In this chapter a brief introduction on seismic wave attenuation is given. Ideally, all the seismic waves of interest propagate in a vertical plane through the line of the ships course. Thus, using wave equation based methods for datuming will. It discretizes the seismic wave equations using a second order accurate numerical method that satisfies the principle of summation by parts, npsk07.
The seismic waves are modeled by the elastic wave equation in a heterogeneous anisotropic material. Ihavepresented atimedomain displacementstress formulation of the anisotropicviscoelastic wave equation, which holds for arbitrarily anisotropic velocity and attenuation 1. Wave equation based traveltime seismic tomography is rooted in the following tomographic equation tobs tsyn d z k. The number of velocity stress equations is dependent on the number of memory variables, e. Comparison of finite difference schemes for the wave. In the general setting the media can be described by piecewise smooth functions with jump discontinuities.
Among the many types of seismic waves, one can make a broad distinction between body waves, which travel through the earth, and surface waves, which travel at the earths surface 4850. Modelling seismic wave propagation for geophysical imaging. This book provides a consistent and thorough development of modelling methods widely used in elastic wave propagation ranging from the whole earth, through re. We propose a new numerical approach for the solution of the 2d acoustic wave equation to model the predicted data in the field of activesource seismic inverse problems.
A local adaptive method for the numerical approximation in. This method consists in using an explicit finite difference technique with an adaptive order of approximation of the spatial derivatives that takes into account the local. Modeling of seismic wave propagation at the scale of the. The velocity of the wave is determined by the physical properties of the material through which it propagates. Here we will discuss a series of matlab files from 6, which. Derivation of the wave equation in 1d and 3d with the help of puzzle pieces. 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. Pdf numerical simulation of elastic wave equation and analysis. Vti media, elastic wave, numerical simulation, thomsen parameter. An unsplit convolutional perfectly matched layer improved. Finally, the velocity contrast between the near surface and the substratum is a relative term. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract. Seismic rays are used instead of the wave front to describe the wave propagation.
Raypaths are lines that show the direction that the seismic wave is propagating. Electric and magnetic polarization in sourced media, 18 b. The wave equation is an important secondorder linear partial differential equation for the. To motivate our discussion, consider the one dimensional wave equation. Basic principles of the seismic method in this chapter we introduce the basic notion of seismic waves. 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. Virieux 1986, which is solved by finitedifferences on a staggeredgrid. Greens function there are many ways of solving the elastic wave equation for different types of initial conditions, boundary conditions, sources, etc. In the first lecture, we saw several examples of partial differential equations that arise in. We present the stability analysis of the three schemes and derive dispersion characteristics for these schemes. 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. Modeling wave propagation in anisotropic media is the key element of seismic imaging and fullwaveform inversion of anisotropic parameters. Wave equation datuming is more accurate way for datuming. Waveequationbased traveltime seismic tomography part.
These boundary conditions are based on replacing the wave equation in the boundary region by oneway wave equations which do not permit energy to propagate from the boundaries into the numerical mesh. Seismic ray tracing and wavefront tracking in laterally. This plane is called the plane of the seismic section. Wc61wc72 november 2015 with 164 reads how we measure reads.
The seismic wave equation rick aster february 15, 2011 waves in one dimension. The wave equation is a partial di erential equation that relates second time and spatial derivatives of propagating wave disturbances in a simple way. Lawrence livermore national high order accurate finite. Numerical simulation of seismic wave propagation in.
For nondispersive body waves, the seismic velocity is equal to both the phase and group velocities. Other applications include waves in plates, beams and solid material structures. Introduction to seismology the wave equation and body. The wave phenomena occurring at a boundary between two layers are discussed, such as snells. The mathematics of pdes and the wave equation michael p. Simulating seismic wave propagation in 3d elastic media using staggeredgrid finite differences by robert w. More recently, nonreflecting boundary conditions were introduced for the finitedifference method clayton and enquist, 1977.
Considering only velocity anisotropy, wave propagation is well modeled by the anisotropic elastic wave equation. Extended waveequation imaging conditions for passive seismic data article pdf available in geophysics 806. Wpp wave propagation program is the predecessor of sw4. Each of these methods requires a specific approach so that a complete solution of the wave equation would be necessary for every different. Basic principles of the seismic method tu delft ocw. Comparison of the mathematical formulations, 27 part iii present applications of the geophysical methods 8. By compensating for amplitude attenuation with a model of the viscoelastic attenuation model type, seismic data can provide true relativeamplitude information for amplitude inversion and. Graves abstract this article provides an overview of the application of the staggered grid finitedifference technique to model wave.
The elastic wave equation governs the propagation of seismic waves resulting from earthquakes and other seismic events. Pade based cpd and noncompact pade based ncpd schemes for the acoustic wave equation. Higher order finite difference discretization for the wave equation the two dimensional version of the wave equation with velocity and acoustic pressure v in homogeneous mu edia can be written as 2 22 2 2 22, u uu v t xy. For free space, the onedimensional wave equation is derived. A general solution of the wave equation can be obtained by using the acoustic reciprocity equation of convolution. For a nondispersive system where all frequencies of excitation. Every point on the wave front is a source of a new wave that travels out of it in the form of spherical shells. Crustal attenuation in the region of the maltese islands. The seismic wave propagation in a geological medium is often modeled by the acoustic 3d equation fichner,2010fichner, 2010 with p acoustic pressure of the wave, f seismic source and c acoustic wave speed. The wave propagation is based on the firstorder acoustic wave equation in stressvelocity formulation e. A nonreflecting boundary condition for discrete acoustic.
A realistic range for wave speed can be between 1500 ms water and 7000 ms granite. The total energy of the transmitted and reflected waves must be equal to the energy of the incident ray. The wave solutions in the background medium admit a geometrical optics representation. The constant c2 comes from mass density and elasticity, as expected in newton s and hookes laws. Seismology and the earths deep interior the elastic wave equation solutions to the wave equation solutions to the wave equation ggeneraleneral let us consider a region without sources. Examples of this would include many applications of. The speed with which an elastic wave propagates through a medium.
A discussion on coda waves and their properties is also included in this chapter. Introduction to seismology the wave equation and body waves by peter m. This is a collection of matlab and python scripts to simulate seismic wave propagation in 1d and 2d. We now consider a few examples that build up to the notion of characteristic curvesurface. Seismic inverse scattering in the waveequationapproach. The full elastic anisotropy wave equation is often used to model the complexity of velocity anisotropy, but it ignores attenuation anisotropy.
Although it is customary to treat seismic waves as if they satisfy the equations of. As such, both seismic migration and seismic wavefield modeling algorithms are based on the wave equation. The demo also shows how to speed up the solution of the wave equation finite difference pde using a custom cuda kernel. Read the readme file to locate the public data sources on the internet. S wave propagation is pure shear with no volume change, whereas p waves involve both a volume change and shearing change in shape in the material. 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. For a seismic p wave in an isotropic medium, the elastic wave equation can be written chapman, 2004. The back scattering model aki and chouet, 1975 is also discussed, which is a way to model coda wave excitation. Seismic inverse qfiltering employs a wave propagation reversal procedure that compensates for energy absorption and corrects wavelet distortion due to velocity dispersion. Here we describe the wave propagation in the background medium by a oneway wave equation. Abstractwe present a new modified wave equation and apply it to develop a smoothing scheme for seismic wave propagation simulations. Seismology is a datadriven science and its most important discoveries usually result from analysis of new data. It takes care of the raypath bending when the contrast between the overburden and the substratum is small. We model acoustic wave propagation by solving the linearized euler equations of compressible.
A production of the geophysical institute gpi in collaboration with. A modified wave equation with diffusion effects and its. The relative amplitudes of the transmitted and reflected waves depend on. Displacements occurring from a harmonic plane p wave top and s wave bottom traveling horizontally across the page. This is a solution to the wave equation in which the displacement varies only in the direction of propagation, e. In particular, we examine questions about existence and.
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