
Working Papers • March/April 2011
Integer Aperture EstimationA Framework for GNSS Ambiguity Acceptance TestingWorking Papers explore the technical and scientific themes that underpin GNSS programs and applications. This regular column is coordinated by Prof. Dr.Ing. Günter Hein. Contact Prof. Hein at Guenter.Hein@unibwmuenchen.de Tracking the carrier phase of GNSS signals has evolved into a widespread practice for achieving rapid and very accurate positioning. A key to this process is implementing a robust method for determining the number of carrier waves between a GNSS satellite and receiver, including any fractional wavelength, in a given signal transmission — socalled integer ambiguity resolution. Researchers have developed a variety of approaches for calculating the number of integers, but reliable means for testing and accepting the results of such calculations — a crucial factor for ensuring the integrity of such measurements — are not as well developed. This column introduces the principle of integer aperturte estimation and show how it can accomplish this goal. Share via: Slashdot Technorati Twitter Facebook For the complete story, including figures, graphs, and images, please download the PDF of the article, above. Integer carrierphase ambiguity resolution is the key to fast and highprecision GNSS positioning and navigation. It is the process of resolving the unknown cycle ambiguities of the carrierphase data as integers. Once this has been done successfully, the very precise carrierphase data will act as pseudorange data, thus making very precise positioning and navigation possible. Procedures for carrierphase ambiguity resolution not only consist of integer ambiguity estimation, but usually also include ambiguity acceptance testing. Such testing is important, in particular in light of the ever increasing integrity demands on GNSS solutions. Although the statistical theory of integer ambiguity estimation is reasonably well established, this cannot yet be said of ambiguity acceptance testing. The aim of this article, therefore, is to present a unifying theoretical framework for ambiguity estimation and testing. It provides the tools for comparing and evaluating current procedures for acceptance testing and creates the possibility to devise new tests that are better than existing ones. We will begin with a review of the fourstep procedure for integer ambiguity resolution, including acceptance testing. Next, we will introduce the principle of integer aperture (IA) estimation and explain how and why this estimation principle provides us the framework we are looking for. The following section will describe how we can evaluate the quality of IA estimation. Finally, we will discuss the way in which to define optimal IA estimators. Two such optimal IA estimators are presented, the failrate constrained maximum successrate estimator and the minimum mean penalty estimator.
The Four Steps of Ambiguity Resolution E(y) = Aa + Bb, D(y) = Q_{yy}, a ∈ Z^{n}, b ∈ R^{p} where E(.) and D(.) denote expectation and dispersion, respectively, and where the mvector y contains the “observed minus computed,” single, dual or multifrequency carrierphase and pseudorange (code) observables. The nvector a contains the integer doubledifferenced (DD) ambiguities, and the realvalued pvector b contains the remaining unknown parameters, such as baseline components (coordinates), atmospheric delay parameters (troposphere, ionosphere), and possibly (receiver, satellite) clock parameters and instrumental delays. Note that the parameterization in DD ambiguities does not require y to be in DD form. Thus, y may be in undifferenced or singledifferenced form as well. The m × (n+p) matrix (A,B) contains the given design matrices of a and b, respectively, and the m×m positive definite matrix Q_{yy} is the variance matrix of y. The process of solving the GNSS model is usually divided into the following four steps (Figure 1, see inset photo, above right). 1. Float Solution. In the first step, the integer nature of the ambiguities is discarded and a standard leastsquares (LS) parameter estimation is performed. As a result, one obtains the socalled float solution, together with its variancecovariance matrix ... 2. Integer solution: In the second step, the realvalued float ambiguity solution is further mapped to an integer solution ... 3. Accept/Reject. Once integer estimates of the ambiguities have been computed, the third step consists of deciding whether or not to accept the integer solution. Several such tests have been proposed in the literature and are currently in use in practice. Examples include the ratiotest, the Fratio test, the differencetest and the projectortest (See the 2005 article by S. Verhagen referenced in Additional Resources for a fuller discussion of these tests.) ... 4. Fixed Solution. Once the integer solution has been accepted, the float estimator is further adjusted to obtain the socalled fixed estimator ... Although the statistical theory of integer ambiguity estimation (second step) is reasonably well established, for a long time no such theory was available for the third step. Consequently, a proper treatment of the accept/reject decision lags behind the level at which the second step is treated in practice. This has resulted in ad hoc approaches to the third step, in an absence of a proper quality description, and sometimes even in a misunderstanding of its essence. The goal of our discussion here, therefore, is to present a unifying framework for ambiguity resolution — one that has Step 3 integrally included. The framework is based on the principle of integer aperture estimation as introduced by P. J. G. Teunissen in his 2003 Journal of Planetary Geodesy article cited in Additional Resources.
This framework allows one to answer such questions as:
Integer Aperture Estimation … Importantly, the principle of IAestimation also generalizes all current ambiguity acceptance tests. That is, each such test procedure, such as the ratiotest, the Fratio test, the differencetest, or the projectortest, is a member of the class of IAestimators … … Note that testing the closeness of the float solution to its nearest integer is not the same as testing the correctness of the ILS solution. Thus the ratiotest does not test for the correctness of the ILS solution, as is sometimes erroneously stated in the literature …
Quality of IAEstimation … The above foregoing probabilities may now be used to evaluate any IAestimator, including those that are currently in use. Additionally, these probabilities may be used to develop new strategies for current IAestimators … … This fixed failrate strategy gives users control over the failrate of their ambiguity resolution. And when applied to current IAestimators, it provides an improvement over the way the tolerance values are selected …
Optimal IAEstimation To determine which of the IAestimators performs best, we first need to formulate an optimality criterium. Two such optimal estimators were introduced by P. Teunissen in his 2004 and 2005 articles cited in Additional Resources. They are the constrained maximum successrate (CMS) estimator and the minimum mean penalty (MMP) estimator.
Constrained Maximum SuccessRate (CMS) Estimator . . .
Minimum Mean Penalty (MMP) Estimator . . .
The Computational Steps The steps for computing the CMS and MMPestimator are:
. . .
Summary
1. What is the exact role played by the acceptance test (Step 3) of ambiguity resolution?
2. How can we describe and evaluate the performance of integer aperture estimation? When a fixed failrate is set, the ambiguity resolution procedure automatically adapts the size of the aperture pullin region to the strength of the underlying GNSS model. (We provide an example in the following section.) When the model gets stronger as time progresses, the size gets larger. With a sudden drop in tracked satellites, however, the size gets reduced again.
3. How do the different current procedures for acceptance testing compare?
4. Do tests exist that are better than the current ones?
IA Estimation Example … Simulated float ambiguities are now shown for two different model strengths. For the weaker model (left) the spread in the float solutions is larger and consequently the aperture pullin region should be chosen small to obtain a failrate of 0.1 percent in this case. For the stronger model (right), the same failrate is obtained with a much larger pullin region. Hence, in this case the success and fixrates are also larger … For the complete story, including figures, graphs, and images, please download the PDF of the article, above.
Acknowledgment
Additional Resources Copyright © 2018 Gibbons Media & Research LLC, all rights reserved. 
