<p>PPP–RTK integrates the strengths of precise point positioning (PPP) and real-time kinematic (RTK), representing the state-of-the-art in global navigation satellite system (GNSS) positioning. The use of undifferenced and uncombined (UDUC) observations offers flexibility in processing multi-frequency and multi-GNSS data, motivating extensive research on UDUC PPP–RTK. A key task in developing a UDUC PPP–RTK model is resolving rank deficiencies by constraining a minimum set of parameters, known as the S-basis. The choice of S-basis varies, particularly when addressing rank deficiencies among satellite phase biases, receiver phase biases, and ambiguities. In principle, the biases or ambiguities of any receiver or satellite, at any epoch, can serve as the S-basis, provided that the resulting ambiguities remain integer-estimable. This contribution analyzes how spatially dependent (satellite- and receiver-dependent) and temporally dependent (epoch-dependent) S-basis choices influence the estimability of satellite phase biases. We show that different spatially dependent S-basis choices yield distinct estimable satellite phase biases and ambiguities, with the biases absorbing different S-basis ambiguities and the corresponding estimable ambiguities being double-differenced or linear combinations thereof. Since the analytical forms of these satellite phase biases differ, their precision varies, leading to inconsistent positioning performance across S-basis choices. However, since these differences stem from the absorbed S-basis ambiguities, fixing the double-differenced ambiguities enhances product precision and ensures consistent PPP–RTK performance across spatially dependent choices. By contrast, adopting different temporally dependent S-basis choices causes the estimable satellite phase biases to absorb the ambiguity of the selected epoch along with the estimable between-epoch single-differenced (BESD) ambiguities. We demonstrate that these BESD ambiguities are also integer-estimable, and consistent PPP–RTK performance can be maintained once they can be reliably fixed. In the case of failed ambiguity fixing, an alternative way is to transform the S-basis from the previous to the current epoch whenever a cycle slip occurs, such that the satellite phase biases absorb the ambiguities of the current epoch, and no BESD ambiguities need to be estimated for that epoch. However, this strategy introduces discontinuities in the estimable satellite phase bias since it absorbs multiple ambiguities, and any cycle slip in one of these absorbed ambiguities will cause discontinuities. Therefore, it is essential to minimize the number of ambiguities absorbed by the satellite phase bias to reduce the risk of such discontinuities. We show that this condition can be satisfied in single-station PPP–RTK and in common-view networks where all receivers track the same satellites, but it remains a challenge in all-in-view networks where satellite visibility varies across receivers, such as global networks.</p>

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Multi-epoch estimability of satellite phase biases in undifferenced and uncombined PPP-RTK: impact of S-basis choices

  • Pengyu Hou,
  • Cheng Ke,
  • Baocheng Zhang

摘要

PPP–RTK integrates the strengths of precise point positioning (PPP) and real-time kinematic (RTK), representing the state-of-the-art in global navigation satellite system (GNSS) positioning. The use of undifferenced and uncombined (UDUC) observations offers flexibility in processing multi-frequency and multi-GNSS data, motivating extensive research on UDUC PPP–RTK. A key task in developing a UDUC PPP–RTK model is resolving rank deficiencies by constraining a minimum set of parameters, known as the S-basis. The choice of S-basis varies, particularly when addressing rank deficiencies among satellite phase biases, receiver phase biases, and ambiguities. In principle, the biases or ambiguities of any receiver or satellite, at any epoch, can serve as the S-basis, provided that the resulting ambiguities remain integer-estimable. This contribution analyzes how spatially dependent (satellite- and receiver-dependent) and temporally dependent (epoch-dependent) S-basis choices influence the estimability of satellite phase biases. We show that different spatially dependent S-basis choices yield distinct estimable satellite phase biases and ambiguities, with the biases absorbing different S-basis ambiguities and the corresponding estimable ambiguities being double-differenced or linear combinations thereof. Since the analytical forms of these satellite phase biases differ, their precision varies, leading to inconsistent positioning performance across S-basis choices. However, since these differences stem from the absorbed S-basis ambiguities, fixing the double-differenced ambiguities enhances product precision and ensures consistent PPP–RTK performance across spatially dependent choices. By contrast, adopting different temporally dependent S-basis choices causes the estimable satellite phase biases to absorb the ambiguity of the selected epoch along with the estimable between-epoch single-differenced (BESD) ambiguities. We demonstrate that these BESD ambiguities are also integer-estimable, and consistent PPP–RTK performance can be maintained once they can be reliably fixed. In the case of failed ambiguity fixing, an alternative way is to transform the S-basis from the previous to the current epoch whenever a cycle slip occurs, such that the satellite phase biases absorb the ambiguities of the current epoch, and no BESD ambiguities need to be estimated for that epoch. However, this strategy introduces discontinuities in the estimable satellite phase bias since it absorbs multiple ambiguities, and any cycle slip in one of these absorbed ambiguities will cause discontinuities. Therefore, it is essential to minimize the number of ambiguities absorbed by the satellite phase bias to reduce the risk of such discontinuities. We show that this condition can be satisfied in single-station PPP–RTK and in common-view networks where all receivers track the same satellites, but it remains a challenge in all-in-view networks where satellite visibility varies across receivers, such as global networks.