Offset‐dependent geometrical spreading in a layered medium

Geophysics ◽  
1990 ◽  
Vol 55 (4) ◽  
pp. 492-496 ◽  
Author(s):  
Bjørn Ursin
Keyword(s):  
Point Source ◽  
Taylor Series ◽  
Layered Medium ◽  
Exact Expression ◽  
Second Order ◽  
Geometrical Spreading ◽  
Approximate Expressions

The geometrical spreading for a point source in a horizontally layered medium has been computed by Ursin (1978) and Hubral (1978) as a Taylor series in the offset coordinate. The coefficients in the Taylor series depend on the thicknesses and the velocities of the layers. Here, I start with the exact expression for geometrical spreading and show that it can be expressed as a function of the velocity in the first layer, the offset, and the first‐ and second‐order traveltime derivatives. A shifted hyperbolic traveltime approximation (Castle, 1988) and the usual hyperbolic traveltime approximation are used to derive approximate expressions for geometrical spreading. These expressions can also be derived from a truncated Taylor series by making additional approximations, but this procedure is not so obvious.

On: “Wavefront Curvature in a Layered Medium” by Bjørn Ursin (GEOPHYSICS, August 1978, p. 10011–1013)

Geophysics ◽  
1978 ◽  
Vol 43 (7) ◽  
pp. 1551-1552 ◽  
Author(s):  
Peter Hubral
Keyword(s):  
Point Source ◽  
Layered Medium ◽  
Fast Computation ◽  
Geometrical Spreading ◽  
Ray Method ◽  
Seismic Parameters ◽  
Zero Offset

It is interesting that not only traveltime, but also wavefront radius and geometrical spreading (and possibly also amplitude and thereby the entire synthetic point‐source seismogram) can be expanded by the ray method in terms of offset X and seismic parameters related to the vertical zero‐offset ray. Such expansions provide a fast computation of these quantities if they are to be obtained for many shot‐receiver offsets.

Study on the Method of the Damage Identification for Frame Structure

2012 ◽  
Vol 204-208 ◽  
pp. 2824-2831
Author(s):  
You Fa Yang ◽  
Shuai Li ◽  
Ling Ling
Keyword(s):  
Analytical Solution ◽  
Iterative Algorithm ◽  
Taylor Series ◽  
Structural Damage ◽  
Damage Identification ◽  
Second Order ◽  
Bias Error ◽  
Dynamic Data ◽  
First Order ◽  
Sensitivity Equations

First order iterative algorithm, mixed iterative algorithm, structural damage identification using static and dynamic data were put forward. The first and second order sensitivity matrixes of modal parameters that respect to the damage member were derived, and the modal truncation error which produced during the derivation of modal mode sensitivity was improved. The first and second order sensitivity equations were established respectively based on the principle of Taylor series expansion. And the solving method of these sensitivity equations was studied. Mixed iterative algorithm took up the second order nonlinear analytical solution as the first substituting value, and then the first substituting value was modified based on the Taylor series bias error using the solution of the first order sensitivity equation. It showed that the mixed iterative algorithm in this paper had a better convergence and a faster iteration speed because the higher precision second order nonlinear analytical solution was adopted. Because the method using static and dynamic data combined the static information and dynamic information of the structure, it could react the inside information of the structure more comprehensively, the result of damage identification was more accurate and it would be adapted more widely.

Quadratic wavefront and traveltime approximations in inhomogeneous layered media with curved interfaces

Geophysics ◽  
1982 ◽  
Vol 47 (7) ◽  
pp. 1012-1021 ◽  
Author(s):  
Bjørn Ursin
Keyword(s):  
Taylor Series ◽  
Layered Medium ◽  
Source Region ◽  
Three Dimensional ◽  
Homogeneous Medium ◽  
Quadratic Approximation ◽  
Reference Source ◽  
Layered Model ◽  
Zero Offset ◽  
Receiver Point

A quadratic approximation for the square of the traveltime from a source region to a receiver region is given for a three‐dimensional (3-D) medium consisting of inhomogeneous layers with curved interfaces. The square of the traveltime, being a function of source and receiver coordinates, is developed in a Taylor series around a reference source and receiver point. The relationships of the traveltime parameters to the ray parameters and the wavefront curvature matrices are shown. Using midpoint, half‐offset coordinates gives a simplified traveltime function compared to using source‐receiver coordinates only in the case that the reference source point and the reference receiver point coincide (zero‐offset approximation). For a medium consisting of homogeneous layers with plane dipping interfaces, the traveltime approximation is further simplified. The derived traveltime approximation is shown to be exact for a reflection from a plane dipping interface in a homogeneous medium. Explicit expressions for the traveltime parameters in terms of the layer parameters are derived for a horizontally layered medium. The traveltime errors of two different approximations are compared for a given layered model in a numerical example.

scholarly journals Perturbation of the Non-Radial Oscillations of a Gaseous Star by an Axial Rotation, a Tidal Action or a Magnetic Field

1974 ◽  
Vol 59 ◽  
pp. 197-198
Author(s):  
P. Smeyers
Keyword(s):  
Magnetic Field ◽  
Homogeneous Model ◽  
Axial Rotation ◽  
Second Order ◽  
Third Order ◽  
Polytropic Model ◽  
Rotational Perturbation ◽  
Tidal Perturbation ◽  
Approximate Expressions ◽  
Gaseous Star

The study of the linear and adiabatic oscillations of a gaseous star gives rise to an eigenvalue problem for the pulsation σ, if perturbations proportional to eiσt are considered. In the presence of a rotation, a tidal action or a magnetic field, the equations are not separable in spherical coordinates. To get approximate expressions for the influence of these factors on the non-radial oscillations of a star, the author and his collaborators J. Denis and M. Goossens have used a perturbation method (Smeyers and Denis, 1971; Denis, 1972; Goossens, 1972; Denis, 1973). Their procedure corresponds to a generalization of the method proposed by Simon (1969) to study the second order rotational perturbation of the radial oscillations of a star.Two types of perturbations are taken into account: volume perturbations due to the local variations of the equilibrium quantities and to the presence of a supplementary force in the equation of motion (Coriolis force, Lorentz force); surface perturbations related to the distortion of the equilibrium configuration and to the change of the condition at the surface in the presence of a magnetic field. The resulting expressions are accurate up to the second order in the angular velocity in the case of a rotational perturbation, to the third order in the ratio of the mean radius of the primary to the distance of the secondary in the case of a tidal perturbation, and to the second order in the magnetic field in the case of a perturbing magnetic field. These expressions can in principle be applied to any mode.Numerical results have been obtained for a homogeneous model and for a polytropic model n = 3. In particular, the splitting of the frequencies of the fundamental radial mode and of the f-mode belonging to l = 2 and m = 0 has been studied for the critical value of y, in the case of a component of a synchronously rotating binary system.

First- and Second-Order Design Sensitivity Analysis Scheme Based on Trefftz Method

1999 ◽  
Vol 121 (1) ◽  
pp. 84-91 ◽  
Author(s):  
E. Kita ◽  
Y. Kataoka ◽  
N. Kamiya
Keyword(s):  
Sensitivity Analysis ◽  
Design Sensitivity ◽  
Second Order ◽  
Design Sensitivity Analysis ◽  
Design Parameters ◽  
Trefftz Method ◽  
Analysis Scheme ◽  
Complete Functions ◽  
Approximate Expressions ◽  
Physical Quantities

This paper presents a new scheme for the first- and second-order design sensitivity analysis of the two-dimensional elastic problem by using Trefftz method. In the Trefftz method, the physical quantities are approximated by superposition of regular T-complete functions. Therefore, direct differentiation of the approximate expressions with respect to design parameters leads to the regular expressions of the sensitivities. The present schemes are applied to some examples in order to confirm the validity.

A THEOREM ON NEUTRON MULTIPLICATION

1947 ◽  
Vol 25a (4) ◽  
pp. 276-292 ◽  
Author(s):  
G. Placzek ◽  
G. Volkoff
Keyword(s):  
Point Source ◽  
Asymptotic Form ◽  
Homogeneous Medium ◽  
Integral Form ◽  
Exact Expression ◽  
Neutron Distribution ◽  
Chain Reaction ◽  
Neutron Multiplication ◽  
Fission Neutrons ◽  
The Fourier Transform

The asymptotic behaviour of the neutron distribution due to a point source in an infinite homogeneous medium in which a convergent chain reaction (multiplication constant k<1) takes place is investigated without special assumptions about the properties of the medium and the mechanism of neutron diffusion. It is shown under very general assumptions that at large distances r from the point source the neutron distribution always has the form[Formula: see text]General expressions for the constants μ and A of this asymptotic form of the distribution are given for any k<1 in terms of the Fourier transform of the spatial distribution of primary fission neutrons. These expressions reduce to particularly simple form for (1 − k) << 1. The exact expression for the neutron distribution throughout the medium is given in integral form. Four special frequently occurring cases are discussed as illustrations of the general result.

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