Load Love Numbers Research Articles

Coupled poroelastic and gravitational deformation of a layered spherical Earth – I: load Love numbers

SUMMARY In this paper, we propose a fully coupled two-phase poroelastic deformation theory for a spherically layered and self-gravitating Earth. The earth model consists of a solid inner core, a fluid outer core and poroelastic mantle and crust. The boundary-value problems are posed in the Laplace-transformed domain using the spherical system of vector functions, and analytical solutions are obtained in each layer using the dual-variable and position matrix method. As an application, the surface loading problem is considered. The undrained and drained limits are discussed with the corresponding governing equations, which are different from the conventional ones where body force is ignored. Numerical examples show that the poroelastic effect can be significant since the difference between the results for undrained and drained limits are large with obvious time-dependent deformation. This newly derived theory for the coupled boundary-value problem will have broad applications, such as displacement and gravity change due to groundwater depletion, poroelastic rebound after an earthquake and risk evaluation of earthquakes induced by water injection.

A commercial finite element approach to modelling Glacial Isostatic Adjustment on spherical self-gravitating compressible earth models

SUMMARYThis paper presents a method that modifies commercial engineering-oriented finite element packages for the modelling of Glacial Isostatic Adjustment (GIA) on a self-gravitating, compressible and spherical Earth with 3-D structures. The approach, called the iterative finite element body and surface force (FEMIBSF) approach, solves the equilibrium equation for deformation using the ABAQUS finite element package and calculates potential perturbation consistently with finite element theory, avoiding the use of spherical harmonics. The key to this approach lies in computing the mean external body forces for each finite element within the Earth and pressure on Earth's surface and core–mantle boundary (CMB). These quantities, which drive the deformation and stress perturbation of GIA but are not included in the equation of motion of commercial finite element packages, are implemented therein. The method also demonstrates how to calculate degree-1 deformation directly in the spatial domain and Earth-load system for GIA models. To validate the FEMIBSF method, loading Love numbers (LLNs) for homogeneous and layered earth models are calculated and compared with three independent GIA methodologies: the normal-mode method, the iterative body force method and the spectral-finite element method. Results show that the FEMIBSF method can accurately reproduce the unstable modes for the homogeneous compressible model and agree reasonably well with the Love number results from other methods. It is found that the accuracy of the FEMIBSF method increases with higher resolution, but a non-conformal mesh should be avoided due to creating the so-called hanging nodes. The role of a potential force at the CMB is also studied and found to only affect the long-wavelength surface potential perturbation and deformation in the viscous time regime. In conclusion, the FEMIBSF method is ready for use in realistic GIA studies, with modelled vertical and horizontal displacement rates in a disc load case showing agreement with other two GIA methods within the uncertainty level of GNSS measurements.

Constraining the Venus Interior Structure with Future VERITAS Measurements of the Gravitational Atmospheric Loading

The complex dynamics of the Venus atmosphere produces a periodic mass redistribution pattern that creates a time-variable modulation of the gravity field of Venus. This gravity signal depends on the net transport of mass across the globe and on the response of the solid body to the normal loading of its crust imparted by the atmosphere. In this work, we explore the possibility of measuring this phenomenon with VERITAS, a NASA Discovery-class mission. By simulating the gravity science experiment, we explore the possibility of measuring the response of Venus to the atmospheric loading, parametrized by the loading Love numbers (), and assess the dependence of these parameters on fundamental interior structure properties. Using the most recent models of Venus’ interior, we compute the Venus Love numbers in a compressible viscoelastic setting and compare them with the predicted uncertainty of the VERITAS measurements. We show that VERITAS will measure at the 4% level and that this measurement could possibly help to distinguish between different equally plausible interior structure models, especially allowing us to distinguish different rheological laws. We also show that a measurement campaign such as the VERITAS gravity science investigation has the potential of measuring not only at the loading forcing frequency, but also at the tidal frequency, ultimately providing a way to probe the response of the planet at different forcing periods.

Free decay and excitation of the chandler wobble: self-consistent estimates of the period and quality factor

The period TCW and quality factor QCW of the Chandler wobble (CW) as well as polar motion (PM) transfer functions are all determined by the Earth’s layered structure, mass distribution, elasticity, rheology and energy dissipation, via the Earth’s dynamic figure parameters and complex degree-2 Love numbers. However, most previous studies used geophysical excitations derived from real-valued PM transfer functions to invert for TCW and QCW, thus leading to results that are not self-consistent. By separating the observed PM into the freely decaying CW and the excited PM, a traverse-based method is proposed to search values of TCW and QCW that can fit both sides simultaneously, yielding the self-consistent estimates of TCW = 430.4 mean solar days and QCW = 130. This implies the degree-2 tidal Love number k = 0.35011 − 0.00226i and load Love number k' = − 0.36090 + 0.00233i, and the PM transfer functions TNL = 1.80001 − 0.00692i (non-loading) and TL = 1.15040 − 0.00023i (loading) valid at the Chandler period.

Scalable self attraction and loading calculations for unstructured ocean tide models

Self attraction and earth-loading effects are important for accurately modeling global tides. A common approach of handling this forcing is to expand mass anomalies into spherical harmonics, which are scaled by load Love numbers to account for elastic earth deformation. We investigate two different approaches to perform these calculations for ocean models that employ unstructured meshes and distributed memory parallelization. The first approach leverages a highly efficient spherical harmonics library, but requires all-to-one and one-to-all communications and interpolation operations between the unstructured and a structured mesh. This approach is compared to a parallel algorithm that computes the spherical harmonic transformations directly on the unstructured mesh with an all-reduce communication. Our results show that although the unstructured mesh calculations are more expensive, the scalability of the unstructured mesh approach allows for more efficient spherical harmonics transforms for high-resolution meshes and large processor counts. This methodology enables the efficient inclusion of tidal dynamics large-scale Earth system model simulations.

Contribution of GRACE Satellite Mission to the Determination of Orthometric/Normal Heights Corrected for Their Dynamics—A Case Study of Poland

Physical heights were traditionally determined without considering the dynamic processes of the Earth induced from temporal mass variations. The Gravity Recovery and Climate Experiment (GRACE) mission provided valuable data that allow the estimation of geoid/quasigeoid height changes and vertical deformations of the Earth’s surface induced from temporal mass loading, and thereby temporal variations of physical heights. The objective of this investigation is to discuss the determination of orthometric/normal heights considering mass transports within the Earth’s system. An approach to determine such heights was proposed. First, temporal variations of orthometric/normal heights (ΔH/ΔH*) were determined using the release 6 GRACE-based Global Geopotential Models together with load Love numbers obtained from the preliminary reference Earth model. Then, those variations were modelled and predicted using the seasonal decomposition (SD) method. The proposed approach was tested on the territory of Poland. The main results obtained reveal that ΔH/ΔH* over the area investigated are at the level of a couple of centimetres and that they can be modelled and predicted with a millimetre accuracy using the SD method. Orthometric/normal heights corrected for their dynamics can be determined by combining modelled ΔH/ΔH* with orthometric/normal heights referred to a specific reference epoch.

Constraining the Internal Structures of Venus and Mars from the Gravity Response to Atmospheric Loading

The gravity fields of celestial bodies that possess an atmosphere are periodically perturbed by the redistribution of fluid mass associated with atmospheric dynamics. A component of this perturbation is due to the gravitational response of the body to the deformation of its surface induced by the atmospheric pressure loading. The magnitude of this effect depends on the relation between the loading and the response in terms of geopotential variations measured by the load Love numbers. In this work, we simulate and analyze the gravity field generated by the atmospheres of Venus and Mars by accounting for different models of their internal structure. By precisely characterizing the phenomena that drive the mass transportation in the atmosphere through general circulation models, we determine the effect of the interior structure on the response to the atmospheric loading. An accurate estimation of the time-varying gravity field, which measures the atmospheric contribution, may provide significant constraints on the interior structure through the measurement of the load Love numbers. A combined determination of tidal and load Love numbers would enhance our knowledge of the interior of planetary bodies, providing further geophysical constraints in the inversion of internal structure models.

Modeling gravimetric signatures of third-degree ocean tides and their detection in superconducting gravimeter records

We employ the barotropic, data-unconstrained ocean tide model TiME to derive an atlas for degree-3 tidal constituents including monthly to terdiurnal tidal species. The model is optimized with respect to the tide gauge data set TICON-td that is extended to include the respective tidal constituents of diurnal and higher frequencies. The tide gauge validation shows a root-mean-square (RMS) deviation of 0.9–1.3 mm for the individual species. We further model the load tide-induced gravimetric signals by two means (1) a global load Love number approach and (2) evaluating Greens-integrals at 16 selected locations of superconducting gravimeters. The RMS deviation between the amplitudes derived using both methods is below 0.5 nGal (1 nGal = 0.01 frac{text {nm}}{text {s}^2}) when excluding near-coastal gravimeters. Utilizing ETERNA-x, a recently upgraded and reworked tidal analysis software, we additionally derive degree-3 gravimetric tidal constituents for these stations, based on a hypothesis-free wave grouping approach. We demonstrate that this analysis is feasible, yielding amplitude predictions of only a few 10 nGal, and that it agrees with the modeled constituents on a level of 63–80% of the mean signal amplitude. Larger deviations are only found for lowest amplitude signals, near-coastal stations, or shorter and noisier data sets.

The Crustal Vertical Deformation Driven by Terrestrial Water Load from 2010 to 2014 in Shaanxi–Gansu–Ningxia Region Based on GRACE and GNSS

The terrestrial water resources in Shaanxi–Gansu–Ningxia (SGN) region are relatively scarce, and its climate change is unstable. Research on the deformation driven by terrestrial water load is of great significance to the dynamic maintenance of reference station networks. In this paper, data derived from Gravity Recovery and Climate Experiment (GRACE) and Global Navigation Satellite System (GNSS) from 2010 to 2014 were combined to monitor the spatiotemporal characteristics of surface vertical deformation caused by terrestrial water load change. The single scale factor was calculated by comparing CPC, WGHM, and GLDAS hydrological model to restore filtering leakage signal. The singular spectrum analysis (SSA) method was used to extract the principal component of temporal vertical deformation, and its spatial distribution was analyzed. At the same time, in order to study the relationship between the terrestrial water load deformation from GRACE and that from GNSS, the first-order term correction, the Atmosphere and Ocean De-aliasing Level-1B product (GAC) correction, and the first-order load LOVE number correction for GRACE were adopted in this paper. In addition, a quantitative comparative analysis of both the monitoring results was carried out. The results show that the time-variable characteristics of surface vertical deformation characterized by the filtered three hydrological models were consistent with those of GRACE. The correlation coefficient and Nash–Sutcliffe efficiency coefficient (NSE) values were the highest in the GLDAS model and the GRACE model, respectively; the former index is 0.93, while the latter is 0.85. The crustal vertical deformation from terrestrial water load showed a declining rate from 2010 to 2014. Its spatial change rate showed an obvious ladder distribution, with the surface subsidence rate gradually decreasing from south to north. In addition, weighted root mean square (WRMS) contribution rate of the crustal vertical deformation resulting from GRACE with GAC correction between the different GNSS stations ranged from 18.52% to 54.82%. The correlation coefficient between them was close to 0.70. After deducting the mass load impact of GRACE only, the WRMS contribution rate of the corresponding stations decreased from −8.42% to 21.18%. The correlation coefficient between them reduced noticeably. Adding GAC back can increase the comparability with GRACE and GNSS in terms of monitoring the crustal vertical deformation. The annual amplitude and phase of surface vertical deformation resulting from GRACE with GAC correction were close to those of GNSS. The research results can help to explore the motion mechanism between water migration and surface deformation, which is of benefit in the protection of the water ecological environment in the region.

Load Love numbers for various rheological models of Venus

Load Love numbers for various rheological models of Venus

Effect of Earth Models on Coulomb Stress Change Caused by Surface Load

Based on the elastic spherical Earth models, this study presents the radial Love numbers and Green’s functions of the surface load problem, then evaluates the effect of different Earth models on the Coulomb stress changes due to elastic deformation by surface load. First of all, the radial load Love numbers are introduced and verified by comparing them with the results of an analytical method. Second, the formulae of the radial Green’s functions of the internal displacements and strain tensors are obtained by summing the Love numbers. The results are compared with Green’s function of half-space to verify the correctness. Then, the stress tensors are obtained by the constitutive relation. Finally, the Coulomb stress changes due to elastic deformation around the Wenchuan Mw7.9 earthquake by the impoundment load of the Zipingpu reservoir are calculated based on five Earth models. The results show that the elastic deformations caused by the impoundment load do not increase the seismicity rate in the study area. At the epicenter of the great event, the maximum effect due to the Earth models is about 38%. The stratification of the Earth is the main effect factor on the Coulomb stress change, and it should not be ignored in actual studies. The biggest difference of Coulomb stress changes based on the different stratified Earth models is close to 15%, which also cannot be ignored.

Deformation due to surface temperature variation on a spherically layered, transversely isotropic and self-gravitating Earth

SUMMARYWe present a theory of modern, thermally induced deformation in a realistic Earth. The heat conduction equation is coupled with standard elastic deformation theory to construct a boundary-value problem comprised of eighth-order differential equations. The accurate and stable dual variable and position propagating matrix technique is introduced to solve the boundary-value problem. The thermal load Love numbers are defined to describe the displacements and potential changes driven by thermally induced deformation. The proposed analytical method is validated by comparing the present results with exact solutions for a homogeneous sphere, which are also derived in this paper. The analytical method is then applied to a realistic Earth model to evaluate the effects of layering and self-gravitation of the Earth on displacement and changes of potential. Furthermore, the frequency dependence in the thermal load is illustrated by invoking different thermal periodicities in the computation. Thermal anisotropy is also considered by comparing the results obtained using isotropic and transversely isotropic Earth models. Results show that, when simulating thermally induced deformation, invoking a homogeneous spherical Earth leads to results that substantially differ from those obtained using a more realistic Earth model.

Oceanic Tide Model FES2014b: Comparison with Gravity Measurements

The oceanic gravimetric effect is calculated by expanding the cotidal maps of the FES2014 oceanic tide model into spherical functions with the use of load Love numbers from the work (Spiridonov and Vinogradova, 2017). The calculation results for the amplitudes and phases of the oceanic gravimetric effect for eight main tidal waves (Q1, O1, P1, K1, N2, M2, S2, and K2) at nine European gravity observation stations of the Global Geodynamic Project (GGP) network are presented. The amplitude delta factors and phase shifts of these tidal waves for the Earth with the ocean are predicted and compared with observations. The difference vectors (prediction minus observations) are analyzed in detail with the use of both different theoretical tide models and oceanic-effect calculation versions. The FES2014 model is included in the ATLANTIDA3.1 tidal prediction software. The results of the work show, first, that the FES2014 is one of the best oceanic models today. The amplitude delta factors and phase shifts of tidal waves calculated with its help for the Earth with the ocean are generally in better agreement with observations when analyzed separately than the FES2012 model results. This is especially evident at Strasbourg and Vienna sites. The analysis of difference vectors implies that FES2014 is slightly inferior to FES2012 in terms of proximity to observations. This fact still requires an explanation. However, the amplitudes of the difference vectors calculated with the FES2012 and FES2014 models usually differ by no more than a few hundredths of a percent of the tidal wave amplitudes. Such small differences can be due to a number of factors unrelated to the oceanic effect.

Atmospheric Loading Deformations

The paper presents a method for calculating the Earth’s surface deformations due to the atmospheric loading effect. The load displacements are calculated using the expansion coefficients of the surface atmospheric pressure fields of the European ERA5 reanalysis project for the spherical functions, as well as the load Love numbers that we calculated earlier. The results of calculating atmospheric deformations for Baksan station are presented.

Atmospheric Loading Tilts

A method is presented for calculating tilts of the Earth’s surface due to the atmospheric loading effect The loading tilts, taking into account the effect of direct Newtonian attraction, are calculated by the expansion coefficients for the spherical functions of the surface atmospheric pressure fields from the European reanalysis project ERA5, as well as the load Love numbers that we calculated earlier. The obtained time series of tilts (in the north and east directions) are compared with studies by other authors.

Estimation of physical height changes from GRACE satellite mission data and WGHM over Turkey

Gravity Recovery and Climate Experiment (GRACE) satellite mission had provided gravity data for ca. 15 years, completed its mission in 2017 and handed over to GRACE Follow-On. During its active life, GRACE contributed to the understanding of temporal variations of geoid/quasigeoid heights as well as vertical and horizontal displacements of the Earth’s surface induced by mass loading, and thereby the physical height (e.g. the orthometric/normal height) changes. The main objective of this research is to estimate physical height changes over the area of Turkey. The release 6 GRACE-based global geopotential models and WaterGAP (Water—a Global Assessment and Prognosis) Global Hydrology Model products as well as load Love numbers from the Preliminary Reference Earth Model were used as input data. Physical height changes over Turkey were estimated using three different approaches: the GRACE-based Global Geopotential Models, the Green’s function, and the Terzaghi’s principle. The estimated physical height changes were analysed and modelled by means of the seasonal decomposition method and the Principal Component Analysis/Empirical Orthogonal Function method. The main findings reveal that physical height changes over Turkey reach the level of a few centimetres during years 2004–2010.

Atmospheric Loading Displacements

A method for calculating of the Earth’s surface displacements due to atmospheric loading effect is presented. The load displacements are calculated by the expansion coefficients of the surface atmospheric pressure fields of the European ERA5 reanalysis project for the spherical functions, as well as the load Love numbers that we calculated earlier. The obtained time series of displacements along three axes (vertical, in the direction to the north and east) are compared with the works of other authors.

Calculation of Tidal Displacements and Tilts for a Nonelastic Rotating Earth

A technique for calculating tidal displacements and tilts for the Earth both with the ocean and without it is presented in this work. This technique mainly follows the technique described in the International Earth Rotation Service Conventions and differs from it in small corrections in formulas for the calculation of displacements for diurnal (21) and semidiurnal (22) waves, as well as in an another set of usual and load Love numbers. Values of the gamma factors for the Earth without the ocean are also presented. Our results can be used for the processing of GNSS and tilt-measuring observations. The obtained results are compared with works by other authors, and the calculation technique is implemented in the new version of the ATLANTIDA3.1_2017 program for predicting parameters of Earth tides.

Benchmarked computation of time-domain viscoelastic Love numbers for adiabatic mantles

SUMMARYThe viscoelastic load Love numbers encapsulate the Earth’s rheology in a remarkably efficient fashion. When multiplied by a sudden increment of spherical harmonic load change, they give the horizontal and vertical surface displacements and gravity change at all later times. Incremental glacial load changes thus need only be harmonically decomposed, multiplied by the Love numbers and summed to predict the Earth’s response to glacial load redistributions. The computation of viscoelastic Love numbers from the elastic, viscous and adiabatic profiles of the Earth is thus the foundation upon which many glacial isostatic adjustment models are based. Usually, viscoelastic Love numbers are computed using the Laplace transform method, employing the correspondence principle to convert the viscoelastic equations of motion into the elastic equations with complex material parameters. This method works well for a fully non-adiabatic Earth, but can accommodate realistic partially adiabatic and fully adiabatic conditions only by changing the Earth’s density profile. An alternative method of Love number computation developed by Cathles (1975) avoids this dilemma by separating the elastic and viscous equations of motion. The separation neglects a small solid-elastic/fluid-elastic transition for compressible deformation, but allows freely defining adiabatic, partially adiabatic or fully non-adiabatic profiles in the mantle without changing the Earth’s density profile. Here, we update and fully describe this method and show that it produces Love numbers closely similar to those computed for fully non-adiabatic earth models computed by the correspondence principle, finite element and other methods. The time-domain method produces Love numbers as good as those produced by other methods and can also realistically accommodate any degree of mantle adiabaticity. All method implementations are available open source.

Earth rotation deceleration/acceleration due to semidiurnal oceanic/atmospheric tides: Revisited with new calculation

The global oceanic/atmospheric tides exert decelerating/accelerating secular torques on the Earth rotation. We developed new formulations to accurately calculate amounts of two kinds of secular tidal torques. After Melchior, we found that an additional factor 1+k-l = 1.216, which has been formerly neglected, must be multiplied unto the tidal torque integral. By using our refined formulations and the recent oceanic/atmospheric global tide models, we found that: (i) semidiurnal oceanic lunar/solar tides exert decelerating torques of about −4.462×1016/−0.676×1016Nm respectively and (ii) atmospheric S2 tide exerts accelerating torque of 1.55×1015Nm. Former estimates of the atmospheric S2 tidal torque were twice as large as our estimate due to improper consideration of loading effect. We took the load Love number for atmospheric loading effect from Guo et al. (2004). For atmospheric loading of spherical harmonic degree two, the value of k'=−0.6031 is different from that for ocean loading as k'=−0.3052, while the latter is currently used for both cases – ocean/atmospheric loading – without distinction. We discuss (i) the amount of solid Earth tidal dissipation (which has been left most uncertain) and (ii) secular changes of the dynamical state of the Earth-Moon-Sun system. Our estimate of the solid Earth tidal torque is −4.94×1015Nm.