Application of Kalman filter in wharf surface displacement of Beidou automatic monitoring

To meet the requirements for real-time dynamic monitoring of wharf surface displacement, an automatic monitoring system based on Beidou satellite navigation was constructed, and it was applied to monitor the surface displacement of a wharf in Tianjin Port. Next, the sensible system noise of the Kalman filter state equation was determined by the simulation experiment. Finally, the monitoring data were processed and analyzed by Kalman filter. The results show that the accuracy of the monitoring data can be improved by more than 30% by Kalman filtering, particularly the elevation direction accuracy can reach more than 40%. The plane and elevation accuracy can be less than 1.5mm, after the Beidou automatic monitoring data has been processed by Kalman filtering, and this meets the needs for most engineering deformation monitoring. In a word, the Beidou automatic monitoring system is characterized by continuity, real-time and high precision, and can be applied to most engineering deformation monitoring requirements.


Introduction
In recent years, with the implementation of the national maritime strategy, China's maritime economy has grown quickly and the scale of trade has continued to grow.As a transport hub, ports have contributed greatly to the development of the marine industry.The wharf is a major port infrastructure and the stability of the wharf structure is linked to the safe operation of the port.Therefore, one efficient way to ensure the safe operation of the wharf is to monitor the displacement of the wharf surface.
Beidou automatic monitoring technology has the features of without intervisibility between stations, all-weather observation, high degree of automation, monitoring the three-dimensional displacement of points, and can achieve mm level accuracy [1][2].It has been widely used to monitor the deformation of the slope, bridge, and lock.
The application of Beidou automatic monitoring system has been studied by relevant scholars.JIANG Jun-ping [3] used GPS to monitor the deformation of landslide mass, and the deformation of the landslide was found, which proved that applying GPS to monitoring the deformation of the landslide mass is feasible.LI Peng [4] applied the Beidou satellite navigation and positioning system to monitor the bridge deformation.the positioning accuracy was equivalent to that of the GPS, which can meet the demands of bridge deformation monitoring.AN Qing [5] designed a Beidou-based deformation

Introduction of Beidou Satellite Navigation System
The Beidou satellite navigation and positioning system began in the 1980s.In China, the construction of the system was carried out according to the "three-step" strategy.The Beidou-1 system was built in 2000 to provide services to China.The Beidou-2 system was completed in 2012 to deliver services to the Asia Pacific region.The Beidou-3 system was completed in 2020 to serve the world [1][2].

Composition of Beidou Automatic Monitoring System
The Beidou automatic monitoring system is mainly composed of three parts.
(1) Sensor system:it is mainly composed by Beidou receiver, 5G communication module and Internet, which collects real-time displacement data of the wharf and transmits it to the data center through the network.
(2) Data center: the data center is mainly composed by computer room, work station and server, which can store the signals collected and transmitted by the sensors in real time.
(3) Data processing and control system: it is mainly composed of Beidou satellite navigation and positioning solution software, database software, data control software and other software.It can receive and process data collected in real time, and display the original data and processed data, perform on-line assessment and early warning.

Introduction of Kalman Filtering
Kalman filtering is one of the methods commonly used to process data in the current field of engineering monitoring.It provides an efficient and reliable method for the best real-time estimation of the system status.
Random interference is controlled, and the data is processed in real time by constructing a state equation and an observation equation.It is applicable for the data processing of the Beidou automatic monitoring process [8].

Equation construction
The monitoring data of the wharf has discrete characteristics, so the discrete state equation is constructed, and the discrete process is described by the following formula.
=  −1 +  −1 +  −1 (1)   =   + (2) Where:   represents the system state variable at the moment , A is the system state transition matrix,  −1 is system variable representing at the moment  − 1,  represents the control input gain,  −1 represents the input variable,  −1 represents the excitation noise during observation,   represents the observation vector,  represents the gain of the state vector   to the observation vector   ,   stands for observation noise.
Equation ( 1) is the discrete stochastic process equation, Equation ( 2) is the measuring equation.
The cumulative displacement of the wharf (cumulative displacement X, cumulative displacement Y, cumulative displacement Z) is taken as the research object at this time, as the wharf itself tends to be stable, and there is no obvious input variable for the wharf.Thus, the system state transition matrix  is defined as the unit matrix, the input variable  −1 is defined as 0, The gain matrix  is the identity matrix.
Next the Kalman filter is constructed with Formula (1) and Formula (2).Predictive and correcting equations are formed.The predictive equation is responsible for forward calculation, and the correcting equation is responsible for feedback.The details are as follows: Formula (3) to Formula (7).
The estimated equation is: Where:  ̂ − represents the prior state estimation of system vector,  ̂−1 、 ̂ represents posterior state estimation of system vector,   − represents the covariance of prior estimation error,   ,  −1 represents the posterior estimation error covariance,  is the covariance matrix of process excitation noise,   is the Kalman gain,  is the covariance matrix of observation noise,   represents the observation vector [9].

Project introduction
A company in Tianjin Port has four operating berths of ten thousand tons deep-water general cargo wharf, the wharf was completed in the 1990s and upgraded in 2004.In the process of long-term use, the wharf has experienced aging.To ensure the safe operation of the wharf, it is necessary to monitor the displacement of the wharf surface in real time.
The Beidou automatic monitoring points were placed at both ends of the wharf, named JCD1 and JCD2.At the same time, a Beidou GNSS receiver is deployed at the top of the company's office building, it serves as a reference station and data centre.The monitoring system has been operational since January 2022.It works 24 hours per day.The survey data is transmitted back in real time via the 5G communication network.Satellite data collected include GPS, GLONASS and Beidou.The coordinates of monitoring points are calculated using the base station restricted network adjustment method.The computing interval is 1h.It performed the automatic monitoring 7 × 24h based on Beidou.
Figure 1 shows the base station of the Beidou Automatic Monitoring System in the project, and Figure 2 shows the Beidou automatic monitoring system monitoring station in the project.

Data processing and analysis
Since its introduction, the system has been functioning continuously.The Beidou automatic monitoring equipment is affected by the troposphere, ionosphere and buildings and structures.Thus, there are corresponding errors in the results of the displacement calculation, therefore, measures should be taken to weaken the observation error.The Kalman filter can compensate for the effect of noise on the observation data.As such, the Kalman filter is used to process the monitoring data of the wharf to improve the monitoring accuracy.

Determination of system noise.
System noise  is an major parameter in the construction process of the state equation.Small system noise means that the model data is more reliable; Large system noise means that the observed data are more reliable.Observation noise  can be calculated and determined by statistics.
The system noise  can be dynamically adjusted by Kalman adaptive filtering [9], or determined by test [10].the system noise  was determined by tests in this article.
200 groups of data were selected from a monitoring station as the real value, and then the noise interval (-2,2) was added.Different system noises  were adjusted, the data was filtered by the Kalman filter, and the most appropriate system noise  was determined.
Figure 3 shows the Kalman filtering effect of the accumulated displacement in the north direction X when the system noise is set to 0.1, 1, 10. Figure 3 shows that, when the value of  was 0.1, the value after the Kalman filter deviated from the real value and the analog value of adding noise.When the value of  10, the value after the Kalman filtering was closer to the analog value of adding noise; When  was taken as 1, the Kalman filtered value was close to the real value.Therefore, it will be more appropriate for the displacement of the wharf that the system noise value is set to 1.
The above description shows that when the smaller value of  is 0.1, the filtered value deviates from the real value and the analog value of adding noise, the model value are more trusted.when the larger value of  is 10, the filtered value is closer to the value of adding noise, the observation data is more trusted.It also proves that the small system noise represents that model data are more reliable; Large system noise means that the observed data are more reliable.
The system noise  can also be determined by automatic or semi-automatic methods, such as Kalman's adaptive dynamic filter adjustment, which will be studied in further work.

Wharf monitoring data filtering.
Based on the determined system noise value, the Kalman filter was used to filter the accumulated displacement of the wharf in all directions, and the filtering effect was analyzed.Figure 4 shows the filtering effect.Table 1 shows the standard deviation of a posterior estimation of Kalman filtering and the standard deviation of the initial observation noise.Figure 4 shows that Kalman filtering has a good effect, which can significantly reduce the error due to observation noise.
Table 1 shows that after Kalman filtering, the posterior estimated standard deviation of the monitoring data is 1.08mm in the north direction X, 1.13mm in the east direction Y, and 1.43mm in the elevation H, all less than 1.5mm.And compared to the original observation noise standard deviation, the posterior estimation standard deviation of the monitoring data is a significant reduction.the minimum reduction is X of the north direction, which is 32% lower, and the maximum reduction is in the elevation direction, which is 43%.
By applying the Kalman filter to the measurement data processing of the Beidou automatic monitoring system, it can minimize the interference from random errors in the measurement process.Taking it as an example that the real-time dynamic monitoring of a wharf surface displacement in Tianjin Port.after Kalman filtering, the plane and elevation accuracy can reach within 1.5mm, which can meet the accuracy requirements of most engineering deformation monitoring.Comparison of the standard deviation of the posterior estimation with the standard deviation of the initial observation noise, the accuracy of the observation is significantly improved, with 30% improvement in plane direction and 40% improvement in the elevation direction.

Discuss
In this article, the real-time dynamic monitoring of the displacement of a wharf in Tianjin Port was taken as the research object, and the Kalman filter is applied to the measurement data processing.The results of the application and analysis show that the Beidou automatic monitoring system can meet the requirements of dynamic, continuous and real-time monitoring of wharf surface displacement, and the monitoring data can reach high precision after Kalman filtering.
Compared to noise reduction methods such as time series and BP neural network, Kalman filtering requires a small amount of historic data, the observation value can be processed dynamically with the observation process [11].The study confirms these advantages of that Kalman filtering.
Use of Beidou automatic monitoring system to monitor the surface displacement of the wharf in real time and dynamically, and the precision of the monitoring is improved by Kalman filtering.This is very important for the safety of wharf to be in motion and do business.Long term, large-scale, and high accurate monitoring data of wharf displacement provide important support for thorough understanding of wharf surface displacement and analysis of wharf displacement mechanism.These will provide engineering basis for safe operation of the wharf.
In determining the system noise  , this paper determines it by tests.It may also be determined by automatic or semi-automatic methods, such as Kalman's adaptive filtering, which will be investigated later by the author.
The application of Operational Mode Analysis (OMA) to the data processing of the Beidou monitoring system is also a possible future research direction.Especially the research based on open source software of OMA, such as PyOMA and PyOMA_GUI [12], which is the open source OMA software based on python.When applied to the automated processing and analysis of wharf monitoring data, it will be able to analyse the modal parameters of the wharf, determine the corresponding state of the wharf and prevent major safety accidents.

Conclusions
In this paper, the Beidou automatic monitoring system is briefly introduced, and applied to the displacement monitoring of the wharf surface.Next the monitoring data are processed and analyzed by Kalman filter.The conclusions are as follows.
(1) The Beidou automatic monitoring system has the characteristics of continuity, real-time and high precision, and can be applied to the displacement monitoring of the wharf surface.
(2) The system noise is a very important Kalman filter parameter, the small system noise indicates that the model data are more reliable; Large system noise means that the observation data are more reliable.And the system noise has to be reasonably determined by experiments.
(3) The accuracy of the monitoring data can be significantly improved by Kalman filtering, the accuracy of the observation can be increased by over 30%, especially in the elevation direction, up to 40%.
(4) The plane and elevation accuracy of the automatic monitoring data of Beidou can be less than 1.5mm by Kalman filtering, and this will satisfy the requirements of most engineering deformation monitoring.
In short, in monitoring the displacement of wharf surface, the Beidou automatic monitoring system is applied, this enhances the efficiency of monitoring the wharf.and Kalman filtering is used to process monitoring data, which greatly improve the accuracy of Beidou automatic monitoring data.these are very important for the safe operation of the wharf.
In the future, the automatic and semi-automatic determination of the system noise, and combined with the source software PyOMA, PyOMA_GUI to analysis the wharf monitoring data will be studied.

Figure 1 .
Figure 1.Base Station of Beidou Automatic Monitoring System.

Figure 2 .
Figure 2. Monitoring Station of Beidou Automatic Monitoring System.

Figure 3 .
Figure 3. Kalman filtering effect in the north direction X under different values  .

Figure 4 .
Figure 4. Kalman filtering effect in different directions.