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1.
Most previous earth models used to calculate viscoelastic relaxation after the removal of the Late Pleistocene ice loads implicitly assume that there is no exchange of mass across the mantle density discontinuities on periods of tens of thousands of years (the material boundary formulation). In the present study, simple incompressible models are used to determine the Earth's behaviour in the case where the density discontinuity remains at a constant pressure rather than deforming with the material (the isobaric boundary formulation). The calculation of the movement of the boundary is more rigorous than in earlier studies and uses the local incremental pressure calculated at the depth of the boundary and allows for the vertical deformation caused by the change in volume as material changes phase. It is shown that the buoyancy modes associated with the density discontinuities decrease in strength and increase in relaxation time analogous to what results when the density contrast is reduced. Also, two viscoelastic modes arise from an isobaric boundary, which is also predicted when there is a contrast in rigidity or viscosity across a material boundary. The difference in predicted radial deformation between the isobaric boundary model and the material boundary model is largest for long-wavelength loads for which the material incremental pressure at depth is largest. If the isobaric boundary model is appropriate for the treatment of the mineral phase changes in the mantle on glacial rebound timescales, then previous inferences of the deep-mantle to shallow-mantle viscosity ratio based on large-scale deformation (spherical harmonic degree < 10) of the Earth and including data from the early part of the glacio-isostatic uplift are too small.  相似文献   

2.
The degree-one deformation of the Earth (and the induced discrepancy between the figure centre and the mass centre of the Earth) is computed using a theoretical approach (Love numbers formalism) at short timescales (where the Earth has an elastic behaviour) as well as at long timescales (where the Earth has a viscoelastic or quasi-fluid behaviour). For a Maxwell model of rheology, the degree-one relaxation modes associated with the viscoelastic Love numbers have been investigated: the Mo mode does not exist and there is only one transition mode (instead of two) generated by a viscosity discontinuity.
The translations at each interface of the incompressible layers of the earth model [surface, 670 km depth discontinuity, core-mantle boundary (CMB) and inner-core boundary (ICB)] are computed. They are elastic with an order of magnitude of about 1 mm when the excitation source is the atmospheric continental loading or a magnetic pressure acting at the CMB. They are viscoelastic when the earth is submitted to Pleistocene deglaciation, with an order of magnitude of about 1 m. In a quasi-fluid approximation (Newtonian fluid) because of the mantle density heterogeneity their order of magnitude is about 100 m (except for the ICB, which is in quasi-hydrostatic equilibrium at this timescale).  相似文献   

3.
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   

4.
Analytical approach for the toroidal relaxation of viscoelastic earth   总被引：1，自引：0，他引：1  
This paper is concerned with post-seismic toroidal deformation in a spherically symmetric, non-rotating, linear-viscoelastic, isotropic Maxwell earth model. Analytical expressions for characteristic relaxation times and relaxation strengths are found for viscoelastic toroidal deformation, associated with surface tangential stress, when there are two to five layers between the core–mantle boundary and Earth's surface. The multilayered models can include lithosphere, asthenosphere, upper and lower mantles and even low-viscosity ductile layer in the lithosphere. The analytical approach is self-consistent in that the Heaviside isostatic solution agrees with fluid limit. The analytical solution can be used for high-precision simulation of the toroidal relaxation in five-layer earths and the results can also be considered as a benchmark for numerical methods. Analytical solution gives only stable decaying modes—unstable mode, conjugate complex mode and modes of relevant poles with orders larger than 1, are all excluded, and the total number of modes is found to be just the number of viscoelastic layers between the core–mantle boundary and Earth's surface—however, any elastic layer between two viscoelastic layers is also counted. This confirms previous finding where numerical method (i.e. propagator matrix method) is used. We have studied the relaxation times of a lot of models and found the propagator matrix method to agree very well with those from analytical results. In addition, the asthenosphere and lithospheric ductile layer are found to have large effects on the amplitude of post-seismic deformation. This also confirms the findings of previous works.  相似文献   

5.
Summary. The Backus–Gilbert theory is extended to the case when the models are piecewise continuous vector functions of depth with variable discontinuity locations. In addition, some of the layers may be represented by linear combinations of known functions. For such layers only a finite number of discrete parameters is to be determined. The iterative process for obtaining a model satisfying the data is convergent, the numerical procedure by which the iterations are performed being equivalent to the method of spectral decomposition for continuous structures. The method of obtaining the Fréchet kernels by using the first perturbation of the differential system satisfied by the corresponding functionals is shown to be valid.
The theory is applied to the magnetotelluric problem, Fréchet kernels being calculated for isotropic and non-isotropic structures.
A few numerical examples are described.  相似文献   

6.
A time-varying spherical harmonic model of the palaeomagnetic field for 0–7 ka is used to investigate large-scale global geomagnetic secular variation on centennial to millennial scales. We study dipole moment evolution over the past 7 kyr, and estimate its rate of change using the Gauss coefficients of degree 1 (dipole coefficients) from the CALS7K.2 field model and by two alternative methods that confirm the robustness of the predicted variations. All methods show substantial dipole moment variation on timescales ranging from centennial to millennial. The dipole moment from CALS7K.2 has the best resolution and is able to resolve the general decrease in dipole moment seen in historical observations since about 1830. The currently observed rate of dipole decay is underestimated by CALS7K.2, but is still not extraordinarily strong in comparison to the rates of change shown by the model over the whole 7 kyr interval. Truly continuous phases of dipole decrease or increase are decadal to centennial in length rather than longer-term features. The general large-scale secular variation shows substantial changes in power in higher spherical harmonic degrees on similar timescales to the dipole. Comparisons are made between statistical variations calculated directly from CALS7K.2 and longer-term palaeosecular variation models: CALS7K.2 has lower overall variance in the dipole and quadrupole terms, but exhibits an imbalance between dispersion in   g 12  and   h 12  , suggestive of long-term non-zonal structure in the secular variations.  相似文献   

7.
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8.
b
We present expressions in a spherical harmonic framework for the gravitational potential of discrete point, surface, and volume mass elements located at any depth within a sphere. Through analysis of the spherical harmonic spectrum, insight is gained into the properties of the potentials arising from a variety of mass distributions. A point mass at the surface of a sphere displays the richest harmonic spectrum in all degrees; spectra become increasingly reddened as the source mass is distributed through larger elements of area or volume, or is located at greater depths below the surface of the reference sphere. The spectra of dipolar distributions, useful in representing compensated masses, are depressed, especially in the low harmonic degrees, relative to the spectra of monopole elements.  相似文献   

9.
We combine Global Positioning System (GPS) measurements with forward modelling of viscoelastic relaxation and after-slip to study the post-seismic deformation of the 1997 Umbria-Marche (Central Apennines) moderate shallow earthquake sequence. Campaign GPS measurements spanning the time period 1999–2003 are depicting a clear post-seismic deformation signal. Our results favour a normal faulting rupture model where most of the slip is located in the lower part of the seismogenic upper crust, consistent with the rupture models obtained from the inversion of strong motion data. The preferred rheological model, obtained from viscoelastic relaxation modelling, consists of an elastic upper crust, underlain by a transition zone with a viscosity of 1018 Pa s, while the rheology of deeper layers is not relevant for the observed time-span. Shallow fault creep and after-slip at the base of the seismogenic upper crust are the first order processes behind the observed post-seismic deformation. The deep after-slip, below the fault zone at about 8 km depth, acting as a basal shear through localized time-dependent deformation, identifies a rheological discontinuity decoupling the seismogenic upper crust from the low-viscosity transition zone.  相似文献   

10.
The modern geomagnetic field is usually expressed as a spherical harmonic expansion. Although the palaeomagnetic record is very incomplete in both space and time, sufficient data are available from a span of ages to generate time-averaged spherical harmonic field models with many degrees of freedom. Here three data sets are considered: directional measurements from lavas, inclination measurements from ocean sediments, and intensity measurements from lavas. Individual data are analysed, as well as site-averages, using the same methods that have been developed for the modern field, to give models for the past 5 Myr. The normal-polarity field model has an axial-dipole intensity similar to that of the modern-day field, whilst the equatorial-dipole component is very much smaller. The field is not axisymmetric, but shows flux concentrations at the core's surface under Canada and Siberia similar to those observed in the field over historical timescales. Tests on synthetic data show that it is unlikely that these similarities result from the overprinting of the palaeomagnetic field due to inadequate cleaning of the samples. The reverse-polarity field model does not show such obvious features, but this may be due to the sparsity of the data.
The patterns observed in the normal-polarity field, with persistent features in the northern hemisphere and a smooth southern hemisphere, could be explained if the present pattern of secular variation is typical of the past several million years. This would reveal itself as large variations over time in the direction of the magnetic vector in regions of high secular variation, with relatively little change over quieter regions. However, we have been unable to find any evidence for a geographical pattern of secular variation in the data.  相似文献   

11.
The technique of bootstrapped discrete scale invariance allows multiple time-series of different observables to be normalized in terms of observed and predicted characteristic timescales. A case study is presented using the SINT2000 time-series of virtual axial dipole moment, which spans the past 2 Myr. It is shown that this sequence not only bears a clear signature of a preferred timescale of about 55.6 Ka, but additionally predicts similar features (of shorter and longer duration) that are actually observed on the timescales of historical secular variation and dipole reversals, respectively. In turn, the latter two empirical sources both predict the characteristic timescale found in the dipole intensity sequence. These communal scaling characteristics suggest that a single underlying process could be driving dynamo fluctuations across all three observed timescales, from years to millions of years.  相似文献   

12.
Summary. Finite element models for shallow subduction produce realistic behaviour for a wide variety of mechanical strength and density distributions. Characteristic displacements are found to occur even without a discrete low-strength megathrust if there is a high-density subducted plate to localize lithospheric compression. A high-density plate is itself unnecessary in the presence of a low-strength megathrust and regional compression.
Successful finite element models produce an outer arc at the top of the trench slope, and forearc basin with geometry characteristic of natural analogues. These structural features occur by upward inelastic bending of the lithospheric wedge overlying the megathrust. This mechanically unstable behaviour may dissipate significant energy and cause the megathrust to migrate continuously by accretion, tectonic erosion, or abandonment and reinitiation farther offshore. Upward bending in the overriding plate is promoted by low megathrust dip, low megathrust shear strength, and high horizontal compression in the overriding plate.  相似文献   

13.
The geodynamo exhibits a bewildering gamut of time-dependent fluctuations, on timescales from years to at least hundreds of millions of years. No framework yet exists that comprises all and relates each to all others in a quantitative sense. The technique of bootstrapped discrete scale invariance quantifies characteristic timescales of a process, based upon log-periodic fits of modulated power-law scaling of size-ranked event durations. Four independent geomagnetic data sets are analysed therewith, each spanning different timescales: the sequence of 332 known dipole reversal intervals (0–161 Ma); dipole intensity fluctuations (0–2 Ma); archeomagnetic secular variation (5000 B.C.–1950 A.D.); and historical secular variation (1590–1990 A.D.).
Six major characteristic timescales are empirically attested: circa 1.43 Ma, 56 Ka, and 763, 106, 21 and 3 yr. Moreover, all detected wavelengths and phases of the detected scaling signatures are highly similar, suggesting that a single process underlies all. This hypothesis is reinforced by extrapolating the log-periodic scaling signal of any particular data set to higher timescales than observed, through which predictions are obtained for characteristic scales attested elsewhere. Not only do many confirm one another, they also predict the typical duration of superchrons and geomagnetic jerks. A universal scaling bridge describes the complete range of geodynamo fluctuation timescales with a single power law.  相似文献   

14.
The   M w γ 9.0  2004 December 26 Sumatra-Andaman and   M w = 8.7  2005 March 28 Nias earthquakes, which collectively ruptured approximately 1800 km of the Andaman and Sunda subduction zones, are expected to be followed by vigorous viscoelastic relaxation involving both the upper and lower mantle. Because of these large spatial dimensions it is desirable to fully account for gravitational coupling effects in the relaxation process. We present a stable method of computing relaxation of a spherically-stratified, compressible and self-gravitating viscoelastic Earth following an impulsive moment release event. The solution is cast in terms of a spherical harmonic expansion of viscoelastic normal modes. For simple layered viscoelastic models, which include a low-viscosity oceanic asthenosphere, we predict substantial post-seismic effects over a region several 100s of km wide surrounding the eastern Indian Ocean. We compare observed GPS time-series from ten regional sites (mostly in Thailand and Indonesia), beginning in 2004 December, with synthetic time-series that include the coseismic and post-seismic effects of the 2004 December 26 and 2005 March 28 earthquakes. A viscosity structure involving a biviscous (Burgers body) rheology in the asthenosphere explains the pattern and amplitude of post-seismic offsets remarkably well.  相似文献   

15.
Reduced-complexity models have considerable potential as tools for elucidating river behaviour over periods of 100–104 years and, consequently, for addressing fundamental questions concerning the scale-dependent nature of explanation in geomorphology. This paper proposes a simple subdivision of reduced-complexity models of river behaviour into two categories that mirror methodological developments in fluvial geomorphology over the past 50 years. First, high-resolution cellular approaches that are implemented within a framework that resolves process-form feedbacks at small time and space scales. Second, models that incorporate section-averaged representations of channel geometry and processes, and that are typically underpinned by regime theory and equilibrium concepts. Examples of both model types are presented here, in the form of a cellular representation of stream braiding and a combined lattice-network model of alluvial fan evolution. Simulations conducted using these models demonstrate how small-scale process-form interactions determine the emergence of larger-scale channel and fan morphology and, in so doing, regulate system response to external forcing. In this sense, both models demonstrate that internal feedbacks play a critical role in controlling river responses to environmental change over historic and Holocene timescales. However, both classes of model are characterised by uncertainty in their parameterisation of geomorphic processes, such that internal feedbacks and thresholds for channel response to external forcing may vary substantially between competing models. Methods of refining both approaches are considered, and hybrid models based on lattice-network structures and mechanistic representations of channel process-form interactions are identified as a means of addressing the shortcomings of existing strategies.  相似文献   

16.
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   

17.
Most seismic hazard estimations are based on the assumption of a Poisson process for earthquake occurrence, even though both observations and models indicate a departure of real seismic sequences from this simplistic assumption. Instrumental earthquake catalogues show earthquake clustering on regional scales while the elastic rebound theory predicts a periodic recurrence of characteristic earthquakes on longer timescales for individual events. Recent implementations of time-dependent hazard calculations in California and Japan are based on quasi-periodic recurrences of fault ruptures according to renewal models such as the Brownian Passage Time model. However, these renewal models neglect earthquake interactions and the dependence on the stressing history which might destroy any regularity of earthquake recurrences in reality. To explore this, we investigate the (coupled) stress release model, a stochastic version of the elastic rebound hypothesis. In particular, we are interested in the time-variability of the occurrence of large earthquakes and its sensitivity to the occurrence of Gutenberg–Richter type earthquake activity and fault interactions. Our results show that in general large earthquakes occur quasi-periodically in the model: the occurrence probability of large earthquakes is strongly decreased shortly after a strong event and becomes constant on longer timescales. Although possible stress-interaction between adjacent fault zones does not affect the recurrence time distributions in each zone significantly, it leads to a temporal clustering of events on larger regional scales. The non-random characteristics, especially the quasi-periodic behaviour of large earthquakes, are even more pronounced if stress changes due to small earthquakes are less important. The recurrence-time distribution for the largest events is characterized by a coefficient of variation from 0.6 to 0.84 depending on the relative importance of small earthquakes.  相似文献   

18.
summary . Treatments of geophysical inverse problems have tended to polarize into approaches intended to generate models either described by piecewise continuous functions or with some prior discretization. The two approaches are here developed in parallel, and the ideas of a trade-off between the anticipated error and the attainable level of detail in the model estimate are extended to the discrete case, either with even or uneven discretization.
An alternative approach to specifying the potential resolution of a model is to establish upper and lower bounds on parameter values. Linear programming methods are extended to determine bounds which allow for subjective limits on parameter values.
For a non-linear system the possible resolution may be investigated by estimation procedures based on the full set of successful solutions obtained by Monte-Carlo inversion.  相似文献   

19.
Synthesis of seismic surface waves   总被引：1，自引：0，他引：1  
Summary. The reflection method is used to produce complete sets of Rayleigh wave dispersion curves in a given phase velocity—frequency window for horizontally stratified media, including modes of very high numerical order, and theoretical surface wave seismograms are then synthesized.
The difficulties encountered when attempting to complete large numbers of dispersion curves are discussed. Particular problems arise from models with a low-velocity zone, when curves in a certain portion of the dispersion diagram split into two families, the crustal and the channel modes. The reflection method provides a convenient framework in which to examine this phenomenon heuristically and so devise a method to overcome the difficulties.
Seismograms are produced by mode summation and it is found that body-wave behaviour, as well as surface-wave features can be synthesized. The effect of truncating the mode series at a number comparable with the number of modes used in previous studies is examined. It is found that although the S -coda at longer ranges is not adversely affected, the arrivals attributable to body-waves are severely distorted. This must call into question the validity of synthetic seismograms generated by summation of only a few modes.  相似文献   

20.
Year-round temperature measurements at 1600 m elevation during 1994 in the Asgard Range Antarctica, indicate that the mean annual frost point of the ice-cemented ground, 25 cm below the surface, is -21.7 +/- 0.2 degrees C and the mean annual frost point of the atmosphere is -27.5 +/- 1.0 degrees C. The corresponding mean annual temperatures are -24.9 degrees C and -23.3 degrees C. These results imply that there is a net flux of water vapour from the ice to the atmosphere resulting in a recession of the ice-cemented ground by about 0.4-0.6 mm yr-1. The level of the ice-cemented permafrost is about 12 cm below the level of dry permafrost. The summer air temperatures would have to increase about 7 degrees C for thawing temperatures to just reach the top of the subsurface ice. Either subsurface ice at this location is evaporating over time or there are sporadic processes that recharge the ice and maintain equilibrium over long timescales.  相似文献   

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