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