A new Method for Computing



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A New Method for Computing


the Ellipsoidal Correction for Stokes’s Formula

Z.L. Fei and Michael G. Sideris

Department of Geomatics Engineering

University of Calgary

2500 University Drive NW

Calgary, Alberta

Canada, T2N 1N4

Zhiling Fei

Phone: ++1-403-220-4113

Fax: ++1-403-285-1980

E-mail: zlfei@ucalgary.ca


M.G. Sideris

Phone: ++1-403-220-4985

Fax: ++1-403-285-1980

E-mail: sideris@ucalgary.ca



ABSTRACT: This paper generalizes the Stokes formula from the spherical boundary surface to ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N0 evaluated from the Stokes formula and the ellipsoidal correction N1, makes the relative geoidal height error decrease from O(e2) to O(e4), which can be neglected for most practical purposes. The ellipsoidal correction N1 is expressed as a sum of an integral about the spherical geoidal height N0 and a simple analytical function of N0 and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this solution and the solutions in Molodensky et al. (1962), Moritz (1980) and Martinec and Grafarend (1997) shows that this solution is more effective than the solutions in Molodensky et al. (1962) and Moritz (1980) and, when the evaluation of the ellipsoidal correction N1 is done in an area where the spherical geoidal height N0 has already been evaluated, it is also more effective than the solution in Martinec and Grafarend (1997).

Keywords: Geoidal height; Stokes’s formula; ellipsoidal correction.



  1. Introduction

The determination of the geoid is a basic task of physical geodesy. The approaches used to determine the geoid are usually classified into the model approach and the operational approach (Moritz, 1978). Stokes’s formula is playing a key role in the model approach. Rigorously, Stokes’s formula is the solution of the spherical boundary value problem. It holds only on a spherical reference, i.e. the input data must be given on the sphere. However, the input data (gravity anomalies) can only be observed on the earth’s surface. These data can be reduced to the geoid or a local level surface via orthometric (or normal) heights. For example, in a remove-restore technique, the gravity anomalies are reduced to a level surface via terrain reductions, and in Moritz’s approach (Moritz, 1980) the gravity anomalies are analytically downward continued onto the geoid (or a point level surface) via a Taylor series expansion. The geoid and the local level surface can be approximated respectively by the reference ellipsoid and the local reference ellipsoid (on which the normal potential equals to the potential of the gravity on the local level surface). Since the flattening of the ellipsoid is very small (about 0.003), in practical computation the ellipsoid is treaded as a sphere so that Stokes’s formula can be applied on it. The systematic error caused from neglecting the flattening of the ellipsoid is about 0.003N. This magnitude, amounting up to several tens of centimeters, is quite considerable. So it becomes very important to evaluate the effect of the flattening on the Stokes formula. In other words, we should investigate the third ellipsoidal geodetic boundary value problem (GBVP).


The mathematical description of the third ellipsoidal GBVP is to determine the disturbing potential function T satisfying

(1.1)

where is the reference ellipsoid or the local reference ellipsoid rP is the radius of point P and is the length along the normal direction of P on .


As discussed above, the ellipsoidal GBVP represents reality better than the spherical GBVP from which Stokes’s formula is obtained. However, it is hard to describe its solution via a rigorous analytic closed formula like Stokes’s formula. Various approaches have been employed to solve the ellipsoidal GBVP (Molodensky et al., 1962; Moritz, 1980; Cruz, 1986; Sona, 1995; Thông, 1996; Yu and Cao, 1996; Martinec and Grafarend, 1997; Martinec and Matyska, 1997; Martinec, 1998; Ritter, 1998). Usually, there exist two main approaches for solving the ellipsoidal GBVP: One directly represents the disturbing potential T in terms of an ellipsoidal harmonic series, which is rigorous but very complicated because it requires the introduction of Legendre functions of the second kind (Sona, 1995); Another one regards Stokes’s formula as the first approximation of the solution of (1.1) and pushes the approximation up to the term of O(e2), where e is the first eccentricity of the ellipsoid. This term, called the ellipsoidal correction, is expressed in terms of closed integral formulas, such as the solutions described in Molodensky et al. (1962), Moritz (1980) and Martinec and Grafarend (1997).
In this paper, we will follow the work done by Molodensky et al. (1962) to obtain a closed integral formula for computing the ellipsoidal correction that is different from the ellipsoidal corrections given by Molodensky et al. (1962), Moritz (1980) and Martinec and Grafarend (1997). A brief comparison will also be given among the four ellipsoidal corrections.



  1. Derivation of the ellipsoidal correction

In the following, to solve equation (1.1), we will (a) establish an integral equation, and (b) solve the integral equation to get Stokes’s formula plus its ellipsoidal correction.




    1. Establishment of the integral equation

According to Moritz (1980), it is easy to prove that for an arbitrarily point P0 given inside Se, the generalized Stokes function



(2.1)

satisfies



(2.2)
From Green’s second identity (Heiskanen and Moritz, 1962), we obtain that

(2.3)
It follows from the third formula in (1.1) that

(2.4)
By differentiating (2.1) we get







(2.5)

where






(2.6)
Following Moritz (1980), when P0 goes to P (the projection of P0 on Se) from the inner of Se, then

(2.7)

and


(2.8)

(2.9)

(2.10)
So for any given point P on Se, we obtain by letting in (2.4) that

(2.11)

where


(2.12)
Equation (2.11) is the integral equation that will be used for determining T on Se.
2.2. Determination of the geoidal height
Denoting the projection of the surface element onto the unit sphere by , we have

(2.13)

where is the angle between the radius vector of Q and the surface normal of the surface Se at point Q. Then, for any given point P on Se, (2.11) becomes



(2.14)
With b the semiminor axis and e the first eccentricity of the reference ellipsoid, and and respectively the complements of the geocentric latitudes of P and Q, we have

(2.15a)

(2.15b)

(2.15c)

(2.15d)
Furthermore, from Molodensky et al. (1962), we have

(2.16a)

(2.16b)

(2.16c)

(2.16d)
It then follows from (2.1) that

(2.17)

where is the Stokes function



(2.18)

and


(2.19)
So





(2.20)

where


(2.21)
From (2.5), we obtain

(2.22)

where














(2.23)
Therefore from (2.12), (2.13) and (2.14), we obtain



(2.24)
The second condition in (1.1) means that the disturbing potential T does not contain the spherical harmonics of degrees one and zero. So we have

(2.25)

(2.26)
Thus (2.24) becomes



(2.27)
Let

(2.28)
Then from (2.27), we obtain

(2.29)

(2.30)

where




















(2.31)
According to the Bruns formula, we obtain

(2.32)

with the spherical geoidal height



(2.33)

and its ellipsoidal correction



(2.34)

where f1 is defined by (2.19) and f0 is obtained from (2.31), (2.21) and (2.23).


Now we further discuss the first term of :

(2.35)
From (2.19), (2.33), (2.25) and (2.26), we have that

(2.36)

According to Heiskanen and Moritz (1967), we have that







(2.37)

We now represent by the spherical harmonic expansion



(2.38)

where G is the mean gravity, are the fully normalized geopotential coefficients of the disturbing potential and , are the fully normalized Ledendre harmonics. Then from the orthogonality of the harmonics and, we obtain





(2.39)

So, finally, the ellipsoidal correction is expressed as



(2.40)
Equations (2.32) plus (2.33), (2.34), (2.40) and (2.31) are the formulas for computing the ellipsoidal geoidal height with an accuracy of the order of O(e4).
Rigorously, equation (2.33) is not the same as Stokes’s formula because the semiminor axis b is used instead of the mean radius R. After a little modification for (2.32), (2.33), (2.34) and (2.40), we finally obtain the formulas for computing the ellipsoidal geoidal height corresponding to Stokes’s formula

(2.41)

(2.42)

(2.43)

where f0 is still defined by (2.31) and





(2.44)



  1. Practical considerations on the integral component of the ellipsoidal correction

In the above section, we obtained the Stokes formula (2.42) and its ellipsoidal correction (2.43). The component N11 of the ellipsoidal correction N1 is a simple analytical function about the spherical geoidal height N0 and the first three degree spherical harmonic coefficients of the disturbing potential. It is easy to be evaluated from the equation (2.44). Here we give some details on the integral component of the ellipsoidal correction N1.


(i) The kernel function f0 of the integral component is singular at =0 because it contains the factors

(3.3)
From the spherical triangle of figure 1, we have

(3.4)
So

(3.5)
It then follows from (2.31) that

(3.6)
This means that the kernel function f0 has the same degree of singularity at as the Stokes function . So the integral component in (2.43) is a weakly singular integral and the singularity can be treated by the method used for Stokes’s integral (see Heiskanen and Moritz, 1967).


  1. From the definitions (2.31) of f0, we know that like the Stokes function, f0 quickly decreases when goes from 0 to . Therefore in practical evaluation of the integral component, we divide into two parts: and , where the area is usually a spherical cap containing the computation point P as its center. Since the kernel function is larger over , the integral over should be carefully computed using a high resolution and high accuracy spherical geoid model obtained from the ground gravity data by means of Stokes’s formula (2.42) if a high accuracy geoid is required. The area is far from the computation point P, so the kernel function is relatively small over . Therefore in the computation of the integral over we can use the spherical geoidal height data N0 computed from a global geopotential model.




  1. For the US geoid, the integral component ranges from –0.8 to 15.0 cm while N1 ranges from 0.4 to 14.2 cm and, if the required accuracy is of the order of 1 cm, the global spherical geoid model with the resolution of 1 degree is good enough for the computation of the integral component over .



4. A brief comparison among Molodensky's method, Moritz’s method, Martinec and Grafarend's method and the method in this paper
The following Tables 1, 2 and 3 show the differences of the solutions in Molodensky et al. (1962), Moritz (1980), Martinec and Grafarend (1997) and this paper.

Notes:
(i) The method used in this paper follows the method used in Molodensky et al. (1962). The difference is that instead of using the general Stokes function S(P, P0) (see (2.1)), Molodensky et al. (1962) used the function as the kernel function of the equation (2.3).


(ii) The regularity condition used in Molodensky et al. (1962), Moritz (1980) and this paper is the same as that used in the derivation of Stokes’s formula (see Heiskanen and Moritz, 1967). This condition is stronger than that used in Martinec and Grafarend (1997).
From the derivations in section 2, we see that the regularity condition in (1.1) is used to make (2.25) and (2.26) hold so that we can get (2.27) from (2.24). However, if we substitute in (2.3) by , which is also harmonic outside Se according to Moritz (1980), then the term will disappear in an equation corresponding to (2.24). What we still need to do is to make (2.26) hold. Obviously, this can be guaranteed by the more general regularity condition used in Martinec and Grafarend (1997). The spherical geoidal height will be given by the general Stokes formula (see Heiskanen and Moritz, 1967) and the ellipsoidal correction will be a little bit different than that given in section 2.
At present time, the mass of the earth can be estimated very accurately. By properly selecting the normal gravity field, we can easily make the disturbing potential T satisfy the regular condition in (1.1). So the difference between the two regularity conditions is not a key problem.
(iii) The boundary condition used in Martinec and Grafarend (1997) is a bit different than the boundary condition used in Molodensky et al. (1962), Moritz (1980) and this paper. The difference is

(4.1)
From the derivations in Section 2, we see that the effect of this difference on the ellipsoidal correction is

(4.2)
Martinec (1998) has shown that the term above has a small impact on the ellipsoidal correction because it is characterized by an integration kernel with a logarithmic singularity at =0 and it can be neglected in cm geoid computation if a higher-degree reference field is introduced as a reference potential according to the numerical demonstration given by Cruz (1986).
(iv) All four solutions express the ellipsoidal geoidal height by the spherical geoidal height N0 given by Stokes’s formula plus the ellipsoidal correction N1 given by closed integral formulas. The relative errors of the solutions are O(e4).
(v) In Molodensky et al. (1962), to evaluate N1 at a single point from , we need two auxiliary data sets T0 and . First we integrate to get auxiliary data set T0; then we integrate T0 to get another auxiliary data set ; finally, we obtain N1 from integrating , T0 and . That is:
g T0  N1
The auxiliary data sets T0 and , except for T0 at the computation point which can be further used to compute the final ellipsoidal geoidal height, are useless after computing N1. So the solution in Molodensky et al. (1962) is very computation-intensive even through the kernel functions fM1, fM2 and fM3 are simple analytical functions.


  1. In Moritz (1980), only one auxiliary data set is needed and the kernel function is also a simple analytical function, but is only expressed by an infinite summation of the coefficients of the spherical harmonic expansion of the disturbing potential T.

The coefficients , however, are what we want to know. They are not the coefficients of the spherical harmonic expansion of the spherical approximation disturbing potential T0 corresponding to N0, which can be computed from the gravity data by means of the spherical approximation formulas.


In the practical evaluation, is approximately computed using truncated spherical harmonic coefficients .

(vii) In Martinec and Grafarend (1997), no auxiliary data set is needed. We can directly integrate g to obtain N1:


g N1
So this solution is much simpler than that in Molodensky et al. (1962) and Moritz (1980). However, the simplicity is obtained by complicating the kernel function in its solution. From the table 3, we see that the kernel function fMG contains the series of Legendre polynomials and their derivatives, so it is obviously more complicated than the kernel functions in Molodensky et al. (1962) and Moritz (1980).
(viii) In this paper, one auxiliary data set N0 is needed to evaluate N1 from . We first integrate to get N0, the 'auxiliary' data set; then we obtain N1 from N0 and the first 3 degree harmonic coefficients plus an integral about N0. That is:

g N0 N1
Like Molodensky et al. (1962), the kernel function in the integral of this solution is a simple analytical function. So this solution is simpler than the solution of Molodensky et al. (1962) in the sense that only one auxiliary data set is needed for the evaluation of N1 from .
Because of the need of an auxiliary set data, it is seems that this solution is more complex than the solution in Martinec and Grafarend (1997). However, the 'auxiliary' data N0 are nothing else but the spherical geoidal heights, which are already available in many areas of the world such as in Europe and North America. When we evaluate N1 in such areas, this solution is simpler than the solution in Martinec and Grafarend (1997) in the sense that we can directly evaluate N1 from N0 with a simple analytical function.

This solution is similar to the solution in Moritz (1980). They both need an auxiliary data set. Their kernel functions are simple analytical functions and have the same degree of singularity at the origin. However, the auxiliary data set N0 in this solution is much simpler than the auxiliary data set in Moritz (1980) in the sense that:



  1. N0 can be computed directly from gravity anomaly data by means of Stokes’s formula;

  2. N0 can also be computed approximately from the geopotential model

(4.3)

Obviously, this formula is simpler than that used for computing (see Table 2) and N0 is less sensitive to the high degree coefficients than is;



  1. N0 is already available globally with the resolutions of less than 1 degree and locally with higher resolutions.



5. Conclusions
This paper investigates the ellipsoidal geodetic boundary value problem and gives a solution for the ellipsoidal geoidal heights.


  • The solution, as the already available ones, generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface by adding an ellipsoidal correction to the Stokes formula. It makes the error of geoidal height decrease from O(e2) to O(e4), which can be neglected for most practical purposes.




  • The ellipsoidal correction N1 in the solution involves the spherical geoidal height N0 and a kernel function which is a simple analytical function that has the same degree of singularity at the origin as the Stokes function.




  • The solution is simpler than the solutions in Molodensky et al. (1962) and Moritz (1980). It is also simpler than the solution in Martinec and Grafarend (1997) when evaluating the ellipsoidal correction N1 in an area where the spherical geoidal height N0 has already been evaluated.


Acknowledgments: Financial support for this research has been provided by a Geodetic Survey of Canada contract, a National Sciences and Engineering Research Council (NSERC) research grant, and an Alexander von Humbold International Fellowship to the second author. The constructive comments of Prof. Martinec have been highly appreciated.

Reference

Cruz JY (1986) Ellipsoidal corrections to potential coefficients obtained from gravity anomaly data on ellipsoid. Report No. 371, Department of Geodetic Science and Surveying, Ohio State University

Heiskanen WA, Moritz H (1967) Physical Geodesy. W.H. Freeman and Co., San Francisco and London

Martinec Z (1998) Construction of Green’s function for the Stokes boundary-value problem with ellipsoidal corrections in the boundary condition. Journal of Geodesy 72: 460-472

Martinec Z, Grafarend EW (1997) Solution to the Stokes boundary-value problem on an ellipsoid of revolution. Stud. Geoph. Geod. 41: 103-129
Martinec Z, Matyska C (1997) On the solvability of the Stokes pseudo-boundary-value problem for geoid determination. Journal of Geodesy 71: 103-112

Molodensky MS, Eremeev VF, Yurkina MI (1962) Methods for study of the external gravitation field and figure of the earth. Transl. from Russian (1960), Jerusalem, Israel Program for Scientific Translations

Moritz H (1978) The operational approach to physical geodesy, Rep. 277, Dept. of Geod. Sci., Ohio State University

Moritz H (1980) Advanced Physical Geodesy, Herbert Wichmann Verlag Karlsruhe, Abacus Press, Tunbridge Wells Kent

Ritter S (1998) The nullfield method for the ellipsoidal Stokes problem. Journal of Geodesy 72: 101-106

Sona G (1995) Numerical problems in the computation of ellipsoidal harmonics, J of Geodesy, 70: 117-126

Thông NC (1996) Explicit expression and regularization of the harmonic reproducing kernels for the earth's ellipsoid. Journal of Geodesy 70(9): 533-538

Yu JH, Cao HS (1996) Ellipsoid harmonic series and the original Stokes problem with the boundary of the reference ellipsoid. J of G 70: 431-439


Table 1. Differences among the solutions in Molodensky et al. (1962), Moritz (1980),

Martinec and Grafarend (1997) and this paper








Molodensky et al.

Moritz

Martinec & Grafarend

This paper

Regularity

condition



()



()



()



()



Boundary condition









Ellipsoidal correction







where








where


and is given in Table 2








where is given by (2.44) and


Kernel functions

, and are given in Table 3.

is the Stokes function.

is given in Table 3.

is given by (2.31).


Table 2. The definition of in Moritz (1980)



where are Legendre surface harmonics and



where are the coefficients of the spherical harmonic expansion of the disturbing potential T and










Table 3. The definitions of the kernel functions


in Molodensky et al. (1962) and Martinec and Grafarend (1997)



; ;

;





where is the reduced latitude of P and KI (i=1,2,3,4) are given as follows:



; ;

; .


PN






P



Q


Q

QP

P


Q P

Figure 1. Spherical triangle







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