## 1 Introduction  
For target tracking in the ocean environment, two approaches are generally used. Linear Kalman filter (KF) is a first and foremost approach, and the extended Kalman filter(EKF) is the second approach [1; 2; 3; 4; 5]. KF/EKF is highly sensitive to the accuracy of the noise statistics. In practical applications, as is often the case, the covariance matrixes of the process noise and the measurement noise may be unknown or inaccurate. In the case of unknown covariance matrixes, the state estimation performance of KF/EKF will be decreased and it may even diverge [6]. More and more adaptive filters with model uncertainty are developed to solve this problem. In this manuscript, we focus on the model uncertainty of the process noise, specifically, the process noise is unknown and non-Gaussian.  
A variety of filters were proposed to address the state estimation problem with unknown noise covariance matrixes, such as robust KF [7; 8], adaptive KF [9; 10; 11; 12; 13; 14; 15], and others [16]. Generally Speaking, the adaptive KF methods can be divided into covariance matching, maximum likelihood (ML), Bayesian methods and so on. More recently, the variational Bayesian (VB) methods wasn applied into the filtering framework [17; 18; 19]. In [14; 20], a VB adaptive KF are proposed to jointly estimate the state and the measurement noise covariance matrix (MNCM) where the prior distribution of the covariance is governed by an inverse-Gamma distribution and inverse-Wishart distribution, respectively.  
To the best of our knowledge, the VB-based adaptive filters are only used for online state estimation with unknown MNCM, and rarely used for the online state estimation with unknown process noise covariance matrix (PNCM). In [13], the online state estimation with inaccurate PNCM is treated, in which the prediction error covariance matrix rather than PNCM is approximated by the VB method. It cannot give the estimated value of the unknown PNCM. Besides, the authors of [21] experimentally show that the state estimation performance of the proposed adaptive filter in [13] may be worse than that of the traditional KF in some cases. In [22], a new adaptive KF algorithm is proposed to cope with the unknown PNCM for the linear discrete-time systems. The PNCM is estimated by the proposed algorithm based on the measurement sequence. In [23], an adaptive KF algorithm in presence of PNCM based on black box variational inference is proposed. The black box variational inference method is recently introduced to conduct the approximate Bayesian inference for the non-conjugate probabilistic model. They structured a posterior model of the system state and PNCM firstly, and then, the black box variational inference online inference for the posterior distribution of the PNCM is derived.  
In this paper, we address the online state estimation problem in the presence of the unknown and non-Gaussian PNCM. For the target tracking in the ocean environment, the process noise of AUV is unknown or inaccurate, and the process noise of MRS is unknown totally. Due to the complexity of an underwater environment, the actual noise of AUV speed sensor is non-Gaussian. The main contributions of this paper are summarized as follows:  
1. We propose a target tracking method based on EKF (TT-EKF) to simultaneously estimate the states of an AUV and an MRS.
2. We estimate the noise statistics of the heading and forward speed of an AUV and an MRS by using the variational Bayesian algorithm. The noisy forward speed is modeled by the Gaussian mixture distribution (GMD). The noisy heading angle is modeled by a von Mises (vM) distribution.
3. We build a neural network compensator to make up for the prediction error caused by the process noise.