A hybrid algorithm based on TDOA and DOA for underwater target localization

To address the problems of complex scenarios, low accuracy of a single localization algorithm, and high requirements for localization system equipment, this paper proposes a Taylor-weighted least squares algorithm based on Time Differences of Arrival (TDOA) and Direction of Arrival (DOA) joint localization method. The method first performs DOA estimation of the target by the Bayesian algorithm with off-grid sparse reconstruction and converts the obtained orientation information into position coordinates, and finds the final position of the target through iterative operations. The proposed algorithm is compared with the existing TDOA method based on the Chan algorithm and the Taylor algorithm based on the TDOA with the Chan algorithm as the initial value and the Two-Step Weighted Least Squares (TSWLS) algorithm in a simulation environment. The results show that the proposed algorithm has better performance in terms of localization accuracy, noise immunity, stability and is suitable for target localization in underwater.


Introduction
TDOA localization [1][2] has received much attention from scholars in underwater target localization because it does not require time synchronization between the target and the sensor.A Taylor-weighted least squares algorithm based on TDOA and FDOA was proposed in the literature [3] and applied underwater to estimate the position and velocity of the target.In the literature [4], the authors proposed a hybrid TDOA and AOA-based localization algorithm using two base stations for target localization, which achieved better results with small measurement errors and was able to achieve the Cramer-Rao Lower Bound (CRLB).
This paper proposes a hybrid TDOA and DOA localization method based on the Taylor-weighted least squares algorithm.Firstly, using the off-grid sparse reconstruction Bayesian estimation method for DOA estimation of submerged targets; Secondly, converting the obtained directional information into position coordinates Then using the position coordinates as the initial value of the TDOA-based Taylorweighted least-square algorithm.The Taylor expansion of TDOA measurements is used to construct the localization error equation, and the local least-square solution of the error equation is derived to continuously iterate and update the target position until it reaches the set iteration threshold algorithm stops iterating.Finally, it obtains the exact position of the target.

Cross-domain co-location scenarios
The cross-domain cooperative communication network architecture based on the joint positioning method of TDOA and DOA consists of a shore-based command centre, Unmanned Aerial Vehicle

Taylor series expansion method
Assuming that the initial position of the localization target is 00 Also based on the principle of TDOA's positioning algorithm, the following set of equations can be obtained (2) and (3).
) ) ) ) where ( , ) xy denotes the coordinates of the target, the location of the hydrophone is ( , ) ii xy.It is assumed that the first hydrophone is the reference hydrophone and the distance from the source to the i hydrophone is i R .Then the 11 ii R R R means the difference between the distance from the target to the i hydrophone and the distance to the first hydrophone.The function is constructed from the above equation as follows (4).
Subjecting the function ( , , , ) The weighted least-squares solution of the above equation is as follows (7).
where Q denotes the covariance matrix of the TDOA measurements.In the next recursive operation, assume 00 , x x x y y y        , update the coordinate values of the target for iterative operation.
The above process is repeated until the calculation stops when the error satisfies xy      and finally obtain the corrected target estimated position.

TDOA and DOA hybrid positioning method
According to the DOA estimation model, we can obtain (8).
()   Y A X E (8) where [ ( 1), ( 2), ( 3      defined as the mismatch error between the target position and the grid point, and the division distance of the adjacent grid is r.The grid correction vector can be obtained as in (12)., 0, , {1, 2, , } = 0, 0, Thus, the off-grid sparse DOA estimation model can be obtained as in (13).= Z   (13) Finally, sparse Bayesian learning is used to solve the global optimal solution of DOA and obtain the DOA estimate.The estimated angle of DOA is used to convert the position coordinates of the target, which is taken as the original value of the TDOA-based Taylor method for iteration.

Theoretical model of Acoustic propagation
According to the literature [8], obtain the shallow acoustic field sound pressure normal mode solution in (14).
is the eigenfunction of the normal mode, which satisfies the characteristic equation of the normal.In order to obtain the propagation time difference of the received signal, the Warping transform-based time-frequency analysis method is used to separate the normal mode from obtaining each order of simple normal modes.
The results of the Warping transform are obtained as (15).where r is the propagation distance and the average speed of sound in the waveguide is c.

Normal mode separation based on the improved Warping transform
The simulation uses the following ocean environment: the seafloor is a liquid seafloor with a sound speed of 1700 m/s, a density of 1.5 g/cm3, absorption of 0.5 dB and a water depth of 25 m.The sound source is located at a depth of 20 m underwater, and the hydrophone is situated at a depth of 24 m underwater.The amplitude modulated LFM signal with a Blackman window of 3s length at 800-1200Hz is used to calculate the sound field using the Kraken model of simple regular waves.The transmit signal and its spectrum used in the simulation as show in figure 2. From figure 3, the separation results of the 1st and 2nd order normal waves are given.From figure 4, the correlation peak between the 1st order normal wave and the 2nd order normal wave is 0.8821.Therefore, the time delay difference between the two orders of the normal wave is obtained as 0.8821s.

DOA estimation results
The number of uniform line array elements is

Hybrid localization algorithm results
The function of RMSE is ,where L is the number of Monte Carlo experiments, MS is the true coordinate of the target, and MS is the estimated position coordinate.
As shown in figure 6 (a), the RMSE of several algorithms gradually increases in the range of measurement noise standard deviation of 1m to 10m.The RMSE of the Chan algorithm is the largest and the TSWLS algorithm is the second largest.The RMSE of Taylor's algorithm with Chan's algorithm as the initial value is closer to that of the proposed algorithm, and the RMSE of the proposed algorithm is always smaller than that of Taylor's algorithm with Chan's algorithm as the initial value as the measurement noise increases.In order to observe the localization accuracy of several algorithms more intuitively, 1000 Monte Carlo random experiments were conducted under the condition that the standard deviation of the measurement noise is 10m.From figure 6 (b), we can see that the estimated position of the proposed algorithm is (99.9053,199.7864), which is closer to the real target position (100,200).

Conclusion
This paper introduces a hybrid TDOA and DOA-based localization algorithm.The method integrates ranging and directional information, uses Bayesian algorithm with off-grid sparse reconstruction to estimate DOA, then converts the orientation information into position information as the initial value based on Taylor's algorithm, and finds the final position of the target through iterative operations.
Simulation experiments show that the proposed algorithm outperforms the existing algorithm in localization accuracy, noise resilience and stability.This offers a novel approach for accurate and stable underwater target positioning, contributing to joint positioning technology research.Finally, this paper acknowledges support from the Kunming AI Computing Centre.

Figure 1 .
Figure 1.The architecture diagram of a cross-domain collaborative communication system.
, then the Taylor series expansion of the function is (1).
matrix.According to the literature[5][6][7], we can calculate the fourth-order accumulation solution of the received signal, as follows in (9).

.
  denotes the conjugate, () ik a  denotes the i row and the k column of array manifold matrix ()  A , cum denotes the cumulative quantity and matrix is constructed twice to reduce its dimensionality, thus removing the redundant elements and keeping only the non-zero elements with information to obtain a sparse representation of the model in (10).manifold matrix of the spatial domain, and  denotes the sparse vector.There are few non-zero elements in the sparse vector, and these elements are the grid points corresponding to the target positions.This paper uses the first-order Taylor formula to approximate the real off-grid points and constructs a new array manifold overcomplete matrix, which is represented as in (11).The grid correction vector is  , T 12 =[ , , , ] [ 1/ 2 ,1/ 2 ]

6 M
 , the array element spacing is 0BS .The DOA estimation results are shown in figure 5(a), the estimated angle of the measured orientation is [44.5071,63.4708] .The estimated DOA azimuth measured in this paper is converted to the position coordinates of the target.The conversion equation is azimuth angle of the target reaching the two hydrophones, respectively, ( , ) xyis the true position of the target. 13, BS BS are the position coordinates of the two hydrophones .In turn, the estimated position of the target is obtained, as shown in figure 5 (b).

Figure 5 .
Figure 5. DOA estimation and estimated position.As shown in figure5(b), it can be seen that the target position estimated by the DOA estimation algorithm proposed in this paper is (98.7191,197.6131),which is closer to the real target position.Thus, it is chosen as the initial location of the target of Taylor's algorithm for iterative calculation.

Figure 6 .
Figure 6.The results of several algorithms for