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1.
Okada (1992) provided expressions for the displacement and strain fields due to a finite rectangular source in an elastic, homogeneous and isotropic half-space. Starting with these results, we applied the correspondence principle of linear viscoelasticity to derive the quasi-static displacement, strain and stress fields in a viscoelastic, homogeneous and isotropic half-space. We assume that the medium deforms viscoelastically with respect to both the shear and the normal stresses but keeps a constant bulk modulus; in particular, the shear modulus relaxes as Maxwell fluid. We presented the viscoelastic effect on displacement, displacement gradient and stress fields, for a choice of parameter values. The viscoelastic effect due to the sudden dislocation reaches a limit value after about 10 times the Maxwell time. The expressions obtained here provide tools for the study of viscoelastic relaxation of lithosphere associated with seismic and volcanic phenomena.  相似文献   

2.
We investigated time dependent piezomagnetic fields due to volcanic sources embedded in a viscoelastic, homogeneous half-space. Especially in volcanic areas, the presence of inhomogeneous materials and high temperatures produce a lower effective viscosity of the Earth's crust that calls for considering anelastic properties of the medium. Piezomagnetic properties are carried by grains of titano-magnetite, which occupy only a small fraction of ordinary rock volume and are supposed to be elastic, while the non-magnetic surrounding matrix is assumed to be viscoelastic. From all the possible rheological models, we investigated two cases in which the bulk modulus is purely elastic and the shear modulus relaxes as: (i) a Maxwell solid and (ii) a standard linear solid (SLS). We applied the Correspondence Principle to the analytical elastic solutions for pressurized spherical sources and dislocation sources in order to determine the time dependent piezomagnetic fields in a viscoelastic medium. The piezomagnetic field completely vanishes after the relaxation process for a Maxwell rheology, whereas it is found to decrease over time and reach some finite offset value for a SLS rheology. These different behaviours provide helpful hints in understanding the temporal evolution of piezomagnetic anomalies in volcanic regions.  相似文献   

3.
We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P , SV and SH waves.  相似文献   

4.
Using the viscoelastic correspondence principle, we utilize the surface coseismic spheroidal deformation fields (i.e. vertical displacements, potential perturbations and gravity changes) of SNREI earth models caused by four typical types of point dislocation, derived by Sun & Okubo (1993 ), to deduce the fundamental formulas for spheroidal fields relevant to viscoelastic earth models. In computations, we employ a strike-slip dislocation on a vertical plane buried at the bottom of the lithosphere to estimate the maximal viscous relaxation responses to this kind of source that possibly exist on the surface of the earth. We take the seismic moment as 1022  N  m, which is characteristic of an average large earthquake. The numerical results demonstrate that, if we take the viscosity as 1019  Pa  s in the asthenosphere, and 1021  Pa  s in the other mantle layers, the rates of surface vertical displacements and gravity changes within about 2.5° for the 10 postseismic years are respectively 1.5–8.1  cm  yr−1 and 4.0–14.9  μgal  yr−1 : the viscous relaxation for this mantle viscosity profile proceeds much faster than for a constant mantle viscosity of 1021  Pa  s.  相似文献   

5.
Summary. The tidal deformation of a homogeneous viscoelastic sphere due to the gravitational attraction of an external body is calculated. The sphere is modelled as an incompressible Kelvin-Voigt solid. An equation for the displacement field is obtained assuming that strains are small and inertia is negligible. This equation has a series solution in terms of Legendre polynomials. The resulting expression for the displacement field reduces to that for an elastic solid and a viscous fluid in the appropriate limits of the material constants. The first term in the viscoelastic solution is used to calculate the moments induced by tidal deformation assuming a circular orbit. In the absence of obliquity and precession, these moments reduced to a torque about the spin axis. This torque is compared to that predicted by a phase lag analysis. These two approaches are formally equivalent if the tidal dissipation function Q −1 depends in a specific way on the difference of the spin and orbital angular velocities.  相似文献   

6.
We obtain and compare analytical and numerical solutions for ground displacement caused by an overpressurized magma chamber placed in a linear viscoelastic medium composed of a layer over a half-space. Different parameters such as size, depth and shape of the chamber, crustal rheology and topography are considered and discussed. Numerical solutions for an axisymmetric extended source are computed using a finite element method (FEM). Analytical solutions for a point source are obtained using the dislocation theory and the propagator matrix technique. In both cases, the elastic solutions are used together with the correspondence principle of linear viscoelasticity to obtain the solution in the Laplace transform domain. Viscoelastic solutions in the time domain are derived inverting the Laplace transform using the Prony series method. The differences between the results allow us to constrain the applicability of the point source and the flat surface hypothesis, which are usually implicitly assumed when analytical solutions are derived. The effect of the topography is also considered. The results obtained show that neglecting the topographic effects may, in some cases, introduce an error greater than that implicit in the point-source hypothesis. Therefore, for an adequate modelling and interpretation of the time-dependent displacements, topography must be considered.  相似文献   

7.
Effects of mechanical layering on volcano deformation   总被引：1，自引：0，他引：1  
The migration and accumulation of magma beneath volcanoes often causes surface displacements that can be measured by geodetic techniques. Usually, deformation signals are explained using models with uniform mechanical properties. In this paper, we study surface displacements due to magma chamber inflation, using heterogeneous finite element models. We first present a systematic analysis of the influence of mechanical layering, showing that the stiffness contrast significantly affects the entity and the pattern of vertical and radial displacements. Second, as an example we apply the models to interpret ground displacements at Darwin volcano (Galápagos Islands) as revealed by InSAR data in the period 1992–1998. The considered models suggest that geodetic data interpreted using homogeneous models leads to underestimation of the source depth and volume change. Thus, we propose correction factors for the source parameters estimated by homogeneous models, in order to consider a range of variation due to mechanical layering as analysed in this study. The effect of the mechanical heterogeneities affects the correct understanding of geodetic data and also influences the evaluation of a volcanic hazard potential.  相似文献   

8.
Summary. We give a systematic formulation and a rigorous justification of a perturbation technique for the computation of the eigenvalues and eigenfunctions of Love waves (and toroidal oscillations by an appropriate change for variables) in an anelastic medium with a constitutive law modelling geophysical media of current interest such as the Kelvin—Voigt Solid, the Maxwell Solid, the Standard Linear Solid, and the Standard Linear Solid with a continuous spectrum of relaxation times. We develop expressions relating the eigenvalues of eigenfunctions for Love waves in a continuously varying vertically stratified anelastic half-space to the corresponding elastic eigenvalues and eigenfunctions. Analytically, our correspondence principle has the form of a regular perturbation expansion in terms of a parameter for both the eigenvalues and eigenfunctions. The identification of ɛ is motivated by the dissipativity principle of viscoelasticity theory. Moreover, we show that our correspondence principle applies respectively only in the high and low frequency range for the Maxwell and Kelvin—Voigt Solids. Outside of the applicable range of frequencies, our correspondence principle yields no useful information. For the family of Standard Linear Solids it is uniformly applicable for all non-zero frequencies.
We also derive an explicit formula to estimate the radius of convergence of our perturbation expansions. This estimate of the radius of convergence for each eigenvalue and eigenfunction is functionally defined by the constitutive model for the anelastic medium. The estimate is frequency dependent and depends on the separation distance between the eigenvalue and the remainder of the spectrum of the corresponding elastic problem.  相似文献   

9.
We present a semi-analytical solution to the 2-D forward modelling of viscoelastic relaxation in a heterogeneous model consisting of eccentrically nested spheres. Several numerical methods for 2-D and 3-D viscoelastic relaxation modelling have been applied recently, including finite-element and spectral-finite-difference schemes. The present semi-analytical approach provides a model response against which more general numerical algorithms can be validated. The eccentrically nested sphere solution has been tested by comparing it with the analytical solutions for viscoelastic relaxation in a homogeneous sphere and in two concentrically nested spheres, and good agreement was obtained.  相似文献   

10.
论山地气候带和气候型   总被引：4，自引：4，他引：0  
毛政旦 《地理研究》1989,8(3):21-29
本文论述了山地高度气候带与纬度气候带在形式上的联系和物理成因上的区别;在达种认识的基础上提出了划分山地高度气候带的基本原则,并提出山地最基本的气候型有四个,即高山气候,谷地气候,坡地气侯和高原气候。  相似文献   

11.
The energy–flux vector and other energy-related quantities play an important role in various wave propagation problems. In acoustics and seismology, the main attention has been devoted to the time-averaged energy flux of time-harmonic wavefields propagating in non-dissipative, isotropic and anisotropic media. In this paper, we investigate the energy–flux vector and other energy-related quantities of wavefields propagating in inhomogeneous anisotropic viscoelastic media. These quantities satisfy energy-balance equations, which have, as we show, formally different forms for real-valued wavefields with arbitrary time dependence and for time-harmonic wavefields. In case of time-harmonic wavefields, we study both time-averaged and time-dependent constituents of the energy-related quantities. We show that the energy-balance equations for time-harmonic wavefields can be obtained in two different ways. First, using real-valued wavefields satisfying the real-valued equation of motion and stress–strain relation. Second, using complex-valued wavefields satisfying the complex-valued equation of motion and stress–strain relation. The former approach yields simple results only for particularly simple viscoelastic models, such as the Kelvin–Voigt model. The latter approach is considerably more general and can be applied to viscoelastic models of unrestricted anisotropy and viscoelasticity. Both approaches, when applied to the Kelvin–Voigt viscoelastic model, yield the same expressions for the time-averaged and time-dependent constituents of all energy-related quantities and the same energy-balance equations. This indicates that the approach based on complex-valued representation of the wavefield may be used for time harmonic waves quite universally. This study also shows importance of joint consideration of time-averaged and time-dependent constituents of the energy-related quantities in some applications.  相似文献   

12.
We present an analytical form of the layer propagator matrix for the response of a locally incompressible, layered, linear‐viscoelastic sphere to an external load assuming that the initial density stratification ϱ 0( r ) within each layer is parametrized by Darwin's law. From this, we show that the relaxation of a sphere consisting of locally incompressible layers is governed by a discrete set of viscous modes. The explicit dependence of the layer propagator matrix on the Laplace transform variable allows us to determine the amplitudes of the viscous modes analytically. Employing Darwin's parametrization, we construct three simplified earth models with different initial density gradients that are used to compare the effects of the local incompressibility constraint, div ( ϱ 0 u )=0, and the material incompressibility constraint, div  u =0, on viscoelastic relaxation. We show that a locally incompressible earth model relaxes faster than a materially incompressible model. This is a consequence of the fact that the perturbations of the initial density are zero during viscoelastic relaxation of a locally incompressible medium, so that there are no internal buoyancy forces associated with the continuous radial density gradients, only the buoyancy forces generated by internal density discontinuities. On the other hand, slowly decaying internal buoyancy forces in a materially incompressible earth model cause it to reach the hydrostatic equilibrium after a considerably longer time than a locally incompressible model. It is important to note that the approximation of local incompressibility provides a solution for a compressible earth model that is superior to the conventional solutions for a compressible earth with homogeneous layers because it is based on an initial state that is consistent with the assumption of compressibility.  相似文献   

13.
The post-seismic response of a viscoelastic Earth to a seismic dislocation can be computed analytically within the framework of normal-modes, based on the application of propagator methods. This technique, widely documented in the literature, suffers from several shortcomings; the main drawback is related to the numerical solution of the secular equation, whose degree increases linearly with the number of viscoelastic layers so that only coarse-layered models are practically solvable. Recently, a viable alternative to the standard normal-mode approach, based on the Post–Widder Laplace inversion formula, has been proposed in the realm of postglacial rebound models. The main advantage of this method is to bypass the explicit solution of the secular equation, while retaining the analytical structure of the propagator formalism. At the same time, the numerical computation is much simplified so that additional features such as linear non-Maxwell rheologies can be simply implemented. In this work, for the first time, we apply the Post–Widder Laplace inversion formula to a post-seismic rebound model. We test the method against the standard normal-mode solution and we perform various benchmarks aimed to tune the algorithm and to optimize computation performance while ensuring the stability of the solution. As an application, we address the issue of finding the minimum number of layers with distinct elastic properties needed to accurately describe the post-seismic relaxation of a realistic Earth model. Finally, we demonstrate the potentialities of our code by modelling the post-seismic relaxation after the 2004 Sumatra–Andaman earthquake comparing results based upon Maxwell and Burgers rheologies.  相似文献   

14.
The interpretation of geodetic data in volcanic areas is usually based on analytical deformation models. Although numerical finite element (FE) modelling allows realistic features such as topography and crustal heterogeneities to be included, the technique is not computationally convenient for solving inverse problems using classical methods. In this paper, we develop a general tool to perform inversions of geodetic data by means of 3-D FE models. The forward model is a library of numerical displacement solutions, where each entry of the library is the surface displacement due to a single stress component applied to an element of the grid. The final solution is a weighted combination of the six stress components applied to a single element-source. The pre-computed forward models are implemented in a global search algorithm, followed by an appraisal of the sampled solutions. After providing extended testing, we apply the method to model the 1993–1997 inflation phase at Mt Etna, documented by GPS and EDM measurements. We consider four different forward libraries, computed in models characterized by homogeneous/heterogeneous medium and flat/topographic free surface. Our results suggest that the elastic heterogeneities of the medium can significantly alter the position of the inferred source, while the topography has minor effect.  相似文献   

15.
We investigate the effects of lateral heterogeneities in the upper mantle on the calculation of postglacial land uplift. For the model calculations we use a commercial finite-element code, which enables us to solve the equations governing a layered, isotropic. incompressible, Maxwell-viscoelastic half-space with laterally varying layer thicknesses and physical properties. Following previous investigations performed by Sabadini. Yuen & Portney (1986) and Gasperini & Sabadini (1989), we extend their results using a more realistic loading history and different earth models. We then focus our attention on the question whether lateral heterogeneities in the upper mantle can be modelled correctly using a set of homogeneous earth models. To this end, a comparison of model calculations using both laterally homogeneous and heterogeneous earth models is performed.
We find that lateral heterogeneities in the upper mantle significantly influence the calculated postglacial land uplift. The resolving power of relative sea-level observations for the prescribed lateral heterogeneities used in this study is mainly focused on observations around the load margin and outside the glaciated areas, where differences in predicted land uplift between individual models are large enough to be resolved by observations.
We can qualitatively determine lateral heterogeneities in the upper mantle using a set of laterally homogeneous earth models, if the geological structure, for example a continental margin, is known. However, in order to infer the correct values of lithospheric thickness and asthenospheric viscosity, we need to use laterally heterogeneous models.  相似文献   

16.
Previous studies of the wander of the rotation pole associated with the Late Pleistocene glacial cycles indicate that the predicted polar wander speed is sensitive to the density jump at the 670 km discontinuity, the thickness of the elastic lithosphere, and the lower mantle viscosity. In particular, the M1 mode related to the density jump at 670 km depth has been shown to contribute a dominant portion of predicted polar wander speed for sufficiently small lower mantle viscosities. In this study, we examine the sensitivity of polar wander to variations in the viscosity of the viscoelastic lithosphere using simplified compressible Maxwell viscoelastic earth models. Model calculations for earth models with a viscoelastic lithosphere of finite viscosity indicate that the contribution of the M1 mode is similar to those associated with the density discontinuity at the core–mantle boundary (C0 mode) and the lithosphere (L0 mode). We speculate that this is due to the interaction between the M1 mode and the transient mode associated with the viscoelastic lithosphere, which reduces the magnitude of polar wander rates. Therefore, the M1 mode does not contribute a dominant portion of the predicted polar wander speed for earth models with a viscoelastic lithosphere of finite viscosity. In this case, predictions of polar wander speed as a function of lower mantle viscosity exhibit the qualitative form of an 'inverted parabola', as predicted for the J ˙2 curve. We caution, however, that these results are obtained for simplified earth models, and the results for seismological earth models such as PREM may be complicated by the interaction between the M1 mode and the large set of transient modes.  相似文献   

17.
Diatom and geochemical data from Crawford Lake, Ontario, have been used to document limnological responses to periods of cultural disturbance resulting from native Iroquoian occupation of the watershed (1268–1486 AD) and Euro-Canadian agriculture and deforestation (1867 AD–present). Here, we further develop the high-resolution nature of the Crawford Lake sediment record to examine the physical, chemical and biological aspects of limnological response to human disturbances in the lake catchment area with exceptional detail. We report detailed diatom abundance and flux data for individual taxa from Crawford Lake, and further describe the relationship between assemblage composition and environmental conditions using canonical correspondence analysis (CCA). Diatom assemblage data are used to calculate diatom inferred-total phosphorus (DI-TP) concentrations for the past ∼1,000 years. We also examine the diatom community response during and after periods of disturbance by Iroquoian and Euro-Canadian populations, and compare this response to existing geochemical proxies of lake production and new elemental geochemical indicators of catchment area erosion. In particular, we explore the differing limnological response to the two distinct periods of cultural eutrophication and examine the limnological processes that occurred during the period of␣low (or no) human activity (1487–1866 AD), when geochemical indicators of lake production recovered to pre-disturbance conditions, but diatom assemblages notably did not. Our results illustrate the highly susceptible nature of diatom communities to periods of anthropogenic disturbance, and emphasize that ecological indicators (such as diatom assemblages) should be included with other proxies (such as nutrient concentrations and physical characteristics) when assessing disturbance and recovery in lake systems.  相似文献   

18.
We identified, enumerated, and interpreted the diatom assemblages preserved in the surface sediments of 59 lakes located between Whitehorse in the Yukon and Tuktoyaktuk in the Northwest Territories (Canada). The lakes are distributed along a latitudinal gradient that includes several ecoclimatic zones. It also spans large gradients in limnological variables. Thus, the study lakes are ideal for environmental calibration of modern diatom assemblages. Canonical correspondence analysis, with forward selection and Monte Carlo permutation tests, showed that maximum lake depth and summer surface-water temperature were the two environmental variables that accounted for most of the variance in the diatom data. The concentrations of sodium and calcium were also important explanatory variables. Using weighted-averaging regression and calibration techniques, we developed a predictive statistical model to infer lake surface-water temperature, and we evaluated the feasibility of using diatoms as paleoclimate proxies. This model may be used to derive paleotemperature inferences from fossil diatom assemblages at appropriate sites in the western Canadian Arctic.  相似文献   

19.
Summary. It is shown that the formulas for the calculation of surface displacement due to a volume source in a half-space may be derived from a potential of the same form as the gravitational potential. The similarity between these expressions allows the use of methods developed for gravity in the study of surface displacement. As an example of the applicability of the correspondence, an efficient algorithm is developed for the calculation of uplift due to volume expansion within a homogeneous half-space. The method may be generalized to three-dimensional volumes undergoing expansion in a half-space.  相似文献   

20.
Summary. We derive the frequency equation and the condition of existence of Stoneley type waves with antisymmetric vibrations which can be propagated along the interfaces between an internal stratum and two adjacent halfspaces of identical elastic properties, all perfectly elastic, homogeneous and isotropic. The ranges of existence of such waves are next obtained by numerical computation and the results are presented both in tabular form and graphically.
For large values of the frequency, the frequency equation for these waves, like the one for waves with symmetric vibrations discussed in an earlier paper*, reduces to the velocity equation of Stoneley waves propagated along the interface between two halfspaces.
For a low-velocity stratum, the results are similar to those for waves with symmetric vibrations. It is found that, for a pair of media for which waves of some frequency can exist, as the frequency of the waves or the thickness of the stratum is decreased, there is a cut-off value of either below which such waves cannot be propagated.
For a high-velocity stratum, the results are in sharp contrast with those for waves with symmetric vibrations. Stoneley waves of all frequencies can exist for some pairs of materials. As the frequency of the waves or the thickness of the stratum is decreased, that is, as the ratio of the wave-length to the thickness of the stratum is increased, the regions of existence expand until a certain limiting region of existence is obtained when this ratio tends to infinity.
Another difference between waves with symmetric and antisymmetric vibrations is that whereas for the former the phase velocity could not be less than the smaller of the two Rayleigh wave velocities, for the latter there is no theoretical lower limit.  相似文献   

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