Transient analytic point‐source response of a layered acoustic medium: Part II

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1478-1487 ◽  
Author(s):  
Martin Tygel ◽  
Peter Hubral
Keyword(s):  
Point Source ◽  
Plane Waves ◽  
Integral Solution ◽  
Acoustic Medium ◽  
Transient Solution ◽  
Full Wave ◽  
Starting Point ◽  
Infinite Integral ◽  
The Time Domain ◽  
Root Operator

In part I (this issue) analytic methods are described for propagating directly in the time domain the full wave field of transient plane waves at arbitrary incidence angles through parallel acoustic homogeneous layers. The theory of part I is used here to compute transient point‐source responses by way of integrating the transient plane‐wave responses over various (real and nonreal) incidence angles. At first the complete transient point‐source reflection (transmission) responses are formulated with the help of the transient Sommerfeld‐Weyl integral that was developed previously by us. This leads to a transient solution in form of an infinite integral over incidence angles involving an (angle‐dependent) integrand that is a time‐convolution between a transient reflectivity (transmissivity) function and an inverse transient analytic square root operator. The transient Sommerfeld‐Weyl integral solution is then proven to be causal. This causality provides the starting point for formulating exact transient point‐source responses in terms of integrals over a finite range of incidence angles only.

Transient analytic point‐source response of a layered acoustic medium: Part I

Geophysics ◽  
1985 ◽  
Vol 50 (9) ◽  
pp. 1466-1477 ◽  
Author(s):  
Martin Tygel ◽  
Peter Hubral
Keyword(s):  
Wave Propagation ◽  
Plane Wave ◽  
Point Source ◽  
Time Domain ◽  
Plane Waves ◽  
Natural Extension ◽  
Acoustic Medium ◽  
Angle Of Incidence ◽  
The Time Domain ◽  
Plane Wave Propagation

The exact transient responses (e.g., reflection or transmission responses) of a transient point source above a stack of parallel acoustic homogeneous layers between two half‐spaces can be analytically obtained in the form of a finite integral strictly in the time domain. (The theory is presented in part II of this paper, this issue.) The transient acoustic potential of the point source is decomposed into transient plane waves, which are propagated through the layers at any angle of incidence as well in the time domain; finally, they are superposed to obtain the total point‐source response. The theory dealing with transient analytic plane wave propagation is described here. It constitutes an essential part of computing the synthetic seismogram by the new transient method proposed in part II. The plane‐wave propagation is achieved by an exact discrete recursion that automatically handles the conversion of homogeneous waves into inhomogeneous transient plane waves at layer boundaries. A particularly efficient algorithm is presented, that can be viewed as a natural extension of the popular normal‐incidence Goupillaud (1961)-type algorithm to the nonnormal incidence case.

Transient representation of the Sommerfeld‐Weyl integral with application to the point source response from a planar acoustic interface

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1495-1505 ◽  
Author(s):  
Martin Tygel ◽  
Peter Hubral
Keyword(s):  
Point Source ◽  
Plane Waves ◽  
Transient Waves ◽  
Layer Boundary ◽  
Spherical Source ◽  
Time Harmonic ◽  
Starting Point ◽  
Inhomogeneous Plane ◽  
Acoustic Interface ◽  
Inhomogeneous Plane Waves

Point source responses from a planar acoustic and/or elastic layer boundary (as well as from a stack of planar parallel layers) are generally obtained by using as a starting point the Sommerfeld‐Weyl integral, which can be viewed as decomposing a time‐harmonic spherical source into time‐harmonic homogeneous and inhomogeneous plane waves. This paper gives a powerful extension of this integral by providing a direct decomposition of an arbitrary transient spherical source into homogeneous and inhomogeneous transient plane waves. To demonstrate with an example the usefulness of this new point source integral representation, a transient solution is formulated for the reflected/transmitted response from a planar acoustic reflector. The result is obtained in the form of a relatively simple integral and essentially corresponds to the solution obtained by Bortfeld (1962). It, however, is arrived at in a physically more transparent way by strictly superimposing the reflected/transmitted transient waves leaving the interface in response to the incident transient homogeneous and inhomogeneous plane waves coming from the center of the point source.

Beam stacking: A generalized preprocessing technique

Geophysics ◽  
1987 ◽  
Vol 52 (9) ◽  
pp. 1199-1210 ◽  
Author(s):  
Shalom Raz
Keyword(s):  
Frequency Domain ◽  
Plane Waves ◽  
Intrinsic Property ◽  
Beam Width ◽  
Propagation Path ◽  
Gaussian Beams ◽  
Starting Point ◽  
The Time Domain ◽  
Preprocessing Technique ◽  
Spatial Support

Gaussian beams are well understood frequency‐ domain entities combining the directional properties of plane waves with an effectively finite region of support. These outstanding properties are retained not only on a prescribed observation plane, but throughout the propagation path. A preprocessing sequence aimed at transforming raw seismic data into beam stacks is proposed. That is, time‐harmonic Gaussian beams are synthesized, replacing the plane waves generated by conventional slant‐stacking procedures. The suggested scheme is characterized by an open parameter, essentially the beam width, whose selection is critical to ultimate success. Specific criteria for choosing this parameter can be given. In the limits of zero and infinite beam widths, beam stacks degenerate to the original raw data and to the conventional slant stacks, respectively. Although beam stacking is basically a frequency‐domain procedure, a transformation into the time domain, using frequency constituents within selected bands, may be accomplished without losing finite spatial support. Advantages of choosing beam‐stacked data as a starting point for subsequent inversion may be cited on two levels. The intrinsic property of finite spatial support overcomes edge effects. In addition, the degree of localization achieved by beam stacking may point the way to new approaches to seismic imaging.

scholarly journals In-situ monitoring for liquid metal jetting using a millimeter-wave impedance diagnostic

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Tammy Chang ◽  
Saptarshi Mukherjee ◽  
Nicholas N. Watkins ◽  
David M. Stobbe ◽  
Owen Mays ◽  
...  
Keyword(s):  
Liquid Metal ◽  
Millimeter Wave ◽  
Manufacturing Systems ◽  
Wave Impedance ◽  
In Situ Monitoring ◽  
Full Wave ◽  
Drop On Demand ◽  
The Time Domain ◽  
Metal Droplets

AbstractThis article presents a millimeter-wave diagnostic for the in-situ monitoring of liquid metal jetting additive manufacturing systems. The diagnostic leverages a T-junction waveguide device to monitor impedance changes due to jetted metal droplets in real time. An analytical formulation for the time-domain T-junction operation is presented and supported with a quasi-static full-wave electromagnetic simulation model. The approach is evaluated experimentally with metallic spheres of known diameters ranging from 0.79 to 3.18 mm. It is then demonstrated in a custom drop-on-demand liquid metal jetting system where effective droplet diameters ranging from 0.8 to 1.6 mm are detected. Experimental results demonstrate that this approach can provide information about droplet size, timing, and motion by monitoring a single parameter, the reflection coefficient amplitude at the input port. These results show the promise of the impedance diagnostic as a reliable in-situ characterization method for metal droplets in an advanced manufacturing system.

Migration‐based traveltime waveform inversion of 2-D simple structures: A synthetic example

Geophysics ◽  
2001 ◽  
Vol 66 (3) ◽  
pp. 845-860 ◽  
Author(s):  
François Clément ◽  
Guy Chavent ◽  
Susana Gómez
Keyword(s):  
Least Squares ◽  
Waveform Inversion ◽  
Domain Of Attraction ◽  
Optimization Methods ◽  
Data Sets ◽  
Seismic Reflection Data ◽  
Prestack Migration ◽  
Reflection Data ◽  
Full Wave ◽  
The Time Domain

Migration‐based traveltime (MBTT) formulation provides algorithms for automatically determining background velocities from full‐waveform surface seismic reflection data using local optimization methods. In particular, it addresses the difficulty of the nonconvexity of the least‐squares data misfit function. The method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity. We present an implementation of the MBTT approach for a 2-D finite‐difference (FD) full‐wave acoustic model. Numerical analysis on a 2-D synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least‐squares approach, even when starting from very poor initial guesses. This enlargement of the domain of attraction for the global minima of the least‐squares misfit has a price: each evaluation of the new objective function requires, besides the usual FD full‐wave forward modeling, an additional full‐wave prestack migration. Hence, the FD implementation of the MBTT approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size.

scholarly journals Modeling nonlinear compressional waves in marine sediments

2009 ◽  
Vol 16 (1) ◽  
pp. 151-157 ◽  
Author(s):  
B. E. McDonald
Keyword(s):  
Marine Sediments ◽  
Plane Waves ◽  
High Stress ◽  
Nonlinear Behavior ◽  
Analytic Solutions ◽  
Compressional Waves ◽  
Series Of Experiments ◽  
Model Equations ◽  
The Time Domain ◽  
Water Saturated

Abstract. A computational model is presented which will help guide and interpret an upcoming series of experiments on nonlinear compressional waves in marine sediments. The model includes propagation physics of nonlinear acoustics augmented with granular Hertzian stress of order 3/2 in the strain rate. The model is a variant of the time domain NPE (McDonald and Kuperman, 1987) supplemented with a causal algorithm for frequency-linear attenuation. When attenuation is absent, the model equations are used to construct analytic solutions for nonlinear plane waves. The results imply that Hertzian stress causes a unique nonlinear behavior near zero stress. A fluid, in contrast, exhibits nonlinear behavior under high stress. A numerical experiment with nominal values for attenuation coefficient implies that in a water saturated Hertzian chain, the nonlinearity near zero stress may be experimentally observable.

scholarly journals Computing the T-matrix of a scattering object with multiple plane wave illuminations

2017 ◽  
Vol 8 ◽  
pp. 614-626 ◽  
Author(s):  
Martin Fruhnert ◽  
Ivan Fernandez-Corbaton ◽  
Vassilios Yannopapas ◽  
Carsten Rockstuhl
Keyword(s):  
Plane Waves ◽  
The Finite Element Method ◽  
Full Wave ◽  
Electric Dipoles ◽  
Complicated Object ◽  
T Matrix ◽  
Multiple Plane ◽  
Optical Nanostructures ◽  
Scattered Fields ◽  
Matrix Calculation

Given an arbitrarily complicated object, it is often difficult to say immediately how it interacts with a specific illumination. Optically small objects, e.g., spheres, can often be modeled as electric dipoles, but which multipole moments are excited for larger particles possessing a much more complicated shape? The T-matrix answers this question, as it contains the entire information about how an object interacts with any electromagnetic illumination. Moreover, a multitude of interesting properties can be derived from the T-matrix such as the scattering cross section for a specific illumination and information about symmetries of the object. Here, we present a method to calculate the T-matrix of an arbitrary object numerically, solely by illuminating it with multiple plane waves and analyzing the scattered fields. Calculating these fields is readily done by widely available tools. The finite element method is particularly advantageous, because it is fast and efficient. We demonstrate the T-matrix calculation at four examples of relevant optical nanostructures currently at the focus of research interest. We show the advantages of the method to obtain useful information, which is hard to access when relying solely on full wave solvers.

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