An asf estimation method based on trend kriging for improving positioning and navigation accuracy of rowland system

Abstract

The application discloses an ASF estimation method based on trend Kriging for improving positioning and navigation precision of a Loran system, and belongs to the field of positioning and navigation. The method is based on a Kriging model, selects effective base functions based on limited real ASF data to construct an optimal trend function, and comprises two nested tasks, namely, selecting an optimal base function and searching for optimal correlation parameters. In order to overcome the challenge that the nested nature of the two tasks is easy to lead to local optimization, the two tasks are integrated into an overall optimization framework. Under the framework, an NDE algorithm with strong global search capability is used to simultaneously search for optimal solutions of the base function and residual correlation parameters. In addition, residual covariance minimization is used as the optimization target of the two tasks, so that the geographical spatial correlation of the sampling data is ensured.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of positioning and navigation, and more particularly relates to a trend Kriging-based ASF estimation method for improving the positioning and navigation accuracy of Loran system. BACKGROUND

[0002] Global Navigation Satellite System (GNSS) is an important component of Positioning and Navigation and Timing (PNT) system, but it is vulnerable to interference or attack, leading to failure or even paralysis, and making safety face extreme risk. Therefore, a backup positioning and navigation system with strong anti-interference ability and independent of GNSS should be constructed and developed. Loran system (including enhanced Loran system) is internationally recognized as the most powerful backup of GNSS because of its stable ground wave phase, strong anti-interference ability and large coverage.

[0003] At present, it is difficult for Loran system to achieve high-precision positioning, which is mainly due to the influence of complex topography and various meteorological factors on the actual propagation time of Loran signal during ground propagation, resulting in unpredictable changes in the actual propagation time of the signal with space and time. Usually, the signal is assumed to propagate uniformly and fixedly in ideal atmosphere for positioning. When the transmitting and receiving ends are fixed, there is a changing propagation time error between the actual propagation time of the signal and the fixed and predictable theoretical propagation time. This error is usually referred to as Additional Secondary Phase Factor (ASF), which is the main error source hindering the improvement of Loran positioning accuracy. In order to improve the positioning accuracy, ASF must be determined and used to correct the observation value. A small deviation of ASF can lead to a positioning error of several kilometers. Therefore, accurate estimation of ASF is crucial to improve the positioning accuracy of Loran. SUMMARY

[0004] In view of the above defects or improvement needs of the prior art, the present application provides a trend Kriging-based ASF estimation method for improving the positioning and navigation accuracy of Loran system, which can improve the ASF estimation accuracy based on small sample ASF measured data.

[0005] To achieve the above-mentioned purpose, according to the first aspect of the present application, a trend Kriging-based ASF estimation method for improving the positioning and navigation accuracy of Loran system is provided, comprising:

[0006] S1, measuring the actual ASF data by using a Loran receiver at the sampling point x in the target area; wherein the actual ASF data includes the coordinates of the sampling point x and the corresponding ASF value y(x), the coordinates of the sampling point x=[x1, x2, …, xn], and the corresponding ASF value y(x)=[y1, y2, …, yn]. n ] T n is the number of sampling points, and the coordinates of the t(th) (t=1, 2, …, n) sampling point are represented as x t =[x t,1 ,x t,2 ,…,x t,mm is the coordinate dimension of the sampling points;

[0007] S2, constructing the base function selection parameter vector H of the Kriging model i the residual related parameter vector θ i , H i is combined with θ i ; i , initialize the population Q = [Q1, Q2, …, Q K ], i = 1, 2, …, K, iteratively update the population using the mutation, crossover and selection operations of the NDE algorithm to obtain the optimal population Q', and select the optimal individual Q opt from Q' according to the objective function;

[0008] The selection operation is realized based on the objective function, Q i = [H i , θ i ] = [H i,1 , H i,2 , …, H i,nf , θ i,1 , θ i,2 , …, θ i,m ], H i,x is used to represent whether the base function included in the trend function of the Kriging model constructed according to the individual Q i includes the x-th base function in the candidate base function set, x = 1, 2, …, nf, θ i,y is the y-th residual related parameter of the Kriging model constructed according to the individual Q i , y = 1, 2, …, m;

[0009] S3, obtaining the optimal base function f opt from the optimal base function selection parameter vector H opt in Q opt ; constructing the Kriging model according to y (x), f opt and the residual related parameter vector θ opt in Q opt , to obtain the ASF estimation value of the unknown point x0

[0010] F' = [f opt (x1), f opt (x2), …, f opt (x n )] T and f opt (x0) T are obtained by bringing the coordinates of the sampling points x and the unknown point x0 into the optimal base function f opt , r'(x0) = [R(θ opt,x1,x0),R(θ opt ,x2,x0),…,R(θ opt ,x n ,x0)] T is the correlation vector between x0 and x,

[0011] is the Gaussian correlation model, t2=1,2,…,n, Indicates sampling point With sampling point The distance between

[0012] According to a second aspect of the present invention, there is provided an electronic device comprising: a computer-readable storage medium and a processor;

[0013] The computer-readable storage medium is used to store executable instructions;

[0014] The processor is configured to read the executable instructions stored in the computer-readable storage medium and execute the method according to the first aspect.

[0015] According to a third aspect of the present invention, a computer-readable storage medium is provided, wherein the computer-readable storage medium stores computer instructions, and the computer instructions are used to enable a processor to execute the method according to the first aspect.

[0016] According to a fourth aspect of the present invention, there is provided a computer program product comprising a computer program or instructions, which implement the method according to the first aspect when executed by a processor.

[0017] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects compared with the prior art:

[0018] The method provided by the application integrates the base function selection of the Kriging model and the residual correlation parameter search into an overall optimization framework, defines the base function selection task as a real coding continuous optimization task, allows the two tasks to be simultaneously optimized in the same coding manner, simplifies the optimization process, effectively solves the interdependence between the two tasks, and improves the prediction accuracy and reliability of the Kriging model; meanwhile, the application adopts residual covariance minimization as the optimization target of the two tasks, ensuring the consistency of the residual space correlation characteristics. The real number coding continuous representation of the base function selection is mapped into a binary coding discrete form, the residual covariance fitting degree is used to evaluate the quality of the local Kriging model composed of the discrete variables and the continuous variables, and the spatial distance between the to-be-estimated point and the adjacent known sample points and the positional relationship between the adjacent sample points are fully considered; in addition, the application simultaneously searches for the optimal combination of the base function and the residual correlation parameter by using the newly proposed strong global random optimization algorithm NDE, avoids the problem of falling into a local optimum, greatly improves the probability of finding a global optimal solution, and thus improves the accuracy of the ASF estimation.

[0019] To sum up, the method provided by the application can obtain high-precision ASF values based on a limited actual ASF data set, solves the problems that the existing theoretical calculation method cannot be used for large-area ASF prediction, and that high-density actual ASF measurement is long in period and consumes manpower and material resources, and the like, does not need to consider related information such as geological parameters and propagation environment, and is more universal in application scenarios. BRIEF DESCRIPTION OF DRAWINGS

[0020] Figure 1 A flowchart of an ASF estimation method based on trend Kriging for improving the positioning and navigation accuracy of a Loran system is provided for the embodiments of the application.

[0021] Figure 2 A schematic diagram of the hyperbolic positioning principle of a Loran system.

[0022] Figure 3 A schematic diagram of an actual data waveform collected at one site is provided for the embodiments of the application.

[0023] Figure 4 A schematic diagram of the positioning results after correction by the method provided by the embodiments of the application and the existing method; wherein the vertical coordinate is the positioning accuracy, and the horizontal coordinate is the method type.

[0024] Figure 5 A box plot of the positioning error of all sampling data after correction by the method provided by the embodiments of the application and the existing method; wherein the vertical coordinate is the positioning accuracy, and the horizontal coordinate is the method type. DETAILED DESCRIPTION

[0025] In order to make the objects, technical solutions and advantages of the present application clearer, further detailed description will be made to the present application in combination with the accompanying drawings and examples. It should be understood that the specific examples described herein are only used to explain the present application and not used to limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0026] The ASF estimation methods mainly include prediction based on a theoretical ground wave propagation model and interpolation based on actual measurement. By establishing a propagation model based on different ground types (plane, sphere, uniform smooth, non-homogeneous complex terrain, etc.), various ASF theoretical prediction methods are continuously developed. The method of theoretical calculation is relatively easy to implement compared with actual measurement, and the prediction accuracy thereof mainly depends on the geological parameters of the divided path. Due to the complexity of the terrain and the secrecy of the special area, it is difficult to accurately obtain the real geological parameters, which leads to low prediction accuracy of the theoretical prediction method. Compared with the theoretical calculation method, the interpolation based on extensive actual measurement data can obtain higher accuracy of the ASF value, but the cost of manpower and material resources is huge. Therefore, among the numerous ASF estimation methods, the interpolation based on a small amount of ASF actual measurement data is the most practical and economical. At this time, the performance of the interpolation method directly affects the estimation accuracy of the ASF.

[0027] Since the ASF of different locations in a local area does not change linearly with the geographical position, but changes complexly and unpredictably according to the terrain and weather on the propagation path, it is difficult for the traditional linear interpolation method to obtain high-precision ASF under limited measurement data. The prior art provides several spatial interpolation methods to improve the estimation accuracy of the ASF, which can be divided into two categories: one is to directly apply the traditional Kriging method for ASF estimation, including ordinary Kriging (OK) and universal Kriging (UK); the other is to improve the Kriging method according to the characteristics of the actual ASF data to obtain high-precision ASF. These studies show that the Kriging method is a kind of ASF estimation method suitable for the case of limited measurement data. However, these Kriging-based ASF estimation methods all have a common shortcoming that reduces their performance, that is, they cannot adaptively capture the actual trend of the actual ASF data. This is because their trend function is fixed and composed of all candidate basis functions of a specified order. Therefore, selecting appropriate basis functions to form the best trend function can effectively improve the prediction accuracy of the Kriging method.

[0028] Based on this, the embodiment of the present application provides an ASF estimation method based on trend Kriging for improving the positioning and navigation accuracy of the Loran system, as shown in Figure 1 , which comprises:

[0029] S1, the actual ASF data is obtained by using a Loran receiver to measure at a sampling point x in a target area; wherein the actual ASF data comprises coordinates of the sampling point x and a corresponding ASF value y(x), and the sampling point x = [x1, x2, …, xn] (n is a number of sampling points). n ] T The t-th (t = 1, 2, …, n) sampling point is represented as x t = [x t,1 1, x t,2 2, …, x t,m m is a coordinate dimension of the sampling point, and m = 2 is usually set in the Loran system (including longitude and latitude).

[0030] Specifically, the actual ASF data is obtained by using a Loran receiver to accurately measure at multiple locations in a specified actual area, including sampling point coordinates and corresponding ASF values.

[0031] Taking an enhanced Loran system as an example, in step S1, the Loran receiver receives time differences of arrival (TDOA) of multiple Loran signals at each test point, and the point ASF value is calculated by using measurement results of the Loran receiver and a GNSS receiver. The Loran system is based on hyperbolic positioning principle for positioning, as shown in FIG. 1, and the specific calculation process of the ASF is described by taking this as an example. Figure 2

[0032] Figure 2 In the formula, M is a main station, S1 and S2 are sub-stations, and U is a user receiver, i.e., a sampling point. Therefore, distances from the two sub-stations to the main station are known, and are respectively represented as d M1 and d M2 Distances from the three Loran stations S1, S2 and M to U are unknown, and are respectively represented as d1, d2 and d M Further, a difference between d1 and d M is defined as Δd1, and a difference between d2 and d M is defined as Δd2. Based on Δd1 and Δd2, two hyperbolic position lines can be obtained, and U is at an intersection of the two hyperbolic position lines.

[0033] A time difference of arrival of signals from S h (h = 1, 2) and M to U is defined as TDOA h In the Loran system, a time at which S h transmits a signal is delayed relative to M by a fixed and known delay, which is defined as τ h Therefore, an actual propagation time difference Δt p,h of the signals from S h and M to U is represented as:

[0034] Δt p,h = TDOA h ​-τ h

[0035] In actual engineering, since Δt p,h is variable, it is usually assumed that the signal propagates uniformly at the speed of light between the transceiver, and a fixed predictable theoretical propagation time difference is used for positioning, so the theoretical time difference of the signal propagating from S h and M to U at the speed of light c is defined as Δt l,h , which is expressed as:

[0036]

[0037] Δt l,h can be calculated by the high-precision position coordinates provided by the GNSS receiver and the above formula, and the error between the actual propagation time and the theoretical propagation time of the signal on the same path is the ASF. Here, the difference between the ASFs of the two paths is calculated, and the ASF from S h and M to U is defined as Δt ASF,h , which can be calculated by the following formula:

[0038] Δt ASF,h =Δt p,h -Δt l,h =TODA h -τ h -Δt l,h

[0039] The above calculation process is also applicable to other Loran systems.

[0040] S2, construct the base function selection parameter vector H i and the residual related parameter vector θ i , combine H i and θ i into an individual Q i , initialize the population Q = [Q1, Q2, …, Q K ], i = 1, 2, …, K, use the mutation, crossover and selection operations of the NDE algorithm to iteratively update the population, obtain the optimal population Q`, and select the optimal individual Q opt from Q` according to the objective function; wherein the selection operation is realized based on the objective function.

[0041] Specifically, in step S2, a Kriging model is constructed using the actual ASF data collected in S1, and theoretical calculation formulas for each parameter are obtained based on the constructed Kriging model; then, the base function selection vector and the residual related vector in the Kriging model are defined to facilitate subsequent solving; finally, the mutation, crossover and selection operations of the NDE algorithm are used to obtain the optimal population Q`, and the optimal individual Q optThat is, step S2 includes:

[0042] S21, using the actually collected ASF data to construct a Kriging model.

[0043] The model includes a trend term and a residual error, and the relationship between the coordinates x of the sampling points and the ASF values y(x) is expressed as:

[0044] y(x) = f(x) T + β + z(x)

[0045] Where f(x) = [f1(x), f2(x), …, fn(x)] is the trend function composed of k base functions; β = [β1, β2, …, βk] is the coefficient of the trend function; z(x) = [z(x1), z(x2), …, zn] is the residual error term, and n is the number of sampling points. k T k T n T

[0046] z(x) is usually assumed to be a stationary Gaussian process, which has:

[0047] E(z(x t )) = 0

[0048]

[0049] where σ 2 is the process variance; is a correlation model of unknown parameters θ = [θ1, θ2, …, θn m ] T , which reflects the geographical correlation between and , t1 = 1, 2, …, n, t2 = 1, 2, …, n. The Gaussian correlation model is expressed as:

[0050]

[0051] where denotes the distance between and .

[0052] Based on the constructed Kriging model, the parameters β, θ and σ 2 are expressed as

[0053]

[0054] S22, define the base function selection vector and the residual error correlation vector in the Kriging model to facilitate subsequent solution.

[0055] ​​​​​​The selection of base functions and the residual correlation parameters in the Kriging model are binary discrete optimization and real continuous optimization problems, respectively. The solutions of the two problems are nested, i.e., the optimization of the former needs to determine the latter, and the optimization of the latter also needs to determine the former. In order to solve the problem of falling into local optimum caused by the nested nature, an optimization framework is designed to solve the two problems as one. That is, considering the mutual transformation of binary discrete optimization and real optimization, the base function selection parameter vector and the residual correlation parameter vector are defined as continuous variables H i and θ i , combined into an individual, denoted as:

[0056] Q i =[H i ,θ i ]=[H i,1 ,H i,2 ,…,H i,nf ,θ i,1 ,θ i,2 ,…,θ i,m ]

[0057] where nf is the number of candidate base functions. H i,x is used to represent whether the xth base function in the candidate base function set is included in the base function of the trend function of the Kriging model constructed according to the individual Q i , and θ i,y is the yth residual correlation parameter of the Kriging model constructed according to the individual Q i , y = 1, 2, …, m.

[0058] In the optimized base function set, the given base function is a polynomial, denoted as is an integer power of x t , e = 1, 2, …, m. P is the highest order of the trend function in the Kriging model, and the total number of candidate base functions is For x t , the candidate base function set f is:

[0059]

[0060] In order to facilitate the selection of base functions, preferably, the value range of H i,x is set to [0, 1], H i,x is the continuous representation of whether to select the xth base function from the candidate base function set, and the value range of θ i,y is (0, 20]. H i,j is mapped to a binary coded discrete vector, i.e.:

[0061]

[0062] wherein, denotes whether the x-th basis function is selected from the set of candidate basis functions, and is equal to 1 if this basis function is selected as a basis function in the trend function, and is equal to 0 if this basis function is not selected.

[0063] According to the above equation, a new binary vector can be obtained, i.e. This vector and the residual related vector can be combined into a binary-real encoded vector, denoted as:

[0064]

[0065] Therefore, the vector Q i is essentially the same as I i , only the encoding method of the basis function selection vector is different, wherein, in the subsequent calculation process, Q i is updated using a random search algorithm, while I i is used to construct the Kriging model. By optimizing Q i , the essence is to solve the two problems simultaneously, which can well solve the local optimization problem caused by the nesting property of the two problems.

[0066] S23, use the mutation and crossover operations of the NDE algorithm to generate different two vector combinations, use the selection operation of the NDE algorithm to select the optimal combination from multiple vector combinations, obtain the optimal population Q`, and select the optimal individual Q opt from Q` according to the objective function.

[0067] Initialize the population Q = [Q1, Q2, …, Q K ]: randomly take values in the range of each dimension of H i and each dimension of θ i ; take the value range of H i,x as [0, 1] and the value range of θ i,y as (0, 20] for example, then randomly take values in the range [0, 1] for each dimension of H i , and randomly take values in the range (0, 20] for each dimension of θ i . Therefore, the matrix Q and the corresponding matrix I can be obtained. Wherein, Q is a real number encoded matrix, and I is a binary-real encoded matrix. The NDE algorithm (niche differential evolution algorithm) is a random optimization algorithm with strong global search ability proposed in recent years, which can well solve the high-dimensional real number coding problem. The mutation operation and the crossover operation of the NDE algorithm are used to update Q multiple times.

[0068] Among them, the minimization of the residual covariance model fitting is used as the objective function to evaluate the Kriging model composed of each vector combination.

[0069] Evaluate the quality of I in each two-vector combination to evaluate the vector combination (Q i , I i ) in I i For example: Based on I i The constructed Kriging model includes the trend term (i.e. f(x) = [f1(x), f2(x), ..., f k (x)] T and β = [β1, β2, ..., β k ] T The product of) and the residual term, according to I i in The value of , selects the corresponding subset from the candidate basis function set to form the trend function f i , according to I i The residual term is calculated by the residual correlation vector in , so as to obtain the residual term according to the vector combination (Q i , I i ) constructed using the Kriging model.

[0070] The residual covariance model is used to evaluate the kriging model composed of each vector combination. That is, the optimization target is the actual average covariance of the residuals. According to the covariance model function R(θ, x i , x j ) The theoretical mean covariance C of the residuals calculated k Minimize the difference between .

[0071] First, calculate the distance between the two sample points and the actual covariance of the ASF value to get n 2 A vector of distance-covariance relationships d s and are the actual covariance of the distance and ASF value of the sth point pair, n is the number of sampling points, and by grouping the above vectors according to distance, we can get q groups of distance-covariance relationships. For the kth group (k = 1, 2, ..., q), the average distance of all points in the statistical group is and the actual mean covariance Will The residual correlation vector in the generated vector combination is brought into the Gaussian correlation model According to the formula The corresponding theoretical covariance C can be calculated k .

[0072] The objective function is expressed as:

[0073]

[0074] where the actual decision variables include the basis function set f and the related parameters θ.

[0075] where λ k is the weight coefficient, expressed as:

[0076] where v is the power coefficient, which can be set to 0.3, is the average distance of the gth group of sample points, N k and N g represent the number of the kth group and the gth group of sample points, respectively, g = 1, 2, …, q.

[0077] The selection operation of the NDE algorithm selects the optimal combination from multiple vector combinations to obtain the optimal population Q`, and selects the optimal individual Q opt from Q` according to the objective function.

[0078] S3, according to the optimal basis function in Q opt , the parameter vector H opt is obtained. opt ; according to y(x), f opt , and the residual correlation parameter vector θ opt in Q opt , the estimated value of the ASF of the unknown point x0 is obtained.

[0079] Specifically, in step S3, the Kriging model of Q opt is constructed based on the optimal combination to estimate the ASF value of the unknown position.

[0080] According to the basis function selection parameter vector in Q opt , an optimal subset f opt is selected from the complete candidate basis function set f, and the total number of possible candidates of the basis function nf is less than the number of samples.

[0081] According to S21, the Kriging model includes the theoretical calculation formula of the three parameters β, θ, and σ 2 , and β and σ 2 are determined based on the value of θ. The objective function is used instead of the objective function for finding the optimal value of θ in the original Kriging When calculating the theoretical covariance in the objective function, β is used.

[0082] Let x0 be the unknown point to be estimated, and the estimated ASF value of x0 is expressed as:

[0083]

[0084] Where, F=[f(x1),f(x2),…,f(x n )] T and f(x0) T By substituting the coordinates of the sampling point x and the unknown point x0 into the basis function f, we can obtain r(x0)=[R(θ, x1, x0), R(θ, x2, x0), ..., R(θ, x n ,x0)] T is the correlation vector between x0 and x, and R is expressed as

[0085]

[0086] The optimal basis function set f opt and the optimal correlation parameter θ opt Substituting this into the formula, we can obtain the estimated ASF value at x0, which is:

[0087]

[0088] Where, F'=[f opt (x1),f opt (x2),…,f opt (x n )] T , r'(x0) and R' are related to θ opt The relevant vector, r'(x0) = [R(θ opt ,x1,x0),R(θ opt ,x2,x0),…,R(θ opt ,x n ,x0)] T , is a Gaussian correlation model, t1=1,2,…,n, t2=1,2,…,n, Indicates sampling point With sampling point The distance between the y-th dimension coordinates,

[0089] Cross-validation can be used to verify the performance of the ASF estimation method provided by the present invention.

[0090] Cross-validation is used to fully verify the performance of the method using a limited dataset. Cross-validation involves temporarily removing known sample points from the dataset and then using an interpolation algorithm based on the remaining sample points to estimate the ASF for the removed sample points. This process is repeated until all sample points have been estimated. Based on the estimated ASF, the position results for all sample points can be corrected.

[0091] Multiple ASF values can be obtained at each test point, and the original positioning result is corrected based on the estimated ASF value. The positioning accuracy is used to compare the performance of each method, which can comprehensively reflect the accuracy of multiple ASFs. The positioning accuracy refers to the distance (absolute error) between the reference coordinates of the GNSS receiver and the coordinates of the Loran receiver, and is defined as ε. In order to comprehensively evaluate the absolute error result, the mean absolute error (MAE) and the root mean square error (RMSE) are used as performance test standards, and the calculation formula is as follows:

[0092]

[0093] wherein, ε t is the absolute error of the tth sampling point, MAE and RMSE are comprehensive indexes for measuring interpolation accuracy in statistics. The smaller the MAE and RMSE values, the higher the positioning accuracy of the method.

[0094] The method provided by the present application will be further described below with a specific example.

[0095] (1) Actual data collection and waveform explanation

[0096] In order to evaluate the performance of the method provided by the present application truly and reliably, field tests are conducted to collect ASF actual data. The test area is located in Yangjiang City, Guangdong Province, China, in an open environment with an altitude of about 4 meters, surrounded by flat terrain without significant height fluctuations, far away from any natural obstacles that may interfere with the signal. The field test was conducted in sunny conditions in December 2023, with no precipitation and an average temperature of 25 degrees Celsius. Magnetic antennas and electric antennas were deployed at selected test sites, and Loran receivers were used to receive signals from Loran transmitting stations. At the same time, high-precision GNSS receivers were equipped to provide high-precision positioning data as a measurement reference.

[0097] In order to ensure the quality of the test environment and the reliability of the data, the closest station chain to the test area was used for positioning, i.e. the South China Sea station chain. Figure 3 The partial waveform of the ground wave signal of the Loran receiver at the test site was drawn. From Figure 3 It can be seen that the received signal is clean and noise-free. In China's Loran system, each station chain is composed of 1 main station and 2 auxiliary stations, and a main station and each auxiliary station in the same station chain can form a station pair.

[0098] The application adopts hyperbolic positioning, randomly selects 21 sites in a designated area, measures the time difference of arrival (TDOA) data of signals from two pairs of stations in the South China Sea station chain at each test site, and collects 10 minutes of data at each static survey site. The ASF is composed of a constant related to the terrain and an instantaneous quantity related to the weather, of which the constant is the main component. The instantaneous quantity fluctuates with temperature and humidity and the daily changes of seasonal climate. The application is mainly based on the ASF estimation of the measured ASF static data, without considering the fluctuations of the ASF instantaneous quantity. In order to reduce the influence of meteorological factors on the test data, several days with stable and similar weather conditions and small daily fluctuations are selected for continuous measurement, and data is collected within the same period of each day.

[0099] (2) Performance verification

[0100] When positioning using raw data, the MAE and RMSE of the absolute error of all sample data are 1535.30 m and 1538.04 m respectively. In order to verify the performance of the method provided by the application, the ASF estimation method proposed by the application is quantitatively compared with four representative ASF estimation methods, namely inverse distance weighting (IDW), ordinary kriging (OK), first-order universal kriging (UK1) and second-order universal kriging (UK2). Among them, the parameter settings are as follows: the number of groups q = 15, the highest order of the basis function P = 2, the population size K = 50, the number of sampling points n = 21, the dimension of the sampling points m = 2, and the number of iterations G max = 100.

[0101] Figure 4 The MAE and RMSE of the positioning results after correction by different ASF estimation methods are compared. It can be seen from Figure 4 that the corrected positioning results are significantly better than the initial positioning results without correction, and the positioning accuracy is improved by an order of magnitude. Compared with the other four methods, the method proposed by the application has a significant improvement in position accuracy. Specifically, compared with IDW, OK, UK1 and UK2, the MAE is reduced by 17.33%, 25.95%, 12.72% and 20.73% respectively, and the RMSE is reduced by 21.68%, 29.26%, 10.53% and 23.45% respectively.

[0102] In order to compare the positioning error distribution of all sample points, Figure 5 the positioning error box plot after correction based on different ASF estimation methods is given. The box plot of each method includes a box, a red median line and two dashed lines. Specifically, the box shows the middle 50% of the positioning error, i.e. the range between the first quartile and the third quartile of the positioning accuracy. The red line represents the median value of the positioning error, and the two whiskers are lines extending from the box to the minimum and maximum values. From Figure 5It can be seen that the overall position error of the method proposed in the application is smaller than that of the other four ASF estimation methods.

[0103] In summary, the method provided by the application integrates the selection of base functions and the search of related parameters into an overall optimization framework. The selection of base functions is defined as a continuous optimization task with real coding, allowing the two tasks to be optimized simultaneously in the same coding manner, simplifying the optimization process, effectively solving the interdependence between the two tasks, and improving the prediction accuracy and reliability of the Kriging model. The strong global random optimization algorithm NDE recently proposed is used to simultaneously search for the optimal combination of base functions and related parameters, avoiding the problem of falling into local optimum and greatly improving the probability of finding the global optimal solution, thereby improving the accuracy of ASF estimation. Residual covariance minimization is used as the optimization objective of the two tasks, ensuring the consistency of the spatial correlation characteristics of the residuals. The continuous representation of the real number coding of the base function selection is mapped to the discrete form of binary coding, and the residual covariance fitting degree is used to evaluate the quality of the local Kriging model composed of discrete variables and continuous variables, fully considering the spatial distance between the estimated point and the neighboring known points and the positional relationship between the neighboring points.

[0104] An electronic device is provided in an embodiment of the application, including: a computer readable storage medium and a processor.

[0105] The computer readable storage medium is configured to store executable instructions.

[0106] The processor is configured to read the executable instructions stored in the computer readable storage medium and execute the method according to any one of the above embodiments.

[0107] A computer readable storage medium is provided in an embodiment of the application, which stores computer instructions. The computer instructions are configured to cause a processor to execute the method according to any one of the above embodiments.

[0108] A computer program product is provided in an embodiment of the application, including a computer program or instructions. The computer program or instructions are executed by a processor to implement the method according to any one of the above embodiments.

[0109] Those skilled in the art will readily understand that the above description is only the preferred embodiment of the application and is not intended to limit the application. Any modification, equivalent replacement and improvement made within the spirit and principle of the application shall be included in the protection scope of the application.

Claims

1. A trend kriging-based ASF estimation method for improving the positioning and navigation accuracy of the Loran system, characterized by: include: S1, sampling points in the target area using Loran receivers x Measure and obtain actual ASF data; wherein, the actual ASF data includes sampling points x Coordinates and corresponding ASF values y ( x ), the coordinates of the sampling points , n is the number of sampling points, t ( The coordinates of the sampling points are expressed as , m is the coordinate dimension of the sampling point; S2, the basis function selection parameter vector for constructing the Kriging model Parameter vector related to the residual ,Will and Combined into individuals , initialize the population , i =1,2,…, K , use the mutation, crossover and selection operations of the NDE algorithm to iteratively update the population to obtain the optimal population Q `, and according to the objective function from Q `Select the best individual Q opt ; Wherein, the selection operation is implemented based on the objective function, , H i,x Used to characterize individual Does the basis function in the trend function of the constructed Kriging model include the first x basis functions, x =1,2,…, , θ i,y According to individual The constructed Kriging model y Residual correlation parameters, y =1,2,…, m ; H i,x The value range of is [0,1], H i,x >0.5 means based on individual The basis functions in the trend function of the constructed Kriging model include the first x basis functions, otherwise it means using individual The basis function in the trend function of the constructed Kriging model does not include the first x basis functions; S3, according to Q opt The optimal basis function selection parameter vector in Get the optimal basis function ;according to y ( x ), and Q opt The residual correlation parameter vector in Construct a Kriging model to obtain unknown points ASF estimate ; in, and By placing the sampling point x and unknown points The coordinates of the optimal basis function Obtained in yes and The correlation vector between , is the Gaussian correlation model, , , Indicates sampling point With sampling point The distance between .

2. The method according to claim 1, wherein The objective function is: in, C k Based on Theoretical covariance of the residual terms of the constructed Kriging model, is the weight coefficient, is the actual mean covariance of the residual terms of the Kriging model; C k and The way to obtain is: calculate the distance between the two sampling points and the actual covariance of the ASF value, and get A vector of distance-covariance relationships , and are the distance between the sth point pair and the actual covariance of the ASF value; grouping the above vectors according to the distance, we can get q Group distance-covariance relationship; for k Group, the average distance of all sampling points in the statistical group and the actual mean covariance ,Will and in Bringing in the Gaussian correlation model, the corresponding theoretical covariance can be calculated C k ; f is the basis function set, are related parameters.

3. The method according to claim 2, wherein ,in, v is the power coefficient, It is g The average distance of the sample points in the group, N k and N g Respectively represent k Group and g The number of sample points in a group, g=1,2,…,q .

4. The method according to claim 1, wherein θ i,y The value range is (0,20].

5. An electronic device, characterized in that: include: Computer-readable storage medium and processor; The computer-readable storage medium is used to store executable instructions; The processor is configured to read the executable instructions stored in the computer-readable storage medium and execute the method according to any one of claims 1 to 4.

6. A computer-readable storage medium, characterized in that The computer-readable storage medium stores computer instructions, and the computer instructions are used to enable a processor to execute the method according to any one of claims 1 to 4.

7. A computer program product comprising a computer program or instructions, characterized in that When the computer program or instruction is executed by a processor, the method according to any one of claims 1 to 4 is implemented.

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