Numerical simulation of seismic wave propagation produced. A oneway wave equation based, generalized screen propagator is used to extrapolate the wave fields from sources and. This approximation is often sufficient to produce reasonable p wave structural images. Direct wave attenuation, intrinsic attenuation, scattering attenuation, medium heterogeneity, coda normalization method. Chapter 2 the seismic wave equation the classical physics behind most of seismology starts with newtons second law of motion f ma 2. Generalized to a continuous medium1, this equation becomes. The seismic wave equation x 1 x 2 x 3 t x 1 t x 1 dx 1 dx 2 dx 3 figure 3. The full wave finitedifference method is used to calculate the waves from the source and receiver to the subsurface. Elastic waveequation migration for isotropic and hti media. Seismic rays are used instead of the wave front to describe the wave propagation. So the solution is 12 of a delta function thats traveling. Claerbout 1970 and methods that factorize the wave equation e.
Im given an initial distribution of the wall of water, shall we say, and its velocity, the normal thing. Least squares migration of converted wave seismic data. The seismic wave equation in isotropic media if velocity is only a function of depth, then the material can be modeled as a series of homogeneous layers. Examples of such signals bold lines and corresponding hilbert. Extending this approximation to converted wave imaging requires some troubling assumptions about amplitude and polarization. A similar equation should be written for each atom in the lattice, resulting in n coupled differential equations, which should be solved simultaneously n total. Wave equation datuming applied to marine obs data and to land. Raypaths are lines that show the direction that the seismic. In reflection seismology, the anelastic attenuation factor, often expressed as seismic quality factor or q which is inversely proportional to attenuation factor, quantifies the effects of anelastic attenuation on the seismic wavelet caused by fluid movement and grain boundary friction. Regional pn wave teleseismic p wave and s wave propagation surface wave propagation at different scales psvrayleigh wave propagation plane sh love wave propagation plane technologies for imaging the moho. The total energy of the transmitted and reflected waves must be equal to the energy of the incident ray. A oneway waveequationbased, generalized screen propagator is used.
Rochester2 2 1 york university, earth sciences, downsview, ontario m3j 1p3 canada department of physics, memorial university of newfoundland, st. Waveequationbased traveltime seismic tomography part. Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano. In both cases, factorization involves choosing a preferred axis or direction of propagation followed by splitting the. S wave propagation is pure shear with no volume change, whereas pwaves involve both a volume change and shearing change in shape in the material. 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. The threedimensional current and surface wave equations.
For p wave seismology it is common to approximate wave propagation using the scalar wave equation. Seismology and the earths deep interior elasticity and seismic waves linearlinear and nonlinear stress and strainand nonlinear stress and strain stress vs. First, a linear relationship between the traveltime residual 1tdtobs tsyn and the relative velocity perturbation c. The seismic wave equation rick aster february 15, 2011 waves in one dimension. Seismic wave animations are used to illustrate different types of wave propagation through elastic materials. We will see that the fundamental theory of wave propagation is primarily based on two equations. And those waves are 12 of a delta function each way. Wave equation datuming applied to marine obs data and to. Animations are provided to view wave propagation in a 3dimensional solid for compressional p, shear s, rayleigh r and love l waves. Introduction to the theory of seismic wave propagation indico. Teleseismic waveform modelling with a oneway wave equation. In the earth, seismic waves can propagate as longitudinal p or as shear s waves. The standard form for seismic wave equation in homogeneous media is.
Elastic wave propagation and generation in seismology. Seismic scalar wave equation modeling by a convolutional differentiator article pdf available in bulletin of the seismological society of america 821. Critical refraction is also recorded at distance beyond which angle of. To generate images m pp and m ps simultaneously, we can write. The derivative matrix defines the direction of plane wave propagation.
A factor defining an exponential decrease with frequency f and propagation time t of a seismic body wave. Velocity tends to increase with depth through earths crust and mantle, but drops sharply going from the mantle to the outer core. As a seismic wave propagates through a medium, the elastic energy associated with the wave is gradually. The algorithms based on the elastic wave equation section l. The wave equation based approach can properly handle forward multiplescattering phenomena, including focusingdefocusing, diffraction, and interference effects. Basic principles of the seismic method in this chapter we introduce the basic notion of seismic waves. For a nondispersive system where all frequencies of excitation. Crustal attenuation in the region of the maltese islands. Matrix transformation and solutions of wave equation of free. A oneway waveequationbased, generalized screen propagator is used to extrapolate the wave. Multiplying f0 1 by f 00 1 results in a function f that is identical to the solution of the generalized wave equation. Pdf in this paper, we propose a waveequationbased traveltime seismic tomography method with a detailed description of its stepbystep. A display in the y, t plane is called a seismic section. Displacements occurring from a harmonic plane p wave top and s wave bottom traveling horizontally across the page.
Direct wave reflected wave refracted wave geophones laid out in a line to record arrivals from a shot. For wave propagation problems assuming linear elasticity is usually sufficient. This equation is simply the elastodynamic wave equation fourier transformed over space and time. Within each layer, there are no gradients in the lames parameters and so these terms go to zero. Yang university of california summary we propose a full wave equation based method for seismic illumination analysis. A fullwave equation based seismic illumination analysis method. 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.
Pdf waveequationbased traveltime seismic tomography part. To contrast the developments in this paper with conventional logic and to simplify discussion, i will temporarily address deep water kh k1 propagating waves such that hx, t, c 5 ax, t cosc, where c5 kx 2 st. For example, for a moderate number of threecomponent seismograms, 0. The velocities of these waves are given in terms of the bulk and shear modulii, k, and density. Teleseismic waveform modelling with a oneway wave equation pascal audet, michael g. Solving the eigensystem in equation is straightforward. For a nondispersive system where all frequencies of excitation propagate at the same velocity, the formula for sinusoidal or harmonic waves. This type of wave forces the point in earth from where it passes to vibrate back and forth in the direction in which the wave is travelling. Seismology and the earths deep interior elasticity and seismic waves stressstrain regimesstressstrain regimes linear elasticity teleseismic waves rupture b, reaknig stable slip aseismic. Wave equation based seismic illumination analysis xiaobi xie1, shengwen jin2, and rushan wu1 abstract we present a wave equation based method for seismic illumination analysis. Since then, surface waves have been used to treat many specific problems. Waveequationbased traveltime seismic tomography part 1.
Berryhill, 1979, berryhill, 1984 can provide a basis computation to deal with irregular surfaces wiggins, 1984 and variable velocities. A display in the f, t plane is called a common midpoint gather or a common reflection point gather, which. In these cases, more complex methods, such as wave equation datuming wed, are adopted. Introduction the very modern technology nonlinear realtime expert seismology is the main and best tool which can be used by the hydrocarbon industry to map petroleum and gas deposits in the earth. When p waves originating from a point source, such as an explosion or a vibroseis truck, travel into the ground, they are reflected back from bedding surfaces as a function of the impedence of the beds, which is a measure of the density of the rock times the velocity of the wave acoustic velocity.
The wave equation is a partial differential equation that relates second. Well, a wave goes to the right, and a wave goes to the left. In the last section, we transform the the generalized wave equation and. Matrix transformation and solutions of wave equation of. Seismic reflection imaging acoustic pressure source is set off near the surface sound waves propagate in all directions from the source 0.
The p wave is a longitudinal wave made up of a series of compressions and rarefactions. Separation of the elastodynamic equation in a homogeneous isotropic. Pdf waveequationbased seismic illumination analysis. A robust and efficient iterative method for the numerical solution of the helmholtz equation phd thesis, delft, 2005. Seismic wave fields are recorded by a seismometer, hydrophone in water, or accelerometer. Nonlinear seismic wave motion in elastodynamics with. Modeling based on oneway wave equations does not include multiples, while modeling based on twoway wave equations includes.
Engineering, helmholtz differential equation, wave equation, real time expert system. Furthermore, the slow wind, tide, and densitydriven horizontal velocities are often stipulated to be independent of z a priori. Basic principles of the seismic method tu delft ocw. 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. Elastic wave equation has been widely used to describe wave propagation in an elastic medium, such as seismic waves in earth and ultrasonic waves in human body. A oneway wave equation based, generalized screen propagator is used to extrapolate the wave. After wavefield separation we extrapolate p and s wave scalar potentials using a shot profile wave equation migration operator with an angle gather imaging condition. A fullwave equation based seismic illumination analysis.
So for the wave equation, what comes out of a delta function in 1d. Compressibility, core dynamics and the subseismic wave. The relative amplitudes of the transmitted and reflected waves depend on. I see that let me write down the other half thats traveling the other way. Indeed, when the medium is smoothly heterogeneous and does not contain scales smaller than the minimum wavelength of the wave. Abstract prestack wave equation migration of isotropic or anisotropic elastic seismic data is described as vector wavefield extrapolation, plus an imaging condition for combinations of shot and receiver wave modes. In this paper, we propose a waveequationbased traveltime seismic tomography method with a detailed description of its stepbystep process. In the prestack domain, wed is applied in two steps. We present a wave equation based method for seismic illumination analysis. I want to solve the wave equation, study it further for other initial conditions. We studied two models a and b of an elastic homogeneous and isotropic medium.
Recording at each geophone is a waveform called a seismogram. It specifies the propagation velocity and particlemotion also called polarization direction for each plane wave component in the fourier domain. The acoustic, sourcefree, wave equation is given by the helmholtz equation and is written as neglecting density. Bostock and jeanphilippe mercier department of earth and ocean sciences, university of british columbia, 6339 stores road, vancouver, bc, canada v6t 1z4. Least squares migration of converted wave seismic data cseg. The derivations of oneway or parabolic wave equations can be split into two categories. The p waves are the fastest moving waves and are simply sound waves. The elastic wave equation equation of motion the force balance equation can be written as. We discussed two types of waves pwavescompressional swavesshear finally, if we assume no shearing then we reduced it to an acoustic wave equation. It is integrated upward with 1 112 where the product of the reflection coefficient c and the downgoing wave d act as a source for the upgoing wave. Sauter we now consider the wave equation, where, for simplicity, we restrict ourselves to a homogeneous righthand side. Mulder pdf, bibtex a robust iterative solver for the twoway wave equation based on a complex shiftedlaplace operator. Elastic wave equation university of calgary in alberta.
We simplify it to the standard form by modeling the material as series of homogeneous layers. Pdf we present a waveequationbased method for seismic illumination analysis. Numerical simulation of seismic wave propagation produced by. Direct signal from shot travels along top of first layer. A full wave equation based seismic illumination analysis method x. The propagation velocity of seismic waves depends on density and elasticity of the medium as well as the type of wave. For free space, the onedimensional wave equation is derived. The wave equation is a partial di erential equation that relates second time and spatial derivatives of propagating wave disturbances in a simple way. The wave phenomena occurring at a boundary between two layers are discussed, such as snells. The kirchhoff integral solution to the scalar wave equation using both nearfield and farfield terms. Every point on the wave front is a source of a new wave that travels out of it in the form of spherical shells. Elastic wave equation migration for isotropic and hti media richard a. And of course, initial conditions plural because the wave equation is second order in time. On the other hand, model b has a pwave velocity of v p 4522 m s.
The seismic wave equation the classical physics behind most of seismology starts with newtons second law of motion f ma 2. Waveequationbased seismic illumination analysis xiaobi xie1, shengwen jin2, and rushan wu1 abstract we present a waveequationbased method for seismic illumination analysis. Seismic phases and 3d seismic waves main seismic phases. Mar 24, 2018 the nature of general solutions to the wave equation are discussed along with the significance of a constant that emerges between the second spatial and temporal derivatives. This develops asymptotic ray theory in threedimensional acoustic and anisotropic elastic media, and then specializes these results to isotropic elastic media and one and twodimensional models. Seismic data interpreters usually work with wave amplitude in both the y, t plane and the f, t plane. The christoffel equation takes the form of a simple eigenvalueeigenvector problem, as follows. The wave equation to motivate our discussion, consider the onedimensional wave equation. Seismic wave propagation in stratified media has been studied for a long time aki and. Seismology is a datadriven science and its most important discoveries usually result from analysis of new data.
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