Accurate computation of gravitational field of a tesseroid

Noname manuscript No. (will be inserted by the editor)

Accurate computation of gravitational field of a tesseroid Toshio Fukushima

Received: / Accepted:

Abstract We developed an accurate method to compute the gravitational field of a tesseroid. The method numerically integrates a surface integral representation of the gravitational potential of the tesseroid by conditionally splitting its line integration intervals and by using the double exponential quadrature rule. Then, it evaluates the gravitational acceleration vector and the gravity gradient tensor by numerically differentiating the numerically integrated potential. The numerical differentiation is conducted by appropriately switching the central and the single-sided second order difference formulas with a suitable choice of the test argument displacement. If necessary, the new method is extended to the case of a general tesseroid with the variable density profile, the variable surface height functions, and/or the variable intervals in longitude or in latitude. The new method is capable of computing the gravitational field of the tesseroid independently on the location of the evaluation point, namely whether outside, near the surface of, on the surface of, or inside the tesseroid. The achievable precision is 14–15 digits for the potential, 9–11 digits for the acceleration vector, and 6–8 digits for the gradient tensor in the double precision environment. The correct digits are roughly doubled if employing the quadruple precision computation. The new method provides a reliable procedure to compute the topographic gravitational field, especially that near, on, and below the surface. Also, it could potentially serve as a sure reference to complement and elaborate the existing approaches using the Gauss-Legendre quadrature or other standard methods of numerical integration. Keywords gravitational field; numerical differentiation; numerical integration; split quadrature; tesseroid

1 Introduction The accurate computation of the gravitational field of the Earth is an important issue in physical geodesy (Heiskanen and Moritz, 1967). This is true especially when the evaluation point is inside the Earth, on and near the surface of the Earth, or outside the Earth but inside its Brillouin sphere or spheroid (Jekeli, 2007). Indeed, the computation of the internal T. Fukushima National Astronomical Observatory / SOKENDAI, Ohsawa, Mitaka, Tokyo 181-8588, Japan E-mail: [email protected]

2

Toshio Fukushima

gravitational potential is essential in the determination of the geoid since the geoid is usually below the ground surface in the land area (Vanicek and Krakiwsky, 1982). However, it is also true that this issue had been difficult to resolve (Klees and Lehmann, 1998). This was because of the algebraic singularities of orders 1 through 3 in the Newton integrals of the gravitational potential V , the gravitational acceleration vector g, and the gravity gradient tensor Γ , respectively (Kellogg, 1929): V (X) ≡ G g(X) ≡

Γ (X) ≡

∂ 2V (X) =G ∂ X∂ X

∫

ρ (x) 3 d x, |x − X|

(1)

∫

∂ V (X) ρ (x) (x − X) 3 d x, =G ∂X |x − X|3 [ ] ∫ ρ (x) 3 (x − X) ⊗ (x − X) − |x − X|2 I |x − X|5

(2)

d3 x,

(3)

where G is Newton’s constant of universal attraction, X is the position vector of the evaluation point, x is the position vector of a mass element of the Earth, ρ (x) is the volume mass density of the Earth, I is the unit matrix, and the integration domain covers all the components of the Earth including the atmosphere, the sea and land water, the ice sheet, the crust, the mantle, the outer fluid core, and the inner solid core (Stacey and Davis, 2008). In order to avoid the integration difficulties, past focus was the approximation methods using a simple body with the known analytical solution as a building block (Heck and Seitz, 2007; Wild-Pfeiffer, 2008). Especially, the rectangular prism model (MacMillan, 1930) and its natural extension, the polyhedron model (Waldvogel, 1979), have been extensively discussed. For example, Nagy et al. (2000) examined the issue of the singularities and GarciaAbdeslem (2005) extended the homogeneous prism model by allowing its vertical density profile as a cubic polynomial. See a comprehensive list of the references provided in a recent work (Conway, 2015). However, the polyhedron model merely provides a first order approximation (Fukushima, 2017b). In fact, its approximation error decreases only linearly with respect to the number of polyhedron faces, and therefore with respect to the computational cost (D’Urso, 2014). Also, the inaccuracy of the model significantly increases near the surface of the polyhedrons (Yu and Baoyin, 2015). This is inconvenient for the precise computation of the gravitational field close to the ground surface. Therefore, in geodesy, intensively investigated are the high order numerical quadrature of the tesseroid model (Hirt and Kuhn, 2014; Wu, 2016) and its spheroidal extension (Novak and Grafarend, 2005; Roussel et al., 2015). For example, Grombein et al. (2013) proposed an adaptive approach to increase the horizontal resolution to assure the computational accuracy in the very near zone. See also a comprehensive list of references in a recent work (Uieda et al., 2016). Nevertheless, the existing methods have not succeeded in providing the reliable gravitational field of a tesseroid when the evaluation point is near the surface of, on the surface of, and inside the tesseroid (Kuhn and Hirt, 2016). Recently, we developed a series of accurate methods to compute the gravitational and electromagnetic potentials and the associated acceleration vector field of general objects (Fukushima, 2016a,b,c,d, 2017a,b). Among them, the last method was designed for a smooth boundary layer of a non spherical but radially unique finite body such as peculiar shaped asteroids like Eros (Fukushima, 2017b). In short, it is a combination of the numerical integration of a surface integral expression of the gravitational potential and the numerical differentiation of the numerically integrated potential to compute the gravitational acceleration vector. Of course, the method is directly applicable to the mass layers of nearly spherical

Accurate computation of gravitational field of a tesseroid

3

objects such as the Earth and other heavy solar system objects like planets, massive satellites, and large asteroids. This is feasible if their density profile and surface height functions are provided in a sufficiently smooth manner, say by the spherical harmonic expansion (Hirt and Rexer, 2015) or by the piecewise bicubic and higher polynomial approximation (Press et al., 2007). Nevertheless, in the case of the Earth, the models of the shape functions and of the density profile of its layers are usually not given smoothly but provided in gridded forms of discontinuous nature (Tachikawa et al., 2011; Laske et al., 2013). Therefore, in this article, we modify the previous method of ours (Fukushima, 2017b) so as to be applicable to a tesseroid. Also, we add the procedure to compute the gravity gradient tensor, which are not usually demanded in celestial mechanics and dynamical astronomy. Below, we explain the modified and enhanced method in Sect. 2 and describe the results of its numerical experiments in Sect. 3. Also, the Appendices provides the detailed expression of the kernel function, a short summary of the numerical quadrature method used, and the adopted finite difference formulae. Furthermore, the Electronic Supplementary Materials (ESM) contain the explanation of the algorithm, the resulting Fortran programs, and a number of graphs illustrating the feature of the gravitational field of the standard homogeneous tesseroid, its triangular variation, and a polar cap slab as its special case.

2 Method 2.1 Numerical integration of gravitational potential Throughout this article, we adopt the spherical limit of the Gauss normal coordinates, (ϕ , λ , h), as the basic coordinates. Here ϕ is the geocentric latitude, λ is the longitude, and h is the height measured from the reference spherical surface along the surface normal. As a result, the geocentric rectangular coordinates (x, y, z) are expressed in terms of them as     x cos ϕ cos λ  y  = (R0 + h)  cos ϕ sin λ  , (4) sin ϕ z where R0 is the radius of the reference spherical surface. The reason why we prefer h to the ordinary choice of the radius, r ≡ R0 + h, is in order to avoid the unnecessary loss of information amounting to 2–4 digits when encountered with using r instead of h. In any case, a tesseroid (Anderson, 1976) can be regarded as a rectangular parallelepiped in these basic coordinates. Namely, it is a three-dimensional volume defined as

ΦS ≤ ϕ ≤ ΦN , ΛW ≤ λ ≤ ΛE , HB ≤ h ≤ HT ,

(5)

where ΦN and ΦS are the north and south end points of the latitude interval of the tesseroid, ΛE and ΛW are the east and west end points of the longitude interval of the tesseroid, and HT and HB are the top and bottom end points of the interval of the height from the reference sphere of the tesseroid, respectively. After analytically integrating the radial part of the original volume integral expression (Martinec, 1988; Heck and Seitz, 2007; Wild-Pfeiffer, 2008), we express the gravitational potential of the tesseroid as a surface integral (Fukushima, 2017b) as ) ∫ Φ N (∫ Λ E V (Φ , Λ , H) = Gρ R20 (6) K (ϕ , λ , HB , HT , Φ , Λ , H) dλ dϕ , ΦS

ΛW

4

Toshio Fukushima

Conditional Split Quadrature Method 90

φ (degree)

45 X 0

-45

1

2

3

4

-90 0

45

90

135 180 225 270 315 360 λ (degree)

Fig. 1 Diagram of conditional split quadrature method. Illustrated is the manner of the conditional split quadrature method. The X mark indicates the angular coordinates of the evaluation point, (Φ , Λ ). The integration domain is split along the two lines, ϕ = Φ and λ = Λ , indicated by broken lines: (i) split into four parts when ΦS < Φ < ΦN and ΛW < Λ < ΛE as in the grid 1, (ii) split into two parts in latitude when ΦS < Φ < ΦN but Λ ≤ ΛW as in the grid 2, (iii) split into two parts in longitude when ΛW < Λ < ΛE but ΦN ≤ Φ as in the grid 3, but (iv) not split when ΦN ≤ Φ and Λ ≤ ΛW as in the grid 4.

where (Φ , Λ , H) are the spherical Gauss normal coordinates of the evaluation point, ρ is the volume mass density of the tesseroid, which we assume to be constant for simplicity, and K is the kernel function explained in Appendix A. In the above, we first integrate the integrand with respect to the longitude. This is in order to reduce the total number of the transcendental function evaluations such as cos ϕ . We evaluate V by conditionally using the split quadrature method in two dimensions. The original split quadrature method (Fukushima, 2014) was invented to resolve the computational difficulties in the numerical quadrature of a general line integral when its integrand has an integrable singularity or a sharp peak in the middle of the given integration


5

interval. More precisely speaking, we shift the location of the logarithmically blowing-up singularity or of the sharp peak of the kernel function K to the end points of the line integrals (Fukushima, 2016a,d, 2017a). Once we split the interval of the line integrals contained in V at the point where ϕ = Φ and λ = Λ (Fukushima, 2017b). At the beginning of the current study, we presumed that the splitting is not necessary if the evaluation point is outside the tesseroid and thus the previous method (Fukushima, 2017b) is directly applicable to the present case without any change. However, this expectation was betrayed when the evaluation point is close to the tesseroid. After a number of numerical experiments, we realized that the splitting is always effective when the integrand has a peak in the integration interval. Therefore, we modified the previous method such that the splitting is applied even if the evaluation point is outside the tesseroid. In order to minimize the effect of precision loss caused by computing the difference of large quantities during the numerical integration procedure, we introduced the new integration variables as ξ ≡ ϕ − Φ, η ≡ λ − Λ . (7) Also, we precomputed some associated constants as

ξS ≡ ΦS − Φ , ξN ≡ ΦN − Φ , ηW ≡ ΛW − Λ , ηE ≡ ΛE − Λ .

(8)

Then, the integration is split conditionally as follows: (i) if ξS < 0 < ξN and ηW < 0 < ηE , then we split V into four pieces, namely two pieces in ξ and two pieces in η as ) ∫ 0 (∫ 0 ∫ ηE P(ξ , η )dη dξ V (Φ , Λ , H) = P(ξ , η )dη + ξS

+

∫ ξN 0

(∫

0

ηW

ηW

0

P(ξ , η )dη +

∫ ηE 0

) P(ξ , η )dη dξ ,

(9)

(ii) else if ξS < 0 < ξN but 0 ≤ ηW or ηE ≤ 0, then we split V into two pieces in ξ as ) ) ∫ 0 (∫ ηE ∫ ξN (∫ ηE V (Φ , Λ , H) = (10) P(ξ , η )dη dξ + P(ξ , η )dη dξ , ξS

ηW

ηW

0

(iii) else if ηW < 0 < ηE but 0 ≤ ξS or ξN ≤ 0, then we split V into two pieces in η as ) ∫ ξN (∫ 0 ∫ ηE V (Φ , Λ , H) = P(ξ , η )dη dξ , (11) P(ξ , η )dη + ξS

ηW

0

and (iv) else, then we do not split V as V (Φ , Λ , H) =

∫ ξN (∫ ηE ξS

ηW

) P(ξ , η )dη dξ .

(12)

In the above, P is an abbreviation of the integrand of the surface integral as P(ξ , η ) ≡ Gρ R20 K (Φ + ξ , Λ + η , HB , HT , Φ , Λ , H) .

(13)

Refer to Fig. 1 illustrating the manner of splitting. The above conditional 4-piece splitting is a significant improvement of the previous treatment (Fukushima, 2017b), which uses only 2-piece splitting at most.

6

Toshio Fukushima

As for the numerical integration of the line integrals, we adopt the double exponential (DE) rule (Takahashi and Mori, 1973, 1974), which is compactly summarized in Appendix B. This is because the DE rule is known to be able to integrate the weakly singular integrands when their singularities are located at the end points of the integration interval (Mori, 1985). Also, its cost performance is regarded as the best in the existing methods of numerical quadrature (Bailey et al., 2005; Press et al., 2007). Refer to Appendix C of Fukushima (2017b) showing the high cost performance and the good fidelity of dqde, a Fortran 90 subroutine to execute the DE rule in the double precision environment. Its quadruple precision extension, qqde, is provided in the ESM.

2.2 Numerical computation of gravitational acceleration vector and gravity gradient tensor Let us move to the computation of the gravitational acceleration vector, g, and the gravity gradient tensor, Γ . We evaluate them by the numerical partial differentiation of the numerically integrated potential, V . As for the numerical differentiation method, we adopt a conditional switch of the second order difference formulae as presented in Appendix C. In most cases, the expressions of the first order partial derivatives are obtained by solving the Taylor series expansion of the gravitational potential around the evaluation point as ( ) ( ) ∂V V (Φ + ∆1 Φ , Λ , H) −V (Φ − ∆1 Φ , Λ , H) (∆1 Φ )2 ∂ 3V = − − · · · , (14) ∂ Φ Λ ,H 2∆1 Φ 6 ∂ Φ3 (

) ∂ 3V − · · · , (15) ∂Λ 3 Φ ,H ) ( ) ( V (Φ , Λ , H + ∆1 H) −V (Φ , Λ , H − ∆1 H) ∂V (∆1 H)2 ∂ 3V = .− − · · · . (16) ∂ H Φ ,Λ 2∆1 H 6 ∂ H3

∂V ∂Λ

)

=

V (Φ , Λ + ∆1Λ , H) −V (Φ , Λ − ∆1Λ , H) (∆1Λ )2 − 2∆1Λ 6

(

If the second and higher order terms are ignored, the second order central difference formulae are derived. Here the test argument displacements are chosen as

∆1 H = Lδ1 , ∆1 Φ =

∆1 H ∆1 H , ∆1Λ = , R R cos Φ

(17)

δ1 is a relative test displacement the definition of which will be explained later, R is the radius vector of the evaluation point computed as R ≡ R0 + H,

(18)

L ≡ max (ℓ0 , ℓ) ,

(19)

and L is a scale factor defined as where ℓ0 is one half of the nominal linear size of the tesseroid expressed as √ [ ] 1 (∆ H)2 + R2C (∆ Φ )2 + (cos ΦC )2 (∆Λ )2 , ℓ0 ≡ 2

(20)

and ℓ is the distance between the evaluation point and the geometrical center of the tesseroid expressed as √ ℓ ≡ R2 + R2C − 2RRC [sin Φ sin ΦC + cos Φ cos ΦC cos (Λ − ΛC )], (21)


7

while ∆ Φ , ∆Λ , and ∆ H are the grid sizes of the tesseroid defined as

∆ Φ ≡ ΦN − ΦS , ∆Λ ≡ ΛE − ΛW , ∆ H ≡ HT − HB .

(22)

and RC , ΦC , and ΛC are the spherical polar coordinates of the geometrical center of the tesseroid expressed as RC ≡ R0 +

ΦN + ΦS ΛE + ΛW HT + HB , ΦC ≡ , ΛC ≡ . 2 2 2

(23)

Refer to Appendix C for the cases when the other type of difference formulae must be used. At any rate, once these partial derivatives are computed, all the components of the gravitational acceleration vector are explicitly obtained by normalizing them as ( ) ( ) ( ) 1 ∂V 1 ∂V ∂V , gΛ = , gH = . (24) gΦ = R ∂ Φ Λ ,H R cos Φ ∂Λ Φ ,H ∂ H Φ ,Λ Similarly, the second order partial derivatives of V are obtained by combining the solutions of the same Taylor expansion in most cases. Sample expressions of the diagonal and nondiagonal components are ( 2 ) ∂ V V (Φ , Λ , H + ∆2 H) − 2V (Φ , Λ , H) +V (Φ , Λ , H − ∆2 H) = ∂ H 2 Φ ,Λ (∆2 H)2 ) ( (∆2 H)2 ∂ 4V + +···, (25) 12 ∂ H4 ( 2 ) ∂ V = [V (Φ + ∆2 Φ , Λ + ∆2Λ , H) −V (Φ + ∆2 Φ , Λ − ∆2Λ , H) ∂ Φ∂Λ H −V (Φ − ∆2 Φ , Λ + ∆2Λ , H) +V (Φ − ∆2 Φ , Λ − ∆2Λ , H)] / [4 (∆2 Φ ) (∆2Λ )] ( ) ∂ 4V (∆2 Φ ) (∆2Λ ) +···. + 36 ∂ Φ 2 ∂Λ 2

(26)

Again, the second order central difference formulae are derived by neglecting the second and higher order terms. This time, the test argument displacements, ∆2 H, ∆2 Φ , and ∆2Λ , are set as ∆2 H ∆2 H ∆2 H = Lδ2 , ∆2 Φ = , ∆2Λ = , (27) R R cos Φ where δ2 is another relative test displacement. Its definition will be explained later. Refer to Appendix C again for the cases when the other type of difference formulae must be used. In any case, once these partial derivatives are computed, the gravity gradient tensor is constructed from these second-order partial derivatives and the gravitational acceleration vector (Tscherning, 1976) as ( ) 1 ∂ 2V gH ΓΦΦ = 2 + , (28) R ∂ Φ 2 Λ ,H R

ΓΛΛ =

1 2 R cos2 Φ

) ∂ 2V gH − gΦ tan Φ + , 2 ∂Λ Φ ,H R ( 2 ) ∂ V = , ∂ H 2 Φ ,Λ

(

ΓHH

(29) (30)

8

Toshio Fukushima

ΓΦΛ = ΓΛ Φ =

1 R2 cos Φ

ΓΦ H = ΓH Φ = ΓΛ H = ΓH Λ =

1 R

1 R cos Φ

( ( (

∂ 2V ∂ Φ∂Λ

∂ 2V ∂ Φ∂ H ∂ 2V ∂Λ ∂ H

) )

)

+

gΦ tan Φ , R2 cos Φ

(31)

−

gΦ , R

(32)

−

gH . R2 cos Φ

(33)

H

Λ

Φ

As for the relative test argument displacements, δ1 and δ2 , we set them as the functions of δ , the relative error tolerance in integrating the potential, as √ √ 3 4 δ1 = δ , δ2 = δ . (34) These are the optimal choices determined by considering the balance of the truncation and round-off errors (Fukushima, 2017b). Anyhow, the resulting relative accuracy of the par√ 3 tially differentiated√quantities are δ12 = δ 2 for the gravitational acceleration vector components, and δ22 = δ for the gravity gradient tensor components. For example, if the gravitational potential is computed by setting δ = 10−33 , then the accuracy of the gravitational acceleration and the gravity gradient tensor are expected to be 22 and 16.5 digits, respectively. On the other hand, the computational labor of each component of g is typically twice as large as that of V . Thus, that of the full evaluation of g costs six times as large. Also, that of the diagonal and non-diagonal components of the second order partial derivatives of V usually costs three and four times as large as that of V . If the potential value at the center is shared, the ratio of the additional cost of the diagonal component reduces from three to two. Furthermore, one can save the computation of one of the diagonal components of Γ by using Poisson’s equation expressed as

ΓΦΦ + ΓΛΛ + ΓHH = −4π Gρ .

(35)

Therefore, a rough estimate of the ratio of the total computation labor of V , g, and Γ to that of V becomes 1 + 3 × 2 + 2 × 2 + 3 × 4 = 23. 2.3 Special case The procedures described in the previous subsections can be simplified in some special cases. For instance, in the case of zonal slabs, namely when the integration interval in longitude is [0, 2π ), we can half the number of split pieces (Fukushima, 2017b). In fact, if ξS < 0 < ξN , then we split V into two pieces in ξ as ) ) ∫ 0 (∫ 2π ∫ ξN (∫ 2π V (Φ , Λ , H) = P(ξ , η )dη dξ + P(ξ , η )dη dξ , (36) ξS

0

0

0

else we do not split V as V (Φ , Λ , H) =

∫ ξN (∫ 2π ξS

0

) P(ξ , η )dη dξ .

(37)

Notice that we shifted the integration interval with respect to η to [0, 2π ) by using the 2π periodicity of the integrand. We used this technique in computing the gravitational field of a polar cap slab presented in the ESM.


9

2.4 Generalization So far, we have posed some restrictions on the tesseroid nature for simplicity. They are the constant density, the constant surface heights, the constant longitude interval, and the constant latitude interval. If necessary, these restrictions can be removed (Fukushima, 2017b). One such direction is to allow the volume density function vary laterally and/or vertically as

ρ=

N

∑ ρn (ϕ , λ ) (R0 + h)n ,

(38)

n=0

where N is a small integer such as 3 in the case of the popular PREM and other interior models of the Earth (Dziewonski and Anderson, 1981; Kennett, 1998) and 5 for the standard atmosphere (Karcol, 2011). Another generalization is to permit the surface radius functions to vary laterally (Smith et al., 2001, Fig. 4) as HT = HT (ϕ , λ ), HB = HB (ϕ , λ ), (39) which are suitable in treating the spheroidal surface or the surfaces of constant geodetic heights. Also, the longitude intervals can vary in latitude as

ΛW = ΛW (ϕ ), ΛE = ΛE (ϕ ).

(40)

Furthermore, the latitude intervals may vary in longitude as

ΦS = ΦS (λ ), ΦN = ΦN (λ ).

(41)

The last two cases enable us to handle the mass layers with a complicated boundary. Anyhow, in the first two cases, one only has to change P as P(ξ , η ) = GR20

N

∑ [ρn (Φ + ξ , Λ + η )×

n=0

Kn (Φ + ξ , Λ + η , HT (Φ + ξ , Λ + η ), HB (Φ + ξ , Λ + η ), Φ , Λ , H)] ,

(42)

where Kn is the kernel function of degree n described in Fukushima (2017b). This device enables us to compute the gravitational field of an inhomogeneous tesseroid or of a spheroidal or geodetic tesseroid accurately. Also, in the third case, we regard the shifted longitude end points as functions of the shifted latitude as

ηW (ξ ) ≡ ΛW (Φ + ξ ) − Λ , ηE (ξ ) ≡ ΛE (Φ + ξ ) − Λ .

(43)

On the other hand, in the last case, we first exchange the order of the line integrations with respect to longitude and latitude as ) ∫ ηE (∫ ξN V (Φ , Λ , H) = P(ξ , η )dξ dη . (44) ηW

ξS

Next, we apply the conditional splitting in a similar manner as described already. Finally, we regard the shifted latitude end points as functions of the shifted longitude as

ξS (η ) ≡ ΦS (Λ + η ) − Φ , ξN (η ) ≡ ΦN (Λ + η ) − Φ .

(45)

This trick simplifies the treatment of the density layers with a limited surface area of an irregular shape such as the sea and land water area and the ice sheets with a complicated coastal line. In any case, these modifications are so simple that no significant alterations are required in the computational procedure of the gravitational field.

10

Toshio Fukushima

3 Numerical experiments 3.1 Accuracy of quadruple precision computation Let us examine the computational accuracy of the new method conducted in the quadruple precision environment. For this purpose, we select a spherical shell as a sample object with the analytical closed-form solution of the gravitational field. In fact, the gravitational potential, V , the radial component of the gravitational acceleration vector, gH , and the radial-radial component of the gravity gradient tensor, ΓHH , of the shell are explicitly described as  2π Gρ (RT + (H < HB ) ,   [{RB ) ∆ H, }  (4π Gρ /3) R2T (H − HB ) + (RT + RB ) RB ∆ H /R Vanalytical = (46) + (RT + R) (HT − H) /2] , (HB ≤ H ≤ HT ) ,    GM/R, (HT < H) ,   0, [ ] (H < HB ) , gH,analytical = −(4π Gρ /3) (H − HB ) 1 + RB (R + RB ) /R2 , (HB ≤ H ≤ HT ) ,  −GM/R2 , (HT < H) ,   0, ( ) (H < HB ) , ΓHH,analytical = (4π Gρ /3) 1 + 2R3B /R3 , (HB < H < HT ) ,  2GM/R3 , (HT < H) , where M is the mass of the shell expressed as ( ) M ≡ (4π /3)ρ R2T + RT RB + R2B ∆ H,

(47)

(48)

(49)

and RT , RB , R, and ∆ H are the parameters and the variables defined as RT ≡ R0 + HT , RB ≡ R0 + HB , R ≡ R0 + H, ∆ H ≡ HT − HB .

(50)

Notice that ΓHH becomes indefinite on the surfaces of the shell, namely when H = HB or H = HT exactly. In those cases, we shift the evaluation points slightly inside the shell such as H = (1 + εQ ) HB or H = (1 − εQ ) HT where εQ ≡ 2−113 ≈ 9.63 × 10−35 is the quadruple precision machine epsilon. The outlook of the above formulae, Eqs (46)–(48), are significantly different from the standard expressions in the literature. This is because we rewrote the existing formulae into cancellation-free forms. The rewriting was needed to reduce the round-off errors of the formulae down to the level of a few machine epsilons, which amount to a few times 10−34 in the quadruple precision environment. Without rewriting, the reliable examination of the new numerical method is not possible. At any rate, by using the new method with a tiny error tolerance as δ = 10−33 and by choosing the unit system such that Gρ = 1 for simplicity, we obtain V , gH , and ΓHH of the shell and compare them with the above analytical solutions. As an example of the spherical shell, we set its top and bottom heights as HT = +10 km and HB = −40 km and made the radius of the reference sphere to be R0 = 6380 km. Fig. 2 shows the relative errors of the gravitational potential defined as

δ ∗V ≡ V /Vanalytical − 1, as a function of the height from the reference sphere, H, in the range |H| ≤ 100 km.

(51)


11

Gravitational Potential Error of Spherical Shell 2.5

δ = 10−33

Shell

2

1033δ*V

1.5 1 0.5 0 -0.5 -1 -100 -80 -60 -40 -20 0 20 40 60 80 100 H (km) Fig. 2 Integration error of spherical shell potential. Shown are the relative errors of the gravitational potential of a spherical shell integrated by the new method. Here the shell is of the constant density, of the top and bottom surface heights as HT = +10 km and HB = −40 km, and the reference radius is R0 = 6380 km. Along a straight radial line on the equatorial plane where Φ = 0◦ and Λ = 180◦ , the errors are measured by comparing with the analytical solution as δ ∗V ≡ V /Vanalytical − 1 and plotted as a function of H in the range |H| ≤ 100 km when the input relative error tolerance of the DE rule is set as tiny as δ = 10−33 . Notice that the potential value is constant inside the void sphere surrounded by the shell and monotonically decreasing with respect to the radius otherwise. Its relative variation is small in this range of H as VH=+100km /VH=−100km ≈ 0.982. The obtained errors are of the order of δ independently on the location of the evaluation point.

Obviously, the obtained errors are randomly scattered. Also, they are sufficiently small, say at the level of 10−33 . These features are independent on the location of the evaluation point, namely (i) in the void sphere surrounded by the shell, (ii) exactly on the inner surface of the shell, (iii) inside the shell, (iv) exactly on the outer surface of the shell, and (v) outside the shell. Notice that this feature of the location independency is kept even when δ is

12

Toshio Fukushima

Gravitational Potential Error of Spherical Shell 0 -4

log10|δ*V|

-8 -12 -16

4 8 12 16

-24

20 24

-28

28

-32

32

-20

-36 -100 -80 -60 -40 -20

0 20 40 60 80 100 120 H (km)

Fig. 3 Integration error of spherical shell potential: δ -dependence. Same as Fig. 2 but plotted are the base-10 logarithm of δ ∗V for various values of δ , the input relative error tolerance, as δ = 10−2 , 10−4 , . . . , and 10−34 . The digits indicated in the figure are − log10 δ for the multiples of 4, which correspond to the curves with filled circles. On the other hand, those with open circles are the results when − log10 δ = 4n − 2.

changed. In fact, Fig. 3 reveals that the magnitude of the achieved errors, |δ V |, are in good fidelity with δ in the sense that |δ V | ≈ δ . Let us move on to the computation of the gravitational acceleration and the gravity gradient tensor. Fig. 4 illustrates δ ∗ gH and δ ∗ΓHH , the scaled errors of gH and ΓHH defined as ( ) δ ∗ gH ≡ gH − gH,analytical / max |gH | , (52) ( ) ∗ δ ΓHH ≡ ΓHH − ΓHH,analytical / max |ΓHH | (53) This time, we scaled the measured errors by max |gH | and max |ΓHH |, the maximum value of the magnitude of gH or ΓHH , respectively. This is because both of gH and ΓHH completely vanish in the inner void sphere where H < HB , and therefore the usual relative error such as


13

Acceleration/Gradient Error of Spherical Shell -15

δ = 10

log10|Scaled Error|

-16

−33

-17 -18

δ*ΓHH

-19 -20 -21 -22

δ*gH

-23 -100 -80 -60 -40 -20 0 20 40 60 80 100 H (km) Fig. 4 Computation error of spherical shell acceleration and gravity gradient. Same as Fig. 2 but for gH , the radial component of the gravitational acceleration vector, and for ΓHH , the radial-radial component of the gravity gradient tensor of the spherical shell computed by the new method. The(errors are measured ) by comparing with the analytical solution and scaled by its maximum value as δ ∗ gH ≡ gH − gH,analytical / max |gH | ) ( and δ ∗ ΓHH ≡ ΓHH − ΓHH,analytical / max |ΓHH |. For showing both of them simultaneously, the base-10 logarithms of the errors are plotted as a function of H. The obtained errors are significantly larger when inside the spherical shell. Also, the errors in the inner void sphere are somewhat smaller than outside the shell. This difference in the computed values of gH and ΓHH comes from that in their nonlinearity with respect to H.

δ ∗V in the above is not appropriate. In any case, this figure roughly confirms our anticipation that √ |δ ∗ gH | ≈ δ 2/3 = 10−22 , |δ ∗ΓHH | ≈ δ ≈ 3 × 10−17 , (54) In the figure, the error curves exhibit no random feature and resemble step functions. This is because the observed errors are mostly due to the truncation errors of the adopted finite difference formulae.

14

Toshio Fukushima

Error of Density Distribution Recovery: H = 0 km -15

−33

δ = 10

Tesseroid

log10|δρ|

-16

-17

-18

-19 85

85.5

86

86.5 Λ (deg)

87

87.5

88

Fig. 5 Longitude dependence of scaled error of density recovered from gravity gradient tensor of tesseroid. Shown are the longitude dependence of the errors of ρG , the density determined from the gravity gradient tensor of the tesseroid computed by the new method executed in the quadruple precision environment with an input relative error tolerance as tiny as δ = 10−33 . Actually shown are the base-10 logarithm of the scaled errors of ρG when H = 0 km. They are plotted as a function of Λ in the longitude interval, 85◦ ≤ Λ ≤ 88◦ . Overlapped are the results when Φ is changed in the latitude interval, +26◦ ≤ Φ ≤ +29◦√since there are no evident dependence on the adopted value of Φ . The obtained errors are of the order of δ ≈ 3 × 10−17 independently on the location of the evaluation point.

3.2 Internal consistency of quadruple precision computation Let us move on to the computational accuracy of the gravitational field of a tesseroid. In this case, no analytical solution is known. Therefore, we can not compare the computed value with the exact solution. Instead, we examine the internal consistency of the quadruple precision computation of the new method. More specifically speaking, we compute ρG , the


15

density recovered from the diagonal components of the computed gravity gradient tensor as

ρG ≡ (ΓΦΦ + ΓΛΛ + ΓHH ) /(−4π G),

(55)

which is nothing but a rewriting of Poisson’s equation. This quantity must be the same as ρ , the density distribution used in the numerical integration of V , and therefore, in the numerical differentiation to derive Γ . Before going further, let us specify a test tesseroid as a volume of 1◦ × 1◦ × 50 km of the geographical area covering Mt. Everest and some high mountains in the Himalayas. More specifically speaking, we set the tesseroid defined as a triple product of the intervals in the spherical Gauss normal coordinates (Φ , Λ , H) as +27◦ ≤ Φ ≤ +28◦ , 86◦ ≤ Λ ≤ 87◦ , and −40 km ≤ H ≤ +10 km while the reference radius is chosen as R0 = 6380 km. Then, we introduced the scaled errors of ρG defined as

δ ρ ≡ (ρG − ρ ) /ρ0 ,

(56)

where ρ0 is the nonzero constant value of ρ . Next, we prepared Fig. 5 plotting the errors as a function of Λ for various values of Φ in the range, +26◦ ≤ Φ ≤ +29◦ while H is fixed as H = 0 km. Notice that the corresponding spherical surface is sufficiently large to cover all the horizontal cross section of the tesseroid and passes through the tesseroid. As a result, ρ behaves like a step function in the sense ρ = ρ0 when inside the tesseroid and ρ = 0 otherwise. In the figure, the base-10 logarithm of δ ρ is plotted as a function of Λ in the interval, 85◦ ≤ Λ ≤ 88◦ . Overlapped are the results when Φ is moved in the interval, +26◦ ≤ Φ ≤ +29◦ . Obviously, this result guarantees around 16 digit consistency of the gravity gradient tensor computed by the new method executed in the quadruple precision environment when δ is set as tiny as 10−33 .

3.3 Precision of double precision computation Let us examine the computational precision of the new method conducted in the double precision environment, which will be the main procedure for the practical computations. In the previous subsections, we confirmed the accuracy of the new method executed in the quadruple precision environment. Therefore, we measure the error of the double precision computation by comparing it with the quadruple precision computation with a sufficiently small error tolerance, say δ = 10−24 . Of course, thus measured errors show only the computational precision of the new method since they were calculated from the comparison of the results obtained by the same method but in different computing precisions. However, since we confirmed the high accuracy and the high internal consistency of the reference method, we may anticipate that the examined double precision computation has the same amount of accuracy as the computation precision confirmed in the followings. Refer to Fig. 6 illustrating the height variation of V , gH , and ΓHH of the sample tesseroid introduced in the previous subsection. Figs 7 and 8 represent the computational errors of V , gH , and ΓHH in the double precision environment. Here the errors are displayed as the scaled errors defined such as ( ) δ V ≡ Vdouble −Vquadruple /Vmax , (57) where Vdouble and Vquadruple are the potential values computed by the new method in the double and quadruple precision environments, respectively, and Vmax is the maximum value of V with respect to H shown in Fig. 6.

16

Toshio Fukushima

Height Variation of Gravitational Field of Tesseroid 4

Φ=+27.5o, Λ=86.5o

Scaled Values

3 2

V

1

ΓHH

0 -1

gH

-2 -3 -4 -100 -50

(+27o,+28o)x(86o,87o)x(−40km,+10km) 0

50

100 150 200 250 300 H (km)

Fig. 6 Height variation of gravitational field of tesseroid. For the test tesseroid introduced in Sect. 3.2, we obtained its V , gH , and ΓHH by the new method along a straight radial line passing through the horizontal geometric center of the tesseroid such that Φ = +27.5◦ and Λ = 86.5◦ . Notice that the tesseroid occupies the height range as −40 km ≤ H ≤ +10 km. After appropriate scalings, we plotted them as functions of H in the range, −100 km ≤ H ≤ +300 km. This selection of Φ and Λ makes some components of g and Γ completely vanish as gΦ = gΛ = ΓΦΛ ( = ΓΦ H )= 0. Also, the other components of Γ are related to gH as ΓΦΦ = ΓΛΛ = gH /R and ΓΛ H = −gH / R2 cos Φ where R = R0 + H.

At any rate, the achieved precision of the gravitational quantities computed by the new method is high. For instance, the amount of the topographic gravity anomaly does not exceed that of a spherical shell with a typical thickness of the continental crust. If we approximate the Earth as a sphere of the radius 6380 km, and assume the mean density and thickness of the crust as 2700 kg m−3 and 44 km, respectively, then the relative contribution of the gravity by the crust becomes around 1% on the surface of the reference sphere. This amounts to 10 gal or so. Therefore, the 10 digit relative precision of the tesseroid gravity computation shown in Fig. 7 means the computing precision of the topographic gravity anomaly at the


17

log10|Scaled Error|

Computational Error of Gravitational Field of Tesseroid -5 -6 ΓHH -7 -8 -9 -10 -11 -12 -13 gH -14 -15 -16 V -17 -100 -50

δ=10

0

50

−16

100 150 200 250 300 H (km)

Fig. 7 Computational error of gravitational field of tesseroid. Same as Fig. 6 but plotted are the base-10 logarithm of the scaled errors of V , gH , and ΓHH of the tesseroid. They are marked by filled circles for V , open circles for gH , and crosses for ΓHH , respectively. Here the errors are measured as the diferences from the quadruple precision computation with a sufficiently small error tolerance, δ = 10−24 , and scaled by the maximum value seen in Fig. 6. The measureed errors are mostly independent on the height of the evaluation point, namely whether it is outside, near the surfaces of, on the surfaces of, and inside the tesseroid. The observed scatter is mainly due to the round-off error accumulation during the double precision integration of V . When the potential integration is conducted with the relative error tolerance as small as δ = 10−16 , the computational precision becomes 15, 10, and 7 digits in the case of V , gH , and ΓHH , respectively.

level of nano gal, which is comparable to the measurement precision of a typical superconducting gravimeter. Now that we examined the height variation of the computational errors in Figs 7 and 8, we depict the other aspects below. Namely, in order to examine the dependence on the angular coordinates, we measured the errors of V as a function of Φ and Λ by fixing H. Refer to Fig. 9 showing the bird’s-eye view and the associated contour map of the scaled errors of the integrated values of V when H = 0 km. Again, there is no significant dependence

18

Toshio Fukushima

log10|Scaled Error|

Computational Error of Gravitational Field of Tesseroid -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17

ΓHH

δ=10

−16

gH

V

-6 -5 -4 -3 -2 -1 0 1 2 3 log10 [(H−HT)/km]

4

5

6

Fig. 8 Computational error of gravitational field of tesseroid: a wide dynamic range. Same as Fig. 7 but plotted as functions of the base-10 logarithm of the altitude, H − HT , for their wide dynamic range as 1 mm to 106 km.

on the location of the evaluation point. Also, we experimentally learned that this feature is independent on the size of the tesseroid and the location of the tesseroid on the unit spherical surface. Their results are omitted for saving the space.

3.4 Computational labor Let us move to the aspect of the computational labor. Fig. 10 displays Neval , the number of the innermost integrand evaluations to compute V of the spherical shell introduced in Fig. 2, as a function of H. Clearly, Neval is almost independent on the location of the evaluation points.


19

Integration Error of Tesseroid Potential: H = 0 km δ=10−15

15

10 δV 15 10 5 0 -5 -10 85

86 87 Λ(deg)

29 28 27 Φ(deg) 88 26

Fig. 9 Bird’s-eye view of integration error of gravitational potential of tesseroid: H = 0 km. Presented are the bird’s-eye view and the associated contour map of the scaled errors of the gravitational potential of the tesseroid introduced in Fig. 6. Illustrated are the results on the reference spherical surface, namely when H = 0 km. The observed zigzag feature in the longitude direction is due to the adaptive nature of the adopted numerical integration method: the double exponential rule (Fukushima, 2017b).

20

Toshio Fukushima

log10Neval

Computational Cost of Spherical Shell Potential 8.5 8 7.5 7 6.5 6 5.5 5 4.5 4 3.5 3 -100 -80 -60 -40 -20

32 28 24 20 16 12 8 4 0 20 40 60 80 100 120 H (km)

Fig. 10 Computational cost of spherical layer potential. Same as Fig. 3 but for the number of integrand evaluations required in the gravitational potential integration.

Also, Fig. 11 displays the H-dependence of Neval to compute V , gH , and ΓHH of the test tesseroid presented in Fig. 6. Again, the counted values of Neval are mostly constant with respect to the height of the evaluation point. Typically, constant is the ratio of Neval among V , gH , and ΓHH as 1:2:3. This is because the central difference formulae are chosen in the most cases. On the other hand, Fig. 12 presents a two-dimensional view and map of Neval required for the potential integration of the tesseroid as a function of the angular coordinates, Φ and Λ , when H is fixed as H = 0 km. The observed step-function-like change is the direct result of the conditional splitting. Indeed, the splitting itself results a 100% increase when either Φ or Λ is within the defining domain of the tesseroid, and a 300% increase when both of Φ and Λ are inside the defining domain.


21

Computational Cost of Gravitational Field of Tesseroid 6 δ=10

−16

Neval/105

5 4

ΓHH

3

gH

2 V 1 0 -100 -80 -60 -40 -20 0 20 40 60 80 100 H (km) Fig. 11 Computational cost of gravitational field of tesseroid. Same as Fig. 7 but plotted are Neval , the number of the innermost integrand evaluations required in the procedure to integrate or compute V , gH , and ΓHH of the tesseroid, respectively. This time, we restricted the range of H as |H| ≤ 100 km in order to emphasize the feature close to the boundary surfaces at H = −40 km and H = 10 km.

One may think that the computational labor will decrease quadratically with respect to the angular size of the tesseroid. Unfortunately, this optimistic expectation is betrayed. Refer to Fig. 13 illustrating the grid size dependence of Neval of the tesseroid potential computation for a wide variety of δ , the input relative error tolerance of the DE rule. Of course, Neval decreases when the grid size decreases but not quadratically as anticipated. Rather, Neval decreases in proportion to a power law with the power index as small as ≈ 0.28. Finally, we examine the total effectiveness of the new method. Fig. 14 indicates the cost performance of the new method, namely the achieved errors as functions of Neval . This figure tells us that, around 2 × 104 integrands must be evaluated if the 9-digit precision is required in integrating V , and then the 6-digit precision in g, and therefore the 4-digit precision in Γ . This amount of the integrand evaluations is significantly large. In fact, it

22

Toshio Fukushima

Integration Cost of Tesseroid Potential: H = 0 km Neval/10

δ=10−15

4

10 8 6 4 2 0 85

86 87 Λ(deg)

29 28 27 Φ(deg) 88 26

Fig. 12 Bird’s-eye view of computational cost to integrate gravitational potential of tesseroid: H = 0 km. Same as Fig. 9 but for the number of innermost integrand evaluations, Neval , to obtain V .

is roughly the same as the cost of (i) the two-dimensional numerical integration using the 140-point Gauss-Legendre quadrature (GLQ) (ii) the two-dimensional numerical integration of 20 × 20 subdivisions by using the 7-point GLQ, or (iii) the three-dimensional numerical integration using the 27-point GLQ. Of course, none of these methods using the GLQ do not provide meaningful results, and therefore can not be used instead of the new method. At any rate, the above large computational labor is the cost we must pay in order to obtain a reliable result.


23

log10 Neval

Grid Size Dependence of Potential Integration Cost 5.2 5 4.8 4.6 4.4 4.2 4 3.8 3.6 3.4 3.2 3

δ = max[10−2, δ0/(∆Φ)2] o ΦC = +27.5 ∆Λ = ∆Φ

δ0=10−16 10−14 −12

10 10−10 10−8 10−6 10−4 10−2

-2

-1

0

1 2 log10 (∆Φ/arcmin)

3

4

Fig. 13 Grid size dependence of potential integration cost of new method. Displayed is Neval as a function of angular grid size, ∆ Φ , for various values of δ as δ = 10−2 , 10−4 , . . . , 10−16 . We set the angular intervals of the tesseroid as |Φ − ΦC | ≤ ∆ Φ and |Λ − ΛC | ≤ ∆ Φ , the coordinates of the evaluation point as Φ = ΦC = +27.5◦ , Λ = ΛC = 86.5◦ , and H = 0 km.

3.5 Computer programs For the readers’ convenience, we prepared xtess.txt and xqtess.txt, a couple of text files including the full implementation of the Fortran programs realizing the new method in the double and quadruple precision environments, respectively. The ESM includes the explanation of their basic algorithm as well as the explicit presentation of the Fortran subprograms consisting xqtess.txt. The above text files are found in the author’s web site: https : //www.researchgate.net/profile/Toshio Fukushima/

The averaged CPU time of these programs providing all of V , g, and Γ are 0.197 s per evaluation point when δ = 10−15 in the double precision environment, and 135 s per evaluation

24

Toshio Fukushima

Cost Performance of New Method 0

H = 0 km

log10|Scaled Error|

-2 -4 -6

ΓHH

-8 gH

-10 -12

V

-14 -16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Neval/104 Fig. 14 Cost performance of new method. Same as Fig. 7 but plotted are the measured errors when H = 0 km as functions of Neval . The results are obtained by varying δ as δ = 2−1 , 2−2 , . . . , 2−53 ≈ 1.11 × 10−16 .

point when δ = 10−33 in the quadruple precision environment at a consumer PC with an Intel Core i7-4600U CPU running at 2.10 GHz clock.

4 Conclusion By modifying and extending the latest method to compute the gravitational field of a general finite body (Fukushima, 2017b), we obtained a numerical method to evaluate accurately the gravitational potential V , the gravitational acceleration vector g, and the gravity gradient tensor Γ of any homogeneous spherical tesseroid. The key point of the method is a precise numerical integration of V by combining the split quadrature method (Fukushima, 2014) and the double exponential quadrature rule (Takahashi and Mori, 1973, 1974). Once V is determined, then g and Γ are evaluated by the numerical differentiation of thus-integrated


25

V . This is done by a pair of the second order difference formulae plus a policy to set the test argument displacement in order to balance the truncation and round-off errors during the process of the numerical differentiation. The new method enables us to compute the gravitational field of any kind of homogeneous spherical tesseroid whether the evaluation point is outside, near the surface of, on the surface of, or inside the tesseroid. In this sense, it will serve as a reliable tool to evaluate the surface and/or internal gravitational field. Also, the method can be sufficiently precise, say assuring 14–15 digits for V , 9–11 digits for g, and 6–8 digits for Γ in the double precision environment. In terms of the gravity anomaly, this corresponds to 0.1–10 nano gal on the ground surface. If demanded, one may execute these processes in the quadruple precision environment. Then, these correct digits are roughly doubled. It is noteworthy that the new method can be generalized to the cases with any varying density profile, any variable surface height functions, and/or any non-constant latitude or longitude boundaries. Thus, the new method will be a sure method to compute the topographic gravitational field. Also, it could potentially serve as a reference method to complement and elaborate the existing approaches using the Gauss-Legendre quadrature or other standard methods of numerical integration. Acknowledgements The author appreciates valuable suggestions and fruitful comments by Prof. Uieda and two anonymous referees to improve the quality of the article.

Appendices A Kernel function The kernel function used in the main text is that of degree 0 introduced in our previous work (Fukushima, 2017b). After omitting the argument dependence for simplicity, and choosing the unit system such that G = ρ = 1, we write it compactly as K = σ [C log1p(Q) + (DT + SB /2) β ] ,

(58)

where Q is a quantity defined as { Q = (1 + T )β ×

1/ (ζB + B + SB ) , (ζB + B > 0) , (−ζB − B + SB ) /[B(2α − B)], (ζB + B ≤ 0) .

(59)

Here σ , C, and other quantities are computed by the following sequence. First of all, we define the surface heights of the tesseroid not by a pair of HT and HB but by HB and the thickness, ∆ H ≡ HT − HB . Then, we compute HT and a constant specific to the tesseroid only as HT = HB + ∆ H, β =

∆H . R0

(60)

Next, prior to the whole process of the numerical integration, we calculate some parameters from the given coordinates of the evaluation point, (Φ , Λ , H), as

α = 1+

H HB − H , γ = cos Φ , ζB = , ζT = ζB + β . R0 R0

(61)

Thirdly, in the outer integration, namely that with respect to ξ , we evaluate the following η -independent variables beforehand: ξ (62) σ = cos(Φ + ξ ), µ = sin . 2 Fourthly, in the inner integration, namely that with respect to η , the variables are computed from the input value of η in the order of the evaluation sequence as

ν = sin

( ) 3 3 η 1 , B = 2α µ 2 + γσ ν 2 , C = α 2 − 3α B + B2 , D = 2α − B + ζT , 2 2 2 2

26

Toshio Fukushima √

√ ζT + ζB + 2B ζB2 + 2BζB + 2α B, T = . ST + SB Finally, Q is computed from these by using Eq. (59), and then, K is obtained by evaluating Eq. (58). In the above, log1p(x) is a special logarithm function defined as ST =

ζT2 + 2BζT + 2α B, SB =

log1p(x) ≡ ln(1 + x) = x +

x 2 x3 + +···. 2 3

(63)

(64)

The function is available from the standard mathematical function library of C, C++, and MATLAB but not of Fortran. A double precision implementation in Fortran, dlog1p, is already provided in Appendix D of Fukushima (2017b). Its quadruple precision extension is provided in the ESM. The above expression of the kernel function is significantly different from those provided in the geodetic literature (Heck and Seitz, 2007; Wild-Pfeiffer, 2008). Especially, we conditionally evaluate Q in two different forms. They are mathematically equivalent with each other but computationally different if round-off errors are taken into consideration. The introduction of a switch is required so as to avoid unnecessary loss of information due to cancellations. At any rate, the rewriting of the kernel function into a cancellation-free form is the main trick to guarantee the proper convergence of the Romberg sequence adopted in the double exponential quadrature rule.

B Double exponential quadrature rule Here we present a brief summary of the double exponential (DE) quadrature rule for a line integral with a finite integration interval (Takahashi and Mori, 1973, 1974). The variations for other type of integration interval are explained in Press et al. (2007, Sect. 4.5). Let us consider the numerical evaluation of a general line integral expressed as I≡

∫ b

f (x)dx.

(65)

a

First, we transform this finite interval integral into an infinite interval integral by the integration variable transformation as ∫ ∞ I = I∗ ≡ F(t)dt, (66) −∞

where t is a new integration variable related to x as x = ψ (t) ≡

(π ) b+a b−a sinht , + tanh 2 2 2

(67)

and therefore, the transformed integrand, F(t), is written as F(t) ≡ f (ψ (t)) ψ ′ (t), where

ψ ′ (t) ≡

dψ = dt

(

(b − a)π 4

)

cosht . cosh2 [(π /2) sinht]

(68)

(69)

Next, we approximate the transformed integral, I ∗ , by truncating the infinite integration interval as I∗ ≈ J ≡

∫ hmax −hmax

F(t)dt,

(70)

where hmax is a certain numerical value such as 2.75 to 6.5 (Fukushima, 2017b, Appendix C). Finally, we numerically evaluate the truncated integral, J, by the Romberg method (Press et al., 2007, Sect. 4.3), namely the Richardson extrapolation applied to the trapezoidal rule. The method is called ‘double exponential’ because the weight factor of the transformed integrand, ψ ′ (t), decreases like a doubly exponential function as

ψ ′ (t) ≈ ((b − a)π /2) exp[t − (π /2) exp(t)], (t ≫ 1).

(71)

Its decreasing manner is so rapid that any kind of blowing-up but integrable singularity of the original integrand, f (x), can be effectively suppressed as long as the location of the singularity coincides with one of the


27

end points, x = a or x = b, and therefore, t = −∞ or t = +∞ (Mori, 1985). Thus, the DE rule is one of rare quadrature rules which can evaluate the improper but integrable integrals in a reliable manner. Also, the convergence of the Romberg sequence adopted in the DE rule is quite rapid. Indeed, it converges exponentially in many cases (Press et al., 2007, Sect. 4.5.1). This results a very high cost performance of the DE rule (Bailey et al., 2005). See also Fukushima (2017b, Appendix C). Another good feature of the DE rule is its adaptive nature inherited from that of the Romberg method. Consequently, the end users do not have to worry about setting many method-specific parameters except the only one: the input relative error tolerance, δ . For example, if one wants an integral value with the 10 digit precision, one may simply set δ = 10−10 and call the computer program of the DE rule. Then, the program automatically takes care of the number of trial integrations by the trapezoidal rule. However, there is a pitfall to be warned. We have experienced that, if the integrand can not be very precisely computed near the endpoints, the convergence of the Romberg sequence in the DE rule degrades significantly, and thus ruins the appropriateness of the DE rule as a whole. In many cases, naive transformations from f (x) to F(ψ (t)) suffer from the information loss due to cancellations of similar terms. Therefore, it is a key point to examine the chance of cancellation in the process of the integrand transformation and, if necessary, to rewrite the transformed integrand into a cancellation-free form as we did for the kernel function, K, in Appendix A. At any rate, as far as we know, the DE rule is an all-purpose quadrature rule with a great cost performance and an ease of use. A double precision implementation of the DE rule as a Fortran subroutine is found in Fukushima (2017b, Appendix C). Its quadruple precision extension is given in the ESM.

C Second order finite difference formulae C.1 First order derivative Let us consider the second order finite difference formulae to compute numerically the partial derivatives of a given function. We begin with the case when the function is of a single variable as f (t). If the function is analytic in the neighborhood of the given argument, t, then we use the central difference formula written as f ′ (t) ≈ g0 (t) ≡

f (t + ∆ t) − f (t − ∆ t) . 2∆ t

This formula is of the second order. Namely, its relative error is roughly in proportion to (∆ t)2 as ( ′′′ ) g0 (t) − f ′ (t) f (t) δ g0 (t) ≡ ≈ (∆ t)2 . f ′ (t) 6 f ′ (t)

(72)

(73)

On the other hand, if f (t) is not fully analytic in the closed interval, [t − ∆ t,t + ∆ t], say if the function has a discontinuity, a kink, or a singularity within the interval, the above formula can not be used. In the case of the gravitational potential of a tesseroid, the surrounding boundaries of the tesseroid correspond to the location of such non analyticity. At any rate, in these cases, we obtain a second order formula by using only the single sided values as − f (t ± 2∆ t) + 4 f (t ± ∆ t) − 3 f (t) g± (t) ≡ , (74) ±2∆ t where the double sign must be appropriately chosen depending on the analyticity of f (t). More specifically speaking, we have to choose the positive sign (+) if f (t) is analytic in the interval [t,t + 2∆ t] and the negative sign (−) if f (t) is analytic in the interval [t − 2∆ t,t]. In any case, the relative error of the single sided formula is estimated as ( ′′′ ) g± (t) − f ′ (t) f (t) δ g± (t) ≡ ≈− (∆ t)2 . (75) ′ f (t) 3 f ′ (t) Its magnitude is twice as large as that of the central difference formula.

C.2 Second order derivative Let us move to the second order derivative. As long as the function is sufficiently smooth, we recommend the usage of the central difference formula expressed as f ′′ (t) ≈ h0 (t) ≡

f (t + ∆ t) − 2 f (t) + f (t − ∆ t) . (∆ t)2

(76)

28

Toshio Fukushima

This is of the second order since

δ h0 (t) ≡

h0 (t) − f ′′ (t) ≈− f ′′ (t)

(

f (4) (t) 12 f ′′ (t)

) (∆ t)2 .

(77)

Meanwhile, if f (t) is not analytic in the interval, [t − ∆ t,t + ∆ t], then we use the second order single-sided formula as − f (t ± 3∆ t) + 4 f (t ± 2∆ t) − 5 f (t ± ∆ t) + 2 f (t) f ′′ (t) ≈ h± (t) ≡ , (78) (∆ t)2 where the double sign must be appropriately chosen depending on the analyticity of f (t) again. Its relative error is estimated as ( ) h± (t) − f ′′ (t) 11 f (4) (t) δ h± (t) ≡ ≈− (∆ t)2 , (79) f ′′ (t) 12 f ′′ (t) the magnitude of which is 11 times larger than that of the central difference formula. We must say that this 1 digit loss is a significant degradation.

C.3 Extension to multiple variable functions Let us extend the formulae in the previous subsections to the case when the function is dependent on multiple variables. For simplicity, we assume that the function is a two-argument functions as f (u, v). Then, if the function is sufficiently smooth around the evaluation point such that the central difference formulae are applicable, the first and the non-mixed second order partial derivatives are approximated in the same manner as in the previous subsections, respectively. Also, the mixed second order partial derivatives are obtained by separately applying the first order central difference formula to the both variables as

∂2 f 1 ≈ [ f (u + ∆ u, v + ∆ v) − f (u + ∆ u, v − ∆ v) ∂ u∂ v 4(∆ u)(∆ v) − f (u − ∆ u, v + ∆ v) + f (u − ∆ u, v − ∆ v)] .

(80)

On the other hand, if f (u, v) is analytic not with respect to u but with respect to v, then the mixed second order partial derivatives are computed by mixing the single sided first order difference formula with respect to u and the central first order difference formula with respect to v as ±1 ∂2 f ≈ [− f (u ± 2∆ u, v + ∆ v) + f (u ± 2∆ u, v − ∆ v) ∂ u∂ v 4(∆ u)(∆ v) +4 f (u ± ∆ u, v + ∆ v) − 4 f (u ± ∆ u, v − ∆ v) − 3 f (u, v + ∆ v) + 3 f (u, v − ∆ v)] ,

(81)

where the double sign must be appropriately chosen depending on the situation. We omit the case when f (u, v) is analytic not with respect to v but with respect to u because the recipe is the same as in the above after exchanging the role of u and v. Finally, if f (u, v) is not analytic with respect to both of u and v, then the mixed second order partial derivatives are approximated by repeatedly using the single sided first order difference formulae as

∂2 f εu εv ≈ [ f (u + 2εu ∆ u, v + 2εv ∆ v) − 4 f (u + εu ∆ u, v + 2εv ∆ v) ∂ u∂ v 4(∆ u)(∆ v) +3 f (u, v + 2εv ∆ v) − 4 f (u + 2εu ∆ u, v + εv ∆ v) + 16 f (u + εu ∆ u, v + εv ∆ v) −12 f (u, v + εv ∆ v) + 3 f (u + 2εu ∆ u, v) − 12 f (u + εu ∆ u, v) + 9 f (u, v)] , where the sign factors εu = ±1 and εv = ±1 must be appropriately chosen depending on the situation.

(82)


29

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