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. Synopsis apparent seismic wave velocities are studied by comparing the stress results obtained by. This is a collection of matlab and python scripts to simulate seismic wave propagation in 1d and 2d. In particular, we examine questions about existence and. Equations of motion and boundary conditions for the propagation of seismic waves. In other words, the in other words, the material was not and is not intended as a standard introductory text on theoretical seismology. Waveequation datuming also is applicable in seismic modeling. Velickovic where l is the distance that seismic wave covers from the source location to the actual sensor, v is the seismic wave propagation velocity, and t is the time of seismic wave propagation from the source of excitation to the sensor.
A discussion on coda waves and their properties is also included in this chapter. Advanced finitedifference methods for seismic modeling. The velocity of pwaves in a homogeneous isotropic medium is given by equation 1. An unsplit convolutional perfectly matched layer improved. All wave types are designed to propagate in the x direction illustrated in figure 1 and parallel to the earths surface. These waves propagate through a 3d geologic model, and are simulated as synthetic seismograms or other graphical output.
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. Numerical simulation of seismic wave propagation in. Elastic wave equation university of calgary in alberta. In this chapter a brief introduction on seismic wave attenuation is given. Comparison of finite difference schemes for the wave. Modelling seismic wave propagation for geophysical imaging.
Observations on seismic wave equation and reflection symmetries in stratified media article pdf available in geophysical journal international 862. We present an improved crustal vs model and moho depth map using ambient noise waveequation tomography. The demo also shows how to speed up the solution of the wave equation finite difference pde using a custom cuda kernel. Seismic wave animations for the p, s, rayleigh and love waves have been created using a 3d grid shown in figure 1. The mathematics of pdes and the wave equation michael p. 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. Displacements occurring from a harmonic plane pwave top and swave bottom traveling horizontally across the page.
Joint waveequation inversion of timelapse seismic data. Implementation of an organized,parallelized and verified code that numerically approximates a solution to a 3d elastic seismic wave equation with a point source. Seismic waves are waves of energy that travel through the earth, and are a result of an earthquake, explosion, or a volcano. The method developed for the measurement of the seismic wave propagation velocity at the actual terrain consists of the. The number of velocity stress equations is dependent on the number of memory variables, e. Swave propagation is pure shear with no volume change, whereas pwaves involve both a volume change and shearing change in shape in the material. The back scattering model aki and chouet, 1975 is also discussed, which is a way to model coda wave excitation. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract. Seismic waves are initiated by earthquake, explosive, andor other sources. Pdf observations on seismic wave equation and reflection. R ar vp br vs cr d, the akirichards equation says that the reflectivity at angle is the weighted sum of the v p, v s and density reflectivities. E3d is capable of simulating seismic wave propagation in a 3d heterogeneous earth. Correlation of seismic pwave velocities with engineering.
We will show that two types of solutions are possible, corresponding to compressional p and shear s waves, and we will derive the equations for their velocities that we presented in the. The wave equation is a partial di erential equation that relates second time and spatial derivatives of propagating wave disturbances in a simple way. A oneway wave equation is a partial differential equation used in fields such as geophysics whose solutions include only waves that propagate in a single direction on one of the axes. Pdf deriving seismic avo intercept and gradient equations. The seismic wave equation rick aster february 15, 2011 waves in one dimension. The seismic wave equation x 1 x 2 x 3 t x 1 t x 1 dx 1 dx 2 dx 3 figure 3. A local adaptive method for the numerical approximation in. The glauconitic sandstone reservoir shows significantly higher vs values than from the offreservoir shales at the same depth. For wave propagation problems assuming linear elasticity is usually sufficient. We extend the idea for twodimensional case as discussed below. Schmidts experiment and the theory of characteristics 25 3. A factor defining an exponential decrease with frequency f and propagation time t of a seismic.
Rays are the normals to the wavefronts, and they point in the direction of the wave propagation. Modeling of seismic wave propagation at the scale of the. The application methodologies are categorised into convolutional and waveequation based groups. 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. Read the readme file to locate the public data sources on the internet. We introduce some basic characteristics of wave propagation, express them quantitatively and combine them into the wave equation. As such, both seismic migration and seismic wavefield modeling algorithms are based on the wave equation. 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. 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. We restrict our discussion to acoustic wave equation because of the ease of algebra.
In the onedimensional case, the oneway wave equation allows the calculation of wave propagation without the complication of having both an outgoing and incoming wave e. While the mathematical description of the wavefronts is rather complex, that of the rays is simple. 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. Our task in this section is to put the wave equation into some prospectors coordinate frames. P and swave seismic reflection and refraction measurements at ccoc by robert a. The socalled ambient noise waveequation tomography is a method to invert seismic ambient noise phase dispersion data based on elastic waveform simulation, which accounts for 3d and finite frequency effects. The kirchhoff integral formulation of the wave equation can provide a basis for computation to deal with the irregular surfaces and variable velocities that are central to the problem. Seismology and the earths deep interior elasticity and seismic waves linearlinear and nonlinear stress and strainand nonlinear stress and strain stress vs. Crustal attenuation in the region of the maltese islands.
The nature of general solutions to the wave equation are discussed. An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation dimitri komatitsch1 and roland martin1 abstract the. Reflection and transmission of seismic waves in layered media. It is observed that seismic waves decrease in amplitude due to spherical spreading and due to mechanical or other loss mechanisms in the rock units that the wave passes through. The computations required in waveequation datuming are related to those of migration. Imaging alpine crust using ambient noise waveequation. The akirichards equation any other angle is modelled with the akirichards equation, a linearized form of the zoeppritz equations which is written and is the basis of virtually all avo methods. Seismic rays are used instead of the wave front to describe the wave propagation. Seismic waves and snells law a wave front is a surface connecting all points of equal travel time from the source. The attenuation for a sinusoidal propagating wave is defined formally as the energy loss per cycle wave length. Building on the basic theory of linear inverse problems, the methodologies of seismic inversion are explained in detail, including rayimpedance inversion and waveform tomography etc. The wave propagation is based on the firstorder acoustic wave equation in stressvelocity formulation e. A fourth order accurate implicit finite difference scheme for one dimensional wave equation is presented by smith 9.
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