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.

Head waves from a transition layer

1968 ◽  
Vol 58 (3) ◽  
pp. 963-976
Author(s):  
Yosio Nakamura
Keyword(s):  
Transition Zone ◽  
Transition Layer ◽  
High Frequency ◽  
Cutoff Frequency ◽  
Head Wave ◽  
Seismic Model ◽  
Frequency Range ◽  
Low Frequencies ◽  
Sharp Discontinuity ◽  
Head Waves

abstract Ultrasonic, seismic-model experiments have been performed to re-examine the nature of head waves from a transition layer. Two-dimensional, layered structures, some with a transition zone and some with a sharp discontinuity, constructed by lead-aluminum laminations have served as the models. Amplitude and phase responses have been measured in a frequency range of 25 kHz to 200 kHz. At low frequencies, where the wavelength is much longer than the thickness of the transition zone, little difference is observed between head waves from a transition zone and those from a sharp discontinuity. At a certain frequency range where the wavelength is close to the thickness of the transition zone, the measured head-wave amplitude from a transition zone becomes a few times larger than that from a sharp discontinuity. This is characteristic of head waves from a poorly defined boundary and may be used to estimate the thickness of a general transition layer. A sharp, high-frequency cutoff is again observed, but the cutoff frequency is not consistent with previous studies.

ON THE THEORY OF HEAD WAVES

Geophysics ◽  
1953 ◽  
Vol 18 (4) ◽  
pp. 871-893 ◽  
Author(s):  
Patrick A. Heelan
Keyword(s):  
Order Term ◽  
Grazing Incidence ◽  
Second Order ◽  
P Wave ◽  
Plane Boundary ◽  
Head Wave ◽  
Wave System ◽  
Localized Source ◽  
Head Waves ◽  
Lower Medium

When a combined longitudinal and transverse disturbance, diverging from a localized source, strikes a plane boundary between two solid elastic media, several systems of head waves and second‐order boundary waves are generated, each associated with grazing incidence of one or the other of the reflected or refracted waves. Associated with grazing incidence of [Formula: see text], the refracted P‐wave, is the head wave system comprising [Formula: see text] (the “refracted wave” of seismic prospectors), and [Formula: see text] (a transverse head wave) in the upper medium, and [Formula: see text] (a transverse head wave) in the lower medium. There is no boundary wave in the lower medium. These three waves, with the second‐order term of [Formula: see text] (the first‐order term is zero on the boundary) satisfy conditions of continuity of stress and displacement at the boundary. Moreover, the energy of the three head waves is derived completely from the second‐order component of [Formula: see text], which possesses a component of energy flow normal to the boundary. The amplitudes of [Formula: see text] [Formula: see text] and [Formula: see text] are calculated for certain cases.

A simple method for the computation of body-wave seismograms

1978 ◽  
Vol 68 (6) ◽  
pp. 1577-1593
Author(s):  
S. K. Dey-Sarkar ◽  
C. H. Chapman
Keyword(s):  
Upper Mantle ◽  
High Frequency ◽  
Asymptotic Form ◽  
Legendre Function ◽  
Body Wave ◽  
Computational Procedure ◽  
Head Wave ◽  
Earth Model ◽  
Simple Method ◽  
Head Waves

abstract The computational procedure for a new method of computing high-frequency body-wave seismograms for a spherically symmetric earth model is presented. The seismograms are accurate provided that the WKBJ approximation can be used for the transformed response and the asymptotic form of the Legendre function is applicable. The method remains valid when geometrical ray theory breaks down at caustics and for head waves. The computation of the seismograms is extremely easy and rapid as both inverse transforms are evaluated analytically. Seismograms are presented to show how, for a discrete time series, the method is insensitive to small features in the model. The method is illustrated for a reflected signal plus a head wave and for turning rays in three very different upper mantle models.

REFRACTION ALONG A LAYER

Geophysics ◽  
1965 ◽  
Vol 30 (3) ◽  
pp. 369-388 ◽  
Author(s):  
T. W. Spencer
Keyword(s):  
High Frequency ◽  
High Speed ◽  
Shear Mode ◽  
Ray Theory ◽  
Dominant Wavelength ◽  
Layer Depth ◽  
Time Intervals ◽  
Reflected Waves ◽  
Head Waves ◽  
Compressional Mode

High‐frequency geometric ray theory is used to investigate the refracted arrival from a high‐speed layer embedded in an infinite medium. The effect of changing the layer thickness to dominant wavelength ratio [Formula: see text] and the range to depth ratio (ρ/H) is analyzed for a point compressional source. The results approximate the exact solution when [Formula: see text]. The theory predicts shingling and shows that it is range‐limited. Factors which improve the resolution between reflected arrivals increase the range over which shingling occurs. As the range increases, the traveltime curves for all the multiply reflected rays which cross the layer the same number of times in the shear mode approach the same asymptote (regardless of the number of crossings in the compressional mode). When the layer is thick compared to the dominant wavelength, the refracted arrival may consist of a series of events separated by equal time intervals. Each event is produced by the superposition of reflected waves which cross the layer the same number of times in the shear mode. The amplitude of each event satisfies [Formula: see text], where H is the layer depth. Because the head waves decay like [Formula: see text], the reflected waves predominate at large ranges.

scholarly journals Exact solution and high frequency asymptotic methods in the wedge diffraction problem

2016 ◽  
Vol 14 (38) ◽  
pp. 9-28
Author(s):  
Hernan G. Triana ◽  
Andrés Navarro Cadavid
Keyword(s):  
Numerical Simulation ◽  
Exact Solution ◽  
Asymptotic Solution ◽  
High Frequency ◽  
Asymptotic Methods ◽  
Diffraction Problem ◽  
Approximate Solutions ◽  
Point Of View ◽  
Simulation Tool ◽  
Computational Point

AbstractThe Sommerfeld exact solution for canonical 2D wedge diffraction problem with perfectly conducting surfaces is presented. From the integral formulation of the problem, the Malyuzhinets solution is obtained and this result is extended to obtain the general impedance solution of canonical 2D wedge problem. Keller’s asymptotic solution is developed and the general formulation of exact solution it’s used to obtain general asymptotic methods for approximate solutions useful from the computational point of view. A simulation tool is used to compare numerical calculations of exact and asymptotic solutions. The numerical simulation of exact solution is compared to numerical simulation of an asymptoticmethod, and a satisfactory agreement found.  Accuracy dependence with frequency is verified.

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.

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.

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