  Localization is one of the pivotal issues in wireless sensor network
applications. In 3D localization studies, most algorithms focus on enhancing
the location prediction process, lacking theoretical derivation of the
detection distance of an anchor node at the varying hops, engenders a
localization performance bottleneck. To address this issue, we propose a
probability-based average distance estimation (PADE) model that utilizes the
probability distribution of node distances detected by an anchor node. The aim
is to mathematically derive the average distances of nodes detected by an
anchor node at different hops. First, we develop a probability-based maximum
distance estimation (PMDE) model to calculate the upper bound of the distance
detected by an anchor node. Then, we present the PADE model, which relies on
the upper bound obtained of the distance by the PMDE model. Finally, the
obtained average distance is used to construct a distance loss function, and it
is embedded with the traditional distance loss function into a multi-objective
genetic algorithm to predict the locations of unknown nodes. The experimental
results demonstrate that the proposed method achieves state-of-the-art
performance in random and multimodal distributed sensor networks. The average
localization accuracy is improved by 3.49\%-12.66\% and 3.99%-22.34%,
respectively.
