The influence of earthquakes on the Chandler wobble during 1977–1983

Summary. A direct calculation is made of the effect on the Chandler wobble of 1287 earthquakes that occurred during 1977–1983. The hypocentral parameters (location and origin time) and the moment tensor representation of the best point source for each earthquake as determined by the 'centroidmoment tensor' technique were used to calculate the change in the Chandler wobble's excitation function by assuming this change is due solely to the static deformation field generated by that earthquake. The resulting theoretical earthquake excitation function is compared with the 'observed' excitation function that is obtained by deconvolving a Chandler wobble time series derived from LAGEOS polar motion data. Since only 7 years of data are available for analysis it is not possible to resolve the Chandler band and determine whether or not the theoretical earthquake excitation function derived here is coherent and in phase with the 'observed' excitation function in that band. However, since the power spectrum of the earthquake excitation function is about 56 dB less than that of the 'observed' excitation function at frequencies near the Chandler frequency, it is concluded that earthquakes, via their static deformation field, have had a negligible influence on the Chandler wobble during 1977–1983. However, fault creep or any type of aseismic slip that occurs on a time-scale much less than the period of the Chandler wobble could have an important (and still unmodelled) effect on the Chandler wobble.     

Random walk of the Earth's pole related to the Chandler wobble excitation

Summary. The equation governing the polar motion shows that the polar secular drift and the Chandler wobble amplitude are related to each other. In particular, a drift of the mean pole position comes out as a consequence of the maintenance of the Chandler wobble by possible step perturbations of the Earth's inertia tensor.
The minimum excitation functions necessary to explain the Chandler wobble amplitude variations for the period 1901–84 are derived from the Chandler term, with the hypothesis that the excitations follow a uniform random distribution in time. It is shown that they have the statistical properties of the steps of a two-dimensional random walk. These functions are then used to derive, from a statistical simulation, a lower limit of the secular drift which may result from the excitation of the Chandler wobble.
The drift generated by the random walk is of the same order of magnitude as the observed secular drift for the period 1901–84, but their time dependence is different. This indicates that the observed secular drift cannot be explained as the consequence of an excitation of the Chandler wobble by random steps of the Earth's inertia tensor. However, the possible contribution of the Chandler wobble excitation to the polar drift has to be taken into account when other mechanisms, such as lithospheric rebound related to deglaciation, are proposed.     

An optimal complex AR.MA model of the Chandler wobble

summary . The character of the Chandler wobble suggests that the optimal representation of the observed polar motion is a complex AR.MA model. This paper develops the theory of such a model and presents a modification of the scalar AR.MA computer program of Akaike, Arahata & Ozaki. The complex AR.MA model is applied to the ILS data covering the period 1900–1975. An optimal AR.MA (1,4) model is obtained. The model parameters are interpreted to give a Chandler frequency of 0.8400 ± 0.0039 cpy a Q value of 50 < Q < 300, with the most probable value of 96 and the power density of the excitation to be about 0.51 × (0".01)²/cpy. This result leads to the conclusion that the contribution of meteorological sources to the excitation of the Chandler wobble is about 11 to 19 per cent.     

Comments on the Chandler wobble Q

Summary. The Chandler wobble Q _w, as obtained from the astronomical data cannot be equated with the Q _m of the source of damping, as an examination of Chandler wobble energetics reveals. We find that if dissipation occurs in the mantle then Q _w≃ 9 Q _m, implying that either the mantle Q is frequency dependent or the wobble Q is much larger than 100. If the dissipation is in the oceans then Q _w≃ 20 Q _o, and the pole tide must be far from equilibrium.     

The Passive Influence of the Oceans upon the Rotation of the Earth

A general theory is developed which allows the exact numerical computation of the static equilibrium response of a non-rotating spherically symmetric Earth model covered by thin oceans with geometrically irregular coastlines to the action of an imposed static tidal or centrifugal potential. The theory is self-consistent, and takes into account the gravitational self-attraction of the oceans and the elastic-gravitational response of the Earth model to both the applied potential and the equilibrium oceanic tidal load on the surface. The results are used to determine the influence of an equilibrium pole tide on the free period and the associated rotational eigenfunction of the Chandler wobble. If the pole is globally well represented by this equilibrium approximation, its effect is to increase the Chandler wobble period by 27·6 days. It is shown that a fully self-consistent theory of the rotation of an Earth model with oceans predicts that changes in spin and wobble will be coupled, and that the Chandler wobble should, as a result, be accompanied by an associated periodic change in the length of day. The consequences of spin-wobble coupling are explored quantitatively, and found to be slight.     

A new derivation of the ratio Q(wobble)/Qμ (mantle)

Summary An extension of the Love-Larmor theory to a low-loss unelastic earth model, leads to the surprisingly simple approximation
   
where τ_s= 447.4 sidereal day is the static wobble period, τ_R= 306 sidereal day is the rigid-earth wobble period and τ_w= 433 sidereal day is the observed Chandler period. Q _W, Q _μ are the respective average Q values of the wobble and the Earth's mantle at τ_W. The known numerical factor F is only slightly dependent on the Earth structure.     

Period and Qw of The Chandler Wobble

A splined ILS/IPMS data set (1900–1973) from the most homogeneous values available has been analysed by the maximum entropy method of Burg. Principal conclusions are: (1) the spectral character of the Chandler wobble is a single broad peak, (2) the period is 432·95 ± 1·02 mean solar days and, (3) the Q _w is 36±10. Measurements indicate that Q _w is non- stationary in time.     

Modelling the pole tide and its effect on the Earth's rotation

Summary. The pole tide is the response of the ocean to incremental centrifugal forces associated with the Chandler wobble. The tide has a potentially important effect on the period and damping of the wobble, but it is at present not well constrained by observations. Here, we construct both analytical and numerical models for the pole tide. The analytical models consider the tide first in a global ocean and then in an enclosed basin on a beta-plane. The results are found to approach equilibrium linearly with decreasing frequency and inversely with increasing basin depth. The numerical models solve Laplace's tidal equations over the world's oceans using realistic continental boundaries and bottom topography. The results indicate that the effects of the non-equilibrium portion of the deep ocean tide on the Chandler wobble period and damping are negligible.     

Estimation of period and Q of the Chandler wobble

The period P and Q -value of the Chandler wobble are two fundamental functional of the Earth's internal physical properties and global geodynamics. We revisit the problem of the estimation of P and Q , using 10.8 yr of modern polar motion as well as contemporary atmospheric angular momentum (AAM) data. We make full use of the knowledge that AAM is a major broad-band excitation source for the polar motion. We devise two optimization criteria under the assumption that, after removal of coherent seasonal and long-period signals, the non-AAM excitation is uncorrelated with the AAM. The procedures lead to optimal estimates for P and Q. Our best estimates, judging from comprehensive sets of Monte Carlo simulations, are P = 433.7 ± 1.8 (1σ) days, Q =49 with a la range of (35, 100). In the process we also obtain (as a by-product) an estimate of roughly 0.8 for a 'mixing ratio' of the inverted-barometer (IB) effect in the AAM pressure term, indicating that the ocean behaves nearly as IB in polar motion excitation on temporal scales from months to years     

Seismic excitation of the Chandler wobble revisited

Summary. An overview is taken of the last decade of studies of the effect of earthquakes on the polar motion. The treatment of the liquid outer core in static deformation is reviewed and some misconceptions in a number of papers are pointed out. Volterra's formula is generalized to the case of a liquid core which does not obey the highly idealized Adams—Williamson density law. The focal mechanism representation of Smylie & Mansinha (1971) is corrected for neglected terms arising from coordinate curvature, bringing the computed polar shifts into near numerical agreement with those of other workers. On the basis of the comparison of the observed and computed polar shifts for the Chile 1960 and Alaska 1964 events, it is suggested that the observed polar shifts for large earthquakes may be useful as discriminators in selecting focal mechanism parameters. The observed level of Chandler wobble excitation provides a constraint on some of the more extreme values of seismic moment recently proposed, unless these are supposed to depend only weakly on magnitude. The cumulative effect of the 30 largest earthquakes in the period 1901–64, recently examined by O'Connell & Dziewonski, is found to yield a rms Chandler wobble excitation of 0".10, using the random walk theory of Mansinha & Smylie (1967). This is close to the observed level (∼ 0".15). In addition to yielding the solution to a very long-standing geophysical puzzle, the study of the effect of earthquakes on the polar motion over the last decade may have produced a useful tool for the elucidation of seismic mechanism.     

Polarization analysis in the wavelet domain based on the adaptive covariance method

We present an improved method for computing polarization attributes of particle motion from multicomponent seismic recordings in the time–frequency domain by using the continuous wavelet transform. This method is based on the analysis of the covariance matrix. We use an approximate analytical formula to compute the elements of the covariance matrix for a time window which is derived from an averaged instantaneous frequency of the multicomponent record. The length of the time-window is automatically and adaptively set to match the dominant period of the analysing wavelet at each time–frequency point. Then the eigenparameters are estimated for each time–frequency point without interpolation. With these key features, our method provides a suitable approach for polarization analysis of dispersive signals or overlapping seismic arrivals in multicomponent seismic data. For polarization analysis in the time domain, we show that the proposed method is consistent with existing polarization analysis methods. We apply the method to real data sets from exploration and earthquake seismology to illustrate some filtering applications and wave type characterizations.     

The period and Q of the Chandler wobble

Summary. We have extended our calculation of the theoretical period of the Chandler wobble to account for the non-hydrostatic portion of the Earth's equatorial bulge and the effect of the fluid core upon the lengthening of the period due to the pole tide. We find the theoretical period of a realistic perfectly elastic Earth with an equilibrium pole tide to be 426.7 sidereal days, which is 8.5 day shorter than the observed period of 435.2 day. Using Rayleigh's principle for a rotating Earth, we exploit this discrepancy together with the observed Chandler Q to place constraints on the frequency dependence of mantle anelasticity. If Qμ in the mantle varies with frequency σ as σα between 30 s and 14 months and if Qμ in the lower mantle is of order 225 at 30 s, we find that 0.04 ρ^α≤ 0.11; if instead Qμ in the lower mantle is of order 350 near 200 s, we find that 0.11 ≤α≤ 0.19. In all cases these limits arise from exceeding the 68 per cent confidence limits of ± 2.6 day in the observed period. Since slight departures from an equilibrium pole tide affect the Q much more strongly than the period we believe these limits to be robust.     

Inversion of controlled-source seismic tomography and gravity data with the self-adaptive wavelet parametrization of velocities and interfaces

A self-adaptive automated parametrization approach is suggested for the sequential inversion of controlled-source seismic tomography and gravity data. The velocities and interfaces are parametrized by their Haar wavelet expansion coefficients. Only those coefficients that are well constrained by the data, as measured by the number of rays that cross the corresponding wavelet function support area and their angular coverage, are inverted for, others are set to zero. This approach results in a reasonable distribution of resolution throughout the model even in cases of irregular ray coverage and does overcome the trade-off between different types of model parameters. A modified sequential inversion approach is suggested to join the traveltimes and gravity anomalies inversion. An algorithm is developed that inverts for smooth velocity and density variations inside the seismic layer, the position of its bottom interface as well as for optimal values of the velocity-to-density regression coefficients. The algorithm makes use of direct (diving), reflected and head (critically refracted) wave traveltimes. The algorithm workflow is demonstrated on a synthetic data example.     

On the amplitude of the Chandler wobble

The change in the inertia tensor of the Earth, due to the mass shift following a seismic event, has been computed by several authors for non-rotating earth models. Rotation is taken into account in the present paper, and the additional change in the inertia tensor is computed for an equivalent earth model, in which the axis of geometrical symmetry becomes tilted instead of the axis of greatest inertia. Rotation is thus seen to produce an increase by a factor 1.4 in the amplitude variation of the Chandler wobble, with respect to the non-rotating case, which, when added to the 1.4 amplitude increase due to the precessional re-adjustment of the equatorial bulge, gives a factor of 2 increase of the Chandler wobble amplitude with respect to the case of a rigid earth model.     

New aspects of the equilibrium pole tide

Summary. We have developed a new spherical harmonic algorithm for the calculation of the loading and self-gravitating equilibrium pole tide. Based on a suggestion of Dahlen, this approach minimizes the distortions in tide height caused by an incomplete representation of the ocean function. With slight modification our approach easily could be used to compute self-gravitating and loading luni-solar tides as well.
Using our algorithm we have compared the static pole tide with tide observations at a variety of locations around the world. We find statistically significant evidence for pole tide enhancements in mid-ocean as well as the shallow seas.
We have also re-investigated the effect of the static tide on the Chandler wobble period. The difference between the wobble period of an oceanless, elastic earth with a fluid core (Smith & Dahlen) and the period of an earth minus static oceans yields a 7.4-day discrepancy. We conclude from tide observations that much of the discrepancy can probably be accounted for by non-equilibrium pole tide behaviour in the deep oceans.     

Normal modes of the viscoelastic earth

Summary A uniformly valid linear viscoelastic rheology is described which takes the form of a 'generalized' Burgers' body and which appears capable of reconciling the behaviour of the Earth's mantle across the complete spectrum of geodynamic time-scales. This spectrum is bracketed by the short time-scales of body wave and free oscillation seismology on which anelastic effects are dominant, and the long time-scale of mantle convection on which the Earth behaves viscously. The parameters of the model which control the viscous response are fixed by post-glacial rebound data whereas those which govern the anelasticity are to be determined by fitting the model to observations of seismic Q. The paper is concerned primarily with a discussion of the normal mode spectrum of the Earth as a generalized Burgers' body. Focusing upon the homogeneous model, it includes an initial analysis of the accuracy of first-order perturbation theory as a method of calculating the respective Q s of the elastic gravitational free oscillations. Also considered are the quasi-static modes of relaxation which only exact eigenanalysis can reveal. The importance of these modes is assessed within the context of a discussion of the effect of viscoelasticity upon the efficiency of Chandler wobble excitation.     

Teleseismic shear wave splitting measurements in noisyenvironments

High noise levels hamper teleseismic shear wave splitting measurements, which bandpass filtering does not always help. To investigate how robust splitting measurements are to noise, we analysed a set of synthetic records with known splitting parameters and added fixed levels of noise. In the presence of weak anisotropy, single-waveform splitting measurements are unreliable when operating with noisy data sets. A practical rule in terms of S/N ratio and splitting delay time parameters is that splitting is confidently detectable at S/N > 8, regardless of the wave's original polarization orientation. However, for the evidence of weak anisotropy to be detectable and measurable at an S/N value of 4, the backazimuth separation of the phases from the fast polarization direction needs to be higher than 20°. Stacks of individual measurements consistently yield reliable results down to S/N values of 4. Applying stacking to data from DSB (Dublin, Ireland), the fast polarization direction φ and lag time δt are 58° and 0.95  s. This orientation reflects surface trends of deformation in the area, as found elsewhere in the UK. Our result thus reinforces the proposed model that the detected anisotropy in the British Isles originates from lithospheric coherent deformation preserved from the last main tectonic episode.     

CLIMATE OSCILLATIONS AS RECORDED IN SVALBARD ICE CORE ω18O RECORDS BETWEEN AD 1200 AND 1997

ABSTRACT. We apply two different time series analytical tools to ω¹⁸O records from two Svalbard ice cores. One ice core is from Lomonosovfonna at 1250 ma.s.l. and the other from Austfonna at 750 m a.s.l. These cores are estimated to cover at least the past 800 years and have been dated using a combination of known reference horizons and glacial modelling. Wavelet analysis reveals low frequency oscillations on the 60–120–year scale on the lower elevation site Austfonna while the higher altitude site on Lomonosovfonna does not reveal such variability throughout the record. The second method, Significant Zero Crossing of Derivates (SiZer) does not resolve the low-frequency periodicity seen in the wavelet analysis. The low-frequency variability resolved by the wavelet analysis is similar to what has been found in various climate records including instrumental temperatures and tree-rings, and has been proposed as the most important oscillation for the observed trends in Arctic air temperatures.     

基于小波分析的土壤速效K含量高光谱反演

选取新疆奇台县的134个土壤样本,利用土壤反射率对数的一阶导数光谱分别对4 种小波函数进行多层离散分解,采用PLSR方法分别建立了土壤速效钾含量的反演模型,并对其精度值进行检验。结果表明：小波分解获得的各层低频系数以1～3层较高,而其余各层则较低。所有函数分解的6层中,均以第2层低频系数建模的精度最高,随着分解层数（>2层）的增加,其精度值和显著性明显降低。相同尺度下,采用4种小波函数的低频系数构建的反演模型的精度差异较小,而Bior1.3为最优函数;基于Bior 1.3分解的ca2低频系数建模的R²达0.964,RMSE仅为8.19 mg·kg^-1,且为极显著水平,为最佳反演模型,经样本检验后发现,此模型可用以快速、准确估算土壤高光谱速效钾含量。     

Variationsl solution of long-period oscillations of the Earth

Summary The linearized equation of motion for the slightly elliptical rotating earth is obtained and using Phinney & Burridge's generalized spherical harmonics, the variational principle is derived for the normal mode oscillations of the Earth. The numerical solutions of two earth models 1066B and B1S6 are searched by minimizing the energy functional for the terrestrial spectral range longer than the lowest order free oscillation. The periods of core modes computed for the earth model B1S6, with stably stratified outer core, ranges from about 4 to 13hr and the periods for the 1066B are much more spread without clustering around the periods of 6 and 12 hr as in B1S6. The results for the earth model 1066B indicate that an outer core can support long-period oscillations even when it is not stably stratified. The Chandler wobble periods obtained are 402.3 day for B1S6 and 402.7 day for 1066B.