Weak-contrast approximation of the elastic scattering matrix in anisotropic media

The plane-wave reflection and transmission coefficients at a plane interface between two anisotropic media constitute the elements of the elastic scattering matrix. For a 1-D anisotropic medium the eigenvector decomposition of the system matrix of the transformed elasto-dynamic equations is used to derive a general expression for the scattering matrix. Depending on the normalization of the eigenvectors, the expressions give scattering coefficients for amplitudes or for vertical energy flux.Computing the vertical slownesses and the corresponding polarizations, the eigenvector matrix and its inverse can be found. We give a simple formula for the inverse, regardless of the normalization of the eigenvectors. When the eigenvectors are normalized with respect to amplitudes of displacement (or velocity), the calculation of the scattering matrix for amplitudes is simplified.When the relative changes in all parameters are small, a weak-contrast approximation of the scattering matrix, based on the exactly determined polarization vectors in an average medium, is obtained. The same approximation is also derived directly from the transformed elasto-dynamic equations for a smooth vertically inhomogeneous medium, proving the consistency of the approximation.For monoclinic media, with the mirror symmetry plane parallel to the interface, the approximative scattering matrix is given in terms of analytic expressions for the non-normalized eigenvectors and vertical slownesses. For transversely isotropic media with a vertical axis of symmetry (VTI) and isotropic media, explicit solutions for the weak-contrast approximations of the scattering matrices have been obtained. The scattering matrix for amplitudes for isotropic media is well known. The scattering matrix for vertical energy flux may have applications in AVO analysis and inversion due to the reciprocity of the reflection coefficients for converted waves.Numerical examples for monoclinic and VTI media provide good agreement between the approximative and the exact reflection matrices. It is, however, expected that the approximations cannot be used when the symmetry properties of the two media are very different. This is because the approximation relies on a small relative contrast between the eigenvectors in the two media.Presented at the Workshop Meeting on Seismic Waves in Laterally Inhomogeneous Media, Castle of Trest, Czech Republic, May 22–27, 1995.     

The elastic properties of single-crystal fayalite as determined by dynamical measurement techniques

We present new elasticity measurements on single-crystal fayalite and combine our results with other data from resonance, pulse superposition interferometry, and Brillouin scattering to provide a set of recommended values for the adiabatic elastic moduliC _ij and their temperature variations. We use a resonance method (RPR) with specimens that were previously investigated by pulse superposition experiments. The nineC _ij of fayalite are determined from three new sets of measurements. One set of our newC _ij data is over the range 300–500 K. We believe that the relatively large discrep ncies found in someC _ij are due in large part to specimen inhomogeneities (chemical and microstructural) coupled with differences in the way various techniques sample, rather than only systematic errors associated with experimental procedures or in the preparations of the specimens.Our recommendeaC _ij's (GPa) and (C _ij/T) _p (GPa/K) are: The resulting values for the isotropic bulk and shear moduli,K _s and , and their temperature derivatives are:K _s=134(4) GPa; =50.7(0.3) GPa; (K _s/T) _p =–0.024(0.005) GPa/K; and (/T) _p =–0.013(0.001) GPa/K. An important conclusion is thatK _s increases as the Fe/(Fe+Mg) ratio in olivine is increased.     

Scattering of elastic waves by a fracture zone containing randomly distributed cracks

We theoretically study the scattering ofP, SV andSH waves by a zonal distribution of cracks, which simulates a fault fracture zone. An investigation is conducted how the geometrical properties of the crack distribution and the frictional characteristics of the crack surface are reflected in the attenuation and dispersion of incident waves, as well as in the amplitudes of the transmitted and reflected waves from the zone. If the crack distribution within the fault zone changes temporally during the preparation process of the expected earthquake, it will be important for earthquake prediction to monitor it, utilizing the scattering-induced wave phenomena.We consider the two-dimensional problem. Aligned cracks with the same length are assumed to be randomly distributed in a zone with a finite width, on which elastic waves are assumed to be incident. The distribution of cracks is assumed to be homogeneous and sparse. The crack surface is assumed to be stress-free, or to undergo viscous friction; the latter case simulates fluid-filled cracks. The opening displacement of the crack is assumed to be negligibly small. The idea of the mean wave formalism is employed in the analysis, and Foldy's approximation is assumed.When the crack surface is stress-free, it is commonly observed for every wave mode (P, SV andSH) that the attenuation coefficientQ ^–1 peaks aroundka1, the phase velocity is almost independent ofk in the rangeka<1 and it increases monotonically withk in the rangeka>1, wherek is the intrinsicS wavenumber anda is the half length of the crack. The effect of the friction is to shift the peak ofQ ^–1 and the corner of the phase velocity curve to the low wavenumber range. The high wavenumber asymptote ofQ ^–1 is proportional tok ^–1 independently of model parameters and the wave modes. If the seismological observation thatQ ^–1 ofS waves has a peak at around 0.5 Hz in the earth's crust is combined with our results, the upper limit of crack size within the crust is estimated about 4 km. The information regarding the transmitted and reflected waves, such as the high wavenumber limit of the amplitude of the transmitted wave etc., allows estimation of the strength of the friction.     

Break out resistance of inclined anchors in sand

Inclined anchors are widely used in a variety of civil engineering problems to resist oblique loads, most relevant being in transmission towers and in rocks and dams for structural strengthening. In the present study, the breakout resistance of the inclined anchors in sand has been worked out using limit equilibrium approach. The breakout resistance has been calculated for different soil friction angles with varying relative depth ratio and anchor inclination. The break out factor increases continuously with the inclination of the anchor. A comparison of predicted values of break out resistance of anchors by the proposed analysis with the experimental values as reported by the other researchers showed reasonably good agreement.     

加固含水平软弱层高陡边坡预应力锚索倾角的取值范围

用直线和曲线滑面两种模型，模拟预应力锚索加固含有水平软弱层的高陡边坡的潜在滑动面，通过对锚索加固的力学机理分析来阐述预应力在滑动面主滑方向上产生分力的原因。从锚索受力角度上对锚索倾角的取值范围进行探讨，结合实际造孔过程有不同程度的孔斜或弯曲以及获取较好的锚固灌浆效果，就仰角锚固和俯角锚固两种锚固条件分别得出锚索倾角的取值范围，并以此来判定锚索的预应力是否会在滑动面走向方向上产生不利于坡体稳定的下滑力。     

Retrieval of stratospheric O3 and NO2 vertical profiles using zenith scattered light observations

Daily zenith scattered light intensity observations were carried out in the morning twilight hours using home-made UV-visible spectrometer over the tropical station Pune (18‡31′, 73‡51′) for the years 2000–2003. These observations are obtained in the spectral range 462–498 nm for the solar zenith angles (SZAs) varying from 87‡ to 91.5‡. An algorithm has been developed to retrieve vertical profiles of ozone (O₃) and nitrogen dioxide (NO₂) from ground-based measurements using the Chahine iteration method. This retrieval method has been checked using measured and recalculated slant column densities (SCDs) and they are found to be well matching. O₃ and NO₂ vertical profiles have been retrieved using a set of their air mass factors (AMFs) and SCDs measured over a range of 87–91.5‡ SZA during the morning. The vertical profiles obtained by this method are compared with Umkehr profiles and ozonesondes and they are found to be in good agreement. The bulk of the column density is found near layer 20–25 km. Daily total column densities (TCDs) of O₃ and NO₂ along with their stratospheric and tropospheric counterparts are derived using their vertical profiles for the period 2000–2003. The total column, stratospheric column and tropospheric column amounts of both trace gases are found to be maximum in summer and minimum in the winter season. Increasing trend is found in column density of NO₂ in stratospheric, tropospheric and surface layers, but no trend is observed in O₃ columns for above layers during the period 2000–2003