Source wavelet estimation by upward extrapolation

Geophysics ◽  
1988 ◽  
Vol 53 (2) ◽  
pp. 158-166 ◽  
Author(s):  
V. Shtivelman ◽  
D. Loewenthal
Keyword(s):  
Wave Equation ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Homogeneous Layer ◽  
Inhomogeneous Wave ◽  
Wavelet Estimation ◽  
One Dimensional ◽  
Data Application ◽  
Inhomogeneous Wave Equation ◽  
Synthetic Models

A new deterministic technique for wavelet estimation and deconvolution of seismic traces was recently introduced. This impedance‐type technique was developed for a marine environment where both the source and the receivers are located inside a homogeneous layer of water. In this work, an extension of the theory of source wavelet estimation is proposed. As in previous publications, this method is based on extrapolation of the wave field measured at depth, upward to the free surface. The extrapolation is performed by using the finite‐difference approximation to the full inhomogeneous wave equation. The extrapolation results in a wavelet which generally includes ghosts and can be used for source signature deconvolution and deghosting. The method needs two closely spaced receivers and is applicable for arbitrary locations of the source and the receivers in one‐dimensional multilayered models, provided the source is above the receivers; furthermore, it can be applied to both marine and land data. Application of the proposed method to a number of synthetic models shows that it gives a good estimate of the source wavelet.

scholarly journals New Nonpolynomial Spline in Compression Method of for the Solution of 1D Wave Equation in Polar Coordinates

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Venu Gopal ◽  
R. K. Mohanty ◽  
Navnit Jha
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Mathematical Formulation ◽  
Finite Difference Approximation ◽  
Polar Coordinates ◽  
Difference Approximation ◽  
Compression Method ◽  
One Dimensional ◽  
Spline In Compression ◽  
The Stability

We propose a three-level implicit nine point compact finite difference formulation of order two in time and four in space direction, based on nonpolynomial spline in compression approximation in -direction and finite difference approximation in -direction for the numerical solution of one-dimensional wave equation in polar coordinates. We describe the mathematical formulation procedure in detail and also discussed the stability of the method. Numerical results are provided to justify the usefulness of the proposed method.

scholarly journals Stable and Efficient Modeling of Anelastic Attenuation in Seismic Wave Propagation

2012 ◽  
Vol 12 (1) ◽  
pp. 193-225 ◽  
Author(s):  
N. Anders Petersson ◽  
Björn Sjögreen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Half Space ◽  
Material Model ◽  
Second Order ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Space Problem ◽  
Standard Linear Solid ◽  
Standard Linear

AbstractWe develop a stable finite difference approximation of the three-dimensional viscoelastic wave equation. The material model is a super-imposition of N standard linear solid mechanisms, which commonly is used in seismology to model a material with constant quality factor Q. The proposed scheme discretizes the governing equations in second order displacement formulation using 3N memory variables, making it significantly more memory efficient than the commonly used first order velocity-stress formulation. The new scheme is a generalization of our energy conserving finite difference scheme for the elastic wave equation in second order formulation [SIAM J. Numer. Anal., 45 (2007), pp. 1902-1936]. Our main result is a proof that the proposed discretization is energy stable, even in the case of variable material properties. The proof relies on the summation-by-parts property of the discretization. The new scheme is implemented with grid refinement with hanging nodes on the interface. Numerical experiments verify the accuracy and stability of the new scheme. Semi-analytical solutions for a half-space problem and the LOH.3 layer over half-space problem are used to demonstrate how the number of viscoelastic mechanisms and the grid resolution influence the accuracy. We find that three standard linear solid mechanisms usually are sufficient to make the modeling error smaller than the discretization error.

Unsteady disturbances of streaming motions around bodies

1989 ◽  
Vol 209 ◽  
pp. 385-403 ◽  
Author(s):  
H. M. Atassi ◽  
J. Grzedzinski
Keyword(s):  
Wave Equation ◽  
Body Surface ◽  
Source Term ◽  
The Body ◽  
Entire Body ◽  
Two Dimensional ◽  
Mean Flow ◽  
Inhomogeneous Wave ◽  
Inhomogeneous Wave Equation ◽  
The Mean

For small-amplitude vortical and entropic unsteady disturbances of potential flows, Goldstein proposed a partial splitting of the velocity field into a vortical part u(I) that is a known function of the imposed upstream disturbance and a potential part ∇ϕ satisfying a linear inhomogeneous wave equation with a dipole-type source term. The present paper deals with flows around bodies with a stagnation point. It is shown that for such flows u(I) becomes singular along the entire body surface and its wake and as a result ∇ϕ will also be singular along the entire body surface. The paper proposes a modified splitting of the velocity field into a vortical part u(R) that has zero streamwise and normal components along the body surface, an entropy-dependent part and a regular part ∇ϕ* that satisfies a linear inhomogeneous wave equation with a modified source term.For periodic disturbances, explicit expressions for u(R) are given for three-dimensional flows past a single obstacle and for two-dimensional mean flows past a linear cascade. For weakly sheared flows, it is shown that if the mean flow has only a finite number of isolated stagnation points, u(R) will be finite along the body surface. On the other hand, if the mean flow has a stagnation line along the body surface such as in two-dimensional flows then the component of u(R) in this direction will have a logarithmic singularity.For incompressible flows, the boundary-value problem for ϕ* is formulated in terms of an integral equation of the Fredholm type. The theory is applied to a typical bluff body. Detailed calculations are carried out to show the velocity and pressure fields in response to incident harmonic disturbances.

Time‐domain behavior of wide‐angle one‐way wave equations

Geophysics ◽  
1991 ◽  
Vol 56 (3) ◽  
pp. 382-384
Author(s):  
A. H. Kamel
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Source Function ◽  
Plane Waves ◽  
Wide Angle ◽  
Wave Processes ◽  
Inhomogeneous Wave ◽  
Dirac Delta ◽  
Inhomogeneous Wave Equation ◽  
Standard Tool

The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide‐angle” one‐way wave equation is designed to be accurate over nearly the whole 180‐degree range of permitted angles. Such formulas can be systematically constructed by drawing upon the connection with the mathematical field of approximation theory (Halpern and Trefethen, 1988).

scholarly journals On the Solutionn-Dimensional of the Product⊗kOperator and Diamond Bessel Operator

2010 ◽  
Vol 2010 ◽  
pp. 1-20
Author(s):  
Wanchak Satsanit ◽  
Amnuay Kananthai
Keyword(s):  
Wave Equation ◽  
Positive Integer ◽  
Nonlinear Equation ◽  
Unknown Function ◽  
Bessel Operator ◽  
Generalized Function ◽  
Inhomogeneous Wave ◽  
Inhomogeneous Wave Equation ◽  
Unknown Unknown

Firstly, we studied the solution of the equation⊗k◊Bku(x)=f(x)whereu(x)is an unknown unknown function forx=(x1,x2,…,xn)∈ℝn,f(x)is the generalized function,kis a positive integer. Finally, we have studied the solution of the nonlinear equation⊗k◊Bku(x)=f(x,□k−1LkΔBk□Bku(x)). It was found that the existence of the solutionu(x)of such an equation depends on the condition offand□k−1LkΔBk□Bku(x). Moreover such solutionu(x)is related to the inhomogeneous wave equation depending on the conditions ofp,q, andk.

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