Physical constraints for zenith wet delay estimation via inequality constrained least squares in real-time PPP

Abstract

In current conventional precise point positioning (PPP) processing strategies, the tropospheric zenith wet delay (ZWD) is usually dynamically estimated as a stochastic parameter. During the convergence period, ZWD estimates can appear to be negative or unrealistically large due to the low estimation precision, which adversely affects the estimation of other state parameters, especially the Up component of coordinates. To address this issue, we propose a method that incorporates physical constraints on ZWD in PPP processing. This method employs the inequality constrained least squares (ICLS), utilizing Karush–Kuhn–Tucker (KKT) conditions to add boundary conditions on ZWD. The boundary conditions of ZWD are calculated based on the relation between ZWD and relative humidity (RH). The use of physical constraints does not rely on external products or space state representation (SSR) corrections for ZWD during PPP processing and can improve the short-term accuracy of ZWD and coordinate Up component. The efficiency of this approach has been validated using GNSS data and products from GFZ operational networks. For real-time PPP solutions, there is a 30%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$30\%$$\end{document} improvement in short-term accuracy of Up component; for post-processing solutions, the short-term RMSE improvement is about 20%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$20\%$$\end{document}. After convergence, the ZWD upper bound is no longer applied as an ICLS constraint, but is instead used as a diagnostic indicator to identify ZWD anomalies. This indicator demonstrates high sensitivity and reliability under extreme weather conditions, highlighting its potential for application in meteorological hazard early-warning systems.