Computing true amplitude reflections in a laterally inhomogeneous earth

Geophysics ◽  
1983 ◽  
Vol 48 (8) ◽  
pp. 1051-1062 ◽  
Author(s):  
Peter Hubral
Keyword(s):  
Three Dimensional ◽  
Physical Experiment ◽  
Reflection Coefficients ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Seismic Section ◽  
Earth Models ◽  
Reflecting Boundaries ◽  
The Common ◽  
Zero Offset

Recently Bortfeld (1982) gave a cursory nonmathematical introduction to a procedure for computing the geometrical spreading factor of a primary zero‐offset reflection from the common datum point traveltime measurements of the event. To underline the significance and consequences of this method, a derivation and discussion of geometrical spreading factors is now given for two‐ and three‐dimensional earth models with curved reflecting boundaries. The spreading factors can be used easily to transform primary reflections in a zero‐offset seismic section to true amplitude reflections. These permit an estimation of interface reflection coefficients, either directly or in connection with a true amplitude migration. A seismic section with true amplitude reflections can be described by one physical experiment: the tuned reflector model. Hence the application of the wave equation (in connection with a migration after stack) is justified on such a seismic section. Also the geometrical spreading factors that are derived can be looked upon as a generalization of a well‐known formula (Newman, 1973), which is commonly used in true amplitude processing and trace inversion in the presence of a vertically inhomogeneous earth.

2.5-D true‐amplitude Kirchhoff migration to zero offset in laterally inhomogeneous media

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 557-573 ◽  
Author(s):  
Martin Tygel ◽  
Jörg Schleicher ◽  
Peter Hubral ◽  
Lúcio T. Santos
Keyword(s):  
Seismic Source ◽  
Velocity Model ◽  
Dynamic Effects ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Seismic Section ◽  
Kirchhoff Migration ◽  
Reflection Angle ◽  
Earth Models ◽  
Zero Offset

The proposed new Kirchhoff‐type true‐amplitude migration to zero offset (MZO) for 2.5-D common‐offset reflections in 2-D laterally inhomogeneous layered isotropic earth models does not depend on the reflector curvature. It provides a transformation of a common‐offset seismic section to a simulated zero‐offset section in which both the kinematic and main dynamic effects are accounted for correctly. The process transforms primary common‐offset reflections from arbitrary curved interfaces into their corresponding zero‐offset reflections automatically replacing the geometrical‐spreading factor. In analogy to a weighted Kirchhoff migration scheme, the stacking curve and weight function can be computed by dynamic ray tracing in the macro‐velocity model that is supposed to be available. In addition, we show that an MZO stretches the seismic source pulse by the cosine of the reflection angle of the original offset reflections. The proposed approach quantitatively extends the previous MZO or dip moveout (DMO) schemes to the 2.5-D situation.

3-D true‐amplitude finite‐offset migration

Geophysics ◽  
1993 ◽  
Vol 58 (8) ◽  
pp. 1112-1126 ◽  
Author(s):  
Jorg Schleicher ◽  
Martin Tygel ◽  
Peter Hubral
Keyword(s):  
Weight Function ◽  
A Priori ◽  
Three Dimensional ◽  
Reflection Coefficients ◽  
Ray Theory ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Prestack Migration ◽  
The One ◽  
Paraxial Ray

Compressional primary nonzero offset reflections can be imaged into three‐dimensional (3-D) time or depth‐migrated reflections so that the migrated wavefield amplitudes are a measure of angle‐dependent reflection coefficients. Various migration/inversion algorithms involving weighted diffraction stacks recently proposed are based on Born or Kirchhoff approximations. Here a 3-D Kirchhoff‐type prestack migration approach is proposed where the primary reflections of the wavefields to be imaged are a priori described by the zero‐order ray approximation. As a result, the principal issue in the attempt to recover angle‐dependent reflection coefficients becomes the removal of the geometrical spreading factor of the primary reflections. The weight function that achieves this aim is independent of the unknown reflector and correctly accounts for the recovery of the source pulse in the migrated image irrespective of the source‐receiver configurations employed and the caustics occurring in the wavefield. Our weight function, which is computed using paraxial ray theory, is compared with the one of the inversion integral based on the Beylkin determinant. It differs by a factor that can be easily explained.

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.

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.

True‐amplitude transformation to zero offset of data from curved reflectors

Geophysics ◽  
1999 ◽  
Vol 64 (1) ◽  
pp. 112-129 ◽  
Author(s):  
Norman Bleistein ◽  
Jack Cohen ◽  
Herman Jaramillo
Keyword(s):  
General Context ◽  
Reflection Coefficients ◽  
Geometrical Spreading ◽  
Curvature Effects ◽  
Multiplicative Factor ◽  
Amplitude Factor ◽  
Amplitude Versus Offset ◽  
Zero Offset ◽  
The Cost ◽  
Amplitude Versus

Transformation to zero offset (TZO), alternatively known as migration to zero offset (MZO), or the combination of normal moveout and dip moveout (NMO/DMO), is a process that transforms data collected at finite offset between source and receiver to a pseudozero offset trace. The kinematic validity of NMO/DMO processing has been well established. The TZO integral operators proposed here differ from their NMO/DMO counterparts by a simple amplitude factor. (The form of the operator depends on how the input and output variables are chosen from among the combinations of midpoint or wavenumber with time or frequency.) With this modification in place, the dynamical validity for planar reflectors of the proposed TZO operators of this paper have been established in earlier studies. This means that the traveltime and geometrical spreading terms of the finite offset data are transformed to their counterparts for zero offset data, while the finite offset reflection coefficient is preserved. The main purpose of this study is to show that dynamical validity of the TZO operator extends to the case of curved reflectors in the 2.5-D limit. Thus, at the cost of a simple additional multiplicative factor in any standard NMO/DMO operator to produce the corresponding TZO operator, the amplitude factor attributed to curvature effects in finite offset data is transformed by this TZO processing to the corresponding curvature factor for zero offset data. This problem has also been addressed in a more general context by Tygel and associates. However, in the generality, some of the specifics and interpretations of the simpler problem are lost. Thus, we see some value in presenting this analysis where one can carry out all calculations explicitly and see specific quantities that are more familiar and accessible to users of DMO. Furthermore, in this paper, we show how processing of the input data with a second TZO operator allows for the extraction of the cosine of the preserved specular angle, a necessary piece of information for amplitude versus angle (AVA) analysis. We then discuss the possibility of using the output of our processing formalism at multiple offsets to create a table of angularly dependent reflection coefficients and attendant incidence angles as a function of offset. This is the basis of a proposed amplitude versus offset/amplitude versus angle (AVO/AVA) analysis of the pseudozero offset traces. Finally, we describe the modifications of Hale DMO and Gardner/Forel DMO to obtain true amplitude output equivalent to ours and also how to extract the cosine of the specular angle for these forms of DMO. This last does not depend on true amplitude processing, but only on processing two DMO operators with slightly different kernels and then taking the quotient of their peak amplitudes.

Geometrical spreading corrections of offset reflections in a laterally inhomogeneous earth

Geophysics ◽  
1992 ◽  
Vol 57 (8) ◽  
pp. 1054-1063 ◽  
Author(s):  
M. Tygel ◽  
J. Schleicher ◽  
P. Hubral
Keyword(s):  
Three Dimensional ◽  
Velocity Model ◽  
Earth Model ◽  
Geometrical Spreading ◽  
Spreading Factor ◽  
Depth Migration ◽  
Avo Analysis ◽  
Amplitude Versus Offset ◽  
One Step

Compressional primary seismic nonzero offset reflections are the most essential wavefield attributes used in seismic parameter estimation and imaging. We show how the determination of angle‐dependent reflection coefficients can be addressed from identifying such events for arbitrarily curved three‐dimensional (3-D) subsurface reflectors below a laterally inhomogeneous layered overburden. More explicitly, we show how the geometrical‐spreading factor along a reflected primary ray with offset can be calculated from the identified (i.e., picked) traveltimes of offset primary reflections. Seismic traces in which all primary reflections are corrected with the geometrical‐spreading factor are, as is well‐known, referred to as true‐amplitude traces. They can be constructed without any knowledge of the velocity distribution in the earth model. Apart from possibly finding a direct application in an amplitude‐versus‐offset (AVO) analysis, the theory developed here can be of use to derive true‐amplitude time‐ and depth‐migration methods for various seismic data acquisition configurations, which pursue the aim of performing the wavefield migration (based upon the use of a macro‐velocity model) and the AVO analysis in one step.

Synthetic Strategies to Access Heteroatomic Spirocentres Embedded in Natural Products

Synthesis ◽  
2021 ◽  
Author(s):  
Michael P. Badart ◽  
Bill C. Hawkins
Keyword(s):  
Natural Products ◽  
Heterocyclic Compound ◽  
Chemical Space ◽  
Three Dimensional ◽  
Coupling Reactions ◽  
Conjugate Additions ◽  
Biological Targets ◽  
Compound Libraries ◽  
The Common ◽  
Significant Attention

AbstractThe spirocyclic motif is abundant in natural products and provides an ideal three-dimensional template to interact with biological targets. With significant attention historically expended on the synthesis of flat-heterocyclic compound libraries, methods to access the less-explored three-dimensional medicinal-chemical space will continue to increase in demand. Herein, we highlight by reaction class the common strategies used to construct the spirocyclic centres embedded in a series of well-studied natural products.1 Introduction2 Cycloadditions3 Palladium-Catalysed Coupling Reactions4 Conjugate Additions5 Imines, Aminals, and Hemiaminal Ethers6 Mannich-Type Reactions7 Oxidative Dearomatisation8 Alkylation9 Organometallic Additions10 Conclusions

scholarly journals Crystal structure of a new monoclinic polymorph ofN-(4-methylphenyl)-3-nitropyridin-2-amine

2014 ◽  
Vol 70 (8) ◽  
pp. 58-61
Author(s):  
Aina Mardia Akhmad Aznan ◽  
Zanariah Abdullah ◽  
Vannajan Sanghiran Lee ◽  
Edward R. T. Tiekink
Keyword(s):  
Three Dimensional ◽  
Basis Set ◽  
Dihedral Angles ◽  
Asymmetric Unit ◽  
Acta Cryst ◽  
Common Features ◽  
Compound C ◽  
B3lyp Level ◽  
The Common ◽  
Independent Molecules

The title compound, C12H11N3O2, is a second monoclinic polymorph (P21, withZ′ = 4) of the previously reported monoclinic (P21/c, withZ′ = 2) form [Akhmad Aznanet al.(2010).Acta Cryst.E66, o2400]. Four independent molecules comprise the asymmetric unit, which have the common features of asyndisposition of the pyridine N atom and the toluene ring, and an intramolecular amine–nitro N—H...O hydrogen bond. The differences between molecules relate to the dihedral angles between the rings which range from 2.92 (19) to 26.24 (19)°. The geometry-optimized structure [B3LYP level of theory and 6–311 g+(d,p) basis set] has the same features except that the entire molecule is planar. In the crystal, the three-dimensional architecture is consolidated by a combination of C—H...O, C—H...π, nitro-N—O...π and π–π interactions [inter-centroid distances = 3.649 (2)–3.916 (2) Å].

The crystal structure of myoglobin. VI. Seal myoglobin

1960 ◽  
Vol 258 (1293) ◽  
pp. 181-201 ◽  
Keyword(s):  
Tertiary Structure ◽  
Heavy Atom ◽  
Three Dimensional ◽  
Sperm Whale ◽  
Fourier Synthesis ◽  
Heavy Atoms ◽  
Isomorphous Replacement ◽  
Unit Cells ◽  
The Common ◽  
Sperm Whale Myoglobin

Myoglobin from the common seal ( Phoca vitulina ) when crystallized from ammonium sulphate forms monoclinic crystals with space group the unit cell, a = 57·9Å, b = 29·6Å, c = 106·4Å, β = 102°15', contains four molecules. The method of isomorphous replacement has been used in an investigation of the centrosymmetric b -axis projection in which it has been possible to determine signs for nearly all the h0l reflexions having spacings greater than 4Å. Three independent heavy-atom derivatives were employed and the signs so determined have been used to compute a map of the electron density projected on the (010) plane. This projection has been interpreted in terms of the molecule of sperm-whale myoglobin, as deduced by Bodo, Dintzis, Kendrew & Wyckoff (1959) from a three-dimensional Fourier synthesis to 6Å resolution. The results of the interpretation show that the two myoglobin molecules are very similar in form (tertiary structure) in spite of the differences in their amino-acid composition. The relative orientation of the two unit cells with respect to the myoglobin molecule is given and a comparison is made of the positions of the heavy atoms in each molecule.

Mechanism Design and Kinematics Simulation of Massage Mechanical Arm

2011 ◽  
Vol 101-102 ◽  
pp. 279-282 ◽  
Author(s):  
Jun Xie ◽  
Jun Zhang ◽  
Jie Li
Keyword(s):  
Mechanism Design ◽  
Three Dimensional ◽  
Forward Kinematics ◽  
Kinematics Analysis ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Kinematics Model ◽  
Kinematics Simulation ◽  
The Common ◽  
Executive Mechanism

Based on the characteristics and the common massage manipulations of Chinese medical massage, a practical series mechanical arm was presented to act the manipulations with the parallel executive mechanism. Forward kinematics was solved by the Denavit-Hartenberg transformation after the kinematics model of the arm was established. And the three-dimensional model of the arm was created by Pro/E and was imported into ADAMS for the kinematics analysis. The results indicated that the common massage manipulations could be simulated by the arm correctly and flexibly, and it verified the accuracy of the mechanism design of the arm.

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