Trend Kriging-based ASF estimation method for improving positioning and navigation precision of Rowland system

Abstract

The invention discloses a trend Kriging-based ASF estimation method for improving the positioning and navigation precision of a Rowland system, and belongs to the field of positioning and navigation.The method is based on a Kriging model, an effective primary function is selected based on limited real ASF data to construct an optimal trend function, and the method comprises two nesting tasks; the optimal basis function is selected, and the optimal relevant parameters are searched; in order to overcome the challenge that the nesting property of the two tasks is easy to cause local optimum, the two tasks are integrated into an integral optimization framework; under the framework, the optimal solutions of the primary function and the residual related parameters are searched at the same time by adopting an NDE algorithm with high global search capability. In addition, residual covariance minimization is used as an optimization target of the two tasks, and geospatial correlation of sampling data is guaranteed.

Description

Technical Field

[0001] The present invention belongs to the field of positioning and navigation, and more specifically, relates to an ASF estimation method based on trend Kriging for improving the positioning and navigation accuracy of a Loran system. Background Art

[0002] The Global Navigation Satellite System (GNSS) is an important component of the Positioning, Navigation and Timing (PNT) system, but it is vulnerable to interference or attack, resulting in failure or even paralysis, which puts safety at extreme risk. Therefore, it is necessary to build and develop a backup positioning and navigation system that has strong anti-interference capabilities and is independent of GNSS. The Loran system (including the enhanced Loran system) is internationally recognized as the most powerful backup of GNSS due to its stable ground wave phase, strong anti-interference capabilities and large coverage.

[0003] At present, it is difficult for the Loran system to achieve high-precision positioning. This is mainly because the Loran signal is affected by complex terrain and various meteorological factors during ground propagation, resulting in unpredictable changes in the actual propagation time of the signal over space and time. It is usually assumed that the signal is positioned with uniform and fixed propagation in an ideal atmosphere. When the transmitting and receiving ends are fixed, there is a variable propagation time error between the actual propagation time of the signal and the fixed and predictable theoretical propagation time. This error is usually called the additional quadratic phase factor (ASF), which is the main error source that hinders the improvement of Loran positioning accuracy. In order to improve positioning accuracy, the ASF must be determined and used to correct the observation value. A very small deviation in the ASF may result in a positioning error of up to several kilometers. Therefore, accurate estimation of the ASF is crucial to improving the positioning accuracy of Loran. Summary of the invention

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

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

[0006] S1, using a Loran receiver to measure at a sampling point x in the target area to obtain actual ASF data; 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 = [x 1 ,x 2 ,…,x n ] T , n is the number of sampling points, and the coordinates of the tth (t=1,2,…,n) sampling point are represented by x t =[x t,1 ,xt,2 ,…,x t,m ], m is the coordinate dimension of the sampling point;

[0007] S2, the basis function selection parameter vector H for constructing the Kriging model i The parameter vector θ related to the residual i , H i With θ i Combined into individual Q i , initialize the population Q = [Q 1 ,Q 2 ,…,Q K ], 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 select the optimal individual Q from Q' according to the objective function opt ;

[0008] Wherein, the selection operation is implemented 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 Used to characterize the individual Q i Whether the basis function in the trend function of the constructed Kriging model includes the xth basis function in the candidate basis function set, x = 1, 2, ..., nf, θ i,y According to individual Q i The yth residual correlation parameter of the constructed Kriging model, y = 1, 2, ..., m;

[0009] S3, according to Q opt The optimal basis function selection parameter vector H in opt Get the optimal basis function f opt ; According to y(x), f opt and Q opt The residual correlation parameter vector θ in opt Construct a Kriging model to get the unknown point x 0 ASF estimate

[0010] Where F'=[f opt (x 1 ),f opt (x 2 ),…,f opt (x n )] T With fopt (x 0 ) T By dividing the sample point x and the unknown point x 0 Substitute the coordinates of into the optimal basis function f opt In the obtained 0 )=[R(θ opt ,x 1 ,x 0 ),R(θ opt ,x 2 ,x 0 ),…,R(θ opt ,x n ,x 0 )] T is x 0 The correlation vector between and x,

[0011] is the Gaussian correlation model, t 2 =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 used to read the executable instructions stored in the computer-readable storage medium and execute the method as described in 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 cause a processor to execute the method as described in 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, wherein when the computer program or instructions are executed by a processor, the method according to the first aspect is implemented.

[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 present invention integrates the basis function selection of the Kriging model and the residual related parameter search into an overall optimization framework, defines the basis function selection task as a real-coded continuous optimization task, allows the two tasks to be optimized simultaneously in the same coding mode, simplifies the optimization process, can effectively solve the mutual dependence between the two tasks, and improves the prediction accuracy and reliability of the Kriging model; at the same time, the present invention adopts the minimization of residual covariance as the optimization target of the two tasks, ensuring the consistency of the residual spatial correlation characteristics. The real number coding continuous representation of the basis function selection is mapped to a binary coded discrete form, 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 adjacent known sample points and the positional relationship between each adjacent sample point; in addition, the present invention adopts the recently proposed strong global random optimization algorithm NDE to simultaneously search for the optimal combination of basis functions and residual related parameters, avoiding falling into the local optimal problem, greatly improving the probability of finding the global optimal solution, and thus improving the accuracy of ASF estimation.

[0019] In summary, the method provided by the present invention can obtain high-precision ASF values ​​based on a limited actual ASF data set, solving the problems that the existing theoretical calculation methods cannot be used for large-area ASF prediction, and that high-density measured ASF has a long cycle and consumes manpower and material resources. It does not need to consider relevant information such as geological parameters and propagation environment, and is more universal in application scenarios. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 A flow chart of an ASF estimation method based on trend Kriging for improving the positioning and navigation accuracy of a Loran system provided in an embodiment of the present invention.

[0021] Figure 2 Schematic diagram of the hyperbolic positioning principle of the Roland system.

[0022] Figure 3 A schematic diagram of actual data waveforms collected at a location provided by an embodiment of the present invention.

[0023] Figure 4 It is a schematic diagram of positioning results after correction using the existing method and the method provided by the embodiment of the present invention; wherein the ordinate is the positioning accuracy and the abscissa is the method type.

[0024] Figure 5 It is a box plot of positioning errors of all sampling data after correction by the existing method and the method provided by the embodiment of the present invention; wherein the ordinate is the positioning accuracy and the abscissa is the method type. DETAILED DESCRIPTION

[0025] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0026] ASF estimation methods mainly include prediction based on theoretical ground wave propagation models and interpolation based on actual measurements. Various ASF theoretical prediction methods have been continuously developed by establishing propagation models based on different ground types (plane, spherical, uniform and smooth, heterogeneous and complex terrain, etc.). Theoretical calculation methods are easier to implement than actual measurements, and their prediction accuracy mainly depends on the geological parameters of the path division. However, due to the complexity of the terrain and the confidentiality of special areas, it is difficult to accurately obtain the real geological parameters, which leads to the low prediction accuracy of the theoretical prediction method. Compared with the theoretical calculation method, interpolation based on extensive measured data can obtain ASF values ​​with higher accuracy, but the cost of manpower and material resources is huge. Therefore, among the many ASF estimation methods, interpolation based on a small amount of ASF measured data is the most practical and economical. At this time, the performance of the interpolation method directly affects the estimation accuracy of ASF.

[0027] Since the ASF at different locations in a local area does not change linearly with the geographical location, but presents complex and unpredictable changes according to the terrain and meteorology on the propagation path, it is difficult for traditional linear interpolation methods to obtain high-precision ASF under limited measurement data. The prior art provides several spatial interpolation methods to improve the estimation accuracy of 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 actual ASF data to obtain high-precision ASF. These studies have shown that the Kriging method is a type of ASF estimation method suitable for cases with limited measurement data. However, these Kriging-based ASF estimation methods all have a common disadvantage 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 is composed of all candidate basis functions of a specified order. Therefore, selecting appropriate basis functions to form the optimal trend function can effectively improve the prediction accuracy of the Kriging method.

[0028] Based on this, the embodiment of the present invention provides an ASF estimation method based on trend Kriging to improve the positioning and navigation accuracy of the Loran system. Figure 1 As shown, including:

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

[0030] Specifically, a Loran receiver is used to perform precise measurements at multiple locations in a specified actual area to obtain actual ASF data, including sampling point coordinates and corresponding ASF values.

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

[0032] Figure 2 In the middle, M is the main station, S 1 and S 2 is the secondary station, and U is the user receiver, i.e., the sampling point. Therefore, the distances from the two secondary stations to the primary station are known and are expressed as d M1 and d M2 And from the three Roland platforms S 1 , S 2 and the distance from M to U are unknown, denoted as d 1 ,d 2 and d M . Further, define d 1 and d M The difference between them is Δd 1 , d 2 and d M The difference between them is Δd 2 Based on Δd 1 and Δd 2 , we can get two hyperbolic position lines, and U is at their intersection.

[0033] From S h The time difference between the signal from (h=1,2) and M and the arrival of U is defined as TDOA h, in the Roland system, S h The time of transmitting the signal has a fixed and known delay relative to M, which is defined as τ h , therefore, the signal from S h The actual propagation time difference Δt from M to U p,h It is expressed as:

[0034] Δt p,h =TDOA h -τ h

[0035] In actual engineering, due to Δt p,h It is variable. It is usually assumed that the signal propagates at the speed of light between the transmitter and the receiver, and a fixed and predictable theoretical propagation time difference is used for positioning. Therefore, the signal is moved from S h The theoretical time difference from M to U at the speed of light c is defined as Δt l,h , expressed as:

[0036]

[0037] Δt l,h The error between the actual propagation time and the theoretical propagation time of the signal on the same path is the ASF, which can be calculated using the high-precision position coordinates provided by the GNSS receiver and the above formula. What is calculated here is the ASF difference on the two paths, defined as S h and the ASF from M to U is Δt ASF,h , 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, the basis function selection parameter vector H for constructing the Kriging model i The parameter vector θ related to the residual i , H i With θ i Combined into individual Q i , initialize the population Q = [Q 1 , Q 2 , …, Q K ], 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 select the optimal individual Q from Q' according to the objective function opt; wherein the selection operation is implemented based on the objective function.

[0041] Specifically, in step S2, the actual ASF data collected in S1 is used to construct a Kriging model, and the theoretical calculation formulas of various parameters are obtained based on the constructed Kriging model; then, the basis function selection vector and residual correlation vector in the Kriging model are defined to facilitate subsequent solutions; finally, the mutation, crossover and selection operations of the NDE algorithm are used to obtain the optimal population Q`, and the optimal individual Q is selected from Q` according to the objective function. opt , that is, step S2 includes:

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

[0043] The model includes trend terms and residuals. The relationship between the coordinate x of the sampling point and the ASF value y(x) is expressed as:

[0044] y(x)=f(x) T β+z(x)

[0045] where f(x) = [f 1 (x), f 2 (x), …, f k (x)] T is a trend function, which is composed of k basis functions; β = [β 1 , β 2 , …, β k ] T is the coefficient of the trend function; z(x)=[z(x 1 ), z(x 2 ), …, z(x n )] T is the residual term, and n is the number of sampling points.

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

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

[0048]

[0049] In the formula, σ 2 is the process variance; is the unknown parameter θ=[θ 1 ,θ 2 ,…,θ m ] T The relevant model reflects and Geographical relevance, t 1 =1,2,…,n,t 2=1,2,…,n. The Gaussian correlation model is expressed as:

[0050]

[0051] in, express and The distance between.

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

[0053]

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

[0055] The basis function selection and residual related parameters in the Kriging model are binary discrete optimization and real continuous optimization problems, respectively, and the solutions of these two problems are nested, that is, the optimization of the former requires the determination of the latter first, and the optimization of the latter also requires the determination of the former first. In order to solve the problem that this nested nature easily leads to falling into the local optimal problem, an optimization framework is designed to unify the two problems into one problem for solution. That is, considering the interconversion nature of binary discrete optimization problems and real optimization problems, the basis function selection parameter vector and the residual related parameter vector are defined as continuous variables H i and θ i , combined into an individual, expressed 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 basis functions. i,x Used to characterize the individual Q i Whether the basis function in the trend function of the constructed Kriging model includes the xth basis function in the candidate basis function set, θ i,y According to individual Q i The yth residual correlation parameter of the constructed Kriging model, y=1,2,…,m.

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

[0059]

[0060] To facilitate the selection of basis functions, preferably, H is set i,x The value range of H is [0,1]. i,x is the continuous representation of whether to select the xth basis function from the set of candidate basis functions, θ i,y The value range is (0,20]. i,j Mapped to a binary coded discrete vector, namely:

[0061]

[0062] In the formula, Indicates whether to select the discrete representation of the xth basis function from the set of candidate basis functions. When it is 1, it means that this basis function is selected as the basis function in the trend function, and when it is 0, it means that this basis function is not selected.

[0063] According to the above formula, we can get a new binary vector, namely This vector and the residual correlation vector can be combined into a binary-real coded vector, expressed as:

[0064]

[0065] Therefore, the vector Q i with I i They are essentially the same, except that the encoding method of the basis function selection vector is different. In the subsequent calculation process, a random search algorithm is used to select Q i To update, I i Used to build Kriging models. i The essence of optimization is to solve these two problems at the same time, which can effectively solve the problem of falling into local optimality due to the nested nature of these 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 from Q` according to the objective function opt .

[0067] Initialize the population Q = [Q 1 , Q 2 , …, Q K ]:For H i Each dimension component, θi Each dimension in H takes a random value within its own range; i,x The value range of θ is [0,1]. i,y The value range of H is (0,20], for example. i Each dimension in θ takes a random value in the range [0,1]. i Each dimensional component in takes a random value in the range (0,20]. Therefore, the matrix Q and the corresponding matrix I can be obtained. Among them, Q is a real-coded matrix, and I is a binary-real-coded matrix. The NDE algorithm (niche differential evolution algorithm) is a random optimization algorithm with strong global search capability proposed in recent years, which can solve the high-dimensional real coding problem well. The mutation operation and crossover operation of the NDE algorithm are used to update Q multiple times.

[0068] Among them, the residual covariance model fitting minimization 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 ) i For example: Based on I i The constructed Kriging model includes a trend term (i.e., f(x) = [f 1 (x), f 2 (x), …, f k (x)] T and β = [β 1 , β 2 , …, 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 the equation, so that the residual term is obtained according to the vector combination (Q i , I i ) constructed 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 average covariance C of the residuals calculated k Minimize the difference between .

[0071] First, calculate the distance between 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] Among them, the actual decision variables include the basis function set f and related parameters θ.

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

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

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

[0078] S3, according to Q opt The optimal basis function selection parameter vector H in opt Get the optimal basis function f opt ; According to y(x), f opt and Q opt The residual correlation parameter vector θ in opt , get the unknown point x 0 The estimated value of ASF.

[0079] Specifically, in step S3, Q is constructed based on the optimal combination opt The kriging model estimates the ASF value at the unknown location.

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

[0081] According to S21, the Kriging model includes β, θ and σ 2 Theoretical calculation formulas for three parameters, β and σ 2 The objective function is used to replace the objective function of finding the optimal value of θ in the original Kriging. The theoretical covariance is calculated using In the subsequent unknown point ASF estimation formula, only β in the Kriging model is involved.

[0082] Define x 0 is the unknown point to be estimated, and its estimated ASF value is expressed as:

[0083]

[0084] In the formula, F=[f(x 1 ),f(x 2 ),…,f(x n )] T With f(x 0 ) T By dividing the sample point x and the unknown point x 0 Substitute the coordinates of into the basis function f to obtain r(x 0 )=[R(θ,x 1 , x 0 ), R(θ, x 2 , x 0 ),…,R(θ,x n , x 0 )] T is x 0 The correlation vector between and x, R is expressed as

[0085]

[0086] The optimal basis function set f opt and the optimal correlation parameter θ opt Substitute it into the formula to get x 0 The estimated ASF at is:

[0087]

[0088] In the formula, F'=[f opt (x 1 ),f opt (x2 ),…,f opt (x n )] T ,r'(x 0 ) and R' are related to θ opt The associated vector, r'(x 0 )=[R(θ opt ,x 1 ,x 0 ),R(θ opt ,x 2 ,x 0 ),…,R(θ opt ,x n ,x 0 )] T , is the Gaussian correlation model, t 1 =1,2,…,n,t 2 =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 based on a limited data set. Cross-validation is to temporarily remove known sample points in the data set, and then use an interpolation algorithm based on the remaining sample points to estimate the ASF of the removed sample points. This process is repeated until all sample points are estimated. Based on the estimated ASF, the position results of all sample points can be corrected.

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

[0092]

[0093] Among them, ε t is the absolute error of the tth sampling point. MAE and RMSE are comprehensive indicators 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 invention is further illustrated by taking a specific example as follows.

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

[0096] In order to truly and reliably evaluate the performance of the method provided by the present invention, field tests were conducted to collect actual ASF data. The test area is located in Yangjiang City, Guangdong Province, China, in an open environment at an altitude of about 4 meters, with a flat terrain around it, no obvious height fluctuations, and away from any natural obstacles that may interfere with the signal. The field test was conducted in December 2023 under sunny conditions, no precipitation, and an average temperature of 25 degrees Celsius. Magnetic antennas and electric antennas were deployed at the selected test sites, and a Loran receiver was used to receive signals from the Loran transmitter. It is also equipped with a high-precision GNSS receiver 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 station chain closest to the test area is used for positioning, namely the South China Sea station chain. Figure 3 Plotted part of the ground wave signal waveform of the Loran receiver at the test site. Figure 3 It can be seen that the received signal is clean and noise-free. In China's Roland system, each station chain consists of 1 main station and 2 secondary stations. A main station and each secondary station in the same station chain can form a station pair.

[0098] The present invention adopts hyperbolic positioning, randomly selects 21 locations in the designated area, measures the signal arrival time difference (TDOA) data from two pairs of stations in the South China Sea station chain for each test location, and collects 10 minutes of data at each static survey location. ASF is composed of constants related to topography and instantaneous quantities related to weather, wherein constants are the main components. Instantaneous quantities fluctuate with daily changes in temperature and humidity as well as seasonal climate. The present invention is mainly based on ASF estimation of measured ASF static data, without considering fluctuations in ASF instantaneous quantities. In order to reduce the impact of meteorological factors on test data, several days with stable and similar weather conditions and small daily fluctuations were selected for continuous measurement, and data were collected during the same period of each day.

[0099] (2) Performance Verification

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

[0101] Figure 4 The MAE and RMSE after correction by different ASF estimation methods are compared. Figure 4 It can be seen that the corrected positioning result is significantly better than the uncorrected initial positioning result, and the positioning accuracy is improved by an order of magnitude. Compared with the other four methods, the method proposed in this invention has significantly improved the positioning accuracy. Specifically, compared with IDW, OK, UK1 and UK2, MAE is reduced by 17.33%, 25.95%, 12.72% and 20.73%, respectively, and 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 box plots of the corrected positioning errors based on different ASF estimation methods are given. The box plot of each method consists of a box, a red center line, and two dashed lines. Specifically, the box shows the middle 50% of the positioning error, that is, the range between the first quartile and the third quartile of 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. Figure 5 It can be seen that compared with the other four ASF estimation methods, the method proposed in the present invention has a smaller overall position error.

[0103] In summary, the method provided by the present invention integrates basis function selection and related parameter search into an overall optimization framework. The basis function selection task is defined as a real-coded continuous optimization task, allowing the two tasks to be optimized simultaneously in the same coding manner, simplifying the optimization process, being able to effectively solve the mutual dependence between the two tasks, and improving the prediction accuracy and reliability of the Kriging model; the recently proposed strong global random optimization algorithm NDE is used to simultaneously search for the optimal combination of basis functions and related parameters, avoiding falling into the local optimal problem, greatly improving the probability of finding the global optimal solution, and thus improving the accuracy of ASF estimation; the residual covariance minimization is used as the optimization goal of the two tasks to ensure the consistency of the residual space correlation features. The real-coded continuous representation of the basis function selection is mapped into a binary-coded discrete form, 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 adjacent known sample points and the positional relationship between each adjacent sample point.

[0104] An embodiment of the present invention provides an electronic device, including: a computer-readable storage medium and a processor;

[0105] The computer-readable storage medium is used to store executable instructions;

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

[0107] An embodiment of the present invention provides a computer-readable storage medium, wherein the computer-readable storage medium stores computer instructions, and the computer instructions are used to enable a processor to execute the method described in any of the above embodiments.

[0108] An embodiment of the present invention provides a computer program product, including a computer program or instructions, and when the computer program or instructions are executed by a processor, the method described in any of the above embodiments is implemented.

[0109] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A trend kriging-based ASF estimation method for improving the positioning and navigation accuracy of the Loran system, characterized in that: include: S1, using a Loran receiver to measure at a sampling point x in the target area to obtain actual ASF data; wherein the actual ASF data includes the coordinates of the sampling point x and the corresponding ASF value y(x), and the coordinates of the sampling point x = [x1, x2, ..., x n ] T , n is the number of sampling points, and the coordinates of the tth (t=1,2,…,n) sampling point are represented by x t =[x t,1 ,x t,2 ,…,x t,m ], m is the coordinate dimension of the sampling point; S2, the basis function selection parameter vector H for constructing the Kriging model i The parameter vector θ related to the residual i , H i With θ i Combined into 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 to obtain the optimal population Q', and select the optimal individual Q from Q' according to the objective function opt ; Wherein, the selection operation is implemented 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 Used to characterize the individual Q i Whether the basis function in the trend function of the constructed Kriging model includes the xth basis function in the candidate basis function set, x = 1, 2, ..., nf, θ i,y According to individual Q i The yth residual correlation parameter of the constructed Kriging model, y = 1, 2, ..., m; S3, according to Q opt The optimal basis function selection parameter vector H in opt Get the optimal basis function f opt ; According to y(x), f opt and Q opt The residual correlation parameter vector θ in opt Construct a Kriging model to obtain the ASF estimate of the unknown point x0 Where F'=[f opt (x1),f opt (x2),…,f opt (x n )] T With f opt (x0) T By substituting the coordinates of the sampling point x and the unknown point x0 into the optimal basis function f opt In the above equation, we obtain 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, is a Gaussian correlation model, t1=1,2,…,n, t2=1,2,…,n, Indicates sampling point With sampling point The distance between 2. The method according to claim 1, characterized in that The objective function is: Among them, C k According to Q i Theoretical covariance of the residual term of the constructed kriging model, λ k is the weight coefficient, is the actual average covariance of the residual terms of the Kriging model; C k and The method of obtaining is: calculate the distance between the two sampling points and the actual covariance of the ASF value, and get n 2 A vector of distance-covariance relationships d s and are the actual covariance of the distance between the sth point pair and the ASF value; by grouping the above vectors according to the distance, we can get q groups of distance-covariance relationships; for the kth group, the average distance of all sampling points in the statistical group and the actual mean covariance Will and Q i θ i Substituting into the Gaussian correlation model, the corresponding theoretical covariance C can be calculated k .

3. The method according to claim 2, characterized in that Among them, v is the power coefficient, is the average distance of the sample points in the gth group, N k and N g They represent the number of pairs of sample points in the kth group and the gth group respectively, g=1,2,…,q.

4. The method according to claim 1, characterized in that H i,x The value range of H is [0,1]. i,x >0.5 means based on individual Q i The basis function in the trend function of the constructed Kriging model includes the xth basis function in the candidate basis function set, otherwise it represents the use of individual Q i The basis function in the trend function of the constructed Kriging model does not include the xth basis function in the candidate basis function set; θ i,y The value range is (0,20].

5. An electronic device, characterized in that: include: A computer readable storage medium and a processor; The computer-readable storage medium is used to store executable instructions; The processor is used 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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