Finite‐difference migration derived from the Kirchhoff‐Helmholtz integral

Geophysics ◽  
1996 ◽  
Vol 61 (5) ◽  
pp. 1394-1399 ◽  
Author(s):  
Thomas Rühl
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Continued Fraction ◽  
Velocity Fields ◽  
Finite Difference Schemes ◽  
Square Root ◽  
Continued Fraction Expansion ◽  
The One ◽  
Root Operator ◽  
Approximate Version

Finite‐difference (FD) migration is one of the most often used standard migration methods in practice. The merit of FD migration is its ability to handle arbitrary laterally and vertically varying macro velocity fields. The well‐known disadvantage is that wave propagation is only performed accurately in a more or less narrow cone around the vertical. This shortcoming originates from the fact that the exact one‐way wave equation can be implemented only approximately in finite‐difference schemes because of economical reasons. The Taylor or continued fraction expansion of the square root operator in the one‐way wave equation must be truncated resulting in an approximate version of the one‐way wave equation valid only for a restricted angle range.

3-D implicit finite‐difference migration by multiway splitting

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 554-567 ◽  
Author(s):  
Dietrich Ristow ◽  
Thomas Rühl
Keyword(s):  
Finite Difference ◽  
Power Series ◽  
Taylor Expansion ◽  
Optimization Technique ◽  
Optimization Procedure ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Square Root ◽  
Implicit Finite Difference ◽  
Root Operator

We show that 3-D implicit finite‐difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome. The basic idea is as follows. We approximate the 3-D square root operator by a sequence of 2-D operators in three, four, or six directions to solve the azimuth symmetry problem. Each 2-D square root operator is then approximated by a sequence of implicit 2-D operators to improve steep dip accuracy. This sequence contains some unknown coefficients, which are calculated by a Taylor expansion technique or by an optimization technique. In the Taylor expansion method, the square root and its approximation are expanded into power series. By comparing the terms, the unknown coefficients are calculated. The more 2-D finite‐difference operators for cascading are taken and the more directions for downward continuation are chosen, the more terms from power series can be compared to obtain a higher‐degree migration operator with better circular symmetry. In the second method, optimized coefficients are calculated by an optimization procedure whereby a variation of all unknown coefficients is performed, in such a way that both the sum of all deviations between the correct square root and its approximation and the sum of all deviations from azimuth symmetry are minimized. A mathematical criterion for azimuth symmetry has been defined and incorporated into the opfimization procedure.

VALIDITY OF ONE-WAY MODELS IN THE WEAK RANGE DEPENDENCE LIMIT

2004 ◽  
Vol 12 (01) ◽  
pp. 55-66 ◽  
Author(s):  
JIANXIN ZHU ◽  
YA YAN LU
Keyword(s):  
Helmholtz Equation ◽  
Numerical Solutions ◽  
Underwater Acoustics ◽  
Square Root ◽  
Orthogonal Transform ◽  
The Difference ◽  
The One ◽  
Dirichlet To Neumann Map ◽  
Root Operator ◽  
Marching Method

Numerical solutions of the Helmholtz equation and the one-way Helmholtz equation are compared in the weak range dependence limit, where the overall range distance is increased while the range dependence is weakened. It is observed that the difference between the solutions of these two equations persists in this limit. The one-way Helmholtz equation involves a square root operator and it can be further approximated by various one-way models used in underwater acoustics. An operator marching method based on the Dirichlet-to-Neumann map and a local orthogonal transform is used to solve the Helmholtz equation.

RELATION BETWEEN A CLASS OF QUASIPERIODIC LATTICES GENERATED BY THE SUBSTITUTION METHOD AND THE GENERALIZED CONTINUED FRACTION EXPANSION

1996 ◽  
Vol 10 (17) ◽  
pp. 2081-2101
Author(s):  
TOSHIO YOSHIKAWA ◽  
KAZUMOTO IGUCHI
Keyword(s):  
Continued Fraction ◽  
Positive Real Number ◽  
Real Numbers ◽  
Continued Fraction Expansion ◽  
Positive Real ◽  
Periodic Sequences ◽  
One Dimensional ◽  
Finite Period ◽  
The One ◽  
Positive Integers

The continued fraction expansion for a positive real number is generalized to that for a set of positive real numbers. For arbitrary integer n≥2, this generalized continued fraction expansion generates (n−1) sequences of positive integers {ak}, {bk}, … , {yk} from a given set of (n−1) positive real numbers α, β, …ψ. The sequences {ak}, {bk}, … ,{yk} determine a sequence of substitutions Sk: A → Aak Bbk…Yyk Z, B → A, C → B,…,Z → Y, which constructs a one-dimensional quasiperiodic lattice with n elements A, B, … , Z. If {ak}, {bk}, … , {yk} are infinite periodic sequences with an identical period, then the ratio between the numbers of n elements A, B, … , Z in the lattice becomes a : β : … : ψ : 1. Thereby the correspondence is established between all the sets of (n−1) positive real numbers represented by a periodic generalized continued fraction expansion and all the one-dimensional quasiperiodic lattices with n elements generated by a sequence of substitutions with a finite period.

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