For timedependent problems like the heat equation and the wave equations, it is a good idea to treat the time variable separately. The wave solutions in the background medium admit a geometrical optics representation. We then apply the proposed approach for efficient implementation of the wave equation tomography of the firstarrival seismic waveforms. The chapter presents the basics of the theory of elasticity required for seismological applications, including the concepts of strain and stress, the equations of equilibrium and motion, and the fundamental nature of solutions to the equations of motion. In the earth, seismic waves can propagate as longitudinal p or as shear s waves. Seismology and the earths deep interior elasticity and seismic waves elasticity and seismic waveselasticity and seismic waves some mathematical basics straindisplacement relation linear elasticity strain tensor meaning of its elements. How do we describe and input seismic sources superposition principle. Viscous seismic wave propagation simulation by using fractional order equation has attracted much recent attention. Shear waves cant travel through any liquid medium, so the absence of s wave in earths outer core suggests a liquid state. We will show that two types of solutions are possible, corresponding. Mar 24, 2018 we introduce some basic characteristics of wave propagation, express them quantitatively and combine them into the wave equation. The anelastic properties of real materials, partic ularly of.
In this representation, the displacement is the sum of the. Seismic wave extrapolation using lowrank symbol approximation. The basic principle of the seismic reflection technique application is to measure the time taken for a seismic wave that travels from a source down into the ground where it is reflected back to the surface where it can be detected by a receiver geophone. The constant density acoustic wave equation has been used most widely in seismic modeling, migration introduction. Taking advantage of the fact that the christoffel equation has the same form in the symmetry planes of orthorhombic and transversely isotropic ti media, we can replace. In particular, we examine questions about existence and. Seismic waves practice questions base your answer to question 1 on the diagram below, which shows models of two types of earthquake waves. The local energy conservation law, the local momentum evolution equations, and the multisymplectic form are. Waveequationbased traveltime seismic tomography part 2.
Seismic displays visualization is key to seismic data analysis 3d displays are mostly done interactively using workstations in this plot. Seismic surface waves travel along the earths surface. What are consequences of the reciprocity principle. Kennett title page contents jj ii j i page 3 of 497 back full screen close quit rs s e anu 2 coupled equations for seismic waves 46. Jackson school of geosciences, the university of texas at austin, university. Seismic inverse scattering in the wave equation approach christiaan c. The propagation velocity of seismic waves depends on density and elasticity of the medium as well as the type of wave. Based on this we derive the doublesquarerootequation, which is a. Reflection frequencies decrease with depth because of attenuation of seismic waves, and this also decreases the resolution at depth. Abstract in this paper we use methods from partial differential equations, in particular microlo. The acoustic wave equation is a secondorder linear hyperbolic partial differential equation.
However, conventional finitedifference fd methods of fractional partial difference equation adopt a global difference operator to. Given a seismic wavefield p x, z 0, t recorded over time t, at the surface z 0, and along the spatial axis x, seismic migration yields the earths reflectivity p x, z, t 0 based on a process of wavefield extrapolation in. Modelling seismic wave propagation for geophysical imaging 3 in the entire domain or at the scale of an elementary element of a given mesh describing the medium structure. And ignoring body force assuming farfield from an earthquake.
Basic principles of the seismic method tu delft ocw. The wave equation is a partial di erential equation that relates second time and spatial derivatives of propagating wave. Seismic waves are formed when there is a sudden movement or slip between layers in the earths crust. Hybrid kinematicdynamic approach to seismic waveequation. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. Reflection and transmission of seismic waves in layered media.
The wave equation is a partial differential equation that relates second. Simulating seismic wave propagation in 3d elastic media using staggeredgrid finite differences by robert w. The mathematical equation for describing the motion of the rope has the same form as the equation for describing the motion of a vibration propagating through the earth. A factor defining an exponential decrease with frequency f and propagation time t of a seismic body wave given by the expression exp. Apr 15, 2004 the multisymplectic geometry for the seismic wave equation is presented in this paper. Graves abstract this article provides an overview of the application of the staggered grid finitedifference technique to model wave propagation problems in 3d elastic media. Derivation of the wave equation in 1d and 3d with the help of puzzle pieces. The multisymplectic geometry for the seismic wave equation is presented in this paper. Viscoacoustic wave propagation simulation in the earth. Despite some similarities which water waves and seismic surface waves display, there are substantial differences in the. In the above table, we saw that seismic waves travel faster in solids than they do in fluids. The seismic wave equation using the stress and strain theory developed in the previous chapter, we now construct and solve the seismic wave equation for elastic wave propagation in a uniform whole space.
Abstractwe present a new modified wave equation and apply it to develop a smoothing scheme for seismic wave propagation simulations. Proposed approach appears to be specifically useful for constructing the socalled sensitivity kernels widely used for tomographic velocity update from seismic data. Seismic resolution is controlled by the bandwidth of the signal, which is typically f30140 hz. In nearly all areas of science and therefore real life energy is transferred via the vibrations that make up waves. The data of a survey are the arrival times of the wave fronts at various distances from the source. Seismic rays are used instead of the wave front to describe the wave propagation. Examples of wave motion include waves on strings, water waves, seismic waves. Nov 20, 2018 how does a string and a seismic wave oscillate. This is a general wave equation for anisotropic elastic media. In the general setting the media can be described by piecewise. Each of these methods requires a specific approach so that a complete solution of the wave equation would be necessary for every different. Pdf simulation of surface seismic waves propagation by 2d. Pdf 1d nonperiodic homogenization for the seismic wave. Correlation of seismic pwave velocities with engineering.
We get the seismic wave equation for an isotropic medium. An unsplit convolutional perfectly matched layer improved. Seismic inverse scattering in the waveequationapproach. Fast pwavefronts show an elliptical anisotropic feature. Wave motion and sound the general discussion of wave motion is important because the ideas of wave propagation are ubiquitous. Viscoacoustic wave propagation simulation in the earth jose m. The perfectly matched layer absorbing boundary condition has proven to be very efficient for the elastic wave equation written as a firstorder system in velocity and stress. Seismology and the earths deep interior elasticity and seismic waves linearlinear and nonlinear stress and strainand nonlinear stress and strain stress vs.
Simulating seismic wave propagation in 3d elastic media. In twophase anisotropic media, fast p waves, slow p waves and sv waves are coupled with each other. The acoustic lagrangian density and the full waveform inversion gradient kris innanen abstract the eulerlagrange equations relate the lagrangian density lfor a system of particles or. The wave phenomena occurring at a boundary between two layers are discussed, such as snells. In principle, what we need is a formulation of the seismic source, equations to describe elastic wave propagation once motion has started somewhere, and a theory for coupling the source description to the solution for the equations of motion. This is a solution to the wave equation in which the displacement varies only in the direction of propagation, e. For free space, the onedimensional wave equation is derived. Realistic examples for acoustic and elastic propagations using fd formulations. It arises in fields like acoustics, electromagnetics, and fluid dynamics.
As such, both seismic migration and seismic wavefield modeling algorithms are based on the wave equation. We have devised extended imaging conditions for passive seismic wave equation imaging algorithms, and we have found that they represent a key step toward verifying and updating elastic velocity. Seismic migration is a process of estimating earths reflectivity from a recorded seismic wavefield using a velocitydepth model. Introduction to the theory of seismic wave propagation indico. A modified wave equation with diffusion effects and its. The most important information on the constitution of the earths interior has been obtained from studies of seismic body waves longitudinal and. Greens function there are many ways of solving the elastic wave equation for different types of initial conditions, boundary conditions, sources, etc.
Although the wavelet phase of the fast pwavefront in a solid and that in a fluid are the. Whitham published for the tata institute of fundamental research, bombay. The elastic wave equation governs the propagation of seismic waves resulting from earthquakes and other seismic events. Advanced finitedifference methods for seismic modeling. We introduce some basic characteristics of wave propagation, express them quantitatively and combine them into the wave equation. Modelling seismic wave propagation for geophysical imaging.
This manuscript has its emphasis in the wave seismic treatment of elastic. Numerical methods for partial di erential equations. The mathematics of pdes and the wave equation michael p. Vertical axis is the twoway reflection travel time, horizontal axes give the collocated sourcereceiver coordinates, color represents reflection amplitudes the colorcoded upper surface is the water bottom. With mathematical rigour we show that the solution of the new equation, which is derived as an analog of the advectiondiffusion equation, can be obtained by the spatial convolution between a solution of the wave equation and the heat kernel and has a finite. What are the most fundamental results in simple media. The wave equation is an important secondorder linear partial differential equation for the description of waves as they occur in classical physicssuch as mechanical waves e. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract. Therefore, seismic wavefield modeling may be viewed as the reverse process of seismic migration. For wave propagation problems assuming linear elasticity is usually sufficient. When ocean waves get to a shelving beach the speed of the waves changes relative to each other and circles go to ellipses and then the wave breaks. Pdf seismic wave propagation in media with complex. Greens function of the wave equation the fourier transform technique allows one to obtain greens functions for a spatially homogeneous in. In other words, the material was not and is not intended as a standard introductory text on theoretical seismology.
Here we describe the wave propagation in the background medium by a oneway wave equation. First, a linear relationship between the traveltime residual 1tdtobs tsyn and the relative velocity perturbation c. The nature of general solutions to the wave equation are discussed. Seismic wave fields are recorded by a seismometer, hydrophone in water, or accelerometer. The local energy conservation law, the local momentum evolution equations. Our approach uses a staggeredgrid finitedifference al gorithm to model the firstorder elastodynamic equations of motion expressed in terms of velocity and. This may happen anywhere between several km and several 100s km down from the surface. Every point on the wave front is a source of a new wave that travels out of it in the form of spherical shells. S waves are slower than p waves, and speeds are typically around 60% of that of p waves in any given material. Waveequationbased traveltime seismic tomography part. Examples of this would include many applications of ray theory which do. Basic principles of the seismic method in this chapter we introduce the basic notion of seismic waves.
Seismic data processing with the wave equation the coordinate frames used by theoreticians to describe wave propagation do not include frames in common use by geophysical prospectors to describe observations. Seismic waves are studied by geophysicists called seismologists. Pdf synthetic seismic data with realistic information about the. Seismology and the earths deep interior the elastic wave equation seismic velocities seismic velocities material and source pwave velocity ms shear wave velocity ms water 1500 0 loose sand 1800 500 clay 11002500 sandstone 14004300 anhydrite, gulf coast 4100 conglomerate 2400 limestone 6030 3030 granite 5640 2870 granodiorite 4780 3100. The wave equation is a partial di erential equation that relates second time and spatial derivatives of propagating wave disturbances. Seismic waves arise in solid media due to elastic forces. Finite element sch emes are based on solving the weak. This article is an 44 45 attempt of a pedagogical introduction to nonperiodic homogenization in 1d, allowing to 46 47 find the effective wave equation and effective physical properties, of the elastodynamics 48 49 equation in a highly heterogeneous medium. Examples of such signals bold lines and corresponding hilbert. Introduction to seismology body waves berkeley seismology lab. Pdf multisymplectic geometry for the seismic wave equation.
The most significant factor affecting seismic wave velocities is porosity. Refraction timedistance plots the raytracing algorithm in surface seismic surveys the point source is located on the surface and detectors of the resulting seismic waves are located on the surface. In the twentyfirst century, the purpose of analytic solutions of the wave equation is not improving the seismic data acquisition any more but mainly focuses on information retrieval and inversion of the seismic reflection data. A primer computational seismology 1 what are the governing equations for elastic wave propagation. The acoustic lagrangian density and the full waveform. Stable and efficient modeling of anelastic attenuation in. In this paper, we propose a waveequation based traveltime seismic tomography method with a detailed description of its stepbystep process. Seismic wave extrapolation using lowrank symbol approximation sergey fomel1. The velocity of pwaves in a homogeneous isotropic medium is given by equation 1. We demonstrate how to use this condition for the same equation written as a secondorder system in displacement. Pdf extended waveequation imaging conditions for passive.
Read pdf the seismic wave equation the seismic wave equation as recognized, adventure as well as experience very nearly lesson, amusement, as well as harmony can be gotten by just checking out a book the seismic wave equation furthermore it is not directly done, you could tolerate even more as regards this life, regarding the world. An attractive approach is the method of lines that uses a discretization in space to obtain a system of ordinary di erential equations that can be. Carcione, dan kosloff, and ronnie kosloffl anelasticity of earth materials produces significant changes in the amplitude and phase spectra of seismic waves. The input to seismic modeling is a representation of the earths reflectivity and a velocitydepth model. Despite some similarities which water waves and seismic surface waves display, there are substantial differences in the forces producing them. Model a best represents the motion of earthquake waves called 1 pwaves compressional waves that travel faster than swaves shear waves shown in model b. Modeling of seismic wave propagation at the scale of the. The center for wave phenomena, colorado school of mines. The seismic wave equation rick aster february 15, 2011 waves in one dimension.
Pdf seismic wave simulation by velocitystress wave. An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation dimitri komatitsch1 and roland martin1 abstract the. The anelastic properties of real materials, partic ularly of porous rocks, are described using the theory. Although examples of transverse waves were already known earlier, e. Other applications include waves in plates, beams and solid material structures. The wave equation for seismic vibrations is discussed in the next section. Thus as the porosity of a rock increases, both the p wave and s wave. Anisotropic parameters and pwave velocity for orthorhombic media. Seismic wave propagation and earth models gfzpublic. The measured time is known as the two way time twt.
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