Function Of Ray Parameter Research Articles

Path-average kernels for long wavelength traveltime tomography

SUMMARY Although much effort goes into improving the resolution of tomographic models, investigating their quality has only just started. Probabilistic tomography provides a framework for the quantitative assessment of uncertainties of long-wavelength tomographic models. So far, this technique has been used to invert maps of surface wave phase velocities and normal-mode splitting functions. Including body waves would substantially increase the depth resolution in the lowermost mantle. In surface wave tomography, the construction of phase velocity maps and splitting functions is a well-defined inverse problem, and the depth inversion is less well constrained but characterized by a small number of dimensions suitable for a Monte Carlo search. Traveltime tomography is mostly based on ray theory that covers the 3-D Earth, thus the dimension of the inverse problem is too large for a Monte Carlo search. The ray-mode duality suggests to apply the path-average approximation to body wave traveltimes. In this way the measured traveltime residual as a function of ray parameter can be inverted using path-average kernels, which depend on depth only, similar to surface wave tomography. We investigate the validity of the path-average approximation for delay times in both the forward and the inverse problem using the velocity model S20RTS as well as random models. We numerically illustrate the precision of such kernels compared with ray-theoretic and finitefrequency ones. We further invert traveltime residuals, calculated from Fermat rays, using the path-average kernels. We find that the agreement between classical ray theory and path-average theory is good for long wavelength structures. We suggest that for mapping long wavelength structures, body waves can be inverted in two steps, similar to surface waves, where the ray parameter and the vertical traveltime play the role of frequency and phase velocity, respectively.

Upper mantle discontinuity structure beneath East Anatolian Plateau (Turkey) from receiver functions

Imaging of seismic velocity discontinuities with receiver functions suggests a complex mantle structure beneath the East Anatolian Plateau. By stacking all the receiver functions as a function of ray parameter, we examined the moveout characteristics of the data set and tested different velocity models to fit the arrival times of converted phases from the Moho and the lithosphere–asthenosphere boundary and their multiples. The best fitting model shows anomalously thin crust (40 km) and lithosphere (65 km), and low uppermost mantle velocities causing systematic delays on later discontinuity arrivals. These findings are consistent with the high plateau supported by partially molten and buoyant asthenosphere. The upper mantle discontinuities are all visible in different frequencies (0.15 and 0.4 Hz). Especially, the 410 km discontinuity does not show frequency dependence in amplitude and suggests a sharpness of 10 km or less. Spatial variations of the 410 and 660 km discontinuities reveal anticorrelated topography and distinct zones of diminished amplitude associated with a detached slab and delaminated fragments of lithospheric mantle. In the south, the slab becomes deeper and horizontally deflected towards the east suggesting westward migration of slab detachment and resistance to slab penetration at the 660 km discontinuity. In the north, the mantle transition zone is characterized by two spatially limited lithospheric fragments with complex geometry. At the center of the study area, the transition zone is thin (230 km) suggesting excess temperature of 120 °C. This warm mantle anomaly most likely is due to hot material trapped within the upper mantle between slab in the south and delaminated fragments in the north.

The application of two-parameter velocity and slowness functions in approximating seismic reflection travel times

Seismic reflection travel time–offset curves are complicated, and depend on many parameters. There is a need for simplified equations involving a reduced number of parameters that can be estimated from data and used for moveout correction and a simplified velocity model of the Earth. Many authors have derived explicit equations for travel time as a function of offset, involving several parameters which depend on velocity. Some authors have applied the approach in which velocity varies with offset (velocity with high-order terms or anisotropic properties). Here we present a new approach in which we employ velocity variation with depth instead of offset. A simple linear variation of velocity is used to derive expressions for travel time and offset as a function of ray parameter, from which the variation of travel time with offset can be obtained. Only two parameters are involved in defining the velocity–depth profile, but the resulting travel time–offset curves are good approximations. Two-parameter linear variations of slowness with depth, and velocity and slowness with depth and zero-offset travel time, have also been derived and the relationships between them are described. Comparison of the new approximations with those of Taner and Koehler (1969 Geophysics 34 859–81), and the large-offset approximations of Causse et al (2000 Geophys. Prospect. 48 763–78) was performed using 1D plane-layered models with different velocity profiles. Taner and Koehler and Causse et al approximations are inaccurate for large and small offsets respectively, while the new approximations show improved accuracy and can be applied over the whole range of offsets. The velocity and slowness expressions can also be used for moveout correction. Synthetic and real records are used to demonstrate their effectiveness.

Traveltimes for infrasonic waves propagating in a stratified atmosphere

SUMMARY The tau‐p method of Buland & Chapman (1983) is reformulated for sound waves propagating in a stratified atmosphere under the influence of a height-dependent wind velocity profile. For a given launch angle along a specified azimuth, the ray parameter is redefined to include the influence of the horizontal wind component along the direction of wave propagation. Under the assumption of negligible horizontal wind shear, the horizontal wind component transverse to the ray propagation does not aVect the direction of the wave normal, but displaces the reference frame of the moving wavefront, thus altering the observed incidence azimuth. Expressions are derived for the time, horizontal range, and transverse range of the arriving waves as a function of ray parameter. Algorithms for the location of infrasonic wave sources using the modified tau‐p formulation in conjunction with regional atmospheric wind and temperature data are discussed.

A convenient expression for the NMO velocity function in terms of ray parameter

Tsvankin (1995) derived an expression for the normal‐moveout (NMO) velocity from horizontal or dipping reflectors that is valid in symmetry planes of anisotropic media. This equation is written in terms of the phase velocity and its derivatives with respect to the phase angle with vertical. The phase angle, however, is not a seismic observable, while the ray parameter p is. Indeed, Alkhalifah and Tsvankin (1995) pointed out the importance of representing the NMO velocity function as a function of ray parameter and obtained a number of significant results via an indirect use of this variable. Subsequently, Cohen (1997) directly transformed the Tsvankin expression for the moveout velocity obtaining an expression involving the phase velocity expressed as a function of ray parameter. Applying this result to the normal moveout of P-waves in transversely isotropic media with a vertical symmetry axis (VTI media), he was able to get various approximations for the ray‐parameter form of the moveout function and thus give additional analytic insight into the success of the velocity‐analysis methodology Alkhalifah and Tsvankin.

Analytic study of the effective parameters for determination of the NMO velocity function in transversely isotropic media

In their studies of transversely isotropic media with a vertical symmetry axis (VTI media), Alkhalifah and Tsvankin observed that, to a high numerical accuracy, the normal moveout (NMO) velocity for dipping reflectors as a function of ray parameter p depends mainly on just two parameters, each of which can be determined from surface P‐wave observations. They substantiated this result by using the weak‐anisotropy approximation and exploited it to develop a time‐domain processing sequence that takes into account vertical transverse isotropy. In this study, the two‐parameter Alkhalifah‐Tsvankin result was further examined analytically. It was found that although there is (as these authors already observed) some dependence on the remaining parameters of the problem, this dependence is weak, especially in the practically important regimes of weak to moderately strong transverse isotropy and small ray parameter. In each of these regimes, an analytic solution is derived for the anisotropy parameter η required for time‐domain P‐wave imaging in VTI media. In the case of elliptical anisotropy (η = 0), NMO velocity expressed through p is fully controlled just by the zero‐dip NMO velocity—one of the Alkhalifah‐ Tsvankin parameters. The two‐parameter representation of NMO velocity also was shown to be exact in another limit—that of the zero shear‐wave vertical velociy. The analytic results derived here are based on new representations for both the P‐wave phase velocity and normal moveout velocity in terms of the ray parameter, with explicit expressions given for the cases of vanishing onaxis shear speed, weak to moderate transverse isotropy, and small to moderate ray parameter. Using these formulas, I have rederived and, in some cases, extended in a uniform manner various results of Tsvankin, Alkhalifah, and others. Examples include second‐order expansions in the anisotropy parameters for both the P‐wave phase‐velocity function and NMO‐velocity function, as well as expansions in powers of the ray parameter for both of these functions. I have checked these expansions against the corresponding exact functions for several choices of the anisotropy parameters.

Dip‐moveout processing by Fourier transform in anisotropic media

Conventional dip‐moveout (DMO) processing is designed for isotropic media and cannot handle angle‐dependent velocity. We show that Hale's isotropic DMO algorithm remains valid for elliptical anisotropy but may lead to serious errors for nonelliptical models, even if velocity anisotropy is moderate. Here, Hale's constant‐velocity DMO method is extended to anisotropic media. The DMO operator, to be applied to common‐offset data corrected for normal moveout (NMO), is based on the analytic expression for dip‐dependent NMO velocity given by Tsvankin. Since DMO correction in anisotropic media requires knowledge of the velocity field, it should be preceded by an inversion procedure designed to obtain the normal‐moveout velocity as a function of ray parameter. For transversely isotropic models with a vertical symmetry axis (VTI media), P‐wave NMO velocity depends on a single anisotropic coefficient (η) that can be determined from surface reflection data. Impulse responses and synthetic examples for typical VTI media demonstrate the accuracy and efficiency of this DMO technique. Once the inversion step has been completed, the NMO-DMO sequence does not take any more computing time than the genetic Hale method in isotropic media. Our DMO operator is not limited to vertical transverse isotropy as it can be applied in the same fashion in symmetry planes of more complicated anisotropic models such as orthorhombic.

Fowler DMO and time migration for transversely isotropic media

The main advantage of Fowler’s dip‐moveout (DMO) method is the ability to perform velocity analysis along with the DMO removal. This feature of Fowler DMO becomes even more attractive in anisotropic media, where imaging methods are hampered by the difficulty in reconstructing the velocity field from surface data. We have devised a Fowler‐type DMO algorithm for transversely isotropic media using the analytic expression for normal‐moveout velocity. The parameter‐estimation procedure is based on the results of Alkhalifah and Tsvankin showing that in transversely isotropic media with a vertical axis of symmetry (VTI) the P‐wave normal‐moveout (NMO) velocity as a function of ray parameter can be described fully by just two coefficients: the zero‐dip NMO velocity [Formula: see text] and the anisotropic parameter η (η reduces to the difference between Thomsen parameters ε and δ in the limit of weak anisotropy). In this extension of Fowler DMO, resampling in the frequency‐wavenumber domain makes it possible to obtain the values of [Formula: see text] and η by inspecting zero‐offset (stacked) panels for different pairs of the two parameters. Since most of the computing time is spent on generating constant‐velocity stacks, the added computational effort caused by the presence of anisotropy is relatively minor. Synthetic and field‐data examples demonstrate that the isotropic Fowler DMO technique fails to generate an accurate zero‐offset section and to obtain the zero‐dip NMO velocity for nonelliptical VTI models. In contrast, this anisotropic algorithm allows one to find the values of the parameters [Formula: see text] and η (sufficient to perform time migration as well) and to correct for the influence of transverse isotropy in the DMO processing. When combined with poststack F-K Stolt migration, this method represents a complete inversion‐processing sequence capable of recovering the effective parameters of transversely isotropic media and producing migrated images for the best‐fit homogeneous anisotropic model.

Velocity analysis for transversely isotropic media

The main difficulty in extending seismic processing to anisotropic media is the recovery of anisotropic velocity fields from surface reflection data. We suggest carrying out velocity analysis for transversely isotropic (TI) media by inverting the dependence of P‐wave moveout velocities on the ray parameter. The inversion technique is based on the exact analytic equation for the normal‐moveout (NMO) velocity for dipping reflectors in anisotropic media. We show that P‐wave NMO velocity for dipping reflectors in homogeneous TI media with a vertical symmetry axis depends just on the zero‐dip value [Formula: see text] and a new effective parameter η that reduces to the difference between Thomsen parameters ε and δ in the limit of weak anisotropy. Our inversion procedure makes it possible to obtain η and reconstruct the NMO velocity as a function of ray parameter using moveout velocities for two different dips. Moreover, [Formula: see text] and η determine not only the NMO velocity, but also long‐spread (nonhyperbolic) P‐wave moveout for horizontal reflectors and the time‐migration impulse response. This means that inversion of dip‐moveout information allows one to perform all time‐processing steps in TI media using only surface P‐wave data. For elliptical anisotropy (ε = δ), isotropic time‐processing methods remain entirely valid. We show the performance of our velocity‐analysis method not only on synthetic, but also on field data from offshore Africa. Accurate time‐to‐depth conversion, however, requires that the vertical velocity [Formula: see text] be resolved independently. Unfortunately, it cannot be done using P‐wave surface moveout data alone, no matter how many dips are available. In some cases [Formula: see text] is known (e.g., from check shots or well logs); then the anisotropy parameters ε and δ can be found by inverting two P‐wave NMO velocities corresponding to a horizontal and a dipping reflector. If no well information is available, all three parameters ([Formula: see text], ε, and δ) can be obtained by combining our inversion results with shear‐wave information, such as the P‐SV or SV‐SV wave NMO velocities for a horizontal reflector. Generalization of the single‐layer NMO equation to layered anisotropic media with a dipping reflector provides a basis for extending anisotropic velocity analysis to vertically inhomogeneous media. We demonstrate how the influence of a stratified anisotropic overburden on moveout velocity can be stripped through a Dix‐type differentiation procedure.

A Program for Finding Seismic Reflection Times for Layered Media

The BASIC computer program given here calculates travel times and distances as a function of ray parameter for seismic reflections in a flat layered earth. Several exercises are presented for student use with the program.

The computation of seismic travel times

We have developed a method for estimating travel times as a function of epicentral range and hypocentral depth which is faster than ray tracing or direct evaluation of ray integrals yet more compact and general than the interpolation of traditional travel-time tables. In this method, delay or intercept time (tau) as a function of ray parameter and source depth is tabulated. We show that direct manipulation of delay time yields travel time as an explicit function of range eliminating the iteration required to determine the proper ray parameter in ray tracing or the evaluation of ray integrals. Travel-time versus range tables exhibit the same explicit dependence by virtue of having eliminated the common independent variable, ray parameter. However, delay-time branches are monotonic, single-valued, and have fixed ray parameter limits. In contrast, travel-time branches may be nonmonotonic and multi-valued and generally have range limits which vary with source depth. Thus, delay-time tables are simpler to generate and interpolate than are travel-time tables and so result in travel-time estimates which are both more reliable and more precise for a given table size. Further, by retaining the explicit ray parameter dependence and hence a more direct relationship with the ray integrals, the delay-time method offers more flexibility in organizing suites of related phases and in tailoring the algorithm to meet specific computational requirements than does the travel-time table approach. Additionally, we show how the delay-time method can be extended to laterally inhomogeneous media.

ON THE CORRESPONDENCE RELATION BETWEEN NORMAL MODES AND RAYS

This paper deals with the torsional vibration of radial higher modes of a spherical earth consisting of a homogeneous elastic mantle and a liquid core with special reference to the correspondence relation between normal modes and body waves.The normal modes and the rays assigned by the mode-ray correspondence relation originally derived by Ben-Menahem are related very closely in several aspects even for the normal modes of the low radial modes having small colatitudinal order numbers. This relation connects modes and rays through the identity of the phase velocity of normal modes and the apparent velocity of body waves. Group velocities of the normal modes associated with a certain ray show nearly constant values independent of radial mode numbers, constants being a function of ray parameter. The travel time of surface waves specified by this constant group velocity gives almost the same value as that of the corresponding body wave. In addition, there are close correlations between the two radii, one to the lowermost maximum or zero point of the radial distribution of azimuthal displacement of the normal modes and the other to the deepest point of the ray.The criterion is established which assigns normal modes of minimum numbers required for the construction of respective body wave phases. Direct, reflected and diffracted SH pulses produced by a point source at zero depth are synthesized by summing up contributions from a limited number of normal modes assigned by this criterion, which reveal almost identical patterns to those obtained by the summation of contributions from a large number of modes.It is found that the separation of body wave phases is possible even from the standpoint of the normal mode theory except for pulses having mutually close ray parameters and that normal modes with very large order numbers are needed for the precise construction of pulses traveling the shallow region and that pulses penetrating to the inner region require normal modes up to much higher radial modes with relatively small order numbers.