scholarly journals A Low-Order Series Approximation of Thin-Bed PP-Wave Reflections

2019 ◽  
Vol 9 (4) ◽  
pp. 709 ◽  
Author(s):  
Chun Yang ◽  
Yun Wang ◽  
Jun Lu ◽  
Benchi Chen ◽  
Lei Shi
Keyword(s):  
Approximate Formula ◽  
Expansion Method ◽  
Wave Impedance ◽  
P Wave ◽  
Reflection Coefficients ◽  
Single Interface ◽  
Series Approximation ◽  
Amplitude Versus Offset ◽  
Small Incidence ◽  
Pp Wave

The study of thin-bed seismic phenomena is important in crustal, exploration and engineering seismology. Presently, seismic reflectivity theories based on single-interface assumption are widely used though they are only suitable for thick deposits. Thin-bed reflectivity theories are established on complex propagator matrices and are difficult to be applied to reveal thin-bed properties directly. Therefore, an approximation of thin-bed PP-wave reflection coefficients (RPP) is derived in this paper. First, the relationship between thin-bed RPP and incidence angles is analyzed through series expansion method. For PP-wave, its reflection coefficients are even power series functions of sine incidence angles. Then, for small incidence, RPP of the thin bed is further simplified into a second-order series approximation with respect to the sine incidence angles. Simulations and accuracy analyses of the approximate formula show that approximation errors are smaller than 5% as the incidence angles smaller than 20 degrees. Based on this approximate formula, an approach is given for estimating thin-bed properties including P-wave impedance ratios and thickness. The estimation approach is applied in properties estimation of a thin bed model. Perfect performances of the model example show the future potentiality of the approximate formula in thin-bed Amplitude-Versus-Offset (AVO) analysis and inversion.

scholarly journals P-Wave Reflection Approximation of a Thin Bed and Its Application

2020 ◽  
Vol 10 (22) ◽  
pp. 8061
Author(s):  
Chun Yang ◽  
Yun Wang ◽  
Shu Xiong ◽  
Zikun Li ◽  
Hewei Han
Keyword(s):  
Approximate Formula ◽  
Synthetic Data ◽  
Analytic Solutions ◽  
P Wave ◽  
Seismic Exploration ◽  
Inversion Method ◽  
Analytic Approximation ◽  
Zoeppritz Equations ◽  
Single Interface ◽  
Amplitude Versus

“Thin-bed” reservoirs have become important targets of seismic exploration and exploitation. However, traditional amplitude versus offset/amplitude versus angle (AVO/AVA) technologies, for example, those based on Zoeppritz equations and their approximations for a single interface, are not sufficiently accurate for thin-bed stratigraphy. Analytic solutions of thin-bed reflectivity may become practical for thin-bed AVO analysis and inversion. Therefore, a linear analytic approximation of thin-bed P-wave reflectivity is developed under small-incidence and thin-bed assumptions. Numerical simulations show that the amplitude approximation errors are usually smaller than 10% for incidence angles less than 20 degrees, and the thin-bed thicknesses are less than one-tenth of the P-wave wavelength. Based on the least-squares approach, the inversion strategy is proposed using the approximate formula. A synthetic data test shows that the proposed inversion method can produce more accurate thin-bed properties than that based on the Zoeppritz equations, which reveals the potential of the inversion method based on the linear analytic approximate formula in the fine characterization of thin reservoirs.

PP-wave reflection coefficients in weakly anisotropic elastic media

Geophysics ◽  
1998 ◽  
Vol 63 (6) ◽  
pp. 2129-2141 ◽  
Author(s):  
Václav Vavryuk ◽  
Ivan Peník
Keyword(s):  
Reflection Coefficient ◽  
Approximate Formula ◽  
Wave Reflection ◽  
Transversely Isotropic ◽  
Reflection Coefficients ◽  
Transversely Isotropic Media ◽  
Isotropic Media ◽  
Pp Wave ◽  
Axis Of Symmetry ◽  
Amplitude Versus

Approximate PP-wave reflection coefficients for weak contrast interfaces separating elastic, weakly transversely isotropic media have been derived recently by several authors. Application of these coefficients is limited because the axis of symmetry of transversely isotropic media must be either perpendicular or parallel to the reflector. In this paper, we remove this limitation by deriving a formula for the PP-wave reflection coefficient for weak contrast interfaces separating two weakly but arbitrarily anisotropic media. The formula is obtained by applying the first‐order perturbation theory. The approximate coefficient consists of a sum of the PP-wave reflection coefficient for a weak contrast interface separating two background isotropic half‐spaces and a perturbation attributable to the deviation of anisotropic half‐spaces from their isotropic backgrounds. The coefficient depends linearly on differences of weak anisotropy parameters across the interface. This simplifies studies of sensitivity of such coefficients to the parameters of the surrounding structure, which represent a basic part of the amplitude‐versus‐offset (AVO) or amplitude‐versus‐azimuth (AVA) analysis. The reflection coefficient is reciprocal. In the same way, the formula for the PP-wave transmission coefficient can be derived. The generalization of the procedure presented for the derivation of coefficients of converted waves is also possible although slightly more complicated. Dependence of the reflection coefficient on the angle of incidence is expressed in terms of three factors, as in isotropic media. The first factor alone describes normal incidence reflection. The second yields the low‐order angular variations. All three factors describe the coefficient in the whole region, in which the approximate formula is valid. In symmetry planes of weakly anisotropic media of higher symmetry, the approximate formula reduces to the formulas presented by other authors. The accuracy of the approximate formula for the PP reflection coefficient is illustrated on the model with an interface separating an isotropic half‐space from a half‐space filled by a transversely isotropic material with a horizontal axis of symmetry. The results show a very good fit with results of the exact formula, even in cases of strong anisotropy and strong velocity contrast.

An exact anelastic model for the free-surface relfection of P and S-I waves

1989 ◽  
Vol 79 (3) ◽  
pp. 842-859
Author(s):  
R. D. Borcherdt ◽  
G. Glassmoyer
Keyword(s):  
Free Surface ◽  
Volumetric Strain ◽  
Major Axis ◽  
P Wave ◽  
Reflection Coefficients ◽  
S Wave ◽  
Low Loss ◽  
Surface Reflection ◽  
Pierre Shale ◽  
Homogeneous Wave

Abstract Exact anelastic solutions incorporating inhomogeneous waves are used to model numerically S-I and P waves incident on the free surface of a low-loss anelastic half-space. Anelastic free-surface reflection coefficients are computed for the volumetric strain and displacement components of inhomogeneous wave fields. For the problem of an incident homogeneous S-I wave in Pierre shale, the largest strain and displacement amplitudes for the reflected P wave occur at angles of incidence for which the particle motion for the reflected inhomogeneous P wave is elliptical (minor/major axis = 0.6), the specific absorption (QP−1) is greater (300 per cent) and the velocity is less (25 per cent) than those for a corresponding homogeneous P wave, the direction of phase propagation is not parallel to the free surface, and the amplitude of the wave shows a significant increase with depth (6 per cent in one wavelength). Energy reflection coefficients computed for this low-loss anelastic model show that energy flow due to interaction of the incident and reflected waves reach maxima (30 per cent of the incident energy) near large but nongrazing angles of incidence. For the problem of an incident homogeneous P wave in Pierre shale, the inhomogeneity of the reflected S wave is shown not to contribute to significant variations in wave field characteristics over those that would be expected for a homogeneous wave.

P-wave backazimuth anomalies observed by a small-aperture seismic array at Pinyon Flat, southern California: Implications for structure and source location

1996 ◽  
Vol 86 (2) ◽  
pp. 470-476 ◽  
Author(s):  
Cheng-Horng Lin ◽  
S. W. Roecker
Keyword(s):  
Southern California ◽  
Seismic Array ◽  
P Wave ◽  
Small Aperture ◽  
Data Set ◽  
Single Interface ◽  
Seismic Networks ◽  
Dense Seismic Array ◽  
Beamforming Technique

Abstract Seismograms of earthquakes and explosions recorded at local, regional, and teleseismic distances by a small-aperture, dense seismic array located on Pinyon Flat, in southern California, reveal large (±15°) backazimuth anomalies. We investigate the causes and implications of these anomalies by first comparing the effectiveness of estimating backazimuth with an array using three different techniques: the broadband frequency-wavenumber (BBFK) technique, the polarization technique, and the beamforming technique. While each technique provided nearly the same direction as a most likely estimate, the beamforming estimate was associated with the smallest uncertainties. Backazimuth anomalies were then calculated for the entire data set by comparing the results from beamforming with backazimuths derived from earthquake locations reported by the Anza and Caltech seismic networks and the Preliminary Determination of Epicenters (PDE) Bulletin. These backazimuth anomalies have a simple sinelike dependence on azimuth, with the largest anomalies observed from the southeast and northwest directions. Such a trend may be explained as the effect of one or more interfaces dipping to the northeast beneath the array. A best-fit model of a single interface has a dip and strike of 20° and 315°, respectively, and a velocity contrast of 0.82 km/sec. Application of corrections computed from this simple model to ray directions significantly improves locations at all distances and directions, suggesting that this is an upper crustal feature. We confirm that knowledge of local structure can be very important for earthquake location by an array but also show that corrections computed from simple models may not only be adequate but superior to those determined by raytracing through smoothed laterally varying models.

Nonlinear two‐dimensional elastic inversion of multioffset seismic data

Geophysics ◽  
1987 ◽  
Vol 52 (9) ◽  
pp. 1211-1228 ◽  
Author(s):  
Peter Mora
Keyword(s):  
Wave Equation ◽  
Wave Velocity ◽  
A Priori ◽  
Gaussian Model ◽  
Wave Impedance ◽  
P Wave ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Gradient Direction ◽  
P Wave Velocity

The treatment of multioffset seismic data as an acoustic wave field is becoming increasingly disturbing to many geophysicists who see a multitude of wave phenomena, such as amplitude‐offset variations and shearwave events, which can only be explained by using the more correct elastic wave equation. Not only are such phenomena ignored by acoustic theory, but they are also treated as undesirable noise when they should be used to provide extra information, such as S‐wave velocity, about the subsurface. The problems of using the conventional acoustic wave equation approach can be eliminated via an elastic approach. In this paper, equations have been derived to perform an inversion for P‐wave velocity, S‐wave velocity, and density as well as the P‐wave impedance, S‐wave impedance, and density. These are better resolved than the Lamé parameters. The inversion is based on nonlinear least squares and proceeds by iteratively updating the earth parameters until a good fit is achieved between the observed data and the modeled data corresponding to these earth parameters. The iterations are based on the preconditioned conjugate gradient algorithm. The fundamental requirement of such a least‐squares algorithm is the gradient direction which tells how to update the model parameters. The gradient direction can be derived directly from the wave equation and it may be computed by several wave propagations. Although in principle any scheme could be chosen to perform the wave propagations, the elastic finite‐ difference method is used because it directly simulates the elastic wave equation and can handle complex, and thus realistic, distributions of elastic parameters. This method of inversion is costly since it is similar to an iterative prestack shot‐profile migration. However, it has greater power than any migration since it solves for the P‐wave velocity, S‐wave velocity, and density and can handle very general situations including transmission problems. Three main weaknesses of this technique are that it requires fairly accurate a priori knowledge of the low‐ wavenumber velocity model, it assumes Gaussian model statistics, and it is very computer‐intensive. All these problems seem surmountable. The low‐wavenumber information can be obtained either by a prior tomographic step, by the conventional normal‐moveout method, by a priori knowledge and empirical relationships, or by adding an additional inversion step for low wavenumbers to each iteration. The Gaussian statistics can be altered by preconditioning the gradient direction, perhaps to make the solution blocky in appearance like well logs, or by using large model variances in the inversion to reduce the effect of the Gaussian model constraints. Moreover, with some improvements to the algorithm and more parallel computers, it is hoped the technique will soon become routinely feasible.

scholarly journals Seismic AVOA Inversion for Weak Anisotropy Parameters and Fracture Density in a Monoclinic Medium

2020 ◽  
Vol 10 (15) ◽  
pp. 5136 ◽  
Author(s):  
Zijian Ge ◽  
Shulin Pan ◽  
Jingye Li
Keyword(s):  
Fractured Rock ◽  
Synthetic Data ◽  
Transversely Isotropic ◽  
P Wave ◽  
Inversion Method ◽  
Fracture Density ◽  
Weak Anisotropy ◽  
Amplitude Versus Offset ◽  
Porosity And Permeability ◽  
Transverse Isotropic

In shale gas development, fracture density is an important lithologic parameter to properly characterize reservoir reconstruction, establish a fracturing scheme, and calculate porosity and permeability. The traditional methods usually assume that the fracture reservoir is one set of aligned vertical fractures, embedded in an isotropic background, and estimate some alternative parameters associated with fracture density. Thus, the low accuracy caused by this simplified model, and the intrinsic errors caused by the indirect substitution, affect the estimation of fracture density. In this paper, the fractured rock of monoclinic symmetry assumes two non-orthogonal vertical fracture sets, embedded in a transversely isotropic background. Firstly, assuming that the fracture radius, width, and orientation are known, a new form of P-wave reflection coefficient, in terms of weak anisotropy (WA) parameters and fracture density, was obtained by substituting the stiffness coefficients of vertical transverse isotropic (VTI) background, normal, and tangential fracture compliances. Then, a linear amplitude versus offset and azimuth (AVOA) inversion method, of WA parameters and fracture density, was constructed by using Bayesian theory. Tests on synthetic data showed that WA parameters, and fracture density, are stably estimated in the case of seismic data containing a moderate noise, which can provide a reliable tool in fracture prediction.

Convolutional neural network for seismic impedance inversion

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R869-R880 ◽  
Author(s):  
Vishal Das ◽  
Ahinoam Pollack ◽  
Uri Wollner ◽  
Tapan Mukerji
Keyword(s):  
Rock Physics ◽  
Normal Incidence ◽  
Wave Impedance ◽  
Training Data ◽  
P Wave ◽  
Computation Method ◽  
Elastic Model ◽  
Spatial Correlations ◽  
Data Set ◽  
Earth Models

We have addressed the geophysical problem of obtaining an elastic model of the subsurface from recorded normal-incidence seismic data using convolutional neural networks (CNNs). We train the network on synthetic full-waveform seismograms generated using Kennett’s reflectivity method on earth models that were created under rock-physics modeling constraints. We use an approximate Bayesian computation method to estimate the posterior distribution corresponding to the CNN prediction and to quantify the uncertainty related to the predictions. In addition, we test the robustness of the network in predicting impedances of previously unobserved earth models when the input to the network consisted of seismograms generated using: (1) earth models with different spatial correlations (i.e. variograms), (2) earth models with different facies proportions, (3) earth models with different underlying rock-physics relations, and (4) source-wavelet phase and frequency different than in the training data. Results indicate that the predictions of the trained network are susceptible to facies proportions, the rock-physics model, and source-wavelet parameters used in the training data set. Finally, we apply CNN inversion on the Volve field data set from offshore Norway. P-wave impedance [Formula: see text] inverted for the Volve data set using CNN showed a strong correlation (82%) with the [Formula: see text] log at a well.

scholarly journals Hadronic Paschen-Back effect in P-wave charmonia under strong magnetic fields

2019 ◽  
Vol 49 ◽  
pp. 1960002
Author(s):  
Sachio Iwasaki ◽  
Makoto Oka ◽  
Kei Suzuki ◽  
Tetsuya Yoshida
Keyword(s):  
Magnetic Fields ◽  
Mass Spectra ◽  
Constituent Quark ◽  
Expansion Method ◽  
Wave Functions ◽  
P Wave ◽  
Strong Magnetic Fields ◽  
Mixing Ratios ◽  
Angular Momenta ◽  
Quark Models

The Hadronic Paschen-Back effect (HPBE) is a new phenomenon induced by the interplay between finite orbital angular momenta of hadrons and external strong magnetic fields. We review the HPBE in P-wave charmonia and show its mass spectra, deformed wave functions, and mixing ratios, which are evaluated by the constituent quark models in a magnetic field and the cylindrical Gaussian expansion method.

Quantitative synthesis of head and foreleg arterial pulses in the dog

1965 ◽  
Vol 208 (5) ◽  
pp. 968-983 ◽  
Author(s):  
John W. Remington
Keyword(s):  
Pulse Height ◽  
Pressure Change ◽  
Wave Impedance ◽  
Reflection Coefficients ◽  
Reflected Waves ◽  
Basic Assumptions ◽  
Junction Points ◽  
Quantitative Synthesis ◽  
In Series ◽  
Arterial Pulses

The carotid and the subclavian arterial-systems were treated as though composed of a series of homogeneous distensible tubes joined in series or in parallel. With minor corrections, the contour of the aortic arch pulse was taken as that of a single transient incident wave entering the proximal end of these systems. Pulse height changes were localized to the junctions between coupled tubes of dissimilar wave impedance values. The amount of pressure change at these junctions was calculated by comparing arterial pulses recorded from various sites in these systems with the aortic pulse. An attempt was made to keep a strict accounting of all reflected waves arising either from the junction points or from peripheral drainage beds. The pulse constructions provided evidence for appreciable junctional positive reflections which alter the contour of the peripheral pulses. Because returning reflected waves from these sites were reflected negatively at each junction, their amplitude on entering the aorta was relatively small. The good-to-excellent reconstructions of the actual pulse contours suggest that the method of analysis used has promise, and that the basic assumptions on which the calculation of reflection coefficients was based did not impose errors of critical importance.

Seismic reflectivity of hydraulic fractures approximated by thin fluid layers

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. T79-T87 ◽  
Author(s):  
A. Oelke ◽  
D. Alexandrov ◽  
I. Abakumov ◽  
S. Glubokovskikh ◽  
R. Shigapov ◽  
...  
Keyword(s):  
Analytical Solutions ◽  
Fluid Layer ◽  
Synthetic Data ◽  
P Wave ◽  
Hydraulic Fractures ◽  
Reflection Coefficients ◽  
S Wave ◽  
Reservoir Properties ◽  
Thin Fluid ◽  
Theoretical Results

We have analyzed the angle-dependent reflectivity of microseismic wavefields at a hydraulic fracture, which we modeled as an ideal thin fluid layer embedded in an elastic, isotropic solid rock. We derived full analytical solutions for the reflections of an incident P-wave, the P-P and P-S reflection coefficients, as well as for an incident S-wave, and the S-S and S-P reflection coefficients. The rather complex analytical solutions were then approximated and we found that these zero-thickness limit approximations are in good agreement with the linear slip model, representing a fracture at slip contact. We compared the analytical solutions for the P-P reflections with synthetic data that were derived using finite-difference modeling and found that the modeling confirmed our theoretical results. For typical parameters of microseismic monitoring by hydraulic fracturing, e.g., a layer thickness of [Formula: see text] and frequencies of [Formula: see text], the reflection coefficients depend on the Poisson’s ratio. Furthermore, the reflection coefficients of an incident S-wave are remarkably high. Theoretical results suggested that it is feasible to image hydraulic fractures using microseismic events as a source and to solve the inverse problem, that is, to interpret reflection coefficients extracted from microseismic data in terms of reservoir properties.

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