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In short-timespan processing of GPS observations the combination of code and carrier phase observations has the shortcoming that the normal matrix is often ill-conditioned and giving unstable computation. The regularized least-squares (RLS) and the iterative least-squares (ILS) methods are often proposed as alternatives to the conventional least-squares (CLS) method. The RLS are claimed to give better reliability of GPS ambiguity solving for short-period observations and to improve the quality of the normal equation, also reducing the condition number of the normal matrix. The regularization parameter is determined by minimizing the trace of the mean square error matrix. However, the regularization induces a biased estimation and its benefits are difficult to be confirmed. On the other hand, the ILS do not improve the estimate results and their stochastic properties, apart from stabilizing the normal matrix and reducing its condition number: this, however, depends on the given initial estimate vector. In this work we investigate the performance of RLS and ILS as compared to CLS when applied to single-frequency epoch-by-epoch processing with ambiguity solving by LAMBDA method. Results show that, while the investigated methods do not produce significant differences in terms of estimated baseline precision, improvements are instead observed in the condition number of the normal matrix, with ILS producing the best results when using estimated initial values from CLS. On the other hand, the RLS method fails to improve the condition number for epoch-by-epoch strategy. All methods also give practically equal reliability for ambiguity resolution, where the evaluation is taken in terms of success rate.

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... In case of short timespan epoch observations (eg. ≥ 1Hz), which potentially give ill-posed normal matrices, one could employ an iterative procedure to solve (5) [16]- [19]. ...

... The iterative solution demonstrated by [16]- [19], however, aims at stabilizing the ill-posed normal matrix. It does not need to improve precision of the parameters. ...

The GNSS carrier phase observation allows for high precision positioning and attitude determination, as long as its integer ambiguity is correctly resolved. Solving integer ambiguities in attitude determination, using multiple antennae installed on the moving body, has some advantages: antennae are separated by relatively short distances, and their geometry is fixed, thus allowing precise baseline lengths in the body reference frame to be known beforehand. From the mathematical point of view, the predefined precise baseline lengths will lead to strong functional and stochastic models, which in turn, will allow for a higher success rate for the integer ambiguity resolution. However, solving several integer ambiguities may also be computationally demanding, especially for real-time applications. The usage of the partial ambiguity resolution method solves this drawback. It was reported that this strategy not only gives higher ambiguity success rate compared to the full ambiguity resolution approach, but also decreases the computational cost. [1]–[5]. Besides that, [6] suggested the so-called affine constrained least squares method, which avoids the complexity of the (multi-) constrained least-squares problem as well as the complexity of the ambiguity search space. This contribution examines the partial ambiguity resolution method for two different search models: the affine constrained least squares model and the gradient iterative least squares model. The two methods are compared by means of the ambiguity dilution of precision, the success rate, and the number of fixed ambiguities. We show that each method has benefit for a particular purpose. keywords: Baseline constraint, Partial ambiguity search strategy, Affine constrained least squares model, Iterative baseline-constrained least squares model, Success rate, Number of fixed ambiguities BIOGRAPHIES Hendy F. Suhandri holds a master's degree in 2008 from the Geoengine program at

... goGPS applies the LAMBDA method either in the epoch-by-epoch processing modes or in the EKF processing modes. In the latter case, the Kalman filter keeps continuous float ambiguity estimation; LAMBDA is then applied after the filter update step (Suhandri and Realini 2013). ...

goGPS is a positioning software application designed to process single-frequency code and phase
observations for absolute or relative positioning. Published under a free and open-source license, goGPS
can process data collected by any receiver, but focuses on the treatment of observations by low-cost
receivers. goGPS algorithms can produce epoch-by-epoch solutions by least squares adjustment, or multiepoch
solutions by Kalman filtering, which can be applied to either positions or observations. It is possible to aid the positioning by introducing additional constraints, either on the 3D trajectory such as a railway, or on a surface, e.g., a digital terrain model. goGPS is being developed by a collaboration of different research groups, and it can be downloaded from http://www.gogps-project.org. The version used in this manuscript can be also downloaded from the GPS Toolbox Web site http://www.ngs.noaa.gov/gpstoolbox. This software is continues to evolve, improving its functionalities according to the updates introduced by the collaborators. We describe the main modules of goGPS along with some examples to show the user how the software works.

... goGPS applies the LAMBDA method either in the epoch-by-epoch processing modes or in the EKF processing modes. In the latter case, the Kalman filter keeps continuous float ambiguity estimation; LAMBDA is then applied after the filter update step (Suhandri and Realini 2013). ...

goGPS is a positioning software application designed to process single-frequency code and phase observations for absolute or relative positioning. Published under a free and open-source license, goGPS can process data collected by any receiver, but focuses on the treatment of observations by low-cost receivers. goGPS algorithms can produce epoch-by-epoch solutions by least squares adjustment, or multi-epoch solutions by Kalman filtering, which can be applied to either positions or observations. It is possible to aid the positioning by introducing additional constraints, either on the 3D trajectory such as a railway, or on a surface, e.g., a digital terrain model. goGPS is being developed by a collaboration of different research groups, and it can be downloaded from http:// www. gogps-project. org. The version used in this manuscript can be also downloaded from the GPS Toolbox Web site http:// www. ngs. noaa. gov/ gps-toolbox. This software is continues to evolve, improving its functionalities according to the updates introduced by the collaborators. We describe the main modules of goGPS along with some examples to show the user how the software works.

Carrier phase ambiguity resolution is the key to fast and high precision GPS kinematic position-ing. Critical in the application of ambiguity resolution is the quality of the computed integer ambiguities. Unsuccessful ambiguity resolution, when passed unnoticed, will too often lead to unacceptable errors in the positioning results. In order to describe the quality of the integer ambiguities, their distributional properties need to be known. This contribution introduces the probability mass function of the integer least-squares ambiguities. This integer normal distribu-tion is needed in order to infer objectively whether or not ambiguity resolution can expected to be successful. Some of its properties are discussed. Attention is given in particular to the prob-ability of correct integer estimation. Various diagnostic measures are presented for evaluating this probability.

The probability of correct integer estimation, the success rate, is an important measure when the goal is fast and high precision positioning with a Global Nav- igation Satellite System. Integer ambiguity estimation is the process of mapping the least-squares ambiguity esti- mates, referred to as the float ambiguities, to an integer value. It is namely known that the carrier phase ambigu- ities are integer-valued, and it is only after resolution of these parameters that the carrier phase observations start to behave as very precise pseudorange measurements. The success rate equals the integral of the probability den- sity function of the float ambiguities over the pull-in region centered at the true integer, which is the region in which all real values are mapped to this integer. The success rate can thus be computed without actual data and is very valuable as an a priori decision parameter whether successful ambi- guity resolution is feasible or not. The pull-in region is determined by the integer estimator that is used and therefore the success rate also depends on the choice of the integer estimator. It is known that the in- teger least-squares estimator results in the maximum suc- cess rate. Unfortunately, it is very complex to evaluate the integral when integer least-squares is applied. Therefore, approximations have to be used. In practice, for example, the success rate of integer bootstrapping is often used as a lower bound. But more approximations have been pro- posed which are known to be either a lower or upper bound of the actual integer least-squares success rate. In this contribution an overview of the most important lower and upper bounds will be given. These bounds are compared theoretically as well as based on their perfor- mance. The performance is evaluated using simulations, since it is then possible to compute the 'actual' success rate. Simulations are carried out for the two-dimensional case, since its simplicity makes evaluation easy, but also for the higher-dimensional geometry-based case, since this gives an insight to the performance that can be expected in practice.

In rapid static or kinematic positioning, especially with single frequency and shorter period GPS phase observa- tions, the normal matrix may be weak multicollinear since the GPS satellite's positions and geometries are changing slightly. Therefore the conventional least squares float solution may cause large biases due to the weak multicol- linearity and thus make the ambiguity search techniques fail to work. Here the newly developed method of deter- mining the optimal regularization parameter α in uniform Tykhonov-Phillips regularization (α-weighted S- homBLE) by A-optimal design (minimizing the trace of the Mean Square Error matrix MSE) is reviewed. This new algorithm with A-optimal Regularization can be ap- plied to overcome this kind of problem in both GPS rapid static and real time kinematic positioning with single or dual frequency measurements, especially for the shorter observation period. In the case study, the estimate method is applied to process the two-epoch L1 data in single fre- quency GPS rapid static positioning. A detailed discus- sion about effects of the initial coordinate accuracy to the determination of the A-optimal regularization parameter will also be presented. The results show that a new algo- rithm with optimal regularization can significantly im- prove the reliability of the GPS ambiguity resolution in shorter observation periods.

In rapid global positioning systems (GPS) positioning one of the key problems is to quickly determine the ambiguities of GPS carrier phase observables. Since carrier phase observations are generally collected only for a few minutes in the mode of rapid GPS positioning, the least squares floating solution of the ambiguities will be highly correlated and the decorrelation approach has often been used in order to reduce the search space of integer ambiguities. In this paper we propose a regularized algorithm as an alternative approach to decorrelation, and compute the regularization parameter by minimizing the trace of mean squared errors. Since regularization has been essential to solve inverse ill-posed problems and shown to be very significant in reducing the condition number of normal matrices, we will explore possible applications of regularization for improving the high correlation of the estimated float ambiguities. Numerical experiments with 50 epochs of single frequency observations show that the condition number after regularization reduces to half of that of the floating solution if the ambiguities could be known to 2-3 cycles. If better knowledge about the ambiguities could be obtained to within 1cycle, further improvement can be achieved. The results indicate that regularization could be used for fast GPS ambiguity resolution. Our experiments also demonstrate that a scale factor of about 8 is needed to multiply the estimated variance of unit weight for obtaining a reasonable estimator for the accuracy of float ambiguities.

The rational function model (RFM) utilized for high resolution satellit e imagery (HRSI) provides a transformation from image to object space coordinates in a geographic reference system. Compared with the rigorous model based on the collinearity condition equation or the affine model, the RFM with 80 coefficients would be over parameterized. That would result in an ill-conditioned normal equation. Tikhonov regularization is often used to resolve this problem, and many applications have verified its serviceability. This paper will detail the method for regularization parameter selection. However, Tikhonov regularization makes the two sides of equation unequal, resulting in a biased solution. An unbiased method - The Iteration by Correcting Characteristic Value (ICCV) was introduced, and a strategy to resolve the ill-conditioned problem for solving rational polynomial coefficients (RPCs) was discussed in this paper. The tests with SPOT-5 and QuickBird imagery were accomplished. The empirical results have shown that our methodology can effectively improve the condition of the normal equations.

The coming decade will bring a proliferation of Global Navigation Satellite Systems (GNSS) that are likely to revolutionize society in the same way as the mobile phone has done. The promise of a broader multi-frequency, multi-signal GNSS “system of systems” has the potential of enabling a much wider range of demanding applications compared to the current GPS-only situation. In order to achieve the highest accuracies one must exploit the unique properties of the received carrier signals. These properties include the multi-satellite system tracking, the mm-level measurement precision, the frequency diversity, and the integer ambiguities of the carrier phases. Successful exploitation of these properties results in an accuracy improvement of the estimated GNSS parameters of two orders of magnitude. The theory that underpins this ultraprecise GNSS parameter estimation and validation is the theory of integer inference. This theory is the topic of the present chapter.

The fast ambiguity resolution method, especially that using single epoch single frequency data, has been a challenging issue in the kinematic GPS positioning. In this paper, artificial immune algorithm (AIA) was introduced to search GPS carrier phase integer ambiguities. It is a new optimization method through simulating biological immune systems to search the best solution. AIA can succeed in improving the poor efficiency and stability of the genetic algorithm (GA). If the normal equation is consisted only by single epoch carrier phase, it is always rank deficient. The general method is to combine the C/A code and the L1 carrier phase together, so as to eliminate rank deficiency. In this situation, the normal equation is ill-conditioned. In addition, the pseudo-range is often influenced by high noise, which would result in lower reliability of the float ambiguity solutions. So, the efficiency of kinematic positioning is influenced because the differences are large between the float ambiguity solution and the correct integer ambiguity. To improve the float solution, an unbiased estimation method named Iteration by Modifying Normal Equation (IMNE) is applied. IMNE can efficiently improve the ill-condition of normal equation. By using it, more precise ambiguity float solution can be acquired and the search space of ambiguity can be greatly reduced. The improved float solution increases the success rate of fixing the carrier phase integer ambiguities. After obtaining more accurate ambiguity float solution, AIA was used to search GPS carrier phase integer ambiguity. The results of some examples show that the new approach (INME-AIA) based on the improving float solution is efficient.

In this paper, a new algorithm is employed in global positioning system (GPS) rapid positioning using several-epoch single-frequency phase data. First, we define the double-k-type ridge estimator (DKRE) based on the structure characteristics of multicollinearities of the normal equations matrix in the double-difference (DD) model, and prove when the DKRE is not worse than the least-squares estimator (LSE) in the sense of a reduced MSEM. Taking into account how the ridge parameter in the ordinary ridge estimator is confirmed based on the generalized ridge estimator (GRE), we propose a method of estimating two ridge parameters for the DKRE. Second, we improve the LAMBDA method through replacing the cofactor matrix computed by the LSE with the cofactor matrix computed by the DKRE. The ambiguities-fixed solution is found by the sequential LSE. A theorem stating that the success rate of the improved LAMBDA method is bigger than the original LAMBDA method is given. Finally, through the GPS positioning tests, it is shown that the present method is highly efficient and reliable, and some very valuable conclusions are obtained by analyzing the computation results.

In a linear Gauss–Markov model, the parameter estimates from BLUUE (Best Linear Uniformly Unbiased Estimate) are not robust
against possible outliers in the observations. Moreover, by giving up the unbiasedness constraint, the mean squared error
(MSE) risk may be further reduced, in particular when the problem is ill-posed. In this paper, the α-weighted S-homBLE (Best homogeneously Linear Estimate) is derived via formulas originally used for variance component estimation on
the basis of the repro-BIQUUE (reproducing Best Invariant Quadratic Uniformly Unbiased Estimate) principle in a model with
stochastic prior information. In the present model, however, such prior information is not included, which allows the comparison
of the stochastic approach (α-weighted S-homBLE) with the well-established algebraic approach of Tykhonov–Phillips regularization, also known as R-HAPS (Hybrid APproximation Solution), whenever the inverse of the “substitute matrix” S exists and is chosen as the R matrix that defines the relative impact of the regularizing term on the final result.