Determination of Synthetic Covariance Matrices - an ... - eurasip

15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP

DETERMINATION OF SYNTHETIC COVARIANCE MATRICES - AN APPLICATION TO GPS MONITORING MEASUREMENTS Volker Schwieger University Stuttgart, Institute for Applications of Geodesy to Engineering Geschwister-Scholl-Str. 24D, 70174, Stuttgart, Germany phone: + (49) 71168584064, fax: + (49) 71168584044, email: [email protected] web: www.uni-stuttgart.de/iagb/

ABSTRACT This paper deals with a method to model variances and correlations for measurement quantities. The model of elementary errors is the base for this approach that leads to synthetic covariance matrices. Frequently measurements are correlated and these correlations influence the following processing and the respective results. The modelling and the influence on the results are presented on the base of precise measurements using the Global Positioning System (GPS). Due to the fact that monitoring measurements should lead to the detection of displacements and deformations within the level of cm up to mm the correct modelling of the correlations is important. The standard deviation of the displacement vector using GPS measurements for the detection of tectonic movements in Romania may be smaller up to 69 %. This allows the significant detection of smaller displacements. 1. INTRODUCTION Engineers as well as scientists frequently have the task to model the stochastic characteristics of measurement quantities, meaning the parameters describing the probability distribution. Normal distributed measurement values are completely described by their expected value and their variance. For multi-dimensional normal distributed data the stochastic dependencies among the different measurement quantities are described by correlations. The influence of these correlations on the results of the processing of the measurements may be quite strong. This is the reason that they have to be modelled, too. One possibility to model variances as well as correlations is the model of elementary errors [11] and [14]. It originates from the assumption that any measurement process is influenced by different error sources that may be identified, modelled and quantified: the hypothesis of elementary errors. This approach is widespread in geodesy, especially in engineering geodesy. Besides the model of elementary errors other approaches regarding the analysis of the measurement process are in use. One well-known possibility to model the variances of measurements is the Guide for the expression of Uncertainty of Measurement (GUM) [5]. This guide is introduced by international organisations in the fields of electrical engineering, chemistry, applied physics and metrology. The variances determined by the method described in [5] are not stochasti-

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cally interpretable as the authors also specially express. Besides, the GUM does not treat correlations among the measurement quantities. Both disadvantages are overcome by the model of elementary errors. Another approach developed in the last years uses interval mathematics to model influences of systematic errors on the measurement quantities. The respective errors are modelled distribution free. For further details is referred to [9] and [13]. The decision for one of the approaches has to be made according to theoretical considerations and is based on the applications [10]. In the following it will be dealt with the model of elementary errors. 2. HYPOTHESIS AND MODEL OF ELEMENTARY ERRORS Geodesists have to deal with measurement quantities and values like angles, distances and coordinates among others. As a rule, the measurements are called observations due to the fact that historically the measurements are really observed by the geodesist using optical methods. The observation respectively measurement quantities are treated as random quantities. The measurement values are respective realisations. Any realisation l of a measured random quantity L shows a random deviation ε with respect to its expected value µl . According to [2] and [6] any random deviation may be presented as a sum of v very small elementary errors di

µ l = l − ε and ε =

v

∑d

i

.

(1)

i =1

The individual elementary errors are assumed to show the same absolute value. The probability for negative and positive sign should be identical. Therefore the expected value of the random deviation µε is determined to zero according to

µ ε = E (ε ) =

v

∑ E (d ) = 0 . i

(2)

i =1

If the analysis of a measurement process gets more detailed, the number of elementary errors increases and their absolute values decrease. If infinite elementary errors are assumed, their absolute values are infinitely small. In this case the assumption of standard normal distribution is justified for

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the standardized random deviation ε . As a consequence the normal distribution is valid for the measurement quantity L , with standard deviation σ and variance σ 2 ,

ε ~ N (0,1) and L ~ N ( µ l , σ 2 ) . (3) σ In general a sum of many small elementary errors with changing signs are sufficient to model normal distributed measurements. No elementary error should dominate the error budget. Two specialisations have to be made. On the one side the measurements and their random deviations may be multidimensional. In this case the scalars in equations (1) and (2) are n-dimensional vectors

The random deviation ε may be computed as the sum of all elementary errors taking into account the projection into the observation space

µl = l − ε

and

E (ε ) =

k =1

h =1

(6)

3. DETERMINATION OF THE SYNTHETIC COVARIANCE MATRIX

(4)

i =1

On the other side a classification of elementary errors is needed for multi-dimensional data and the influence on the measurement has to be modelled by influence factors that are integrated into influence matrices. Most important difference among the types of elementary errors is their effect on correlations. As pointed out in [14] three types have to be distinguished. They are given as vectors in the following: - p non-correlating errors δk - m functional correlating errors ξ and - q stochastic correlating errors γ h . In general a model describing the influence of the elementary error is available. The influence on the measurements is modelled using partial derivatives showing the derivative of one measurement quantity with respect to one elementary error. The derivatives are determined on the basis of the model assumption and may be determined analytical or numerical. The derivatives are integrated into influencing matrices that describe the influences of different elementary errors on the observations. Different influencing matrices have to be calculated for the different error types: - p matrices Dk for non-correlating errors, - matrix F for m functional correlating errors and - q matrices G h for stochastic correlating errors. The influencing matrices describe the projection of the elementary errors into the observation space. The matrices Dk and G h are diagonal, since each elementary error influences exactly one measurement quantity functionally. In contrast the matrix F is completely filled, since one functional correlating error may influence more than one measurement quantity. As an example the structure of matrix F is given in the following ∂ξ m ⎞ ⎛ ∂ξ1 ∂ξ 2 L ⎜ ∂l ∂l1 ∂l1 ⎟⎟ 1 ⎛ f11 f12 L f1n ⎞ ⎜ ⎜ ⎟ ⎜ ∂ξ ∂ξ 2 L ∂ξ m ⎟ f 21 f 22 L f 2 n ⎟ ⎜ 1 ⎟ ⎜ ∂l2 ⎟ . (5) = ∂l2 ∂l2 F= ⎜ M M M ⎟ ⎜ M M ⎟ ⎜ ⎟ ⎜ M ⎜ ⎟ L f f f m2 mn ⎠ ⎝ m1 ⎜ ∂ξ1 ∂ξ 2 L ∂ξ m ⎟ ⎜ ∂l ∂ln ⎟⎠ ⎝ n ∂ln

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q

As written above the determination of the influencing matrices may be realised without problems. But for the elementary errors for one respectively for all measurements no information is available due to the fact that they are random. If we would know them, the random deviation may be computed directly and one may resign on the stochastic modelling as shown in equation (7).

v

∑ E (d) = 0 .

p

ε = ∑ Dk ⋅ δ k + F ⋅ ξ + ∑ G h ⋅ γ h .

ε=

The aim of this paper is the determination of variances and covariances respectively correlations for the measurement vector l . This is realised within a covariance matrix, the socalled synthetic covariance matrix Σ ll . This matrix may be determined applying the law of error propagation on equation (6) yielding to p

q

k =1

h =1

Σ ll = ∑ Dk ⋅ Σδδ,k ⋅DkT + F ⋅ Σξξ ⋅ F T + ∑ G h ⋅Σ γγ,h ⋅ G hT .

(7)

The influencing matrices are known from the previous chapter. To determine the covariance matrices of the elementary errors first of all one has to specify the structure of these matrices. The aim of the synthesis is the separation of the measurement process into elementary errors that have no relationship among each other. This implies stochastic independence, too. Hence the covariance matrices of the noncorrelating Σ δδ,k and the functional correlating elementary errors Σ ξξ are diagonal to avoid the modelling of correlations between the elementary errors. In [11] and [12] the use of these elementary error types for the generation of synthetic covariance matrices is established. In the last years more sophisticated possibilities to analyse measurements lead to the detection of more error sources that may be not separated stochastically. Thus in [14] the stochastic correlating elementary errors are introduced. As written in chapter 2 no functional dependence is modelled by the matrices G h . The modelling of the correlations is achieved in the way that the respective covariance matrices Σ γγ,h contain covariances among the elementary errors of one type. Figure 1 gives an overview about all types of elementary errors and the respective matrices that were explained in the proceeding sections. The remaining problem is the determination of the variances and, in the case of stochastic correlating errors, the covariances respectively correlations, too. These stochastic quantities may be defined on the base of empirical investigations, information of manufacturers, experience values or by estimation of maximum errors [12]. In any case this definition is the least reliable part of the process for the determination of a synthetic covariance matrix. The model of elementary errors has been applied to different measurement quantities in geodesy. In [8] and [1] the

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model is applied to distances measurements. In [17] a simplified example on the same topic is given. [7] presents a sophisticated application on a horizontal geodetic net and [14] applies the model on GPS measurements used for monitoring tasks. In the following the paper will give an extract of these investigations.

non-correlating p errors δ k

correlating functional

stochastic

m errors ξ j

q errors γ h

diagonal Σδδ ,k

diagonal Σξξ

non - diagonal Σ γγ ,h

diagonal

non - diagonal

diagonal

Dk

F

Gh

covariance matrix of observatio ns : non - diagonal Σ ll

Fig.1: General overview about model of elementary errors; model elements responsible for correlations shaded in grey

4. ELEMENTARY ERROR CONCEPT FOR GPS DEFORMATION MEASUREMENTS In general the aim of monitoring surveys is the determination of rigid body movements and deformations of objects. These objects may be constructions like bridges and tunnels as well as natural phenomena like landslides or tectonic plates. To detect these movements and deformations geodesists use a technique called deformation analysis that uses statistical methods to detect the movements. For this task the geodesist has to model the dynamic behaviour of the monitored object. Different levels of complexity and actuality exist for the modelling approach. The most simple and used model is the congruency model that takes into account the geometry of a measured geodetic network only. In this case the geometry of a measured network is compared with the same measured at another point in time, that is called epoch. With other words a vector of deformations d is computed on the base of the coordinates x 1 respectively x 2 determined in the epochs. The standard deviation of one deformation respectively the covariance matrix of deformations Σ dd in general are computed using the law of error propagation on the assumption that no correlations between the measurements of the different epochs exist d = x 2 − x 1 and Σ dd = Σ x1x1 + Σ x 2 x 2 . (8) If interepochal correlations exist, the covariance matrix of deformations Σ dd is calculated in another way Σ dd = Σ x1x1 + Σ x 2 x 2 − Σ x1x 2 − Σ x 2 x1 . (9) Existing interepochal correlations improve the detection of point movements, since the test quantity T that is relevant to determine the movements for a whole network is given by

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T=

−1 d ⋅ Σ dd ⋅ dT

σ 02

.

(10)

If the elements in the matrices Σ x1 x2 and Σ x2 x1 show positive signs, the elements in Σ dd gets smaller and as a consequence the test quantity T gets larger. In this case the detection of smaller movements is possible. For negative signs in Σ x1 x2 and Σ x2 x1 the reverse case occurs. For further details regarding the estimation of covariance matrices of coordinates using least-square adjustment and deformation analysis many text books and research exist, here it is referred to [14]. The measurements to determine the movements may be carried trough using rather different instruments. Typical geodetic instruments are tachymeters that measure distances and horizontal as well as vertical angles, level instruments that measure height differences and GPS receivers that derive the coordinates of the antenna by the use of measured distances between the antenna and the satellites. These GPS distances are erroneous due to different error sources, e.g. ionosphere, troposphere, satellite orbits and satellite and receiver clock. Most of the effects may be corrected by differential observation techniques, the use of the carrier phases as measurement quantities and sophisticated models, e.g for the troposphere. Nevertheless remaining systematic effects are still present leading to physical correlations between the measurements as well as between the determined coordinates. For more general information regarding GPS measurement technique is referred to [18]. The quantites introduced into least square adjustment of geodetic networks are the coordinate differences between two measured points (relative or differential GPS). The vector of these three coordinate differences is called baseline. Table 1: Interepochal correlations for GPS measurements error source interepochal remark correlation satellite orbit no satellite clock no ionosphere no measurement on two frequencies troposphere stochastic multipath functional homogeneous reflector stochastic identical antennaantenna phase centre types in different epochs phase noise no centring error no Table 1 gives a summary of the error sources that have to be taken into account for GPS measurements and shows the occurrence of interepochal correlations. Special attention is given to correlations covering years of observations, because the investigation are focussed on this topic. Remarks are given regarding the conditions for the occurrence of correlations respectively of no correlations.

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In the application presented in this paper, GPS measurements are the choice to determine movements between measured epochs. The measurement area Vrancea is located in the Carpathian arc in Romania. This measurement area was chosen due to the fact that horizontal and vertical movements up to 2 mm were expected ([3] and [4]), that cause earthquakes in the region, too. This was the reason to establish a GPS network based on an existing levelling network used for vertical deformation analysis, meaning that the height movements were analysed. Figure 2 presents the location of the network as well as the assumed movement directions. Figure 3 shows the GPS measured points of the network and the selected baselines measured in both epochs. Ukrainia

Un ga

ria

via lda Mo

East Carpathians

time differs between 1 hour for short baselines and 3.5 hours for long baselines. Additionally variations were simulated with respect to observation time, satellite configuration etc. to show the external influences that effect the magnitude of these correlations. These simulations have shown that the interepochal correlations essentially depend on the baseline length, the height difference of the stations and the individual satellite configurations [14]. The influence of the height difference for baseline 39-36 is presented in figure 5. The important influence of the day time and the season is documented in table 2 for the same baseline. The interepochal correlation coefficient reaches 0.38 for identical day time, season and satellite configuration. The reason for this high value is the influence of the troposphere that correlates by a 24 hour and a 1 year period for long time scales as available for this simulation. In any case the actual and the simulated correlations coefficients are of the magnitude that they have to be taken into account for the determination of standard deviations of the deformation vector as presented in the following.

Vrancea - Region

Romania

0.200 Brasov

0.180 0.160

South Carpathians Danub e

interepochal correlation

Turnu Severin Constanta

Ju go sla

Bucharest via Danube

Bulgaria 0 Km

0.140 0.120 height - correlation 0.100

north - correlation east - correlation

0.080 0.060

50 Km 100 Km

0.040

Fig. 2: Location of measured network and movement directions [16]

0.020 0.000

111

1.9

north 106

100

3.6

5.9

15.5 15.6 25.8 27.2 30.8

baseline length [km] 087 082

Fig. 4: Interepochal correlation in dependence of the baseline length for selected baselines [14]

Valea Sarii 036 029

Odobesti 018 GPS-stations 1992 und 1993 GPS-stations 1992 and 1993 GPS-stations 1993 limit of vertical movement fault and limit of veritcal movement

007

selected baseline

Focsani

Fig. 3: Monitoring points and selected baselines

0.20 0.15

height - correlation

0.10

north - correlation

0.05

east - correlation

0.00 10 00

152

40 0

039 154

25 0

001 159

interepochal correlation

162 160

15 0

064 165

50

070

Tulnici

height difference [m]

In [14] the interepochal correlations are simulated for all measured baselines. In Figure 4 the interepochal correlations simulated for selected baselines of the network are shown. The computations result in maximum correlation coefficients of approximately 0.19 for actually measured baselines. Figure 4 presents the baselines 36-39, 106-111, 1-64, 39-64, 139, 7-39, 1-106 and 1-111 from left to right. The observation

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Fig. 5: Interepochal correlation in dependence of height difference; height component for selected baseline 36-39 [14]

The covariance matrix of deformations is calculated according to equation (8) as well as according to (9) leading to different test quantities (10). Table 2 shows the alternations of standard deviations of height displacements respectively

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deformations. The influence of day time, season and satellite configuration is obvious (compare table 2). The gain for the standard deviation of the height deformation (baseline 36-39) reaches an amount of 69 %, if identical satellite configurations, season and day time are simulated. Obviously the measurements in different seasons lead to a loose of 7 % in accuracy for equation (9) when compared to the calculation using equation (8). Nevertheless the interepochal correlations lead to a more realistic model for the stochastic of the deformation vector and therefore have to be integrated into the analysis process. Table 2: Interepochal correlation coefficients and alternation of standard deviation; selected baseline 36-39, height component [16]

interepochal correlation alternation of standard deviation

original, identical seasons

simulated, day time identical, too 0.23

simulated, identical satellite configuration, too 0.38

0.19 +33 %

simulated, different seasons

-0.08

+42 %

+69 %

-7 %

5. FINAL REMARKS Within this paper the model of elementary errors is presented as a method for determination of a synthetic covariance matrix based on the assumption that multiple error sources exist, the influence of the elementary errors is of random nature and no error dominates the respective budget. The concept was applied to the determination of synthetic covariance matrices for GPS measurements. The correlations between deformation epochs, the interepochal correlations reached values up to 0.38. The gain for standard deviations of deformations may reach up to 69 % caused by these interepochal correlations. Finally it should be mentioned that the occurrence, the periods and the magnitude of these correlations are proven partly in empirical correlation analysis [16]. REFERENCES [1] W. Augath, Lagenetze. In: H. Pelzer (Ed.), Geodätische Netze in Landes- und Ingenieurvermessung. Konrad Wittwer Verlag, Stuttgart, 1985. [2] F.W. Bessel, Untersuchungen über die Wahrscheinlichkeit von Beobachtungsfehlern. Astronomische Nachrichten, Vol. 15, 1837. [3] I. Cornea, I. Dragoescu, M,N. Popescu, M. Visarion, Map of recent crustal vertical movements in the S.R of Romania. Revue Roumaine de Geologie, Geophysique et Geographie, Geophysique, 1979.

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[4] M. Grigore, M. Biter, F. Radulescu, V. Nacu, Utilizarea masuratorilor GPS pentru determinarea miscarilor crustale actual in zona dobrogea. Proceedings of International Symposium GPS Technology Applications, Bucharest, 1995. [5] Guide for the expression of uncertainty in measurement (GUM). ISO, 1995. [6] G. Hagen, Grundzüge der Wahrscheinlichkeitsrechnung. Berlin, 1837. [7] O. Heunecke, Nochmals über Korrelationen in der Messtechnik. In: Festschrift Univ. Prof. H. Pelzer zur Emeritierung, Schriftenreihe der Fachrichtung Vermessungswesen der Universität Hannover, Vol. 120, 2004. [8] W. Höpcke, Eine Studie über die Korrelation elektromagnetisch gemessener Strecken. Allgemeine Vermessungsnachrichten, 1965. [9] H. Kutterer, Zum Umgang mit der Ungewissheit in der Geodäsie. Deutsche Geodätische Kommission, Reihe C, Vol 553, München, 2002. [10] H. Kutterer and S. Schön, Alternativen bei der Modellierung der Unsicherheit im Messen. Zeitschrift für Vermessungswesen, 2004. [11] H. Pelzer, Zur Behandlung singulärer Ausgleichungsaufgaben. Zeitschrift für Vermessungswesen, 1974. [12] H. Pelzer, Grundlagen der mathematischen Statistik und Ausgleichungsrechnung. In: H. Pelzer (Ed.), Geodätische Netze in Landes- und Ingenieurvermessung. Konrad Wittwer Verlag, Stuttgart, 1985. [13] S. Schön, Analyse und Optimierung geodätischer Messanordnungen unter besonderer Berücksichtigung des Intervallansatzes. Deutsche Geodätische Kommission, Reihe C, Vol 567, München, 2004. [14] V. Schwieger, Eine Elementarfehlermodell für GPS Überwachungsmessungen. Schriftenreihe der Fachrichtung Vermessungswesen der Universität Hannover, Vol. 231, 1999. [15] V. Schwieger, The effect of interepochal correlations in the analysis of monitoring surveys. Proceedings on 9th international FIG symposium on deformation measurements, Olsztyn, 1999. [16] V. Schwieger, Time dependent correlations using the GPS. Proceedings on FIG Working Week, Seoul, Korea, 2001. [17] V. Schwieger, Zur Konstruktion synthetischer Kovarianzmatrizen. Zeitschrift für Vermessungswesen, pp 143-149, 2001. [18] G. Seeber, Satellite Geodesy. Walter de Gruyter, Berlin New York, 2003. [19] W. Welsch, O. Heunecke, H. Kuhlmann, Auswertung geodätischer Überwachungsmessungen. Handbuch Ingenieurgeodäsie, H. Wichmann Verlag, Heidelberg, 2000.

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