High-Precision Astrogeodetic Determination of a Local Geoid Profile Using the Digital Zenith Camera System TZK2-D Christian Hirt Institut fur¨ Erdmessung, Universitat¨ Hannover, Schneiderberg 50, 30167 Hannover, Germany E-mail: [email protected] Fax: +49 511 762 4006 Birger Reese Institut fur¨ Photogrammetrie und GeoInformation, Universitat¨ Hannover, Nienburger Str. 1, 30167 Hannover, Germany E-mail: [email protected] Fax: +49 511 762 2483

Abstract. At the University of Hannover, the trans- technology provides the observation data directly af- portable and automated Digital Zenith Camera Sys- ter acquisition, thus enabling astrogeodetic measure- tem TZK2-D has been developed for the fast and ments of the direction of the plumb line and vertical high-precision astrogeodetic measurement of verti- deflections in real-time. Due to the simplicity and au- cal deflections. Meanwhile the new astrogeodetic in- tomation of observation zenith cameras are of partic- strumentation has been extensively tested in several ular interest in the digital era of geodetic astronomy. field projects. One main application for the TZK2- The Digital Zenith Camera System TZK2-D has D is the astrogeodetic geoid determination in local been developed at the University of Hannover bet- areas. In this paper first results of a highly accu- ween 2001 and 2003 and was already introduced rate high-resolution local geoid profile determination at the 3rd Meeting of the International Gravity and and some experiences using the system TZK2-D for Geoid Commission (Hirt and Burki¨ 2002). A de- vertical deflection measurements are presented. As tailed and comprehensive description of the system test area a particular site near Hannover has been se- TZK2-D is given by Hirt (2004), a shorter depiction lected where a salt dome causes a local gravity field can be found in Hirt (2001). The system TZK2-D perturbation. Over 5 nights during spring 2004 the (Fig. 1) consists of two major components: Firstly, system TZK2-D was used to collect vertical deflec- a CCD sensor is applied for the automatic determi- tion data at 39 stations. The geoid undulation is ob- nation of the direction of the plumb line (Φ, Λ). For tained applying the well-known classical method of the data processing the new star catalogues UCAC astronomical levelling. Due to the highly accurate (Zacharias et al. 2004) and Tycho-2 (Høg et al. determined vertical deflections (000.10 - 000.15) and the 2000) serve as celestial reference. Secondly, a GPS- dense arrangement of measurement stations (approx- receiver is used for precise timing and measurement imately 350 m), the geoid undulation is derived at the of ellipsoidal coordinates (ϕ, λ). Combining both mm-accuracy-level over a distance of 9 km. The re- components the TZK2-D provides vertical deflec- sults obtained are very promising since they demon- tions (ξ, η): strate the potential of modern astrogeodetic measure- ment systems, like the TZK2-D, for high-precision ξ = Φ − ϕ η = (Λ − λ) cos ϕ (1) local geoid determinations. Since the whole process of producing vertical de- flections is automated, from observation via data Keywords. Digital Zenith Camera System, verti- transfer to data processing, vertical deflections can cal deflection, astronomical levelling, astrogeodetic be determined with utmost efficiency if compared to geoid determination instrumentations from the analogue era of geodetic astronomy (cf. Gessler 1975, Wissel 1982, Chesi 1 Introduction 1984, Burki¨ 1989). By now, the TZK2-D has be- come an operational system which has been exten- Since the start of the 21st century, the availability and sively used in several field projects in Northern Ger- application of digital sensors (CCD, charge-coupled many (Hirt et al. 2004, Hirt 2004) and Switzerland device) for imaging lead to a revitalization of astro- (Muller¨ et al. 2004, Brockmann et al. 2004). The geodetic instruments and methods for the local and aim of this paper is to demonstrate the potential of regional determination of the Earth’s gravity field. the Digital Zenith Camera System TZK2-D as one In contrast to analogue photographic media CCD representative of modern astrogeodetic instrumenta- tion for the high-precision local geoid determination, reactivating the classical method of astronomical lev- elling. 2 Description of Test Area and Mea- surements A particular site near Hannover has been selected as test area. Here, a salt dome called ”Benther Salz- stock” forms an extended geophysical anomaly and significantly perturbs the local gravity field. Due to former observations using photographic zenith cam- eras, 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 verti- cal deflection stations using the Digital Zenith Cam- era System TZK2-D. Within 5 nights during spring 2004, 39 independent vertical deflection determina- tions have been carried out at 26 stations homoge- Fig. 1. The transportable Digital Zenith Camera System neously distributed over the course of the profile. At TZK2-D each station usually 40-60 single measurements have been performed. Thus a total of approximately 2000 single solutions of vertical deflections could be col- Z n ∆N1n = − dN − E1n (4) lected at the 26 stations. 1 In average, vertical deflections were collected at about 8 stations per night. In order to estimate the ac- where ds is the distance between neighbouring sta- curacy level achieved a second set of measurements tions and E1n the orthometric correction taking the have been performed at 10 stations. In addition, a se- curvature of the plumb line into account. For reasons lected station (no. 14) has been repeatedly occupied of simplification the orthometric correction E1n is in four nights (Sec. 4.1). The analysis of 26 measure- neglected. For details on the computation of E1n the ment stations clearly shows a variation of about 400 in reader is referred to Heiskanen and Moritz (1967). In the deflection data (ξ, η) due to the gravitational im- practice, the deflection data is not continuously avail- pact of the salt dome (Fig. 2). able. Hence the integral from Eq. 4 is replaced by the sum of increments of geoid undulation. Using the av- 3 Evaluation and Interpretation erage of the deflections at every pair of neighboured stations i and i + 1, it follows: Applying the well-known classical formulae of astro- nomical levelling, the geoid undulation ∆N in the course of the profile is obtained (cf. Torge 2001, ξ + ξ η + η ε = i i+1 cos α + i i+1 sin α (5) Heiskanen and Moritz 1967). Starting from the ver- i 2 2 tical deflection ε i=n−1 X ∆N = − εi · dsi. (6) ε = ξ cos α + η sin α, (2) i=1 being the tilt of the equipotential surface in the az- Fig. 3 (a) shows the result of the geoid profile imuth α, integration of increments of geoid undula- computation: The geoid undulation changes by 8 cm tion over a distance of 9 km. Obviously a striking short- wave gravity field structure superposes the general dN = ε · ds (3) behaviour of the profile. This short-wave structure between two neighboured stations leads to the differ- can be extracted from the profile applying a regres- ence of geoid undulation ∆N1n between the starting sion as high-frequency filter. point no. 1 and the endpoint no. n 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). The resulting profile depicted in Fig. 3 (b) lucidly ∆N. Along the profile, a total of n = 25 increments reveals a depression of nearly 2 cm of the local grav- of geoid undulation ∆N is summed up (Eq. 6). It ity field. A clear density contrast between the salt follows dome and the denser surrounding masses induces this √ typical shape of the profile obtained. σ∆N = n · σdN (8) 4 Accuracy Aspects as standard deviation of the geoid undulation ∆N. Applying equation 8, the standard deviation σ∆N is 4.1 Accuracy of the Deflection Data found to be about 0.85 mm (1.3 mm) over a profile Comparative and repeated observations have been length of 9 km. Due to neglecting the orthometric extensively carried out at selected stations in Ham- correction, this computation is certainly slightly too burg and Hannover, described in Hirt (2004) and Hirt optimistic. Nevertheless this estimation clearly illus- et al. (2004). Based on these investigations a rea- trates that the astrogeodetic method can be used for sonable estimate for the external accuracy level of the determination of the geoid over distances of a few the deflection data is 000.10 - 000.15. In the course kilometers with millimeter-accuracy. The main rea- of the profile determination independent double ob- sons for this high accuracy level are firstly the very servations have been carried out at 10 stations in dense distribution of measurement stations and sec- different nights. Considering the residuals of the ondly the highly-accurate determined deflection data double measurements, standard deviations of 000.08 used for the geoid computation. ξ 00 η for and 0.09 for are obtained. Station no. 14 5 Efficiency Aspects has been selected for repeated observations during four nights. They show standard deviations of 000.11 On the strength of the high degree of automation, the for the ξ-component and 000.05 for the η-component system TZK2-D provides vertical deflections at sin- (Tab. 1). These results underline the high accuracy gle stations in the order of half an hour. This time level reached for the deflection data. period includes set up time, acquisition time for 60 repeated measurements and processing time. Due to Tab. 1. Repeated observations at station no. 14 - vertical the short station spacing the Digital Zenith Camera deflections (ξ, η) and residual errors (rξ, rη) System TZK2-D could be used for collecting deflec- 00 00 00 00 tion data at 8 stations per night. Depending on the date ξ [ ] η [ ] rξ [ ] rη [ ] 20040331 6.92 2.70 0.15 0.01 season (length of night), station spacing and obser- 20040414 7.06 2.78 0.01 -0.07 vation time per station, 15 or more vertical deflection 20040416 7.16 2.66 -0.09 0.05 stations can be observed per night. Hence, in com- 20040423 7.13 2.68 -0.06 0.03 parison with formerly used conventional – analogue – zenith cameras (e.g. Seeber and Torge 1985, Wissel Mean 7.07 2.71 1982) the whole process of providing vertical deflec- Std.dev. 0.11 0.05 tion data has been accelerated considerably using the digital system TZK2-D. 4.2 Accuracy of the Geoid Determination 6 Conclusions and Outlook Simple error propagation can be applied in order to estimate the accuracy of the computed geoid undu- In this paper the new Digital Zenith Camera System lation ∆N in the course of the profile. An angle of TZK2-D has been presented as a powerful tool for 100 corresponds to a length of arc of 4.8 mm over a the local gravity field determination in terms of ac- distance of 1000 m. Hence a standard deviation σdN curacy and efficiency. The profile determination ex- is obtained depending on the uncertainty σε of the emplarily performed above a salt dome demonstrates deflection data and the distance ds between neigh- the potential of astrogeodetic measurements for the boured stations: high-resolution and high-precision geoid determina- tion in local areas. Along profile lines of a few kilo- ds [m] σ [00] ε meters length, the astrogeodetic method can provide σdN = 4.8 mm · · 00 . (7) 1000 [m] 1[ ] geoid information with millimeter-accuracy as such fulfilling even highest accuracy requirements for lo- An external accuracy σε for the deflections of 000.10 (000.15) and a mean distance ds of 350 m as cal gravity field determination. Astrogeodetic observations carried out econom- station spacing leads to σdN ≈ 0.17 mm (0.25 mm) being the accuracy estimate for a single increment ically with modern instruments like the TZK2-D are very well suited for the determination of high- Combined Geodetic Network (CH-CGN). EUREF’04 Sym- frequent gravity field structures with wavelengths of posium of the IAG Commission 1 - Reference Frames, Sub- a few 100 meter up to kilometers being the fre- commission 1-3a Europe (EUREF), Bratislava, Slovakia. quency domain of the Earth’s gravity field where still Chesi, G. (1984). Entwicklung einer tragbaren Zenitkammer only little empirical knowledge is available. High- und ihr Einsatz im 47. Parallel. Dissertation an der Fakultat¨ resolution information of the fine structure of the fur¨ Bauingenieurwesen der Technischen Universitat¨ Graz. Deutsche Geodatische¨ Kommission C 287. gravity field, for example required in geophysical and engineering projects, can easily be provided by ver- Gessler, J. (1975). Entwicklung und Erprobung einer trans- portablen Zenitkamera fur¨ astronomisch-geodatische¨ Orts- tical deflection measurements at densely distributed bestimmungen. Wissenschaftliche Arbeiten der Lehrstuhle¨ stations (e.g. 50 or 100 m station spacing). fur¨ Geodasie,¨ Photogrammetrie und Kartographie an der The system TZK2-D can also be applied for the Technischen Universitat¨ Hannover Nr. 60. local validation of gravity field models basing mostly Heiskanen, W. A. and Moritz, H. (1967). Physical Geodesy. on gravity data. Here, astrogeodetically determined W.H. Freeman and Company, San Francisco. vertical deflections serve as completely independent Hirt, C. (2001). Automatic Determination of Vertical Deflec- observables of the gravity field. As such vertical de- tions in Real-Time by Combining GPS and Digital Zenith flections allow a reliable control of existing geoid Camera for Solving the GPS-Height-Problem. Proc. 14th models. International Technical Meeting of The Satellite Division In the future, attention has to be focussed on the of the Institute of Navigation: 2540-2551, Alexandria, Vir- ginia. development of appropriate methods for the highly precise computation of the orthometric correction Hirt, C. (2004). Entwicklung und Erprobung eines digitalen Zenitkamerasystems fur¨ die hochprazise¨ Lotabweichungs- E 1n neglected in this contribution. This could be bestimmung. Wissenschaftliche Arbeiten der Fachrich- done on the basis of high-resolution digital elevation tung Geodasie¨ und Geoinformatik der Universitat¨ Han- data, density models and optionally gravity measure- nover Nr. 253. ments. In order to determine highly precise geoid Hirt, C. and Burki,¨ B. (2002). The Digital Zenith Camera - profiles it is of prime importance to provide the E1n A New High-Precision and Economic Astrogeodetic Obser- correction on an accuracy level comparable to that of vation System for Real-Time Measurement of Deflections of the deflection data. the Vertical. Proc. of the 3rd Meeting of the International An extensive application of the Digital Zenith Gravity and Geoid Commission of the International Asso- ciation of Geodesy, Thessaloniki, Greece (ed. I. Tziavos): Camera System TZK2-D is planned for further high- 161-166. resolution gravity field determinations similar to the Hirt, C., Reese, B. and Enslin, H. (2004). On the Accu- one presented. Firstly, intended projects in differ- racy of Vertical Deflection Measurements Using the High- ent European regions aim at the better understand- Precision Digital Zenith Camera System TZK2-D. Proc. ing of the fine structure of the Earth’s gravity field. IAG GGSM2004 Symposium, Porto, Portugal. Secondly, currently used geoid models will be val- Høg, E., Fabricius, C., Makarov, V. V., Urban, S., Corbin, T., idated astrogeodetically. The measurements will be Wycoff, G., Bastian, U., Schwekendiek, P. and Wicenec, A. supported by the Deutsche Forschungsgemeinschaft (2000). The Tycho-2 Catalogue of the 2.5 Million Brightest (DFG, German National Research Foundation). Stars. Astronomy and Astrophysics 355: L27-L30. Muller,¨ A., Burki,¨ B., Hirt, C., Marti, U. and Kahle, H.-G. 7 Acknowledgement (2004). First Results from New High-precision Measure- ments of Deflections of the Vertical in Switzerland. Proc. The development of the Digital Zenith Camera Sys- IAG GGSM2004 Symposium, Porto, Portugal. tem TZK2-D has been supported by the DFG from Seeber, G. and Torge, W. (1985). Zum Einsatz trans- 2001 - 2004. The authors are grateful to Rene´ Kaker¨ portabler Zenitkameras fur¨ die Lotabweichungsbestim- and Tobias Kromer¨ for their unrestless support of the mung. 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