic wave propagation within a randomly-heterogeneous, isotropic elastic medium is simulated with an explicit, time-domain, finite-difference (FD) algorithm based on the velocity-stress system of elastodynamics. Geologically realistic earth models, containing spatially-correlated random variations in material properties, are readily generated with a wavenumber-domain algorithm. Calculated synthetic seismic data clearly show numerous effects associated with scattering of compressional, shear, and surface waves by the randomly distributed perturbations.

Seismic reciprocity rules

The reciprocity principle implies that a synthetic seismic trace is invariant when the roles (positions, orientations, magnitude/sensitivity) of a point source and point receiver are interchanged. Many reciprocity studies assume that the seismic wavefields involved are generated by body forces (Knopoff and Gangi, 1959; White, 1960; Gangi, 2000). Although some restricted treatments involving body moment sources do exist (de Hoop and Stam, 1988; Nyitrai et al., 1996; Nowack and Chen, 1999; Arntsen and Carcione, 2000) these can be extended and unified in four important ways: i) wave propagation in a general anelastic medium is considered, ii) unrestricted body moment sources are introduced, iii) combinations of different source types (forces, moments, tractions) are allowed , and iv) particular source attributes (magnitudes, waveforms, orientations) are explicitly included in the analysis. The reciprocity rules developed from such a general treatment have numerous applications in seismic data acquisition, analysis, modeling, and inversion, as well as strong theoretical utility.