The density and Q-factor models based on the new data on the nutations and overtones of the free oscillations of the earth: 2. The results of the numerical modeling of the solution of the inverse problem

In the first part of the paper, we obtained the refined estimates for the periods and Q-factors of the fundamental modes and overtones of spherical and toroidal oscillations with periods longer than 3 min from the data on the free oscillations of the Earth, which were excited by the earthquakes with magnitude 9 that occurred in Sumatra, Japan, and the Sea of Okhotsk. In (Molodenskii et al., 2013), we analyzed the limits of the admissible density distributions in the mantle and liquid core of the Earth, using the data on the amplitudes and phases of the forced nutations, as well as the periods and attenuation factors of the fundamental modes of the free spheroidal and toroidal oscillations of the Earth. These studies were conducted with the fixed values of the total mass and total moment of inertia of the Earth and the fixed distributions of the body seismic waves in the mantle and in the core. The solution was obtained by orthogonalizing the kernels of the integral equations that link the residuals of the observed frequencies and attenuation factors of the free oscillations, as well as the amplitudes and phases of the forced nutations, with the sought densities and Q-factors of the mantle and liquid core. Below, we present the solution of the same problem with allowance for the results obtained in the first part of this paper, namely, the new data on the periods and attenuation factors of the fundamental modes of free oscillations of the Earth and on the periods of the first four overtones of the free spheroidal and toroidal oscillations. Despite the involvement of the new data on the overtones, which have not been considered in our calculations, the weighted root mean square deviations of the theoretical predictions from the observed periods and attenuation factors of the free oscillations, as well as the amplitudes and phases of the forced nutations, have significantly decreased. This is due to (1) the noticeable reduction of the real errors in estimating the parameters of the free oscillations described in the first part of the paper and (2) the inclusion of the quantities determining the depth- and frequency dependences of the Q-factor in the mantle in the set of the independently varied parameters.     

Models of density and mechanical <Emphasis Type="Italic">Q</Emphasis>-factor distributions according to new data on the nutations and free oscillations of the Earth: 2. Comparison with astrometric data

In the first part of the paper [Molodenskii, 2011], we considered the problem of ambiguity in the solution of the inverse problem of retrieval of density distribution in the Earth’s core and mantle and determination of the Q factors in the mantle from the entire set of modern data on seismic velocities (V _p and V _S ), the frequencies f _i and quality factors Q _i of free oscillations of the Earth, and the amplitudes and phases of its forced nutations. We have constructed the model distributions of these parameters, in which the root-meansquared (rms) deviations of all observed values from the predicted ones are much smaller than in the PREM model. Below, we compare the observed amplitudes of the forced nutation with the values predicted by our model. In order to understand how rigid are the constraints imposed by the amplitudes of forced nutation, we not only calculate the deviations of the observed amplitudes of nutation from the predictions by our model but also estimate the changes in these deviations caused by small variations in several parameters of the model. To the parameters to be varied we refer those which have no or barely any effect on the periods and damping constants of free oscillations but have a pronounced effect on the amplitudes of forced nutation. These parameters include (1) the rheological properties of the mantle in the interval of periods from an hour to a day; (2) the dynamical flattening of the liquid core; (3) the dynamic flattening of the solid inner core; (4) the viscosity of the liquid core; and (5) the moment of inertia of the solid inner core. In addition, we estimate the effects of variations in the moment of inertia of the liquid core to be small (±0.2%) and not to affect, within the observation error, the periods of free oscillations. We show that the uncertainty in the model depth distributions of density considerably decreases when the new data on the amplitudes and phases of the forced nutation of the Earth are taken into account. With these data, it is possible to estimate the creep function for the lower mantle in a wide range of periods from a second to a day.     

Models of density and mechanical Q-factor distributions according to new data on the nutations and free oscillations of the Earth: 1. Ambiguity in the inverse problem

Ambiguity in the inverse problem of retrieval of the mechanical parameters of the Earth’s shell and core from the set of data on the velocities V _p and V _S , of longitudinal and transverse seismic body waves, the frequencies f _i and quality factors Q _i , of free oscillations, and the amplitudes and phases of forced nutation is considered. The numerical experiments show that the inverse problem of simultaneous retrieval of the density profile ρ in the mantle-liquid core system and the mechanical quality factor Q _μ of the mantle (if the total mass M and the total mean moment of inertia I of the Earth, and V _p and V _S are constant at all depths) has most unstable solutions. An example of depth distributions of ρ and Q _μ which are alternative to the well-known PREM model is given. In these distributions, the values of M and I and the velocities V _p and V _S at all depths for the period of oscillations T = 1 s exactly coincide with their counterparts yielded by PREM model (T = 1 s); however, the maximum deviations of the ρ and Q _μ profiles from those in the PREM model are about 3% and 40%, respectively; the mass and the moment of inertia of the liquid core are smaller than those for the PREM model by 0.75% and 0.63%, respectively. In this model, the root mean square (rms) deviations of all the measured values of f _i and Q _i from their values predicted by theory are half to third the corresponding values in the PREM model; the values of Δ for natural frequencies of the fundamental tone and overtones of radial oscillations, the fundamental tones of torsional oscillations, and the fundamental tones of spheroidal oscillations, which are measured with the highest relative accuracy, are smaller by a factor of 30, 6.6, and 2 than those in the PREM model, respectively. Such a large ambiguity in the solution of the inverse problem indicates that the current models of the depth distribution of density have relatively low accuracy, and the models of the depth distribution of the mechanical Q in the mantle are extremely unreliable. It is shown that the ambiguity in the models of depth distribution of density considerably decreases after the new data on the amplitudes and phases of the forced nutation of the Earth are taken into account. Using the same data, one may also refine by several times the recent estimates of the creep function for the lower mantle within a wide interval of periods ranging from a second to a day.     

On the admissible range for the mass and inertia moments of the liquid core: I. Solving the inverse problem of nutations and free oscillations of the Earth by decomposition of mechanical parameters in an orthogonalized basis

The results of solving the inverse problem of forced nutations and free oscillations of the Earth by decomposing the Q-factor and small depth variations in density in a system of orthogonal functions are considered. These functions are determined by orthogonalization of the functional derivatives of the observed parameters with respect to the depth distributions of the sought parameters (assuming there are no distributions of the velocities of body seismic waves V _p and V _S with depth and unchanged total mass M and inertia moments I of the Earth). The examples are presented to illustrate the numerical solution of the inverse problem on finding the density distributions in the mantle and core of the Earth using orthogonalization of the integral constraints for the probable depth distributions of density describing the conditions of unchanged M and I, as well as the constraints posed by the data on the periods of the free low-order oscillations of the Earth.     

Preliminary reference Earth model

A large data set consisting of about 1000 normal mode periods, 500 summary travel time observations, 100 normal mode Q values, mass and moment of inertia have been inverted to obtain the radial distribution of elastic properties, Q values and density in the Earth's interior. The data set was supplemented with a special study of 12 years of ISC phase data which yielded an additional 1.75 × 10⁶ travel time observations for P and S waves. In order to obtain satisfactory agreement with the entire data set we were required to take into account anelastic dispersion. The introduction of transverse isotropy into the outer 220 km of the mantle was required in order to satisfy the shorter period fundamental toroidal and spheroidal modes. This anisotropy also improved the fit of the larger data set. The horizontal and vertical velocities in the upper mantle differ by 2–4%, both for P and S waves. The mantle below 220 km is not required to be anisotropic. Mantle Rayleigh waves are surprisingly sensitive to compressional velocity in the upper mantle. High S_n velocities, low P_n velocities and a pronounced low-velocity zone are features of most global inversion models that are suppressed when anisotropy is allowed for in the inversion.The Preliminary Reference Earth Model, PREM, and auxiliary tables showing fits to the data are presented.     

The admissible interval for the Q factor in the solid inner core: Constraints from new data on the nutation and free oscillations of the earth

The models of the mechanical Q factor of the inner solid core of the Earth are reconstructed from the amplitudes and phases of forced nutation and the periods and damping constants of the high-order overtones of free radial modes. The admissible range of the Q-factor in the solid core is estimated and the stability of the obtained distributions is analyzed. The real accuracy of the obtained model distributions is estimated on the basis of the previous conclusions concerning the uncertainty in the solution of the inverse problem of reconstructing the internal structure of the Earth in the low-frequency range.     

On the admissible range of the radial temperature gradient and Brünt-Väisäla frequency in the mantle and core: I. Main relations

The question of ambiguity in the solution of the inverse problem for determining the Brünt-Väisäla frequency in the Earth’s mantle from the entire set of the up-to-date data on seismicity, free oscillations, and forced nutations of the Earth, as well as the data on the Earth’s total mass and total moment of inertia, is considered. Based on the results of a series of numerical experiments, the band of admissible distributions of the Brünt-Väisäla frequency and mantle density with depth is calculated. This estimate is used for investigating the convective and gravitational stability of the different regions of the mantle against relatively small adiabatic and nonadiabatic perturbations. The generalization of the known Rayleigh criterion of convective stability of homogeneous and a nonself-gravitating incompressible viscous fluid for the case of a compressible self-gravitating fluid is given. A system of the ordinary eight-order differential equations with complex coefficients and homogeneous boundary conditions, whose eigenvalues determine the transition from the stable state to instability, is obtained. Examples of the numerical determination of these eignevalues are presented. For interpreting the data about the band of the admissible distributions of the Brünt-Väisäla frequency with depth, the notion of the effective bulk modulus of the medium at different depths is introduced. This quantity governs the depth changes in temperature in a convecting mantle and allows us to make a conclusion about the role of heat conduction and the radial heterogeneity of the mantle composition without imposing any constraints on the convection mechanism. It is shown that within the present-day observation errors in the frequencies of the Earth’s free oscillations, the simplest reasonable model is that in which the ratio of the effective bulk modulus to its adiabatic value in the lower and middle mantle is 1.043 ± 0.05. The closeness of this value to unity indicates that convection in the lower and middle mantle is fairly close to adiabatic. At the same time, when the analysis only relies on seismic data and on the information about the periods of the free oscillations of the Earth, there is a significant uncertainty in the models of the effective bulk modulus distribution in the upper mantle and crust. This uncertainty precludes us from making purely empirically derived conclusions that reliably and unambiguously describe the role of the effects of heat conduction and radially heterogeneous material composition in the convection in the upper mantle.     

Comparison between inelastic earth models constructed from astrometric and tidal gravity data

As was shown in [Molodensky, 2004a, 2004b], modern very long base interferometer (VLBI) data on the amplitudes and phases of the Earth’s forced nutation can provide significantly more rigid constraints on possible values of the quality factor of the lower mantle Q _μ and on the dynamic flattening of the liquid core e _lc as compared with seismic evidence and data on damping of the free oscillations of the Earth. On the other hand, the accuracy of modern tidal gravity data (obtained from twenty-year series of observations with a cryogenic gravimeter) is also very high and these data must be taken into account while estimating the parameters Q _μ and e _lc . The paper presents comparative estimates of the determination accuracy of the parameters Q _μ and the dynamic flattening of the liquid core from VLBI and the aforementioned tidal gravity data.     

Estimation of inner-core rotation rate by using the Earth’s free oscillation

We estimate a rate of inner-core differential rotation from time variations of splitting functions of seven core modes of the Earth’s free oscillations excited by eight large earthquakes in a period of 1994–2003. The splitting functions and moment tensor elements are simultaneously determined for each core mode by a spectral fitting technique. The estimated moment tensor well agrees with Harvard CMT solution. The splitting functions are corrected for the effect of mantle heterogeneity using a 3D mantle velocity model. Inner-core rotation angle about the Earth’s spin axis is determined for each core mode as a function of event year by comparison of the corrected and reference splitting functions. Mean rotation rate of six core modes is estimated at 0.03±0.18° per year westward, and this value is insignificantly different from zero. Therefore, the inner core is not rotating at a significant rate relatively to the crust and mantle.     

Internal constitution and evolution of the moon

The composition, structure and evolution of the moon's interior are narrowly constrained by a large assortment of physical and chemical data. Models of the thermal evolution of the moon that fit the chronology of igneous activity on the lunar surface, the stress history of the lunar lithosphere implied by the presence of mascons, and the surface concentrations of radioactive elements, involve extensive differentiation early in lunar history. This differentiation may be the result of rapid accretion and large-scale melting or of primary chemical layering during accretion; differences in present-day temperatures for these two possibilities are significant only in the inner 1000 km of the moon and may not be resolvable. If the Apollo 15 heat-flow result is representative of the moon, the average uranium concentration in the moon is 0.05–0.08 p.p.m.Density models for the moon, including the effects of temperature and pressure, can be made to satisfy the mass and moment of inertia of the moon and the presence of a low-density crust inferred from seismic refraction studies only if the lunar mantle is chemically or mineralogically inhomogeneous. The upper mantle must exceed the density of the lower mantle at similar conditions by at least 5%. The average mantle density is that of a pyroxenite or olivine pyroxenite, though the density of the upper mantle may exceed 3.5 g/cm³. The density of the lower mantle is less than that of the combined crust and upper mantle at similar temperature and pressure, thus reinforcing arguments for early moon-wide differentiation of both major and minor elements. The suggested density inversion is gravitationally unstable and implies stresses in the mantle 2–5 times those associated with the lunar gravitational field, a difficulty that can be explained or avoided by: (1) adopting lower values for the moment of inertia and/or crustal thickness, or (2) postulating that the strength of the lower mantle increases with depth or with time, either of which is possible for certain combinations of composition and thermal evolution.A small iron-rich core in the moon cannot be excluded by the moon's mass and moment of inertia. If such a core were molten at the time lunar surface rocks acquired remanent magnetization, then thermal-history models with initially cold interiors strongly depleted in radioactive heat sources as a primary accretional feature must be excluded. Further, the presence of ～||pre|40 K in a FeFeS core could significantly alter the thermal evolution and estimated present-day temperatures of the deep lunar interior.     

Low Q zone at the base of the mantle: Evidence for lower mantle convection?

Temperatures in the lower mantle of the Earth are estimated from the observed Q distribution. A thermal boundary-layer where temperatures rise rapidly is found at the base of the mantle, corresponding to the low Q zone described by Anderson and Hart (1978a,b). The existence of this thermal boundary layer indicates that the lower mantle participates in convection, and also that some of the energy driving the convection is coming from the core.     

Inversion of seismic and gravity data for the composition and core sizes of the Moon

We model the internal structure of the Moon, initially homogeneous and later differentiated due to partial melting. The chemical composition and the internal structure of the Moon are retrieved by the Monte-Carlo inversion of the gravity (the mass and the moment of inertia), seismic (compressional and shear velocities), and petrological (balance equations) data. For the computation of phase equilibrium relations and physical properties, we have used a method of minimization of the Gibbs free energy combined with a Mie-Gr@uneisen equation of state within the CaO-FeO-MgO-Al₂O₃-SiO₂ system. The lunar models with a different degree of constraints on the solution are considered. For all models, the geophysically and geochemically permissible ranges of seismic velocities and concentrations in three mantle zones and the sizes of Fe-10%S core are estimated. The lunar mantle is chemically stratified; different mantle zones, where orthopyroxene is the dominant phase, have different concentrations of FeO, Al₂O₃, and CaO. The silicate portion of the Moon (crust + mantle) may contain 3.5–5.5% Al₂O₃ and 10.5–12.5% FeO. The chemical boundary between the middle and the lower mantle lies at a depth of 620–750 km. The lunar models with and without a chemical boundary at a depth of 250–300 km are both possible. The main parameters of the crust, the mantle, and the core of the Moon are estimated. At the depths of the lower mantle, the P and S velocities range from 7.88 to 8.10 km/s and from 4.40 to 4.55 km/s, respectively. The radius of a Fe-10%S core is 340 ± 30 km.     

Equation of state fits to the lower mantle and outer core

The lower mantle and outer core are subjected to tests for homogeneity and adiabaticity. An earth model is used which is based on the inversion of body waves and Q-corrected normal-mode data. Homogeneous regions are found at radii between 5125 and 4825 km, 4600 and 3850 km, and 3200 and 2200 km. The lower mantle and outer core are inhomogeneous on the whole and are only homogeneous in the above local regions.Finite-strain and atomistic equations of state are fit to the homogeneous regions. The apparent convergence of the finite-strain relations is examined to judge their applicability to a given region. In some cases the observed pressure derivatives of the elastic moduli are used as additional constraints. The effect of minor deviations from adiabaticity on the extrapolations is also considered. An ensemble of zero-pressure values of the density and seismic velocities are found for these regions. The range of extrapolated values from these several approaches provides a measure of uncertainties involved.     

The earth's thermal profile: Is there a mid-mantle thermal boundary layer?

Shock observations on melting of iron by Brown and McQueen with the inner core boundary (ICB) density contrast estimated by Masters are used with the assumption that the light ingredient of the outer core is oxygen to calculate the boundary temperature T_ICB = (5000 ± 900) K. Adiabatic extrapolation to the core-mantle boundary (CMB) gives T_ICB = (3800 ± 800) K. The temperature increment across the D″ layer is not well constrained, but is estimated to be T_D″ = (800 ± 400) K and a slightly superadiabatic extrapolation to 670 km gives T_{670 +} = (2300 ± 950) K. This is only about 300 K higher than the extrapolation to the same level from the upper mantle, T_670? = (1970 ± 150) K. The difference is far too small to make a viable mid-mantle boundary layer. Remaining unceertainties are too large to discount such a boundary layer with certainty, but agreement of our new temperature profile with temperatures deduced from equation of state studies on the lower mantle and core encourages the view that we are converging to a well-determined temperature profile for the Earth.     

Thermal regime of the Earth's core

The outer core is assumed to consist of iron and sulfur, with a small amount of potassium that generates heat by radioactive decay of sim||pre|40 K. Two cases are considered, corresponding respectively to a high rate of heat production (Q = 2 · 10¹² cal./sec, about 0.1% K), and to a low rate (Q = 2 · 10¹¹ cal./sec). The temperature at a depth of 2800 km in the mantle is taken to be 3300°K (Wang, 1972). The temperature T_c at the core-mantle boundary depends on whether or not a density gradient in the lowermost layer D″ of the mantle prevents convection in that layer. In the first case, and for high Q, T_c = 4500–5000°K. In the second case, or for low Q, T_c ≈ 3500°K.The heat-conduction equation is used to calculate the temperature T_i at the inner-core boundary in the absence of convection. For high Q, T_i ? T_c ≈ 1600°K; for low Q, T_i ? T_c ≈ 160°K. Corresponding temperature gradients at r = r_c and r = r_i are listed in Table I.The adiabatic gradient at the top of the core is calculated by the method of Stewart (1970). It strongly depends on the parameters (ρ₀, c₀, γ₀, etc.) that characterize core material at low pressure. Stewart has drawn graphs that allow the selection of sets of parameters that are consistent with seismic velocities and a given density distribution in the core. Some acceptable sets of parameters are listed in Table II. Many sets yield temperatures T_c in the range 3500–5000°K; some give an adiabatic gradient steeper than the conductive gradient and are compatible with convection; others do not. Since properties of FeS melts remain unknown, there is at present no way of selecting any set in preference to another.Properties of the FeS system at low pressure suggest the possible appearance of immiscibility at high temperature in liquids of low sulfur content; accordingly, the inner-core boundary is thought to represent equilibrium between a solid (FeNi) inner core and a liquid layer containing only a small amount of sulfur; layer F in turn is in equilibrium with another liquid (forming layer E) containing more sulfur, and slightly less dense, than F. The temperature T_i at the inner-core boundary is about 6000–6500°K for high Q and T_c ≈ 4500–5000°K. It is consistent with Alder's (1966) and Leppaluoto's (1972) estimates of the melting point of iron at 3.3 Mbar, but not with that of Higgins and Kennedy (1971).     

Modeling of the Earth’s pole motion from data on the atmospheric and oceanic angular momenta over 1980–2002

The model values of the mantle quality factor Q=40±20 and the Chandler wobble period T=435–436 days are obtained by numerical modeling of the yearly and Chandler components in the pole motion from data on the angular momenta of the atmosphere and the ocean. The oceanic and the atmospheric excitations account for about 65–70% of the dispersion of the observed pole motion.     

应用Lane-Emden方程分析下月幔厚度与月核半径大小

文中取圈层结构和球对称形态为月球的基本结构假设,并以月球平均密度和无量纲惯性矩作为约束,数值求解月球Lane-Emden方程,得到下月幔厚度和月核大小的变化范围.结果表明月核的密度在4.7 ~7.0 g/cm3范围内变化时,月核半径的变化范围为704~356 km,相应的月幔厚度的变化范围约为33～381 km,月核占月球总质量的百分比在0.6%～7%之间变化.所得结果可为后续的关于月球内部结构的研究提供一定的参考.     

Attenuation models of the earth

Free oscillation and body wave data are used to construct average Q models for the earth. The data set includes fundamental and overtone observations of the radial, spheroidal and toroidal modes, ScS observations and amplitudes of body waves as a function of distance. The preferred model includes a low-Q zone at both the top and the bottom of the mantle. In these regions the seismic velocities are likely to be frequency dependent in the “seismic” band. Absorption in the mantle is predominantly due to losses in shear. Compressional absorption may be important in the inner core.A grain-boundary relaxation model is proposed that explains the dominance of shear over compressional dissipation, the roughly frequency independent average values for Q and the variation of Q with depth. In the high-Q regions, the lithosphere and the midmantle (200–2000 km), Q is predicted to be frequency dependent. However, the low-Q regions of the earth, where Q is roughly frequency independent, dominate the observations of attenuation.     

On the effect of oceanic tides on the inertial coupling between the liquid core and the mantle

The amplitudes and phases of forced nutation and diurnal earth tides depend significantly on the moment of forces between the liquid core and mantle of the Earth, resulting from the differential rotation of the core. The solution to the dynamic problem of rotation of an imperfectly elastic mantle with an imperfectly liquid core and an ocean indicates that the predominant role is played by the so-called core-mantle inertial coupling (related to the effect of hydrodynamic pressure in the liquid core on the ellipsoidal core-mantle boundary). The magnitude of this coupling depends significantly not only on the dynamic flattening of the liquid core but also on the elastic and inelastic properties of the mantle, as well as on the amplitudes and phases of oceanic tides. In this paper, the effects of oceanic tides on the magnitude of inertial coupling between the liquid core and the mantle and on the period and damping decrement of free nearly diurnal nutation are estimated.     

Gravity compensation in the mantle beneath the neovolcanic zone of Iceland

An enormous component of isostatic compensation(～?430±40mgal) is provided by subcrustal material beneath the neovolcanic zone of Iceland. Previously published values for the component of gravity contributed by anomalous mantle material beneath this region range from ?250 to ?140 mgal. These values are only a partial indication of the magnitude of the compensation mechanism. If one takes into account constraints on crustal thickness from seismic refraction studies and compares Iceland to less active tectonic regions near by, one obtains a mantle gravity effect of approximately?430±40mgal, which for a simple slab model leads to a vertically integrated mass deficit per unit surface area of 10⁶ g/cm². The effects of thermal expansion, solid-solid phase transitions and partial fusion can provide significant contributions to the total mass deficiency; however, none of these mechanisms alone seems sufficient to account for the entire anomaly.The relation of this mass anomaly to the evolution of the Iceland-Faeroe ridge is considered by examining two extreme end-members of a suite of possible evolutionary models for this region. The first of these is the case where, in time, the 10⁶ g/cm² mass deficit will be resorbed into the mantle with the effects of chemical segregation playing a minor role. The second case, which is preferred, involves a significant redistribution of material from the mantle such that basaltic melt segregated from high levels in the mantle accumulates and crystallizes in a zone at the base of the crust. In this latter model, if the Iceland-Faeroe ridge is considered to have evolved under a steady rate of magma production over the last 30–40 × 10⁶ years, then underplating of the crust may account for its increasing thickness as it matures from 8–10 km in the neovolcanic zone to a value of approximately 32 km for the Iceland-Faeroe ridge. If we assume a 10% increase in density upon crystallization, thickening of the crust by 22 km through underplating by magma accumulation leads to an increase in mass per unit surface area of 0.6 × 10⁶ g/cm², and accounts for approximately 60% of the total mass difference in the mantle between Iceland and the Iceland-Faeroe ridge.