The mathematics of pdes and the wave equation mathtube. Since then, surface waves have been used to treat many specific problems. For a nondispersive system where all frequencies of excitation. Pdf matrix methods for wave equations researchgate. Introduction to seismology the wave equation and body waves by peter m.
The observed data has higher frequency than in synthetic example, thus the locations are more intensive. These functions read and write seismic data in standard segy format, read and write well logs in lasformat 2. Elastic wave propagation and generation in seismology. Pdf in this paper, we propose a waveequationbased traveltime seismic tomography method with a detailed description of. The p waves are the fastest moving waves and are simply sound waves. We present a waveequationbased method for seismic illumination analysis. For physical examples of nonspherical wave solutions to the 3d wave equation that do possess. Wave equation based microseismic source location and. 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.
These new equations only have terms which contain therefractiveindexn2 and e or h,thus. Seismic viscoelastic attenuation university of connecticut. To motivate our discussion, consider the one dimensional wave equation. There were challenging logistics in planting and cabling them, and in their recording with three times the normal number of channels. The seismic wave equation rick aster february 15, 2011 waves in one dimension. Displacements occurring from a harmonic plane pwave top and swave bottom traveling horizontally across the page. The dashed line represents the pick of the wave arrival time in this noise free case. Seismic wave propagation and earth models gfzpublic. Generation of finite difference formulas on arbitrarily spaced grids, numerical modeling of seismic wave propagation. 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.
Thus, the attempt to compress threedimensional reality into a one or twodimen. Animations are provided to view wave propagation in a 3dimensional solid for compressional p. Introduction to the theory of seismic wave propagation indico. If, on the other hand, we have a free end to the string, the physical constraint could. A fullwave equation based seismic illumination analysis. Results from our numerical simulations showed good agreement with those from fdm. A factor defining an exponential decrease with frequency f and propagation time t of a seismic. Displacements occurring from a harmonic plane p wave top and s wave bottom traveling horizontally across the page.
Elasticity and seismic waveselasticity and seismic waves some mathematical basics straindisplacement relation linear elasticity strain tensor meaning of its elements stressstrain relation hookes law stress tensor symmetry elasticity tensor lames parameters equation of motion p and s waves plane wave solutions. Matrix transformation and solutions of wave equation of free. An impulsive wave from a line source the line would be on the oceans surface perpendicular to the ships course has an amplitude decay proportional to t12. Seismic waveforms and velocity model heterogeneity. The derivative matrix defines the direction of planewave propagation. The wave equation is a partial di erential equation that relates second time and spatial derivatives of propagating wave disturbances in a simple way. Introduction to seismology the wave equation and body. The numerical modeling of 3d elastic wave equation using a high. Least squares migration of converted wave seismic data cseg. Wave equation based microseismic source location and velocity. The elastic wave equation equation of motion the force balance equation can be written as.
The field data is collected to monitor hydraulic fracturing experiment. Seismic waveform modeling and surface wave tomography in. Furthermore, the slow wind, tide, and densitydriven horizontal velocities are often stipulated to be independent of z a priori. The recursion begins with the wave field values known at two successive time steps t0 and t the wavefield values at a future time step at all the spatial locations are. A oneway waveequationbased, generalized screen propagator is used to extrapolate the wave. Hence, attenuation studies for these waves are pretty much important in seismic hazard estimation. 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. Crustal attenuation in the region of the maltese islands. The oneway wave equation is computationally more efficient than full waveform solutions e. The nature of general solutions to the wave equation are discussed. Elastic wave equation university of calgary in alberta. A oneway wave equation based, generalized screen propagator is used to extrapolate the wave fields from sources and. Laplaces equation, you solve it inside a circle or inside some closed region. Teleseismic waveform modelling with a oneway wave equation.
We present a wave equation based method for seismic illumination analysis. Various approaches had been developed worldwide to study. Numerical simulation is an effective way to study the propagation laws of seismic wave. Nonlinear seismic wave motion in elastodynamics with. In order to test pml absorbing effect based on finite element. Engineering, helmholtz differential equation, wave equation, real time expert system. 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 given by the expression exp. Seismic rays are used instead of the wave front to describe the wave propagation. Earth or for extremely longperiod freeoscillation modes see 2.
We will see that the fundamental theory of wave propagation is primarily based on two equations. Chapter 2 the seismic wave equation the classical physics behind most of seismology starts with newtons second law of motion f ma 2. Towards a fullwaveform 3 microseismic location algorithm. Introduction to seismology the wave equation and body waves. Seismic waves arise in solid media due to elastic forces. A set of about 170 functions plus support functions called by them for analysis and display of explorationseismic data and well logs. Equation 7 is the recursion formula for solving 1d wave equation by a finitedifference method. Download fulltext pdf matrix methods for wave equations article pdf available in mathematische zeitschrift 2534. And the wave equation, the fullscale wave equation, is second order in time. Pdf waveequationbased seismic illumination analysis. For p wave seismology it is common to approximate wave propagation using the scalar wave equation. The p wave is a longitudinal wave made up of a series of compressions and rarefactions.
The first arrival p wave part is extracted from the data as we use acoustic equation in the wavefield modeling. An alternative and more direct derivation is given by angus et althe propagation direction is taken to be more or less along the x 1 axis and the lateral coordinates x 2,3 and slownesses p 2,3 are denoted with greek subscripts e. The threedimensional current and surface wave equations. Numerical simulation of seismic wave propagation produced by.
A oneway waveequationbased, generalized screen propagator is used to extrapolate the wavefields from sources and. Examples of this would include many applications of. The wave equation is an important secondorder linear partial differential equation for the. We simplify it to the standard form by modeling the material as series of homogeneous layers. Matrix transformation and solutions of wave equation of. This approximation is often sufficient to produce reasonable p wave structural images.
The black triangles represent the receiver array placed in the right monitoring well and the two red stars represent the perforation positions placed in the left fracturing well. The velocities of these waves are given in terms of the bulk and shear modulii, k, and density. A fullwave equation based seismic illumination analysis method. Our simulation results showed accurate surface wave propagation on the complex ground surface. Extending this approximation to converted wave imaging requires some troubling assumptions about amplitude and polarization. The seismic wave equation x 1 x 2 x 3 t x 1 t x 1 dx 1 dx 2 dx 3 figure 3. Waveequationbased seismic illumination analysis xiaobi xie1, shengwen jin2, and rushan wu1 abstract we present a waveequationbased method for seismic illumination analysis. Solving the eigensystem in equation is straightforward. We discussed two types of waves pwavescompressional swavesshear finally, if we assume no shearing then we reduced it to an acoustic wave equation.
The location result using initial velocity is shown in fig. Finally, we calculated the surface wave propagation at a free surface with arbitrary topography to demonstrate the advantage of hpm. For a narrow range of propagation, the 15 wave equation is an efficient tool for simulating elastic waves for an assortment of 3d seismic diffraction and scattering problems and is capable of incorporating the frequencydependent effects of wave propagation due to smooth variations in the medium down to the subfresnel scale. Ppt seismic waves and inversion powerpoint presentation. 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 slope of the arrivals across the array is an indication of the waves velocity. Pdf waveequationbased traveltime seismic tomography part. In the last section, we transform the the generalized wave equation and. Yang university of california summary we propose a full wave equation based method for seismic illumination analysis.
The heat equation and the wave equation, time enters, and youre going forward in time. The fullwave finitedifference method is used to calculate the waves from the source and receiver to the subsurface. Advanced finitedifference methods for seismic modeling. 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. 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. Our task in this section is to put the wave equation into some prospectors coordinate frames. As such, both seismic migration and seismic wavefield modeling algorithms are based on the wave equation. Differential equations and linear algebra gilbert strang, massachusetts institute of technology mit the wave equation. Yang university of california summary we propose a fullwave equation based method for seismic illumination analysis.
Raypaths are lines that show the direction that the seismic. A full wave equation based seismic illumination analysis method x. This article provides the application of the highorder, staggeredgrid, finite difference scheme to model elastic wave propagation in 3d isotropic media. The full wave finitedifference method is used to calculate the waves from the source and receiver to the subsurface. The initial model is established according to the log data. Examples of such signals bold lines and corresponding hilbert. At free surface all stresses applied to the surface socalled tractions vanish, i. Perfectly matched layer absorbing boundary condition of. Generation of finite difference formulas on arbitrarily. A brief summary is presented in the next subsection. Every point on the wave front is a source of a new wave that travels out of it in the form of spherical shells. A fullwave equation based seismic illumination analysis method x. In the first lecture, we saw several examples of partial differential equations that arise.
96 533 1654 1227 737 1380 1098 635 1392 1141 1315 219 175 1365 1028 1075 192 437 853 1154 1230 251 43 993 1189 1155 26 1432 1434 1285 538 702 1262 445 258 327 584 111