## 1 Introduction  
The ultra-short baseline positioning system (USBL) is based on the installation of an ultra-short baseline array on a ship bottom and a transponder mounted on an underwater target. The transmitter emits an acoustic signal, and the underwater target receives the transmitted acoustic signal and sends back an answer signal. After receiving the signal, the USBL calculates the azimuth and distance [1] of the target using the time-delay difference or phase difference between each receiving hydrophone [2], and calculates the position coordinates of the underwater target. The ultra-short baseline positioning system (USBL) is generally used for underwater positioning in shallow sea area [1]. Due to the complexity of the actual marine environment, the accuracy of an ultra-short baseline positioning system is limited mainly due to the existence of the ranging error, phase difference error, transmission and reception delay estimation error, and acoustic noise in the communication. The main reason for the above errors is as follows. When an acoustic signal propagates underwater, the sound speed is affected by the influence of the marine environment (such as temperature, depth of the seabed, salinity, etc.). Namely, the sound ray will propagate along a curved trajectory, which aggravates the prediction of the ray propagation paths such as refraction and reflection, making it difficult to implement the sound-tracking and compensation measures effectively. The acoustic propagation path bending caused by the sound speed profile affects the accuracy of distance measurement in the underwater acoustic propagation and causes errors which make a deviation from the true value large [3]. Therefore, it is necessary to optimize the sound speed profile in the shallow sea area and to compensate the sound ray in the underwater acoustic propagation, to improve the positioning accuracy of the ultra-short baseline. At present, there are many international studies on the methods of correcting the model of sound speed profile such as the empirical sound velocity [4], weighted average sound velocity method [5], polynomial fitting method [6; 7], the method of equivalent speed profile [8], effective speed of sound see table and look-up table method [9; 10], etc. The above methods are based on the original sound velocity data. Although these methods can simulate the sound speed profile well, they abandon the measured sound velocity profile data completely, which easily leads to a poor sound speed correction in the shallow water areas. However, it is difficult to provide the effective data of sound speed profile in complex shallow water areas. Therefore, in order to accurately restore the sound speed profile in the actual environment and improve the ray tracing method, this paper uses an empirical orthogonal function (EOF) to invert the sound speed profile [11]. This paper proposes a combined ray tracing method that can be used in the USBL to adaptively determine whether to use the constant sound speed ray tracing method or the equal gradient ray tracing method. The simulation experiments show that the proposed method has better flexibility than the traditional ray tracing and positioning method, and the positioning accuracy is improved keeping the amount of calculation iterations basically the same. It should be claimed that, in this work, the amount of iterations is not same as the amount of calculation. If the work done in each iteration is different between the compared processes, the amount of calculation compared between several methods could be different. This paper is divided into eight parts. Section 1 describes the principle of ultra-short baseline positioning optimization and the reasons for the positioning error, and briefly introduces the existing technology of sound ray tracing method. Section 2 introduces the specific methods and principles of ultra-short baseline positioning. Section 3 introduces the principle method of empirical orthogonal function. Section 4 introduces the basic theory of ray acoustics in layered media.