Frequency Ratio Research Articles

Spatial mapping of the seismic vulnerability index in Kathmandu Valley: insight from dominant frequency and amplification factor

Abstract Kathmandu Valley situated in Nepal has experienced several earthquakes in the past. The expanding population of Kathmandu Valley is driving a significant acceleration in building construction reaching record levels and posing risks to both human lives and the economy, despite this surge in development. This study aimed to spatially map the dominant frequency, amplification factor, and vulnerability index map of Kathmandu Valley, Nepal using ArcGIS. This paper presents the nonlinear response analysis of 225 boreholes using the Gorkha earthquake as input motion and DEEPSOIL software for determining the amplification ratio and dominant frequency. The results revealed a range of values for dominant frequency from 0.08 to 7.65 Hz and soil amplification ratio from 0.13 to 8.14. Furthermore, the seismic vulnerability index was derived from the dominant frequency and soil amplification ratio ranging from 0.22 to 21.84. The resulting maps acquired through this study enable the decision-makers and stakeholders, to prioritize resources and implement targeted measures for enhancing seismic resilience in the region.

Experiments on High-Resolution Digitizer Accuracy in Measuring Voltage Ratio and Phase Difference of Distorted Harmonic Waveforms above 2 kHz

High-resolution multi-channel digitizers are used extensively for precision low voltage measurements in numerous applications and allow the simultaneous measurement of voltage magnitude ratio and phase difference between two different waveforms in power system applications. Delta–sigma-based analog-to-digital conversion enables the use of sampling frequencies in the range of megahertz, which provides accurate measurement bandwidths for transformed high-frequency, high-voltage signals. With the increased use of power electronic converters contributing to high-frequency harmonic emissions in power systems, there is a growing interest in developing calibration systems to measure voltage ratio and phase difference of distorted fundamental frequency waveforms consisting of superimposed, high-frequency harmonics. However, information regarding the accuracy of the high-resolution digitizers in the measurement of distorted voltage waveforms is limited as characterization is typically performed under sinusoidal voltage waveform conditions. This paper presents the details of the accuracy characterization of a 24-bit resolution digitizer under both sinusoidal and distorted waveform conditions for measuring complex voltage ratio and phase error for frequencies up to 10 kHz. The detailed experimental results and the measurement uncertainty evaluations show that increased voltage ratio and phase difference errors should be allocated when these high-resolution digitizers are used to measure distorted voltage waveforms. The estimated expanded uncertainties of complex voltage ratio measurement and phase error measurement for harmonic frequencies up to 10 kHz are ±260 ppm and ±100 µrad, respectively.

Landslide Susceptibility Assessment by Machine Learning and Frequency Ratio Methods Using XRAIN Radar-Acquired Rainfall Data

This study is an efficiency comparison between four methods for the production of landslide susceptibility maps (LSMs), which include random forest (RF), artificial neural network (ANN), and logistic regression (LR) as the machine learning (ML) techniques and frequency ratio (FR) as a statistical method. The study area is located in the Southern Hiroshima Prefecture in western Japan, a locality known to suffer from rainfall-induced landslide disasters, the most recent one in July 2018. The landslide conditioning factors (LCFs) considered in this study are lithology, land use, altitude, slope angle, slope aspect, distance to drainage, distance to lineament, soil class, and mean annual precipitation. The rainfall LCF data comprise XRAIN (eXtended RAdar Information Network) radar records, which are novel in the task of LSM production. The accuracy of the produced LSMs was calculated with the area under the receiver operating characteristic curve (AUROC), and an automatic hyperparameter tuning and result comparison system based on AUROC scores was utilized. The calculated AUROC scores of the resulting LSMs were 0.952 for the RF method, 0.9247 for the ANN method, 0.9016 for the LR method, and 0.8424 for the FR. It is also noteworthy that the ML methods are substantially swifter and more practical than the FR method and allow for multiple and automatic experimentations with different hyperparameter settings, providing fine and accurate outcomes with the given data. The results evidence that ML techniques are more efficient when dealing with hazard assessment problems such as the one exemplified in this study. Although the conclusion that the RF method is the most accurate for LSM production as found by other authors in the literature, ML method efficiency may vary depending on the specific study area, and thus the use of an automatic multi-method LSM production system with hyperparameter tuning such as the one utilized in this study is advised. It was also found that XRAIN radar-acquired mean annual precipitation data are effective when used as an LCF in LSM production.

Assessing Machine Learning and Statistical Methods for Rock Glacier‐Based Permafrost Distribution in Northern Kargil Region

ABSTRACTEstimating permafrost distribution in high‐mountain areas is challenging. In these situations, rock glaciers, provide valuable insights into permafrost distribution and are often used as proxies for identifying permafrost occurrence. Integrating various climatological and topographical conditioning factors with rock glaciers enables inferring the distribution of permafrost in these environments. This study utilized three machine learning models such as random forest (RF), support vector (SVM), and artificial neural network (ANN), and one statistical model, namely, the frequency ratio (FR), to assess the permafrost probability over the northern Kargil region of Indian Himalayas. Among 198 rock glaciers identified through high‐resolution images from Google Earth, 70% are used as training dataset, rest 30% as testing dataset. The study considered eight factors: slope, aspect, elevation, curvature, mean annual land surface temperature (MA‐LST), mean annual normalized difference snow index (MA‐NDSI), mean annual normalized difference water index (MA‐NDWI), and lithology for mapping. Furthermore, the SHapley Additive exPlanations (SHAP) test assessed the variable importance for model performance. The results revealed that the RF model performs best for permafrost probability mapping, followed by the SVM, FR, and ANN models. The study also found that 11% of the total geographic area has a high and very high probability of permafrost occurrence.

Correction: Landslide Susceptibility Mapping Using Probabilistic Frequency Ratio and Shannon Entropy for Chamoli, Uttarakhand Himalayas

Correction: Landslide Susceptibility Mapping Using Probabilistic Frequency Ratio and Shannon Entropy for Chamoli, Uttarakhand Himalayas

24-h continuous non-invasive multiparameter home monitoring of vitals in patients with Rett syndrome by an innovative wearable technology: evidence of an overlooked chronic fatigue status.

Sleep is disturbed in Rett syndrome (RTT), a rare and progressive neurodevelopmental disorder primarily affecting female patients (prevalence 7.1/100,000 female patients) linked to pathogenic variations in the X-linked methyl-CpG-binding protein 2 (MECP2) gene. Autonomic nervous system dysfunction with a predominance of the sympathetic nervous system (SNS) over the parasympathetic nervous system (PSNS) is reported in RTT, along with exercise fatigue and increased sudden death risk. The aim of the present study was to test the feasibility of a continuous 24 h non-invasive home monitoring of the biological vitals (biovitals) by an innovative wearable sensor device in pediatric and adolescent/adult RTT patients. A total of 10 female patients (mean age 18.3 ± 9.4 years, range 4.7-35.5 years) with typical RTT and MECP2 pathogenic variations were enrolled. Clinical severity was assessed by validated scales. Heart rate (HR), respiratory rate (RR), and skin temperature (SkT) were monitored by the YouCare Wearable Medical Device (Accyourate Group SpA, L'Aquila, Italy). The average percentage of maximum HR (HRmax%) was calculated. Heart rate variability (HRV) was expressed by consolidated time-domain and frequency-domain parameters. The HR/LF (low frequency) ratio, indicating SNS activation under dynamic exercise, was calculated. Simultaneous continuous measurement of indoor air quality variables was performed and the patients' contributions to the surrounding water vapor partial pressure [PH2O (pt)] and carbon dioxide [PCO2 (pt)] were indirectly estimated. Of the 6,559.79 h of biovital recordings, 5051.03 h (77%) were valid for data interpretation. Sleep and wake hours were 9.0 ± 1.1 h and 14.9 ± 1.1 h, respectively. HRmax % [median: 71.86% (interquartile range 61.03-82%)] and HR/LF [median: 3.75 (interquartile range 3.19-5.05)] were elevated, independent from the wake-sleep cycle. The majority of HRV time- and frequency-domain parameters were significantly higher in the pediatric patients (p ≤ 0.031). The HRV HR/LF ratio was associated with phenotype severity, disease progression, clinical sleep disorder, subclinical hypoxia, and electroencephalographic observations of multifocal epileptic activity and general background slowing. Our findings indicate the feasibility of a continuous 24-h non-invasive home monitoring of biovital parameters in RTT. Moreover, for the first time, HRmax% and the HR/LF ratio were identified as potential objective markers of fatigue, illness severity, and disease progression.

Effect of loading frequency and cyclic stress ratio on undrained cyclic behaviors of clay-structure interface with various overconsolidation ratios

The widely used suction caissons in offshore engineering are commonly subject to cyclic loads including the wind, wave, and current. For proper design and sustainable maintenance of the suction caisson foundations, it is essential to investigate the dynamic response of the clay-structure interface. This research conducted a series of clay-structure interface cyclic shear tests under constant volume equivalent undrained state using a modified direct simple shear device to fully investigate the effects of cyclic stress ratio (CSR), loading frequency, and overconsolidation ratio (OCR) on the dynamic response of the interface. The interface shear strength, relative shear displacement, the number of cycles to failure (Nf), and interface pore water pressure were analyzed. Results show that the Nf is strongly affected by CSR and loading frequency. A model is proposed to describe the decrease in the Nf with increasing CSR under a given loading frequency and OCR. The Nf increases approximately linearly with loading frequency on a double-logarithmic scale. Additionally, the contour diagrams, defining the Nf as a function of interface average and cyclic shear stress under various OCRs, are proposed to predict the failure behaviors of clay-structure interface subjected to various loading combinations. Besides, the development models are established respectively to describe the evolutions of accumulated interface pore pressure (Uia) against the number of cycles (N) and cyclic relative shear displacement (Wcy), presenting better capability to describe the development of Uia. The findings in this study hold potential implications for determining whether the failure occurs under corresponding loading conditions and predicting interface pore pressure development.

Deep Learning based Channel Estimation and Hybrid Beamforming for 5G Massive MIMO Wireless Communications

Hybrid beamforming (BF), which divides beamforming operation into radio frequency (RF) and baseband (BB) domains, will play a critical role in MIMO communication at millimeter-wave(mmWave) frequencies. This paper also introduce offline training and prediction schemes for channel estimation and hybrid beamforming. The aim of this paper is that to increase spectral efficiency over more data streams by leveraging the deep learning based LSTM network. The LSTM network is used to train the numeric values from sequence data and predict on new sequence data. The performance is evaluated under different parameters including number of data streams (1, 2, 3 and 4) with different signal-to-noise ratio (SNR) for different carrier frequencies (28GHz, 38GHz, 60GHz and 73GHz) through computer simulation using MATLAB. The simulation results verified that the proposed method can achieve higher spectral efficiency when the number of data streams increases and the value of SNR-Test increases too.

A novel machine learning framework informed by the fractional calculus dynamic model of hybrid glass/jute woven composite

AbstractThis study investigates the influence of glass‐jute fiber hybridization on the dynamic viscoelastic performance of woven natural fiber composites. Experiments elucidated the effects of low‐frequency vibration and glass/jute ratio on the storage modulus and loss factor. Results exhibited nonlinear escalations in storage modulus and loss factor with increasing frequency and glass fiber content. The loss factor also demonstrated nonlinear rises with frequency and jute content. A fractional calculus‐based two‐branch model successfully captured these behaviors by incorporating frequency and hybrid ratio as key variables, showing excellent agreement with measurements. To further improve predictive accuracy, an artificial neural network informed by the fractional calculus dynamic model is implemented by enforcing physical constraints during training. The physics‐informed artificial neural network achieved higher correlation to experiments than unconstrained models, affirming the value of fusing physics knowledge into data‐driven models. This study highlights the promise of physics‐guided machine learning for predicting the intricate dynamics of sustainable natural fiber hybrid composites. The integrated analytical and data‐driven techniques provide a pathway to comprehensively model the mechanical performance of advanced multiphase materials over diverse conditions.

Cyclic behavior of port-said soft clay in cyclic direct simple shear tests

A thick deposit of soft clay, known as “Port-Said Clay” (PSC), is commonly encountered in the northern formations close to the Mediterranean shores of Egypt. The growing level of development, including transportation corridors like highways and tunnels, in areas rich with PSC necessitated an in-depth study to characterize its dynamic behavior. This study presents a first comprehensive experimental investigation of the dynamic behavior of the soft PSC using Cyclic Direct Simple Shear Tests. The effects of plasticity index (PI), loading frequency (f), and cyclic stress ratio on the cyclic resistance, shear modulus, shear modulus reduction ratio, pore water pressure generation, and damping ratio was investigated. The results showed that all the investigated behavior parameters are primarily sensitive to PI and f. The conclusions drawn from this study on PSC open the way to further laboratory-based investigations of the dynamic behavior of PSC, in addition to numerical modelling of the seismic interaction between PSC and different types of structures.

Ultrathin absorber with independently tunable dual-resonances for terahertz sensing

A technique is implemented for the generation and alteration of two independently tunable resonance peaks in a graphene based ultrathin absorber structure designed for the terahertz spectrum. The proposed structure is a multi-layered stacked arrangement of two resonator layers of graphene grown over the top of two isolated dielectric (SiO2) substrate layers shielded by a gold metal reflector at the bottom. The structure exhibits very low thickness profile and can be implemented with a thickness equivalent to λ/19.6 where λ is the free space wavelength. Two patterned graphene layers contributes to two independently tunable narrowband/near-perfect absorption peaks at 9.56 THz and 10.96 THz. This polarisation insensitive absorber structure also exhibits frequency ratio tunability of higher order mode with respect to lower order mode by distinctly/separately varying the Fermi level of graphene layers. The response is stable for a wider angle of wave incidence up to 70°. Its narrowband response makes it a suitable candidate to be utilized as biosensor for refractive index sensing.

Planar oscillation-enabled deterministic magnetorheological micro-polishing

Micro-optics are attracting ever-increasing attention in advanced optical systems owing to their merits of high optical performance and high structure compactness. Although deterministic polishing is crucial for obtaining optically qualified components, the decreased micro-optic feature size imposes significant challenges on the polishing process. To proceed the post-processing technique for micro-optics, we developed a novel magnetorheological micro-polishing (μMRP) method, which features a non-contact and deterministic removal of the workpiece materials. An electromagnetic polishing tool for stiffening the MR slurry was specially designed to tightly focus the high-gradient magnetic field within a small region around the tip, thereby enabling the material removal to occur at the micro-scale. The pulse-width modulation (PWM) control strategy for the excitation currents could continuously refresh the MR slurry in the polishing, as well as precisely control the material removal. Moreover, a planar dual-axial oscillation was employed to move the tool to generate relative motions between the abrasives and workpieces for removing the materials. In practice, smooth Gaussian-shape footprints with full width at half-maximum (FWHM) around hundreds of micrometers were achieved, and the influence of the working gap, oscillation frequency, and duty ratio on the material removal was further investigated. Finally, the deterministic material removal capability was demonstrated by generating a sinusoidal micro-grid surface with an amplitude of 80 nm and a spatial period of 600 μm. The surface roughness and form error were about Sa = 1.027 nm and 6.5 nm (RMS), demonstrating that the developed μMRP is promising for figuring micro-optics.

High Frequency of Ancestral Haplotype A of Fatty Acid Desaturase Genes in the Yakut Population.

Aims: The purpose of this study was to study the correlation of the body weight of Yakuts with the variability of polymorphisms rs174537, rs174546 and rs3834458 of the FADS1 - FADS2 region to identify the connection of certain genotypes with obesity. Materials and Methods: For genotyping, classical methods of PCR-RFLP analysis were used. A sample of 446 DNA samples from Yakut volunteers without chronic diseases (143 women and 303 men) was studied. Results: The predominance of the ancestral alleles of SNPs rs174537, rs174546 and rs3834458 was established in all of our studied groups. Analysis of the odds ratio of allele and genotype frequencies in patients with normal BMI, high BMI and obesity did not show statistically significant values. We did not find an association between rs174537, rs174546 and rs3834458 with obesity, but we did not take into account the diet of the subjects, which may have had a stronger effect on BMI. Analysis of pairwise linkage disequilibrium and assessment of haplotypes for 3 SNPs in the FADS1 and FADS2 genes showed strong linkage of all three SNPs to each other (r2 = 0.93-0.96). Conclusions: According to the result of genotyping of SNP rs174537, the frequency of haplotype A in the Yakut population was 0.76 and, in comparison with other world data, is quite high. Which in turn is associated with lower conversion of short-chain polyunsaturated fatty acid to long-chain polyunsaturated fatty acid. Accordingly, a shift in nutrition towards more plant foods can negatively impact the health of the Yakuts. At the moment, the exact dosage of polyunsaturated fatty acids (PUFAs) for humans has not yet been established, but judging by the fact that all recommendations are mainly made on the basis of European populations, in connection with the results of the study, the Yakuts have a particularly high need for PUFAs.

Analytical optimization of an inerter‐based dynamic vibration absorber for suppressing plate vibration

AbstractAn inerter‐based dynamic vibration absorber (IDVA) is proposed and applied to suppress the vibration of a plate. The dynamic model of a thin rectangular plate attached with an IDVA is established and the frequency response function of the transverse displacement of the plate is derived through the classical vibration theory and the analysis in the domain. In addition, the equation for frequency ratios of fixed points is obtained, and the condition for the existence of four fixed points is found by investigating the problem of solving quartic equations. Moreover, the extended fixed‐point method is employed to analytically obtain the optimal parameters of the IDVA attached to a plate, the expressions of the optimal damping ratio and other dimensionless qualities are presented.

Altered cardiac autonomic functioning associates with self-disorders in schizophrenia

Elevated resting heart rate (RHR) and reduced heart rate variability (HRV) are signs of autonomic nervous system dysfunction identified in schizophrenia (SCZ). This dysfunction has been found to manifest prior to the onset of the clinical diagnosis. Yet whether such autonomic dysfunction is associated with vulnerability to schizophrenia remains unknown.This case-control study included recent onset SCZ patients (n = 35) and healthy controls (HC) (n = 33). Patients were scored for self-disorders (SD's) using the EASE manual and all participants underwent a 5-minute resting state electrocardiogram (ECG) recording. Patients were included from outpatient clinics in Denmark. The main measures comprised EASE total scores (SDs), RHR (beats per minute) and three standard HRV measures usually included in testing autonomic nervous system dysfunction: root mean squared of successive differences (RMSSD), standard deviation of normal-to-normal interval (SDNN) and high-frequency/ low frequency ratio (HF/LF). Pearson correlations and linear regression models adjusted for age, sex and medication were used in the SCZ group.The main finding was a positive moderate association between SDs and RHR (r = 0.463; p = 0.005) and a negative association between SDs and HRV (RMSSD) (r = −0.440; p = 0.008) in the SCZ group. Linear regression models found SDs to explain 22 % of the variance of RHR and 19 % in RMSSD. SDs correlated with LF/HF (r = 0.434; p = 0.009), but non-significantly with SDNN.The study provides evidence of an intriguing link between SDs as a susceptibility trait for schizophrenia spectrum disorders and altered cardiac autonomic functioning.

Geological hazard susceptibility assessment based on different mathematical statistical models: a case study of Zhejiang province

Geological disaster is a great threat to people's life and property safety, how to predict it reliably and divide it according to the susceptibility of geological disaster is of great significance. Based on the historical collapse data of Zhejiang Province, this paper selects the evaluation indexes of water flow intensity index (SPI), terrain wetness index (TWI), land use type, slope aspect, slope, annual rainfall, river network density, elevation, road density, plane curvature, normalized difference vegetation index (NDVI) and other factors. Based on the frequency ratio model (FR), random forest model (RF) and logistic regression model (LR), the susceptibility zoning of geological disasters in Zhejiang Province is carried out. The results show that: (1) The high and extremely high risk areas in Zhejiang province are mainly distributed near the river system and road network in the high altitude area; (2) In terms of the evaluation effect of comprehensive PR curve, the random forest model has a better evaluation effect than other models; (3) With the increase of the vulnerability level, the proportion of disasters under each model zoning is gradually increasing, indicating that the zoning effect of each model is good; 4) Quzhou city, Jinhua City, Wenzhou city and Shaoxing city in Zhejiang Province account for a large proportion of the high susceptibility areas and above. The results of vulnerability zoning can provide reference for geological hazard prevention and land planning in Zhejiang province.

Comparative study of construction land development suitability assessment: Insights from multiple models and driving forces

Construction land development suitability (CLDS) assessment holds paramount importance in contemporary urban planning and management. Previous CLDS studies have faced challenges such as insufficient evaluation indicators, model complexity, and limited interpretability. To address these gaps, this study applies three simplistic yet mature models from the field of geological hazard assessment - Weight of Evidence (WOE), Information Value (IV), and Frequency Ratio (FR) - to evaluate the CLDS in Wuhan, China, and examine their feasibility. In parallel, a comparative analysis with the Patch-generating Land Use Simulation (PLUS) model was conducted. The WOE, IV, and FR models demonstrate a gradual decrease in CLDS from the city center and the Yangtze and Han rivers towards the periphery. Conversely, the PLUS model indicates the highest suitability around the city's development boundaries, followed by the city center, with suburbs displaying lower suitability. In terms of accuracy, the PLUS model's CARS module outperforms others, particularly for micro-level urban expansion simulations. The IV and WOE models follow closely, manifesting a preference for macro-level urban planning. Additionally, this study pinpoints key drivers influencing CLDS, notably population density, distance to social service Points of Interest (POI) and proximity to road networks. The cross-validation of the WOE, IV, and FR models underscores their reliability, with their straightforward principles positioning them as valuable tools in urban development planning. This comparative study validates the application of geological hazard models in urban contexts and diversifies CLDS assessment methodologies.

Integration of geospatial analysis, frequency ratio, and analytical hierarchy process for landslide susceptibility assessment in the maze catchment, omo valley, southern Ethiopia

This investigation was conducted in southern Ethiopia's Maze watershed in the Omo River Valley. Frequency ratio (FR) and analytic hierarchy process (AHP) techniques were used to assess landslide susceptibility in the region. Identifying causative components and landslide inventory data achieved the goal. Remote sensing and on-site investigations found 793 landslide polygons. To assess vulnerability, the landslide inventory information is categorized into two groups: the training dataset (70%) and the validation dataset (30%). This study examined “slope, aspect, curvature, lithology, land use and cover, normalized vegetation index, and proximity to fault lines, rivers, and distance to road as landslide controlling factors”. The spatial analysis capabilities in Arc GIS were used to overlay the weights of all landslide-causing components to create the susceptibility map. A final landslide susceptibility map is produced using FR and AHP methods and categorized as “very low,” “low,” “moderate,” “high,” and “very high.” The frequency ratio method divides the region into susceptibility classes by frequency. The very low, low, medium, high, and very high susceptibility groups cover 25%, 20%, 18%, and 19% of the territory. The analytical hierarchical process technique shows that 3%, 7%, 26%, 36%, and 28% of the area are very low, low, medium, moderate, and very high landslide susceptibility. The receiver operating characteristic curve was employed to validate the area-underlayer susceptibility maps. The success rates were determined using the FR and AHP approaches, resulting in AUC numbers of 0.873 and 0.87. Similarly, the prediction rates were determined to be 0.81 and 0.80. The landslide susceptibility maps will significantly influence land resource allocation.

Tomographic entanglement indicators in a coupled oscillator model

We study entanglement in a simple model comprising two coupled linear harmonic oscillators of the same natural frequency. The system is separable in the center of mass (COM) and relative coordinates into two oscillators of frequency ω c and ω r. We compute standard entanglement measures (subsystem linear entropy and subsystem von Neumann entropy) as well as several tomographic entanglement indicators (Bhattacharyya distance, Kullback–Leibler divergence, and inverse participation ratio) as functions of the frequency ratio η = ω c/ω r, keeping the COM oscillator in its ground state. We demonstrate that, overall, the entanglement indicators reflect quite faithfully the variations in the standard measures. The entanglement is shown to be minimum at η = 1 and maximum at η → 0 or ∞.

The Xe isotope nuclear resonance frequency shift of NMR angular velocity sensors due to transverse gradient magnetic fields

The nuclear magnetic resonance (NMR) angular velocity sensor detects angular velocity by measuring the shift in the nuclear spin precession frequency, which is of fundamental interest. Recent studies have revealed the parameter dependency of nuclei frequency shifts induced by non-uniformly distributed polarized pump beams, temperature, and magnetic field properties. In this study, we investigated the effect of the linear transverse gradient magnetic field on the frequency ratio shift of the Xe isotope nuclear resonance frequency in the NMR sensors. A theoretical analysis method was proposed based on the Fermi contact interaction in atomic polarization and the spin-diffusion relaxation of 129Xe and 131Xe nuclei. The frequency ratio shift of the Xe isotope under different x-axis gradient magnetic fields was measured experimentally. Furthermore, we eliminated the equivalent residual magnetic field through a feedback system and compensated for the original gradient magnetic field in the system, which contributed to accurately revealing the frequency shift induced by the magnetic field gradient. The results indicate that the frequency ratio shift of the Xe isotope is proportional to the strength of the second-order linear transverse gradient magnetic field. This study provides a reference for the analysis and evaluation of the presence of the gradient magnetic field in the NMR angular velocity sensor.