Global Navigation Satellite Systems(GNSSs)are vulnerable to both unintentional interference and intentional attacks,making it difficult to meet the stringent safety requirements of railway train control systems.The gr...Global Navigation Satellite Systems(GNSSs)are vulnerable to both unintentional interference and intentional attacks,making it difficult to meet the stringent safety requirements of railway train control systems.The growing threat to information security posed by spoofing attacks has received limited attention.This study investigates the impact of GNSS spoofing attacks on train positioning,emphasizing their detrimental effects on the accuracy and availability of train location report functions for train operation control.To explore the antispoofing performance of typical GNSS-based train positioning schemes,specific approaches,and system architectures are designed under two GNSS-alone and two GNSS-integrated train positioning schemes.Field data are utilized to establish spoofing attack scenarios for GNSS-based train positioning,with which the anti-spoofing capabilities of different train positioning schemes are evaluated.Experimental results indicate that under specific conditions,the GNSS-integrated positioning schemes demonstrate superior GNSS spoofing suppression capabilities.Results of the tests present valuable guidance for designers and manufacturers in developing more advanced and resilient train positioning solutions and equipment for the next generation of train control systems,thereby promoting the applications of GNSS technology in railway systems.展开更多
Global Navigation Satellite System(GNSS)-based continuous and accurate train positioning is one of the key technologies for advanced train operations such as train virtual coupling.However,GNSS-based train positioning...Global Navigation Satellite System(GNSS)-based continuous and accurate train positioning is one of the key technologies for advanced train operations such as train virtual coupling.However,GNSS-based train positioning faces significant challenges in real-world scenarios due to environmental complexities and signal interferences.Considering this issue,this paper presents an approach for modeling and performance analysis of GNSS-based train positioning systems using Colored Petri Nets(CPNs).By systematically modeling the GNSS signal reception and processing process,the performance of the positioning system under various environment scenarios is evaluated.The system model integrates three types of interference signals(i.e.,Amplitude Modulation(AM)signals,Frequency Modulation(FM)signals,and pulse signals)while incorporating environmental factors such as terrain obstructions and tunnel shielding.Additionally,the Extended Kalman Filter(EKF)algorithm is employed to process GNSS observation data,providing accurate train position estimations.The simulation results demonstrate that signal interferences and complex environmental conditions significantly affect the GNSS-based positioning accuracy.This study offers a comprehensive framework for evaluating the performance of GNSS-based train positioning systems in different scenarios,highlighting critical factors that influence positioning accuracy and stability.展开更多
In this paper,a novel train positioning method considering satellite raw observation data was proposed,which aims to promote train positioning performance from an innovative perspective of the train satellite-based po...In this paper,a novel train positioning method considering satellite raw observation data was proposed,which aims to promote train positioning performance from an innovative perspective of the train satellite-based positioning error sources.The method focused on overcoming the abnormal observations in satellite observation data caused by railway environment rather than the positioning results.Specifically,the relative positioning experimental platform was built and the zero-baseline method was firstly employed to evaluate the carrier phase data quality,and then,GNSS combined observation models were adopted to construct the detection values,which were applied to judge abnormal-data through the dual-frequency observations.Further,ambiguity fixing optimization was investigated based on observation data selection in partly-blocked environments.The results show that the proposed method can effectively detect and address abnormal observations and improve positioning stability.Cycle slips and gross errors can be detected and identified based on dual-frequency global navigation satellite system data.After adopting the data selection strategy,the ambiguity fixing percentage was improved by 29.2%,and the standard deviation in the East,North,and Up components was enhanced by 12.7%,7.4%,and 12.5%,respectively.The proposed method can provide references for train positioning performance optimization in railway environments from the perspective of positioning error sources.展开更多
To solve the problem of data fusion for prior information such as track information and train status in train positioning,an adaptive H∞filtering algorithm with combination constraint is proposed,which fuses prior in...To solve the problem of data fusion for prior information such as track information and train status in train positioning,an adaptive H∞filtering algorithm with combination constraint is proposed,which fuses prior information with other sensor information in the form of constraints.Firstly,the train precise track constraint method of the train is proposed,and the plane position constraint and train motion state constraints are analysed.A model for combining prior information with constraints is established.Then an adaptive H∞filter with combination constraints is derived based on the adaptive adjustment method of the robustness factor.Finally,the positioning effect of the proposed algorithm is simulated and analysed under the conditions of a straight track and a curved track.The results show that the positioning accuracy of the algorithm with constrained filtering is significantly better than that of the algorithm without constrained filtering and that the algorithm with constrained filtering can achieve better performance when combined with track and condition information,which can significantly reduce the train positioning error.The effectiveness of the proposed algorithm is verified.展开更多
The traditional train positioning methods suffer from inadequate accuracy and high maintenance costs,rendering them unsuitable for the development requirements of lightweight and intelligent train positioning technolo...The traditional train positioning methods suffer from inadequate accuracy and high maintenance costs,rendering them unsuitable for the development requirements of lightweight and intelligent train positioning technology.To address these restraints,the BeiDou navigation satellite system/strapdown inertial navigation system(BDS/SINS)integrated train positioning system based on an adaptive unscented Kalman filter(AUKF)is proposed.Firstly,the combined denoising algorithm(CDA)and Lagrange interpolation algorithm are introduced to preprocess the original data,effectively eliminating the influence of noise signals and abnormal measurements on the train positioning system.Secondly,the innovation theory is incorporated into the unscented Kalman filter(UKF)to derive the AUKF,which accomplishes an adaptive update of the measurement noise covariance.Finally,the positioning performance of the proposed AUKF is contrasted with that of conventional algorithms in various operation scenes.Simulation results demonstrate that the average value of error calculated by AUKF is less than 1.5 m,and the success rate of positioning touches 95.0%.Compared to Kalman filter(KF)and UKF,AUKF exhibits superior accuracy and stability in train positioning.Consequently,the proposed AUKF is well-suited for providing precise positioning services in variable operating environments for trains.展开更多
Virtual coupling(VC) is an emerging technology for addressing the shortage of rail transportation capacity. As a crucial enabling technology, the VC-specific acquisition of train information, especially train followin...Virtual coupling(VC) is an emerging technology for addressing the shortage of rail transportation capacity. As a crucial enabling technology, the VC-specific acquisition of train information, especially train following distance(TFD), is underdeveloped.In this paper, a novel method is proposed to acquire real-time TFD by analyzing the vibration response of the front and following trains, during which only onboard accelerometers and speedometers are required. In contrast to the traditional arts of train positioning, this method targets a relative position between two adjacent trains in VC operation, rather than the global positions of the trains. For this purpose, an adaptive system containing three strategies is designed to cope with possible adverse factors in train operation. A vehicle dynamics simulation of a heavy-haul railway is implemented for the evaluation of feasibility and performance. Furthermore, a validation is conducted using a set of data measured from in-service Chinese high-speed trains. The results indicate the method achieves satisfactory estimation accuracy using both simulated and actual data. It has favorable adaptability to various uncertainties possibly encountered in train operation. Additionally, the method is preliminarily proven to adapt to different locomotive types and even different rail transportation modes. In general, such a method with good performance, low-cost, and easy implementation is promising to apply.展开更多
Train positioning is the key to ensure the transportation and efficient operation of the railway.Due to the low accuracy and the poor real-time of the train positioning,a train positioning system based on global navig...Train positioning is the key to ensure the transportation and efficient operation of the railway.Due to the low accuracy and the poor real-time of the train positioning,a train positioning system based on global navigation satellite system/inertial measurement unit/odometer(GNSS/IMU/ODO)combination framework and a train integrated positioning method based on grey neural network are put forward.A data updating method based on the established grey prediction model of train positioning is put forward,which uses the accumulation and summary of the grey theory for the rough prediction of the data.The purpose of the method is to reduce the noise of the original data.Moreover,the radial basis function(RBF)neural network is introduced to correct residual sequence of the grey prediction model.Compared with the single model calibration,this method can make full use of the advantages of each model,thus getting a high positioning accuracy in the case of small samples and poor information.Experiments show that the method has good real-time performance and high accuracy,and has certain application value.展开更多
This paper addresses the challenge of accurately and timely determining the position of a train,with specific consideration given to the integration of the global navigation satellite system(GNSS)and inertial navigati...This paper addresses the challenge of accurately and timely determining the position of a train,with specific consideration given to the integration of the global navigation satellite system(GNSS)and inertial navigation system(INS).To overcome the increasing errors in the INS during interruptions in GNSS signals,as well as the uncertainty associated with process and measurement noise,a deep learning-based method for train positioning is proposed.This method combines convolutional neural networks(CNN),long short-term memory(LSTM),and the invariant extended Kalman filter(IEKF)to enhance the perception of train positions.It effectively handles GNSS signal interruptions and mitigates the impact of noise.Experimental evaluation and comparisons with existing approaches are provided to illustrate the effectiveness and robustness of the proposed method.展开更多
This paper presents a novel Autonomous Integrity Monitoring and Assurance (AIMA) scheme for integrity assurance of the GNSS-based train integrated positioning system. In this scheme, integrity assurance strategies a...This paper presents a novel Autonomous Integrity Monitoring and Assurance (AIMA) scheme for integrity assurance of the GNSS-based train integrated positioning system. In this scheme, integrity assurance strategies are combined with a three-stage hierarchical architecture, considering the coupling effects among sensor collection, sensor fusion and matching decision level in train integrated positioning. In sensor collecting stage, the AIMA scheme deals with sensor faults and failures with a PCA-based fault detection, diagnosis and isolation approach. In multi-sensor fusion stage, a novel cubature point H0o filter is presented to enhance the fault tolerance capability, and a hybrid approach is applied to indicating and monitoring the protection level of position estimation, concerning both the estimating covariance and measurement slopes. In map matching stage, hypothesis testing with specific test statistic is carried out to determine effectiveness of positioning results. Position calculation will be invalid with an alarm triggered if the specific integrity criterion is not satisfied in any stage. Since independent solutions are applied in AIMA, integrity assurance is closely coupled with information processing in train integrated positioning. Numerical results of the three cases correspond to the hierarchical architecture with field data and simulations are presented to illustrate features and applicability of the proposed AIMA scheme and specific solutions.展开更多
To overcome the problem that the confusion between texts limits the precision in text re- trieval, a new text retrieval algorithm that decrease confusion (DCTR) is proposed. The algorithm constructs the searching te...To overcome the problem that the confusion between texts limits the precision in text re- trieval, a new text retrieval algorithm that decrease confusion (DCTR) is proposed. The algorithm constructs the searching template to represent the user' s searching intention through positive and negative training. By using the prior probabilities in the template, the supported probability and anti- supported probability of each text in the text library can be estimated for discrimination. The search- ing result can be ranked according to similarities between retrieved texts and the template. The com- plexity of DCTR is close to term frequency and mversed document frequency (TF-IDF). Its distin- guishing ability to confusable texts could be advanced and the performance of the result would be im- proved with increasing of training times.展开更多
At 13:46 on March 11, 2011(Beijing time), an earthquake of Mw=9.0 occurred in Japan. By comparing the tsunami data from Guanhekou marine station with other tsunami wave observation gathered from southeast coastal a...At 13:46 on March 11, 2011(Beijing time), an earthquake of Mw=9.0 occurred in Japan. By comparing the tsunami data from Guanhekou marine station with other tsunami wave observation gathered from southeast coastal area of China, it was evident that, only in Guanhekou, the position of the maximum wave height appeared in the middle part rather than in the front of the tsunami wave train. A numerical model of tsunami propagation based on 2-D nonlinear shallow water equations was built to study the impact range and main causes of the special tsunami waveform discovered in Jiangsu coastal area. The results showed that nearly three-quarters of the Jiangsu coastal area, mainly comprised the part north of the radial sand ridges, reached its maximum tsunami wave height in the middle part of the wave train. The main cause of the special waveform was the special underwater topography condition of the Yellow Sea and the East China Sea area, which influenced the tsunami propagation and waveform significantly. Although land boundary reflection brought an effect on the position of the maximum wave height to a certain extent, as the limits of the incident waveform and distances between the observation points and shore, it was not the dominant influence factor of the special waveform. Coriolis force's impact on the tsunami waves was so weak that it was not the main cause for the special phenomenon in Jiangsu coastal area. The study reminds us that the most destructive wave might not appear in the first one in tsunami wave train.展开更多
基金the National Key Research and Development Program of China(2023YFB3907300)the National Natural Science Foundation of China(U2268206,T2222015)the Beijing Natural Science Foundation(4232031).
文摘Global Navigation Satellite Systems(GNSSs)are vulnerable to both unintentional interference and intentional attacks,making it difficult to meet the stringent safety requirements of railway train control systems.The growing threat to information security posed by spoofing attacks has received limited attention.This study investigates the impact of GNSS spoofing attacks on train positioning,emphasizing their detrimental effects on the accuracy and availability of train location report functions for train operation control.To explore the antispoofing performance of typical GNSS-based train positioning schemes,specific approaches,and system architectures are designed under two GNSS-alone and two GNSS-integrated train positioning schemes.Field data are utilized to establish spoofing attack scenarios for GNSS-based train positioning,with which the anti-spoofing capabilities of different train positioning schemes are evaluated.Experimental results indicate that under specific conditions,the GNSS-integrated positioning schemes demonstrate superior GNSS spoofing suppression capabilities.Results of the tests present valuable guidance for designers and manufacturers in developing more advanced and resilient train positioning solutions and equipment for the next generation of train control systems,thereby promoting the applications of GNSS technology in railway systems.
基金supported by the National Key Research and Development Program of China(2023YFB3907300)the Fundamental Research Funds for the Central Universities(2024JBMC002)the National Natural Science Foundation of China(T2222015,U2268206).
文摘Global Navigation Satellite System(GNSS)-based continuous and accurate train positioning is one of the key technologies for advanced train operations such as train virtual coupling.However,GNSS-based train positioning faces significant challenges in real-world scenarios due to environmental complexities and signal interferences.Considering this issue,this paper presents an approach for modeling and performance analysis of GNSS-based train positioning systems using Colored Petri Nets(CPNs).By systematically modeling the GNSS signal reception and processing process,the performance of the positioning system under various environment scenarios is evaluated.The system model integrates three types of interference signals(i.e.,Amplitude Modulation(AM)signals,Frequency Modulation(FM)signals,and pulse signals)while incorporating environmental factors such as terrain obstructions and tunnel shielding.Additionally,the Extended Kalman Filter(EKF)algorithm is employed to process GNSS observation data,providing accurate train position estimations.The simulation results demonstrate that signal interferences and complex environmental conditions significantly affect the GNSS-based positioning accuracy.This study offers a comprehensive framework for evaluating the performance of GNSS-based train positioning systems in different scenarios,highlighting critical factors that influence positioning accuracy and stability.
基金Project(52272339)supported by the National Natural Science Foundation of ChinaProject(2023YFB390730303)supported by the National Key Research and Development Program of China+2 种基金Project(L2023G004)supported by the Science and Technology Research and Development Program of China State Railway Group Co.,Ltd.Project(QZKFKT2023-005)supported by the State Key Laboratory of Heavy-duty and Express High-power Electric Locomotive,ChinaProject(2022JZZ05)supported by the Open Foundation of MOE Key Laboratory of Engineering Structures of Heavy Haul Railway(Central South University),China。
文摘In this paper,a novel train positioning method considering satellite raw observation data was proposed,which aims to promote train positioning performance from an innovative perspective of the train satellite-based positioning error sources.The method focused on overcoming the abnormal observations in satellite observation data caused by railway environment rather than the positioning results.Specifically,the relative positioning experimental platform was built and the zero-baseline method was firstly employed to evaluate the carrier phase data quality,and then,GNSS combined observation models were adopted to construct the detection values,which were applied to judge abnormal-data through the dual-frequency observations.Further,ambiguity fixing optimization was investigated based on observation data selection in partly-blocked environments.The results show that the proposed method can effectively detect and address abnormal observations and improve positioning stability.Cycle slips and gross errors can be detected and identified based on dual-frequency global navigation satellite system data.After adopting the data selection strategy,the ambiguity fixing percentage was improved by 29.2%,and the standard deviation in the East,North,and Up components was enhanced by 12.7%,7.4%,and 12.5%,respectively.The proposed method can provide references for train positioning performance optimization in railway environments from the perspective of positioning error sources.
基金the National Natural Science Fund of China(61471080)Training Plan for Young Backbone Teachers in Colleges and Universities of Henan Province(2018GGJS171).
文摘To solve the problem of data fusion for prior information such as track information and train status in train positioning,an adaptive H∞filtering algorithm with combination constraint is proposed,which fuses prior information with other sensor information in the form of constraints.Firstly,the train precise track constraint method of the train is proposed,and the plane position constraint and train motion state constraints are analysed.A model for combining prior information with constraints is established.Then an adaptive H∞filter with combination constraints is derived based on the adaptive adjustment method of the robustness factor.Finally,the positioning effect of the proposed algorithm is simulated and analysed under the conditions of a straight track and a curved track.The results show that the positioning accuracy of the algorithm with constrained filtering is significantly better than that of the algorithm without constrained filtering and that the algorithm with constrained filtering can achieve better performance when combined with track and condition information,which can significantly reduce the train positioning error.The effectiveness of the proposed algorithm is verified.
基金supported by Project Fund of China National Railway Group Co.,Ltd.(No.N2022G012)Natonal Natural Science Foundation of China(No.61661027)。
文摘The traditional train positioning methods suffer from inadequate accuracy and high maintenance costs,rendering them unsuitable for the development requirements of lightweight and intelligent train positioning technology.To address these restraints,the BeiDou navigation satellite system/strapdown inertial navigation system(BDS/SINS)integrated train positioning system based on an adaptive unscented Kalman filter(AUKF)is proposed.Firstly,the combined denoising algorithm(CDA)and Lagrange interpolation algorithm are introduced to preprocess the original data,effectively eliminating the influence of noise signals and abnormal measurements on the train positioning system.Secondly,the innovation theory is incorporated into the unscented Kalman filter(UKF)to derive the AUKF,which accomplishes an adaptive update of the measurement noise covariance.Finally,the positioning performance of the proposed AUKF is contrasted with that of conventional algorithms in various operation scenes.Simulation results demonstrate that the average value of error calculated by AUKF is less than 1.5 m,and the success rate of positioning touches 95.0%.Compared to Kalman filter(KF)and UKF,AUKF exhibits superior accuracy and stability in train positioning.Consequently,the proposed AUKF is well-suited for providing precise positioning services in variable operating environments for trains.
基金supported by the National Natural Science Foundation of China(52222217,52388102,52372435)the Major Science and TechnologyProject of China Energy(GJNY-22-7)
文摘Virtual coupling(VC) is an emerging technology for addressing the shortage of rail transportation capacity. As a crucial enabling technology, the VC-specific acquisition of train information, especially train following distance(TFD), is underdeveloped.In this paper, a novel method is proposed to acquire real-time TFD by analyzing the vibration response of the front and following trains, during which only onboard accelerometers and speedometers are required. In contrast to the traditional arts of train positioning, this method targets a relative position between two adjacent trains in VC operation, rather than the global positions of the trains. For this purpose, an adaptive system containing three strategies is designed to cope with possible adverse factors in train operation. A vehicle dynamics simulation of a heavy-haul railway is implemented for the evaluation of feasibility and performance. Furthermore, a validation is conducted using a set of data measured from in-service Chinese high-speed trains. The results indicate the method achieves satisfactory estimation accuracy using both simulated and actual data. It has favorable adaptability to various uncertainties possibly encountered in train operation. Additionally, the method is preliminarily proven to adapt to different locomotive types and even different rail transportation modes. In general, such a method with good performance, low-cost, and easy implementation is promising to apply.
基金Gansu Province Basic Research Innovation Group Plan(No.1606RJIA327)Natural Science Foundation of Gansu Province(No.1606RJYA225)+1 种基金Lanzhou Jiaotong University Youth Fund(No.2014031)Longyuan Youth Innovative Support Program(No.2016-43)
文摘Train positioning is the key to ensure the transportation and efficient operation of the railway.Due to the low accuracy and the poor real-time of the train positioning,a train positioning system based on global navigation satellite system/inertial measurement unit/odometer(GNSS/IMU/ODO)combination framework and a train integrated positioning method based on grey neural network are put forward.A data updating method based on the established grey prediction model of train positioning is put forward,which uses the accumulation and summary of the grey theory for the rough prediction of the data.The purpose of the method is to reduce the noise of the original data.Moreover,the radial basis function(RBF)neural network is introduced to correct residual sequence of the grey prediction model.Compared with the single model calibration,this method can make full use of the advantages of each model,thus getting a high positioning accuracy in the case of small samples and poor information.Experiments show that the method has good real-time performance and high accuracy,and has certain application value.
基金supported by the National Natural Science Foundation of China(Nos.61925302,62273027)the Beijing Natural Science Foundation(L211021).
文摘This paper addresses the challenge of accurately and timely determining the position of a train,with specific consideration given to the integration of the global navigation satellite system(GNSS)and inertial navigation system(INS).To overcome the increasing errors in the INS during interruptions in GNSS signals,as well as the uncertainty associated with process and measurement noise,a deep learning-based method for train positioning is proposed.This method combines convolutional neural networks(CNN),long short-term memory(LSTM),and the invariant extended Kalman filter(IEKF)to enhance the perception of train positions.It effectively handles GNSS signal interruptions and mitigates the impact of noise.Experimental evaluation and comparisons with existing approaches are provided to illustrate the effectiveness and robustness of the proposed method.
基金supported by the National Natural Science Foundation of China (Grant Nos. 60634010, 60736047, 60870016)
文摘This paper presents a novel Autonomous Integrity Monitoring and Assurance (AIMA) scheme for integrity assurance of the GNSS-based train integrated positioning system. In this scheme, integrity assurance strategies are combined with a three-stage hierarchical architecture, considering the coupling effects among sensor collection, sensor fusion and matching decision level in train integrated positioning. In sensor collecting stage, the AIMA scheme deals with sensor faults and failures with a PCA-based fault detection, diagnosis and isolation approach. In multi-sensor fusion stage, a novel cubature point H0o filter is presented to enhance the fault tolerance capability, and a hybrid approach is applied to indicating and monitoring the protection level of position estimation, concerning both the estimating covariance and measurement slopes. In map matching stage, hypothesis testing with specific test statistic is carried out to determine effectiveness of positioning results. Position calculation will be invalid with an alarm triggered if the specific integrity criterion is not satisfied in any stage. Since independent solutions are applied in AIMA, integrity assurance is closely coupled with information processing in train integrated positioning. Numerical results of the three cases correspond to the hierarchical architecture with field data and simulations are presented to illustrate features and applicability of the proposed AIMA scheme and specific solutions.
文摘To overcome the problem that the confusion between texts limits the precision in text re- trieval, a new text retrieval algorithm that decrease confusion (DCTR) is proposed. The algorithm constructs the searching template to represent the user' s searching intention through positive and negative training. By using the prior probabilities in the template, the supported probability and anti- supported probability of each text in the text library can be estimated for discrimination. The search- ing result can be ranked according to similarities between retrieved texts and the template. The com- plexity of DCTR is close to term frequency and mversed document frequency (TF-IDF). Its distin- guishing ability to confusable texts could be advanced and the performance of the result would be im- proved with increasing of training times.
基金financially supported by the Fundamental Research Funds for the Central Universities,Hohai University(Grant No.2011B06014)the Fundamental Research Funds for the Central Public Welfare Research Institutes,Nanjing Hydraulic Research Institute(Grant No.YN912001)+2 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK2012411)the National Science & Technology Pillar Program(Grant No.2012BAB03B01)the Cultivation of Jiangsu Province Graduate Innovation Project(Grant No.KYZZ_0151)
文摘At 13:46 on March 11, 2011(Beijing time), an earthquake of Mw=9.0 occurred in Japan. By comparing the tsunami data from Guanhekou marine station with other tsunami wave observation gathered from southeast coastal area of China, it was evident that, only in Guanhekou, the position of the maximum wave height appeared in the middle part rather than in the front of the tsunami wave train. A numerical model of tsunami propagation based on 2-D nonlinear shallow water equations was built to study the impact range and main causes of the special tsunami waveform discovered in Jiangsu coastal area. The results showed that nearly three-quarters of the Jiangsu coastal area, mainly comprised the part north of the radial sand ridges, reached its maximum tsunami wave height in the middle part of the wave train. The main cause of the special waveform was the special underwater topography condition of the Yellow Sea and the East China Sea area, which influenced the tsunami propagation and waveform significantly. Although land boundary reflection brought an effect on the position of the maximum wave height to a certain extent, as the limits of the incident waveform and distances between the observation points and shore, it was not the dominant influence factor of the special waveform. Coriolis force's impact on the tsunami waves was so weak that it was not the main cause for the special phenomenon in Jiangsu coastal area. The study reminds us that the most destructive wave might not appear in the first one in tsunami wave train.