Power-law Noise Model Research Articles

The Relationship of Time Span and Missing Data on the Noise Model Estimation of GNSS Time Series

Accurate noise model identification for GNSS time series is crucial for obtaining a reliable GNSS velocity field and its uncertainty for various studies in geodynamics and geodesy. Here, by comprehensively considering time span and missing data effect on the noise model of GNSS time series, we used four combined noise models to analyze the duration of the time series (ranging from 2 to 24 years) and the data gap (between 2% and 30%) effects on noise model selection and velocity estimation at 72 GNSS stations spanning from 1992 to 2022 in global region together with simulated data. Our results show that the selected noise model have better convergence when GNSS time series is getting longer. With longer time series, the GNSS velocity uncertainty estimation with different data gaps is more homogenous to a certain order of magnitude. When the GNSS time series length is less than 8 years, it shows that the flicker noise and random walk noise and white noise (FNRWWN), flicker noise and white noise (FNWN), and power law noise and white noise (PLWN) models are wrongly estimated as a Gauss–Markov and white noise (GGMWN) model, which can affect the accuracy of GNSS velocity estimated from GNSS time series. When the GNSS time series length is more than 12 years, the RW noise components are most likely to be detected. As the duration increases, the impact of RW on velocity uncertainty decreases. Finally, we show that the selection of the stochastic noise model and velocity estimation are reliable for a time series with a minimum duration of 12 years.

Improving the sensitivity of liver tumor classification in ultrasound images via a power-law shot noise model.

Power laws have been observed in various fields and help us understand natural phenomena. Power laws have also been observed in ultrasound images. This study used the power spectrum of the signal identified from the reflected ultrasound signal observed in ultrasonography based on the power-law shot noise (PLSN) model. The power spectrum follows a power law, which has a scaling factor that depends on the characteristics of the tissue in the region where the ultrasound wave propagates. To distinguish between a tumor and blood vessels in the liver, we propose a classification model that includes a scaling factor based on ResNet, a deep learning model for image classification. In a task to classify 6 types of tissue - a tumor, the inferior vena cava, the descending aorta, the Gleason sheath, the hepatic vein, and small blood vessels - tumor sensitivity increased 3.8% and the F-score for a tumor improved 2% while precision was maintained. The scaling factor obtained using the PLSN model was validated for classification of liver tumors.

Common-mode signals and vertical velocities in the greater Alpine area from GNSS data

Abstract. We study the time series of vertical ground displacements from continuous global navigation satellite system (GNSS) stations located in the European Alps. Our goal is to improve the accuracy and precision of vertical ground velocities and spatial gradients across an actively deforming orogen, investigating the spatial and temporal features of the displacements caused by non-tectonic geophysical processes. We apply a multivariate statistics-based blind source separation algorithm to both GNSS displacement time series and ground displacements modeled from atmospheric and hydrological loading, as obtained from global reanalysis models. This allows us to show that the retrieved geodetic vertical deformation signals are influenced by environment-related processes and to identify their spatial patterns. Atmospheric loading is the most important process, reaching amplitudes larger than 2 cm, but hydrological loading is also important, with amplitudes of about 1 cm, causing the peculiar spatial features of GNSS ground displacements: while the displacements caused by atmospheric and hydrological loading are apparently spatially uniform, our statistical analysis shows the presence of N–S and E–W displacement gradients. We filter out signals associated with non-tectonic deformation from the GNSS time series to study their impact on both the estimated noise and linear rates in the vertical direction. Taking into account the long time span of the time series considered in this work, while the impact of filtering on rates appears rather limited, the uncertainties estimated from filtered time series assuming a power law plus white noise model are significantly reduced, with an important increase in white noise contributions to the total noise budget. Finally, we present the filtered velocity field and show how vertical ground velocity spatial gradients are positively correlated with topographic features of the Alps.

Updated GNSS velocity solution in the Nordic and Baltic countries with a semi-automatic offset detection method

In Fennoscandia, the Glacial Isostatic Adjustment (GIA) causes intraplate deformations that affect the national static reference frames. The GNSS-determined velocities are important data for constraining the GIA models, which are necessary for maintaining the national reference frames. The Nordic Geodetic Commission (NKG) has published a dense and consistent GNSS station velocity solution in 2019, and we present now an update of the solution covering additional 3.5 years of data. Undetected positional offsets are the main factor decreasing the accuracy of the velocity estimates. We developed a method for the semi-automatic offset detection to improve the quality of our solution. The results show that we could correctly detect 74% of the manually determined offsets, and the undetected offsets would have caused a median 0.1 mm/y bias in trend. The method pointed out some otherwise unnoticed offsets and will decrease the need for manual analysis in the future. The updated velocity solution especially improves the velocity estimates of the newly established stations and the quality of the velocity estimates in Baltic countries. The formal uncertainties estimated using the power-law plus white noise model were at a median of 0.06 and 0.15 mm/y for horizontal and vertical velocities, respectively. However, we concluded that the systematic velocity uncertainties due to the reference frame alignment were approximately at the same level.

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.

Analysis of Noise and Velocity in GNSS EPN-Repro 2 Time Series

This study evaluates the EUREF Permanent Network (EPN) station position time series of approximately 200 GNSS stations subject to the Repro 2 reprocessing campaign in order to characterize the dominant types of noise and amplitude and their impact on estimated velocity values and associated uncertainties. The visual inspection on how different noise model represents the analysed data was done using the power spectral density of the residuals and the estimated noise model and it is coherent with the calculated Allan deviation (ADEV)-white and flicker noise. The velocities resulted from the dominant noise model are compared to the velocity obtained by using the Median Interannual Difference Adjusted for Skewness (MIDAS). The results show that only 3 stations present a dominant random walk noise model compared to flicker and powerlaw noise model for the horizontal and vertical components. We concluded that the velocities for the horizontal and vertical component show similar values in the case of MIDAS and maximum likelihood estimation (MLE), but we also found that the associated uncertainties from MIDAS are higher compared to the uncertainties from MLE. Additionally, we concluded that there is a spatial correlation in noise amplitude, and also regarding the differences in velocity uncertainties for the Up component.

Assessment of geocenter motion estimates from the IGS second reprocessing

We investigate geocenter motion time series derived from the combined solutions and six individual analysis center (AC) solutions of the International GNSS Service (IGS) second reprocessing campaign using the network shift approach, in terms of noise content, long-term trends, periodic and aperiodic variations. We assess these GNSS geocenter motion estimates by comparison with independent estimates from satellite laser ranging (SLR). The GNSS geocenter time series exhibit correlated noise which is better represented by a white plus power–law noise model in the X and Y directions, and by a white plus first-order autoregressive (or generalized Gauss–Markov) noise model in the Z direction. The GNSS geocenter time series include expected seasonal variations, but also spurious draconitic signals, particularly in the Z direction. GNSS annual geocenter motion estimates are in reasonable agreement with SLR estimates in the X and Y directions. In the Z direction, however, the annual signals derived from the IGS solutions disagree with SLR estimates, except for three particular ACs. This suggests that the different orbit modeling strategies used by these ACs may constitute an improvement over the conventional strategy employed by the other ACs. The background noise in GNSS and SLR geocenter time series finally appears to be correlated, suggesting that it might partly reflect real, aperiodic geocenter motion.

Correlated timing noise and high-precision pulsar timing: measuring frequency second derivatives as an example

Abstract We investigate the impact of noise processes on high-precision pulsar timing. Our analysis focuses on the measurability of the second spin frequency derivative $\ddot{\nu }$. This $\ddot{\nu }$ can be induced by several factors including the radial velocity of a pulsar. We use Bayesian methods to model the pulsar times-of-arrival in the presence of red timing noise and dispersion measure variations, modelling the noise processes as power laws. Using simulated times-of-arrival that both include red noise, dispersion measure variations, and non-zero $\ddot{\nu }$ values, we find that we are able to recover the injected $\ddot{\nu }$, even when the noise model used to inject and recover the input parameters are different. Using simulations, we show that the measurement uncertainty on $\ddot{\nu }$ decreases with the timing baseline T as Tγ, where γ = −7/2 + α/2 for power-law noise models with shallow power-law indices α (0 < α < 4). For steep power-law indices (α > 8), the measurement uncertainty reduces with T−1/2. We applied this method to times-of-arrival from the European Pulsar Timing Array and the Parkes Pulsar Timing Array and determined $\ddot{\nu }$ probability density functions for 49 millisecond pulsars. We find a statistically significant $\ddot{\nu }$ value for PSR B1937+21 and consider possible options for its origin. Significant (95 per cent C.L.) values for $\ddot{\nu }$ are also measured for PSRs J0621+1002 and J1022+1001, thus future studies should consider including it in their ephemerides. For binary pulsars with small orbital eccentricities, such as PSR J1909−3744, extended ELL1 models should be used to overcome computational issues. The impacts of our results on the detection of gravitational waves are also discussed.

Investigation of the noise properties at low frequencies in long GNSS time series

The accuracy by which velocities can be estimated from GNSS time series is mainly determined by the low-frequency noise, below 0.2–0.1 cpy, which are normally described by a power-law model. As GNSS observations have now been recorded for over two decades, new information about the noise at these low frequencies has become available and we investigate whether alternative noise models should be considered using the log-likelihood, Akaike and Bayesian information criteria. Using 110 globally distributed IGS stations with at least 12 years of observations, we find that for 80–90% of them the preferred noise models are still the power law or flicker noise with white noise. For around 6% of the stations, we found the presence of random-walk noise, which increases the linear trend uncertainty when taken into account in the stochastic noise model of the time series by about a factor of 1.5 to 8.4, in agreement with previous studies. Next, the Generalised Gauss–Markov with white noise model describes the stochastic properties better for 4% and 5% of the stations for the East and North component, respectively, and 13% for the vertical component. For these stations, the uncertainty associated with the tectonic rate is about 2 times smaller compared to the case when the standard power-law plus white noise model is used.

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.

Random deposition with a power-law noise model: Multiaffine analysis.

We study the random deposition model with power-law distributed noise and rare-event dominated fluctuation. In this model instead of particles with unit sizes, rods with variable lengths are deposited onto the substrate. The length of each rod is chosen from a power-law distribution P(l)∼l^{-(μ+1)}, and the site at which each rod is deposited is chosen randomly. The results show that for μ<μ_{c}=3 the log-log diagram of roughness, W(t), versus deposition time, t, increases as a step function, where the roughness in each interval acts as W_{loc}(t)≈t^{β_{loc}}. The local growth exponent, β_{loc}, is zero for μ=1. By increasing the μ exponent, the value of β_{loc} is increased. It tends to the growth exponent of the random distribution model with Gaussian noise, β=1/2, at μ_{c}=3. The fractal analysis of the height fluctuations for this model was performed by multifractal detrended fluctuation analysis algorithm. The results show multiaffinity behavior for the height fluctuations at μ<μ_{c} and the multiaffinity strength is greater for smaller values of the μ exponent.

GPS Time Series Analysis from Aboa the Finnish Antarctic Research Station

Continuous Global Positioning System (GPS) observations have been logged at the Finnish Antarctic research station (Aboa) since February 2003. The station is located in Dronning Maud Land, East Antarctica. Almost 5000 daily observation files have been archived based on yearly scientific expeditions. These files have not been fully analysed until now. This study reports for the first time on the consistent and homogeneous data processing and analysis of the 15-year long time series. Daily coordinates are obtained using Precise Point Positioning (PPP) processing based on two approaches. The first approach is based on the Kalman filter and uses the RTKLIB open source library to produce daily solutions by unconventionally running the filter in the forward and backward direction. The second approach uses APPS web service and is based on GIPSY scientific processing engine. The two approaches show an excellent agreement with less than 3 mm rms error horizontally and 6 mm rms error vertically. The derived position time series is analysed in terms of trend, periodicity and noise characteristics. The noise of the time series was found to be power-law noise model with spectral index closer to flicker noise. In addition, several periodic signals were found at 5, 14, 183 and 362 days. Furthermore, most of the horizontal movement was found to be in the North direction at a rate of 11.23 ± 0.09 mm/y, whereas the rate in the East direction was estimated to be 1.46 ± 0.05 mm/y. Lastly, the 15-year long time series revealed a movement upwards at a rate of 0.79 ± 0.35 mm/y. Despite being an unattended station, Aboa provides one of the most continuous and longest GPS time series in Antarctica. Therefore, we believe that this research increases the awareness of local geophysical phenomena in a less reported area of the Antarctic continent.

Common mode error in Antarctic GPS coordinate time-series on its effect on bedrock-uplift estimates

Spatially correlated common mode error (CME) always exists in regional, or-larger, global positioning system (GPS) networks. We applied independent component analysis (ICA) to GPS vertical coordinate time-series in Antarctica from 2010 to 2014 and made a comparison with the principal component analysis (PCA). Using PCA/ICA, the time-series can be decomposed into a set of temporal components and their spatial responses. We assume the components with common spatial responses are CME. An average reduction of ∼40 per cent about the root-mean-square (RMS) values was achieved in both PCA and ICA filtering. However, the common mode components obtained from the two approaches have different spatial and temporal features. ICA time-series present interesting correlations with modelled atmospheric and non-tidal ocean loading displacements. A white noise plus power law (PL) noise model was adopted in the GPS velocity estimation using maximum likelihood estimation analysis, with ∼55 per cent reduction of the velocity uncertainties after filtering using ICA. Meanwhile, spatiotemporal filtering reduces the amplitude of PL and periodic terms in the GPS time-series. Finally, we compare the GPS uplift velocities, after correction for elastic effects, with recent models of glacial isostatic adjustment (GIA). The agreements of the GPS observed velocities and four GIA models are generally improved after the spatiotemporal filtering, with a mean reduction of ∼0.9 mm yr−1 of the weighted RMS values, possibly allowing for more confident separation of various GIA model predictions.

A data-driven approach for denoising GNSS position time series

Global navigation satellite system (GNSS) datasets suffer from common mode error (CME) and other unmodeled errors. To decrease the noise level in GNSS positioning, we propose a new data-driven adaptive multiscale denoising method in this paper. Both synthetic and real-world long-term GNSS datasets were employed to assess the performance of the proposed method, and its results were compared with those of stacking filtering, principal component analysis (PCA) and the recently developed multiscale multiway PCA. It is found that the proposed method can significantly eliminate the high-frequency white noise and remove the low-frequency CME. Furthermore, the proposed method is more precise for denoising GNSS signals than the other denoising methods. For example, in the real-world example, our method reduces the mean standard deviation of the north, east and vertical components from 1.54 to 0.26, 1.64 to 0.21 and 4.80 to 0.72 mm, respectively. Noise analysis indicates that for the original signals, a combination of power-law plus white noise model can be identified as the best noise model. For the filtered time series using our method, the generalized Gauss–Markov model is the best noise model with the spectral indices close to − 3, indicating that flicker walk noise can be identified. Moreover, the common mode error in the unfiltered time series is significantly reduced by the proposed method. After filtering with our method, a combination of power-law plus white noise model is the best noise model for the CMEs in the study region.

Uncertainty in geocenter estimates in the context of ITRF2014

AbstractUncertainty in the geocenter position and its subsequent motion affects positioning estimates on the surface of the Earth and downstream products such as site velocities, particularly the vertical component. The current version of the International Terrestrial Reference Frame, ITRF2014, derives its origin as the long‐term averaged center of mass as sensed by satellite laser ranging (SLR), and by definition, it adopts only linear motion of the origin with uncertainty determined using a white noise process. We compare weekly SLR translations relative to the ITRF2014 origin, with network translations estimated from station displacements from surface mass transport models. We find that the proportion of variance explained in SLR translations by the model‐derived translations is on average less than 10%. Time‐correlated noise and nonlinear rates, particularly evident in the Y and Z components of the SLR translations with respect to the ITRF2014 origin, are not fully replicated by the model‐derived translations. This suggests that translation‐related uncertainties are underestimated when a white noise model is adopted and that substantial systematic errors remain in the data defining the ITRF origin. When using a white noise model, we find uncertainties in the rate of SLR X, Y, and Z translations of ±0.03, ±0.03, and ±0.06, respectively, increasing to ±0.13, ±0.17, and ±0.33 (mm/yr, 1 sigma) when a power law and white noise model is adopted.

Investigation of long-range dependencies in the stochastic part of daily GPS solutions

The long-range dependence (LRD) of the stochastic part of GPS-derived topocentric coordinates change (North, East, Up) results with relatively high autocorrelation values with a focus on self-similarity. One of the reasons for such self-similarity in the GPS time series are noises that are commonly recognised to prevail in the form of the flicker noise model. To prove the self-similarity of the stochastic part of GPS time series we used more than 130 ASG-EUPOS (active geodetic network EUPOS) stations from an area of Poland with a 5-year span of the daily topocentric coordinate changes. The deterministic part of time series was removed by means of the least-squares (LS) method, median absolute deviation (MAD) criterion and the sequential t-test algorithm, respectively. Then the self-similarity of the residue was proved by the results of the Ljung–Box test, whose values close to zero showed the dependence of the stochastic part of the GPS time series. The residue was analysed by means of the rescaled range (R/S) method with the H (Hurst) parameter and the detrended fluctuation analysis (DFA) with the scaling exponent α. Both H and α values ranged within assumed LRD limits of 0.5 and 1. This analysis was followed by noise investigation with a maximum likelihood estimation (MLE). The white plus the power-law noise models were assumed a priori, which gave us a spectral indices κ between − 0.4 and − 1.2 for all of the time series. It proved that fractional white noise outweighs other types of noises in GPS time series. Authors found here, that the LRD methods by omitting the noise amplitude data led to an underestimation of H values and their misinterpretation. The larger the omitted amplitude is, the greater the difference between the noise characters estimated with R/S values in comparison to the reference values of κ are. Some of these differences exceed even the value of 0.6, which may result in the estimation of false noise character in GPS data thus eliciting wrong conclusions.

Vertical crustal movement around the southeastern Tibetan Plateau constrained by GPS and GRACE data

Using measurements of continuous GPS stations and GRACE across the southeastern Tibetan Plateau, we found that the GRACE-derived vertical displacements are highly correlated with GPS-modeled vertical annual and semiannual displacements, which demonstrates that the vertical seasonal variations on ground surface are mainly caused by hydrological mass loading. After removing GRACE-modeled seasonal variations from the GPS time series, we used a stacking technique to filter out the common mode errors. Then we estimated the best noise model for the filtered time series, which can be best characterized by power law noise model. Finally, we determined the rates of vertical crustal movement of southeastern Tibetan Plateau. Our results suggest that the southeastern Tibetan Plateau is undergoing uplift, whereas the southern Sichuan–Yunnan fragment is subsiding with respect to the region to the north. The assumption of uniform extension throughout the crust does not explain the subsidence of southern Sichuan–Yunnan fragment.

Investigation of noises in the EPN weekly time series

The constantly growing needs of permanent stations’ velocities users cause their stability level to increase. To this research we included more than 150 stations located across Europe operating within the EUREF Permanent Network (EPN) with weekly changes in the ITRF2005 reference frame. The obvious long-range dependencies in the stochastic part of GPS time series were proven by Ljung-Box test. Moreover, the results of Detrended Fluctuation Analysis (DFA) comprised in 90 % in the long-time range dependency, indicated an autocorrelation of the signal with a possible coloured noise, similar to the flicker one. The noise in the GPS time series was analysed here with the Maximum Likelihood Estimation (MLE). The amplitudes of white noise for the white, flicker and random-walk combination vary between 0.5 to 3 mm for horizontal components and between 1.5 up to 9 mm for the vertical component. The amplitudes of white and flicker noise show a clear geographical latitude dependence with a correlation with its minimum centred at approximately 50°N. The assumption of white plus flicker plus randomwalk noise led to stations’ velocity errors from 0 to 3 mm/year for the flicker noise and from 0 up to 7 mm/year for the random-walk. The spectral indices for an approach assuming the occurrence of the white and power-law noise model in weekly time series varied between –1.7 to –0.2 and –1.4 to –0.2 for horizontal and vertical changes. The performed spatial filtering (weighted stacking method) resulted in decreasing the power-law amplitudes by averagely 1-2 mm/year -κ/4 and moving the spectral indices closer to zero, with amplitudes and indices median decreased by 5 %, compared to the time series before stacking. ARTICLE INFO

Effect of the span of Australian GPS coordinate time series in establishing an optimal noise model

The span of coordinate time series affects the determination of an optimal noise model. We analyzed position data recorded for 10 continuous Global Positioning System (GPS) sites from 1998.0 to mid-2009 on the Australian Plate to estimate the best noise model and thereafter obtain the true uncertainties of the velocity, employing the maximum likelihood estimation (MLE) method. MLE was employed to analyze the data in four ways. In the first two analyses, the noise was assumed to be a combination of flicker noise and white noise for the raw time series and spatially filtered time series. In the final two analyses, the spectral indices and amplitudes were simultaneously estimated for a power law noise plus white noise model for the raw time series and spatially filtered time series. We conclude that the noise model of GPS time series in Australia can be best described as the combination of flicker noise and white noise. Velocity uncertainties fall below ∼0.2 mm/yr when the time span exceeds ∼9.5 years. A comparison of noise amplitudes and maximum likelihood estimation values between the raw and spatially filtered time series suggests that traditional spatial filtering to remove common-mode errors might not be applicable to the raw time series of this region.

Velocity autocorrelation of a free particle driven by a Mittag-Leffler noise: fractional dynamics and temporal behaviors.

We investigate the dynamical phase diagram of the generalized Langevin equation of the free particle driven by a Mittag-Leffler noise and show critical curves and a critical value of the exponent parameter of the Mittag-Leffler function that mark different dynamical regimes. By considering that the modeling of a Mittag-Leffer memory kernel corresponds to a power-law second-order memory kernel, we show that the generalized Langevin equation of the velocity autocorrelation function (VACF) is transformed in a fractional Langevin equation. In the superdiffusive case our results exhibit oscillations and negative correlations of the VACF that are not provided by the usual power-law noise model.