Reflection Traveltimes Research Articles

Inversion for elliptically anisotropic velocity using VSP reflection traveltimes

ABSTRACTThis paper presents a traveltime inversion approach, using the reflection traveltimes from offset VSP data, to reconstruct the horizontal and vertical velocities for stratified anisotropic media. The inverse problem is reduced to a set of linear equations, and solved by the singular value decomposition (SVD) technique. The validity of this inversion scheme is verified using two sets of synthetic data simulated using a finite‐difference method, one for an isotropic model and the other for an elliptically anisotropic model. The inversion result demonstrates that our anisotropic velocity inversion scheme may be applied to both isotropic and anisotropic media. The method is finally applied to a real offset VSP data set, acquired in an oilfield in northwestern China.

Anisotropic semblance analysis and NMO corrections for long-offset data

Sedimentary rocks can be regarded as transversely isotropic (TI) media. P-wave reflections from horizontal reflectors in transversely isotropic media have non-hyperbolic moveout. One difficulty in seismic processing is how to flatten such reflection events for long-offset data. For a multi-layered model, the reflection moveout formula is usually expressed as a Taylor series with higher order terms ignored. Alkhalifah and Tsvankin (1995) developed a three-term Taylor-series formula to calculate reflection travel times from a horizontal reflector in TI media with a vertical symmetry axis (VTI). Using this formula, NMO correction works well for short spread lengths, but not so well for long spreads.Zhang and Uren (2001a, b) developed an approximate explicit analytical P-wave ray velocity function for transversely isotropic (TI) media. From this ray velocity function, a reflection travel-time formula for a horizontal reflector in TI media was derived. This formula can be used for anisotropic NMO correction. It works well for both large offsets and small offsets. In order to obtain the unknown parameters required for seismic processing, a 3D-semblance analysis technique has been developed. We tested this method on numerical data from TI models with a single horizontal reflector and from multiple horizontal reflector models. The method was also tested on an isotropic model with multiple horizontal reflectors. The results show that the events can be completely flattened even for very large offsets, and that both multi-layered TI and isotropic models may appear to be anisotropic.

Segment-scale variations in the crustal structure of 150-300 kyr old fast spreading oceanic crust (East Pacific Rise, ) from wide-angle seismic refraction profiles

Summary We have simultaneously inverted seismic refraction and wide-angle Moho reflection traveltimes for the 2-D crustal thickness and velocity structure of 150–300 kyr old crust along the East Pacific Rise (EPR) between the Siqueiros and Clipperton fracture zones (FZs). Our results show a strong correlation between ridge segmentation and upper- and mid-crustal seismic velocities, with higher velocities near segment centres and lower velocities near segment ends. Low crustal velocities at the Clipperton and Siqueiros FZs are interpreted as fracturing resulting from brittle deformation of the crust in the transform domain. A relict overlap basin left on the Pacific Plate by the 9°03′N overlapping spreading centre (OSC) as it propagated southward is associated with a large (∼1 km s−1), negative upper- and mid-crustal velocity anomaly. This anomaly is consistent with the presence of an unusually thick extrusive section within the basin and with tectonic alteration, fracturing and shearing arising from rotation of the basin as it was formed. The discordant zone left by this OSC on the Cocos Plate is characterized by moderately low crustal velocities, probably because of crustal fracturing as the OSC propagated into older crust. Higher crustal velocities near segment centres may reflect a higher ratio of dikes to extrusives in the upper crust, and lower-intensity tectonic alteration of the crust, than near segment ends. The mean crustal thickness along the EPR between the Siqueiros and Clipperton FZs is 6.7–6.8 km. The thickest crust is found beneath the Lamont seamounts (∼9 km), and in a southward-pointing, V-shaped band located just north of the off-axis trace of the 9°03′N OSC (7.3–7.8 km). The thinnest crust (<6 km) is found proximal to the Clipperton and Siqueiros FZs. The crust associated with the off-axis trace of the 9°03′N OSC is not anomalously thin, suggesting that magma supply beneath the OSC is similar to that of the northern and southern segments. We see a similar pattern of crustal thickness variation to that determined using multichannel reflection data, including a gradual thickening of the crust from north to south along the northern ridge segment, and the location of the thickest crust just north of the 9°03′N OSC. However, the magnitude of the along-axis crustal thickness variation we observe along the northern ridge segment between 9°50′N and 9°15′N(∼1.3–1.8 km, excluding the Lamont seamounts) is significantly less than the 2.3 km of variation previously reported, weakening the case for the existence of a low-density mantle diapir at 9°50′N inferred from gravity data. The band of thick crust located just north of the off-axis trace of the 9°03′N OSC suggests a close genetic link between this feature and the OSC. Thus we attribute the pattern of crustal thickness variations along the northern segment to the kinematics of the southward-propagating 9°03′N OSC over the past 0.5 Myr, and not to along-axis melt migration away from a mantle diapir as previously proposed.

PP + PS = SS

Converted (PS) waves can provide important information about shear‐wave velocity and, in the presence of anisotropy, about the medium parameters responsible for both P‐ and S‐wave propagation. Kinematics and amplitudes of reflected PS‐waves, however, possess such undesirable features as moveout asymmetry, reflection point dispersal, and polarity reversal, which preclude application of conventional velocity‐analysis methods to mode conversions.Rather than using PS‐wave kinematics directly, here we propose a method for reconstructing SS‐wave reflection traveltimes from PP and PS data. The required preprocessing steps include picking of PP and PS traveltimes on prestack data and identification (correlation) of the PP and PS events from the same interfaces. The key idea of the method is to match the reflection slopes (horizontal slownesses) on common‐receiver PP and PS gathers. This procedure allows us to find the coordinates of receivers that record PP‐ and PS‐waves reflected at exactly the same (albeit unknown) subsurface points and to determine the shear‐wave reflection traveltime tSSas a simple combination of the PP and PS traveltimes. The reconstructed SS‐wave moveout can then be processed by velocity‐analysis methods designed for pure reflection modes. The developed technique, however, cannot be used to compute the true amplitudes of the reflected SS‐waves.Our method has the following attractive features:1) No information about the velocity field or anisotropic parameters is required to obtain SS‐wave traveltimes from PP and PS data. If the input PP and PS arrivals are picked correctly and correspond to the same reflector, the method produces exact traveltimes tSS.2) The estimates of traveltimes and reflection slopes are local, which makes reflection‐point dispersal irrelevant.3) Another consequence of the local nature of this procedure is that the portion of PS data in the vicinity of the polarity reversal (where the PS amplitudes are small) can be muted out without compromising the quality of tSSestimates for source–receiver pairs with high PS amplitudes.

The joint nonhyperbolic moveout inversion of PP and PS data in VTI media

Nonhyperbolic moveout of P‐waves in horizontally layered transversely isotropic media with a vertical symmetry axis (VTI) can be used to estimate the anellipticity coefficient η in addition to the NMO velocity Vnmo,P. Those two parameters are sufficient for time processing of P‐wave data (despite a certain instability in the inversion for η), but they do not constrain the vertical velocity VP0and the depth scale of the model. It has been suggested in the literature that this ambiguity in the depth‐domain velocity analysis for layer‐cake VTI media can be resolved by combining long‐spread reflection traveltimes of P‐waves and mode‐converted PSV‐waves.Here, we show that reflection traveltimes of horizontal PSV events help to determine the ratio of the P‐ and S‐wave vertical velocities and the NMO velocity of SV‐waves, and they give a more accurate estimate of η. However, nonhyperbolic moveout of PSV‐waves turns out to be mostly controlled by wide‐angle P‐wave traveltimes and does not provide independent information for the inversion. As a result, even for a single‐layer model and uncommonly large offsets, traveltimes of P‐ and PSV‐waves cannot be inverted for the vertical velocity and anisotropic parameters ε and δ. To reconstruct the horizontally layered VTI model from surface data, it is necessary to combine long‐spread traveltimes of pure P and SV reflections.

Divergent and asymptotic nature of the time‐offset Taner–Koehler series in reflection seismics

An extension of the hyperbolic approximation for reflection traveltime (Dix, 1955), in the context of isotropic and horizontally layered media, leads to the time-offset power series (Taner and Koehler, 1969)

Seismic imaging of complex structures by non-linear traveltime inversion of dense wide-angle data: application to a thrust belt

We have used a dense wide-angle data set to test a two-step procedure for the separate inversion of first-arrival and reflection traveltimes. Data were collected in a complex thrust belt environment (southern Italy) along a 14-km line, with closely spaced sources (60 m) and receivers (90 m). We have applied a fully non-linear tomographic technique, specially designed to image complex structures, to over 6400 first-arrival traveltimes in order to determine a detailed velocity model. A bi-cubic spline velocity model parametrization is used. The inversion strategy follows a multiscale approach, and employs a non-linear velocity optimization scheme. The tomographic velocity model is adopted as the background reference medium for a subsequent interface inversion aimed at imaging a target upper-crust reflector. The interface inversion method is also based on a multiscale approach and uses a non-linear technique for model parameters (interface position nodes) optimization. We have applied the interface inversion method to over 1600 reflection traveltimes of a target event picked both in the near- and in the wide-angle offset range. The retrieved interface is well resolved in the central part of the model, where ray coverage mainly includes clear post-critical reflections and the background velocity model is accurate in depth thanks to large offset deep turning rays. The velocity and interface models thus determined are consistent with Vertical Seismic Profiling data and correlate well with the geometry of known geological structures. This study shows that the used inversion approach is efficient for target-orientated investigations in complex geological environments.

Joint inversion of PP and PS reflection data for VTI media: A North Sea case study

Despite the significant advantages of combining PP and PS reflection data in anisotropic parameter estimation, application of this approach has been hindered by the inherent complexity of PS‐wave moveout. To overcome this problem, V. Grechka and I. Tsvankin suggested a model‐independent procedure to construct the traveltimes of pure SS‐wave reflections from PP and PS data. Here, we apply their method and anisotropic multicomponent stacking‐velocity tomography to a 2‐D line acquired over the lower Tertiary Siri reservoir in the North Sea.The computed traveltimes of SS reflections from several horizons are used to estimate the effective SS‐wave normal‐moveout (NMO) velocities, which are combined with the NMO velocities of PP‐waves in the tomographic velocity analysis. Comparison of the vertical and NMO velocities of the PP‐ and SS‐waves provides clear evidence of anisotropy in the section above the reservoir. The interval parameter estimation is performed under the assumption that the section is composed of transversely isotropic layers with a vertical symmetry axis (VTI media). Since the subsurface structure is close to horizontally layered, the reflection data cannot be uniquely inverted for the VTI parameters without additional information (e.g., the vertical velocities found from borehole data).The parameter‐estimation algorithm produces a family of equivalent VTI models that fit the PP and PS (or SS) traveltimes equally well. Although the range of variations in ε and δ for the equivalent models is rather wide, it does not include isotropic media (ε = δ = 0), which implies that accurate matching of both PP and PS data is impossible without accounting for anisotropy. To overcome the nonuniqueness in the inversion of reflection data and build a VTI depth model, we use P‐wave check shots acquired in the only borehole drilled on the processed line. Throughout the section ε &gt; δ, with the largest values of both anisotropic coefficients observed in the depth range 0.7–1.5 km where ε reaches almost 0.25. Time sections of the original PS data computed using the VTI model have a much higher quality than the conventional isotropic sections.

A simple, exact solution for the P‐SV wave conversion point via prestack migration

The conversion point (CP) is the lateral position of the reflection point, relative to the source-receiver midpoint, for a laterally homogeneous earth (horizontal reflectors) and the P-SV converted-wave (C-wave) geometry depicted in Figure 1. The source emits a P-wave, which propagates to the horizontal reflector where it reflects as an SV-wave and then propagates to the receiver. Common conversion point (CCP) binning and stacking are tools that use CP modeling for processing C-wave data. Applications include iterative statics and velocity analysis procedures. Acquisition design methods also use CP modeling (see Thomsen, 1999). Calculation of the CP is complicated, unfortunately, and exact solutions exist only for homogeneous and isotropic media. Thomsen (1999) developed asymptotic expansions that approximate the CP in vertically inhomogeneous isotropic and anisotropic media. Yuan and Li (2001) extended these expressions, improving their accuracy for large source-receiver offsets. Exact solutions for the CP exist only in simple media, and that is the subject of this paper. Tessmer and Behle (1988) used Fermat9s principle to find a quartic equation for the exact CP in homogeneous, isotropic media. I show that the equation for the C-wave prestack migration impulse response yields a cubic equation for the exact CP in homogeneous, isotropic media. Analytically solving the cubic equation is less cumbersome than analytically solving the quartic equation, and I have obtained the exact solution of the new cubic equation in simple trigonometric form. The Tessmer and Behle (1998) quartic CP equation depends upon the reflector depth, whereas the new cubic CP equation depends instead upon the two-way reflection traveltime of the C-wave. Thus, the new result directly specifies the CP for each sample of a prestack seismic trace (assuming reflectors are horizontal).

2-D crustal structure of Changbaishan-Tianchi volcanic region determined by seismic traveltime inversion

2-D velocity structure and tectonics of the crust and upper mantle is revealed by inversion of seismic refraction and wide-angle reflection traveltimes acquired along the profile L1 in the Changbaishan-Tianchi volcanic region. It is used in this study that seismic traveltime inversion for simultaneous determination of 2-D velocity and interface structure of the crust and upper mantle. The result shows that, under Changbaishan-Tianchi crater, there exists a low-velocity body in the shape of an inverted triangle, and the crustal reflecting boundaries and Moho all become lower by a varying margin of 2–6 km, forming a crustal root which is assumed to be the Changbaishan-Tianchi volcanic system. Finally, we make a comparison between our 2-D velocity model and the result from the studies by using trial-and-error forward modeling with SEIS83.

Explicit Anisotropic P-wave Ray Velocity Functions

Sedimentary rocks encountered in seismic exploration frequently exhibit directional seismic wave velocity dependence. These anisotropic rocks often can be regarded as transversely isotropic (TI) media in which the ray velocity surface of SH-waves is elliptical, while those of P- and SV-waves are non-elliptical. Transversely isotropic phase velocity expressions exist for P-, SH- and SV-waves. Unfortunately, until now no explicit exact analytical ray velocity functions exist for P- or SV-waves propagating in TI media. In exploration seismology, explicit ray velocity functions are needed for efficient migration, velocity analysis and other seismic processing procedures.Approximate explicit analytical P-wave ray velocity functions in transversely isotropic media have now been developed. These ray velocity functions include a new dimensionless constant, A, which depends on elastic parameters C11, C33, C44, and C13 or Thomsen's parameters ε, δ, α0 and (β0. The accuracy of the functions has been tested for all the materials published by Thomsen (1986). From these ray velocity functions, the travel time between any two points in homogeneous transversely isotropic media may be calculated. Reflection travel times from a single horizontal reflector overlain by transversely isotropic media with vertical symmetrical axis (VTI) are also calculated and compared with those obtained by Tsvankin (1996). The results show that the proposed explicit ray velocity functions have high accuracy.

New evidence of Tasmania's tectonic history from a novel seismic experiment

In March 1995, 44 land‐based recorders were deployed throughout Tasmania, SE Australia, to record seismic energy from an encircling array of marine normal‐incidence reflection shot lines. We invert refraction and wide‐angle reflection traveltimes for crustal structure, with the principal outcome being a map of the Tasmanian Moho. Key tectonic inferences from this map include: (1) the Arthur Lineament metamorphic belt in NW Tasmania overlies a major change in crustal thickness (over 5 km) and probably represents the NW limit of deformation in Tasmania during the Mid‐Late Cambrian Tyennan Orogeny, (2) thickening of the crust beneath central northern Tasmania may be associated with the juxtaposition of the Eastern and Western Tasmania Terranes during the Mid‐Devonian Tabberabberan Orogeny, and (3) the difference in crustal thickness between the east and west coasts reflects the presence of differing strain regimes during the Cretaceous break‐up of Gondwana.

Interval velocity and thickness estimate from wide‐angle reflection data

A method to estimate interval velocities and thickness in a horizontal isotropic layered medium from wide‐angle reflection traveltime curves is presented. The method is based on a relationship between the squared reflection traveltime differences and the squared offset differences relative to two adjacent reflectors. The envelope of the squared‐time versus offset‐difference curves, for rays with the same ray parameter, is a straight line, whose slope is the inverse of the square of the interval velocity and whose intercept is the square of the interval time. The method yields velocity and thickness estimates without any knowledge of the overlying stratification. It can be applied to wide‐angle reflection data when either information on the upper crust and/or refraction control on the velocity is not available. Application to synthetic and real data shows that the method, used together with other methods, allows us to define a reliable 1D starting model for estimating a depth profile using either ray tracing or another technique.

2D inversion of refraction traveltime curves using homogeneous functions

A method using simple inversion of refraction traveltimes for the determination of 2D velocity and interface structure is presented. The method is applicable to data obtained from engineering seismics and from deep seismic investigations. The advantage of simple inversion, as opposed to ray‐tracing methods, is that it enables direct calculation of a 2D velocity distribution, including information about interfaces, thus eliminating the calculation of seismic rays at every step of the iteration process. The inversion method is based on a local approximation of the real velocity cross‐section by homogeneous functions of two coordinates. Homogeneous functions are very useful for the approximation of real geological media. Homogeneous velocity functions can include straight‐line seismic boundaries. The contour lines of homogeneous functions are arbitrary curves that are similar to one another. The traveltime curves recorded at the surface of media with homogeneous velocity functions are also similar to one another. This is true for both refraction and reflection traveltime curves. For two reverse traveltime curves, non‐linear transformations exist which continuously convert the direct traveltime curve to the reverse one and vice versa. This fact has enabled us to develop an automatic procedure for the identification of waves refracted at different seismic boundaries using reverse traveltime curves. Homogeneous functions of two coordinates can describe media where the velocity depends significantly on two coordinates. However, the rays and the traveltime fields corresponding to these velocity functions can be transformed to those for media where the velocity depends on one coordinate. The 2D inverse kinematic problem, i.e. the computation of an approximate homogeneous velocity function using the data from two reverse traveltime curves of the refracted first arrival, is thus resolved. Since the solution algorithm is stable, in the case of complex shooting geometry, the common‐velocity cross‐section can be constructed by applying a local approximation. This method enables the reconstruction of practically any arbitrary velocity function of two coordinates. The computer program, known as godograf, which is based on this theory, is a universal program for the interpretation of any system of refraction traveltime curves for any refraction method for both shallow and deep seismic studies of crust and mantle. Examples using synthetic data demonstrate the accuracy of the algorithm and its sensitivity to realistic noise levels. Inversions of the refraction traveltimes from the Salair ore deposit, the Moscow region and the Kamchatka volcano seismic profiles illustrate the methodology, practical considerations and capability of seismic imaging with the inversion method.

Improved crosshole radar tomography by using direct and reflected arrival times

Traveltime tomography by inverting direct arrival times is the most common technique in crosshole radar surveying. In a standard crosshole experiment, the inversion is a mathematically nonunique process due to the limited range of viewing angles producing ambiguous and smeared results. If possible, additional information should be extracted from the data or added from the outside to improve and constrain the inversion and the interpretation. In low attenuation media, reflected wave energy can also be present in the crosshole data. Reflections are often caused by well-known interfaces, such as the free earth surface or the groundwater table. Because of their differing travel paths, these reflected events contain additional information about subsurface structure and parameter distribution. We present a technique to include reflection traveltimes associated with known interfaces into tomographic traveltime inversion of crosshole radar data. If in a crosshole radar survey such reflections can be clearly identified, the total number of rays and the range of raypath angles are increased by the presented inversion strategy. Applications to a synthetic and a real field data set show the practicability and the improvements offered by this method. The results show that horizontal smearing is reduced and local anomalies are better contoured compared to standard crosshole tomography using only direct arrivals.

Inversion of seismic refraction and wide-angle reflection traveltimes for three-dimensional layered crustal structure

We present a method for the determination of crustal structure by simultaneous inversion of seismic refraction and wide-angle reflection traveltimes for 3-D interface geometry and layer velocity. Crustal structure is represented by layers in which velocity varies linearly with depth, separated by smooth interfaces with a cubic B-spline parametrization. Lateral variations in structure are therefore represented by variations in interface depth only. The model parametrization we have chosen means that ray paths consist of piecewise circular arc segments for which analytic expressions of trajectory and traveltime are calculated. The two-point problem of finding the first-arrival ray path and traveltime of a specified phase between a given source and receiver is solved using a shooting technique. A subspace inversion method is used to solve the inverse problem, which is formulated as a non-linear optimization problem in which we seek to minimize an objective function that consists of a data residual term and a regularization term. Before performing the inversion, each data pick must be assigned as a refraction or reflection from a particular layer or interface. Since our method represents structure in terms of interfaces, fewer parameters would generally be used in a reconstruction compared to an equivalent 3-D variable-velocity inversion. The method is well suited to wide-angle surveys that consist of many sources and relatively few receivers (or vice versa), such as marine shot lines used in conjunction with land-based receivers. Data coverage in this kind of survey is often sparse and, especially if near-offset ray paths are unavailable, highly variable. A 3-D synthetic test with an array consisting of eight sources lying within a three-sided square of 79 receivers is described. The test model consists of a three-interface structure that includes a layer pinch-out, and the synthetic data set comprises 987 refraction and 930 reflection travel times contaminated with 75 ms of data noise. Six iterations of an 18-D subspace method demonstrate that the method can produce an accurate reconstruction that satisfies the data from a 1-D starting model. We also find that estimates of a posteriori model covariance and resolution obtained from linear theory are useful in analysing solution reliability despite the non-linear nature of the problem. Application of the method to data collected as part of the 1995 TASGO project in Tasmania shows that the method can satisfy 1345 refraction and reflection traveltime picks with a geologically reasonable and robust 254-parameter three-interface model. The inversion results indicate that the Moho beneath NW Tasmania varies in depth from 27 km near the coast to 37 km near central Tasmania, with the major increase in depth occurring across the Arthur Lineament.

Depth‐domain velocity analysis in VTI media using surfaceP-wave data: Is it feasible?

The main difficulties in anisotropic velocity analysis and inversion using surface seismic data are associated with the multiparameter nature of the problem and inherent trade‐offs between the model parameters. For the most common anisotropic model, transverse isotropy with a vertical symmetry axis (VTI media), P-wave kinematic signatures are controlled by the vertical velocity V0and the anisotropic parameters ε and δ. However, only two combinations of these parameters—NMO velocity from a horizontal reflector Vnmo(0) and the anellipticity coefficient η—can be determined from P-wave reflection traveltimes if the medium above the reflector is laterally homogeneous. While Vnmo(0) and η are sufficient for time‐domain imaging in VTI media, they cannot be used to resolve the vertical velocity and build velocity models needed for depth migration. Here, we demonstrate that P-wave reflection data can be inverted for all three relevant VTI parameters (V0, ε and δ) if the model contains nonhorizontal intermediate interfaces. Using anisotropic reflection tomography, we carry out parameter estimation for a two‐layer medium with a curved intermediate interface and reconstruct the correct anisotropic depth model. To explain the success of this inversion procedure, we present an analytic study of reflection traveltimes for this model and show that the information about the vertical velocity and reflector depth was contained in the reflected rays which crossed the dipping intermediate interface. The results of this work are especially encouraging because the need for depth imaging (such as prestack depth migration) arises mostly in laterally heterogeneous media. Still, we restricted this study to a relatively simple model and constrained the inversion by assuming that one of the layers is isotropic. In general, although lateral heterogeneity does create a dependence of P-wave reflection traveltimes on the vertical velocity, there is no guarantee that for more complicated models all anisotropic parameters can be resolved in a unique fashion.

Estimation of fracture parameters from reflection seismic data—Part II: Fractured models with orthorhombic symmetry

Existing geophysical and geological data indicate that orthorhombic media with a horizontal symmetry plane should be rather common for naturally fractured reservoirs. Here, we consider two orthorhombic models: one that contains parallel vertical fractures embedded in a transversely isotropic background with a vertical symmetry axis (VTI medium) and the other formed by two orthogonal sets of rotationally invariant vertical fractures in a purely isotropic host rock. Using the linear‐slip theory, we obtain simple analytic expressions for the anisotropic coefficients of effective orthorhombic media. Under the assumptions of weak anisotropy of the background medium (for the first model) and small compliances of the fractures, all effective anisotropic parameters reduce to the sum of the background values and the parameters associated with each fracture set. For the model with a single fracture system, this result allows us to eliminate the influence of the VTI background by evaluating the differences between the anisotropic parameters defined in the vertical symmetry planes. Subsequently, the fracture weaknesses, which carry information about the density and content of the fracture network, can be estimated in the same way as for fracture‐induced transverse isotropy with a horizontal symmetry axis (HTI media) examined in our previous paper (part I). The parameter estimation procedure can be based on the azimuthally dependent reflection traveltimes and prestack amplitudes of P-waves alone if an estimate of the ratio of the P- and S-wave vertical velocities is available. It is beneficial, however, to combine P-wave data with the vertical traveltimes, NMO velocities, or AVO gradients of mode‐converted (PS) waves. In each vertical symmetry plane of the model with two orthogonal fracture sets, the anisotropic parameters are largely governed by the weaknesses of the fractures orthogonal to this plane. For weak anisotropy, the fracture sets are essentially decoupled, and their parameters can be estimated by means of two independently performed HTI inversions. The input data for this model must include the vertical velocities (or reflector depth) to resolve the anisotropic coefficients in each vertical symmetry plane rather than just their differences. We also discuss several criteria that can be used to distinguish between the models with one and two fracture sets. For example, the semimajor axis of the P-wave NMO ellipse and the polarization direction of the vertically traveling fast shear wave are always parallel to each other for a single system of fractures, but they may become orthogonal in the medium with two fracture sets.

Crustal structure of the southeast Greenland margin from joint refraction and reflection seismic tomography

We present results from a combined multichannel seismic reflection (MCS) and wideangle onshore/offshore seismic experiment conducted in 1996 across the southeast Greenland continental margin. A new seismic tomographic method is developed to jointly invert refraction and reflection travel times for a two‐dimensional velocity structure. We employ a hybrid ray‐tracing scheme based on the graph method and the local ray‐bending refinement to efficiently obtain an accurate forward solution, and we employ smoothing and optional damping constraints to regularize an iterative inversion. We invert 2318 Pg and 2078 PmP travel times to construct a compressional velocity model for the 350‐km‐long transect, and a long‐wavelength structure with strong lateral heterogeneity is recovered, including (1) ∼30‐km‐thick, undeformed continental crust with a velocity of 6.0 to 7.0 km/s near the landward end, (2) 30‐ to 15‐km‐thick igneous crust within a 150‐km‐wide continent‐ocean transition zone, and (3) 15‐ to 9‐km‐thick oceanic crust toward the seaward end. The thickness of the igneous upper crust characterized by a high‐velocity gradient also varies from 6 km within the transition zone to ∼3 km seaward. The bottom half of the lower crust generally has a velocity higher than 7.0 km/s, reaching a maximum of 7.2 to 7.5 km/s at the Moho. A nonlinear Monte Carlo uncertainty analysis is performed to estimate the a posteriori model variance, showing that most velocity and depth nodes are well determined with one standard deviation of 0.05–0.10 km/s and 0.25–1.5 km, respectively. Despite significant variation in crustal thickness, the mean velocity of the igneous crust, which serves as a proxy for the bulk crustal composition, is surprisingly constant (∼7.0 km/s) along the transect. On the basis of a mantle melting model incorporating the effect of active mantle upwelling, this velocity‐thickness relationship is used to constrain the mantle melting process during the breakup of Greenland and Europe. Our result is consistent with a nearly constant mantle potential temperature of 1270–1340°C throughout the rifting but with a rapid transition in the style of mantle upwelling, from vigorous active upwelling during the initial rifting phase to passive upwelling in the later phase.

Large-offset approximation to seismic reflection traveltimes

Conventional approximations of reflection traveltimes assume a small offset-to-depth ratio, and their accuracy decreases with increasing offset-to-depth ratio. Hence, they are not suitable for velocity analysis and stacking of long-offset reflection seismic data. Assuming that the offset is large, rather than small, we present a new traveltime approximation which is exact at infinite offset and has a decreasing accuracy with decreasing offset-to-depth ratio. This approximation has the form of a series containing powers of the offset from 1 to −∞. It is particularly accurate in the presence of a thin high-velocity layer above the reflector, i.e. in a situation where the accuracy of the Taner and Koehler series is poor. This new series can be used to gain insight into the velocity information contained in reflection traveltimes at large offsets, and possibly to improve velocity analysis and stacking of long-offset reflection seismic data.