On: “Resolution limits of migrated images” (J. Chen and G. T. Schuster, GEOPHYSICS, 64, 1046–1053).

Geophysics ◽  
2001 ◽  
Vol 66 (2) ◽  
pp. 691-692
Author(s):  
Hocine Tabti ◽  
Leiv‐J. Gelius ◽  
Isabelle Lecomte
Keyword(s):  
Half Space ◽  
Impulse Response ◽  
Horizontal Resolution ◽  
Point Of View ◽  
Migration Process ◽  
Prestack Migration ◽  
Homogeneous Half Space ◽  
Resolution Limits ◽  
Analytical Formulas ◽  
Quick Estimate

The paper by Chen and Schuster is an attempt to provide simplified analytical formulas that can be used for a quick estimate of the horizontal resolution that might be obtained by the “diffraction stack” migration process. To this end the authors used the impulse response of a point diffractor buried in a homogeneous half‐space with sources and receivers at the surface. The paper also provides to some extent an analysis and a discussion of the resolution “limits” achieved by 2-D and 3-D poststack and prestack migration. Some questions arise from the results and conclusions given in the paper which, from our point of view, deserve to be clarified.

Resolution limits of migrated images

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1046-1053 ◽  
Author(s):  
Jing Chen ◽  
Gerard T. Schuster
Keyword(s):  
Dynamic Range ◽  
Horizontal Resolution ◽  
Far Field ◽  
Aperture Size ◽  
Resolution Limit ◽  
Point Scatterer ◽  
Zero Crossing ◽  
Prestack Migration ◽  
Numerical Tests ◽  
Resolution Limits

Far‐field formulas are derived for the point scatterer responses, the horizontal resolution limits, and the dynamic ranges of 2-D and 3-D migrated images. Numerical tests verify the validity of these formulas for practical CDP geometries. Our results show that in the far field, where the depth is much greater than the width of the recording aperture, the horizontal resolution limit for migrated images is proportional to the depth of the scatterer and inversely proportional to the wavenumber and aperture size. Given the same recording aperture size, the 2-D prestack migration image has the same horizontal resolution limit as the corresponding 2-D poststack image. For the same aperture size, the 3-D poststack migration image has twice the resolution as the 3-D prestack migration with sources and receivers distributed in fixed areas. Also, prestack migration images have a better dynamic range than poststack migration images, and dynamic range improves with an increase in width of the recording aperture. The resolution limits are based on a Rayleigh‐like zero crossing criterion.

scholarly journals The mechanics of Campi Flegrei unrests as related to plastic behaviour of the caldera borders

1999 ◽  
Vol 42 (3) ◽  
Author(s):  
S. M. Petrazzuoli ◽  
C. Troise ◽  
F. Pingue ◽  
G. De Natale
Keyword(s):  
Shear Stress ◽  
Half Space ◽  
Ground Deformation ◽  
Point Of View ◽  
Campi Flegrei ◽  
Caldera Collapse ◽  
Homogeneous Half Space ◽  
Ground Deformations ◽  
Plastic Zones ◽  
Differential Uplift

We present here a model which explains the mechanism of generation of unrest episodes at Campi Flegrei caldera from a mechanical point of view. The mechanism involves the effects of plastic zones at the borders of the inner collapsed area on both static deformations and seismicity. The large amount of ground uplift observed necessarily calls for plastic effects. These effects are interpreted as concentrated at the caldera borders: the generation of such plastic zones is simulated in terms of the mechanisms leading to the caldera collapse. In order to simulate the influence of such plastic zones on both ground deformations and seismicity we model them as surfaces of discontinuities free from shear stress within an elastic homogeneous half-space. The presence of such discontinuities allows the inner caldera block to move differentially from the outer areas, by slip along the plastic bordering zones. Such a differential uplift of the central block causes the concentration of the ground deformation. Our model explains a lot of puzzling observations at Campi Flegrei in terms of the effects of the caldera structure. The model is applicable to other caldera areas, which show typical evidence for the effects of such discontinuity zones, during unrest episodes.

A STUDY OF TWO‐DIMENSIONAL HEAD WAVES IN FLUID AND SOLID SYSTEMS

Geophysics ◽  
1963 ◽  
Vol 28 (4) ◽  
pp. 563-581 ◽  
Author(s):  
John W. Dunkin
Keyword(s):  
Half Space ◽  
Vertical Displacement ◽  
Point Of View ◽  
Head Wave ◽  
Apparent Velocity ◽  
Two Dimensional ◽  
Multiple Reflections ◽  
Transient Wave ◽  
Line Load ◽  
Head Waves

The problem of transient wave propagation in a three‐layered, fluid or solid half‐plane is investigated with the point of view of determining the effect of refracting bed thickness on the character of the two‐dimensional head wave. The “ray‐theory” technique is used to obtain exact expressions for the vertical displacement at the surface caused by an impulsive line load. The impulsive solutions are convolved with a time function having the shape of one cycle of a sinusoid. The multiple reflections in the refracting bed are found to affect the head wave significantly. For thin refracting beds in the fluid half‐space the character of the head wave can be completely altered by the strong multiple reflections. In the solid half‐space the weaker multiple reflections affect both the rate of decay of the amplitude of the head wave with distance and the apparent velocity of the head wave by changing its shape. A comparison is made of the results for the solid half‐space with previously published results of model experiments.

Inversion of slingram electromagnetic induction data using a Born approximation

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. E201-E212 ◽  
Author(s):  
Jochen Kamm ◽  
Michael Becken ◽  
Laust B. Pedersen
Keyword(s):  
Half Space ◽  
Born Approximation ◽  
High Efficiency ◽  
Three Dimensions ◽  
Background Model ◽  
Desktop Computer ◽  
Near Surface ◽  
Homogeneous Half Space ◽  
Integral Equation Formulation ◽  
Forward Operator

We present an efficient approximate inversion scheme for near-surface loop-loop EM induction data (slingram) that can be applied to obtain 2D or 3D models on a normal desktop computer. Our approach is derived from a volume integral equation formulation with an arbitrarily conductive homogeneous half-space as a background model. The measurements are not required to fulfill the low induction number condition (low frequency and conductivity). The high efficiency of the method is achieved by invoking the Born approximation around a half-space background. The Born approximation renders the forward operator linear. The choice of a homogeneous half-space yields closed form expressions for the required electromagnetic normal fields. It also yields a translationally invariant forward operator, i.e., a highly redundant Jacobian. In connection with the application of a matrix-free conjugate gradient method, this allows for very low memory requirements during the inversion, even in three dimensions. As a consequence of the Born approximation, strong conductive deviations from the background model are underestimated. Highly resistive anomalies are in principle overestimated, but at the same time difficult to resolve with induction methods. In the case of extreme contrasts, our forward model may fail in simultaneously explaining all the data collected. We applied the method to EM34 data from a profile that has been extensively studied with other electromagnetic methods and compare the results. Then, we invert three conductivity maps from the same area in a 3D inversion.

Numerical seismograms obtained using ω-κ integrals, for a point P source in a vertically inhomogeneous anelastic model

1996 ◽  
Vol 86 (3) ◽  
pp. 750-760
Author(s):  
F. Abramovici ◽  
L. H. T. Le ◽  
E. R. Kanasewich
Keyword(s):  
Half Space ◽  
Homogeneous Layer ◽  
Variable Step Size ◽  
Step Size ◽  
Adaptive Process ◽  
Cpu Time ◽  
Homogeneous Half Space ◽  
Linear Layer ◽  
Variable Step ◽  
Viscoelastic Layers

Abstract This article presents some numerical experiments in using a computer program for calculating the displacements due to a P source in a vertically inhomogeneous structure, based on the Fourier-Bessel representation. The structure may contain homogeneous, inhomogeneous, elastic, or viscoelastic layers. The source may act in any type of sublayer or in the half-space. Synthetic results for the simple case of a homogeneous layer overlaying a homogeneous half-space compare favorably with computations based on the Cagniard method. Numerical seismograms for an elastic layer having velocities and density varying linearly with depth were computed by integrating numerically the governing differential systems and compared with results based on the Haskell model of splitting the linear layer in homogeneous sublayers. Even an adaptive process with a variable step size based on the Haskell model has a poorer performance on the accuracy-cpu time scale than numerical integration.

Surface-wave dispersion computations

1970 ◽  
Vol 60 (2) ◽  
pp. 321-344 ◽  
Author(s):  
Fred Schwab ◽  
Leon Knopoff
Keyword(s):  
Half Space ◽  
Rayleigh Waves ◽  
Wave Dispersion ◽  
Reduction Technique ◽  
Surface Wave Dispersion ◽  
Single Precision ◽  
Homogeneous Half Space ◽  
Full Control ◽  
Significant Figures ◽  
Rayleigh Wave Dispersion

abstract Fundamental-mode Love- and Rayleigh-wave dispersion computations for multilayered, perfectly-elastic media were studied. The speed of these computations was improved, and the accuracy brought under full control. With sixteen decimal digits employed in these computations, fifteen significant-figure accuracy was found possible with Love waves and twelve to thirteen figure accuracy with Rayleigh waves. In order to ensure that the computed dispersion is correct to a specified accuracy, say σ significant figures, (σ + 1)/4 wavelengths of layered structure must be retained above a homogeneous half-space. To this accuracy, the homogeneous half-space is a sufficient model of the true layering it replaces. Using this result, it was possible to refine the usual layer-reduction technique so as to ensure retention of the specified accuracy while employing reduction. With this reduction technique in effect, and with σ specified below single-precision accuracy, the program can be run entirely in single precision; the specified accuracy is maintained without overflow or loss-of-precision problems being encountered during calculations.

scholarly journals Numerical Prediction of Background Buildup of Salinity Due to Desalination Brine Discharges into the Northern Arabian Gulf

Water ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 2284 ◽  
Author(s):  
Aaron C. Chow ◽  
Wilbert Verbruggen ◽  
Robin Morelissen ◽  
Yousef Al-Osairi ◽  
Poornima Ponnumani ◽  
...  
Keyword(s):  
Near Field ◽  
Regional Scale ◽  
Arabian Gulf ◽  
Potential Effect ◽  
Horizontal Resolution ◽  
Point Of View ◽  
Far Field ◽  
Exchange Flow ◽  
3 Dimensional ◽  
Desalination Plants

Brine discharges from desalination plants into low-flushing water bodies are challenging from the point of view of dilution, because of the possibility of background buildup effects that decrease the overall achievable dilution. To illustrate the background buildup effect, this paper uses the Arabian (Persian) Gulf, a shallow, reverse tidal estuary with only one outlet available for exchange flow. While desalination does not significantly affect the long-term average Gulf-wide salinity, due to the mitigating effect of the Indian Ocean Surface Water inflow, its resulting elevated salinities, as well as elevated concentrations of possible contaminants (such as heavy metals and organophosphates), can affect marine environments on a local and regional scale. To analyze the potential effect of background salinity buildup on dilutions achievable from discharge locations in the northern Gulf, a 3-dimensional hydrodynamic model (Delft3D) was used to simulate brine discharges from a single hypothetical source location along the Kuwaiti shoreline, about 900 km from the Strait of Hormuz. Using nested grids with a horizontal resolution, comparable to a local tidal excursion (250 m), far field dilutions of about 28 were computed for this discharge location. With this far field dilution, to achieve a total dilution of 20, the near field dilution (achievable using a submerged diffuser) would need to be increased to approximately 70. Conversely, the background build-up means that a near field dilution of 20 yields a total dilution of only about 12.

scholarly journals Simple Approximate Formulas for Postbuckling Deflection of Heavy Elastic Columns

2020 ◽  
Vol 10 (20) ◽  
pp. 7163
Author(s):  
Hiroyuki Shima
Keyword(s):  
Mathematical Expression ◽  
Point Of View ◽  
Mathematical Methods ◽  
Concentrated Load ◽  
Combined Loads ◽  
Living Things ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Analytical Formulas ◽  
Elastic Column

Columnar buckling is a ubiquitous phenomenon that occurs in both living things and man-made objects, regardless of the length scale ranging from macroscopic to nanometric structures. In general, analyzing the post-buckling behavior of a column requires the application of complex mathematical methods because it involves nonlinear problem solving. To complement these complex methods, this study presents simple analytical formulas for the large deflection of a heavy elastic column under combined loads. The analytical formulas relate the concentrated load acting on the tip of the column, the column’s own weight, and the deflection angle of the column through a simple mathematical expression. This can assist in obtaining an overall picture of the post-buckling behavior of heavy columns from an application point of view.

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