Accurate computation of gravitational field of a tesseroid

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.

Appendices

Appendices

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 $G = \rho = 1$, we write it compactly as

$$\begin{aligned} K = \sigma \left[ C \; \mathtt{log1p} (Q) + \left( D T + S_\mathrm{B}/2 \right) \beta \right] , \end{aligned}$$

(58)

where Q is a quantity defined as

$$\begin{aligned} Q= & {} (1 + T) \beta \nonumber \\&\times \left\{ \begin{array}{ll} 1/\left( \zeta _\mathrm{B} + B + S_\mathrm{B} \right) , &{} \left( \zeta _\mathrm{B} + B > 0 \right) , \\ \left( - \zeta _\mathrm{B} - B + S_\mathrm{B} \right) /[B (2\alpha - B)], &{} \left( \zeta _\mathrm{B} + B \le 0 \right) . \\ \end{array} \right. \end{aligned}$$

(59)

Here $\sigma$, 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 $H_\mathrm{T}$ and $H_\mathrm{B}$ but by $H_\mathrm{B}$ and the thickness, $\varDelta H \equiv H_\mathrm{T} - H_\mathrm{B}$. Then, we compute $H_\mathrm{T}$ and a constant specific to the tesseroid only as

$$\begin{aligned} H_\mathrm{T} = H_\mathrm{B} + \varDelta H, \;\; \beta = {\frac{\varDelta H}{R_0}}. \end{aligned}$$

(60)

Next, prior to the whole process of the numerical integration, we calculate some parameters from the given coordinates of the evaluation point, $(\varPhi , \varLambda , H)$, as

$$\begin{aligned} \alpha = 1 + {\frac{H}{R_0}}, \;\; \gamma = \cos \varPhi , \;\; \zeta _\mathrm{B} = {\frac{H_\mathrm{B}-H}{R_0}}, \;\; \zeta _\mathrm{T} = \zeta _\mathrm{B} + \beta . \end{aligned}$$

(61)

Thirdly, in the outer integration, namely that with respect to $\xi$, we evaluate the following $\eta$-independent variables beforehand:

$$\begin{aligned} \sigma = \cos (\varPhi +\xi ), \;\; \mu = \sin {\frac{\xi }{2}}. \end{aligned}$$

(62)

Fourthly, in the inner integration, namely that with respect to $\eta$, the variables are computed from the input value of $\eta$ in the order of the evaluation sequence as

$$\begin{aligned} \nu= & {} \sin {\frac{\eta }{2}}, \;\; B = 2 \alpha \left( \mu ^2 + \gamma \sigma \nu ^2 \right) ,\nonumber \\ C= & {} \alpha ^2 - 3 \alpha B + {\frac{3}{2}} B^2, D = 2 \alpha - {\frac{3}{2}} B + {\frac{1}{2}} \zeta _\mathrm{T},\nonumber \\ S_\mathrm{T}= & {} \sqrt{\zeta _\mathrm{T}^2 + 2B \zeta _\mathrm{T} + 2 \alpha B}, \;\; S_\mathrm{B} = \sqrt{\zeta _\mathrm{B}^2 + 2B \zeta _\mathrm{B} + 2 \alpha B}, \nonumber \\ T= & {} {\frac{\zeta _\mathrm{T} + \zeta _\mathrm{B} + 2 B}{S_\mathrm{T} + S_\mathrm{B}}}. \end{aligned}$$

(63)

Finally, Q is computed from these by using Eq. (59), and then, K is obtained by evaluating Eq. (58).

In the above, $\mathtt{log1p}(x)$ is a special logarithm function defined as

$$\begin{aligned} \mathtt{log1p}(x) \equiv \ln ( 1 + x ) = x + {\frac{x^2}{2}} + {\frac{x^3}{3}} + \cdots . \end{aligned}$$

(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, $\mathtt{dlog1p}$, 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). In particular, 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 cancelations. At any rate, the rewriting of the kernel function into a cancelation-free form is the main trick to guarantee the proper convergence of the Romberg sequence adopted in the double exponential quadrature rule.

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

$$\begin{aligned} I \equiv \int _a^b f(x) \mathrm{d}x. \end{aligned}$$

(65)

First, we transform this finite interval integral into an infinite interval integral by the integration variable transformation as

$$\begin{aligned} I = I^* \equiv \int _{-\infty }^{\infty } F(t) \mathrm{d}t, \end{aligned}$$

(66)

where t is a new integration variable related to x as

$$\begin{aligned} x = \psi (t) \equiv {\frac{b+a}{2}} + {\frac{b-a}{2}} \tanh \left( {\frac{\pi }{2}} \sinh t \right) , \end{aligned}$$

(67)

and therefore, the transformed integrand, F(t), is written as

$$\begin{aligned} F(t) \equiv f \left( \psi (t) \right) \psi '(t), \end{aligned}$$

(68)

where

$$\begin{aligned} \psi '(t) \equiv {\frac{d \psi }{dt}} = \left( {\frac{(b-a) \pi }{4}} \right) {\frac{\cosh t}{\cosh ^2 \left[ (\pi /2) \sinh t \right] }}. \end{aligned}$$

(69)

Next, we approximate the transformed integral, $I^*$, by truncating the infinite integration interval as

$$\begin{aligned} I^* \approx J \equiv \int _{-h_\mathrm{max}}^{h_\mathrm{max}} F(t) \mathrm{d}t, \end{aligned}$$

(70)

where $h_\mathrm{max}$ 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, $\psi '(t)$, decreases like a doubly exponential function as

$$\begin{aligned} \psi '(t) \approx ((b-a)\pi /2) \exp [ t - (\pi /2) \exp (t)], \;\; ( t \gg 1 ).\nonumber \\ \end{aligned}$$

(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 end points, $x=a$ or $x=b$, and therefore, $t=-\,\infty$ or $t=+\,\infty$ (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, $\delta$. For example, if one wants an integral value with the 10 digit precision, one may simply set $\delta =10^{-10}$ 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 $F(\psi (t))$ suffer from the information loss due to cancelations of similar terms. Therefore, it is a key point to examine the chance of cancelation in the process of the integrand transformation and, if necessary, to rewrite the transformed integrand into a cancelation-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.

Second-order finite difference formulae

1.1 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

$$\begin{aligned} f' \left( t \right) \approx g_0 \left( t \right) \equiv {\frac{f \left( t+\varDelta t \right) - f \left( t-\varDelta t \right) }{2 \varDelta t}}. \end{aligned}$$

(72)

This formula is of the second order. Namely, its relative error is roughly in proportion to $(\varDelta t)^2$ as

$$\begin{aligned} \delta g_0 \left( t \right) \equiv {\frac{g_0 \left( t \right) - f' \left( t \right) }{f' \left( t \right) }} \approx \left( {\frac{f''' \left( t \right) }{6 f' \left( t \right) }} \right) (\varDelta t)^2. \end{aligned}$$

(73)

On the other hand, if f(t) is not fully analytic in the closed interval, $\left[ t-\varDelta t, t+\varDelta t \right]$, 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

$$\begin{aligned} g_{\pm } \left( t \right) \equiv {\frac{- f \left( t \pm 2 \varDelta t \right) + 4 f \left( t \pm \varDelta t \right) - 3 f \left( t \right) }{\pm 2 \varDelta t}}, \end{aligned}$$

(74)

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 $[t, t+2\varDelta t]$ and the negative sign $(-)$ if f(t) is analytic in the interval $[t-2\varDelta t, t]$. In any case, the relative error of the single-sided formula is estimated as

$$\begin{aligned} \delta g_{\pm } \left( t \right) \equiv {\frac{g_{\pm } \left( t \right) - f' \left( t \right) }{f' \left( t \right) }}\approx - \left( {\frac{f''' \left( t \right) }{3 f' \left( t \right) }} \right) (\varDelta t)^2. \end{aligned}$$

(75)

Its magnitude is twice as large as that of the central difference formula.

1.2 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

$$\begin{aligned} f'' \left( t \right) \approx h_0 \left( t \right) \equiv {\frac{f \left( t+\varDelta t \right) - 2 f \left( t \right) + f \left( t-\varDelta t \right) }{(\varDelta t)^2}}. \end{aligned}$$

(76)

This is of the second order since

$$\begin{aligned} \delta h_0 \left( t \right) \equiv {\frac{h_0 \left( t \right) - f'' \left( t \right) }{f'' \left( t \right) }}\approx - \left( {\frac{f^{(4)} \left( t \right) }{12 f'' \left( t \right) }} \right) (\varDelta t)^2. \end{aligned}$$

(77)

Meanwhile, if f(t) is not analytic in the interval, $[ t-\varDelta t, t+\varDelta t ]$, then we use the second-order single-sided formula as

$$\begin{aligned} f'' \left( t \right)\approx & {} h_{\pm } \left( t \right) \nonumber \\\equiv & {} {\frac{- f \left( t \pm 3\varDelta t \right) + 4 f \left( t \pm 2\varDelta t \right) - 5 f \left( t \pm \varDelta t \right) + 2 f \left( t \right) }{(\varDelta t)^2}},\nonumber \\ \end{aligned}$$

(78)

where the double sign must be appropriately chosen depending on the analyticity of f(t) again. Its relative error is estimated as

$$\begin{aligned} \delta h_{\pm } \left( t \right) \equiv {\frac{h_{\pm } \left( t \right) - f'' \left( t \right) }{f'' \left( t \right) }}\approx - \left( {\frac{11 f^{(4)} \left( t \right) }{12 f'' \left( t \right) }} \right) (\varDelta t)^2, \end{aligned}$$

(79)

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.

1.3 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

$$\begin{aligned} {\frac{\partial ^2 f}{\partial u \partial v}}\approx & {} {\frac{1}{4(\varDelta u)(\varDelta v)}} \left[ f \left( u+\varDelta u, v+\varDelta v \right) \right. \nonumber \\&- f \left( u+\varDelta u, v-\varDelta v \right) \nonumber \\&\left. - f \left( u-\varDelta u, v+\varDelta v \right) + f \left( u-\varDelta u, v-\varDelta v \right) \right] .\nonumber \\ \end{aligned}$$

(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

$$\begin{aligned} {\frac{\partial ^2 f}{\partial u \partial v}}\approx & {} {\frac{\pm 1}{4(\varDelta u)(\varDelta v)}} \left[ - f \left( u \pm 2 \varDelta u, v + \varDelta v \right) \right. \nonumber \\&+\, f \left( u \pm 2 \varDelta u, v - \varDelta v \right) \nonumber \\&+\, 4f \left( u \pm \varDelta u, v + \varDelta v \right) - 4f \left( u \pm \varDelta u, v - \varDelta v \right) \nonumber \\&\,\left. - 3f \left( u, v + \varDelta v \right) + 3f \left( u, v - \varDelta v \right) \right] , \end{aligned}$$

(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

$$\begin{aligned} {\frac{\partial ^2 f}{\partial u \partial v}}\approx & {} {\frac{\epsilon _u \epsilon _v}{4 (\varDelta u)(\varDelta v)}} \left[ f \left( u + 2 \epsilon _u \varDelta u, v + 2 \epsilon _v \varDelta v \right) \right. \nonumber \\&\quad -\,4 f \left( u + \epsilon _u \varDelta u, v + 2 \epsilon _v \varDelta v \right) \nonumber \\&\quad +\,3 f \left( u, v + 2 \epsilon _v \varDelta v \right) - 4 f ( u + 2 \epsilon _u \varDelta u, v \nonumber \\&\quad +\,\epsilon _v \varDelta v )+ 16 f \left( u + \epsilon _u \varDelta u, v + \epsilon _v \varDelta v \right) \nonumber \\&\quad -\,12 f \left( u, v + \epsilon _v \varDelta v \right) + 3 f \left( u + 2 \epsilon _u \varDelta u, v \right) \nonumber \\&\quad \,\left. -\, 12 f \left( u + \epsilon _u \varDelta u, v \right) + 9 f \left( u, v \right) \right] , \end{aligned}$$

(82)

where the sign factors $\epsilon _u = \pm \, 1$ and $\epsilon _v = \pm \, 1$ must be appropriately chosen depending on the situation.