Offset‐dependent geometrical spreading in isotropic laterally homogeneous media using constant velocity gradient models

Geophysics ◽  
2002 ◽  
Vol 67 (5) ◽  
pp. 1612-1615 ◽  
Author(s):  
Norman Ettrich
Keyword(s):  
Velocity Gradient ◽  
Constant Velocity ◽  
Geometrical Spreading ◽  
Homogeneous Media ◽  
Gradient Models

Geometrical spreading and Q of Pn waves: An investigative study in eastern Canada

1991 ◽  
Vol 81 (3) ◽  
pp. 882-896 ◽  
Author(s):  
Tianfei Zhu ◽  
Kin-Yip Chun ◽  
Gordon F. West
Keyword(s):  
Velocity Gradient ◽  
Geometrical Spreading ◽  
Eastern Canada ◽  
Frequency Dependent ◽  
Study Region ◽  
Amplitude Data ◽  
Short Period ◽  
Head Waves ◽  
Geometric Spreading ◽  
Spreading Function

Abstract Station site effects, uncertainties in seismic source spectrum, and instrument response errors are among the well-known frequency-dependent contaminating factors that limit the reliability of short-period Q measurements of regional phases. For the Pn wave, a regional phase of importance for both magnitude determination and nuclear test ban verification, the problem is made worse by the added uncertainty of its geometric spreading function. For realistic earth models, the Pn geometric spreading function is likely to depart drastically from that expected of canonical head waves. The extent of this departure is sensitively dependent upon the regional crust/mantle structure, making geometric spreading assumption a conspicuous source of disagreement among the published Pn attenuation (QPn) estimates. We describe a technique, referred to here as the extended reversed two-station method (RTSM), for simultaneous determination of QPn and geometrical spreading function. The formulation, being designed to bring about direct cancellation of the contaminating source, station and instrument effects, is a reliable tool for mapping the Pn propagation characteristics over continental paths, long and short. The extended RTSM has been tested using Pn spectral amplitude data derived from seismic records of the Eastern Canada Telemetered Network (ECTN). We find the spreading rate coefficient n in the power-law representation of the geometric spreading (d−n, d being epicentral distance) to be frequency dependent, increasing from 1.11 at 1 Hz to 1.77 at 20 Hz. Our QPn model in eastern Canada takes the form of QPn = 189f0.87. The results from eastern Canada suggest that: (a) there exists a significant positive velocity gradient in the uppermost mantle (≧ 0.0037 sec−1); (b) the regionally recorded Pn waves are dominated by the superposition of a series of interfering diving waves bent by the velocity gradient and internally reflected at the underside of the Moho discontinuity; and (c) the very strong frequency-dependence of QPn we found in this study region may not be unique among low-attenuating shield and platform regions.

Fresnel-zone binning: Fresnel-zone shape with offset and velocity function

Geophysics ◽  
2010 ◽  
Vol 75 (1) ◽  
pp. T9-T14 ◽  
Author(s):  
David J. Monk
Keyword(s):  
Velocity Gradient ◽  
Constant Velocity ◽  
Fresnel Zone ◽  
Geometric Approach ◽  
Velocity Function ◽  
Zero Offset ◽  
Limiting Case ◽  
Fresnel Zones

The concept of the Fresnel zone has been explored by many workers; most commonly, their work has involved examining the Fresnel zone in the limiting case of zero offset and constant velocity. I have examined the shape of the Fresnel zone for nonzero offset and in the situation of constant velocity gradient. Finite-offset Fresnel zones are not circular but are elliptical and may be many times larger than their zero-offset equivalents. My derivation takes a largely geometric approach, and I suggest a useful approximation for the dimension of the Fresnel zone parallel to the shot-receiver azimuth. The presence of a velocity gradient (velocity increasing with depth) in the subsurface leads to an expansion of the Fresnel zone to an area that is far larger than may be determined through a more usual straight-ray determination.

Topography can affect linearization in tomographic inversions

Geophysics ◽  
1997 ◽  
Vol 62 (6) ◽  
pp. 1797-1803 ◽  
Author(s):  
Sean M. Wiggins ◽  
LeRoy M. Dorman ◽  
Bruce D. Cornuelle
Keyword(s):  
Surface Topography ◽  
Velocity Gradient ◽  
Radius Of Curvature ◽  
Model Data ◽  
Velocity Models ◽  
Gradient Model ◽  
Small Model ◽  
Tomographic Inversions ◽  
Gradient Models ◽  
Constant Gradient

Linearized inverse techniques commonly are used to solve for velocity models from traveltime data. The amount that a model may change without producing large, nonlinear changes in the predicted traveltime data is dependent on the surface topography and parameterization. Simple, one‐layer, laterally homogeneous, constant‐gradient models are used to study analytically and empirically the effect of topography and parameterization on the linearity of the model‐data relationship. If, in a weak‐velocity‐gradient model, rays turn beneath a valley with topography similar to the radius of curvature of the raypaths, then large nonlinearities will result from small model perturbations. Hills, conversely, create environments in which the data are more nearly linearly related to models with the same model perturbations.

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.

Reflection of rays in a constant gradient medium: CRP geometry

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 707-712
Author(s):  
Franklyn K. Levin
Keyword(s):  
Velocity Gradient ◽  
Seismic Data ◽  
Constant Velocity ◽  
Finite Range ◽  
Velocity Increase ◽  
Rate Of Increase ◽  
Constant Gradient ◽  
Gradient Medium

In a medium having a velocity that increases linearly with depth (constant gradient), rays are arcs of circles (Slotnick, 1936). A constant gradient medium is not a good approximation to a real subsurface. Not only does velocity increase without limit with depth, but the rate of increase is constant. Nonetheless, over a finite range of depths, a constant gradient medium is closer to reality than a medium having constant velocity down to reflector of interest. For that reason, a number of investigators have considered the changes in processes applied to seismic data when a constant velocity gradient other than zero is assumed.

Sign in / Sign up

Export Citation Format

Share Document