A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE NETWORKS

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1 2007/4 PAGES RECEIVED ACCEPTED M. KOVÁČ, J. HEFTY A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE NETWORKS ABSTRACT Marián Kováč, Ing., PhD. Research field: geodesy, software modeling, geoinformatics Ján Hefty, Prof., Ing., PhD. Research field: geodesy, geodynamics, satellite geodesy, adjustment of observations Department of Theoretical Geodesy, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, Bratislava, Slovak Republic marian.kovac@stuba,sk, jan.hefty@stuba.sk KEY WORDS Mathematical models for the combination of a multi-stage epoch or permanent GPS geodetic networks as well as the models for the combination of satellite and terrestrial geodetic observations are evaluated in the paper. We describe the software package aimed at the processing of geodetic networks with an analytically defined mathematical model. The main feature of our method is its universality. The software option developed in the paper is an analytical definition of the model. This is a different approach when compared to standard geodetic software, which specialize in a one-purpose processing of geodetic networks. The universality of our software is demonstrated with a practical example of the processing of a multi-epoch geodetic network. The processing is realized as a case study starting from the elementary combination of GPS epoch networks that take the covariance matrices into account, through the common processing of GPS and terrestrial observations, up to a complex approach considering the time dependence and transformation parameters among epoch networks in a unique mathematical model. Geodetic multi-stage networks Mathematical modelling of adjustment of observations Software modelling Four-dimensional GPS networks 1. INTRODUCTION One of the fundamental tasks of geodesy is the building and maintenance of geodetic reference networks. Geodetic networks are comprised of a set of well-defined and monumented geodetic markers distributed on Earth s surface. They form the basis for investigations of the shape, dimension and gravity field on the Earth. All these quantities have to be considered as time dependent. Geodetic reference networks also are the basis for all technical and construction work in building as well as the reference frame for monitoring the stability of various large constructions. The definition and practical realization of geodetic reference networks changed due to progress in geodetic observation techniques. A typical approach to the building and processing of geodetic networks was in the separate treatment of horizontal, vertical and gravity observations and is known as 1D (onedimensional) and 2D (two-dimensional or plane) geodesy. With the development of satellite surveying methods and their availability and high degree of accuracy, the problem arose, how to use these three-dimensional (3D) observations in their complexity and not to lose or reduce the information about the spatial point positions included in the measurements. Also, the problem of how to merge these observations with terrestrial and gravity observations without losing any information became very important. These requirements need to be integrated with all available observations in a common mathematical model. The terms four-dimensional (4D) geodesy and integrated geodesy are used for geodetic theory and the methods of processing and interpretation of observations which result in the determination of spatial point positions by also considering the time evolution SLOVAK UNIVERSITY OF TECHNOLOGY Kovac.indd :05:02

2 2. MATHEMATICAL MODELLING OF GEODETIC NETWORKS This section is aimed at a description of mathematical models for processing multistage and epoch GPS observations, mathematical models for processing of permanent GPS observations and also mathematical models for combination of terrestrial and GPS observations. These mathematical models correctly describe the stochastic properties of observations and enable to formulation of procedures for the adjustment of unknown parameters. Definition of the mathematical model of a multi-stage network The general mathematical model is based on the idea of multistage geodetic network (Dobeš et al., 1990, Hefty, 2003) (a) stage-by-stage geodetic networks with parameters which are time-independent. The reference sites are assumed to be stable in time, and no site velocities have to be introduced. (b) epoch networks with parameters which are time-dependent. Both, the site coordinates and velocities of the single points have to be subjects of adjustment. An approach considering the time factor in the evaluation of a 3D geodetic network is often denoted as a four-dimensional (4D) model of a geodetic network. According to the various relations of an actual network to the reference frames in other epochs or stages. we can distinguish: (a) networks reduced to a common reference frame (b) networks reduced to various reference frames. The following formulae describe a mathematical model of the most complex situation - epoch geodetic networks related to various reference frames (Hefty, 2003): where x (i) is the vector of realization of i-th stage; A i is the design matrix of the i-th stage, C i, j is the incident matrix, which defines the dependence between the parameters of the i-th and j-th stage; y (i) is the vector of the parameters of the i-th stage, which includes only the new points of the i-th stage; and 0 is the zero matrix. The random errors of observations in the i-th stage are denoted as ε (i). As the stochastic independency among the stages is assumed, it is expected that the covariance matrix of the observations can be defined for a multiepoch model as (1) where is the vector of the observed coordinates in the i-th epoch; v ref are the velocities of the reference sites; is the matrix relating the coordinate observations in the i-th epoch to the estimated coordinates y 0 related to the epoch t 0. is the diagonal matrix relating the observed coordinate changes to the estimated velocities v y ; is the matrix relating the coordinate observations to the transformation parameters ; and E is matrix relating the reference and estimated velocities. The random errors ε ref of the reference velocities are also considered in the model. The covariance matrix of the observed coordinates and reference velocities is assumed in the form (3) (2) Here Σ (i) means the covariance matrix of the realization of the i-th stage. Combination of geodetic networks measured by GPS We describe here two kinds of networks according to their time dependence (4) where is the covariance matrix of the observations in the i-th epoch and is the covariance matrix of the reference velocities. A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE Kovac.indd :05:07

3 Processing of permanent GPS networks Permanent observations provide more valuable information about position variations when compared to epoch observations. These observations allow for the determining of not only the mean positions and velocities, but also other factors which are functions of time like seasonal periodic variations, magnitude of discontinuities in the series, etc. A model of the common processing of a time series comprising the n epoch t i has the form (Hefty, 2003): where is the matrix of relations among the coordinates in epoch t 0 and in epoch t i ; is a diagonal matrix with the elements (t i t 0 ), which defines the relation between the velocities v x and the observations in epoch t i. is the matrix of arguments of the seasonal terms and discontinuities relating the observations to α - amplitudes of the seasonal variations and magnitudes of jumps in the series. The other parameters are coordinates x 0 in epoch t 0 and v x - adjusted site velocities. If is the covariance matrix of the coordinates for observations in epoch t i, the covariance matrix of the whole period of permanent observations (when considering the epochs as mutually independent) will be Common processing of GPS and terrestrial observations Finally, we will discuss a model for the common processing of GPS and terrestrial observations. This is the case when besides the GPS observations terrestrial observations in the geodetic networks are also available. Processing of such a situation is possible in two variants. (5) (6) Variant A. The independent processing of GPS and terrestrial observations resulting in two independent sets of Cartesian coordinates x (GPS) and x (TER) and their covariance matrices Σ (GPS) and Σ (TER). Their combination results then in a parametric hypervector consisting of the combined coordinates y and transformation parameters Θ. A necessary requirement for this approach is that the terrestrial observations are sufficient to yield 3D coordinates of the whole network. Then the combination is realized by the model (Hefty, 2003) where elements in J are the coefficient relating the individual coordinates of the two networks to the common coordinates y; and T is the matrix with coefficients for transformation among terrestrial and epoch reference frames. Variant B. The terrestrial observations l (TER) are inserted directly into the solution of satellite network. The advantage of this processing mode is that the terrestrial observations need not be complete to form an independent solution x (TER). The terrestrial observations are used to strengthen the GPS solution. The linearized mathematical model of the integration of the terrestrial observations into the GPS network is then where x (GPS) is the vector of the coordinates of the geodetic network determined solely by GPS and y = y 0 + Δy is the vector of the final combined coordinates of the spatial network, A (GPS) is the design matrix which relates the coordinates of the satellite network to the adjusted coordinates y and A (TER) is the design matrix relating terrestrial observations to the adjusted Cartesian coordinates y. Σ (GPS) is the covariance matrix of the GPS network and Σ (TER) is the covariance matrix of terrestrial observations. The linearized functional relation of the terrestrial observations to the parameters y is I (TER) = f(y) = f(y 0 ) + A (TER) Ay. It is worth mentioning that the 4D approach is fully applicable to a combination of the GPS coordinates x (GPS) and terrestrial network solution x (TER) or the terrestrial observations l (TER) if the time factor of the observations is considered. All estimating models are using the least squares method procedures with applying the effective approach considering the full stochastic information represented by variance covariance matrices (Kubáčková, 1990). All the mentioned situations are solved as indirect observations of vector parameter. (7) (8) 12 A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE... Kovac.indd :05:09

4 3. SOFTWARE APPLICATION All the models presented here are incorporated in a software package designed to estimate the parameters of various kinds of geodetic networks. The software presented is composed of a modular system where the fundamental application can be expanded with plug-ins and scripts written in the Python language. It is is named as SoNet (acronym for Solution of Networks). The software is available on the request at the address marian.kovac@stuba.sk. The numerical procedures for matrix operations are adopted according to (Golub and Van Loan, 1989). The input data format of the programs is in the XML language (Harold and Means, 2002). The input data file includes two kinds of information: (a) a part containing the mathematical model of the geodetic network (b) a part containing the processed observations. Geodetic observations The geodetic observations which can be processed in the program (the XML element is quoted in the round bracket) are as follows: geocentric Cartesian coordinates (coordinate), shifts of spatial positions in time annual velocities (velocity), horizontal angles (angle), zenith angles (z-angle), spatial distances (distance), and vertical distances - elevations (diffh). An example of the horizontal angle (from the standpoint A): <from name= A > <angle to= B next= C value= /> </from> The program allows for the processing of multiepoch geodetic networks. In this case the individual epochs are marked as a unit. Each unit encapsulates the observations grouped in the block. Each block also contains, apart from the observations, their covariance matrix. As example of one unit with a block with one distance and covariance matrix is: <unit id= 1 > <block id= 1.1 > <from name= A > <distance to= B value= /> </from> <link href= c.cova /> </block> </unit> Mathematical model The mathematical model is defined analytically. It contains: (a) observations (observations) integrated into the processing (b) unknowns - adjusted parameters (unknowns) in symbolic forms (c) observation equations (equations) in symbolic forms which associate the observations with unknown parameters. Selection of unknown parameters The element unknown allows for the definition of unknown parameters estimated by adjustments. Each element unknown contains the element group, which includes the elements point. The element point defines points to which the selected unknowns are related. If the group (element group) has the attribute name, then the group is called a named group; otherwise, it is called an anonymous group. The significance of an anonymous group is that for each point defined inside this group, the program creates a stand-alone unknown. An example of an anonymous group can be, e.g., the definition of adjusted coordinates. Otherwise in the named group one unknown is created which is tied to all the points included in the named group. An example of a named group can be, for instance, a definition of the transformation parameters which are tied to more than one point. An example of a named group: <unknowns> <unknown type= omega[ \\omega,g, ,cc,100,5]: psi[ \\psi,g, ,cc,100,5]: epsilon[ \\epsilon,g, ,cc,100,5] > <group name= second > <point name=.* /> </group> </unknown> </unknowns> Observation equations The definition of the observation equations relating the unknowns and the observations represents the basic concept of the application. The observation equations provide the deterministic relation among the observations which are the subject of the measurement and the estimated unknown parameters which are subject of the estimation. Generally, the mathematical model is composed of a set of observation equations. The bservation equations can be freely modified or defined in the application according to the requirements desired. A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE Kovac.indd :05:11

5 The observation equations are recorded in the program in a symbolic form. All the mathematical operators and standard mathematical functions can be used in the observation equations. An example of a simple observation equation of levelling is: h{i,j} = H{j} - H{i}; The indices of the relevant observations or unknown parameters are in curly brackets. Except for the observations and unknown parameters meta-information and the parameters of the reference ellipsoids loaded from external files can also be used in the observation equations. Derivation of observation equations. The program automatically performs the expansion of the observation equations into the Taylor series. In other words, the linearization of the observation equation is solved purely analytically. Meta-information The meta-information allows for the inclusion of some numeric values for processing in a way that they can be used in a symbolic form in the observation equations. Examples of meta-information include time information, temperature, pressure, height of the observing instrument, the position of the GPS antenna phase centre, etc. Meta-information is included in the element meta. The following example shows the use of some attributes in the element meta: Figure 1 Terrestrial observations in the Mochovce network: Symbol O means the observation of the horizontal angles, and the symbol shows the spatial distances observed. <meta value= alias= t0 label= refepoch /> <meta value= alias= t label= epoch /> 4. EXAMPLE Geodetic network of the Mochovce power-station The potential of the software will be demonstrated on the 3D and subsequently the 4D network of the Mochovce Nuclear power station. Repeated isolated terrestrial measurements in the local geodetic network were performed in two epochs in 1988 and 1989 by conventional geodetic techniques - measurement of slope distances by the EMD Mekometer 3000 and horizontal angles by the theodolite WILD T3. The distribution of these observations is shown in Figure 1. The GPS observations were performed after more than ten years in three epochs, namely in 2001, 2002 and 2003 (Hefty, 2002). The distribution of the GPS monitored points is in Figure 2. Figure 2 Points measured by GPS. The symbol highlights the points measured in at least two stages. Mathematical model The geodetic network of the Mochovce power plant described was processed using various variants of the estimated parameters. The most complex is the model of the common processing of GPS with terrestrial observations with an estimation of all the coordinates and 14 A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE... Kovac.indd :05:13

6 velocities as well as the parameters of the transformations. The time factor is considered too. If we restrict ourselves to a static 3D model without considering the time factor, it is sufficient to omit the vector v y from the parameters. geocentric Cartesian coordinates obtained by GPS are (10) The corresponding record in the input data file is then <equations> <eq form= x{i} = X{i} + tx{...} + vx{i}*(t-t0); /> <eq form= y{i} = Y{i} + ty{...} + vy{i}*(t-t0); /> <eq form= z{i} = Z{i} + tz{...} + vz{i}*(t-t0); /> </equations> The mathematical formula for the spatial distance observed by EDM The record in the input data file (12) Here, is the matrix relating the observations in the i-th epoch to the estimated coordinate; is the matrix relating the terrestrial observations in the i-th epoch with the estimated geocentric Cartesian coordinates; and, are diagonal matrices relating the velocities with the observations in the i-th epoch for the GPS and terrestrial observations. is the matrix relating the observations in the various epochs (3D transformations); is the vector of the realized coordinate observations in the i-th epoch; and is the vector of the terrestrial observations in the i-th epoch. The covariance matrices are: is the covariance matrix of the coordinates determined by GPS in the i-th epoch; is the covariance matrix of the terrestrial observations in the i-th epoch. Finally, y are the adjusted coordinates; v y are the adjusted velocities of the monitored points, and are the adjusted transformation parameters. Matematical model assembled in the input data file of the program The mathematical formulae for the observation equations for the (9) <eq form= s{i,j} = sqrt((x{j}+vx{j}*(t-t0) - X{i}- vx{i}*(t-t0))^2 + Y{j}+vY{j}*(t-t0) - Y{i}-vY{i}*(t-t0))^2 + (Z{j}+vZ{j}*(t-t0) - Z{i} - vz{i}*(t-t0))^2); /> The mathematical formula of the horizontal angle The record in the input data file is (13) <eq form= a{i,j,k} = (atan2((-sin(gl(x{i},y{i},z{i})) * (X{k}+vX{k}*(t-t0)-X{i}-vX{i}*(t-t0)) + cos(gl(x{i},y{i},z{i})) * (Y{k}+vY{k}*(t-t0)-Y{i}-vY{i}*(t- 0))), (-sin(gb(x{i},y{i},z{i})) * cos(gl(x{i},y{i},z{i})) * (X{k}+vX{k}*(t-t0)-X{i}-vX{i}*(t-t0)) - sin(gb(x{i},y{i},z{i}))* sin(gl(x{i},y{i},z{i})) * (Y{k}+vY{k} * (t-t0)-y{i}-vy{i}*(t-t0)) + cos(gb(x{i},y{i},z{i})) * (Z{k}+vZ{k} * (t-t0)-z{i}-vz{i}*(t-t0))))) - (atan2((-sin(gl(x{i},y{i},z{i})) * A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE Kovac.indd :05:15

7 (X{j}+vX{j}* (t-t0)-x{i}-vx{i}*(t-t0)) + cos(gl(x{i},y{i},z{i})) * (Y{j}+vY{j}*(t-t0)-Y{i}-vY{i}* (t-t0))), (-sin(gb(x{i},y{i},z{i})) * cos(gl(x{i},y{i},z{i})) * (X{j}+vX{j}*(t-t0)-X{i}-vX{i}* (t-t0)) - sin(gb(x{i},y{i},z{i})) * sin(gl(x{i},y{i},z{i})) * (Y{j}+vY{j} * (t-t0)-y{i}-vy{i}* (t-t0)) + cos(gb(x{i},y{i},z{i})) * (Z{j}+vZ{j}*(t-t0)-Z{i}-vZ{i}*(t-t0))))); /> Results of the common processing The results of the common processing of the GPS and terrestrial observations with an estimation of the coordinates and the velocities of the monitored points are summarized in Tables 1 3. The results presented of 4D approach are based on the terrestrial observations in 1988 and 1989, namely, 16 spatial distances and 8 plane angles and GPS observations performed in 2001, 2002, and 2003 on 20 points. Table 1 shows the resulting coordinates in the Cartesian system, the following two tables show the transformation of the estimated geocentric Cartesian site coordinates and velocities into ellipsoidal coordinates and local velocity components. 5. CONCLUSIONS We have introduced a general description of a universal software package oriented towards the modelling, analysis and processing of geodetic 4D networks. The theoretical background uses the knowledge gained from mathematics, numerical processing, informatics and geodesy. The software package can be used not Table 1 Estimated geocentric Cartesian coordinates of the points of the Mochovce geodetic network Point X (m) σ X (m) Y (m) σ Y (m) Z (m) σ Z (m) MO MO MO MO MO MO Table 2 Geographic coordinates of the points of the Mochovce geodetic network related to the WGS-84 ellipsoid Point Latitude B ( ) σ B (mm) Longitude L ( ) σ L (mm) Heihgt H (m) σ H (mm) MO MO MO MO MO MO Table 3 Horizontal velocities v n, v e and the vertical velocity v H of thed points of the Mochovce geodetic network Point v n (mm/year) σ vn (mm/year) v e (mm/year) σ ve (mm/year) v H (mm/year) σ vh (mm/year) MO MO MO MO MO MO A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE... Kovac.indd :05:18

8 only with the mathematical models presented in this article, but also with whatever mathematical models are used for the adjustment of parameters of geodetic networks. The program allows for the processing of stage-by-stage, epoch and permanent GPS networks as well as a combination of GPS and terrestrial observations. The integration of the terrestrial observations to a GPS network allows for the improvement of many issues, such as the geometry, scale, and time dependence of coordinates, vertical components, etc.. The variability in the definition of the mathematical models allows for the modification of a mathematical model easily and quickly. This variability also allows not only for the separate processing and analysis of 1D, 2D, 3D and 4D geodetic networks, but also for their mutual combination with existing reference frames when using a global covariance matrix. The mathematic model, or rather the observation equations which create the mathematical model, is implemented in a symbolic form. The functionality of the software package was demonstrated in the solution of a multi-epoch heterogeneous geodetic network. Acknowledgement: This work was supported by Grant No. 1/4089/ 07 of the Grant Agency of Slovak Republic VEGA. REFERENCES DOBEŠ, J. et al. (1990) Precise local geodetic networks. Research Institute of Geodesy and Cartography in Bratislava, Bratislava, (in Slovak). GOLUB, G. H., VAN LOAN, CH. F. (1989) Matrix computations The John Hopkins University Press. HAROLD, E. R., MEANS, W. S. (2002) XML in a Nutshell 2nd Edition, O Reilly. HEFTY, J. (2002) Monitoring recent lithospheric movements in the area of the Mochovce power plant using geodetic methods. Research report, unpublished. SUT Bratislava, (in Slovak) HEFTY, J. (2003) Global positioning system in 4D geodesy Bratislava SUT, (in Slovak). KUBÁČKOVÁ, L. (1990) Methods of experimental data processing Bratislava, Veda, (in Slovak). A UNIVERSAL APPROACH TO PROCESSING 4-DIMENSIONAL GEODETIC REFERENCE Kovac.indd :05:20