Double-differenced Carrier Phase Research Articles

Risk overbounding for ionospheric gradient monitor using geometry-free double differenced carrier phase measurements

Ionosphere anomaly can cause large spatial gradients in the double-differenced carrier phase (DDCP) measurements. Taking advantage of this characteristic, the ionospheric gradient monitor (IGM) is designed with multiple ground reference stations to detect threatening ionospheric gradients for safety of life applications. An optimal IGM should be most sensitive to ionospheric gradients and least sensitive to other errors. However, current IGM suffers from the influence of tropospheric error, which can degrade monitor performance under extreme weather conditions by causing risks of false alarms and missed detections. To address this issue, an alternative IGM is proposed using the geometry-free combination of DDCP as the test statistic. Ambiguity resolution procedure is designed for estimating the ambiguity term in the test statistic. The risk induced by the wrong ambiguity fix and tropospheric error are analyzed and bounded with required averaging period and minimum baseline length. The results show that the proposed theoretical IGM is capable of achieving probability of false alarm of 10–8 and probability of missed detection of 10–6 with a filtering period of 612 s and baseline length of 371.7 m or a filtering period of 544 s and baseline length of 384.4 m. The experimental results using data collected from Hong Kong Satellite Positioning Reference Station Network demonstrate that the performance of the proposed theoretical IGM is comparable to that of the existing detection algorithm.

A method to build feature descriptor for GNSS spoofing detection by carrier phase double difference measurement

GNSS spoofing is a type of attack that aims to deceive GNSS receivers by transmitting fake signals that imitate the authentic ones. To detect such attacks, a possible solution is to calculate the double difference (DD) of carrier phase measurements between two antennas of two separated receivers. This DD measurement represents the angle of arrival (AoA) information of the signal. In the case of authentic signals, each signal comes from a different satellite, so the AoA values are different for each satellite. However, in the case of a spoofing attack, all the counterfeit signals from the same broadcaster come from the same direction, and thus the AoA values are the same for all DD measurements. Sum of Squares (SoS) detector and Dispersion of Double Difference (D3) detector are known as simple but effective methods for spoofing detection. However, these detectors are not robust in some special regions where cycle slips are large and vary rapidly. This paper proposes a new method for detecting GNSS spoofing by using carrier phase double difference measurements to generate a feature vector as a descriptor for each satellite in a set of observed satellites. The feature vectors are formed by statistical parameters of a series of consecutive double difference values computed for each reference satellite. Our proposed feature descriptor has been evaluated on a dataset containing both real and fake signals using an SVM classifier. The study shows that the proposed feature descriptor is highly effective in detecting spoofing satellite signals.

Improving Single-Frequency Multi-GNSS RTK/INS Tightly Coupled Integration by Mixing Time-Differenced and Double-Differenced Carrier Phase for Vehicle-borne Positioning

Improving Single-Frequency Multi-GNSS RTK/INS Tightly Coupled Integration by Mixing Time-Differenced and Double-Differenced Carrier Phase for Vehicle-borne Positioning

Localization of GNSS Spoofing Interference Source Based on a Moving Array Antenna

GNSS spoofing interference utilizes falsified navigation signals to launch attacks on GNSS systems, posing a significant threat to applications that rely on GNSS signals for positioning, navigation, and time services. Therefore, achieving effective localization of the sources causing spoofing interference is crucial in ensuring the secure operation of GNSS. This article proposes a method for locating GNSS spoofing interference sources using a moving array antenna. Firstly, the proposed method utilizes the inherent characteristics of the double-differenced carrier phase from the deception signals received by the array antenna to effectively extract the spoofing signals. Subsequently, by moving the antenna array, the original carrier phase single-difference data of multiple observation points for deception signals are fused to provide a cost function for direct localization of spoofing interference, and a solution method for the cost function is designed. The proposed method addresses the challenge of extracting and localizing GNSS spoofing interference weak signals, effectively avoiding the data correlation of traditional two-step methods for DOA estimation parameters and ensuring the location accuracy of spoofing interference and the robustness of the method. The effectiveness of the proposed method has been validated through simulation experiments, and its adaptability to factors such as errors in carrier phase measurements has been examined. The method exhibits strong applicability and is well-suited for the hardware platform of the GNSS nulling antenna, thereby enabling it to possess simultaneous capabilities in both anti-interference and spoofing interference localization.

GNSS Receiver Antenna Absolute Field Calibration System Development: Testing and Preliminary Results

For high-precision Global Navigation Satellite Systems (GNSS) positioning based on carrier-phase measurements, knowledge of the GNSS receiver antenna electrical signal reception characteristics, i.e., phase center, is crucial. Numerous studies have led to the understanding of the influence of GNSS receiver antenna phase center corrections (PCCs) on GNSS positioning accuracy and other estimated parameters (e.g., receiver clock estimates, ambiguities, etc.). With the goal of determining the PCC model of GNSS receiver antennas, only a few antenna calibration systems/facilities are in operation or under development worldwide. The International GNSS Service (IGS) publishes type-mean PCC models for almost all geodetic-grade GNSS antennas. However, the type-mean models are not perfect and do not fully reflect the signal reception properties of individual GNSS receiver antennas. Relevant published scientific research has shown that the application of individual PCC models significantly improves the accuracy of GNSS positioning and other estimated parameters. In this article, the new automated GNSS antenna calibration system, recently developed at the Laboratory for Measurements and Measuring Technique (LMMT) of the Faculty of Geodesy of the University of Zagreb in Croatia, is presented. The developed system is an absolute field calibration system based on the utilization of a Mitsubishi MELFA 6-axis industrial robot. During calibration, the robot tilts and rotates the GNSS antenna under test (AUT) around a fixed point within the antenna. The antenna PCC modelling is based on time-differenced double-difference carrier-phase observations. Our preliminary results for the Global Positioning System (GPS) L1 (G01) frequency show a submillimeter repeatability of the estimated PCC model and a submillimeter agreement with the Geo++ GmbH calibration results.

An Efficient Fault Detection and Exclusion Method for Ephemeris Monitoring

The ephemeris fault needs to be detected and mitigated in the ground-based augmentation system to provide precision approach for the aircraft. In the current fault detection and exclusion (FDE) method, the double-differenced carrier phase (DDCP) observation is used as a test statistic to detect a faulty satellite caused by an ephemeris fault, taking advantage of the residual spatial gradient. However, the current FDE method cannot distinguish whether the fault comes from a reference satellite (RS) or a non-reference satellite (NRS) in DDCP. One way to address this issue is to pre-validate the RS before it can be used to form a DDCP test statistic for detecting ephemeris fault on the NRS. The RS is pre-validated using the previous ephemeris for any newly acquired and re-acquired satellite. This method is developed in detail to present the shortcomings. A more efficient FDE method using multiple hypothesis testing to detect ephemeris fault on both the RS and NRS simultaneously in real time is proposed. Moreover, to facilitate the application in integrity monitoring, the test risks and minimum detectable error are analyzed. The numerical results of the proposed FDE method show an improved performance in detecting ephemeris fault on the RS and a comparable performance on the NRS compared with the current FDE method.

Time-differenced double difference method for measurement of Navigation with Indian Constellation (NavIC) receiver differential phase bias

The pseudo-range (PR) and carrier phase (CP) observations of Global Navigation Satellite System (GNSS)/Navigation with Indian Constellation (NavIC) receivers are influenced due to various errors like ionosphere delay, troposphere delay, receiver and satellite clock error, multipath, integer ambiguity, receiver hardware noise etc. However, the real-time positioning and navigation applications are in quest of high-quality carrier phase measurements with zero bias. The GNSS/NavIC receiver hardware noise is one of the critical error sources that degrade the quality of carrier phase measurements. And it can be measured by computing the differential phase bias or carrier phase bias employing the double-difference method. This research article proposes a novel approach called Time Differenced Double Difference Carrier Phase (TDDDCP) for estimating differential phase bias or carrier phase bias of NavIC receiver. Moreover, the proposed TDDDCP technique has the advantage over the Double Difference Carrier Phase (DDCP) because it requires data from only one satellite with two zero base length receivers for which the differential phase bias is to be measured. The experimental results show that the differential phase bias value differs for different satellites, ranging from −0.3904 m to 0.3168 m. The bias-free carrier phase observations based on position coordinates (X, Y, Z) found improved precision of standard deviation of 0.4176 m, 0.645 m, and 0.328 m, respectively.

Global Navigation Satellite System Real-Time Kinematic Positioning Framework for Precise Operation of a Swarm of Moving Vehicles.

The global navigation satellite system (GNSS) real-time kinematic (RTK) technique is used to achieve relative positioning centimeter levels among multiple agents on the move. A typical GNSS RTK estimates the relative positions of multiple rover receivers with respect to a single-base receiver. In a fleet of rover GNSS receivers, this approach is inefficient because each rover receiver only uses GNSS measurements of its own and those sent from a single-base receiver. In this study, we propose a novel GNSS RTK framework that facilitates the precise positioning of a swarm of moving vehicles through the GNSS measurements of multiple receivers and broadcasts fixed-integer ambiguities of GNSS carrier phases. The proposed framework not only provides efficient RTK positioning but also reliable performance with a limited number of GNSS satellites in view. Our experimental flight tests with six GNSS receivers showed that the systematic procedure of the proposed framework could maintain lower than 6 cm of 3D RMS positioning errors, whereas the conventional RTK failed to resolve the correct integer ambiguities of double difference carrier phase measurements more than 13% in five out of nine total baselines.

Evaluation of the performance of GNSS-based velocity estimation algorithms

Global Navigation Satellite System (GNSS) based velocity estimation is one of the most cost-effective and widely used methods in determining velocity in geodesy and transport applications. Highly accurate and reliable velocity measurements can be obtained by exploiting the raw Doppler, carrier phase, and pseudorange measurements with a GNSS receiver. There are several approaches to GNSS-based velocity determination. This paper investigates the characteristics of the approaches which are currently popular and applicable to the observations of Global Positioning System (GPS), BeiDou Navigation Satellite System (BDS), and their combination (GPS/BDS). Specifically, it evaluates the performance of the velocity estimated based on the Raw Doppler method, the Time-Differenced Pseudorange method, the Time-Differenced Carrier Phase method, and the Double-Differenced Carrier Phase method, in both static and dynamic modes and in open and urban scenarios. The experiments show that BDS has the advantages in delivering accurate velocity determinations over GPS in the Asia–Pacific region, and the effectiveness of the GPS/BDS in improving the overall accuracy of velocity determination in complex urban scenarios.

Adaptive weighted particle swarm optimization (AWPSO) attitude determination algorithm based on Chi-square test

The optimal search algorithm based on double-difference carrier phase is one of the methods to solve the attitude angle. An adaptive weighted particle swarm optimization (AWPSO) algorithm based on the Chi-square test is proposed to solve the attitude angle of ultra-short baseline. We establish the fitness function by introducing the relationship between attitude angle and baseline vector into the observation equations of double-difference carrier phase. Searching the attitude according to the fitness model instead of calculating the angle directly, which avoids solving the integer ambiguity. Using adaptive inertial weights and synchronous adaptive learning factors to speed up the convergence of attitude search. Constructing the candidate solution sequence to save the search result and calculating the Mahalanobis distance of the candidate solution. The local optimal solution is eliminated by the Chi-square test, and the attitude angle is gained by weighting the filtered sequence. Various static experimental results show that the new algorithm performs better than the direct solution method, the least square estimation method, and the PSO-based attitude solution method. The root mean square error of yaw angle and pitch angle is 0.245° and 0.236° directly at 0.575 m baseline.

Noises in Double-Differenced GNSS Observations

Precise data processing from the Global Navigation Satellite Systems (GNSS) reference station network is mainly based on a combination of double-differenced carrier phase and code observations. This approach allows most of the measurement errors to be removed or reduced and is characterized as the most accurate method. However, creating observation differences between two receivers and two satellites increases the measurement noise of the observations by a factor of 2. As a result, it increases the impact of the incorrect definition of the noise characteristic on the results of the estimation of the unknowns in the positioning model. This is especially important in Multi-GNSS solutions, which integrate measurements from different systems, for which the stochastic parameters of observation may differ significantly. In this paper, the authors prepared a complex analysis of the noise type in double-differenced GNSS (GPS, GLONASS and Galileo) observations, both carrier phase and code ones, with a 1 s sampling interval. The Autocorrelation Function (ACF) method, the Lomb–Scargle (L-S) periodogram method, and the Allan variance (AVAR) method were used. The results that were obtained for the weekly set of measurement data showed that, depending on the system and type of observation, the noise level and its type are significantly different. Among the code measurements, the lowest noise levels were obtained for the GPS C5Q and Galileo C7Q/C8Q observations, with the standard deviations not exceeding ±10 cm, while the noisiest observations were for the GLONASS C1C and C2C signals, which had standard deviations of about ±90 cm and ±45 cm, respectively. For the carrier phase observations, each signal type was characterized by very similar noise levels of ±1.5–3.5 mm. The ACF analysis showed that 1 Hz double-differenced GNSS data can only be treated as being not correlated to time for carrier phase observations; for code observations, an irrelevant autocorrelation may be considered for measurement intervals greater than 20 s. Depending on the GNSS signals, the spectral index k varies in a range from −1.3 to −0.2 for code data and k = 0.0 in the case of phase data. Using the modified Allan deviation (MDEV) allows for specific noise types for each signal and GNSS system to be determined. All of the code observations were characterized by either flicker PM or white PM. In the case of the phase observations, they were all uniquely characterized by white PM (GPS and Galileo or by white PM and flicker PM (GLONASS).

Improvement of Multi-GNSS Precision and Success Rate Using Realistic Stochastic Model of Observations

With the advancement of multi-constellation and multi-frequency global navigation satellite systems (GNSSs), more observations are available for high precision positioning applications. Although there is a lot of progress in the GNSS world, achieving realistic precision of the solution (neither too optimistic nor too pessimistic) is still an open problem. Weighting among different GNSS systems requires a realistic stochastic model for all observations to achieve the best linear unbiased estimation (BLUE) of unknown parameters in multi-GNSS data processing mode. In addition, the correct integer ambiguity resolution (IAR) becomes crucial in shortening the Time-To-Fix (TTF) in RTK, especially in challenging environmental conditions. In general, it is required to estimate various variances for observation types, consider the correlation between different observables, and compensate for the satellite elevation dependence of the observable precision. Quality control of GNSS signals, such as GPS, GLONASS, Galileo, and BeiDou can be performed by processing a zero or short baseline double difference pseudorange and carrier phase observations using the least-squares variance component estimation (LS-VCE). The efficacy of this method is investigated using real multi-GNSS data sets collected by the Trimble NETR9, SEPT POLARX5, and LEICA GR30 receivers. The results show that the standard deviation of observations depends on the system and the observable type in which a particular receiver could have the best performance. We also note that the estimated variances and correlations among different observations are also dependent on the receiver type. It is because the approaches utilized for the recovery techniques differ from one type of receiver to another kind. The reliability of IAR will improve if a realistic stochastic model is applied in single or multi-GNSS data processing. According to the results, for the data sets considered, a realistic stochastic model can increase the computed empirical success rate to 100% in multi-GNSS as well as a single system. As mentioned previously, the realistic precision of the solution can be achieved with a realistic stochastic model. However, using the estimated stochastic model, in fact, leads to better precision and accuracy for the estimated baseline components, up to 39% in multi-GNSS.

Instantaneous, Dual-Frequency, Multi-GNSS Precise RTK Positioning Using Google Pixel 4 and Samsung Galaxy S20 Smartphones for Zero and Short Baselines.

The recent development of the smartphone Global Navigation Satellite System (GNSS) chipsets, such as Broadcom BCM47755 and Qualcomm Snapdragon 855 embedded, makes instantaneous and cm level real-time kinematic (RTK) positioning possible with Android-based smartphones. In this contribution we investigate the instantaneous single-baseline RTK performance of Samsung Galaxy S20 and Google Pixel 4 (GP4) smartphones with such chipsets, while making use of dual-frequency L1 + L5 Global Positioning System (GPS), E1 + E5a Galileo, L1 + L5 Quasi-Zenith Satellite System (QZSS) and B1 BeiDou Navigation Satellite System (BDS) code and phase observations in Dunedin, New Zealand. The effects of locating the smartphones in an upright and lying down position were evaluated, and we show that the choice of smartphone configuration can affect the positioning performance even in a zero-baseline setup. In particular, we found non-zero mean and linear trends in the double-differenced carrier-phase residuals for one of the smartphone models when lying down, which become absent when in an upright position. This implies that the two assessed smartphones have different antenna gain pattern and antenna sensitivity to interferences. Finally, we demonstrate, for the first time, a near hundred percent (98.7% to 99.9%) instantaneous RTK integer least-squares success rate for one of the smartphone models and cm level positioning precision while using short-baseline experiments with internal and external antennas, respectively.

Multipath Error Fusion Modeling Methods for Multi-GNSS

The multipath error is considered to be the most limiting factor for high precision positioning applications. The sidereal filtering (SF) method can be used to mitigate the multipath error in the observation domain, and it has been successfully applied in the multipath mitigation in global positioning systems (GPS) and regional BeiDou navigation satellite systems (BDS2). However, there are few reports on the SF method in other systems. The performance of the SF method relies on the explicit orbit repeat periods of satellites in diverse systems or even different types of constellations. It is therefore inconvenient to utilize the SF method for multi-GNSS multipath error mitigation. Alternatively, a space domain multipath error reduction method, which establishes the multi-point hemispherical grid model (MHGM) using the residuals of the double-differenced carrier phase observations in the ambiguity-fixed period, has been modified. It is an integrated model for multi-GNSS, without considering the diversity of different systems and constellations. To compare the performance of MHGM and SF from a multi-GNSS point of view, the determination method of orbit repeat periods via the broadcast ephemerides is summarized, and the SF method is extended to the global BeiDou navigation satellite system (BDS3) and Galileo navigation satellite system. Further test results show that the performance of MHGM and SF are comparable from the perspective of root mean squares (RMS) and the power spectrum analysis of double-differenced residuals, as well as the static positioning results. This implies that the space domain MHGM can obtain similar correction effects as the SF method in the observation domain, but the former is more flexible for modeling with various systems’ data. In addition, the established MHGM using the data of multi orbit periods demonstrates a better performance compared with that of only one orbit period, and an average improvement of 13.1% in the RMS of the double-differenced residuals can be achieved.

Raw GNSS observations from Android smartphones: characteristics and short-baseline RTK positioning performance

The release of raw global navigation satellite system (GNSS) observations by Google Android makes high-precision positioning possible with low-cost smart devices. This study contributes to this research trend by constructing a stochastic model based on raw GNSS observation characteristics from Android smartphones and verifying the feasibility of smartphone-based ambiguity fixing in the short-baseline real-time kinematic (RTK) case. This study uses the raw observation standard deviations (ROSTDs) delivered by the Android application programming interface (API) as a stochastic model and takes advantage of the multipath index from the API to rule out unusable observations. As well as these, the ambiguity integer property is investigated by analyzing the residuals of double-differenced carrier-phase observations associated with one smartphone and one geodetic-grade receiver. Furthermore, we note that the carrier-phase observations collected by tested smartphones do not have the integer property but for the Huawei P30 and Xiaomi 8 devices, such an integer property can be successfully recovered by means of detrending. With the use of ROSTD-dependent weighting, we first perform single-point positioning (SPP) and real-time differentition (RTD) using pseudorange observations delivered by the Huawei P30 and Xiaomi 8 devices. The results show that the stochastic model is applicable to the Xiaomi 8. Moreover, the three-dimensional root-mean-square (3D-RMS) errors of the two smartphones for SPP are 1.28 m and 1.96 m, and the 3D-RMS errors for RTD are 0.79 m and 1.64 m, respectively. We next test the RTK positioning performance based on a short-baseline of 882 m using carrier-phase observations with recovered integer ambiguities. For the Huawei P30, the positioning errors achieved were 7.8, 2.4, 1.1 mm for the east, north, and up (ENU) components at the time of first fix while for the Xiaomi 8, the positioning errors achieved were 4.3, 4.2, 4.2 mm for the ENU components at the time of first fix.

A New Train Integrity Resolution Method Based on Online Carrier Phase Relative Positioning

Train integrity resolution is an important part of a train operation control system. In current train operation control systems, train integrity resolution is achieved by using information on the track circuit or the wind pressure. However, both approaches have limitations; the utilization of the track circuits requires high construction and maintenance costs, and the wind pressure approach has to overcome the problem of low update rate and false alarm caused by the incident pipe leakage. In this paper a new train integrity resolution method based on double-differenced carrier phase relative positioning is proposed. To improve the system performance, an online integer ambiguity fixing method is applied to overcome the frequent integer ambiguity re-fixing problem caused by the high train speed and the complex operational environment. Moreover, the elevation angle constraint optimal distribution based reference satellite selection method is applied to guarantee optimal satellite geometry distribution during train operation. To evaluate the performance of the proposed train integrity resolution method, a real train test was carried out on the Beijing-Shenyang high speed railway. The experimental results show that the proposed train integrity resolution method can provide train integrity solutions with an accuracy of better than 0.5 m, with STD and RMS of 0.15 m and 0.17 m respectively. Compared with the conventional GNSS single point positioning based train integrity method, the proposed method can achieve a much better performance, with an improvement, in this test, of 77%.

Ephemeris monitor with ambiguity resolution for CAT II/III GBAS

In safety-critical applications, such as the Ground-Based Augmentation System for precision approaches in civil aviation, it is important to safeguard users under the case of ephemeris failures. For CAT II/III approaches, different ephemeris monitors with approaches for ambiguity resolution are proposed with the double differenced carrier phase as the test statistics. The continuity risks introduced by the ambiguity resolution are addressed by deriving the required averaging time for new, acquired, and re-acquired satellites. Since the ephemeris fault is closely related with the baseline length between ground stations, the minimum baseline length is derived to meet the probability of missed detection region. Current methods are compared with both the averaging time and the ground baseline length. It is demonstrated that a combination of two methods is able to achieve the best performance with 94 averaging epochs and 218 m ground baseline length.

A New Method of Real-Time Kinematic Positioning Suitable for Baselines of Different Lengths

This study introduces a new real-time kinematic (RTK) positioning method which is suitable for baselines of different lengths. The method merges carrier-phase wide-lane, and ionosphere-free observation combinations (LWLC) instead of using pseudo-range, and carrier-phase ionosphere-free combination (PCLC), or single-frequency pseudo-range and phase combination (P1L1). In a first step, the double-differenced wide-lane ambiguities were calculated and fixed using the pseudo-range and carrier-phase wide-lane combination observations. Once the double-differenced wide-lane integer ambiguities were known, the wide-lane combined observations were regarded as accurate pseudo-range observations. Subsequently, the carrier-phase wide-lane, and ionosphere-free combined observations were used to fix the double-differenced carrier-phase integer ambiguities, achieving the final RTK positioning. The RTK positioning analysis was performed for short, medium, and long baselines, using the P1L1, PCLC, and LWLC methods, respectively. For a short baseline, the LWLC method demonstrated positioning accuracy similar to the P1L1 method, and performed better than the PCLC method. For medium and long baselines, the positioning accuracy of the LWLC method was slightly higher than those of the PCLC and P1L1 methods. In conclusion, the LWLC method provided high-precision RTK positioning results for baselines with different lengths, as it used high-precision carrier-phase observations with fixed ambiguities instead of low-precision pseudo-range observations.

BDS RTK High-precision Positioning Algorithm for Deformation Solution

In view of the increasing scope, scale and quantity of landslide, debris flow and other geological disasters and without achieving long-term stable high-precision monitoring in the exist, a solution algorithm on RTK high-precision differential positioning deformation is presented in this paper. The algorithm uses high-precision board receiver as the satellite positioning reference station, the u-blox receiver as the monitoring station. The carrier phase double difference relative positioning model is based on the carrier phase observation and pseudo range observation, combined with the RTK unscented Kalman filter model and smooth filtering algorithm, and then more accurate position results are obtained. This method can be used to calculate the position of deformation monitoring body with high precision, which is simple, easy to realize and practical. The positioning accuracy of deformation monitoring is higher. After smoothing, the horizontal precision can reach 3mm, and the elevation precision can reach 5mm.

BDS triple-frequency carrier phase combination smoothing pseudo-range algorithm

The carrier phase smoothing pseudo-range algorithm can effectively reduce the influence of multi-path effect and observation noise on pseudo-range positioning and improve the accuracy of pseudo-range positioning. Aiming at the problem that the existing classic Hatch filter smoother is susceptible to the cumulative interference of ionospheric delay in the pseudo-range smoothing process, the ionospheric delay and observed noise of the original double-difference observation are further weakened by constructing the optimal Beidou triple-frequency linear combination. The triple-frequency double-difference carrier phase combination are used to smooth the pseudo-range combination to replace the original observations. Finally, four groups of Beidou triple-frequency measured data with different baseline lengths, including 4m, 14.4km, 61.7km and 131.6km, are used to verify the algorithm. The theoretical derivation and effectiveness of the algorithm are verified by comparing and analyzing the smoothing effect of the algorithm in the following four groups of baseline conditions.