A passive inverse filter for greens function retrieval. Solution for the elastodynamic green function unam. Elastodynamic greens function retrieval through singlesided. The fullspace elastodynamic greens functions for time.
In this paper, the response of a vertically heterogeneous elastic halfspace with a smooth modulus variation under a set of timeharmonic ring and pointsources is derived analytically. When a part of the enclosing surface is the earths free surface, the integral needs only to be evaluated over the remaining part of the closed surface. Acoustic and elastodynamic raytheory greens functions 2. The elastodynamic greens function for the halfspace is fundamental to the application of boundary element method bem to situations involving semiinfinite media. However, for anisotropic solids, numerical integration is required for the elastodynamic greens function computation.
Fast computation of elastodynamic halfspace greens. From this extension it follows, for example, that the crosscorrelation of passive elastodynamic and electric noise observations at two di. Exact elastodynamic green functions for simple types of anisotropy. Furthermore, the elastodynamic greens functions were calculated using the. Discussion on the timeharmonic elastodynamic halfspace. The concept of greens function is one of the most powerful mathematical tools to solve boundary value problems. To that end, we model the elastodynamic homogeneous greens function. Equation 3 implies that we can retrieve the greens function of any linear response medium. Examples we present a numerical example of elastodynamic singlesided homogeneous green s function repre sentation following the theory shown above. It presents the green s functions for calculating the wave. Quantification of material degradation and its behavior of. The greens functions are found to have the same logarithmic singularities as the legendre functions of positive and negative halfdegree of the. In both monolayer and bilayer rves with fiber breakage, c 31, c 32, and c 33 were degraded compared to others. An improved technique for elastodynamic greens function.
This is called the fundamental solution for the greens function of the laplacian on 2d domains. Various derivations of the elastic displacement due to a subsurface transient or timeharmonic point force can be found in the literature. Elastodynamic greens function retrieval through single. A greens function timedomain bem of elastodynamics book. Greens function consisted of a reg ula r elastodynamic term and a singular ela stostatic term, with both terms being integrals over a unit sphere. Greens function g x,t is the response to a concentrated impulsive force. Unlike in other derivations, which apply to diffuse wave fields in. Show that this new function is the negative of the gibbs energy use the relation u s, where p, and that the two differential relations correspond to two of the relations in table 4.
The greens function is purely algebraic without any integrals and is presented in a numerically applicable form for the first time. Their local, pointwise behavior in spacetime is governed by a hyperbolic system of firstorder partial differential equations that are representative for the physical phenomena involved on a local scale. Retrieving the elastodynamic greens function of an. Here we extend this approach to 3d vectorial wave fields that satisfy the elastodynamic wave equation and recover greens functions from points. This paper presents an analytical solution, together with explicit expressions, for the steady state response of a homogeneous threedimensional halfspace subjected to a spatially sinusoidal, harmonic line load. Weaver and lobkis 4 retrieved greens functions from natural thermal. An explicit analytical solution is presented for elastodynamic greens functions of an orthotropic elastic half plane subjected to timeharmonic buried loadings. For 3d domains, the fundamental solution for the greens function of the laplacian is. I a derivation of the elastodynamic greens function unbounded isotropic and homogeneous medium elastodynamic navier equations. Here we study the canonical problem of the retrieval of 2d elastodynamic green function in an infinite space containing a cylinder inclusion. Pdf some properties of the greens function of simplified. Greens functions for beams, plates and acoustic media are also shown, along with their mathematical derivations.
For isotropic materials, the elastodynamic greens function can be obtained analytically. Notes on elastodynamics, greens function, and response to. Acoustic, elastodynamic, and electromagnetic wavefields share a number of common features in their mathematical description. In a diffuse regime the net energy flux is null and this lead to equipartitioned fields. The method proposed by guzina and pak is adopted here to develop the fundamental solution by including the boundary conditions of the plate, and the speci. Greens function representation we now develop an integral relationship between the greens functions from sources inside d, to greens functions with sources outside of it, as well as to the focusing functions.
Threedimensional timeharmonic elastodynamic greens. Greens functions in a free solid layer formation 3. We evaluate the accuracy of the elastodynamic singlesided homogeneous greens function representation by comparing it to the exact elastodynamic homogeneous greens function. Retrieving the greens function in an open system by cross. The tool we use is the green function, which is an integral kernel representing the inverse operator l1. Exact elastodynamic green functions for simple types. Box 5028, 2600 ga delft, the netherlands received 18 august 2004. Recall that this function represents the displacement field ux,t due to a body force fx,t applied impulsively i. Recall that this function represents the displacement field ux,t due to a body force fx,t. However, the underlying assumption of the theory that the sources are in the farfield limits the technology to extracting only the highfrequency part of the greens function in an open system.
The main idea is to find a function g, called greens function, such that the solution of the above differential equation can be. Greens function for the wave equation duke university. The superscripts represent the observed quantity and the source quantity, respectively v standing for particle velocity, t for. First, governing euqations are established in terms of displacements, and a general solution is obtained. Greens function retrieval from crosscorrelations of acoustic ambient noise recordings and to use the results to form acoustic images of the surrounding medium. Cwp501 retrieving the greens function in an open system by crosscorrelation. Electromagnetic greens functions retrieval by cross.
Elastodynamic fundamental solutions for anisotropic solids. Greens function gf is a basic solution to a linear differential equation, a building block that can be used to construct many useful solutions. The greens function is purely algebraic without any integrals and is presented in numerically applicable form. The saw dispersion relations are obtained from the poles of the greens function without any need for numerical integration. Some properties of the greens function of simplified elastodynamic problems.
This paper deals with the greens function for the timeharmonic elastodynamic problem in a plate domain, including damping e. This is a vital step in the derivation of the 3d elastodynamic marchenko equation. Retrieving the elastodynamic greens function of an arbitrary inhomogeneous medium by cross correlation. Pe281 greens functions course notes stanford university. The rayleighbetti reciprocity theorem leads to a similar representation of the elastodynamic greens function. A pomerantz, timedistance helioseismology, nature london 362, 430 432 1993. The method uses the pee and the partition factors arise explicitly. This disk demonstrates how the transient greens function of the elastodynamic 2d lambs problem is derived and used to develop a fast and accurate timedomain bem. Secondly, we derive the elastodynamic greens function and show that it is not very complicated as compared to the greens function in the homogeneous medium. Slowness surfaces of all three waves are spheroids with axis of rotation along the. Using changed material properties, elastodynamic full space stress greens function in the degraded composites was calculated to report the differences. Theory and applications in exploration geophysics thesis pdf available september 2017 with 1 reads how we measure reads. Exact elastodynamic green functions for simple types of. Elastodynamic 2d green function retrieval from cross.
Timeharmonic elastodynamic greens functionfor the half. Mathematical, physical and engineering sciences, 466. A correlationtype reciprocity theorem is used to show that the elastodynamic greens function of any inhomogeneous medium random or deterministic can be retrieved from the cross correlation of two recordings of a wave field at different receiver locations at the free surface. Elastodynamic singlesided homogeneous greens function.
Passive imaging methods are based on the extraction of information contained in an ambient background wavefield through reconstructing the greens function between the sensors. Let us derive our first solution for the elastodynamic green function. We assume the spectra for both p and sv uniform and such that the energy ratio e s e p. Campillo and paul 5 applied greens function retrieval to the di. Polarization vectors near a kiss singularity in the x 1 x 2 plane. It is used to develop a greens function bem in which surface discretizations on the traction. Chapter 5 green functions in this chapter we will study strategies for solving the inhomogeneous linear di erential equation ly f. We illuminate isotropically the space with plane waves. In practice, not all sources are equally important. Limit forms for the greens functions are derived for both zero. The two coordinate vectors between the brackets represent the observation point and the source point, respectively. Here we present a practical approach for fast and accurate computation of the elastodynamic greens functions of a halfspace. Solution for the elastodynamic green function to generalize this formula for any direction x j of the bodyforce x 0t, we use the direction cosines. Elastodynamic greens function for reinforced concrete.
Derive also the two differential relations analogous to eqns. These methods were first developed in the field of helioseismology 1 1. Fundamental solutions in the frequency domain follow directly by a subsequent evaluation of the fourier transforms of the timedomain solutions. Elastodynamic greens function for reinforced concrete beams article in international journal of solids and structures 4215. Since it is our aim to retrieve the green s function from the cross correlation of observed wave. Greensfunction method for modeling surface acoustic wave.
The greens functions in the righthand side of equations 5 and 6 represent measured electric fields at the points x and x b generated by electric and magnetic sources located on the surface. Apart from their use in solving inhomogeneous equations, green functions play an. The elastodynamic greens functions for timeharmonic radial and axial ring sources in a homogeneous, isotropic, linear elastic fullspace medium are derived using the fourierhankel transform. Delft university of technology elastodynamic singlesided. I a derivation of the elastodynamic greens function. Vavryiuk and yomogida 1995 showed that the elastodynamic green function for isotropic, homogeneous and unbounded media can be calculated using higher. These equations are of great importance in the formulation of threedimensional elastodynamic problems in a halfspace by means of integral transform methods andor.
Suppose, we have a linear differential equation given by. The cagniardde hoop method for double inversion is described in detail and 2d and 3d elastodynamic problems are treated in full. Greens functions for a volume source in an elastic halfspace. Application of empirical greens functions in earthquake. This example is a feasibility test using the 2d lossless model shown in figure 2a.1020 1039 609 1069 282 215 1264 1322 341 1142 542 1229 1226 1468 990 985 545 402 1636 206 391 930 1173 1151 852 216 1293 236 1403 133