Ambiguity Resolution Algorithm Research Articles

Single Epoch Ambiguity Resolution Algorithms for Single Frequency Gps with Baseline Length and Pitch Constraints

In some applications of GPS kinematic orientation and attitude determination, the length and pitch of the baseline can be measured by other means. Methods of single epoch single frequency ambiguity resolution with these constraints have been proposed. This paper analyzes the influence of the accuracy of the length and pitch to ambiguity resolution, and several examples are given to test Ambiguity Resolution (AR) performance of different length and pitch weight strategies. The analysis and the test results show that only proper weight of the baseline length and pitch can improve the performance of ambiguity resolution. Finally, a best weight strategy is given and a two-day long data set is used to test the method. The results show that the AR success rate is correlated to the number of satellites and the accuracy of the given pitch. The AR success rate is more than 95% and AR wrong rate is less than 0.005% if the error of pitch is less than 1.0 degree.

Increasing GNSS RTK availability with a new single-epoch batch partial ambiguity resolution algorithm

GPS Single-epoch Real-Time Kinematic positioning is immune to cycle slips and can be immediately re-initialized after loss-of-lock, providing high availability. This technique requires reliable ambiguity resolution: incorrect ambiguities can cause position errors of several meters, and failed ambiguity resolution reduces availability. However, a bias or inaccuracy in a single phase observation can prevent successful resolution of the whole set of ambiguities. Partial ambiguity resolution allows a subset of ambiguities to be resolved with greater probability of success than the full set. A new algorithm for resolving a subset of ambiguities with validation from previous epochs is described. If normal ambiguity resolution fails, all ambiguity subsets are generated and ordered with the best subsets first. Each subset is then resolved in turn. Fixed subsets are validated against values from previous epochs; this validation procedure greatly reduces the proportion of epochs with incorrect ambiguities. An additional algorithm is described that uses the fixed ambiguities as precise ranges to resolve the remaining unfixed ambiguities. In order to test these new algorithms, GPS data were collected from static and ship-based GPS receivers around Harwich harbor and processed from reference stations at distances up to 111 km. In the static tests the distance over which a 90% ambiguity resolution success rate for dual-frequency data was achieved was increased from 15 to 76 km. However, in some cases, the processing time was too long for this algorithm to be practical without a time-based cut-off. There is also a risk of incorrect ambiguities being propagated, particularly for single-frequency processing. In a ship-based test, the distance over which sufficient availability to support harbor navigation was achieved using single-epoch dual-frequency RTK was increased from 1 to 66 km.

Code-Aided Maximum-Likelihood Ambiguity Resolution Through Free-Energy Minimization

In digital communication receivers, ambiguities in terms of timing and phase need to be resolved prior to data detection. In the presence of powerful error-correcting codes, which operate in low signal-to-noise ratios (SNR), long training sequences are needed to achieve good performance. In this contribution, we develop a new class of code-aided ambiguity resolution algorithms, which require no training sequence and achieve good performance with reasonable complexity. In particular, we focus on algorithms that compute the maximum-likelihood (ML) solution (exactly or in good approximation) with a tractable complexity, using a factor-graph representation. The complexity of the proposed algorithm is discussed and reduced complexity variations, including stopping criteria and sequential implementation, are developed.

A New Method For DGPS Ambiguity Resolution

This paper deals with the problem of determining the baseline vector between two GPS receivers in single frequency (L1) using the basic principles of interferometric Differential GPS, therefore using the interferometric relations between the two receivers and the satellites visible to both receivers. As a preliminary step, ambiguity identification was solved using the results provided by the Kalman filter; these results were optimized by evaluating the Dilution Of Precision indexes for satellites in view of the receivers. Results achieved by applying this first procedure to data collected are discussed. To increase the accuracy of the results, a new, computationally fast algorithm for carrier phase ambiguity resolution on data collected from static and dynamics acquisitions was developed, implemented and tested. The new algorithm permitted an increase of accuracy of about two orders of magnitude with respect to results given by filtered Double Difference in the resolution of baseline vector.

Performance Assessment of a Long Range Reference Station Ambiguity Resolution Algorithm for Network RTK GPS Positioning

The authors firstly propose a new ambiguity resolution (AR) method for processing long range reference station baselines in Network Real-Time Kinematic (NRTK) GPS positioning. This method has the following advantages: no need for observational equation linearization and resolution, satellite by satellite based ambiguity resolution and long range baseline ambiguity resolution of up to a few hundred kilometres. In order to assess the performance of this method, several sub-networks of Ordnance Survey's OSNetTM in which the baseline lengths vary from 200km to 400km are formed. The corresponding initialization times and Double-Differenced (DD) biases are estimated. The results show that even if the inter-station distances are up to 400km the proposed long range AR algorithm can quickly and accurately resolve the ambiguities. According to the performance of ambiguity resolution and the character of DD biases between reference stations, the authors conclude that it is possible to implement highly accurate NRTK GPS positioning based on much sparse OSNetTM with inter-station distances up to 400km and this will significantly reduce the investment in the establishment and maintenance of a network of reference stations.

Single receiver phase ambiguity resolution with GPS data

Global positioning system (GPS) data processing algorithms typically improve positioning solution accuracy by fixing double-differenced phase bias ambiguities to integer values. These “double-difference ambiguity resolution” methods usually invoke linear combinations of GPS carrier phase bias estimates from pairs of transmitters and pairs of receivers, and traditionally require simultaneous measurements from at least two receivers. However, many GPS users point position a single local receiver, based on publicly available solutions for GPS orbits and clocks. These users cannot form double differences. We present an ambiguity resolution algorithm that improves solution accuracy for single receiver point-positioning users. The algorithm processes dual- frequency GPS data from a single receiver together with wide-lane and phase bias estimates from the global network of GPS receivers that were used to generate the orbit and clock solutions for the GPS satellites. We constrain (rather than fix) linear combinations of local phase biases to improve compatibility with global phase bias estimates. For this precise point positioning, no other receiver data are required. When tested, our algorithm significantly improved repeatability of daily estimates of ground receiver positions, most notably in the east component by approximately 30% with respect to the nominal case wherein the carrier biases are estimated as real values. In this “static” test for terrestrial receiver positions, we achieved daily repeatability of 1.9, 2.1 and 6.0 mm in the east, north and vertical (ENV) components, respectively. For kinematic solutions, ENV repeatability is 7.7, 8.4, and 11.7 mm, respectively, representing improvements of 22, 8, and 14% with respect to the nominal. Results from precise orbit determination of the twin GRACE satellites demonstrated that the inter-satellite baseline accuracy improved by a factor of three, from 6 to 2 mm up to a long-term bias. Jason-2/Ocean Surface Topography Mission precise orbit determination tests results implied radial orbit accuracy significantly below the 10 mm level. Stability of time transfer, in low-Earth orbit, improved from 40 to 7 ps. We produced these results by applying this algorithm within the Jet Propulsion Laboratory’s (JPL’s) GIPSY/OASIS software package and using JPL’s orbit and clock products for the GPS constellation. These products now include a record of the wide-lane and phase bias estimates from the underlying global network of GPS stations. This implies that all GIPSY–OASIS positioning users can now benefit from this capability to perform single-receiver ambiguity resolution.

Robust Doppler ambiguity resolution using multiple paired pulse repetition frequencies

In the present study, a robust algorithm for Doppler ambiguity resolution is proposed. This new algorithm is based on a recently developed robust phase unwrapping (RPUW) algorithm and the Ferrari-Be-renguer-Alengrin (FBA) method using multiple pulse repetition frequencies (PRF). In the FBA method, the PRFs are grouped into pairs and each paired PRF values are symmetric about 1. Then the Doppler frequency estimation is divided into two steps (folded frequency, i.e. the fractional part and ambiguity order, i.e. the integer part): the folded frequency is estimated by circularly averaging the folded frequency estimates for each pair PRF and the ambiguity order is obtained by searching the finite integers based on a quasi-maximum-likelihood criterion using the estimated folded frequency. By observing that the folded frequency estimates for each pair PRF may have errors because of the finite FFT implementations, noise and interference, the circular mean of the erroneous folded frequency estimates may be erroneous too, which may lead to an erroneous ambiguity order estimate, that is, a large error. In this study, the authors replace the integer ambiguity order searching by a recently developed RPUW algorithm that is robust to the frequency estimates from the FFT implementations. The simulation results of this study show that the newly proposed algorithm significantly outperforms the FBA method and is also better than the RPUW algorithm.

Resolving the 180° Ambiguity in Solar Vector Magnetic Field Data: Evaluating the Effects of Noise, Spatial Resolution, and Method Assumptions

The objective testing of algorithms for performing ambiguity resolution in vector magnetic field data is continued, with an examination of the effects of noise in the data. Through the use of analytic magnetic field models, two types of noise are “added” prior to resolving: noise to simulate Poisson photon noise in the observed polarization spectra, and a spatial binning to simulate the effects of unresolved structure. The results are compared through the use of quantitative metrics and performance maps. We find that while no algorithm severely propagates the effects of Poisson noise beyond very local influences, some algorithms are more robust against high photon-noise levels than others. In the case of limited spatial resolution, loss of information regarding fine-scale structure can easily result in erroneous solutions. Our tests imply that photon noise and limited spatial resolution can act so as to make assumptions used in some ambiguity resolution algorithms no longer consistent with the observed magnetogram. We confirm a finding of the earlier comparison study that results can be very sensitive to the details of the treatment of the observed boundary and the assumptions governing that treatment. We discuss the implications of these findings, given the relative sensitivities of the algorithms to the two sources of noise tested here. We also touch on further implications for interpreting observational vector magnetic field data for general solar physics research.

Applied Interval Based Integer Ambiguity Resolution

This paper provides improvements on the interval integer ambiguity resolution algorithm BOUNDS, which can give theoretical guarantees on finding the correct integers. The BOUNDS algorithm is validated by applying it to real GPS data and comparing it to the LAMBDA method. The first improvement is a transformation of the search space, which reduces dependency. Secondly the process of using multiple frequencies is explained and demonstrated. The third improvement, called contractions, quickly reduces the search space and increases solution accuracy. Finally, matrix computations are introduced to decrease the computational load. Previous research into BOUNDS indicated that defining correct noise bands is an important step of the algorithm. This is investigated in more detail, discussing several noise iteration schemes and noise models based on the signal-to-noise ratio. Application of the improved BOUNDS algorithm to the attitude determination problem for a static baseline shows considerable reduction in computation time and improvement in accuracy.

On the performance of a maximum likelihood method with delayed correlation for the coarse carrier frequency offset estimation of OFDM signals with multiple preamble symbols

Estimation of carrier frequency offset (CFO) is an important issue in the design of a wireless receiver that employs orthogonal frequency division multiplexing (OFDM) techniques. In this paper, using the ten short training symbols specified in the signal format of the IEEE 802.11a WLAN, we investigate the performance of a coarse CFO estimation scheme for OFDM signals with multiple preamble symbols. This scheme, which we call DC-ML, employs the maximum likelihood (ML) method with delayed correlation (DC). For AWGN channels under moderate signal to noise ratio (SNR) conditions, we develop an analysis to evaluate the variance of estimation error (VEER). The analysis is corroborated in light of simulations, and compared with the formulated Cramer-Rao lower bound (CRLB). VEER of the DC-ML in a multipath environment is studied via simulations. Numerical results show that a certain parameter combination can result in minimum VEER. Simulation results justify that the probability of estimation error (PEER) approximates Gaussian distribution in both AWGN and multipath scenarios. We also present a Two-Branch DC-ML (TBDC-ML) scheme, which comprises two correlation branches of DC-ML, and an associative ambiguity resolution algorithm. Numerical examples reveal that TBDC-ML outperforms DC-ML in both VEER and PEER. Assuming that the estimation error resulting from the two branches is jointly Gaussian, we derive the joint probability density function (pdf) and validate it via simulations.

Fixed point theorems of GPS carrier phase ambiguity resolution and their application to massive network processing: Ambizap

Precise point positioning (PPP) has become popular for Global Positioning System (GPS) geodetic network analysis because for n stations, PPP has O(n) processing time, yet solutions closely approximate those of O(n3) full network analysis. Subsequent carrier phase ambiguity resolution (AR) further improves PPP precision and accuracy; however, full‐network bootstrapping AR algorithms are O(n4), limiting single network solutions to n < 100. In this contribution, fixed point theorems of AR are derived and then used to develop “Ambizap,” an O(n) algorithm designed to give results that closely approximate full network AR. Ambizap has been tested to n ≈ 2800 and proves to be O(n) in this range, adding only ∼50% to PPP processing time. Tests show that a 98‐station network is resolved on a 3‐GHz CPU in 7 min, versus 22 h using O(n4) AR methods. Ambizap features a novel network adjustment filter, producing solutions that precisely match O(n4) full network analysis. The resulting coordinates agree to ≪1 mm with current AR methods, much smaller than the ∼3‐mm RMS precision of PPP alone. A 2000‐station global network can be ambiguity resolved in ∼2.5 h. Together with PPP, Ambizap enables rapid, multiple reanalysis of large networks (e.g., ∼1000‐station EarthScope Plate Boundary Observatory) and facilitates the addition of extra stations to an existing network solution without need to reprocess all data. To meet future needs, PPP plus Ambizap is designed to handle ∼10,000 stations per day on a 3‐GHz dual‐CPU desktop PC.

Investigation of Instantaneous Carrier Phase Ambiguity Resolution with the GPS/Galileo Combination using the General Ambiguity Search Criterion

The success of high-precision Global Navigation Satellite Systems (GNSS) kinematic positioning depends partly on the ability to resolve the integer phase ambiguities. In this paper, we propose a new algorithm for instantaneous kinematic ambiguity resolution, for present and modernised GPS and for GALILEO. This approach - the GEneral Criterion Cascading Ambiguity Resolution (GECCAR) - selects the integer set of ambiguities using the General Ambiguity Search Criterion (GASC). Simulation runs have shown that single-epoch L1/E1 frequency ambiguity resolution was possible 99% of the considered epochs, when the three frequencies from both systems were used together. This new approach shows an improvement in the selection of the correct set of ambiguities when compared with the selection made by the Integer Least Squares Criterion (ILSC). We conclude that the GECCAR approach is a very promising algorithm for instantaneous ambiguity resolution.

Analysis of the narrowband interference effect on frequency ambiguity resolution for OFDM systems

AbstractIn orthogonal frequency division multiplexed (OFDM) systems affected by carrier frequency offsets, frequency ambiguity resolution, i.e. the estimation of the part of the frequency offset corresponding to an integer times the carrier spacing is a crucial issue. The proper action of frequency ambiguity resolution algorithms can be strongly affected by the presence of disturbances, like narrowband interference (NBI). In this paper, the susceptibility of the blind and data aided maximum likelihood (ML) frequency ambiguity estimators to NBI signals, which may arise in the OFDM band as the spectrum becomes more crowded, is investigated in an analytical way. The analytical results are verified by means of simulations. Although the estimators turn out to be essentially independent of the bandwidth of the interferers and the number of interferers, the performance of the estimators is very sensitive to the positions of the interferers when the interferers are strong. Copyright © 2008 John Wiley & Sons, Ltd.

Low-cost relative navigation sensing: GNSS-like devices hosted on deployed tethers

Currently, the possibility to perform very precise GNSS-based relative navigation determination among two or more spacecraft in rendezvous, docking and/or formation flying scenarios is deemed as the baseline for several current and future missions. Due to the required centimetre-level accuracy, the pseudorange measurements accuracy is not sufficient, and carrier phase measurements shall be employed. Those algorithms making exclusive use of carrier phase measurements are usually called “kinematic” algorithms, and require for dedicated integer ambiguity resolution algorithms. Obviously, this relative navigation determination concept is only suitable when the spacecraft is in the range of the GNSS constellations coverage. In situations where the GNSS constellations do not provide coverage (from MEO/GEO orbits to non-Earth planetary orbits and passing through inter-planetary missions), systems based on radio frequency (RF) signal combining emitters and receivers on-board the same platform have been proposed and are currently being studied (i.e. DARWIN RF subsystem). Nevertheless, DARWIN scenario takes advantage of the multi-spacecraft scenario characteristic (four, at least, spacecraft hosting an emitter are required to provide three-dimensional navigation). Innovative approaches are required when considering three or just two spacecraft in formation flying. This paper illustrates the concept and supplies simulation-based accuracy results of a full GNSS-like-based relative navigation determination with RF emitters/receivers hosted on board two spacecraft flying in close formation or performing a RDV and docking manoeuvre, a target and a chaser respectively named. To fulfil concept requirements, at least three RF pseudolites shall be placed on-board or around the target spacecraft so to provide the required RF coverage meant for the chaser spacecraft. A key point to achieve the required system performance is related to the relative geometry between the emitting pseudolites and the receiving antennas. The ideal being the emitters uniformly distributed in the receiver antenna field of view. It is analysed how to achieve this spatial distribution, that cannot be accomplished by placing multiple emitters on-board the same vehicle structure (characteristic spacecraft dimension will be much smaller than the inter-spacecraft distance) by using a tether-based deployed pseudolites system. Tether deployment mechanisms, particularly gravity gradient assisted, are assessed in the paper, in order to achieve the best RF emitter pseudolites distribution for a maximum formation flying navigation determination accuracy.

Improved slope estimation for SAR Doppler ambiguity resolution

The idea of using the Radon transform to measure the alignment of linear features in synthetic aperture radar (SAR) data has breathed new life into the class of Doppler ambiguity resolution algorithms. In these algorithms, the slope of target energy is estimated to obtain the satellite beam pointing angle accurately enough to resolve the Doppler ambiguity. After explaining the method and adding some minor improvements, it is shown how it can work well on satellite SAR data. Then, an alternate method is developed that combines the ideas of the Radon and look displacement algorithms to obtain a computationally simpler and more accurate algorithm. In addition, the quality checks of the spatial diversity approach are used to increase the robustness of the algorithm. Even though the algorithm was conceived for high-contrast scenes, it works remarkably well in low to medium contrast scenes as well.

On accuracy and reliability of instantaneous network RTK as a function of network geometry, station separation, and data processing strategy

The network-based approach to kinematic GPS positioning significantly increases the distance, over which carrier-phase ambiguity resolution can be performed. This can be achieved either by introducing geometric conditions based on the fixed reference locations, and/or through the use of reference network data to estimate the corrections to GPS observations that can be broadcast to the users. The Multi Purpose GPS Processing Software (MPGPS) developed at The Ohio State University uses the multiple reference station approach for wide area and regional differential kinematic GPS positioning. The primary processing algorithm uses the weighted free-net (WFN) approach with the distance-dependent weighting scheme to derive optimal estimates of the user coordinates and realistic accuracy measures. The WFN approach, combined with the single epoch (instantaneous) ambiguity resolution algorithm is presented here as one approach to real-time kinematic (RTK) GPS. Since for baselines exceeding ~100 km, the instantaneous ambiguity resolution may not always be possible due to the increasing observation noise and insufficient number of observations to verify the integer selection, an alternative approach, based on a single- (or multiple-) baseline solution, supported by a double-difference (DD) ionospheric delay propagated from the previous epoch is also presented. In this approach, some data accumulation, supported by the network-derived atmospheric corrections, is required at the beginning of the rover data processing to obtain the integer ambiguities; after this initialization period, the processing switches to the instantaneous RTK positioning mode. This paper presents a discussion on the effects of the network geometry, station separation and the data reduction technique on the final quality and reliability of the rover positioning solution. A 24-h data set of August 31, 2003, collected by the Ohio Continuously Operating Reference Station (CORS) network was processed by both techniques under different network geometry and reference station separation. Various solutions, such as (1) single-baseline solution for varying base-rover separation, (2) multi-baseline solution with medium-range base separation (over 100 km), and (3) multi-baseline solution with long-range base separation (up to 377 km), were obtained and compared for accuracy and consistency. The horizontal positioning accuracy achieved in these tests, expressed as the difference between the estimated coordinates and the known rover coordinates, is at the sub-decimeter level for the first approach, and at the centimeter-level for the second method, for baselines over 100 km. In the vertical coordinate, decimeter- and sub-decimeter levels were achieved for the two approaches, respectively. Even though all the results presented here were obtained in post-processing, both algorithms are suitable for real-time applications.

Efficient Attitude Determination Using GPS Multiple Antennas-Geometrical Concept-

We developed an ambiguity resolution algorithm for attitude determination using multiple GPS antennas. It is focused on reducing computational loads using a geometrical approach. By defining a new decision variable for stochastic verification, overall computational loads are effectively reduced with respect to the chi-square model. This paper also uses the first base vector estimated as line-of-sight vector to obtain a second base vector. The additional 1-D search space is sufficient to resolve cycle ambiguity of the second base vector for complete attitude determination. Consequently it is possible to resolve the multi-baseline cycle ambiguity using a single baseline algorithm load.

Blind Subcarrier Frequency Ambiguity Resolution for OFDM Signals Over Selective Channels

Many frequency-recovery algorithms for orthogonal frequency-division multiplexing systems are geared to recover offsets up to half the distance of the subcarriers. If the error is larger, an ambiguity of an integer number of subcarrier spacings is left. This paper develops a blind algorithm for ambiguity resolution that works effectively over frequency-selective radio channels. Simulation results show a sizable improvement over existing techniques, expecially at high signal-to-noise ratios.

Impact of Real-Time Ionospheric Determination on Improving Precise Navigation with GALILEO and Next-Generation GPS

: Ionospheric differential refraction at distances of tens of kilometers or more is one of the main problems affecting the capability of instantaneous carrier-phase ambiguity resolution. As a consequence, the ionosphere affects the feasibility of centimeter-accuracy navigation with dual-frequency global navigation satellite systems (GNSSs), such as present-day GPS, as well as future systems with three frequencies, such as GALILEO and modernized GPS. We have shown in previous work that a real-time tomographic model of the ionosphere, computed from GPS reference network data, is accurate enough to allow GPS real-time carrier-phase ambiguity resolution for a rover at hundreds of kilometers from the nearest reference site. In this work, we study the extension of such techniques to the improvement of instantaneous three-carrier-phase ambiguity resolution algorithms, such as three-carrier ambiguity resolution (TCAR), at medium and long distances (from tens to hundreds of kilometers) and with a minimum of geodetic computation. The datasets used were generated with a modified GNSS satellite signal generator. In particular, the success rate of instantaneous ambiguity resolution with three frequencies can be improved from about 60 percent of all attempts in previous work to about 100 percent with the proposed technique at distances of about 60 km and with low ionospheric values, and to more than 90 percent at distances of more than 100 km under solar maximum conditions.

An integer ambiguity resolution algorithm for real-time GPS attitude determination

The problem of integer ambiguity resolution must be solved to fully exploit the information contained in the GPS carrier phase measurements for real-time attitude determination. A new algorithm for integer ambiguity resolution that uses single-difference smoothed pseudorange measurements is proposed that has several advantages over the conventional algorithms such as the search methods and the motion-based algorithms. These advantages include minimal computational effort at each epoch and an analytically derived equation which calculates the expected time to ambiguity resolution. The validity and the performance of the new algorithm are investigated through simulations although processing real data will be necessary to verify the usability of the proposed algorithm in real situations.