Most underwater target tracking algorithms are based on noisy bearing-only measurements. The bearing measurements are actually bearing estimates obtained from acoustic field measurements made by an array of sensors placed on a manoeuvrable platform. In this paper, we investigate the problem of tracking a manoeuvring target in the ocean directly from the acoustic field measurements. The use of acoustic field measurements eliminates the need for the use of a manoeuvrable platform, and also eliminates the intermediate step of bearing estimation. We consider the vector acoustic field measurements by an array of acoustic vector sensors (AVS), which are more informative than the scalar acoustic field measurements by an array of acoustic pressure sensors (APS). We model the target dynamics by a piecewise-constant turn rate model. The resultant nonlinear estimation problem is solved using the Gaussian approximation filtering approach. Within this framework, we consider the quasi-Monte Carlo Kalman filtering (QMC-KF) algorithm that employs an advanced numerical integration procedure to solve the multi-dimensional integrals appearing in the filtering process. We also derive the posterior Cramer-Rao lower bound (PCRLB) on the root mean square error (RMSE) of the sequential Bayesian estimator, and show that the PCRLB for acoustic field measurements is lower than that for bearing-only measurements. Simulation results indicate that acoustic field measurements yield a significantly better performance than bearing-only measurements, the QMCKF performs better than other deterministic filters, and an AVS array provides a much better performance than an APS array.