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Chapter 7 Orbit Determination Using Carrier Phase Observation order to use carrier phase observation for GEO satellite orbit determination, other independent methods such as CCD camera and SLR could be used in data preprocessing. Using these techniques, the precise distance between satellite and tracking station is obtained and the initial ambiguities of carrier phase observations can be computed. Because the tracking stations for orbit determination are located in the open area, the locked signals in the tracking stations will not be easily interrupted. Therefore initial ambiguities will not be easily changed. 7.3 Ambiguity Resolution Approaches There are many ambiguity resolution approaches for GPS. Some typical methods may also be suitable for GNSS-2/Galileo orbit determination when using carrier phase measurements. Following Teunissen (1994), the ambiguity approach can be described as follows. The carrier phase measurement can be represented as ϖ ϖ Φ= r − R + c( dt− dT) + λN + ε (7-15) where ϖ r unknown receiver antenna position vector at signal reception time ϖ R given satellite antenna position vector at signal transmission time c speed of light in vacuum cdt receiver clock range offset cdT satellite clock range offset λ carrier wavelength of the signal N carrier phase ambiguity The term ε represents carrier phase measurement noise and biases such as satellite ephemeris errors, tropospheric and ionospheric delays, and ranging errors caused by multipath effects. In the processing of carrier phase data it is common to differentiate the carrier phase measurements between satellites and between receivers to eliminate some common errors. This gives the single and double differenced carrier phase observation equations ∇Φ = ∇ r− R + c∇ dt+ λ∇ N+ ∇ε (7-16) ∆∇Φ = ∆∇ r− R + λ∆∇N+ ∆∇ε (7-17) ∇ and ∆∇ stand for the single and double difference operator. In the following discussion a two-receiver situation is assumed. The term ∇dt then denotes the single differenced clock error between the reference receiver and the remote receiver. Because of double differencing this error vanishes in Eq.(7-17). In both equations the ambiguity-terms ∇N and ∆∇N are known to be integers. Linearization of the observation equations with respect to the unknown parameters and a collection of these linearized equations into a linear system of equations gives: ϖ ϖ ϖ ϖ y = Aa+ Bb +ε (7-18) ϖ ϖ ϖ with E{} ε = 0, D{} ε = D{} y = Qy where ϖ y vector of observed minus computed differenced carrier phase observations ϖ a vector of unknown ambiguities ϖ b vector of unknown coordinates of baseline or satellite position, velocity and related dynamic parameters A,B design matrices for unknown ambiguities and parameters ϖ ε vector of unmodeled errors (in the following ϖ Qy ε is sufficiently small to be neglected) variance-covariance matrix of the observations ϖ y E{.} expectation operator D{.} dispersion operator Because ϖ a should be integers, it can be shown, that Eq. (7-18) may be solved in three steps. Firstly, estimating a double difference float solution (real-valued ambiguities ϖ ϖ a ∃ ∃b with ), i.e. ϖ ϖ ϖ 2 min y− Aa−Bb with ab , Qy ϖ ϖ ab R n , ∈ (7-19) 90
Chapter 7 Orbit Determination Using Carrier Phase Observation where 2 T −1 Q y y . = (.) Q (.) Q y R n variance-covariance matrix of double-difference carrier phase observations n-dimensional space of reals Secondly, solving the following minimization problem, integer ambiguity estimation: min( ∃ ) ( ∃ ϖ ϖT −1 ϖ ϖ a−a Qϖa−a) with ϖ a Z m ∈ (7-20) a where Z m a∃ m-dimensional space of integers The final step is called fixed solution, once the integer least square ambiguity vector ϖ( m a ∈ Z has been obtained, the residual ( ∃ ϖ ϖ( a−a) is used to adjust the solution ϖ∃ b . As a result, the final solution is obtained as ϖ( ϖ∃ − ϖ ϖ( b = b −QϖϖQϖ( a−a ∃ ) (7-21) ∃∃ ba a∃ 1 The difficulty of integer ambiguity solution is in the second step, the integer ambiguity estimation. There are many ambiguity resolution approaches to solve this problem, two of them are LAMBDA and TCAR. 7.3.1 LAMBDA Method LAMBDA stands for Least-squares AMBiguity Decorrelation Adjustment method. Its two main features are • sequential conditional least squares estimation • preceded by a decorrelation of the ambiguities The method contains a strict extension of standard least-squares to the integer domain. The novelty of the method is a decorrelating reparametrization of the ambiguities, by which the integer least-squares estimates can be computed very fast and efficiently. This new method is proposed by Teunissen (1993, 1994). The basic LAMBDA idea is that integer ambiguity solution will become easy once the confidence ellipsoid of the ambiguities equals a sphere. In the case of GPS, however, the confidence ellipsoid is usually rotated with respect to the coordinate axes and extremely elongated, particularly for short observational time-spans and without P-code data. Teunissen (1993, 1994, 1995) introduced a one-to-one transformation from the original set of ambiguities to a new set of ambiguities, of which the confidence ellipsoid has more sphere-like properties and therefore the ambiguities are more decorrelated. LAMBDA method consists of two steps. The first, the ambiguity transformation, i.e. ambiguity decorrelation, and the second, ambiguity search based on transformed ambiguity space. The first step is unique for LAMBDA method. The second step may use any other ambiguity search approach (Hatch, 1990; Frei et al, 1990, 1993; Euler and Landau, 1992 etc). Therefore in the following we focus on the first step, ambiguity transformation used by LAMBDA. Since in Eq.(7-20) the constraint is an integer-constraint ϖ a Z m ∈ , it is also called integer least-squares estimation. The key problem is the minimization of Eq.(7-20). All possible ambiguity vectors with integer elements that can solve Eq.(7-20) belong to a confidence ellipsoid that is given by: ϖ m ϖ ϖ ϖT − ϖ ϖ { ϖ a∃ m, − } 1 2 1 E: a R | Q( a) ( a a Q a a ∃ ) ( ∃ = ∈ = − − ) ≤ χ α (7-22) where, E ellipsoid set of points ϖ Qa ( ) a in ϖ quadratic form in ϖ a R m 2 χ m,1−α chi-squares percentile for m degrees of freedom and confidence level 1- α α error probability (significance level) LAMBDA objective is to reparametrize Eq.(7-20) in such a way, that an equivalent formulation is obtained, but one that is easily solved. The simplest integer estimation method is "rounding to the nearest integer" and applied 91
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PRECISE ORBIT DETERMINATION OF GLOB
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Summary GNSS-2 (Global Navigation S
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3.2.2.1 Principle..................
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9.3.3 The Effect of Distribution of
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Chapter 1 Introduction 1.2 Developm
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Chapter 2 Observations of Orbit Det
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Chapter 3 Orbit Tracking System and
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Chapter 4 Major Error Sources of Sa
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Page 48 and 49: Chapter 4 Major Error Sources of Sa
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Page 104 and 105: Chapter 8 Geostationary Orbit Deter
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Chapter 9 Software and Simulation R
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Chapter 9 Software and Simulation R
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Conclusion 144
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References Carpino M. (1995): SAEG
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References Klobuchar, J. A.(1996):
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References Traquilla, J. M. and J.
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Acknowledgments 152
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