RELIABILITY OF THE GPS CARRIER-PHASE FIX SOLUTION UNDER HARSH CONDITION

Once the unknown integer ambiguity values are resolved, the GPS carrier phase observation will be transformed into a millimeter-level precision measurement. However, GPS observation are prone to a variety of errors, making it a biased measurement. There are two components in identifying integer ambiguities: estimation and validation. The estimation procedure aims to determine the ambiguity's integer values, and the validation step checks whether the estimated integer value is acceptable. Even though the theory and procedures for ambiguity estimates are well known, the topic of ambiguity validation is still being researched. The dependability of computed coordinates will be reduced if a false fixed solution emerges from an incorrectly estimated ambiguity integer value. In this study, the reliability of the fixed solution obtained by using several base stations in GPS positioning was investigated, and the coordinates received from these bases were compared. In a conclusion, quality control measures such as employing several base stations will improve the carrier phase measurement's accuracy.

Performance of GPS Precise Time and Frequency Transfer with Integer Ambiguity Resolution

The global positioning system (GPS) carrier-phase (CP) technique is a widely used spatial tool for remote precise time and frequency transfer. However, the performance of traditional GPS time and frequency transfer has been limeted because the ambiguity paramter is still the float solution. This study focuses on the performance of GPS precise time and frequency transfer with integer ambiguity resolution and discusses the corresponding mathematical model. Fractional-cycle bias (FCB) products were estimated by using an ionosphere-free combination. The results show that the satellite wide-lane (WL) FCB products are stable, with a standard deviation (STD) of 0.006 cycles. The narrow-lane (NL) FCB products were estimated over 15 min with the STD of 0.020 cycles. More than 98% of the WL and NL residuals are smaller than 0.25 cycles, which helps to fix the ambiguity into integers during the time and frequency transfer. Subsequently, the performances of the time transfers with integer ambiguity resolution at two time links between international laboratories were assessed in real-time and post-processing modes and compared. The results show that fixing the ambiguity into an integer in the real-time mode significantly decreases the convergence time compared with the traditional float approach. The improvement is ~49.5%. The frequency stability of the fixed solution is notably better than that of the float solution. Improvements of 48.15% and 27.9% were determined for the IENG–USN8 and WAB2–USN8 time links, respectively.

A mean-squared-error condition for weighting ionospheric delays in GNSS baselines

Although ionosphere-weighted GNSS parameter estimation is a popular technique for strengthening estimator performance in the presence of ionospheric delays, no provable rules yet exist that specify the needed weighting in dependence on ionospheric circumstances. The goal of the present contribution is therefore to develop and present the ionospheric conditions that need to be satisfied in order for the ionosphere-weighted solution to be mean squared error (MSE) superior to the ionosphere-float solution. When satisfied, the presented conditions guarantee from an MSE performance view, when (a) the ionosphere-fixed solution can be used, (b) the ionosphere-float solution must be used, or (c) an ionosphere-weighted solution can be used.

A Comparative Study of BDS Triple-Frequency Ambiguity Fixing Approaches for RTK Positioning

Concerning the triple-frequency ambiguity resolution, in principle there are three different realizations. The first one is to fix all the ambiguities of the original frequencies together. However, it is also believed that fixing the combined integer ambiguities with longer wavelength, such as extra-wide-lane (EWL), wide-lane (WL), should be advantageous. Also, it is demonstrated that fixing sequentially EWL, WL and one type of original ambiguities provides better results, as the previously fixed ambiguities increase parameters’ precision for later fixings. In this paper, we undertake a comparative study of the three fixing approaches by means of experimental validation. In order to realize the three fixing approaches from the same information in terms of adjustment, we developed a processing strategy to provide fully consistent normal equations. We first generate the normal equation with the original undifferentiated carrier phase ambiguities, then map it into that with the combined and double-differenced ambiguities required by the individual approach for fixing. Four baselines of 258 m, 22 km, 47 km and 53 km are selected and processed in both static and kinematic mode using the three ambiguity-fixing approaches. Indicators including time of first fixed solution (TFFS), the correct fixing rate, positioning accuracy and RATIO are used to evaluate and investigate results. We also made a preliminary theoretical explanation of the results by looking into the decorrelation procedure of the ambiguity searching algorithm and the intermediate results. As conclusions, integrated searching of original ambiguities or combined ambiguities has almost the same fixing performance, whereas the sequential fixing of EWL, WL and B1 ambiguities overperforms the integrated searching. By the way, the third-frequency data can shorten the TFFS significantly but can hardly improve the positioning.

Global, non-scattering solutions to the energy critical Yang–Mills problem

We consider the Yang–Mills problem on [Formula: see text] with gauge group [Formula: see text]. In an appropriate equivariant reduction, this Yang–Mills problem reduces to a single scalar semilinear wave equation. This semilinear equation admits a one-parameter family of solitons, each of which is a re-scaling of a fixed solution. In this work, we construct a class of solutions, each of which consists of a soliton whose length scale is asymptotically constant, coupled to large radiation, plus corrections which slowly decay to zero in the energy norm. Our class of solutions includes ones for which the radiation component is only “logarithmically” better than energy class. As such, the solutions are not constructed by a priori assuming the length scale to be constant. Instead, we use an approach similar to a previous work of the author regarding wave maps. In the setup of this work, the soliton length scale asymptoting to a constant is a necessary condition for the radiation profile to have finite energy. An interesting point of our construction is that, for each radiation profile, there exist one-parameter families of solutions consisting of the radiation profile coupled to a soliton, which has any asymptotic value of the length scale.

On one Saddle Point Search Algorithm for Continuous Linear Games as Applied to Information Security Problems

The paper introduces a game formulation of the problem of two players: the defender determines the security levels of objects, and the attacker determines the objects for attack. Each of them distributes his resources between the objects. The assessment of a possible damage to the defender serves as an indicator of quality. The problem of a continuous zero-sum game under constraints on the resources of the players is formulated so that each player must solve his own linear programming problem with a fixed solution of the other player. The purpose of this research was to develop an algorithm for finding a saddle point. The algorithm is approximate and based on reducing a continuous problem to discrete or matrix games of high dimension, since the optimal solutions are located at the vertices or on the faces of the simplices which determine the sets of players' admissible solutions, and the number of vertices or faces of the simplices is finite. In the proposed algorithm, the optimization problems of the players are sequentially solved with the accumulated averaged solution of the other player, in fact, the ideas of the Brown --- Robinson method are used. An example of solving the problem is also given. The paper studies the dependences of the number of algorithm steps on the relative error of the quality indicator and on the dimension of the problem, i.e., the number of protected objects, for a given relative error. The initial data are generated using pseudo-random number generators

PPP-AR with GPS and Galileo: Assessing diverse approaches and satellite products to reduce convergence time

<p><span>Precise Point Positioning (PPP) is one of the most promising processing techniques for Global Navigation Satellite System (GNSS) data. By the use of precise satellite products (orbits, clocks and biases) and sophisticated algorithms applied on the observations of a multi-frequency receiver, coordinate accuracies at the decimetre/centimetre level for a float solution and at the centimetre/millimetre level for a fixed solution can be achieved. In contrast to relative positioning methods (e.g. RTK), PPP does not require nearby reference stations or a close-by reference network. On the other hand PPP has a non-negligible convergence time. To make PPP more competitive against other high-precision GNSS positioning techniques, scientific research focuses on </span><span>reducing the convergence time of PPP. </span></p><p> </p><p><span>In this contribution, we present results of PPP with focus on integer</span><span> ambiguity resolution (PPP-AR) using satellite products from different analysis centers. </span><span>The resulting coordinate accuracy and convergence behaviour are evaluated in various test scenarios. In these test cases we distinguish between the use of satellite products from Graz University of Technology, which are calculated using a raw observation approach, and nowadays publicly available satellite products of different analysis centers (e.g. CNES, CODE). All those products enable PPP-AR in different approaches. To shorten the convergence time, we investigate and compare different PPP processing approaches using GPS and Galileo observations. The use of 2+ frequencies and alternatives to the classical PPP model, which is based on two frequencies and the ionosphere-free linear combination are discussed (e.g. uncombined model with ionospheric constraint). The PPP calculations are performed with the in-house software raPPPid, which has been developed at the research division Higher Geodesy of TU Vienna and is part of the Vienna VLBI and Satellite Software (VieVS PPP).</span></p>

Real-Time Kinematic Precise Orbit Determination for LEO Satellites Using Zero-Differenced Ambiguity Resolution

The rapid growing number of earth observation missions and commercial low-earth-orbit (LEO) constellation plans have provided a strong motivation to get accurate LEO satellite position and velocity information in real time. This paper is devoted to improve the real-time kinematic LEO orbits through fixing the zero-differenced (ZD) ambiguities of onboard Global Navigation Satellite System (GNSS) phase observations. In the proposed method, the real-time uncalibrated phase delays (UPDs) are estimated epoch-by-epoch via a global-distributed network to support the ZD ambiguity resolution (AR) for LEO satellites. By separating the UPDs, the ambiguities of onboard ZD GPS phase measurements recover their integer nature. Then, wide-lane (WL) and narrow-lane (NL) AR are performed epoch-by-epoch and the real-time ambiguity–fixed orbits are thus obtained. To validate the proposed method, a real-time kinematic precise orbit determination (POD), for both Sentinel-3A and Swarm-A satellites, was carried out with ambiguity–fixed and ambiguity–float solutions, respectively. The ambiguity fixing results indicate that, for both Sentinel-3A and Swarm-A, over 90% ZD ambiguities could be properly fixed with the time to first fix (TTFF) around 25–30 min. For the assessment of LEO orbits, the differences with post-processed reduced dynamic orbits and satellite laser ranging (SLR) residuals are investigated. Compared with the ambiguity–float solution, the 3D orbit difference root mean square (RMS) values reduce from 7.15 to 5.23 cm for Sentinel-3A, and from 5.29 to 4.01 cm for Swarm-A with the help of ZD AR. The SLR residuals also show notable improvements for an ambiguity–fixed solution; the standard deviation values of Sentinel-3A and Swarm-A are 4.01 and 2.78 cm, with improvements of over 20% compared with the ambiguity–float solution. In addition, the phase residuals of ambiguity–fixed solution are 0.5–1.0 mm larger than those of the ambiguity–float solution; the possible reason is that the ambiguity fixing separate integer ambiguities from unmodeled errors used to be absorbed in float ambiguities.

The Unified Form of Code Biases and Positioning Performance Analysis in Global Positioning System (GPS)/BeiDou Navigation Satellite System (BDS) Precise Point Positioning Using Real Triple-Frequency Data

Multi- system and multi-frequency are two key factors that determine the performance of precise point positioning. Both multi-frequency and multi-system lead to new biases, which are not solved systematically. This paper concentrates on mathematical models of biases, influences of these biases, and positioning performance analysis of different observation models. The biases comprise the inter-frequency clock bias in multi-frequency and the inter-system clock bias in multi-system. The former is the residual differential code biases (DCBs) from receiver clock and satellite clock and usually occurs at the third frequency, the latter is the deviation of the receiver clock errors in different systems. Unified mathematical models of the biases are presented by analyzing the general formula of observation equations. The influences of these biases are validated by experiments with corresponding observation models. Subsequently, the experiments, which are based on the data at five globally distributed stations in Multi-Global Navigation Satellite System (GNSS) Experiment (MGEX) on day of year 100, 2018, assess positioning performance of different observation models with combination of frequencies (dual-frequency or triple- frequency) and systems (BeiDou Navigation Satellite System (BDS) or Global Positioning System (GPS)). The results show that the performances of triple-frequency models are almost as the same level as the dual-frequency models. They provide scientific support for the triple-frequency ambiguity-fixed solution which has a better convergence characteristic than dual-frequency ambiguity-fixed solution. Furthermore, the biases are expressed as an unified form that gives an important and valuable reference for future research on multi-frequency and multi-system precise point positioning.

Saddle Point Search Algorithm for the Problem of Site Protection Level Assignment Based on Search of Simplices

This paper deals with a continuous zero-sum game with constraints on resources between a defender allocating resources for protection of sites and an attacker choosing sites for attack. The problem is formulated so that each player would have to solve its own linear program with a fixed solution of the other player. We show that in this case the saddle point is located on the faces of simplices defining feasible solutions. We propose an algorithm of saddle point search based on search of the simplices' faces on hyperplanes of equal dimension. Each possible face is defined using a boolean vector defining states of variables and problem constraints. The search of faces is reduced to the search of feasible boolean vectors. In order to reduce computational complexity of the search we formulate the rules for removing patently unfeasible faces. Each point of a face belonging to an (m--1)-dimensional hyperplane is defined using m points of the hyperplane. We created an algorithm for generating these points. Two systems of linear equations must be solved in order to find the saddle point if it located on the faces of simplices belonging to hyperplanes of equal dimension. We created a generic algorithm of saddle point search on the faces located on hyperplanes of equal dimension. We present an example of solving a problem and the results of computational experiments