Theory of 2-D complex seismic trace analysis

Geophysics ◽  
1996 ◽  
Vol 61 (1) ◽  
pp. 264-272 ◽  
Author(s):  
Arthur E. Barnes
Keyword(s):  
Phase Velocity ◽  
Group Velocity ◽  
Hilbert Transform ◽  
Instantaneous Frequency ◽  
Trace Analysis ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Instantaneous Amplitude ◽  
Dip Angle

The ideas of 1-D complex seismic trace analysis extend readily to two dimensions. Two‐dimensional instantaneous amplitude and phase are scalars, and 2-D instantaneous frequency and bandwidth are vectors perpendicular to local wavefronts, each defined by a magnitude and a dip angle. The two independent measures of instantaneous dip correspond to instantaneous apparent phase velocity and group velocity. Instantaneous phase dips are aliased for steep reflection dips following the same rule that governs the aliasing of 2-D sinusoids in f-k space. Two‐dimensional frequency and bandwidth are appropriate for migrated data, whereas 1-D frequency and bandwidth are appropriate for unmigrated data. The 2-D Hilbert transform and 2-D complex trace attributes can be efficiently computed with little more effort than their 1-D counterparts. In three dimensions, amplitude and phase remain scalars, but frequency and bandwidth are 3-D vectors with magnitude, dip angle, and azimuth.

Local phase velocity from complex seismic data

Geophysics ◽  
1988 ◽  
Vol 53 (12) ◽  
pp. 1503-1511 ◽  
Author(s):  
Tim Ellis Scheuer ◽  
D. W. Oldenburg
Keyword(s):  
Phase Velocity ◽  
Instantaneous Frequency ◽  
Trace Analysis ◽  
Random Noise ◽  
Limited Range ◽  
Two Dimensions ◽  
New Method ◽  
Complex Data ◽  
Local Phase ◽  
Seismic Record

We propose a new method that uses the concepts of complex trace analysis for the automatic estimation of local phase velocity. A complex seismic record is obtained from a real seismic record by extending complex trace analysis into higher dimensions. Phase velocity is estimated from the complex data by finding trajectories of constant phase. In two dimensions, phase velocity calculation reduces to a ratio of instantaneous frequency and wavenumber, and thus provides a measure of the dominant plane‐wave component at each point in the seismic record. The algorithm is simple to implement, and computational requirements are small, partly due to a new method for computing instantaneous frequency and wavenumber which greatly simplifies these calculations for 2-D and 3-D complex records. In addition, our approach has the advantage that no a priori velocity input is needed; however, optimum stability is achieved when a limited range of dipping events is considered. Preconditioning the record with an appropriate velocity filter helps reduce the detrimental effects of crossing events, spatial aliasing, and random noise contamination.

scholarly journals Averaging over Narain moduli space

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Alexander Maloney ◽  
Edward Witten
Keyword(s):  
Einstein Gravity ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Simons Theory ◽  
Two Dimensional ◽  
Dual Theory ◽  
Recent Developments ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT’s to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain’s family of two-dimensional CFT’s obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like U(1)2D Chern-Simons theory than like Einstein gravity.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

[Ni(cyclam)(OCOR)2], a finite molecular complex: hydrogen-bonded supramolecular aggregation in one, two and three dimensions, and coordination polymers in one and two dimensions

2001 ◽  
Vol 58 (1) ◽  
pp. 78-93 ◽  
Author(s):  
Choudhury M. Zakaria ◽  
George Ferguson ◽  
Alan J. Lough ◽  
Christopher Glidewell
Keyword(s):  
Hydrogen Bonding ◽  
Coordination Polymer ◽  
Hydrogen Bonds ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Compound I ◽  
Supramolecular Aggregation ◽  
Linear Coordination

In the complexes [Ni(cyclam)(OCOR)2] (cyclam = 1,4,8,11-tetraazacyclotetradecane), where (RCOO)− is 2-naphtho-ate [bis-(2-naphthoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (I), monoclinic P21/c, Z′ = 0.5], 3,5-dinitrobenzoate [bis-(3,5-dinitrobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (II), triclinic P\bar 1, Z′ = 0.5], 4-nitrobenzoate [bis-(4-nitrobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (III), monoclinic P21/n, Z′ = 0.5], 3-hydroxybenzoate [bis-(3-hydroxybenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (IV), monoclinic P21/c, Z′ = 0.5] and 4-aminobenzo-ate [bis-(4-aminobenzoato)-1,4,8,11-tetraazacyclotetradecanenickel(II), (V), monoclinic C2/c, Z′ = 0.5], the Ni lies on a centre of inversion with monodentate carboxylato ligands occupying trans sites. Compound (I) consists of isolated molecules. In (II) and (III), N—H...O hydrogen bonds link the complexes into chains. Compounds (IV) and (III) form two- and three-dimensional structures generated entirely by hard hydrogen bonds. The 5-hydroxyisophthalate(2−) anion forms a hydrated complex, [Ni(cyclam)(5-hydroxyisophthalate)(H2O)]·4H2O {[aqua-(5-hydroxyisophthalato)-1,4,8,11-tetraazacyclotetradecanenickel(II)] tetrahydrate, (VI), monoclinic Cc, Z′ = 1}, in which the monodentate carboxylato ligand and a water molecule occupy trans sites at Ni: extensive hydrogen bonding links the molecular aggregates into a three-dimensional framework. The terephthalate(2−) anion forms a hydrated linear coordination polymer {catena-poly[terephthalato-1,4,8,11-tetraazacyclotetradecanenickel(II)] monohydrate, (VII), monoclinic C2/c, Z′ = 0.5}. In 1,2,4,5-benzenecarboxylate tris[1,4,8,11-tetraazacyclotetradecanenickel(II)] diperchlorate hydrate (VIII), [Ni(cyclam)]3·[1,2,4,5-benzenetetracarboxylate(4−)]·[ClO4]2·-[H2O]3, there are two distinct Ni sites: [Ni(cyclam)]2+ and centrosymmetric [C10H2O8]4− units form a two-dimensional coordination polymer, whose sheets are linked by centrosymmetric [Ni(cyclam)(H2O)2]2+ cations.

Instability of pressure-driven gas–liquid two-layer channel flows in two and three dimensions

2018 ◽  
Vol 849 ◽  
pp. 1-34 ◽  
Author(s):  
Lennon Ó Náraigh ◽  
Peter D. M. Spelt
Keyword(s):  
Flow Pattern ◽  
Linear Theory ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Channel Flows ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Vorticity Field ◽  
Flow Pattern Map

We study unstable waves in gas–liquid two-layer channel flows driven by a pressure gradient, under stable stratification, not assumed to be set in motion impulsively. The basis of the study is direct numerical simulation (DNS) of the two-phase Navier–Stokes equations in two and three dimensions for moderately large Reynolds numbers, accompanied by a theoretical description of the dynamics in the linear regime (Orr–Sommerfeld–Squire equations). The results are compared and contrasted across a range of density ratios $r=\unicode[STIX]{x1D70C}_{liquid}/\unicode[STIX]{x1D70C}_{gas}$. Linear theory indicates that the growth rate of small-amplitude interfacial disturbances generally decreases with increasing $r$; at the same time, the cutoff wavenumbers in both streamwise and spanwise directions increase, leading to an ever-increasing range of unstable wavenumbers, albeit with diminished growth rates. The analysis also demonstrates that the most dangerous mode is two-dimensional in all cases considered. The results of a comparison between the DNS and linear theory demonstrate a consistency between the two approaches: as such, the route to a three-dimensional flow pattern is direct in these cases, i.e. through the strong influence of the linear instability. We also characterize the nonlinear behaviour of the system, and we establish that the disturbance vorticity field in two-dimensional systems is consistent with a mechanism proposed previously by Hinch (J. Fluid Mech., vol. 144, 1984, p. 463) for weakly inertial flows. A flow-pattern map constructed from two-dimensional numerical simulations is used to describe the various flow regimes observed as a function of density ratio, Reynolds number and Weber number. Corresponding simulations in three dimensions confirm that the flow-pattern map can be used to infer the fate of the interface there also, and show strong three-dimensionality in cases that exhibit violent behaviour in two dimensions, or otherwise the development of behaviour that is nearly two-dimensional behaviour possibly with the formation of a capillary ridge. The three-dimensional vorticity field is also analysed, thereby demonstrating how streamwise vorticity arises from the growth of otherwise two-dimensional modes.

scholarly journals THE “DUAL” VARIABLES OF YANG-MILLS THEORY AND LOCAL GAUGE-INVARIANT VARIABLES

1996 ◽  
Vol 11 (32) ◽  
pp. 5701-5728 ◽  
Author(s):  
ORI GANOR ◽  
J. SONNENSCHEIN
Keyword(s):  
Degrees Of Freedom ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Six Degrees Of Freedom ◽  
Local Gauge ◽  
Dual Variables ◽  
Topological Part

After adding auxiliary fields and integrating out the original variables, the Yang-Mills action can be expressed in terms of local gauge-invariant variables. This method reproduces the known solution of the two-dimensional SU (N) theory. In more than two dimensions the action splits into a topological part and a part proportional to αs. We demonstrate the procedure for SU (2) in three dimensions where we reproduce a gravitylike theory. We discuss the four-dimensional case as well. We use a cubic expression in the fields as a space-time metric to obtain a covariant Lagrangian. We also show how the four-dimensional SU (2) theory can be expressed in terms of a local action with six degrees of freedom only.

scholarly journals Relaxation of some multi-well problems

2001 ◽  
Vol 131 (2) ◽  
pp. 279-320 ◽  
Author(s):  
Kaushik Bhattacharya ◽  
Georg Dolzmann
Keyword(s):  
Thin Films ◽  
Phase Transitions ◽  
Young Measure ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Convex Hulls ◽  
Quasiconvex Functions ◽  
Two Dimensional ◽  
Existence Of Minimizers

Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫Ω W (Du) dx, where W has a multi-well structure, i.e. W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e. for K = SO(2)U1 ∪ … ∪SO(2)Uk or K = O(2)U1 ∪ … ∪ O(2)Uk for U1, … , Uk ∈ M2×2 with det Ui = δ in three dimensions when the matrices Ui are essentially two-dimensional and also for K = SO(3)Û1 ∪ … ∪ SO(3)Ûk for U1, … , Uk ∈ M3×3 with , which arises in the study of thin films. Here, Ûi denotes the (3×2) matrix formed with the first two columns of Ui. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose ‘approximate relaxed energies’, quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.

DONOR BINDING ENERGY IN TWO DIMENSIONS

1994 ◽  
Vol 08 (17) ◽  
pp. 1041-1043 ◽  
Author(s):  
E. S. ACKLEH ◽  
G. D. MAHAN ◽  
JI-WEI WU
Keyword(s):  
Binding Energy ◽  
Electron Density ◽  
Electron Gas ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Dimensional Electron ◽  
Two Dimensional Electron Gas ◽  
Donor Binding Energy ◽  
Dimensional Electron Gas

We calculate numerically the binding energy of an electron bound to a proton in two-dimensional electron gas. Unlike the case in three dimensions, in 2D the binding energy is nonzero for all values of electron density.

3D diffraction imaging with Kirchhoff time migration using vertical traveltime difference gathers

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. S555-S566 ◽  
Author(s):  
Zhengwei Li ◽  
Jianfeng Zhang
Keyword(s):  
Fresnel Zone ◽  
Lateral Position ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Small Scale ◽  
Imaging Method ◽  
Time Migration ◽  
Diffraction Imaging ◽  
Dip Angle ◽  
Angle Gather

We have built a vertical traveltime difference (VTD) gather to image diffractions in the 3D time domain. This significantly improves detection of small-scale faults and heterogeneities in 3D seismic data. The VTD gather is obtained using 3D Kirchhoff prestack time migration based on the traveltime-related inline and crossline dip angles, which is closely related to the 2D dip-angle gather. In VTD gathers, diffraction events exhibit flattening, whereas reflection events have convex upward-sloping shapes. Different from the 2D dip-angle gather, Fresnel zone-related specular reflections are precisely focused on the given regions over all offsets and azimuths, thus leaving more diffraction energy after muting. To image linear diffractors, such as faults in three dimensions, the VTD gather can be extended into two dimensions by adding a dip-azimuth dimension. This makes it possible to correct phases of edge diffractions and detect the orientations of the linear diffractors. The memory requirement of the VTD or VTD plus azimuth gathers is much less than that of the 2D dip-angle gathers. We can store the gathers at each lateral position and then correct the phase and enhance the weak diffractions in 3D cases. Synthetic and field data tests demonstrate the effectiveness of our 3D diffraction imaging method.

scholarly journals Instantaneous Amplitude-Frequency Feature Extraction for Rotor Fault Based on BEMD and Hilbert Transform

2019 ◽  
Vol 2019 ◽  
pp. 1-19 ◽  
Author(s):  
Chuanjin Huang ◽  
Haijun Song ◽  
Wenping Lei ◽  
Zhanya Niu ◽  
Yajun Meng
Keyword(s):  
Feature Extraction ◽  
Hilbert Transform ◽  
Vibration Signal ◽  
Three Dimensions ◽  
Intrinsic Mode Functions ◽  
Instantaneous Amplitude ◽  
Vibration Signals ◽  
Feature Extraction Method ◽  
Fault Features ◽  
The Hilbert Transform

The vibration signals propagating in different directions from rotating machines can contain a variety of characteristic information. A novel feature extraction method based on bivariate empirical mode decomposition (BEMD) for rotor is proposed to comprehensively extract the fault features. In this work, the number of signal projection directions is determined through simulation, and the energy end condition based on the energy threshold is increased using BEMD to enhance the decomposition quality. Mixed vibration signals are generated along two orthogonal directions. Then, the acquired vibration signal can be decomposed into several intrinsic mode functions (IMFs) at the rotational speed using the BEMD method. Furthermore, the instantaneous frequency and instantaneous amplitude of the real signals and the imaginary part of the IMF signals are obtained using the Hilbert transform. The fault features along two and three dimensions can be investigated, providing more comprehensive information to aid in the fault diagnosis of rotor. Experimental results on oil film oscillation, the oil whirl, the bistability of the rotor, and looseness and rotor rubbing composite fault indicate the effectiveness of the proposed method.

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