On GNSS Ambiguity Acceptance Tests

Transcription

1 International Global Navigation Satellite Systems Society IGNSS Symposium 27 The University o New South Wales, Sydney, Australia 4 6 December, 27 On GNSS Ambiguity Acceptance Tests P.J.G. Teunissen Delt Institute o Earth Observation and Space Systems Delt University o Technology, The Netherlands Phone: , Fax: , p.j.g.teunissen@tudelt.nl S. Verhagen Delt Institute o Earth Observation and Space Systems Delt University o Technology, The Netherlands Phone: , Fax: , a.a.verhagen@tudelt.nl ABSTRACT Integer carrier phase ambiguity resolution is the key to ast and highprecision global navigation satellite system (GNSS) positioning and application. Apart rom integer estimation, also acceptance tests are part o 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 o the ratio-test. We discuss some misconceptions o the ratio-test and in particular show that the ratio-test is not a test or testing the correctness o the integer least-squares solution. We also show that the common usage o the ratio-test with a ixed critical value has shortcomings. Instead, the ixed ailure rate approach is recommended. This approach, which is part o the more general theory o integer aperture estimation, has the advantage that the times to irst ix are reduced, while it is guaranteed that the ailure rate does not exceed a user-deined value. Results o the ixed ailure-rate ratiotest are illustrated with a number o examples. KEYWORDS: GNSS, integer ambiguity resolution, ratio test, integer aperture estimation, ixed ailure rate.

2 . INTRODUCTION Integer carrier phase ambiguity resolution is the key to ast and high-precision GNSS positioning and navigation. It is the process o resolving the unknown cycle ambiguities o the double-dierenced carrier phase data as integers. Once this has been done successully, the very precise carrier phase data will act as pseudo range data, thus making very precise positioning and navigation possible. GNSS ambiguity resolution applies to a great variety o current and uture models o GPS, modernized GPS and Galileo, with applications in surveying, navigation, geodesy and geoscience in general. These models may dier greatly in complexity and diversity. They range rom single-baseline models used or kinematic positioning to multi-baseline models used as a tool or studying geodynamic phenomena. The models may or may not have the relative receiver-satellite geometry included. They may also be discriminated on the basis o whether the slave receiver(s) is stationary or in motion, or whether or not the dierential atmospheric delays (ionosphere and troposphere) are included as unknowns. An overview o these models can be ound in textbooks such as Strang and Borre (997), Teunissen and Kleusberg (998), Homann-Wellenho et al. (2), Leick (23), and Misra and Enge (26). GNSS ambiguity resolution can conceptually be divided into our steps: In the irst step, one disregards the integer nature o the ambiguities and perorms a standard least-squares adjustment. As a result one obtains the so-called loat solution o all the parameters (i.e. ambiguities, baseline components, and possibly additional parameters such as atmospheric delays), together with their variancecovariance matrix. In the second step, the real-valued loat solution o the ambiguities is urther adjusted, so as to take the integer constraints into account. As a result one obtains an integer solution or the ambiguities. Integer rounding, integer bootstrapping and integer leastsquares are dierent methods or obtaining the integer solution. Integer least-squares (ILS) is optimal, as it can be shown to maximize the probability o correct integer estimation (Teunissen, 999). In contrast to rounding and bootstrapping, an integer search is needed to compute the ILS solution. This can eiciently be done with the LAMBDA method. Once the integer ambiguities are computed, they are used in the third step as input to decide whether or not to accept the integer solution. Several such tests have been proposed in the literature and are currently in use in practice (Abidin 993, Chen 997, Euler and Scharin 99, Frei and Beutler 99, Han 997, Han and Rizos 996, Landau and Euler 992, Tiberius and de Jonge 995, Wang et al. 998). Examples are the ratio-test, the distance-test and the projector-test. A review and evaluation o these tests can be ound in Verhagen (25). Once the integer solution is accepted, the ourth step consists o correcting the loat solution o all other parameters by virtue o their correlation with the ambiguities. As a result one obtains the so-called ixed solution. Provided a correct decision has been made in the third step, the ixed solution will have a precision that is commensurate with the high precision o the phase data. In this contribution we ocus attention on the third step, and in particular study the properties and use o the popular ratio-test.

3 2. RATIO TEST 2. Deinition and Misconceptions In this section we give a deinition o the popular ratio-test and point to some o the misconceptions that are linked to this test. The ratio-test is deined as ollows. Let the loat ambiguity vector and its variance matrix be given as â and, respectively. Furthermore, let a ( be the ILS solution, i.e. the integer Qa ˆ a ˆ T minimizer o q( a) = ( aˆ a) Qaˆ aˆ ( aˆ a),and let a ( ' be the integer vector that returns the second smallest value o the quadratic orm q (a). Then the ratio-test reads as: Accept a ( i: ( q( a ') ( c q( a) () where c is a tolerance value, to be selected by the user. Thus only i q (a' ( ) is suiciently larger than q (a ( ), will the decision be made to accept the ILS solution. Otherwise, the ILS solution is rejected in avour o the loat solution. Questions that need to be addressed when using the above test are:. What does the ratio-test actually test? 2. What errors can be made with the ratio-test? 3. What value or c should be chosen? Answers to these questions are needed in order to have a proper understanding o the ratiotest. One motivation that is oten given or the use o the ratio-test is that it tests the correctness o the ILS solution. With reerence to the theory o hypothesis testing, the ratio o the two quadratic orms, q (a ( ) and q (a' ( ), is then assumed to have a Fisher-distribution, rom which c can be computed, once the level o signiicance has been set. The problem with this approach is, however, that the ratio o the two quadratic orms is not Fisher-distributed. Even i one was allowed to assume that q (a ( ) and q (a' ( ) are Chi-square distributed (which is not true, since also the uncertainty o the integer vectors needs to be taken into account), then their ratio would still not be Fisher-distributed. The two quadratic orms are namely not independent. Also, the ratio-test is not a test or testing the correctness o the ILS solution. In act, one can add an arbitrary integer vector to the loat solution, without altering the outcome o the ratio test. Hence, biases o arbitrary size (provided they are integer) can be present in the loat solution, without them ever being noticed by the ratio-test. In many o the existing sotware packages a ixed value or c is chosen, no matter the strength o the underlying GNSS model (Leick, 23). This is strange, since one would expect that with a varying strength o the GNSS model or with varying degrees o reedom, one also would use varying values or c. The use o a ixed value can however be explained by a lack o a proper theory. That is, by not knowing how to rigorously compute a critical value, one adopts the value that, on the basis o empirical evidence, seems to give reasonable results.

4 Indeed, the popular usage o the value 3 seems to be based on various empirical studies that have shown, although not conclusively, that a workable value or c will lie somewhere around this value. Wei and Schwarz (995), or instance, proposed to use the ratio-test with a critical value o 2. Han and Rizos (996) showed that good results can be obtained with the ratio-test with a critical value o.5, provided that one has conidence in the stochastic model, while Euler and Scharin (99) used test computations, rom which a value o c between 5 and ollowed. The outcomes o these studies are, however, diicult to generalize, since they are based on dierent GNSS measurement scenarios. 2.2 What Does the Ratio-Test Test? To understand what the ratio-test tests, we need to get a better insight into its acceptance region and rejection region. We already remarked that the outcome o the ratio-test remains unchanged when an arbitrary integer vector, say z, is added to the loat solution. This implies that its acceptance region, denoted as Ω, must be a region which is z-translational invariant. That is, i the acceptance region is translated over an arbitrary integer vector, then the same acceptance region is recovered again. Since the rejection region is complementary to the acceptance region, also the rejection region o the ratio-test is z-translational invariant. The z-translational invariance o the acceptance region implies that it must equal the union o z-translated copies o a smaller region Ω. Thus: Ω = Ω where Ω = Ω z (2) Un z Ζ z z + A closer look at the ratio-test shows that Ω is given by the set: Ω = n T T { x R x Q x ( x u) Q ( x u), u Z n } aa ˆ ˆ c aa ˆ ˆ (3) This region, which is called an aperture pull-in region, is symmetric with respect to the origin and its shape is governed by the variance matrix o the loat solution, while its size is governed by the value o c. Each integer vector z has its own pull-in region. They are translated copies o Ω, i.e. Ω z = Ω + z. We have: ( a = z i aˆ Ω z (4) Thus i the loat solution resides in Ω z, the ratio-test leads to acceptance and the ILS solution is equal to z. Now we are in a position to understand what the ratio-test actually tests. The ratio-test tests the closeness o the loat solution to its nearest integer vector. I it is close enough, the test leads to acceptance o a (. I it is not close enough, then the test leads to rejection in avour o the loat solution â. The size or aperture o the pull-in region provides the largest distance one is willing to accept. The value or c can be used to tune this aperture. Note that testing the closeness o the loat solution to its nearest integer is not the same as testing the correctness o the ILS solution. The outcome o the ILS solution is correct i it would equal the unknown integer mean o â, a = E(aˆ ). But the closeness o â to the integer vector a is not tested by the ratio-test. 2.3 Relevance o the Ratio-Test Now that we know what the ratio-test actually does, one may ask in what way this test helps us gain conidence in the outcome o ambiguity resolution. In order to answer this question,

5 we have to realize that acceptance o the ILS solution by the ratio-test can be correct or incorrect. We thereore have to distinguish between the ollowing three cases: aˆ Ω a success: correct integer estimation aˆ Ω \{ Ωa} ailure: incorrect integer estimation â Ω undecided: ambiguity not ixed to an integer where Ω \{ Ωa} means that Ω a is deleted rom the set Ω, with a being the unknown integer ambiguity vector. The corresponding probabilities o success (s), ailure () and undecidedness (u) are given by: P ( x dx = Ωa s aˆ ) z Z P = dx (5) P u = s n \{ } ˆ ( ) a Ω a x P P z where a ( x) is the probability density unction (PDF) o a ˆ ~ N( a, ). The irst two ˆ Q a ˆaˆ probabilities are reerred to as success rate and ailure rate, respectively. Thus probability o acceptance o the ratio-test and P u is its probability o rejection. P + P is the The above probabilities all depend on the shape and size o Ω and on the PDF o â. Thus by changing Ω and/or the PDF o â, one can inluence the above probabilities. Changing the PDF will not be possible, once the measurement scenario is given (this will be dierent i one was designing a measurement scenario). Changing the shape o Ω is also not possible, since the shape is determined by the ratio-test. This leaves us with the size o Ω, which is determined by c. Hence, by changing c one can inluence the above probabilities. Thus through the choice o c, the user is able to have control over the ailure rate, i.e. the probability o incorrect integer estimation. This is a very important result because it gives the user the necessary lexibility over what he/she inds an acceptable risk to take with integer ambiguity resolution. This is the relevance o having the ratio-test included as the third step in the our-step procedure o ambiguity resolution. 2.4 Fixed Failure Rate Should Be Used The above discussion makes clear that the common practice o using a ixed value or c is not the way to go. By using a ixed value or c, the user is deprived rom any control over the ailure rate. The ailure rate will then be dierent or dierent measurement scenarios. Already in a kinematic or navigation scenario, where data are collected on an epoch by epoch basis, the ailure rate will change rom epoch to epoch i a ixed value or c is used. As an illustration o the dierence between the traditional ratio-test and our approach, ive dual-requency GPS models are considered. Based on Monte-Carlo simulations the success and ailure rates as unction o µ = / c are determined or each o the models, see Figure. It can be seen that with a ixed value o µ =. 3 or most o the models considered here a very low ailure rate is obtained, but that this is not guaranteed. This seems good, but at the same time also the corresponding success rate is low. I the threshold value would have been based on a ixed ailure rate o e.g..5, the corresponding µ = / c would have been very s

6 dierent or each o the models, and in most cases larger than.3, and thus a higher success rate and higher probability o a ix (probability o acceptance P + P ) would be obtained. s success rate ailure rate µ µ Figure. Success and ailure rates as unction o the threshold value µ = / c or 5 GPS models. In order to execute the ratio-test with a ixed ailure rate, one should be able to compute c rom the chosen ailure rate P. This is, unortunately, a rather computationally demanding task. It involves the 'inversion' o the integral equation that links the ailure rate to the size o the aperture pull-in region. A practical solution to this problem is thereore to work with lookup tables, that allows one to select (i needed through interpolation) the proper value or c. In Section 3.3 we will give an example o such a look-up table. 2.5 What About the Success Rate? Changing c will aect the ailure rate as well as the success rate. The larger c is chosen, the smaller the aperture o Ω and thereore the smaller the ailure rate, but also the smaller the success rate. So, what have we gained? To understand what we have gained, we have to consider the success rate o acceptance o the ratio-test and not the overall success rate. The success rate o acceptance is the requency with which correct integer outcomes are realized, when the outcome o the ratio-test is to accept the integer solution. It is the probability o having successul ixes, denoted as P s, and it is given by the ratio o the success rate and the probability o acceptance: Ps Ps = (6) P + P s This probability will be close to one, i the ailure rate is close to zero. Thus i the ailure rate is small, one can be very conident about the correctness o the integer solutions that are accepted by the ratio-test. 2.6 Is the Ratio-Test Optimal? As we have argued, the ratio-test should be used with a ixed ailure rate instead o with a

7 ixed value or c. In the next section, some urther examples will be given that show the dierence between these two approaches. Despite the preerence or the ixed ailure rate approach, when using the ratio-test, one may pose the question whether the use o the ratiotest is the best one can do. That is, given the ailure rate, is the ratio-test the test that results in the largest success rate? The answer is no. It can be shown that the ratio-test is a member o the class o tests as given by the theory o integer aperture estimation developed by Teunissen (23). Members rom this class dier in the way the shape o the aperture pull-in region Ω is deined. Hence, within this class, one can, by ixing the ailure rate, solve or the aperture pull-in region that maximizes the success rate (Figure 2 gives a two-dimensional example o the optimal aperture pull-in regions). The optimal test so obtained diers rom the ratio-test. Since the discussion o the optimal test and its relation to the ratio-test is outside the scope o the present contribution, we reer the reader or more details to (Teunissen 23, Verhagen and Teunissen 26) Figure 2. Two-dimensional example o optimal aperture pull-in regions, together with the ILS pull-in regions (hexagons). 3. THE FIXED FAILURE RATE RATIO-TEST In this section we illustrate the improved perormance o the ixed ailure rate approach. We also show how the value o c can be computed rom a user-deined ailure rate. 3. A short baseline example We irst consider a short GNSS baseline based on simulated data. The data set contains Hz code and phase observations on the L, E6 and E5 Galileo requencies, with known multipath errors in order to demonstrate some robustness against biases. The data set was processed on an epoch-by-epoch basis. Figure 3 shows the errors in the loat and ixed estimates o the position components (East, North, Up). The top panels show the results i the traditional ratio-test when a ixed critical value o c=2 is used; the bottom panels i the ratio-test with ixed ailure rate is used ( P =.). Note that i the ratio-test is rejected, the loat solution is used. In act, the

8 ambiguities were estimated correctly in all epochs. However, with the traditional ratio-test the ixed solution was unnecessarily rejected 2% o the time. Obviously, this deteriorates the position solutions in those epochs x E [m] x N [m] x U [m] x E [m] x N [m] x U [m] Figure 3. East, North, Up position errors, based on ratio-test with ixed critical value c=5 (top panels) and based on ratio-test with ixed ailure rate P =. (bottom panels). 3.2 A long baseline example This next example concerns a longer baseline (baseline Delt-Brussels, 32 km) or which real data was used. The characteristics o the data and the processing mode are as ollows: Dual-requency phase and code (L,L2,C,P2); cut-o elevation deg Standard deviations phase 3mm; code 5 cm (undierenced) Standard deviation o ionospheric corrections (zero sample values): cm (undierenced) 288 epochs with 3 sec interval (whole day) Tropospheric zenith delay estimation (positions kept ixed) A priori tropospheric corrections using Saastamoinen model Epoch-by-epoch processing (Kalman iltering) over whole time span (ambiguities assumed constant) LAMBDA ambiguity resolution Ratio-test with ixed critical value c=2 and with ixed ailure rate P =. Veriication o results ('ground truth') based on batch solution o whole day The results are given in Figure 4. Shown are the epoch-by-epoch values o the ratio-test,

9 together with the rejected values or the ixed critical value (top panel) and the rejected values using the ixed ailure rate. (bottom panel). With the ixed ailure rate, the ratio-test is not passed only during 2 epochs, while the ambiguities would have been correct. Hence, during.7% o the time the ixed solution is unnecessarily rejected (alse alarm). The ambiguities are always correctly ixed (success rate is ). In case o the ixed critical value c=2, the ratio-test is not passed during 544 epochs, while the ambiguities would have been correct. Hence, during 9% o the time the ixed solution is unnecessarily rejected (alse alarm) ratio epoch ratio epoch ( ( Figure 4. Epoch-by-epoch values o the ratio q( a) / q( a' ) ; top panel: acceptance i ( ( q ( a) / q( a') / c, with c=2; bottom panel: P =..

10 3.3 Determining the Critical Value We already remarked that in order to execute the ratio-test with a ixed ailure rate, one has to compute c rom the chosen ailure rate P. This is a nontrivial task, as it involves the 'inversion' o the integral equation that links the ailure rate to the size o the aperture pull-in region. To determine c, one needs simulations based on the variance matrix o the loat ambiguities. This may lead to a high computational burden. To lessen this burden, the idea is that look-up tables are created rom which the appropriate critical value c can be determined based on the variance matrix o the loat ambiguities. This means that the only input or the complete ambiguity resolution kernel (LAMBDA + ratio-test) would consist o the loat ambiguities and their variance matrix. For P =., Table gives an example o how to look up the /c values. It works as ollows. The user irst computes the ILS ailure rate P, ( minus the ILS success rate) rom the n n variance matrix o the loat ambiguities. Then rom n and P, ILS, the /c value that corresponds with, in this case P =., is obtained rom the table. ILS P, n = n = 6 n = 7 n = 8 n = 9 n = n = ILS M M M M M M M M Table. Example o part o the look-up table or /c, given P =. (values are indicative). One practical problem with this approach is, o course, the computation o the ILS ailure rate. Exact computation would again require simulation. Hence, an approximation is needed. Several approximations are available, most o which are known to be either an upper bound or lower bound. Obviously, an upper bound should be used in order to guarantee that the actual ailure rate is lower than the maximum allowable value. To show how well this works, the /c values obtained with this upper bound approach are then used to determine the corresponding ailure rates and ix probabilities based on simulated data. Ideally, the ailure rates should be very close to the ixed value.. The results are shown in Figure 5 or two models. The actual ailure rate as a unction o time is shown in the let panels. The ix probability is shown in the right panels. The black solid line shows the ailure rate obtained by using the approximated /c value with the look-up table. The grey lines show the ailure rates obtained by using a ixed /c value o.5. The dashed black lines show the true values i the /c value corresponding to a ixed ailure rate o. was used. Note that as soon as the ILS ailure rate is smaller than., the threshold value becomes equal to, and hence the ailure rate becomes equal to the ILS ailure rate.

11 ailure rate ix probability epoch epoch ailure rate ix probability epoch epoch Figure 5. Failure rates and ix probabilities with approximated threshold value using a look-up table (black solid). True values (black dashed) and values with ixed threshold value o c=2 (grey) are also shown. Let panels: 3-requency GPS, 5 ambiguities. Right panels: 2-requency Galileo/GPS, 2 ambiguities. It ollows that the approximation o the /c value using the look-up table works very well, even though the upper bound o the ILS ailure rate was used. In general the ailure rates are somewhat lower than the required value, which is good. This implies that also the ix probabilities are somewhat lower, since a smaller ailure rate means that the acceptance region is smaller. However, the dierence compared to the probabilities obtained with the 'true' /c value is small. Obviously, using the ixed ailure rate determination o the /c value gives much better perormance as compared to using a ixed c value, as is done with the traditional ratio-test.

12 4. CONCLUSIONS In this contribution we showed what the popular ratio-test does and how it should be used. The ratio-test does not test, as is oten believed, the correctness o the integer least-squares solution. Also, the ratio-test should not be used, as is commonly done, with a ixed critical value. The ratio-test should be used with a ixed ailure rate, thus giving the user control over the succes rate o ixing. Examples were given that illustrate the improved perormance o our ixed ailure rate ratio-test over the traditional ratio-test. It was also shown how the critical value can be computed rom a user-deined ailure rate by means o look-up tables. Readers interested in this approach can contact the authors or more details on the construction o these look-up tables. Finally, it was pointed out that the ratio-test is member o the class o tests provided by the theory o integer aperture estimation. With this theory available, there is no need anymore to make incorrect assumptions on the distribution o the parameters or test statistics (i.e. assuming that the estimated integer ambiguities are deterministic, or that the quadratic orm ratio has a Fisher-distribution). With the help o this theory, it can also be shown that the ratio-test is not optimal. Other tests exist that have a larger success rate or a given ixed ailure rate (Teunissen, 23). ACKNOWLEDGEMENTS The research o the irst author was done in the ramework o his ARC International Linkage Proessorial Fellowship, at the Curtin University o Technology, Perth, Australia, with Proessor Will Featherstone as his host. This support is greatly acknowledged. REFERENCES Abidin HA (993) Computational and geometrical aspects o on-the-ly ambiguity resolution, PhD thesis, Dept. o Surveying Eng, Tech. Report No 4, University o New Brunswick, Canada, 34 p. Chen, Y (997) An approach to validate the resolved ambiguities in GPS rapid positioning, Proc. o the International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Ban, Canada, pp Euler HJ, Scharin B (99) On a measure or the discernibility between dierent ambiguity solutions in the static-kinematic GPS-mode, IAG Symposia no.7, Kinematic Systems in Geodesy, Surveying, and Remote Sensing, Springer-Verlag, New York, pp Frei E, Beutler G (99) Rapid static positioning based on the ast ambiguity resolution approach FARA: theory and irst results, Manuscripta Geodaetica, Vol. 5, No. 6, pp Han S, Rizos C (996) Validation and rejection criteria or integer least-squares estimation, Survey Review, Vol. 33, No. 26, pp Han S (997) Quality control issues relating to instantaneous ambiguity resolution or real-time GPS kinematic positioning, Journal o Geodesy, Vol. 7, No. 6, pp Homann-Wellenho B, Lichtenegger H, Collins J (2) Global Positioning System: Theory and Practice (5 th edition), Springer-Verlag, Berlin.

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