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.     

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.     

Estimation of the pole tide gravimetric factor at the chandler period through wavelet filtering

Wavelet analysis for filtering is used to improve estimation of gravity variations induced by Chandler wobble. This method eliminate noise in superconducting gravimeter (SG) records with bandpass filters derived from Daubechies wavelet. The SG records at four European stations (Brussels, Membach, Strasbourg and Vienna) are analysed in this study. First, the earth tidal constituents are removed from the observed data by using synthetic tides, then the gravity residuals are filtered into a narrow period band of 256–512 d by a wavelet bandpass filter. These data are submitted to three regression analysis methods for estimating the gravimetric factor of the Chandler wobble. After processing by wavelet filtering, SG records can provide amplitude factors δ and phase lags κ of the Chandler wobble with much smaller mean square deviation (MSD) than these provided by former studies. It is mainly because the wavelet method can effectively eliminate instrumental drift and provide smoothed data series for the regression analysis.     

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.     

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.     

The effects of post-seismic motions on the moment of inertia of a stratified viscoelastic earth with an asthenosphere

Summary. We give the analytical formulation for calculating the transient displacement of fields produced by earthquakes in a stratified, selfgravitating, incompressible, viscoelastic earth. We have evaluated the potential of viscous creep in the asthenosphere in exciting the Chandler wobble by a four-layer model consisting of an elastic lithosphere, a two-layer Maxwell viscoelastic mantle, and an inviscid core. The seismic source is modelled as an inhomogeneous boundary condition, which involves a jump condition of the displacement fields across the fault in the lithosphere. The response fields are derived from the solution of a two-point boundary value problem, using analytical propagator matrices in the Laplace-transformed domain. Transient flows produced by post-seismic rebound are found to be confined within the asthenosphere for local viscosity values less than 10²⁰P. The viscosity of the mantle below the low-viscosity channel is kept at 10²²P. For low-viscosity zones with widths greater than about 100 km and asthenospheric viscosities less than 10¹⁸P, we find that viscoelasticity can amplify the perturbations in the moment of inertia by a factor of 4–5 above the elastic contribution within the time span of the wobble period. We have carried out a comparative study on the changes of the inertia tensor from forcings due to surface loading and to faulting. In general the global responses from faulting are found to be much more sensitive to the viscosity structure of the asthenosphere than those produced from surface loading.     

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.     

Local and regional components of western Aegean deformation extracted from 100 years of geodetic displacement measurements

Our objectives are as follows. First, we wish to develop a methodology to recover the long-term component of deformation from any set of distributed, time-averaged geodetic strain measurements that were subject to seismic disturbance, given a catalogue of local seismicity that occurred during the measurement period. Second, using seismic and geodetic data sets that span approximately 100 years, we apply this technique in the western Aegean to assess the role of local seismicity in regional deformation. The methodology is developed using a model for crustal deformation constructed from a long-term, smooth regional strain field combined with instantaneous, local perturbations from upper-crustal earthquakes approximated by static elastic dislocations. By inverting geodetic displacements for the smooth field while simultaneously floating influential but uncertain earthquake source parameters, an estimate of the regional component of deformation that is approximately independent of the seismicity can be made. In the western Aegean we find that the horizontal component of regional deformation can be described with minor inaccuracy by a quadratic relative displacement field. The principal horizontal extensional axes calculated from the regionally smooth displacement field agree in orientation with the T-axes of earthquakes in the region. These observations indicate that the instantaneous elastic strain of the 10 km thick seismogenic layer is driven by a stress field that is smooth on the scale of the geodetic network as a whole, 200-300 km.     

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.     

Stress accumulation between volcanoes: an explanation for intra-arc earthquakes in Nicaragua?

Destructive upper crustal earthquakes in Central America are often located between active volcanic centres—a geometric relationship that we study using finite element Coulomb failure stress (CFS) models that incorporate the rheologically heterogeneous nature of the volcanic arc. Volcanoes are simulated as mechanically weak zones within a stronger crust. We find that deformation of the volcanic centres within a regional stress field dominated by dextral shear causes stress increases in surrounding crust, with a maximum CFS change between neighbouring volcanoes. This increase in CFS enhances the probability of fault slip on arc-normal faults that are located between volcanic centres; for example, the Tiscapa fault, which ruptured during the 1972 December 13,   M _s   6.2 Managua earthquake. The amount of stress increase due to long-term (100 yr) volcano shearing is on the order of 0.1–0.6 bars, similar to values estimated for subduction zone earthquakes.     

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.     

Simulations of strong ground motion for earthquakes in the Mexicali–Imperial Valley region

Summary. In this paper computer modelling is used to test simple approximations for simulating strong ground motions for moderate and large earthquakes in the Mexicali–Imperial Valley region. Initially, we represent an earthquake rupture process as a series of many independent small earthquakes distributed in a somewhat random manner in both space and time along the rupture surface. By summing real seismograms for small earthquakes (used as empirical Green's functions), strong ground motions at specific sites near a fault are calculated. Alternatively, theoretical Green's functions that include frequencies up to 20 Hz are used in essentially similar simulations. The model uses random numbers to emulate some of the non-deterministic irregularities associated with real earthquakes, due either to complexities in the rupture process itself and/or strong variations in the material properties of the medium. Simulations of the 1980 June 9 Victoria, Baja California earthquake ( M _L= 6.1) approximately agree with the duration of shaking, the maximum ground acceleration, and the frequency content of strong ground motion records obtained at distances of up to 35 km for this moderate earthquake. In the initial stages of modelling we do not introduce any scaling of spectral shape with magnitude, in order to see at what stage the data require it. Surprisingly, such scaling is not critical in going from M = 4–5 events to the M = 6.1 Victoria earthquake. However, it is clearly required by the El Centro accelerogram for the Imperial Valley 1940 earthquake, which had a much higher moment ( Ms ∼ 7). We derive the spectral modification function for this event. The resulting model for this magnitude ∼ 7 earthquake is then used to predict the ground motions at short distances from the fault. Predicted peak horizontal accelerations for the M ∼ 7 event are about 25–50 per cent higher than those observed for the M = 6.1 Victoria event.     

Rayleigh-wave amplitudes from earthquakes in the range 0–150°

Summary . Vertical component Rayleigh-wave amplitudes from 1461 shallow earthquakes recorded in the distance range 0–150° are analysed to separate the effects of earthquake size, epicentral distance (Δ) and recording station.
The estimated decay of amplitude with distance has the form of a theoretical curve for the decay of Rayleigh waves with distance if the assumption is made that the decay due to dispersion for the data analysed is that of an Airy phase. Writing the decay due to anelastic attenuation as exp (- k Δ), k is estimated to be 0.676/rad over the whole range of distance. If the distance effects are represented by a straight line of the form h log Δ+ constant, h is estimated to be 1.15. The calibration function for computing M _s derived from the estimated distance effects is very similar to that of Marshall & Basham.
Station effects on Rayleigh-wave amplitudes though statistically significant are small, and can probably be ignored in the computation of M _s.
Comparing the estimated surface-wave magnitudes (earthquake size) obtained in this study with the long and short period body-wave magnitudes ( m ^LP_b and m ^SP_b respectively) obtained by Booth, Marshall & Young for the same earthquake shows that m ^LP_b is about equal to M _s over the magnitude range of interest (˜4.0–7.0). The m ^LP_b and M_s relationship shows that the greater the long-period energy radiated by an earthquake the smaller proportionately is the short-period energy.     

Application of the negative binomial to earthquake occurrences in the Alpide-Himalayan belt

Summary . Shallow focus earthquakes ( h ≤ 60 km) of magnitude range M = 4.0–6.0, occurred during 1954–75 in various high seismicity zones of the Alpide-Himalayan belt have been tested by the Poisson and the negative binomial laws. When the clustering of events make the simple Poisson model inapplicable in most of the high seismicity zones of the Alpide-Himalayan belt it has been shown that the negative binomial entries provide an excellent model for describing the earthquake occurrences. The chi-square ( X ²) test is employed for testing the actual observations with theoretical distributions.     

The effects of the atmosphere and oceans on the Earth's wobble — I. Theory

Summary The theory of wobble excitation for a non-rigid earth is extended to include the effects of the earth's fluid core and of the rotationally induced pole tide in the ocean. The response of the solid earth and oceans to atmospheric loading is also considered. The oceans are shown to be affected by changes in the gravitational potential which accompany atmospheric pressure disturbances and by the load-induced deformation of the solid earth. These various improvements affect the excitation equations by about 10 per cent. Atmospheric and oceanic excitation can be computed using either an angular momentum or a torque approach. We use the dynamical equations for a thin fluid to relate these two methods and to develop a more general, combined approach. Finally, geostrophic winds and currents are shown to be potentially important sources of wobble excitation, in contrast to what is generally believed.     

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     

Anomalous variations of the geomagnetic field in East Fergana — magnetic precursor of the May earthquake with M= 7.0 (1978 November 2)

Summary. A secular variation anomaly has been discovered at the north-east part of the Fergana vdey by repeated measurements every year or less. The change of total field Δ F at the 'magnetic epicentre' was 9 nT in 1977 and 16 nT in 1978 relative to the level of 1973. In 1977 an anomalous region was recognized, where according to the data from 25 observation points Δ F increased in the northern part up to 5.2 nT, and decreased by 4.7nTin the southern part according to a further 22 points. Permanent observations were begun at the epicentre in 1978 October. We normally observed variations of Δ F differences with magnitude ± 2–3 nT, which were not correlated with worldwide magnetic activity. Anomalous variations appeared on October 26 and rose to a maximum value of + 23 nT on October 30. The decrease of this anomalous field began on October 31. This made it possible to predict a potential earthquake. The Alay earthquake with M = 7.0 occurred on November 2 six hours after the prediction was issued; Δ F then returned to the initial level. Thus, using the geomagnetic field variations in the Fergana region, geophysicists were able to predict the moment of a strong earthquake.     

Seismic reflection studies using local earthquake sources

Summary. A method is presented for processing three-component digital recordings of micro-earthquakes to obtain near-vertical reflection profiles in regions of shallow seismicity. The processing includes magnitude and focal-depth normalization and event stacking, where stacking is by small localized groups, with ray theoretical time and distance corrections applied to compensate for varying focal depths. In areas with high seismicity, this procedure allows earthquakes to be treated as "controlled" sources to probe layered structures of the deep crust and upper mantle. The validity of our approach is demonstrated using S-waves from aftershocks of the Borah Peak, Idaho, earthquake (Ms = 7.3) of 1983.