Ricker wave equation pdf

The intensity of waves called irradiance in optics is defined as the power delivered per unit area. Pdf frequencies of the ricker wavelet yanghua wang. Chapter 2 the wave equation after substituting the. Perhaps there is nothing more happening here than an issue with notation. Chapter maxwells equations and electromagnetic waves. So how did the authors generate the wavelets in figure 1 of the paper. Secondorder wave equation here, we now examine the second order wave equation. Broader band wavelets will require more computer time but should still be. In the generalized wave equation, the degree of viscoelasticity varies between zero and unity, and is defined by a controlling parameter. The ricker wavelet is a theoretical waveform obtained by solving the stokes differential equation and it has been widely used on fdtd simulations, for example, it is applicable to seismic wave. Examplesincludewaterwaves,soundwaves,electromagneticwavesradiowaves. Chapter 4 derivation and analysis of some wave equations wavephenomenaareubiquitousinnature.

Theoretically, it is a solution of the stokes differential equation, including the effect of newtonian viscosity ricker 1943, 1944. The 2d isotropic elastic wave equation is considered throughout this work. In particular, we examine questions about existence and. Pdf the frequency band and the center of the ricker wavelet. The constant c gives the speed of propagation for the vibrations. Through a series of manipulations outlined in table 2. The justification for using any reproduction curve must in the long run come from observation. It is usually only referred to as the mexican hat wavelet in the americas, due to taking the shape of a sombrero when used as a 2d image processing kernel. More specifically, it explores the viscoelastic behavior of a medium, whether gaseous, liquid, or solid, for verysmallamplitude disturbances. Looking at another paper estimation of quality factors from cmp records, the frequency spectrum of a ricker wavelet is computed using equation 1.

Although we will not discuss it, plane waves can be used as a basis for. Outline introduction equations for water waves linear wave theory classi. Combining finite element and finite difference methods for isotropic. Wave equation in 1d part 1 derivation of the 1d wave equation vibrations of an elastic string solution by separation of variables three steps to a solution several worked examples travelling waves more on this in a later lecture dalemberts insightful solution to the 1d wave equation. University of calgary seismic imaging summer school august 711, 2006, calgary abstract abstract.

A solution to the wave equation in two dimensions propagating over a fixed region 1. One of typical examples of hyperbolic partial differential equations is the wave equation with a spatial variable and a time variable, where is a constant, whose solution is a scalar function describing the propagation of a wave at a speed in the spatial direction. Its temporal profile is chosen as the ricker wavelet with central frequency of 5hz and. More specifically, it explores the viscoelastic behavior of a medium, whether gaseous, liquid, or. The ricker wavelet is frequently employed to model seismic data, and as a broad spectrum source term in computational electrodynamics. We will see this again when we examine conserved quantities energy or wave action in wave systems. Finite di erence methods for wave motion github pages. Frequencies of the ricker wavelet spiral imperial college london. The wave equation is an important secondorder linear partial differential equation for the description of wavesas they occur in classical physicssuch as mechanical waves e. This section presents a range of wave equation models for different physical phenomena. A ricker wavelet is often used as a zerophase embedded wavelet in modeling and synthetic seismogram manufacture.

Ricker wavelet and the lambert w function geophysical journal. The wave equation is a secondorder linear hyperbolic pde that describesthe propagation of a variety of waves, such as sound or water waves. Progressive wave solutions ux,t superimposed at four consecutive times, starting with a gaussian initial condition. Ricker compliant deconvolution jon claerbout and antoine guitton abstract ricker compliant deconvolution spikes at the center lobe of the ricker wavelet. It enables deconvolution to preserve and enhance seismogram polarities. The calculated csg is then shifted by the traveltime from the shot to the scatterer. The intensity, impedance and pressure amplitude of a wave. Frequencies of the ricker wavelet geophysics seg library. The seismic wave equation using the stress and strain theory developed in the previous chapter, we now construct and solve the seismic wave equation for elastic wave propagation in a uniform whole space. Pdf frequencies of the ricker wavelet researchgate. Stokes differential equation, which takes into account the effect of. Notes on the algebraic structure of wave equations.

Wave equation the purpose of these lectures is to give a basic introduction to the study of linear wave equation. It is applicable to seismic wave propagation through viscoelastic homogeneous media, that is. Ricker wavelet and the lambert w function geophysical. The r0 is the maximum amplitude of the ricker wavelet and fs is the principal frequency. Transient waves in viscoelastic media deals with the propagation of transient elastic disturbances in viscoelastic media. Constant qwave propagation and attenuation einar kjartansson rock physics project, department of geophysics, stanford university, stanford, california 94305 a linear model for attenuation of waves is presented, with q, or the portion of energy lost during each. Waves in water 1 governing equations for waves on the sea surface in this chapter we shall model the water as an inviscid and incompressible. We will show that two types of solutions are possible, corresponding. When applied to linear wave equation, twostep laxwendroff method. Relativistiv wave equations from klein gordon equation to dirac theory.

Although many wave motion problems in physics can be modeled by the standard linear wave equation, or a similar formulation with a system of firstorder equations, there are some exceptions. It arises in fields like acoustics, electromagnetics, and fluid dynamics. Wave trains will always exhibit irregularities in amplitude between the. Modified equation and amplification factor are the same as original laxwendroff method. Two analytical expressions for the amplitude response as a function of the thickness are. Chapter 4 the wave equation another classical example of a hyperbolic pde is a wave equation. Pdf the ricker wavelet is theoretically a solution of the stokes differential equation, which takes into account the effect of newtonian viscosity. Smith oregon center for optics and department of physics university of oregon, eugene, oregon 97403 abstract james clerk maxwell unknowingly discovered a correct relativistic, quantum theory for the light quantum, fortythree years before einstein postulated the photons existence. Expressing the phase spectrum as a function of lag, it works by suppressing the phase at small lags. The water velocity is 1500 ms and the source is a zero phase ricker. Is something other than equation 4 being used here. Ricker 1954 the ricker model, a scalar difference equation with densitydependent growth, is one of the most widely known models for modeling fish populations.

When this viscoelastic controlling parameter equals to 0, the viscous property vanishes and the generalized wave equation becomes a pure elastic wave equation. The ricker wavelet is theoretically a solution of the stokes differential equation, which takes into account the effect of newtonian viscosity, and is applicable to seismic waves propagated through viscoelastic homogeneous media. We now extend the wave equation to threedimensional space and look at some basic solutions to the 3d wave equation, which are known as plane waves. Wave equations on rn solutions of this equation describe the propagation of light, of sound waves in a gas or a. Abstractthe ricker wavelet is theoretically a solution of the. Elastic wave finite difference modelling as a practical. The method were going to use to solve inhomogeneous problems is captured in the elephant joke above. Elastic wave finite difference modelling as a practical exploration tool peter m. In this paper, we defined the timedomain breadth and the frequencydomain bandwidth. The fractional wave equation is obtained from the classical wave equation by replacing the secondorder derivative with a fractional derivative of order. In a recent publication, ricker 1977 gave integral representations for his generating, displacement, velocity, and acceleration functions pertaining to the propagation of a seismic pulse in a viscoelastic medium.

Modeling timevarying sea surface e ects on seismic data. The ricker wavelet is the second derivative of a gaussian function. Mathematically, the ricker wavelet is the second derivative of a gaussian function. Figure shows examples of waves when k is the gaussian kernel and the ricker wavelet. It is applicable to seismic wave propagation through viscoelastic homogeneous media, that is the voigt model. Generalized viscoelastic wave equation geophysical journal. For example, pressure is the intensity of force as it is forcearea. The mathematics of pdes and the wave equation michael p.