Two-dimensional PP/PS-stereotomography: P- and S-waves velocities estimation from OBC data

We present the extension of stereotomography to P - and S -wave velocity estimation from PP - and PS -reflected/diffracted waves. In this new context, we greatly benefit from the use of locally coherent events by stereotomography. In particular, when applied to S -wave velocity estimation from PS -data, no pairing of PP - and PS -events is a priori required. In our procedure the P -wave velocity model is obtained first using stereotomography on PP -arrivals. Then the S -wave velocity model is obtained using PS -stereotomography on PS -arrivals fixing the P -wave velocity model. We present an application to an 'ideal' synthetic data set demonstrating the relevance of the approach, which allows us to recover depth consistent P - and S -waves velocity models even if no pairing of PP - and PS -events is introduced. Finally, results to a real data set from the Gulf of Mexico are presented demonstrating the potential of the method in a noisy data context.     

Non-linear Bayesian joint inversion of seismic reflection coefficients

Inversion of seismic reflection coefficients is formulated in a Bayesian framework. Measured reflection coefficients and model parameters are assigned statistical distributions based on information known prior to the inversion, and together with the forward model uncertainties are propagated into the final result. This enables a quantification of the reliability of the inversion. Quadratic approximations to the Zoeppritz equations are used as the forward model. Compared with the linear approximations the bias is reduced and the uncertainty estimate is more reliable. The differences when using the quadratic approximations and the exact expressions are minor. The solution algorithm is sampling based, and because of the non-linear forward model, the Metropolis–Hastings algorithm is used. To achieve convergence it is important to keep strict control of the acceptance probability in the algorithm. Joint inversion using information from both reflected PP waves and converted PS waves yields smaller bias and reduced uncertainty compared to using only reflected PP waves.     

Determination of Moho dip using PS reflections

Summary. A clear PS reflection was recorded in the northern section of the LISPB line (shot-point N1, profile ALPHA). Travel times of PS and PP phases reflected at the same part of the Moho were used to model the lower crustal structure. The observed high apparent velocity of the PS phase (8.4 km/s) is interpreted as due to a localized Moho dip of 7 per cent (4°) towards the north. This is supported by observations of Pn refractions from other shots. It is shown that PS reflections could be very useful for determination of Moho dips.     

3-D seismic velocities calculated from lattice-preferred orientation and reflectivity of a lower crustal section: examples of the Val Sesia section (Ivrea zone, northern Italy)

To quantify the seismic properties of lower crustal rocks and to better constrain the origin of the lower crustal seismic reflectivity, we determined the complete 3-D seismic properties of a lower crustal section. Eight representative samples of the main lithologic and structural units outcropping in the Val Sesia (Ivrea zone) were studied in detail. The seismic velocities were calculated using the single crystal stiffness coefficients and the lattice preferred orientation (LPO) of each mineral in all samples. The 21 stiffness coefficients characterizing the elastic behaviour of each rock are determined. Mafic and ultramafic rocks such as pyroxenite and pyroxene-bearing gabbros display complex shear wave properties. These rocks are weakly birefringent (maximum 0.1 kms⁻¹) and it is difficult to find consistent relationships between the seismic properties and the rock structure. On the other hand, seismic properties of deformed felsic rocks are essentially controlled by mica. They display strong S -wave birefringence (0.3 km s⁻¹) and relatively high V _p anisotropy (7.6 per cent). Amphibole also strongly influences the rock birefringence patterns. For both kind of rocks, the foliation is highly birefringent and the fast polarized shear wave is systematically oriented parallel to the foliation. We show that the number of mineral phases in the rock strongly controls the anisotropy. The seismic anisotropy has a complex role in the P -wave reflectivity. Compared to the isotropic case, anisotropy enhances the reflection coefficient for about 60 per cent of the possible lithological interfaces. For 40 per cent of the interfaces, the reflection coefficient is much lower when one considers the medium as anisotropic.     

Asymptotic edge-and-vertex diffraction theory

Uniformly asymptotic formulae for edge-and-vertex diffraction in the time-domain, involving elementary functions of time, traveltimes and GTD amplitudes, are derived. Explicit expressions for diffraction at a pyramid and a triangle are constructed. They can be applied to the numerical calculation of a field reflected and diffracted at 3-D objects with sharp edges and to reflection from triangulated surfaces. the computational cost is very low.     

Seismic reflection and transmission coefficients of a self-similar interface

The derivation of seismic reflection and transmission coefficients is generally based on the assumption that the medium parameters behave as step functions of depth, at least in a finite region around the interface. However, outliers observed in well logs generally behave quite differently from step functions. In this paper we represent an interface by a self-similar singularity, embedded between two homogeneous half-spaces, and we derive its frequency-dependent normal-incidence reflection and transmission coefficients. For ω  → 0 the expressions for the coefficients reduce to those for a discrete boundary between two homogeneous half-spaces; for ω → ∞ they become frequency-independent. These asymptotic expressions have a relatively simple form and depend on the singularity exponent α .
  The exact as well as the asymptotic expressions are used to evaluate the time-domain reflection and transmission responses of a self-similar interface. Finally, we use a numerical method to model the response of a smoothed version of a self-similar interface (note that the velocity of a smoothed singularity remains finite). It turns out that smoothing has hardly any effect on the response, provided that the smoothing does not affect the scales corresponding to the seismic frequency range.     

Asymptotic eigenfrequencies of the Earth's normal modes

We derive asymptotic formulae for the toroidal and spheroidal eigenfrequencies of a SNREI earth model with two discontinuities, by considering the constructive interference of propagating SH and P-SV body waves. For a model with a smooth solid inner core, fluid outer core and mantle, there are four SH and 10 P-SV ray parameters regimes, each of which must be examined separately. The asymptotic eigenfrequency equations in each of these regimes depend only on the intercept times of the propagating wave types and the reflection and transmission coefficients of the waves at the free surface and the two discontinuities. If the classical geometrical plane-wave reflection and transmission coefficients are used, the final eigenfrequency equations are all real. In general, the asymptotic eigenfrequencies agree extremely well with the exact numerical eigenfrequencies; to illustrate this, we present comparisons for a crustless version of earth model 1066A.     

电导法测定LiCl在水溶液中的活度系数

应用电导法测定了LiCl水溶液的活度系数,首先在288~308 K温度范围内测定LiCl在极性水溶剂中的电导率,利用公式计算LiCl的摩尔电导率,应用Debye-Hücker和Osager-falkenhangen公式计算LiCl在水中的活度系数,并讨论了温度和浓度对LiCl水溶液活度系数的影响。     

The explicit crustal transfer function for SH-waves by ray theory — the case of one and two layers

Summary. Previous theoretical studies on the transfer function of crustal plane surface layers have been primarily based on applying recursion formulae to the solutions of wave equations. In this way the detailed physical insight of the transfer function is often obscure and it is very difficult to express the transfer function in an explicit form. To show that this situation can be improved by an alternative approach, we demonstrate in this paper with models of one and two surface layers that the explicit transfer function can be derived by summing multiple-reflected rays between interfaces. Since the derived transfer function is related explicitly to some physical parameters, such as wave travel times between interfaces, reflection and transmission coefficients at the interfaces, and attenuation and dispersion of waves in each layer of the model, it is more flexible than the traditional recursion form when applied to different models. This suggests that the ray theory can play an important role in problems of layered media.     

Seismic reflection traveltimes in two-dimensional statistically anisotropic random media

Velocity estimation remains one of the main problems when imaging the subsurface with seismic reflection data. Traveltime inversion enables us to obtain large-scale structures of the velocity field and the position of seismic reflectors. However, as the media currently under study are becoming more and more complex, we need to know the finer-scale structures. The problem is that below a certain range of velocity heterogeneities, deterministic methods become difficult to use, so we turn to a probabilistic approach. With this in view, we characterize the velocity field as a random field defined by its first and second statistical moments. Usually, a seismic random medium is defined as a homogeneous velocity background perturbed by a small random field that is assumed to be stationary. Thus, we make a link between such a random velocity medium (together with a simple reflector) and seismic reflection traveltimes. Assuming that the traveltimes are ergodic, we use 2-D seismic reflection geometry to study the decrease in the statistical traveltime fluctuations as a function of the offset (the source–receiver distance). Our formulae are based on the Rytov approximation and the parabolic approximation for acoustic waves. The validity and the limits are established for both of these approximations in statistically anisotropic random media. Finally, theoretical inversion procedures are developed for the horizontal correlation structure of the velocity heterogeneities for the simplest case of a horizontal reflector. Synthetic seismograms are then computed (on particular realizations of random media) by simulating scalar wave propagation via finite difference algorithms. There is good agreement between the theoretical and experimental results.     

The fine structure of the normal incidence synthetic seismogram

Summary . The spectral function of a perfectly elastic, horizontally stratified medium has been demonstrated previously to provide an attractive formulation to describe the properties of the one-dimensional synthetic seismogram (Robinson & Treitel). Here we examine the mathematical framework of the Model in still greater detail. Knowledge of this fine structure of the synthetic seismogram leads to the solution of two particular seismic inverse problems. First, we consider a layered medium with an arbitrary surface reflection coefficient c _o, where | c _o|<1, and which contains an impulsive source immediately above the surface. Given the corresponding synthetic seismogram, we develop an inverse, or backward recursion formalism which recovers the entire series of original reflection coefficients. Second, we consider a similar problem for an impulsive source located just below the surface. Both inversion procedures constitute a continuation of the work of Goupillaud and of Sherwood & Trorey, and represent a generalization of the classical technique originally proposed by Kunetz which, however, only holds for the marine case, c_o =±1. The present approach is not so constrained and thereby becomes applicable to land seismograms as well.
If products of third or higher order in the reflection coefficients can be neglected, significant simplifications arise in the theory. In that event the usual representation of the synthetic seismogram as a ratio of two polynomials in the complex variable z becomes particularly revealing. The numerator polynomial is then approximately equal to the z transform of the reflection coefficient series, while the denominator polynomial is approximately equal to the z transform of the autocorrelation of these reflection coefficients. The resulting simplified theory affords important computational savings in the appropriate backward recursion algorithms.     

On the relative scattering of P- and S-waves

Summary. Using a single scattering approximation, we derive equations for the scattering attenuation coefficients of P- and S -body waves. We discuss our results in the light of some recent energy renormalization approaches to seismic wave scattering. Practical methods for calculating the scattering attenuation coefficients for various earth models are emphasized. The conversions of P - to S -waves and S- to P -waves are included in the theory. The earth models are assumed to be randomly inhomogeneous, with their properties known only through their average wavenumber power spectra. We approximate the power spectra with piecewise constant functions, each segment of which contributes to the net, frequency-dependent, scattering attenuation coefficient. The smallest and largest wavenumbers of a segment can be plotted along with the wavevectors of the incident and scattered waves on a wavenumber diagram. This diagram gives a geometric interpretation for the frequency behaviour associated with each spectral segment, including a 'transition' peak that is due entirely to the wavenumber limits of the segment. For regions of the earth where the inhomogeneity spectra are concentrated in a band of wavenumbers, it should be possible to observed such a peak in the apparent attenuation of seismic waves. We give both the frequency and distance limits on the accuracy of the theoretical results.     

First-order P-wave ray synthetic seismograms in inhomogeneous weakly anisotropic media

We propose approximate equations for P -wave ray theory Green's function for smooth inhomogeneous weakly anisotropic media. Equations are based on perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. For evaluation of the approximate Green's function, earlier derived first-order ray tracing equations and in this paper derived first-order dynamic ray tracing equations are used.
The first-order ray theory P -wave Green's function for inhomogeneous, weakly anisotropic media of arbitrary symmetry depends, at most, on 15 weak-anisotropy parameters. For anisotropic media of higher-symmetry than monoclinic, all equations involved differ only slightly from the corresponding equations for isotropic media. For vanishing anisotropy, the equations reduce to equations for computation of standard ray theory Green's function for isotropic media. These properties make the proposed approximate Green's function an easy and natural substitute of traditional Green's function for isotropic media.
Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of the approximate Green's function on inhomogeneity of the medium. Accuracy depends more strongly on strength of anisotropy in general and on angular variation of phase velocity due to anisotropy in particular. For example, for anisotropy of about 8 per cent, considered in the examples presented, the relative errors of the geometrical spreading are usually under 1 per cent; for anisotropy of about 20 per cent, however, they may locally reach as much as 20 per cent.     

The mean transmission properties of a fault with imperfect facial contact

A model of a fault where the two faces do not exactly conform is characterized by a distribution of approximately circular contacts; elsewhere, the faces are stress-free. This contrasts with most earlier models, which have assumed the contact geometry to be equivalent to a plane distribution of approximately circular cracks. The contact regions in the present model are taken to be sparsely distributed, and averaged interface conditions are derived that are accurate to second order. At lowest order they agree with established formulae for the normal stiffness of non-conforming surfaces. These averaged, or mean, conditions are expected to hold at wavelengths long compared with the radii and spacing distance of the contact points. Unsurprisingly, they are equivalent to the continuity conditions for a thin elastic layer whose properties are given here in terms of the parameters of the contact surface.     

Computation of high-frequency seismic wavefields in 3-D laterally inhomogeneous anisotropic media

Summary. An algorithm for the computation of travel times, ray amplitudes and ray synthetic seismograms in 3-D laterally inhomogeneous media composed of isotropic and anisotropic layers is described. All 21 independent elastic parameters may vary within the anisotropic layers. Rays and travel times are evaluated by numerical solution of the ray tracing equations. Ray amplitudes are determined by evaluating reflection/ transmission coefficients and the geometrical spreading along individual rays. The geometrical spreading is computed approximately by numerical measurement of the cross-sectional area of the ray tube formed by three neighbouring rays. A similar approximate procedure is used for the determination of the coefficients of the paraxial ray approximation. The ray paraxial approximation makes computation of synthetic seismograms on the surface of the model very efficient. Examples of ray synthetic seismograms computed with a program package based on the described algorithm are presented.     

Polarization and amplitude attributes of reflected plane and spherical waves

The characteristics of a reflected spherical wave at a free surface are investigated by numerical methods; in particular, the polarization angles and amplitude coefficients of a reflected spherical wave are studied. The classical case of the reflection of a plane P wave from a free surface is revisited in order to establish our terminology, and the classical results are recast in a way which is more suited for the study undertaken. The polarization angle of a plane P wave, for a given angle of incidence, is shown to be 90° minus twice the angle of reflection of the reflected S wave. For a Poisson's ratio less than 1/3, there is a non-normal incident angle for which both amplification coefficients are 2 precisely; for this incident angle the direction of the particle motion at the free surface is also the direction of the incident wave. For a wave emanating from a spherical source, the polarization angle, for all angles of incidence, is always less than, or equal to, the polarization angle of a plane P wave. The vector amplification coefficient of a spherical wave, for all angles of incidence, is always greater than the vector amplification coefficient of a plane P wave. As expected, the results for a spherical wave approach the results for a plane P wave in the far field. Furthermore, there was a good agreement between the theoretical modelling and the numerical modelling using the dynamic finite element method (DFEM).     

The effect of a second-order velocity discontinuity on elastic waves near their turning point

Summary. The rather abrupt changes in velocity gradient which have sometimes been proposed, notably in the upper mantle and near the base of the mantle, have an effect equivalent to that of one or more second-order discontinuities, where partial reflection occurs due to a change in curvature of the wavefront across these discontinuities. The effect is ignored in the classical WKBJ approximation to the wave functions, but it can be explicitly demonstrated by applying the extended WKBJ method (Langer's approximation) to a piecewise smooth layered model. For the purpose of this study it is convenient to represent the response of such a modelby a generalized reflection coefficient. For a model of one or a system of several second-order discontinuities (approximating a change in velocity gradient over a finite depth interval), the reflection coefficient can be perhaps surprisingly large for long-period waves near their turning point. It is shown that this effect can significantly alter the amplitude decay of SH waves diffracted around the core, in models where a change in velocity gradient near the core—mantle boundary constitutes a low-velocity zone at the base of the mantle; such models have recently been proposed. With the same velocity gradients, the effect on P diffraction is less important. The results for SH diffraction in these models support the conclusion that a small amplitude decay must be explained by a velocity decrease with depth, i.e. a low-velocity zone at the base of the mantle.     

Guided wave propagation in laterally varying media - I. Theoretical development

Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed to describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution of the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH -waves, P - SV -waves and full anisotropy.
The reflected and transmitted wavefields for a region of heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields to those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able to derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg - and Sn -waves in a multilayered model for which there are 19 coupled modes.
The method is applicable to three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse to the propagation path, leads to a net rate of increase of the transverse components of the seismogram at the expense of the other components.     

The S reflector west of Galicia: the seismic signature of a detachment fault

The so-called S reflector is a group of bright, continuous reflections underlying the landward-tilted fault blocks of the west Galicia rifted margin, S has been interpreted as the brittle-ductile transition, the lop of an intrusion, a detachment fault, and the crust-mantle boundary. To constrain the internal structure of the reflector, we have carried out detailed analyses of these reflections. We compare the waveforms of the seafloor reflection and its first multiple, both to determine the amplitude of the seafloor reflection and to show (hat the seafloor is in effect a spike in the reflectivity series so that the seafloor reflection can be used as the far-field wavelet, including both source and receiver ghosts. We compare (he waveform of the seafloor and 5 and show that, within the resolution of our data, S is a reflection from a step increase in acoustic impedance. This result is confirmed through complex trace analysis, and in particular the determination of the apparent polarity of S, and the examination of the instantaneous frequency function: S is consistently positive polarity, and shows no significant frequency anomaly. Simple modelling shows that S is very unlikely to be a reflection from a thin layer. We thus conclude that S is probably a single steplike interface. From the varying frequency content of the data, we determine a value for the effective Q between S and the seafloor, and use this to assess the amplitude loss due to attenuation and scattering. We use a comparison between the seafloor and the S reflection to constrain the amplitude of S, and estimate a reflection coefficient for S of at least 0.2 in places, decreasing landwards. By analogy with structures developed in the highly extended regions of the western United States, we consider that the most likely interpretation of S is as a sharp west-dipping detachment fault separating a 'granitic' upper plate from a higher-velocity lower plate, locally probably serpentinized mantle.