DIVERGENCE EFFECTS IN A LAYERED EARTH

Geophysics ◽  
1973 ◽  
Vol 38 (3) ◽  
pp. 481-488 ◽  
Author(s):  
P. Newman
Keyword(s):  
Normal Incidence ◽  
Wave Theory ◽  
Diagnostic Value ◽  
Earth Model ◽  
Correction Factors ◽  
Amplitude Correction ◽  
Multiple Attenuation ◽  
Seismic Recording ◽  
Zero Offset ◽  
Special Case

Of the various factors which influence reflection amplitudes in a seismic recording, divergence effects are possibly of least direct interest to the interpreter. Nevertheless, proper compensation for these effects is mandatory if reflection amplitudes are to be of diagnostic value. For an earth model consisting of horizontal, isotropic layers, and assuming a point source, we apply ray theory to determine an expression for amplitude correction factors in terms of initial incidence, source‐receiver offset, and reflector depth. The special case of zero offset yields an expression in terms of two‐way traveltime, velocity in the initial layer, and the time‐weighted rms velocity which characterizes reflections. For this model it follows that information which is needed for divergence compensation in the region of normal incidence is available from the customary analysis of normal moveout (NMO). It is hardly surprising that NMO and divergence effects are intimately related when one considers the expanding wavefront situation which is responsible for both phenomena. However, it is evident that an amplitude correction which is appropriate for the primary reflection sequence cannot in general be appropriate for the multiples. At short offset distances the disparity in displayed amplitude varies as the square of the ratio of primary to multiple rms velocities, and favors the multiples. These observations are relevant to a number of concepts which are founded upon plane‐wave theory, notably multiple attenuation processes and record synthesis inclusive of multiples.

Magnitude from short-period P-wave data

1972 ◽  
Vol 62 (2) ◽  
pp. 435-452
Author(s):  
K. F. Veith ◽  
G. E. Clawson
Keyword(s):  
P Wave ◽  
Earth Model ◽  
Correction Factors ◽  
Surface Source ◽  
Distance Curve ◽  
Amplitude Correction ◽  
Short Period ◽  
Geometric Spreading ◽  
Deep Focus ◽  
Q Factors

abstract An empirical surface-focus amplitude-distance curve, based on approximately 2400 short-period P-wave amplitudes from 43 large explosions at 19 different sites, and a corresponding curve based on geometrical spreading in the Herrin earth model are developed. The relative amplitude differences are inverted to obtain the Q structure of the mantle. Theoretical, deep-focus, amplitude-distance curves incorporating geometric spreading and attenuation are developed. Corresponding distance and depth amplitude-correction P factors are given for computing magnitude (mp). The correction factors are evaluated and are shown to provide a significantly improved basis for computing magnitude relative to the presently used Q factors which yield mb. Values of mp were normalized to be equal on the average to values of mb for a surface source; differences in attenuation factors generally make mb larger than mp for deep earthquakes.

Three‐dimensional true‐amplitude zero‐offset migration

Geophysics ◽  
1991 ◽  
Vol 56 (1) ◽  
pp. 18-26 ◽  
Author(s):  
Peter Hubral ◽  
Martin Tygel ◽  
Holger Zien
Keyword(s):  
Time Derivative ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Earth Model ◽  
Ray Theory ◽  
Transmission Losses ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Time Migration ◽  
Zero Offset

The primary zero‐offset reflection of a point source from a smooth reflector within a laterally inhomogeneous velocity earth model is (within the framework of ray theory) defined by parameters pertaining to the normal‐incidence ray. The geometrical‐spreading factor—usually computed along the ray by dynamic‐ray tracing in a forward‐modeling approach—can, in this case, be recovered from traveltime measurements at the surface. As a consequence, zero‐offset reflections can be time migrated such that the geometrical‐spreading factor for the normal‐incidence ray is removed. This leads to a so‐called “true‐amplitude time migration.” In this work, true‐amplitude time‐migrated reflections are obtained by nothing more than a simple diffraction stack essentially followed by a time derivative of the diffraction‐stack traces. For small transmission losses of primary zero‐offset reflections through intermediate‐layer boundaries, the true‐amplitude time‐migrated reflection provides a direct measure of the reflection coefficient at the reflecting lower end of the normal‐incidence ray. The time‐migrated field can be easily transformed into a depth‐migrated field with the help of image rays.

scholarly journals Mathematical analogies in physics. Thin-layer wave theory

2014 ◽  
Vol 57 (1) ◽  
Author(s):  
José M. Carcione ◽  
Vivian Grünhut ◽  
Ana Osella
Keyword(s):  
Quantum Mechanics ◽  
Thin Layer ◽  
Quantum Physics ◽  
Constitutive Relations ◽  
Normal Incidence ◽  
Wave Theory ◽  
Transverse Magnetic ◽  
Momentum Conservation ◽  
Tunnel Effect ◽  
Basic Equations

<p>Field theory applies to elastodynamics, electromagnetism, quantum mechanics, gravitation and other similar fields of physics, where the basic equations describing the phenomenon are based on constitutive relations and balance equations. For instance, in elastodynamics, these are the stress-strain relations and the equations of momentum conservation (Euler-Newton law). In these cases, the same mathematical theory can be used, by establishing appropriate mathematical equivalences (or analogies) between material properties and field variables. For instance, the wave equation and the related mathematical developments can be used to describe anelastic and electromagnetic wave propagation, and are extensively used in quantum mechanics. In this work, we obtain the mathematical analogy for the reflection/refraction (transmission) problem of a thin layer embedded between dissimilar media, considering the presence of anisotropy and attenuation/viscosity in the viscoelastic case, conductivity in the electromagnetic case and a potential barrier in quantum physics (the tunnel effect). The analogy is mainly illustrated with geophysical examples of propagation of S (shear), P (compressional), TM (transverse-magnetic) and TE (transverse-electric) waves. The tunnel effect is obtained as a special case of viscoelastic waves at normal incidence.</p>

Dip selective 2-D multiple attenuation in the plane‐wave domain

Geophysics ◽  
2000 ◽  
Vol 65 (1) ◽  
pp. 264-274 ◽  
Author(s):  
Faqi Liu ◽  
Mrinal K. Sen ◽  
Paul L. Stoffa
Keyword(s):  
Plane Wave ◽  
Source Function ◽  
Real Data ◽  
Wave Theory ◽  
Geological Structure ◽  
Multiple Attenuation ◽  
Invariant Embedding ◽  
Free Data ◽  
Air Water Interface ◽  
Lateral Variations

In many geological settings, strong reflections at the air‐water interface contribute to most of the multiple energy in the recorded seismograms. Here, we describe a method for free‐surface multiple attenuation using a reflection operator model of a seismic record, derived using the well‐known invariant embedding technique. We implement this method in the 2-D plane‐wave domain, where lateral variation of the geological structure of the earth is taken into account by the coupling of different ray parameters. In situations where the lateral variations are smooth, the data are well compressed in the 2-D plane‐wave domain and the resultant bandlimited matrices significantly reduce the computation cost. One important feature of the proposed method is its flexibility, which allows for the removal of multiples from selected reflections. To generate multiple free data, wave‐theory‐based multiple attenuation methods attempt to estimate either the source function or the subsurface reflectivity. Our method takes advantage of both approaches, such that we initially predict multiple traveltime using the reflectivity approach and then seek a source function to predict the amplitudes. Synthetic and real data examples show that this method is stable and successful in attenuating the surface multiples.

Application of raypath modelling to verify the mapping of Gippsland gas fields

1989 ◽  
Vol 20 (2) ◽  
pp. 325
Author(s):  
M. Megallaa
Keyword(s):  
Oil And Gas ◽  
Synthetic Data ◽  
Production Capacity ◽  
Normal Incidence ◽  
Wave Theory ◽  
Seismic Profiles ◽  
Well Control ◽  
Depth Maps ◽  
Petroleum Resources ◽  
Gas Fields

One of the Victorian Government's policies in the oil and gas area is to enhance the benefits to the State in the energy sector by assessing the nature and extent of the petroleum resources. To evaluate the production capacity of developed and undeveloped gas fields, a comprehensive study was commissioned by the DITR in 1988. The first step in a study of this type is to check the accuracy of the depth maps, to see if they adequately describe the reservoir geometry. Raypath modelling, using the Advanced Interpretation Mapping System (AIMS ? Version III), was carried out by Geophysical Services International (GSI), Sydney, on a number of selected profiles over the Snapper, Marlin-Turrum, Barracouta, Kipper and Emperor fields for the DITR. Input data for the models were extracted from the operator's maps. The software simulates the normal incidence raypaths (or wave theory solution) for all shotpoints, and from this information it generates gather records and/or synthetic seismic profiles. By comparing the model data with those from data acquisition, processing and interpretation, it was possible to check the validity of the interpretation of the reservoir's geometry. This modelling work showed that the synthetic data were comparable with the acquisition and processing data, confirming that the depth maps (tied to well control) produced by the operator using its proprietary software are adequate and most likely to represent subsurface configuration of the reservoirs.

Quarter mode rectangular cavity-based perfect metamaterial absorber in the terahertz region

2019 ◽  
Vol 33 (36) ◽  
pp. 1950460 ◽  
Author(s):  
Xiaojie Lu ◽  
Zhongyin Xiao ◽  
Mingming Chen
Keyword(s):  
Normal Incidence ◽  
Wave Theory ◽  
Metamaterial Absorber ◽  
Metal Plate ◽  
Rectangular Cavity ◽  
Perfect Absorption ◽  
Analysis And Design ◽  
Terahertz Region ◽  
Perfect Metamaterial Absorber ◽  
Good Agreement

In this work, we present the analysis and design of a perfect metamaterial absorber (MA) based on quarter mode rectangular cavity in the terahertz region. This structure is consisted of a metal plate where three different size quarter mode rectangular cavities are vertically placed on. Based on rectangular-cavity-theory, a formula is proposed to calculate the resonant frequency, which provides a guidance for designing MAs of such type. In terms of normal incidence, the simulated results show that the MA has three resonance points on 4.1301 THz, 4.6051 THz and 5.1088 THz, respectively, which is in good agreement with the calculated results. Furthermore, we present the distribution of E-field in the cavity and use the standing wave theory to explain the physical mechanism of the perfect absorption. These results verify the application of resonant cavities in the field of MA.

Elastic least-squares reverse time migration using the energy norm

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. S237-S248 ◽  
Author(s):  
Daniel Rocha ◽  
Paul Sava
Keyword(s):  
Least Squares ◽  
Normal Incidence ◽  
Wave Mode ◽  
Energy Norm ◽  
Earth Model ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration ◽  
Mode Decomposition ◽  
High Convergence Rate

Incorporating anisotropy and elasticity into least-squares migration is an important step toward recovering accurate amplitudes in seismic imaging. An efficient way to extract reflectivity information from anisotropic elastic wavefields exploits properties of the energy norm. We derive linearized modeling and migration operators based on the energy norm to perform anisotropic least-squares reverse time migration (LSRTM) describing subsurface reflectivity and correctly predicting observed data without costly decomposition of wave modes. Imaging operators based on the energy norm have no polarity reversal at normal incidence and remove backscattering artifacts caused by sharp interfaces in the earth model, thus accelerating convergence and generating images of higher quality when compared with images produced by conventional methods. With synthetic and field data experiments, we find that our elastic LSRTM method generates high-quality images that predict the data for arbitrary anisotropy, without the complexity of wave-mode decomposition and with a high convergence rate.

Three‐dimensional primary zero‐offset reflections

Geophysics ◽  
1993 ◽  
Vol 58 (5) ◽  
pp. 692-702 ◽  
Author(s):  
Peter Hubral ◽  
Jorg Schleicher ◽  
Martin Tygel
Keyword(s):  
Ray Tracing ◽  
Large Scale ◽  
Initial Conditions ◽  
Fresnel Zone ◽  
Three Dimensional ◽  
Normal Incidence ◽  
Careful Analysis ◽  
Point Sources ◽  
Dynamic Ray Tracing ◽  
Zero Offset

Zero‐offset reflections resulting from point sources are often computed on a large scale in three‐dimensional (3-D) laterally inhomogeneous isotropic media with the help of ray theory. The geometrical‐spreading factor and the number of caustics that determine the shape of the reflected pulse are then generally obtained by integrating the so‐called dynamic ray‐tracing system down and up to the two‐way normal incidence ray. Assuming that this ray is already known, we show that one integration of the dynamic ray‐tracing system in a downward direction with only the initial condition of a point source at the earth’s surface is in fact sufficient to obtain both results. To establish the Fresnel zone of the zero‐offset reflection upon the reflector requires the same single downward integration. By performing a second downward integration (using the initial conditions of a plane wave at the earth’s surface) the complete Fresnel volume around the two‐way normal ray can be found. This should be known to ascertain the validity of the computed zero‐offset event. A careful analysis of the problem as performed here shows that round‐trip integrations of the dynamic ray‐tracing system following the actually propagating wavefront along the two‐way normal ray need never be considered. In fact some useful quantities related to the two‐way normal ray (e.g., the normal‐moveout velocity) require only one single integration in one specific direction only. Finally, a two‐point ray tracing for normal rays can be derived from one‐way dynamic ray tracing.

THE METHOD OF GENERALIZED REFLECTION AND TRANSMISSION COEFFICIENTS

Geophysics ◽  
1960 ◽  
Vol 25 (3) ◽  
pp. 625-641 ◽  
Author(s):  
T. W. Spencer
Keyword(s):  
Integral Representation ◽  
Wave Theory ◽  
Earth Model ◽  
Reflection And Transmission ◽  
Localized Source ◽  
Transmission Coefficients ◽  
The Laplace Transform ◽  
Normal Mode Theory ◽  
Ray Path ◽  
Irrotational Wave

The objective of this work is to provide a method for predicting the surface response of a stratified half space to the radiation from a localized source when neither the assumptions of the plane wave theory nor the assumptions of the normal mode theory are valid. The earth model consists of a finite number of perfectly elastic, homogeneous, isotropic layers separated by interfaces which are plane and parallel to one another. The method leads to an infinite series for the Laplace transform of the response function (displacement, velocity, stress, etc.) in a multi‐interface system. Each term in the series describes all the energy which traverses a particular generalized ray path between the source and the receiver. The specification of the mode of propagation across each stratum (either as an irrotational wave or as an equivoluminal wave) and of the sequence in which the strata are traversed serve to define a generalized ray path. A prescription is given for constructing the integral representation for the disturbance which has traversed such a path directly from the integral representation for the source radiation. The method therefore obviates the necessity for solving a tedious boundary value problem. The time function associated with each term can be obtained by using Cagniard’s method.

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