Surface Mass Loading Research Articles

Basis functions for the consistent and accurate representation of surface mass loading

SUMMARY Inversion of geodetic site displacement data to infer surface mass loads has previously been demonstrated using a spherical harmonic representation of the load. This method suffers from the continent-rich, ocean-poor distribution of geodetic data, coupled with the predominance of the continental load (water storage and atmospheric pressure) compared with the ocean bottom pressure (including the inverse barometer response). Finer-scale inversion becomes unstable due to the rapidly increasing number of parameters which are poorly constrained by the data geometry. Several approaches have previously been tried to mitigate this, including the adoption of constraints over the oceanic domain derived from ocean circulation models, the use of smoothness constraints for the oceanic load, and the incorporation of GRACE gravity field data. However, these methods do not provide appropriate treatment of mass conservation and of the ocean’s equilibrium-tide response to the total gravitational field. Instead, we propose a modified set of basis functions as an alternative to standard spherical harmonics. Our basis functions allow variability of the load over continental regions, but impose global mass conservation and equilibrium tidal behaviour of the oceans. We test our basis functions first for the efficiency of fitting to realistic modelled surface loads, and then for accuracy of the estimates of the inferred load compared with the known model load, using synthetic geodetic displacements with real GPS network geometry. Compared to standard spherical harmonics, our basis functions yield a better fit to the model loads over the period 1997‐2005, for an equivalent number of parameters, and provide a more accurate and stable fit using the synthetic geodetic displacements. In particular, recovery of the low-degree coefficients is greatly improved. Using a nine-parameter fit we are able to model 58 per cent of the variance in the synthetic degree-1 zonal coefficient time-series, 38‐41 per cent of the degree-1 non-zonal coefficients, and 80 per cent of the degree-2 zonal coefficient. An equivalent spherical harmonic estimate truncated at degree 2 is able to model the degree-1 zonal coefficient similarly (56 per cent of variance), but only models 59 per cent of the degree-2 zonal coefficient variance and is unable to model the degree-1 non-zonal coefficients.

Glacial-Isostatic Adjustment-I. The Forward Problem

Summary The isostatic adjustment of a radially stratified visco-elastic spheroid is treated using space-time Green functions for the associated surface mass load boundary value problem. These impulse response functions are convolved with a Heaviside function to give the time dependent deformation of the planet which would be produced by a unit point mass brought up from infinity at t= 0 and allowed to remain on the surface. The resulting ’ Heaviside Green functions ’ can be employed to simulate all of the important signatures of glacial isostatic adjustment. Given a space-time dependent surface mass load consisting of ice sheet ablation histories and a model of the simultaneous filling of the ocean basins, these source terms are simply convolved with the Heaviside Green function appropriate to a specific response signature. A realistic model of the spatial distribution of the main late Pleistocene ice loads and of their temporal disintegration is constructed. The response of two visco-elastic earth models is computed and compared to a global set of relaxation data (relative sea-level curves). On the basis of this initial comparison of theory and observation the possibility that the lower mantle has a viscosity which is significantly in excess of the viscosity of the upper mantle is excluded. In addition, clear evidence of the presence of the lithosphere has been found in relaxation data from sites which were near the edge of the Laurentide ice sheet. Such data should therefore prove useful as a basis for analysis of lateral variations in lithosphere thickness. Further extensions of the calculation are suggested.

Glacial-Isostatic Adjustment-II. The Inverse Problem

Summary The principle of correspondence is invoked to determine Laplace transform domain solutions to the surface mass loading problem for radially stratified visco-elastic (Maxwell) spheroids. These Laplace transform domain solutions are expressed in terms of visco-elastic analogues of the conventional surface load Love numbers of elasticity. These visco-elastic Love numbers may be approximately transformed to the time domain using an extremal technique. Application of this technique shows that the Love number time histories may be well approximated by the linear superposition of a discrete set of purely exponential relaxations plus a constant term. Alternatively the discrete spectrum of relaxation times involved in the synthesis of each Love number time history may be found exactly by solving the associated homogeneous problem. Such solutions determine the set of eigen-decay times associated with the normal modes of viscous gravitational relaxation of the visco-elastic planetary model. The solution of the inhomogeneous problem may be expressed in the form of a normal mode expansion. This normal mode expansion is employed as the basis for the construction of a rigorous first-order perturbation theory for the inference of the viscosity of the deep interior of the planet. A variational principle is derived which determines to first order that shift in position of a free decay pole in the relaxation spectrum which is forced by the addition of a radially-distributed perturbation of viscosity to the starting model. This determines the differential kernels required for the solution of the inverse problem. The uniqueness of the state of isostatic equilibrium for the viscously incompressible Maxwell models employed in this analysis is demonstrated and this uniqueness allows us to calculate the shift in the residue which is associated with the shift in position of a free decay pole for the inhomogeneous problem. The inhomogeneous problem is thus closed to first order. A formula is derived for the differential kernels appropriate to the inhomogeneous problem. The time domain form of these kernels may be calculated analytically. The structure of the full inverse theory is sufficiently simple that it may be employed to rigorously test the compatibility of the simple Maxwell model with the observed relaxation of the Earth's shape which accompanied deglaciation at the end of the last ice age.

Rotational stability of dynamic planets with elastic lithospheres

We revisit the classic problem of the secular rotational stability of planets in response to loading using the fluid limit of viscoelastic Love number theory. Gold (1955) and Goldreich and Toomre (1969) considered the stability of a hydrostatic planet subject to an uncompensated surface mass load and concluded that a mass of any size would drive true polar wander (TPW) that ultimately reorients the load to the equator. Willemann (1984) treated the more self‐consistent problem where the presence of a lithosphere leads to both imperfect load compensation and a remnant rotational bulge. Willemann considered axisymmetric loads and concluded that the equilibrium pole location was governed by a balance, independent of elastic lithospheric thickness, between the load‐induced TPW and stabilization by the remnant bulge. Our new analysis demonstrates that the equilibrium pole position is a function of the lithospheric strength, with a convergence to Willemann's results evident at high values of elastic thickness (>400 km for an application to Mars), and significantly larger predicted TPW for planets with thin lithospheres. Furthermore, we demonstrate that nonaxisymmetric surface mass loads and internal (convective) heterogeneity, even when these are small relative to axisymmetric contributions, can profoundly influence the rotational stability. Indeed, we derive the relatively permissive conditions under which nonaxisymmetric forcing initiates an inertial interchange TPW event (i.e., a 90° pole shift). Finally, Willemann's analysis is often cited to argue for a small (<18°) TPW of Mars driven by the development of a Tharsis‐sized load. We show that even in the absence of the destabilizing effects of load asymmetry, the equations governing rotational stability permit higher excursions of the Martian rotation vector than has previously been appreciated.

Oscillating frequency response of a langasite crystal microbalance in liquid phases

Langasite (La 3Ga 5SiO 14) crystal is one of the perspective crystalline materials for various applications in acoustoelectron and piezoelectric devices. In this work, the frequency responses of resonators made from Y-cut langasite crystal in a liquid phase were investigated. It was shown that the oscillating frequency of the langasite resonator decreased with increasing the surface mass loading, viscosity ( η) and density ( ρ) of the liquid phase. The correlation of the frequency shift and the mass loading was similar to the Sauerbrey equation. A linear dependence of the frequency shift on the surface mass loading was obtained within 3% of the intrinsic frequency of the langasite resonator. The frequency–mass coefficient of the langasite crystal microbalance (LCM) was 71.7% of that of the classical quartz crystal microbalance (QCM). The frequency of the LCM decreased linearly with increasing ( ρη) 1/2 The LCM had a low frequency–temperature coefficient in water in the temperature range of 30–50 °C. The LCM possessed excellent oscillating ability in the oscillator method, especially in high viscosity liquids. It could be operated in pure glycerol with two faces in contact with the liquid phase. The LCM was applied to monitor the coagulation process of whole blood.

Description of adhering with saturation using boundary conditions of hysteresis type

The problem considered in this paper is related to the development of biosensors which serve for the quantitative detection of proteins in solutions. An important part of such sensors is a wet cell, say a tiny box, filled with water into which a solution containing a protein to be detected is injected. Special molecules called aptamers are immobilized on the bottom of the wet cell. The aptamers can selectively bind the desired protein from the solution. The change of the surface mass loading can be analyzed using acoustic waves propagating along the aptamer layer. Thus, the concentration of the protein in the solution can be estimated. In this paper, a model describing the propagation of the protein in the wet cell and its adherence to the aptamer is proposed. It is assumed for simplicity that the propagation of the injected protein is governed by a diffusion equation. The central point of the model is a boundary condition of hysteresis type posed on the bottom of the wet cell. This condition provides the monotone growth with saturation of the surface concentration of the deposited protein. The monotonicity reflects non-detachment of already adhered protein molecules, whereas the saturation corresponds to the exhaustion of free aptamer molecules. We prove the existence and the uniqueness of solutions to this problem, study their regularity, and perform numerical simulations that clarify their behavior.

Effect of gravitational consistency and mass conservation on seasonal surface mass loading models

Increasingly, models of surface mass loads are used either to correct geodetic time coordinates by removing seasonal and other “noise”, or for comparison with other geodetic parameters. However, models of surface loading obtained by simply combining the mass redistribution due to individual phenomena will not in general be self‐consistent, in that (i) the implied global water budget will not be mass‐conserving, and (ii) the modelled sea level will not be an equipotential surface of Earth's total gravity field. We force closure of the global water budget by allowing the “passive” ocean to change in mass. This medium‐term passive ocean response will not be a uniform change in non‐steric ocean surface height, but must necessarily be spatially variable to keep the “passive” ocean surface on an equipotential. Using existing load models, we demonstrate the effects of our consistency theory. Geocenter motion is amplified significantly, by up to 43%.

Thin Film Silicon Microbridges for DNA Detection

AbstractThin-film MEMS molecular sensors are fabricated at temperatures below 110°C on glass substrates. The microelectromechanical structure consists of a surface micromachined bilayer bridge of phosphorous-doped hydrogenated amorphous silicon and aluminum with a patterned SiO2 layer on the top. Specific binding of DNA to functionalized SiO2 on the bridge is confirmed using fluorescence microscopy. Microbridges are electrostatically actuated and the resonance frequency measurements are performed in vacuum in the initial state after fabrication, after the chemical functionalization of the SiO2surface and after DNA immobilization. The sensor is able to detect the functionalization molecular layer, the cross-linker molecular layer, and the DNA molecules attached to the surface through a shift in its resonance frequency. The binding of molecules to the surface results in a shift of the resonance frequency due to contributions from surface stresses and mass loading.

Mass sensitivity of dual mode SAW delay lines on AlN∕sapphire structure

A surface acoustic wave (SAW) and a leaky shear horizontal SAW (SH-SAW) have been simultaneously excited along Al2O3[1 1 00] on an AlN(0001)/Al2O3(11 2 0) structure. Both modes are highly sensitive to surface mass loading. The measured mass sensitivities are in good agreement with theoretical prediction, and are comparable or higher than those reported on other SAW devices.

Suspended microchannel resonators for biomolecular detection

We present a resonant mass sensor for specific biomolecular detection in a subnanoliter fluid volume. The sensing principle is based on measuring shifts in resonance frequency of a suspended microfluidic channel upon accumulation of molecules on the inside walls of the device. Confining the fluid to the inside of a hollow cantilever enables direct integration with conventional microfluidic systems, significantly increases sensitivity by eliminating high damping and viscous drag, and allows the resonator to be actuated by electrostatic forces. Fluid density measurements reveal a mass resolution of 10−17 g/μm2 in a 4 mHz–4 Hz bandwidth. To demonstrate biomolecular detection, we present real-time measurements of the specific binding between avidin and biotinylated bovine serum albumin. Based on these measurements, we expect that changes in surface mass loading on the order of 10−19 g/μm2 can be detected in an optimized system.

Investigation of strain in microstructures by a novel moire method

A focused ion beam (FIB) moire method is proposed and demonstrated to measure the strain in microstructures. This technique is based on the advantages of the FIB system in nanofabrication, imaging, selective deposition, and fine adjustment. A nanograting is directly written on the top of the microstructures by ion milling without the requirement of an etch mask. The FIB moire pattern is formed by the interference between a prepared specimen grating and FIB raster scan lines. The strain of the microstructures is derived by calculating the average spacing of moire fringes. The sensitivity and accuracy of FIB moire in strain measurement is subsequently discussed. Since the local strain of a microstructure itself can be monitored during the process, the FIB moire technique has many potential applications in the mechanical metrology of microelectromechanical systems (MEMS). As an example, the strain distribution along the sticking microstructures and the contribution of surface oxidization and mass loading to the cantilever strain is determined by this FIB moire technique.

Site distribution and aliasing effects in the inversion for load coefficients and geocenter motion from GPS data

Precise GPS measurements of elastic relative site displacements due to surface mass loading offer important constraints on global surface mass transport. We investigate effects of site distribution and aliasing by higher‐degree (n ≥ 2) loading terms on inversion of GPS data for n = 1 load coefficients and geocenter motion. Covariance and simulation analyses are conducted to assess the sensitivity of the inversion to aliasing and mismodeling errors and possible uncertainties in the n = 1 load coefficient determination. We found that the use of center‐of‐figure approximation in the inverse formulation could cause 10–15% errors in the inverted load coefficients. n = 1 load estimates may be contaminated significantly by unknown higher‐degree terms, depending on the load scenario and the GPS site distribution. The uncertainty in n = 1 zonal load estimate is at the level of 80–95% for two load scenarios.

Gravitational potential changes of a spherically symmetric earth model caused by a surface load

In this paper we study gravity and potential changes of a spherically symmetric earth model caused by a point surface mass load. We define load Love numbers for any radial distance r, and derive expressions of Green’s functions for calculating the vertical displacement, potential and gravity changes for any r, especially on the undeformed and deformed surface of the earth. Numerical calculations are carried out for the 1066A earth model, using the Runge–Kutta integration method. Load Love numbers are obtained for three depths: r=a, r=6334.4 km and r=3484 km. The results show that the load Love numbers inside the earth decrease very quickly as nincreases. The smaller the radial distance r, the faster the decrease of the load Love numbers with n. As one would expect, the load Love numbers inside the earth for degrees 2 and 20 indicate that a low-degree load Love number has a global contribution and varies gently in magnitude along the radius, while a high-degree load Love number reflects a regional or even a local contribution and decreases very rapidly as the depth increases. The Green’s functions of vertical displacement, potential and gravity changes for the three depths mentioned above are also numerically calculated. All the Green’s functions show that deformation in the near field is much larger than that in the far field and the whole earth deforms under a surface mass load. Since a mass redistribution must occur in the whole earth, a topographic–isostatic potential model obtained by the loading theory must be composed of all harmonic degrees, from 0 to infinity. The immediate application of this work will be to the study of a topographic–isostatic potential model.

Focused Ion Beam Nano-Machined Structures For Strain Analysis By A Moiré Technique

ABSTRACTThe accurate determination of residual stress/strain in thin films is especially important in the emerging field of MicroElectroMechanical Systems (MEMS). In this article, a focused ion beam (FIB) moiré method is proposed and demonstrated to measure the strain in MEMS structures. This technique is based on the advantages of the FIB system in nano-fabrication, imaging, in-situ deposition, and fine adjustment. Nano-grating lines with 70 nm width and 140 nm spacing are directly written on the top of the MEMS structures by ion milling without the requirement of an etch mask. The FIB moiré pattern is formed by the interference between a prepared specimen grating and FIB raster scan lines. The strain of the MEMS structures is derived by calculating the average spacing of moiré fringes. Since the local strain of a MEMS structure itself can be monitored during the process, the FIB moiré technique has many potential applications in the mechanical metrology of MEMS. As an example, the strain distribution along the sticking MEMS structures, and the contribution of surface oxidization and mass loading to the cantilever strain is determined by this FIB moiré technique.

Focused Ion Beam Nano-Machined Structures For Strain Analysis By A Moiré Technique

ABSTRACTThe accurate determination of residual stress/strain in thin films is especially important in the emerging field of MicroElectroMechanical Systems (MEMS). In this article, a focused ion beam (FIB) moiré method is proposed and demonstrated to measure the strain in MEMS structures. This technique is based on the advantages of the FIB system in nano-fabrication, imaging, in-situ deposition, and fine adjustment. Nano-grating lines with 70 nm width and 140 nm spacing are directly written on the top of the MEMS structures by ion milling without the requirement of an etch mask. The FIB moiré pattern is formed by the interference between a prepared specimen grating and FIB raster scan lines. The strain of the MEMS structures is derived by calculating the average spacing of moiré fringes. Since the local strain of a MEMS structure itself can be monitored during the process, the FIB moiré technique has many potential applications in the mechanical metrology of MEMS. As an example, the strain distribution along the sticking MEMS structures, and the contribution of surface oxidization and mass loading to the cantilever strain is determined by this FIB moiré technique.

Spectral-finite element approach to three-dimensional viscoelastic relaxation in a spherical earth

Summary We present a spectral–finite element approach to the forward modelling of the visco-elastic response of a spherical earth with a 3-D viscosity structure to a surface mass load. It represents an alternative to a variety of numerical methods for 2-D and 3-D postglacial rebound modelling used recently (the finite element method, the perturbation method, the semi-analytical approach and the spectral–finite difference method). For a fixed time, the problem is reformulated in a weak sense and parametrized by tensor surface spherical harmonics in the angular direction, whereas piecewise linear finite elements span the radial direction. The solution is obtained with the Galerkin method, which leads to solving a system of linear algebraic equations. The time dependence of the problem is treated directly in the time domain (not in the Laplace domain) as a time evolution problem. The time derivative in the constitutive equation for a Maxwell viscoelastic body is approximated by the explicit Euler time-differencing scheme, which leads to time splitting of the stress tensor. The spectral–finite element method and the associated numerical code have been tested for 2-D (azimuthally symmetric) eccentrically nested spheres models, and good agreement has been obtained.

Surface acoustic wave–liquid drop interactions

Surface acoustic (Rayleigh) waves (SAWs) are highly sensitive to surface mass loading. Traditionally SAWs have been considered unsuitable for liquid sensing applications due to the large attenuation. However if the liquid volume is small, SAWs can provide a powerful tool for investigating the dynamics of viscous liquid spreading. In this work we report experiments in which small droplets of viscous oil were allowed to spread into the path of a SAW device. In all data the SAW signal initially increases while the liquid is still outside the SAW path. This is followed by a rapid attenuation in SAW amplitude as the liquid spreads across the SAW path. In data for larger volume drops the SAW amplitude is observed to pass through a minima. For drops of smaller volume, small features are observed superimposed on the overall attenuation.

A new global topographic-isostatic model

This is a preliminary report of our on-going research on a global topographic-isostatic model. The model comes from a completely new idea — the geoid undulation is the responses of an elastic earth to the topographic mass load. Assuming the topography as a condensed surface mass load, we derive expressions for calculating the vertical displacement, potential and equipotential surface changes, based on the load theory proposed in Sun and Sjöberg (1996). The modeled geoid is composed of three parts: loading potential, surface displacement and mass redistribution. The mass redistribution of the earth compensates to some extent the topography. We mainly calculate and discuss the vertical displacements and equipotential surface changes for three depths: the earth's surface, d = 36 km and the core-mantle boundary. Numerical results show that the displacements at depth 36 km and the earth's surface have the same distribution pattern and magnitude, while the vertical movement of the core-mantle boundary appears much smoother and smaller. The modeled geoid undulations vary between −352 and +555 m. The comparison between the modeled and observed geoid undulations shows that there are strong positive correlations between them, but a compensation only by elastic deformations is not sufficient to explain the observed undulations because of the big difference in magnitude between the two geoids. More geodynamic effects should be considered to better explain the long-wavelength geoid features.

Postglacial variations in the level of the sea: Implications for climate dynamics and solid‐Earth geophysics

Throughout the latter half of the Pleistocene epoch of Earth history, beginning ∼900 kyr ago, the climate system has been dominated by an intense oscillation between full glacial and interglacial conditions. During each glacial stage, global sea level fell by ∼120 m on average, as extensive ice sheets formed and thickened on the surfaces of the continents at high northern (primarily) and southern latitudes. Within each cycle this glaciation phase lasted ∼90 kyr and was followed by a much more rapid deglaciation event which terminated after ∼10 kyr and which returned the system to the interglacial state. The period of the canonical glacial cycle has remained very close to 100 kyr since its inception in mid‐Pleistocene time. Because of the magnitude of the mass that was redistributed over the surface of the Earth during each such glacial cycle and because of the viscoelastic nature of the rheology of the planetary mantle, these shifts in surface mass load induced variations in the shape of the planet that have been indelibly transcribed into the geological record of sea level variability. Indeed, the geological, geophysical, and even astronomical signatures of this process, which is continuing today, are now being measured with unprecedented precision using the methods of space geodesy and have thereby begun to provide important new scientific insight and understanding, both of the interior of the solid Earth and of the climate system variability with which the ice ages themselves are associated. In this article my purpose is to bring together, in a single review, an assessment of where we currently stand scientifically with regard to understanding both of these aspects of the ice ages. Although the discussion will not address in any detail the fascinating issue of ice age climate, since this topic is sufficiently complex of itself to require a detailed review of its own, I will nevertheless attempt to briefly summarize the current state of understanding of the physical processes that are responsible for the occurrence of the ice age cycle, by way of providing a more complete context in which to appreciate the main lines of argument that will be developed.

Response of quartz-crystal resonators to gas and liquid analyte exposure

The electrical response of chemical bulk acoustic wave sensors depends on changes in the surface mass loading and changes in the viscoelastic properties of the coating material. We consider here the acoustic behaviour and the electrical response of a quartz-crystal resonator to changes in surface mass and shear parameters of the coating material. Both a glassy and a rubbery poiymer have been investigated. The frequency changes of an oscillator are calculated from a full transmission-line model, an impedance approximation and Sauerbrey's or Martin's equation.