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.

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.

HalfSeis – Designing a High-Resolution Zero-Offset Quad Point-Source Survey to Capture Shallow Target AVO Effects

2020 ◽  
Author(s):  
P.E. Dhelie ◽  
V. Danielsen ◽  
J.E. Lie ◽  
S.J. Støen ◽  
A. Dustira ◽  
...  
Keyword(s):  
High Resolution ◽  
Point Source ◽  
Zero Offset

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.

A Student’s Guide to Teleseismic Body Wave Amplitudes

1992 ◽  
Vol 63 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Emile A. Okal
Keyword(s):  
Point Source ◽  
Elastic Medium ◽  
Elastic Waves ◽  
Body Wave ◽  
Seismic Sources ◽  
Geometrical Spreading ◽  
Seismic Amplitude ◽  
Anelastic Attenuation ◽  
Algebraic Expressions ◽  
Double Couple

Abstract We discuss the nature of the various factors contributing to the amplitude of a teleseismic body wave in the context of a geometrical ray solution, specifically: the radiation of elastic waves into an elastic medium by a point source; the radiation patterns resulting from the orientation of the double-couple in space; the effect of propagation through a radially heterogeneous Earth, known as geometrical spreading; the effect of anelastic attenuation; the contribution of depth phases to the seismogram; and finally the influence of distance on the receiver response function. For each of these parameters, we emphasize the physical arguments underlying the exact algebraic expressions of the various factors contributing to the seismic amplitude. Finally, we discuss the extension of the geometrical ray solution to deep seismic sources.

Estimation of layer parameters for linear P- and S-wave velocity functions

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. U27-U30 ◽  
Author(s):  
Alexey Stovas ◽  
Bjørn Ursin
Keyword(s):  
Wave Velocity ◽  
Layered Medium ◽  
S Wave ◽  
Velocity Gradients ◽  
Zero Offset ◽  
Seismic Reflections

For a horizontally layered medium with isotropic layers with constant P- and S-wave velocity gradients, it is possible to estimate the velocity functions (gradient and velocity at the top of each layer) and thickness of each layer. From large-offset PP seismic reflections, one can estimate three traveltime parameters: the zero-offset two-way traveltime, the NMO velocity, and a heterogeneity coefficient responsible for the nonhyperbolicity of the traveltime curve, using the different traveltime approximations. From large-offset (offset/depth greater than two) PS seismic reflections, one can estimate two traveltime parameters: zero-offset two-way traveltime and NMO velocity. From the estimated traveltime parameters at the top and bottom of a layer, it is possible to compute the thickness and velocity functions of the layer.

Reflection of waves generated by a point source over a randomly layered medium

Wave Motion ◽  
1991 ◽  
Vol 13 (1) ◽  
pp. 53-87 ◽  
Author(s):  
Werner Kohler ◽  
George Papanicolaou ◽  
Benjamin White
Keyword(s):  
Point Source ◽  
Layered Medium ◽  
Reflection Of Waves

Controlling amplitudes in 2.5D common‐shot migration to zero offset

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1299-1310 ◽  
Author(s):  
Jörg Schleicher ◽  
Claudio Bagaini
Keyword(s):  
Wave Propagation ◽  
Weight Function ◽  
Seismic Data ◽  
Constant Velocity ◽  
A Priori ◽  
Weight Functions ◽  
A Priori Knowledge ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Zero Offset

Configuration transform operations such as dip moveout, migration to zero offset, and shot and offset continuation use seismic data recorded with a certain measurement configuration to simulate data as if recorded with other configurations. Common‐shot migration to zero offset (CS‐MZO), analyzed in this paper, transforms a common‐shot section into a zero‐offset section. It can be realized as a Kirchhoff‐type stacking operation for 3D wave propagation in a 2D laterally inhomogeneous medium. By application of suitable weight functions, amplitudes of the data are either preserved or transformed by replacing the geometrical‐spreading factor of the input reflections by the correct one of the output zero‐offset reflections. The necessary weight function can be computed via 2D dynamic ray tracing in a given macrovelocity model without any a priori knowledge regarding the dip or curvature of the reflectors. We derive the general expression of the weight function in the general 2.5D situation and specify its form for the particular case of constant velocity. A numerical example validates this expression and highlights the differences between amplitude preserving and true‐amplitude CS‐MZO.

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