Allan Variance Analysis Research Articles

Rotational motion investigation in seismology – remote sensing by an optical fiber seismograph

This paper aims to present data recorded by a three-axial Fiber-Optic Rotational Seismograph (FORS). The laboratory and field tests showed a high correlation coefficient above 99% between two seismographs signals during the external disturbance, with an amplitude ranging from 0.5 mrad/s to 1.2 rad/s. This is achieved by ensuring a 100 ns time synchronization in the systems. At the same time, the Allan Variance analysis was applied to determine the basic parameters of random errors of the presented seismographs. The performed analysis indicates the angular random walk of 35–45 nrad/s/√Hz and a bias instability below 50 nrad/s. FORS offers the widest dynamic range available, at 170 dB, which is crucial during rotational seismology exploration due to its wide range of interest from seismology to engineering applications. Finally, a field research is also presented during an explosion in a closed limestone quarry in the border area.

Calibration experiments for dual-comb IPDA XCO2 measurements using a variable pressure absorption cell

Measuring XCO2, the column-averaged dry-air mixing ratio of CO2 in the atmosphere, is essential for monitoring climate change and guiding mitigation efforts. Integrated Path Differential Absorption (IPDA) is a highly effective XCO2 measurement solution for space-borne lidar. However, the limited number of wavelengths limits its capabilities. IPDA, enhanced with Electro-Optical Frequency Comb (EOFC) technology, offers a robust solution. This study validates IPDA measurements using both asymmetric and symmetric Electro-Optic Dual-Comb Spectroscopy (EO-DCS) systems at 1572 nm. This indoor experimental system does not operate in a true lidar mode, but focuses on IPDA measurements of XCO2 through a transmission-style cell. We achieved atmospheric spectral transmittance by a variable pressure CO2 absorption cell, correlating pressures from 425 to 470 mb to XCO2 of 386.81–447.08 ppm. The XCO2 measurements of asymmetric/symmetric systems both exhibit excellent linearity. Their linear fittings yielded the Root Mean Square Error (RMSE) as low as 0.198/0.064 ppm, and the Mean Absolute Percentage Errors (MAPE) was as low as 0.0386%/0.0140%. This level of measurement performance has never been demonstrated in other works. Accuracy was assessed by comparing IPDA results with a pressure gauge, and precision was confirmed through repetitive measurements (1000 times over 10 min) and Allan variance analysis. Notably, our system operates effectively without frequency stabilization, showing tolerance to laser frequency drift—a significant advantage for IPDA applications. The entire experiment was conducted in a comparison between asymmetric and symmetric systems, especially in the analysis of absorption line residuals. We discuss the error transfer from phase to spectral transmittance and its implications and explained why the residuals in the asymmetric system were larger. These results underscore the superior performance of EO-DCS IPDA in measuring XCO2 and its promising potential for applications. To our knowledge, this is the only DCS IPDA calibration demonstration to date.

Noise assessment of cold atom interferometry gradiometry observations

Over the past few decades, the development of interferometric techniques utilizing cold atoms has significantly advanced, particularly in the realm of gravimetry. This paper explores the characterization of noise within atomic gradiometry systems, which is essential for enhancing measurement accuracy as new methodologies and tools are introduced. Specifically, we examine the noise components using advanced techniques such as least squares component estimation and Allan variance analysis. Applying these sophisticated methods to two distinct datasets, our results robustly indicate that white noise is the predominant and most resilient type of noise in gravity gradiometry measurements based on atomic interferometry, with variances of σP12=2.97×10−8 for dataset 1 and σP22=5.66×10−7 for dataset 2. The predominance of white noise simplifies the system dynamics and enhances the reliability of the estimation methods. Furthermore, our calculations yield an uncertainty in the gravity gradient resolution of σ Γ1 = 0.13 E (where 1 E = 10−9 s−2) and σ Γ2 = 3.55 E over time, underscoring the high precision achievable with current interferometric techniques.

Active near‐infrared laser heterodyne system based on supercontinuum laser source for gas measurement

AbstractThe performance of an active near‐infrared laser heterodyne system using a supercontinuum light source as the light signal for trace species detection is experimentally demonstrated. The characteristics of the supercontinuum light source and the data processing methods used in heterodyne detection are described. The measurement of methane (CH₄) and carbon dioxide (CO₂) absorption spectra was carried out in the laboratory to evaluate the active laser heterodyne system, and high‐resolution absorption spectra of CO₂ and CH₄ were recorded simultaneously. The stability of the active laser heterodyne system is analyzed using Allan variance analysis of long‐term observation data. The active near‐infrared laser heterodyne detection system demonstrated in this paper has great potential for the development of industrial parks' gas monitoring.

Accurate drift-invariant single-molecule force calibration using the Hadamard variance

Single-molecule force spectroscopy (SMFS) techniques play a pivotal role in unraveling the mechanics and conformational transitions of biological macromolecules under external forces. Among these techniques, multiplexed magnetic tweezers (MT) are particularly well suited to probe very small forces, ≤1 pN, critical for studying noncovalent interactions and regulatory conformational changes at the single-molecule level. However, to apply and measure such small forces, a reliable and accurate force-calibration procedure is crucial. Here, we introduce a new approach to calibrate MT based on thermal motion using the Hadamard variance (HV). To test our method, we perform bead-tether Brownian dynamics simulations that mimic our experimental system and compare the performance of the HV method against two established techniques: power spectral density (PSD) and Allan variance (AV) analyses. Our analysis includes an assessment of each method’s ability to mitigate common sources of additive noise, such as white and pink noise, as well as drift, which often complicate experimental data analysis. We find that the HV method exhibits overall similar or higher precision and accuracy, yielding lower force estimation errors across a wide range of signal/noise ratios (SNRs) and drift speeds compared with the PSD and AV methods. Notably, the HV method remains robust against drift, maintaining consistent uncertainty levels across the entire studied SNR and drift speed spectrum. We also explore the HV method using experimental MT data, where we find overall smaller force estimation errors compared with PSD and AV approaches. Overall, the HV method offers a robust method for achieving sub-pN resolution and precision in multiplexed MT measurements. Its potential extends to other SMFS techniques, presenting exciting opportunities for advancing our understanding of mechanosensitivity and force generation in biological systems. To make our methods widely accessible to the research community, we provide a well-documented Python implementation of the HV method as an extension to the Tweezepy package.

In Situ High-Precision Measurement of Deep-Sea Dissolved Methane by Quartz-Enhanced Photoacoustic and Light-Induced Thermoelastic Spectroscopy.

Rapid and accurate realization of in situ analysis of deep-sea dissolved gases imperative to the study of ecological geology, oil and gas resource exploration, and global climate change. Herein, we report for the first time the deep-sea dissolved methane (CH4) in situ sensor based on quartz-enhanced photoacoustic and light-induced thermoelastic spectroscopy. The developed sensor system has a volume of φ120 mm × 430 mm and a power consumption of 7.6 W. The sensor, in the manner of frequency division multiplexing, is able to simultaneously measure the photoacoustic signals and light-induced thermoelastic signals, which can accurately correct laser-intensity induced influence on concentration. The spectral response of CH4 concentration varying from 0.01 to 5% is calibrated in detail based on the pressure and temperature in the application environment. The trend of the photoacoustic signal of CH4 at different water molecule (H2O) concentrations is investigated. An Allan variance analysis of several hours demonstrates a minimum detection limit of 0.21 ppm for the CH4 spectrometer. The sensor combined with the gas-liquid separation and enrichment unit is integrated into a compact marine standalone system. Since the specifically designed photoacoustic cell has a volume of only 1.2 mL, the time response for dissolved CH4 detection is reduced to 4 min. Furthermore, the sensor is successfully deployed in the vicinity of the "HaiMa" cold seeps at 1380 m underwater in the South China Sea, completing three consecutive days of measurements of dissolved CH4.

Signal enhancement of the gas detection based on quartz-enhanced photothermal spectroscopy technology.

This paper presents an improved gas sensor based on the dual-excitation of quartz-enhanced photothermal spectroscopy (QEPTS) using a single quartz tuning fork (QTF) for signal detection. The silver coating on one side of the QTF was chemically etched to increase the laser power interacted with QTF for QEPTS signal excitation. By etching the silver coating on one side of QTF, the reflection structure between the silver coating of the other side of QTF and the external flat mirror was established. The device uses an absorption gas cell with an optical range length of 3 m, making the laser beam interact with the gas more completely and posing more gas concentration information. Acetylene was selected as the target gas to verify the performance of the sensor. The experimental results show that the signal amplitude with a flat mirror was 1.41 times that without a flat mirror, and 2.47 times that of traditional QEPTS sensor. The system has a minimum detection limit (MDL) of 1.10 ppmv, corresponding to a normalized noise equivalent absorption coefficient (NNEA) of 7.14 × 10-9 cm-1·W·Hz-1/2. Allan variance analysis results show that when the integration time is 700 s, the MDL of the system is 0.21 ppmv. The proposed gas sensor can play an important role on detecting trace gas in many fields.

Modelling inertial measurement unit error parameters for an unmanned air vehicle

This paper demonstrates a study that focuses on the modeling, design, and realization of an Inertial Measurement Unit (IMU) component for the use of Unmanned Aerial Vehicles (UAV). The experimental data is obtained by multiple flights conducted by the realized UAV (Teknofest–SEMRUK team UAV). The structure is remodeled for increasing the accuracy, and performance of the UAV after the conducted flights. Noise parameters are estimated throughout the Allan variance analysis. MEMS technology-based capacitive-type accelerometers and gyroscopes are preferred. This paper also discusses the error types and compares the real data with the modeled simulation data. Systematic errors of the inertial sensors are simulated according to their datasheet parameters. Sensor filters and noise are modeled and they are also implemented in the simulation. Simulation results and UAV measurements are compared to observe the efficiency of modeling. A complementary filter is presented and combined with a magnetometer, accelerometer, and gyroscope to obtain the ultimate design. The comparison showed a satisfactory agreement among the complementary filter measurements and UAV measurements in the stable position and the results presented.

Simultaneous measurement of 13C-,18O-, and 17O- isotopes of CO2 using a compact mid-infrared hollow waveguide gas sensor

A compact gas sensor has been developed for the first time employing a quantum cascade laser (QCL) with a central wavelength of 4.32 μm, combined with direct absorption spectroscopy technology and a hollow waveguide. The sensor could measure the concentrations of four isotopic molecules (16O12C16O, 18O12C16O, 16O13C16O, and 17O12C16O) in CO2 gas simultaneously, thereby obtaining the relative isotope ratios of δ13C, δ18O, and δ17O. By adjusting the incident light path, the anomalous absorption peaks in the system have been effectively suppressed, and a compact feedback function generator has been integrated to ensure control of the laser output wavelength to eliminate the loss of measurement precision caused by wavelength fluctuations. Under the two conditions of free-running and wavelength-controlled, a standard CO2 sample with a concentration of 5 % was measured continuously. The results showed that under wavelength-controlled conditions, the precision of the four isotopic molecule concentration measurements was improved by a minimum of 1.8 times, and the measurement precision could be further improved after filtering. According to the Allan variance analysis, the detection precision for δ13C, δ18O, and δ17O were 0.7 ‰, 0.62 ‰, and 1.58 ‰, respectively, at the optimal integration time of 98 seconds. This study has the potential for respiratory diagnosis to detect simultaneously the values of δ13C, δ18O, and δ17O in exhaled samples.

High-Sensitivity Differential Helmholtz Photoacoustic System Combined with the Herriott Multipass Cell and Its Application in Seed Respiration.

A highly sensitive photoacoustic detection system using a differential Helmholtz resonator (DHR) combined with a Herriott multipass cell is presented, and its implementation to sub-ppm level carbon dioxide (CO2) detection is demonstrated. Through the utilization of erbium-doped optical fiber amplifier (EDFA), the laser power was amplified to 150 mW. Within the multipass cell, a total of 22 reflections occurred, contributing to an impressive 33.6 times improvement in the system sensitivity. The normalized noise equivalent absorption coefficient (NNEA) was 8.64 × 10-11 cm-1·W·Hz-1/2 [signal-to-noise ratio, (SNR) = 1] and according to the Allan variance analysis, a minimum detection limit of 500 ppb could be achieved for CO2 at 1204 s, which demonstrates the long-term stability of the system. The system was applied to detect the respiration of rice and upland rice seeds. It is demonstrated that the system can monitor and distinguish the respiration intensity and respiration rate of different seeds in real time.

Vertical Distribution Mapping for Methane Fugitive Emissions Using Laser Path-Integral Sensing in Non-Cooperative Open Paths.

Observing the vertical diffusion distribution of methane fugitive emissions from oil/gas facilities is significant for predicting the pollutant's spatiotemporal transport and quantifying the random emission sources. A method is proposed for methane's vertical distribution mapping by combining the laser path-integral sensing in non-non-cooperative open paths and the computer-assisted tomography (CAT) techniques. It uses a vertical-plume-mapping optical path configuration and adapts the developed dynamic relaxation and simultaneous algebraic reconstruction technique (DR-SART) into methane-emission-distribution reconstruction. A self-made miniaturized TDLAS telemetry sensor provides a reliable path to integral concentration information in non-non-cooperative open paths, with Allan variance analysis yielding a 3.59 ppm·m sensitivity. We employed a six-indexes system for the reconstruction performance analysis of four potential optical path-projection configurations and conducted the corresponding validation experiment. The results have shown that that of multiple fan-beams combined with parallel-beam modes (MFPM) is better than the other optical path-projection configurations, and its reconstruction similarity coefficient (ε) is at least 22.4% higher. For the different methane gas bag-layout schemes, the reconstruction errors of maximum concentration (γm) are consistently around 0.05, with the positional errors of maximum concentration (δ) falling within the range of 0.01 to 0.025. Moreover, considering the trade-off between scanning duration and reconstruction accuracy, it is recommended to appropriately extend the sensor measurement time on a single optical path to mitigate the impact of mechanical vibrations induced by scanning motion.

Simultaneous dual-harmonic detection for LITES with a multi-pass configuration based on combined flexural modes of a quartz tuning fork

The simultaneous dual-harmonic detection for light-induced thermoelastic spectroscopy (LITES) with a multi-pass configuration based on combined flexural modes of a quartz tuning fork (QTF) is demonstrated. To enhance the photothermal energy absorbed by a custom QTF, the laser beam is designed to pass through the QTF multiple times. With dual-frequency modulation methodology, the second harmonic (2f_fun) signal and the first harmonic (1f_ove) signal are simultaneously inspired at resonant frequencies of fundamental and overtone flexural modes of the QTF, respectively. Thus, precise light intensity and gas concentration can be retrieved concurrently with corresponding 1f and 1f-normalized 2f signals using frequency division multiplexing technology. In comparison to traditional LITES under the same condition sensor, our triple-pass configuration performs 1.84 times better in signal-to-noise ratio (SNR). According to Allan variance analysis, 1f-normalized 2f signal enables a longer integration time of 1100 s and a lower detection limit of 0.07 ppm. The corresponding normalized noise equivalent absorption coefficient (NNEA) is 0.8 × 10−9 cm−1W/Hz1/2.

Measurement of Ammonia Leakage by TDLAS in Mid-Infrared Combined with an EMD-SG Filter Method

Ammonia (NH3) has been used as the main refrigerant in various types of cold storages, exceeding 90% of cooling applications. However, NH3 leakage occurs frequently due to damage of the refrigeration compressor, resulting in environmental pollution and intoxication accidents. Therefore, reliable and highly sensitive NH3 detection is a critical requirement in cold storage. In this paper, tunable diode laser absorption spectroscopy (TDLAS) is used to measure NH3 leakage, where a new denoising method combining empirical mode decomposition with the Savitzky-Golay smoothing algorithm (EMD-SG) is proposed to improve the signal-to-noise ratio (SNR) of absorbance signals. The results show that a 5.5 times improvement of SNR and 86% reduction of the linear fitting error in calibration is obtained; furthermore, this system achieves a detection limit of 0.2 ppm at an average time of 69 s by Allan variance analysis. Hence, it can be concluded that the EMD-SG algorithm can effectively improve the performance of TDLAS systems to meet the requirements for NH3 leakage detection in cold storages.

Near infrared ammonia sensing system based on on‐axis cavity‐enhanced absorption spectroscopy using board‐level mode‐locked circuit

AbstractWe proposed a near‐infrared gas sensing system based on mode‐locked cavity‐enhanced absorption spectroscopy. A distributed feedback fiber laser (DFB‐FL) with a central wavelength of 1550 nm was employed as the light source, coupling with a Fabry–Perot (F‐P) cavity with an effective optical path length of 42.7 m served as the gas cell. By utilizing a homemade mode‐locked electric circuit and Pound–Drever–Hall (PDH) technique, the mode‐locking between the DFB‐FL and F‐P cavity was achieved with long‐term fluctuation of 3.2%. System performance was verified using ammonia as the analyte gas with a target line at 1550.17 nm. Allan variance analysis revealed that the minimum detection limit was 15.8 ppm with an averaging time of 216 s, corresponding to an absorption coefficient of 2.3 × 10−7 cm−1.

A high-precision online compensation method for random errors of optical gyroscope

The traditional Allan variance cannot reflect the time-varying characteristics of random error signals, and the dynamic Allan variance cannot simultaneously guarantee both the signal dynamic tracking performance and confidence of variance estimation due to adopting the sliding window with fixed window length to intercept and segment random error signals. To address these issues, a dynamic Allan variance analysis method with adaptive window length is proposed. Based on the idea of piecewise stability of time-varying random error signals, a sliding window is introduced to intercept and segment the signal to be analyzed along the time axis, and the Allan variance analysis results of all signal segments are synthesized to track and describe the time-varying characteristics of the signal. Then according to the changes in the statistical characteristics of the random error signal, an adaptive factor is constructed to adaptively adjust the length of the sliding window, thereby guaranteeing both the signal dynamic tracking performance and confidence of variance estimation at the same time. On this basis, this method is applied to the online compensation of optical gyroscope random errors, and then a high-precision online compensation method for random errors of optical gyroscope is proposed. The experimental results show that the proposed method can achieve high-precision online compensation for the time-varying random error signals of optical gyroscope measured in complex environment such as uncertainty and noise interference.

Modeling of nonlinear and nonstationary stochasticity for atomic ensembles

This paper addresses the problem of stochastic modeling of atomic ensembles under multi-source noise and makes the model interpretable. First, based on Itô’s lemma and Allan variance analysis (ITÔ-AVAR), an approach is proposed to model nonstationary stochastic submodels of atomic ensembles. On this basis, the variance decomposition and nonlinear optimization algorithms are utilized to hybridize modeling atomic ensembles with nonlinear and nonstationary properties. Second, an Itô’s lemma dynamic allan variance analysis (ITÔ-DAVAR) approach is developed for online modeling of atomic ensembles. Further, an atomic ensembles sensitivity enhancement scheme based on the proposed approach is given, which effectively promotes the progress of quantum instrument engineering. Finally, the proposed scheme are deployed in the optical pumping magnetometer and spin-exchange relaxation-free comagnetometer, respectively, while experimentally verifying the sensitivity of the spin-exchange relaxation-free comagnetometer reaches 5.36×10−6degs−1Hz−1/2.

Methane gas sensor based on a novel dual‐path multipass cell

AbstractWe have developed a novel dual‐path multipass cell (DP‐MPC) for trace gas detection. The sealed Herriott cell, measuring 16 × 16 × 41.5 cm, has two input/output coupling holes. By adjusting the mirror position and incident angle of the beam, optical paths of 105 and 37.5 m can be obtained, respectively. To test its performance, we setup a methane (CH4) concentration detection system by combining the DP‐MPC with a 1.65 μm distributed feedback laser and carried out a methane gas detection experiment. The Allan variance analysis showed that the limit of detection for the two optical paths were 6 and 30 ppb, respectively.

Analysis of temporal structure of laser chaos by Allan variance.

Allan variance has been widely utilized for evaluating the stability of the time series generated by atomic clocks and lasers, in time regimes ranging from short to extremely long. This multiscale examination capability of the Allan variance may also be beneficial in evaluating the chaotic oscillating dynamics of semiconductor lasers- not just for conventional phase stability analysis. In the present study, we demonstrated Allan variance analysis of the complex time series generated by a semiconductor laser with delayed feedback, including low-frequency fluctuations (LFFs), which exhibit both fast and slow dynamics. While the detection of LFFs is difficult with the conventional power spectrum analysis method in the low-frequency regime, the Allan variance approach clearly captured the appearance of multiple time-scale dynamics, such as LFFs. This study demonstrates that Allan variance can help in understanding and characterizing diverse laser dynamics, including LFFs, spanning a wide range of timescales.

A miniaturized multipass cell for measurement of O2 concentration in vials based on TDLAS

Tunable diode laser absorption spectroscopy (TDLAS) is an effective method to inspect possible leakage during filling. However, the sensitivity of TDLAS is confined by the optical path length of the vial headspace. In this letter, the development of a miniaturized multipass cell (MMPC) for improving the sensitivity of laser spectrometer for detecting the O2 content in vial headspaces is presented. Based on analytical vector calculation, a few types of MMPC have been designed and their corresponding beam distributions are derived. The beam spots on the mirrors are compressed in a line-shape parallel with the vial, given the limitation of vial headspace. The simulated configuration with the minimum open path-length is then used for an experimental validating of the MMPC. The horizontal position of the vial has only little influence on the output laser power in the range of -0.5–0.5 mm relative to the optimum one, which is acceptable for practical applications. Wavelength modulation spectroscopy of calibration-free TDLAS was employed to measure the O2 concentration. It is demonstrated that the signal-to-noise ratio improves by 6 times compared to single pass configuration. The R2 of linear fitting the O2 concentration against signal amplitude could reach 0.998 in the range of 0–21% O2. According to the Allan variance analysis, the minimum detection sensitivity of the system is estimated at 0.05% at 141 s.

A two-stage Kalman filtering approach using GNSS and smartphone sensors for seismogeodetic applications

Recently, an enormous number of concepts have been introduced as new insights into seismogeodetic tools and algorithms. Integration of Global Navigation Satellite Systems (GNSS) with accelerometers is a well-known and widely used approach for estimating co-seismic displacements. In this study, we present a two-stage Kalman Filtering (TSKF) approach for seismogeodetic applications with low-cost accelerometers embedded in smartphones. The TSKF is initialized after a cost-effective and magnetometer-aided attitude estimation of accelerations in the sensor frame with respect to the local geodetic (topocentric) frame using the Singular Value Decomposition (SVD) method. In the first stage of TSKF, correlated noise processes in GNSS positions identified by the Allan Variance (AV) analysis are estimated with a linear Kalman Filter (LKF) and displacements are reflected in the innovation sequence of the LKF. In the second stage, the innovation sequence of the first stage LKF is integrated with the smartphone accelerations by a multi-rate Kalman Filter. The performance of TSKF was evaluated with harmonic motion and earthquake experiments using a single-axis shake table in a high multipath environment. Results of harmonic motion experiments showed that TSKF can produce mm-level amplitude deviations however the effect of phase shift decreases the performance and may cause mm to cm-level root mean square (RMS). On the other hand, the performance of the TSKF in earthquake experiments resulted in mm-level RMS values. In addition, comparisons of TSKF with an existing method i.e. the Multi-Rate Kalman Filter (MKF) show that TSKF may significantly increase the performance of displacement estimations.