Eigenray method: Geometric spreading

2019 ◽  
Author(s):  
Zvi Koren ◽  
Igor Ravve
Keyword(s):  
Geometric Spreading

3D mapping of kinematic attributes in anisotropic media

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. C159-C170 ◽  
Author(s):  
Yuriy Ivanov ◽  
Alexey Stovas
Keyword(s):  
Anisotropic Medium ◽  
Heterogeneous Media ◽  
Homogeneous Medium ◽  
Anisotropic Media ◽  
Transformation Matrix ◽  
3D Mapping ◽  
Slowness Surface ◽  
Point Correspondence ◽  
Geometric Spreading ◽  
Point To Point

Based on the rotation of a slowness surface in anisotropic media, we have derived a set of mapping operators that establishes a point-to-point correspondence for the traveltime and relative-geometric-spreading surfaces between these calculated in nonrotated and rotated media. The mapping approach allows one to efficiently obtain the aforementioned surfaces in a rotated anisotropic medium from precomputed surfaces in the nonrotated medium. The process consists of two steps: calculation of a necessary kinematic attribute in a nonrotated, e.g., orthorhombic (ORT), medium, and subsequent mapping of the obtained values to a transformed, e.g., rotated ORT, medium. The operators we obtained are applicable to anisotropic media of any type; they are 3D and are expressed through a general form of the transformation matrix. The mapping equations can be used to develop moveout and relative-geometric-spreading approximations in rotated anisotropic media from existing approximations in nonrotated media. Although our operators are derived in case of a homogeneous medium and for a one-way propagation only, we discuss their extension to vertically heterogeneous media and to reflected (and converted) waves.

Amplitude, Fresnel zone, and NMO velocity for PP and SS normal-incidence reflections

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. W1-W14 ◽  
Author(s):  
Einar Iversen
Keyword(s):  
Phase Shift ◽  
Ray Tracing ◽  
Explicit Knowledge ◽  
Fresnel Zone ◽  
Normal Wave ◽  
Normal Incidence ◽  
Geometric Spreading ◽  
Dynamic Ray Tracing ◽  
The One ◽  
Paraxial Ray

Inspired by recent ray-theoretical developments, the theory of normal-incidence rays is generalized to accommodate P- and S-waves in layered isotropic and anisotropic media. The calculation of the three main factors contributing to the two-way amplitude — i.e., geometric spreading, phase shift from caustics, and accumulated reflection/transmission coefficients — is formulated as a recursive process in the upward direction of the normal-incidence rays. This step-by-step approach makes it possible to implement zero-offset amplitude modeling as an efficient one-way wavefront construction process. For the purpose of upward dynamic ray tracing, the one-way eigensolution matrix is introduced, having as minors the paraxial ray-tracing matrices for the wavefronts of two hypothetical waves, referred to by Hubral as the normal-incidence point (NIP) wave and the normal wave. Dynamic ray tracing expressed in terms of the one-way eigensolution matrix has two advantages: The formulas for geometric spreading, phase shift from caustics, and Fresnel zone matrix become particularly simple, and the amplitude and Fresnel zone matrix can be calculated without explicit knowledge of the interface curvatures at the point of normal-incidence reflection.

scholarly journals Comparison study between the elevation and datum statics in NC 210, western Libya

2021 ◽  
Author(s):  
Riyadh Alhajni
Keyword(s):  
Conventional Method ◽  
Seismic Data ◽  
Seismic Velocity ◽  
Sand Dunes ◽  
Angle Of Repose ◽  
Amplitude Analysis ◽  
Reflected Waves ◽  
Geometric Spreading ◽  
Amplitude Compensation ◽  
Seismic Lines

Abstract This research compares the results of each method to solve problems caused by sand dunes, In the southwestern region of Libya, the Murzuq basin is covered with sand dunes, which are a significant source of noise in land seismic data, which caused issues in seismic processing, also sand dunes cause increases of travel time of reflected events in seismic data, procuring false structures this problem caused by residual static errors. The presence of extensive sand dunes causes logistic and technical difficulties for seismic reflection prospecting, Due to the steep angle of repose of the sand dunes faces and the low seismic velocity within them, which causes significant time delay to the reflected waves. In this research, three seismic lines (202, 207, 209), of total length 12 km, have been completely reprocessed at Western Geco processing center (Tripoli) using omega software. the methods of gain corrections: time function gain and geometric spreading. Spreading amplitude compensation, has been proceed the results will be compared to another method of gain corrections called residual amplitude analysis compensation (RAAC) which is has better results for static problems the conventional method of computing field statics has been implemented and the result is compared with elevation static. It is obtained by using uphole method (conventional method) yielded a significant improvement over the elevation method.

Propagation of high-frequency (4- to 50-Hz) P waves in the northeastern United States

1995 ◽  
Vol 85 (4) ◽  
pp. 1244-1248
Author(s):  
Eric P. Chael ◽  
Patrick J. Leahy ◽  
Jerry A. Carter ◽  
Noël Barstow ◽  
Paul W. Pomeroy
Keyword(s):  
United States ◽  
Decay Rate ◽  
High Frequency ◽  
P Wave ◽  
Wave Spectra ◽  
Northeastern United States ◽  
P Waves ◽  
Geometric Spreading ◽  
Frequency Independent

Abstract We have measured the decay rate of high-frequency (4- to 50-Hz) P waves in the northeastern United States. We analyzed signals from 28 explosions of a 1988 USGS/AFGL/GSC refraction survey recorded at distances between 30 and 400 km. Over this range, the decay rate steadily increases from Δ−2 at 10 Hz to Δ−4 at 45 Hz. If one assumes geometric spreading of Δ−1.3, then the remaining decay is consistent with a nearly frequency-independent Q of about 1000. The results provide a useful parameterization for predicting P-wave spectra at near-regional ranges.

SS-traveltime parameters from PP and PS reflections

Geophysics ◽  
2009 ◽  
Vol 74 (4) ◽  
pp. R35-R47 ◽  
Author(s):  
Bjørn Ursin ◽  
Martin Tygel ◽  
Einar Iversen
Keyword(s):  
Velocity Model ◽  
Ocean Bottom ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Partial Reconstruction ◽  
Propagator Matrix ◽  
Geometric Spreading ◽  
Derivatives Of ◽  
Full Simulation ◽  
Original Formula

The SS-wave traveltimes can be derived from PP- and PS-wave data with the previously derived [Formula: see text] method. We have extended this method as follows. (1) The previous requirement that sources and receivers be located on a common acquisition surface is removed, which makes the method directly applicable to PS-waves recorded on the ocean bottom and PP-waves recorded at the ocean surface. (2) By using the concept and properties of surface-to-surface propagator matrices, the second-order traveltime derivatives of the SS-waves are obtained. In the same way as for the original [Formula: see text] method, the proposed extension is valid for arbitrary anisotropic media. The propagator matrix and geometric spreading of an SS-wave reflected at a given point on a target reflector are obtained explicitly from the propagators of the PP- and PS-waves reflected at the same point. These additional parameters provided by the extended [Formula: see text] method can be used for a partial reconstruction of the SS-wave amplitude as well as for tomographic estimation of the elastic velocity model. A full simulation of the SS-wave, which includes reflection and transmission coefficients, cannot be obtained directly from recorded PP- and PS-wave amplitudes.

The isochron ray in seismic modeling and imaging

Geophysics ◽  
2004 ◽  
Vol 69 (4) ◽  
pp. 1053-1070 ◽  
Author(s):  
Einar Iversen
Keyword(s):  
Seismic Imaging ◽  
Model Updating ◽  
Velocity Model ◽  
Seismic Modeling ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Geometric Spreading ◽  
Model Independent ◽  
Map Migration ◽  
Upper Level

The isochron, the name given to a surface of equal two‐way time, has a profound position in seismic imaging. In this paper, I introduce a framework for construction of isochrons for a given velocity model. The basic idea is to let trajectories called isochron rays be associated with iso chrons in an way analogous to the association of conventional rays with wavefronts. In the context of prestack depth migration, an isochron ray based on conventional ray theory represents a simultaneous downward continuation from both source and receiver. The isochron ray is a generalization of the normal ray for poststack map migration. I have organized the process with systems of ordinary differential equations appearing on two levels. The upper level is model‐independent, and the lower level consists of conventional one‐way ray tracing. An advantage of the new method is that interpolation in a ray domain using isochron rays is able to treat triplications (multiarrivals) accurately, as opposed to interpolation in the depth domain based on one‐way traveltime tables. Another nice property is that the Beylkin determinant, an important correction factor in amplitude‐preserving seismic imaging, is closely related to the geometric spreading of isochron rays. For these reasons, the isochron ray has the potential to become a core part of future implementations of prestack depth migration. In addition, isochron rays can be applied in many contexts of forward and inverse seismic modeling, e.g., generation of Fresnel volumes, map migration of prestack traveltime events, and generation of a depth‐domain–based cost function for velocity model updating.

Traveltime and relative geometric spreading approximation in elastic orthorhombic medium

Geophysics ◽  
2020 ◽  
Vol 85 (5) ◽  
pp. C153-C162 ◽  
Author(s):  
Shibo Xu ◽  
Alexey Stovas ◽  
Hitoshi Mikada
Keyword(s):  
Seismic Data ◽  
Real Data ◽  
P Wave ◽  
Form Expression ◽  
Amplitude Variation ◽  
Rational Form ◽  
Amplitude Variation With Offset ◽  
Practical Applications ◽  
Numerical Tests ◽  
Geometric Spreading

Wavefield properties such as traveltime and relative geometric spreading (traveltime derivatives) are highly essential in seismic data processing and can be used in stacking, time-domain migration, and amplitude variation with offset analysis. Due to the complexity of an elastic orthorhombic (ORT) medium, analysis of these properties becomes reasonably difficult, where accurate explicit-form approximations are highly recommended. We have defined the shifted hyperbola form, Taylor series (TS), and the rational form (RF) approximations for P-wave traveltime and relative geometric spreading in an elastic ORT model. Because the parametric form expression for the P-wave vertical slowness in the derivation is too complicated, TS (expansion in offset) is applied to facilitate the derivation of approximate coefficients. The same approximation forms computed in the acoustic ORT model also are derived for comparison. In the numerical tests, three ORT models with parameters obtained from real data are used to test the accuracy of each approximation. The numerical examples yield results in which, apart from the error along the y-axis in ORT model 2 for the relative geometric spreading, the RF approximations all are very accurate for all of the tested models in practical applications.

NGA-West2 Equations for Predicting Vertical-Component PGA, PGV, and 5%-Damped PSA from Shallow Crustal Earthquakes

2016 ◽  
Vol 32 (2) ◽  
pp. 1005-1031 ◽  
Author(s):  
Jonathan P. Stewart ◽  
David M. Boore ◽  
Emel Seyhan ◽  
Gail M. Atkinson
Keyword(s):  
Site Response ◽  
Vertical Component ◽  
Site Amplification ◽  
Intensity Measures ◽  
Global Database ◽  
Crustal Earthquakes ◽  
Anelastic Attenuation ◽  
Nonlinear Site Response ◽  
Active Tectonic ◽  
Geometric Spreading

We present ground motion prediction equations (GMPEs) for computing natural log means and standard deviations of vertical-component intensity measures (IMs) for shallow crustal earthquakes in active tectonic regions. The equations were derived from a global database with M 3.0–7.9 events. The functions are similar to those for our horizontal GMPEs. We derive equations for the primary M- and distance-dependence of peak acceleration, peak velocity, and 5%-damped pseudo-spectral accelerations at oscillator periods between 0.01–10 s. We observe pronounced M-dependent geometric spreading and region-dependent anelastic attenuation for high-frequency IMs. We do not observe significant region-dependence in site amplification. Aleatory uncertainty is found to decrease with increasing magnitude; within-event variability is independent of distance. Compared to our horizontal-component GMPEs, attenuation rates are broadly comparable (somewhat slower geometric spreading, faster apparent anelastic attenuation), VS30-scaling is reduced, nonlinear site response is much weaker, within-event variability is comparable, and between-event variability is greater.

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