Far-Zone Effects for Spherical Integral Transformations II: Formulas for Horizontal Boundary Value Problems and Their Derivatives

Abstract

Integral formulas represent a methodological basis for the determination of gravitational fields generated by planetary bodies. In particular, spherical integral transformations are preferred for their symmetrical properties with the integration domain being the entire surface of the sphere. However, global coverage of boundary values is rarely guaranteed. In practical calculations, we therefore split the spherical surface into a near zone and a far zone, for convenience, by a spherical cap. While the gravitational effect in the near zone can be evaluated by numerical integration over available boundary values, the contribution of the far zone has to be precisely quantified by other means. Far-zone effects for the isotropic integral transformations and those depending on the direct azimuth have adequately been discussed. On the other hand, this subject has only marginally been addressed for the spherical integral formulas that are, except for other variables, also functions of the backward azimuth. In this article, we significantly advance the existing geodetic methodology by deriving the far-zone effects for the two classes of spherical integral transformations: (1) the analytical solutions of the horizontal, horizontal–horizontal, and horizontal–horizontal–horizontal BVPs including their generalisations with arbitrary-order vertical derivative of respective boundary conditions and (2) spatial (vertical, horizontal, or mixed) derivatives of these generalised analytical solutions up to the third order. The integral and spectral forms of the far-zone effects are implemented in MATLAB software package, and their consistency is tested in closed-loop simulations. The presented methodology can be employed in upward/downward continuation of potential field observables or for a quantification of error propagation through spherical integral transformations.

Article Highlights

• We complete the mathematical theory for the far-zone contributions of spherical integral formulas that are functions of the attenuation factor, spherical distance, and direct and backward azimuths

• We implement a final set of spherical harmonic series in a MATLAB software package and numerically test its correctness in a closed-loop simulation

• The theory and its implementation are invaluable for an accurate gravitational field modelling and for analysing statistical properties of spherical integral transformations

1 Introduction

Gravitational field is a fundamental property of any mass and in particular of planetary bodies (e.g. McSween et al. 2020; Angermann et al. 2022). Knowledge of this field is indispensable not only for advancing industry and science, but also for addressing broad range of societal issues, such as sustainable energy, environmental aspects, or infrastructure development. It is thus no surprise that gravitation is officially one of the three main pillars of the modern geodesy (e.g. Plag et al. 2009) and its determination is of utmost importance.

A popular mathematical tool for gravitational field modelling is integral transformations (e.g. Banerjea and Mandal 2023). Analytical solutions to boundary value problems (BVPs) of the potential theory form a principal group of integral formulas. Another group can be derived by applying respective differential operators to the analytical solutions of these BVPs. For the spherical boundary, possessing numerous symmetries, the existing integral transformations mutually relate all components of gravitational tensors up to the third order (Novák et al. 2017).

The integration domain of spherical integral formulas is the entire surface of the sphere. Besides this assumption, global coverage of boundary values may be problematic to achieve, either for distribution rights or restrictions in data collection. The spherical surface is therefore divided into a close neighbourhood around a computational point (near zone) and its complement (far zone). The contribution in the near zone can be calculated by numerical integration over the available boundary values (e.g. Huang et al. 2000). The effect of the far zone has to be evaluated by other means and for highly accurate gravitational field modelling its quantification is essential (e.g. Novák et al. 2001).

When a spherical cap separates the near and the far zone, far-zone effects can efficiently be expressed in terms of spherical harmonic series weighted by truncation error coefficients (e.g. Molodenskii et al. 1962). Numerous authors considered this regular delimiting curve and derived respective spherical harmonic expansions of far-zone effects for various integral transformations, either isotropic or those also involving dependencies on the direct azimuth. This effort has culminated in a recent article systematically reviewing the mathematical theory of the far-zone effects for the spherical integrals relating the disturbing gravitational potential or its purely radial derivatives into the disturbing gravitational potential and its vertical, horizontal, or mixed derivatives of the first, second, or third order, see (Šprlák and Pitoňák 2024).

Surprisingly, the same task has poorly been addressed for the spherical integral transformations that also contain backward azimuth dependencies, such as the analytical solutions of the horizontal, gradiometric, or gravitational curvature BVPs (e.g. Martinec 2003; Jekeli 2007; Šprlák and Novák 2016). We have thoroughly reviewed the existing geodetic literature and found only one relevant article by Tóth (2003), who derived the corresponding far-zone effects for the analytical solutions of the vertical–horizontal and horizontal–horizontal gradiometric BVPs.

The purpose of this article is to fill this theoretical gap and to provide computational tools for potential users. Namely, we systematically derive the far-zone effect formulas for a rich class of spherical integral transformations, see Fig. 2, that includes: (1) the analytical solutions of the horizontal, horizontal–horizontal, and horizontal–horizontal–horizontal BVPs including their generalisations for an arbitrary-order vertical derivatives of the corresponding boundary conditions, (2) first-, second-, and third-order spatial derivatives (vertical, horizontal, or mixed) of these generalised analytical solutions. We consider quantities from the disturbing gravitational potential up to the third-order disturbing gravitational tensor, because these can be measured by available instruments (e.g. Rummel 2010; Torge and Müller 2012; Rosi et al. 2015; Denker et al. 2018).

The current study is a continuation of our recent review in (Šprlák and Pitoňák 2024). Both these works form a completion of the mathematical theory for the far-zone effects including its versatile implementation of an unprecedented extent.

2 Nomenclature and Preliminaries

The purpose of this section is to introduce the basic notation and fundamental terms used in this article. By our convention, scalar, vector, and tensor quantities are typed in italics, bold, and sans serif fonts, respectively.

2.1 Reference Frames and Nomenclature

We assume that the gravitational field is generated by a general planetary body. Its masses are located inside the spherical surface ${\mathbb {S}}_R$ of radius R with origin at the centre of the gravitating body. As the spherical approximation is employed, we introduce the triplet of the spherical geocentric coordinates $(r, \Omega )$. r is the geocentric radius and $\Omega = (\varphi , \lambda )$ substitutes the pair of the angular spherical coordinates, i.e. the spherical latitude $\varphi$ and longitude $\lambda$.

Fig. 1

Graphical illustration of basic reference frames and of the decomposition of the sphere into the near zone ${\mathbb {S}}_R^0$ (grey area) and the far zone ${\mathbb {S}}_R^0 - {\mathbb {S}}_R$ (white area)

We distinguish two kinds of points, see Fig. 1: (1) Quantities (of the gravitational field) are calculated at a computational point at $(r, \Omega )$, $r\ge R$. (2) The differential surface element of ${\mathbb {S}}_R$ is an integration point at $(R, \Omega ')$.

The spherical polar coordinates $\alpha$, $\alpha '$, and $\psi$ determine the relative position between the computational point and the integration point, see Fig. 1. $\alpha$ stands for the direct azimuth (measured clockwise from North at the computational point), $\alpha '$ is the backward azimuth (measured clockwise from North at the integration point), and $\psi$ is the spherical distance. The transformation from the angular spherical geocentric coordinates $\Omega$ and $\Omega '$ onto the spherical polar coordinates $\alpha$, $\alpha '$, and $\psi$ can be performed by the rules of spherical trigonometry (Chauvenet 1875, p. 151-154):

$$\begin{aligned} \alpha&= \alpha (\Omega , \Omega ') = \arccos \Bigg \{ \frac{1}{\sqrt{1 - \cos ^2 \psi }} \left[ \sin \varphi ' \cos \varphi - \cos \varphi ' \sin \varphi \cos \left( \lambda ' - \lambda \right) \right] \Bigg \} \nonumber \\&= \arcsin \Bigg [ \frac{1}{\sqrt{1 - \cos ^2 \psi }} \cos \varphi ' \sin \left( \lambda ' - \lambda \right) \Bigg ], \end{aligned}$$

(1)

$$\begin{aligned} \alpha '&= \alpha '(\Omega , \Omega ') = \arccos \Bigg \{ \frac{1}{\sqrt{1 - \cos ^2 \psi }} \left[ \sin \varphi \cos \varphi ' - \cos \varphi \sin \varphi ' \cos \left( \lambda ' - \lambda \right) \right] \Bigg \} \nonumber \\&= \arcsin \Bigg [ - \frac{1}{\sqrt{1 - \cos ^2 \psi }} \cos \varphi \sin \left( \lambda ' - \lambda \right) \Bigg ], \end{aligned}$$

(2)

$$\begin{aligned} \psi&= \psi (\Omega , \Omega ') = \arccos \big [\sin \varphi \sin \varphi ' + \cos \varphi \cos \varphi ' \cos \left( \lambda ' - \lambda \right) \big ]. \end{aligned}$$

(3)

We next define the Local North-Oriented Reference Frame (LNORF). Such Cartesian reference frame has a moving origin that varies with the angular direction. Orthonormal right-handed basis of LNORF is formed by the vector triplet $({\textbf{e}}_x, {\textbf{e}}_y, {\textbf{e}}_z)$. The vector ${{\textbf{e}}}_x$ points to the North, ${{\textbf{e}}}_y$ points to the West, and ${{\textbf{e}}}_z$ is directed vertically outward, see Fig. 1.

Below, we routinely consider a restricted integration in addition to its global counterpart. For convenience, we divide the spherical surface ${\mathbb {S}}_R$ by a spherical cap of the size $\psi _0$ around the projection of the computational point onto ${\mathbb {S}}_R$ into two domains, see Fig. 1: (1) The domain ${\mathbb {S}}_R^0$ bounded by the spherical polar coordinates $\alpha \in [0^{\circ }, 180^{\circ }]$ and $\psi \in [0^{\circ }, \psi _0]$ is the near zone. A gravitational effect in ${\mathbb {S}}_R^0$ is referred to as the effect of the near zone. (2) The far zone ${\mathbb {S}}_R - {\mathbb {S}}_R^0$ is limited by the spherical polar coordinates $\alpha \in [0^{\circ }, 180^{\circ }]$ and $\psi \in (\psi _0, 180^{\circ }]$. A gravitational effect in ${\mathbb {S}}_R - {\mathbb {S}}_R^0$ is called the effect of the far zone (also referred to as the truncation error).

For brevity, we introduce these three substitutions:

$$\begin{aligned} t = t(r, R) = \frac{R}{r}, \quad u = u(\Omega , \Omega ') = \cos \psi , \quad u_0 = \cos \psi _0 = \mathrm {const.} \end{aligned}$$

(4)

Integration over the entire spherical surface ${\mathbb {S}}_R$ is abbreviated as:

$$\begin{aligned} \int \limits _{0}^{2\pi } \int \limits _{-\pi /2}^{\pi /2} (\cdot )\, \cos \varphi '\, \text{d}\varphi ' \text{d}\lambda ' = \int \limits _{0}^{2\pi } \int \limits _{0}^{\pi } (\cdot )\, \sin \psi \, \text{d}\psi \text{d}\alpha = \frac{1}{R^2} \int _{{\mathbb {S}}_R} (\cdot )\, {\text{d}} \omega _R, \end{aligned}$$

(5)

where ${\text{d}} \omega _R = R^2 \cos \varphi '\, \text{d}\varphi ' \text{d}\lambda '$. $(\cdot )$ may replace a multiplication of boundary conditions (gravitational data) by an integral kernel (see below). Integrals over the near zone and over the far zone are denoted as follows:

$$\begin{aligned} \int \limits _{0}^{2\pi } \int \limits _{0}^{\psi _0} (\cdot )\, \sin \psi \, \text{d}\psi \text{d}\alpha = \frac{1}{R^2} \int _{{\mathbb {S}}_R^0} (\cdot )\, {\text{d}} \omega _R, \quad \int \limits _{0}^{2\pi } \int \limits _{\psi _0}^{\pi } (\cdot )\, \sin \psi \, \text{d}\psi \text{d}\alpha = \frac{1}{R^2} \int _{{\mathbb {S}}_R - {\mathbb {S}}_R^0} (\cdot )\, {\text{d}} \omega _R. \end{aligned}$$

(6)

2.2 Disturbing Gravitational Tensors

We denote a disturbing gravitational tensor of order (rank) $s \in {\mathbb {Z}}^{*}$ by the symbol ${\textsf{T}}^{(s)}$. The most simplest disturbing gravitational tensor ${\textsf{T}}^{(0)}$ is the disturbing gravitational potential T. Being a scalar quantity, T is invariant to our choice of a reference frame.

The disturbing gravitational tensor ${\textsf{T}}^{(1)}$ is identical with the disturbing gravitational vector. It results from the application of the gradient operator to T and in LNORF it is defined by the three components $T_{o}$, $o \in \{x,y,z\}$:

$$\begin{aligned} {\textsf{T}}^{(1)} = \text{grad}\ T = \bigg \{T_{z}\, {\textbf{e}}_{z}\bigg \} + \bigg \{T_{x}\, {\textbf{e}}_{x} + T_{y}\, {\textbf{e}}_{y}\bigg \}. \end{aligned}$$

(7)

We use braces on the right-hand sides of Eqs. (7)–(11) to indicate significant tensor elements. The disturbing gravitational vector in Eq. (7) is formed by the vertical part (inside the first braces on the right-hand side) and the horizontal part (inside the second braces on the right-hand side).

The disturbing gravitational tensor ${\textsf{T}}^{(2)}$ in LNORF is generally defined by the nine components $T_{op}$, $o, p \in \{x, y, z\}$. Because T is continuous, $T_{op} = T_{po}$. In other words, ${\textsf{T}}^{(2)}$ is symmetric and completely given by six components. Laplace’s equation $T_{xx} + T_{yy} + T_{zz} = 0$ is valid in the massless space, where ${\textsf{T}}^{(2)}$ is defined by five independent components. Besides this fact, we explicitly work with the six second-order tensor components below.

From the mathematical point of view, ${\textsf{T}}^{(2)}$ follows from the twofold application of the gradient operator onto T and reads:

$$\begin{aligned} {\textsf{T}}^{(2)}&= \text{grad} \otimes \text{grad}\ T = \bigg \{T_{zz}\, {{\textbf{e}}}_{zz}\bigg \} + \bigg \{T_{xz}\, ({\textbf{e}}_{xz} + {\textbf{e}}_{zx}) + T_{yz}\, ({\textbf{e}}_{yz} + {\textbf{e}}_{zy})\bigg \} \nonumber \\&\quad +\ \bigg \{T_{xx}\, {\textbf{e}}_{xx} + T_{xy}\, ({\textbf{e}}_{xy} + {\textbf{e}}_{yx}) + T_{yy}\, {\textbf{e}}_{yy}\bigg \}. \end{aligned}$$

(8)

${\textbf{e}}_{op} = {\textbf{e}}_{o} \otimes {\textbf{e}}_{p}$, $o, p \in \{x, y, z\}$, is the tensor basis uniquely representing ${\textsf{T}}^{(2)}$ and referred to as the spherical dyadic. ${\textsf{T}}^{(2)}$ is split into these three parts differing from each other by the number of vertical (z) and horizontal (x, y) indices:

• 1) vertical–vertical (inside the first braces on the right-hand side),

• 2) vertical–horizontal (inside the second braces on the right-hand side),

• 3) horizontal–horizontal (inside the third braces on the right-hand side).

Equation (8) can equivalently be written in the form:

$$\begin{aligned} {\textsf{T}}^{(2)}&= \bigg \{T_{zz}\, {\textbf{e}}_{zz}\bigg \} + \bigg \{T_{xz} \left( {\textbf{e}}_{xz} + {\textbf{e}}_{zx}\right) + T_{yz} \left( {\textbf{e}}_{yz} + {\textbf{e}}_{zy}\right) \bigg \} \nonumber \\&\quad +\ \bigg \{\frac{1}{2} \big (T_{xx} - T_{yy}\big ) \left( {\textbf{e}}_{xx} - {\textbf{e}}_{yy}\right) + T_{xy} \left( {\textbf{e}}_{xy} + {\textbf{e}}_{yx}\right) \bigg \} \nonumber \\&\quad +\ \bigg \{\frac{1}{2} \big (T_{xx} + T_{yy}\big ) \left( {\textbf{e}}_{xx} + {\textbf{e}}_{yy}\right) \bigg \}. \end{aligned}$$

(9)

Accordingly, ${\textsf{T}}^{(2)}$ can be represented by the four parts (each enclosed inside braces), i.e. vertical–vertical, vertical–horizontal, first type horizontal–horizontal, and second type horizontal–horizontal. The horizontal–horizontal part of the second type can be transformed into its vertical–vertical counterpart by Laplace’s equation.

The quantity ${\textsf{T}}^{(3)}$ is the third-order disturbing gravitational tensor, and in LNORF it is given by the 27 components $T_{opq}$, $o, p, q \in \{x, y, z\}$. Since T is continuous, the symmetries $T_{opq} = T_{poq} = T_{oqp} = T_{qop}$ apply and ${\textsf{T}}^{(3)}$ is defined by ten components only. The three derivatives of the Laplace equation $T_{xxq} + T_{yyq} + T_{zzq} = 0$, $q \in \{x, y, z\}$, are valid in the massless space, in which ${\textsf{T}}^{(3)}$ is fully defined by only seven components. In spite of this fact, we systematically work with ten components of the tensor ${\textsf{T}}^{(3)}$ below.

Mathematically, ${\textsf{T}}^{(3)}$ can be obtained by the triple application of the gradient operator to T:

$$\begin{aligned} {\textsf{T}}^{(3)}&= \text{grad} \otimes \text{grad} \otimes \text{grad}\ T = \bigg \{T_{zzz}\, {\textbf{e}}_{zzz}\bigg \} \nonumber \\&\quad +\ \bigg \{T_{xzz}\, ({\textbf{e}}_{xzz} + {\textbf{e}}_{zxz} + {\textbf{e}}_{zzx}) + T_{yzz}\, ({\textbf{e}}_{yzz} + {\textbf{e}}_{zyz} + {\textbf{e}}_{zzy})\bigg \} \nonumber \\&\quad +\ \bigg \{T_{xxz}\, ({\textbf{e}}_{xxz} + {\textbf{e}}_{xzx} + {\textbf{e}}_{zxx}) + T_{xyz}\, ({\textbf{e}}_{xyz} + {\textbf{e}}_{xzy} + {\textbf{e}}_{yxz} + {\textbf{e}}_{yzx} + {\textbf{e}}_{zxy} + {\textbf{e}}_{zyx}) \nonumber \\&\quad +\ T_{yyz}\, ({\textbf{e}}_{yyz} + {\textbf{e}}_{yzy} + {\textbf{e}}_{zyy})\bigg \} \nonumber \\&\quad +\ \bigg \{T_{xxx}\, {\textbf{e}}_{xxx} + T_{xxy}\, ({\textbf{e}}_{xxy} + {\textbf{e}}_{xyx} + {\textbf{e}}_{yxx}) \nonumber \\&\quad +\ T_{xyy}\, ({\textbf{e}}_{xyy} + {\textbf{e}}_{yxy} + {\textbf{e}}_{yyx}) + T_{yyy}\, {\textbf{e}}_{yyy}\bigg \}. \end{aligned}$$

(10)

The tensor basis, which represents ${\textsf{T}}^{(3)}$ uniquely, is denoted ${\textbf{e}}_{opq} = {\textbf{e}}_{o} \otimes {\textbf{e}}_{p} \otimes {\textbf{e}}_{q}$, $o, p, q \in \{x, y, z\}$, and termed spherical triad. ${\textsf{T}}^{(3)}$ is split into the following parts:

• 1) vertical–vertical–vertical (inside the first braces on the right-hand side),

• 2) vertical–vertical–horizontal (inside the second braces on the right-hand side),

• 3) vertical–horizontal–horizontal (inside the third braces on the right-hand side),

• 4) horizontal–horizontal–horizontal (inside the fourth braces on the right-hand side).

Alternatively, the tensor ${\textsf{T}}^{(3)}$ can be expressed as:

$$\begin{aligned} {\textsf{T}}^{(3)}&= \bigg \{T_{zzz}\, {\textbf{e}}_{zzz}\bigg \} + \bigg \{T_{xzz} ({\textbf{e}}_{xzz} + {\textbf{e}}_{zxz} + {\textbf{e}}_{zzx}) + T_{yzz} ({\textbf{e}}_{yzz} + {\textbf{e}}_{zyz} + {\textbf{e}}_{zzy})\bigg \} \nonumber \\&\quad + \bigg \{\frac{1}{2} \left( T_{xxz} - T_{yyz}\right) ({\textbf{e}}_{xxz} + {\textbf{e}}_{xzx} + {\textbf{e}}_{zxx} - {\textbf{e}}_{yyz} - {\textbf{e}}_{yzy} - {\textbf{e}}_{zyy}) \nonumber \\&\quad + T_{xyz} ({\textbf{e}}_{xyz} + {\textbf{e}}_{xzy} + {\textbf{e}}_{yxz} + {\textbf{e}}_{yzx} + {\textbf{e}}_{zxy} + {\textbf{e}}_{zyx})\bigg \} \nonumber \\&\quad + \bigg \{\frac{1}{4} \left( T_{xxx} - 3\,T_{xyy}\right) ({\textbf{e}}_{xxx} - {\textbf{e}}_{xyy} - {\textbf{e}}_{yxy} - {\textbf{e}}_{yyx}) \nonumber \\&\quad + \frac{1}{4} \left( T_{yyy} - 3\, T_{xxy}\right) ({\textbf{e}}_{yyy} - {\textbf{e}}_{xxy} - {\textbf{e}}_{xyx} - {\textbf{e}}_{yxx}) \bigg \} \nonumber \\&\quad + \bigg \{\frac{1}{2} \left( T_{xxz} + T_{yyz}\right) ({\textbf{e}}_{xxz} + {\textbf{e}}_{xzx} + {\textbf{e}}_{zxx} + {\textbf{e}}_{yyz} + {\textbf{e}}_{yzy} + {\textbf{e}}_{zyy})\bigg \} \nonumber \\&\quad + \bigg \{\frac{1}{4} \left( T_{xxx} + T_{xyy}\right) (3\, {\textbf{e}}_{xxx} + {\textbf{e}}_{xyy} + {\textbf{e}}_{yxy} + {\textbf{e}}_{xyy}) \nonumber \\&\quad + \frac{1}{4} \left( T_{yyy} + T_{xxy}\right) (3\, {\textbf{e}}_{yyy} + {\textbf{e}}_{xxy} + {\textbf{e}}_{xyx} + {\textbf{e}}_{yxx})\bigg \}. \end{aligned}$$

(11)

Equation (11) decomposes ${\textsf{T}}^{(3)}$ into six parts indicated by the braces, namely vertical–vertical–vertical, vertical–vertical–horizontal, first type vertical–horizontal–horizontal, first type horizontal–horizontal–horizontal, second type vertical–horizontal–horizontal, and second type horizontal–horizontal–horizontal. Using derivatives of the Laplace equation, the last two parts can be transferred into the vertical–vertical–vertical and vertical–vertical–horizontal equivalents.

The differential operators relating T to the three components of ${\textsf{T}}^{(1)}$, the six components of ${\textsf{T}}^{(2)}$, and the ten components of ${\textsf{T}}^{(3)}$ in LNORF are listed in Tables 12, 13, and 14, respectively. The operators are defined by the spherical geocentric coordinates $(r,\Omega )$ after the first equalities. The identical operators are expressed by the variables $(R,t,u,\alpha )$ after the second equalities.

The 20 quantities presented here, namely T, the three components $(T_{x}, T_{y}, T_{z})$ of ${\textsf{T}}^{(1)}$, the six components $(T_{xx}, T_{xy}, T_{xz}, T_{yy}, T_{yz}, T_{zz})$ of ${\textsf{T}}^{(2)}$, and the ten components $(T_{xxx}, T_{xxy}, T_{xxz}, T_{xyy}, T_{xyz}, T_{xzz}, T_{yyy}, T_{yyz}, T_{yzz}, T_{zzz})$ of ${\textsf{T}}^{(3)}$ or their far zone equivalents will explicitly occur on the left-hand sides of numerous equations below.

2.3 $\{k,l\}$ Boundary Value Problem

The fundamental integral transformations in this article are the analytical solutions of the external horizontal, horizontal–horizontal, and horizontal–horizontal–horizontal BVPs for the spherical boundary. We extend these three kinds of BVPs for an arbitrary vertical (radial) derivative of respective boundary conditions, see Fig. 2. To reflect this generalisation, we propose the term “$\{k,l\}$ BVP”, where $k \in {\mathbb {Z}}^{*}$ stands for the order of the vertical derivative and $l \in \{1,2,3\}$ indicates the occurrence of ‘horizontal’ (or the number of horizontal indices). In this section, we describe boundary conditions and present formulations and analytical solutions of $\{k,l\}$ BVPs.

2.3.1 Definition of Boundary Conditions

Boundary conditions of $\{k,1\}$ BVPs include components of disturbing gravitational tensors with k vertical indices and one horizontal index and are specified as:

$$\begin{aligned} {\textsf{T}}^{\{k,1\}} = T_{x[k]}\, {\textbf{e}}_{x[k]}^{\{1\}} + T_{y[k]}\, {\textbf{e}}_{y[k]}^{\{1\}}. \end{aligned}$$

(12)

The symbol [k] signifies k occurrences of z (e.g. [0] = Null, $[1] = z$, $[2] = zz$, $[3] = zzz$, etc.). The tensor bases in Eq. (12) are abbreviated by the relation:

$$\begin{aligned} {\textbf{e}}_{o[k]}^{\{1\}} = \frac{1}{\sqrt{k + 1}}\ \sum \ {\textbf{e}}_{o} \otimes \ \overbrace{{\textbf{e}}_{z} \otimes {\textbf{e}}_{z} \otimes \ldots \otimes {\textbf{e}}_{z}}^{{\textbf{e}}_{z}\ k-\text {times}}, \ o \in \{x,y\}. \end{aligned}$$

(13)

The summation on the right-hand side goes over all possible permutations of x or y with [k]. For $k = 0$ and 1, for example, the boundary conditions read:

$$\begin{aligned} {\textsf{T}}^{\{0,1\}}&= T_{x}\, {\textbf{e}}_{x} + T_{y}\, {\textbf{e}}_{y}, \end{aligned}$$

(14)

$$\begin{aligned} {\textsf{T}}^{\{1,1\}}&=\frac{1}{\sqrt{2}} \big [T_{xz} \left( {\textbf{e}}_{xz} + {\textbf{e}}_{zx}\right) + T_{yz} \left( {\textbf{e}}_{yz} + {\textbf{e}}_{zy}\right) \big ]. \end{aligned}$$

(15)

Clearly, ${\textsf{T}}^{\{0,1\}}$ corresponds to the horizontal part of ${\textsf{T}}^{(1)}$ in Eq. (7), while ${\textsf{T}}^{\{1,1\}}$ is the vertical–horizontal part of ${\textsf{T}}^{(2)}$ in Eq. (9) (except for the factor $\frac{1}{\sqrt{2}}$). We can thus say that ${\textsf{T}}^{\{k,1\}}$ is the k-times-vertical–horizontal part of a disturbing gravitational tensor of order $k + 1$.

Boundary conditions of $\{k,2\}$ BVPs compose of the disturbing gravitational tensor components with k vertical and two horizontal indices and are of the form:

$$\begin{aligned} {\textsf{T}}^{\{k,2\}} = \frac{1}{2} \left( T_{xx[k]} - T_{yy[k]}\right) \left( {\textbf{e}}_{xx[k]}^{\{2\}} - {\textbf{e}}_{yy[k]}^{\{2\}}\right) + T_{xy[k]} \left( {\textbf{e}}_{xy[k]}^{\{2\}} + {\textbf{e}}_{yx[k]}^{\{2\}}\right) , \end{aligned}$$

(16)

where the tensor bases are:

$$\begin{aligned} {\textbf{e}}_{op[k]}^{\{2\}} = \sqrt{\frac{4}{(k+1)(k+2)}}\ \sum \ {\textbf{e}}_{o} \otimes {\textbf{e}}_{p} \otimes \ \overbrace{{\textbf{e}}_{z} \otimes {\textbf{e}}_{z} \otimes \ldots \otimes {\textbf{e}}_{z}}^{{\textbf{e}}_{z}\ k-\text {times}}, \ o, p \in \{x,y\}. \end{aligned}$$

(17)

The summation in Eq. (17) extends over all possible orderings of xx, yy, xy, or yx with [k]. Examples of ${\textsf{T}}^{\{k,2\}}$ for $k = 0$ and 1 are:

$$\begin{aligned} {\textsf{T}}^{\{0,2\}}&= \sqrt{2}\, \bigg [\frac{1}{2} \big (T_{xx} - T_{yy}\big ) \left( {\textbf{e}}_{xx} - {\textbf{e}}_{yy}\right) + T_{xy} \left( {\textbf{e}}_{xy} + {\textbf{e}}_{yx}\right) \bigg ], \end{aligned}$$

(18)

$$\begin{aligned} {\textsf{T}}^{\{1,2\}}&= \sqrt{\frac{2}{3}}\, \bigg [\frac{1}{2}\, \big (T_{xxz} - T_{yyz}\big ) ({\textbf{e}}_{xxz} + {\textbf{e}}_{xzx} + {\textbf{e}}_{zxx} - {\textbf{e}}_{yyz} - {\textbf{e}}_{yzy} - {\textbf{e}}_{zyy}) \nonumber \\&\quad +\ T_{xyz} ({\textbf{e}}_{xyz} + {\textbf{e}}_{xzy} + {\textbf{e}}_{yxz} + {\textbf{e}}_{yzx} + {\textbf{e}}_{zxy} + {\textbf{e}}_{zyx})\bigg ]. \end{aligned}$$

(19)

Except for the constant factors $\sqrt{2}$ and $\sqrt{\frac{2}{3}}$, ${\textsf{T}}^{\{0,2\}}$ and ${\textsf{T}}^{\{1,2\}}$ match with the horizontal–horizontal part of the first kind of ${\textsf{T}}^{(2)}$ in Eq. (9) and with the first type vertical–horizontal–horizontal part of ${\textsf{T}}^{(3)}$ in Eq. (11). Thus, ${\textsf{T}}^{\{k,2\}}$ is the k-times-vertical–horizontal–horizontal part of ${\textsf{T}}^{(k+2)}$.

For $\{k,3\}$ BVPs, the corresponding boundary conditions are linear combinations of disturbing gravitational tensor components having k vertical and three horizontal indices and are precisely defined as follows:

$$\begin{aligned} {\textsf{T}}^{\{k,3\}}&= \frac{1}{2} \left( T_{xxx[k]} - 3\, T_{xyy[k]}\right) \left( {\textbf{e}}_{xxx[k]}^{\{3\}} - {\textbf{e}}_{xyy[k]}^{\{3\}} - {\textbf{e}}_{yxy[k]}^{\{3\}} - {\textbf{e}}_{yyx[k]}^{\{3\}}\right) \nonumber \\&\quad + \frac{1}{2} \left( T_{yyy[k]} - 3\, T_{xxy[k]}\right) \left( {\textbf{e}}_{yyy[k]}^{\{3\}} - {\textbf{e}}_{xxy[k]}^{\{3\}} - {\textbf{e}}_{xyx[k]}^{\{3\}} - {\textbf{e}}_{yxx[k]}^{\{3\}}\right) . \end{aligned}$$

(20)

The corresponding tensor bases are:

$$\begin{aligned} {\textbf{e}}_{opq[k]}^{\{3\}}&= \sqrt{\frac{6}{(k+1)(k+2)(k+3)}}\ \sum \ {\textbf{e}}_{o} \otimes {\textbf{e}}_{p} \otimes {\textbf{e}}_{q} \otimes \ \overbrace{{\textbf{e}}_{z} \otimes {\textbf{e}}_{z} \otimes \ldots \otimes {\textbf{e}}_{z}}^{{\textbf{e}}_{z}\ k-\text {times}}, \quad o, p, q \in \{x,y\}, \end{aligned}$$

(21)

where the summation is taken over all possible orderings of xxx, xyy, yxy, yyx, yyy, xxy, xyx, or yxx with [k]. We provide an example of ${\textsf{T}}^{\{k,3\}}$ for $k = 0$, which reads:

$$\begin{aligned} {\textsf{T}}^{\{0,3\}}&= 2\, \bigg [\frac{1}{4} \left( T_{xxx} - 3\, T_{xyy}\right) ({\textbf{e}}_{xxx} - {\textbf{e}}_{xyy} - {\textbf{e}}_{yxy} - {\textbf{e}}_{yyx}) \nonumber \\&\quad + \frac{1}{4} \left( T_{yyy} - 3\, T_{xxy}\right) ({\textbf{e}}_{yyy} - {\textbf{e}}_{xxy} - {\textbf{e}}_{xyx} - {\textbf{e}}_{yxx})\bigg ]. \end{aligned}$$

(22)

Obviously, ${\textsf{T}}^{\{0,3\}}$ differs from the first type horizontal–horizontal–horizontal part of ${\textsf{T}}^{(3)}$ in Eq. (11) by the factor two. In general, ${\textsf{T}}^{\{k,3\}}$ is the k-times-vertical–horizontal–horizontal–horizontal part of ${\textsf{T}}^{(k+3)}$.

The boundary conditions above can be expanded in terms of spherical harmonics. Importantly, the corresponding tensor spherical harmonic base functions are orthogonal over the spherical surface ${\mathbb {S}}_R$. Except for degree-dependent factors, the boundary conditions ${\textsf{T}}^{\{k,1\}}$, ${\textsf{T}}^{\{k,2\}}$, and ${\textsf{T}}^{\{k,3\}}$ are orthogonal in the sense of Eqs. (B24), (B25), and (B26), respectively. The orthogonality property is indispensable when finding solutions of BVPs.

2.3.2 Formulations and Analytical Solutions

$\{k,l\}$ BVPs aim at determining the disturbing gravitational potential from their respective boundary conditions. Tables 1, 2, and 3 summarise formulations of $\{k,1\}$, $\{k,2\}$, and $\{k,3\}$ BVPs. The field equation for T is the Laplace equation (23), restated in Eqs. (32) and (41), and holds outside the spherical surface ${\mathbb {S}}_R$. Equations (24), (33), and (42) are identical to the boundary conditions (12), (16), and (20), and are continuously prescribed on ${\mathbb {S}}_R$. In addition, T is subject to the regularity conditions at infinity, see Eqs. (25), (34), and (43). These mutually differ by the rate of decay, as indicated by the Landau symbol ${\mathcal {O}}$.

The analytical solution of the $\{k,1\}$ BVP formulated in Eqs. (23)–(25) can be found in the form of the integral formula (26). The scalar components of the tensor boundary condition (24) appear under the integral sign, in correspondence with the scalar value of $T^{\{k,1\}}$ on the left-hand side. The components $T_{x[k]}$ and $T_{y[k]}$ are expanded in the spherical harmonic series (27) and (28). The n-th-degree spherical harmonics of $T_{x[k]}$ and $T_{y[k]}$ are defined by Eqs. (29) and (30), where G is the (universal) gravitational constant, M is the mass of the gravitating body, a is a metric scale factor (e.g. the major semi-axis of the reference ellipsoid), $\Delta {\bar{C}}_{n,m}$ is the fully normalised spherical harmonic coefficient of degree n and order m of T, and ${\bar{Y}}_{n,m}$ are the $4\pi$-normalised spherical harmonics of degree n and order m forming an orthogonal basis in the sense of Eq. (B23). The boundary values in Eq. (26) are multiplied by the integral kernel ${\mathcal {H}}^{\{k,1\}}$. This function is given by the infinite series (31), where $P_{n,1}$ stands for the un-normalised associated Legendre function of the first kind of degree n and order 1.

Table 1 Formulation and analytical solution of $\{k,1\}$ BVP

Analytical solutions of $\{k,1\}$ BVPs in Eq. (26) have been found for $k \le 2$, namely:

• the analytical solution of the horizontal BVP (e.g. Grafarend 2001; Jekeli 2007) ($k=0$),

• the analytical solution for the vertical–horizontal gradiometric BVP (e.g. van Gelderen and Rummel 2001; Martinec 2003; Tóth 2003) ($k=1$),

• the analytical solution for the vertical–vertical–horizontal gravitational curvature BVP (Šprlák and Novák 2016) ($k=2$).

The integral transformation (35) is the analytical solution of the $\{k,2\}$ BVP formulated by Eqs. (32)–(34). The components $(T_{xx[k]} - T_{yy[k]})$ and $T_{xy[k]}$ can be identified under the integral sign as these form the scalar part of the tensor boundary condition (33). These two components can be expanded by the spherical harmonic series (36)–(39). The scalar boundary values inside the surface integral (35) are multiplied by the integral kernel ${\mathcal {H}}^{\{k,2\}}$, see Eq. (40).

Table 2 Formulation and analytical solution of $\{k,2\}$ BVP

Analytical solutions of $\{k,2\}$ BVPs have been found for $k \le 1$ and correspond to:

• the analytical solution for the horizontal–horizontal gradiometric BVP (e.g. van Gelderen and Rummel 2001; Martinec 2003; Tóth 2003) ($k=0$),

• the analytical solution for the vertical–horizontal–horizontal gravitational curvature BVP (Šprlák and Novák 2016) ($k=1$).

The integral transformation (44) is the analytical solution of the $\{k,3\}$ BVP of Eqs. (41)–(43). The scalar components $(T_{xxx[k]} - 3\, T_{xyy[k]})$ and $(T_{yyy[k]} - 3\, T_{xxy[k]})$ of the tensor boundary condition (42) can be identified under the surface integral. Equations (45) and (46) are the spherical harmonic expansions of $(T_{xxx[k]} - 3\, T_{xyy[k]})$ and $(T_{yyy[k]} - 3\, T_{xxy[k]})$. Their n-th-degree spherical harmonics are the series (47) and (48). The integral kernel ${\mathcal {H}}^{\{k,3\}}$ in Eq. (49) fulfils the role of a weight function in the integral formula (44).

For a fixed t, the integral kernels ${\mathcal {H}}^{\{k,1\}}$, ${\mathcal {H}}^{\{k,2\}}$, and ${\mathcal {H}}^{\{k,3\}}$ depend only on the spherical distance $\psi$ and are referred to as isotropic kernels.

Šprlák and Novák (2016) found an analytical solution of the horizontal–horizontal–horizontal gravitational curvature BVP, which corresponds to the $\{0,3\}$ BVP in Table 3.

Table 3 Formulation and analytical solution of $\{k,3\}$ BVP

3 Mathematical Derivation of Practical Far-Zone Effect Formulas

We now focus on the far-zone effects for the analytical solution of the $\{k,l\}$ BVPs, $k \in {\mathbb {Z}}^{*}, l \in \{1,2,3\}$, and for the first-, second-, and third-order spatial derivatives of these solutions, see Fig. 2. Our goal is to transform integral forms of the far-zone effects into more efficient external spherical harmonic expansions. To our best knowledge, the generalised theory has not been presented before and is thus original.

Fig. 2

Scheme illustrating relationships of the boundary conditions of $\{k,1\}$ (blue), $\{k,2\}$ (red), and $\{k,3\}$ (green) BVPs with T and with the components of ${\textsf{T}}^{(1)}$, ${\textsf{T}}^{(2)}$, and ${\textsf{T}}^{(3)}$. The boundary conditions are placed at the lower level as functions of $(R,\Omega ')$. T and the components of ${\textsf{T}}^{(1)}$, ${\textsf{T}}^{(2)}$, and ${\textsf{T}}^{(3)}$ are functions of $(r,\Omega )$ and are located at the upper level. The red, blue, and green links represent analytical solutions of $\{k,1\}$, $\{k,2\}$, and $\{k,3\}$ BVPs, respectively. The black links are integral formulas transforming the boundary conditions to the components of ${\textsf{T}}^{(1)}$, ${\textsf{T}}^{(2)}$, and ${\textsf{T}}^{(3)}$

The electronic documentation with step-by-step mathematical derivations takes up approximately 250 pages and its presentation would be a challenging task. Below, we opted for a comprehensive summary of our methodological activities in Tables 4, 5, 6, and 7. An informative text concisely describes the meaning of individual functions and the fundamental derivation steps.

3.1 Far-Zone Effects When Calculating T

We start with the far-zone effects $\delta T^{\{k,l\}}$ for the analytical solutions of the $\{k,l\}$ BVPs in Eqs. (26), (35), and (44). The points of departure in Table 4 are the formulas after the first equal signs in Eqs. (52)–(54) termed integral forms of $\delta T^{\{k,l\}}$. The surface integration (from the products of the boundary values with ${\mathcal {H}}^{\{k,l\}}$) extends over the far zone ${\mathbb {S}}_R - {\mathbb {S}}_R^0$.

The integral forms are transformed into their spectral counterparts by the following procedure:

• 1) The original integral forms of $\delta T^{\{k,l\}}$ are reformulated as surface integrals over the entire spherical surface ${\mathbb {S}}_R$ after the second equal signs in Eqs. (52)–(54). This is permitted by introducing the error kernels $\Delta {\mathcal {H}}^{\{k,l\}}$ parametrised by two mathematically equivalent expressions in Eq. (55). By the spatial form (after the first equality sign in Eq. (55)), $\Delta {\mathcal {H}}^{\{k,l\}}$ are exactly zero in the interval $u \in [u_0,+1]$ and coincide with the isotropic kernels ${\mathcal {H}}^{\{k,l\}}$ of Eqs. (31), (40), and (49) in the interval $u \in [-1,u_0)$. By the spectral form (after the second equality sign in Eq. (55)), $\Delta {\mathcal {H}}^{\{k,l\}}$ are infinite series in terms of $P_{n,l}$. The coefficients $Q_{n}^{\{k,l\}}$ are specified by Eq. (56) and are referred to as the truncation error coefficients. The functions $e_{j,n}^{(l)}$ on the right-hand side are defined by the definite integral or by the recursion of the form:

  $$\begin{aligned} e_{j,n}^{(l)}(u_0)&= \frac{(n-l)!}{(n+l)!} \int _{-1}^{u_0} P_{j,l} (u)\, P_{n,l} (u)\ {\text{d}}u \nonumber \\&= \bigg [\frac{(n-l)!}{(n+l)!} \sqrt{1-u_0^2}\, P_{j,l-1}(u_0)\, P_{n,l} (u_0) + e_{j,n}^{(l-1)}(u_0)\bigg ]. \end{aligned}$$

  (50)

  We list several equivalent mathematical definitions for $Q_{n}^{\{k,l\}}$ to demonstrate their connection to the error isotropic kernel $\Delta {\mathcal {H}}^{\{k,l\}}$ or to the isotropic kernel ${\mathcal {H}}^{\{k,l\}}$. We also include the series form, whose truncated version is employed in the numerical experiments below.

• 2) The infinite series for $\Delta {\mathcal {H}}^{\{k,l\}}$ are substituted into the global integrals (52)–(54).

• 3) Assuming the uniform convergence of the series, the order of integration and summation can be interchanged.

• 4) By using the global integral identities (C35) (for $l=1$), (C43) (for $l=2$), and (C51) (for $l=3$), we finally obtain Eq. (57), i.e. the far-zone effects $\delta T^{\{k,l\}}$ for the analytical solution of the $\{k,l\}$ BVPs, see Eqs. (26), (35), and (44). The n-th-degree spherical harmonic of $T_{[k]}$ in Eq. (57) is:

  $$\begin{aligned} T_{[k],n}(R,\Omega ) = (-1)^{k}\, \frac{GM}{a^{k+1}} \left( \frac{a}{R}\right) ^{n+k+1}\ \frac{(n + k)!}{n!} \sum \limits _{m=-n}^{+n} \Delta {\bar{C}}_{n,m}\ {\bar{Y}}_{n,m}(\Omega ). \end{aligned}$$

  (51)

Table 4 Far-zone effects for $\{k,l\}$ BVP when calculating T

3.2 Far-Zone Effects When Calculating $T_{o}, \, o \in \{x,y,z\}$

We continue our mathematical derivations with the far-zone effects for the first-order derivatives of the analytical solution of the $\{k,l\}$ BVPs. The three components $T_{o}^{\{k,l\}}$, $o \in \{x,y,z\}$, of the disturbing gravitational vector result from the action of the differential operators $\mathcal{{D}}^{o}$, $o \in \{x,y,z\}$, in Table 12 on the integral formulas (26), (35), and (44). The differentiation under the integral sign changes the original sub-integral kernels into the sub-integral kernels ${\mathcal {H}}_{o,{\textsf{A}}}^{\{k,l\}}$, ${\mathcal {H}}_{o,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, and ${\mathcal {H}}_{z}^{\{k,l\}}$.

We define the far-zone effects $\delta T_{o}^{\{k,l\}}$, $o \in \{x,y,z\}$, by Eqs. (58)–(63) in Table 5. For each l, $\delta T_{x}^{\{k,l\}}$ and $\delta T_{y}^{\{k,l\}}$ are concisely embedded in one surface integral and $\delta T_{z}^{\{k,l\}}$ is presented by another integral. The formulas after the first equality signs are the far-zone effects with integration in the domain ${\mathbb {S}}_R - {\mathbb {S}}_R^0$.

The right-hand sides of Eqs. (58)–(63) consider integration over the complete spherical surface ${\mathbb {S}}_R$. The change of the integration domain is possible when using the error sub-integral kernels $\Delta {\mathcal {H}}_{o,{\textsf{A}}}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{o,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, and $\Delta {\mathcal {H}}_{z}^{\{k,l\}}$, see Eqs. (64)–(68). These kernels consist of: 1) the azimuthal components being a function of $\alpha$ and $\alpha '$, and 2) the isotropic components represented by the error isotropic kernels $\Delta {\mathcal {H}}_{t}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{u+}^{\{k,l\}}$, and $\Delta {\mathcal {H}}_{u-}^{\{k,l\}}$.

The chosen symbols and defining formulas of $\Delta {\mathcal {H}}_{u+}^{\{k,l\}}$ and $\Delta {\mathcal {H}}_{u-}^{\{k,l\}}$ differ only by a sign. As such, the two defining equations can be condensed into a single one containing ± and both functions can concisely be denoted $\Delta {\mathcal {H}}_{u\pm }^{\{k,l\}}$ with ± in the subscript. Contrariwise, we can identify two equations from its condensed counterpart, which includes ±, i.e. first is given by considering the upper signs, second is defined by reading the lower signs. We employ this concise notation for numerous functions below.

Two representations of the error isotropic kernels are given in Eqs. (69) and (70). By the spatial representation, the kernels are identical to their isotropic counterparts ${\mathcal {H}}_{t}^{\{k,l\}}$ and ${\mathcal {H}}_{u\pm }^{\{k,l\}}$ in the interval $u \in [-1,u_0)$, and exactly zero in the interval $u \in [u_0,+1]$. Equations (71) and (72) originate from the application of the specified differential operators to ${\mathcal {H}}^{\{k,l\}}$, see Eqs. (31), (40), and (49). The differential operator $\mathcal{{D}}^{1}_{1}$ is defined by Eq. (A4).

The spectral representation of the error isotropic kernels in Eqs. (69) and (70) are the infinite series with the basis functions $P_{n,l}$ and the truncation error coefficients $Q_{t,n}^{\{k,l\}}$ and $Q_{u\pm ,n}^{\{k,l\}}$, see Eqs. (73) and (74). Among the truncation error coefficients, the addition $\left( Q_{u+,n}^{\{k,l\}} + Q_{u-,n}^{\{k,l\}}\right)$ is explicitly provided in Eq. (75). Several equivalent relations are again presented for the truncation error coefficients to show their link to the corresponding error isotropic kernels, isotropic kernels, or truncation error coefficients $Q_{n}^{\{k,l\}}$. Interestingly, only $Q_{t,n}^{\{k,l\}}$ and $\left( Q_{u+,n}^{\{k,l\}} + Q_{u-,n}^{\{k,l\}}\right)$ appear in the final far-zone effect formulas. We therefore supplement their series forms, which are used in the numerical investigations.

Using the mathematical functions above, we can find the spherical harmonic series of the far-zone effects $\delta T_{o}^{\{k,l\}}$, $o \in \{x,y,z\}$. The spectral forms of the error sub-integral kernels $\Delta {\mathcal {H}}_{o,{\textsf{A}}}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{o,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, and $\Delta {\mathcal {H}}_{z}^{\{k,l\}}$ are inserted into the global integrals on the right-hand sides of Eqs. (58)–(63). On account of the uniform convergence, we are permitted to interchange the order of summation and integration. By employing the global integral identities (C35)–(C37), (C42), (C43)–(C45), (C50), (C51)–(C53), and (C58), we get the final spherical harmonic series in Eqs. (76)–(78).

Table 5 Far-zone effects for $\{k,l\}$ BVPs when calculating $T_{o}, o \in \{x,y,z\}$

3.3 Far-Zone Effects When Calculating $T_{op},\, o,p \in \{x,y,z\}$

We can obtain six integral transformations for the second-order derivatives of the analytical solutions of the $\{k,l\}$ BVPs by the application of the differential operators $\mathcal{{D}}^{op}$, $o,p \in \{x,y,z\}$, in Table 13 to Eqs. (26), (35), and (44). The differentiation transforms T on the left-hand sides to the six (scalar) components $T_{op}$, $o,p \in \{x,y,z\}$, of the disturbing gravitational tensor ${\textsf{T}}^{(2)}$. The original isotropic kernels ${\mathcal {H}}^{\{k,l\}}$ on the right-hand sides change to the sub-integral kernels ${\mathcal {H}}_{op,{\textsf{A}}}^{\{k,l\}}$, ${\mathcal {H}}_{op,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, $p \in \{x,y,z\}$, and ${\mathcal {H}}_{zz}^{\{k,l\}}$.

We now aim at deriving the far-zone effects $\delta T_{op}^{\{k,l\}}$, $o,p \in \{x,y,z\}$. These are initially prescribed by the integration in the domain ${\mathbb {S}}_R - {\mathbb {S}}_R^0$, see Eqs. (79)–(84) in Table 6. For further mathematical manipulations, we rewrite the six far-zone effects in terms of global integration on the right-hand sides of Eqs. (79)–(84). This is permitted with the use of the error sub-integral kernels $\Delta {\mathcal {H}}_{op,{\textsf{A}}}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{op,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, $p \in \{x,y,z\}$, and $\Delta {\mathcal {H}}_{zz}^{\{k,l\}}$. Equations (85)–(95) show that the error sub-integral kernels are decomposed into the azimuthal component (depending on $\alpha$ and $\alpha '$) and the error isotropic kernels $\Delta {\mathcal {H}}_{tt}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{tu\pm }^{\{k,l\}}$, and $\Delta {\mathcal {H}}_{uu\pm }^{\{k,l\}}$ given by Eqs. (96)–(98).

According to the spatial representations, the error isotropic kernels are equal to the isotropic kernels ${\mathcal {H}}_{tt}^{\{k,l\}}$, ${\mathcal {H}}_{tu\pm }^{\{k,l\}}$, and ${\mathcal {H}}_{uu\pm }^{\{k,l\}}$ for $u \in [-1,u_0)$ and zero for $u \in [u_0,+1]$. The differential operator $\mathcal{{D}}^{4}_{2}$, application of which provides ${\mathcal {H}}_{tt}^{\{k,l\}}$, is defined by Eq. (A11). Alternatively, the error isotropic kernels are parametrised by the infinite series (with $P_{n,l}$ and its derivatives as basis functions) on the right-hand sides of Eqs. (96)–(98). The expansion coefficients are the truncation error coefficients $Q_{tt,n}^{\{k,l\}}$, $Q_{tu\pm ,n}^{\{k,l\}}$, and $Q_{uu\pm ,n}^{\{k,l\}}$ by the formulas (102)–(104). For practical purposes, we prefer the additions $\left( Q_{tu+,n}^{\{k,l\}} + Q_{tu-,n}^{\{k,l\}}\right)$ and $\left( Q_{uu+,n}^{\{k,l\}} + Q_{uu-,n}^{\{k,l\}}\right)$ summarised by the expressions (105) and (106).

We now take the spectral forms of the error sub-integral kernels $\Delta {\mathcal {H}}_{op,{\textsf{A}}}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{op,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, $p \in \{x,y,z\}$, and $\Delta {\mathcal {H}}_{zz}^{\{k,l\}}$, and insert them into the global integrals (79)–(84). Uniform convergence of the infinite series allows us to change the order of summation and integration. Finally, by considering the integral identities (C35), (C38), (C39), (C42), (C43), (C46), (C47), (C50), (C51), (C54), (C55), (C58), we get the spherical harmonic expansions for $\delta T_{op}^{\{k,l\}}$, $o,p \in \{x,y,z\}$, in Eqs. (107)–(112).

Table 6 Far-zone effects for $\{k,l\}$ BVPs when calculating $T_{op},\, o,p \in \{x,y,z\}$

3.4 Far-Zone Effects When Calculating $T_{opq},\, o,p,q \in \{x,y,z\}$

We get ten components $T_{opq}$, $o,p,q \in \{x,y,z\}$, of the third-order disturbing gravitational tensor ${\textsf{T}}^{(3)}$, when the differential operators $\mathcal{{D}}^{opq}$, $o,p,q \in \{x,y,z\}$, listed in Table 14, act on the integral formulas (26), (35), and (44). The isotropic kernels ${\mathcal {H}}^{\{k,l\}}$ under the integral sign are converted into the sub-integral kernels ${\mathcal {H}}_{opq,{\textsf{A}}}^{\{k,l\}}$, ${\mathcal {H}}_{opq,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, $p,q \in \{x,y,z\}$, and ${\mathcal {H}}_{zzz}^{\{k,l\}}$.

The far-zone effects $\delta T_{opq}^{k,l}$, $o,p,q \in \{x,y,z\}$, are the restricted integrals (113)–(118) after the first equality signs, see Table 7. The identical quantities are alternatively represented by global integrals on the right-hand sides. Essential for transferring the integration in the domain ${\mathbb {S}}_R - {\mathbb {S}}_R^0$ to its counterpart over the entire sphere ${\mathbb {S}}_R$ are the error sub-integral kernels $\Delta {\mathcal {H}}_{opq,{\textsf{A}}}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{opq,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, $p,q \in \{x,y,z\}$, and $\Delta {\mathcal {H}}_{zzz}^{\{k,l\}}$ of Eqs. (119)–(137). These contain products of the error isotropic kernels $\Delta {\mathcal {H}}_{ttt}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{ttu\pm }^{\{k,l\}}$, $\Delta {\mathcal {H}}_{tuu\pm }^{\{k,l\}}$, and $\Delta {\mathcal {H}}_{uuu\pm }^{\{k,l\}}$ by the trigonometric functions of $\alpha$ and $\alpha '$.

Formulas (138)–(141) present both the spatial and spectral forms of the error isotropic kernels. By the spatial forms, these functions are zero in the interval $u \in [u_0,+1]$ and correspond precisely to their isotropic versions ${\mathcal {H}}_{ttt}^{\{k,l\}}$, ${\mathcal {H}}_{ttu\pm }^{\{k,l\}}$, ${\mathcal {H}}_{tuu\pm }^{\{k,l\}}$, and ${\mathcal {H}}_{uuu\pm }^{\{k,l\}}$ for $u \in [-1,u_0)$. The isotropic kernels are characterised by Eqs. (142)–(145), where the differential operator $\mathcal{{D}}_{3}^{6}$ can be found in Eq. (A22). By the spectral forms, the error isotropic kernels are the series expansions containing products of the basis functions $P_{n,l}$ (and their derivatives with respect to the variable u) with the truncation error coefficients $Q_{ttt,n}^{\{k,l\}}$, $Q_{ttu\pm ,n}^{\{k,l\}}$, $Q_{tuu\pm ,n}^{\{k,l\}}$, and $Q_{uuu\pm ,n}^{\{k,l\}}$, see the right-hand sides of Eqs. (138)–(141). Equations (146)–(149) compile several equivalent expressions for the truncations error coefficients. The three additions $\left( Q_{ttu+,n}^{\{k,l\}} + Q_{ttu-,n}^{\{k,l\}}\right)$, $\left( Q_{tuu+,n}^{\{k,l\}} + Q_{tuu-,n}^{\{k,l\}}\right)$, and $\left( Q_{uuu+,n}^{\{k,l\}} + Q_{uuu-,n}^{\{k,l\}}\right)$ are also explicitly provided by formulas (150)–(152), as these are used in practical computations.

After the exhaustive presentation of the mathematical functions above, we can find the spherical harmonic series of $\delta T_{opq}^{\{k,l\}}$, $o,p,q \in \{x,y,z\}$. We insert the error sub-integral kernels $\Delta {\mathcal {H}}_{opq,{\textsf{A}}}^{\{k,l\}}$, $\Delta {\mathcal {H}}_{opq,{\textsf{B}}}^{\{k,l\}}$, $o \in \{x,y\}$, $p,q \in \{x,y,z\}$, and $\Delta {\mathcal {H}}_{zzz}^{\{k,l\}}$ (with the infinite series substituting for the error isotropic kernels) into the global integrals of Eqs. (113)-(118). The desired spherical harmonic series of Eqs. (153)–(162) follow when interchanging the order of summation and integration (with the assumption of the uniform convergence of these series) and using the global integral identities (C35), (C40)–(C42), (C43), (C48)–(C50), (C51), (C56)–(C58).

Table 7 Far-zone effects for $\{k,l\}$ BVPs when calculating $T_{opq},\, o,p,q \in \{x,y,z\}$

3.5 Note on the Derivation of the Far-Zone Effects

The mathematical derivations of the far-zone effects above have been based on a sequence of several steps and may seem cumbersome, yet straightforward. This is obvious for the integral formulas transforming the boundary conditions of $\{k,l\}$ BVPs to T (see Table 4), $T_{z}$ (see Eqs. (59), (61), (63) in Table 5), $T_{zz}$ (see Eqs. (80), (82), (84) in Table 6), or $T_{zzz}$ (see Eqs. (114), (116), (118) in Table 7). For the other integrals, by which we calculate the horizontal $(T_{x}, T_{y})$, vertical–horizontal $(T_{xz}, T_{yz})$, horizontal–horizontal $(T_{xx}, T_{xy}, T_{yy})$, vertical–vertical–horizontal $(T_{xzz}, T_{yzz})$, vertical–horizontal–horizontal $(T_{xxz}, T_{xyz}, T_{yyz})$, or horizontal–horizontal–horizontal $(T_{xxx}, T_{xxy}, T_{xyy}, T_{yyy})$ disturbing gravitational tensor components, the possibility of exploiting the identical derivation steps is permitted only if the corresponding sub-integral kernels have a suitable form.

We demonstrate this point for the $\{1,1\}$ BVP when calculating $T_{x}$ and $T_{y}$. The integral transformations of this type, see Eq. (58) in Table 5, were derived and their mathematical properties were extensively investigated in (Šprlák and Novák 2014a). The corresponding error sub-integral kernels deduced from (Šprlák and Novák 2014a) are presented in the form of Eqs. (163)–(166) in Table 8. These are clearly split into the azimuthal part (depending on $\alpha$ and $\alpha '$) and the error isotropic kernels $\Delta {\mathcal {H}}_{*}^{\{1,1\}}$ and $\Delta {\mathcal {H}}_{u}^{\{1,1\}}$ with the spatial forms (167) and (168). Equations (169) and (170) provide the explicit definition of the isotropic kernels ${\mathcal {H}}_{*}^{\{1,1\}}$ and ${\mathcal {H}}_{u}^{\{1,1\}}$.

Table 8 Definition of the error sub-integral kernels $\Delta {\mathcal {H}}_{o,{\textsf{A}}}^{\{1,1\}}$ and $\Delta {\mathcal {H}}_{o,{\textsf{B}}}^{\{1,1\}}$, $o \in \{x,y\}$, and of the corresponding error isotropic kernels $\Delta {\mathcal {H}}_{*}^{\{1,1\}}$ and $\Delta {\mathcal {H}}_{u}^{\{1,1\}}$ from (Šprlák and Novák 2014a)

Importantly, Table 8 does not include the spectral forms for $\Delta {\mathcal {H}}_{*}^{\{1,1\}}$ and $\Delta {\mathcal {H}}_{u}^{\{1,1\}}$. In fact, the Legendre series of these functions do not exist as the corresponding basis functions are not orthogonal, i.e.

$$\begin{aligned}&\int \limits _{-1}^{+1} \frac{1}{\sqrt{1 - u^2}} P_{n,1}(u)\ \frac{1}{\sqrt{1 - u^2}} P_{n',1}(u)\ \text{d}u \ne N(n)\ \delta _{n,n'}, \end{aligned}$$

(171)

$$\begin{aligned}&\int \limits _{-1}^{+1} \sqrt{1 - u^2}\ \frac{\partial }{\partial u} P_{n,1}(u)\ \sqrt{1 - u^2}\ \frac{\partial }{\partial u} P_{n',1}(u)\ \text{d}u \ne N(n)\ \delta _{n,n'}, \end{aligned}$$

(172)

where N indicates a normalising factor. Because the spectral forms of the isotropic kernels are a necessary element for the derivation of the far-zone effects, such derivation would be impossible when starting from the error sub-integral kernels of Eqs. (163)–(166).

It is thus indispensable to rewrite the error sub-integral kernels (163)–(166) in Table 8 into the equivalent forms (64)–(67) in Table 5 by using identities for the trigonometric functions of $\alpha$ and $\alpha '$. The error isotropic kernels change (compare Eq. (70) in Table 5 with Eqs. (167) and (168) in Table 8), but their basis functions are orthogonal as required. Orthogonality properties of such basis functions for all kernels presented in this article are summarised in Table 16.

4 Numerical Experiments

The expressions presented in Sect. 3 may appear relatively complex and are susceptible to potential errors. Instead of analytical methods, we employed numerical means to check the formulas and exclude possible missteps in the mathematical derivations. Namely, the integral and spectral forms of the far-zone effects in Tables 4-7 were coded in MATLAB and closed-loop simulations were designed for their mutual numerical comparison.

An indispensable tool in our closed-loop simulations was a global gravitational field model (GGFM). It composes of a finite set of spherical harmonic coefficients and allows synthesising self-consistent input and output values. Among numerous GGFMs available, we used the most recent estimate of the Earth’s gravitational field from GRACE (Gravity Recovery and Climate Experiment, Tapley et al. 2004) and GOCE (Gravity field and steady-state Ocean Circulation Explorer, ESA 1999) satellite observations abbreviated Tongji_GMMG2021S (Chen et al. 2022).

Fig. 3

a Bathymetry and topography from ETOPO2v2 (Amante and Eakins 2009), b) Vertical component of the disturbing gravitational vector at $R = 6378136.46$ m. The red line on the left and the green line on the right illustrate the sub-satellite points of the GOCE mission. (1 mGal = 10$^{-5}$ m s$^{-2}$)

Far-zone effects of 20 quantities (i.e. $\delta T^{\{k,l\}}$, $\delta T_{o}^{\{k,l\}}$, $o \in \{x,y,z\}$, $\delta T_{op}^{\{k,l\}}$, $o,p \in \{x,y,z\}$, $\delta T_{opq}^{\{k,l\}}$, $o,p,q \in \{x,y,z\}$) were validated at 539 computational points. These represent the realistic orbit of GOCE and originate from the EGG_TRF_2 data product (EGG-C 2010) and form a segment of the orbital arc on September 13 2013 (first point at GPS time 1063112775.6537980 s). We chose this geometric constellation of computational points for the following reasons: 1) The number of points was a reasonable compromise between the statistical significance and the computational time. 2) The parameter $t = \frac{R}{r} < 1$ and varies along the orbit as $R = 6378136.46$ m and $r \in [6603502.086\ \text{m}, 6612362.349\ \text{m}]$. 3) The selected location is characteristic of extreme changes of topography and of the gravitational field along the track from the Indian Ocean towards and above the Himalayas, see Fig. 3.

In a first step, we calculated the 20 far-zone effects by integrating in the domain ${\mathbb {S}}_R - {\mathbb {S}}_R^0$, see the expressions after the first equal signs in Eqs. (52)–(54), (58)–(63), (79)–(84), and (113)–(118). We set the spherical cap size to $\psi _0 = 5^{\circ }$ that may correspond to a preferred spherical cap size in regional gravitational field modelling. The isotropic kernels under the integrals were calculated by their spectral forms up to degree 250 that is computationally intensive. We intentionally sped up the challenging calculations by using High-Performance Facilities and completed all computations in approximately three weeks. Correspondingly, the functionals of the gravitational field were synthesised up to degree 250 in equiangular grids with sampling $\Delta \varphi = \Delta \lambda =0.05^{\circ }$. We also employed coarser grid resolutions in our initial experiments. However, local details of higher-order disturbing gravitational tensor components may have been lost that severely degraded the precision of the numerical integration. We chose $k=0$, i.e. $(T_{x}, T_{y})$, $(T_{xx}, T_{xy}, T_{yy})$, and $(T_{xxx}, T_{xxy}, T_{xyy}, T_{yyy})$ were generated and integrated over in case of the $\{k,1\}$, $\{k,2\}$, and $\{k,3\}$ BVP, respectively.

In a second step, we directly synthesised the 20 far-zone effects by the formulas (57), (76)–(78), (107)–(112), and (153)–(162). When computing the truncation error coefficients, we considered the same input parameters as employed for calculation of the restricted integrals, namely the spherical cap size $\psi _0 = 5^{\circ }$ and the maximum degree 250 for series representations of the isotropic kernels. The spherical harmonic series of the far-zone effects were truncated at the same harmonic degree 250.

All calculated values represented quantities of the disturbing gravitational field. Thus, we introduced a normal gravitational field, which was generated by the Geodetic Reference System 1980 (GRS80, Moritz 2000).

Table 9 summarises the statistics of the far-zone effects for the $\{0,1\}$ BVP. The signal (synthesised by the spherical harmonic series) reaches tens m$^2$ s$^{-2}$ for $\delta T^{\{0,1\}}$, several mGal for $\delta T_{o}^{\{0,1\}}, o \in \{x,y,z\}$, several tens up to few hundreds mE for $\delta T_{op}^{\{0,1\}}, o,p \in \{x,y,z\}$, and up to hundreds am$^{-1}$ s$^{-2}$ for $\delta T_{opq}^{\{0,1\}}, o,p,q \in \{x,y,z\}$. The mean values of all far-zone effects are significantly different from zero that demonstrates their systematic characters. Variations alter among the components of the disturbing gravitational vector, or among the components of the second- and third-order disturbing gravitational tensors. Horizontal components may have stronger signal variations compared to their purely vertical counterparts. This proves complexity of the gravitational field along the orbit in the latitude and longitude directions. The statistics of the differences between the spectral and spatial forms of the far-zone effects show strong agreement. The mean values are close to zero and the standard deviations are at least three orders of magnitude smaller than those for the signal.

The statistics of the far zones for the $\{0,2\}$ BVP is listed in Table 10. Mean values of the signals for the individual quantities are notably shifted from zero, as for the $\{0,1\}$ BVP in Table 9. On the other hand, their variations are higher compared to those for the $\{0,1\}$ BVP. In particular, the signals of the vertical components $\delta T_{zz}^{\{0,2\}}$ and $\delta T_{zzz}^{\{0,2\}}$ are now the strongest compared to their horizontal or mixed equivalents. The means of differences vary closely around zero. The difference standard deviations are smaller by at least two orders of magnitude compared to the signal standard deviations.

We present the statistics of the far-zone effects related to the $\{0,3\}$ BVP in Table 11. The mean values of signals synthesised by the spherical harmonic series are again different from zero. Their standard deviations and extrema are even higher than those for the $\{0,2\}$ BVP in Table 10, and the signals of the vertical components $\delta T_{zz}^{\{0,3\}}$ and $\delta T_{zzz}^{\{0,3\}}$ dominate in size among the other ones. With larger signal variations and higher extrema, the differences between the integral and the spectral forms of the far-zone effects increase, as compared to those in Table 9 and Table 10. Nevertheless, the difference standard deviations are always lower by at least two orders of magnitude to the signal standard deviations.

Randomly selected far-zone effects were calculated with a larger integration radius of $20^{\circ }$. In addition, we employed k = 1 and 2 for the $\{k,1\}$ BVP and $k = 1$ for $\{k,2\}$ BVP in spot checks. These parameter swaps did not lead to any different conclusions from those observed above. Overall, the results presented in this numerical experiment demonstrate numerical agreement of the integral and the spectral forms of the far-zone effects at the level of 1% and better. We can thus exclude any possible errors in our mathematical derivations and may consider the final spherical harmonic series of the far-zone effects in Tables 4-7 to be correct.

Table 9 Far-zone effects for the $\{0,1\}$ BVP. The attribute ’Signal’ indicates statistics of the far-zone effects synthesised by the spherical harmonic expansions (spectral forms). The attribute ’Differences’ specifies statistics of the differences between the far-zone effects calculated by the spherical harmonic expansions and those from the discretised integration. Units: $\delta T^{\{0,1\}}$ is in m$^2$ s$^{-2}$; $\delta T_{o}^{\{0,1\}}$, $o \in \{x,y,z\}$, are in mGal = 10$^{-5}$ m s$^{-2}$; $\delta T_{op}^{\{0,1\}}$, $o,p \in \{x,y,z\}$, are in mE = 10$^{-12}$ s$^{-2}$; $\delta T_{opq}^{\{0,1\}}$, $o,p,q \in \{x,y,z\}$, are in am$^{-1}$ s$^{-2}$ = 10$^{-18}$ m$^{-1}$ s$^{-2}$

Table 10 Same as in Table 9, but for the $\{0,2\}$ BVP

Table 11 Same as in Table 9, but for the $\{0,3\}$ BVP

5 Conclusions

In this article, we developed a mathematical theory for the far-zone effects of the following spherical integral transformations (see also Fig. 2): (1) the analytical solutions of the horizontal, horizontal–horizontal, and horizontal–horizontal–horizontal BVPs including their generalisations with arbitrary-order vertical derivatives of respective boundary conditions and (2) spatial (vertical, horizontal, or mixed) derivatives of these generalised analytical solutions up to the third order. The delimiting boundary between the near zone and the far zone was the spherical cap. As a result, the final far-zone effect formulas have the form of spherical harmonic series multiplied by truncation error coefficients depending on the spherical harmonic degree, attenuation factor, and the size of the spherical cap, see the Sections ‘Far-zone effects (spectral forms)’ in Tables 4-7. Except for one previous study, the theoretical treatment of the far-zone effects for the spherical integral formulas with dependencies on the backward azimuth has not been presented before and our generalised formulations are thus original.

Correctness of the relatively complex expressions and of the related mathematical derivations was thoroughly validated. The integral and spectral forms of the far-zone effects were implemented in MATLAB software package and compared in closed-loop simulations. The analysis of results confirmed a strong agreement of both far-zone representations at the level of 1% and better and thus proved their equivalence.

This current work together with our recent review in (Šprlák and Pitoňák 2024) enrich the theoretical apparatus used in geodesy and complete the Meissl diagram (e.g. Rummel and van Gelderen 1995) for the far-zone effects. Both these studies also contain software implementation of the mathematical formulations that will be available to potential users upon request. The theory and its implementation are invaluable for the geodetic, geophysical, and planetary science communities in gravitational field modelling studies when solving direct or inverse problems. In addition, the presented formulas do not apply exclusively to gravitation, but may be employed in any potential field. Quantification and accurate parametrisation of far-zone effects are also indispensable when studying statistical properties of spherical integral transformations, e.g. propagation of errors.

Appendices

Appendix A Differential Operators

Relationships among various quantities of the gravitational field are defined by differential operators. In this section, we review differential operators for the components of higher-order disturbing gravitational tensors. For completeness, these are presented in terms of the spherical geocentric coordinates $(r, \Omega )$ and by the variables $(R, t, u, \alpha )$.

Table 12 lists the three differential operators $\mathcal{{D}}^{o}$, $o \in \{x,y,z\}$, for the components of the gravitational gradient ${\textsf{T}}^{(1)}$ (e.g. Heiskanen and Moritz 1967, p. 113; Šprlák and Novák 2014b). While $\mathcal{{D}}^{z}$ is isotropic (for t fixed), $\mathcal{{D}}^{x}$ and $\mathcal{{D}}^{y}$ are split into the products of $\mathcal{{D}}^{2}_{1}$ by the trigonometric functions $\cos \alpha$ and $\sin \alpha$. The subscripts of the isotropic differential operators indicate the order of the disturbing gravitational tensor. The superscripts merely distinguish among these operators.

Table 12 Differential operators for the components of ${\textsf{T}}^{(1)}$ in LNORF

Review of the six differential operators $\mathcal{{D}}^{op}$, $o,p \in \{x,y,z\}$, for the components of the second-order disturbing gravitational tensor ${\textsf{T}}^{(2)}$ is provided in Table 13 (e.g. Wolf 2007; Šprlák et al. 2014). $\mathcal{{D}}^{zz}$ depends on the variable u and is thus isotropic for a fixed t. The operators $\mathcal{{D}}^{xy}$, $\mathcal{{D}}^{xz}$, and $\mathcal{{D}}^{yz}$ are products of the azimuthal part (i.e. $\sin 2\alpha$, $\cos \alpha$, and $\sin \alpha$) with the isotropic differential operators $\mathcal{{D}}^{2}_{2}$ and $\mathcal{{D}}^{3}_{2}$. $\mathcal{{D}}^{xx}$ and $\mathcal{{D}}^{yy}$ represent linear combinations of $\mathcal{{D}}^{1}_{2}$ with $\cos 2\alpha \, \mathcal{{D}}^{2}_{2}$.

Table 13 Differential operators for the components of ${\textsf{T}}^{(2)}$ in LNORF

The differential operators $\mathcal{{D}}^{opq}$, $o,p,q \in \{x,y,z\}$, for the components of the third-order disturbing gravitational tensor ${\textsf{T}}^{(3)}$ are listed in Table 14 (Šprlák and Novák 2015). $\mathcal{{D}}^{xxx}$, $\mathcal{{D}}^{xxy}$, $\mathcal{{D}}^{xyy}$, and $\mathcal{{D}}^{yyy}$ combine the products of the isotropic differential operators $\mathcal{{D}}^{1}_{3}$ and $\mathcal{{D}}^{2}_{3}$ by the trigonometric functions of $\alpha$ and $3\alpha$. $\mathcal{{D}}^{xxz}$ and $\mathcal{{D}}^{yyz}$ are linear combinations of $\mathcal{{D}}^{3}_{3}$ with $\cos 2\alpha \, \mathcal{{D}}^{4}_{3}$. The operators $\mathcal{{D}}^{xyz}$, $\mathcal{{D}}^{xzz}$, and $\mathcal{{D}}^{yzz}$ are products of functions $\sin 2\alpha$, $\cos \alpha$, and $\sin \alpha$ with the isotropic differential operators $\mathcal{{D}}^{4}_{3}$ and $\mathcal{{D}}^{5}_{3}$. $\mathcal{{D}}^{zzz}$ is the simplest operator as it changes only with the variables u and t.

Table 14 Differential operators for the components of ${\textsf{T}}^{(3)}$ in LNORF

The differential operators are symmetric when expressed in the variables $(R, t, u, \alpha )$. Such operators can act on the analytical solutions of $\{k,l\}$ BVPs in Tables 1, 2, and 3, and result in the components of higher-order disturbing gravitational tensors. In contrast, the differential operators are highly asymmetric in the spherical geocentric coordinates $(r, \Omega )$. Besides, when spherical harmonic parametrisation is used, these complex operators are applied to obtain higher-order derivatives of T.

Appendix B Auxiliary Orthogonality Relationships

To get the analytical solutions of the $\{k,l\}$ BVPs in Tables 1–3, we use the orthogonality of the $4\pi$-normalised spherical harmonics of degree n and order m, abbreviated ${\bar{Y}}_{n,m}$, including their derivatives. These orthogonality expressions are summarised in Table 15 and may be found, for example, in geodetic studies on scalar, vector, or tensor spherical harmonics (e.g. Heiskanen and Moritz 1967; Meissl 1971; Rummel 1997; Martinec 2003; Freeden and Schreiner 2009; Šprlák et al. 2016).

Table 15 Orthogonality relationships for ${\bar{Y}}_{n,m}$

The mathematical theory for the far-zone effects presented in Tables 4, 5, 6, and 7 relies on the series expansions of the error isotropic kernels. The basis functions in these series are the (un-normalised) associated Legendre functions of the first kind of degree n and order l and their derivatives. These basis functions are orthogonal by the relationships in Table 16. Some of these formulas, e.g. Eq. (B27), are notoriously known and may be found in the mathematical literature (e.g. Hobson 1965; Samaddar 1974; Freeden and Gutting 2013). The general formulas (B28)–(B30) have independently been derived, and their validity was verified numerically within this study.

Table 16 Orthogonality relationships for $P_{n,l}$

Appendix C Global Integral Identities

We now summarise global integral identities required for the derivations of the far-zone effects in Tables 4, 5, 6, and 7. Detailed mathematical proofs for these identities in Tables 17, 18, and 19 are lengthy and cumbersome. For this reason, we illustrate the basic steps of such derivations for Eq. (C35), while other identities can be derived similarly.

We firstly make use of the formula for the un-normalised Legendre function of degree n and order 1 (e.g. Heiskanen and Moritz 1967, Eq. 1-60):

$$\begin{aligned} P_{n,1}(u) = \sqrt{1 - u^2}\ \frac{\partial }{\partial u} P_{n,0} (u), \end{aligned}$$

(C31)

and the relationships for the two differential operators (e.g. Šprlák and Novák 2016):

$$\begin{aligned} \cos \alpha ' \sqrt{1 - u^2}\ \frac{\partial }{\partial u} = \frac{\partial }{\partial \varphi '}, \quad \sin \alpha ' \sqrt{1 - u^2}\ \frac{\partial }{\partial u} = \frac{1}{\cos \varphi '}\frac{\partial }{\partial \lambda '}. \end{aligned}$$

(C32)

Then it holds that:

$$\begin{aligned} \cos \alpha ' P_{n,1}(u) = \frac{\partial }{\partial \varphi '} P_{n,0}(u), \quad \sin \alpha ' P_{n,1}(u) = \frac{1}{\cos \varphi '}\frac{\partial }{\partial \lambda '} P_{n,0}(u). \end{aligned}$$

(C33)

In addition, we substitute $P_{n,0}$ by the right-hand side of the addition theorem (Heiskanen and Moritz 1967, Eq. 1–82’):

$$\begin{aligned} P_{n,0} (u) = \frac{1}{2n + 1} \sum _{m=-n}^{+n} {\bar{Y}}_{n,m}(\Omega )\, {\bar{Y}}_{n,m}(\Omega '). \end{aligned}$$

(C34)

In a further step, we insert the spherical harmonic series (27)–(30) for $T_{x[k]}$ and $T_{y[k]}$. Assuming the uniform convergence of these expansions, we interchange the order of summation and integration. Finally, using the orthogonality formula (B24), we get the right-hand side of Eq. (C35).

Majority of identities listed in Tables 17, 18, and 19 include ± or ${\mp }$ signs, see, for example, Eq. (C36). Every such expression encodes two global identities, i.e.: 1) the first formula is defined by considering the upper signs, and 2) the second formula is specified by reading the lower signs.

Table 17 Global integral identities for $\{k,1\}$ BVP

Table 18 Global integral identities $\{k,2\}$ BVP

Table 19 Global integral identities $\{k,3\}$ BVP