CONTENTS 2.5.3 Mixed Model with Observation Equations I 30 2.5.4 Sequential Observation Equation Model I 32 2.5.5 Observation Equation Model with Observed Parameters I 32 2.5.6 Mixed Model with Conditions I 34 2.5.7 Observation Equation Model with Conditions I 35 2.6 Minimal and Inner Constraints I 37 2.7 Statistics in Least-Squares Adjustrnent I 42 2.7.1 Fundamental Test I 42 2.7.2 Testing Sequential Least Squares I 48 2.7.3 General Linear Hypothesis I 49 2.7.4 Ellipses as Confidence Regions I 52 2.7.5 Properties of Standard Ellipses I 56 2.7.6 Other Measures of Precision I 60 2.8 Reliability I 62 2.8.1 Redundancy Numbers I 62 2.8.2 Controlling Type-II Error for a Single Blunder I 64 2.8.3 Interna! Reliability I 67 2.8.4 Absorption I 67 2.8.5 External Reliability I 68 2.8.6 Correlated Cases I 69 2.9 Blunder Detection I 70 2.9.1 Tau Test I 71 2.9.2 Data Snooping I 71 2.9.3 Changing Weights of Observations I 72 2.10 Examples I 72 2.11 Kaiman Filtering I 77 RECURSIVE LEAST SQUARES 3.1 Static Parameter I 82 3.2 Static Parameters and Arbitrary Time-Varying Variables I 87 3.3 Dynamic Constraints I 96 3.4 Static Parametersand Dynamic Constraints I 112 3.5 Static Parameter, Parameters Subject to Dynamic Constraints, and Arbitrary Time-Varying Parameters I 125 4 GEODESY 4.1 International Terrestrial Reference Frame I 131 4.1.1 Polar Motion I 132 4.1.2 Tectonic Plate Motion I 133 4.1.3 Solid Earth Tides I 135 81 129 CONTENTS vii 4.1.4 Ocean Loading I 135 4.1.5 Relating of Nearly Aligned Frarnes I 136 4.1.6 ITRF and NAD83 I 138 4.2 International Celestial Reference System I 141 4.2.1 Transforming Terrestrial and Celestial Frames I 143 4.2.2 Time Systems I 149 4.3 Datum I 151 4.3.1 Geoid I 152 4.3.2 Ellipsoid of Rotation I 157 4.3.3 Geoid Undulations and Deftections of the Vertical I 158 4.3.4 Reductions to the Ellipsoid I 162 4.4 3D Geodetic Model I 166 4.4.1 Partial Derivatives I l 69 4.4.2 Repararneterization I 170 4.4.3 Implementation Considerations I 171 4.4.4 GPS Vector Networks I 174 4.4.5 Transforming Terrestrial and Vector Networks I 176 4.4.6 GPS Network Exarnples I 178 4.4.6.1

The integration of GPS with GLONASS may be considered a major milestone in satellite-based positioning, because it can dramatically improve the reliability and productivity of said positioning. However, unlike GPS, GLONASS satellites transmit signals at different frequencies, which result in significant complexity in terms of modeling and ambiguity resolution for integrated GPS and GLONASS positioning systems. In this paper, a variety of mathematical and stochastic modeling methodologies and ambiguity resolution strategies are analyzed, and some remaining research challenges are identified. The exercise, of developing mathematical models and processing methodologies for integrated systems based on more than one satellite system, is a valuable one as it identifies crucial issues concerned with the combination of any two or more microwave positioning systems, be they satellite-based or terrestrial. Hence these are experiences that can be applied to future projects that might integrate GPS with Galileo, or GLONASS and Galileo, or all three.

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