Velocity Anisotropy Parameter Research Articles

Anisotropic full-waveform inversion of crosshole seismic data: A vertical symmetry axis field data application

Anisotropic waveform tomography (AWT) uses anisotropic traveltime tomography followed by anisotropic full-waveform inversion (FWI). Such an approach is required for FWI in cases in which the geology is likely to exhibit anisotropy. An important anisotropy class is that of transverse isotropy (TI), and the special case of TI media with a vertical symmetry axis (VTI) media is often used to represent elasticity in undeformed sedimentary layering. We have developed an approach for AWT that uses an acoustic approximation to simulate waves in VTI media, and we apply this approach to crosshole data. In our approach, the best-fitting models of seismic velocity and Thomsen VTI anisotropy parameters are initially obtained using anisotropic traveltime tomography, and they are then used as the starting models for VTI FWI within the acoustic approximation. One common problem with the acoustic approach to TI media is the generation of late-arriving (spurious) S-waves as a by-product of the equation system. We used a Laplace-Fourier approach that effectively damps the spurious S-waves to suppress artifacts that might otherwise corrupt the final inversion results. The results of applying AWT to synthetic data illustrate the trade-offs in resolution between the two parameter classes of velocity and anisotropy, and they also verify anisotropic traveltime tomography as a valid method for generating starting models for FWI. The synthetic study further indicates the importance of smoothing the anisotropy parameters before proceeding to FWI inversions of the velocity parameter. The AWT technique is applied to real crosshole field gathers from a sedimentary environment in Western Canada, and the results are compared with the results from a simpler (elliptical) anisotropy model. The transversely isotropic approach yields an FWI image of the vertical velocity that (1) exhibits a superior resolution and (2) better predicts the field data than does the elliptical approach.

Experimental investigation of mechanical compaction on the physical and elastic properties of synthetic shales

Three sets of synthetic shale samples with different clay minerals were prepared to study the effects of mechanical compaction stress on the physical and elastic properties of shales. The results suggest that the physical and elastic properties (i.e., porosity, density, acoustic velocity, etc.) of the mudstones vary greatly the compaction stress, and the type of clay minerals. For a given compaction stress, the synthetic shale with kaolinite exhibits the smallest density and a highest porosity, while the sample with smectite shows the highest density and the smallest porosity. Further, the velocity demonstrates a considerable dependence to compaction stress, both P- and S-wave velocities in different directions increase with the compaction stress. We also find that the velocity anisotropy parameters increase with the compaction stress, and the S-wave anisotropy is more sensitive to the compaction stress compared to the P-wave anisotropy. With respect to the mechanical properties, the dynamic Young's modulus increase with the compaction stress, while the Poisson's ratio decrease with the compaction stress. This can be attributed to the decrease in the porosity. The results also indicate that the mechanical properties display obvious anisotropic behaviours. Finally, strong correlations are observed between the porosity and the velocity anisotropy and mechanical anisotropy for all the prepared samples, providing a means of estimating seismic anisotropy from porosity in shales.

Testing Weyl gravity at galactic and extra-galactic scales

We examine the viability of Weyl conformal gravity as an alternative to the general theory of relativity. By using the extended rotation curve of the Milky Way and velocity dispersions of four globular clusters, we show that Weyl gravity predictions without resorting to dark matter comply with observations at the galactic scale. For the Milky Way, we demonstrate that the uncertainty in baryonic modeling results in a bracket of possible rotational velocities which well encompasses the diversity in rotation curve construction. Such diversity generally arises from differences in measurements of velocity anisotropy parameter, and the circular speed and Galactocentric distance of the Sun. Furthermore, we explore the ability of Weyl gravity to account for the inferred acceleration of Abell cluster 1689.

Beta Dips in the Gaia Era: Simulation Predictions of the Galactic Velocity Anisotropy Parameter (β) for Stellar Halos

Abstract The velocity anisotropy parameter, β, is a measure of the kinematic state of orbits in the stellar halo, which holds promise for constraining the merger history of the Milky Way (MW). We determine global trends for β as a function of radius from three suites of simulations, including accretion-only and cosmological hydrodynamic simulations. We find that the two types of simulations are consistent and predict strong radial anisotropy ( ) for Galactocentric radii greater than 10 kpc. Previous observations of β for the MW’s stellar halo claim a detection of an isotropic or tangential “dip” at r ∼ 20 kpc. Using the N-body+SPH simulations, we investigate the temporal persistence, population origin, and severity of “dips” in β. We find that dips in the in situ stellar halo are long-lived, while dips in the accreted stellar halo are short-lived and tied to the recent accretion of satellite material. We also find that a major merger as early as z ∼ 1 can result in a present-day low (isotropic to tangential) value of β over a broad range of radii and angles. While all of these mechanisms are plausible drivers for the β dip observed in the MW, each mechanism in the simulations has a unique metallicity signature associated with it, implying that future spectroscopic surveys could distinguish between them. Since an accurate knowledge of β(r) is required for measuring the mass of the MW halo, we note that significant transient dips in β could cause an overestimate of the halo’s mass when using spherical Jeans equation modeling.

The velocity ellipsoid in the Galactic disc using Gaia DR1

The stellar velocity ellipsoid of the solar neighbour (d < 200 pc) is re-examined using intermediate-old mono-abundance stellar groups with high-quality chemistry data together with parallaxes and proper motions from Gaia DR1. We find the average velocity dispersion values for the three space velocity components for the thin and thick discs of (σU, σV, σW)thin = (33 ± 4, 28 ± 2, 23 ± 2) and (σU, σV, σW)thick = (57 ± 6, 38 ± 5, 37 ± 4) km s−1, respectively. The mean values of the ratio between the semi-axes of the velocity ellipsoid for the thin disc are found to be σV/σU = 0.70 ± 0.13 and σW/σU is 0.64 ± 0.08, while for the thick disc σV/σU = 0.67 ± 0.11 and σW/σU is 0.66 ± 0.11. Inputting these dispersions into the linear Strömberg relation for the thin disc groups, we find the Sun's velocity with respect to the Local Standard of Rest in Galactic rotation to be V⊙ = 13.9 ± 3.4 km s−1. A relation is found between the vertex deviation and the chemical abundances for the thin disc, ranging from −5 to +40° as iron abundance increases. For the thick disc we find a vertex deviation of luv ∼− 15°. The tilt angle (luw) in the U–W plane for the thin disc groups ranges from −10 to +15°, but there is no evident relation between luw and the mean abundances. However, we find a weak relation for luw as a function of iron abundances and α-elements for most of the groups in the thick disc, where the tilt angle decreases from −5 to −20° when [Fe/H] decreases and [α/Fe] increases. The velocity anisotropy parameter is independent of the chemical group abundances and its value is nearly constant for both discs (β ∼ 0.5), suggesting that the combined disc is dynamically relaxed.

ZOMG - III. The effect of halo assembly on the satellite population

We use zoom hydrodynamical simulations to investigate the properties of satellites within galaxy-sized dark-matter haloes with different assembly histories. We consider two classes of haloes at redshift $z=0$: `stalled' haloes that assembled at $z>1$ and `accreting' ones that are still forming nowadays. Previously, we showed that the stalled haloes are embedded within thick filaments of the cosmic web while the accreting ones lie where multiple thin filaments converge. We find that satellites in the two classes have both similar and different properties. Their mass spectra, radial count profiles, baryonic and stellar content, and the amount of material they shed are indistinguishable. However, the mass fraction locked in satellites is substantially larger for the accreting haloes as they experience more mergers at late times. The largest difference is found in the satellite kinematics. Substructures fall towards the accreting haloes along quasi-radial trajectories whereas an important tangential velocity component is developed, before accretion, while orbiting the filament that surrounds the stalled haloes. Thus, the velocity anisotropy parameter of the satellites ($\beta$) is positive for the accreting haloes and negative for the stalled ones. This signature enables us to tentatively categorize the Milky Way halo as stalled based on a recent measurement of $\beta$. Half of our haloes contain clusters of satellites with aligned orbital angular momenta corresponding to flattened structures in space. These features are not driven by baryonic physics and are only found in haloes hosting grand-design spiral galaxies, independently of their assembly history.

The Milky Way's Circular Velocity Curve and Its Constraint on the Galactic Mass with RR Lyrae Stars

Abstract We present a sample of 1148 ab-type RR Lyrae (RRLab) variables identified from Catalina Surveys Data Release 1, combined with SDSS DR8 and LAMOST DR4 spectral data. We first use a large sample of 860 Galactic halo RRLab stars and derive the circular velocity distributions for the stellar halo. With the precise distances and carefully determined radial velocities (the center-of-mass radial velocities) and by considering the pulsation of the RRLab stars in our sample, we can obtain a reliable and comparable stellar halo circular velocity curve. We follow two different prescriptions for the velocity anisotropy parameter β in the Jeans equation to study the circular velocity curve and mass profile. Additionally, we test two different solar peculiar motions in our calculation. The best result we obtained with the adopted solar peculiar motion 1 of (U, V, W) = (11.1, 12, 7.2) km s−1 is that the enclosed mass of the Milky Way within 50 kpc is (3.75 ± 1.33) × 1011 M ⊙ based on β = 0 and the circular velocity 180 ± 31.92 (km s−1) at 50 kpc. This result is consistent with dynamical model results, and it is also comparable to the results of previous similar works.

Reliability of the Measured Velocity Anisotropy of the Milky Way Stellar Halo

Abstract Determining the velocity distribution of halo stars is essential for estimating the mass of the Milky Way and for inferring its formation history. Since the stellar halo is a dynamically hot system, the velocity distribution of halo stars is well described by the three-dimensional velocity dispersions or by the velocity anisotropy parameter . Direct measurements of consistently suggest β = 0.5–0.7 for nearby halo stars. In contrast, the value of β at large Galactocentric radius r is still controversial, since reliable proper motion data are available for only a handful of stars. In the last decade, several authors have tried to estimate β for distant halo stars by fitting the observed line-of-sight velocities at each radius with simple velocity distribution models (local fitting methods). Some results of local fitting methods imply at , which is inconsistent with recent predictions from cosmological simulations. Here we perform mock-catalog analyses to show that the estimates of β based on local fitting methods are reliable only at with the current sample size (∼103 stars at a given radius). As r increases, the line-of-sight velocity (corrected for the solar reflex motion) becomes increasingly closer to the Galactocentric radial velocity, so it becomes increasingly more difficult to estimate the tangential velocity dispersion from the line-of-sight velocity distribution. Our results suggest that the forthcoming Gaia data will be crucial for understanding the velocity distribution of halo stars at .

Deep spectroscopy in nearby galaxy clusters – III. Orbital structure of galaxies in Abell 85

Galaxies in clusters are strongly affected by their environment. They evolve according to several physical mechanisms that are active in clusters. Their efficiency can strongly depend on the orbital configuration of the galaxies. Our aim is to analyse the orbits of the galaxies in the cluster Abell 85, based on the study of the galaxy velocity anisotropy parameter. We have solved the Jeans equation under the assumption that the galaxies in A85 are collisionless objects, within the spherically symmetric gravitational potential of the virialized cluster. The mass of the cluster was estimated with X-ray and caustic analyses. We find that the anisotropy profile of the full galaxy population in A85 is an increasing monotonic function of the distance from the cluster centre: on average, galaxies in the central region (r/r200 < 0.3) are on isotropic orbits, while galaxies in the outer regions are on radial orbits. We also find that the orbital properties of the galaxies strongly depend on their stellar colour. In particular, blue galaxies are on less radial orbits than red galaxies. The different families of cluster galaxies considered here have the pseudo phase-space density profiles Q(r) and Qr(r) consistent with the profiles expected in virialized dark matter halos in $N$-body simulations. This result suggests that the galaxies in A85 have reached dynamical equilibrium within the cluster potential. Our results indicate that the origin of the blue and red colour of the different galaxy populations is the different orbital shape rather than the accretion time.

ON ESCAPING A GALAXY CLUSTER IN AN ACCELERATING UNIVERSE

ABSTRACT We derive the escape velocity profile for an Einasto density field in an accelerating universe and demonstrate its physical viability by comparing theoretical expectations to both light-cone data generated from N-body simulations and archival data on 20 galaxy clusters. We demonstrate that the projection function ( ) is deemed physically viable only for the theoretical expectation that includes a cosmology-dependent term. Using simulations, we show that the inferred velocity anisotropy is more than 6σ away from the expected value for the theoretical profile that ignores the acceleration of the universe. In the archival data, we constrain the average velocity anisotropy parameter of a sample of 20 clusters to be at the 68% confidence level. Lastly, we briefly discuss how our analytic model may be used as a novel cosmological probe based on galaxy clusters.

BAYESIAN MASS ESTIMATES OF THE MILKY WAY: THE DARK AND LIGHT SIDES OF PARAMETER ASSUMPTIONS

ABSTRACT We present mass and mass profile estimates for the Milky Way (MW) Galaxy using the Bayesian analysis developed by Eadie et al. and using globular clusters (GCs) as tracers of the Galactic potential. The dark matter and GCs are assumed to follow different spatial distributions; we assume power-law model profiles and use the model distribution functions described in Evans et al. and Deason et al. We explore the relationships between assumptions about model parameters and how these assumptions affect mass profile estimates. We also explore how using subsamples of the GC population beyond certain radii affect mass estimates. After exploring the posterior distributions of different parameter assumption scenarios, we conclude that a conservative estimate of the Galaxy’s mass within 125 kpc is , with a 50% probability region of . Extrapolating out to the virial radius, we obtain a virial mass for the MW of with 50% credible region of ( . If we consider only the GCs beyond 10 kpc, then the virial mass is ( kpc). We also arrive at an estimate of the velocity anisotropy parameter β of the GC population, which is with a 50% credible region (0.21, 0.35). Interestingly, the mass estimates are sensitive to both the dark matter halo potential and visible matter tracer parameters, but are not very sensitive to the anisotropy parameter.

Frequency dependence of anisotropy in fluid saturated rocks – Part I: aligned cracks case

ABSTRACTA major cause of attenuation in fluid‐saturated media is the local fluid flow (or squirt flow) induced by a passing wave between pores of different shapes and sizes. Several squirt flow models have been derived for isotropic media. For anisotropic media however, most of the existing squirt flow models only provide the low‐ and high‐frequency limits of the saturated elastic properties. We develop a new squirt flow model to account for the frequency dependence of elastic properties and thus gain some insight into velocity dispersion and attenuation in anisotropic media. In this paper, we focus on media containing aligned compliant pores embedded in an isotropic background matrix. The low‐ and high‐frequency limits of the predicted fluid‐saturated elastic properties are respectively consistent with Gassmann theory and Mukerji–Mavko squirt flow model. Results are also expressed in terms of Thomsen anisotropy parameters. It turns out that the P‐wave anisotropy parameter ε tends to zero in the high‐frequency limit, whereas the δ parameter remains the only indicator of P‐S⊥ anisotropy. The S‐wave anisotropy parameter γ is not affected by the presence of fluid and remains the same for all frequency ranges. A new definition for attenuation anisotropy parameters is also proposed to quantify the attenuation anisotropy. In the most important case of liquid saturation, analytical expressions are derived for elastic properties, velocity anisotropy parameters, quality factors, and attenuation anisotropy parameters. A companion paper considers the case of cracks with an ellipsoidal distribution of orientations resulting from the application of anisotropic stress.

Frequency‐dependent anisotropy of porous rocks with aligned fractures

ABSTRACTNaturally fractured reservoirs are becoming increasingly important for oil and gas exploration in many areas of the world. Because fractures may control the permeability of a reservoir, it is important to be able to find and characterize fractured zones. In fractured reservoirs, the wave‐induced fluid flow between pores and fractures can cause significant dispersion and attenuation of seismic waves. For waves propagating normal to the fractures, this effect has been quantified in earlier studies. Here we extend normal incidence results to oblique incidence using known expressions for the stiffness tensors in the low‐ and high‐frequency limits. This allows us to quantify frequency‐dependent anisotropy due to the wave‐induced flow between pores and fractures and gives a simple recipe for computing phase velocities and attenuation factors of quasi‐P and SV waves as functions of frequency and angle. These frequency and angle dependencies are concisely expressed through dimensionless velocity anisotropy and attenuation anisotropy parameters. It is found that, although at low frequencies, the medium is close to elliptical (which is to be expected as a dry medium containing a distribution of penny‐shaped cracks is known to be close to elliptical); at high frequencies, the coupling between P‐wave and SV‐wave results in anisotropy due to the non‐vanishing excess tangential compliance.

An Experimental Study of Wave Velocity Anisotropy in Sandstone Bearing Differential Pore Fluids from the Yanchang Formation

AbstractThis work applied the AutoLab 2000 test system to investigate three kinds of core samples of sandstone from the Yanchang oilfield that were cut parallel and perpendicular to the bedding planes (named H and V, respectively), and obtained accurate P‐ and S‐wave velocities, and wave velocity anisotropy data at pressures up to 180 MPa in dry, water‐ and oil‐saturated states for those sandstone. The results show the wave velocities VP, VSH and VSV for all H and V samples cut from three kinds of samples of sandstone Y1, Y2 and Y3 change basically logarithmically with increasing confining pressure p, and for all VP, the slopes of the oil‐saturated curves are greater than that of the corresponding water‐saturated curves. For all H samples, in dry, water‐ and oil‐saturated conditions, the VP and VSH and VSV are larger than the corresponding ones for V samples. For two kinds of sandstones Y1 and Y2, the wave velocity anisotropy parameters ε, γ and ζ change exponentially or in a quadratic curve with increasing confining pressure, and Y2 has a greater correlation coefficient of anisotropy parameters than Y1, but there is no wave velocities anisotropy in Y3. In addition, in dry, water‐ and oil‐saturated conditions the relationships between ε, γ and p or γ, ζ and p are quite distinct from each other, that is, the anisotropy factors ε, γ and ζ of Y1 and Y2 exhibit a clear correlation with pressure and the fluid property, and ε and γ in oil‐saturated condition are larger than that in the corresponding water‐saturated condition at the experimental pressure. The experimental results can provide an important basis for the interpretation of seismic data, identifying oil and water zones and comparing with well log data in the study area.

ROTATION CURVE OF THE MILKY WAY OUT TO ∼200 kpc

The rotation curve (RC) of our Galaxy, the Milky Way, is constructed starting from its very inner regions (few hundred pc) out to a large Galactocentric distance of $\sim 200$ kpc using kinematical data on a variety of tracer objects moving in the gravitational potential of the Galaxy, without assuming any theoretical models of the visible and dark matter components of the Galaxy. We study the effect on the RC due to the uncertainties in the values of the Galactic Constants (GCs) $R_\odot$ and $V_\odot$ (these being the sun's distance from and circular rotation speed around the Galactic center, respectively) and the velocity anisotropy parameter $\beta$ of the halo tracer objects used for deriving the RC at large Galactocentric distances. The resulting RC in the disk region is found to depend significantly on the choice of the GCs, while the dominant uncertainty in the RC at large distances beyond the stellar disk comes from the uncertainty in the value of $\beta$. In general we find that the mean RC steadily declines at distances beyond $\sim 60$ kpc, independently of the value of $\beta$. Also, at a given radius, the circular speed is lower for larger values of $\beta$ (i.e., for more radially biased velocity anisotropy). Considering that the largest possible value of $\beta$ is unity, which corresponds to stellar orbits being purely radial, our results for the case of $\beta=1$ give a lower limit to the total mass of the Galaxy within $\sim 200$ kpc, $M(200 {\rm kpc}) \gsim (6.8\pm4.1) \times 10^{11} M_\odot$, independently of any model of the dark matter halo of the Galaxy.

Estimation of the Effect of Anisotropy on Young’s Moduli and Poisson’s Ratios of Sedimentary Rocks Using Core Samples in Western and Central Part of Tripura, India

The velocity anisotropy parameters and elastic constants play a very important role to estimate Young’s modulus and Poisson’s ratios accurately. For geomechanics applications such as hydraulic fracturing design, analysis of wellbore stability and rock failure, determination of in situ stress and assessment of the response of reservoirs and surrounding rocks to changes in pore pressure and stress, Young’s modulus and Poisson’s ratios play a very important role. Four rock samples were collected from four different wells situated in study area. The ultrasonic transmission method has been used to measure P-wave, Sh-wave and Sv-wave travel times as a function of orientation and confining pressure. The five independent stiffnesses constants, Young’s moduli, Poisson’s ratios and Bulk moduli of the samples were estimated. The Poisson’s ratios ( and ) are varying as the confining pressure is changed. The axial strain is larger than the lateral strain, resulting . For shales, the Young’s modulus measured parallel to bedding E1 is usually greater than the Young’s modulus measured perpendicular to bedding E3. Through this study it has been observed that, there is a strong effect of anisotropy parameters on Young’s modulus and Poisson’s ratio.

Enhanced sensitivity in H photofragment detection by two-color reduced-Doppler ion imaging

Two-color reduced-Doppler (TCRD) and one-color velocity map imaging (VMI) were used for probing H atom photofragments resulting from the ~243.1 nm photodissociation of pyrrole. The velocity components of the H photofragments were probed by employing two counterpropagating beams at close and fixed wavelengths of 243.15 and 243.12 nm in TCRD and a single beam at ~243.1 nm, scanned across the Doppler profile in VMI. The TCRD imaging enabled probing of the entire velocity distribution in a single pulse, resulting in enhanced ionization efficiency, as well as improved sensitivity and signal-to-noise ratio. These advantages were utilized for studying the pyrrole photodissociation at ~243.1 and 225 nm, where the latter wavelength provided only a slight increase in the H yield over the self-signal from the probe beams. The TCRD imaging enabled obtaining high quality H(+) images, even for the low H photofragment yields formed in the 225 nm photolysis process, and allowed determining the velocity distributions and anisotropy parameters and getting insight into pyrrole photodissociation.

Analytical families of two-component anisotropic polytropes and their relativistic extensions

In this paper, we study a family of two-component anisotropic polytropes which model a wide range of spherically symmetric astrophysical systems such as early-type baryonic galaxies. This family is found to contain a large class of models such as the hypervirial family (which satisfy the virial theorem locally), the Plummer and Hernquist models and Navarro-Frenk-White (NFW) like models. The potential–density pair for these models are derived, as well as their velocity dispersions and anisotropy parameters. The projected quantities are computed and found to reduce to analytical expressions in some cases. The first section of this paper presents an extension of the two-term anisotropic polytropes to encompass a very wide range of potential–density pairs. In the next section, we present the general relativistic extension of the potential–density pair and calculate the stress–energy tensor, the relativistic anisotropy parameter, the velocity of circular orbits and the angular momentum. Remarkably, for the case of the hypervirial family, the relativistic pressure in the Newtonian limit and the relativistic anisotropy parameter are found to coincide with the corresponding Newtonian expressions. The weak, dominant and strong energy conditions are found to be satisfied only for a certain range of the free parameters. We show that the relativistic hypervirial family also has a finite total mass like its Newtonian counterpart. In the first appendix, a relativistic extension of a different hypervirial family of models is studied and the relativistic anisotropy parameter is found to coincide with the Newtonian one. Finally, we present a family of models obtained from our distribution function that are similar to the Ossipkov–Merritt models; by computing their anisotropy parameters, we show that they model systems with isotropic cores and radially anisotropic exteriors.

Stabilization of generalized polytropes

AbstractStabilization of classical generalized polytropic models is studied. We discuss some contradictions found in the literature: on the one hand, calculations of the growth rates in the framework of linear stability theory and N‐body simulations show that these systems are stable if the velocity anisotropy parameter of the stellar distribution is lower than some finite positive value. On the other hand, instability is shown to persist up to the isotropic limit (zero anisotropy parameter). We demonstrate that stabilization of generalized polytropic models cannot be adequately described by approximate equations derived for nearly radial orbits. (© 2013 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Contrasts of seismic velocity, density and strength across the Moho

Abstract We provide an overview of contrast of elastomechanical parameters across the Moho, basically contrasts in seismic wave velocities, density and yield strength. These can be regarded as dynamic and quasi-static endmembers of elastomechanical parameters of the crust–mantle transition. With respect to practical applications they are closely related because seismic investigations play the role of exploring the earth structure and physical state forming the boundary conditions for determining density and rheological models. We address both average global Moho models and petrophysical and structural factors that cause deviations from the global averages, such as variations in Moho depth, temperature and rock composition, fabric (anisotropy) and macro-scale heterogeneity. Besides principle considerations these factors are put in a regional context in order to demonstrate how they are related to past and present tectonic processes. Seismic velocity and density contrasts are found between 0 and 25% whereas strength or viscosity contrasts may be orders of magnitude higher, but may vanish as well or show a discontinuity of higher order only. Especially, oceanic and old cratonic crust and near-Moho mantle may appear as a rheological unit although they are seismically well distinguishable. In terms of seismic velocities the Moho may “vanish” mainly under the influence of serpentinization or eclogitization. Originally defined as a first-order discontinuity in seismic velocity, the Moho has turned out to be an interface across which other seismic properties, such as seismic velocity gradients, anisotropy and heterogeneity scale parameters, can change strongly, too. However, knowledge of these parameters, as well as their relation to the rheology of the crust–mantle transition, is still restricted to local or regional examples so no global conclusions can be drawn.