Integer Aperture Estimation Research Articles

A Controllable Success Fix Rate Threshold Determination Method for GNSS Ambiguity Acceptance Tests

Global navigation satellite system (GNSS) integer ambiguity acceptance test is one of the open problems in GNSS data processing. A number of ambiguity acceptance tests have been proposed from different perspectives and then unified into the integer aperture estimation framework. The existing comparative studies indicate that the impact of test statistics form on the test performance is less critical, while how to construct an efficient, practical test threshold is still challenging. Based on the likelihood ratio test theory, a new computationally efficient ambiguity acceptance test with controllable success fix rate, namely the fixed likelihood ratio (FL-) approach is proposed, which does not require Monte Carlo simulation. The study indicates that the fixed failure rate (FF-) approach can only control the overall failure rate of the acceptance region, but the local failure rate is not controllable. The proposed FL-approach only accepts the fixed solution meeting the likelihood ratio requirement. With properly chosen likelihood ratio threshold, the FL-approach achieves comparable success rate as the FF-approach and even lower failure rate than the FF-approach for the strong underlying model cases. The fixed success fix rate of the FL-approach is verified with both simulation data and real GNSS data. The numerical results indicate that the success fix rate of the FL-approach achieves >98% while the failure rate is <1.5%. The real-time kinematic (RTK) positioning with ambiguities tested by the FL-approach achieved 1–2cm horizontal precision and 2–4 cm vertical precision for all tested baselines, which confirms that the FL-approach can serve as a reliable and efficient threshold determination method for the GNSS ambiguity acceptance test problem.

Improving GNSS Ambiguity Acceptance Test Performance with the Generalized Difference Test Approach.

In Global navigation satellite system (GNSS) data processing, integer ambiguity acceptance test is considered as a challenging problem. A number of ambiguity acceptance tests have been proposed from different perspective and then unified into the integer aperture estimation (IA) framework. Among all the IA estimators, the optimal integer aperture (OIA) achieves the highest success rate with the fixed failure rate tolerance. However, the OIA is of less practical appealing due to its high computation complexity. On the other hand, the popular discrimination tests employ only two integer candidates, which are the essential reason for their sub-optimality. In this study, a generalized difference test (GDT) is proposed to exploit the benefit of including three or more integer candidates to improve their performance from theoretical perspective. The simulation results indicate that the third best integer candidates contribute to more than 70% success rate improvement for integer bootstrapping success rate higher than 0.8 case. Therefore, the GDT with three integer candidates (GDT3) achieves a good trade-off between the performance and computation burden. The threshold function is also applied for rapid determination of the fixed failure rate (FF)-threshold for GDT3. The performance improvement of GDT3 is validated with real GNSS data set. The numerical results indicate that GDT3 achieves higher empirical success rate while the empirical failure rate remains comparable. In a 20 km baseline test, the success rate GDT3 increase 7% with almost the same empirical failure rate.

Integer aperture estimation in the presence of biases

Ambiguity validation is a quality control step for ambiguity resolution. It is realized by integer aperture (IA) estimator and critical value determination approach. During the past decades, most of research about IA estimator were mainly implemented by numerical simulation. The influence of biases to IA estimation has not been studied. Actually, ambiguity resolution is subject to various kinds of biases in practice, which influences the performance of IA estimation. In this contribution, properties of IA estimators are investigated when they are biased. The probability evaluation formulae for IA estimators with bias-affected are derived and verified. The proper ways to evaluate biased-IA estimators are recommended by numerical experiments. In addition to this, the influences of atmospheric biases to IA estimation are analyzed in different experiments. Those results show that under the same influence of biases, all IA estimators have no better positioning precision than integer estimator, and different IA estimators may lead to different positioning precision. A better choice of IA estimator may lead to less loss of positioning precision. However, if biases can be properly separated, positioning precision of integer and IA estimators can be greatly improved. Hence, the adaptation of biases is more important than the choice of IA estimators in positioning and other applications.

An alternative approach to calculate the posterior probability of GNSS integer ambiguity resolution

When precise positioning is carried out via GNSS carrier phases, it is important to make use of the property that every ambiguity should be an integer. With the known float solution, any integer vector, which has the same degree of freedom as the ambiguity vector, is the ambiguity vector in probability. For both integer aperture estimation and integer equivariant estimation, it is of great significance to know the posterior probabilities. However, to calculate the posterior probability, we have to face the thorny problem that the equation involves an infinite number of integer vectors. In this paper, using the float solution of ambiguity and its variance matrix, a new approach to rapidly and accurately calculate the posterior probability is proposed. The proposed approach consists of four steps. First, the ambiguity vector is transformed via decorrelation. Second, the range of the adopted integer of every component is directly obtained via formulas, and a finite number of integer vectors are obtained via combination. Third, using the integer vectors, the principal value of posterior probability and the correction factor are worked out. Finally, the posterior probability of every integer vector and its error upper bound can be obtained. In the paper, the detailed process to calculate the posterior probability and the derivations of the formulas are presented. The theory and numerical examples indicate that the proposed approach has the advantages of small amount of computations, high calculation accuracy and strong adaptability.

Robust and precise baseline determination of distributed spacecraft in LEO

Recent experience with prominent formation flying missions in Low Earth Orbit (LEO), such as GRACE and TanDEM-X, has shown the feasibility of precise relative navigation at millimeter and sub-millimeter levels using GPS carrier phase measurements with fixed integer ambiguities. However, the robustness and availability of the solutions provided by current algorithms may be highly dependent on the mission profile. The main challenges faced in the LEO scenario are the resulting short continuous carrier phase tracking arcs along with the observed rapidly changing ionospheric conditions, which in the particular situation of long baselines increase the difficulty of correct integer ambiguity resolution. To reduce the impact of these factors, the present study proposes a strategy based on a reduced-dynamics filtering of dual-frequency GPS measurements for precise baseline determination along with a dedicated scheme for integer ambiguity resolution, consisting of a hybrid sequential/batch algorithm based on the maximum a posteriori and integer aperture estimators. The algorithms have been tested using flight data from the GRACE, TanDEM-X and Swarm missions in order to assess their robustness to different formation and baseline configurations. Results with the GRACE mission show an average 0.7mm consistency with the K/Ka-band ranging measurements over a period of more than two years in a baseline configuration of 220km. Results with TanDEM-X data show an average of 3.8mm consistency of kinematic and reduced-dynamic solutions in the along-track component over a period of 40days in baseline configurations of 500m and 75km. Data from Swarm A and Swarm C spacecraft are largely affected by atmospheric scintillation and contain half cycle ambiguities. The results obtained under such conditions show an overall consistency between kinematic and reduced-dynamic solutions of 1.7cm in the along-track component over a period of 30days in a variable baseline of approximately 60–175km. An analysis of one orbital period excluding a region where errors due to atmospheric scintillation occur, shows a consistency between kinematic and reduced-dynamic solutions of 3mm in the along-track direction.

Instantaneous and controllable integer ambiguity resolution: review and an alternative approach

In the high-precision application of Global Navigation Satellite System (GNSS), integer ambiguity resolution is the key step to realize precise positioning and attitude determination. As the necessary part of quality control, integer aperture (IA) ambiguity resolution provides the theoretical and practical foundation for ambiguity validation. It is mainly realized by acceptance testing. Due to the constraint of correlation between ambiguities, it is impossible to realize the controlling of failure rate according to analytical formula. Hence, the fixed failure rate approach is implemented by Monte Carlo sampling. However, due to the characteristics of Monte Carlo sampling and look-up table, we have to face the problem of a large amount of time consumption if sufficient GNSS scenarios are included in the creation of look-up table. This restricts the fixed failure rate approach to be a post process approach if a look-up table is not available. Furthermore, if not enough GNSS scenarios are considered, the table may only be valid for a specific scenario or application. Besides this, the method of creating look-up table or look-up function still needs to be designed for each specific acceptance test. To overcome these problems in determination of critical values, this contribution will propose an instantaneous and CONtrollable (iCON) IA ambiguity resolution approach for the first time. The iCON approach has the following advantages: (a) critical value of acceptance test is independently determined based on the required failure rate and GNSS model without resorting to external information such as look-up table; (b) it can be realized instantaneously for most of IA estimators which have analytical probability formulas. The stronger GNSS model, the less time consumption; (c) it provides a new viewpoint to improve the research about IA estimation. To verify these conclusions, multi-frequency and multi-GNSS simulation experiments are implemented. Those results show that IA estimators based on iCON approach can realize controllable ambiguity resolution. Besides this, compared with ratio test IA based on look-up table, difference test IA and IA least square based on the iCON approach most of times have higher success rates and better controllability to failure rates.

Integer aperture estimation comparison between ratio test and difference test: from theory to application

Acceptance testing or ambiguity validation is a key step in global navigation satellite system (GNSS) ambiguity resolution, which combined with integer estimator is the so-called integer aperture (IA) estimator. The difference test and ratio test are the two most popular tests. In order to compare the performances of both IA estimators, their differences in different GNSS models are analyzed from algebraic and geometrical perspectives. Furthermore, both tests are connected by comparing with the optimal acceptance test, and then, they can be transformed each other based on certain numerical conditions. As to the instantaneous applications, both tests are first compared with their corresponding instantaneous approaches, including the fixed failure rate approach for ratio test IA (RTIA) and the instantaneous and controllable approach for difference test IA (DTIA). Advantages and drawbacks of both IA estimators in theory and application are analyzed based on these comparisons. In order to verify these conclusions, typical multi-frequency, multi-GNSS situations are constructed to evaluate the performances of DTIA and RTIA ambiguity resolution. Then, both IA estimators are compared based on their instantaneous approaches in the field test. All the simulation experiments and field test results indicate that DTIA has more advantages and is better than RTIA not only in theory, but also in the practical applications.

Integer aperture ambiguity resolution based on difference test

Carrier-phase integer ambiguity resolution (IAR) is the key to highly precise, fast positioning and attitude determination with Global Navigation Satellite System (GNSS). It can be seen as the process of estimating the unknown cycle ambiguities of the carrier-phase observations as integers. Once the ambiguities are fixed, carrier phase data will act as the very precise range data. Integer aperture (IA) ambiguity resolution is the combination of acceptance testing and integer ambiguity resolution, which can realize better quality control of IAR. Difference test (DT) is one of the most popular acceptance tests. This contribution will give a detailed analysis about the following properties of IA ambiguity resolution based on DT: All these conclusions are verified based on the single and combination GNSS simulation experiments. The experiment results indicate the correctness of these conclusions. These properties demonstrate the special properties of DT-IA estimator, and also provide the research frame to investigate other IA estimators. They are helpful to realize better use of IA estimators in quality control and precise positioning in future.

The inertial attitude augmentation for ambiguity resolution in SF/SE-GNSS attitude determination.

The Unaided Single Frequency/Single Epoch Global Navigation Satellite System (SF/SE GNSS) model is the most challenging scenario for ambiguity resolution in the GNSS attitude determination application. To improve the performance of SF/SE-GNSS ambiguity resolution without excessive cost, the Micro-Electro-Mechanical System Inertial Measurement Unit (MEMS-IMU) is a proper choice for the auxiliary sensor that carries out the inertial attitude augmentation. Firstly, based on the SF/SE-GNSS compass model, the Inertial Derived Baseline Vector (IDBV) is defined to connect the MEMS-IMU attitude measurement with the SF/SE-GNSS ambiguity search space, and the mechanism of inertial attitude augmentation is revealed from the perspective of geometry. Then, through the quantitative description of model strength by Ambiguity Dilution of Precision (ADOP), two ADOPs are specified for the unaided SF/SE-GNSS compass model and its inertial attitude augmentation counterparts, respectively, and a sufficient condition is proposed for augmenting the SF/SE-GNSS model strength with inertial attitude measurement. Finally, in the framework of an integer aperture estimator with fixed failure rate, the performance of SF/SE-GNSS ambiguity resolution with inertial attitude augmentation is analyzed when the model strength is varying from strong to weak. The simulation results show that, in the SF/SE-GNSS attitude determination application, MEMS-IMU can satisfy the requirements of ambiguity resolution with inertial attitude augmentation.

Generalized integer aperture estimation for partial GNSS ambiguity fixing

In satellite navigation, the key to high precision is to make use of the carrier-phase measurements. The periodicity of the carrier-phase, however, leads to integer ambiguities. Often, resolving the full set of ambiguities cannot be accomplished for a given reliability constraint. In that case, it can be useful to resolve a subset of ambiguities. The selection of the subset should be based not only on the stochastic system model but also on the actual measurements from the tracking loops. This paper presents a solution to the problem of joint subset selection and ambiguity resolution. The proposed method can be interpreted as a generalized version of the class of integer aperture estimators. Two specific realizations of this new class of estimators are presented, based on different acceptance tests. Their computation requires only a single tree search, and can be efficiently implemented, e.g., in the framework of the well-known LAMBDA method. Numerical simulations with double difference measurements based on Galileo E1 signals are used to evaluate the performance of the introduced estimation schemes under a given reliability constraint. The results show a clear gain of partial fixing in terms of the probability of correct ambiguity resolution, leading to improved baseline estimates.

GNSS carrier phase ambiguity resolution based on integrity restriction in ambiguity domain

Carrier phase ambiguity resolution of Global Navigation Satellite System (GNSS) is a key technology for high-precision navigation and positioning, and it is a challenge for applications which require both high accuracy and high integrity. This paper proposes efficient ambiguity resolution methods based on integrity restriction using Fixed Failure rate Ratio Test (FF-RT) and Doubly Non-central F-distribution Ratio Test (DNF-RT), and derives the related processing models and numerical algorithms compared with the traditional Ratio Test (RT) method. Firstly, the integer ambiguity resolution and validation procedures, especially the Least squares AMBiguity Decorrelation Adjustment (LAMBDA) estimation and RT validation are analyzed. Then the quality evaluation using success rate, the FF-RT method using Integer Aperture (IA) estimation and the NDF-RT method are proposed. Lastly, the simulation and analysis for LAMBDA using RT, FF-RT and DNF-RT methods are performed. Simulation results show that in case of unbiased scenario FF-RT and DNF-RT have similar performances, which are significantly better than RT. In case of biased scenario it is difficult for FF-RT to predict the biased success rate thus it should not be used for bias detection, while DNF-RT can detect biases in most cases except for the biases are approximate or equal to integer, which has the important benefit for early detection of potential threat to the position solution.

Analysis of the upper bounds for the integer ambiguity validation statistics

Integer ambiguity validation is an essential quality control step for high-precision positioning and navigation with global navigation satellite systems (GNSS). In order to validate the resolved integer ambiguities, statistical tests, such as the R-ratio test, F-ratio test, W-ratio test, difference test, and projector test, have been favored. In practice, the critical values for these statistical tests are determined either empirically or from the assumed distributions. However, previous research has revealed that some of these statistics have upper bounds, which can be obtained from simulations. In this contribution, we find that under the framework of the integer aperture estimation, the upper bounds for these ambiguity validation statistical tests can be derived without actual measurements or simulation. As a result, the assumed distributions for these statistical tests are inappropriate. According to the derivation, it has been concluded that the upper bounds of these ambiguity validation tests depend only on the ambiguity geometry (e.g., the float ambiguity variance---covariance matrix) and can be obtained at the design stage of GNSS positioning. Thus, the critical value for these ambiguity validation statistics has a rigorous range, and it should be chosen to be smaller than a priori derived upper bound. Otherwise, no integer ambiguities can be obtained.

Theoretical Upper Bound and Lower Bound for Integer Aperture Estimation Fail-Rate and Practical Implications

Integer ambiguity validation is pivotal in precise positioning with Global Navigation Satellite Systems (GNSS). Recent research has shown traditionally used ambiguity validation methods can be classified as members of the Integer Aperture (IA) estimators, and by the virtue of the IA estimation, a user controllable IA fail-rate is preferred. However, an appropriately chosen fail-rate is essential for ambiguity validation. In this paper, the upper bound and the lower bound for the IA fail-rate, which are extremely useful even at the designing stage of a GNSS positioning system, have been analysed, and numerical results imply that a meaningful IA fail-rate should be within this range.

Some remarks on GNSS integer ambiguity validation methods

Ambiguity resolution is an indispensable step in fast and high precision Global Navigation Satellite System (GNSS) based positioning. In general, ambiguity resolution consists of three steps. The first step is to estimate the ambiguities using a least-squares estimation process, from which the so called ‘float solution’ or real valued solution is obtained. Then in the second step, the float solution is used to search for the integer ambiguities. Once integer ambiguities are resolved, the last step is to apply the integer ambiguities into the models so that fairly accurate fixed solution can be generated. Owing to the importance of the integer ambiguities, one indispensable procedure that needs to be implemented in the second step is integer ambiguity validation. Over the past decades, considerable work has been concentrated on this procedure and various approaches have been proposed, such as R-ratio test, F-ratio test, W-ratio test, integer aperture estimator, etc. However, their performances are controversial. Therefore, in this contribution, an overview of the existing ambiguity validation methods is firstly presented, and then some numerical analysis is carried out to evaluate their performances.

Ambiguity validation with combined ratio test and ellipsoidal integer aperture estimator

To ensure reliable ambiguity resolution, ambiguity validation is an indispensable step. It has been a challenge for many years and is far from being resolved. Over the past years, various ambiguity validation methods have been proposed, such as F-ratio test, R-ratio test, difference test, projector test, ellipsoidal integer aperture (EIA) estimator and penalized integer aperture (PIA) estimator. In this paper, through analysis and testing, we find that, when the aperture region of EIA is not allowed to be overlapped, the efficiency of ambiguity resolution with EIA is low and it is not applicable to fast static positioning or real-time kinematic (RTK) applications. Then EIA with overlapped aperture regions is recommended and the resulted fail-rate becomes upper bound of the actual one. After that, it is suggested that combined use of overlapped EIA and R-ratio test can increase the reliability of ambiguity resolution. Finally, numerical tests are carried out based on practical buoy data and simulated Galileo data.

The GNSS ambiguity ratio-test revisited: a better way of using it

Integer carrier phase ambiguity resolution is the key to fast and high-precision global navigation satellite system (GNSS) positioning and application. Apart from integer estimation, also acceptance tests are part of the ambiguity resolution process. A popular acceptance test is the so-called ratio-test. In this contribution we study the properties and the underlying concepts of the ratio-test. We discuss some misconceptions of the ratio-test and in particular show that the ratio-test is not a test for testing the correctness of the integer least-squares solution. We also show that the common usage of the ratio-test with a fixed critical value has shortcomings. Instead, the fixed failure rate approach is recommended. This approach is part of the more general theory of integer aperture estimation, and enables that the failure rate does not exceed a user-defined value. Results of the fixed failure-rate ratio-test and its improved performance are illustrated with a number of examples.

On the Feasibility of Adding Carrier Phase - Assistance to Cellular GNSS Assistance Standards

In this invited contribution a brief review will be presented of the integer estimation theory as developed by the author over the last decade and which started with the introduction of the LAMBDA method in 1993. The re- view discusses three different, but closely related classes of ambiguity estimators. They are the integer estimators, the integer aperture estimators and the integer equivariant estimators. Integer estimators are integer aperture estima- tors and integer aperture estimators are integer equivari- ant estimators. The reverse is not necessarily true how- ever. Thus of the three types of estimators the integer es- timators are the most restrictive. Their pull-in regions are translational invariant, disjunct and they cover the ambi- guity space completely. Well-known examples are integer rounding, integer bootstrapping and integer least-squares. A less restrictive class of estimators is the class of inte- ger aperture estimators. Their pull-in regions only obey two of the three conditions. They are still translational invariant and disjunct, but they do not need to cover the ambiguity space completely. As a consequence the inte- ger aperture estimators are of a hybrid nature having either integer or non-integer outcomes. Examples of integer aper- ture estimators are the ratio-testimator and the difference- testimator. The class of integer equivariant estimators is the less restrictive of the three classes. These estimators only obey one of the three conditions, namely the condi- tion of being translational invariant. As a consequence the outcomes of integer equivariant estimators are always real- valued. For each of the three classes of estimators we also present the optimal estimator. Although the Gaussian case is usually assumed, the results are presented for an arbi- trary probability density function of the float solution. The optimal integer estimator in the Gaussian case is the inte- ger least-squares estimator. The optimality criterion used is that of maximizing the probability of correct integer es- timation, the so-called success rate. The optimal integer aperture estimator in the Gaussian case is the one which only returns the integer least-squares solution when the in- teger least-squares residual resides in the optimal aperture pull-in region. This region is governed by the probability density function of the float solution and by the probabil- ity density function of the integer least-squares residual. The aperture of the pull-in region is governed by a user- defined aperture parameter. The optimality criterion used is that of maximizing the probability of correct integer esti- mation given a fixed, user-defined, probability of incorrect integer estimation. The optimal integer aperture estimator becomes identical to the optimal integer estimator in case the success rate and the fail rate sum up to one. The best integer equivariant estimator is an infinite weighted sum of all integers. The weights are determined as ratios of the probability density function of the float so- lution with its train of integer shifted copies. The optimal- ity criterion used is that of minimizing the mean squared error. The best integer equivariant estimator therefore al- ways outperforms the float solution in terms of precision.

New Global Navigation Satellite System Ambiguity Resolution Method Compared to Existing Approaches

Integer carrier phase ambiguity resolution is the process of resolving unknown cycle ambiguities of doubledifferenced carrier phase data as integers, and it is a prerequisite for rapid and high-precision global navigation satellite system positioning and navigation. Besides integer estimation, integer ambiguity resolution also involves validation of the integer estimates. In this contribution a new ambiguity resolution method is presented, based on the class of integer aperture estimators, which for the first time reveals an overall approach to the combined problem of integer estimation and validation. Furthermore, it is shown how the different discrimination tests that are currently in use in practice can be cast into the framework of the new approach.

Integer aperture bootstrapping: a new GNSS ambiguity estimator with controllable fail-rate

In this contribution, we introduce a new bootstrap-based method for Global Navigation Satellite System (GNSS) carrier-phase ambiguity resolution. Integer bootstrapping is known to be one of the simplest methods for integer ambiguity estimation with close-to-optimal performance. Its outcome is easy to compute due to the absence of an integer search, and its performance is close to optimal if the decorrelating Z-transformation of the LAMBDA method is used. Moreover, the bootstrapped estimator is presently the only integer estimator for which an exact and easy-to-compute expression of its fail-rate can be given. A possible disadvantage is, however, that the user has only a limited control over the fail-rate. Once the underlying mathematical model is given, the user has no freedom left in changing the value of the fail-rate. Here, we present an ambiguity estimator for which the user is given additional freedom. For this purpose, use is made of the class of integer aperture estimators as introduced in Teunissen (2003). This class is larger than the class of integer estimators. Integer aperture estimators are of a hybrid nature and can have integer outcomes as well as non-integer outcomes. The new estimator is referred to as integer aperture bootstrapping. This new estimator has all the advantages known from integer bootstrapping with the additional advantage that its fail-rate can be controlled by the user. This is made possible by giving the user the freedom over the aperture of the pull-in region. We also give an exact and easy-to-compute expression for its controllable fail-rate.

Penalized GNSS Ambiguity Resolution

Global Navigation Satellite System (GNSS) carrier phase ambiguity resolution is the process of resolving the carrier phase ambiguities as integers. It is the key to fast and high precision GNSS positioning and it applies to a great variety of GNSS models which are currently in use in navigation, surveying, geodesy and geophysics. A new principle of carrier phase ambiguity resolution is introduced. The idea is to give the user the possibility to assign penalties to the possible outcomes of the ambiguity resolution process: a high penalty for an incorrect integer outcome, a low penalty for a correct integer outcome and a medium penalty for the real valued float solution. As a result of the penalty assignment, each ambiguity resolution process has its own overall penalty. Using this penalty as the objective function which needs to be minimized, it is shown which ambiguity mapping has the smallest possible penalty. The theory presented is formulated using the class of integer aperture estimators as a framework. This class of estimators was introduced elsewhere as a larger class than the class of integer estimators. Integer aperture estimators, being of a hybrid nature, can have integer outcomes as well as non-integer outcomes. The minimal penalty ambiguity estimator is an example of an integer aperture estimator. The computational steps involved for determining the outcome of the minimal penalty estimator are given. The additional complexity in comparison with current practice is minor, since the optimal integer estimator still plays a major role in the solution of the minimal penalty ambiguity estimator.