Power-law Noise Research Articles

Sensitivity of GNSS to vertical land motion over Europe: effects of geophysical loadings and common-mode errors

We perform a statistical sensitivity analysis on a parametric fit to vertical daily displacement time series of 244 European Permanent GNSS stations, with a focus on linear vertical land motion (VLM), i.e., station velocity. We compare two independent corrections to the raw (uncorrected) observed displacements. The first correction is physical and accounts for non-tidal atmospheric, non-tidal oceanic and hydrological loading displacements, while the second approach is an empirical correction for the common-mode errors. For the uncorrected case, we show that combining power-law and white noise stochastic models with autoregressive models yields adequate noise approximations. With this as a realistic baseline, we report improvement rates of about 14% to 24% in station velocity sensitivity, after corrections are applied. We analyze the choice of the stochastic models in detail and outline potential discrepancies between the GNSS-observed displacements and those predicted by the loading models. Furthermore, we apply restricted maximum likelihood estimation (RMLE), to remove low-frequency noise biases, which yields more reliable velocity uncertainty estimates. RMLE reveals that for a number of stations noise is best modeled by a combination of random walk, flicker noise, and white noise. The sensitivity analysis yields minimum detectable VLM parameters (linear velocities, seasonal periodic motions, and offsets), which are of interest for geophysical applications of GNSS, such as tectonic or hydrological studies.

Analysis of short-term polarization stability using Allan variance

The application of Allan variance to characterize the stability of optical signals affected by stochastic polarization fluctuations and the identification of the underlying power law noise processes is explored. Allan variance can ease the comparison regarding polarization stability of optical systems affected by polarization noise and define a near-optimum integration interval to reveal trends. Examples of the application of Allan variance to optical systems with stable polarization conditions show that white noise and random walk terms can be observed. Additionally, experiments show that the three Stokes parameters can exhibit different statistical behaviors in the Brownian-noise regime. Allan analysis can easily be used to define, in real-time systematically, the denoising strategy in polarization-based sensing and for the optimization of polarization-sensitive optical systems instead of the conventional approach relying on heuristics or information criteria.

An Algorithm to Estimate the Power Spectral Density From Allan Deviation.

Complex architectures for wireless communications, digital electronics, and space-based navigation interlink several oscillator-based devices such as clocks, transponders, and synthesizers. Estimators characterizing their stability are critical for addressing the impact of random fluctuations (noise) on the overall system performance. Manufacturers typically specify this as an Allan/Hadamard Variance (AVAR/HVAR) profile in the time domain. However, stochastic processes constituting the noise are more thoroughly described in the frequency domain by the power spectral density function (PSD). Both are second-moment measures of the time series, but it is only possible to translate unambiguously from the PSD to the AVAR/HVAR, not vice versa, except in the case of a single noise type, a rather unrealistic case. This note presents an analytical method to generate an approximated PSD expressed as a set of power-laws defined in specific intervals in the frequency domain, starting from an AVAR/HVAR expressed as a set of power-laws in the time domain. The proposed algorithm is straightforward to implement, applicable to all noise types (and combinations thereof), and can be self-validated by reconstructing the corresponding AVAR/HVAR by direct computation. Coupling with well-established algorithms relying on the PSD for power-law noise generation, the ensuing method encompasses the capability for generating multicolored noise in end-to-end simulations, as demonstrated hereby for NASA's deep space atomic clock. We also report on the limitations of the algorithm and analytical expressions of the continuous version of the algorithm.

Noise analysis of irregularly sampled VLBI position time series using maximum likelihood estimation

Identification of the optimal noise model for space geodetic time series is crucial for various studies in geodynamics and geodesy. However, the position time series of Very Long Baseline Interferometry (VLBI) are typically irregularly sampled. There is no clear evidence from previous studies to support that the Maximum Likelihood Estimate (MLE) method can allow for stable and reliable noise estimates for irregularly sampled data. To address this issue, we first conducted simulation studies. Results indicate that when the number of data points in the VLBI position time series reaches 300, reliable noise estimations result can be obtained using the MLE. We also found that the higher proportion of white noise a station has, the less influences the data gaps would be on noise estimations. Second, we apply the MLE to investigate the noise properties in the 32 VLBI position time series. About 12 (37.5%) stations exhibit pure white noise, and 15 (46.9%) stations display a combination of white noise and flicker noise. For these stations, the median amplitudes of white noise in the East, North, and Up components are 3.9, 4.6, and 11.9 mm, respectively. Moreover, white and flicker noise amplitudes show a latitude-dependent pattern, with stations located in the southern hemisphere and low latitudes (0-30°N) in the northern hemisphere exhibiting higher noise amplitudes. The remaining approximately 5 stations (15.6%) have the optimal noise model of power law noise with the mean spectral index of −0.32, −0.36, and −0.53 for the East, North, and Up components, respectively. Simply assuming pure white noise can significantly affect velocity uncertainty estimates when compared to using a noise model that includes colored noise. The velocity uncertainty will be underestimated by a factor of 2.9 on average. The effects of a small proportion of colored noise in VLBI data on velocity uncertainty estimation can be ignored only when the time series length exceeds 32.6 years. Additionally, the choice of different noise model can lead to a maximum difference of 0.84 mm for the annual signal of the Up component.

Data-driven modeling of noise time series with convolutional generative adversarial networks ∗ ∗U.S. government work not protected by U.S. copyright.

Random noise arising from physical processes is an inherent characteristic of measurements and a limiting factor for most signal processing and data analysis tasks. Given the recent interest in generative adversarial networks (GANs) for data-driven modeling, it is important to determine to what extent GANs can faithfully reproduce noise in target data sets. In this paper, we present an empirical investigation that aims to shed light on this issue for time series. Namely, we assess two general-purpose GANs for time series that are based on the popular deep convolutional GAN architecture, a direct time-series model and an image-based model that uses a short-time Fourier transform data representation. The GAN models are trained and quantitatively evaluated using distributions of simulated noise time series with known ground-truth parameters. Target time series distributions include a broad range of noise types commonly encountered in physical measurements, electronics, and communication systems: band-limited thermal noise, power law noise, shot noise, and impulsive noise. We find that GANs are capable of learning many noise types, although they predictably struggle when the GAN architecture is not well suited to some aspects of the noise, e.g. impulsive time-series with extreme outliers. Our findings provide insights into the capabilities and potential limitations of current approaches to time-series GANs and highlight areas for further research. In addition, our battery of tests provides a useful benchmark to aid the development of deep generative models for time series.

Measuring the environment of a Cs qubit with dynamical decoupling sequences

We report the experimental implementation of dynamical decoupling on a small, non-interacting ensemble of up to 25 optically trapped, neutral Cs atoms. The qubit consists of the two magnetic-insensitive Cs clock states and , which are coupled by microwave radiation. We observe a significant enhancement of the coherence time when employing Carr-Purcell-Meiboom-Gill (CPMG) dynamical decoupling. A CPMG sequence with ten refocusing pulses increases the coherence time of 16.2(9) ms by more than one order of magnitude to 178(2) ms. In addition, we make use of the filter function formalism and utilise the CPMG sequence to measure the background noise floor affecting the qubit coherence, finding a power-law noise spectrum with . This finding is in very good agreement with an independent measurement of the noise in the intensity of the trapping laser. Moreover, the measured coherence evolutions also exhibit signatures of low-frequency noise originating at distinct frequencies. Our findings point toward noise spectroscopy of engineered atomic baths through single-atom dynamical decoupling in a system of individual Cs impurities immersed in an ultracold 87Rb bath.

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.

Porosity and conductivity in ballistic deposition with power-law distributed noise

The ballistic deposition model of rod-like particles with power-law distributed noise is studied by the Monte Carlo simulation. In this modified model instead of particles with fixed unit sizes, vertical rods whose length follows the power-law distribution [Formula: see text] are deposited, where [Formula: see text] denotes the power-law strength exponent. This deposition leads to porous rock structures with varying porosity and conductivity. The time evolution of the surface roughness and the porosity of the resulting structures are studied. The conductivity of the structures is calculated using the parallel resistors in the percolation model using a random walk algorithm. Finally, we discuss the relation between porosity, conductivity and the strength exponent of the power-law noise. The results show that the surface roughness increases as a pseudo-step function versus deposition time for [Formula: see text], which leads to an observable reduction in porosity and conductivity. By increasing the [Formula: see text] exponent, the growth exponent of [Formula: see text] for the Gaussian model appears. The conductivity increases as [Formula: see text] versus porosity, [Formula: see text], and remains constant for [Formula: see text] which the value of [Formula: see text] has been identified for BD.

Quantifying the effect of non-seasonal non-tidal loadings on background noise properties of GPS vertical displacements in mainland China

Global positioning system (GPS) vertical displacements measuring the elastic response of Earth’s crust to tectonic motion have been used to produce vertical velocity estimates. These accurate estimates require minimal contamination by non-tectonic deformations, mainly including atmospheric-oceanic-hydrologic mass loadings. However, the variability of background noise properties in GPS time-series caused by the removal of these loading deformations remains unclear. In this study, we investigate how the non-seasonal parts of two types of combined non-tidal loading deformations (i.e., atmospheric plus non-tidal oceanic loadings (AO) or AO plus hydrological loadings (AOH)) affect the background noise properties. The study region is focused on the mainland China, and the data are 258 GPS vertical land motion time-series from the Crustal Movement Observation Network of China (CMONOC). Before and after the loading correction, we found that the more satisfactory background noise model is white plus power-law noise. The AO correction achieves significant improvement in white noise estimates and drastically reduces power-law noise content. Meanwhile, the estimated spectral index shows the worse stationarity of background noise properties but can be alleviated by additional hydrological loading correction. Besides, the exploited GPS imaging method reveals the spatial heterogeneity of the background noise amplitude, spectral index, and velocity uncertainty.

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.

Synchronized oscillations of carbon nanotubes dispersed in solution

Although synchronized oscillations are found in a variety of systems and living organisms in nature, there has been no report on technologically important materials. We have observed by a fluorescence microscope that a large number of carbon nanotubes (CNTs) dispersed in an aqueous mixture of the surfactant and dye execute synchronized oscillations spontaneously. The movement was quantified to give a power spectrum, revealing a single, sharp synchronization peak at 20 Hz. It was found not to be affected nor created by external vibrations. The surfactant concentration dependence demonstrates that the Kuramoto model is applicable to describe the CNT synchronization. It is always associated with the power-law noise, indicating the presence of complex heterogeneous networks. These results suggest a highly cooperative form of the sparse CNT network connected with variable linkages.

Potential Contributors to CME and Optimal Noise Model Analysis in the Chinese Region Based on Different HYDL Models

Optimizing the noise model for global navigation satellite system (GNSS) vertical time series is vital to obtain reliable uplift (or subsidence) deformation velocity fields and assess the associated uncertainties. In this study, by thoroughly considering the effects of hydrological loading (HYDL) that dominates the seasonal fluctuations and common mode error (CME), we analyzed the optimal noise characteristics of GNSS vertical time series at 39 stations spanning from January 2011 to August 2019 in the Chuandian region, southeast of the Qinghai–Tibet Plateau. Our results showed that the optimal noise models without HYDL correction were white noise plus flicker noise (WN + FN), white noise plus power law noise (WN + PL), and white noise plus Gauss–Markov noise (WN + GGM), which accounted for 87%, 10%, and 3% of GNSS stations, respectively. By contrast, the optimal noise models at all stations were WN + FN and WN + PL after correction by different HYDLs. The correlation between CME and HYDL provided by the School and Observatory of Earth Sciences (EOST), namely EOST_HYDL, was 0.63~0.8 and the value of RMS reduction was 18.9~40.3% after removing EOST_HYDL time series from the CME, with a mean value of 31.8%, there is a good correlation and consistency between CME and EOST_HYDL. The absolute value of vertical velocity and its uncertainty with and without EOST_HYDL correction varied from 0.11 to 0.55 mm/a and 0 to 0.23 mm/a, respectively, implying that the effect of HYDL should not be neglected when performing optimal noise model analysis for GNSS vertical time series in the Chuandian region.

Bidirectional steering, entanglement and coherence of accelerated qubit–qutrit system with a stochastic noise

The influence of classical stochastic noise on three quantum correlation measurements of an accelerated qubit–qutrit system is examined. Bidirectional steerability, entanglement, and quantum coherence are investigated. The classical stochastic noises are investigated using either the Gaussian noise, the power-law noise, or a mixed Gaussian-power-law noise. Our results demonstrate that all quantum correlations decrease when the interaction time between the system and the noise channels advances or the acceleration parameter increases. The bidirectional steerability from qubit to qutrit and reversal is explored. The decoherence imposed by the Unruh effect of the observer is more impressive for qutrit than for the qubit subsystem. In a comparative study of bidirectional steerability, we conclude that the steering from qubit to qutrit is better than the steerability from qutrit to qubit. By mixing the various noises, no matter the order of subsystems, we can control the degradation of entanglement and failure in steerability among subsystems.

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.

Entropic uncertainty and quantum correlations dynamics in a system of two qutrits exposed to local noisy channels

We address the dynamics of the lower bound of geometric quantum discord and quantum-memory-assisted entropic uncertainty in a two-qutrit system when exposed to classical channels characterized by power-law (PL) and random telegraph (RT) noises. The system-channel coupling strategy is examined in two contexts: common qutrit-environment (CQE) and different qutrit-environment (DQE) configurations. We show that the geometric quantum discord functions remain anti-correlated with entropic uncertainty and decline as uncertainty appears in the system. The rate of entropic uncertainty appearance seems more prevalent than the decline rate of quantum discord function, suggesting that uncertainty causes the quantum correlations to fade in quantum systems. We find that non-local correlations estimated by the lower bound of geometric quantum discord are not destroyed even at the maximum entropic disorder and entropic uncertainty. In addition, the efficacy of entropic uncertainty and the lower bound is strongly influenced by the state’s purity factor, with the former being more robust at higher purity values and the latter at lower purity values. All the parameters impact entropic uncertainty, however, the mixedness of the state is noticed to greatly alter the generation of quantum memory. Besides, PL noise caused Markovian behavioral dynamics, and the RT noise allowed non-Markovian dynamics, while the latter remains more resourceful for the quantum correlations preservation and entropic uncertainty suppression. We also demonstrate how to model longer quantum correlations and provide optimal parameter settings for suppressing the dephasing and entropic uncertainty effects.

Spatiotemporal Filtering for Continuous GPS Coordinate Time Series in Mainland China by Using Independent Component Analysis

Continuous Global Positioning Systems (GPS) coordinate time series with a high spatiotemporal resolution, and provide a great opportunity to study their noise models and common mode errors (CMEs), thus making it possible to detect and analyse spatiotemporal characteristics of tectonic and non-tectonic signals in time series, and further to estimate a reliable and accurate velocity field of crustal movement in a region by removing CMEs and using the optimal noise model. In this paper, we used GPS coordinate time series from the Crustal Movement Observation Network of China (CMONOC) with an approximate decadal period from 2010 to 2020, to construct optimal noise models by fitting them with several noise combinations according to the Akaike information criterion (AIC). We further adopted independent component analysis (ICA) to extract CMEs and analysed their spatiotemporal characteristics, and then evaluated their effects on noise models and velocity uncertainties, and finally estimated a decennial velocity field of crustal movement with a higher signal-to-noise ratio (SNR) by applying the CME filtering and considering the optimal noise model in Mainland China. Our results show that optimal noise models are dominated by white noise (WN) plus flicker noise (FN) for both east and north components, and WN plus power law noise (PN) with spectral index close to −1 for up component, respectively. ICA filtering can remove the highly spatially correlated CMEs and decrease the mean RMSEs of the residual time series by about 40–60%, providing a more accurate velocity field with a higher SNR in Mainland China, accordingly.

Mean-field description of aging linear response in athermal amorphous solids

We study the linear response to strain in a mean-field elastoplastic model for athermal amorphous solids, incorporating the power-law mechanical noise spectrum arising from plastic events. In the ``jammed'' regime of the model, where the plastic activity exhibits a nontrivial slow relaxation referred to as aging, we find that the stress relaxes incompletely to an age-dependent plateau, on a timescale which grows with material age. We determine the scaling behavior of this aging linear response analytically, finding that key scaling exponents are universal and independent of the noise exponent $\ensuremath{\mu}$. For $\ensuremath{\mu}>1$, we find simple aging, where the stress relaxation timescale scales linearly with the age ${t}_{\mathrm{w}}$ of the material. At $\ensuremath{\mu}=1$, which corresponds to interactions mediated by the physical elastic propagator, we find instead a ${t}_{\mathrm{w}}^{1/2}$ scaling arising from the stretched exponential decay of the plastic activity. We compare these predictions with measurements of the linear response in computer simulations of a model jammed system of repulsive soft athermal particles, during its slow dissipative relaxation towards mechanical equilibrium, and find good agreement with the theory.

Relating Mandelbrot and Barnes-Jarvis Power Law Noise State Space Models using Partial Fractions

This paper addresses two state space models for generating approximate power law noise and shows that these models are related through partial fractions in frequency domain.

Demonstration of entanglement and coherence in GHZ-like state when exposed to classical environments with power-law noise

We investigate the dynamics and protection of entanglement and coherence in three non-interacting qubits that are initially prepared as mixed entangled GHZ-like state when coupled with external classical fields. A type of Gaussian noise, namely power-law (PL) noise, controls the external local fields in three different configurations with single, double, and triple noise sources. When influenced by a single PL noise source, the GHZ-class state remains partially entangled and coherent indefinitely, in contrast to the multiple PL noise sources. However, long-term non-local correlations and coherence are still possible in the presence of multiple noise sources. In classical fluctuating environments, the PL noise restricts the entanglement sudden death and birth phenomenon, thereby preventing the conversion of free states to resource states. In addition to the noise phase, parameter optimization in local fields regulates disorders, noise effects, and memory properties. Unlike the bipartite and tripartite W-type states, the GHZ-like state has shown positive traits of preserving entanglement and coherence in the presence of PL noise.

A New Spatial Filtering Algorithm for Noisy and Missing GNSS Position Time Series Using Weighted Expectation Maximization Principal Component Analysis: A Case Study for Regional GNSS Network in Xinjiang Province

Common Mode Error (CME) presents a kind of spatially correlated error that is widespread in regional Global Navigation Satellite System (GNSS) networks and should be eliminated during postprocessing of a GNSS position time series. Several spatiotemporal filtering methods have been developed to mitigate the effects of CME. However, such methodologies become inappropriate when missing and noisy data exists. In this research, we introduce a novel spatial filtering algorithm called Weighted Expectation Maximization Principal Component Analysis (WEMPCA) for detecting and removing CME from noisy GNSS position time series with missing values, among which formal errors of daily GNSS solutions are utilized to weight the input data. Compared with traditional PCA and the special case of EMPCA, simulation experiments demonstrate that the new WEMPCA algorithm always has outstanding performance over others. The WEMPCA algorithm was then successfully used to extract the CME from real noisy and missing GNSS position time series in Xinjiang province. Our results show that only the first principal component exhibits significant spatial response, with average values of 70.11%, 66.53%, and 52.45% for North, East, and Up (NEU) components, respectively, indicating that it represents the CME of this region. After removing CME, the canonical correlation coefficients and root mean square error of GNSS residual time series, as well as the amplitudes of power-law noises (PLN), are obviously decreased in all three directions. However, the white noise (WN) amplitudes are found to diminish exclusively in the North and East component, not in the Up components. Moreover, the average velocity differences before and after filtering CME are 0.19 mm/year, 0.03 mm/year, and −0.56 mm/year for the NEU components, respectively, indicating that CME has an influence on the GNSS station velocity estimation. The velocity uncertainty is also reduced by 43.51%, 38.64%, and 40.39% on average for the NEU components, respectively, implying that the velocity estimates are more reliable and accurate after removing CME. Therefore, we conclude that the new WEMPCA approach provides an efficient solution to detect and mitigate CME from the noisy and missing GNSS position time series.