气煤弹性各向异性系数实验测试

气煤裂隙丰富,具有弹性各向异性,确定气煤弹性各向异性类型与大小,对煤田各向异性研究具有重要意义. 本文在对气煤煤样孔隙率测试的基础上,测量了顺煤层面、垂直于煤层面、与煤层面成45°三个方向上纵波速度（V_P）和横波速度（V_SH、V_SV）,计算弹性常数. 对于具有垂向对称轴横向各向同性模型,各向异性系数平均值|ε|≤0.2、|δ|≤0.2和|γ|≤0.2,表明气煤是弱各向异性介质;对于横向对称轴横向各向异性模型,各向异性系数平均值ε(V)≈δ(V),按照Thomsen观点气煤各向异性是由椭圆裂隙引起弹性各向异性.本文还讨论了各向异性系数与孔隙率的关系,表明各向异性的大小与孔隙率之间有一定的关系. 因此当煤层厚度与其顶底板岩性不变或变化很小,可以认为气煤各向异性主要由裂隙引起的,气煤各向异性大小和方向能够反映煤的裂缝密度和方位.     

Quantitative measures of textural anisotropy resulting from magmatic compaction illustrated by a sample from the Palisades sill, New Jersey

Early in the crystallization of many tholeiitic basaltic magmas, plagioclase crystals cluster together into a 3-D cellular network, which forms a passive marker capable of recording the deformation that accompanies compaction of crystal mush. Although irregular in detail, the overall network is initially isotropic and only becomes anisotropic as a result of compaction. We have developed four independent methods to quantify the 3-D textural anisotropy of a basalt sample using at least three non-parallel thin sections. Three of the methods are based on the geometrical properties of digitized maps of the feldspar chain networks. One approach focuses on the angular variation of the mean intercept along parallel traverses through the network, another examines the orientation and size distribution of individual links, and the third considers the average shape of interstitial regions outlined by the plagioclase network. The fourth technique approximates the textural anisotropy by the variogram anisotropy of a scanned thin section image. We illustrate the methods using five oriented non-parallel thin sections from a sample of diabase 146 m above the base of the 300-m-thick Palisades sill of New Jersey. Compaction of crystal mush in this sill has previously been postulated on the basis of chemical evidence. The 3-D feldspar network anisotropy based on the first three approaches suggests nearly uniaxial compaction on the order of 8.6% in a direction within 3° of the intersection of the columnar joints at the sample site. A rigorous statistical test based on the statistics of elliptically contoured non-normal multivariate distributions documents that the link-vector distribution in vertical sections are statistically anisotropic at a 95% confidence level and that the overall compaction is 7.9±2.6%. The orientation and magnitude of the 3-D textural anisotropy determined by the image variogram of the non-opaque minerals is almost identical to the mean feldspar network anisotropy; 8.5% compaction in a direction 10° from the columnar joint intersections. The major silicate textural and feldspar network anisotropy axes both plunge almost directly down dip of the sill. On the other hand, the major axis of the variogram anisotropy of the opaque minerals is approximately parallel to the strike of the sill and to the major axis of the anisotropic magnetic susceptibility. The anisotropy of the silicate mineral fabric may reflect down-dip flow of a deformable melt-rich crystal mush, whereas the AMS and opaque textural anisotropy reflects the influence of gravitational stresses during the growth of magnetite in the final stages of melt crystallization. Evidently the Palisades sill was not originally horizontal but was intruded in an orientation close to its present attitude.     

双相各向异性研究、问题与应用前景

地球内部流体的存在和岩石各向异性是地下介质的两大表征,考虑地下流体和介质各向异性问题的双相各向异性理论是当今地震学和地球物理学理论与应用研究的前沿和难题之一。孔隙流体的存在、固体和流体之间的相互作用会弱化或硬化岩石的力学属性,上起声波或弹性声波速度的频散的振幅的衰减,并产生第二类压缩波。裂缝或裂隙的定向分布、岩层的旋性沉积、应力场的定向排列,都会引起传播速度的各向异性、横波发生分裂等重要现象,这些     

Research on propagation properties of elastic waves in two-phase anisotropic media

IntroductionItiswellknownthatanisotropylieswidelyintheundergroundmedia.Anisotropicmediawhicharemetintheseismicengineeringandseismicexplorationofenergyaremainlycausedbytheperiodicthinlayers(PTL)andextensivedilatancyanisotropy(EDA).Insuchmedia,anisotropyleadstomorecomplicatepropagationofseismicwave,thesignificantfeatureinanisotropicmediaisvelocityanisotropy.Infact,undergroundstrataareverycomplicated,whichareusuallycomposedofsolidframeandfluid(suchasoil,gasesorwater)inpores.Inordertostudyseism…     

Distribution of gas hydrate in fractured reservoirs: Insights from anisotropic seismic measurements

Seismic properties of hydrate-bearing reservoirs that are affected significantly by the hydrate distribution are key for quantitative assessment of the reservoir. The knowledge of hydrate distribution in fractured reservoirs remains poorly understood. To obtain such knowledge, we measured and analyzed five anisotropic velocities needed to fully characterize the seismic anisotropy in an artificial sandstone with aligned fractures during hydrate formation associated with varying distribution. We showed that while the formation of hydrate improved the velocities, the improvement was more significant for hydrate saturation above 10%. We also showed that the increasing trends varied among the anisotropic velocities when hydrate saturation was above 10%. Specifically, the compressional wave velocity travelling vertical to the bedding plane and the shear wave velocity with polarization perpendicular to the bedding plane increased more rapidly than the other compressional and shear wave velocities, respectively. Interpretation of the anisotropic seismic results suggested that the hydrate tends to bind to the grains in the fractures at low hydrate saturation, and becomes to bridge the fracture surfaces when the hydrate saturation exceeds 10%. The results have provided new insights into the hydrate distribution and its resulting anisotropic seismic properties in fractured reservoirs. This will pave the way for the successful assessment of hydrate in fractured reservoirs.     

Poynting and polarization vectors based wavefield decomposition and their application on elastic reverse time migration in 2D transversely isotropic media

With the progress in computational power and seismic acquisition, elastic reverse time migration is becoming increasingly feasible and helpful in characterizing the physical properties of subsurface structures. To achieve high-resolution seismic imaging using elastic reverse time migration, it is necessary to separate the compressional (P-wave) and shear (S-wave) waves for both isotropic and anisotropic media. In elastic isotropic media, the conventional method for wave-mode separation is to use the divergence and curl operators. However, in anisotropic media, the polarization direction of P waves is not exactly parallel to the direction of wave propagation. Also, the polarization direction of S-waves is not totally perpendicular to the direction of wave propagation. For this reason, the conventional divergence and curl operators show poor performance in anisotropic media. Moreover, conventional methods only perform well in the space domain of regular grids, and they are not suitable for elastic numerical simulation algorithms based on non-regular grids. Besides, these methods distort the original wavefield by taking spatial derivatives. In this case, a new anisotropic wave-mode separation scheme is developed using Poynting vectors. This scheme can be performed in the angle domain by constructing the relationship between group and polarization angles of different wave modes. Also, it is performed pointwise, independent of adjacent space points, suitable for parallel computing. Moreover, there is no need to correct the changes in phase and amplitude caused by the derivative operators. By using this scheme, the anisotropic elastic reverse time migration is more efficiently performed on the unstructured mesh. The effectiveness of our scheme is verified by several numerical examples.     

Shear waves at a corrugated interface between anisotropic elastic and visco-elastic solid half-spaces

Rayleigh’s method of approximation is employed to find out the reflection and transmission coefficients due to an incident plane SH wave at a corrugated interface between a laterally and vertically inhomogeneous anisotropic elastic solid half-space and a laterally and vertically inhomogeneous isotropic visco-elastic solid half-space. The lateral and vertical inhomogeneities are described by the exponential variations of elastic parameters. The formulae of reflection and transmission coefficients are derived in closed form for the first-order approximation of the corrugation. The effects of the corrugation of the interface, the inhomogeneity, the anisotropy, the visco-elasticity and the frequency of the incident wave on these coefficients are studied analytically and numerically for a specific model containing a periodic interface. The results of earlier workers have been reduced as particular cases from the present formulation.     

Crustal shear-wave splitting in the epicentral zone of the 2001 Mw 7.7 Bhuj earthquake, Gujarat, India

We report here crustal shear-wave anisotropy, ranging from 1% to 10.76% with an average of 2.4% in the aftershock zone of the 2001 Bhuj earthquake, Gujarat, India, from a study of leading shear-wave polarization directions (LPSDs), which vary on average from NNW–SSE to E–W with a delay of 0.07–0.14 s. The delays in the NNW–SSE to NE–SW directions observed at seven stations, near the seismogenic fault, suggest cracks parallel to the direction of the maximum horizontal regional compressional stress prevailing in the region, suggesting a dilatancy-induced anisotropy resulting from approximately stress-aligned parallel vertical micro-cracks. In contrast, the LPSDs at Ramvav, Rapar and Vondh stations, away from the seismogenic fault, are fault parallel, approximately E–W and almost orthogonal to the stress-aligned polarizations inferred elsewhere. The maximum average time delay of 0.14 s is observed at Lodai, where the fast polarization direction is found to be N338°W. This has been observed from anisotropic poro-elastic (APE) modelling and observations that these are 90° flips in shear-wave polarization, resulting from propagation through micro-cracks containing fluids at critically high pore-fluid pressure surrounding the hypocenter of the 2001 mainshock. The presence of high pore-fluid pressure in the seismogenic fault zone could also explain the observed scatter in shear-wave time delays. Further, the coincidence of the N–S trending intrusive bodies (as inferred from tomographic studies in the area) with the N–S direction of regional maximum horizontal compressional stress supports the interpretation of stress-aligned vertical extensive-dilatant anisotropic (EDA) cracks. The depth distribution of the estimated anisotropy (1–10.76%), b-values and stress drop values suggests an increase at 18–30 km depths, which could be attributed to high pore-fluid pressures resulting from a fluid-filled fractured rock matrix or open micro-cracks (characterized by high crack density and high porosity) coinciding with a low velocity zone (at 18–30 km depths) as delineated from tomographic studies in the area.     

A comparison of cross-hole electrical and seismic data in fractured rock

Cross‐hole anisotropic electrical and seismic tomograms of fractured metamorphic rock have been obtained at a test site where extensive hydrological data were available. A strong correlation between electrical resistivity anisotropy and seismic compressional‐wave velocity anisotropy has been observed. Analysis of core samples from the site reveal that the shale‐rich rocks have fabric‐related average velocity anisotropy of between 10% and 30%. The cross‐hole seismic data are consistent with these values, indicating that observed anisotropy might be principally due to the inherent rock fabric rather than to the aligned sets of open fractures. One region with velocity anisotropy greater than 30% has been modelled as aligned open fractures within an anisotropic rock matrix and this model is consistent with available fracture density and hydraulic transmissivity data from the boreholes and the cross‐hole resistivity tomography data. However, in general the study highlights the uncertainties that can arise, due to the relative influence of rock fabric and fluid‐filled fractures, when using geophysical techniques for hydrological investigations.     

地球内核及其边界的结构特征和动力学过程

现代地震学展现出了一个复杂的地球内核内部和表面结构.地球内核内部结构的主要特征表现为其地震波速度和衰减呈现各向异性,且各种结构（速度、衰减和各向异性）均呈现东西半球差异,而内核表面的新发现则包括其局部区域存在起伏的地形和固液并存的糊状层.地球内核压缩波速度和衰减均呈现以地球旋转轴为轴的柱对称各向异性,沿地球旋转轴方向传播的压缩波比沿赤道方向传播的压缩波传播更快且衰减更强烈.同时,内核各向异性结构随深度而变化：内核顶部约100~400 km接近各向同性,而在内核最深处300~600 km内则可能存在一个具有不同各向异性特征的内内核.地球内核的东西半球差异表现在多方面：在内核顶部~100 km厚度内,东半球的各向同性速度比西半球快约0.8%,东半球具有较强的衰减（Q=250）,而西半球则具有较弱的衰减（Q=600）;西半球的顶部各向同性层厚度约为100 km,而东半球顶部各向同性层厚度则约为400 km;在各向同性层底下,西半球具有较强的各向异性（~4%）,而东半球则具有较弱的各向异性（~0.7%）.地球内核边界在菲律宾海、黄海、西太平洋以及中美洲下方存在1~14 km高的地形起伏,在鄂霍次克海西南部下方存在4~8 km厚的糊状层.地球内核的这些新发现引发了对许多可能的新物理机制的探讨,也促使我们重新评估我们对外核成分、外核热化学对流、内核凝固过程和地球磁场驱动力的认识.这些结果表明内核凝固过程和地球磁场的热和化学驱动力远比传统观念认为的横向均匀分布复杂得多.内核西半球可能不断凝固并释放潜热和轻元素,而东半球则可能不断熔化并吸收潜热和轻元素,外核对流的驱动力在东西半球可能截然不同,甚至呈现相反方向.这些凝固与熔化交替过程也发生在局部地形起伏区域.在糊状层区域,地球内核凝固释放潜热和化学能,而在大部分无糊状地区,内核凝固只释放潜热.     

Quantifying Damage, Saturation and Anisotropy in Cracked Rocks by Inverting Elastic Wave Velocities

Crack damage results in a decrease of elastic wave velocities and in the development of anisotropy. Using non-interactive crack effective medium theory as a fundamental tool, we calculate dry and wet elastic properties of cracked rocks in terms of a crack density tensor, average crack aspect ratio and mean crack fabric orientation from the solid grains and fluid elastic properties. Using this same tool, we show that both the anisotropy and shear-wave splitting of elastic waves can be derived. Two simple crack distributions are considered for which the predicted anisotropy depends strongly on the saturation, reaching up to 60% in the dry case. Comparison with experimental data on two granites, a basalt and a marble, shows that the range of validity of the non-interactive effective medium theory model extends to a total crack density of approximately 0.5, considering symmetries up to orthorhombic. In the isotropic case, Kachanov's (1994) non-interactive effective medium model was used in order to invert elastic wave velocities and infer both crack density and aspect ratio evolutions. Inversions are stable and give coherent results in terms of crack density and aperture evolution. Crack density variations can be interpreted in terms of crack growth and/or changes of the crack surface contact areas as cracks are being closed or opened respectively. More importantly, the recovered evolution of aspect ratio shows an exponentially decreasing aspect ratio (and therefore aperture) with pressure, which has broader geophysical implications, in particular on fluid flow. The recovered evolution of aspect ratio is also consistent with current mechanical theories of crack closure. In the anisotropic cases—both transverse isotropic and orthorhombic symmetries were considered—anisotropy and saturation patterns were well reproduced by the modelling, and mean crack fabric orientations we recovered are consistent with in situ geophysical imaging. Our results point out that: (1) It is possible to predict damage, anisotropy and saturation in terms of a crack density tensor and mean crack aspect ratio and orientation; (2) using well constrained wave velocity data, it is possible to extrapolate the contemporaneous evolution of crack density, anisotropy and saturation using wave velocity inversion as a tool; 3) using such an inversion tool opens the door in linking elastic properties, variations to permeability.     

地球内核顶部300km速度和衰减各向异性的区域变化

衰减结构是地球内核的重要性质,它可以与地球内核的速度结构结合,对内核的形成和演化机制提供更全面的信息.本文系统收集了1991年到2014年全球、区域和临时地震台网的PKPDF和PKPBC数据,研究了澳大利亚、非洲和太平洋中部下方内核顶部300km的速度和衰减各向异性结构.速度结果表明,澳大利亚下方内核的速度没有明显的各向异性,但是非洲和太平洋中部下方的内核具有明显的各向异性,且非洲的速度各向异性强于太平洋中部.同时,相对于AK135模型,澳大利亚的平均速度快0.5%,而非洲和太平洋中部的平均速度与参考模型没有明显差异.对于内核的衰减结构,我们得到以下结果:1)在东西方向,内核顶部200km左右的区域,澳大利亚的衰减最强(Q值在400左右),非洲和太平洋中部的Q值分别在600和500左右.2)澳大利亚下方的内核衰减没有明显的各向异性,非洲和太平洋中部下方的内核衰减存在明显的各向异性.此外,内核在非洲地区的衰减各向异性强于太平洋中部的各向异性.3)最后,内核中三个区域的速度和衰减具有良好的相关性,即高/低速对应于高/低衰减.考虑到以上结果以及三个区域的位置,我们认为内核顶部的速度和衰减结构都存在区域变化,而不是简单的半球变化.这种区域变化很可能是由于核幔边界热结构的不均一性和内核耦合,使得内核顶部的不同区域在形成过程中受不同的变形影响,从而形成铁晶体不同的生长和排列,引发了不同的各向异性特征.     

应力诱导各向异性介质中地震波的传播

主要讨论了应力变化如何影响各向异性介质中波速度的问题。推导了一般各向异性介质在初始应力下的Christoffel方程,得到介质中3种波的相速度和初始应力的关系表达式;通过实验数据验证了单轴应力能够诱导各向异性,当施加单轴应力时,速度在沿应力的方向增加最大,在垂直应力的方向增加最小,实验结果与理论推导一致;用Christoffel方程的数值解模拟在3种对称情况下的弹性各向异性介质中初始应力对波速度的影响。数值结果表明:初始应力对各向异性介质中波传播速度的影响,随着各向异性强度的增加而增大,而且速度越慢,影响越大。     

Magma dynamics, crystallization, and chemical differentiation of the 1959 Kilauea Iki lava lake, Hawaii, revisited

Using constraints from an extensive database of geological and geochemical observations along with results from fluid mechanical studies of convection in magma chambers, we identify the main physical processes at work during the solidification of the 1959 Kilauea Iki lava lakes. In turn, we investigate their quantitative influence on the crystallization and chemical differentiation of the magma, and on the development of the internal structure of the lava lake. In contrast to previous studies, vigorous stirring in the magma, driven predominately by the descent of dense crystal-laden thermal plumes from the roof solidification front and the ascent of buoyant compositional plumes due to the in situ growth of olivine crystals at the floor, is predicted to have been an inevitable consequence of very strong cooling at the roof and floor. The flow is expected to have caused extensive but imperfect mixing over most of the cooling history of the magma, producing minor compositional stratification at the roof and thermal stratification at the floor. The efficient stirring of the large roof cooling is expected to have resulted in significant internal nucleation of olivine crystals, which ultimately settled to the floor. Additional forcing due to either crystal sedimentation or the ascent of gas bubbles is not expected to have increased significantly the amount of mixing. In addition to convection in the magma, circulation driven by the convection of buoyant interstitial melt in highly permeable crystal-melt mushes forming the roof and the floor of the lava lake is envisaged to have produced a net upward flow of evolved magma from the floor during solidification. In the floor zone, mush convection may have caused the formation of axisymmetric chimneys through which evolved magma drained from deep within the floor into the overlying magma and potentially the roof. We hypothesize that the highly evolved, pipe-like ‘vertical olivine-rich bodies’ (VORBs) [Bull. Volcanol. 43 (1980) 675] observed in the floor zone, of the lake are fossil chimneys. In the roof zone, buoyant residual liquid both produced at the roof solidification front and gained from the floor as a result of incomplete convective mixing is envisaged to have percolated or ‘leaked‘ into the overlying highly-permeable cumulate, displacing less buoyant interstitial melt downward. The results from Rayleigh fractionation-type models formulated using boundary conditions based on a quantitative understanding of the convection in the magma indicate that most of the incompatible element variation over the height of the lake can be explained as a consequence of a combination of crystal settling and the extensive but imperfect convective mixing of buoyant residual liquid released from the floor solidification front. The remaining chemical variation is understood in terms of the additional influences of mush convection in the roof and floor on the vertical distribution of incompatible elements. Although cooling was concentrated at the roof of the lake, the floor zone is found to be thicker than the roof zone, implying that it grew more quickly. The large growth rate of the floor is explained as a consequence of a combination of the substantial sedimentation of olivine crystals and more rapid in situ crystallization due to both a higher liquidus temperature and enhanced cooling resulting from imperfect thermal and chemical mixing.     

转换波在储层预测中的研究分析

岩石的各向异性与油气的运移和储集有着密切关系.在油气勘探和油气田开发中,横波对各向异性的敏感性具有重要价值,它可提供一些其他方法无法获取的新信息.纵波在地下传播时,当通过路径的岩石存在各向异性时,会在波阻抗界面产生纵波和横波.这相当于纵波震源同时激发出纵波和横波,利用这些信息就可以对地下岩石的物性进行研究.本文以有限差分为基础,以地下油气田的储藏地质特性为对象,利用弹性波波动方程的传播特性, 研究P_SV转换波,分析其传播特征,用来指导其在储层预测中的应用.通过研究分析可以看出,P_SV转换波通过含油气介质时受影响比较少,能够得到比纵波好的多的成像资料.     

部分熔融强化了青藏高原地壳的各向异性?

深部岩石的弹性波各向异性是人们了解地壳深部构造特征,分析其成因,探讨其动力学意义的重要岩石物理参数。实验结果表明由矿物晶格定向分布(LPO)所引起的地壳岩石平均各向异性强度通常不超过5%,远不足以解释在青藏高原地壳中所观测到的弹性波各向异性之强度。模拟结果显示,熔体的定向分布(MPO)能够引起强烈的弹性波各向异性。例如,当熔体的形态因子(α)值介于0.1～0.5之间,熔融程度为5%～10%时,由定向分布的酸性熔体囊所产生的各向异性强度可以达到2%～10%(P波)、2.2～40%(S波)。众多研究资料显示,青藏高原—川滇西部具有加厚的中、上地壳和高地热梯度,低度部分熔融作用在其深部地壳中广泛存在。低度熔体在构造应力作用下的定向分布可能是造成该地区深部地壳存在异常强的各向异性层的重要原因。这暗示目前在青藏高原—川滇西部探测到的异常强的区域性各向异性层是具有部分熔融成因的强烈构造变形带。该构造变形带具有潜在的"解耦"功能,并作为地壳浅部刚性层(块体)的底界协调着块体与其下伏地壳或岩石圈地幔的差异运动。     

Elastic anisotropy of ferromagnesian post-perovskite in Earth's D” layer

We report experimental observation of a sizable elastic anisotropy in a polycrystalline sample of ferromagnesian silicate in post-perovskite (ppv) structure. Using a novel composite X-ray transparent gasket to contain and synthesize ppv in a panoramic diamond-anvil cell along with oblique X-ray diffraction geometry, we observed the anisotropic lattice strain and {1 0 0} or {1 1 0} slip-plane texture in the sample at 140 GPa. We deduced the elasticity tensor (c_ij), orientation-dependent compressional wave velocities, polarization-dependent shear-wave velocities, and the velocity anisotropy of the silicate ppv. Our results are consistent with calculations and indicate that with sufficient preferred orientation, the elastic anisotropy of this phase can produce large shear-wave splitting.     

A method for inversion of layered shear wavespeed azimuthal anisotropy from Rayleigh wave dispersion using the Neighborhood Algorithm

Seismic anisotropy provides important constraints on deformation patterns of Earth's material. Rayleigh wave dispersion data with azimuthal anisotropy can be used to invert for depth-dependent shear wavespeed azimuthal anisotropy, therefore reflecting depth-varying deformation patterns in the crust and upper mantle. In this study, we propose a two-step method that uses the Neighborhood Algorithm(NA) for the point-wise inversion of depth-dependent shear wavespeeds and azimuthal anisotropy from Rayleigh wave azimuthally anisotropic dispersion data. The first step employs the NA to estimate depthdependent VSV(or the elastic parameter L) as well as their uncertainties from the isotropic part Rayleigh wave dispersion data. In the second step, we first adopt a difference scheme to compute approximate Rayleigh-wave phase velocity sensitivity kernels to azimuthally anisotropic parameters with respect to the velocity model obtained in the first step. Then we perform the NA to estimate the azimuthally anisotropic parameters Gc/L and Gs/L at depths separately from the corresponding cosine and sine terms of the azimuthally anisotropic dispersion data. Finally, we compute the depth-dependent magnitude and fast polarization azimuth of shear wavespeed azimuthal anisotropy. The use of the global search NA and Bayesian analysis allows for more reliable estimates of depth-dependent shear wavespeeds and azimuthal anisotropy as well as their uncertainties.We illustrate the inversion method using the azimuthally anisotropic dispersion data in SE Tibet, where we find apparent changes of fast axes of shear wavespeed azimuthal anisotropy between the crust and uppermost mantle.     

Reference ellipsoids for anisotropic media

Perturbation methods are common tools for describing wave propagation in weakly anisotropic media. The anisotropic medium is replaced by an average isotropic medium where wave propagation can be treated analytically and the correction for the effect of anisotropy is computed by perturbation techniques. This works well for anisotropies of up to 10%. Some materials (e.g. shales), however, can exhibit a much stronger anisotropy. In this case a background is required which still can be treated analytically but is applicable to stronger P-wave anisotropy. We present an averaging technique to compute a best-fitting ellipsoidal medium to an arbitrary anisotropic medium. Ellipsoidal media are sufficiently simple for analytical expressions to be available for many applications and allow consideration of strong P-wave anisotropy. The averaging of the arbitrary anisotropic medium can be carried out globally (i.e. for the whole sphere) or sectorially (e.g. for seismic waves propagating predominantly in the vertical direction). We derive linear relationships for the coefficients of the ellipsoid which depend on the elastic coefficients of the anisotropic medium. We also provide specifications for best-fitting elliptical and best-fitting isotropic media. Numerical examples for different rocks demonstrate the improved approximation of the anisotropic model obtained using the formulae derived, compared with the conventionally used average isotropic medium.