An analytical solution to separate P‐waves and S‐waves in VSP wavefields

Geophysics ◽  
1995 ◽  
Vol 60 (4) ◽  
pp. 955-967 ◽  
Author(s):  
Hiroshi Amano
Keyword(s):  
Analytical Solution ◽  
Formal Solution ◽  
Seismic Profile ◽  
Synthetic Seismograms ◽  
Third Order ◽  
P Waves ◽  
Practical Applications ◽  
S Waves ◽  
The Third ◽  
Z Domain

An analytical solution to separate P‐waves and S‐waves in vertical seismic profile (VSP) wavefields is derived using combinations of certain terms of the formal solution for forward VSP modeling. Some practical applications of this method to synthetic seismograms and field data are investigated and evaluated. Little wave distortion is recognized, and the weak wavefield masked by dominant wavetrains can be extracted with this method. The decomposed wavefield is expressed in the frequency‐depth (f-z) domain as a linear combination of up to the third‐order differential of traces, which is approximated by trace differences in the practical separation process. In general, five traces with single‐component data are required in this process, but the same process is implemented with only three traces in the acoustic case. Two‐trace extrapolation is applied to each edge of the data gather to enhance the accuracy of trace difference. Since the formulas are developed in the f-z domain, the influence of anelasticity can be taken into account, and the calculation is carried out fast enough with the benefit of the fast Fourier transform (FFT).

scholarly journals Estimation of radiated energy using the KiK-net downhole records—old method for modern data

2020 ◽  
Vol 221 (2) ◽  
pp. 1029-1042 ◽  
Author(s):  
Hiroo Kanamori ◽  
Zachary E Ross ◽  
Luis Rivera
Keyword(s):  
Site Response ◽  
Source Model ◽  
Synthetic Seismograms ◽  
S Wave ◽  
P Waves ◽  
S Waves ◽  
Radiated Energy ◽  
Time Integral ◽  
Magnitude Range ◽  
Significant Difference

SUMMARY We use KiK-net (NIED) downhole records to estimate the radiated energy, ER, of 29 Japanese inland earthquakes with a magnitude range from Mw = 5.6 to 7.0. The method is based on the work of Gutenberg and Richter in which the time integral of S-wave ground-motion velocity-squared is measured as a basic metric of the radiated energy. Only stations within a distance of 100 km are used to minimize complex path and attenuation effects. Unlike the teleseismic method that uses mainly P waves, the use of S waves which carry more than 95 per cent of the radiated energy allows us to obtain robust results. We calibrate the method using synthetic seismograms to modernize and improve the Gutenberg–Richter method. We compute synthetic seismograms for a source model of each event with a given source function (i.e. known ER), the actual mechanism and the source-station geometry. Then, we compare the given ER with the computed energy metric to correct for the unknown effect of wave propagation and the mechanism. The use of downhole records minimizes the uncertainty resulting from the site response. Our results suggest that the currently available estimates of ER from teleseismic data are probably within a factor of 3, on average, of the absolute value. The scaled energy eR ( = ER/M0) is nearly constant at about 3 × 10−5 over a magnitude range from Mw = 5.6 to 7.0 with a slight increasing trend with Mw. We found no significant difference in eR between dip-slip and strike-slip events.

scholarly journals Analytical solution of a class of nonlinear differential equations of the third order in the complex area

2021 ◽  
pp. 75-84
Author(s):  
Виктор Николаевич Орлов ◽  
Людмила Витальевна Мустафина
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
A Priori Estimates ◽  
Nonlinear Differential Equations ◽  
A Priori ◽  
A Priori Estimate ◽  
Third Order ◽  
The Third ◽  
Change Of Variables ◽  
Priori Estimates

В работе приводится доказательство теоремы существования и единственности аналитического решения класса нелинейных дифференциальных уравнений третьего порядка, правая часть которого представлена полиномом шестой степени, в комплексной области. Расширен класс рассматриваемых уравнений за счет новой замены переменных. Получена априорная оценка аналитического приближенного решения. Представлен вариант численного эксперимента оптимизации априорных оценок с помощью апостериорных. The article presents a proof of the theorem of the existence and uniqueness of the analytical solution of the class of nonlinear differential equations of the third order, with a polynomial right-hand side of the sixth degree, in the complex domain. The class of the considered equations has been extended by means of a new change of variables. An a priori estimate of the analytical approximate solution is obtained. A variant of the numerical experiment of optimizing a priori estimates using a posteriori estimates is presented.

scholarly journals On The Third-Order Complex Differential Inequalities of ξ-Generalized-Hurwitz–Lerch Zeta Functions

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 845
Author(s):  
Hiba Al-Janaby ◽  
Firas Ghanim ◽  
Maslina Darus
Keyword(s):  
Differential Inequality ◽  
Function Theory ◽  
Zeta Functions ◽  
Differential Subordination ◽  
Geometric Function Theory ◽  
Third Order ◽  
The Third ◽  
Geometric Function ◽  
Z Domain ◽  
Admissible Functions

In the z- domain, differential subordination is a complex technique of geometric function theory based on the idea of differential inequality. It has formulas in terms of the first, second and third derivatives. In this study, we introduce some applications of the third-order differential subordination for a newly defined linear operator that includes ξ -Generalized-Hurwitz–Lerch Zeta functions (GHLZF). These outcomes are derived by investigating the appropriate classes of admissible functions.

GENERATION OF THE TERAHERZ RADIATION USING χ(3) IN SEMICONDUCTOR

1995 ◽  
Vol 04 (01) ◽  
pp. 163-189 ◽  
Author(s):  
J. B. KHURGIN
Keyword(s):  
Energy Coupling ◽  
Difference Frequency Generation ◽  
Difference Frequency ◽  
Alternative Methods ◽  
Third Order ◽  
Photogalvanic Effect ◽  
Practical Applications ◽  
The Third ◽  
Coupling Techniques ◽  
The Difference

A rigorous theory of the difference frequency mixing of two signals, one with frequency ω and the other with frequency near 2ω, in semiconductors is presented. It is shown that a lower-frequency (DC ~ 10THz) directional photocurrent and voltage are generated as a result of this nonlinear interaction. This result conclusively links the 'directional photogalvanic effect' with the third-order nonlinearity. The magnitude of the difference frequency response is evaluated as a function of frequency and the efficiency of the method is examined for various energy coupling techniques. Comparison with alternative methods for difference frequency generation using the second order nonlinearities is made and the practical applications are considered.

A plane‐wave decomposition for elastic wave fields applied to the separation of P‐waves and S‐waves in vector seismic data

Geophysics ◽  
1986 ◽  
Vol 51 (2) ◽  
pp. 419-423 ◽  
Author(s):  
A. J. Devaney ◽  
M. L. Oristaglio
Keyword(s):  
Plane Wave ◽  
Elastic Wave ◽  
Seismic Data ◽  
Seismic Profile ◽  
P Waves ◽  
Wave Fields ◽  
S Waves ◽  
Plane Wave Decomposition ◽  
Wave Decomposition ◽  
Seismic Measurements

We describe a method to decompose a two‐dimensional (2-D) elastic wave field recorded along a line into its longitudinal and transverse parts, that is, into compressional (P) waves and shear (S) waves. Separation of the data into P-waves and S-waves is useful when analyzing vector seismic measurements along surface lines or in boreholes. The method described is based on a plane‐wave expansion for elastic wave fields and is illustrated with a synthetic example of an offset vertical seismic profile (VSP) in a layered elastic medium.

On the Generalization of Displacement-Based Laminate Theories

1989 ◽  
Vol 42 (11S) ◽  
pp. S213-S222 ◽  
Author(s):  
J. N. Reddy
Keyword(s):  
Analytical Solution ◽  
Elasticity Theory ◽  
Shear Deformation ◽  
Single Layer ◽  
Numerical Results ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
Laminate Theory ◽  
Displacement Based

A review and generalization of the displacement-based two-dimensional plate theories is presented. The classical and shear deformation single–layer theories up to the third-order are presented in a single theory through tracers. The layer–wise laminate theory developed by the author is reviewed. Numerical results are presented to illustrate the accuracy of the layer–wise theory by comparison with the analytical solution of the 3–D elasticity theory.

Wave‐field transformations of vertical seismic profiles

Geophysics ◽  
1987 ◽  
Vol 52 (3) ◽  
pp. 307-321 ◽  
Author(s):  
Liang‐Zie Hu ◽  
George A. McMechan
Keyword(s):  
Fourier Transform ◽  
Wave Field ◽  
Seismic Profile ◽  
Seismic Profiles ◽  
P Waves ◽  
S Waves ◽  
Vertical Seismic Profiles ◽  
Slice Theorem ◽  
Vsp Data ◽  
T Domain

Vertical seismic profile (VSP) data may be partitioned in a variety of ways by application of wave‐field transformations. These transformations provide insights into the nature of the data and aid in the design of processing operations. Transformations are implemented in a reversible sequence that takes the observed VSP data from the depth‐time (z-t) domain through the slowness‐time intercept (p-τ) domain (by a slant stack), to the slowness‐frequency (p-ω) domain (by a 1-D Fourier transform over τ), to the wavenumber‐frequency (k-ω) domain (by resampling using the Fourier central‐slice theorem), and finally back to the z-t domain (by an inverse 2-D Fourier transform). Multidimensional wave‐field transformations, combined with k-ω, p-ω, and p-τ filtering, can be applied to wave‐field resampling, interpolation, and extrapolation; separation of P-waves and S-waves; separation of upgoing and downgoing waves; and wave‐field decomposition for isolation, identification, and analysis of arrivals.

Interpretation of a vertical seismic profile conducted in the Columbia Plateau basalts

Geophysics ◽  
1989 ◽  
Vol 54 (10) ◽  
pp. 1258-1266 ◽  
Author(s):  
J. Pujol ◽  
B. N. Fuller ◽  
S. B. Smithson
Keyword(s):  
Large Amplitude ◽  
Volcanic Rocks ◽  
Seismic Energy ◽  
Seismic Profile ◽  
P Wave ◽  
Low Velocity ◽  
P Waves ◽  
Seismic Reflection Data ◽  
S Waves ◽  
Columbia Plateau

Seismic reflection data are often of poor quality when recorded in areas where volcanic rocks are present at or near the surface. In order to investigate this phenomenon, a vertical seismic profiling (VSP) experiment was conducted in the Columbia Plateau basalts so that the behavior of seismic energy in subsurface volcanic rocks could be observed directly, thus giving insight into data acquisition in volcanic terrains. The lithologic section at the VSP site consists of low‐velocity (400 m/s to 900 m/s) alluvium in the uppermost 50 m, beneath which are layers of high‐velocity (about 5800 m/s), high‐density basalts interbedded with clay layers with much lower velocities (about 1700 m/s) and densities. These large velocity and density contrasts dramatically influence wave generation and propagation. In spite of the small source‐borehole offset (61 m), large‐amplitude S waves are generated by the downgoing P waves when they reach a shallow (250 m) clay‐basalt boundary. These S waves, in turn, generate strong reflected P waves when they interact with another clay layer at 500 m. On the other hand, strong primary P‐wave reflections are also present in the data but are affected by various interfering effects which reduce their amplitudes. The VSP data are also characterized by large‐amplitude reverberations caused by seismic energy trapped in the upper 250 m of the lithologic section. Reverberations are also observed in surface data recorded near the VSP site. We conclude from our analysis that volcanic rocks, at least in the Columbia Plateau, do not exhibit unusual energy transmission characteristics and that the observations can be explained in terms of the large contrast in the elastic properties of interbedded clay and basalt.

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