We propose an acoustic-based framework for automatically homing an Autonomous Underwater Vehicle (AUV) to the fixed docking station (F-DS) and mobile docking station (M-DS). The proposed framework contains a simultaneous localization method of AUV and docking station (DS) and a guidance method based on the position information. The Simultaneous localization and mapping (SLAM) algorithm is not available as the statistical characteristics of the measurement error of the observation system are unknown. To solve this problem, we propose a data pre-processing method. Firstly, the measurement error data of acoustic sensor are collected. Then, We propose a Variational Auto-Encoder (VAE) based Gaussian mixture model (GMM) for estimating the statistical characteristics of measurement error. Finally, we propose a support vector regression (SVR) algorithm to fit the non-linear relationship between the statistical characteristics of measurement error and its corresponding working distance. We adopt a guidance method based on line-of-sight (LOS) and path tracking method for homing an AUV to the fixed docking station (F-DS) and mobile docking station (M-DS). The lake experimental data are used to verify the performance of the localization with the estimated statistical characteristics of measurement error.