GPS Positioning by an Estimation of the Measurement Variance Satellite Surveying and Navigation Laboratory Center for Space and Remote Sensing Research National Central University 320 Jhongli, Taiwan, R.O.C. 1 Outline Preface Mathematical Model Least-Squares Adjustment Variance-Component Estimator Whitening Filter Experiments Summary and Outlook 2 Preface (1/2) Ranges between several global positioning system (GPS) satellites and a user can be measured at a signal receiving epoch. The ranges can be processed to determine the user’s 3D position, in either a static or on-the-fly mode. Basically, the quality of GPS positioning depends on the range precision, satellite geometry, etc. Preface Model LeastSquares VarianceComponent Whitening Tests Summary 3 Preface (2/2) Preface Model LeastSquares VarianceComponent Whitening A double-difference combination of the observations is very often carried out. In a local area, such a differencing operation can reduce many errors effectively. As far as a resolution of the whole-cycle ambiguity parameter is concerned, the estimated covariance matrix plays an important role. Tests Summary 4 Mathematical Model (1/4) Provided that unmodeled errors are negligibly small, the pseudorange ρ (m) and the carrier phase (cycles) can be written as: ( ρ ν ρ )ijgh Rijgh 0, Preface Model LeastSquares VarianceComponent Whitening Tests Summary (φ νφ )ijgh 1 gh Rij N ijgh 0, λ where Rijgh Rig R gj Rih R hj , gh ρij g ρi g ρj h ρi h ρj , 5 Mathematical Model (2/4) N ijgh N ig N gj N ih N hj ; and i, j : reference and user receivers, Preface Model LeastSquares VarianceComponent Whitening Tests Summary g, h : λ: R: N: ν ρ , νφ : reference and other satellites, the wavelength (m), Euclidean topocentric distance (m), the integer ambiguity (cycles), the stochastic zero-mean pseudorange and phase errors, respectively. 6 Mathematical Model (3/4) By defining a measurement residual vector, T v (ν ρ g , ν ρ g , νφ g , νφ g , ν ρ h , ν ρ h , νφh , νφh ) , i j i j i j i j and a parameter correction vector, Preface Model LeastSquares VarianceComponent Whitening Tests Summary x (dX j , dY j , dZ j , N ijgh )T , we get Bv Ax l , where B is a linear coefficient matrix, A the design matrix, and l a misclosure vector. 7 Mathematical Model (4/4) Preface Model LeastSquares VarianceComponent Whitening Tests In addition, the covariance matrix of the vvector is a positive definite σ 02 Q matrix, presumably with σ 02 1. 2 σ0 Q If the ranges are independent, is diagonal, but has different V () variances, resulting in σ 02 Q diag(V ( ρig ), V ( ρ gj ), V (φig ), V (φ gj ), V ( ρih ), V ( ρ hj ), V (φih ), V (φ hj )). Summary 8 Least-Squares Adjustment Now, let Qx stand for the scaled covariance matrix of the x-vector, Ql and Qv for the respective covariance matrices of l and v, one gets Preface Model LeastSquares VarianceComponent Whitening x Qx A 1 Ql l , T 1 Ql l . T v Qv B Tests Summary 9 Variance-Component Estimator (1/5) In order to check whether the prior Q is built correctly, an estimator is derived from the expected value of the v T Q 1v quadraticform. Since zero-mean error is assumed, the expectation of v is zero, E (v ) 0. Preface Model LeastSquares VarianceComponent Whitening Tests Summary 10 Variance-Component Estimator (2/5) Preface Model LeastSquares T The expectation of v Q v becomes 1 tr(Qv Q ). A decomposition of the covariance matrix into m components is considered, m σ 02Q σ i2C i , i 1 VarianceComponent Whitening Tests Summary 1 where factor. 2 σ i stands for an unknown scale 11 Variance-Component Estimator (3/5) The four accompanying matrices are: g g C1 diag( V ( ρ1,i ), V ( ρ1, j ), V ( ρ1h,i ), V ( ρ1h, j ), 0, 0, 0, 0 , 0, 0, 0, 0 , 0, 0, 0, 0, ...), Preface Model g g C 2 diag( 0, 0, V ( ρ2 ,i ), V ( ρ2 , j ), 0, 0, 0, 0, Variance0, 0, V ( ρ2h,i ), V ( ρ2h, j ), 0, 0, 0, 0, ...), Component LeastSquares Whitening Tests Summary 12 Variance-Component Estimator (4/5) C 3 diag(0, 0, 0, 0, V (φ1g,i ), V (φ1g, j ), 0, 0, 0, 0, 0, 0, h V (φ1,i ), h V (φ1, j ), 0, 0, ...), Preface Model LeastSquares C 4 diag(0, 0, 0, 0, VarianceComponent 0, 0, 0, 0, 0, 0, g g 0, 0, V (φ2 ,i ), V (φ2 , j ), h h V (φ2 ,i ), V (φ2 , j ), ...). Whitening Tests Summary 13 Variance-Component Estimator (5/5) If one-to-one correspondence holds, an unbiased variance-component estimator can be defined as follows: Preface Model LeastSquares VarianceComponent T σˆi2 1 1 v Q Ci Q v , i 1, 2, ..., m . T tr(BCi B Q k ) Whitening Tests Summary 14 Whitening Filter (1/5) The covariance of an a-vector floatambiguity solution is the Qa matrix. A matrix factorization can lead to the T expression Qa UDU , where U is an upper unit triangular matrix and D is a diagonal one. Preface Model LeastSquares VarianceComponent Whitening Testts Summary 15 Whitening Filter (2/5) Preface Model LeastSquares VarianceComponent An intU operation changes every entry of U to integers. The matrix is transformed such that it, 1 T Q a (intU ) Qa (intU ) , becomes more diagonally dominant. Whitening Tests Summary 16 Whitening Filter (3/5) Preface Model LeastSquares VarianceComponent Whitening Tests Alternatively, a lower triangular L-matrix is used. If at the k-th iteration, either int U k or int Lk changes to an identity matrix, then T 1 (int Lk ) 1 (int U k ) 1 (int Lk 1 ) 1 1 1 1 (int U k 1 ) (int L1 ) (int U1 ) , represents a decorrelating transformation matrix. Summary 17 Whitening Filter (4/5) Preface Model LeastSquares VarianceComponent 1 The determinant of the T matrix is equal to one, leading to a volume-preserving transformation. The ambiguity a-vector and its covariance σ 02 Qa - matrix are mapped into new ones, 1 such that z T a and Q z T QaT 1 T . Whitening Tests Summary 18 Whitening Filter (5/5) Preface Model LeastSquares VarianceComponent Whitening 2 σ z When centered at , the new 0 Q z hyperellipsoid is aligned with the coordinate axis, due to diagonal dominance. Some int ( z ) vectors are chosen and transformed back, in terms of 1 T int ( z ) TT int (a ) int (a ). Tests Summary 19 Single-Epoch Weighting Adjustment (1/3) Preface Model A software, called ManGo (Managing GNSS-data for orientation), has been developed using the parameter estimators mentioned above. LeastSquares VarianceComponent Whitening Tests Summary 20 Single-Epoch Weighting Adjustment (2/3) Preface Model LeastSquares A double-difference ambiguity covariance matrix before a decorrelating transformation will be shown. The decorrelated ambiguity covariance matrix is plotted for comparison. VarianceComponent Whitening Tests Summary 21 Single-Epoch Weighting Adjustment (3/3) The dual-frequency observations were each solved for the variances by using the (unbiased) estimator, on a component-bycomponent basis. The scaling (dimensionless) variancecomponent values indicated that they are time-varying. Preface Model LeastSquares VarianceComponent Whitening Tests Summary 22 Zero-Baseline Test Preface Model ManGo yields single-epoch position solutions. Also plotted are the results from SKI-Pro (the predecessor of LGO, Leica’s proprietary software). LeastSquares VarianceComponent Whitening Tests Summary 23 Effectiveness in Resolving the Ambiguity Preface The effectiveness consists in reducing the number of some candidate integerambiguity vectors. Model LeastSquares VarianceComponent Whitening Tests Summary 24 Local-Area GPS Positioning (1/9) Preface Model LeastSquares VarianceComponent Whitening Tests For ManGo, a radius of about 10 km is considered practical in a local area surrounding a fixed reference receiver. Kinematic experiments were conducted. A dual-frequency Leica SR530 GPS receiver was carried at a walking speed around a track. Summary 25 Local-Area GPS Positioning (2/9) Preface Model LeastSquares VarianceComponent A reference receiver was placed at a geodetic control point, separated from the track by 2.5 km. The cut-off angle was 15º, the sampling rate at 1-Hz, the average positional DOP was 3.7. Whitening Tests Summary 26 Local-Area GPS Positioning (3/9) The trajectory determined by Leica’s LGO software served as a bench-mark. Prior variances were made dependent on the satellite elevation angle. The variances were employed in Bv Ax l to initialize the error 2 covariance matrix σ 0 Q . Preface Model LeastSquares VarianceComponent Whitening Tests Summary 27 Local-Area GPS Positioning (4/9) Preface Model LeastSquares VarianceComponent Both kinematic GPS positioning results and the histograms of the ensemble measurement residuals are illustrated. Next, the estimated pseudorange and carrier-phase variance components are displayed for 250 epochs. Whitening Tests Summary 28 Local-Area GPS Positioning (5/9) Preface Model LeastSquares The figures showed that, with the decorrelating transformation, the estimation of variance factors has made an impact on the ambiguity resolution. VarianceComponent Whitening Tests Summary 29 Local-Area GPS Positioning (6/9) It demonstrated the number of epochs in each dataset required by ManGo for a correct ambiguity-fixed positioning. Preface Model LeastSquares VarianceComponent A distinction between the ambiguity-fixed and -float processing schemes is made. Whitening Tests Summary 30 Local-Area GPS Positioning (7/9) Preface Model LeastSquares VarianceComponent A simultaneous estimation of the coordinate, the ambiguity and the ionospheric delay is often prohibited by high correlation. Despite this, an analyst can gain an insight into the path delay by treating the station coordinates as constant. Whitening Tests Summary 31 κ κijgh f 2 Local-Area GPS Positioning (8/9) The first-order ionospheric path delay (m) 2 gh can be expressed as κij f , and κ : a function of the total electron content, f : the carrier frequency (Hz). Preface Model LeastSquares VarianceComponent Whitening Tests The x vector is arranged as: gh T (κijgh f12 , ..., N1gh , N , ...) , h {1, 2, ..., nh }. ,ij 2 ,ij Summary 32 κ κijgh f 2 Local-Area GPS Positioning (9/9) The parameter estimation and covariancematrix diagonalization remain the same. The time series of the estimated doubledifference ionospheric delays are shown. It becomes evident that some delays have a decimeter-level order of magnitude and reflect no stationary behavior. Preface Model LeastSquares VarianceComponent Whitening Tests Summary 33 Summary and Outlook (1/2) Preface Model LeastSquares VarianceComponent Whitening A variance-component estimator has been formed, under the requirement that the estimator’s expected value should reflect a true scale. The volume-preserving, integer-valued transformation technique can function as a powerful tool for ambiguity resolution. Tests Summary 34 Summary and Outlook (2/2) Preface Single-epoch GPS L1-, L2-carrier phases and code pseudoranges are usually adequate for a short-baseline solution. For medium-length baselines, a doubledifference combination can contain unmodeled errors, which still acts as a challenge. Model LeastSquares VarianceComponent Whitening Tests Summary 35 Thank You for Your Attention 36 References Wu, J. and Hsieh, C.H., 2008. GPS On-TheFly Medium-Length Positioning by an Estimation of the Measurement Variance, Journal of the Chinese Institute of Engineers, Vol. 31, No. 3, pp. 459-468. Wu, J. and Yeh, T.F., 2005. Single-Epoch Weighting Adjustment of GPS Phase Observables, Navigation, Vol. 52, No. 1, pp. 39-47. Time and Setting Preface Model LeastSquares VarianceComponent Whitening Tests Summary Date: June 22, 2002 Time interval: 16:12 - 16:25 Baseline N091-SPP2: 7.5 km Receiver(s): Leica SR530 Sampling rate: 1 Hz Observed GPS satellites: 7 Dilution of precision (DOP): 3.9 - 5.0 Cut-off (elevation mask) angle: 15° Preface Model LeastSquares VarianceComponent Whitening Tests Summary Values of a single-epoch 1212 ambiguity covariance matrix before the decorrelation Preface Model LeastSquares VarianceComponent Whitening Tests Summary Values of a single-epoch 1212 ambiguity covariance matrix after the decorrelation Time and Setting Preface Model LeastSquares VarianceComponent Whitening Tests Summary Date: March 24, 2001 Time interval: 00:36 - 01:04 Baseline N915-SPP1: 1.6 km Receiver(s): Leica SR530 Sampling rate: 1 Hz Observed GPS satellites: 9 DOP: 2.3 - 2.8 Cut-off angle: 15° Preface Model LeastSquares VarianceComponent Whitening Tests Summary Best linear unbiased estimates of scaling factors for the L1 and L2 phases Preface Model LeastSquares VarianceComponent Whitening Tests Summary Best linear unbiased estimates of scaling factors for the C/A- and P-code pseudoranges Time and Setting Preface Model LeastSquares VarianceComponent Whitening Tests Summary Date: February 10, 2003 Time interval: 15:47:20 - 16:13:50 Zero-baseline at SPP1 Receiver(s): Leica SR530 Sampling rate: 0.1 Hz Observed GPS satellites: 9 DOP: 2.3 - 2.8 Cut-off angle: 15° Preface Model LeastSquares VarianceComponent Whitening Tests Summary Northings from the single-epoch GPS positioning of a zero-baseline vector Preface Model LeastSquares VarianceComponent Whitening Tests Summary Eastings from the single-epoch GPS positioning of a zero-baseline vector Preface Model LeastSquares VarianceComponent Whitening Tests Summary Heights from the single-epoch GPS positioning of a zero-baseline vector Preface Model LeastSquares VarianceComponent Whitening Tests Summary Relative frequency of change (decreased: 98, no change: 153, and increased: 44 cases; on a 1.6 km baseline) Preface Model LeastSquares VarianceComponent Whitening Tests Summary The kinematic GPS positioning results with (left) and without (right) the variance-component estimator Preface Model LeastSquares VarianceComponent Whitening Tests Summary The histograms of the residuals with the variancecomponent estimator (left) and without (right) Preface Model LeastSquares VarianceComponent Whitening Tests Summary The histograms of the residuals with the variancecomponent estimator (left) and without (right) ρ1 ρ2 Preface Model LeastSquares VarianceComponent Whitening Tests Summary Variance components versus each 1-s epoch (ensemble average in sigma: 9.3 and 15.5 cm for the pseudoranges) φ1 φ2 Preface Model LeastSquares VarianceComponent Whitening Tests Summary Variance components versus each 1-s epoch (ensemble average in sigma: 0.01 and 0.03 cycles for the carrier phases) Preface Model LeastSquares VarianceComponent Whitening Tests Summary Trial-and-error vector sets of the integer-valued phase ambiguity Preface Model LeastSquares VarianceComponent Whitening Tests Summary Trial-and-error vector sets of the integer-valued phase ambiguity 2-epoch 3-epoch 4-epoch 5-epoch 41% 69% 78% 90% 200 300 400 Sets of multi-epoch data 500 600 Percentage in accumulated success-rate Number of epochs 5 4 3 2 Preface Model LeastSquares VarianceComponent Whitening Tests Summary 1 0 100 Minimum number of epochs that lead to a determination of the whole-cycle ambiguity Preface Model LeastSquares VarianceComponent Whitening Tests Summary GPS positioning precision and accuracy values in north-component for the 29.6 km baseline Preface Model LeastSquares VarianceComponent Whitening Tests Summary GPS positioning precision and accuracy values in eastcomponent for the 29.6 km baseline Preface Model LeastSquares VarianceComponent Whitening Tests Summary GPS precision and accuracy in height for the 29.6 km baseline Preface Model LeastSquares VarianceComponent Whitening Tests Summary Ionospheric path delays for the 29.6 km control baseline, with pseudo-random noise (PRN) 18 acting as a reference satellite Preface Model LeastSquares VarianceComponent Whitening Tests Summary Ionospheric delays for the 29.6 km control baseline

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