Traveltime Equation Research Articles

Seismic Refraction Analysis of Multiple Dipping Interfaces—Revisited

Many geophysical exploration texts do not provide an accurate or complete analysis of planar multiple dipping layers with respect to the determination of seismic-refraction travel times. Often, given equations apply only to special cases and not to the general layered-earth model composed of arbitrarily dipping interfaces. As an extension of these textbook discussions, this paper presents a derivation of a general travel-time equation that not only satisfies the rule of reciprocity for any given model but also provides an accurate representation of layered-earth models when forward and reverse travel times computed for individual geophone positions are graphed as time-distance curves. The general equation is based on the use of layer thicknesses measured perpendicular to the dipping interfaces at refraction points of down-going or up-going ray-path intervals, rather than layer thicknesses measured beneath shot points as in other methods. This detailed approach is readily adaptable to computer manipulation through the use of sequential equations and can be applied to a layer-earth model composed of any number of layers.

Cartesian parametrization of anisotropic traveltime tomography

A new method for inverting P-wave traveltimes for seismic anisotropy on a local scale is presented and tested. In this analysis, direction-dependent seismic velocity is represented by a second- or fourth-order Cartesian tensor, which is shown to be equivalent to decomposing a velocity surface using a basis set of Cartesian products of unit vectors. The new inversion method for P- and S-wave anisotropy from traveltime data is based on the tensor decomposition. The formulation is formally derived from a Taylor series expansion of a continuously extended, 3-D velocity function originally defined on the surface of the unit sphere. This approach allows us to solve a linear inversion instead of the standard non-linear method. The resultant, linearized, fourth-order traveltime equation is similar to a previous fourth-order result (Chapman & Pratt 1992), although our representation offers a natural second-order simpli-fication. Conventional isotropic traveltime tomography is a special case of our tensorial representation of velocities. P-wave velocity can be represented by a second-order tensor (matrix) as a first approximation, although S-wave traveltime tomography is intrinsically fourth order because of S-wave solution duality. Differences between isotropic and anisotropic parametrizations are investigated when velocity is represented by a matrix A. The trade-off between isotropy and anisotropy in practical tomography, which differs from the fundamental deficiency of anisotropic traveltime tomography (Mochizuki 1997), is shown to be ~ 1; that is, their effects are of the same order. We conclude that anisotropic considerations may be important in velocity inversions where ray coverage is less than optimal. On the other hand, when the ray directional coverage is complete and balanced, effects of anisotropy sum to zero and the isotropic part gives the result obtained from inverting for isotropic variations of velocity alone. Synthetic test data sets are inverted, demonstrating the effectiveness of the new inversion approach. When ray coverage is fairly complete, original anisotropy is well recovered, even with random noise introduced, although anisotropy ambiguities arise where ray coverage is limited. Random noise was found to be less important than ray directional coverage in anisotropic inversions.

Velocity Analysis Using Nonhyperbolic Move-Out in Anisotropic Media of Arbitrary Symmetry: Synthetic and Field Data Studies

A robust method for estimating the interval parameters (i. e. the normal move-out velocity Vnmo and the anisotropy parameter h) of horizontally layered transversely isotropic media from reflected P-waves data has been recently proposed by Alkhalifah (1997) based on move-out equation from Tsvankin and Thomsen (1994). The method, tested on synthetic and field data, is based first on semblance analysis on nonhyberbolic (i. e. long spread) move-out for the estimation of the effective parameters, and then on a layer stripping process. Sayers and Ebrom (1997) recently proposed another nonhyperbolic traveltime equation and a corresponding interval velocity analysis which can be used for azimuthally anisotropic layered media. The method was tested on synthetic and physical model data in homogeneous anisotropic media of various symmetry. Here we propose a generalization of the method proposed by Alkhalifah, which can deal with arbitrary, but moderately (i. e. anisotropy strength of roughly 20%), anisotropic layered media. The parametrization is a natural extension of the parametrization used by the previous author and based on generalized Thomsen's parameters (Thomsen, 1986) proposed by Mensch and Rasolofosaon (1997). The method is first applied to synthetic data on a six layer model of contrasted anisotropy (type and magnitude). The robustness of the method is demonstrated. All the interval parameters (here Vnmo and the horizontal velocity Vh) are estimated with reasonable errors (typically < 2%, to be compared with the considered anisotropies of about 15 to 20%) at all azimuths. The method is also tested on field data from the North Sea including three 2D seismic lines intersecting at a well location.

Seismic traveltime analysis for azimuthally anisotropic media: Theory and experiment

Natural fractures in reservoirs, and in the caprock overlying the reservoir, play an important role in determining fluid flow during production. The density and orientation of sets of fractures is therefore of great interest. Rocks possessing an anisotropic fabric and a preferred orientation of fractures display both polar and azimuthal anisotropy. Sedimentary rocks containing several sets of vertical fractures may be approximated as having monoclinic symmetry with symmetry plane parallel to the layers if, in the absence of fractures, the rock is transversely isotropic with symmetry axis perpendicular to the bedding plane. A nonhyperbolic traveltime equation, which can be used in the presence of azimuthally anisotropic layered media, can be obtained from an expansion of the inverse‐squared ray velocity in spherical harmonics. For a single set of aligned fractures, application of this equation to traveltime data acquired at a sufficient number of azimuths allows the strike of the fractures to be estimated. Analysis of the traveltimes measured in a physical model simulation of a reverse vertical seismic profile in an azimuthally anisotropic medium shows the medium to be orthorhombic with principal axes in agreement with those given by an independent shear‐wave experiment. In contrast to previous work, no knowledge of the orientation of the symmetry planes is required. The method is therefore applicable to P‐wave data collected at multiple azimuths using multiple offset vertical seismic profiling (VSP) techniques.

Laterally Varying Velocity Estimation

Lateral velocity variations in the subsurface are a departure from the simple horizontally layered model on which the extraction of NMO velocities and interval velocities are based. Significant differences between stacking velocity and interval velocity may result in complex geological areas when the estimation is based on this horizontally layered media assumption. In this paper methods are presented of velocity estimation that are both quick and accurate in the presence of lateral velocity variations. The method developed here solves an analytic expression by fitting an unknown smooth velocity function, and also determines the reflector depth using only two simple assumptions: (1) that the ray in the laterally varying medium travels along the same path as in the homogeneous medium; and (2) that the subsurface velocity can be approximated by a smoothly varying lateral velocity function. Assumption (1) is shown to introduce only higher order correction terms to the travel time for a given source-receiver pair. It is demonstrated by assumption (2) how the use of a stable, smooth and continuous velocity function across the section results in a better migration velocity than a velocity analysis performed at an isolated CDP location. By estimating the degree of lateral velocity variation across the seismic section at an early stage in the processing sequence, a decision on whether to use pre- or post-stack migration can be made with reliable velocity information. This method has the advantage of using an analytic expression for the travel time instead of complicated ray tracing by solving differential equations. This makes it useable in an interactive velocity analysis mode. An example using field data from the North West Shelf of Australia shows how the new travel time equation can improve the velocity analysis over a fault zone.

WICK SAMPLERS

Wick samplers have the potential to improve solute sampling in the vadose zone, but more information is needed about the effect of soil-wick sampling interface disturbances on measured fluxes and solute concentrations. The objective of this paper is to examine how the wick sampler alters the matric potential, streamlines, and solute concentrations in the native soil. Both theoretical analysis and experimental studies were performed. An equation for travel time was developed that included the effect of changes in moisture content in the soil when a wick sampler is installed. Experimental measurements of pressure head near the soil-wick interface were taken, and a travel time moment analysis of narrow solute pulses was conducted for several steady flow experiments. Model and theory agreed well and showed that even when the capillary length α−1 is similar in the soil and wick, dissimilarity of the hydraulic conductivity and cross-section flow area may significantly affect soil moisture content above the soilwick interface and on the solute pulse travel time. It was also found that in many cases solute pulse travel time was affected more by the pressure head changes that occurred at the soil-wick interface than by the flow through the wick.

High resolution traveltime imaging

It is well known that traveltime tomography does not provide sufficient image quality of the earth. There are two major reasons: (1) resolution in traveltime inversion critically depends on the completeness of the recording aperture; (2) the number of independent traveltime equations is usually much less than the number of unknowns. The latter situation leads to the existence of different solutions which perfectly suit the same traveltime equations. Correct solutions cannot then be distinguished unless independent conditions (constraints) are imposed.

Temperature fluctuation and thermodynamic properties in Earth's lower mantle: An application of the complete travel time equation of state

The complete travel time equation of state (CT-EOS) enables us to obtain all the thermodynamic parameters from travel time or velocity measurements of elastic waves at simultaneously high pressure and temperature. Here we formulate the adiabatic CT-EOS and apply it to the lateral heterogeneity of seismic wave velocity in the lower mantle. We obtained internally consistent results for temperature, thermal expansivity and the Gruneisen constant in the lower mantle; at the depth (2741 km) just above the D″ layer, the most likely results are ∼ 2500 K for temperature, ∼ 1.0 × 10 −5 K −1 for thermal expansivity and ∼ 1.0 for the Gruneisen constant. We also discuss the geophysical implication of the small thermal expansivity suggested in the deep lower mantle.

Performance of the complete travel-time equation of state at simultaneous high pressure and temperature

Performance of the complete travel-time equation of state at simultaneous high pressure and temperature

Traveltime inversion of both direct and reflected arrivals in vertical seismic profile data

Traveltimes from both direct and reflected arrivals in a VSP data set (Bridenstein no. 1 well in Oklahoma) are inverted in a least‐squares sense for velocity structure. By comparing the structure from inversion to the sonic log, we conclude that the accuracy of the reconstructed velocities is greater than that found when only the direct arrivals are used. Extensive tests on synthetic VSP data confirm this observation. Apparently, the additional reflection traveltime equations aid in averaging out the traveltime errors, as well as reducing the slowness variance in reflecting layers. These results are consistent with theory, which predicts a decrease in a layer’s slowness variance with an increase in the number and length of terminating reflected rays. For the Bridenstein data set, 130 direct traveltimes and 399 primary reflection traveltimes were used in the inversion.

An analytic generalized inverse for common‐depth‐point and vertical seismic profile traveltime equations

This paper presents an analytic formula for the generalized inverse matrix associated with vertical seismic profile (VSP) and common‐depth‐point (CDP) traveltime equations. Its functional form explicitly depends upon the source‐receiver geometry and its validity is restricted to horizontally layered earth models where there is negligible ray bending. The importance of the generalized inverse is that it can be used to derive closed‐form expressions for the covariance matrix, condition number, resolution matrix, and average inverse eigenvalues for the traveltime equations; these formulas are useful in designing source‐receiver geometries which optimize velocity reconstruction by traveltime inversion. These formulas also provide a fundamental understanding of some resolution limitations associated with earth tomography experiments. Previously, resolution characteristics of tomographic inversions were usually estimated by empirical computer tests on specialized earth models. Some specific conclusions drawn from these formulas include the following: (1) The elements of the slowness covariance matrix for uncorrelated traveltime errors depend only upon the length and number of ray segments which terminate in a layer, not on the number of rays which intersect a layer. (2) Better velocity resolution in a layer is achieved by increasing the length (larger source offset) and number (more sources) of rays which terminate in that layer. (3) Decreasing the layer thickness will result in a poorly conditioned set of equations; in fact, the [Formula: see text] condition number of the covariance matrix will increase faster than [Formula: see text] as the number of layers N increases. (4) A tradeoff exists between better depth resolution and an increase in the variance of the slowness estimate. (5) The discrete generalized inverse is a back‐projection operator followed by a directional second‐order difference operator. The latter operator is a local and numerically tractable operator, while the equivalent filtering operation for the classical inverse Radon transform is a global Hilbert transform preceded by a first‐order derivative.

Numerical verification and extension of an analytic generalized inverse for common‐depth‐point and vertical‐seismic‐profile traveltime equations

This paper empirically establishes the range of validity for an analytic generalized inverse (assuming negligible ray bending) associated with one‐dimensional vertical seismic profiles (VSP) or common‐depth‐point (CDP) traveltime equations. Computer tests show that the analytic inverse closely predicts the condition number of [Formula: see text] and roughly predicts some features of the unit covariance matrix for a source offset‐to‐well depth ratio less than 1. The analytic inverse is invalid for offset‐to‐depth ratios greater than 1; i.e., when ray bending becomes severe enough to violate the assumption of negligible ray bending. To overcome the restriction of negligible ray bending, we extend the analytic inverse to traveltime equations which honor Snell’s law. Computer tests show this extended inverse is a good approximation to the actual generalized inverse. The extended inverse appears to be valid for any practical source‐receiver offset and any layered velocity structure. The important implication is that, prior to their execution, VSP or CDP experiments (over approximately one‐dimensional structures) can now be designed for optimal velocity resolution. No numerical inverses need to be computed and the condition number, covariance matrix, resolution matrix, and inverse matrix are closely approximated by analytic formulas. This development promises to allow accurate and efficient velocity analysis of VSP data or CDP gathers by least‐squares traveltime inversion.

Three‐dimensional traveltime equation for dipping layers

The traveltime equation for homogeneous layers with two‐dimensionally dipping interfaces is well known, but it has been proven only for a limited variety of profile geometries. Existing proofs do not necessarily lend themselves to use with other types of profiles, or to extension into three dimensions. When the distance that a ray travels and the elapsed time are decomposed into orthogonal components, and each interaction of the ray and the interface is examined separately, the time contributions for ray travel in the two orthogonal directions can be isolated for ray segments between horizontal planes of reference. The terms in which these contributions are expressed are independent of any part of the structure beyond the ray segment under consideration. Therefore, the total traveltime for a ray can be obtained by simple summation of such contributions. This process represents a new proof of the traveltime equation, but, more importantly, it provides a method for constructing the correct equation for virtually any profile geometry and arrival type, including direct, reflected, and refracted waves as well as converted phases. When the rays and interface normals are treated as three‐dimensional vectors, this proof and method can be extended to three dimensions. The result is a general expression describing point‐to‐point traveltimes for any source‐receiver geometry in structures comprising homogeneous layers with interfaces having arbitrary three‐dimensional dip. Since this equation accurately describes the incremental traveltime for every portion of any propagating wave, nearly any three‐dimensional geometry can be accommodated.

Tomographic inversion via the conjugate gradient method

Tomographic inversion of seismic traveltime residuals is now an established and widely used technique for imaging the Earth’s interior. This inversion procedure results in large, but sparse, rectangular systems of linear algebraic equations; in practice there may be tens or even hundreds of thousands of simultaneous equations. This paper applies the classic conjugate gradient algorithm of Hestenes and Stiefel to the least‐squares solution of large, sparse systems of traveltime equations. The conjugate gradient method is fast, accurate, and easily adapted to take advantage of the sparsity of the matrix. The techniques necessary for manipulating sparse matrices are outlined in the Appendix. In addition, the results of the conjugate gradient algorithm are compared to results from two of the more widely used tomographic inversion algorithms.

Lightning Surge Response Of Transmission Towers

A sensitivity study of HV and EHV lightning performance shows that surge response characteristics of towers and lines are as important as the footing response. Tower travel time has a very large sensitivity for taller double-circuit lines. Existing methods for analysis of tower and line response have considerable uncertainty. This uncertainty is resolved by extensions to travelling- wave analysis combined with experimental verification. New equations for tower surge impedance and tower travel time are proposed to deal with the most common situation of a midspan stroke to the skywire.

Stochastic Analysis of Groundwater Traveltime for Long-Term Repository Performance Assessment

ABSTRACTThis paper describes a method of stochastically analyzing groundwater traveltime. The method uses a Monte Carlo technique to generate a suite of random spatial fields that are subsequently input to the groundwater flow and groundwater traveltime equations. Stochastic inputs to these equations can be (1) transmissivity (or hydraulic conductivity), (2) effective thickness (or effective porosity), or (3) boundary conditions. In a transient problem, storage coefficient (or specific storage) could also be stochastically treated. Spatial correlation in the random input fields is accounted for by means of a multivariate random-number generator, which requires the first two statistical moments of these fields to be specified. The output from the Monte Carlo analysis is a suite of groundwater traveltime realizations that can be used to derive exceedance probabilities. These probabilities provide a measure of the degree of confidence in meeting set criteria.A preliminary application of this method using data from the deep basalts beneath the Hanford Site is also presented. The results illustrate how this method can be used to evaluate compliance with a technical criterion relating to groundwater traveltime.

A linear programming approach to time-term analysis

abstractA linear programming approach to time-term analysis of seismic refraction data, one that is more adaptable than previous methods, has been developed. The travel-time equations are constraints to a choice of objective functions, which produce different models that fit the data. As examples: a “best-fit” model can be obtained by minimizing the L1 norm of the misfit errors; by maximizing and minimizing the sum of the time terms, the deepest and shallowest models can be derived. In addition, the method can be adapted to allow for refractor anisotropy without any assumptions on the form of the anisotropy.Examples with sets of theoretical data illustrate the different ways in which the technique can be used. A set of travel times for the Pn phase from the northeast Pacific, previously interpreted to document compressional wave velocity anisotropy, exemplifies the application of the method.

The traveltime equation, tau‐p mapping, and inversion of common midpoint data

Seismic traveltime data can be described in terms of instantaneous slope (dT/dX) and intercept time (τ). With this parameterization, the most general form of the traveltime equation is developed and found to be valid for reflections and refractions, all source‐receiver offsets, and all commonly used experimental geometries. For common source‐receiver experiments, the instantaneous slope of the traveltime data represents the true horizontal ray parameter p, and the intercept time is the sum along the ray of thickness‐vertical slowness products referenced to the fixed surface station. For common midpoint (CMP) experiments, we show that the instantaneous slope is the average slowness of upgoing and downgoing rays at the surface, and the intercept time is again the sum of thickness‐vertical slowness products, but referenced to the common midpoint. This inherent averaging immediately explains the superiority of the CMP experiment in the presence of dip. In addition to providing a general traveltime equation, we find that τ-p parameterization reorganizes traveltime data in a way which greatly facilitates interpretation. For reflections, the τ-p trajectories are sums of ellipses and all turning rays (whether refracted or postcritically reflected) combine to form a well defined and easily identified trajectory. We present a method for deriving a velocity‐depth function from this trajectory which is applicable to both refracted and reflected arrivals.

Reply by the authors to discussion by H. Pascal and D. Rankin

The interesting comment by Pascal and Rankin discusses, inter alia, the physical basis of a equation proposed by Schlumberger. This equation expresses formation traveltimes (e.g., inverse phase velocities) derived from measurements made by Schlumberger’s electromagnetic propagation tool (EPT) as the volume weighted sum of the traveltimes of the formation constituents. Pascal and Rankin interpret this equation as implying a layered medium consisting of layers of different dielectric properties. There is no doubt that the Schlumberger traveltime equation is correct for such a layered medium if the direction of wave propagation is normal to the direction of the layering planes. This does not preclude the possibility, however, that the Schlumberger equation may provide an adequate description, at microwave frequencies, of the composite traveltimes of more complex heterogeneous systems.

Models for transportation level of service

The paper describes the development of models for predicting travel times of door to door trips for both transit and automobile trips. The models for access times have a distribution associated with them and permit a truly disaggregate assignment of travel time components. The equations for linehaul travel time of the highway using modes (bus, auto) are volume dependent and can thus be used in equilibrating travel demand and level of service. All the models are related directly to transportation policy options—changing bus line spacings, bus headways, number of (priority) lanes, etc.—and translate the effects of such policies into specific values of the level-of-service attributes without the need to code networks and run paths. The use of the models in a practical application is also discussed.