CN114035182B - Multi-station time difference multivariable short wave target positioning method based on ionosphere reflection - Google Patents

Multi-station time difference multivariable short wave target positioning method based on ionosphere reflection Download PDF

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CN114035182B
CN114035182B CN202111226181.6A CN202111226181A CN114035182B CN 114035182 B CN114035182 B CN 114035182B CN 202111226181 A CN202111226181 A CN 202111226181A CN 114035182 B CN114035182 B CN 114035182B
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张纪峰
郭健
薛文超
孔垂柳
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    • G—PHYSICS
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Abstract

The invention relates to a multi-station time difference multivariable short wave target positioning method based on ionosphere reflection. And then combining the measured time difference data to obtain an ionospheric short-wave reflection equation set of p sending and q receiving. Secondly, the equation is converted into an optimization problem with constraints by utilizing nonlinear least squares, and different ionospheric virtual heights are introduced to serve as variables to be optimized. And finally, obtaining an upper bound of the target positioning error by using an error transfer formula. The method specifically comprises the following steps: the first step is as follows: establishing a nonlinear equation of a distance measurement value of a short wave signal reflected by an ionized layer and the information of the virtual height of the ionized layer, a transmitting station, a receiving station and a target position; the second step is that: transforming and solving a multivariable optimization problem; the third step: and (4) evaluating the error range of the multivariate multistation time difference positioning method.

Description

Multi-station time difference multivariable short wave target positioning method based on ionosphere reflection
Technical Field
The invention relates to a method for positioning a target by utilizing short wave reflection in an ionosphere under the general condition and a method for evaluating a positioning error range, wherein a detected target is a short wave radiation source positioned on the earth surface, and belongs to the technical field of short wave communication radiation source positioning.
Background
Target localization is a classical problem in radar and sonar research. With the development of global satellite positioning system, map navigation and other technologies, the theory and practical application of wireless signal positioning technology has new requirements. In the civil or military field, the requirements on the positioning accuracy and reliability are higher and higher. The traditional short-wave radiation source positioning methods are many, including GPS positioning in the case of line-of-sight propagation; multi-station time difference localization in non-line-of-sight propagation scenarios (see reference [3 ]). In the case of line-of-sight propagation, common positioning algorithms include time of arrival (TOA), time difference of arrival (TDOA), Received Signal Strength (RSS), and angle of arrival (AOA) based methods, among others. Reference [1] applies a Semi-definite linear Programming algorithm (SDP) to the object localization problem based on range measurements. Firstly, a pseudo-linear equation related to a target position is established, the pseudo-linear equation is converted into an optimization problem by using least square, and the convex problem is relaxed and an SDP algorithm is directly applied to solve the problem. Positioning in non-line-of-sight propagation scenarios utilizes ionospheric reflection or refraction to achieve radio wave propagation. A large number of target positioning algorithms based on the ionosphere model are developed at home and abroad by researching the ionosphere model, pollution removal and other modes, and a series of active radar target detection systems are successfully developed through the algorithms. Reference [2] finds that when the position of a receiving base station is fixed, the ionospheric virtual height of a low-speed moving target reaching the base station is almost unchanged within a period of time, and establishes a positioning and tracking equation of the target position by using the ionospheric virtual height, and implements a corresponding algorithm.
However, the existing methods have the following limitations:
1. the GPS positioning method in the case of line-of-sight propagation is usually based on a time delay model of a straight line propagation path (or called a line-of-sight propagation path), and is only suitable for the case of straight line propagation (see references [7, 8 ]).
2. The methods based on the parameters such as time of arrival (TOA), time difference of arrival (TDOA), Received Signal Strength (RSS) and angle of arrival (AOA) such as semi-positive definite linear programming algorithm (see references [4 and 5]), taylor expansion algorithm (see reference [6]), importance sampling method (see reference [9]), have the problems of being overly dependent on the accuracy of the initial value of the target position and being not suitable for the time difference location model of the ionospheric reflection signal.
3. Conventional moveout location algorithms based on ionospheric reflection signals generally assume that ionospheric pseudo-height information is the same in the received short-wave signals over a certain time or over a certain space (see references [10, 11 ]).
In order to solve the problems, the invention provides a new method for a multi-station time difference positioning model based on ionosphere reflection signals, and the method introduces different ionosphere virtual heights as variables to be optimized, has higher applicability and can realize target positioning. The new positioning method can be applied to the situation that the ionospheric parameters are unknown, and meanwhile, the accurate result is obtained for the situation that the ionospheric parameters are known. The application scenarios of the invention are as follows: the target position and each transmitting or receiving radar base station are located on the surface of the earth.
The invention content is as follows:
the technical problems solved by the invention are as follows: aiming at the situation of a general multi-transmitting multi-receiving radar base station, a method for positioning a target by utilizing short wave reflection in an ionized layer and a method for evaluating a positioning error range are provided, and the method belongs to the technical field of short wave communication radiation source positioning. The method provides an effective solution for the problem of achieving target positioning and error range assessment by reflecting short waves in the ionized layer.
The solution of the invention is as follows: first, a model of the ionosphere and earth's reflection is modeled. And setting the height of the transmitting point of the short wave on the spherical surface of the ionized layer from the ground as the virtual height of the ionized layer. A specific signal propagation scenario for ionospheric reflection signals is shown in fig. 2. The earth is a sphere with a radius R. The distance expression of the short wave transmitted by a single transmitting station and reaching the target position after being reflected by the ionosphere is obtained from the figure 2. Then, the path length of the short wave transmitted by the transmitting station to the target through the ionosphere reflection is expressed, the short wave reflected by the target reaches the receiving station through the ionosphere reflection, and the measurable time difference data is combined to obtain an ionosphere short wave reflection equation set of p-transmission q-reception, wherein the ionosphere short wave reflection equation set comprises hk(1≤k≤p+q)，x1，x2，x3And p + q +3 variables to be estimated. Second using the non-linear maximumThe equation is converted into an optimization problem with constraint by a small second product, and the estimated values of the virtual height and the target position are obtained by utilizing an optimization algorithm.
The specific steps of the method are described below for the short-wave reflection localization problem of multiple ionosphere parameters.
The first step is as follows: and establishing a model of distance measurement values of short wave signals reflected by the ionosphere and information of the ionosphere virtual height, the transmitting station, the receiving station and the target position.
1.1, establishing a single nonlinear relation equation of short-wave signals transmitted by a transmitting station to reach a target position through ionospheric reflection:
Figure BDA0003314409040000031
wherein A is a transmitting station, B is a target point, O is the earth centroid, C is an equivalent reflection point of the short wave signal in the ionosphere, h is a virtual height parameter of the ionosphere, and R is the earth radius.
1.2, establishing a path equation of a short wave signal which is transmitted from a transmitting station, reflected by an ionized layer to reach a target position, reflected by the ionized layer again to reach a receiving station.
Figure BDA0003314409040000032
Figure BDA0003314409040000041
Wherein, diOThe path length of a short-wave signal transmitted by the ith transmitting station to reach an ionosphere reflecting point; dOjThe path length of the short wave signal from the ionosphere reflection point to the jth receiving station is long; h is a total ofiThe ionosphere virtual height corresponding to the ith transmitting station; h isj+pThe ionosphere virtual height corresponding to the jth receiving station; i represents the ith transmitting station, the values are 1, 2,. and p is the number of short wave transmitting stations; j represents the jth receiving station, and the values are 1, 2, ·, q, and q are short wave receiving stationsThe number of (2); (x)iT，yiT，ziT) The position of the ith transmitting station under a rectangular coordinate system is taken as the position of the ith transmitting station; (x)iR，yiR，ziR) Is the position of the jth receiving station under the rectangular coordinate system; s isijFor measuring the total path length from the short-wave signal transmitted by the i transmitting station to the ionosphere and from the short-wave signal to the j receiving station through ionosphere reflection in a certain time, the time difference data can be obtained as follows:
Figure BDA0003314409040000042
wherein, c is the speed of light,
Figure BDA0003314409040000043
the k-th radar transmitting period, the time point of the transmitting station transmitting the signal and the time point of the receiving station receiving the signal are respectively. The formula (2) includes 3 unknowns of the target position information and p + q unknowns of an imaginary height reflected by the ionosphere of each transmitting station or receiving station.
The second step is that: and establishing an optimization problem about p + q +3 variables to be optimized by using a nonlinear least square method and solving the optimization problem.
2.1 establishing a nonlinear optimization problem about a target position variable and an unknown ionospheric virtual high parameter variable.
The formula (2) is a total of p × q nonlinear equations with respect to the target position. Further, the target position is constrained to
Figure BDA0003314409040000044
Wherein a, b and c are constant parameters of the earth ellipsoid model, and specifically are as follows:
a＝6378.137km，
b＝6356.75231414km，
c＝6378.137km.
let n be p + q, the transmitting station denoted T, the receiving station denoted R, and diOFor short-wave signals transmitted by the ith transmitting station to reach ionospheric reflection pointsThe path is long; dOjFor short-wave signals, the path length from the ionospheric reflection point to the jth receiving station, e.g.
Figure BDA0003314409040000051
Figure BDA0003314409040000052
Wherein T isi＝(xiT，yiT，ziT)，Rj＝(xjR，yjR，zjR) The position of the ith transmitting station and the position of the jth receiving station respectively, | | | |, represents the two norms of the vector.
The optimization problem obtained by the formula (2) is
Figure BDA0003314409040000053
2.2 solving the constrained optimization problem (3).
Let x ═ x1，x2，x3，h1，...，hn) Taking into account its unconstrained Lagrange multiplier function
Figure BDA0003314409040000054
Where μ is the lagrange multiplier. Definition of
Figure BDA0003314409040000061
Figure BDA0003314409040000062
Wherein gamma isiO，γOjAnd hi，hj+pIs shot one time and fullFoot
Figure BDA0003314409040000063
Figure BDA0003314409040000064
Solving the following optimization problem
Figure BDA0003314409040000065
And further obtaining a positioning result of the target position and a corresponding possible ionospheric pseudo-height value.
The third step: error range estimation of multivariate multistation time difference positioning algorithm
For the target position location equation set (2), a multivariate vector value function is defined
r(·)＝[r1(·)，…，rp×q+1(·)]T，
Wherein the variable is x ═ x1，x2，x3，h1，…，hn)；r1(·)，...，rp×q(. is) diO+dOj-sijI is more than or equal to 1 and less than or equal to p, j is more than or equal to 1 and less than or equal to q, and p multiplied by q functions are total;
Figure BDA0003314409040000066
is an unknown target constraint. Note the book
x1＝f1(h1，...，hn)
x2＝f2(h1，...，h6)，
x3＝f3(h1，...，h6)
Computing
Figure BDA0003314409040000071
Figure BDA0003314409040000072
Note the book
Figure BDA0003314409040000073
Computing
Jf(h1，...，hn)
＝-(J2r(f1(·)，f2(·)，f3(·)，h1，...，hn))+J1r(f1(·)，f2(·)，f3(·)，h1，...，hn).
Finally, calculate the slave h1，...，hnTo x1，x2，x3Absolute error of, i.e.
Figure BDA0003314409040000074
Figure BDA0003314409040000075
Figure BDA0003314409040000076
Wherein, Δ x1Is from h1，...，hnTo x1Absolute error of (d); Δ x2Is from h1，...，hnTo x2Absolute error of (d); Δ x3Is from h1，...，hnTo x3Absolute error of (d); Δ h is the prior absolute error of the imaginary high variable; (x)1，x2，x3) Coordinates of the target position; l (x) is a Lagrangian multiplier function of the objective function; gamma rayiO，γOjIs pine ofA relaxation variable; r is a target position positioning equation set; j is a unit of1r is a Jacobi matrix of the virtual high variable of the target position positioning equation set; j. the design is a square2r is a Jacobi matrix of the target position positioning equation set to the position variable;
from the second step and the third step, the multi-station short-wave target positioning method based on ionospheric reflection can realize target positioning and give an upper bound of positioning accuracy errors caused by ionospheric errors, and the method has stability and generality.
Compared with the prior art, the invention has the advantages that: first, the present invention does not need to make consistency assumptions on ionospheric pseudo-heights, and the method has stability and more general adaptability. Secondly, the invention provides an error transfer formula from the ionosphere virtual high prior error to the target positioning result, and can obtain the upper bound of the target positioning error.
Drawings
FIG. 1 is a flow chart of a multi-station time difference short-wave target positioning method based on ionospheric reflection in the invention.
Fig. 2 is a schematic diagram of the propagation path of a short wave signal in the ionosphere.
FIG. 3a is a graph of 1000 iterations of estimated position as a function of initial value in a simulation scenario.
FIG. 3b is a graph of estimated position versus iteration number for different initial values in simulation case one.
Fig. 3c is a graph of error as a function of iteration number for different initial values in simulation case one.
Fig. 4a shows the estimation error of 100000 times simulation of the partial virtual high equivalence algorithm in the second simulation scenario. Fig. 4b is a histogram of the error frequency distribution of a part of the pseudo-high equivalence algorithm in simulation case two.
Fig. 4c is a diagram of the positioning error results in the simulation case two.
Fig. 4d is a histogram of the positioning error frequency distribution in the second simulation case.
Detailed Description
The invention provides a multivariable optimization target positioning method with higher adaptability and better stability, aiming at the problem of multi-station time difference target positioning by utilizing short wave reflection in an ionosphere under the general condition. To test the effectiveness and utility of the present invention, we performed a 2-out, 4-receive, simulation of the surface target location of the earth, and considered the following two cases:
simulation situation one: the positions in the subordinate setting are given only the first two variables, the third one being determined by the satisfaction of the ellipsoid equation.
A transmitting station: (0,0),(50,0)
A receiving station: (0, 50),(50, 50),(0, 100),(50, 100)
The virtual height corresponds to six stations respectively as follows: [100, 150, 100, 150, 300, 300]
Target position: (2000,0),
initial value of pseudo-high iteration: (100, 100, 100, 100, 100, 100),
the initial position is (a, 0), and a is 1000, 1500, 2500, 3000.
Simulation situation two: rectangular coordinate system with lower coordinates of geocentric and geostationary coordinates
A transmitting station: (0,0),(50,0)
The receiving station: (0, 50),(50, 50),(0, 100),(50, 100)
Deficiency and height: h ═ 400, 410, 400, 405, 405, 400 for six stations, respectively
Target position: (-3700-2400)
The pseudo-high iteration initial value error is 10 km.
The following are specific steps for carrying out the process of the present invention.
The first step is as follows: establishing a nonlinear equation set of a distance measurement value of a short wave signal reflected by an ionized layer and the virtual height of the ionized layer, and information of a transmitting station, a receiving station and a target position as follows:
Figure BDA0003314409040000091
Figure BDA0003314409040000092
Figure BDA0003314409040000101
Figure BDA0003314409040000102
Figure BDA0003314409040000103
Figure BDA0003314409040000104
Figure BDA0003314409040000105
Figure BDA0003314409040000106
Figure BDA0003314409040000107
wherein, x, y, z, h1，h2，...，h6The target position variable is unknown and ionospheric reflection point virtual height variable corresponding to each transmitting station or receiving station. (x)iT，yiT，ziT) For the position of the ith transmitting station in the rectangular coordinate system, (x)j，yj，zj) Is the position of the jth receiving station in the rectangular coordinate system. We use the case of two settings to illustrate the effectiveness and stability of the invention.
The second step is that: and establishing a nonlinear optimization problem about the 9 variables to be optimized by using a nonlinear least square method. Unconstrained Lagrange multiplier function of
Figure BDA0003314409040000111
For simulation case one: the solution was performed using the Levenberg-Marquardt algorithm.
For simulation case two: the solution was performed using the Levenberg-Marquardt algorithm. Meanwhile, in order to show the stability of the algorithm, the algorithm is compared with an algorithm which uses a prior assumption of high consistency and solves a nonlinear target positioning equation set by using a Newton iteration method.
For simulation case one, the simulation results are shown in fig. 3a, 3b and 3 c. Fig. 3a shows a graph of the estimated position over 1000 iterations as a function of the initial value. Fig. 3b shows a graph of estimated position as a function of iteration number, taking initial position as [ a 0], a being 1000, 1500, 2500, 3000, and each initial value iterates 1000 times. Fig. 3c shows the same initial value setting, error versus iteration number for different initial values. As can be seen from the figure, with the increase of the iteration number, the method of the invention can obtain a convergence result, and if the initial value of the target position is properly selected, the final convergence result can gradually approach the real target position. The effectiveness of the invention is demonstrated.
For the second simulation case, the simulation results are shown in fig. 4a, 4b, 4c and 4 d. In the simulation, the virtual heights of the transmitting station 1 and the receiving stations 1 and 4 are assumed to be the same and are all h1The virtual heights of the ionosphere of the receiving stations 2 and 3 are all h4The virtual height of the ionized layer of the transmitting station 2 is h2. But due to the irregular nature of the ionosphere, the empirical results may be biased. The actual virtual height is set as: h ═ 400, 410, 400, 405, 410]The real position of the target is X ═ Xr，tr，zr) In order to eliminate the influence of initial value errors, points are randomly selected within the maximum error of the true value to be used as initial values for iterative solution, and the initial values are set to be
h0＝h+10E
Figure BDA0003314409040000123
Figure BDA0003314409040000124
Figure BDA0003314409040000121
Wherein E is a random variable uniformly distributed according to U (0, 1),
Figure BDA0003314409040000122
the method comprises the steps of obeying U (0, 2 pi) uniformly distributed random variables, randomly generating 10000 initial values, solving an optimization problem and obtaining an optimization result. In fig. 4a, the horizontal axis corresponds to 100000 simulations, and the vertical axis represents the error of the current simulation, which can be seen to be greater than 1000 km. In fig. 4b, the horizontal axis represents the positioning error, and the vertical axis represents the number of simulations in which the positioning error falls within the segment in 100000 simulations, it can be seen that the mode is 1300 km. It can be seen that the estimation result will be completely incorrect when the pseudo-high assumption is biased. For the method of the present invention, the initial value is uniformly and randomly generated 1000000 times within the error range, the error between the iterated solution and the true solution is calculated, and an error result graph is made as shown in fig. 4c and a frequency distribution histogram is made as shown in fig. 4 d. It can be seen that the error is always within 14km, and more than half of the probability is within 10km, and the mode of the estimation result is around 10 km. Showing the stability of the process of the invention.
In conclusion, in the method, under the condition that the prior assumption of the virtual height is not needed, all the virtual heights are introduced as unknown variables, the nonlinear least square method is used for converting the unknown variables into the optimization problem, the initial value is properly selected, the solution close to the real target position can be obtained, and the method has more general adaptability.
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Claims (3)

1. a multi-station time difference multivariable short wave target positioning method based on ionospheric reflection is characterized in that:
the method comprises the following steps: establishing a model of distance measurement values of short wave signals reflected by an ionized layer and information of the virtual height of the ionized layer, a transmitting station, a receiving station and a target position;
step two: establishing an optimization problem about p + q +3 variables to be optimized by using a nonlinear least square method and solving the optimization problem;
step three: evaluating the error range of a multivariate multistation time difference positioning algorithm;
in the first step, a single nonlinear relation equation of the short-wave signal transmitted by the transmitting station reaching the target position through ionospheric reflection is established:
Figure FDA0003577823520000011
wherein A is a transmitting station, B is a target point, O is the earth centroid, C is an equivalent reflection point of the short-wave signal in the ionosphere, h is a virtual height parameter of the ionosphere, and R is the earth radius;
in the first step, a path equation is established, wherein the short-wave signal is transmitted from a transmitting station, reaches a target position through ionosphere reflection, is reflected by the target position, and then reaches a receiving station through ionosphere reflection;
Figure FDA0003577823520000012
Figure FDA0003577823520000013
diO+dOj＝sij (2)
wherein d isiOFor short-wave signals transmitted by the ith transmitting station toThe path to the ionospheric reflection point is long; dOjThe path length of the short wave signal from the ionosphere reflection point to the jth receiving station is long; h isiThe ionosphere virtual height corresponding to the ith transmitting station; h isj+pThe ionosphere virtual height corresponding to the jth receiving station; i represents the ith transmitting station, the values are 1, 2,. and p is the number of short wave transmitting stations; j represents the jth receiving station, the values are 1, 2, ·, and q are the number of short-wave receiving stations; (x)iT，yiT，ziT) The position of the ith transmitting station under a rectangular coordinate system is taken as the position of the ith transmitting station; (x)iR，yiR，ziR) Is the position of the jth receiving station under the rectangular coordinate system; sijFor measuring the total path length from the short wave signal transmitted by the i transmitting station to the ionosphere and from the short wave signal to the j receiving station by ionosphere reflection in a certain time, the time difference data is obtained as follows:
Figure FDA0003577823520000021
wherein c is the speed of light,
Figure FDA0003577823520000022
respectively setting a kth radar transmission period, a time point of a signal transmitted by an i transmitting station and a time point of a signal received by a j receiving station; the formula (2) comprises 3 unknown quantities of target position information and p + q unknown quantities of virtual height of each transmitting station or receiving station reflected by an ionosphere;
in the second step, establishing a nonlinear optimization problem about a target position variable and an unknown ionospheric virtual high parameter variable; the formula (2) is p × q nonlinear equations about the target position; further, the target position is constrained to
Figure FDA0003577823520000023
Wherein a, b and c are constant parameters of the earth ellipsoid model, and specifically are as follows:
a＝6378.137km，
b＝6356.75231414km，
c＝6378.137km.
let n be p + q, the transmitting station denoted T, the receiving station denoted R, and diOThe path length of a short-wave signal transmitted by the ith transmitting station to reach an ionosphere reflecting point; dOjFor short-wave signals, the path length from the ionospheric reflection point to the jth receiving station, e.g.
Figure FDA0003577823520000031
Figure FDA0003577823520000032
Wherein T isi＝(xiT，yiT，ziT)，Rj＝(xjR，yjR，zjR) The position of the ith transmitting station and the position of the jth receiving station respectively, | | | |, represents the two norms of the vector;
the optimization problem obtained by the formula (2) is
Figure FDA0003577823520000033
Figure FDA0003577823520000034
2. The ionospheric reflection-based multi-station time difference multivariate short wave target positioning method of claim 1, wherein: in the second step, solving a constrained optimization problem (3);
let x ═ x1，x2，x3，h1，...，hn) Taking into account its unconstrained Lagrange multiplier function
Figure FDA0003577823520000035
Where μ is the Lagrangian multiplier; definition of
Figure FDA0003577823520000036
Figure FDA0003577823520000037
Wherein gamma isiO，γOjAnd hi，hj+pIs single shot and satisfies
Figure FDA0003577823520000041
Figure FDA0003577823520000042
Solving the following optimization problem
Figure FDA0003577823520000043
Figure FDA0003577823520000044
And further obtaining a positioning result of the target position and a corresponding possible ionospheric pseudo-height value.
3. The ionospheric reflection-based multi-station time difference multivariate short wave target positioning method of claim 1, wherein: for the target position location equation set (2), a multivariate vector value function is defined
r(·)＝[r1(·)，...，rp×q+1(·)]T，
Wherein the variable is x ═ x1，x2，x3，h1，...，hn)；r1(·)，...，rp×q(. is) diO+dOj-sijI is more than or equal to 1 and less than or equal to p, and j is more than or equal to 1 and less than or equal to q, wherein p is multiplied by q;
Figure FDA0003577823520000045
is an unknown target constraint condition; note the book
Figure FDA0003577823520000046
Computing
Figure FDA0003577823520000047
Figure FDA0003577823520000051
Note the book
Figure FDA0003577823520000052
Computing
Jf(h1，...，hn)
＝-(J2r(f1(·)，f2(·)，f3(·)，h1...，hn))+J1r(f1(·)，f2(·)，f3(·)，h1，...，hn).
Finally, calculate the slave h1，...，hnTo x1，x2，x3Absolute error of, i.e.
Figure FDA0003577823520000053
Figure FDA0003577823520000054
Figure FDA0003577823520000055
Wherein, Δ x1Is from h1，...，hnTo x1Absolute error of (d); Δ x2Is from h1，...，hnTo x2Absolute error of (d); Δ x3Is from h1，...，hnTo x3Absolute error of (d); Δ h is the prior absolute error of the imaginary high variable; (x)1，x2，x3) Coordinates of the target position; l (x) is a Lagrangian multiplier function of the objective function; gamma rayiO，γOjIs a relaxation variable; r is a target position positioning equation set; j. the design is a square1r is a Jacobi matrix of the virtual high variable of the target position positioning equation set; j. the design is a square2r is the Jacobi matrix of the target position location equation set for the position variables.
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