High-Precision Astrogeodetic Determination of a Local Geoid Profile Using the Digital Zenith Camera System TZK2-D

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1 High-Precision Astrogeodetic Determination of a Local Geoid Profile Using the Digital Zenith Camera System TZK2-D Christian Hirt Institut für Erdmessung, Universität Hannover, Schneiderberg 50, Hannover, Germany hirt@mbox.ife.uni-hannover.de Fax: Birger Reese Institut für Photogrammetrie und GeoInformation, Universität Hannover, Nienburger Str. 1, Hannover, Germany reese@ipi.uni-hannover.de Fax: Abstract. At the University of Hannover, the transportable and automated Digital Zenith Camera System TZK2-D has been developed for the fast and high-precision astrogeodetic measurement of vertical deflections. Meanwhile the new astrogeodetic instrumentation has been extensively tested in several field projects. One main application for the TZK2- D is the astrogeodetic geoid determination in local areas. In this paper first results of a highly accurate high-resolution local geoid profile determination and some experiences using the system TZK2-D for vertical deflection measurements are presented. As test area a particular site near Hannover has been selected where a salt dome causes a local gravity field perturbation. Over 5 nights during spring 2004 the system TZK2-D was used to collect vertical deflection data at 39 stations. The geoid undulation is obtained applying the well-known classical method of astronomical levelling. Due to the highly accurate determined vertical deflections ( ) and the dense arrangement of measurement stations (approximately 350 m), the geoid undulation is derived at the mm-accuracy-level over a distance of 9 km. The results obtained are very promising since they demonstrate the potential of modern astrogeodetic measurement systems, like the TZK2-D, for high-precision local geoid determinations. Keywords. Digital Zenith Camera System, vertical deflection, astronomical levelling, astrogeodetic geoid determination 1 Introduction Since the start of the 21st century, the availability and application of digital sensors (CCD, charge-coupled device) for imaging lead to a revitalization of astrogeodetic instruments and methods for the local and regional determination of the Earth s gravity field. In contrast to analogue photographic media CCD technology provides the observation data directly after acquisition, thus enabling astrogeodetic measurements of the direction of the plumb line and vertical deflections in real-time. Due to the simplicity and automation of observation zenith cameras are of particular interest in the digital era of geodetic astronomy. The Digital Zenith Camera System TZK2-D has been developed at the University of Hannover between 2001 and 2003 and was already introduced at the 3rd Meeting of the International Gravity and Geoid Commission (Hirt and Bürki 2002). A detailed and comprehensive description of the system TZK2-D is given by Hirt (2004), a shorter depiction can be found in Hirt (2001). The system TZK2-D (Fig. 1) consists of two major components: Firstly, a CCD sensor is applied for the automatic determination of the direction of the plumb line (Φ, Λ). For the data processing the new star catalogues UCAC (Zacharias et al. 2004) and Tycho-2 (Høg et al. 2000) serve as celestial reference. Secondly, a GPSreceiver is used for precise timing and measurement of ellipsoidal coordinates (ϕ, λ). Combining both components the TZK2-D provides vertical deflections (ξ, η): ξ = Φ ϕ η = (Λ λ) cos ϕ (1) Since the whole process of producing vertical deflections is automated, from observation via data transfer to data processing, vertical deflections can be determined with utmost efficiency if compared to instrumentations from the analogue era of geodetic astronomy (cf. Gessler 1975, Wissel 1982, Chesi 1984, Bürki 1989). By now, the TZK2-D has become an operational system which has been extensively used in several field projects in Northern Germany (Hirt et al. 2004, Hirt 2004) and Switzerland (Müller et al. 2004, Brockmann et al. 2004). The aim of this paper is to demonstrate the potential of the Digital Zenith Camera System TZK2-D as one representative of modern astrogeodetic instrumenta-

2 tion for the high-precision local geoid determination, reactivating the classical method of astronomical levelling. 2 Description of Test Area and Measurements A particular site near Hannover has been selected as test area. Here, a salt dome called Benther Salzstock forms an extended geophysical anomaly and significantly perturbs the local gravity field. Due to former observations using photographic zenith cameras, the approximate location and extension of the salt dome is known (Seeber and Torge 1985). At the surface, an astrogeodetic profile with the total length of 9 km has been set up covering the salt dome completely in one dimension. Every 350 m, the geoidal slope has been sampled by vertical deflection stations using the Digital Zenith Camera System TZK2-D. Within 5 nights during spring 2004, 39 independent vertical deflection determinations have been carried out at 26 stations homogeneously distributed over the course of the profile. At each station usually single measurements have been performed. Thus a total of approximately 2000 single solutions of vertical deflections could be collected at the 26 stations. In average, vertical deflections were collected at about 8 stations per night. In order to estimate the accuracy level achieved a second set of measurements have been performed at 10 stations. In addition, a selected station (no. 14) has been repeatedly occupied in four nights (Sec. 4.1). The analysis of 26 measurement stations clearly shows a variation of about 4 in the deflection data (ξ, η) due to the gravitational impact of the salt dome (Fig. 2). 3 Evaluation and Interpretation Applying the well-known classical formulae of astronomical levelling, the geoid undulation N in the course of the profile is obtained (cf. Torge 2001, Heiskanen and Moritz 1967). Starting from the vertical deflection ε ε = ξ cos α + η sin α, (2) being the tilt of the equipotential surface in the azimuth α, integration of increments of geoid undulation dn = ε ds (3) between two neighboured stations leads to the difference of geoid undulation N 1n between the starting point no. 1 and the endpoint no. n Fig. 1. The transportable Digital Zenith Camera System TZK2-D N 1n = n 1 dn E 1n (4) where ds is the distance between neighbouring stations and E 1n the orthometric correction taking the curvature of the plumb line into account. For reasons of simplification the orthometric correction E 1n is neglected. For details on the computation of E 1n the reader is referred to Heiskanen and Moritz (1967). In practice, the deflection data is not continuously available. Hence the integral from Eq. 4 is replaced by the sum of increments of geoid undulation. Using the average of the deflections at every pair of neighboured stations i and i + 1, it follows: ε i = ξ i + ξ i+1 cos α + η i + η i+1 sin α (5) 2 2 i=n 1 N = ε i ds i. (6) i=1 Fig. 3 (a) shows the result of the geoid profile computation: The geoid undulation changes by 8 cm over a distance of 9 km. Obviously a striking shortwave gravity field structure superposes the general behaviour of the profile. This short-wave structure can be extracted from the profile applying a regression as high-frequency filter.

3 Fig. 2. Vertical deflections (ξ, η) in the course of the astrogeodetic geoid profile Benthe Fig. 3. Astrogeodetic geoid profile Benthe. Fig. (a) shows the original geoid profile. The geoid undulation N at the beginning of the profile (station no. 1) is supposed to be 0 m. The local gravity field perturbation caused by the salt dome is depicted in Fig. (b). Both the course of the profile and the approximate position of the salt dome can be found in Fig. (c).

4 The resulting profile depicted in Fig. 3 (b) lucidly reveals a depression of nearly 2 cm of the local gravity field. A clear density contrast between the salt dome and the denser surrounding masses induces this typical shape of the profile obtained. 4 Accuracy Aspects 4.1 Accuracy of the Deflection Data Comparative and repeated observations have been extensively carried out at selected stations in Hamburg and Hannover, described in Hirt (2004) and Hirt et al. (2004). Based on these investigations a reasonable estimate for the external accuracy level of the deflection data is In the course of the profile determination independent double observations have been carried out at 10 stations in different nights. Considering the residuals of the double measurements, standard deviations of 0.08 for ξ and 0.09 for η are obtained. Station no. 14 has been selected for repeated observations during four nights. They show standard deviations of 0.11 for the ξ-component and 0.05 for the η-component (Tab. 1). These results underline the high accuracy level reached for the deflection data. Tab. 1. Repeated observations at station no vertical deflections (ξ, η) and residual errors (r ξ, r η) date ξ [ ] η [ ] r ξ [ ] r η [ ] Mean Std.dev Accuracy of the Geoid Determination Simple error propagation can be applied in order to estimate the accuracy of the computed geoid undulation N in the course of the profile. An angle of 1 corresponds to a length of arc of 4.8 mm over a distance of 1000 m. Hence a standard deviation σ dn is obtained depending on the uncertainty σ ε of the deflection data and the distance ds between neighboured stations: σ dn = 4.8 mm ds [m] 1000 [m] σε[ ] 1[ ]. (7) An external accuracy σ ε for the deflections of 0.10 (0.15) and a mean distance ds of 350 m as station spacing leads to σ dn 0.17 mm (0.25 mm) being the accuracy estimate for a single increment N. Along the profile, a total of n = 25 increments of geoid undulation N is summed up (Eq. 6). It follows σ N = n σ dn (8) as standard deviation of the geoid undulation N. Applying equation 8, the standard deviation σ N is found to be about 0.85 mm (1.3 mm) over a profile length of 9 km. Due to neglecting the orthometric correction, this computation is certainly slightly too optimistic. Nevertheless this estimation clearly illustrates that the astrogeodetic method can be used for the determination of the geoid over distances of a few kilometers with millimeter-accuracy. The main reasons for this high accuracy level are firstly the very dense distribution of measurement stations and secondly the highly-accurate determined deflection data used for the geoid computation. 5 Efficiency Aspects On the strength of the high degree of automation, the system TZK2-D provides vertical deflections at single stations in the order of half an hour. This time period includes set up time, acquisition time for 60 repeated measurements and processing time. Due to the short station spacing the Digital Zenith Camera System TZK2-D could be used for collecting deflection data at 8 stations per night. Depending on the season (length of night), station spacing and observation time per station, 15 or more vertical deflection stations can be observed per night. Hence, in comparison with formerly used conventional analogue zenith cameras (e.g. Seeber and Torge 1985, Wissel 1982) the whole process of providing vertical deflection data has been accelerated considerably using the digital system TZK2-D. 6 Conclusions and Outlook In this paper the new Digital Zenith Camera System TZK2-D has been presented as a powerful tool for the local gravity field determination in terms of accuracy and efficiency. The profile determination exemplarily performed above a salt dome demonstrates the potential of astrogeodetic measurements for the high-resolution and high-precision geoid determination in local areas. Along profile lines of a few kilometers length, the astrogeodetic method can provide geoid information with millimeter-accuracy as such fulfilling even highest accuracy requirements for local gravity field determination. Astrogeodetic observations carried out economically with modern instruments like the TZK2-D

5 are very well suited for the determination of highfrequent gravity field structures with wavelengths of a few 100 meter up to kilometers being the frequency domain of the Earth s gravity field where still only little empirical knowledge is available. Highresolution information of the fine structure of the gravity field, for example required in geophysical and engineering projects, can easily be provided by vertical deflection measurements at densely distributed stations (e.g. 50 or 100 m station spacing). The system TZK2-D can also be applied for the local validation of gravity field models basing mostly on gravity data. Here, astrogeodetically determined vertical deflections serve as completely independent observables of the gravity field. As such vertical deflections allow a reliable control of existing geoid models. In the future, attention has to be focussed on the development of appropriate methods for the highly precise computation of the orthometric correction E 1n neglected in this contribution. This could be done on the basis of high-resolution digital elevation data, density models and optionally gravity measurements. In order to determine highly precise geoid profiles it is of prime importance to provide the E 1n correction on an accuracy level comparable to that of the deflection data. An extensive application of the Digital Zenith Camera System TZK2-D is planned for further highresolution gravity field determinations similar to the one presented. Firstly, intended projects in different European regions aim at the better understanding of the fine structure of the Earth s gravity field. Secondly, currently used geoid models will be validated astrogeodetically. The measurements will be supported by the Deutsche Forschungsgemeinschaft (DFG, German National Research Foundation). 7 Acknowledgement The development of the Digital Zenith Camera System TZK2-D has been supported by the DFG from The authors are grateful to René Käker and Tobias Krömer for their unrestless support of the field measurements in Benthe. References Bürki, B. (1989). Integrale Schwerefeldbestimmung in der Ivrea-Zone und deren geophysikalische Interpretation. Geodätisch-geophysikalische Arbeiten in der Schweiz, Nr. 40. Schweizerische Geodätische Kommission. Brockmann, E., Becker, M., Bürki, B., Gurtner, W., Haefele, P., Hirt, C., Marti, U., Müller, A., Richard, P., Schlatter, A., Schneider, D. and Wiget, A. (2004). Realization of a Swiss Combined Geodetic Network (CH-CGN). EUREF 04 Symposium of the IAG Commission 1 - Reference Frames, Subcommission 1-3a Europe (EUREF), Bratislava, Slovakia. Chesi, G. (1984). Entwicklung einer tragbaren Zenitkammer und ihr Einsatz im 47. Parallel. Dissertation an der Fakultät für Bauingenieurwesen der Technischen Universität Graz. Deutsche Geodätische Kommission C 287. Gessler, J. (1975). Entwicklung und Erprobung einer transportablen Zenitkamera für astronomisch-geodätische Ortsbestimmungen. Wissenschaftliche Arbeiten der Lehrstühle für Geodäsie, Photogrammetrie und Kartographie an der Technischen Universität Hannover Nr. 60. Heiskanen, W. A. and Moritz, H. (1967). Physical Geodesy. W.H. Freeman and Company, San Francisco. Hirt, C. (2001). Automatic Determination of Vertical Deflections in Real-Time by Combining GPS and Digital Zenith Camera for Solving the GPS-Height-Problem. Proc. 14th International Technical Meeting of The Satellite Division of the Institute of Navigation: , Alexandria, Virginia. Hirt, C. (2004). Entwicklung und Erprobung eines digitalen Zenitkamerasystems für die hochpräzise Lotabweichungsbestimmung. 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