Analysis of the Spatial Distribution and Common Mode Error Correlation in a Small-Scale GNSS Network

When analyzing GPS time series, common mode errors (CME) often obscure the actual crustal movement signals, leading to deviations in the velocity estimates of station coordinates. Therefore, mitigating the impact of CME on station positioning accuracy is crucial to ensuring the precision and reliability of GNSS time series. The current approach to separating CME mainly uses signal filtering methods to decompose the residuals of the observation network into multiple signals, from which the signals corresponding to CME are identified and separated. However, this method overlooks the spatial correlation of the stations. In this paper, we improved the Independent Component Analysis (ICA) method by introducing correlation coefficients as weighting factors, allowing for more accurate emphasis or attenuation of the contributions of the GNSS network's spatial distribution during the ICA process. The results show that the improved Weighted Independent Component Analysis (WICA) method can reduce the root mean square (RMS) of the coordinate time series by an average of 27.96%, 15.23%, and 28.33% in the E, N, and U components, respectively. Compared to the ICA method, considering the spatial distribution correlation of stations, the improved WICA method shows enhancements of 12.53%, 3.70%, and 8.97% in the E, N, and U directions, respectively. This demonstrates the effectiveness of the WICA method in separating CMEs and provides a new algorithmic approach for CME separation methods.

Time-Varying GPS Displacement Network Modeling by Sequential Monte Carlo

Geodetic observations through high-rate GPS time-series data allow the precise modeling of slow ground deformation at the millimeter level. However, significant attention has been devoted to utilizing these data for various earth science applications, including to determine crustal velocity fields and to detect significant displacement from earthquakes. The relationships inherent in these GPS displacement observations have not been fully explored. This study employs the sequential Monte Carlo method, specifically particle filtering (PF), to develop a time-varying analysis of the relationships among GPS displacement time-series within a network, with the aim of uncovering network dynamics. Additionally, we introduce a proposed graph representation to enhance the understanding of these relationships. Using the 1-Hz GEONET GNSS network data of the Tohoku-Oki Mw9.0 2011 as a demonstration, the results demonstrate successful parameter tracking that clarifies the observations' underlying dynamics. These findings have potential applications in detecting anomalous displacements in the future.

Multilayer Model in Soil Moisture Content Retrieval Using GNSS Interferometric Reflectometry

The global navigation satellite system-interferometric reflectometry (GNSS-IR) was developed more than a decade ago to monitor soil moisture content (SMC); a system that is essentially finished has emerged. The standard GNSS-IR model typically considers soil to be a single layer of medium and measures the average SMC between 1 and 10 cm below the soil surface. The majority of the SMC is not distributed uniformly along the longitudinal axis. This study is based on a simulation platform and suggests a SMC-stratified measurement model that can be used to recover the SMC at different depths in the sink and reverse osmosis to address the issue that conventional techniques cannot accurately measure soil moisture at different depths. The soil moisture of each layer was assessed by utilizing the GNSS signals reflected by various soil layers, and this study employed total transmission when the vertical linearly polarized component of the electromagnetic wave was conveyed by the GNSS signal reflected by the soil. This work employed the Hilbert transform to obtain the interference signal envelope, which increases the visibility of the interference signal's "notch" and reduces the burr impact of the interference signal brought on by ambient noise. The accuracy of the SMC measurement at the bottom declines due to the soil's attenuation of the GNSS signal power, but the correlation between the predetermined value and SMC retrieved by the GNSS-IR multilayer SMC measurement model similarly approached 0.92.

A Height Nonlinear Velocity Field Algorithm for CORS Station Based on GARCH Model

In this study, the basic concept of height nonlinear velocity field modeling in the CORS station is described. The noise results in a large deviation between the observation and predicted height. An ARCH testing method for heteroscedasticity of CORS height residual square series was proposed and the non-stationary characteristic of CORS height residual square time series was proved. A CORS height nonlinear velocity field reconstruction method based on the GARCH model was proposed. First, a nonlinear LS periodic fitting model was established for CORS height series data. Then, a GARCH model was established for the fitted non-stationary residual series. Finally, the signal term, linear trend term, and GARCH model noise term of nonlinear LS modeling were combined to reconstruct the nonlinear velocity field of the CORS height. The RMSE of nonlinear LS cycle modeling for 25 CORS stations worldwide ranged from 5 to 10 mm. The differences between the velocity, approximate annual and semi-annual amplitudes, and SOPAC results were 0.73 mm/a, 0.94 mm, and 0.51 mm, respectively. Compared with the centimeter amplitude of the CORS station height, the accuracy of the nonlinear model established in this study met the requirements. The results of height nonlinear velocity field reconstruction at 25 CORS stations worldwide showed that the mean square error of prediction of the one-year height movement reached 9 mm, and the average prediction accuracy of the semi-annual was 7 mm. Compared with the calculation accuracy of the current global CORS elevation component of 3-5 mm, the prediction error in this study was about 3 mm. The expected goal was achieved regarding the accuracy of the CORS station height nonlinear velocity field model.

Identification of Areas of Anomalous Tremor of the Earth’s Surface on the Japanese Islands According to GPS Data

Statistical properties of Earth surface tremors measured by means of GPS were investigated. This article considers measurements of the Earth’s surface displacements in three orthogonal directions relayed by a network of GPS sensors with about 1200 points distributed across Japan in 2009–2021. Next, the following characteristics of the tremors were considered: the entropy of the distribution of squared orthogonal wavelet coefficients, the entropy of the distribution of power spectrum values, and the spectral index. The anomalous regions of maxima of probability densities of the distribution of extreme values of the tremor statistics were determined: entropy minima and spectral index maxima. The average density maps of the distribution of extreme value tremor statistics were found to be highly correlated with one another. This made it possible to consider a weighted average density map and identify five anomalous regions in the center and south of Japan. A trajectory of visiting anomalous regions by a sequence of points realizing local maxima of the average probability density was obtained, for which seasonal periodicity was set. Estimates of changes in the average and maximum values of the correlation coefficients of tremor properties in an auxiliary network of 16 reference points in a semi-annual time window were obtained.

Strain Field Features and Three-Dimensional Crustal Deformations Constrained by Dense GRACE and GPS Measurements in NE Tibet

The continuing impact between the Eurasia Plate and India results in the thickening and shortening of the N-S Tibetan Plateau. There has been strong tectonic movement along the boundary of the zones of deformation of the NE corner of the Tibetan plateau (NET) since the new tectonic period, with its dynamic mechanisms remaining controversial. Here, we use observations of 39 Continuous Global Positioning System (CGPS) gauges and 451 Crustal Movement Observation Network of China (CMONOC) campaign-mode stations to detect the three-dimensional deformation of the crust in the NET. Improved processing procedures are implemented to strengthen the patterns of strain throughout the NET. The principal component analysis (PCA) technique is introduced to decompose the time series into spatial eigenvectors and principal components (PCs), and the first three PCs are used to estimate and rectify common mode errors (CMEs). In addition, GRACE observations are used to detect deformation changes that account for non-tidal oceanic mass loading, hydrological loading, and surface pressure. The rectified deformation of the crust indicates the anisotropic nature of both the subsidence and uplift, and that the highest uplift rate of the Longmen Shan fault uplift reaches 7.13 ± 0.53 mm/yr. Finally, the horizontal velocity is further used to enumerate the strain rates throughout the NET. The results show that the shear band retained property in line with the strike-slip fault along the Altyn Tagh fault, the Qilian Shan faults, the Haiyuan fault, the West Qinling fault, the East Kunlun fault, and the Longmen Shan fault. In addition, the results further indicate that the whole NET shows a strong relationship with the mean principal rates of horizontal shortening strain. Extension and compression of the crust reasonably describe its sinking and uplifting.

Interseismic Fault Coupling and Slip Rate Deficit on the Central and Southern Segments of the Tanlu Fault Zone Based on Anhui CORS Measurements

The Tanlu fault zone, extending over 2400 km from South China to Russia, is one of the most conspicuous tectonic elements in eastern Asia. In this study, we processed the Global Positioning System (GPS) measurements of Anhui Continuously Operating Reference System (AHCORS) between January 2013 and June 2018 to derive a high-precision velocity field in the central and southern segments of the Tanlu fault zone. We integrated the AHCORS data with those publicly available for geodetic imaging of the interseismic coupling and slip rate deficit distribution in the central and southern segments of the Tanlu fault zone. This work aims at a better understanding of strain accumulation and future seismic hazard in the Tanlu fault zone. The result indicates lateral variation of coupling distribution along the strike of the Tanlu fault zone. The northern segment of the Tanlu fault zone has a larger slip rate deficit and a deeper locking depth than the southern segment. Then, we analyzed three velocity profiles across the fault. The result suggests that the central and southern segments of the Tanlu fault zone are characterized by right-lateral strike-slip (0.29–0.44 mm/y) with compression components (0.35–0.76 mm/y). Finally, we estimated strain rates using the least-squares collocation method. The result shows that the dilatation rates concentrate in the region where the principal strain rates are very large. The interface of extension and compression is always accompanied by sudden change of direction of principal strain rates. Especially, in the north of Anhui, the dilatation rate is largest, reaching 3.780×10−8/a. Our study suggests that the seismic risk in the northern segment of the Tanlu fault zone remains very high for its strong strain accumulation and the lack of historical large earthquakes.

Assessment of Contemporary Antarctic GIA Models Using High-Precision GPS Time Series

Past redistributions of the Earth’s mass resulting from the Earth’s viscoelastic response to the cycle of deglaciation and glaciation reflect the process known as glacial isostatic adjustment (GIA). GPS data are effective at constraining GIA velocities, provided that these data are accurate, have adequate spatial coverage, and account for competing geophysical processes, including the elastic loading of ice/snow ablation/accumulation. GPS solutions are significantly affected by common mode errors (CMEs) and the choice of optimal noise model, and they are contaminated by other geophysical signals due primarily to the Earth’s elastic response. Here, independent component analysis is used to remove the CMEs, and the Akaike information criterion is used to determine the optimal noise model for 79 GPS stations in Antarctica, primarily distributed across West Antarctica and the Antarctic Peninsula. Next, a high-resolution surface mass variation model is used to correct for elastic deformation. Finally, we use the improved GPS solution to assess the accuracy of seven contemporary GIA forward models in Antarctica. The results show that the maximal GPS crustal displacement velocity deviations reach 4.0 mm yr−1, and the mean variation is 0.4 mm yr−1 after removing CMEs and implementing the noise analysis. All GIA model-predicted velocities are found to systematically underestimate the GPS-observed velocities in the Amundsen Sea Embayment. Additionally, the GPS vertical velocities on the North Antarctic Peninsula are larger than those on the South Antarctic Peninsula, and most of the forward models underestimate the GIA impact on the Antarctic Peninsula.

Spatial Variations of Stochastic Noise Properties in GPS Time Series

The noise in position time series of 568 GPS (Global Position System) stations across North America with an observation span of ten years has been investigated using solutions from two processing centers, namely, the Pacific Northwest Geodetic Array (PANGA) and New Mexico Tech (NMT). It is well known that in the frequency domain, the noise exhibits a power-law behavior with a spectral index of around −1. By fitting various noise models to the observations and selecting the most likely one, we demonstrate that the spectral index in some regions flattens to zero at long periods while in other regions it is closer to −2. This has a significant impact on the estimated linear rate since flattening of the power spectral density roughly halves the uncertainty of the estimated tectonic rate while random walk doubles it. Our noise model selection is based on the highest log-likelihood value, and the Akaike and Bayesian Information Criteria to reduce the probability of over selecting noise models with many parameters. Finally, the noise in position time series also depends on the stability of the monument on which the GPS antenna is installed. We corroborate previous results that deep-drilled brace monuments produce smaller uncertainties than concrete piers. However, if at each site the optimal noise model is used, the differences become smaller due to the fact that many concrete piers are located in tectonic/seismic quiet areas. Thus, for the predicted performance of a new GPS network, not only the type of monument but also the noise properties of the region need to be taken into account.

Analysis of GNSS Displacements in Europe and Their Comparison with Hydrological Loading Models

Thanks to the increasing number of permanent GNSS stations in Europe and their long records, we computed position solutions for more than 1000 stations over the last two decades using the REPRO3 orbit and clock products from the IGS CNES-CLS (GRGS) Analysis Center. The velocities, which are mainly due to tectonics and glacial isostatic adjustment (GIA), and the annual solar cycle have been estimated using weighted least squares. The interannual variations have been accounted for in the stochastic model or in the deterministic model. We demonstrated that the velocity and annual cycle, in addition to their uncertainties, depend on the estimation method we used and that the estimation of GPS draconitic oscillations minimises biases in the estimation of annual solar cycle displacements. The annual solar cycle extracted from GPS has been compared with that from loading estimates of several hydrological models. If the annual amplitudes between GPS and hydrological models match, the phases of the loading models were typically in advance of about 1 month compared to GPS. Predictions of displacements modelled from GRACE observations did not show this phase shift. We also found important discrepancies at the interannual frequency band between GNSS, loading estimates derived from GRACE, and hydrological models using principal component analysis (PCA) decomposition. These discrepancies revealed that GNSS position variations in the interannual band cannot be systematically interpreted as a geophysical signal and should instead be interpreted in terms of autocorrelated noise.

Potential Contributors to Common Mode Error in Array GPS Displacement Fields in Taiwan Island

The existence of the common mode error (CME) in the continuous global navigation satellite system (GNSS) coordinate time series affects geophysical studies that use GNSS observations. To understand the potential contributors of CME in GNSS networks in Taiwan and their effect on velocity estimations, we used the principal component analysis (PCA) and independent component analysis (ICA) to filter the vertical coordinate time series from 44 high-quality GNSS stations in Taiwan island in China, with a span of 10 years. The filtering effects have been evaluated and the potential causes of the CME are analyzed. The root-mean-square values decreased by approximately 14% and 17% after spatio-temporal filtering using PCA and ICA, respectively. We then discuss the relationship between the CME sources obtained by ICA and the environmental loads. The results reveal that the independent displacements extracted by ICA correlate with the atmospheric mass loading (ATML) and land water storage mass loading (LWS) of Taiwan in terms of both its amplitude and phase. We then use the white noise plus power law noise model to quantitatively estimate the noise characteristics of the pre- and post-filtered coordinate time series based on the maximum likelihood estimation criterion. The results indicate that spatio-temporal filtering reduces the amplitude of the PL and the periodic terms in the GPS time series.

The Ground Deformation History of the Neapolitan Volcanic Area (Campi Flegrei Caldera, Somma–Vesuvius Volcano, and Ischia Island) from 20 Years of Continuous GPS Observations (2000–2019)

The Neapolitan volcanic area includes three active and high-risk volcanoes: Campi Flegrei caldera, Somma–Vesuvius, and Ischia island. The Campi Flegrei volcanic area is a typical example of a resurgent caldera, characterized by intense uplift periods followed by subsidence phases (bradyseism). After about 21 years of subsidence following the 1982–1984 unrest, a new inflation period started in 2005 and, with increasing rates over time, is ongoing. The overall uplift from 2005 to December 2019 is about 65 cm. This paper provides the history of the recent Campi Flegrei caldera unrest and an overview of the ground deformation patterns of the Somma–Vesuvius and Ischia volcanoes from continuous GPS observations. In the 2000–2019 time span, the GPS time series allowed the continuous and accurate tracking of ground and seafloor deformation of the whole volcanic area. With the aim of improving the research on volcano dynamics and hazard assessment, the full dataset of the GPS time series from the Neapolitan volcanic area from January 2000 to December 2019 is presented and made available to the scientific community.

Analysis and Discussion on the Optimal Noise Model of Global GNSS Long-Term Coordinate Series Considering Hydrological Loading

The displacement of Global Navigation Satellite System (GNSS) station contains the information of surface elastic deformation caused by the variation of land water reserves. This paper selects the long-term coordinate series data of 671 International GNSS Service (IGS) reference stations distributed globally under the framework of World Geodetic System 1984 (WGS84) from 2000 to 2021. Different noise model combinations are used for noise analysis, and the optimal noise model for each station before and after hydrologic loading correction is calculated. The results show that the noise models of global IGS reference stations are diverse, and each component has different optimal noise model characteristics, mainly white noise + flicker noise (WN+FN), generalized Gauss–Markov noise (GGM) and white noise + power law noise (WN+PL). Through specific analysis between the optimal noise model and the time series velocity of the station, it is found that the maximum influence value of the vertical velocity can reach 1.8 mm when hydrological loading is considered. Different complex noise models also have a certain influence on the linear velocity and velocity uncertainty of the station. Among them, the influence of white noise + random walking noise is relatively obvious, and its maximum influence value in the elevation direction can reach over 2 mm/year. When studying the impact of hydrological loading correction on the periodicity of the coordinate series, it is concluded whether the hydrological loading is calculated or not, and the GNSS long-term coordinate series has obvious annual and semi-annual amplitude changes, which are most obvious in the vertical direction, up to 16.48 mm.

Improving Real-Time Position Estimation Using Correlated Noise Models

We provide algorithms for inferring GPS (Global Positioning System) location and for quantifying the uncertainty of this estimate in real time. The algorithms are tested on GPS data from locations in the Southern Hemisphere at four significantly different latitudes. In order to rank the algorithms, we use the so-called log-score rule. The best algorithm uses an Ornstein-Uhlenbeck (OU) noise model and is built on an enhanced Kalman Filter (KF). The noise model is capable of capturing the observed autocorrelated process noise in the altitude, latitude and longitude recordings. This model outperforms a KF that assumes a Gaussian noise model, which under-reports the position uncertainties. We also found that the dilution-of-precision parameters, automatically reported by the GPS receiver at no additional cost, do not help significantly in the uncertainty quantification of the GPS positioning. A non-learning method using the actual position measurements and employing a constant uncertainty does not even converge to the correct position. Inference with the enhanced noise model is suitable for embedded computing and capable of achieving real-time position inference, can quantify uncertainty and be extended to incorporate complementary sensor recordings, e.g., from an accelerometer or from a magnetometer, in order to improve accuracy. The algorithm corresponding to the augmented-state unscented KF method suggests a computational cost of O(dx2dt), where dx is the dimension of the augmented state-vector and dt is an adjustable, design-dependent parameter corresponding to the length of "past values" one wishes to keep for re-evaluation of the model from time to time. The provided algorithm assumes dt=1. Hence, the algorithm is likely to be suitable for sensor fusion applications.

Imprints of COVID-19 Lockdown on GNSS Observations: An Initial Demonstration Using GNSS Interferometric Reflectometry

The ongoing coronavirus disease 2019 (COVID-19) pandemic has imposed tight mobility restrictions in urban areas, causing substantial reduction in roadway traffic. Many public parking lots are nearly vacant as people across the world have gone on lockdown since mid-March. This environmental change may have impacts on Global Navigation Satellite System (GNSS) sensors installed on roof of buildings. Here, we use a monitoring site in Boston to exemplify a likely sensitivity of precise GNSS sensors to their nearby dynamic environments including parked vehicles in parking lots. We show that reduced number of parked vehicles since 23 March has decreased the reflector roughness, resulting in an increase in the reflected signal power whose amplitude is quantified by GNSS interferometric reflectometry technique. The uncertainty of retrieved GNSS antenna height drops with beginning of lockdown, allowing more accurate estimate of reflector height, which could have a general implication for better understanding of the fundamental limitations of the technique.

Modeling Seasonal Variations in Vertical GPS Coordinate Time Series Using Independent Component Analysis and Varying Coefficient Regression

Common seasonal variations in Global Positioning System (GPS) coordinate time series always exist, and the modeling and correction of the seasonal signals are helpful for many geodetic studies using GPS observations. A spatiotemporal model was proposed to model the common seasonal variations in vertical GPS coordinate time series, based on independent component analysis and varying coefficient regression method. In the model, independent component analysis (ICA) is used to separate the common seasonal signals in the vertical GPS coordinate time series. Considering that the periodic signals in GPS coordinate time series change with time, a varying coefficient regression method is used to fit the separated independent components. The spatiotemporal model was then used to fit the vertical GPS coordinate time series of 262 global International GPS Service for Geodynamics (IGS) GPS sites. The results show that compared with least squares regression, the varying coefficient method can achieve a more reliable fitting result for the seasonal variation of the separated independent components. The proposed method can accurately model the common seasonal variations in the vertical GPS coordinate time series, with an average root mean square (RMS) reduction of 41.6% after the model correction.

Extracting Seasonal Signals in GNSS Coordinate Time Series via Weighted Nuclear Norm Minimization

Global Navigation Satellite System (GNSS) coordinate time series contains obvious seasonal signals, which mainly manifest as a superposition of annual and semi-annual oscillations. Accurate extraction of seasonal signals is of great importance for understanding various geophysical phenomena. In this paper, a Weighted Nuclear Norm Minimization (WNNM) is proposed to extract the seasonal signals from the GNSS coordinate time series. WNNM assigns different weights to different singular values that enable us to estimate an approximate low rank matrix from its noisy matrix. To address this issue, the low rank characteristics of the Hankel matrix induced by GNSS coordinate time series was investigated first, and then the WNNM is applied to extract the seasonal signals in the GNSS coordinate time series. Meanwhile, the residuals have been analyzed, obtaining the estimation of the uncertainty of velocity. To demonstrate the effectiveness of the proposed algorithm, a number of tests have been carried out on both simulated and real GNSS dataset. Experimental results indicate that the proposed scheme offers preferable performances compared with many state-of-the-art methods.

Quantitative Evaluation of Environmental Loading Induced Displacement Products for Correcting GNSS Time Series in CMONOC

Mass redistribution within the Earth system deforms the surface elastically. Loading theory allows us to predict loading induced displacement anywhere on the Earth’s surface using environmental loading models, e.g., Global Land Data Assimilation System. In addition, different publicly available loading products are available. However, there are differences among those products and the differences among the combinations of loading models cannot be ignored when precisions of better than 1 cm are required. Many scholars have applied these loading corrections to Global Navigation Satellite System (GNSS) time series from mainland China without considering or discussing the differences between the available models. Evaluating the effects of different loading products over this region is of paramount importance for accurately removing the loading signal. In this study, we investigate the performance of these different publicly available loading products on the scatter of GNSS time series from the Crustal Movement Observation Network of China. We concentrate on five different continental water storage loading models, six different non-tidal atmospheric loading models, and five different non-tidal oceanic loading models. We also investigate all the different combinations of loading products. The results show that the difference in RMS reduction can reach 20% in the vertical component depending on the loading correction applied. We then discuss the performance of different loading combinations and their effects on the noise characteristics of GNSS height time series and horizontal velocities. The results show that the loading products from NASA may be the best choice for corrections in mainland China. This conclusion could serve as an important reference for loading products users in this region.

The Noise Properties and Velocities from a Time-Series of Estonian Permanent GNSS Stations

The aim of this study was to estimate the noise properties, velocities, and their uncertainties from a time-series of selected (~9 years long) Estonian continuously operating Global Navigation Satellite System (GNSS) stations. Two software packages based on different processing methods, Gipsy–Oasis and Bernese, were used for daily coordinate calculations. Different methods and software (Tsview, Hector, and MIDAS) were used for coordinate time-series analysis. Outliers were removed using three different strategies. Six different stochastic noise models were used for trend estimation altogether with the analysis of the noise properties of the residual time-series with Hector. Obtained velocities were compared with different land uplift and glacial isostatic adjustment models (e.g., ICE-6G (VM5a), NKG2016LU, etc.). All compared solutions showed similar fit to the compared models. It was confirmed that the best fit to the time-series residuals were with the flicker noise plus white noise model (for the North and East component) and generalized Gauss–Markov model (for Up). Velocities from MIDAS, Tsview, and Hector solutions within the same time-series (Gipsy–Oasis or Bernese) agreed well but velocity uncertainties differed up to four times. The smallest uncertainties were obtained from Tsview; the MIDAS solution produced the most conservative values. Although the East and Up component velocities between Gipsy and Bernese solutions agreed well, the North component velocities were systematically shifted.

Spatiotemporal Filtering and Noise Analysis for Regional GNSS Network in Antarctica Using Independent Component Analysis

The common mode error (CME) and optimal noise model are the two most important factors affecting the accuracy of time series in regional Global Navigation Satellite System (GNSS) networks. Removing the CME and selecting the optimal noise model can effectively improve the accuracy of GNSS coordinate time series. The CME, a major source of error, is related to the spatiotemporal distribution; hence, its detrimental effects on time series can be effectively reduced through spatial filtering. Independent component analysis (ICA) is used to filter the time series recorded by 79 GPS stations in Antarctica from 2010 to 2018. After removing stations exhibiting strong local effects using their spatial responses, the filtering results of residual time series derived from principal component analysis (PCA) and ICA are compared and analyzed. The Akaike information criterion (AIC) is then used to determine the optimal noise model of the GPS time series before and after ICA/PCA filtering. The results show that ICA is superior to PCA regarding both the filter results and the consistency of the optimal noise model. In terms of the filtering results, ICA can extract multisource error signals. After ICA filtering, the root mean square (RMS) values of the residual time series are reduced by 14.45%, 8.97%, and 13.27% in the east (E), north (N), and vertical (U) components, respectively, and the associated speed uncertainties are reduced by 13.50%, 8.06% and 11.82%, respectively. Furthermore, different GNSS time series in Antarctica have different optimal noise models with different noise characteristics in different components. The main noise models are the white noise plus flicker noise (WN+FN) and white noise plus power law noise (WN+PN) models. Additionally, the spectrum index of most PN is close to that of FN. Finally, there are more stations with consistent optimal noise models after ICA filtering than there are after PCA filtering.

Mapping heterogeneities through avalanche statistics

Avalanche statistics of various threshold-activated dynamical systems are known to depend on the magnitude of the drive, or stress, on the system. Such dependences exist for earthquake size distributions, in sheared granular avalanches, laboratory-scale fracture and also in the outage statistics of power grids. In this work, we model threshold-activated avalanche dynamics and investigate the time required to detect local variations in the ability of model elements to bear stress. We show that the detection time follows a scaling law where the scaling exponents depend on whether the feature that is sought is either weaker, or stronger, than its surroundings. We then look at earthquake data from Sumatra and California, demonstrate the trade-off between the spatial resolution of a map of earthquake exponents (i.e. the b-values of the Gutenberg-Richter Law) and the accuracy of those exponents, and suggest a means to maximize both.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Identification of the Noise Model in the Time Series of GNSS Stations Coordinates Using Wavelet Analysis

Analysis of the time series of coordinates is extremely important in geodynamic research. Indeed, the correct interpretation of coordinate changes may facilitate an understanding of the diverse geophysical processes taking place in the earth’s crust. At present, when rigorously processing global navigation satellite system (GNSS) observations, the influence of deformations in the surface of the earth’s crust is not considered. This article presents signal modelling for the influence on the analysis of noise occurring in the time series of GNSS station coordinates. The modelling of coordinate time series was undertaken using the classic least-squares estimation (LSE) method and the inverse continuous wavelet transform (CWT). In order to determine the type of noise character, the coefficient spectral index was used. Analyses have demonstrated that the nature of noise in measurement data does not depend on the signal estimation method. The differences between classic modelling (LSE) of the time series with annual and semiannual oscillation and signal reconstruction are very small ( Δ κ = 0.0 ÷−0.2).

Effects of Spatiotemporal Filtering on the Periodic Signals and Noise in the GPS Position Time Series of the Crustal Movement Observation Network of China

Analysis of Global Positioning System (GPS) position time series and its common mode components (CMC) is very important for the investigation of GPS technique error, the evaluation of environmental loading effects, and the estimation of a realistic and unbiased GPS velocity field for geodynamic applications. In this paper, we homogeneously processed the daily observations of 231 Crustal Movement Observation Network of China (CMONOC) Continuous GPS stations to obtain their position time series. Then, we filtered out the CMC and evaluated its effects on the periodic signals and noise for the CMONOC time series. Results show that, with CMC filtering, peaks in the stacked power spectra can be reduced at draconitic harmonics up to the 14th, supporting the point that the draconitic signal is spatially correlated. With the colored noise suppressed by CMC filtering, the velocity uncertainty estimates for both of the two subnetworks, CMONOC-I (≈16.5 years) and CMONOC-II (≈4.6 years), are reduced significantly. However, the CMONOC-II stations obtain greater reduction ratios in velocity uncertainty estimates with average values of 33%, 38%, and 54% for the north, east, and up components. These results indicate that CMC filtering can suppress the colored noise amplitudes and improve the precision of velocity estimates. Therefore, a unified, realistic, and three-dimensional CMONOC GPS velocity field estimated with the consideration of colored noise is given. Furthermore, contributions of environmental loading to the vertical CMC are also investigated and discussed. We find that the vertical CMC are reduced at 224 of the 231 CMONOC stations and 170 of them are with a root mean square (RMS) reduction ratio of CMC larger than 10%, confirming that environmental loading is one of the sources of CMC for the CMONOC height time series.

Physical applications of GPS geodesy: a review

Geodesy, the oldest science, has become an important discipline in the geosciences, in large part by enhancing Global Positioning System (GPS) capabilities over the last 35 years well beyond the satellite constellation's original design. The ability of GPS geodesy to estimate 3D positions with millimeter-level precision with respect to a global terrestrial reference frame has contributed to significant advances in geophysics, seismology, atmospheric science, hydrology, and natural hazard science. Monitoring the changes in the positions or trajectories of GPS instruments on the Earth's land and water surfaces, in the atmosphere, or in space, is important for both theory and applications, from an improved understanding of tectonic and magmatic processes to developing systems for mitigating the impact of natural hazards on society and the environment. Besides accurate positioning, all disturbances in the propagation of the transmitted GPS radio signals from satellite to receiver are mined for information, from troposphere and ionosphere delays for weather, climate, and natural hazard applications, to disturbances in the signals due to multipath reflections from the solid ground, water, and ice for environmental applications. We review the relevant concepts of geodetic theory, data analysis, and physical modeling for a myriad of processes at multiple spatial and temporal scales, and discuss the extensive global infrastructure that has been built to support GPS geodesy consisting of thousands of continuously operating stations. We also discuss the integration of heterogeneous and complementary data sets from geodesy, seismology, and geology, focusing on crustal deformation applications and early warning systems for natural hazards.

Potential Seasonal Terrestrial Water Storage Monitoring from GPS Vertical Displacements: A Case Study in the Lower Three-Rivers Headwater Region, China

This study uses the observed vertical displacements of Global Positioning System (GPS) time series obtained from the Crustal Movement Observation Network of China (CMONOC) with careful pre- and post-processing to estimate the seasonal crustal deformation in response to the hydrological loading in lower three-rivers headwater region of southwest China, followed by inferring the annual EWH changes through geodetic inversion methods. The Helmert Variance Component Estimation (HVCE) and the Minimum Mean Square Error (MMSE) criterion were successfully employed. The GPS inferred EWH changes agree well qualitatively with the Gravity Recovery and Climate Experiment (GRACE)-inferred and the Global Land Data Assimilation System (GLDAS)-inferred EWH changes, with a discrepancy of 3.2-3.9 cm and 4.8-5.2 cm, respectively. In the research areas, the EWH changes in the Lancang basin is larger than in the other regions, with a maximum of 21.8-24.7 cm and a minimum of 3.1-6.9 cm.

Seasonal Mass Changes and Crustal Vertical Deformations Constrained by GPS and GRACE in Northeastern Tibet

Surface vertical deformation includes the Earth’s elastic response to mass loading on or near the surface. Continuous Global Positioning System (CGPS) stations record such deformations to estimate seasonal and secular mass changes. We used 41 CGPS stations to construct a time series of coordinate changes, which are decomposed by empirical orthogonal functions (EOFs), in northeastern Tibet. The first common mode shows clear seasonal changes, indicating seasonal surface mass re-distribution around northeastern Tibet. The GPS-derived result is then assessed in terms of the mass changes observed in northeastern Tibet. The GPS-derived common mode vertical change and the stacked Gravity Recovery and Climate Experiment (GRACE) mass change are consistent, suggesting that the seasonal surface mass variation is caused by changes in the hydrological, atmospheric and non-tidal ocean loads. The annual peak-to-peak surface mass changes derived from GPS and GRACE results show seasonal oscillations in mass loads, and the corresponding amplitudes are between 3 and 35 mm/year. There is an apparent gradually increasing gravity between 0.1 and 0.9 μGal/year in northeast Tibet. Crustal vertical deformation is determined after eliminating the surface load effects from GRACE, without considering Glacial Isostatic Adjustment (GIA) contribution. It reveals crustal uplift around northeastern Tibet from the corrected GPS vertical velocity. The unusual uplift of the Longmen Shan fault indicates tectonically sophisticated processes in northeastern Tibet.

Application of the Allan Variance to Time Series Analysis in Astrometry and Geodesy: A Review

The Allan variance (AVAR) was introduced 50 years ago as a statistical tool for assessing the frequency standards deviations. For the past decades, AVAR has increasingly been used in geodesy and astrometry to assess the noise characteristics in geodetic and astrometric time series. A specific feature of astrometric and geodetic measurements, as compared with clock measurements, is that they are generally associated with uncertainties; thus, an appropriate weighting should be applied during data analysis. In addition, some physically connected scalar time series naturally form series of multidimensional vectors. For example, three station coordinates time series X, Y, and Z can be combined to analyze 3-D station position variations. The classical AVAR is not intended for processing unevenly weighted and/or multidimensional data. Therefore, AVAR modifications, namely weighted AVAR (WAVAR), multidimensional AVAR (MAVAR), and weighted multidimensional AVAR (WMAVAR), were introduced to overcome these deficiencies. In this paper, a brief review is given of the experience of using AVAR and its modifications in processing astrogeodetic time series.

Properties of GPS noise at Japan islands before and after Tohoku mega-earthquake

The field of 3-components GPS signals is analyzed for the network of 1203 stations at the Japanese islands from January 30 up to March 26, 2011. This time interval includes just over 40 days of observation before the Tohoku mega-earthquake on March 11, 2011 (M = 9.0) and nearly 16 days of observation following this event. The signals from each station are three-component time series with time step 30 minutes. We study the statistical properties of the random fluctuations of GPS signals before and after the seismic catastrophe after transition to increments. The values of wavelet-based spectral index for GPS noise components for each station were estimated separately for pieces of records before and after seismic event. The maps of the noise spectral index are constructed as the values for grid size of 50 × 50 nodes covering the region under study, based on information from 10 stations closest to each node. These maps clearly extract the region of future seismic catastrophe by relatively high noise spectral index. The using of principal components method distinguished this spatial anomaly more explicitly. These results support the hypothesis that statistical properties of random fluctuations of geophysical fields carry important information about earthquake preparation.

Use of terrestrial laser scanning technology for long term high precision deformation monitoring

The paper presents a new methodology for high precision monitoring of deformations with a long term perspective using terrestrial laser scanning technology. In order to solve the problem of a stable reference system and to assure the high quality of possible position changes of point clouds, scanning is integrated with two complementary surveying techniques, i.e., high quality static GNSS positioning and precise tacheometry. The case study object where the proposed methodology was tested is a high pressure underground pipeline situated in an area which is geologically unstable.