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.

翻译：定位是无线传感器网络应用中的关键问题之一。在3D定位研究中，大多数算法集中于改进位置预测过程，缺乏对锚节点在不同跳数下探测距离的理论推导，导致定位性能出现瓶颈。为解决这一问题，我们提出一种基于概率的平均距离估计（PADE）模型，该模型利用锚节点探测到的节点距离的概率分布，旨在通过数学推导得出锚节点在不同跳数下探测到的节点平均距离。首先，我们开发了一种基于概率的最大距离估计（PMDE）模型，用于计算锚节点探测距离的上界。然后，我们提出了PADE模型，该模型依赖于PMDE模型获得的距离上界。最后，将获得的平均距离用于构建距离损失函数，并将其与传统的距离损失函数嵌入到多目标遗传算法中，以预测未知节点的位置。实验结果表明，所提出的方法在随机和多模态分布传感器网络中达到了最先进的性能。平均定位精度分别提升了3.49%-12.66%和3.99%-22.34%。