Existing works have widely studied relay-aided underwater acoustic networks under some specialized relay distributions, e.g., equidistant and rectangular-grid. In this paper, we investigate two fundamental problems that under which conditions a relay should be deployed and where to deploy it if necessary in terms of the energy and delay performance in linear underwater acoustic networks. To address these two problems, we first accurately approximate the complicated effective bandwidth and transmit power in the logarithm domain to formulate an energy consumption minimization problem. By analyzing the formulation, we discover a critical transmission distance, defined as open distance, and explicitly show that a relay should not be deployed if the transmission distance is less than the open distance and should be otherwise. Most importantly, we derive a closed-form and easy-to-calculate expression for the open distance and also strictly prove that the optimal placing position is at the middle point of the link when a relay should be introduced. Moreover, although this paper considers a linear two-hop relay network as the first step, our derived results can be applied to construct energy-efficient and delay-friendly multi-hop networks. Simulation results validate our theoretical analysis and show that properly introducing a relay can dramatically reduce the network energy consumption almost without increasing the end-to-end delay.

Existing works have explored anchor deployment for autonomous underwater vehicles (AUVs) localization under the assumption that sound propagates straightly underwater at a constant speed. Considering that the underwater acoustic waves propagate along bent curves at varying speeds in practice, it becomes much more challenging to determine a proper anchor deployment configuration. In this paper, taking the practical variability of underwater sound speed into account, we investigate the anchor-AUV geometry problem in a 3-D time-of-flight-based underwater scenario from the perspective of localization accuracy. To address this problem, we first rigorously derive the Jacobian matrix of measurement errors to quantify the Cramer–Rao lower bound (CRLB) with a widely-adopted isogradient sound speed profile. We then formulate an optimization problem that minimizes the trace of the CRLB subject to the angle and range constraints to figure out the anchor-AUV geometry, which is multivariate and nonlinear, and thus generally hard to handle. For mathematical tractability, by adopting tools from the estimation theory, we interestingly find that this problem can be equivalently transformed into a more explicit univariate optimization problem. By this, we obtain an easy-to-implement anchor-AUV geometry that yields satisfactory localization performance, referred to as the uniform sea-surface circumference (USC) deployment. Extensive simulation results validate our theoretical analysis and show that our proposed USC scheme outperforms both the cube and the random deployment schemes in terms of localization accuracy under the same parameter settings.