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