Hydraulic models have emerged as a powerful tool for simulating the real behavior of water distribution systems (WDSs). In using the models for estimating nodal water demands, measurement uncertainty must be considered. A common approach is to use the covariance of measurement noises to quantify the measurement uncertainty. The noise covariance is typically assumed constant and estimated a priori. However, such an assumption is frequently misleading as actual measurement accuracies are affected by measuring instruments and environmental noises. In this study, we develop a variational Bayesian approach for real-time estimation of noise covariance and nodal water demands. The approach can adaptively adjust the noise covariance with the variation of the noise intensity, thereby efficiently avoiding model overfitting. The measurement residual decomposition reveals that this new approach is effective in determining model structural errors caused by topological structure parameterization.

水力模型已经成为模拟水分配系统（WDS）真实行为的强大工具。在使用模型估算节点需水量时，必须考虑测量不确定度。一种常见的方法是使用测量噪声的协方差来量化测量不确定度。通常假定噪声协方差为常数，并先验估计。但是，由于实际的测量精度会受到测量仪器和环境噪声的影响，因此这种假设经常会产生误导。在这项研究中，我们开发了一种变分贝叶斯方法来实时估计噪声协方差和节点需水量。该方法可以随着噪声强度的变化而自适应地调整噪声协方差，从而有效地避免了模型的过度拟合。