On the Spectrum of the GPS DD-Ambiguities

On the Spectrum of the GPS DD-Ambiguities Dr. Peter Teunissen, Paul de Jonge, and Christiaan Tiberius, Delft University of Technology (in: Proceedings ION GPS-94, Salt Lake City, Utah, USA, September 20-23, 1994) Biography Peter Teunissen is professor in Mathematical Geodesy and Positioning. Paul de Jonge and Christiaan Tiberius 1 both graduated at the Faculty of Geodetic Engineering of the Delft University of Technology. They are currently engaged in the development of mathematical models for the GPS data processing in surveying and geodesy. Abstract One of the major problems in processing GPS phase observations is estimating the double-difference (DD) ambiguities as integers. Based on carrier phase data only, short observational time spans result in strongly correlated ambiguities and in very elongated ambiguity confidence ellipsoids. As a result the estimation of the integer least-squares ambiguities becomes an extremely time consuming task, when traditional search-methods are applied. In this contribution, it will be shown both analytically as well as numerically, that this can be explained by the distinctivediscontinuitythat is present in the spectrum of conditional variances of the DD-ambiguities. In [1] a method has been introduced, that allows an efficient estimation of the integer ambiguities over short observational time spans. This method removes the discontinuity from the spectrum, thereby returning transformed ambiguities that are much less correlated and that show a dramatic improvement in precision. In this contribution the performance of the method will be demonstrated for a number of specific cases. 1. Introduction As our point of departure we consider the linear(ized) system of carrier phase and/or code observation equations y = Aa + Bb + e (1) with y the vector of observed minus computed data, a the m-vector of unknown integer DD-ambiguities, b the vector that contains the unknown increments to the baseline 1 Supported by the Lely Foundation of Rijkswaterstaat. components, A and B the corresponding design matrices, and e the vector of measurement noise and unmodelled errors. The least-squares criterion for solving the above linear system of observation equations reads min a,b y Aa Bb 2 Q y with a Z m, b R n (2) where. 2 Q y = (.) Q 1 y (.), Q y is the variancecovariance matrix of the observables, R n is the n- dimensional space of real numbers, and Z m is the m- dimensional space of integer numbers. In the tworeceiver situation, the number of ambiguities m, equals the number of satellites minus one, times the number of frequencies used. The number of baseline components n, is minimally three, which is the case when both receivers are stationary, but is a multiple of this when one of the receivers is roving. The solution of the above integer least-squares problem will be denoted as ǎ, ˇb. The solution that follows when the integer constraint a Z m is removed, will be denoted as â, ˆb. It can be shown that ǎ follows from minimizing min â a 2 with a Z m (3) a Qâ in which Qâ is the variance-covariance matrix of the least-squares ambiguities. Due to the presence of the integer constraint a Z m, there are unfortunately no standard techniques available for solving (3) as they are available for solving ordinary least-squares problems. As a consequence one has to resort to methods that in one way or another make use of a discrete search strategy for finding the integer minimizer of (3). The idea is to use the objective function of (3) for introducing an ellipsoidal region in R m, on the basis of which the search can be performed. This ellipsoidal ambiguity search space is determined by (a â) Q 1 â (a â) χ2 (4) This region is centred at â R m, and its size can be controlled through the selection of the positive constant χ 2. It will be assumed that χ 2 has been chosen such that the region at least contains the sought for integer least-squares solution. In order to be able to obtain sharp bounds for the individual ambiguities, we first apply a 115

sequential conditional least-squares adjustment [1]. This allows us to give (4) the following sum-of-squares structure m (a i â i I ) 2 χ 2 (5) σâ(i I,i I) i=1 Where â i I is the (conditional) least-squares estimate for ambiguitya i with ambiguitiesa j, j = 1, i 1 fixed (conditioned) to some value, and σâ(i I,i I) its (conditional) variance. The inequality (5) can now be used to successively determine bounds for the individual ambiguities a 1, a 2,...,a m.thisgives (a i â i I ) 2 l i σâ(i I,i I) χ 2 for i,...,m (6) where l i = (1 χ2 i 1 χ 2 i 1 ) and χ2 i 1 = j=1 (a j â j J ) 2 σâ( j J, j J ) (7) These intervals on the individual ambiguities form the basis of our depth-first search for the integer leastsquares solution. It will be clear that the efficiency of the search depends entirely on the interval-lengths. They on their turn depend on χ 2, which acts as a scale parameter, on the conditional estimates â j J, which are driven by the data, and on the conditional variances σâ( j J, j J ),which do not depend on the data. The purpose of this contribution is to study the characteristics of the spectrum of conditional variances of the ambiguities. This will allow us then to understand why the search for the integer leastsquares DD-ambiguities performs so poorly when short observational time spans are used based on carrier phase data only. It will also be shown for a variety of cases, how the method introduced in [1] allows one to considerably improve upon the efficiency of the search for the ambiguities. 2. Why is the search for the DD-ambiguities so inefficient? In this section we will study the behaviour of the conditional variances of the DD-ambiguities. It will be shown both analytically as well as numerically that there is a distinct discontinuity present in the spectrum of these conditional variances. We assume that a single baseline is measured using only carrier phase data on L 1. The standard single baseline model is used. Hence, it is assumed that the separation of the two receivers is such that the DD-observables are sufficiently insensitive to orbital uncertainties in the fixed orbits and to differential atmospheric delays. Also we assume that the (m + 1) satellites are continuously tracked during the observational time span of k measurement epochs. This is not an unrealistic assumption for the short time spans in the applications where fast ambiguity estimation is required. For the stochastic model we assume that all observables have equal precision σ φ and that correlation in time and correlation between the channels is absent. Based on these assumptions, it can be shown that the variance-covariance matrix of the DD-ambiguities is given as ( ) Qâ = 1 σ 2 φ λ 2 1 k Ds D s + BQˆb B (8) where λ 1 is the carrier wavelength, D s is the m-by- (m + 1) single-to-double difference operator having satellite s as reference, B is the time-averaged version of the m-by-3 design matrix that captures the doubledifference receiver-satellite geometry, and Q ˆb is the variance-covariance matrix of the least-squares solution of the baseline before fixing. Note that the variancecovariance matrix Qâ equals the sum of a rank-m matrix and a rank-3 matrix. Due to the high precision of the phase observables, the entries of the rank-m matrix are very small. The entries of the rank-3 matrix, on the other hand, are very large when k is small; the precision of the real-valued baseline components is namely rather poor when the observational time span is short. And as we will see, it are these characteristics of the rank-m and rank-3 matrix, that play such a crucial role in the search for the integer least-squares DD-ambiguities. In particular they have a very distinctive influence on the spectrum of conditional variances. In order to make this clear, we will first give an example based on a synthetic variance-covariance matrix. example 1: In this example we will consider a synthetic variance-covariance matrix which has been chosen such that its structure is similar to the actual variance-covariance matrix of the DD-ambiguities. The synthetic variance-covariance matrix is given as the sum of a scaled unit matrix and a rank-2 matrix Qâ = σφ 2 I 3 + (β 1 β 2 β 3 ) (β 1 β 2 β 3 ) with β i R 2 for i = 1,...,3 (9) It is furthermore assumed that the diagonal entries of the rank-2 matrix are all of the same order and significantly larger than the scale factor of the scaled unit matrix. Hence, it is assumed that β 1 β 1 β 2 β 2 β 3 β 3 and σ 2 φ β i β i for i = 1,...,3 (10) For this example the scale factor is chosen as σφ 2 =0.04 and the entries of the rank-2 matrix as [ ] [ ] [ ] 0.000 2.165 1.768 β 1 =,β 2.500 2 =,β 1.250 3 = 1.768 Based on these chosen values the variances resp. sequential conditional variances can be computed. They read i 1 2 3 σâ(i,i) 6.290 6.290 6.292 σâ(i I,i I) 6.290 4.737 0.070 116

From these results we see that there is a relatively large drop in value when going from the second to the third conditional variance. The location of this discontinuity is due to the fact that the second matrix in the sum of (9) is of rank 2 and the size of the discontinuity is due to the differences in size between the entries of the two matrices in the sum of (9). 2 The characteristic behaviour of the conditional variances shown in the example above is also present in the actual variance-covariance matrix of the DD-ambiguities. In the case of the actual variance-covariance matrix of the DD-ambiguities however, the discontinuity is located between the third and fourth conditional variance. This is due to the fact that the second matrix in the sum of (8) is not of rank 2 but of rank 3. In order to proof the presence of the discontinuity, the variance-covariance matrix Qâ is partitioned as [ ] Q11 Q Qâ = 12 (11) Q 21 Q 22 in which Q 11 is of order 3 and Q 22 is of order (m 3). It is assumed that m > 3. Hence, we assume that satellite redundancy is present. The overall redundancy equals m(k 1) 3 with k being the number of epochs. Thus, the overall redundancy equals the satellite redundancy, when the minimum number of two epochs is used. Now in order to proof the presence of the discontinuity in the spectrum, we will first give a first order approximation of the conditional variance-covariance matrix Q 22 1 = Q 22 Q 21 Q 1 11 Q 12 (12) This is the variance-covariance matrix of the conditional least-squares estimators of the last (m 3)-number of DD-ambiguities assuming that the first three DDambiguities are fixed. It can be shown that we have to a first order Q 22 1 1 σφ 2 s D λ 2 2 1 k D 2 s (13) where D 2 s = PDs 2, P = I Ds 1 ( B 1 Ds 1 ) 1 B 1, Ds = (D1 s Ds 2 ) and B = ( B 1 B 2 ). When we compare the corresponding diagonal entries of Q 22 and Q 22 1,andassume that the unconditional variances of the ambiguities do not differ to much, i.e. (Qâ) ii for i = 1,...,m are all of the same order, it follows that (Q 22 1 ) ii (Q 22 ) ii (Q 11 ) jj for j = 1,...,3andi = 1,...,m 3 (14) This shows, since σâ(i,i i,i) = (Q 22 1 ) 11 for i = 4and since for i > 4 the variance of the i-th ambiguity conditioned on the previous (i 1)-number of ambiguities is even smaller than the corresponding diagonal entry of Q 22 1,that σâ(i I,i I) (Q 11 ) jj for j = 1,...,3andi = 4,...,m (15) Figure 1: A GPS spectrum of L 1 conditional variances. We also have, when the receiver-satellite geometry is of reasonable extent, that the difference in value between the first three conditional variances and the corresponding unconditional variances will not be as dramatic as the differences shown in (15). Hence, the important conclusion that can be reached from the above is that σâ(i I,i I) σâ( j J, j J ) for j = 1,...,3andi = 4,...,m (16) This shows that the spectrum of conditional variances contains a large discontinuity when passing from the third to the fourth conditional variance. This discontinuity is largely independent of the receiver-satellite geometry and almost solely a consequence of the presence of satellite redundancy. The location of the discontinuity is due to the three dimensions of the baseline. And the size of the discontinuity is mainly governed by the ratio of the poor precision of the baseline before fixing, and the very good precision of the carrier phase observables. Of course, it may happen in particular cases that there are additional discontinuities in the spectrum of conditional variances. But, they will then be a consequence of the particular receiver-satellite geometry. To illustrate (16) graphically, figure 1 shows an example of a spectrum of conditional variances of the L 1 DD-ambiguities. This example is based on a 7 satellite configuration using single frequency carrier phase data for an observational time span of only one second. The standard deviation of the L 1 carrier phase observables was assumed to be σ φ = 3 mm. The conditional variances of the L 1 DD-ambiguities are expressed in (cycles) 2. Note the logarithmic scale along the vertical axis. The figure clearly shows the large drop in value when passing from the third to the fourth variance. There are three large conditional variances and three extremely small ones. These three small conditional variances are due to the presence of satellite redundancy. The consequence of the above discontinuity for the search of the integer least-squares ambiguities, is as follows. Since the first three conditional variances will usually be of a large order, we see with (6) that the bounds for the first three ambiguities will usually be rather loose. This implies that quite a number of integer triples satisfy these bounds. The likelihood of choosing the cor- 117

χ 2 = 1.0 χ 2 =.727 χ 2 =.632 a 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 a 3 3 3-3 2-3 4-3 1-3 5-2 3 6 2 3-2 4-2 2 5 2 2-4 3-4 2-4 4 - #B 6 8 10 12 14 24 26 34 38 40 42 Figure 2: The ellipsoid of example 2, with direction of view along the shortest eigen-axis. Table 1: The integer triples that are encountered during the search in the two times shrunk ellipsoid of example 2. - stands for no integer found. The last column #B gives the cumulative number of bounds that are computed (two times the number of intervals for all three ambiguities that are investigated). variance-covariance matrix as used in example 1. The least-squares estimates of the ambiguities are given as â 1 = 3.0,â 2 = 3.0 andâ 3 = 5.51. The three intervals on the basis of which the search is performed are given as (a 1 â 1 ) 2 σâ(1,1) χ 2 (a 2 â 2 1 ) 2 σâ(2 1,2 1) χ 2 [1 (a1 â1)2 ] σâ(1,1) χ 2 (a 3 â 3 2,1 ) 2 (17) σâ(3 (2,1),3 (2,1)) χ 2 [1 (a1 â1)2 σâ(1,1) χ 2 (a2 â2 1)2 σâ(2 1,2 1) χ 2 ] Figure 3: The ellipsoid of example 2, with direction of view along the longest eigen-axis. rect integer triple is therefore low, even when we follow the principle of choosing the integer nearest to the leastsquares estimate. But, this implies, when we start working with the fourth bound, which is very tight due to the steep decrease in value of the conditional variances, that we have a high likelihood of not being able to find an integer candidate that satisfies the fourth bound. The potential of halting is therefore very significant when one passes from the third to the fourth bound. As a consequence a large number of trials is required, before one is able to move on to the next bound. To illustrate numerically how the discontinuity in the spectrum of conditional variances affects the search, we again consider our example of the synthetic variancecovariance matrix. example 2: This example is based on the same The constant χ 2 is chosen to be equal to one. In figures 2 and 3 the ellipsoid is depicted in which the search for the integer triples is performed. In figure 2 the direction of view is along the shortest eigen-axis, and the eigenellipse belonging to the two longest eigen-axes as well as the perpendicular projection of the ellipsoid onto the 1-2 plane are depicted. In figure 3 where the direction of view is along the longest eigen-axis, this projection is also depicted together with the ellipse belonging to the two shortest eigen-axes. The grid points in this projected ellipse are the integer-tuples (a 1, a 2 ) that satisfy the first two bounds of (17). The with these integer-tuples corresponding intervals for a 3 and the a 3 -integers within them are depicted as resp. the little bars, and the dots inside the eigen-ellipses. The large values of the conditional variances of a 1 and a 2 compared to those of a 3 (see example 1) lead to intervals for a 1 and a 2 that are significantly larger than the intervals for a 3. Moreover, since the intervals for a 3 are small compared to the grid spacing, there is a high probability that there will be no a 3 - integer within them. In the search process the integers are kept as close as possible to their corresponding conditional estimates. We start searching the ellipsoid with a value for χ 2 of 118

(a) Single baseline. (b) Multi baseline. Figure 4: The spectrum of L 1 and L 2 DD-ambiguities. 1.0. We need to investigate six intervals to find the integer triple (2,3,6) with a distance of.727 from â. Setting χ 2 to this value shrinks the ellipsoid, and a second search follows. Again following the strategy to keep as close as possible to the conditional estimates, but disregarding the intervals which in the earlier search turned out to be empty, we find after investigating three intervals, the triple (2,2,5) with a distance of.632. Again the ellipsoid is shrunk, and after investigating four more intervals we may conclude that (2,2,5) is indeed the integer triple that minimizes (3). In table 1 the integer triples that are encountered duringthe search, and the number of bounds that need to be computed, are given. 2 3. On the GPS spectra of ambiguity conditional variances 3.1 Spectra based on carrier phase data only As it was proven in the previous section for the single baseline model, the spectrum of conditional variances of the L 1 DD-ambiguities exhibits a distinct discontinuity when passing from the third to the fourth conditional variance. It is due to this discontinuity that the search for the integer least-squares solution suffers from the potential problem of halting. In the present section it will be shown that a similar discontinuity is also present in case one works with dual-frequency carrier phase data. This will be shown for both the single baseline model as well as for the multi baseline model. Apart from the spectra of the DD-ambiguities, also the spectra of the wide-lane ambiguities and narrow-lane ambiguities will be shown. Although it is possible to proof for each of the above scenarios the location and size of the discontinuity, we will restrict ourselves in this section by only illustrating, through graphical examples, the characteristics of the spectra of conditional variances. All the examples are based on the same configuration of 7 satellites, with a time span of 1 second, as used in the previous section. Figure 4 shows the spectrum of conditional variances for both a single baseline as well as a multi baseline case. In both cases dual-frequency carrier phase data was used. The L 1 and L 2 carrier phase observables were assumed to be equally precise, having a standard deviation of σ φ = 3 mm. Figure 4(a) shows the single baseline case. The figure clearly shows the presence of the discontinuity. As in the single frequency case (compare with figure 1), the discontinuity is found when passing from the third to the fourth conditional variance. Hence, the number of large conditional variances is again equal to three. The number of very small conditional variances however, has now been increased from three to nine. The shape of the spectrum as shown in figure 4(a) is typical for single baseline dual-frequency carrier phase data. That is, for the dual-frequency case, the number of very small conditional variances will always be equal to 2m 3, when (m + 1)-number of satellites are tracked. For the single frequency case this number equals m 3. Figure 4(b) shows the spectrum for the multi baseline case. The number of baselines equals three. It is remarked that these three baselines were not determined separately. They were rigorously adjusted for simultaneously from the carrier phase data. The figure again shows a discontinuity. This time however, it is found when passing from the ninth to the tenth conditional variance. For the multi baseline dual-frequency case, one will generally have (2m 3)b-number of small conditional variances, with b being the number of baselines. It is well known that certain linear combinations of the GPS-observables play a prominent role in the problem of ambiguity fixing. Well-known examples are the narrow-lane, the wide-lane and the extra wide-lane combinations, see e.g. [2-4]. In particular wide-laning techniques have proven to be very successful. It is therefore of interest, assuming that dual frequency carrier phase data is available, to study the spectra of conditional variances of the ambiguities that correspond with these linear combinations. In figure 5 the single baseline spectra of both the widelane and narrow lane ambiguities are shown. In order to facilitate a comparison with the single frequency case, the spectrum of the L 1 DD-ambiguities is also shown (dotted curves). Figure 5(a) shows the spectrum for the wide-lane ambiguities. Again we observe a discontinuity when passing from the third to the fourth conditional variance. The size however, of the discontinuity in the wide-lane spectrum is smaller than that of the L 1 spectrum. That is, the three large conditional variances of the wide-lane ambiguities are smaller than those of the L 1 119

(a) Wide-lane. (a) The code-aided L 1 spectrum. (b) Narrow-lane. (b) The code-aided L 1 / L 2 spectrum. Figure 5: The spectra of the wide-lane (L 5 )andthe narrow-lane ambiguities. ambiguities, and the three small conditional variances of the wide-lane ambiguities are larger than those of the L 1 ambiguities. In other words, the wide-lane spectrum is flatter than that of the L 1 spectrum. This shows, with reference to our earlier discussion on the search for the integer least-squares solution, that the search for the integer wide-lane ambiguities will be less hindered by the potential problem of halting than the search for the integer L 1 DD-ambiguities. Note however, that although the three large conditional variances of the wide-lane ambiguities are very much smaller than those of the L 1 DDambiguities, the difference between the third and fourth conditional variance of the wide-lane ambiguities is still significant. Figure 5(b) shows the spectrum for the narrow-lane ambiguities. Again we observe a discontinuity when passing from the third to the fourth conditional variance. Note however, that the conditional variances of the narrow-lane ambiguities are all larger than those of the L 1 DD-ambiguities. 3.2 Spectra based on both carrier phase and code data It will be clear that the search for the integer least-squares solutionof the ambiguities would benefit from a decrease in size of the discontinuity. One way to achieve this would be to make use of much longer observational time spans. By this we do not mean the increase of the number of observational epochs k, but instead the increase in time interval between the first and last observational epoch. Hence, k can still be equal to two. The time inter- (c) The code-aided wide-lane spectrum. Figure 6: Code-aided spectra of conditional variances. val however, has to be such that a sufficient change in the relative receiver-satellite geometry has been achieved. Although this is one way of flattening the spectrum of conditional variances, it is not a very attractive option if one aims at real-time or near real-time applications of GPS. An alternative option is therefore to include sufficiently precise code observations into the model. In this section it will be shown what the inclusion of the code observations brings in terms of decreasing the size of the discontinuity. The standard deviation of the code observables was chosen to be σ P = 30 cm for both frequencies. It is recognized that this precision estimate is rather optimistic. Code observables of such a precision will probably only be available for civilian users when AS is turned off. The chosen value of σ P allows us however to indicate the lower bounds of the code-aided spectra. Figure 6 shows the code-aided spectra of resp. the L 1 DD-ambiguities, the L 1 /L 2 DD-ambiguities and the wide-lane ambiguities. In all three cases, the correspond- 120

ing spectra based on carrier phase data only, are also shown (dotted curves). As can be clearly seen from these figures, the inclusion of the very precise dual-frequency code observables results in a considerable improvement of the spectra. In all three cases, the first three large conditional variances of the spectra are brought down to much smaller values. The very small conditional variances in the spectra however, are hardly affected. The net result is therefore that the inclusion of the very precise code observables results in quite a lower and more flattened spectrum. 4. On the use of a priori single-channel ambiguity transformations 4.1 Single-channel ambiguity transformations As it was remarked earlier, certain linear combinations of the GPS-observables play a prominent role in the problem of ambiguity fixing. In the study of integer linear combinations however, one usually seems to work from two phase observables, namely on L 1 and on L 2,towards one single derived phase observable. This derived phase observable is then chosen such that it has certain desirable properties. The wide-lane phase observable, of which the spectrum of its ambiguities was shown in figure 5(a), is a prime example in this respect. It will be clear however, that the original two phase observables contain more, or at least the same, information as the single phase observable derived from them. So, why not start from two and end with two? It is this observation which has led to the study of [5], in which integer linear combinations are identified that preserve the information content of the original two phase observables. Let 1 and 2 be the DD-carrier phase observables on L 1 and L 2 expressed in units of range. Their wavelengths are denoted resp. as λ 1 and λ 2. Instead of working with 1 and 2, one might as well work with the derived phase observables αβ = αλ 2 βλ 1 1 + 2 αλ 2 + βλ 1 αλ 2 + βλ 1 γβ = γλ 2 δλ 1 1 + 2 γλ 2 + δλ 1 γλ 2 + δλ 1 (18) provided of course that the transformation is invertible and thus αδ γβ 0 holds. If we apply this transformation to the original carrier phase observation equations 1 = ρ + λ 1 a 1 + ε 1 2 = ρ + λ 2 a 2 + ε 2 (19) where ρ is the DD-form of the range from receiver to satellite, a i is the integer DD L i phase ambiguity and ε i is the L i measurement noise plus remaining unmodelled errors, we get the transformed observation equations αβ = ρ + λ αβ a αβ + ε αβ γδ = ρ + λ γδ a γδ + ε γδ (20) where λ αβ, λ γδ and a αβ, a γδ are resp. the wavelengths and ambiguities of αβ and γδ,andwhereε αβ, ε γδ are the corresponding noise terms. The original and transformed ambiguities are related as [ ] [ ][ ] aαβ α β a1 = (21) a γδ γ δ a 2 For the purpose of aiding the ambiguity fixing process, one may now choose, under certain restrictions, suitable values for the scalars α, β, γ and δ in order to obtain ambiguities with certain desirable properties. As it was shown in [5], the restrictions are that both the transformation matrix of (21) as well as its inverse need to have entries which all are integer. These restrictions then guarantee that the integerness of the ambiguities remains preserved. This shows for instance, that it is not allowed to pair the wide-lane ambiguity with the narrow-lane ambiguity. In (18), the phase observables have been transformed. This however, is not really necessary. If the sole purpose is to work with ambiguities other than the original DD-ambiguities, one can stick to the original phase observables 1 and 2 and simply consider (21) as a reparametrization of the ambiguities. 4.2 Some single-channel transformed spectra It is generally believed that for the purpose of ambiguity fixing, only those integer linear combinations are of value that produce a phase observable which has a relatively long wavelength, a relatively low noise behaviour and a reasonable small ionospheric delay. And indeed, these properties are beneficial to the integer ambiguity fixing process. One should recognize however, that a more complete picture is obtained once one knows how for a particular case, the combination of carrier phase noise and chosen functional model, propagates into the variance-covariance matrix of the ambiguities. Hence, the choice for certain linear combinations should not so much be made on the basis of only phase noise and wavelength, but more on how the variance-covariance matrix of the ambiguities is affected by the choice. This then also allows one to choose integer linear combinations that have a favourable influence on the effect the receiver-satellite geometry has on the noise behaviour of the estimated ambiguities. In order to show how one can influence the spectrum of conditional variances through the use of (21), the following three single-channel ambiguity transformations were chosen as an example, [ 1 1 Z1 = 0 1 ], Z 2 = [ 4 5 1 1 ] [ ] 60 77, Z3 = 7 9 It is easily verified that these transformations are indeed admissible. Note that Z1 transforms the original DD-ambiguities into wide-lane and L 2 ambiguities. The above transformations were not chosen on the basis of a rigorous optimality criterion. Instead, they were found on quite an ad hoc basis. For our 7 satellite example they seemed to be the ones that to a certain extent could improve the spectrum of conditional variances. The results are shown in figure 7. 121

Figure 7: The original and the single-channel transformed spectra; L 1 /L 2 -spectrum = full curve, Z 1 - spectrum = dashed curve, Z 2 -spectrum = dotted curve, Z 3 -spectrum = dash-dotted curve. The figure clearly shows that all of the first three large conditional variances in the transformed spectra are smaller than the ones of the original spectrum. We also observe that the large discontinuity which is present in the original spectrum gets reduced in the transformed spectra. In fact it is absent in the Z 3 -spectrum. This spectrum is reasonably flat in the beginning, but drops when one passes the ninth conditional variance. 5. On the least-squares ambiguity decorrelation adjustment 5.1 Multi-channel ambiguity transformations In the previous section we identified the class of singlechannel ambiguity transformations. It was shown how some of these a priori chosen ambiguity transformations affect the spectrum of conditional variances. Although some improvement could be seen, it is clear that the usage of these single-channel ambiguity transformations is limited and that one cannot expect to obtain results that are overall satisfactory. Moreover, the transformations used were obtained on quite an ad hoc basis. In this section it will be discussed how these single-channel ambiguity transformations can be generalized. In the previous section the four entries of the transformation (21) were assumed to be the same for all channels. This however, is not really necessary. One can in principle choose different sets of values for the different channels. In this way one can accommodate to the different entries in the variance-covariance matrix of the ambiguities. It also seemed in the previous section that the transformation was restricted to the dual-frequency case. That is, a 1 of (21) was assumed to be an L 1 DDambiguity and a 2 an L 2 DD-ambiguity. But this is not necessary either. The transformation can namely also be used in case one is dealing with L 1 data only. In that case, a 1 and a 2 are simply L 1 DD-ambiguities of two different channels. This observation also suggests a generalization to more than two channels. And indeed, there is no reason for restricting the order of the transformation to two. This then brings us to the multi-channel ambiguity transformations. And as it was shown in [5], the necessary and sufficient conditions for preserving the integerness of the ambiguities are again that both the ambiguity transformation and its inverse should have entries which all are integer. The purpose of having multi-channel integer linear combinations available is to be in a better position for the problem of ambiguity fixing. That is, members from the identified class of ambiguity transformations can now be used to aid the ambiguity fixing process and in particular be used to reduce the effect that a slowly changing receiver-satellite geometry has on the condition of the ambiguity variance-covariance matrix. A method to achieve this has been introduced in [1]. The idea is to construct an ambiguity transformation that aims at a decorrelation of the least-squares ambiguities. This method of the least-squares ambiguity decorrelation adjustment (LAMBDA), flattens the spectrum of conditional variances and returns new ambiguities that are largely decorrelated and that show a dramatic improvement in precision. First results of the method were presented in [6] and [7]. In the next section it will be shown how the method performs on the spectra which were discussed earlier in section 3. 5.2 Results obtained with the LAMBDA method Let Z be an admissible ambiguity transformation of order m. The original m-vector of ambiguities, its least-squares estimate and its variance-covariance matrix can then be transformed as z = Z a, ẑ = Z â, and Qẑ = Z Qâ Z. The original search space (5) also transforms. With this transformed search space, the search for the integer least-squares solution can be performed in exactly the same way as for the original search space. Once the integer least-squares solution ž has been found, the corresponding integer estimate ǎ of the original ambiguities follows from invoking ǎ = (Z ) 1 ž. When the ambiguity transformation Z is constructed following [1], the spectrum of conditional variances will be largely flattened implying that the search can be performed in a more efficient manner. In order to demonstrate this, we will first continue with our example. example 3: The decorrelating ambiguity transformation which was found for the synthetic variance-covariance matrix of example 1 reads Z = 1 3 3 1 2 3 0 1 1 (22) It is easily verified that this transformation is indeed admissible. When Z of (22) is used to transform the original variance-covariance matrix Qâ into Qẑ, the variances and conditional variances of ẑ i, i = 1,...,3, follow as i 1 2 3 σẑ(i,i) 3.073 1.601 0.506 σẑ(i I,i I) 3.073 1.389 0.489 122

much more extreme values for the elongation and the ambiguity decorrelation number, and the reduction is considerably larger. The corresponding values for the original ambiguities follow from ǎ = (Z ) 1 ž as ǎ = (2, 2, 5), which is of course identical to the solution found in example 2. 2 In order to demonstrate the performance of the method on actual data, four different spectra were considered: the L 1 spectrum, the L 5 spectrum, the L 1 /L 2 spectrum and the dual frequency code-aided L 1 /L 2 spectrum. The results are shown in figure 9. Figure 8: The ellipsoid of example 2, after the Z- transformation. χ 2 = 1.0 χ 2 =.632 z 1 z 2 z 3 z 1 z 2 z 3 11 13 3 11 13-11 14-10 14-10 13 - #B 12 14 18 20 Table 2: Integer triples encountered during the search in the transformed ellipsoid, see also table 1. Note that the relatively large drop in value which was present in the original spectrum has now been reduced in size in the transformed spectrum. Also note that the new ambiguities are more precise than the original ones. The transformed ambiguities are also less correlated and their search space is less elongated. The decorrelation numbers [6] read râ = 0.092 and rẑ = 0.916, and the elongations read eâ = 19.655 and eẑ = 2.666. The decorrelation number is defined as the square-root of the determinant of the correlation matrix and elongation is defined as the square-root of the condition number of the variance-covariance matrix. In figure 8 the transformed search space is depicted. It is centred at ẑ 1 = 10.53, ẑ 2 = 13.53, ẑ 3 = 2.51, which follows from ẑ = Z â. The direction of view is parallel to that of figure 3. The smaller elongation of the transformed search space can be clearly seen; the intervals for the third ambiguity z 3 are now in general larger than the grid spacing, and we see that several intervals contain more than one integer. Only two intervals fail to contain an integer. Applying the same search strategy as used before, we now find immediately the integer least-squares solution ž = ( 11, 13, 3). After shrinking the ellipsoid we find no more integer triples, so the search is completed. In table 2 the search process is schematized. Comparing it with table 1 learns us that the number of bounds that have to be computed reduces from 42 to 20. In practice, we do encounter It is clear from figure 9 that our method indeed succeeded in lowering and flattening the spectrum of conditional variances in all four cases. In each case the large conditional variances have been pushed down to much smaller values, thus resulting in a spectrum in which the conditional variances are all of about the same small order. And the low levels obtained for each of these four cases can be considered very dramatic indeed. In the sequence of the four cases shown in figure 9, the levels of the transformed conditional variances are of the order 5, 1, 0.03 and 0.003 (cycles) 2. The improvement in the spectrum is largest for the L 1 /L 2 spectrum. This large improvement is due to the fact that the number of very small conditional variances in the original spectrum is larger than those in the L 1 and L 5 spectrum. Here we clearly see the effect of redundancy. In the L 1 case, only satellite redundancy is present. With 7 satellites, this results in three small conditional variances. A similar situation holds for the L 5 case. In this case namely the two available frequencies are used to construct one single wide-lane frequency. In the L 1 /L 2 case however, full use is made of the presence of a second frequency. With 7 satellites, the combination of dual-frequency and satellite redundancy results in a total of nine very small conditional variances. Although the improvement in the spectrum is largest for the L 1 /L 2 case, the best result is of course obtained for the codeaided L 1 /L 2 spectrum. Despite the fact that the original conditional variances are already quite small, our method still succeeds in bringing these values down to significantly lower values. In support of the results shown in figure 9, the elongations, decorrelation numbers and variances before and after the transformationare given in table 3. It shows that the highly elongated ambiguity search spaces are transformed into much more sphere-like search spaces. It also shows that the transformed ambiguities are much less correlated than the original ambiguities. A full decorrelation corresponds with a decorrelation number of one. We also observe the dramatic improvement in precision. Note that the transformed variances and their conditional counterparts are of about the same order. This again accentuates the presence of decorrelation. 123

eâ eẑ -log10 râ -log10 rẑ L 1 23278 2.4 11.5 0.22 L 5 2887 3.1 8.8 0.26 L 1 /L 2 26780 4.2 33.6 0.94 L 1 /L 2 234 5.8 14.9 1.00 (code-aided) σ 2 â (max) σ 2 â (min) σ 2 ẑ (max) σ 2 ẑ (min) (a) L 1 L 1 91825.77 9630.92 7.424 4.053 L 5 1989.36 234.72 1.399 0.807 L 1 /L 2 40812.89 2923.88 0.061 0.027 L 1 /L 2 2.39 0.31 0.005 0.003 (code-aided) Table 3: Elongation (e), decorrelation number (-log10 r) and maximum and minimum variance (σ 2 (max), σ 2 (min)) of the original and transformed ambiguities. (b) L 5 (c) L 1 /L 2 (d) code-aided L 1 /L 2 Figure 9: The original and transformed spectra of conditional variances. References [1] Teunissen, P.J.G. (1993): Least-Squares Estimation of the Integer GPS Ambiguities. Invited Lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing, China, August 1993, 16 pp. Also in: LGR Series, No. 6. [2] Wübbena, G. (1989): The GPS Adjustment Software Package - GEONAP - Concepts and Models. Proceedings 5th Int. Geod. Symp. on Satellite Positioning. Las Cruces, New Mexico, 13-17 March 1989, pp. 452-461. [3] Goad, C.C. (1992): Robust Techniques for Determining GPS Phase Ambiguities. Proceedings 6th Int. Geod. Symp. on Satellite Positioning. Columbus, Ohio, 17-20 March 1992, pp. 245-254. [4] Mervart, L., G. Beutler, M. Rothacher, U. Wild (1994): Ambiguity Resolution Strategies using the Results of the International GPS Geodynamics Service (IGS), Bulletin Geodesique, 68: 29-38. [5] Teunissen, P.J.G. (1993): The Invertible GPS Ambiguity Transformations. Delft Geodetic Computing Centre (LGR), 9 p.. Accepted for publication in Manuscripta Geodaetica. [6] Teunissen, P.J.G. (1994): A New Method for Fast Carrier Phase Ambiguity Estimation. Proceedings IEEE Position Location and Navigation Symposium PLANS94, Las Vegas, 11-15 April 1994, pp. 562-573. [7] Jonge, P.J. de, C.C.J.M. Tiberius (1994): A New GPS Ambiguity Estimation Method based on Integer Least-Squares. Proceedings 3th Int. Conf. on Differential Satellite Navigation Systems DSNS94, London, 18-22 April 1994, paper 73, 9p. 124