Open access peer-reviewed chapter

# Mathematical Analysis of Some Typical Problems in Geodesy by Means of Computer Algebra

By Hou-pu Li and Shao-feng Bian

Submitted: February 27th 2018Reviewed: September 19th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.81586

## Abstract

There are many complicated and fussy mathematical analysis processes in geodesy, such as the power series expansions of the ellipsoid’s eccentricity, high order derivation of complex and implicit functions, operation of trigonometric function, expansions of special functions and integral transformation. Taking some typical mathematical analysis processes in geodesy as research objects, the computer algebra analysis are systematically carried out to bread, deep and detailed extent with the help of computer algebra analysis method and the powerful ability of mathematical analysis of computer algebra system. The forward and inverse expansions of the meridian arc in geometric geodesy, the nonsingular expressions of singular integration in physical geodesy and the series expansions of direct transformations between three anomalies in satellite geodesy are established, which have more concise form, stricter theory basis and higher accuracy compared to traditional ones. The breakthrough and innovation of some mathematical analysis problems in the special field of geodesy are realized, which will further enrich and perfect the theoretical system of geodesy.

### Keywords

• geodesy
• computer algebra
• mathematical analysis
• meridian arc
• singular integration
• mean anomaly

## 1. Introduction

Geodesy is the science of accurately measuring and understanding three fundamental properties of the Earth: its geometric shape, its orientation in space, and its gravity field, as well as the changes of these properties with time. There are many fussy symbolic problems to be dealt with in geodesy, such as the power series expansions of the ellipsoid’s eccentricity, high order derivation of complex and implicit functions, expansions of special functions and integral transformation. Many geodesists have made great efforts to solve these problems, see [1, 2, 3, 4, 5, 6, 7, 8]. Due to historical condition limitation, they mainly disposed of these problems by hand, which were not perfectly solved yet. Traditional algorithms derived by hand mainly have the following problems: (1) The expressions are complex and lengthy, which makes the computation process very complicated and time-consuming. (2) Some approximate disposal is adopted, which influences the computation accuracy. (3) Some formulae are numerical and only apply to a specific reference ellipsoid, which are not convenient to be generalized.

In computational mathematics, computer algebra, also called symbolic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Software applications that perform symbolic calculations are called computer algebra systems, which are more popular today. Computer algebra systems, like Mathematica, Maple and Mathcad, are powerful tools to solve some mathematical derivation in geodesy, see [9, 10, 11]. By means of computer algebra analysis method and computer algebra system Mathematica, we have already solved many complicated mathematical problems in special fields of geodesy in the past few years; see [12, 13, 14, 15].

The main contents and research results presented in this chapter are organized as follows: In Section 2, we discuss the forward and inverse expansions of the meridian arc often used in geometric geodesy. In Section 3, the nonsingular expressions of singular integration in physical geodesy are derived. In Section 4, we discuss series expansions of direct transformations between three anomalies in satellite geodesy. Finally in Section 5, we make a brief summary.

## 2. The forward and inverse expansions of the meridian arc in geometric geodesy

The forward and inverse problem of the meridian arc is one of the fundamental problems in geometric geodesy, which seems to be a solved one. Briefly reviewing the existing methods, however, one will find that the inverse problem was not perfectly and ideally solved yet. This situation is due to the complexity of the problem itself and the lack of advanced computer algebra systems. Yang had given the direct expansions of the inverse transformation by means of the Lagrange series method, but their coefficients are expressed as polynomials of coefficients of the forward expansions, which are not convenient for practical use, see [6, 7]. Adams expressed the coefficients of inverse expansions as a power series of the eccentricity eby hand, but expanded them up to eighth-order terms of eat most, see . Due to these reasons, the forward and inverse expansions of the meridian arc are discussed by means of Mathematica in the following sections. Their coefficients are uniformly expressed as a power series of the eccentricity and extended up to tenth-order terms of e.

### 2.1. The forward expansion of the meridian arc

The meridian arc from the equator where the latitude is from B=0to Bis

X=a1e20B1e2sin2B3/2dBE1

where Xis the meridian arc; Bis the geodetic latitude; ais the semi-major axis of the reference ellipsoid; eis the first eccentricity of the reference ellipsoid.

Expanding the integrand in Eq. (1) and integrating it item by item using Mathematica as follows:      Then one arrives at

X=a1e2K0B+K2sin2B+K4sin4B+K6sin6B+K8sin8B+K10sin10BE2

where

K0=1+34e2+4564e4+175256e6+1102516384e8+4365965536e10K2=38e21532e45251024e622054096e872765131072e10K4=15256e4+1051024e6+220516384e8+1039565536e10K6=353072e61054096e810395262144e10K8=315131072e8+3465524288e10K10=6931310720e10E3

Eqs. (2) and (3) can also be derived using the binomial theorem by hand which consumes much more time, see [6, 7, 8, 9]. The denominator 693 of the last coefficient K10is mistaken as 639 in Ref. .

### 2.2. The inverse expansion of the meridian arc using the Hermite interpolation method

Differentiation to the both sides of Eq. (1) yields:

dXdB=a1e21e2sin2B3/2E4

Define ψas

ψ=Xa1e2K0E5

Substituting Eq. (5) into Eq. (4) yields:

dBK0=1e2sin2B3/2E6

Suppose that the inverse solution of Eq. (6) has the following form:

B=ψ+a2sin2ψ+a4sin4ψ+a6sin6ψ+a8sin8ψ+a10sin10ψE7

Eq. (7) has five coefficients to be determined. Once these coefficients are known, the inverse problem can be solved.

We consider that the values of differentiation Eq. (6) at the beginning and end points can be treated as interpolation constraints. It is generally known that

B0=K0E8
Bπ2=K01e23/2E9

The further derivation of Eq. (6) with respect to ψyields Bψ. Unfortunately, Bψis equal to zero at ψ=0, ψ=π2. Hence, one differentiates Eq. (6) twice and it yields Bψ. Omitting the derivative procedure by means of Mathematica, one arrives at

B0=3e2274e472964e64329256e838164516384e10E10
Bπ2=3e2154e4+5764e6+3256e85116384e10E11

The further derivation of Bψwith respect to ψyields B4ψ, but B4ψis equal to zero at ψ=0, ψ=π2. Hence, one differentiates Bψtwice and it yieldsB5ψ. Then one arrives at

B50=12e2+90e4+445516e6+2014532e8+49249354096e10E12

Making use of the five interpolation constraints in Eqs. (8–12) and differentiating Eq. (7) correspondingly, one arrives at a set of linear equations for the unknown coefficients

24681024681086421651210008642165121000321024777632768100000a2a4a6a8a10=B01Bπ21B0Bπ2B50E13

The solution to Eq. (13) is

a2a4a6a8a10=246810246810864216512100086421651210003210247776327681000001B01Bπ21B0Bπ2B50E14

Omitting the main operations by means of Mathematica, one arrives at

{a2=38e2+316e4+2132048e6+2554096e8+20861524288e10a4=21256e4+21256e6+5338192e8+1974096e10a6=1516144e6+1514096e8+5019131072e10a8=1097131072e8+109765536e10a10=80112621440e10E15

### 2.3. The inverse expansion of the meridian arc using Lagrange’s theorem method

Suppose that

y=x+fxE16

with fx<<xand yx. Lagrange’s theorem states that in a suitable domain the solution of Eq. (16) is

x=y+n=11nn!dn1dyn1fynE17

Supposefxis defined by the following finite trigonometric series:

fx=αsin2x+βsin4x+γsin6x+δsin8x+εsin10xE18

where the coefficients α=Oe2, β=Oe4, γ=Oe6, δ=Oe8, ε=Oe10are small enough for the condition fx<<x. In deriving the inversion we shall truncate the infinite Lagrange expansion at eighth-order terms of eand drop higher powers. Inserting Eq. (18) into Eq. (17), one arrives at

x=yfy+12!ddyfy213!d2dy2fy3+14!d3dy3fy415!d4dy4fy5E19

One should make use of several trigonometric identities to calculate the derivatives, substituting them into Eq. (19) and grouping terms according to the trigonometric functions. It is a difficult and time-consuming work to do by hand, but could be easily realized by means of Mathematica. Omitting the main procedure, one arrives at

x=y+d2sin2y+d4sin4y+d6sin6y+d8sin8y+d10sin10yE20

where

d2=ααββγ+12α3+αβ212α2γ+13α3β112α5d4=β+α22αγ+4α2β43α4d6=γ+3αβ3αδ32α3+92αβ2+9α2γ272α3β+278α5d8=δ+2β2+4αγ8α2β+83α4d10=ε+5αδ+5βγ252αβ2252α2γ+1256α3β12524α5E21

Substituting xfor Band yfor ψin Eq. (16), the coefficients α, β, γ, δ, εare consistent with α2, α4, α6, α8in Eq. (3). According to Eq. (20) and denoting a2, a4, a6, a8, a10as the new coefficients, the inverse expansion of the meridian arc can be written as

B=ψ+a2sin2ψ+a4sin4ψ+a6sin6ψ+a8sin8ψ+a10sin10ψE22

where

a2=α2α2α4α4α6+12α23+α2α4212α22α6+13α23α4112α25a4=α4+α222α2α6+4α22α443α24a6=α6+3α2α43α2α832α23+92α2α42+9α22α6272α23α4+278α25a8=α8+2α42+4α2α68α22α4+83α24a10=α10+5α2α8+5α4α6252α2α42252α22α6+1256α23α412524α25E23

The coefficients in Eq. (23) are also easily expressed in a power series of the eccentricity by means of Mathematica. Omitting the main procedure, one arrives at

a2=38e2+316e4+2132048e6+2554096e8+20861524288e10a4=21256e4+21256e6+5338192e8+1974096e10a6=1516144e6+1514096e8+5019131072e10a8=1097131072e8+109765536e10a10=80112621440e10E24

The results in Eq. (24) are consistent with the coefficients in Eq. (15), which substantiates the correctness of the derived formula.

### 2.4. The accuracy of the inverse expansion of the meridian arc

In order to validate the exactness of the inverse expansions of meridian arc derived by the author, one has examined its accuracy choosing the CGCS2000 reference ellipsoid with parametersa=6378137, e=0.08181919104281579. The accuracy of the inverse expansions derived by Yang (see [6, 7]) is also examined for comparison.

One makes use of Eq. (1) and Eq. (5) to obtain the theoretical value ψ0at given geodetic latitude B0. Then one makes use of the inverse expansions derived by Yang (see [6, 7]) to obtain the computation value B1. Substituting ψ0into Eq. (22), one arrives at the computation value B1. The differences ΔBi=BiB0, ΔBi=BiB0i=12indicate the accuracies of the inverse expansions derived by Yang (see [6, 7]) and the author respectively. These errors are listed in Table 1.

B0/20406080
ΔB1/3.2×1071.6×1061.8×1061.4×106
ΔB1/2.7×1098.6×1091.3×1081.7×108

### Table 1.

Errors of the inverse expansions of meridian arc.

From Table 1, one could find that the accuracy of the inverse expansion of meridian arc derived by Yang (see [6, 7]) is higher than 10−5″, and the accuracy of the inverse expansion Eq. (22) derived by the author is higher than 10−7″. The accuracy is improved by 2 orders of magnitude by means of computer algebra.

## 3. The singular integration in physical geodesy

Singular integrals associated with the reciprocal distance usually exist in the computations of physical geodesy and geophysics. For example, the integral expressions of height anomaly, deflections of the vertical and vertical gradient of gravity anomaly can be written in planar approximation as

ζ=12πγΔgrdxdyE25
ξ=12πγxΔgr3dxdyE26
η=12πγyΔgr3dxdyE27
L=12πΔgΔg0r3dxdyE28

where r=x2+y2, Δg0is the gravity anomaly at the computation point. When r0, the above integrals become singular and need special treatment at the computation point. The past treatments are with respect to template computations, which regards the innermost area as a circle, see [3, 16]. But the gravity anomalies are given in the rectangular grids(such as 2×2), if the approximate disposal is used, some significant error may be introduced. Sunkel and Wang expressed the gravity anomalies block by block as an interpolation polynomial and derived the analytic values of the integrals, see [17, 18]. However, the integrals of the rational functions are very complicated, especially when related interpolation polynomials contain many terms. One only can give the analytic values of the corresponding linear approximation. In this chapter, we use the nonsingular integration transformations proposed by Bian (see ) to compute the above integrals precisely.

As shown in Figure 1, let the innermost area be the rectangular σa<x<ab<y<b(a>0, b>0) due to the convergence of meridian, and the gravity anomaly is expressed as double quadratic polynomial:

Δgxy=i=02xaij=02αijybjE29

Inserting the innermost area into Eqs. (25)–(28), and let the contributions be Δζ, Δξ, Δηand ΔL, one arrives at

Δζ=12πγσΔgxyrdxdyE30
Δξ=12πγσxΔgxyr3dxdyE31
Δη=12πγσyΔgxyr3dxdyE32
ΔL=12πσΔgxyΔg00r3dxdyE33

The following transformation is introduced for σ

u=xav=ybE34

Using the properties of the integration for even/odd functions and exploiting the symmetry of the integration area, one arrives at

Δζ=4ab2πγ0101α00+α20u2+α02v2+α22u2v2a2u2+b2v2dudvE35
Δξ=4a2b2πγ0101u2α10+α12v2a2u2+b2v23/2dudvE36
Δη=4ab22πγ0101v2α01+α21v2a2u2+b2v23/2dudvE37
ΔL=4ab2π0101α20u2+α02v2+α22u2v2a2u2+b2v23/2dudvE38

Drawing a line from the origin to the upper right corner, it divides the upper right quadrant.

into σ10<u<10<v<u, σ20<u<v0<u<1.

The following nonsingular integration transformation is introduced for σ1

u=vk=vuE39

The following nonsingular integration transformation is introduced for σ2

v=vλ=uvE40

Insertingv=ku(oru=λv)into Eqs. (35)(37),one arrives at

Δζ=4ab2πγ01α00+13α20+13α02k2+15α22k2dka2+b2k2+4ab2πγ01α00+13α02+13α20λ2+15α22λ2b2+a2λ2E41
Δξ=4a2b2πγ01α10+13α12k2dka2+b2k23/24a2b2πγ01α10+13α12λ2b2+a2λ23/2E42
Δη=4ab22πγ01α01+13α21k2dka2+b2k23/24ab22πγ01α01+13α21λ2b2+a2λ23/2E43
ΔL=4ab2π01α20+α02k2+13α22k2dka2+b2k23/2+4ab2π01α00+α20λ2+13α22λ2b2+a2λ23/2E44

Now we can see that the denominators are greater than zero after transformation Eq. (39) and Eq. (40), and the singularities have been eliminated. The integrals in xand ydirections are converted to the integrals of the powers of kandλ. This basically changes the double integrals to single variable integrals, which could easily be calculated in Mathematica as follows: E45 E46 E47 E48

In case of a square grid with a unit length, Eqs. (45)–(48) can be simplified in Mathematica as. E49 E50 E51 E52

One could find that it is greatly fussy to complete these integrals by hand, which could be easily realized using some commands of computer algebra system.

## 4. The series expansions of direct transformations between three anomalies in satellite geodesy

The determination of satellite orbit is one of the fundamental problems in satellite geodesy. A graphical representation of the Keplerian orbit is given in Figure 2, see .

Eccentric, mean and true anomalies are used to describe the movement of satellites. Their transformations are often to be dealt with in satellite ephemeris computation and orbit determination of the spacecraft. In Figure 2, Eis the Eccentric anomaly, υis the true anomaly. In order to realize the direct transformations between these anomalies, the series expansions of their transformations are derived using the power series method with the help of computer algebra system Mathematica. Their coefficients are expressed in a power series of the orbital eccentricity eand extended up to eighth-order terms of the orbital eccentricity.

### 4.1. The series expansions of the direct transformation between eccentric and mean anomalies

Let the mean anomaly be M. Mcan be expressed by Eas follows:

M=EesinEE53

Differentiating the both sides of Eq. (53) yields

dEdM=11ecosEE54

To expand Eq. (54) into a power series of cosM, we introduce the following new variable

t=cosME55

therefore

dMdt=1sinME56

and then denote

ft=dEdM=11ecosEE57

Substituting E0=π2into Eq. (53) yields

M0=π2eE58

Substituting Eq. (59) into Eq. (55), one arrives at

t0=sineE59

Making use of the chain rule of implicit differentiation

ft=dfdEdEdMdMdtft=dfdEdEdMdMdt+dfdMdMdt

It is easy to expand Eq. (58) into a power series of t0

ft=dEdM=ft0+ft0tt0+12!ft0tt02+13!ft0tt03+E60

One can imagine that these procedures are too complicated to be realized by hand, but will become much easier and be significantly simplified by means of Mathematica. Omitting the detailed procedure in Mathematica, one arrives at

dEdM=1+b1cosMsine+b22!cosMsine2+b33!cosMsine3+b44!cosMsine4+b55!cosMsine5+b66!cosMsine6+b77!cosMsine7+b88!cosMsine8E61

where

b1=e+12e3+124e+561720e7b2=4e2+133e4+4715e6+12163e8b3=27e3+912e5+112724e7b4=256e4+29375e6+82771105e8b5=3125e5+181732e7b6=46656e6+11505937e8b7=823543e7b8=16777216e8E62

Integrating at the both sides of Eq. (62) gives the series expansion

E=M+α1sinM+α2sin2M+α3sin3M+α4sin4M+α5sin5M+α6sin6M+α7sin7M+α8sin8ME63

where

α1=e18e3+1192e519216e7α2=12e216e4+148e61720e8α3=38e327128e5+2435120e7α4=13e4415e6+445e8α5=125384e531259216e7α6=2780e6243560e8α7=1680746080e7α8=128315e8E64

### 4.2. The series expansions of the direct transformation between eccentric and true anomalies

The true anomaly υcan be expressed by Eas follows:

tanυ2=1+e1etanE2E65

Therefore, it holds

υ=2arctan1+e1etanE2E66

One could expand υas a power series of the eccentricity at e=0in order to obtain the direct series expansion of the transformation from Eto υ. Omitting the detailed procedure in Mathematica, one arrives at

υ=E+β1sinE+β2sin2E+β3sin3E+β4sin4E+β5sin5E+β6sin6E+β7sin7E+β8sin8EE67

where

β1=e+14e3+18e5+564e7β2=14e2+18e4+564e6+7128e8β3=112e3+116e5+364e7β4=132e4+132e6+7256e8β5=180e5+164e7β6=1192e6+1128e8β7=1448e7β8=11024e8E68

From Eq. (67), one knows

E=2arctan1e1+etanυ2E69

Expanding Eas a power series of the eccentricity at e=0by means of Mathematica yields the direct series expansion of the transformation from υto E

E=υ+γ1sinυ+γ2sin2υ+γ3sin3υ+γ4sin4υ+γ5sin5υ+γ6sin6υ+γ7sin7υ+γ8sin8υE70

where

γ1=e14e318e5564e7γ2=14e2+18e4+564e6+7128e8γ3=112e3116e5364e7γ4=132e4+132e6+7256e8γ5=180e5164e7γ6=1192e6+1128e8γ7=1448e7γ8=11024e8E71

### 4.3. The series expansions of the direct transformation between mean and true anomalies

The whole formulae for the transformation from Mto υare as follows

E=M+α1sinM+α2sin2M+α3sin3M+α4sin4M+α5sin5M+α6sin6M+α7sin7M+α8sin8Mυ=E+β1sinE+β2sin2E+β3sin3E+β4sin4E+β5sin5E+β6sin6E+β7sin7E+β8sin8EE72

Since the coefficients αi,βi(i=1,2,8) are expressed in a power series of the eccentricity, one could expand υas a power series of the eccentricity at e=0in order to obtain the direct expansion of the transformation from Mto υ. Omitting the main operations by means of Mathematica, one arrives at the direct expansion of the transformation from Mto υ

υ=M+δ1sinM+δ2sin2M+δ3sin3M+δ4sin4M+δ5sin5M+δ6sin6M+δ7sin7M+δ8sin8ME73

where

δ1=2e14e3+596e5+1074608e7δ2=54e21124e4+17192e6+435760e8δ3=1312e34364e5+95512e7δ4=10396e4451480e6+412311520e8δ5=1097960e559574608e7δ6=1223960e679134480e8δ7=4727332256e7δ8=556403322560e8E74

The whole formulae for the transformation from υto Mare as follows

E=υ+γ1sinυ+γ2sin2υ+γ3sin3υ+γ4sin4υ+γ5sin5υ+γ6sin6υ+γ7sin7υ+γ8sin8υM=EesinEE75

Expanding Mas a power series of the eccentricity at e=0by means of Mathematica yields the direct series expansion of the transformation from υto M

M=υ+ε1sinυ+ε2sin2υ+ε3sin3υ+ε4sin4υ+ε5sin5υ+ε6sin6υ+ε7sin7υ+ε8sin8υE76

where

ε1=2eε2=34e2+18e4+364e6+3128e8ε3=13e318e5716e7ε4=532e4+332e6+15256e6ε5=340e5116e7ε6=7192e6+5128e6ε7=156e7ε8=91024e8E77

### 4.4. The accuracy of the derived series expansions

In order to validate the exactness of the derived series expansions, one has examined their accuracies when the orbital eccentricity eis respectively equal to 0.01, 0.05, 0.1 and 0.2.

One makes use of Eq. (53) and Eq. (67) to obtain the theoretical value M0 and υ0at given geodetic latitude E0. Substituting M0 into Eq. (64) and Eq. (74), one arrives at the computation value E1 and υ1. Substituting υ0into Eq. (71) and Eq. (77), one arrives at the computation value E2 and M1. Substituting E0 into Eq. (68), one arrives at the computation value υ2. The differences between the computation and theoretical values indicate the accuracies of the derived series expansions, which are denoted as ΔE1, Δυ1/, ΔM1/,ΔE2/,Δυ2/. Due to limited space, these errors when eis equal to 0.05 are only listed in Table 2.

E0/20406080
ΔE1/8.2×1081.7×1071.6 × 10−72.2×108
Δυ1/3.5×1077.4×1076.8×1079.9×108
ΔM1/3.5 × 10−85.2×1097.1×1093.7×109
ΔE2/2.8×1082.1×1081.4×1081.2×108
Δυ2/2.9×1082.4×1081.5×1081.3×108

### Table 2.

Errors of the derived series expansions.

From Table 2, one could find that the accuracy of derived series expansions is higher than 105, which could satisfy practical application. Other numerical examples indicate that when the orbital eccentricity eis respectively equal to 0.01, 0.01 and 0.2, the accuracy of derived series expansions is correspondingly higher than 1010, 103and 0.1.

## 5. Conclusions

Some typical mathematical problems in geodesy are solved by means of computer algebra analysis method and computer algebra system Mathematica. The main contents and research results presented in this chapter are as follows:

1. The forward and inverse expansions of the meridian arc often used in geometric geodesy are derived. Their coefficients are expressed in a power series of the first eccentricity of the reference ellipsoid and extended up to its tenth-order terms.

2. The singularity existing in the integral expressions of height anomaly, deflections of the vertical and gravity gradient is eliminated using the nonsingular integration transformations, and the nonsingular expressions are systematically derived.

3. The series expansions of direct transformations between three anomalies in satellite geodesy are derived using the power series method. Their coefficients are expressed in a power series of the orbital eccentricity eand extended up to eighth-order terms of the orbital eccentricity.

## Acknowledgments

This work was financially supported by National Natural Science Foundation of China (Nos. 41631072, 41771487, 41571441).

## Conflict of interest

There are no conflicts of interest.

## Thanks

We would like to express our great appreciation to the editor and reviewers. Thanks very much for the kind work and consideration of Ms. Kristina Jurdana on the publication of this chapter.

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Hou-pu Li and Shao-feng Bian (November 5th 2018). Mathematical Analysis of Some Typical Problems in Geodesy by Means of Computer Algebra, Trends in Geomatics - An Earth Science Perspective, Rifaat Abdalla, IntechOpen, DOI: 10.5772/intechopen.81586. Available from:

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