High-Precision Astrogeodetic Determination of a Local Geoid
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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.