SIGMA-F: Variances of GPS Observations Determined by a Fuzzy System

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1 SIGMA-F: Variances of GPS Observations Determined by a Fuzzy System A. Wieser and F.K. Brunner Engineering Surveying and Metrology, Graz University of Technology, Steyrergasse 3, A-8 Graz, Austria Keywords. GPS phase observations, stochastic model, fuzzy control Abstract. The determination of a realistic variance model is still an important issue in GPS data proc- set up using their measured signal-to-noise ratios (C/N ). We present an adaptive variance model for GPS carrier phase observations (SIGMA-F) tobe used with least squares estimation (LS). It is based on a fuzzy system which combines robust estimation and data quality assessment. The processing results obtained are more stable than the purely robust estimates, and they are more reliable than LS estimates using purely C/N - based weights. We present results using actually measured GPS data from two different projects. We show that SIGMA-F successfully mitigates biases with kinematic processing of short baselines. essing. An a-priori model of the variances of the double differenced phase observations is ξˆ ( A' V A) A' V y = (3) ~ e = Aξ ˆ y (4) E{ }... expectation operator D{ }... dispersion operator y... vector of reduced observations e... vector of observation errors A... Jacobi-matrix ξ... vector of unknown parameters σ... a-priori variance factor V... cofactor matrix of observations (regular) ξˆ... vector of estimated parameters ~ e... vector of predicted residuals Introduction The functional model for processing GPS phase observations is well understood, but the establishment of an appropriate stochastic model (variancecovariance matrix) is still an issue. This is mainly due to physical correlations and unmodeled systematic effects, e.g., signal distortion effects. In this paper we present the SIGMA-F variance model which covers random observation noise, algebraic correlations, and the automatic detection and mitigation of outliers. Currently, we do not take physical correlations into account, but the model can be extended if necessary. The processing of the (double differenced) GPS phase observations is based on a linearized Gauss-Markov model, eq. () and (), and the leastsquares formalism (LS), eq. (3) and (4). E{ y } = y + e = Aξ () E{ e} =, D{ e} = D{ y} = σ V () The cofactor matrix of the observations is computed using the newly developed SIGMA-F variance model. If there is any evidence that an observation is an outlier or it is contaminated by unmodeled effects, its variance is inflated. Consequently, a deteriorating effect on the processing results is suppressed. The decision about variance inflation is based on the predicted residuals and on the measured signal-to-noise ratio (C/N ), which is an independent signal quality indicator. The combination of both quantities requires simultaneous handling of different types of uncertainty. This is accomplished by using a fuzzy system as kernel of SIGMA-F. The concept of SIGMA-F The SIGMA-F variance of an observation incorporates three contributions, see eq. (5): an a-priori variance model, PREPRINT of paper to be published in the Proceedings of the IAG Scientific Assembly, Budapest, -9-4

2 with variance inflation as a function of the signal quality, and variance inflation as a function of the predicted residuals. σ σ f α, ) g( β, e~ ) (5) k ) = ) ( k ) i k ) k ) k ) k ) α, β (6) α / 3 f ( α, ) = (7) æ ~ ei ö g( β, e~ = + ç i ) β exp (8) è c σ i ø σ k )... variance of i-th observation in k- th iteration σ ) f ( ), g( ) ~ e α c i k ) β k ), k )... a-priori variance of i-th observation... variance inflation functions... difference between measured and expected C/N of i-th observation (db-hz)... predicted residual after (k )-th iteration... influence quantities... constant (we use c=3) The a-priori variance model describes the random noise. The corresponding value σ is a lower ) bound to the final variance of the observations. We use the SIGMA-ε variance model, Hartinger and Brunner (999), which computes σ by a function ) of the measured C/N values, and is closely related to the satellite elevation. Anon-zerodifference between the measured C/N valueandtheexpectedvalueforasignalarriving at the same elevation (template value) often indicates signal distortion. This information has beenexploitedbythesigma- variance model, Brunner et al. (999). It is incorporated in SIGMA- F by means of the variance inflation function f, see eqs. (5) and (7). f increases monotonously with and α. It yields the minimum value f= only if = or α=. So, if is close to zero, the a-priori variance will not be increased significantly by f, butif is high, an increase is indicated. The coefficient /3 in eq. (7) allows a maximum variance inflation of 4 with very poor signal quality ( = db-hz), which we have found to be sufficient. However, the single C/N value cannot describe all the physical phenomena of signal scattering and interference. So, the C/N is an intrinsically imprecise quantity, and may not always correctly indicate bad observations. The influence quantity α then reduces (or suppresses) the variance inflation, in case high values are contradicted by other quality indicators. The function g of eq. (8) provides variance inflation if the corresponding residual indicates an outlier. This function is similar to the Danish Method of robust estimation, Krarup et al. (98). Again, there is a possibility to reduce the actual amount of variance inflation by means of an influence quantity, β. This may be useful if the residuals are considered unreliable, e.g., in case of poor redundancy and significant contamination of the observations. Since the computation of the residuals requires the stochastic model to be established, and the stochastic model is a function of these residuals, eq. (5) is solved iteratively. Initially the residuals are computed using the a-priori variance model only. Then the influence quantities are determined, the variances are updated, and the computation of the residuals is repeated. This iterative process terminates once the maximum change of the variances between two iteration steps is below a suitable threshold, e.g..%. SIGMA-F combines the idea of traditional robust estimation with quality assessment based on the measured signal-to-noise ratio. The key parameters are the influence quantities α and β. They determine the extent to which different information is used for variance inflation. The decision is mainly based on a test statistic for outlier detection and two quantities derived from the C/N. These indicators represent very different types of uncertainties which may be conveniently handled by a fuzzy system. So, thecoreofsigma-f is a fuzzy controller which computes appropriate values of α and β. As stated before, currently we do not account for the physical correlations. However, we handle the algebraic correlations properly: if a reference satellite is selected for all double differenced phase observations (DD) of an epoch, then the covariance matrix has a special structure, i.e., all the offdiagonal elements depend on the variances of the reference satellite observations only, Wieser and Brunner (). With careful reference satellite selection, an indicated need for variance inflation may be attributed to the rover-satellite observations and affects only the diagonal entries of the covariance matrix. So, we compute the full covariance PREPRINT of paper to be published in the Proceedings of the IAG Scientific Assembly, Budapest, -9-4

3 HUGE EARLY LATE T Iteration MaxDelta Alpha MEDIUM MaxDDelta Beta Fig.: SIGMA-F - fuzzy variables, terms and membership functions. matrix of the DD using the a-priori variance model only once. There is no need to modify its offdiagonal entries thereafter. If a DD is subject to variance inflation, the corresponding diagonal element of the VCM is replaced. A more detailed description of the procedure is given by Wieser (a). 3 The fuzzy controller A fuzzy controller maps numeric input values into numeric output values. However, as opposed to a mathematical transfer system, it operates with linguistic variables and terms, e.g., "high" and "low", rather than with numeric variables. The transfer properties are defined by rules. A review of the well established theory of fuzzy systems is outside the scope of this paper. However, we have found Cox (998) useful as an introduction, and Pedrycz and Gomide (998) as a comprehensive text on fuzzy logic and fuzzy set theory. An overview of fuzzy systems with a focus on GPS data processing applications is given by Wieser (a). This section rather focuses on the actual input variables, rules and output variables of the fuzzy kernel of SIGMA-F. The numeric input variables of the fuzzy controller are: the value of a test statistic T i for outlier detection the maximum value max of, i.e., the worst signal quality, of all four contributions to a DD observation the maximum max of the between-station difference of, i.e., the maximum difference in signal quality between the two stations of a baseline the iteration count k The test statistic T i is computed from the predicted residuals, the cofactor matrix of the observations, the a-priori variance factor, and the cofactor matrix V~ of the predicted residuals, see Teunissen e (998): η[i] T η' V ~ e [ i] i = (9) σ η' [ i ] V V~ ev η[ i] is a unit vector with as the i-th entry. If the observation errors are normally distributed with first and second moments according to eq. (), T i has a standard normal distribution. If the i-th observation is an outlier, then this is indicated by a high value of T i. The interpretation of T i as or is resembled by the fuzzy set representation of the variable T, Fig.. The fuzzy set representations of the other variables are also shown in Fig.. Thecoreofafuzzycontrolleraretherules which describe linguistically the relations between PREPRINT of paper to be published in the Proceedings of the IAG Scientific Assembly, Budapest,

4 input and output variables. Tab. shows the fuzzy rules of SIGMA-F. These rules are based on our experience in interpreting the input parameters, and reflect the way, we would decide about variance inflation manually. # Rule Table : Fuzzy rules of SIGMA-F. If Iterationis EARLY then Alpha is and Beta is. If Iteration is EARLY and T is then Alpha is. 3 If Iteration is LATE then Alpha is. 4 If Iteration is LATE and T is then Beta is. 5 If Iteration is LATE and T ishuge then Alpha is and Beta is. 6 If Iteration is LATE and T is and Max- Deltais and MaxDDelta is then Beta is MEDIUM. 7 If Iteration is LATE and T is and Max- Delta is then Beta is. 8 If Iteration is LATE and T is and MaxDDelta is then Beta is. Note: The rules with multiple consequents ( and ) are evaluated separately for each output variable. The selected rules, the number of terms, i.e., possible values of the fuzzy variables, and the shape of the membership functions are design problems of a fuzzy system. They are usually determined heuristically. Different experts would perhaps come to different terms, different membership functions and different collections of rules. However, fuzzy systems are quite tolerant with respect to approximation, Cox (998, p.). So, these different systems yield similar results, if all experts have captured the main parameters and relations underlying the problem. The rules show that during the initial iterations, possible variance inflation is mainly driven by. Towards the end of the iterations usually 5 if there are outliers and less otherwise the residuals take over. This behavior is motivated by the fact that outliers must stand out by the (normalized) residuals in order to be clearly identified. However, an initial down-weighting of suspect observations may help to compute good residuals, and may therefore support the robust estimation part of SIGMA-F in correctly identifying the outliers. The evaluation of all rules yields a fuzzy set for each of the output variables Alpha and Beta. Finally, distinct numerical values of α and β are computed from these fuzzy sets by centroid defuzzification, see Cox (998), which facilitates the output values to vary smoothly between and. North [mm] - North [mm] - East[mm] - East[mm] - - : : - : : Fig.: LS and SIGMA-ε, epoch-by-epoch coordinate Fig.3: LS and SIGMA-F, epoch-by-epoch coordinate PREPRINT of paper to be published in the Proceedings of the IAG Scientific Assembly, Budapest,

5 North [mm] - North [mm] - East [mm] - -4 East [mm] : 8: 9: 7: 8: 9: Fig.4: LS and SIGMA-ε, epoch-by-epoch coordinate Fig.5: LS and SIGMA-F, epoch-by-epoch coordinate 4 Application of SIGMA-F Slim obstacle Fig. shows the results of the epoch-to-epoch L processing of a short baseline (baseline length 3m). The coordinates of the rover station have been estimated using a standard LS algorithm and the SIGMA-ε variance model. The ambiguities have been fixed in advance using the observations of the whole session, and are therefore not part of the GMM. A slim obstacle close to the rover antenna causes diffraction effects which distort the time series of the estimated east coordinate significantly. The time series show an apparent site motion with an amplitude of approximately mm, while the site is in fact static. Fig. 3 shows the corresponding processing results obtained from epoch-by-epoch LS processing using the SIGMA-F variance model. Clearly, the bias of the east coordinate is almost completely removed. Generally, the gain in precision and accuracy is most pronounced for kinematic processing or short static sessions, because for sufficiently long session times the signal distortion effects are mitigated by averaging. There are a few epochs left, in which SIGMA- F is not successful. SIGMA-F currently uses the data of a single epoch to compute the variances of this epoch. The process may sometimes fail therefore if the bias is small and the redundancy poor. When using SIGMA-F in combination with a Kalman Filter, the predicted residuals (of the observations of one epoch) will not be severely contaminated by a few outliers, and the results should improve. The height variations shown in Fig. are mainly due to noise amplification by geometry. These effects cannot be reduced by variance inflation, see Fig. 3. So, in this example, SIGMA-F does not improve the height results. Building Figs. 4 and 5 show the epoch-by-epoch processing results of another short baseline ( m). A building in the vicinity of the rover antenna (site EXT) causes signal diffraction affecting the signals of one satellite. The time series of the standard processing results is contaminated by these signal distortion effects from 7: to 7:3. Due to the receiverobstacle-satellite geometry, again mainly the east coordinate is affected. The maximum bias exceeds mm in this case. The strong offsets of the solutions from 8:5 to 8:3 are due to erroneously fixed ambiguity parameters corresponding to a low elevation satellite with several loss-of-lock occurrences. Using the SIGMA-F variance model, the biases are successfully mitigated, Fig. 5. This is also confirmed by the standard deviations and ranges of thetimeseries,givenintab..sigma-f mitigates PREPRINT of paper to be published in the Proceedings of the IAG Scientific Assembly, Budapest,

6 not only signal distortion effects but also biases caused by other unmodeled effects (e.g. erroneous ambiguity fixing) provided these effects are significant with respect to the random noise level. With a moderate percentage of outlying or contaminated observations, SIGMA-F also increases the reliability of the solutions. This may be demonstrated using the success-rate computed by a quality assessment of the results, Wieser (b). Table : Standard deviation and range computed from epoch-by-epoch coordinate results of rover site EXT [mm]. standard deviation range SIGMA: ε F ε F North 4 43 East 9 93 Up Conclusion In this paper, we have presented SIGMA-F, a variance model for robust estimation, which automatically detects and compensates unmodeled effects like signal distortions. The model combines the idea of traditional robust estimation based on the residualswithanempiricalc/n based variance model. The resulting variance inflation approach is well suited for use in static and kinematic processing of GPS data. However, its application is generally more beneficial for kinematic processing. A fuzzy systems approach has been chosen to handle the uncertainties represented by the stochastics of the residuals and the imprecision of the C/N values. The main advantage of such systems with respect to traditional stochastic approaches is their ability to handle different types of uncertainty simultaneously. GPS data obtained from two different projects were used to investigate the performance of the algorithm. In both cases, a part of the observations is contaminated by significant signal distortion effects. The positioning results obtained from kinematic processing may be biased by a few cm if these effects are not taken into account. These biases are almost completely removed by SIGMA-F. These examples are characteristic for situations with strong and distinct signal distortion, which affects a small number of observations only, i.e., certainly less than 5%. Such situations may be provoked for instance by a small obstacle close to a GPS antenna. However, if signal distortions affect most of the observations, this can not be compensated by variance inflation. SIGMA-F will not perform superior to any other variance model in these cases. In many practical situations only a small percentage of the observations are affected significantly. In these situations the newly developed variance model increases the precision and the reliability of the results. References Brunner FK, Hartinger H, Troyer L (999) GPS Signal Diffraction Modelling: the stochastic SIGMA- Model. Journal of Geodesy 73: Cox E (998) The Fuzzy Systems Handbook. nd edn, Academic Press, San Diego Hartinger H, Brunner FK (999) Variances of GPS Phase Observations: The SIGMA-ε Model. GPS Solutions /4: Krarup T, Juhl J, Kubik K (98) Götterdämmerung over least squares adjustment. 4 th Congress of the International Society of Photogrammetry. Vol B3, Hamburg, pp Pedrycz W, Gomide F (998) An Introduction to Fuzzy Sets. MIT Press, Cambridge, Massachusetts Teunissen PJ (998) Quality Control and GPS. In: Teunissen PJ, Kleusberg A (eds), GPS for Geodesy. Springer, Berlin-Heidelberg-New York, pp 7-38 Wieser A (a) Robust and fuzzy techniques for parameter estimation and quality assessment in GPS. Ph.D. dissertation, Engineering Surveying and Metrology, Graz University of Technology, Shaker Verlag, Aachen (in print) Wieser A (b) A Fuzzy System for Robust Estimation and Quality Assessment of GPS Data for Real- Applications. In Proc: ION GPS, US Institute of Navigation, Salt Lake City (in print) Wieser A, Brunner FK () Robust estimation applied to correlated GPS phase observations. In: Carosio A, Kutterer H (eds) Proc: First International Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GIS, IAG SSG 4.9, Zürich, pp PREPRINT of paper to be published in the Proceedings of the IAG Scientific Assembly, Budapest,