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**Gravity Field and Geodynamics**

**Geodesy Group**

Welcome!

The Geodesy Group at UNB is composed of a group of professors, students and visiting researchers who are interested in geodesy in its broad mathematical and physical aspects. A list of current and past members can be found here.

## Story of the informal "Geodesy Group" in GGE

It all started somewhere in 1985 with the arrival of Dr Alfred Kleusberg from Germany as a Research Associate in geodesy [Vaníček and Kleusberg, 1985]. **Geodesy** being the theoretical foundation of surveying/ geomatics, and as such being rooted in physics and mathematics, was not the prime subject of research in an engineering department. Yet, there was a demonstrable need for establishing a research group focused on geodesy.

The main topic of research for this group became the **geoid**, the horizontal surface to which heights are referred. A horizontal surface is easy to visualize, particularly at the sea or a lake level, but is considerably more difficult to describe on land, where the heights are really needed. First Canadian geoid of decent accuracy was produced [Vaníček, et al., 1986] and the Stokes-Helmert's theory of geoid determination was launched [Vaníček and Kleusberg, 1987; Vaníček and Sjoberg, 1991].

A close relative to the problem of reference surface for heights is the problem of definition of **heights** to be used, i.e., the evaluation of orthometric, a.k.a. the "practical heights" from observed height differences on the Earth's surface. Together, the choice of the reference system and the definition of heights constitute the package of "Height system". At the end of 20th century, only approximate values of orthometric heights could be obtained.

The geoid research got a new impetus in 1992-3, when Prof. Z. Martinec spent a year with our group and to work on the Stokes-Helmert's theory conceived in 1985. As a result, the **Stokes-Helmert theory** was reformulated basically the way it still holds today [Vaníček,and Martinec, 1994; Martinec and Vaníček, 1994]. A sequence of post-doctoral fellows (Sun Wenke, J. Janak, R. Tenzer and A.Ellmann) visitors (Prof. W. Featherstone, Prof. Prof. L. Sjoberg, M. Veronneau) and PhD students (S. Pagiatakis, P. Vajda, M. Santos, M. Craymer, M. Najafi, P. Novák, J. Huang, R. Kingdon) - only those who contributed to geoid research are listed here but D. Avalos should be mentioned - then worked on various aspects of the research between 1993 and 2006. This research is reflected in a host of seminal papers published during this era and can be, perhaps, divided in the following topics, each with its own raison d'etre.

1) **Downward continuation** of gravity field from the Earth's surface down (or up) to the geoid [Vaníček, et al., 1996] has to be carried out to render gravity anomalies on the geoid needed to solve the boundary value problem in Stokes-Helmert's (S-H) theory. It is generally an inverse problem for a potential field which, if done in a physically meaningful way is inherently numerically unstable and thus troublesome [Sun and Vaníček,1998; Huang et al., 2003].

2) **Modification of Stokes's integration kernel** is an integral part of S-H theory [Featherstone and Vaníček,1997]. Various approaches were studied with the aim of improving the accuracy of the S-H technique [Vaníček and Featherstone, 1998].

3) The choice of **global gravity model for a reference surface** is another choice one has to make when using the S-H approach [Vaníček et al, 1996; Martinec and Vaníček,1996]. The main issue here was the transformation of the available real Earth's gravity model (EGM) from the real space to Helmert's space defined by the distribution of topographical masses implied by Helmert's second condensation technique, where all of the mathematical processing takes place.

4) An integral part of the transformation from real to Helmert's space is the **modelling of topographical density** to come up with as good an approximation of an empty (harmonic) space above the geoid as we can. The first iteration of this task was to model just lateral variations of topographical density using surface geological maps [Martinec, 1993; Huang et al., 2001]. It turned out that the lateral variations of topographical density contribute significantly to the accuracy of the resulting geoid model but are considerably less significant than originally thought by Molodenskij. Attempts were made to estimate what the vertical topographical in-homogeneities would contribute to the geoid model [Kingdon et al., 2009]. It looks as if the vertical variations will contribute even less than the lateral variations but the jury is still out.

5) Our investigation of the only physically meaningful downward continuation process as formulated by Poisson led us to the realization that only **solid gravity anomalies** (anomalies that can be described by solid spherical harmonics) can be continued downwards and upwards. This realization resulted, among other things, in a rigorous formulation of spherical complete Bouguer anomaly [Vaníček et al, 2004], a.k.a., "No Topography" (NT) anomaly, a solid anomaly that belongs to the NT space, i.e., the space characterized by the absence of topography . The NT anomalies are now used in the S-H approach for prediction of gravity anomalies on a regular grid at the Earth surfasse.

6) As part of the "Height System" package, orthometric heights had to be put also on solid physical foundations, i.e., **rigorous orthometric heights** had to be formulated [Tenzer et al., 2005; Santos et al., 2006; Ellmann et al., 2007]]. The effect of switching to rigorous orthometric heights in Canada was also investigated [Kingdon et al., 2005].

7) To test the theory and the computational algorithms, the International Association of Geodesy decided to establish a Study Group charged with the construction of a **synthetic gravity field**. This attempt resulted in the production of Australian synthetic field [Baran et al., 2006] in which our group played some role. Tests of the S-H technique on the Australian synthetic field did not come out as expected because the synthetic field had not been constructed rigorously [Vaníček et al., 2013]. We were able to conclude nevertheless that the S-H theory and the accompanying numerical algorithms are at present accurate to a standard deviation of 2 to 2.5 cm. That estimate has now been confirmed by the standard deviations of geoidal heights; this work still continues.

8) Finally we should mention our first excursion into the realm of the presently **largest geodetic controversy**: should the "classical height system" of geoid and orthometric heights be used or would it be correct to switch to the "Russian system" consisting of the quasi-geoid and normal heights, as offered by M.S. Molodensky [Molodensky et al., 1960]. Molodenskij started from the premise that topographical density would never be known accurately enough to allow us to compute a precise geoid. Instead of using the geoid, which is an equipotential surface of the gravity field, he suggested to use the quasi-geoid, a surface not too different from the geoid but having no clear physical meaning and being much more complicated than the geoid and normal heights (referred to the quasi-geoid). Molodensky's approach does not require the knowledge of topographical density but has its own problems of different kind. It is not clear if the quasi-geoid is describable in the mathematical sense and it requires integration on a surface that is not integrable. Our contention is that the classical system is more practical and offers a clear geoid improvement in the future as topographical density becomes better known, while Molodensky system has and will remain haunted by "blind spots/areas" where the quasi-geoid is too complex to be used as a reference surface [Vaníček et al., 2012]. Presently, the Russian system is used in some European countries and in Russia, the classical system is used in North and South Americas and elsewhere.

Our group is viewed internationally as being the strongest supporter of the classical system and our publications are closely followed as witnessed by the number of citations our publications get (e.g., PV, 1762 citation as of Sept.27, 2016; source: ResearchGate GmbH, Invalidenstr. 115, 10115 Berlin, Germany). Perhaps the inclusion of Prof. Vanicek in the "List of Geophysicists" as one of 163 total dead or alive listed (this is "a **list of geophysicists**, people who made notable contributions to geophysics, whether or not geophysics was their primary field should be also mentioned. These include historical figures who laid the foundations for the field of geophysics", source en.wikipedia.org/wiki/List_of_geophysicists).

The research described herein had been supported from three main sources: NSERC personal research and strategic grants, grants from the GEOIDE, a Canadian National Centre of Excellence and research contracts with Geodetic Survey Division of Canadian Federal Government in Ottawa. When Prof. Vanicek retired from teaching at UNB in 1999, the research support from the latter two sources gradually dwindled out, but 14 years later, he received a decent 5-year NSERC discovery grant (the largest in the GGE department in 2013). The grant is for studying the downward continuation and formulating physically meaningful constraints for the iterative process.

With this background information let us turn to present situation. The latest NSERC discovery grant has allowed PV to finance two PhD students, M. Sheng and I. Foroughi (Sheng eventually got a full NSERC PhD scholarship). The "Geodesy Group", composed now of these two graduate students, Dr. Santos, Dr, Kingdon, PV and, for a year, one MScE student, M. Klu, complemented occasionally during the Summer months by Prof. Janak and by Prof. Novák, thus keeps on functioning and producing good quality research. The list of recent publications by the group is shown below.

Recently, Michael Sheng spent a month in Ireland on a research stay with Prof. Martinec at the Dublin Institute for Advanced Studies. Ismael Foroughi spent twoo months in Czech Republic at NTIS - New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia, Plze?, doing research with Prof. Novák.

It should be mentioned that the research done by the "Geodesy Group" has had many spinoffs. First, a private company, FUGRO, bought a license to our geoid software (SHGeo) and is using it commercially. Drs. Kingdon and Vanicek are consultants to FUGRO. One of our former members, D. Avalos, is spearheading an effort on behalf of Mexico, to produce a common geoid model for Central America. In this effort, the Central American countries are using our SHGeo software. Prof. Martinec is using his version of the Stokes-Helmert approach to compute the geoid in Ireland. Saudi Arabia, Egypt and Israel have obtained our software to produce geoid in their respective countries.

## Selected publications:

- Molodensky, M.S., Yeremeev, V.F., Yurkina, M.I, 1960. Methods for Study of the External Gravitational Field and Figure of the Earth. TRUDY TsNIIGAiK, 131, Geodezizdat, Moscow. English translat.: Israel Program for Scientific Translation, Jerusalem 1962.
- Vaníček,P. and A. Kleusberg, 1985. What an external gravitational potential can really tell us about mass distribution.
*Bollettino di Geofisica Teorica ed Applicata*, Vol. XXCII, No. 108, December, pp. 243-250. - Vaníček,P., A. Kleusberg, R.G. Chang, H. Fashir, N. Christou, M. Hofman, T. Kling, T. Arsenault 1986. The Canadian Geoid. Geodetic Survey of Canada; Energy, Mines and Resources Canada, Ottawa, Technical Report No.???. Also Department of Surveying Engineering, University of New Brunswick, Fredericton, Technical Report No. 129, pp. 123.
- Vaníček,P. and A. Kleusberg, 1987. The Canadian geoid-Stokesian approach.
*Manuscripta Geodaetica*, 12(2), pp. 86-98. - Vaníček,P. and L.E. Sjöberg 1991. Reformulation of Stokes's Theory for Higher than Second-Degree Reference Field and Modification of Integration Kernels.
*JGR*, 96(B4), pp. 6529-6539. - Martinec, Z., 1993. Effect of lateral density variations of topographical masses in improving geoid model accuracy over Canada. Research Report for Geodetic Survey of Canada, Ottawa.
- Vaníček,P. and Z. Martinec, 1994. Stokes-Helmert scheme for the evaluation of a precise geoid
*Manuscripta Geodaetica*19 pp. 119-128. - Martinec, Z. and P. Vaníček,1994. Direct topographical effect of Helmert's condensation for a spherical geoid.
*Manuscripta Geodaetica*19(3), pp. 257-268. - Martinec, Z. and P. Vaníček,1994. The indirect effect of Stokes-Helmert's technique for a spherical approximation of the geoid.
*Manuscripta Geodaetica*19(2), pp.213-219. - Martinec, Z. and P. Vaníček,1996. Formulation of the boundary-value problem for geoid determination with a higher-order reference field.
*Geophysical Journal International*, 126, pp.219-228. - Vaníček,P., W. Sun, P. Ong, Z. Martinec, P. Vajda and B. ter Horst, 1996. Downward continuation of Helmert's gravity,
*Journal of Geodesy*71(1), pp.21-34. - Vaníček,P., M. Najafi, Z. Martinec, L. Harrie and L.E.Sjöberg, 1996. Higher-order reference field in the generalized Stokes-Helmert scheme for geoid computation.
*Journal of Geodesy*, 70 (3), pp. 176-182 . - Featherstone, W.E. and P. Vaníček,1997. To modify or not to modify?, IAG General Assembly, Rio de Janeiro, Sept. 3 to 9.
- Vaníček,P. and W. E. Featherstone, 1998. Performance of three types of Stokes's kernel in the combined solution for the geoid,
*Journal of Geodesy*, 72, 12, pp. 684-697. - Sun, W. and P. Vaníček,1998. On some problems of the downward continuation of 5' x 5' mean Helmert's gravity disturbance.
*Journal of Geodesy*, 72, 7-8, pp. 411- 420. - Najafi, M., P. Vaníček, P. Ong and M.R. Craymer, 1999. Accuracy of a regional geoid,
*Geomatica*53,3, pp. 297-305. - Huang, J., P. Vaníček and P. Novák, 2000. An alternative algorithm to FFT for the numerical evaluation of Stokes's integral.
*Studia Geophysica et Geodaetica*44, pp. 374-380. - Véronneau, M., S.D. Pagiatakis, P. Vaníček, P. Novák, J. Huang, J. Janák, M.G. Sideris and O. Esan, 2000. Canadian Gravimetric Geoid Model 2000 (CGG2000): Preliminary results. GGG2000 conference, Banff, July 31 -August 4.
- Huang, J., P. Vaníček, S. Pagiatakis and W. Brink, 2001. Effect of topographical mass density variation on gravity and the geoid in the Canadian Rocky mountains.
*Journal of Geodesy*74 (11-12), pp. 805-815. - Novák, P., P. Vaníček, Z. Martinec and M. Véronneau, 2001. The effect of distant terrain on gravity and the geoid.
*Journal of Geodesy*75 (9-10), pp. 491-504. - Huang, J., M.G.Sideris, P. Vaníček and I.N.Tsiavos, 2003. Numerical investigation of downward continuation techniques for gravity anomalies.
*Bollettino di Geofisica Teorica ed Applicata*LXII, No.1, pp. 34-48. - Vaníček,P., R.Tenzer, L.E. Sjöberg, Z. Martinec and W.E.Featherstone, 2004. New views of the spherical Bouguer gravity anomaly.
*Journal of Geophysics International*159(2), pp. 460-472. - Tenzer, R., P. Vaníček, M. Santos, W. E. Featherstone, and M. Kuhn, 2005. Rigorous orthometric heights.
*Journal of Geodesy*79, pp. 1432-1394. DOI: 10.1007/s001-005-0445-2. - Kingdon, R., P. Vaníček, M. Santos, A. Ellmann and R. Tenzer, 2005. Toward an improved height system for Canada.
*Geomatica*, Vol. 59, No. 3, pp. 241 to 249. - Baran, I., M. Kuhn, S.J. Claessens, W.E. Featherstone, S.A Holmes and P. Vaníček, 2006. A synthetic Earth's gravity model designed specifically for testing regional gravimetric geoid determination algorithms,
*Journal of Geodesy*80(1):1-16. - Santos, M., P. Vaníček, W. E. Featherstone, R. Kingdon, A. Ellmann, B.-A. Martin, M. Kuhn and R. Tenzer, 2006. The relation between rigorous and Helmert's definitions of orthometric heights.
*Journal of Geodesy*, Vol. 80, pp. 691-704. - Ellmann, A., P. Vaníček, M. Santos and R. Kingdon, 2007. Interrelation between the geoid and orthometric heights, First International Symposium of The International Gravity Field Service (IGFS), oral presentation, August 28 - September 1, 2006, Istanbul, Turkey. Forsberg R.; Kilicoglu, A. (Eds.). Proceedings of the 1st International Symposium of the International Gravity Field Service "Gravity Field of the Earth", pp. 130 - 135. General Command of Mapping, Ankara, Turkey.
- Kingdon, R., P. Vaníček and M. Santos 2009. Effects of hypothetical complex mass-density distributions on geoidal height. Oral presentation and
*Proceedings of IAG Scientific Assembly*, Buenos Aires, Argentina, August 31-September 4, Springer, New York. - Vaníček, P., R. Kingdon and M. Santos, 2012. Geoid versus quasi-geoid: a case of physics versus geometry,
*Contributions to Geophysics and Geodesy*, Vol. 42, No. 1, pp. 101-119. - Vaníček, P., R. Kingdon, M. Kuhn, A. Ellmann, W. E. Featherstone, M. C. Santos, Z. Martinec, C. Hirt, D. Avalos, 2013. Testing Stokes-Helmert geoid model computation on a synthetic gravity field: experiences and shortcomings,
*Studia Geophysica et Geodaetica*57 (2013), 369?400, DOI: 10.1007/s11200-012-0270-z.

## Most recent publications

- Vaníček P., and R.W.Kingdon, 2014. Gravimetry, a reference module in
*Earth Systems and Environmental Sciences*, Elsevier, doi: 10.1016/B978-0-12-409548-9.09145-4. - Foroughi, I., J. Janak, R.W. Kingdon, M. Sheng, M. Santos and P. Vaníček, 2015. Illustration of how satellite global field should be used in regional precise geoid modelling, oral presentation at EGU meeting, Vienna, May 4 to 8, 2015.
- Foroughi, I., P. Vaníček, R.W. Kingdon, M. Sheng and M. Santos, 2015. Assessment of discontinuity of Helmert's gravity anomalies on the geoid, poster at AGU/CGU meeting, Montreal, May 4 to 8, 2015.
- Sheng, M., P. Vaníček and R.W. Kingdon, 2015. Concerning the roundabout method of downward continuation and the evaluation of the far zone contribution, poster at AGU/CGU meeting, Montreal, May 4 to 8, 2015.
- Kingdon, R. W. and P.Vaníček, 2015. Continuation of surface gravity anomalies to the geoid in regions of negative heights, poster at AGU/CGU meeting, Montreal, May 4 to 8, 2015.
- Klu, M., P. Vaníček, R. Kingdon, M. Sheng, I. Foroughi, M. Santos, P. Dare, 2015. Towards the computation of a geoid model of Ghana, poster at AGU/CGU meeting, Montreal, May 4 to 8, 2015.
- Kingdon, R., P.Vaníček and D. Zhong, 2015. Least squares downward continuation, fusion and gridding of airborne and terrestrial gravity observations, presented at IUGG General Assembly, Prague, June 22 to July 2.
- Foroughi, I., M. Sheng, R. Kingdon, J. Huang, Z. Martinec, P.Vaníček and M. Santos, 2015. The effect of lateral topographical density variations on the geoid in Auvergne, poster at IUGG General Assembly, Prague, June 22 to July 2.
- Tavakoli, A., A. Safari and P. Vaníček, 2016. A special case of the Poisson PDE for formulated Earth's surface and its capability to approximate the terrain mass density employing land-based gravity data, a case study in the south of Iran.
*Geophysical Journal International*(Accepted for publication in September 2016). - Avalos, D. S. Pagiatakis, P. Vaníček, M. Santos and A. Alvarado, 2016. Towards the re-definition of the gravity network in Mexico, Presented as a poster at GGHS Symposium, Thessaloniki, Sept. 19 to 23, Greece.
- Avalos, D. P. Vaníček, M. Santos and J. Huang, 2016. Construction of the regional geoid model GGM-CA-2015, Presented as a poster at GGHS Symposium, Thessaloniki, Sept. 19 to 23, Greece.
- Foroughi, I., P. Vaníček, R. Kingdon, P. Novák, M. Sheng and M. Santos, 2016. Poisson downward continuation of scattered Helmert's gravity anomalies to mean values on a raster on the geoid using least squares. Poster presentation to GGHS Symposium, Thessaloniki, Sept. 19 to 23, Greece.
- Sheng, M.B., P. Vaníček, R. Kingdon and I. Foroughi, 2016. Rigorous evaluation of a satellite-only gravitational model, Presented as a poster at GGHS Symposium, Thessaloniki, Sept. 19 to 23, Greece.
- Vaníček, P, P. Novák, M. Sheng, R. Kingdon, J. Janák, I. Foroughi, Z. Martinec, and M. Santos, 2016. Does Poisson's downward continuation give physically meaningful results? (Submitted to
*Studia Geophysica et Geodaetica*, June 2016).

Current members:

Dr. Marcelo Santos (Professor and primary contact)

Dr. Petr Vaníček (Professor Emeritus)

Robert Kingdon (Instructor)

Ismael Foroughi (PhD. student)

Michael Sheng (PhD. student)

Past Members (in alphabetical order):

Alexander Garcia

Artu Ellmann

Azadeh Koohzare

David Avalos-Naranjo

James D. Mtamakaya

Jeff Wong

Jianliang Huang

Juraj Janak

Mehdi Najafi

Mensur Omerbasic

Michael Adjei Klu

Mustafa Berber

Pavel Novak

Peng Ong

Peter Vajda

Robert Tenzer

W. Sun