Modeling the geoid and polar motion in geological tim

Amplitude-frequency analysis of the Earth's rotaryoscillatory motion due to lunisolar gravita� tional and tidal perturbing torques is performed using methods of classical mechanics. Dynamic processes that occur in Earth's rotation parameters (ERP) and Earth's potential due to tidal deformations lead to distortions in Earth's shape and fluctuations in the gravitational field (1, 2). The results of the numerical modeling of oscillatory processes in the Earth's pole motion and variations in the second zonal harmonic δc20 of the geopotential are discussed. The amplitude and phase of oscillations of the Earth's pole are determined using the dynamic Euler-Liou� ville equations. The power spectral densities of the time series of Earth's polar coordinates are compared with those of δc20 variations in the geopotential. This harmonic is found to be a function of the amplitude and phase of the oscillatory process of the Earth's pole. 1. The study of temporal variations in the geopo� tential (2, 3) due to Earth's rotaryoscillatory motion is of scientific and practical interest. The observed ERP variations and variations in the Earth's gravita� tional field are highly interrelated; dynamic processes that lead to appreciable changes in both ERPs and geophysical phenomena have acquired the most detailed reflection in diurnal oscillations (4). The refinement of the Earth's gravitational field involves replacing the statistical geoid by a geoid that corresponds to the deformable shape of the Earth (due to oceanic and rigidbody tides and other factors). Due to increasing demands on the accuracy of the coordinatetime maintenance of navigation systems, investigations related to the refinement of the gravita� tional field and the shape of the Earth have developed intensively (2). Note that the modeling of tidal deformations on a rotating geoid can be used to reach high characteristics of atomic clock synchronization (the stability of atomic frequency is 10 -14 ) for objects that are located on different continents of the globe. Differential equations of the Earth's rotaryoscilla� tory motions with allowance for the harmonic struc� ture of the tidal coefficients were derived from the dynamic Euler-Liouville equations in the problem considering "a deformable Earth-Moon" system in the solar gravitational field (4):

(TYPE=abstract)This study is concerned with the forward modelling of the present-day glacial-isostatic adjustment (GIA) of the earth to present and past changes of the Antarctic ice sheet (AIS). We predict temporal variations in the geoid height and topographic height within the context of the Gravity Recovery and Climate Experiment (GRACE) satellite mission and terrestrial Global Positioning System (GPS) stations in Antarctica. We first adopt an elastic-compressible earth model and calculate the earth's response to seasonal (< 1 a) and secular (~ 100 a) ice-mass changes. Then, we consider a viscoelastic-incompressible earth model and determine the earth's response to the melting of the AIS since the last glacial maximum (LGM), i.e. the post-glacial rebound (PGR). Both models are radially symmetric and self gravitating. A range of viscosity values account for the uncertainty in the viscosity stratification and the lateral heterogeneity of the mantle beneath Antarctica. We employ six load models simulating the most important temporal aspects of the AIS's evolution. Finally, we predict the geoid-height change and the land uplift and compare them with recent GRACE observations and determinations of the uplift rate for the permanent GPS stations along the Antarctic coast.

Ocean contribution to co-seismic crustal deformation and geoid anomalies: Application to the 2004 December 26 Sumatra–Andaman earthquake

The Gravity Recovery and Climate Experiment (GRACE) and its Follow-On mission (GRACE-FO) have greatly improved our ability to monitor changes in Earth's mass distribution, providing unprecedented insights into variations in total water storage (TWS). These variations can be expressed as equivalent water thickness, and they can also be derived from other geopotential variations, such as changes in geoid height, gravity anomalies, or vertical displacements. Understanding these variations is essential for comprehending regional hydrology and solid Earth dynamics.In this study, we use DDK2-filtered solutions from GRACE and GRACE-FO spherical harmonics to compute geoid height variations over the T&amp;#252;rkiye region, based on roughly one hundred grid points. The trend in the geoid height changes for this region is approximately at the millimeter level. We also derive TWS time series from these DDK2-filtered spherical harmonics to compare the changes in geoid height with the corresponding equivalent water thickness values, aiming to explore the functional relationship and correlation coefficients between these two geopotential variations. In addition to time-domain analysis, we apply spectral analysis to examine the power spectrum of geoid height and TWS variations in the frequency-domain. This approach helps us understand the spatial and temporal diversity of geoid height changes across T&amp;#252;rkiye and provides insights into their underlying patterns and trends. The preliminary results of this study offer an overview of geoid height changes in T&amp;#252;rkiye and highlight the potential of GRACE and GRACE-FO data for monitoring mass redistribution on a regional scale.

The forthcoming GRAV-D gravimetric geoid model over the United States is to be updated regularly to account for changes in geoid height. Its baseline precision is to be at the 10–20 mm level over non-mountainous regions. The aim of this study is to provide an estimate of the magnitude, time scale, and spatial footprint of geoid height change over North America, from mass redistribution processes of hydrologic, cryospheric and solid Earth nature. Geoid height changes from continental water storage changes over the past 50 years and predicted over the next century are evaluated and are highly dependent on the used model. Groundwater depletion from anthropogenic pumping in regional scale aquifers may lead to geoid changes of 10 mm magnitude every 50–100 years. The GRACE time varying gravity fields are used to (i) assess the errors in a glacial isostatic adjustment model, which, if used to correct the GRAV-D model, may induce errors at the 10 mm geoid height level after ~20 years, (ii), evaluate geoid height change over ice mass loss regions of North America, which, if they remain unchanged in the future, may lead to geoid height changes at the 10 mm level in under a decade and (iii), compute sea level rise and its effect on the geoid, which is found to be negligible. Coseismic gravitational changes from past North American earthquakes are evaluated, and lead to geoid change at the 10-mm level for only the largest thrust earthquakes. Finally, geoid change from volcanic processes are assessed and found to be significant with respect to the GRAV-D geoid model baseline precision for cataclysmic events, such as that of the 1980 Mt. St. Helens eruption. Recommendations on how to best monitor geoid change in the future are given.

The rotational behaviour of a stratified visco-elastic planet submitted to changes in its inertia tensor is studied in a viscous quasi-fluid approximation. This approximation allows for large displacements of the Earth rotation axis with respect to the entire mantle but is only valid for mass redistribution within the planet occurring on the time scale of a few million years. Such a motion, called true polar wander (TPW), is detected by palaeomagneticiens assuming that the Earth's magnetic field remains on average aligned with the spin axis. Our model shows that a downgoing cold slab induces a TPW which quickly brings this slab to the pole for a mantle of uniform viscosity. The same slab is slowly moved toward the equator when a large viscosity increase with depth takes place in the mantle. Our model is also suitable to investigate the effects of a non-steady-state convection on the Earth's rotation. We discuss these effects using a simple mass redistribution model inspired by the pioneering paper of Goldreich & Toomre (1969). It consists of studying the TPW induced by a random distribution of slabs sinking into the mantle. For such a mass redistribution, only a strongly stratified mantle can reduce the Earth's pole velocity below 1d Ma-1, which is the upper bound value observed by palaeomagnetic investigations for the last 200 Ma. Our model also shows that when corrected for the hydrostatic flattening, the Earth's polar inertia generally corresponds to the maximum inertia, as it is presently observed. However, this may not be the case during some short time periods. We also discuss the amount of excess polar flattening that can be related to tidal deceleration. This frozen component is found to be negligible. Copyright © 1993, Wiley Blackwell. All rights reserved

THERE is accumulating evidence1,2 that the average location of the Earth's magnetic pole has moved relative to the deep mantle over the past 200 Myr. This phenomenon of 'true polar wander' may be caused by the advance and retreat of ice sheets3 or by mass redistribution in the Earth's interior due to changes in the pattern of mantle convection4,5. New analyses1,2,6 of polar-wander data show a significant shift of the pole in the late Cretaceous, but this period is thought to have been too warm for glaciation to have occurred. Thus, most of the true polar wander must be due to mass movements in the mantle. Here we show, by analysing the appropriate equations for polar wander, that both viscosity and chemical stratification in the mantle are important in determining the rate of polar wander. The dynamical effects of chemical stratification and high viscosity (>1023 poise) in the lower mantle are to decrease polar-wander speed to a level consistent with the averaged velocities inferred from palaeomagnetic data. Whole-mantle convection models, without non-adiabatic density jumps at 670 km depth and with viscosities less than 1023 P, would produce wandering rates in excess of 5° per Myr.

True polar wander and long-wavelength dynamic topography

A new class of multilayered, viscoelastic Earth models based on PREM is applied to the modeling of Earth's rotation instabilities and associated sea‐level changes, induced by the occurrence of Pleistocene ice‐age cycles that match the oxygen isotope records over the last 0.8 Myr. The novelty of our approach stands on the usage, for the first time in post‐glacial rebound induced sea‐level studies, of a fully analytical scheme based on normal mode theory that allows to deal with complexities of the real Earth, such as lithospheric and mantle layering, sphericity and self‐gravitation. The ice models are based on ICE‐3G.Our results show that differences in true polar wander (TPW) and TPW‐induced sea‐level changes between Earth models with a few layers and models having enough layers so that saturated continuum limits are reached, can amount factors 2 to 3. This may change conclusions derived from some earlier studies on polar wander induced changes in climate, that were based on Earth models with a limited amount of layers. The results indicate that models containing about 15 layers with a stratified lower mantle and transition zone have reached saturated continuum responses for both TPW and TPW‐induced sea‐level changes. Stratification of the lithosphere is not important. This is in contrast with the sensitivity of post‐seismic deformation models. This difference in sensitivity on the radial profile of the Earth model can be explained with simple arguments on the dependence on zonal degree.

[1] Satellite laser ranging (SLR) data were used to determine the variations in the Earth's principal figure axis represented by the degree 2 and order 1 geopotential coefficients: C21 and S21. Significant variations at the annual and Chandler wobble frequencies appear in the SLR time series when the rotational deformation or “pole tides” (i.e., the solid Earth and ocean pole tides) were not modeled. The contribution of the ocean pole tide is estimated to be only ∼8% of the total annual variations in the normalized coefficients: / based on the analysis of SLR data. The amplitude of the nontidal annual variation of is only ∼ 30% of from the SLR time series. The estimates of the annual variation in from SLR, the Gravity Recovery and Climate Experiment (GRACE) and polar motion excitation function, are in a good agreement. The nature of the linear trend for the Earth's figure axis determined by these techniques during the last several years is in general agreement but does not agree as well with results predicted from current glacial isostatic adjustment (GIA) models. The “fluid Love number” for the Earth is estimated to be ∼0.9 based on the position of the mean figure axis from the GRACE gravity model GGM03S and the mean pole defined by the IERS 2003 conventions. The estimate of / from GRACE and SLR provides an improved constraint on the relative rotation of the core. The results presented here indicate a possible tilt of the inner core figure axis of ∼2° and ∼3 arc sec displacement for the figure axis of the entire core.

We introduce a new method by which to compute global post-seismic deformation (PSD) in a spherically symmetric, self-gravitating viscoelastic earth model. Previous methods are based on simplified earth models that neglect compressibility and/or the continuous variation of the radial structure of Earth. This is because the previous mode summation technique cannot avoid intrinsic numerical difficulties caused by the innumerable poles that appear in a realistic earth model that considers such effects. In contrast, the proposed method enables both of these effects to be taken into account simultaneously. We carry out numerical inverse Laplace integration, which allows evaluation of the contribution from all of the innumerable modes of the realistic earth model. Using this method, a complete set of Green's functions is obtained, including functions of the time variation of the displacement, gravity change, and the geoid height change at the surface for strike-slip, dip-slip, horizontal and vertical tensile point dislocations. As an earth model, we employ the preliminary reference earth model (PREM) and a convex viscosity profile. Further, we investigate the effects of fine layering of the viscoelastic structure and compressibility on a time-series of PSD using the Green's function for a dip-slip fault. The result indicates that the effect of increasing number of layers is saturated at several tens of layers even when compressibility is taken into account and that the effect of compressibility is detectable with modern observational techniques for a shallower large earthquake (Mw∼ 8). As an application, the PSD due to the Sumatra-Andaman Islands earthquake (Mw= 9.3) is estimated. We show that the rate of post-seismic vertical displacement and gravity change is possibly detected in the far field where the epicentral distance exceeds 400 km.

The amplitude of true polar wander events is shown to occur in cycles out of phase with the formation of supercontinents over the past 3 Gyr. Associated with small-amplitude true polar wander, supercontinents act to stabilize the spin axis. Stabilization can be explained by reduced lithospheric elasticity and/or the triaxial (oblate) figure of the Earth, both of which are legacies of the supercontinent cycle. An excess triaxial ellipticity would only be expected to affect the first transition between supercontinents, whereas decreased lithospheric elasticity would have also influenced formation of the first supercontinent, if sizable enough. My analysis indicates the presence of 4 supercontinents since 3 Ga and proposes that the triaxial Earth originates from the supercontinent cycle.

A Mw 9.0 earthquake occurred off the coast of Japan in the Pacific Ocean on March 11, 2011. At shorter time and spatial scales, mass redistribution caused by an earthquake produces local variations of the geoid reaching a few centimeters. In this paper, we apply the bilinear wavelets analysis based on the Abel-Poisson scale and its corresponding wavelet functions to GRACE monthly gravity data in order to detect coseismic changes of the geoid heights due to the Tohoku-Oki earthquake. As a result, we can clearly distinguish that multiscale geoid heights are decreased temporarily in the order about -4 meter on average throughout the region between February 2011 and March 2011 when compared with monthly geoid heights observed by GRACE satellite.

[1] The present-day velocity of true polar wander (TPW) and the displacement of the axis of rotation of the Earth in response to ice ages, resulting from stratified, viscoelastic Earth models, are sensitive to the nonadiabatic density gradient in the mantle. Previous studies, based on a fully nonadiabatic, or chemically stratified mantle, overestimated the present-day TPW for lower mantle viscosities in the range 1021–1022 Pa s. For a density profile in agreement with the reference seismological model, where nonadiabaticity is confined to the transition zone between 420 and 670 km, the predicted present-day TPW for viscosities on the order of 1021 Pa s is 0.65–0.9 deg/Myr, substantially lower than the 3.0 deg/Myr obtained for the chemical mantle. This decrease is due to the lack of an isostatic restoring force in the adiabatic case or to a global reduction of the buoyancy, which favors the attainment of rotational equilibrium. The correctness of this physical interpretation is demonstrated by the behavior of a fully adiabatic phase change that can be satisfactorily modeled by deleting the buoyancy restoring modes due to chemical density contrasts. This finding provides quantitative support to the procedure used in previous studies to simulate the rotational behavior of an adiabatic transition zone by simply deleting the buoyancy modes triggered by the (chemical) density contrasts at 670 km, as done by Mitrovica and Milne [1998] for present-day TPW and by Ricard and Sabadini [1990] for long-term TPW driven by mantle density anomalies. The reduction of present-day TPW induced by the Pleistocenic deglaciation, for an adiabatic mantle with nonadiabatic density gradients due to phase changes localized in the transition zone, impacts the inversion of the lower mantle viscosity. Predictions based on such a model are characterized by two best fit values in proximity of 1021 and 1022 Pa s, resembling the behavior of the time derivative of the second-degree component of the gravity field. The reduction of predicted present-day TPW due to Pleistocene deglaciation suggests that other mechanisms, such as present-day ice mass instability in Antarctica and Greenland, are presently contributing to the drift of 0.9 deg/Myr of the axis of rotation toward Newfoundland. The secular drift of the adiabatic mantle model during the continuous occurrence of ice ages is increased by 50% with respect to the chemically stratified one, showing a longer decay time after termination of the ice cycles. This mantle model confirms that ice cycles cannot be the major source of TPW for timescales of 106–107 years.

In recent years a number of studies have investigated the influence of compressibility on geophysical observables such as postglacial rebound deformation rates and the geoid. Some of these studies indicate that long-term signatures such as the geoid might be sensitive to compressibility. As both load relaxation and tidal-effective relaxation of the equatorial bulge are operative in a dependent way, polar wander can potentially be more sensitive to compressible rheologies if the interference between the two relaxation mechanisms is constructive. This has motivated us to study the influence of compressibility on true polar wander by means of spherical, laterally homogeneous, self-gravitating analytical earth models. As we wish to study both short-term rotational changes and polar wander on geological time-scales, we employ a Maxwell viscoelastic model instead of a Newtonian viscous model. The latter is commonly used in geoid modelling. The purpose of this paper is to concentrate on the basic physical aspects of the differences between compressible and incompressible rotational deformation, rather than applying the procedures to fine-graded multi-layered PREM models with realistic forcing functions. An important issue of our method concerns the analytical instead of numerical way of solving the differential equations by the propagator matrix method. Compressible viscoelastic relaxation has usually been treated numerically until now. The results show that homogeneous earth models do not have significant differences on long time-scales between compressible and the corresponding incompressible cases. Compressibility introduces a denumerably infinite set of short-time relaxation modes. The relaxation times of these dilatation modes can be approximated analytically. Two-layer core-mantle models show relatively large differences between incompressible and compressible Maxwell rheologies. Simplified models of true polar wander triggered by Heaviside loads show that differences of several tens of per cent between incompressible and compressible Maxwell rheologies are possible. True polar wander is decreased in the compressible case on both short and long time-scales, which means that smaller viscosities are required to explain polar-wander measurements than in the incompressible case.