ALBERT

Description: By introducing two types of zenith troposphere delay (ZTD) products in precise point positioning (PPP), we developed the ZTD-corrected PPP and the ZTD-constrained PPP, both of them reduced the PPP convergence time. Both enhanced PPP methods are examined by global empirical ZTD models and regional ZTD corrections. For global ZTD models, we verified that ZTD-corrected PPP will deviate the positioning results, while ZTD-constrained PPP could produce unbiased estimations. Therefore, the latter is utilized to study the performance of global ZTD models (ITG, GPT2w, GZTD and UNB3m). After numerous experiments, we found that the performance of ZTD models was positively related to the real ZTD accuracy, and we proposed a universal tropospheric stochastic model 2SQR(9rms) which denotes double the square of nine times ZTD rms, to constrain ZTD in PPP. The proposed model subsequently was validated by real-time static and kinematic ZTD-constrained PPP on the premise that the ZTD rms on every station was known. Compared with traditional PPP, in static PPP, the number of improved stations is increased by 15.5 per cent (ITG), 14.4 per cent (GPT2w), 11.1 per cent (GZTD) and 8.3 per cent (UNB3m). For kinematic PPP, PPP constrained by ITG model still had the best performance, the number of improved stations is increased by 14.4 per cent, after 30 min of initialization time, 13.4 cm east, 13.4 cm north and 11.7 cm up positioning accuracy was obtained, compared with 15.3 cm east, 15.3 cm north and 14.3 cm up accuracy by traditional PPP. In addition, experiments using regional ZTD corrections to enhance real-time PPP showed that both ZTD-corrected PPP and ZTD-constrained PPP can notably reduce the convergence time on the vertical component (within 15 cm).

Description: We study fluctuations in the degree-2 zonal spherical harmonic coefficient of the Earth's gravity potential, C 20 , over the period 2003–2015. This coefficient is related to the Earth's oblateness and studying its temporal variations, C 20 , can be used to monitor large-scale mass movements between high and low latitude regions. We examine C 20 inferred from six different sources, including satellite laser ranging (SLR), GRACE and global geophysical fluids models. We further include estimates that we derive from measured variations in the length-of-day (LOD), from the inversion of global crustal displacements as measured by GPS, as well as from the combination of GRACE and the output of an ocean model as described by Sun et al. We apply a sequence of trend and seasonal moving average filters to the different time-series in order to decompose them into an interannual, a seasonal and an intraseasonal component. We then perform a comparison analysis for each component, and we further estimate the noise level contained in the different series using an extended version of the three-cornered-hat method. For the seasonal component, we generally obtain a very good agreement between the different sources, and except for the LOD-derived series, we find that over 90 per cent of the variance in the seasonal components can be explained by the sum of an annual and semiannual oscillation of constant amplitudes and phases, indicating that the seasonal pattern is stable over the considered time period. High consistency between the different estimates is also observed for the intraseasonal component, except for the solution from GRACE, which is known to be affected by a strong tide-like alias with a period of about 161 d. Estimated interannual components from the different sources are generally in agreement with each other, although estimates from GRACE and LOD present some discrepancies. Slight deviations are further observed for the estimate from the geophysical models, likely to be related to the omission of polar ice and groundwater changes in the model combination we use. On the other hand, these processes do not seem to play an important role at seasonal and shorter timescales, as the sum of modelled atmospheric, oceanic and hydrological effects effectively explains the observed C 20 variations at those scales. We generally obtain very good results for the solution from SLR, and we confirm that this well-established technique accurately tracks changes in C 20 . Good agreement is further observed for the estimate from the GPS inversion, showing that this indirect method is successful in capturing fluctuations in C 20 on scales ranging from intra- to interannual. Obtaining accurate estimates from LOD, however, remains a challenging task and more reliable models of atmospheric wind fields are needed in order to obtain high-quality C 20 , in particular at the seasonal scale. The combination of GRACE data and the output of an ocean model appears to be a promising approach, particularly since corresponding C 20 is not affected by tide-like aliases, and generally gives better results than the solution from GRACE, which still seems to be of rather poor quality.

Description: The geocentre motion is the motion of the centre of mass of the entire Earth, considered an isolated system, in a terrestrial system of reference. We first derive a formula relating the harmonic degree-1 Lagrangian variation of the gravity at a station to both the harmonic degree-1 vertical displacement of the station and the displacement of the whole Earth's centre of mass. The relationship is independent of the nature of the Earth deformation and is valid for any source of deformation. We impose no constraint on the system of reference, except that its origin must initially coincide with the centre of mass of the spherically symmetric Earth model. Next, we consider the geocentre motion caused by surface loading. In a system of reference whose origin is the centre of mass of the solid Earth, we obtain a specific relationship between the gravity variation at the surface, the geocentre displacement and the load Love number $h^{\prime }_1$ , which demands the Earth's structure and rheological behaviour be known. For various networks of real or fictitious stations, we invert synthetic signals of surface gravity variations caused by atmospheric loading to retrieve the degree-1 variation of gravity. We then select six well-distributed stations of the Global Geodynamics Project, which is a world network of superconducting gravimeters, to invert actual gravity data for the degree-1 variations and determine the geocentre displacement between the end of 2004 and the beginning of 2012, assuming it to be due to surface loading. We find annual and semi-annual displacements with amplitude 0.5–2.3 mm.

Description: We review the theory of the Earth's elastic and gravitational response to a surface disk load. The solutions for displacement of the surface and the geoid are developed using expansions of Legendre polynomials, their derivatives and the load Love numbers. We provide a matlab  function called diskload that computes the solutions for both uncompensated and compensated disk loads. In order to numerically implement the Legendre expansions, it is necessary to choose a harmonic degree, n max , at which to truncate the series used to construct the solutions. We present a rule of thumb (ROT) for choosing an appropriate value of n max , describe the consequences of truncating the expansions prematurely and provide a means to judiciously violate the ROT when that becomes a practical necessity.

Description: Two types of signals are clearly visible in continuous GPS (cGPS) time-series in Iceland, in particular in the vertical component. The first one is a yearly seasonal cycle, usually sinusoid-like with a minimum in the spring and a maximum in the fall. The second one is a trend of uplift, with higher values the closer the cGPS stations are to the centre of Iceland and ice caps. Here, we study the seasonal cycle signal by deriving its average at 71 GPS sites in Iceland. We estimate the annual and semi-annual components of the cycle in their horizontal and vertical components using a least-squares adjustment. The peak-to-peak amplitude of the cycle of the vertical component at the studied sites ranges from 4 mm near the coastline up to 27 mm at the centre of the Vatnajökull, the largest ice cap in Iceland. The minimum of the seasonal cycle occurs earlier in low lying areas than in the central part of Iceland, consistent with snow load having a large influence on seasonal deformation. Modelling shows that the seasonal cycle is well explained by accounting for elastically induced surface displacements due to snow, atmosphere, reservoir lake and ocean variations. Model displacement fields are derived considering surface loads on a multilayered isotropic spherical Earth. Through forward and inverse modelling, we were able to reproduce a priori information on the average seasonal cycle of known loads (atmosphere, snow in non-glaciated areas and lake reservoir) and get an estimation of other loads (glacier mass balance and ocean). The seasonal glacier mass balance cycle in glaciated areas and snow load in non-glaciated areas are the main contributions to the seasonal deformation. For these loads, induced seasonal vertical displacements range from a few millimetres far from the loads in Iceland, to more than 20 mm at their centres. Lake reservoir load also has to be taken into account on local scale as it can generate up to 20 mm of vertical deformation. Atmosphere load and ocean load are observable and generate vertical displacements in the order of a few millimetres. Inversion results also shows that the Iceland crust is less rigid than the world average. Interannual deviation from the GPS seasonal cycle can occur and are caused by unusual weather conditions over extended period of time.

Description: Seismic waves produced by fault ruptures give rise to gravity perturbations. So far, these perturbations have either been modelled as permanent coseismic gravity change in a half-space or spherical Earth model, or as full time-domain model in infinite space. In this paper, we present the explicit solution of gravity perturbations in time domain produced by a double-couple buried in a homogeneous half-space. This result is especially suited to study gravity perturbations up to a few hundreds of kilometres from the epicentre. It facilitates detailed parametric studies of gravity perturbations from fault rupture, and predicts gravity perturbations of real earthquakes with greatly improved accuracy. The results may serve to develop first designs of gravity-assisted earthquake early-warning systems, made possible by a new generation of ultrasensitive gravity gradiometers, which is currently under development.

Description: Continuous gravimetric observations have been made with three successive generations of superconducting gravimeter over 20 yr at Syowa Station ( $39.6\deg$ E, $69.0\deg$ S), East Antarctica. The third-generation instrument, OSG#058, was installed in January 2010 and was calibrated by an absolute gravimeter during January and February, 2010. The estimated scale factor was –73.823 ± 0.053 μGal V –1 (1 μGal = 10 –8 m s –2 ). The first 5 yr of OSG#058 data from 2010 January 7 to 2015 January 10 were decomposed into tidal waves (M3 to Ssa) and other non-tidal components by applying the Bayesian tidal analysis program BAYTAP. Long-term non-tidal gravity residuals, which were obtained by subtracting annual and 18.6 year tidal waves and the predicted gravity response to the Earth's variable rotation, showed significant correlation with the accumulated snow depth measured at Syowa Station. The greatest correlation occurred when the gravity variations lagged the accumulated snow depth by 21 d. To estimate the gravitational effect of the accumulated snow mass, we inferred a conversion factor of 3.13 ± 0.08 μGal m –1 from this relation. The accumulated snow depth at Syowa Station was found to represent an extensive terrestrial water storage (the snow accumulation) around Syowa Station, which was estimated from the Gravity Recovery and Climate Experiment satellite gravity data. The snow accumulation around Syowa Station was detectable by the superconducting gravimeter.

Description: The new release AIUB-RL02 of monthly gravity models from GRACE GPS and K-Band range-rate data is based on reprocessed satellite orbits referring to the reference frame IGb08. The release is consistent with the IERS2010 conventions. Improvements with respect to its predecessor AIUB-RL01 include the use of reprocessed (RL02) GRACE observations, new atmosphere and ocean dealiasing products (RL05), an upgraded ocean tide model (EOT11A), and the interpolation of shallow ocean tides (admittances). The stochastic parametrization of AIUB-RL02 was adapted to include daily accelerometer scale factors, which drastically reduces spurious signal at the 161 d period in C 20 and at other low degree and order gravity field coefficients. Moreover, the correlation between the noise in the monthly gravity models and solar activity is considerably reduced in the new release. The signal and the noise content of the new AIUB-RL02 monthly gravity fields are studied and calibrated errors are derived from their non-secular and non-seasonal variability. The short-period time-variable signal over the oceans, mostly representing noise, is reduced by 50 per cent with respect to AIUB-RL01. Compared to the official GFZ-RL05a and CSR-RL05 monthly models, the AIUB-RL02 stands out by its low noise at high degrees, a fact emerging from the estimation of seasonal variations for selected river basins and of mass trends in polar regions. Two versions of the monthly AIUB-RL02 gravity models, with spherical harmonics resolution of degree and order 60 and 90, respectively, are available for the time period from March 2003 to March 2014 at the International Center for Global Earth Models or from ftp://ftp.unibe.ch/aiub/GRAVITY/GRACE (last accessed 22 March 2016).

Description: We document two kinds of traveling ionospheric disturbances, namely, CTIDs (Co-tsunami-Traveling-Ionospheric-disturbances) and ATIDs (Ahead-of-Tsunami-Traveling-Ionospheric-disturbances) related to the Tohoku-Oki tsunami of 2011 March 11. They are referred to the disturbances that remain behind and ahead of the principal tsunami wave front, respectively. We first note their presence in a numerical experiment performed using a simulation code coupling the tsunami, atmosphere and ionosphere. This code uses the tsunami wavefield as an input and simulates acoustic-gravity waves (AGWs) in the atmosphere and TIDs, in the form of total electron content (TEC) disturbance, in the ionosphere. The simulated TEC reveals the excitation of CTIDs (at about 2 TECU) and ATIDs (at about 1 TECU), representing up to 5 per cent disturbance over the ambient electron density, and they arise from the dissipation of AGWs in the thermosphere. A novel outcome is that during the tsunami passage between ~6° and 12° of epicentral distance, strong ATIDs arrive ~20–60 min ahead of the tsunami wave front covering ~3°–10° of distance from the tsunami location. Simulation results are compared with the far-field observations using GNSS satellites and confirm that ATIDs are the first detected TEC maximum, occurring 20–60 min ahead of the tsunami arrival. Our simulation also confirms the presence of largest TEC maximum representing CTIDs, 10–20 min after the first tsunami wave. ATIDs reported in this study have characteristics that can be potentially used for the early warning of the tsunami.

Description: Apparent acceleration in Gravity Recovery and Climate Experiment (GRACE) Antarctic ice mass time-series may reflect both ice discharge and surface mass balance contributions. However, a recent study suggests there is also contamination from errors in atmospheric pressure de-aliasing fields [European Center for Medium-Range Weather Forecast (ECMWF) operational products] used during GRACE data processing. To further examine this question, we compare GRACE atmospheric pressure de-aliasing (GAA) fields with in situ surface pressure data from coastal and inland stations. Differences between the two are likely due to GAA errors, and provide a measure of error in GRACE solutions. Time-series of differences at individual weather stations are fit to four presumed error components: annual sinusoids, a linear trend, an acceleration term and jumps at times of known ECMWF model changes. Using data from inland stations, we estimate that atmospheric pressure error causes an acceleration error of about +7.0 Gt yr –2 , which is large relative to prior GRACE estimates of Antarctic ice mass acceleration in the range of –12 to –14 Gt yr –2 . We also estimate apparent acceleration rates from other barometric pressure (reanalysis) fields, including ERA-Interim, MERRA and NCEP/DOE. When integrated over East Antarctica, the four mass acceleration estimates (from GAA and the three reanalysis fields) vary considerably (by ~2–16 Gt yr –2 ). This shows the need for further effort to improve atmospheric mass estimates in this region of sparse in situ observations, in order to use GRACE observations to measure ice mass acceleration and related sea level change.

Description: The long-wavelength gravity field contains information about processes in the sublithospheric mantle. As satellite-derived gravity models now provide the long to medium-wavelength gravity field at unprecedented accuracy, techniques used to process gravity data need to be updated. We show that when determining these long-wavelengths, the treatment of topographic-isostatic effect (TIE) and isostatic effects (IE) is a likely source of error. We constructed a global isostatic model and calculated global TIE and IE. These calculations were done for ground stations as well as stations at satellite height. We considered both gravity and gravity gradients. Using these results, we determined how much of the gravity signal comes from distant sources. We find that a significant long-wavelength bias is introduced if far-field effects on the topographic effect are neglected. However, due to isostatic compensation far-field effects of the topographic effect are to a large degree compensated by the far-field IE. This means that far-field effects can be reduced effectively by always considering topographic masses together with their compensating isostatic masses. We show that to correctly represent the ultra-long wavelengths, a global background model should be used. This is demonstrated both globally and for a continental-scale case area in North America. In the case of regional modelling, where the ultra-long wavelengths are not of prime importance, gravity gradients can be used to help minimize correction errors caused by far-field effects.

Description: Constraining laterally varying structures in planetary interiors is important for understanding both the composition and the internal dynamics of a planet. Recognizing that seismic imaging technique is currently only viable for studying the Earth's interior structures, methods that can be supported by advanced space geodetic techniques may become alternatives to ‘image’ the interiors of other planets. The method of tidal tomography is one possibility, and it relies on high precision measurement of the response of a planet to its body tide. However, it is essential to develop an efficient analytical tool that computes the dependence of tidal response to 3-D interior structures. In this paper, we present a complete formulation of such an analytical tool, which calculates to high accuracy the tidal response of a terrestrial planet with lateral heterogeneities in its elastic and density structures. We treat the lateral heterogeneities as small perturbations and derive the governing equations based on the perturbation theory. In a spherical harmonic representation, equations at each order of perturbation are reduced into multiple matrix equations at harmonics that are allowed by mode couplings, and the total response equals the sum of all those single-harmonic responses, which can be solved semi-analytically. We test our perturbation method by applying it to the Moon with a harmonic degree-1 mantle structure for which the perturbation solutions of the tidal response are compared with those from a fully numerical method. The remarkable agreement between results from these two methods validates the perturbation method. As an example, we then use the perturbation method to evaluate the impact of lunar crustal thickness variations on tidal response of the Moon. We find that lunar crust produces much smaller degree-3 tidal responses than a relatively weak degree-1 structure in the deep lunar mantle. Our calculations show that degree-3 tidal response measurements may hold key constraints on possible degree-1 mantle structure of the Moon, as suggested from previous modelling results.

Description: Seismic data are primarily used in studies of the Earth's lithospheric structure including the Moho geometry. In regions, where seismic data are sparse or completely absent, gravimetric or combined gravimetric-seismic methods could be applied to determine the Moho depth. In this study, we derive and present generalized expressions for solving the Vening Meinesz–Moritz's (VMM) inverse problem of isostasy for a Moho depth determination from gravity and vertical gravity-gradient data. By solving the (non-linear) Fredholm's integral equation of the first kind, the linearized observation equations, which functionally relate the (given) gravity/gravity-gradient data to the (unknown) Moho depth, are derived in the spectral domain. The VMM gravimetric results are validated by using available seismic and gravimetric Moho models. Our results show that the VMM Moho solutions obtained by solving the VMM problem for gravity and gravity-gradient data are almost the same. This finding indicates that in global applications, using the global gravity/gravity-gradient data coverage, the spherical harmonic expressions for the gravimetric forward and inverse modelling yield (theoretically) the same results. Globally, these gravimetric solutions have also a relatively good agreement with the CRUST1.0 and GEMMA GOCE models in terms of their rms Moho differences (4.7 km and 4.1 km, respectively).

Description: Geophysical techniques are widely used to monitor volcanic unrest. A number of studies have also demonstrated that hydrological processes can produce or trigger geophysical signals. Hydrologically induced gravity signals have previously been recorded by specifically designed gravity surveys as well as, inadvertently, by volcano monitoring studies. Water table corrections of microgravity surveys are commonplace. However, the fluctuations of the water table beneath survey locations are often poorly known, and such a correction fails to account for changes in water-mass storage in the unsaturated zone. Here, we combine 2-D axis-symmetrical numerical fluid-flow models with an axis-symmetric, distributed-mass, gravity calculation to model gravity changes in response to fluctuating hydrological recharge. Flow simulations are based on tropical volcanic settings where high surface permeabilities promote thick unsaturated zones. Our study highlights that mass storage (saturation) changes within the unsaturated zone beneath a survey point can generate recordable gravity changes. We show that for a tropical climate, recharge variations can generate gravity variations of over 150 μGal; although, we demonstrate that for the scenarios investigated here, the probability of recording such large signals is low. Our modelling results indicate that microgravity survey corrections based on water table elevation may result in errors of up to 100 μGal. The effect of inter-annual recharge fluctuations dominate over seasonal cycles which makes prediction and correction of the hydrological contribution more difficult. Spatial hydrogeological heterogeneity can also impact on the accuracy of relative gravity surveys, and can even result in the introduction of additional survey errors. The loading fluctuations associated with saturation variations in the unsaturated zone may also have implications for other geophysical monitoring techniques, such as geodetic monitoring of ground deformation.

Description: Several attempts have been made to obtain a radiographic image inside volcanoes using cosmic-ray muons (muography). Muography is expected to resolve highly heterogeneous density profiles near the surface of volcanoes. However, several prior works have failed to make clear observations due to contamination by background noise. The background contamination leads to an overestimation of the muon flux and consequently a significant underestimation of the density in the target mountains. To investigate the origin of the background noise, we performed a Monte Carlo simulation. The main components of the background noise in muography are found to be low-energy protons, electrons and muons in case of detectors without particle identification and with energy thresholds below 1 GeV. This result was confirmed by comparisons with actual observations of nuclear emulsions. This result will be useful for detector design in future works, and in addition some previous works of muography should be reviewed from the view point of background contamination.

Description: Essential to understanding sea level change and its causes during the last interglacial (LIG) is the quantification of uncertainties. In order to estimate the uncertainties, we develop a statistical framework for the comparison of palaeoclimatic sea level index points and GIA model predictions. For the investigation of uncertainties, as well as to generate better model predictions, we implement a massive ensemble approach by applying a data assimilation scheme based on particle filter methods. The different runs are distinguished through varying ice sheet reconstructions based on oxygen-isotope curves and different parameter selections within the GIA model. This framework has several advantages over earlier work, such as the ability to examine either the contribution of individual observations to the results or the probability of specific input parameters. This exploration of input parameters and data leads to a larger range of estimates than previously published work. We illustrate how the assumptions that enter into the statistical analysis, such as the existence of outliers in the observational database or the initial ice volume history, can introduce large variations to the estimate of the maximum highstand. Thus, caution is required to avoid overinterpreting results. We conclude that there are reasonable doubts whether the data sets previously used in statistical analyses are able to tightly constrain the value of maximum highstand during the LIG.

Description: A 3-D density model of the crust and upper mantle beneath the Karoo basin is presented here. The model is constrained using potential field, borehole and seismic data. Uplift of the basin by the end of the Cretaceous has resulted in an unusually high plateau (〉1000 m) covering a large portion of South Africa. Isostatic studies show the topography is largely compensated by changes in Moho depths (~35 km on-craton and 〉45 km off-craton) and changes in lithospheric mantle densities between the Kaapvaal Craton and surrounding regions (~50 kg m –3 increase from on- to off-craton). This density contrast is determined by inverted satellite gravity and gravity gradient data. The highest topography along the edge of the plateau (〉1200 m) and a strong Bouguer gravity low over Lesotho, however, can only be explained by a buoyant asthenosphere with a density decrease of around 40 kg m –3 .

Description: A method to estimate the rotation change in the orientation of the centre-of-figure (CF) frame caused by earthquakes is proposed for the first time. This method involves using the point dislocation theory based on a spherical, non-rotating, perfectly elastic and isotropic (SNREI) Earth. The rotation change in the orientation is related solely to the toroidal displacements of degree one induced by the vertical dip slip dislocation, and the spheroidal displacements induced by an earthquake have no contribution. The effects of two recent large earthquakes, the 2004 Sumatra and the 2011 Tohoku-Oki, are studied. Results showed that the Sumatra and Tohoku-Oki earthquakes both caused the CF frame to rotate by at least tens of μas (micro-arc-second). Although the visible co-seismic displacements are identified and removed from the coordinate time-series, the rotation change due to the unidentified ones and errors in removal is non-negligible. Therefore, the rotation change in the orientation of the CF frame due to seismic deformation should be taken into account in the future in reference frame and geodesy applications.

Description: In this study, we propose an approach for determining the geopotential difference using high-frequency-stability microwave links between satellite and ground station based on Doppler cancellation system. Suppose a satellite and a ground station are equipped with precise optical-atomic clocks (OACs) and oscillators. The ground oscillator emits a signal with frequency f a towards the satellite and the satellite receiver (connected with the satellite oscillator) receives this signal with frequency f b which contains the gravitational frequency shift effect and other signals and noises. After receiving this signal, the satellite oscillator transmits and emits, respectively, two signals with frequencies f b and f c towards the ground station. Via Doppler cancellation technique, the geopotential difference between the satellite and the ground station can be determined based on gravitational frequency shift equation by a combination of these three frequencies. For arbitrary two stations on ground, based on similar procedures as described above, we may determine the geopotential difference between these two stations via a satellite. Our analysis shows that the accuracy can reach 1 m 2 s – 2 based on the clocks’ inaccuracy of about 10 –17 (s s –1 ) level. Since OACs with instability around 10 –18 in several hours and inaccuracy around 10 –18 level have been generated in laboratory, the proposed approach may have prospective applications in geoscience, and especially, based on this approach a unified world height system could be realized with one-centimetre level accuracy in the near future.

Description: A sequence of large earthquakes occurred along the North Anatolian fault in the 20th century. These earthquakes, including the 1999 Izmit/Düzce earthquakes, generally propagated westward towards the Marmara Sea, defining the Main Marmara fault as a potential seismic gap. It is important to conduct a detailed assessment of the seismic hazards along the main Marmara fault because the megacity Istanbul lies only approximately 10 km north of the eastern segment of the Main Marmara fault, which is referred to as the Princes’ Islands Fault segment (PIF). Here, we study the locking status of this fault segment to evaluate the seismic hazard potential. For the first time, combined ascending and descending Interferometric Synthetic Aperture Radar and Global Positioning System observations were used to investigate the crustal deformation associated with the PIF. After careful corrections of the estimated ground velocity, a deformation pattern relating to fault locking near the Princes’ Islands was identified. The modeling results revealed that the slip rate and locking depth of the fault segment show a clear trade-off, which were estimated as 18.9 ± 7.2 mm yr –1 and 12.1 ± 7.0 km, respectively. With a moment accumulation rate of 1.7 ± 0.4  x  10 17 Nm yr –1 (proportional to the product of slip rate and locking depth), our results imply a build-up of a geodetic moment on the PIF and therefore a potential for earthquake hazards in the vicinity of the Istanbul megacity.

Description: We present a distributed slip model for the 1999 M w 6.3 Chamoli earthquake of north India using interferometric synthetic aperture radar (InSAR) data from both ascending and descending orbits and Bayesian estimation of confidence levels and trade-offs of the model geometry parameters. The results of fault-slip inversion in an elastic half-space show that the earthquake ruptured a $9 _{ - 2.2}^{\circ + 3.4}$ northeast-dipping plane with a maximum slip of ~1 m. The fault plane is located at a depth of ~ $15.9_{ - 3.0}^{ + 1.1}$ km and is ~120 km north of the Main Frontal Thrust, implying that the rupture plane was on the northernmost detachment near the mid-crustal ramp of the Main Himalayan Thrust. The InSAR-determined moment is 3.35 x 10 18 Nm with a shear modulus of 30 GPa, equivalent to M w 6.3, which is smaller than the seismic moment estimates of M w 6.4–6.6. Possible reasons for this discrepancy include the trade-off between moment and depth, uncertainties in seismic moment tensor components for shallow dip-slip earthquakes and the role of earth structure models in the inversions. The released seismic energy from recent earthquakes in the Garhwal region is far less than the accumulated strain energy since the 1803 M s 7.5 earthquake, implying substantial hazard of future great earthquakes.

Description: We use a Bayesian formalism combined with a grid node discretization for the linear inversion of gravimetric data in terms of 3-D density distribution. The forward modelling and the inversion method are derived from seismological inversion techniques in order to facilitate joint inversion or interpretation of density and seismic velocity models. The Bayesian formulation introduces covariance matrices on model parameters to regularize the ill-posed problem and reduce the non-uniqueness of the solution. This formalism favours smooth solutions and allows us to specify a spatial correlation length and to perform inversions at multiple scales. We also extract resolution parameters from the resolution matrix to discuss how well our density models are resolved. This method is applied to the inversion of data from the volcanic island of Basse-Terre in Guadeloupe, Lesser Antilles. A series of synthetic tests are performed to investigate advantages and limitations of the methodology in this context. This study results in the first 3-D density models of the island of Basse-Terre for which we identify: (i) a southward decrease of densities parallel to the migration of volcanic activity within the island, (ii) three dense anomalies beneath Petite Plaine Valley, Beaugendre Valley and the Grande-Découverte-Carmichaël-Soufrière Complex that may reflect the trace of former major volcanic feeding systems, (iii) shallow low-density anomalies in the southern part of Basse-Terre, especially around La Soufrière active volcano, Piton de Bouillante edifice and along the western coast, reflecting the presence of hydrothermal systems and fractured and altered rocks.

Description: We compute the gravimetric factor at the Chandler wobble (CW) frequency using time-series from superconducting gravimeters (SG) longer than a decade. We first individually process the polar motion and data at each individual gravity station to estimate the gravimetric factor amplitude and phase, then we make a global analysis by applying a stacking method to different subsets of up to seven SG stations. The stacking is an efficient way of getting rid of local effects and improving the signal-to-noise ratio of the combined data sets. Using the stacking method, we find a gravimetric factor amplitude and phase of 1.118 ± 0.016 and –0.45 ± 0.66 deg, respectively, which is smaller in amplitude than expected. The sources of error are then carefully considered. For both local and global analyses, the uncertainties on our results are reliably constrained by computing the standard deviation of the estimates of the gravimetric factor amplitude and phase for increasing length of the time-series. Constraints on the CW anelastic dissipation can be set since any departure of the gravimetric factor from its elastic value may provide some insights into the dissipative processes that occur at the CW period. In particular, assuming given rheological models for the Earth's mantle enables us to make the link between the gravimetric factor phase and the CW quality factor.

Description: Analysing independent 1-yr data sets of 10 European superconducting gravimeters (SG) reveals statistically significant temporal variations of M2 tidal parameters. Both common short-term (〈2 yr) and long-term (〉2 yr) features are identified in all SG time-series but one. The averaged variations of the amplitude factor are about 0.2. The path of load vector variations equivalent to the temporal changes of tidal parameters suggests the presence of an 8.85 yr modulation (lunar perigee). The tidal waves having the potential to modulate M2 with this period belong to the 3rd degree constituents. Their amplitude factors turn out to be much closer to body tide model predictions than that of the main 2nd degree M2, which indicates ocean loading for 3rd degree waves to be less prominent than for 2nd degree waves within the M2 group. These two different responses to the loading suggest that the observed modulation is more due to insufficient frequency resolution of limited time-series rather than to time variable loading. Presently, SG gravity time-series are still too short to prove if time variable loading processes are involved too as in case of the annual M2 modulation known to appear for analysis intervals of less than 1 yr. Whatever the variations are caused by, they provide the upper accuracy limit for earth model validation and permit estimating the temporal stability of SG scale factors and assessing the quality of gravity time-series.

Description: A sequence of large earthquakes occurred along the North Anatolian fault in the 20th century. These earthquakes, including the 1999 Izmit/Düzce earthquakes, generally propagated westward towards the Marmara Sea, defining the Main Marmara fault as a potential seismic gap. It is important to conduct a detailed assessment of the seismic hazards along the main Marmara fault because the megacity Istanbul lies only approximately 10 km north of the eastern segment of the Main Marmara fault, which is referred to as the Princes’ Islands Fault segment (PIF). Here, we study the locking status of this fault segment to evaluate the seismic hazard potential. For the first time, combined ascending and descending Interferometric Synthetic Aperture Radar and Global Positioning System observations were used to investigate the crustal deformation associated with the PIF. After careful corrections of the estimated ground velocity, a deformation pattern relating to fault locking near the Princes’ Islands was identified. The modeling results revealed that the slip rate and locking depth of the fault segment show a clear trade-off, which were estimated as 18.9 ± 7.2 mm yr –1 and 12.1 ± 7.0 km, respectively. With a moment accumulation rate of 1.7 ± 0.4  x  10 17 Nm yr –1 (proportional to the product of slip rate and locking depth), our results imply a build-up of a geodetic moment on the PIF and therefore a potential for earthquake hazards in the vicinity of the Istanbul megacity.

Description: Deformation analysis in general and strain analysis in particular using permanent GPS networks require proper analysis of time-series in which all functional effects are taken into consideration and all stochastic effects are captured using an appropriate noise model. This contribution addresses both issues when considering the strain parameters of a GPS network. Estimates of spatial correlation, time correlated noise, and multivariate power spectrum for daily position time-series of the Southern California Integrated GPS Network (SCIGN) stations collected between 1996 and 2011 are obtained. Significant signals with periods of 13.63 d and those related to the GPS draconitic year are identified in these time-series. We aim to assess the effect of a realistic noise model of the series on the uncertainties of the strain parameters including displacements, normal and shear strains, and rotations. For the SCIGN network considered, the following results are highlighted. Contrary to the common belief, the uncertainties of the displacements parameters become smaller when taking a realistic noise model into account. This however was not the case when assessing the noise characteristics of the normal and shear strain, and rotation parameters. The uncertainties increase nearly by a factor of two, in agreement to what is expected. Some of the significant deformation parameters of the white noise model become less significant in case of the realistic noise model.

Description: We have extended backwards from 2001 to 1979 the current release 05 (RL05) of the Gravity Recovery and Climate Experiment (GRACE) Atmospheric and Oceanic De-aliasing Level-1B (AOD1B) product and studied the impact of this and a previous release 04 (RL04) of the AOD1B product on precise orbits of five altimetry satellites (ERS-1, ERS-2, TOPEX/Poseidon, Envisat and Jason-1) for the time span 1991–2012, as compared to the case when no AOD1B product is used. We have found that using AOD1B RL05 product reduces root mean square (RMS) fits of satellite laser ranging (SLR) observations by about 1.0–6.4 per cent, 2-d arc overlaps in radial, cross-track and along-track directions by about 1.3–12.0, 0.3–10.0 and 2.0–10.0 per cent, respectively, for various satellites tested, as compared to the case without AOD1B product. Using AOD1B RL05 product instead of RL04 one reduces SLR RMS fits by 0.1–0.7 per cent, 2-d arc overlaps in radial, cross-track and along-track directions by 0.1–0.6, 0.1–1.3 and 0.2–1.2 per cent, respectively, for the satellite orbits tested. The multi-mission crossover analysis shows that the application of an AOD1B product reduces the scatter of radial errors by 0.4–2.8 per cent for the satellite missions studied. At the regions with the most pronounced changes the use of the AOD1B products improves the consistency between the sea level as measured by the TOPEX and ERS-2 missions and by the Jason-1 and Envisat missions by 5 to 10 per cent (globally by about 2 per cent). The results of our study show that extended AOD1B RL05 product performs better than the AOD1B RL04 and improves orbits of altimetry satellites and consistency of sea level products.

Description: A new analytical method for the computation of a truncated series of solid spherical harmonic coefficients (HCs) from data on a spheroid (i.e. an oblate ellipsoid of revolution) is derived, using a transformation between surface and solid spherical HCs. A two-step procedure is derived to extend this transformation beyond degree and order (d/o) 520. The method is compared to the Hotine–Jekeli transformation in a numerical study based on the EGM2008 global gravity model. Both methods are shown to achieve submicrometre precision in terms of height anomalies for a model to d/o 2239. However, both methods result in spherical harmonic models that are different by up to 7.6 mm in height anomalies and 2.5 mGal in gravity disturbances due to the different coordinate system used. While the Hotine–Jekeli transformation requires the use of an ellipsoidal coordinate system, the new method uses only spherical polar coordinates. The Hotine–Jekeli transformation is numerically more efficient, but the new method can more easily be extended to cases where (a linear combination of) normal derivatives of the function under consideration are given on the surface of the spheroid. It therefore provides a solution to many types of ellipsoidal boundary-value problems in the spectral domain.

Description: We explore Earth's elastic deformation response to ocean tidal loading (OTL) using kinematic Global Positioning System (GPS) observations and forward-modelled predictions across South America. Harmonic coefficients are extracted from up to 14 yr of GPS-inferred receiver locations, which we estimate at 5 min intervals using precise point positioning. We compare the observed OTL-induced surface displacements against predictions derived from spherically symmetric, non-rotating, elastic and isotropic (SNREI) Earth models. We also compare sets of modelled predictions directly for various ocean-tide and Earth-model combinations. The vector differences between predicted displacements computed using separate ocean-tide models reveal uniform-displacement components common to all stations in the South America network. Removal of the network-mean OTL-induced displacements from each site substantially reduces the vector differences between observed and predicted displacements. We focus on the dominant astronomical tidal harmonics from three distinct frequency bands: semidiurnal (M 2 ), diurnal (O 1 ) and fortnightly (M f ). In each band, the observed OTL-induced surface displacements strongly resemble the modelled displacement-response patterns, and the residuals agree to about 0.3 mm or better. Even with the submillimetre correspondence between observations and predictions, we detect regional-scale spatial coherency in the final set of residuals, most notably for the M 2 harmonic. The spatial coherency appears relatively insensitive to the specific choice of ocean-tide or SNREI-Earth model. Varying the load model or 1-D elastic structure yields predicted OTL-induced displacement differences of order 0.1 mm or less for the network. Furthermore, estimates of the observational uncertainty place the noise level below the magnitude of the residual displacements for most stations, supporting our interpretation that random errors cannot account for the entire misfit. Therefore, the spatially coherent residuals may reveal deficiencies in the a priori SNREI Earth models. In particular, the residuals may indicate sensitivity to regional deviations from standard globally averaged Earth structure due to the presence of the South American craton.

Description: Planar faults are widely adopted during inversions to determine slip distributions and fault geometries using geodetic observations; however, little research has been conducted with respect to curved faults. We attribute this to the lack of an appropriate parameterized modelling method. In this paper, we present a curved-fault modelling method (CFMM) that describes a curved fault according to specific parameters, and we also develop a corresponding hybrid iterative inversion algorithm (HIIA) to perform inversions for parametric curved-fault geometries and slips. The results of the strike-component and dip-component synthetic tests show that a complex S-shaped fault surface and a circular slip distribution are successfully recovered, indicating the strong performance of the CFMM and HIIA methods. In addition, we describe and verify a scenario for determining the number of necessary geometrical parameters for the HIIA and examine the case study of the Wenchuan earthquake, which occurred on a complex listric fault surface. During the iteration process of the HIIA, both the fault geometry and slip distribution of the Beichuan and Pengguan faults converge to optimal values, indicating a Beichuan fault (BCF) model with a continuous listric shape and gradual steepening from the southwest to the northeast, which is highly consistent with geological survey results. Both the synthetic and real-world case studies show that the HIIA and the CMFF are superior to the conventional fault modelling method based on rectangular planes and that these models have the potential for use in more integrated research involving inversion studies, such as joint slip/curved-fault-geometry inversions that take into account data resolving power.

Description: Globally gridded estimates of monthly-mean anomalies of terrestrial water storage (TWS) are estimated from the most recent GRACE release 05a of GFZ Potsdam in order to provide non-geodetic users a convenient access to state-of-the-art GRACE monitoring data. We use an ensemble of five global land model simulations with different physics and different atmospheric forcing to obtain reliable gridded scaling factors required to correct for spatial leakage introduced during data processing. To allow for the application of this data-set for large-scale monitoring tasks, model validation efforts, and subsequently also data assimilation experiments, globally gridded estimates of TWS uncertainties that include (i) measurement, (ii) leakage and (iii) re-scaling errors are provided as well. The results are generally consistent with the gridded data provided by Tellus, but deviate in some basins which are largely affected by the uncertainties of the model information required for re-scaling, where the approach based on the median of a small ensemble of global land models introduced in this paper leads to more robust results.

Description: Regional recovery of the disturbing gravitational potential in the area of Central Europe from satellite gravitational gradients data is discussed in this contribution. The disturbing gravitational potential is obtained by inverting surface integral formulas which transform the disturbing gravitational potential onto disturbing gravitational gradients in the spherical local north-oriented frame. Two numerical approaches that solve the inverse problem are considered. In the first approach, the integral formulas are rigorously decomposed into two parts, that is, the effects of the gradient data within near and distant zones. While the effect of the near zone data is sought as an inverse problem, the effect of the distant zone data is synthesized from the global gravitational model GGM05S using spectral weights given by truncation error coefficients up to the degree 150. In the second approach, a reference gravitational field up to the degree 180 is applied to reduce and smooth measured gravitational gradients. In both cases we recovered the disturbing gravitational potential from each of the four well-measured gravitational gradients of the GOCE satellite separately as well as from their combination. Obtained results are compared with the EGM2008, DIR-r2, TIM-r2 and SPW-r2 global gravitational models. The best fit was achieved for EGM2008 and the second approach combining all four well-measured gravitational gradients with rms of 1.231 m 2  s –2 .

Description: We present efficient Fourier-domain algorithms for modelling gravity effects due to topographic masses. The well-known Parker's formula originally based on the standard fast Fourier transform (FFT) algorithm is modified by applying the Gauss–FFT method instead. Numerical precision of the forward and inverse Fourier transforms embedded in Parker's formula and its extended forms are significantly improved by the Gauss–FFT method. The topographic model is composed of two major aspects, the geometry and the density. Versatile geometric representations, including the mass line model, the mass prism model, the polyhedron model and smoother topographic models interpolated from discrete data sets using high-order splines or pre-defined by analytical functions, in combination with density distributions that vary both laterally and vertically in rather arbitrary ways following exponential or general polynomial functions, now can be treated in a consistent framework by applying the Gauss–FFT method. The method presented has been numerically checked by space-domain analytical and hybrid analytical/numerical solutions already established in the literature. Synthetic and real model tests show that both the Gauss–FFT method and the standard FFT method run much faster than space-domain solutions, with the Gauss–FFT method being superior in numerical accuracy. When truncation errors are negligible, the Gauss–FFT method can provide forward results almost identical to space-domain analytical or semi-numerical solutions in much less time.

Description: The gravity gradient tensor (GGT) has been increasingly used in practical applications, but the advantages and the disadvantages of the analysis of GGT components versus the analysis of the vertical component of the gravity field are still debated. We analyse the performance of joint inversion of GGT components versus separate inversion of the gravity field alone, or of one tensor component. We perform our analysis by inspection of the Picard Plot, a Singular Value Decomposition tool, and analyse both synthetic data and gradiometer measurements carried out at the Vredefort structure, South Africa. We show that the main factors controlling the reliability of the inversion are algebraic ambiguity (the difference between the number of unknowns and the number of available data points) and signal-to-noise ratio. Provided that algebraic ambiguity is kept low and the noise level is small enough so that a sufficient number of SVD components can be included in the regularized solution, we find that: (i) the choice of tensor components involved in the inversion is not crucial to the overall reliability of the reconstructions; (ii) GGT inversion can yield the same resolution as inversion with a denser distribution of gravity data points, but with the advantage of using fewer measurement stations.

Description: This paper compares GRACE (Gravity Recovery and Climate Experiment) and ICESat (Ice, Cloud and land Elevation Satellite) observations to confirm whether the observed gravity increase in the Tibetan Plateau (TP) was primarily caused by lake storage gain, and comprehensively analyses the changing pattern of lake level over 2003–2009. An improved automated method was used to obtain lake-level changes and the underestimation of lake water storage was considered due to lake area expansion and lake density. The result demonstrates that GRACE recorded a mass gain (16.43 ± 1.65/11.79 ± 1.25 gt a –1 ) in the total/inner TP, of which lake storage increase accounts for (8.78 ± 0.75/7.53 ± 0.56 gt a –1 ) based on ICESat. The northwestern residual may be stored in new lakes and soil moisture as a result of net precipitation gain. According to the character of the lake-level changes, we divide the TP into four subregions. Generally, the changing pattern of lake level concurs with the distribution of precipitation, which is increasing in the inner TP and decreasing in the upstream area of the Indus and Brahmaputra Rivers. An excess of rainfall in the northeastern TP in the summer of 2005 and 2009 caused a simultaneous large increase in water level in many lakes. The correlation of lake changes with precipitation demonstrates that precipitation rather than glacial melt is the main cause of lake-level change in most places. Nonetheless, the meltwater is a considerable supplement for lakes near glaciers such as Selin Co and Nam Co, which partly explains why GRACE indicates a much weaker signal in this region.

Description: Traditional processing of Global Navigation Satellite System (GNSS) data using dedicated scientific software has provided the highest levels of positional accuracy, and has been used extensively in geophysical deformation studies. To achieve these accuracies a significant level of understanding and training is required, limiting their availability to the general scientific community. Various online GNSS processing services, now freely available, address some of these difficulties and allow users to easily process their own GNSS data and potentially obtain high quality results. Previous research into these services has focused on Continually Operating Reference Station (CORS) GNSS data. Less research exists on the results achievable with these services using large campaign GNSS data sets, which are inherently noisier than CORS data. Even less research exists on the quality of velocity fields derived from campaign GNSS data processed through online precise point positioning services. Particularly, whether they are suitable for geodynamic and deformation studies where precise and reliable velocities are needed. In this research, we process a very large campaign GPS data set (spanning 10 yr) with the online Jet Propulsion Laboratory Automated Precise Positioning Service. This data set is taken from a GNSS network specifically designed and surveyed to measure deformation through the central North Island of New Zealand. This includes regional CORS stations. We then use these coordinates to derive a horizontal and vertical velocity field. This is the first time that a large campaign GPS data set has been processed solely using an online service and the solutions used to determine a horizontal and vertical velocity field. We compared this velocity field to that of another well utilized GNSS scientific software package. The results show a good agreement between the CORS positions and campaign station velocities obtained from the two approaches. We discuss the implications of these results for how future GNSS campaign field surveys might be conducted and how their data might be processed.

Description: We have developed a method to estimate the geometry, location and densities of anomalies coming from 2-D gravity data based on compact gravity inversion technique. Compact gravity inversion is simple, fast and user friendly but severely depends on the number of model parameters, that is, by increasing the model parameters, the anomalies tend to concentrate near the surface. To overcome this ambiguity new weighting functions based on density contrast, depth, and compactness models have been introduced. Variable compactness factors have been defined here to get either a sharp or a smooth model based on the depth of the source or existence of prior information. Depth weighting derived from one station of gravity data whereas the effect of gravity data is 2-D and 3-D. To compensate this limitation an innovating weighting function namely kernel function has been introduced which multiplies with weight and compactness matrixes to yield a general model weighting function. The method is tested using three different sets of synthetic examples: a body at various depths (20, 40, 80 and 140 m), two bodies at the same depth but various distances to estimate lateral resolution and three bodies with negative and positive density contrast in different depths. The method is also applied to three real gravity data of Woodlawn massive sulphide body, sulphides mineralization of British Colombia and iron ore body of Missouri. The method produces solutions consistent with the known geologic attributes of the gravity sources, illustrating its potential practicality.

Description: This paper resurrects a version of Poisson's Partial Differential Equation (PDE) associated with the gravitational field at the Earth's surface and illustrates how the PDE possesses a capability to extract the mass density of Earth's topography from land-based gravity data. Herein, first we propound a theorem which mathematically introduces this version of Poisson's PDE adapted for the Earth's surface and then we use this PDE to develop a method of approximating the terrain mass density. Also, we carry out a real case study showing how the proposed approach is able to be applied to a set of land-based gravity data. In the case study, the method is summarized by an algorithm and applied to a set of gravity stations located along a part of the north coast of the Persian Gulf in the south of Iran. The results were numerically validated via rock-samplings as well as a geological map. Also, the method was compared with two conventional methods of mass density reduction. The numerical experiments indicate that the Poisson PDE at the Earth's surface has the capability to extract the mass density from land-based gravity data and is able to provide an alternative and somewhat more precise method of estimating the terrain mass density.

Description: The Moho surface can be determined according to isostatic theories, and among them, the recent Vening Meinesz-Moritz (VMM) theory of isostasy has been successfully applied for this purpose. In this paper, this method is studied from a theoretical prospective and its connection to the Airy–Heiskanen (AH) and Vening Meinesz original theories are presented. Jeffrey's inverse solution to isostasy is developed according to the recent developments of the VMM method and both are compared in similar situations. It is shown that they are generalizations of the AH model in a global and continuous domain. In the VMM spherical harmonic solution for Moho depth, the mean Moho depth contributes only to the zero-degree term of the series, while in Jeffrey's solution it contributes to all frequencies. In addition, the VMM spherical harmonic series is improved further so that the mean Moho can contribute to all frequencies of the solution. This modification makes the VMM global solution superior to the Jeffrey one, but in a global scale, the difference between both solutions is less than 3 km. Both solutions are asymptotically convergent and we present two methods to obtain smooth solutions for Moho from them.

Description: In a pioneering study, Wahr & Bergen developed the widely adopted, pseudo-normal mode framework for predicting the impact of anelastic effects on the Earth's body tides. Lau et al. have recently derived an extended normal mode treatment of the problem (as well as a minor variant of the theory known as the direct solution method) that makes full use of theoretical developments in free oscillation seismology spanning the last quarter century and that avoids a series of assumptions and approximations adopted in the traditional theory for predicting anelastic effects. There are two noteworthy differences between these two theories: (1) the traditional theory only considers perturbations to the eigenmodes of an elastic Earth, whereas the new theory augments this set of modes to include the relaxation modes that arise in anelastic behaviour; and (2) the traditional theory approximates the complex perturbation to the tidal Love number as a scaled version of the complex perturbation to the elastic moduli, whereas the new theory computes the full complex perturbation to each eigenmode. In this study, we highlight the above differences using a series of synthetic calculations, and demonstrate that the traditional theory can introduce significant error in predictions of the complex perturbation to the Love numbers due to anelasticity and the related predictions of tidal lag angles. For the simplified Earth models we adopt, the computed lag angles differ by ~20 per cent. The assumptions in the traditional theory have important implications for previous studies that use model predictions to correct observables for body tide signals or that analyse observations of body tide deformation to infer mantle anelastic structure. Finally, we also highlight the fundamental difference between apparent attenuation (i.e. attenuation inferred from observations or predicted using the above theories) and intrinsic attenuation (i.e. the material property investigated through experiments), where both are often expressed in terms of lag angles or Q –1 . In particular, we demonstrate the potentially significant (factor of two or more) bias introduced in estimates of Q –1 and its frequency dependence in studies that have treated Q –1 determined from tidal phase lags or measured experimentally as being equal. The observed or theoretically predicted lag angle (or apparent Q –1 ) differs from the intrinsic, material property due to inertia, self-gravity and effects associated with the energy budget. By accounting for these differences we derive, for a special case, an expression that accurately maps apparent attenuation predicted using the extended normal mode formalism of Lau et al. into intrinsic attenuation. The theory allows for more generalized mappings which may be used to robustly connect observations and predictions of tidal lag angles to results from laboratory experiments of mantle materials.

Description: We consider a new approach to both the forward and inverse problems in post-seismic deformation. We present a method for forward modelling post-seismic deformation in a self-gravitating, heterogeneous and compressible earth with a variety of linear and nonlinear rheologies. We further demonstrate how the adjoint method can be applied to the inverse problem both to invert for rheological structure and to calculate the sensitivity of a given surface measurement to changes in rheology or time-dependence of the source. Both the forward and inverse aspects are illustrated with several numerical examples implemented in a spherically symmetric earth model.

Description: The Seiland Igneous Province (SIP) is the largest complex of mafic and ultramafic intrusions in northern Fennoscandia intruded at ca . 580–560 Ma. The depth extent and the deep structure of the SIP are mainly unknown apart from three profiles modelled by gravity and refraction seismic data. Utilizing 3-D gravity modelling, a complex model of the deep subsurface structure of the SIP has been developed. The structure is presented in a multiprofile model ranging from the surface to the Moho. The mafic/ultramafic rocks of the SIP are modelled with densities of 3100 and 3300 kg m –3 , the surrounding rocks by densities of 2700 and 2900 kg m –3 for upper and lower crust, respectively. This density model explains the pronounced positive Bouguer gravity anomaly of up to 100 mGal above background. Its minimum volume is estimated from the subsurface model to 17 000 km 3 and as such we revise downwards the earlier estimations of 25 000 km 3 . The new subsurface model suggests that most of the SIP has a thickness between 2 and 4 km. An area with roots in an annular pattern is found and two deep-reaching roots have been identified located below the islands of Seiland and Sørøy. The depth of these roots is estimated to approximatively 9 km. The SIP is presently interpreted to be in the Caledonian Kalak Nappe Complex and the roots depth constrains its minimum thickness which is larger than earlier estimated. Furthermore, the rather undisturbed shape of the annular root pattern indicates that the SIP has not been subjected to strong tectonic reworking during the Caledonian orogeny.

Description: Estimating the relief of the Moho from gravity data is a computationally intensive nonlinear inverse problem. What is more, the modelling must take the Earths curvature into account when the study area is of regional scale or greater. We present a regularized nonlinear gravity inversion method that has a low computational footprint and employs a spherical Earth approximation. To achieve this, we combine the highly efficient Bott's method with smoothness regularization and a discretization of the anomalous Moho into tesseroids (spherical prisms). The computational efficiency of our method is attained by harnessing the fact that all matrices involved are sparse. The inversion results are controlled by three hyperparameters: the regularization parameter, the anomalous Moho density-contrast, and the reference Moho depth. We estimate the regularization parameter using the method of hold-out cross-validation. Additionally, we estimate the density-contrast and the reference depth using knowledge of the Moho depth at certain points. We apply the proposed method to estimate the Moho depth for the South American continent using satellite gravity data and seismological data. The final Moho model is in accordance with previous gravity-derived models and seismological data. The misfit to the gravity and seismological data is worse in the Andes and best in oceanic areas, central Brazil and Patagonia, and along the Atlantic coast. Similarly to previous results, the model suggests a thinner crust of 30–35 km under the Andean foreland basins. Discrepancies with the seismological data are greatest in the Guyana Shield, the central Solimões and Amazonas Basins, the Paraná Basin, and the Borborema province. These differences suggest the existence of crustal or mantle density anomalies that were unaccounted for during gravity data processing.

Description: Geophysical parameters of the deep Earth's interior can be evaluated through the resonance effects associated with the core and inner-core wobbles on the forced nutations of the Earth's figure axis, as observed by very long baseline interferometry (VLBI), or on the diurnal tidal waves, retrieved from the time-varying surface gravity recorded by superconducting gravimeters (SGs). In this paper, we inverse for the rotational mode parameters from both techniques to retrieve geophysical parameters of the deep Earth. We analyse surface gravity data from 15 SG stations and VLBI delays accumulated over the last 35 yr. We show existing correlations between several basic Earth parameters and then decide to inverse for the rotational modes parameters. We employ a Bayesian inversion based on the Metropolis–Hastings algorithm with a Markov-chain Monte Carlo method. We obtain estimates of the free core nutation resonant period and quality factor that are consistent for both techniques. We also attempt an inversion for the free inner-core nutation (FICN) resonant period from gravity data. The most probable solution gives a period close to the annual prograde term (or S 1 tide). However the 95 per cent confidence interval extends the possible values between roughly 28 and 725 d for gravity, and from 362 to 414 d from nutation data, depending on the prior bounds. The precisions of the estimated long-period nutation and respective small diurnal tidal constituents are hence not accurate enough for a correct determination of the FICN complex frequency.

Description: We evaluate the benefit of different global geophysical loading products on the internal scatter of GPS position time-series from 240 globally distributed sites. We focus on the non-tidal atmospheric pressure loading predicted from NASA's Modern-Era Retrospective Analysis for Research and Applications (MERRA-NATML) and the European Center for Medium-Range Weather Forecasts operational model (ECMWF-NATML), non-tidal ocean loading predicted from the Ocean Model for Circulation and Tides model (OMCT-NTOL), and the continental water storage loading predicted from the MERRA model (MERRA-CWSL) and the GFZ's Land Surface Discharge Model (LSDM-CWSL), respectively. The result shows that the root mean square (RMS) discrepancy of different CWSL models is larger than that of NATML models in the vertical component due to the varying model parameters and input data choices. We discuss the performance of different loading models and their combination to reduce the weighted RMS of GPS up-coordinates. MERRA-NATML & OMCT-NTOL & MERRA-CWSL reduced the weighted RMS (WRMS) in 96 per cent (JPL solutions) and 86 per cent (SOPAC solutions) of the cases, and ECMWF-NATML & OMCT-NTOL & LSDM-CWSL reduced the WRMS in 95 per cent (JPL solutions) and 88 per cent (SOPAC solutions) of the cases. The result shows that local effects and technical uncertainties in GPS time-series hamper the meaningful comparison between GPS observations and mass loading models. Hence, simply using the RMS of the time-series as the assessment criteria may lead to biased comparison results. Nonetheless, we give a detailed comparison (differences in phase and amplitude at seasonal timescales) for eight representative stations located adjacent to great rivers, lakes and reservoirs. We find that LSDM can provide a complementary model to study the small-scale hydrological loading like loading extremes along river channels. However, such small-scale hydrological loading effects are still instable to be modelled in some regions with its current accuracy. Finally, we discuss the impacts of mass loading corrections on the velocity and noise estimates. The noise reductions have the consistent performance as WRMS reductions for most sites, whereas some sites have their noise increased even though GPS signal WRMS is decreased there, suggesting that our posterior correction is potentially feasible, but not sufficient.

Description: Although an amount of hydrocarbon has been discovered in the West Korea Bay Basin (WKBB), located in the North Korean offshore area, geophysical investigations associated with these hydrocarbon reservoirs are not permitted because of the current geopolitical situation. Interpretation of satellite-derived potential field data can be alternatively used to image the 3-D density distribution in the sedimentary basin associated with hydrocarbon deposits. We interpreted the TRIDENT satellite-derived gravity field data to provide detailed insights into the spatial distribution of sedimentary density structures in the WKBB. We used 3-D forward density modelling for the interpretation that incorporated constraints from existing geological and geophysical information. The gravity data interpretation and the 3-D forward modelling showed that there are two modelled areas in the central subbasin that are characterized by very low density structures, with a maximum density of about 2000 kg m –3 , indicating some type of hydrocarbon reservoir. One of the anticipated hydrocarbon reservoirs is located in the southern part of the central subbasin with a volume of about 250 km 3 at a depth of about 3000 m in the Cretaceous/Jurassic layer. The other hydrocarbon reservoir should exist in the northern part of the central subbasin, with an average volume of about 300 km 3 at a depth of about 2500 m.

Description: We develop a high-resolution regional gravity field model by a combination of spherical harmonics, band-limited spherical radial basis functions (SRBFs) and the residual terrain model (RTM) technique. As the main input data set, we employ a dense terrestrial gravity database (3–6 stations km –2 ), which enables gravity field modelling up to very short spatial scales. The approach is based on the remove–compute–restore methodology in which all the parts of the signal that can be modelled are removed prior to the least-squares adjustment in order to smooth the input gravity data. To this end, we utilize degree-2159 spherical harmonic models and the RTM technique using topographic models at 2 arcsec resolution. The residual short-scale gravity signal is modelled via the band-limited Shannon SRBF expanded up to degree 21 600, which corresponds to a spatial resolution of 30 arcsec. The combined model is validated against GNSS/levelling-based height anomalies, independent surface gravity data, deflections of the vertical and terrestrial vertical gravity gradients achieving an accuracy of 2.7 cm, 0.53 mGal, 0.39 arcsec and 279 E in terms of the RMS error, respectively. A key aspect of the combined approach, especially in mountainous areas, is the quality of the RTM. We therefore compare the performance of two RTM techniques within the innermost zone, the tesseroids and the polyhedron. It is shown that the polyhedron-based approach should be preferred in rugged terrain if a high-quality RTM is required. In addition, we deal with the RTM computations at points located below the reference surface of the residual terrain which is known to be a rather delicate issue.

Description: In this paper we present the potential of a new compact superconducting gravimeter (GWR iGrav) designed for groundwater monitoring. At first, 3 yr of continuous gravity data are evaluated and the performance of the instrument is investigated. With repeated absolute gravity measurements using a Micro-g Lacoste FG5, the calibration factor (–894.8 nm s –2 V –1 ) and the long-term drift of this instrument (45 nm s –2 yr –1 ) are estimated for the first time with a high precision and found to be respectively constant and linear for this particular iGrav. The low noise level performance is found similar to those of previous superconducting gravimeters and leads to gravity residuals coherent with local hydrology. The iGrav is located in a fully instrumented hydrogeophysical observatory on the Durzon karstic basin (Larzac plateau, south of France). Rain gauges and a flux tower (evapo-transpiration measurements) are used to evaluate the groundwater mass balance at the local scale. Water mass balance demonstrates that the karst is only capacitive: all the rainwater is temporarily stored in the matrix and fast transfers to the spring through fractures are insignificant in this area. Moreover, the upper part of the karst around the observatory appears to be representative of slow transfer of the whole catchment. Indeed, slow transfer estimated on the site fully supports the low-flow discharge at the only spring which represents all groundwater outflows from the catchment. In the last part of the paper, reservoir models are used to characterize the water transfer and storage processes. Particular highlights are done on the advantages of continuous gravity data (compared to repeated campaigns) and on the importance of local accurate meteorological data to limit misinterpretation of the gravity observations. The results are complementary with previous studies at the basin scale and show a clear potential for continuous gravity time-series assimilation in hydrological simulations, even on heterogeneous karstic systems.

Description: A thorough understanding of time-dependent noise in Global Navigation Satellite System (GNSS) position time-series is necessary for computing uncertainties in any signals found in the data. However, estimation of time-correlated noise is a challenging task and is complicated by the difficulty in separating noise from signal, the features of greatest interest in the time-series. In this paper, we investigate how linear trends affect the estimation of noise in daily GNSS position time-series. We use synthetic time-series to study the relationship between linear trends and estimates of time-correlated noise for the six most commonly cited noise models. We find that the effects of added linear trends, or conversely de-trending, vary depending on the noise model. The commonly adopted model of random walk (RW), flicker noise (FN) and white noise (WN) is the most severely affected by de-trending, with estimates of low-amplitude RW most severely biased. FN plus WN is least affected by adding or removing trends. Non-integer power-law noise estimates are also less affected by de-trending, but are very sensitive to the addition of trend when the spectral index is less than one. We derive an analytical relationship between linear trends and the estimated RW variance for the special case of pure RW noise. Overall, we find that to ascertain the correct noise model for GNSS position time-series and to estimate the correct noise parameters, it is important to have independent constraints on the actual trends in the data.

Description: Previous formalisms for determining the static perturbation of spherically symmetric self-gravitating elastic Earth models due to displacement dislocations deal with each infinitesimal element of the fault system in its epicentral reference frame. In this work, we overcome this restriction and present novel and compact formulas for obtaining the perturbation due to the whole fault system in an arbitrary and common reference frame. Furthermore, we show that, even in an arbitrary reference frame, it is still possible to discriminate the contributions associated with the polar, bipolar and quadrupolar patterns of the seismic source response, as well as their relation with the along strike, along dip and tensile components of the displacement dislocation. These results allow a better understanding of the relation between the static perturbation and the whole fault system, and find direct applications in geodetic problems, like the modelling of long-wavelength geoid or gravity data from GRACE and GOCE space missions and of the perturbation of the deviatoric inertia tensor of the Earth.

Description: Sensitive instruments like strainmeters and tiltmeters are necessary for measuring slowly varying low amplitude Earth deformations. Nonetheless, laser and fibre interferometers are particularly suitable for interrogating such instruments due to their extreme precision and accuracy. In this paper, a practical design of a simple pendulum borehole tiltmeter based on laser fibre interferometric displacement sensors is presented. A prototype instrument has been constructed using welded borosilicate with a pendulum length of 0.85 m resulting in a main resonance frequency of 0.6 Hz. By implementing three coplanar extrinsic fibre Fabry-Perot interferometric probes and appropriate signal filtering, our instrument provides tilt measurements that are insensitive to parasitic deformations caused by temperature and pressure variations. This prototype has been installed in an underground facility (Rustrel, France) where results show accurate measurements of Earth strains derived from Earth and ocean tides, local hydrologic effects, as well as local and remote earthquakes. The large dynamic range and the high sensitivity of this tiltmeter render it an invaluable tool for numerous geophysical applications such as transient fault motion, volcanic strain and reservoir monitoring.

Description: Data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission can be used to estimate the mass change rate for separate drainage systems (DSs) of the Greenland Ice Sheet (GrIS). One approach to do so is by inversion of the level-2 spherical harmonic data to surface mass changes in predefined regions, or mascons. However, the inversion can be numerically unstable for some individual DSs. This occurs mainly for DSs with a small mass change signal that are located in the interior region of Greenland. In this study, we present a modified mascon inversion approach with an improved implementation of the constraint equations to obtain better estimates for individual DSs. We use separate constraints for mass change variability in the coastal zone, where run-off takes place, and for the ice sheet interior above 2000 m, where mass changes are smaller. A multi-objective optimization approach is used to find optimal prior variances for these two areas based on a simulation model. Correlations between adjacent DSs are suppressed when our optimized prior variances are used, while the mass balance estimates for the combination of the DSs that make up the GrIS above 2000 m are not affected significantly. The resulting mass balance estimates for some DSs in the interior are significantly improved compared to an inversion with a single constraint, as determined by a comparison with mass balance estimates from surface mass balance modelling and discharge measurements. The rate of mass change of the GrIS for the period of January 2003 to December 2012 is found to be –266.1 ± 17.2 Gt yr –1 in the coastal zone and areas below 2000 m, and +8.2 ± 8.6 Gt yr –1 in the interior region.

Description: The static and transient deformations produced by earthquakes cause density perturbations which, in turn, generate immediate, long-range perturbations of the Earth's gravity field. Here, an analytical solution is derived for gravity perturbations produced by a point double-couple source in homogeneous, infinite, non-self-gravitating elastic media. The solution features transient gravity perturbations that occur at any distance from the source between the rupture onset time and the arrival time of seismic P waves, which are of potential interest for real-time earthquake source studies and early warning. An analytical solution for such prompt gravity perturbations is presented in compact form. We show that it approximates adequately the prompt gravity perturbations generated by strike-slip and dip-slip finite fault ruptures in a half-space obtained by numerical simulations based on the spectral element method. Based on the analytical solution, we estimate that the observability of prompt gravity perturbations within 10 s after rupture onset by current instruments is severely challenged by the background microseism noise but may be achieved by high-precision gravity strainmeters currently under development. Our analytical results facilitate parametric studies of the expected prompt gravity signals that could be recorded by gravity strainmeters.

Description: The relative gravimeter is the primary terrestrial instrument for measuring spatially and temporally varying gravitational fields. The background noise of the instrument—that is, non-linear drift and random tares—typically requires some form of least-squares network adjustment to integrate data collected during a campaign that may take several days to weeks. Here, we present an approach to remove the change in the observed relative-gravity differences caused by hydrologic or other transient processes during a single campaign, so that the adjusted gravity values can be referenced to a single epoch. The conceptual approach is an example of coupled hydrogeophysical inversion, by which a hydrologic model is used to inform and constrain the geophysical forward model. The hydrologic model simulates the spatial variation of the rate of change of gravity as either a linear function of distance from an infiltration source, or using a 3-D numerical groundwater model. The linear function can be included in and solved for as part of the network adjustment. Alternatively, the groundwater model is used to predict the change of gravity at each station through time, from which the accumulated gravity change is calculated and removed from the data prior to the network adjustment. Data from a field experiment conducted at an artificial-recharge facility are used to verify our approach. Maximum gravity change due to hydrology (observed using a superconducting gravimeter) during the relative-gravity field campaigns was up to 2.6 μGal d –1 , each campaign was between 4 and 6 d and one month elapsed between campaigns. The maximum absolute difference in the estimated gravity change between two campaigns, two months apart, using the standard network adjustment method and the new approach, was 5.5 μGal. The maximum gravity change between the same two campaigns was 148 μGal, and spatial variation in gravity change revealed zones of preferential infiltration and areas of relatively high groundwater storage. The accommodation for spatially varying gravity change would be most important for long-duration campaigns, campaigns with very rapid changes in gravity and (or) campaigns where especially precise observed relative-gravity differences are used in the network adjustment.

Description: In the present work we illustrate a new local inversion algorithm to retrieve the Moho depth from GOCE (Gravity field and steady-state Ocean Circulation Explorer) gravity field. In details the proposed procedure can be divided into two main steps: the first one consists in recognizing and isolating the different geological provinces in the study area by exploiting information coming from the GOCE global gravity field model. Once the main geological provinces are defined, a function relating the crust density of each province with depth is built and used to reduce the data. The gravitational effects of sediments, topography, bathymetry and upper mantle are also removed. In the second step the residual gravitational field is inverted to retrieve the Moho depth and some information on the crustal density. In particular, the clustering of geological province is performed by means of an automatic Bayesian classification algorithm while the inversion of GOCE residual field is performed by adapting the global algorithm developed in the framework of the GEMMA project to the local scale. The procedure, based on an iterative Wiener filter, allows to compute the Moho depth considering lateral as well as radial variations of crustal density. The algorithm has been applied to the fifth release of GOCE time-wise global gravity field model to infer information on the crustal structure in the Western Balkan area, that is, the region laying between Bulgaria and the Adriatic Sea. This region is one of the most complex and active, from the tectonic point of view, in the whole Europe and it is characterized by the presence of the Alpine-Himalayan orogenic belt, formed by the collision between the African and Eurasian plates, and by the opening of the Pannonian Basin. Results show a good agreement between the obtained geological provinces with the actual knowledge on the region. The resulting Moho depth ranges between about 20 km beneath the Adriatic Sea and 45 km in the Dinarides. Comparisons with available seismic data show differences smaller than 1 km (standard deviation).

Description: Gravity variations associated with Earth's oblateness ( J 2 ) have been observed by satellite laser ranging (SLR) since 1976. The J 2 time-series has been used to measure and help understand many geophysical processes within the Earth system ranging from the mantle to the atmosphere. While post glacial rebound and the Earth climate system are believed to be the primary driving forces of long-term and seasonal J 2 variations, the physical cause of decadal and longer timescale J 2 variations has remained uncertain, although recent evidence indicates that polar ice mass changes are important. In this study, we estimate a variety of climate contributions to J 2 over the period 1979–2010, and find that ice mass variations in Greenland and Antarctica are the dominant cause of observed decadal and longer J 2 variations. Residual variations at periods near 10–11 years may reflect limitations of numerical climate models in estimating mass change variability at long periods, but are also suggestive of potential contribution related to variable solar activity.

Description: Interferometric synthetic aperture radar (InSAR) technology provides a valuable tool for obtaining Earth surface deformation and topography at high spatial resolution for crustal deformation studies. Similar to global positioning system (GPS), InSAR measurements are affected by the Earth's ionospheric and tropospheric layers as the electromagnetic signals significantly refract while propagating through the different layers. While GPS signals propagating through the neutral atmosphere are affected primarily by the distribution, pressure and temperature of atmospheric gases, including water vapour, the propagation through the ionosphere is mainly affected by the number of free electrons along the signal path. Here, we present the use of dense regional GPS networks for extracting tropospheric zenith delays and ionospheric total electron content (TEC) maps in order to reduce the noise levels in InSAR images. The results show significant reduction in the root mean square (RMS) values when simultaneously combining the two corrections, both at short time periods where no surface deformation is expected, and at longer periods, where imaging of localized subsidence and fault creep is enhanced.

Description: The main objective of the Gravity Recovery and Climate Experiment (GRACE) Atmospheric and Oceanic De-Aliasing Level-1B product (AOD1B) is the removal of high-frequency non-tidal mass variations due to sub-monthly mass transport in the atmosphere and oceans. Application of AOD1B shall avoid aliasing of these high-frequency signals into monthly gravity models derived from modern gravity missions and shall help to derive consistent orbit solutions for altimetry and Satellite Laser Ranging missions. The AOD1B 6-h series of spherical harmonic coefficients up to degree and order 100 are routinely generated at the German Research Centre for Geoscience and distributed to the GRACE Science Data System and the user community. Inputs for this product are acquired from numerical weather prediction models which are regularly revised and consequently not stable in time. The latest AOD1B release 5 (RL05) is based, as all other releases, on input from ECMWF and does not resolve this problem of discontinuities present in the surface pressure and surface geopotential input data. This might contaminate the gravity field variations derived from atmospheric mass variations. In this paper we present a method to overcome this problem during future AOD1B product generation, as well as two new Level-2 products (GAE and GAF) that, over land, fix a posteriori the two jumps present in the already distributed Level-2 RL05 monthly gravity models which were based on AOD1B RL05. The impact of the proposed correction on the variations and long-term trend of the total mass of the atmosphere and on the ice mass balance over Antarctica and over Greenland is also illustrated. We found that the GAE/GAF-corrected trend of the global atmospheric mass over the GRACE mission lifetime significantly decreased from –0.05 to –0.02 mm yr –1 in terms of geoid height. A considerable effect (33 per cent) was also found in the quadratic term of ice mass loss over Antarctica which results in an acceleration of 3.2 Gt yr –1  yr –1 smaller than without applying this correction.

Description: Previous studies of Earth rotation perturbations due to ice-age loading have predicted a slow secular drift of the rotation axis relative to the surface geography (i.e. true polar wander, TPW) of order of several degrees over the Plio-Pleistocene. It has been argued that this drift and the change in the geographic distribution of solar insolation that it implies may have been responsible for important transitions in ice-age climate, including the termination of ice-age cycles.We use a revised rotational stability theory that incorporates a more accurate treatment of the Earth's background ellipticity to reconsider this issue, and demonstrate that the net displacement of the pole predicted in earlier studies disappears. This more muted polar motion is due to two factors: first, the revised theory no longer predicts the permanent shift in the rotation axis, or the so-called ‘unidirectional TPW’, that appears in the traditional stability theory; and, second, the increased background ellipticity incorporated in the revised predictions acts to reduce the normal mode amplitudes governing the motion of the pole. We conclude that ice-age-induced TPW was not responsible for the termination of the ice age. This does not preclude the possibility that TPW induced by mantle convective flow may have played a role in major Plio-Pleistocene climate transitions, including the onset of Northern Hemisphere glaciation.

Description: A new approach based on energy conservation principle for satellite gravimetry mission has been developed and yields more accurate estimation of in situ geopotential difference observables using K-band ranging (KBR) measurements from the Gravity Recovery and Climate Experiment (GRACE) twin-satellite mission. This new approach preserves more gravity information sensed by KBR range-rate measurements and reduces orbit error as compared to previous energy balance methods. Results from analysis of 11 yr of GRACE data indicated that the resulting geopotential difference estimates agree well with predicted values from official Level 2 solutions: with much higher correlation at 0.9, as compared to 0.5–0.8 reported by previous published energy balance studies. We demonstrate that our approach produced a comparable time-variable gravity solution with the Level 2 solutions. The regional GRACE temporal gravity solutions over Greenland reveals that a substantially higher temporal resolution is achievable at 10-d sampling as compared to the official monthly solutions, but without the compromise of spatial resolution, nor the need to use regularization or post-processing.

Description: We interpreted the TRIDENT satellite derived gravity field to provide detailed insights into the spatial distribution of the crustal density structures in the area of the Yellow Sea. We used 3-D forward density modelling for the interpretation that incorporated constraints from existing geological and geophysical information. A gravity stripping method is used to separate out the gravity effects of different geological crustal structures. From this analysis we see that (1) the Gunsan sedimentary basin is isostatically compensated. (2) The satellite-derived Bouguer anomalies ranging from 15 to –30 x 10 –5 m s –2 are linked to basin thicknesses in the Yellow Sea. (3) The calculated Moho depth in the Yellow Sea varies from 27 km beneath the deep sedimentary basin to 34 km in the uplifted zones. (4) Moho depth calculations show two distinct areas, characterized by the deepest Moho depths and the largest crustal thicknesses in the Yellow Sea. The one region extends along the Qianliyan Uplift Zone from Jiaodong to Hongsung while the other area extends from southeastern China to Hongsung in the Korean peninsula. Compared to previous works we suggest that they are the part of the collisions zone between North and South China Blocks extending from China to the Korean peninsula via the Yellow Sea.

Description: Regional refinement of the gravity field models from satellite data using spherical radial base functions (SRBF) is an ill-posed problem. This is mainly due to the regional confinement of the data and the base functions, which leads to severe instabilities in the solutions. Here, this ill-posedness as well as the related regularization process are investigated. We compare three methods for the choice of the regularization parameter, which have been frequently used in gravity modelling. These methods are (1) the variance component estimation (VCE), (2) the generalized cross validation (GCV) and (3) the L-curve criterion. A particular emphasis is put on the impact of the SRBF type on the regularization parameter. To do this, we include two types of SRBF which are often used for regional gravity field modelling. These are the Shannon SRBF or the reproducing kernel and the Spline SRBF. The investigations are performed on two months of the real GOCE ultrasensitive gravity gradients over Central Africa and Amazon. The solutions are validated against a state-of-the-art global gravity solution. We conclude that if a proper regularization method is applied, both SRBF deliver more or less the same accuracy. We show that when the Shannon wavelet is used, the L-curve method gives the best results, while with the Spline kernel, the GCV outperforms the other two methods. Moreover, we observe that the estimated coefficients for the Spline kernel cannot be spatially interpreted. In contrast, the coefficients obtained for the Shannon wavelet reflect the energy of the recovered gravity field with a correlation factor of above 95 per cent. Therefore, when combined with the L-curve method, the Shannon SRBF is advantageous for regional gravity field estimation, since it is one of the simplest band-limited SRBF. In addition, it delivers promising solutions and the estimated coefficients represent the characteristics of the gravity field within the target region.

Description: This paper describes an alternative acceleration approach for determining GRACE monthly gravity field models. The main differences compared to the traditional acceleration approach can be summarized as: (1) The position errors of GRACE orbits in the functional model are taken into account; (2) The range ambiguity is eliminated via the difference of the range measurements and (3) The mean acceleration equation is formed based on Cowell integration. Using this developed approach, a new time-series of GRACE monthly solution spanning the period January 2003 to December 2010, called Tongji_Acc RL01, has been derived. The annual signals from the Tongji_Acc RL01 time-series agree well with those from the GLDAS model. The performance of Tongji_Acc RL01 shows that this new model is comparable with the RL05 models released by CSR and JPL as well as with the RL05a model released by GFZ.

Description: In this paper, we present a method for incorporating prior geological information into potential field data inversion problem. As opposed to the traditional inverse algorithm, our proposed method takes full advantage of prior geological information as a constraint and thus obtains a new objective function for inversion by adding Lagrangian multipliers and slack variables to the traditional inversion method. These additional parameters can be easily solved during iterations. We used both synthetic and observed data sets to test the stability and validity of the proposed method. Our results using synthetic gravity data show that our new method predicts depth and density anomalies more efficiently and accurately than the traditional inversion method that does not include prior geological constraints. Then using observed gravity data in the Three Gorges area and geological constraint information, we obtained the density distribution of the upper and middle crust in this area thus revealing its geological structure. These results confirm the proposed method's validity and indicate its potential application for magnetism data inversion and exploration of geological structures.

Description: It remains enigmatic how slow slip events (SSEs) interact with other slow seismic events and large distant earthquakes at many subduction zones. Here we model the spatiotemporal slip evolution of the most recent long-term SSE in 2009–2011 in the Bungo Channel region, southwest Japan using GEONET GPS position time-series and a Kalman filter-based, time-dependent slip inversion method. We examine the space-time relationship between the geodetically determined slow slip transient and seismically observed low frequency earthquakes (LFEs) and very-low frequency earthquakes (V-LFEs) near the Nankai trough. We find a strong but distinct temporal correlation between transient slip and LFEs and V-LFEs, suggesting a different relationship to the SSE. We also find the great Tohoku-Oki earthquake appears to disrupt the normal source process of the SSE, probably reflecting large-scale stress redistribution caused by the earthquake. Comparison of the 2009–2011 SSE with others in the same region shows much similarity in slip and moment release, confirming its recurrent nature. Comparison of transient slip with plate coupling shows that slip transients mainly concentrate on the transition zone from strong coupling region to downdip LFEs with transient slip relieving elastic strain accumulation at transitional depth. The less consistent spatial correlation between the long-term SSE and seismic slow earthquakes, and susceptibility of these slow earthquakes to various triggering sources including long-term slow slip, suggests caution in using the seismically determined slow earthquakes as a proxy for slow slip.

Description: Normal mode treatments of the Earth's body tide response were developed in the 1980s to account for the effects of Earth rotation, ellipticity, anelasticity and resonant excitation within the diurnal band. Recent space-geodetic measurements of the Earth's crustal displacement in response to luni-solar tidal forcings have revealed geographical variations that are indicative of aspherical deep mantle structure, thus providing a novel data set for constraining deep mantle elastic and density structure. In light of this, we make use of advances in seismic free oscillation literature to develop a new, generalized normal mode theory for the tidal response within the semi-diurnal and long-period tidal band. Our theory involves a perturbation method that permits an efficient calculation of the impact of aspherical structure on the tidal response. In addition, we introduce a normal mode treatment of anelasticity that is distinct from both earlier work in body tides and the approach adopted in free oscillation seismology. We present several simple numerical applications of the new theory. First, we compute the tidal response of a spherically symmetric, non-rotating, elastic and isotropic Earth model and demonstrate that our predictions match those based on standard Love number theory. Second, we compute perturbations to this response associated with mantle anelasticity and demonstrate that the usual set of seismic modes adopted for this purpose must be augmented by a family of relaxation modes to accurately capture the full effect of anelasticity on the body tide response. Finally, we explore aspherical effects including rotation and we benchmark results from several illustrative case studies of aspherical Earth structure against independent finite-volume numerical calculations of the semi-diurnal body tide response. These tests confirm the accuracy of the normal mode methodology to at least the level of numerical error in the finite-volume predictions. They also demonstrate that full coupling of normal modes, rather than group coupling, is necessary for accurate predictions of the body tide response.

Description: Crustal vertical deformation (CVD) observed by continuous GPS height time-series can be explained largely by surface loading effects recovered from both Gravity Recover and Climate Experiment (GRACE) and General Circulation Models (GCMs) data. We first show that lower degree CVD spatial spectrum due to the Earth's elastic response to a uniform surface loading plays more important roles than that of high-degree case. We then demonstrate that GRACE data with 300–400 km spatial resolution have the ability to detect 99 per cent power of global and regional CVD in spatial spectrum domain using a global frequency–wavenumber spectrum method. We can just use either GRACE or GCMs 36 degree/order (d/o) spherical harmonic coefficients (SHCs) which correspond to 500 km spatial resolution to acquire more than 90 per cent variance of total CVD modeled by up to 180 d/o SHCs at 98 per cent global gridpoints. Globally, CVD modeled by GRACE loading can explain 72 per cent annual amplitude and 69 per cent variance of GPS observed height time-series, which is better than the GCMs results of 64 per cent for annual amplitude and 41 per cent for variance. Using a three cornered hat method, we also show that the noise level of monthly averaged CVD is about 3 mm for both GPS height time-series and GRACE loading result, while that of GCMs result is only 1.3 mm.

Description: We present a new method to derive 3-D surface deformation from an integration of interferometric synthetic aperture radar (InSAR) images and Global Navigation Satellite System (GNSS) observations based on Akaike's Bayesian Information Criterion (ABIC), considering relationship between deformations of neighbouring locations. This method avoids interpolated errors by excluding the interpolation of GNSS into the same spatial resolution as InSAR images and harnesses the data sets and the prior smooth constraints of surface deformation objectively and simultaneously by using ABIC, which were inherently unresolved in previous studies. In particular, we define surface roughness measuring smoothing degree to evaluate the performance of the prior constraints and deduce the formula of the covariance for the estimation errors to estimate the uncertainty of modelled solution. We validate this method using synthetic tests and the 2008 M w 7.9 Wenchuan earthquake. We find that the optimal weights associated with ABIC minimum are generally at trade-off locations that balance contributions from InSAR, GNSS data sets and the prior constraints. We use this method to evaluate the influence of the interpolated errors from the Ordinary Kriging algorithm on the derivation of surface deformation. Tests show that the interpolated errors may contribute to biasing very large weights imposed on Kriged GNSS data, suggesting that fixing the relative weights is required in this case. We also make a comparison with SISTEM method, indicating that our method allows obtaining better estimations even with sparse GNSS observations. In addition, this method can be generalized to provide a solution for situations where some types of data sets are lacking and can be exploited further to account for data sets such as the integration of displacements along radar lines and offsets along satellite tracks.

Description: While it has been known for some time that offsets in the time-series of Global Navigation Satellite System (GNSS) position estimates degrade station velocity determinations, the magnitude of the effect has not been clear. Using products of the International GNSS Service (IGS), we assess the impact empirically by injecting progressively larger numbers of artificial offsets and solving for a series of long-term secular GNSS frames. Our results show that the stability of the IGS global frame datum is fairly robust, with significant effects at the formal error level only for the R x (and Y-pole) and R z rotational orientations. On the other hand, station velocity estimates are more seriously affected, especially the vertical component. For the typical IGS station, the mean vertical rate uncertainty is already limited to 0.34 mm yr –1 for the current set of position discontinuities. If the number of breaks doubles, which might occur using newer detection schemes, then that uncertainty will worsen by ~40 per cent to 0.48 mm yr –1 . This error source is generally a more important component of realistic velocity uncertainties than any other, including accounting for temporal correlations in the GNSS data. The only way to improve future GNSS velocity estimates is to severely limit manmade displacements at the tracking stations.

Description: In general, observations are normally considered to refer to an epoch in time, however, observations take time. During this time span temporal variations of the observable alias the measurement. Similar phenomenon can be defined in the space domain as well: data treated to refer to a geographical location often contains integrated information of the surroundings. In each case the appropriate signal content can partially be recovered by desmoothing the averaged data. The present study delivers the theoretical foundation of a desmoothing method, and suggests its use on different applications in geodesy. The theoretical formulation of the desmoothing has been derived for 1-D and 2-D signals, the latter is interpreted on a plain and also on a sphere. The presented case studies are less elaborated, but intended to demonstrate the need and usefulness of the desmoothing tool.

Description: Extracting geophysical signals from Global Positioning System (GPS) coordinate time-series is a well-established practice that has led to great insights into how the Earth deforms. Often small discontinuities are found in such time-series and are traceable to either broad-scale deformation (i.e. earthquakes) or discontinuities due to equipment changes and/or failures. Estimating these offsets accurately enables the identification of coseismic deformation estimates in the former case, and the removal of unwanted signals in the latter case which then allows tectonic rates to be estimated more accurately. We develop a method to estimate accurately discontinuities in time series of GPS positions at specified epochs, based on a so-called ‘offset series’. The offset series are obtained by varying the amount of GPS data before and after an event while estimating the offset. Two methods, a mean and a weighted mean method, are then investigated to produce the estimated discontinuity from the offset series. The mean method estimates coseismic offsets without making assumptions about geophysical processes that may be present in the data (i.e. tectonic rate, seasonal variations), whereas the weighted mean method includes estimating coseismic offsets with a model of these processes. We investigate which approach is the most appropriate given certain lengths of available data and noise within the time-series themselves. For the Sumatra–Andaman event, with 4.5 yr of pre-event data, we show that between 2 and 3 yr of post-event data are required to produce accurate offset estimates with the weighted mean method. With less data, the mean method should be used, but the uncertainties of the estimated discontinuity are larger.

Description: We present an analytical solution for the elastic deformation of an elastic, transversely isotropic, layered and self-gravitating Earth by surface loads. We first introduce the vector spherical harmonics to express the physical quantities in the layered Earth. This reduces the governing equations to a linear system of equations for the expansion coefficients. We then solve for the expansion coefficients analytically under the assumption (i.e. approximation) that in the mantle, the density in each layer varies as 1/ r (where r is the radial coordinate) while the gravity is constant and that in the core the gravity in each layer varies linearly in r with constant density. These approximations dramatically simplify the subsequent mathematical analysis and render closed-form expressions for the expansion coefficients. We implement our solution in a MATLAB code and perform a benchmark which shows both the correctness of our solution and the implementation. We also calculate the load Love numbers (LLNs) of the PREM Earth for different degrees of the Legendre function for both isotropic and transversely isotropic, layered mantles with different core models, demonstrating for the first time the effect of Earth anisotropy on the LLNs.

Description: Moment accumulation rate (also referred to as moment deficit rate) is a fundamental quantity for evaluating seismic hazard. The conventional approach for evaluating moment accumulation rate of creeping faults is to invert for the slip distribution from geodetic measurements, although even with perfect data these slip-rate inversions are non-unique. In this study, we show that the slip-rate versus depth inversion is not needed because moment accumulation rate can be estimated directly from surface geodetic data. We propose an integral approach that uses dense geodetic observations from Interferometric Synthetic Aperture Radar (InSAR) and the Global Positioning System (GPS) to constrain the moment accumulation rate. The moment accumulation rate is related to the integral of the product of the along-strike velocity and the distance from the fault. We demonstrate our methods by studying the Creeping Section of the San Andreas fault observed by GPS and radar interferometry onboard the ERS and ALOS satellites. Along-strike variation of the moment accumulation rate is derived in order to investigate the degree of partial locking of the Creeping Section. The central Creeping Segment has a moment accumulation rate of 0.25–3.1  x  10 15 Nm yr –1 km –1 . The upper and lower bounds of the moment accumulation rates are derived based on the statistics of the noise. Our best-fitting model indicates that the central portion of the Creeping Section is accumulating seismic moment at rates that are about 5 per cent to 23 per cent of the fully locked Carrizo segment that will eventually be released seismically. A cumulative moment budget calculation with the historical earthquake catalogue ( M  〉 5.5) since 1857 shows that the net moment deficit at present is equivalent to a M w 6.3 earthquake.

Description: The increasing availability of geophysical models of the Earth's lithosphere and mantle has generated renewed interest in computation of theoretical gravity effects at global and regional scales. At the same time, the increasing availability of gravity gradient anomalies derived from satellite measurements, such as those provided by GOCE satellite, requires mathematical methods that directly model the gravity gradient anomalies in the same reference frame as GOCE gravity gradients. Our main purpose is to interpret these anomalies in terms of source and density distribution. Numerical integration methods for calculating gravity gradient values are generally based on a mass discretization obtained by decomposing the Earth's layers into a finite number of elementary solid bodies. In order to take into account the curvature of the Earth, spherical prisms or ‘tesseroids’ have been established unequivocally as accurate computation tools for determining the gravitational effects of large-scale structures. The question which then arises from, is whether gravity calculation methods using spherical prisms remain valid when factoring in the ellipticity of the Earth. In the paper, we outline a comprehensive method to numerically compute the complete gravity field with the help of the Gauss–Legendre quadrature involving ellipsoidal shaped prisms. The assessment of this new method is conducted by comparison between the gravity gradient values of simple sources obtained by means of numerical and analytical calculations, respectively. A comparison of the gravity gradients obtained from PREM and LITHO1.0 models using spherical- and ellipsoidal-prism-based methods is also presented. Numerical results indicate that the error on gravity gradients, caused by the use of the spherical prism instead of its ellipsoidal counterpart to describe an ellipsoidally shaped Earth, is useful for a joint analysis with those deduced from GOCE satellite measurements. Provided that a suitable scaling of prism densities has been performed, the spherical approximation error at GOCE height hardly reaches 1 mE for the entire Earth's lithosphere. The error attains 6 mE at a peak for a complete modeling of the Earth, from the crust down to the internal core.

Description: This paper presents efficient numerical schemes for 3-D gravity field inversion. We propose a 2-D multilayer model to approximate a 3-D density distribution, and prove that the solution of the multilayer model will converge to the discretized 3-D solution. Differed from the conventional fast Fourier transform (FFT) based methods in which FFT is applied to the kernel, the proposed approach directly generates the Block-Toeplitz Toeplitz-Block (BTTB) structure by discretizing the multilayer model and the BTTB matrix is embedded into a Block-Circulant Circulant-Block (BCCB) matrix such that FFT can be utilized. In this approach, both regularization and optimal pre-conditioning operator can be constructed in the form of BTTB matrix. Consequently, very efficient solvers can be developed, and tremendous reduction in storage requirement and computing time can be achieved. To validate the new approach, numerical simulations using synthetic and real field data are reported, and numerical analysis is carried out for the inversion problems. Based on this study, we conclude that the proposed methods are capable of performing large-scale 3-D density inversions with a modest computing resource.

Description: In this study, we propose a method to determine dislocation Love numbers using co-seismic gravity changes from GRACE measurements. First, we present an observation equation to model GRACE observations taking into account the effect of ocean water mass redistribution. The L-curve method was used to determine the regulation parameter in the inversion of the geopotential dislocation Love numbers constrained by an a priori preliminary reference Earth (PREM) model. Then, the GRACE data error was estimated in the study area to evaluate the uncertainty of our inversion, and our inverted Love numbers are significantly deviated from the PREM ones even the uncertainty is considered. Finally, GRACE data observed for the 2011 Tohoku-Oki earthquake ( M w  = 9.0) were used to estimate the gravity dislocation Love numbers, considering three different fault-slip models. The results show that the inverted dislocation Love numbers deviate from PREM model, especially for $k_{l1}^{32}$ and $k_{l0}^{33} - k_{l0}^{22}$ , which indicates that the inverted dislocation Love numbers can reflect the local structure that is different from the global average. This inconsistency is possibly because that the cold denser oceanic slab dives from the Japanese Trench into the softer asthenosphere, and then changes the local density here higher than the global average. And with these sets of Love numbers, we can invert for more accurate fault model and analyse focal rupture mechanism when some other earthquake in this area occurs in the future. This study provides a new approach to invert for dislocation Love numbers linked with local geological information.

Description: The intra-plate deformation of the Upper Rhine Graben (URG) located in Central Europe is investigated using geodetic measurement techniques. We present a new approach to calculate a combined velocity field from InSAR, levelling and GNSS measurements. As the expected tectonic movements in the URG area are small (less than 1 mm a –1 ), the best possible solutions for linear velocity rates from single-technique analyses are estimated in a first step. Second, we combine the velocity rates obtained from InSAR (line of sight velocity rates in ascending and descending image geometries), levelling (vertical velocity rates) and GNSS (horizontal velocity rates) using least-squares adjustment (LSA). Focusing on the Northern URG area, we analyse SAR data on four different image stacks (ERS ascending, ERS descending, Envisat ascending, Envisat descending) using the Persistent Scatterer (PS) approach. The linear velocity rates in ascending and descending image geometries, respectively, are estimated in an LSA from joint time-series analysis of ERS and Envisat data. Vertical velocity rates from levelling are obtained from a consistent adjustment of more than 40 000 measured height differences using a kinematic displacement model. Horizontal velocity rates in east and north direction are calculated from a time-series analysis of daily coordinate estimates at 76 permanently operating GNSS sites in the URG region. As the locations, at which the measurement data of PS-InSAR, levelling and GNSS reside, do not coincide, spatial interpolation is needed during several steps of the rigorous processing. We use Ordinary Kriging to interpolate from a given set of data points to the locations of interest with a special focus on the modeling and propagation of errors. The final 3-D velocity field is calculated at a 200 m grid, which carries values only close to the location of PS points, resulting in a mean horizontal and vertical precision of 0.30 and 0.13 mm a –1 , respectively. The vertical component of the combined velocity field shows a significant subsidence of about 0.5 mm a –1 in the northern part of the graben coinciding with a well-known quaternary basin structure. Horizontal displacement rates of up to 0.8 mm a –1 in southeast direction are observed outside the graben, in reasonable alignment with the average direction of maximum horizontal stress. Within the graben, the velocity directions rotate toward east in the non-subsiding part, while an opposite trend is observed in the subsiding part of the graben. The complexities of the observed velocity field are compatible to the geomechanical situation in our investigation area which is characterized by a transition from a restraining to a releasing bend setting. Glacial isostatic adjustment is another potential source influencing the observed velocity field, as well as anthropogenic signals due to mining, oil exploration and groundwater usage that have been identified in some places.

Description: The measurement of ongoing ice-mass loss and associated melt water contribution to sea-level change from regions such as West Antarctica is dependent on a combination of remote sensing methods. A key method, the measurement of changes in Earth's gravity via the GRACE satellite mission, requires a potentially large correction to account for the isostatic response of the solid Earth to ice-load changes since the Last Glacial Maximum. In this study, we combine glacial isostatic adjustment modelling with a new GPS dataset of solid Earth deformation for the southern Antarctic Peninsula to test the current understanding of ice history in this region. A sufficiently complete history of past ice-load change is required for glacial isostatic adjustment models to accurately predict the spatial variation of ongoing solid Earth deformation, once the independently-constrained effects of present-day ice mass loss have been accounted for. Comparisons between the GPS data and glacial isostatic adjustment model predictions reveal a substantial misfit. The misfit is localized on the southwestern Weddell Sea, where current ice models under-predict uplift rates by approximately 2 mm yr –1 . This under-prediction suggests that either the retreat of the ice sheet grounding line in this region occurred significantly later in the Holocene than currently assumed, or that the region previously hosted more ice than currently assumed. This finding demonstrates the need for further fieldwork to obtain direct constraints on the timing of Holocene grounding line retreat in the southwestern Weddell Sea and that GRACE estimates of ice sheet mass balance will be unreliable in this region until this is resolved.

Description: Density is a key driver of tectonic processes, but it is a difficult property to define well in the lithosphere because the gravity method is non-unique, and because converting to density from seismic velocity models, themselves non-unique, is also highly uncertain. Here we use a new approach to define the lithospheric density field of Australia, covering from 100°E to 165°E, from 5°N to 55°S and from the crust surface to 300 km depth. A reference model was derived primarily from the recently released Australian Seismological Reference Model, and refined further using additional models of sedimentary basin thickness and crustal thickness. A novel form of finite-element method based deterministic gravity inversion was applied in geodetic coordinates, implemented within the open-source escript modelling environment. Three spatial resolutions were modelled: half-, quarter- and eighth-degree in latitude and longitude, with vertical resolutions of 5, 2.5 and 1.25 km, respectively. Parameter sweeps for the key inversion regularization parameters show that parameter selection is not scale dependent. The sweep results also show that finer resolutions are more sensitive to the uppermost crust, but less sensitive to the mid- to lower-crust and uppermost mantle than lower resolutions. All resolutions show similar sensitivity below about 100 km depth. The final density model shows that Australia's lithospheric density field is strongly layered but also has large lateral density contrasts at all depths. Within the continental crust, the structure of the middle and lower crust differs significantly from the crystalline upper crust, suggesting that the tectonic processes or events preserved in the deep crust differ from those preserved in the shallower crust. The lithospheric mantle structure is not extensively modified from the reference model, but the results reinforce the systematic difference between the density of the oceanic and continental domains, and help identify subdivisions within each. The lithospheric static pressure field was resolved in 3D from the gravity and density fields. The pressure field model also highlights the fundamental difference between the oceanic and continental domains, with the former possessing lower pressure through most of the model. Overall pressure variability is large in the upper crust (60 MPa) but reduces significantly by –30 km elevation (20–30 MPa). By –50 km elevation, thick lower-crust generates further disequilibria (25–35 MPa) that are not compensated until –125 km elevation (10–20 MPa). Beneath –125 km elevation higher pressure is observed in the continental domain, extending to the base of the model. This indicates a lithosphere that is to a large degree isostatically compensated near the base of the felsic-intermediate continental crust, and again near the theoretical base of mature oceanic lithosphere.

Description: Measurements of ground deformation can be used to identify and interpret geophysical processes occurring at volcanoes. Most studies rely on a single geodetic technique, or fit a geophysical model to the results of multiple geodetic techniques. Here we present a methodology that combines GPS, Total Station measurements and InSAR into a single reference frame to produce an integrated 3-D geodetic velocity surface without any prior geophysical assumptions. The methodology consists of five steps: design of the network, acquisition and processing of the data, spatial integration of the measurements, time series computation and finally the integration of spatial and temporal measurements. The most significant improvements of this method are (1) the reduction of the required field time, (2) the unambiguous detection of outliers, (3) an increased measurement accuracy and (4) the construction of a 3-D geodetic velocity field. We apply this methodology to ongoing motion on Arenal's western flank. Integration of multiple measurement techniques at Arenal volcano revealed a deformation field that is more complex than that described by individual geodetic techniques, yet remains consistent with previous studies. This approach can be applied to volcano monitoring worldwide and has the potential to be extended to incorporate other geodetic techniques and to study transient deformation.

Description: In autumn 2012, the new release 05 (RL05) of monthly geopotencial spherical harmonics Stokes coefficients (SC) from Gravity Recovery and Climate Experiment (GRACE) mission was published. This release reduces the noise in high degree and order SC, but they still need to be filtered. One of the most common filtering processing is the combination of decorrelation and Gaussian filters. Both of them are parameters dependent and must be tuned by the users. Previous studies have analyzed the parameters choice for the RL05 GRACE data for oceanic applications, and for RL04 data for global application. This study updates the latter for RL05 data extending the statistics analysis. The choice of the parameters of the decorrelation filter has been optimized to: (1) balance the noise reduction and the geophysical signal attenuation produced by the filtering process; (2) minimize the differences between GRACE and model-based data and (3) maximize the ratio of variability between continents and oceans. The Gaussian filter has been optimized following the latter criteria. Besides, an anisotropic filter, the fan filter, has been analyzed as an alternative to the Gauss filter, producing better statistics.

Description: The paper in question by Van Camp and co-authors [MVC] challenges previous work showing that ground gravity data arising from hydrology can provide a consistent signal for the comparison with satellite gravity data. The data sets used are similar to those used previously, that is, the gravity field as measured by the GRACE satellites versus ground-based data from superconducting gravimeters (SGs) over the same continental area, in this case Central Europe. One of the main impediments in this paper is the presentation that is frequently confusing and misleading as to what the data analysis really shows, for example, the irregular treatment of annual components that are first subtracted then reappear in the analysis. More importantly, we disagree on specific points. Two calculations are included in our comment to illustrate where we believe that the processing in [MVC] paper is deficient. The first deals with their erroneous treatment of the global hydrology using a truncated spherical harmonic approach which explains almost a factor 2 error in their computation of the loading. The second shows the effect of making the wrong assumption in the GRACE/hydrology/surface gravity comparison by inverting the whole of the hydrology loading for underground stations. We also challenge their claims that empirical orthogonal function techniques cannot be done in the presence of periodic components, and that SG data cannot be corrected for comparisons with GRACE data. The main conclusion of their paper, that there is little coherence between ground gravity stations and this invalidates GRACE comparisons, is therefore questionable. There is nothing in [MVC] that contradicts any of the previous papers that have shown clearly a strong relation between seasonal signals obtained from both ground gravity and GRACE satellite data.

Description: The influence of changes in surface ice-mass redistribution and associated viscoelastic response of the Earth, known as glacial isostatic adjustment (GIA), on the Earth's rotational dynamics has long been known. Equally important is the effect of the changes in the rotational dynamics on the viscoelastic deformation of the Earth. This signal, known as the rotational feedback, or more precisely, the rotational feedback on the sea level equation, has been mathematically described by the sea level equation extended for the term that is proportional to perturbation in the centrifugal potential and the second-degree tidal Love number. The perturbation in the centrifugal force due to changes in the Earth's rotational dynamics enters not only into the sea level equation, but also into the conservation law of linear momentum such that the internal viscoelastic force, the perturbation in the gravitational force and the perturbation in the centrifugal force are in balance. Adding the centrifugal-force perturbation to the linear-momentum balance creates an additional rotational feedback on the viscoelastic deformations of the Earth. We term this feedback mechanism, which is studied in this paper, as the rotational feedback on the linear-momentum balance. We extend both the time-domain method for modelling the GIA response of laterally heterogeneous earth models developed by Martinec and the traditional Laplace-domain method for modelling the GIA-induced rotational response to surface loading by considering the rotational feedback on linear-momentum balance. The correctness of the mathematical extensions of the methods is validated numerically by comparing the polar-motion response to the GIA process and the rotationally induced degree 2 and order 1 spherical harmonic component of the surface vertical displacement and gravity field. We present the difference between the case where the rotational feedback on linear-momentum balance is considered against that where it is not. Numerical simulations show that the resulting difference in radial displacement and sea level change between these situations since the Last Glacial Maximum reaches values of ±25 and ±1.8 m, respectively. Furthermore, the surface deformation pattern is modified by up to 10 per cent in areas of former or ongoing glaciation, but by up to 50 per cent at the bottom of the southern Indian ocean. This also results in the movement of coastlines during the last deglaciation to differ between the two cases due to the difference in the ocean loading, which is seen for instance in the area around Hudson Bay, Canada and along the Chinese, Australian or Argentinian coastlines.

Description: During megathrust earthquakes, great ruptures are accompanied by large scale mass redistribution inside the solid Earth and by ocean mass redistribution due to bathymetry changes. These large scale mass displacements can be detected using the monthly gravity maps of the GRACE satellite mission. In recent years it has become increasingly common to use the long wavelength changes in the Earth's gravity field observed by GRACE to infer seismic source properties for large megathrust earthquakes. An important advantage of space gravimetry is that it is independent from the availability of land for its measurements. This is relevant for observation of megathrust earthquakes, which occur mostly offshore, such as the $M_{\text{w}} \sim 9$ 2004 Sumatra–Andaman, 2010 Maule (Chile) and 2011 Tohoku-Oki (Japan) events. In Broerse et al. , we examined the effect of the presence of an ocean above the rupture on long wavelength gravity changes and showed it to be of the first order. Here we revisit the implementation of an ocean layer through the sea level equation and compare the results with approximated methods that have been used in the literature. One of the simplifications usually lies in the assumption of a globally uniform ocean layer. We show that especially in the case of the 2010 Maule earthquake, due to the closeness of the South American continent, the uniform ocean assumption is not valid and causes errors up to 57 per cent for modelled peak geoid height changes (expressed at a spherical harmonic truncation degree of 40). In addition, we show that when a large amount of slip occurs close to the trench, horizontal motions of the ocean floor play a mayor role in the ocean contribution to gravity changes. Using a slip model of the 2011 Tohoku-Oki earthquake that places the majority of slip close to the surface, the peak value in geoid height change increases by 50 per cent due to horizontal ocean floor motion. Furthermore, we test the influence of the maximum spherical harmonic degree at which the sea level equation is performed for sea level changes occurring along coastlines, which shows to be important for relative sea level changes occurring along the shore. Finally, we demonstrate that ocean floor loading, self-gravitation of water and conservation of water mass are of second order importance for coseismic gravity changes. When GRACE observations are used to determine earthquake parameters such as seismic moment or source depth, the uniform ocean layer method introduces large biases, depending on the location of the rupture with respect to the continent. The same holds for interpreting shallow slip when horizontal motions are not properly accounted for in the ocean contribution. In both cases the depth at which slip occurs will be underestimated.

Description: Long-term volcanic subsidence provides insight into intereruptive processes, which comprise the longest portion of the eruptive cycle. Ground-based geodetic surveys of Medicine Lake Volcano (MLV), northern CA, document subsidence at rates of ~–10 mm yr –1 between 1954 and 2004. The long observation period plus the duration and stable magnitude of this signal presents an ideal opportunity to study long-term volcanic deformation, but this first requires accurate knowledge of the geometry and magnitude of the source. Best-fitting analytical source models to past levelling and GPS data sets show conflicting source parameters—primarily the model depth. To overcome this, we combine multiple tracks of InSAR data, each with a different look angle, to improve upon the spatial resolution of ground-based measurements. We compare the results from InSAR to those of past geodetic studies, extending the geodetic record to 2011 and demonstrating that subsidence at MLV continues at ~–10 mm yr –1 . Using geophysical inversions, we obtain the best-fitting analytical source model—a sill located at 9–10 km depth beneath the caldera. This model geometry is similar to those of past studies, providing a good fit to the high spatial density of InSAR measurements, while accounting for the high ratio of vertical to horizontal deformation derived from InSAR and recorded by existing levelling and GPS data sets. We discuss possible causes of subsidence and show that this model supports the hypothesis that deformation at MLV is driven by tectonic extension, gravitational loading, plus a component of volume loss at depth, most likely due to cooling and crystallization within the intrusive complex that underlies the edifice. Past InSAR surveys at MLV, and throughout the Cascades, are of variable success due to dense vegetation, snow cover and atmospheric artefacts. In this study, we demonstrate how InSAR may be successfully used in this setting by applying a suite of multitemporal analysis methods that account for atmospheric and orbital noise sources. These methods include: a stacking strategy based upon the noise characteristics of each data set; pixelwise rate-map formation (-RATE) and persistent scatterer InSAR (StaMPS).

Description: In the literature, the inverted coseismic slip models from seismological and geodetic data for the 2011 Tohoku-Oki earthquake portray significant discrepancies, in particular regarding the intensity and the distribution of the rupture near the trench. For a megathrust earthquake, it is difficult to discern the slip along the shallow part of the fault from the geodetic data, which are often acquired on land. In this paper, we discuss the uncertainties in the slip distribution inversion using the geodetic data for the 2011 Tohoku earthquake and the Fully Bayesian Inversion method. These uncertainties are due to the prior information regarding the boundary conditions at the edges of the fault, the dip subduction angle and the smoothing operator. Using continuous GPS data from the Japan Island, the results for the rigid and free boundary conditions show that they produce remarkably different slip distributions at shallow depths, with the latter producing a large slip exceeding 30 m near the surface. These results indicate that the smoothing operator (gradient or Laplacian schemes) does not severely affect the slip pattern. To better invert the coseismic slip, we then introduce the ocean bottom GPS (OB-GPS) data, which improve the resolution of the shallow part of the fault. We obtain a near-trench slip greater than 40 m that reaches the Earth's surface, regardless of which boundary condition is used. Additionally, we show that using a mean dip angle for the fault as derived from subduction models is adequate if the goal is to invert for the general features of the slip pattern of this megathrust event.

Description: Complications arise in the interpretation of gravity fields because of interference from systematic degradations, such as boundary blurring and distortion. The major sources of these degradations are the various systematic errors that inevitably occur during gravity field data acquisition, discretization and geophysical forward modelling. To address this problem, we evaluate deconvolution method that aim to detect the clear horizontal boundaries of anomalous sources by the suppression of systematic errors. A convolution-based multilayer projection model, based on the classical 3-D gravity field forward model, is innovatively derived to model the systematic error degradation. Our deconvolution algorithm is specifically designed based on this multilayer projection model, in which three types of systematic error are defined. The degradations of the different systematic errors are considered in the deconvolution algorithm. As the primary source of degradation, the convolution-based systematic error is the main object of the multilayer projection model. Both the random systematic error and the projection systematic error are shown to form an integral part of the multilayer projection model, and the mixed norm regularization method and the primal-dual optimization method are therefore employed to control these errors and stabilize the deconvolution solution. We herein analyse the parameter identification and convergence of the proposed algorithms, and synthetic and field data sets are both used to illustrate their effectiveness. Additional synthetic examples are specifically designed to analyse the effects of the projection systematic error, which is caused by the uncertainty associated with the estimation of the impulse response function.

Description: We propose to test if gravimetry can prove useful in discriminating different models of long-term deep crustal processes in the case of the Taiwan mountain belt. We discuss two existing tectonic models that differ in the deep processes proposed to sustain the long-term growth of the orogen. One model assumes underplating of the uppermost Eurasian crust with subduction of the deeper part of the crust into the mantle. The other one suggests the accretion of the whole Eurasian crust above crustal-scale ramps, the lower crust being accreted into the collisional orogen. We compute the temporal gravity changes caused only by long-term rock mass transfers at depth for each of them. We show that the underplating model implies a rate of gravity change of –6 x 10 –2 μGal yr –1 , a value that increases to 2 x 10 –2 μGal yr –1 if crustal subduction is neglected. If the accretion of the whole Eurasian crust occurs, a rate of 7 x 10 –2 μGal yr –1 is obtained. The two models tested differ both in signal amplitude and spatial distribution. The yearly gravity changes expected by long-term deep crustal mass processes in Taiwan are two orders of magnitude below the present-day uncertainty of land-based gravity measurements. Assuming that these annually averaged long-term gravity changes will linearly accumulate with ongoing mountain building, multidecadal time-series are needed to identify comparable rates of gravity change. However, as gravity is sensitive to any mass redistribution, effects of short-term processes such as seismicity and surface mass transfers (erosion, sedimentation, ground-water) may prevent from detecting any long-term deep signal. This study indicates that temporal gravity is not appropriate for deciphering the long-term deep crustal processes involved in the Taiwan mountain belt.

Description: The computation of quasi-static deformation for axisymmetric viscoelastic structures on a gravitating spherical earth is addressed using the spectral element method (SEM). A 2-D spectral element domain is defined with respect to spherical coordinates of radius and angular distance from a pole of symmetry, and 3-D viscoelastic structure is assumed to be azimuthally symmetric with respect to this pole. A point dislocation source that is periodic in azimuth is implemented with a truncated sequence of azimuthal order numbers. Viscoelasticity is limited to linear rheologies and is implemented with the correspondence principle in the Laplace transform domain. This leads to a series of decoupled 2-D problems which are solved with the SEM. Inverse Laplace transform of the independent 2-D solutions leads to the time-domain solution of the 3-D equations of quasi-static equilibrium imposed on a 2-D structure. The numerical procedure is verified through comparison with analytic solutions for finite faults embedded in a laterally homogeneous viscoelastic structure. This methodology is applicable to situations where the predominant structure varies in one horizontal direction, such as a structural contrast across (or parallel to) a long strike-slip fault.

Description: The geodetic rates for the gravity variation and vertical uplift in polar regions subject to past and present-day ice-mass changes (PDIMCs) provide important insight into the rheological structure of the Earth. We provide an update of the rates observed at Ny-Ålesund, Svalbard. To do so, we extract and remove the significant seasonal content from the observations. The rate of gravity variations, derived from absolute and relative gravity measurements, is –1.39 ± 0.11 μGal yr –1 . The rate of vertical displacements is estimated using GPS and tide gauge measurements. We obtain 7.94 ± 0.21 and 8.29 ± 1.60 mm yr –1 , respectively. We compare the extracted signal with that predicted by GLDAS/Noah and ERA-interim hydrology models. We find that the seasonal gravity variations are well-represented by local hydrology changes contained in the ERA-interim model. The phase of seasonal vertical displacements are due to non-local continental hydrology and non-tidal ocean loading. However, a large part of the amplitude of the seasonal vertical displacements remains unexplained. The geodetic rates are used to investigate the asthenosphere viscosity and lithosphere/asthenosphere thicknesses. We first correct the updated geodetic rates for those induced by PDIMCs in Svalbard, using published results, and the sea level change due to the melting of the major ice reservoirs. We show that the latter are at the level of the geodetic rate uncertainties and are responsible for rates of gravity variations and vertical displacements of –0.29 ± 0.03 μGal yr –1 and 1.11 ± 0.10 mm yr –1 , respectively. To account for the late Pleistocene deglaciation, we use the global ice evolution model ICE-3G. The Little Ice Age (LIA) deglaciation in Svalbard is modelled using a disc load model with a simple linear temporal evolution. The geodetic rates at Ny-Ålesund induced by the past deglaciations depend on the viscosity structure of the Earth. We find that viscous relaxation time due to the LIA deglaciation in Svalbard is more than 60 times shorter than that due to the Pleistocene deglaciation. We also find that the response to past and PDIMCs of an Earth model with asthenosphere viscosities ranging between 1.0 and 5.5 x 10 18 Pa s and lithosphere (resp. asthenosphere) thicknesses ranging between 50 and 100 km (resp. 120 and 170 km) can explain the rates derived from geodetic observations.

Description: The 2 principle and the unbiased predictive risk estimator are used to determine optimal regularization parameters in the context of 3-D focusing gravity inversion with the minimum support stabilizer. At each iteration of the focusing inversion the minimum support stabilizer is determined and then the fidelity term is updated using the standard form transformation. Solution of the resulting Tikhonov functional is found efficiently using the singular value decomposition of the transformed model matrix, which also provides for efficient determination of the updated regularization parameter each step. Experimental 3-D simulations using synthetic data of a dipping dike and a cube anomaly demonstrate that both parameter estimation techniques outperform the Morozov discrepancy principle for determining the regularization parameter. Smaller relative errors of the reconstructed models are obtained with fewer iterations. Data acquired over the Gotvand dam site in the south-west of Iran are used to validate use of the methods for inversion of practical data and provide good estimates of anomalous structures within the subsurface.

Description: Global navigation satellite systems (GNSSs) have revealed that a mega-thrust earthquake that occurs in an island-arc trench system causes post-seismic crustal deformation. Such crustal deformation data have been interpreted by combining three mechanisms: afterslip, poroelastic rebound and viscoelastic relaxation. It is seismologically important to determine the contribution of each mechanism because it provides frictional properties between the plate boundaries and viscosity estimates in the asthenosphere which are necessary to evaluate the stress behaviour during earthquake cycles. However, the observation sites of GNSS are mostly deployed over land and can detect only a small part of the large-scale deformation, which precludes a clear separation of the mechanisms. To extend the spatial coverage of the deformation area, recent studies started to use satellite gravity data that can detect long-wavelength deformations over the ocean. To date, compared with theoretical models for calculating the post-seismic crustal deformation, a few models have been proposed to interpret the corresponding gravity variations. Previous approaches have adopted approximations for the effects of compressibility, sphericity and self-gravitation when computing gravity changes. In this study, a new spectral-finite element approach is presented to consider the effects of material compressibility for Burgers viscoelastic earth model with a laterally heterogeneous viscosity distribution. After the basic principles are explained, it is applied to the 2004 Sumatra–Andaman earthquake. For this event, post-seismic deformation mechanisms are still a controversial topic. Using the developed approach, it is shown that the spatial patterns of gravity change generated by the above three mechanisms clearly differ from one another. A comparison of the theoretical simulation results with the satellite gravity data obtained from the Gravity Recovery and Climate Experiment reveals that both afterslip and viscoelastic relaxation are occurring. Considering the spatial patterns in satellite gravity fields is an effective method for investigating post-seismic deformation mechanisms.

Description: Some of the major geothermal anomalies in central Europe are linked to tectonic structures within the top of crystalline basement, which modify strongly the top of this basement. Their assessment is a major challenge in exploration geophysics. Gravity has been proven to be suitable for the detection of mainly large scale lithological and structural inhomogeneities. Indeed, it is well known and proven by different wells that, for example, in northern Switzerland extended negative anomalies are linked to such structures. Due to depth limitation of wells, there vertical extension is often unknown. In this study, we have investigated the potential of gravity for the geometrical characterization of such basement structures. Our approach consists in the combination of the series of Butterworth filters, geological modelling and best-fitting between observed and computed residual anomalies. In this respect, filters of variable wavelength are applied to observed and computed gravity data. The geological model is discretized into a finite element mesh. Near-surface anomalies and the effect of the sedimentary cover were eliminated using cut-off wavelength of 10 km and geological and seismic information. We analysed the potential of preferential Butterworth filtering in a sensitivity study and applied the above mentioned approach to part of the Swiss molasses basin. Sensitivity analyses reveal that such sets of residual anomalies represents a pseudo-tomography revealing the distribution of different structures with depth. This finding allows for interpreting negative anomalies in terms of 3-D volumes. Best-fitting then permits determination of the most likely 3-D geometries of such basement structures. Our model fits both, geological observations and gravity: among 10 deep boreholes in the studied area, six reach the respective units and confirm our distribution of the negative (and positive) anomalies.

Description: On 2008 October 5, a magnitude 6.6 earthquake struck the eastern termination of the intermontane Alai valley between the southern Tien Shan and the northern Pamir of Kyrgyzstan. The shallow thrust earthquake occurred in the footwall of the Main Pamir thrust, where the Pamir orogen is colliding with the southern Tien Shan mountains. We measure the coseismic surface displacements using SAR (Synthetic Aperture RADAR) data; the results show clear gradients in the vertical and horizontal directions along a complex pattern of surface ruptures and active faults. To integrate and to interpret these observations in the context of the regional tectonics, we complement the SAR data analysis with seismological data and geological field observations. While the main moment release of the Nura earthquake appears to be on the Pamir Frontal thrust, the main surface displacements and surface rupture occurred in the footwall along the NE–SW striking Irkeshtam fault. With InSAR data from ascending and descending tracks along with pixel offset measurements, we model the Nura earthquake source as a segmented rupture. One fault segment corresponds to high-angle brittle faulting at the Pamir Frontal thrust and two more fault segments show moderate-angle and low-friction thrusting at the Irkeshtam fault. Our integrated analysis of the coseismic deformation argues for rupture segmentation and strain partitioning associated to the earthquake. It possibly activated an orogenic wedge in the easternmost segment of the Pamir-Alai collision zone. Further, the style of the segmentation may be associated with the presence of Palaeogene evaporites.

Description: The terrestrial reference frame is a cornerstone for modern geodesy and its applications for a wide range of Earth sciences. The underlying assumption for establishing a terrestrial reference frame is that the motion of the solid Earth's figure centre relative to the mass centre of the Earth system on a multidecadal timescale is linear. However, past international terrestrial reference frames (ITRFs) showed unexpected accelerated motion in their translation parameters. Based on this underlying assumption, the inconsistency of relative origin motions of the ITRFs has been attributed to data reduction imperfection. We investigated the impact of surface mass loading from atmosphere, ocean, snow, soil moisture, ice sheet, glacier and sea level from 1983 to 2008 on the geocentre variations. The resultant geocentre time-series display notable trend acceleration from 1998 onward, in particular in the z -component. This effect is primarily driven by the hydrological mass redistribution in the continents (soil moisture, snow, ice sheet and glacier). The acceleration is statistically significant at the 99 per cent confidence level as determined using the Mann–Kendall test, and it is highly correlated with the satellite laser ranging determined translation series. Our study, based on independent geophysical and hydrological models, demonstrates that, in addition to systematic errors from analysis procedures, the observed non-linearity of the Earth-system behaviour at interannual timescales is physically driven and is able to explain 42 per cent of the disparity between the origins of ITRF2000 and ITRF2005, as well as the high level of consistency between the ITRF2005 and ITRF2008 origins.

Description: This paper presents a novel mathematical reformulation of the theory of the free wobble/nutation of an axisymmetric reference earth model in hydrostatic equilibrium, using the linear momentum description. The new features of this work consist in the use of (i) Clairaut coordinates (rather than spherical polars), (ii) standard spherical harmonics (rather than generalized spherical surface harmonics), (iii) linear operators (rather than J-square symbols) to represent the effects of rotational and ellipticity coupling between dependent variables of different harmonic degree and (iv) a set of dependent variables all of which are continuous across material boundaries. The resulting infinite system of coupled ordinary differential equations is given explicitly, for an elastic solid mantle and inner core, an inviscid outer core and no magnetic field. The formulation is done to second order in the Earth's ellipticity. To this order it is shown that for wobble modes (in which the lowest harmonic in the displacement field is degree 1 toroidal, with azimuthal order m  = ±1), it is sufficient to truncate the chain of coupled displacement fields at the toroidal harmonic of degree 5 in the solid parts of the earth model. In the liquid core, however, the harmonic expansion of displacement can in principle continue to indefinitely high degree at this order of accuracy. The full equations are shown to yield correct results in three simple cases amenable to analytic solution: a general earth model in rigid rotation, the tiltover mode in a homogeneous solid earth model and the tiltover and Chandler periods for an incompressible homogeneous solid earth model. Numerical results, from programmes based on this formulation, are presented in part II of this paper.

Description: Numerical solutions are presented for the formulation of the linear momentum description of Earth's dynamics using Clairaut coordinates. We have developed a number of methods to integrate the equations of motion, including starting at the Earth's centre of mass, starting at finite radius and separating the displacement associated with the primary rigid rotation. We include rotation and ellipticity to second order up to spherical harmonic T $_5^m$ , starting with the primary displacement T $_1^m$ with m  = ±1. We are able to confirm many of the previous results for models PREM (with no surface ocean) and 1066A, both in their original form and with neutrally stratified liquid cores. Our period search ranges from the near-seismic band [0.1 sidereal days (sd)] to 3500 sd, within which we have identified the four well-known wobble-nutation modes: the Free Core Nutation (retrograde) at –456 sd, the Free Inner Core Nutation (FICN, prograde) at 468 sd, the Chandler Wobble (prograde) at 402 sd, and the Inner Core Wobble (ICW, prograde) at about 2842 sd (7.8 yr) for neutral PREM. The latter value varies significantly with earth model and integration method. In addition we have verified to high accuracy the tilt-over mode at 1 sd within a factor 10 –6 . In an exhaustive search we found no additional near-diurnal wobble modes that could be identified as nutations. We show that the eigenfunctions for the as-yet-unidentified ICW are extremely sensitive to the details of the earth model, especially the core stability profile and there is no well-defined sense of its wobble relative to the mantle. Calculations are also done for a range of models derived from PREM with homogeneous layers, as well as with incompressible cores. For this kind of model the ICW ceases to have just a simple IC rigid motion when the fluid compressibility is either unchanged or multiplied by a factor 10; in this case the outer core exhibits oscillations that arise from an unstable fluid density stratification. For the FICN our results for the truncation at harmonic T 5 show less change from the T 3 truncation than a similar result reported elsewhere. Finally, we give a thorough discussion of the complete spectrum of the characteristic determinant including the location of poles and non-wobble gravity modes, and discuss in general the dynamics of the inviscid core at periods short compared to those involved in the geodynamo.

Description: Considering the drawback of existing global weighted mean temperature model, this paper uses 2006–2012 NCEP reanalysis data to establish global empirical model for mapping zenith wet delays onto precipitable water—GTm_N, takes the influence of half-year periodicity of Tm into account when modelling and estimate the initial phase of each cycle. In order to evaluate the precision of GTm_N, we use three different Tm data sets from the NCEP during 2013, 650 radiosonde stations and COSMIC occultation in 2011 to test this model. The results show that GTm_N has higher precision in both ocean and continental area in every moment of every day. The accuracy of GTm_N is higher than Bevis formulas and GTm_II models. In addition, the actual surface temperature is not required in GTm_N model, and it will have wide application in GPS meteorology.