Astro Geodetic Determinations For Geoid Modeling

ASTRO-GEODETIC DETERMINATIONS FOR GEOID MODELING

Octavian Bădescu, Assoc. professor – Faculty of Geodesy, Bucharest Technical University of Civil Engineering, [anonimizat]

Paul Daniel Dumitru, Assist. Professor – Faculty of Geodesy, Bucharest Technical University of Civil Engineering, [anonimizat]

Alexandru Călin, Assist. Professor – Faculty of Geodesy, Bucharest Technical University of Civil Engineering, [anonimizat]

Marin Plopeanu, Assist. Professor – Faculty of Geodesy, Bucharest Technical University of Civil Engineering, [anonimizat]

Dan Alin Nedelcu, Dr. Researcher II – Astronomical Institute of the Romanian Academy, [anonimizat]

Abstract: The geoid knowing is of equal scientifically and practical importance for geodesy, geophysics, management of natural resources and risks (gravitational masses transport, heights systems unification, etc.). Today are available several global geoid models realized by satellite observations (Gravity field and steady-state Ocean Circulation Explorer – GOCE, CHAllenging Minisatellite Payload – CHAMP, Gravity Recovery and Climate Experiment – GRACE) or by world-wide terrestrial measurements (EGM2008 – Earth Gravitational Model 2008). These global models are long wave surfaces, having a certain resolution and accuracy. At the international level runs several studies to assess the quality of these global models by different methods, such as gravimetric, geometric (GNSS-geodetic levelling), astro-geodetic or combination of these. Our researches are focused on astro-geodetic method, both for geoid determination and evaluation of existing models. We work to develop a new astro-geodetic method, based on present high accuracy geodetic instruments as well as on an improved mathematical algorithms and procedures. The paper present first results obtained in our research on the use of astro-geodetic determinations for the evaluation of the existing global geoid models or new geoid determination in Romania.

Keywords: astronomical geodesy, geoid, vertical deviation.

Introduction

Geoid represents the physical figure of the Earth and, in the same time, the reference equipotential surface, usual named in land surveying mean sea level. This surface closely approximates sea level in the absence of disturbing forces as winds, ocean currents and tides. In the every point on the Earth’s surface we have a reference direction, tangent to the force line of the terrestrial gravity field. This direction is named local vertical or plumb line and it is normal to the geoid. Therefore, the geoid is a close figure, on which permanently, the verticals are normal.

The ellipsoid is a geometrical figure, with an easy mathematical description and which, theoretically, approximates as good as possible the geoid. Through the same point on the Earth we have the second reference direction, normal to the ellipsoid. Angular difference between normal to the geoid and normal to the ellipsoid, in the same point on the Earth’s topographical surface, represents the vertical deviation (Helmert definition). In a similar way is defined the vertical deviation at the geoid surface (Pizetti definition), but this cannot be directly determined because of terrestrial relief (topographic masses).

Practically, the vertical deviation angles indicate the relative position ellipsoid-geoid. The astro-geodetic method of vertical deviation determination is the only one that can directly provide vertical deviation angle at the Earth's surface, by comparisons between astronomical and geodetic coordinates. This method provides the meridian and prime vertical components of the vertical deviation, quantities which give the total vertical deviation :

,

,

,

where , represent the astronomical latitude and longitude and , the geodetic latitude and longitude. On an arbitrary direction of geodetic azimuth , the vertical deviation has the expression:

.

Vertical deviation and its components in all above relations are under the Helmert definition, i.e. determined by astro-geodetic methods. The configuration of the vertical and the shape of the geoid are correlated and the astronomical coordinates are determined with respect to the local vertical from observation's point [9], [15], [22].

The vertical deviation at geoid surface, in general, is not equal with vertical deviation at Earth's surface. For linking these two quantities it is necessary to know the curvature of the local vertical (or more exactly the deflection caused by the curvature, named in the scientific literature vertical deflection) between geoid and topographical Earth's surface [21]. Unfortunately, the deflection cannot be observed because of terrestrial masses, it can be only estimated at a low accuracy level:

,

where represent the difference between vertical deviations in Pizetti and Helmert definition, i.e. difference between vertical deviation on the geoid minus vertical deviation at the Earth’s surface. Relation (5) shows that only meridian component of the vertical deviation, , is affected and is the orthometric height taken in kilometers [15], [21].

Another quantity involved in the geoid determination is the geoid–ellipsoid separation, N (also named geoid undulation) which represents the distance along the vertical (or plumb line) between the geoid and ellipsoid. The separation can be obtained by gravimetric measurements, combined GNSS (Global Navigation Satellite Systems) observations with geodetic levelling measurements (geometric method) or from gravity potential coefficients determined by satellite measurements. Knowing the geoid-ellipsoid separations it is possible to derive vertical deviations at the geoid surface, which is relatively useless since the almost all terrestrial measurements (except spatial distances) are realized towards local vertical at the Earth's topographic surface [7], [15], [21]. The geoid–ellipsoid separation can be derived from astro-geodetic determinations, by astro-geodetic levelling, if it is known the separation at least one astro-geodetic point. For instance, this can be done by using global geoid models, or GNSS combined with geometrical levelling measurements. Supplementary, it is important to ensure an adequate density of the astro-geodetic points for admitting a linear variation of the vertical deviation between points: generally, 1 point at 10-20 km for flat areas and 1 point at 3-5 kilometers for rugged area [9].

The astro-geodetic levelling formula is:

,

where is the ellipsoid-geoid separation to be determined, is one known ellipsoid-geoid separation, has the expression (4) where is the geodetic azimuth from the point to the point and is the distance between two adjacent astro-geodetic point [15].

Motivations and needs

The geoid reflects the anomalies in the masses distribution and the density inhomogeneity in the terrestrial crust over a certain zone or global, for the whole Earth. Satellite missions as CHAMP (CHAllenging Minisatellite Payload), GOCE (Gravity field and steady-state Ocean Circulation Explorer) or GRACE (Gravity Recover and Climate Experiment) provide global long–wave global geoid models (smooth surfaces) which can serve as reference for a better global or zonal geoid model. There are also global non-satellite models as EGM2008 (Earth Gravitational Model 2008), resulted from world-wide terrestrial, altimetry-derived, and airborne gravity data. Thus, a combination between global models and regional data from terrestrial measurements is the suitable solution for obtaining a high resolution geoid. Terrestrial methods as the astro-geodetic one are able to detect short wavelength structures of the geoid beside gravimetric, GNSS and levelling. All this terrestrial methods have both advantages and disadvantages, general rules being their use in various combination.

Regionally or globally geoid determination is a problem of equal scientific and practical interest. Geoid knowing is important not only for geodesy, where almost all measuring data on the Earth surface have to be corrected by vertical deviation, but also in studies regarding geodynamics processes, mantle composition and rheology, uplift and subduction processes. Also, the geoid play a crucial role for realizing a global reference surface as height-reference system, for topographic studies, sea-level change and unification of worldwide height systems.

The determination of a precise and high-resolution geoid is a subject of many research projects, the combination of different geodetic networks and observation types being of great importance on a national and on a regional and continental level. The compatibility of spatial reference frames, the height systems and the geoid are main objectives for all national authorities in the field of geodesy. In the last years, some theoretical and practical methods for geoid determination were developed and an accuracy of a few centimeters and even millimeters is feasible, depending on time and effort [11], [12], [23].

Classical astro-geodetic observations were characterized by several inconveniences: always expensive, as a result of low productivity, frequently ambiguous accuracy through the ignorance both of the weights’ observations and rigorous method of processing to allow a scientific estimation of the accuracy of the determinations. At these is added the missing of the coherence of the determinations as a result of a long duration of stars' observations, often one month or even two months for a single astronomical point.

For the above reasons, for a long time, astro-geodetic observations have been avoided. But in the last decade, the astro-geodetic method back into focus for at least three reasons: a) the development of the CCD technology which, to a large extent, allowed the elimination of inherent human error, bringing a substantial accuracy and efficiency increase; b) the development of new astronomical or geodetic instruments; c) portable computers and software able to take the entire astrometric calculations, inclusive accurate time signals, since the field [12].

In Romania, there are over 100 year (1859-1999) of astro-geodetic determinations mainly realized by Topographical Military Directorate (TMD) and Military Astronomical Observatory (MAO). Most and relevant determinations was performed in the 1965-1975 decade, resulting over the whole national territory 146 Laplace points (points with complete astro-geodetic determination: astronomical latitude, longitude and azimuth) and 118 astro-gravimetric points (points with determinations of astronomical latitude, longitude and gravimetric determinations). The observations time were usually of 24 nights long for one Laplace point and 15 nights long for one astro–gravimetric point, for a team formed by one geodesist, one technician and 15 soldiers. These astro-gravimetric determinations were the basis of the first geoid model over the national territory. Methods used in the army campaigns were generally based on the zenithal observations on a relative small star number, usually in vicinity of the meridian. Theo 010A, Wild T2, Wild T3, Kern DKM3A was the geodetically instruments used for astro-geodetic determination of the vertical deviation. Another characteristic of the classical determinations is that each astronomical coordinates (astronomical latitude, longitude or azimuth) was separately determinate. Reported precision of these measurements was of for , for and for the astronomical azimuth [20]. Today, these values seems to be quite optimistic, but we have to take into account the methodology used at the time to achieve both measurements and their processing. However, the situation in our country is similar to that of most countries, relatively few in number, who were able to realize such astro-geodetic networks. In these circumstances, the national geoid model is an old one, pre-1980, obtained by combination of astro–geodetic and gravimetric measurements.

The geoid determination represents a scientific problem with practical and economic implications. In Romania, after the return to democracy in 1990, although modern geodetic instruments entered in use, both in public and private sector, many of them are not suitable for very precise measurements such as implied in geoid determination. Considerably GNSS and levelling measurements were made over national territory but, most of them, are disparate, without common reference and of an inferior quality. Until now, only few singular experiments were made to obtain geoid models, especially from GNSS – geodetic levelling combination using existing data sets [2], [3], [4], [5]. Also, in the last years were achieved some gravimetric measurements but over limited areas and unincorporated at national level.

Following a protocol of cooperation between Technical University of Civil Engineering Bucharest – Faculty of Geodesy (TUCEB-FG) and TMD, were found in the MAO database a huge number of determinations (over 3000 points with astronomical latitude and longitude determinations which uniformly cover the national territory), apparently unused until now. Thus, it has started a study on the recovery and validation of these determinations of great historical and scientific value [8].

An important advantage of the astro–geodetic method is that a local geoid can be determined directly from pointy (un-relative) determination, within the interest area. This is in contrast with the gravimetric method, where the application of the well-known Stokes formula theoretically requires coverage over the whole Earth. Due to the technological evolution within last years (CCD observations, satellite time transmissions, etc.), the astro-geodetic method is now much less time-consuming and expensive than in the past [10], [11], [12], [13], [17].

Data, methods and results

3.1 Romanian historical data and global geoid models (GOCE, EGM2008)

The first question was to what extent the old astro-geodetic determinations of the vertical deviation (Helmert deviations) can be used to the geoid determination or to the global geoid models validation. For this we selected 300 astro-geodetic points out of 3000, which covers a test area of approximately (north-south) (east-west) in the south-east part of Romania (about km), including the capital Bucharest (Fig. 1).

Fig. 1. South-east test area of the Romanian territory covered by 300 astro-geodetic points.

For these points, table 1 gives the extreme values ​​of components of the vertical deviations and the total vertical deviations. It should be noted that the original values of geodetic coordinates and were referred to the national reference datum (ellipsoid Krasovski 1942). For our goal, original geodetic coordinates was transformed and referred to GRS80 ellipsoid (ETRS89 – European Terrestrial Reference System) and the vertical deviations components recalculated.

Figures 2 and 3 show the vertical deviations components computed as mention above, from Romanian historical astronomical observations, using a grid obtained by triangulation with linear interpolation from Romanian historic scattered data.

For the same area, using GUT (GOCE User Toolbox) software was calculated the Molodensky vertical deviations components [6]. Molodensky deviation differ not only quantitatively but also in definition from Helmert and Pizetti, representing an intermediate conception between the last two [16]. For Molodenski deviations, the term in the relation (3) become , representing normal height (km). Figures 4 and 5 shows the vertical deviations components derived from GOCE global geoid model, over the area of interest, using GUT with input files containing spherical harmonic potential, DEM altitude grid function, GRS80 ellipsoid and mean-tide options and Surfer v.12 software for visualization. Because the relief over the testing area is relative plane and low altitude, the difference between Helmert and Molodensky deviations is small (less than one arc-second), in comparison with the estimated standard deviation of the astro-geodetic deviations (for highest point with m and , results ). Consequently, this difference was neglected at the moment [8].

A similar procedure as in the case of the GOCE global model was applied for the EGM2008 global model. In the case of EGM2008, the geodetic coordinates and are referred to WGS84 ellipsoid. However, because the difference between GRS80 and WGS84 ellipsoids is small, differences between corresponding geodetically coordinates are enough small to be neglected (less than 10-15 cm). Figures 6 and 7 show the vertical deviations components obtained from EGM2008 global model. Here was used EGM2008 global grids of the vertical deviations components for the interpolation of the Romanian historical scattered data and the resulted vertical deviations components was used to generate components grids by triangulation with linear interpolation.

3.2 The latest Romanian astro-geodetic determinations

In Romania, latest astro-geodetic vertical deviations (Helmert deviations) was performed jointly at TUCEB-FG and Astronomical Institute of the Romanian Academy, Bucharest Observatory (AIRA-BO). At AIRA-BO was used the Danjon Astrolabe (DA), placed on a very stable concrete pilaster. The instrument was upgraded by mounting a CCD camera and using a time-image receiver network board in a computer which leading on the observational process [18], [19]. DA is un-transportable, heavy and very stable astronomical instrument and which measure only at fixed zenithal distance of . Horizontal position of the instrument was ensuring by a mercury bath. Were performed 25 night of observations, the resulting values are shown in Table 2, where , , represent the individual standard deviations of , and for each determination.

Aiming to increase the accuracy and efficiency of the astro-geodetic determinations, it was developed a new methodology for conducting and processing the astronomical observations, designed for modern surveying instruments (electronically total stations or theodolites). These instruments are portable, relatively easy to use and although are not intended for astronomical observations, have a number of features to achieve them. For testing the methodology (measurement technique and functional-stochastic model), were performed several nights of observations both at TUCEB-FG and AIRA-BO.

In the first stage, we used a geodetic Electronic Total Station (ETS), Leica TC2002, gifted by bent eyepiece, liquid biaxial automate compensator, illuminated reticular wires, display and keyboard. This ETS is not motorized one but is from the superior class having the resolution of and the small value displayed is . The work domain of the compensator is and the magnification . Before starting observations, the ETS was calibrated according with manufacturer indications, in terrain, not in laboratory. For all observations, the biaxial compensator of the instrument was activated and the instrument was set up to automatically correct all angular measurements by instrumental errors. All angular observations was made visually and times measurements by a manual portable electronic chronometer with minimum display of . The chronometer was manual synchronized with UTC timescale via internet.

Table 1. Extreme values of the Romanian historical astro-geodetic vertical deviations on the test area, referred to GRS ellipsoid (ETRS89).

Table 2. Results obtained with Danjon Astrolabe at AIRA-BO (Popescu et al., 1997; Popescu&Paraschiv, 1998).

Table 3. Results obtained with Leica TC2002 at TUCEB-FG point.

Table 4. Results obtained with Leica TCRP1201+ at TUCEB-FG point.

Table 5. Results obtained with Leica TC2002 at AIRA-BO point.

It was inconsecutively made 6 nights of observations on the Faculty of Geodesy pilaster situated on the roof building. Observations was effectuated, for every star, in the next order: in the first position of the instrument 5 azimuthal measurements, next 5 zenithal measurements, next, once again, in the second position of the instrument 5 azimuthal measurements and 5 zenithal measurements; therefore a total of 20 angular measurement accompanied by corresponding times (20 times values). In this way each star observed contributed with two equations, one corresponding to azimuthal observations, and the second one to zenithal observations. Applying least square adjustment, the resulting value are shown in the Table 3.

In the second stage, we use a motorized ETS, Leica TCRP1201+ (with technical characteristics similar to Leica TC2002), and making only zenithal observations (3 in position I and 3 in the second position). There were made 5 series of observations in 2 nights at the pilaster of TUCEB-FG. In this case each observed star contributed with one equation, corresponding to zenithal observations. Applying a different functional-stochastic model from the first stage, the resulting values are shown in the Table 4.

For already mentioned reasons we decided to perform observations by Leica TC2002 at the astrolabe location. Therefore, we set coaxially the ETS on the astrolabe and perform 4 nights of observations in the same system as to TUCE-FG (visual observations, manual timing and synchronization). Results are shown in Table 5.

For all vertical deviations determinations with ETS, average time observations were about 3 hours per one night of observation. All observations in all nights were made by the same operator, except for the two nights of observations made ​​by Leica TCRP1201+. We ensured a uniform azimuthal distribution of the observed star and the zenithal distances domain was between and . All zenithal angular measurements were corrected by astronomical refraction. All vertical deviations (Tab. 1, 2, 3, 4 and 5) are referred to the GRS80 ellipsoid, the geodetic coordinates being obtained by GNSS measurements.

We mention that the final values , and (Tab. 1, 2, 3, 4 and 5) are weighted averages depending on individual standard deviation of each determination. Also, the final values , , represent standard deviations corresponding to the , and quantities, as weighted averages of the individual standard deviations, depending on the number of determinations.

3.3 Punctual comparisons between vertical deviations obtained from the historical data, latest data and global geoid models (GOCE, EGM2008)

For the two points TUCEB-FG and AIRA-BO, using Surfer v.12 software, the values of vertical deviations components was extracted (Grid Menu/Residuals) by bilinear interpolation in the grid nodes, from GOCE global geoid model. Similarly, was extracted the values of the vertical deviations components, from EGM2008 global geoid, as mentioned in the sub-chapter 3.1.

Reminding that from GOCE and EGM2008 results Molodensky vertical deviations, table 6 shows all values of the vertical deviations resulting from historical data, latest observations, GOCE and EGM2008 global geoid model at observation points TUCEB-FG.

Table 7 shows the same data as table 6 but for the observation point AIRA-BO. In this point, the astrolabe results represent the reference data taking into account the measurements methodology where there are no observer personal errors.

Table 6. Values of the vertical deviations resulting from historical data, latest observations, GOCE and EGM2008 global geoid models at observation points TUCEB-FG.

Table 7. Values of the vertical deviations resulting from historical data, latest observations, GOCE and EGM2008 global geoid model at observation points AIRA-BO.

(*) As mentioned above, reported precision of the astronomical coordinates were of for , for , that seem to be too optimistic. A reasonable precision for both vertical deviations components may be

(**) The comparisons effectuated between the astro-geodetic and EGM2008 vertical deflections showed an agreement of about (root mean square, RMS) over some European areas. It has been demonstrated that adding omission error estimates from residual terrain model data to EGM2008, reduces the discrepancies from astro-geodetic deflections to RMS. Depending on the region, the RMS errors vary between and [14].

Remarks and conclusions

As already mentioned, the main goal of our study was to test a new methodology for astro-geodetic vertical deviations determination. For this we realize a few comparisons: a) between results obtained by different instruments and measurements adjusted by different functional-stochastic models (Tab. 8) and b) between results obtained at a) and results obtained from global geoid models (GOCE and EGM2008).

Table 8. Differences between used instruments and functional-stochastic models.

As can be seen we use 2 similar geodetic instruments and one astronomical instruments. Also we realize three adjustment types: different azimuthal and zenithal distances adjusted in block (Leica TC2002), different zenithal distances (Leica TCRP1201+) and fixed zenithal distances (DA), the last two being adjusted by different functional-stochastic models. In the case of geodetic instruments, taking into account how to make the measurements (visual observations, manual timing and synchronization), it is clear that all results are contaminated by observer personal errors and the synchronization error. It is difficult to appreciate the observer personal errors, but by comparison with the astrolabe results (which from already mentioned reasons are not affected by observer personal errors) they were kept within acceptable limits (about under). Inside of the effectuated measurements limit, specific statistical tests do not relieve systematic errors at a significant level. Inside of these observations we remark (that in fact we already known) a strong correlation between meteorologically conditions and results quality (Table 3, night no. 6, unusual dark and clear sky). Also we mention that all observations were carried out in the Bucharest metropolitan area, in relatively bad astronomical conditions (light and atmospheric pollution). The developed method is stable, ensuring results uniformity and coherence. We performed tests regarding conditional number and conditional level of the equations system which demonstrated the connection between these and the uniform azimuthal distribution of the observed stars (geometric conditioning). Despite of limited measurements, we took a lot of useful information regarding our methodology, taking into account that between astrolabe and Leica TC2002 as well as between two methods, there are significant conceptual differences [1].

Regarding comparisons between results obtained from our measurements and results obtained from global geoid models, tables 6 and 7 reveals an acceptable agreement with both global geoid models, especially gravimetric EGM2008 global model. This also results from figures 2, 3, 6 and 7, confirming the close connection between astro-geodetic and gravimetric results. So, within the area tested and the number of astro-geodetic points used, it can be said that the old astro-geodetic measurements can be used for geoid modelling as well as for other geodetic purposes. The results obtained until now demonstrate that performing both azimuthal and zenithal measurements (Leica TC2002) seems to be a good solution for getting the closest results of GOCE (Tab. 6 and 7), despite the fact that we obtain relatively worse standard deviations than only performing zenithal observations. The interpolation process play a role in the quality of results which cannot be neglected. For this reasons, in the near future we will perform both type of observations that will be in block adjusted, in more points from the testing area. Furthermore, are planned both azimuthal and zenithal observations, in the old astro-geodetic points, to extract more rigorous conclusions regarding the developed methodology as well as the quality of the Romanian historical astro-geodetic determinations.

Acknowledgements. This work was supported by a grant of the Romanian National Authority for Scientific Research, Program for research – Space Technology and Advanced Research – STAR, project number 216. The authors express their gratitude to the Topographical Military Directorate and Military Astronomical Observatory for the cooperation and provided support. Also, the authors tank to postgraduate Eng. Maxim G.A. for her support to data acquisitions and adjusting.

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