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

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Helical antennas have been developed that feature a cutoff pattern and are suitable for practical positioning at millimeter level. The antennas are in the form of a tube with 3 cm in diameter and 40 cm in height. The antennas provide 20-dB suppression of the reflections from the earth surface, starting from low elevations. Open sky tests have demonstrated that the multipath contribution to the real-time positioning error falls below thermal noise. By smoothing the noise, these antennas achieve a positioning accuracy of 1 mm RMS in the vertical coordinate and 0.7 mm in the horizontal in almost real time.

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