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# STOKES-HELMERT'S SOLUTION TO GEODETIC BOUNDARY VALUE PROBLEM

## PAPERS IN “PUBLIC BINDER”

### 1. Stokes-Helmert technique

**Vaníček P. and A. Kleusberg, 1987.**

The Canadian geoid - Stokesian approach, *Manuscripta Geodaetica* 12, pp. 86-98.

**Martinec, Z., C. Matyska, E.W. Grafarend and P. Vaníček, 1993.**

On Helmert's 2nd condensation method, *Manuscripta Geodaetica*, # 18, pp. 417-421.

**Vaníček, P. and Z.Martinec, 1994.**

The Stokes-Helmert scheme for the evaluation of a precise geoid, *Manuscripta Geodaetica* 19, pp.119-128.

**Vaníček, P., A. Kleusberg, Z. Martinec, W. Sun, P. Ong, M. Najafi, P. Vajda, L. Harrie, P. Tomasek and B. ter Horst, 1995.**

Compilation of a precise regional geoid, DSS Contract # 23244-1-4405/01-SS Report for Geodetic Survey Division, Ottawa, 45 pp.

**Vaníček, P. and J. Janák, 2000.**

The UNB technique for precise geoid determination, CGU meeting, Banff, May 24-26.

**Tenzer,R. and J. Janák, 2002.**

Stokes-Helmert's scheme for precise geoid determination, *Revista Cartografica*, 74-75, pp. 135-145.

**Ellman, A., P.Vaníček, M. Santos, 2005.**

Precise geoid determination for geo-referencing and oceanography, Poster presented at the annual meeting of the GEOIDE, Quebec City, May 29-31.

**Ellmann, A., and P Vaníček (2007).**

"UNB application of Stokes–Helmert's approach to geoid computation." *Journal of Geodynamics*, Vol. 23, pp. 200–213.

### 2. Reference field

**Vaníček P. and L.E. Sjøberg, 1991.**

Reformulation of Stokes's theory for higher than second degree reference field and modification of integration kernels, *JGR* 96 (B4), pp. 6529-6539.

**Vaníček, P., M. Najafi, Z. Martinec, L. Harrie and L.E.Sjöberg, 1996.**

Higher-degree reference field in the generalized Stokes-Helmert scheme for geoid computation. *Journal of Geodesy* , 70 (3), pp. 176-182 .

**Martinec, Z. and P. Vaníček, 1996.**

Formulation of the boundary-value problem for geoid determination with a higher-degree reference field, *Geophys. J. Int.* 126, pp. 219-228.

**Najafi, M., 1996.**

Contributions towards the computation of a precise regional geoid. Doctoral thesis, University of New Brunswick, Fredericton, Canada.

### 3. Topographical effects

**Martinec, Z., 1993.**

Effect of lateral density variations of topographical masses in improving geoid model accuracy over Canada. Contract report for Geodetic Survey of Canada, Ottawa.

**Martinec, Z. and P. Vaníček, 1994.**

The indirect effect of topography in the Stokes-Helmert technique for a spherical approximation of the geoid. *Manuscripta Geodaetica* 19, pp. 213-219.

**Martinec, Z. and P. Vaníček, 1994.**

Direct topographical effect of Helmert's condensation for a spherical approximation of the geoid. *Manuscripta Geodaetica*, # 19, pp. 257-268.

**Sjöberg, L.E. 1994.**

The total terrain effects in geoid and quasigeoid determinations using Helmert's second condensation method. Dept of Geodesy and Photogrammetry, KTH, Stockholm. 9p.

**Martinec, Z., P. Vaníček, A. Mainville and M. Véronneau, 1995.**

The effect of lake water on geoidal heights, *Manuscripta Geodaetica* , 20, pp. 193-203.

**Martinec, Z., P. Vaníček, A. Mainville and M. Véronneau, 1996.**

Evaluation of topographical effects in precise geoid determination from densly sampled heights, *Journal of Geodesy* , 70(11), pp. 746-754.

**Sjöberg, L.E. and H. Nahavandchi, 1999.**

On the indirect effect in the Stokes-Helmert method of geoid determination, *Journal of Geodesy*, 73, pp. 87-93.

**Vaníček, P., and P. Novák, 1999.**

Comparison between planar and spherical models of topography. CGU Annual Meeting, Banff, May 9 -13, 1999.

**Novák, P. (2000).**

Evaluation of Gravity Data for the Stokes-Helmert Solution to the Geodetic Boundary-Value Problem. Ph.D. dissertation, Department of Geodesy and Geomatics Engineering Technical Report No. 207, University of New Brunswick, Fredericton, New Brunswick, Canada, 133 pp. (http://gge.unb.ca/Pubs/TR207.pdf)

**Huang, J., P. Vaníček, S. Pagiatakis and W. Brink, 2001**.

Effect of topographical mass density variation on geoid in the Canadian Rocky Mountains. *Journal of Geodesy*, 74, pp. 805-815.

**Nahavandchi, H. and L.E. Sjöberg, 2001.**

Two different views of topographical and downward-continuation corrections in the Stokes-Helmert approach to geoid computation, *Journal of Geodesy*, 74, pp. 816-822.

**Tenzer, R., P. Vaníček, S. van Eck der Sluijs and A.Hernandez, 2004**.

On some numerical aspects of primary indirect topographical effect computation in the Stokes-Helmert theory of geoid determination. *Revista Cartografica*, Vol. 76-77, pp. 71-77.

**Janák J., Vaníček P., Alberts B. (2011)**.

Numerical evaluation of mean values of topographical effects. *Journal of Geodetic Science*, 1, 2, pp. 89-93, DOI: 10.2478/v10156-010-01106.

### 4. Downward continuation

**Sjöberg, L.E., 1995.**

On the Downward Continuation Error in Geoid Computation from Satellite Derived Geopotential Models. EGS XX General Assembly, Hamburg, 3-7 April 1995, 8p.

**Vaníček, P. and W. Sun, 1995.**

Downward continuation of Helmert's gravity. Paper presented at CGU Annual meeting, Banff, May ABSTRACT ONLY.

**Sun, W. and P. Vaníček, 1996.**

On the discrete problem of downward continuation of Helmert's gravity. Proceedings of Session G7 (Techniques for local geoid determination), Annual meeting of European Geophysical Society, The Hague, May 6-10, 1996, Reports of the Finnish Geodetic Institute, 96:2, pp. 29-34.

**Martinec, Z., 1996.**

Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains, *Journal of Geodesy*, 70/11, 805-828.

**Vaníček, P., W. Sun, P. Ong, Z. Martinec, M. Najafi, P. Vajda and B ter Horst, 1996**.

Downward continuation of Helmert's gravity, *Journal of Geodesy* 71, pp. 21-34.

**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.

**Vaníček, P. and G.Wong, 1999.**

On the downward continuation of Helmert's gravity anomalies. Paper presented at AGU Spring meeting, Boston. ABSTRACT ONLY.

**Huang, J., S. Pagiatakis and P. Vaníček, 2001.**

On some numerical aspects of downward continuation of gravity anomalies, Proceedings of IAG General Assembly, Budapest, Sept. 3 to 7, Paper #58BD.

**Huang.,J. 2001.**

Computational methods for the discrete downward continuation of the earth gravity. The 5th Canadian geoid workshop, Ottawa, May 16-17

**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.

**Tenzer, R, and P. Vaníček, 2005**.

Proof of equivalence between Poisson's and analytical continuation and the equivalence of integral mean of gravity and the integral mean of analytically continued gravity *Studia Geodaetica et Geofysica* (submitted in June 2004).

**Huang, Jianliang (2002).**

Computational Methods for the Discrete Downward Continuation of the Earth Gravity and Eff ects of Lateral Topographical Mass Density Variation on Gravity and the Geoid. Ph.D. dissertation, Department of Geodesy and Geomatics Engineering, Technical Report No. 216, University of New Brunswick, Fredericton, New Brunswick, Canada, 141 pp.(http://gge.unb.ca/Pubs/TR216.pdf)

**Vaníček P. and M. C. Santos (2010).**

“Can mean values of Helmert's gravity anomalies be continued downward directly?” *Geomatica*, Vol. 64, No. 2, pp. 245-251.

### 5. Stokes's integration

**Vaníček, P. and W. E. Featherstone, 1997.**

To modify or not to modify? Paper presented at IAG General Assembly, Rio de Janeiro, September 2-8. ABSTRACT ONLY.

**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.

**Huang, J. and P. Vanicek, 1999**

A faster algorithm for numerical Stokes's integration. CGU annual conference, Banff, May ABSTRACT ONLY.

**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.

### 6. Far zone contributions

**Novák P. and P. Vaníček, 1999.**

Effect of distant topographical masses on geoid determination. CGU annual conference, Banff, May. ABSTRACT ONLY.

**Vaníček, P. and J. Janák, 2000.**

Truncation of spherical convolution integration with an isotropic kernel, Algorithms 2000 conference, Tatranska Lomnica, Slovakia, September 15-18.

**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.

**Hernandez, N.A., M.R. Gomez and P. Vaníček, 2002.**

The far zone contribution in spherical Stokes's integration, *Revista Cartografica* 74-75, pp. 61-74.

**Tenzer, R., P. Vaníček and P. Novák, 2003.**

Far-zone contribution to the topographical effects in the Stokes-Helmert method of geoid determination. *Studia Geophysica et Geodaetica* 47, pp. 467-480.

**Tenzer, R., P. Vaníček and S. van Eck der Sluijs, 2003**.

The far-zone contribution to upward continuation of gravity anomalies. *Revista Brasileira de Cartografia*, 55(2), pp. 48-54.

**Vaníček, P., J. Janák and W.E. Featherstone, 2003.**

Truncation of spherical convolution integration with an isotropic kernel, *Studia Geophysica et Geodaetica*, 47 (3), pp. 455-465.

### 7. Accuracy

**Sjöberg, L.E., 1996.**

On the error of analytical continuation in physical geodesy,* Journal of Geodesy*, 70, pp. 724-730.

**Najafi, M., P. Vaníček, P. Ong and M.R. Craymer, 1999.**

On the accuracy of a regional geoid, *Geomatica* 53,3, pp. 297-305.

**Vaníček P., J. Janák and M. Véronneau, 2000.**

Impact of Digital Elevation Models on Geoid Modelling, Geomatics 2000, Montreal, March 8.

**Novák, P., P. Vaníček, M. Véronneau, S.A. Holmes and W.E. Featherstone , 2001.**

On the accuracy of modified Stokes's integration in high-frequency gravimetric geoid determination. *Journal of Geodesy* 74, pp. 644-654.

**Janák, J. and P. Vaníček, 2001.**

Systematic error of the geoid model in the Rocky Mountains, CGU annual conference, Ottawa, May 15-17, 2001. ABSTRACT ONLY.

### 8. Results

**Vaníček, P., A. Kleusberg, Z. Martinec, W. Sun, P. Ong, M. Najafi, P. Vajda, L. Harrie, P. Tomasek and B ter Horst, 1995.**

Compilation of a precise regional geoid, Final report on DSS contract # 23244-1-4405/01- SS, Fredericton.

**Janák J. and P. Vaníček, (2000).**

UNB North-American geoid 2000 model: theory, intermediate and final results. Second Annual GEOIDE Conference, Calgary. May 25-26, 2000. Oral.

**Vaníček, P. and J. Janák, 2001.**

Refinement of the UNB geoid model: progress report for proj.#10, poster presentation at GEOIDE annual meeting, Fredericton, June 21-22. ABSTRACT ONLY

**Janák, J. and P. Vaníček, 2001.**

Improvement of the University of New Brunswick's gravimetric geoid model for Canada, poster presentation at IAG General Assembly, Budapest, Sept. 3 to 7.

**Hernandez-Navarro, A., 2004.**

The submetric geoid of Mexico. Paper presented at the FIG Working Week, Athens, Greece, May 22-27.

**Ellmann, A., P. Vaníček and M. Santos, 2005.**

Report on precise geoid determination for geo-referencing and oceanography, Poster presentation at GEOIDE annual meeting in Quebec City, June.

### 9.Other topics

**Vaníček P., Zhang Changyou and L.E. Sjøberg, 1992.**

A comparison of Stokes's and Hotine's approaches to geoid computation, *Manuscripta Geodaetica* 17, pp. 29-35.

**Vaníček P., M. Véronneau and Z. Martinec, 1997.**

Determination of mean Helmert's anomalies on the geoid. Paper presented at IAG General Assembly, Rio de Janeiro, September 2-8. ABSTRACT ONLY.

**Martinec, Z., 1998.**

Construction of Green's function to the Stokes boundary-value problem with ellipsoidal corrections in boundary condition, *Journal of Geodesy*, 72, pp. 460-472.

**Vaníček, P.,J. Huang, P. Novák, M. Véronneau, S. Pagiatakis, Z. Martinec and W. E. Featherstone, 1999.**

Determination of boundary values for the Stokes-Helmert problem. *Journal of Geodesy* 73, pp.180-192.

**Vaníček, P., J. Janák and J. Huang, 2000.**

Mean Vertical Gradient of Gravity, Poster presentation at GGG2000 conference, Banff, July 31 – August 4. GGG2000 Proceedings (Ed. M.Sideris), pp. 259-262.

**Featherstone, W.E. and J.F. Kirby, 2000.**

The reduction of aliasing in gravity anomalies and geoid heights using digital terrain data, *Geophysical Journal International*, 141, pp. 204-212.

**Vaníček, P., P. Novák and Z. Martinec, 2001.**

Geoid, topography, and the Bouguer plate or shell. *Journal of Geodesy*, 75 (4), pp. 210-215.

**Tenzer, R. and P. Vaníček, 2003.**

Geoid-quasigeoid correction in the formulation of the fundamental formula of physical geodesy. *Revista Brazileira de Cartografica*, No.55(1), pp. 57 – 61.

**Huang, J., M. Véronneau and S. Pagiatakis, 2003.**

On the ellipsoidal correction to the spherical Stokes solution of the gravimetric geoid. *Journal of Geodesy*, 77, pp. 171-181.

**Vaníček, P., R.Tenzer, and J. Huang, 2003.**

Role of “No Topography space” in Stokes-Helmert's scheme for geoid determination, paper presented at CGU annual meeting, Banff, May.

**Vaníček, P., R.Tenzer, and M. Santos, 2003.**

New look at spherical Bouger anomaly. Poster presentation at IUGG Assembly in Sapporo, Japan,

**Vajda, P., P.Vaníček, P. Novák, and B. Meurers, 2004.**

On the evaluation of Newton integrals in geodetic coordinates: Exact formulation and spherical approximation. *Contributions to Geophysics and Geodesy*, 34 (4), pp. 289–314.

**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.

**Janák, J. and P. Vaníček, 2004.**

Mean free-air gravity anomalies in the mountains. *Studia Geophysica et Geodaetica*, 49(1), pp. 31-42.

**P. Novák and E.W.Grafarend, 2005.**

Ellipsoidal representation of the topographical potential and its vertical gradient. *Journal of Geodesy*, 79.

**Sjoberg, L.E., 2005.**

A discussion on the approximations made in the practical implementation of the remove-compute-restore technique in regional geoid modeling. *Journal of Geodesy*, 79.

**Wong, Jeff Chak Fu (2002).**

On Picard Criterion and the Well-Posed Nature of Harmonic Downward Continuation. M.Sc.E. thesis, Department of Geodesy and Geomatics Engineering Technical Report No. 213, University of New Brunswick, Fredericton, New Brunswick, Canada, 85 pp. (http://gge.unb.ca/Pubs/TR213.pdf)