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.

Velocity estimates derived from three‐dimensional seismic data

Geophysics ◽  
1983 ◽  
Vol 48 (11) ◽  
pp. 1486-1497 ◽  
Author(s):  
Kwame Owusu ◽  
G. H. F. Gardner ◽  
Wulf F. Massell
Keyword(s):  
Seismic Data ◽  
Constant Velocity ◽  
Three Dimensional ◽  
Logarithmic Scale ◽  
Model Data ◽  
Reconstruction Process ◽  
Input Field ◽  
Scattering Center ◽  
Straight Line ◽  
The Matrix

A new computer algorithm is described by which velocity estimates can be derived from three‐dimensional (3-D) multifold seismic data. The velocity estimate, referred to as “imaging velocity,” is that which best describes the diffraction hyperboloid due to a scatterer. The scattering center is best imaged when this velocity is used in the reconstruction process. The method is based on the 3-D Kirchhoff summation migration before stack. The implementation consists of two basic phases: (1) differentiating the input field traces and resampling them to a logarithmic time scale, and (2) shifting, weighting, and summing each resampled trace to a range of depth levels also chosen on a logarithmic scale. Peak amplitudes in the resulting image matrix give a time T and depth Z from which velocity is obtained using the relation [Formula: see text] The locus of constant velocity is a slanted straight line in the coordinate system of the matrix. In the usual application of migration for velocity analysis, each input trace of N samples is migrated for each of M constant velocity functions requiring [Formula: see text] moveout shift calculations. In the new method presented here, a constant shift is calculated for a given resampled trace, for each depth into which it is summed. This reduces the number of calculations per trace to about N, resulting in a significant improvement in computing efficiency. The operation of the algorithm is illustrated using synthetic and physical model data.

Transformation to zero offset for mode‐converted waves

Geophysics ◽  
1992 ◽  
Vol 57 (3) ◽  
pp. 474-477 ◽  
Author(s):  
Mohammed Alfaraj ◽  
Ken Larner
Keyword(s):  
Seismic Data ◽  
Constant Velocity ◽  
Reflection Point ◽  
P Waves ◽  
S Waves ◽  
Nmo Correction ◽  
Converted Waves ◽  
Zero Offset ◽  
Correct Data

The transformation to zero offset (TZO) of prestack seismic data for a constant‐velocity medium is well understood and is readily implemented when dealing with either P‐waves or S‐waves. TZO is achieved by inserting a dip moveout (DMO) process to correct data for the influence of dip, either before or after normal moveout (NMO) correction (Hale, 1984; Forel and Gardner, 1988). The TZO process transforms prestack seismic data in such a way that common‐midpoint (CMP) gathers are closer to being common reflection point gathers after the transformation.

scholarly journals Analytical approximations of diving-wave imaging in constant-gradient medium

Geophysics ◽  
2014 ◽  
Vol 79 (4) ◽  
pp. S131-S140 ◽  
Author(s):  
Alexey Stovas ◽  
Tariq Alkhalifah
Keyword(s):  
Initial Velocity ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Image Point ◽  
Practical Applications ◽  
Analytical Approximations ◽  
Wave Imaging ◽  
Analytical Formulas ◽  
Constant Gradient ◽  
Gradient Medium

Full-waveform inversion (FWI) in practical applications is currently used to invert the direct arrivals (diving waves, no reflections) using relatively long offsets. This is driven mainly by the high nonlinearity introduced to the inversion problem when reflection data are included, which in some cases require extremely low frequency for convergence. However, analytical insights into diving waves have lagged behind this sudden interest. We use analytical formulas that describe the diving wave’s behavior and traveltime in a constant-gradient medium to develop insights into the traveltime moveout of diving waves and the image (model) point dispersal (residual) when the wrong velocity is used. The explicit formulations that describe these phenomena reveal the high dependence of diving-wave imaging on the gradient and the initial velocity. The analytical image point residual equation can be further used to scan for the best-fit linear velocity model, which is now becoming a common sight as an initial velocity model for FWI. We determined the accuracy and versatility of these analytical formulas through numerical tests.

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.

Joint probability density analysis of the structure and dynamics of the vorticity field of a turbulent boundary layer

1998 ◽  
Vol 367 ◽  
pp. 291-328 ◽  
Author(s):  
LAWRENCE ONG ◽  
JAMES M. WALLACE
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Probability Density ◽  
Velocity Gradient ◽  
Joint Probability ◽  
Simulated Data ◽  
Wall Region ◽  
Rate Of Increase ◽  
Density Analysis ◽  
Joint Probability Density

An experimental study of a turbulent boundary layer at Rθ≈1070 and Rτ≈543 was conducted. Detailed measurements of the velocity vector and the velocity gradient tensor within the near-wall region were performed at various distances from the wall, ranging from approximately y+=14 to y+=89. The measured mean statistical properties of the fluctuating velocity and vorticity components agree well with previous experimental and numerically simulated data. These boundary layer measurements were used in a joint probability density analysis of the various component vorticity and vorticity–velocity gradient products that appear in the instantaneous vorticity and enstrophy transport equations. The vorticity filaments that contribute most to the vorticity covariance Ω[bar]xΩ [bar]y in this region were found to be oriented downstream with angles of inclination to the wall, when projected on the streamwise (x, y)-plane, that decrease with distance moving from the buffer to the logarithmic layer. When projected on the planview (x, z)- and cross-stream (y, z)-planes, the vorticity filaments that most contribute to the vorticity covariances Ω [bar]xΩ [bar]z and Ω [bar]yΩ [bar]z have angles of inclination to the z-ordinate axis that increase with distance from it. All the elements of the ΩiΩj ∂Ui/∂xj term in the enstrophy transport equation, i.e. the term that describes the rate of increase or decrease of the enstrophy by vorticity filament stretching or compression by the strain-rate field, have been examined. On balance, the average stretching of the vorticity filaments is greater than compression at all y+ locations examined here. However, some individual velocity gradient components compress the vorticity filaments, on average, more than they stretch them.

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