Fitting a Triaxial Ellipsoid to a Geoid Model

J. Geod. Sci. 2020; 10:69–82

Research Article Open Access

G. Panou*, R. Korakitis, and G. Pantazis Fitting a triaxial ellipsoid to a geoid model

DOI: https://doi.org/10.1515/jogs-2020-0105 comprehensive list by Heiskanen (1962). Also, recent de- Received March 10, 2020; accepted July 8, 2020 terminations of the parameters of triaxiality (semi-axes ax, Abstract: The aim of this work is the determination of the ay and b, longitude of major semi-axis λ0) are presented parameters of Earth’s triaxiality through a geometric t- in Table 1. However, the dierence between the equatorial ting of a triaxial ellipsoid to a set of given points in space, semi-axes ax and ay, as determined in the satellite era, is as they are derived from a geoid model. Starting from a smaller than the one previously proposed. Some of the rea- Cartesian equation of an ellipsoid in an arbitrary refer- sons have been explained in Burša (1972). ence system, we develop a transformation of its coe- Generally, there are three methods for the determina- cients into the coordinates of the ellipsoid center, of the tion of the parameters of triaxiality: i) the astrogeodetic, three rotation angles and the three ellipsoid semi-axes. ii) the gravimetric and iii) the satellite method. The gravi- Furthermore, we present dierent mathematical models metric method is given in detail by Heiskanen (1962). In for some special and degenerate cases of the triaxial el- this method, the parameters of triaxiality are determined lipsoid. We also present the required mathematical back- using the least-squares method by minimizing the sum ground of the theory of least-squares, under the condition of squares of gravity anomalies ∆g. However, in the pre- of minimization of the sum of squares of geoid heights. satellite era, as mentioned by Heiskanen (1962), the in- Also, we describe a method for the determination of the ability to determine the triaxiality parameters with the re- foot points of the set of given space points. Then, we pre- quired accuracy is due to the lack of enough data. The pare suitable data sets and we derive results for various satellite method is described in several papers, e.g. Burša geoid models, which were proposed in the last fty years. (1970, 1971, 1972) and Burša and Šíma (1980). In this Among the results, we report the semi-axes of the triax- method, the requested parameters are determined using ial ellipsoid of geometric tting with four unknowns to be the least-squares method by minimizing the radial dis- 6378171.92 m, 6378102.06 m and 6356752.17 m and the equa- tance between a given geoidal point and the correspond- torial longitude of the major semi-axis –14.9367 degrees. ing point on the surface of the ellipsoid. On the other hand, Also, the parameters of Earth’s triaxiality are directly esti- once some gravity eld model is adopted, from the second- mated from the spherical harmonic coecients of degree degree coecients of spherical harmonics, the orienta- and order two. Finally, the results indicate that the geoid tions of the principal axes and the magnitudes of the prin- heights with reference to the triaxial ellipsoid are smaller cipal moments may be determined by a well-known theory than those with reference to the oblate spheroid and the (Chen and Shen, 2010). improvement in the corresponding rms value is about 20 Nowadays, we can obtain all the data required for per cent. applying any of the aforementioned methods. Indeed, there are several satellite-only or combined global grav- Keywords: algebraic tting, coordinates conversion, geo- ity eld models available by the International Cen- metric tting, least squares method, principal axes tre for Global Earth Models (ICGEM) (http://icgem.gfz- potsdam.de/home.html). Also, according to Barthelmes 1 Introduction (2014), the satellite-only models are computed from satellite measurements alone, whereas ground and altimetry The problem of determining the parameters of Earth’s tri- data are additionally used for the combined models. Fur- axiality has been recognized since 1859 and various treat- thermore, the spatial resolution of the satellite-only mod- ments by dierent scientists have been reported in the els is lower, but the accuracy is nearly uniform over the Earth. In this work, we determine the parameters of Earth’s *Corresponding Author: G. Panou: Department of Surveying En- triaxiality through a least-squares tting of a triaxial el- gineering, National Technical University of Athens, 15780 Athens, Greece, E-mail: [email protected] lipsoid to a set of ‘almost equally spaced’ geoid points, R. Korakitis, G. Pantazis: Department of Surveying Engineering, whose coordinates are of equal precision and are derived National Technical University of Athens, 15780 Athens, Greece

Open Access. © 2020 G. Panou et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. 70 Ë G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model

Table 1. Parameters of Earth’s triaxiality in the satellite era

◦ Reference ax (m) ay (m) b (m) λ0 ( ) ax − ay (m) Burša and Fialová (1993) 6378171.36 ±0.30 6378101.61 ±0.30 6356751.84 ±0.30 –14.93 ±0.05 69.75 ±0.42 Burša and Šíma (1980) 6378172 6378102.7 6356752.6 –14.9 69.3 Eitschberger (1978) 6378173.435 6378103.9 6356754.4 –14.8950 69.5 Burša (1972) 6378173 6378104 6356754 –14.8 69 Burša (1970) 6378173 ±10 6378105 ±18 6356754 ±10 –14.8 ±5 68 ±21 from a global gravity eld model. The condition for the of the triaxial ellipsoid, including the oblate spheroid tting is the minimization of the sum of squares of geoid and the sphere. heights N, known as geometric tting (non-linear), using – the problem of geometric tting is solved following approximate values for the unknowns derived from an al- an iterative approach, as a least-squares problem with gebraic tting (linear). Theoretically, the results of geo- constraints involving additional parameters and us- metric tting are dierent from those of algebraic tting ing a very large data set. In contrast, Bektas (2015) and some of the advantages of the geometric tting are re- uses the Levenberg-Marquardt algorithm for solving ferred to in Bektas (2015). The required mathematical back- the non-linear least-squares problem of geometric t- ground of the theory of least-squares is included in Mikhail ting, in order to determine its parameters. In addition, (1976) and Dermanis (2017). Furthermore, in order to illu- he uses a general algorithm for the determination of minate the dierences between the triaxial ellipsoid and the distance between a data point and its closest point the oblate spheroid, which has been adopted in geodesy, on the ellipsoid, as presented in Bektas (2014). Finally, we consider special and degenerate cases of the triaxial el- as an application of his method, he presents a numer- lipsoid in the tting. For this scope, we start from a Carte- ical example for a virtual body using only 12 points. sian equation of an ellipsoid in an arbitrary reference sys- – the presented procedure can be also used for a geomet- tem, as given in Hirvonen (1971), and we develop a trans- ric tting of the physical surface of the Earth, as men- formation of the unknowns of the tting into parameters, tioned in Tserklevych, Zaiats and Shylo (2016), as well following the algorithm of determining an error ellipsoid as of the physical surfaces of other celestial bodies, from a 3 × 3 variance-covariance matrix (for the details see as presented in Karimi, Ardalan and Farahani (2016, Hirvonen (1971)). Taking into account the improvement of 2017), who use an algebraic tting in their work. This the global gravity eld models in the last fty years, we capability is of special importance for other celestial also derive results for several past geoid models. Finally, bodies, where the lack of gravity eld models does not we compare the results with those directly derived from the permit the estimation of their triaxial parameters. coecients of a global gravity eld model. – it vividly demonstrates and veries the numerical sta- The main contributions of the present work can be bility of the values of the triaxial parameters of the summarized as follows: Earth. As pointed out in Heiskanen (1962), the triaxial- – it oers updated values for the triaxial parameters of ity is geophysically interesting and important. For ex- the Earth, derived from a large volume of contempo- ample, it is important in modifying the Earth rotation rary data, lling the gap in the relevant geodetic lit- theory, since it is closely related to the principal mo- erature since the seventies, which was mostly due to ments of inertia of the Earth (Chen and Shen, 2010). the use of the oblate spheroid as the adopted refer- Hence, this task should be a challenge for the follow- ence surface. Nowadays, however, the accuracy re- ing years. quirements are higher and the computing capabilities are greatly improved. – starting from a Cartesian equation of an ellipsoid in an arbitrary reference system, we develop a transformation of its polynomial coecients into geometrical parameters, like the coordinates of the ellipsoid center, the three rotation angles and the three ellipsoid semi-axes. Furthermore, we present dierent mathematical models for some special and degenerate cases G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model Ë 71

2 Cartesian equation of an ellipsoid 2.1 Triaxial ellipsoid and special cases in an arbitrary reference system 2.1.1 General case T1

The Cartesian equation of an ellipsoid, in an arbitrary ref- Equation (1) is equivalently written as erence system xyz, may be given by (Hirvonen 1971)   pxx pxy pxz  −1 h i q q q x − tx y − ty z − tz  pxy pyy pyz  h i xx xy xz     p p p f (x, y, z) ˙= x − tx y − ty z − tz  qxy qyy qyz  xz yz zz   qxz qyz qzz x − tx     = 1 (4) x − tx  y − ty    z − t  y − ty  = 1 (1) z z − tz or where the elements of vector pxx pyy pzz 2pxy 2pxz 2pyz x2 + y2 + z2 + xy + xz + yz   d d d d d d tx 2 2   − (pxx tx + pxy ty + pxz tz) x − (pxy tx + pyy ty + pyz tz) y t =  ty  (2) d d tz 2 − (pxz tx + pyz ty + pzz tz) z − 1 = 0 (5) d are the coordinates of the ellipsoid center and the elements where of symmetric matrix 2 2 2   d = 1− pxx tx − pyy ty − pzz tz −2pxy tx ty −2pxz tx tz −2pyz ty tz q q q xx xy xz (6) Q =  q q  (3)  yy yz  Equation (5) is of the form sym. qzz 2 2 2 cxx x +cyy y +czz z +cxy xy+cxz xz+cyz yz+cx x+cy y+cz z = 1 are functions of the three rotation angles θx, θy and θz, (7) and the three ellipsoid semi-axes ax, ay and b (see subsection 2.4). Thus, the mathematical model given by Eq. (1), includes nine parameters corresponding to the spatial 2.1.2 Special cases T2 – T6 properties of an ellipsoid (translation, orientation, semi- axes). Also, we can consider dierent cases of an ellipsoid In the following, we present the equations of special cases (T1 – S4), as presented in Table 2. In this table, the symbol of a triaxial ellipsoid, which correspond to the forms T2 – “x” refers to a parameter that will be determined, whereas T6 of Table 3. the symbol “0” indicates that the corresponding parame-  −1   ter is not included in the mathematical model. Thus, apart q q q x h i xx xy xz from the general case (T1) of a triaxial ellipsoid, we con-     T2 : x y z  qxy qyy qyz   y  = 1 (8) sider some special cases (T2-T6), two cases (B3 and B4) of qxz qyz qzz z an oblate spheroid without rotations and two cases (S3 and S4) of a sphere.  −1 For each of the above mentioned cases, we present the q 0 0 h i xx   corresponding mathematical model, as follows. T3 : x − tx y − ty z − tz  0 qyy 0  0 0 qzz   x − tx    y − ty  = 1 (9) z − tz

 −1   q 0 0 x h i xx     T4 : x y z  0 qyy 0   y  = 1 0 0 qzz z (10) 72 Ë G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model

Table 2. Parameters for dierent cases of a triaxial ellipsoid and degenerate cases

Case Parameters

tx ty tz θx θy θz ax ay b

T1 x x x x x x x x x T2 0 0 0 x x x x x x T3 x x x 0 0 0 x x x T4 0 0 0 0 0 0 x x x T5 x x x 0 0 x x x x T6 0 0 0 0 0 x x x x B3 x x x 0 0 - x x B4 0 0 0 0 0 - x x S3 x x x --- x S4 0 0 0 --- x

 −1 q q 0 2.3 Sphere h i xx xy   T5 : x − tx y − ty z − tz  qxy qyy 0  0 0 qzz In the degenerate case of a sphere, Eq. (1) is written as    −1 x − tx q 0 0 h i  y − ty  = 1 (11)     S3 : x − tx y − ty z − tz  0 q 0  z − tz 0 0 q    −1   x − tx qxx qxy 0 x h i  y − ty  = 1 (15) T6 : x y z  qxy qyy 0   y  = 1 (12)       z − tz 0 0 qzz z which corresponds to the form S3 of Table 3. Finally, in the special case of a sphere without translation, Eq. (1) is writ- 2.2 Oblate spheroid without rotations ten as  −1   q 0 0 x We now present the equations of special cases of an oblate h i     spheroid without rotations, which correspond to the forms S4 : x y z  0 q 0   y  = 1 (16) B3 and B4 of Table 3. 0 0 q z  −1 q 0 0 which corresponds to the form S4 of Table 3. h i   The determination of m0 elements of vector c (i.e. of B3 : x − tx y − ty z − tz  0 q 0  0 0 qzz the unknowns) may be accomplished by either an alge-   braic or a geometric t to a given set of k points, where x − t x k > m , as presented in Sections 3 and 5, respectively.   0  y − ty  = 1 (13) z − tz

 −1   2.4 Transformation of vector c into vector t q 0 0 x h i     and matrix Q (or into parameters) B4 : x y z  0 q 0   y  = 1 (14) 0 0 q z zz In this subsection, we present the transformation of the elements of the vector c into the vector t and the matrix Q for the general case of a triaxial ellipsoid (Case T1). From Eqs. (5) to (7), we have    −1   tx 2cxx cxy cxz cx       t =  ty  = − cxy 2cyy cyz   cy  (17) tz cxz cyz 2czz cz G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model Ë 73

Table 3. Mathematical models for dierent cases of a triaxial ellipsoid and degenerate cases

Case m0 Mathematical model 2 2 2 T1 9 cxx x + cyy y + czz z + cxy xy + cxz xz + cyz yz + cx x + cy y + cz z = 1

2 2 2 T2 6 cxx x + cyy y + czz z + cxy xy + cxz xz + cyz yz = 1

2 2 2 T3 6 cxx x + cyy y + czz z + cx x + cy y + cz z = 1

2 2 2 T4 3 cxx x + cyy y + czz z = 1

2 2 2 T5 7 cxx x + cyy y + czz z + cxy xy + cx x + cy y + cz z = 1

2 2 2 T6 4 cxx x + cyy y + czz z + cxy xy = 1

2 2 2 B3 5 c x + y + czz z + cx x + cy y + cz z = 1

2 2 2 B4 2 c x + y + czz z = 1

2 2 2 S3 4 c x + y + z + cx x + cy y + cz z = 1 S4 1 c x2 + y2 + z2 = 1 and i.e. the semi-axes of the ellipsoid −1  cxy cxz  cxx 2 2 p p p ax = λ1, ay = λ2, b = λ3 (24) Q = D cxy cyz  (18)  2 cyy 2  cxz cyz We note that the rows of matrix R are the components 2 2 czz of the unit vectors that correspond to the semi-axes of the where ellipsoid. Also, from the 2 × 2 sub-matrices of matrix Q 2 2 2 D = 1+cxx tx +cyy ty +czz tz +cxy tx ty +cxz tx tz +cyz ty tz (19) we obtain ellipses on the three planes containing the axes of the xyz system. This facilitates the investigation of the Once the matrix Q is determined, we apply the Singu- proper sign of the rotation angles. Obviously, we apply the lar Value Decomposition (SVD is a well-known procedure, appropriate parts of the transformation in all special and included in most mathematical software packages) and we degenerate cases. obtain Q = RTΛR (20) which gives directly the orthogonal rotation matrix 3 Algebraic (linear) tting   r11 r12 r13   In this section, we present the process of the algebraic t- R =  r21 r22 r23  r31 r32 r33 ting in the general case of a triaxial ellipsoid (T1), in order   to obtain the best estimates of elements of vector c. We use cy cz cx sz + sx sy cz sx sz − cx sy cz   the method of least squares, assuming that the Cartesian =  −cy sz cx cz − sx sy sz sx cz + cx sy sz  (21) coordinates of the given points (Xi , Yi , Zi) are of equal pre- sy −sx cy cx cy cision, uncorrelated and are identical to the coordinates of where, for instance, cx sz = cosθx sinθz, i.e. the rotation foot points (xi , yi , zi). Thus, we can write the linear Eq. (7) angles in the form   2 2 2 cxx xi + cyy yi + czz zi + cxy xi yi + cxz xi zi + cyz yi zi + cx xi + cy yi −r32 r31 θx = atan , θy = atan q  , r33 2 2 + cz zi = 1 + υi , i = 1,..., k (25) r11 + r21 which can be represented in matrix form as −r21 θz = atan (22) r11 Dc = i + υ (26) and the diagonal matrix where  2 2    x1 y1 ··· z1 λ 0 0 1  . . .. .    D =  . . . .  (27) Λ =  0 λ2 0  (23)  . .  2 2 0 0 λ3 xk yk ··· zk 74 Ë G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model

is a k × m0 matrix containing the coecients of the ele- Also, we can compute the statistical indexes min (N), ments of vector c (design matrix), i is a k × 1 vector with max (N), k all elements equal to 1 and υ is a k × 1 vector of residuals. 1 X mean (N) = N (32) Then, the solution vector is given by k i i=1 −1 c = DTD DTi (28) and the root mean square (rms) v u k Subsequently, we can compute the vector t, the rota- u1 X rms (N) = t N2 (33) tion matrix R, i.e. the rotation angles and the matrix Λ, i.e. k i i=1 the semi-axes of the ellipsoid, following the descriptions in subsection 2.4. All special and degenerate cases can be easily obtained by suitable modications of the matrix D. 5 Geometric (non-linear) tting

In a similar manner, in this section we present the proce- 4 Determination of the foot points dure of the geometric tting in the general case of a triaxial and statistical indexes ellipsoid (T1). For the given points (Xi , Yi , Zi) (assuming unweighted and uncorrelated data) and their foot points Before applying the geometric (non-linear) tting, at rst (xi , yi , zi), i = 1,..., k, the ‘observation’ equations are we compute the projections (xi , yi , zi) of the given points xi = Xi + υχi (34) (Xi , Yi , Zi) on the surface of the ellipsoid with the estimated parameters, in the same reference system xyz. We yi = Yi + υYi (35) rst convert the coordinates of the given points (Xi , Yi , Zi) to the reference system uvw, aligned to the axes of the zi = Zi + υZi (36) estimated ellipsoid, using the known translation vector t while the constraints are (Eq. (2)) and the rotation matrix R (Eq. (21)), as follows: 2 2 2       gi ˙=cxx xi + cyy yi + czz zi + cxy xi yi + cxz xi zi + cyz yi zi + cx xi Ui r11 r12 r13 Xi − tx       + cy yi + cz zi = 1 (37)  Vi  =  r21 r22 r23   Yi − ty  , i = 1,..., k Wi r31 r32 r33 Zi − tz In the above formulation, we have n = 3k mea- (29) surements (observations), for the determination of 3k + For every point (U , V , W ), the desired projection i i i m0 unknowns (3k for (xi , yi , zi) and m0 for c) sub- (ui , vi , wi) can be computed by any of the relevant meth- ject to k constraints. Thus, the actual number of un- ods of conversion of Cartesian to geodetic coordinates (e.g. knowns is m = (3k + m0) − k = 2k + m0. Since Ligas (2012), Panou and Korakitis (2019)), using the known the constraints are non-linear equations, we need to semi-axes ax, ay and b. Finally, the foot points (ui , vi , wi) linearize all equations using approximate values r0 = T are expressed in the reference system xyz, as follows: h 0 0 0 0 0 0 i 0 x1 y1 z1 ··· xk yk zk and c for the un-         0 0 xi tx r11 r21 r31 ui knowns ˆr = r + δr and ˆc = c + δc. The approximate   =   +     , 0 0 0  yi   ty   r12 r22 r32   vi  values xi , yi , zi may be computed as foot points on the 0 zi tz r13 r23 r33 wi ellipsoid with parameters c resulting from the algebraic i = 1,..., k (30) tting (Section 3), using the approach proposed in Section 4. Thus, the linear approximation of Eqs. (34) to (36) and since R−1 = RT. From the foot points, we can compute the (37) can be represented in matrix form as geoid heights by δr = δl + υ (38) q N = (U − u )2 + (V − v )2 + (W − w )2 i i i i i i i and q 2 2 2 Cδr + Dδc = w (39) = (Xi − xi) + (Yi − yi) + (Zi − zi) , i = 1,..., k (31) where G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model Ë 75

 0 0 0  ∂g1 ∂g1 ∂g1 0 0 0 ··· 0 0 0 ∂x1 ∂y1 ∂z1  0 0 0   ∂g2 ∂g2 ∂g2   0 0 0 ∂x ∂y ∂z ··· 0 0 0  C = J0 =  2 2 2  (40) gr  ......   ......   ......   0 0 0  0 0 0 0 0 0 ··· ∂gk ∂gk ∂gk ∂xk ∂yk ∂zk is a k × n matrix containing the partial derivatives (Jacobian matrix)

∂gi = 2cxx xi + cxy yi + cxz zi + cx (41) ∂xi

∂gi = 2cyy yi + cxy xi + cyz zi + cy (42) ∂yi

∂gi = 2czz zi + cxz xi + cyz yi + cz (43) ∂zi 0 0 0 0 D = Jgc is a k × m0 matrix, as in the linear tting (Eq. (27)) for the approximate values of the foot points xi , yi , zi ,

 0  X1 − x1  0   Y1 − y1   0   Z1 − z1     .  δl =  .  (44)    X − x0   k k   0   Yk − yk  0 Zk − zk is a n × 1 vector, υ is a n × 1 vector of residuals and

 0 0 0 0  1 − g1 x1, y1, z1, c  .  w =  .  (45)  .  0 0 0 0 1 − gk xk , yk , zk , c is a k × 1 vector. The above problem constitutes a least squares problem with constraints which involve additional parameters (see e.g. Mikhail (1976)). Then, the corrections δc and δr to the approximate values, in our case where A = I and the weight matrix P = I, are δc = L−1DTK−1 (w − Cδl) (46) and δr = δl + CTK−1 (w − Cδl − Dδc) (47) where K = CCT (48) is a diagonal matrix and L = DTK−1D (49) 2 Furthermore, the a-posteriori variance factor σˆ0 is computed by

T T 2 υ υ (δr − δl) (δr − δl) σˆ0 = = (50) n − m k − m0 where n − m represents the degrees of freedom, while the vector of residuals υ is computed from Eq. (38). Finally, the a-posteriori variance-covariance matrix of the vector ˆc is given by

ˆ 2 −1 Vˆc = σˆ0L (51) 76 Ë G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model

We then compute the coordinates of the ellipsoid cen- 6 Data and numerical results ter, the rotation angles and the semi-axes of the ellipsoid, according to the description in subsection 2.4. Theoretically, the solution process is iterative until the 6.1 Data corrections to the approximate values become negligible. It is worth emphasizing that the minimization problem The methodology described above was applied to an ex- solved in the geometric approach satises the equation tensive set of points in space, corresponding to points of Pk 2 the geoid, at particular values of geodetic coordinates φ i=1 Ni = min. This can be checked by repeating the procedure described in Section 4 and computing new statistic and λ, using values of the geoid undulation N provided by indexes min Nˆ , max Nˆ , mean Nˆ and rms Nˆ . the ICGEM service for several gravity eld models. The grid of points used was roughly equidistant, with a spatial resolution determined by the maximum degree of 5.1 Uncertainty estimates the gravity eld model. For example, the model GOCO06s has a maximum degree 300, so the equivalent resolution (0.6 degrees) corresponds to a spatial resolution (half Assuming that cxy ∼= cxz ∼= cyz ≈ 0 and using Eq. (17), the wavelength) of about 67 km at the equator. coordinates of the ellipsoid’s center are computed by The geodetic coordinates of the grid were determined 1 cx by setting up a number of points along circles of equal lat- tx = − (52) 2 cxx itude, separated by the given resolution. The angular sep-

1 cy aration along these circles (longitude dierence) was in- ty = − (53) 2 cyy creased at each latitude step (kept constant at the value of the angular resolution of the model) so that the spatial dis- 1 cz tz = − (54) 2 czz tance of points along each latitude circle remained roughly constant. In this way, the resulting grid has a uniform sur- Thus, their uncertainties σ may be computed by ap- face density of points, in contrast to a regular grid of φ, λ plying the error propagation law ( ) values, which is denser towards the poles (dataset D1.1 in s 2 2 1 c 1 Table 4). The geodetic coordinates φi , λi , Ni of a point σ = ± x σ2 + − σ2 (55) tx 2 cxx cx on the geoid are related to the corresponding Cartesian 2 cxx cxx coordinates (Xi , Yi , Zi) by well-known expressions, where s 2 2 h ≡ N . In the case of WGS 84 we have: major semiaxis 1 cy 1 i i σ = ± σ2 + − σ2 (56) 1 ty 2 cyy cy a = 6378137.0 m and attening f = (minor 2 cyy cyy 298.257223563 semiaxis b = a (1 − f )) (NIMA (2000)). s 1 c 2 1 2 In the ellipsoid tting we used six dierent grav- σ = ± z σ2 + − σ2 (57) tz 2 czz cz 2 czz czz ity eld models, which represent dierent stages of our knowledge about the gravity eld in the last fty years, as where σc are the uncertainties of the elements of vector c, follows: as computed from the tting. We note that the uncertainty – GOCO06s, described by Kvas et al. (2019) of the ellipsoid semi-axes can be approximated by – EGM2008, described by Pavlis et al. (2012) ∼ ∼ σˆ0 – EGM96, described by Lemoine et al. (1998) σax = σay = σb ≈ √ (58) k – OSU86f, described by Rapp and Cruz (1986) – GEM8, described by Wagner et al. (1977) where k is the number of points of tting. Finally, the un- – SE1, described by Lundquist and Veis (1966) certainty of the orientation of the ellipsoid, can be approximated by For more details regarding the evaluation of these models, ∼ σˆ0 σθ = σθ ≈ √ (59) x y b k please visit http://icgem.gfz-potsdam.de/home.html. and Table 4 summarizes the relevant data for each data σˆ set, as well as some important statistical properties of each √0 σθz ≈ (60) ay k model for the particular grid of points used. We should note that the great values of the mean and rms quantities for the SE1 model are due to the fact that the oblate spheroid used at the time was dierent from G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model Ë 77

Table 4. Characteristics of the data sets used

Dataset Geoid model Spatial resolution (km) Number of points Statistics mean (N) max (N) min (N) (m) rms (N) (m) (m) (m) D1.1 GOCO06s Variable 179402 -0.90 -106.27 84.93 29.28 D1.2 (±0.15 m) 67 114446 -0.05 -106.33 85.31 30.59 D2.1 56 164836 -0.06 -106.43 85.92 30.59 D2.2 EGM2008 67 114446 -0.06 -106.26 85.31 30.60 D2.3 (±0.15 m) 111 41164 -0.06 -105.93 85.92 30.60 D2.4 1334 286 -0.13 -94.59 72.39 30.59 EGM96 D3 56 164836 -0.05 -106.48 85.44 30.59 (±0.50 m) OSU86f D4 56 164836 0.02 -107.21 82.20 30.53 (±1.5 m) GEM8 D5 801 769 0.13 -102.14 74.42 30.58 (±3 m) D6 SE1 (±10 m) 1334 286 44.12 -34.74 101.20 51.83 the WGS 84. In any case, the particulars of the reference our decision to use the geometric tting for the study of all spheroid do not aect the preparation of the data sets. subsequent cases and geoid models. Therefore, we decided to use WGS 84 as a common refer- From the results presented in Table 6 it is apparent that ence for all models, since it is also more realistic than the the translation parameters tx, ty and tz, as well as the ro- older reference surfaces. tation angles θx and θy, have a very small impact on the In all numerical computations that are presented in statistics of the tting. Therefore, we added the row ‘T6’ in the following sections, we used a personal computer run- both Tables 5 and 6, which represent rounded values of the ning a 64-bit Linux Debian operating system, with an Intel parameters. We also note the agreement of the statistics of Core i5-2430M CPU (clocked at 2.4 GHz) and 6 GB RAM. All the WGS 84 initial data and the statistics of tting an oblate algorithms were coded in C++ and compiled by the open- spheroid (case G - B4 above). In contrast, we note the im- source GNU GCC compiler (Level 2 optimization). The com- provement of the statistics when tting a triaxial ellipsoid putations for the ttings (both algebraic and geometric) (case G - T6 above). This improvement is visualized in the were performed at quad precision (33 digits accuracy) us- following Figures 1 to 4. Figures 1 to 3 were created by the ing the open-source “libquadmath”, the GCC Quad Preci- free software GMT 6 (Wessel et al. (2019)). Figures 1 and 2 sion Math Library. The computations for the foot points, present the same quantity (geoid height) with respect to following the method presented in Panou and Korakitis two dierent reference surfaces, while Figure 3 depicts the (2019), were performed using the “long double” data type, dierence between the reference surfaces themselves. which provides an accuracy of 18 digits.

6.2 Results and statistics for the geoid model GOCO06s

Using the data sets derived from the GOCO06s geoid model (a satellite-only model) we obtained the results for the ellipsoid parameters and the relevant statistics for all cases as presented in Tables 5 and 6, respectively. Fig. 1. Geoidal heights (in meters) of model GOCO06s with respect Looking at the dierences in the parameter values to the oblate spheroid WGS 84 in Mollweide projection. mean = from the algebraic tting in the T1 case between the data −0.05 m, min = −106.33 m, max = 85.31 m, rms = 30.59 m. sets D1.1 and D1.2, one realizes the importance of using a data set of points with uniform surface density (almost equally spaced points). We also note some minor dierences in the values for the rotation angles, which justies 78 Ë G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model

Table 5. Resulting values of the parameters of tting for the geoid model GOCO06s

Dataset Fitting – Case Parameters tx ty tz θ (◦) θ (◦) θ (◦) a (m) a (m) b (m) (m) (m) (m) x y z x y D1.1 A - T1 5.43 0.55 6.93 -0.0048 -0.0056 -13.9666 6378173.07 6378102.94 6356749.49

A - T1 -0.08 -0.05 -0.04 -0.000020 -0.000056 -14.9283215 6378171.918 6378102.064 6356752.169 -0.11 -0.05 -0.06 0.0000043 -0.000083 -14.9366733 6378171.915 6378102.061 6356752.175 G - T1 ±0.13 ±0.13 ±0.13 ±0.0000007 ±0.0000007 ±0.0000007 ±0.07 ±0.07 ±0.07 0.0000043 -0.000083 -14.9366732 6378171.92 6378102.06 6356752.17 G - T2 0 0 0 ±0.0000007 ±0.0000007 ±0.0000007 ±0.07 ±0.07 ±0.07 D1.2 -0.11 -0.05 -0.06 6378167.27 6378106.70 6356752.17 G - T3 0 0 0 ±0.13 ±0.13 ±0.13 ±0.08 ±0.08 ±0.08 6378167.27 6378106.70 6356752.17 G - T4 0 0 0 0 0 0 ±0.08 ±0.08 ±0.08 -0.11 -0.05 -0.06 -14.9366733 6378171.92 6378102.06 6356752.17 G - T5 0 0 ±0.13 ±0.13 ±0.13 ±0.0000007 ±0.07 ±0.07 ±0.07 -14.9366732 6378171.92 6378102.06 6356752.17 G - T6 0 0 0 0 0 ±0.0000007 ±0.07 ±0.07 ±0.07 T6 0 0 0 0 0 -14.9367 6378171.92 6378102.06 6356752.17 -0.11 -0.05 -0.06 6378136.99 6356752.17 G - B3 0 0 - ±0.16 ±0.16 ±0.01 ±0.09 ±0.09 6378136.99 6356752.17 G - B4 0 0 0 0 0 - ±0.09 ±0.09 -0.08 -0.05 -0.04 6371037 G - S3 --- ±33 ±33 ±33 ±19 6371037 G - S4 0 0 0 --- ±19

Table 6. Statistics of tting for the geoid model GOCO06s

Statistics Dataset Fitting – Case ax − ay (m) mean Nˆ (m) min Nˆ (m) max Nˆ (m) rms Nˆ (m) D1.1 A - T1 70.13 -0.0002 -74.48 71.73 24.29 A - T1 69.854 -0.0002 -71.90 70.30 24.70 G - T1 69.854 0.000001 -71.89 70.28 24.70 G - T2 69.86 0.000001 -71.96 70.29 24.70 G - T3 60.57 0.000001 -78.36 86.29 26.29 G - T4 60.57 0.000001 -78.43 86.30 26.29 D1.2 G - T5 69.86 0.000001 -71.88 70.28 24.70 G - T6 69.86 0.000001 -71.96 70.29 24.70 T6 69.86 0.0001 -71.96 70.29 24.70 G - B3 0 0.000001 -106.24 85.25 30.59 G - B4 0 0.000001 -106.32 85.33 30.59 G - S3 0 -0.00006 -14314 7171 6365 G - S4 0 -0.00006 -14314 7177 6365 G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model Ë 79

ferent data sets, all based on the geoid model EGM2008, at dierent resolutions, corresponding to the ones of the other geoid models. The results of the geometric tting and the relevant statistics (cases G - T6 and G - B4) are presented in Table 7. From the above values we conclude that the eect of the spatial resolution is insignicant, in the sense that the values are in agreement considering their statistical error Fig. 2. Geoidal heights (in meters) of model GOCO06s with respect bounds. to the computed triaxial ellipsoid of case G-T6 in Mollweide projection. mean = 0.000001 m, min = −71.96 m, max = 70.29 m, rms = 24.70 m. 6.4 Results and statistics for various geoid models

In order to study the performance of dierent geoid models, going back roughly 50 years ago, we prepared data sets for four geoid models of the past, which were proposed about every 10 years. The results of the geometric tting and the relevant statistics (cases G - T6 and G - B4) are presented in Table 8. Fig. 3. Geoidal heights given in Fig. 2 minus geoidal heights given in From the above values we conclude that the estimates Fig. 1 (in meters) in Mollweide projection. mean = 0.05 m, min = for the dierence between the equatorial semi-axes, as −34.92 m, max = 34.94 m. well as for the longitude of the semi-major axis θz, are almost the same.

6.5 Comparison with results from spherical harmonic coecients

For a near-spherical body, using the spherical harmonic coecients of degree and order two of a global gravity eld model, we can compute the longitude of the principal axis of the least moment of inertia by: 1 −1 S22 λ0 = tan (61) 2 C22 and using the reference radius R (which only has the meaning of a scale factor) one gets an expression for the (positive) dierence of the two equatorial semi-axes (Liu Fig. 4. Geoidal heights (in meters – vertical axis) in a cross section and Chao (1991)) along equator (in degrees – horizontal axis) of model GOCO06s √ q with respect to: i) the oblate spheroid WGS 84 (red line) and ii) the 2 2 ax − ay = R 15 C22 + S22 (62) computed triaxial ellipsoid of case G-T6 (blue line). The green line illustrates the dierence (in meters) of the two heights (heights The uncertainties σ of the parameters in Eqs. (61) and given in blue line minus heights given in red line). (62) can be computed by applying the error propagation law: q 6.3 Results and statistics for the geoid 1 2 2 2 2 σλ = ± S σ + C σ (63) 0 2 C2 + S2 22 C22 22 S22 model EGM2008 22 22 and In order to study the eect of the spatial resolution of the √ R 15 q 2 2 2 2 σa −a = ± C σ + S σ (64) data on the parameters of the tting, we prepared four dif- x y q 22 C22 22 S22 2 2 C22 + S22 80 Ë G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model

Table 7. Resulting values of the parameters and statistics of tting for the geoid model EGM2008

Dataset Fitting – Case Parameters Statistics a − a ◦ x y mean Nˆ min Nˆ max Nˆ rms Nˆ θz ( ) ax (m) ay (m) b (m) (m) (m) (m) (m) (m) -14.9356740 6378171.88 6378102.03 6356752.24 G - T6 69.85 0.000001 -71.97 70.49 24.70 D2.1 ± 0.0000005 ± 0.06 ± 0.06 ± 0.06 6378136.96 6356752.24 G - B4 - 0 0.00001 -106.38 85.97 30.59 ± 0.08 ± 0.08 -14.9369405 6378171.90 6378102.04 6356752.20 G - T6 69.86 0.000001 -71.87 70.45 24.71 D2.2 ± 0.0000005 ± 0.07 ± 0.07 ± 0.07 6378136.97 6356752.20 G - B4 - 0 0.000001 -106.23 85.34 30.60 ± 0.09 ± 0.09 -14.939350 6378171.88 6378102.03 6356752.24 G - T6 69.85 0.000001 -71.46 67.43 24.70 D2.3 ±0.000001 ±0.12 ±0.12 ±0.12 6378136.95 6356752.24 G - B4 - 0 0.000001 -105.89 85.97 30.60 ±0.15 ±0.15 -15.12522 6378172.0 6378102.1 6356751.8 G - T6 69.9 0.000001 -64.19 60.03 24.75 D2.4 ± 0.00001 ±1.5 ±1.5 ±1.5 6356751.8 G - B4 - 6378137.1 ±1.8 0 0.000001 -94.65 72.33 30.59 ±1.8

Table 8. Resulting values of the parameters and statistics of tting for various geoid models

Dataset Fitting – Case Parameters Statistics ◦ ax −ay mean Nˆ min Nˆ max Nˆ rms Nˆ θz ( ) ax (m) ay (m) b (m) (m) (m) (m) (m) (m) -14.9366367 6378171.88 6378102.03 6356752.23 G - T6 69.85 0.000001 -72.03 69.85 24.70 D3 ± 0.0000005 ± 0.06 ± 0.06 ± 0.06 6378136.96 6356752.23 G - B4 - 0 0.000001 -106.44 85.48 30.59 ± 0.08 ± 0.08 -14.9328972 6378171.95 6378102.13 6356752.30 G - T6 69.82 0.000001 -72.90 66.13 24.63 D4 ± 0.0000005 ± 0.06 ± 0.06 ± 0.06 6378137.04 6356752.30 G - B4 - 0 0.000001 -107.25 82.16 30.53 ± 0.08 ± 0.08 -14.929005 6378171.7 6378102.0 6356753.0 G - T6 69.7 0.000001 -67.88 68.04 24.59 D5 ± 0.000008 ± 0.9 ± 0.9 ± 0.9 6378136.9 6356753.0 G - B4 - 0 0.000001 -102.01 74.55 30.58 ± 1.1 ± 1.1 -14.82805 6378215.3 6378147.4 6356796.0 G - T6 67.9 0.000001 -66.82 58.78 20.84 D6 ± 0.00001 ± 1.2 ±1 .2 ± 1.2 6378181.4 6356796.0 G - B4 - 0 0.000001 -79.10 57.23 27.20 ± 1.6 ± 1.6 G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model Ë 81

Table 9. Data and estimated values for the parameters for the geoid models

Geoid ◦ R (m) C22 S22 λ0 ( ) ax − ay (m) model 2.439370388690 · 10−6 −1.400307620664 · 10−6 −14.9288750 69.480959 GOCO06s 6378136.3 ±8.092959988764 · 10−13 ±9.226405195253 · 10−13 ±0.0000091 ±0.000021 2.43938357328313 · 10−6 −1.40027370385934 · 10−6 −14.928509 69.48082 EGM2008 6378136.3 ±7.230231722 · 10−12 ±7.425816951 · 10−12 ±0.000075 ±0.00018 2.43914352398 · 10−6 −1.40016683654 · 10−6 −14.92878 69.4744 EGM96 6378136.3 ±5.3739154 · 10−11 ±5.4353269 · 10−11 ±0.00055 ±0.0013 OSU86f 6378136.3 2.43834012895 · 10−6 −1.39928194222 · 10−6 −14.92504 69.4463

GEM8 6378136.3 2.4345 · 10−6 −1.3953 · 10−6 −14.90929 69.3151

SE1 6378165 2.379 · 10−6 −1.351 · 10−6 −14.79581 67.5823

where σC and σS are the uncertainties of the coecients, the geometric tting has the added advantage of a straight- as given in a global gravity eld model. forward geometric interpretation of the residuals and the Table 9 presents the data and the estimated values for a-posteriori variance factor, something that is dicult to the above parameters for each one of the geoid models achieve with the algebraic tting. used. We should note that, the procedure to derive the ellip- From the estimates in Table 9 we note the high degree soidal parameters from the polynomial coecients (sub- of precision of these values, as well as their consistency section 2.4) is simpler than the corresponding method pre- and the very small dierence from the results of the corre- sented in Bektas (2015). However, it is important to com- sponding geometric tting. Finally, we note that all geoid pare the performance of the geometric tting presented models do not include the spherical harmonic coecients here with the corresponding tting presented in Bektas. of degree one, implying that the translations are assumed A further study of the problem of estimating the tri- zero. axiality parameters of the Earth could involve other minimization conditions, such as the gravity anomalies (gravimetric method) or the deections of the vertical. Finally, 7 Conclusions and future work using the parameters of the triaxial ellipsoid one could also study the dierences between this surface and the cur- rently adopted oblate spheroid, as well as their eects, not In this work, we present a procedure for the estimation of only regarding the geoid heights but other geometric en- parameters of Earth’s triaxiality through a geometric t- tities (e.g. geodesic lines) or dynamical characteristics of ting of a triaxial ellipsoid to a set of given points in space, the Earth. derived from and representing various geoid models. The condition for the tting is the minimization of the sum of the squared geoid heights. Taking into consideration the advantages of a satellite-only model (GOCO06s), we con- References clude that the optimum semi-axes of the triaxial ellipsoid with four parameters are equal to 6378171.92 m, 6378102.06 Barthelmes F., 2014. Global Models. In: Grafarend E. (Ed.) En- m and 6356752.17 m, while the longitude of the equatorial cyclopedia of Geodesy, Springer International Publishing. https://doi.org/10.1007/978-3-319-02370-0_43-2 major semi-axis is –14.9367 degrees. This reference surface Bektas S., 2015. Least squares tting of ellipsoid using leads to signicantly reduced values of the geoid heights: orthogonal distances. Bol Ciênc Geod 21, 329–339. the range is reduced from 191.64 m to 142.25 m (25%) and https://doi.org/10.1590/S1982-21702015000200019 rms value is reduced from 30.59 m to 24.70 m (19%). Bektas S., 2014. Orthogonal distance from an ellipsoid. Bol The geometric tting presented in this work (concern- Ciênc Geod 20, 970–983. https://doi.org/10.1590/S1982- ing orthogonal distances) is dierent from the tting used 21702014000400053 Burša M., 1972. Fundamental geodetic parameters of the earth’s by Burša (concerning radial distances). The two methods gure and the structure of the earth’s gravity eld de- give the same results for a spherical body or quite simi- rived from satellite data. Stud Geophys Geod 16, 10–29. lar results for a nearly spherical one, like the Earth. Also, https://doi.org/10.1007/BF01614229 82 Ë G. Panou, R. Korakitis, and G. Pantazis, Fitting a triaxial ellipsoid to a geoid model

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