Amplitude responses of thin beds: Sinusoidal approximation versus Ricker approximation

Geophysics ◽  
1995 ◽  
Vol 60 (1) ◽  
pp. 223-230 ◽  
Author(s):  
Hai‐Man Chung ◽  
Don C. Lawton
Keyword(s):  
Numerical Modeling ◽  
Normal Incidence ◽  
Reflection Coefficients ◽  
Amplitude Dependence ◽  
Dominant Wavelength ◽  
Amplitude Response ◽  
Analytical Expression ◽  
Ricker Wavelet ◽  
Analytical Expressions

The amplitude response of a thin bed with arbitrary upper and lower normal incidence reflection coefficients is studied. Two analytical expressions for the normal incidence amplitude response as a function of the thickness are derived and are both valid for weak reflectivities and for thicknesses below [Formula: see text], where [Formula: see text] is the dominant wavelength. The first expression is based on the substitution of a cosine wave for the source wavelet, and the second is based directly on the analytical expression for a Ricker wavelet. The results calculated from these two expressions are compared to numerical modeling results for a Ricker wavelet for several models. We found that the differences between the two expressions are small, and both are good approximations. Above the [Formula: see text] thickness, the percentage differences increase rapidly for both expressions, implying that the thin‐bed assumptions in both derivations break down rapidly beyond the [Formula: see text] thickness. Below the [Formula: see text] thickness, except in the case where the two reflection coefficients are equal in magnitude but opposite in sign, the amplitude dependence on the thickness is nonlinear.

scholarly journals Filtering of a Ricker wavelet induced by anelastic seismic wave propagation and reflection

2020 ◽  
Vol 17 (5) ◽  
pp. 838-851
Author(s):  
Stephan Ker ◽  
Yves Le Gonidec
Keyword(s):  
Seismic Wave ◽  
Normal Incidence ◽  
Peak Frequency ◽  
Seismic Wave Propagation ◽  
Cumulative Effects ◽  
Seismic Attribute ◽  
Analytical Expression ◽  
Ricker Wavelet ◽  
Analytical Quantification ◽  
Q Factors

Abstract A varying Q factor with depth induces modifications of seismic wave features due to anelastic propagation but also reflections at the discontinuities. Standard signal analysis methods often neglect the reflection contribution when assessing Q values from seismic data. We have developed an analytical quantification of the cumulative effects of both the propagation and reflection contributions by considering Kjartansson's model and a seismic plane wave at normal incidence on a step-like discontinuity. We show that the cumulative effects are equivalent to a frequency filter characterised by a bandform and phase that both depend on the ratio between the elastic and anelastic contrasts. When considering this filter applied to a Ricker wavelet, we establish an analytical expression of the peak-frequency attribute as a function of propagation and reflection properties. We demonstrate that this seismic attribute depends on the anelastic contrast, which cannot be neglected when assessing Q factors: the error in Q estimate is not linearly dependent on the anelastic contrast and we establish an analytical expression for the case where this contrast is weak. An unexpected phenomenon for a step-like interface is an increase in the peak frequency that is observed when the anelastic and elastic contrasts have opposite signs, with a constraint on the anelastic propagation properties. This behaviour allows for assessment of the elastic and anelastic parameters.

scholarly journals Strains analysis at the wave propagation throught he contact layer of the elastic solids

2020 ◽  
pp. 89-96
Author(s):  
N.B. Chertova ◽  
◽  
Yu.V. Grinyaev ◽  
Keyword(s):  
Wave Propagation ◽  
Strain Distribution ◽  
Strain State ◽  
Strain Analysis ◽  
Contact Layer ◽  
Reflection Coefficients ◽  
Block Medium ◽  
Elastic Solids ◽  
Elastic Parameters ◽  
Analytical Expressions

The stress-strain state on the interface of the elastic solids is investigated. The studied interface presents a contact layer which is characterized by dimension and the set of physics mechanical parameters. The models of layered and block medium are used for the description this boundary. In the framework of these models the problem of elastic wave propagation through the interface is considered. Analytical expressions for the refraction and reflection coefficients allowing us to determine the strains on the interface and strains distribution in the contact layer are found. Corresponding strains amplitudes depending on the layer thickness are calculated at the different elastic parameters of contacting solids and boundary. The strain laws on the interface which is described by the layered and block medium models are analyzed. The regions of equivalent use these models are determined in the case of strain analysis on the boundary and the strain distribution in the contact layer.

On the Frequency and Amplitude Dependence of the Payne Effect: Theory and Experiments

2003 ◽  
Vol 76 (2) ◽  
pp. 533-547 ◽  
Author(s):  
A. Lion ◽  
C. Kardelky ◽  
P. Haupt
Keyword(s):  
Time Scale ◽  
Time Derivative ◽  
Amplitude Dependence ◽  
Deformation History ◽  
Payne Effect ◽  
Physical Time ◽  
Starting Point ◽  
Effect Theory ◽  
Fractional Time Derivative ◽  
Analytical Expressions

Abstract In this paper, we develop a physical approach to represent the Payne effect which is observed in filler-reinforced elastomers. The starting point for the constitutive model is the well-known theory of linear viscoelasticity, where the stress is a linear functional of the deformation history. Since the corresponding relations are unable to describe any kind of amplitude dependence, we introduce a nonlinearity into the model. To this end, we replace the physical time, t, by a modified time scale, z, which is a functional of the deformation history. This approach was originally introduced by Valanis in the context of rate-independent plasticity and applied in a modified form; for example, by Haupt and Sedlan to represent process-dependent relaxation properties of rubber. The modified time, z, is a monotonic function of the physical time, t, and can be interpreted as an intrinsic time scale of the material. The rate of this time scale is non-negative and depends on the process history. We propose a constitutive relation for the variable z(t), which is driven by a fractional time derivative of the deformation, calculate analytical expressions for the storage and dissipation moduli and show that such phenomena as the frequency and the amplitude dependence, observed in experiments, are well represented.

REFLECTION COEFFICIENTS AND BOTTOM LOSSES AT NORMAL INCIDENCE COMPUTED FROM PACIFIC SEDIMENT PROPERTIES

Geophysics ◽  
1970 ◽  
Vol 35 (6) ◽  
pp. 995-1004 ◽  
Author(s):  
Edwin L. Hamilton
Keyword(s):  
Normal Incidence ◽  
Silty Sand ◽  
Reflection Coefficients ◽  
Silty Clay ◽  
Sediment Properties ◽  
Compressional Waves ◽  
Sea Floor ◽  
Sediment Types ◽  
The Pacific ◽  
Clayey Silt

Rayleigh reflection coefficients and bottom losses of compressional waves at normal incidence on the water‐sediment interface are computed with values of density and velocity measured in sea‐floor sediment samples; main sediment types in three major environments of the Pacific and adjacent areas are included. Some typical average computed values of acoustic bottom loss at normal incidence in db are (1) continental shelf: sands, 8; silty sand, 10; sandy silt, 14; silty clay, 16; (2) abyssal plain: clayey silt, 17; silty clay and clay, 21; and (3) abyssal hill: silty clay and clay, 17. Comparisons with actual measurements at sea by several investigators demonstrate the validity of the approach.

Numerical modeling and interpretation of drillstring waves

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1569-1581 ◽  
Author(s):  
Flavio Poletto ◽  
Massimo Malusa ◽  
Francesco Miranda
Keyword(s):  
Signal Processing ◽  
Numerical Modeling ◽  
Boundary Conditions ◽  
Transmission Line ◽  
Reflection Coefficients ◽  
Drill Bit ◽  
Noise Sources ◽  
Numeric Modeling ◽  
Joint Contribution ◽  
Group Velocities

Drill‐bit seismic and pilot seismograms contain drillstring periodicities from signal and environmental noise sources. These coherent components are similar in seismic‐while‐drilling correlations, and their joint contribution may cause distortions in signal processing. Numeric modeling of the drillstring transmission line is used to correctly interpret the axial and torsional events with similar propagating modes of signal and noise. The reflection coefficients are computed for drillstrings of arbitrary materials, and their average mechanical features are related to the axial and torsional group velocities obtained by the dispersion equations. Fitting of the periodical components in the real seismograms is used to automatically correct the drill‐bit pilot delays and to estimate the surface and downhole boundary conditions.

Theory of electrochemical impedance of surface reactions: second-harmonic and large-amplitude response

1997 ◽  
Vol 75 (11) ◽  
pp. 1508-1517 ◽  
Author(s):  
David A. Harrington
Keyword(s):  
Hydrogen Evolution ◽  
Hydrogen Evolution Reaction ◽  
Electrochemical Impedance ◽  
Second Harmonic ◽  
Surface Reactions ◽  
Nonlinear Effects ◽  
Amplitude Dependence ◽  
Adsorbed Species ◽  
Amplitude Response ◽  
Potential Perturbation

The theory for the electrochemical impedance of surface reactions involving a single adsorbed species is presented. A new methodology is used, in which many harmonics are considered, and the differential equations are reduced to algebraic matrix equations. The amplitude of the ac potential perturbation is not assumed to be small, and nonlinear effects are taken into account. The amplitude dependence of the impedance and the second-harmonic response are investigated. The quasi-reversible electrosorption reaction and the hydrogen evolution reaction are considered in detail, assuming that the adsorbed species obeys the Langmuir isotherm. Keywords: electrochemistry, impedance, adsorption, hydrogen evolution reaction, second harmonic.

High Multiple Radio Reflections from the F2 Layer of the Ionosphere at Brisbane

1954 ◽  
Vol 7 (1) ◽  
pp. 165 ◽  
Author(s):  
K Baird
Keyword(s):  
Normal Incidence ◽  
Reflection Coefficients ◽  
Fixed Frequency ◽  
Night Time ◽  
Variable Gain

Continuous night-time records of multiple F2 reflections at normal incidence have been made at a fixed frequency. The echo patterns have been classified, and qualitative explanations given in terms of humped ionization contours, extending the work of Pierce and Mimno (1940). These patterns have been studied also by a variable gain technique. It is concluded that accurate measurements of reflection coefficients cannot be made by this means.

Three-qubit Groverian measure

2008 ◽  
Vol 8 (10) ◽  
pp. 925-942
Author(s):  
E.-L. Jung ◽  
M.-R. Hwang ◽  
D. Park ◽  
L. Tamaryan ◽  
S. Tamaryan
Keyword(s):  
Geometric Interpretation ◽  
Qubit State ◽  
Phase Factor ◽  
Analytical Expression ◽  
Near Future ◽  
Analytical Expressions ◽  
Qubit States ◽  
Local Unitary Invariance ◽  
Unitary Invariance

The Groverian measures are analytically computed in various types of three-qubit states. The final results are also expressed in terms of local-unitary invariant quantities in each type. This fact reflects the manifest local-unitary invariance of the Groverian measure. It is also shown that the analytical expressions for various types have correct limits to other types. For some types (type 4 and type 5) we failed to compute the analytical expression of the Groverian measure in this paper. However, from the consideration of local-unitary invariants we have shown that the Groverian measure in type 4 should be independent of the phase factor $\varphi$, which appear in the three-qubit state $|\psi \rangle$. This fact with geometric interpretation on the Groverian measure may enable us to derive the analytical expressions for general arbitrary three-qubit states in near future.

Fast analytical models for texture evolution in anisotropic polycrystals

2020 ◽  
Author(s):  
Neil Ribe ◽  
Olivier Castelnau ◽  
Neil Goulding ◽  
Ralf Hielscher ◽  
Andrew Walker ◽  
...  
Keyword(s):  
Finite Strain ◽  
Seismic Anisotropy ◽  
Texture Evolution ◽  
Analytical Models ◽  
Mantle Flow ◽  
Analytical Expression ◽  
Arbitrary Amplitude ◽  
Individual Grains ◽  
Weighting Coefficients ◽  
Analytical Expressions

<p>To use observations of seismic anisotropy to constrain mantle flow patterns, we need a model for how progressive deformation of a rock leads to preferred orientation (CPO) of its constituent crystals. An important class of such models comprises so-called `self-consistent' (SC) models such as VPSC (viscoplastic SC) and SOSC (second-order SC). However, calculations based on SC models are far too costly for use in 3-D time-dependent convection simulations. To overcome this difficulty, we have developed two new analytical models that combine the accuracy of SC models with a greatly enhanced (by orders of magnitude) computational efficiency. The basis of our new models is the discovery that the crystallographic spin predicted by SC models as a function of crystal orientation is always a generalized spherical harmonic of degree 2, regardless of the CPO of the aggregate. This fact allows us to find an analytical expression for the spin to within an arbitrary amplitude, which we then determine by fitting to the predictions of the SOSC model.  Our first new model, ANPAR, uses the analytical expression for the spin to calculate evolving CPO in an aggregate comprising many (typically 2000) individual grains. The resulting CPO is visually indistinguishable from the SOSC predictions, but is ~ 50000 times faster to compute. Our second model, SBFTEX, is based on a more economical representation of CPO as a weighted sum of a small number of  analytical `structured basis functions' (SBFs), each of which represents the virtual CPO that would be produced by one intracrystalline slip system acting alone. The model consists of analytical expressions for the weighting coefficients of the SBFs as functions of the finite strain experienced by the aggregate. While somewhat less accurate than ANPAR, SBFTEX is ~ 2000 times faster, or ~ 10<sup>8</sup> times faster than SOSC. We will illustrate the predictions of ANPAR and SBFTEX for pure olivine polycrystals, a simple model for the upper 400 km of the mantle.</p>

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