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<p>In GNSS precise point positioning (PPP), a user commonly estimates a troposphere parameter representing the zenith wet delay (ZWD). The ZWD is non-negative and generally assumes values of up to around 30&#xa0;cm. In the initial epochs of a PPP solution, however, the precision of the estimated ZWD parameter can be on the several meter level, most likely leading to unrealistic ZWD estimates outside of this interval, and large errors in the up component. In this contribution, we introduce and discuss a troposphere-bounded GNSS model, in which the ZWD parameter is constrained to an a-priori defined feasible interval. This translates to an observation model with linear inequality constraints, for which we derive and discuss two solutions, the constrained float solution and the constrained integer least-squares (ILS) solution. We analyze the positioning capabilities of the troposphere-bounded model as compared to the unconstrained case by means of simulated multi-frequency GPS and Galileo PPP examples and exemplary real-data experiments. With the constrained ambiguity-float solution, the RMS error of the up component is improved by up to 75% for GPS and up to 65% for GPS + Galileo solutions in the first observation epochs, even for relatively loose ZWD constraint intervals of 20&#xa0;cm or more. The constrained ILS solution leads to higher ambiguity success rates, so that for instance ZWD constraint intervals of five and 20&#xa0;cm reduce the average times-to-first-fix the ambiguities by around 17.5% and 10%.</p>

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Inequality-constrained integer least-squares and its application to troposphere-bounded GNSS PPP

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  • 发表日期：

    2025年11月12日

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  • Andreas Brack,
  • Shengping He

摘要

In GNSS precise point positioning (PPP), a user commonly estimates a troposphere parameter representing the zenith wet delay (ZWD). The ZWD is non-negative and generally assumes values of up to around 30 cm. In the initial epochs of a PPP solution, however, the precision of the estimated ZWD parameter can be on the several meter level, most likely leading to unrealistic ZWD estimates outside of this interval, and large errors in the up component. In this contribution, we introduce and discuss a troposphere-bounded GNSS model, in which the ZWD parameter is constrained to an a-priori defined feasible interval. This translates to an observation model with linear inequality constraints, for which we derive and discuss two solutions, the constrained float solution and the constrained integer least-squares (ILS) solution. We analyze the positioning capabilities of the troposphere-bounded model as compared to the unconstrained case by means of simulated multi-frequency GPS and Galileo PPP examples and exemplary real-data experiments. With the constrained ambiguity-float solution, the RMS error of the up component is improved by up to 75% for GPS and up to 65% for GPS + Galileo solutions in the first observation epochs, even for relatively loose ZWD constraint intervals of 20 cm or more. The constrained ILS solution leads to higher ambiguity success rates, so that for instance ZWD constraint intervals of five and 20 cm reduce the average times-to-first-fix the ambiguities by around 17.5% and 10%.