Synthetic seismograms in attenuating media

Geophysics ◽  
1983 ◽  
Vol 48 (12) ◽  
pp. 1575-1587 ◽  
Author(s):  
Thierry Bourbié ◽  
Alfonso Gonzalez‐Serrano
Keyword(s):  
Wave Theory ◽  
Relative Increase ◽  
Synthetic Seismograms ◽  
Head Wave ◽  
Complex Frequency ◽  
Two Dimensional ◽  
Borehole Data ◽  
Solid Interface ◽  
Working Space ◽  
Head Waves

Synthetic seismograms are computed in a two‐dimensional medium that contains contrasts both in the elastic and in the attenuation (Q) properties with depth. The case of a liquid‐solid interface, with and without a Q contrast between the two media, is examined in detail. We show the results when the S velocity in the solid is greater than the P velocity in the liquid. The synthetic seismograms show that the effect of the attenuation contrast is not noticeable on precritical reflections when the change in elastic properties is large even in the case of large Q contrast. When this situation occurs, there is a relative increase in the amplitude of the postcritical reflections and a decrease in amplitude of the head waves. It is interesting to notice that when two head waves are generated, the P head wave always has a very low amplitude with respect to the S head wave. This result agrees with what has been observed in borehole data. We also show the results obtained at a solid‐solid interface with and without Q contrast. The synthetic seismogram shows the same type of results as in the previous case but less pronounced due to the smaller portion of the total path occurring in attenuating media. It also shows that the converted waves (S — P and P — S) are of opposite polarities and so nearly cancel each other. The synthetic seismograms are calculated in the horizontal wavenumber‐frequency [Formula: see text] domain, and Fourier transformed back into offset‐time (x, t) domain. The choice of the [Formula: see text] domain as a working space is motivated by the fact that it allows us to include easily the effects of attenuation by specifying complex frequency‐dependent elastic moduli. The reflection and transmission coefficients in the ([Formula: see text], ω) domain are derived from a plane wave theory. Accurate geometric divergence is represented by using the exact Green’s functions in a two‐dimensional medium. The constant Q model for attenuation, derived by Kjartansson, is used.

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.

MULTI‐REFLECTED HEAD WAVES IN A SINGLE‐LAYERED MEDIUM

Geophysics ◽  
1966 ◽  
Vol 31 (5) ◽  
pp. 927-939 ◽  
Author(s):  
Yosio Nakamura
Keyword(s):  
Layered Medium ◽  
Fourier Transforms ◽  
Finite Thickness ◽  
Single Layer ◽  
Head Wave ◽  
Two Dimensional ◽  
Propagation Characteristics ◽  
Dimensional Model ◽  
Dispersive Effect ◽  
Head Waves

The propagation characteristics of surface‐reflected head waves have been studied experimentally using two‐ dimensional, Plexiglas‐aluminum and polystyrene‐aluminum, single‐layer models. In addition to the contributions from the reflected refractions (with reflections near the source) and the refracted reflections (with reflections near the receiver) those from the refraction‐reflection intermixtures (with reflections midway in the refraction path) have been found to be significant, resulting in a slight reinforcement of the surface‐reflected head waves with increasing distance. The phase relations of the arrivals have been examined by computing the Fourier transforms of the observed waveforms. It has been found that, relative to the principal head‐wave arrival, the surface‐reflected head waves exhibit an anomalous phase shift, which is due probably to the finite thickness of the model materials. For the dispersive effect to be negligible, the thickness of the two‐dimensional model must be less than approximately one‐fifteenth of the wavelength of interest.

Dynamic properties of reflected and head waves near the critical point

1979 ◽  
Vol 16 (7) ◽  
pp. 1388-1401 ◽  
Author(s):  
Larry W. Marks ◽  
F. Hron
Keyword(s):  
Exact Solution ◽  
High Frequency ◽  
Classical Problem ◽  
Plane Boundary ◽  
Head Wave ◽  
Disk File ◽  
Ray Theory ◽  
Computational Point ◽  
Spherical Waves ◽  
Head Waves

The classical problem of the incidence of spherical waves on a plane boundary has been reformulated from the computational point of view by providing a high frequency approximation to the exact solution applicable to any seismic body wave, regardless of the number of conversions or reflections from the bottoming interface. In our final expressions the ray amplitude of the interference reflected-head wave is cast in terms of a Weber function, the numerical values of which can be conveniently stored on a computer disk file and retrieved via direct access during an actual run. Our formulation also accounts for the increase of energy carried by multiple head waves arising during multiple reflections of the reflected wave from the bottoming interface. In this form our high frequency expression for the ray amplitude of the interference reflected-head wave can represent a complementary technique to asymptotic ray theory in the vicinity of critical regions where the latter cannot be used. Since numerical tests indicate that our method produces results very close to those obtained by the numerical integration of the exact solution, its combination with asymptotic ray theory yields a powerful technique for the speedy computation of synthetic seismograms for plane homogeneous layers.

Exact seismograms for a point force using generalized ray theory

Geophysics ◽  
1983 ◽  
Vol 48 (11) ◽  
pp. 1421-1427 ◽  
Author(s):  
E. R. Kanasewich ◽  
P. G. Kelamis ◽  
F. Abramovici
Keyword(s):  
Half Space ◽  
Near Field ◽  
Sedimentary Basins ◽  
Transverse Component ◽  
Point Force ◽  
Seismic Exploration ◽  
Head Wave ◽  
Vertical Force ◽  
Low Velocity ◽  
Head Waves

Exact synthetic seismograms are obtained for a simple layered elastic half‐space due to a buried point force and a point torque. Two models, similar to those encountered in seismic exploration of sedimentary basins, are examined in detail. The seismograms are complete to any specified time and make use of a Cagniard‐Pekeris method and a decomposition into generalized rays. The weathered layer is modeled as a thin low‐velocity layer over a half‐space. For a horizontal force in an arbitrary direction, the transverse component, in the near‐field, shows detectable first arrivals traveling with a compressional wave velocity. The radial and vertical components, at all distances, show a surface head wave (sP*) which is not generated when the source is compressive. A buried vertical force produces the same surface head wave prominently on the radial component. An example is given for a simple “Alberta” model as an aid to the interpretation of wide angle seismic reflections and head waves.

scholarly journals Microseismic hydraulic fracture imaging in the Marcellus Shale using head waves

Geophysics ◽  
2018 ◽  
Vol 83 (2) ◽  
pp. KS1-KS10 ◽  
Author(s):  
Zhishuai Zhang ◽  
James W. Rector ◽  
Michael J. Nava
Keyword(s):  
Marcellus Shale ◽  
P Wave ◽  
Head Wave ◽  
Wave Polarization ◽  
Horizontal Section ◽  
Location Accuracy ◽  
Arrival Times ◽  
Microseismic Data ◽  
Head Waves ◽  
Event Location

We have studied microseismic data acquired from a geophone array deployed in the horizontal section of a well drilled in the Marcellus Shale near Susquehanna County, Pennsylvania. Head waves were used to improve event location accuracy as a substitution for the traditional P-wave polarization method. We identified that resonances due to poor geophone-to-borehole coupling hinder arrival-time picking and contaminate the microseismic data spectrum. The traditional method had substantially greater uncertainty in our data due to the large uncertainty in P-wave polarization direction estimation. We also identified the existence of prominent head waves in some of the data. These head waves are refractions from the interface between the Marcellus Shale and the underlying Onondaga Formation. The source location accuracy of the microseismic events can be significantly improved by using the P-, S-wave direct arrival times and the head wave arrival times. Based on the improvement, we have developed a new acquisition geometry and strategy that uses head waves to improve event location accuracy and reduce acquisition cost in situations such as the one encountered in our study.

Experiments and Computations for KCS Added Resistance for Variable Heading

2015 ◽  
Author(s):  
Hamid Sadat-Hosseini ◽  
Serge Toxopeus ◽  
Dong Hwan Kim ◽  
Teresa Castiglione ◽  
Yugo Sanada ◽  
...  
Keyword(s):  
Surface Profile ◽  
Engine Performance ◽  
Head Wave ◽  
Added Resistance ◽  
Local Flow ◽  
Wake Field ◽  
Head Waves ◽  
Calm Water ◽  
Free Surface Profile ◽  
Peak Location

Experiments, CFD and PF studies are performed for the KCS containership advancing at Froude number 0.26 in calm water and regular waves. The validation studies are conducted for variable wavelength and wave headings with wave slope of H/λ=1/60. CFD computations are conducted using two solvers CFDShip-Iowa and STAR-CCM+. PF studies are conducted using FATIMA. For CFD computations, calm water and head wave simulations are performed by towing the ship fixed in surge, sway, roll and yaw, but free to heave and pitch. For variable wave heading simulations, the roll motion is also free. For PF, the ship model moves at a given speed and the oscillations around 6DOF motions are computed for variable wave heading while the surge motion for head waves is restrained by adding a very large surge damping. For calm water, computations showed E<4%D for the resistance,<8%D for the sinkage, and <40%D for the trim. In head waves with variable wavelength, the errors for first harmonic variables for CFD and PF computations were small, <5%DR for amplitudes and <4%2π for phases. The errors for zeroth harmonics of motions and added resistance were large. For the added resistance, the largest error was for the peak location at λ/L=1.15 where the data also show large scatter. For variable wave heading at λ/L=1.0, the errors for first harmonic amplitudes were <17%DR for CFD and <26%DR for PF. The comparison errors for first harmonic phases were E<24%2π. The errors for the zeroth harmonic of motions and added resistance were again large. PF studies for variable wave headings were also conducted for more wavelength condition, showing good predictions for the heave and pitch motions for all cases while the surge and roll motions and added resistance were often not well predicted. Local flow studies were conducted for λ/L=1.37 to investigate the free surface profile and wake field predicted by CFD. The results showed a significant fluctuation in the wake field which can affect the propeller/engine performance. Additionally it was found that the average propeller inflow to the propeller is significantly higher in waves.

Steady State Motion and Surface Waves

1996 ◽  
Author(s):  
T. T. C. Ting
Keyword(s):  
Steady State ◽  
Surface Wave ◽  
Surface Waves ◽  
Existence And Uniqueness ◽  
Wave Speed ◽  
Wave Theory ◽  
Two Dimensional ◽  
Elastic Materials ◽  
Steady State Motion ◽  
Anisotropic Elastic

The Stroh formalism for two-dimensional elastostatics can be extended to elastodynamics when the problem is a steady state motion. Most of the identities in Chapters 6 and 7 remain applicable. The Barnett-Lothe tensors S, H, L now depend on the speed υ of the steady state motion. However S(υ), H(υ), L(υ) are no longer tensors because they do not obey the laws of tensor transformation when υ≠0. Depending on the problems the speed υ may not be prescribed arbitrarily. This is particularly the case for surface waves in a half-space where υ is the surface wave speed. The problem of the existence and uniqueness of a surface wave speed in anisotropic materials is the crux of surface wave theory. It is a subject that has been extensively studied since the pioneer work of Stroh (1962). Excellent expositions on surface waves for anisotropic elastic materials have been given by Farnell (1970), Chadwick and Smith (1977), Barnett and Lothe (1985), and more recently, by Chadwick (1989d).

scholarly journals Harnessing by a diacetylene unit: a molecular design for porous two-dimensional network formation at the liquid/solid interface

2014 ◽  
Vol 50 (22) ◽  
pp. 2831 ◽  
Author(s):  
Kazukuni Tahara ◽  
Jinne Adisoejoso ◽  
Koji Inukai ◽  
Shengbin Lei ◽  
Aya Noguchi ◽  
...  
Keyword(s):  
Network Formation ◽  
Molecular Design ◽  
Two Dimensional ◽  
Solid Interface ◽  
Dimensional Network

Preface

1997 ◽  
Vol 11 (07) ◽  
pp. 1023-1024
Author(s):  
Serge Miguet ◽  
Annick Montanvert ◽  
P. S. P. Wang
Keyword(s):  
Finite State Machine ◽  
Finite Automata ◽  
Upper Bounds ◽  
Two Dimensional ◽  
Turing Machines ◽  
Closure Properties ◽  
Working Space ◽  
Finite State ◽  
The Individual ◽  
Alternating Turing Machines

Several nonclosure properties of each class of sets accepted by two-dimensional alternating one-marker automata, alternating one-marker automata with only universal states, nondeterministic one-marker automata, deterministic one-marker automata, alternating finite automata, and alternating finite automata with only universal states are shown. To do this, we first establish the upper bounds of the working space used by "three-way" alternating Turing machines with only universal states to simulate those "four-way" non-storage machines. These bounds provide us a simplified and unified proof method for the whole variants of one-marker and/or alternating finite state machine, without directly analyzing the complex behavior of the individual four-way machine on two-dimensional rectangular input tapes. We also summarize the known closure properties including Boolean closures for all the variants of two-dimensional alternating one-marker automata.

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