## 1 Introduction  
In the last years, sea protection and monitoring have been central topics for the underwater community. However, because of the difficulties and risks involved in underwater missions, most part of the marine environment still remains unexplored. Thanks to recent developments in technology and automation, _Autonomous Underwater Vehicles_ (AUVs) can now play a key role in underwater exploration. Indeed, AUVs represent effective tools for very different purposes, such as glacier inspection [1], monitoring of marine habitats [2] and climate changes effects [3], and for seafloor mapping [4]. In particular, the exploitation of AUVs for monitoring purposes makes data collection faster and certainly safer [5]. Traditionally, in _Information Gathering_ (IG) operations, where underwater exploration entails environmental data collection, AUVs follow predefined geometric routes, typically with lawnmower patterns. However, this approach is not efficient as the AUV moves along a path which can lead to areas of low interest, wasting time and energy. On the contrary, optimal path planning algorithms generate paths that optimise an objective function representing specific criteria, such as energy consumption, survey duration or distance travelled [6]. Moreover, by benefiting from the measurements coming from the sensors on-board the vehicle, the planner can employ online replanning strategies to avoid collisions with potential obstacles along the way. Among the various optimal planning strategies, _Informative Path Planning_ (IPP) focuses on increasing the knowledge about environmental variables of interest while respecting some constraints on the robot and path features [7]. A model of the environmental variable of interest, based on the _a priori_ knowledge about the area to be inspected, is crucial to achieve efficient informative paths and a successful completion of missions. An efficient way for modelling the environment is applying a _Gaussian Process_ (GP) [8]. GPs are powerful nonparametric techniques that can handle a large variety of problems, having the ability to learn spatial correlation with noisy measured data [9]. The key feature of GPs for IG scenarios is their capability to handle both data uncertainty and data incompleteness effectively. Thus, they allow to perform dense estimation of environmental variables for coverage-based exploration purposes, giving a clear idea of which are the places that need to be inspected to reduce model uncertainty. However, the weakness of these techniques is related to their high computational time. Some works overcome this issue by proposing the use of a sparse version of the GP [10; 11], or local maps fusion using Bayesian Committee Machine (BCM) [12], to reduce computational costs and enable online execution. The estimate of the variable of interest provided by the GP, in terms of mean and covariance, can be exploited to quantitatively describe the informativeness associated with each position within the target area, using the so called information function. In particular, higher values on the covariance map indicate points where the GP estimates are more uncertain and, thus, more informative: collecting data in those areas means reducing the uncertainty of the model, and thus increasing knowledge about the environmental variable of interest. While the authors of [13] propose the use of an interpolated version of the GP model predicted variance, most of the methods use either _Differential Entropy_ (DE) or _Mutual Information_ (MI) as information function. Using DE results in lower computation time and higher uncertainty reduction than using MI, when computed from GP predictions of stationary setups [14]. Despite the submodularity property of MI [15], the computational time of such information function results prohibitive in many applications. The IPP approaches used for IG purposes can be categorised into two groups: myopic and non-myopic. Myopic strategies only plan one step forward and they work following greedy heuristics [16].