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Bucha B, Hirt C, Kuhn M (2019) Cap integration in spectral gravity forward modelling up to the full gravity tensor,

Journal of Geodesy, DOI: 10.1007/s00190-019-01277-3

Note: This is a preprint (author’s own manuscript that has not been peer reviewed) of an article accepted for publication in

Journal of Geodesy. The ﬁnal authenticated version is available online at https://doi.org/10.1007/s00190-019-01277-3.

Cap integration in spectral gravity forward modelling up to the full

gravity tensor

Blaˇ

zej Bucha ·Christian Hirt ·Michael Kuhn

Received: / Accepted:

Abstract Cap-modiﬁed spectral gravity forward modelling

is extended in this paper to the full gravity vector and ten-

sor expressed in the local north-oriented reference frame.

This is achieved by introducing three new groups of altitude-

dependent Molodensky’s truncation coefﬁcients. These are

given by closed-form and inﬁnite spectral relations that are

generalized for i) an arbitrary harmonic degree, ii) an ar-

bitrary topography power, iii) an arbitrary radial derivative,

iv) any radius larger than the radius of the reference sphere,

and v) for both near- and far-zone gravity effects. Thanks

to the generalization for an arbitrary radial derivative, the

cap-modiﬁed technique can efﬁciently be combined with the

gradient approach for harmonic synthesis on irregular sur-

faces. In a numerical study, we exemplarily apply the new

technique by forward modelling Earth’s degree-2159 topog-

raphy up to degree 21,590, employing 30 topography pow-

ers. The experiment shows that near- and far-zone gravity

effects can be synthesized on the topography with an ac-

curacy (RMS) of 0.005 – 0.03 m2s−2(potential), 0.8 –

20 µGal (gravity vector) and 0.1 mE – 1 E (gravity tensor).

The numerical experiment also shows that the divergence ef-

fect of spherical harmonics comes into play around degree

10,795 when evaluating the series on the Earth’s surface.

Blaˇ

zej Bucha

Department of Theoretical Geodesy, Slovak University of Technology

in Bratislava, Radlinsk´

eho 11, 81005 Bratislava, Slovak Republic

E-mail: blazej.bucha@stuba.sk

Christian Hirt

Institute for Astronomical and Physical Geodesy & Institute for

Advanced Study, Technische Universit¨

at M¨

unchen, Arcisstr 21,

80333 M¨

unchen, Germany

E-mail: c.hirt@tum.de

Michael Kuhn

School of Earth and Planetary Sciences & Western Australian Geodesy

Group, Curtin University, GPO Box U1987, Perth, WA 6845, Australia

E-mail: m.kuhn@curtin.edu.au

The difﬁcult-to-compute truncation coefﬁcients that are em-

ployed in the study are made freely available at

http://edisk.cvt.stuba.sk/~xbuchab/ and are accom-

panied by Matlab-based routines to evaluate them. Enclosed

is also a Matlab-based package to perform ultra-high-degree

surface spherical harmonic analysis, a step of central impor-

tance in spectral gravity forward modelling techniques.

Keywords Spectral gravity forward modelling ·Topo-

graphic potential ·Spherical harmonics ·Molodensky’s

truncation coefﬁcients ·Divergence effect

1 Introduction

Spectral gravity forward modelling (e.g., Rummel et al, 1988;

Martinec and Pˇ

eˇ

c, 1989; Balmino, 1994; Wieczorek and Phillips,

1998; Hirt and Kuhn, 2014) is a technique to deliver the

gravitational ﬁeld induced by a topographic mass distribu-

tion using spherical (or other) harmonics. Its recent applica-

tions include computations of Bouguer anomalies (Balmino

et al, 2012; Hirt et al, 2016), quasigeoid-to-geoid separation

(Tenzer et al, 2016), studying the gravity effect due to the

Earth’s ﬂattening (Wang and Yang, 2013; Rexer et al, 2016),

investigation of the convergence/divergence behaviour of spher-

ical harmonics on planetary surfaces (Hirt et al, 2016; Hirt

and Kuhn, 2017; Bucha et al, 2019b) or the exploration and

mitigation of the harmonic correction issue and the spec-

tral ﬁlter problem of residual terrain modelling (RTM, Rexer

et al, 2018; Hirt et al, 2019).

Common to these studies is that they forward model to-

pographic masses over the entire globe (global integration).

While this is required by potential theory in most instances,

practical evaluations may necessitate a restriction of the in-

tegration domain, for instance, to masses inside/outside a

spherical cap centred at the evaluation point. The resulting

2 Bucha B., Hirt C. and Kuhn M.

cap integration then delivers near- or far-zone gravity ef-

fects, respectively. Perhaps most frequently, the restriction

is done in order to lower the computational burden associ-

ated with the evaluation of the Newton integral in the spatial

domain. To enable the inside- and outside-cap integration

also for the spectral domain, Bucha et al (2019a) modiﬁed

spectral gravity forward modelling by introducing Moloden-

sky’s truncation coefﬁcients (Molodensky et al, 1962). In

this paper, this technique is denoted as cap-modiﬁed spec-

tral gravity forward modelling or simply as cap-modiﬁed

spectral technique. Currently, it enables the computation of

an arbitrary radial derivative of the gravitational potential at

any point above the ﬁeld-generating masses, provided that

the spherical harmonic series converges which holds true

at least at points above the limit sphere encompassing all

gravitating masses (the sphere of convergence; e.g., Hotine

1969).

The cap-modiﬁed spectral technique is particularly suited

when a speciﬁc spectral band of near- and/or far-zone grav-

ity effects is sought. A prominent example is the RTM tech-

nique, in which gravity effects due to the reference (smooth)

topography need to be band-limited in the spectral domain

and at the same time spatially restricted to inside-cap masses

(cf. Bucha et al, 2019a). Other applications include an ef-

ﬁcient development of high-resolution global gravity maps

or investigations of the near- and/or far-zone gravity spectra

(ibid.).

In this paper, we extend the cap-modiﬁed spectral tech-

nique to the full gravity vector and tensor expressed in the

local north-oriented reference frame (LNOF). In addition,

we also provide means to evaluate an arbitrary radial deriva-

tive of these quantities, thereby enabling to compute, for

instance, 6 out of 10 components of the third-order gravi-

tational tensor (after considering its symmetry). The present

study thus extends cap-modiﬁed spectral gravity forward mod-

elling to the large palette of commonly used gravity ﬁeld

quantities such as the height anomalies, the gravity, the de-

ﬂections of the vertical or the gravity tensor.

In a numerical case study, we apply the new technique to

gravity effects implied by the Earth’s degree-2159 topogra-

phy and validate the results against an independent spatial-

domain Newtonian integration that provides accurate refer-

ence values. In particular, we model 10 quantities: the grav-

itational potential, the three elements of the gravitational

vector and the six elements of the gravitational tensor. All

of them are evaluated up to degree 21,590 relying on 30

powers of the topography. As an additional outcome of the

experiment, these rather advanced settings allow us to pro-

vide further insights into the divergence effect of spherical

harmonics on planetary surfaces (e.g., Jekeli, 1981, 1983;

Moritz, 1980; Hu and Jekeli, 2015; Hirt et al, 2016; Hirt and

Kuhn, 2017; Rexer, 2017; Bucha et al, 2019b; Chen et al,

2019).

The paper is organized as follows. After a brief recapit-

ulation of the basic principles of global and cap-modiﬁed

spectral gravity forward modelling in Section 2, we proceed

with extending the cap-modiﬁed technique up to the second-

order derivatives of the gravitational potential in Section 3.

The derivations (Appendices A to C) are then validated in

Section 4 in a controlled environment using synthetic grav-

ity ﬁeld implied by the Earth’s degree-2159 topography. In

Section 5, we summarize the main conclusions of the pa-

per and discuss topics that may be relevant to future investi-

gations. The full statistical information from our validation

experiments and selected visualisations are provided in the

Electronic Supplementary Materials (ESM) to this paper.

2 Global and cap-modiﬁed spectral gravity forward

modelling

Let the shape of the gravitating topographic masses be given

by topographic heights ˆ

Hmeasured in the radial direction

from a reference sphere Ωhaving the radius R(spherical ar-

rangement of the topographic masses). Next, the topographic

heights ˆ

Hare transformed into a dimensionless topographic

height function

H(ϕ,λ) = ˆ

H(ϕ,λ)

R,(1)

which can be approximated by a ﬁnite surface spherical har-

monic expansion

H(ϕ,λ) =

nmax

∑

n=0

n

∑

m=−n

¯

Hnm ¯

Ynm(ϕ,λ).(2)

Here, (ϕ,λ)are the spherical latitude and longitude, respec-

tively, ¯

Ynm(ϕ,λ)are the fully normalized spherical harmonic

functions of degree nand order m(e.g., Heiskanen and Moritz,

1967), ¯

Hnm are the spherical harmonic coefﬁcients of the to-

pographic height function and nmax is the maximum degree

of the expansion.

Assuming a constant mass density ρ, spectral gravity

forward modelling approximates the implied gravitational

potential Vby a solid spherical harmonic expansion of the

form (e.g., Balmino, 1994; Wieczorek and Phillips, 1998)

V(r,ϕ,λ) = 2πGρR2

pmax

∑

p=1

p×nmax

∑

n=0

Snp (r)

×

n

∑

m=−n

¯

Hnmp ¯

Ynm(ϕ,λ),

(3)

where ris the spherical radius of the evaluation point, Gis

the gravitational constant, pis the integer power of the to-

pography, ¯

Hnmp are the fully normalized spherical harmonic

Cap integration in spectral gravity forward modelling up to the full gravity tensor 3

coefﬁcients of the pth power of the topographic height func-

tion,

Hp(ϕ,λ) = ˆ

H(ϕ,λ)

Rp

=

p×nmax

∑

n=0

n

∑

m=−n

¯

Hnmp ¯

Ynm(ϕ,λ),

(4)

and, ﬁnally, the term Snp(r)is given as

Snp (r) = 2

2n+1

p

∏

i=1

(n+4−i)

p!(n+3)R

rn+1

.(5)

For the explanation of the maximum degree p×nmax in Eq. (4),

see, for instance, Freeden and Schneider (1998), Hirt and

Kuhn (2014) or Bucha et al (2019a). Note that both nmax

and pmax are generally inﬁnite in case of real-world objects,

but are truncated here for practical reasons. The convergence

of the inﬁnite series in Eq. (3) is guaranteed for evalua-

tion points satisfying the condition r>max(R+ˆ

H(ϕ,λ)).

Otherwise, the inﬁnite series may converge or diverge (e.g.,

Rummel et al, 1988; Wieczorek and Phillips, 1998; Balmino,

1994; Hirt and Kuhn, 2017; Bucha et al, 2019a,b).

The gravitational potential Vfrom Eq. (3) is induced

by topographic masses all around the globe. To restrict the

integration from the whole sphere to a spherical cap, cap-

modiﬁed gravity forward modelling introduced by Bucha

et al (2019a) can be employed. With ψ0being the spheri-

cal radius of the cap, this leads to

Vj(r,ϕ,λ) = 2πGρR2

pmax

∑

p=1

p×nmax

∑

n=0

Qj

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp ¯

Ynm(ϕ,λ),

(6)

where the variable j={‘In’,‘Out’}denotes either near-zone

effects (inside-cap integration, j=‘In’) or far-zone effects

(outside-cap integration, j=‘Out’) and the symbol Qj

np (r,ψ0)

stands for Molodensky’s truncation coefﬁcients, which are

deﬁned in Appendix A of Bucha et al (2019a). For the nu-

merical evaluation of the Qj

np (r,ψ0)coefﬁcients and their

radial derivatives, either inﬁnite spectral relations or recur-

rence relations with a ﬁxed number of terms can be used (cf.

Appendices B, C and D of Bucha et al, 2019a). Needless to

say, it holds that

VIn(r,ϕ,λ) +VOut (r,ϕ,λ) = V(r,ϕ,λ).(7)

3 Extension of cap-modiﬁed spectral gravity forward

modelling up to the second-order potential derivatives

in LNOF

Throughout the paper, the directional derivatives of the grav-

itational potential are expressed in LNOF, which is a right-

handed orthogonal coordinate system, whose origin is at the

evaluation point P(r,ϕ,λ)and its axes are deﬁned as fol-

lows: the x-axis points to the north, the y-axis points to the

west and the z-axis points radially outwards.

3.1 First-order potential derivatives in LNOF

Applying the gradient operator expressed in LNOF to Eq. (6),

the gravitational vector induced by the near- and far-zone to-

pographic masses is given as (Appendix A)

gj(r,ϕ,λ) = ∇Vj(r,ϕ,λ) =

Vx,j(r,ϕ,λ)

Vy,j(r,ϕ,λ)

Vz,j(r,ϕ,λ)

,(8)

where

Vx,j(r,ϕ,λ) = −2πGρR2

pmax

∑

p=1

p×nmax

∑

n=1

Q1,1,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp

∂¯

Ynm(ϕ,λ)

∂ ϕ ,

(9)

Vy,j(r,ϕ,λ) = 2πGρR2

cosϕ

pmax

∑

p=1

p×nmax

∑

n=1

Q1,1,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp

∂¯

Ynm(ϕ,λ)

∂ λ

(10)

and

Vz,j(r,ϕ,λ) = 2πGρR2

pmax

∑

p=1

p×nmax

∑

n=0

Q1,0,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp ¯

Ynm(ϕ,λ).

(11)

The newly introduced truncation coefﬁcients, Q1,0,j

np (r,ψ0)

and Q1,1,j

np (r,ψ0), are deﬁned in Eqs. (45) and (54) of Ap-

pendix A. The ﬁrst superscript next to Q(here 1) indicates

that the truncation coefﬁcients relate to the ﬁrst-order deriva-

tives of the topographic potential and the second superscript

(here either 0 or 1) stands for the order of the derivative with

respect to the spherical distance ψ(cf. Eq. 38). The practical

evaluation of these coefﬁcients via inﬁnite series and closed

relations is discussed in Appendices A.1 and A.2, respec-

tively. From (59) and (60), it follows that Q1,0,j

np (r,ψ0)and

Q1,1,j

np (r,ψ0)are related to Qj

np (r,ψ0)from Eq. (6). Worth

noting is that two types of truncation coefﬁcients are used

to compute three elements the gravitational vector. Finally,

Eqs. (9) and (10) are singular at the poles. As a remedy, the

strategy by, for instance, Petrovskaya and Vershkov (2006)

could be investigated to avoid the issue.

4 Bucha B., Hirt C. and Kuhn M.

3.2 Second-order potential derivatives in LNOF

Continuing the differentiation in LNOF, the near- and far-

zone effects on the gravitational tensor read (Appendix B)

Vj(r,ϕ,λ) = ∇⊗∇Vj(r,ϕ,λ)

=

Vxx,j(r,ϕ,λ)Vxy,j(r,ϕ,λ)Vxz,j(r,ϕ,λ)

Vyx,j(r,ϕ,λ)Vyy,j(r,ϕ,λ)Vyz,j(r,ϕ,λ)

Vzx,j(r,ϕ,λ)Vzy,j(r,ϕ,λ)Vzz,j(r,ϕ,λ)

,

(12)

where

Vxx,j(r,ϕ,λ) = Vxx,j

1(r,ϕ,λ) + Vxx,j

2(r,ϕ,λ)

+Vxx,j

3(r,ϕ,λ),(13)

Vxx,j

1(r,ϕ,λ) = −πGρR2

pmax

∑

p=1

p×nmax

∑

n=0

Q2,0,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp ¯

Ynm(ϕ,λ),

(14)

Vxx,j

2(r,ϕ,λ) = 2πGρR2

pmax

∑

p=1

p×nmax

∑

n=2

n(n+1)Q2,2,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp ¯

Ynm(ϕ,λ),

(15)

Vxx,j

3(r,ϕ,λ) = 4πGρR2

pmax

∑

p=1

p×nmax

∑

n=2

Q2,2,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp

∂2¯

Ynm(ϕ,λ)

∂ ϕ 2,

(16)

Vxy,j(r,ϕ,λ) = Vxy,j

1(r,ϕ,λ) + Vxy,j

2(r,ϕ,λ),(17)

Vxy,j

1(r,ϕ,λ) = −4πGρR2

cosϕtan ϕ

pmax

∑

p=1

p×nmax

∑

n=2

Q2,2,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp

∂¯

Ynm(ϕ,λ)

∂ λ ,

(18)

Vxy,j

2(r,ϕ,λ) = −4πGρR2

cosϕ

pmax

∑

p=1

p×nmax

∑

n=2

Q2,2,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp

∂2¯

Ynm(ϕ,λ)

∂ λ ∂ ϕ ,

(19)

Vxz,j(r,ϕ,λ) = 2πGρR2

pmax

∑

p=1

p×nmax

∑

n=1

Q2,1,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp

¯

Ynm(ϕ,λ)

∂ ϕ ,

(20)

Vyy,j(r,ϕ,λ) = Vxx,j

1(r,ϕ,λ)−Vxx,j

2(r,ϕ,λ)

−Vxx,j

3(r,ϕ,λ),(21)

Vyz,j(r,ϕ,λ) = −2πGρR2

cosϕ

pmax

∑

p=1

p×nmax

∑

n=1

Q2,1,j

np (r,ψ0)

×

n

∑

m=−n

¯

Hnmp

∂¯

Ynm(ϕ,λ)

∂ λ ,

(22)

Vzz,j(r,ϕ,λ) = −2Vxx,j

1(r,ϕ,λ).(23)

Since the gravitational tensor is symmetric, it holds that

Vxy(r,ϕ,λ) = Vyx(r,ϕ,λ), etc. The truncation coefﬁcients

Q2,0,j

np (r,ψ0),Q2,1,j

np (r,ψ0)and Q2,2,j

np (r,ψ0)are deﬁned in Eqs. (78)

and (85) and formulae suitable for their practical evaluation

are discussed in Appendices B.1 and B.2. Note that only

three groups of truncation coefﬁcients enters the evaluation

of six unique elements of the gravitational tensor. Similarly

as in the previous section, these truncation coefﬁcients are

related to Qj

np (r,ψ0)via Eqs. (90), (91) and (92), and singu-

lar expressions occur as well (Eqs. 16, 18, 19, 20, 22).

3.3 Efﬁcient spherical harmonic synthesis at grids residing

on the irregular Earth’s surface

From the numerical point of view, Eqs. (6), (8) and (12)

are computationally intensive to evaluate at densely spaced

grids that refer to an irregular surface (e.g., the Earth’s to-

pography as in this study). This is caused by the altitude-

dependency of the truncation coefﬁcients, implying that dif-

ferent coefﬁcients are needed for points with different ele-

vations. When further combined with the numerical issues

associated with the evaluation of Qj

np (r,ψ0)(cf. Bucha et al,

2019a) as well as with ultra-high-degree spherical harmonic

expansions (say, beyond degree 10,800), the direct point-

wise evaluation does not appear to be currently possible (even

in case of a few hundreds of computation points).

To overcome these difﬁculties, Bucha et al (2019a) pro-

posed to apply the gradient approach for spherical harmonic

synthesis at regular grids residing on irregular surfaces (Holmes,

2003; Balmino et al, 2012; Hirt, 2012). This technique relies

on i) an analytical upward/downward continuation from a

regular surface to the irregular surface using a Taylor series

Cap integration in spectral gravity forward modelling up to the full gravity tensor 5

and ii) numerically efﬁcient FFT-based algorithms for spher-

ical harmonic synthesis on the regular surface (e.g., sphere

or ellipsoid of revolution). In the case of cap-modiﬁed spec-

tral modelling, this means that some tens of successive radial

derivatives of Eqs. (6), (8) and (12) need to be evaluated for

the continuation process (cf. Eqs. 9 and 10 of Bucha et al

2019a). Since the truncation coefﬁcients are the only radius-

dependent terms in these relations, we provide in Appen-

dices A and B formulae for an arbitrary radial derivative of

all the truncation coefﬁcients from Eqs. (9) – (23). More

speciﬁcally, we provide spectral relations (Sections A.3 and

B.3) and closed relations with a ﬁxed number of terms (A.4

and B.4) for near- and far-zone effects and i) an arbitrary

radius r, ii) an arbitrary harmonic degree n, iii) an arbitrary

topography power pand iv) an arbitrary order of the radial

derivative k. For the sake of brevity, the formulae resulting

from the application of the gradient approach to Eqs. (6) –

(23) are omitted here, but can readily be obtained analo-

gously to Eqs. (9) and (10) of Bucha et al (2019a).

Finally, Eqs. (6) – (23) can be rewritten such that only a

single spherical harmonic synthesis is needed for their eval-

uation instead of repeating it pmax times, each time with a

different maximum degree p×nmax . Taking Eq. (6) as an

example, all these relations can be rewritten into a single

spherical harmonic synthesis,

Vj(r,ϕ,λ) = 2πGρR2N

∑

n=0

n

∑

m=−n

¯

Vj

nm(r,ψ0)¯

Ynm(ϕ,λ),(24)

where Nis the maximum degree of the synthesis, say, N=

pmax ×nmax, and the coefﬁcients ¯

Vj

nm(r,ψ0)can be prepared

prior to the synthesis via

¯

Vj

nm(r,ψ0) =

pmax

∑

p=1

Qj

np (r,ψ0)¯

Hnmp ,(25)

after realizing that (cf. Eq. 53)

¯

Hnmp =0 for n>p×nmax .(26)

Furthermore, when combining Eq. (24) with the gradient ap-

proach, we found it efﬁcient to compute the ¯

Vj

nm(r,ψ0)coef-

ﬁcients together with their radial derivatives (necessary for

the continuation process) beforehand and store all of them in

RAM during the entire synthesis. Although this may require

several tens of GBs of RAM, which is certainly true for the

synthesis up to degree 21,590 as in our numerical study (cf.

Section 4), the gain in computational efﬁciency may easily

outperform costs associated with that amount of RAM. In

other words, the time-consuming evaluation of ultra-high-

degree fully-normalized Legendre functions needs to be per-

formed only once per synthesis which is a substantial com-

putational acceleration in case of high values of nmax,pmax

and N, say, nmax =2159, pmax =30 and N=21,590 as in

our numerical study.

4 Numerical experiments

4.1 Truncation coefﬁcients

In this section, we numerically examine the three newly de-

rived groups of truncation coefﬁcients: Q1,1,j

np (r,ψ0),Q2,1,j

np (r,ψ0)

and Q2,2,j

np (r,ψ0). The coefﬁcients Qj

np (r,ψ0)and their higher-

order radial derivatives (hence, including Q1,0,j

np (r,ψ0),Q2,0,j

np (r,ψ0),

cf. Eqs. 59 and 90, respectively) were discussed in detail in

Bucha et al (2019a). In this experiment, we designed pa-

rameters of the truncation coefﬁcients such that a high ac-

curacy level could be achieved later in the validation of the

cap-modiﬁed spectral technique (Section 4.2). By high ac-

curacy, we mean ∼0.001 m2s−2,∼µGal and ∼E for the

gravitational potential and the components of the gravita-

tional vector and tensor, respectively. The following input

parameters are studied here:

•n=0,...,21600 (spherical harmonic degree),

•p=1,...,30 (topography power),

•k=0,...,40 (order of the radial derivative),

•j={‘In’,‘Out’}(inside- and outside-cap integration),

•R=6,378.137 km (radius of the reference sphere),

•r=6,378.137 km+7 km (radius of the evaluation sphere

to be used as an auxiliary sphere in the gradient ap-

proach),

•ψ0=100 km/R≈0.90◦(integration radius).

Note that the evaluation radius ris chosen such that the

evaluation sphere passes above all the gravitating masses.

This is done in order to avoid possible issues with the slow

convergence of the gradient approach in Section 4.2 (see

Appendix B of Bucha et al, 2019b). The integration radius

ψ0, separating the inside- and outside-cap masses, is chosen

as 100 km which seems to be a reasonable choice, for in-

stance, for future RTM applications that utilize topography

expanded to degree ∼2159.

For the numerical evaluation of Q1,1,j

np (r,ψ0),Q2,1,j

np (r,ψ0)

and Q2,2,j

np (r,ψ0), we rely in this work on the closed relations

from Appendices A.2, A.4, B.2 and B.4 (Eqs. 66, 98 and 99).

This because once the coefﬁcients Qj

np (r,ψ0)from Eq. (6)

and their radial derivatives are computed, which is perhaps

the most difﬁcult part discussed in Bucha et al (2019a), the

truncation coefﬁcients introduced in this paper (Q1,1,j

np (r,ψ0),

Q2,1,j

np (r,ψ0)and Q2,2,j

np (r,ψ0)) and their radial derivatives

can be obtained without any substantial difﬁculties. Opposed

to this, the spectral relations from Appendices A.1, A.3, B.1

and B.3 are here considered as more time-consuming, as

they involve i) inﬁnite sums that need to be truncated at

ultra-high degrees (e.g., 100,000 in the study of Bucha et al,

2019a) and ii) integrals of products of two ultra-high-degree

Legendre functions over a restricted domain.

Following Bucha et al (2019a), we evaluate the Qj

np (r,ψ0)

coefﬁcients and their radial derivatives through recurrence

6 Bucha B., Hirt C. and Kuhn M.

relations (cf. Bucha et al, 2019a) with 256 signiﬁcant dig-

its. This rather large number of digits is employed in or-

der to mitigate numerical inaccuracies that are associated

with the recurrence relations to evaluate the truncation co-

efﬁcients (ibid.). The computation was conduced in Mat-

lab (www.mathworks.com) using the ADVANPIX toolbox

(www.advanpix.com), the latter of which enables to extend

the number of signiﬁcant digits.

Having the Qj

np (r,ψ0)coefﬁcients and their radial deriva-

tives, we evaluated the coefﬁcients Q1,1,j

np (r,ψ0),Q2,1,j

np (r,ψ0)

and Q2,2,j

np (r,ψ0)and their radial derivatives. The entire com-

putation was performed using 256 signiﬁcant digits and was

followed by an indirect accuracy check based on Appendix C.

Using 256 signiﬁcant digits, the validation revealed that the

worst agreements for the left-hand and the right-hand sides

of Eqs. (107), (108) and (109) were, respectively, 28, 24 and

28 correct digits for the studied values of n,p,k,R,rand

ψ0(see the ﬁrst paragraph of this section). To this end, we

utilized the measure

δA=

A−Areference

Areference

,(27)

where Areference represents the right-hand side of Eqs. (107),

(108) and (109) and Astands for the left-hand sides of the

same equations.

After this, the truncation coefﬁcients were converted into

double precision, ensuring an excellent accuracy down to the

15th or 16th digit.

Fig. 1, showing the dependence of Q1,1,j

np (r,ψ0),Q2,1,j

np (r,ψ0)

and Q2,2,j

np (r,ψ0)on harmonic degree nand topography power

p, indicates some important properties of near-zone coefﬁ-

cients (left panels in Fig. 1).

•The magnitude of the truncation coefﬁcients grows with

increasing p(cf. the normalization factors in the sub-

plots of Fig. 1). This could be somewhat anticipated based

on the previous experiments with Qj

np (r,ψ0)and their

high-order radial derivatives (Bucha et al, 2019a).

•Contrary to Bucha et al (2019a), where the QIn

np (r,ψ0)

coefﬁcients exhibit a single-wave pattern for p≥2, the

curves for the near-zone coefﬁcients Q1,1,In

np (r,ψ0),Q2,1,In

np (r,ψ0)

and Q2,2,In

np (r,ψ0)show multiple short-wavelength waves

after, say p≈12.

•Beyond p≈12, the near-zone coefﬁcients of lower har-

monic degrees are of higher magnitudes and dominate

over high-degree coefﬁcients (in a relative sense).

•Within the same type of near-zone coefﬁcients, very sim-

ilar curves are seen beyond power p≈12 (up to their

signs and the normalization factors, which change sub-

stantially).

These observations imply that the near-zone gravity effects

will feature strong spatial variations and high maximum de-

grees will be needed in Section 4.2 for their accurate evalu-

ation.

On the other hand, far-zone coefﬁcients (the right panels

in Fig. 1)

•show a much reduced variation with respect to harmonic

degree (note the different horizontal axes for left and

right panels in Fig. 1),

•take the largest values (in a relative sense) in low spectral

bands, and

•similarly as the near-zone coefﬁcients, grow in magni-

tude as the topography power pincreases (cf. the nor-

malization factors).

Translated into gravity effects, the ﬁrst and the second items

conﬁrm the known fact that the far-zone gravity effects are

mostly of long-wavelength character, and thus can be accu-

rately evaluated with much lower maximum harmonic de-

grees, provided that ψ0is large enough.

In Fig. 2, we show the newly derived truncation coefﬁ-

cients for a ﬁxed topography power p=1 and varying order

of the radial derivative k. It reveals that the maximum magni-

tude of the near-zone coefﬁcients moves towards higher har-

monic degrees as the order of the derivative grows. Opposed

to this, the far-zone coefﬁcients and their radial derivatives

take the largest values in the lower part of the spectrum (say,

up to degree 400).

4.1.1 Errors of the indirect check performed in double

precision

Despite the high accuracy achieved in the computation of

the truncation coefﬁcients in the previous section (∼15 – 16

digits after conversion into double precision), a catastrophic

cancellation may be encountered when evaluating the left-

hand side of Eqs. (104), (105), (107), (108), (109) in double

precision. Fig. 3 shows that the sum of near- and far-zone co-

efﬁcients in double precision may not necessarily yield the

correct value, even when the coefﬁcients are accurate down

to the 16th digit. This is because for some speciﬁc combi-

nations of n,pand k, the near- and far-zone coefﬁcients are

equal up to many digits (even up to the 16th and beyond)

but are of opposite signs. As a result, summing them in dou-

ble precision may produce, for instance, zero output values,

which is apparently an incorrect result when considering the

right-hand side of these equations. Taking Q2,2,j

np (r,ψ0)and

p=20 from Fig. 3 as an example, this procedure yields zero

correct digits up to harmonic degree ∼2500, after which the

accuracy improves. For p=30, the catastrophic cancellation

is observed over the entire studied interval of harmonic de-

grees (0 - 21,600). These errors then massively deteriorate

the accuracy of spherical harmonic synthesis, in which the

coefﬁcients are involved. In our numerical study from Sec-

tion 4.2, prone to these issues are especially the newly de-

rived truncation coefﬁcients Q1,1,j

np (r,ψ0),Q2,1,j

np (r,ψ0)and

Cap integration in spectral gravity forward modelling up to the full gravity tensor 7

-1

-0.5

0

0.5

1

Qnp

1,1,j (Normalized)

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Qnp

2,1,j (Normalized)

-1

-0.5

0

0.5

1

0 5000 10000 15000 20000

Spherical harmonic degree

-1

-0.5

0

0.5

1

Qnp

2,2,j (Normalized)

0 200 400 600 800 1000

Spherical harmonic degree

-1

-0.5

0

0.5

1

p = 1 p = 5 p = 10 p = 15 p = 20 p = 25 p = 30

Near zone

Norm. factors from 1.1e-09 (p=1)

to 1.1e+97 (p=30)

Far zone

Norm. factors from 8.5e-08 (p=1)

to 1.1e+97 (p=30)

Norm. factors from 2.2e-14 (p=1)

to 3.0e+93 (p=30)

Norm. factors from 2.7e-14 (p=1)

to 3.0e+93 (p=30)

Norm. factors from 5.9e-17 (p=1)

to 6.9e+90 (p=30)

Norm. factors from 4.4e-15 (p=1)

to 6.9e+90 (p=30)

Fig. 1 Normalized truncation coefﬁcients Q1,1,j

np (r,ψ0)(upper row), Q2,1,j

np (r,ψ0)(middle row) and Q2,2,j

np (r,ψ0)(bottom row) evaluated for R=

6,378,137 m, r=6,378,137 m +7000 m, ψ0=100 km/R, ﬁxed k=0 and varying pas a function of the harmonic degree. For visualization

purposes, the coefﬁcients are normalized by the maximum of their absolute value from the depicted interval. The ranges of the normalization

factors are shown in the plots. Note that the curves representing the near-zone coefﬁcients are overlapped at this scale for pbeyond, say, 12. Also

note that the far-zone coefﬁcients are depicted only up to degree 1000, because they show much reduced variation. The values were prepared in

Matlab with 256 digits using the ADVANPIX toolbox

Q2,2,j

np (r,ψ0), implying that a decreased numerical accuracy

can be expected for Vx,j,Vy,j,Vxx,j,Vxy,j,Vxz,j,Vyy,j,Vyz,j

when n,pand kexceed some critical values, which will be

determined in the next section.

To overcome this, extended number of signiﬁcant dig-

its could be employed also in the entire synthesis in cap-

modiﬁed spectral gravity forward modelling. However, for

our multiple ultra-high-degree expansions from Sections 2

and 3, spherical harmonic synthesis at millions of points

with, say, 256 digits is far beyond our current computational

capabilities. An alternative approach would be to derive nu-

merically more efﬁcient formulae, but this is left for future

work.

In the numerical experiments presented in the next sec-

tion, we therefore study the effect of the maximum topog-

raphy power on the ﬁnal results by using various values of

pmax (5, 10, 15, 20 and 30) and N(nmax, 2 nmax ,..., 10 nmax

with nmax =2159) (cf. Eq. 24). In all computations, the trun-

cation coefﬁcients are stored in double precision (16 digits)

and the entire cap-modiﬁed spectral gravity forward mod-

8 Bucha B., Hirt C. and Kuhn M.

-1

-0.5

0

0.5

1

Qnp

1,1,j (Normalized)

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

Qnp

2,1,j (Normalized)

-1

-0.5

0

0.5

1

0 5000 10000 15000 20000

Spherical harmonic degree

-1

-0.5

0

0.5

1

Qnp

2,2,j (Normalized)

0 200 400 600 800 1000

Spherical harmonic degree

-1

-0.5

0

0.5

1

k = 0 k = 5 k = 10 k = 15 k = 20 k = 25 k = 30 k = 35 k = 40

Near zone

Norm. factors from 1.1e-09 (k=0) to 5.7e-121 (k=40)

Far zone

Norm. factors from 1.0e-07 (k=0)

to 1.2e-162 (k=40)

Norm. factors from 2.2e-14 (k=0) to 1.9e-123 (k=40) Norm. factors from 2.7e-14 (k=0)

to 4.5e-167 (k=40)

Norm. factors from 5.9e-17 (k=0) to 4.5e-128 (k=40) Norm. factors from 4.8e-15 (k=0)

to 9.8e-169 (k=40)

Fig. 2 The same as Fig. 1, but for a ﬁxed p=1 and varying k

elling, that is, the spherical harmonic analysis and the syn-

thesis, are performed in double precision.

4.2 Validation of cap-modiﬁed spectral forward modelling

using the Earth’s degree-2159 topography

To check the correctness of the newly derived equations from

Section 3 and Appendices A and B, we designed a numer-

ical test, in which cap-modiﬁed spectral modelling is vali-

dated against an independent spatial-domain Newtonian in-

tegration as a reference. In the experiment, the gravity ﬁeld

is implied by RET2014 (Hirt and Rexer, 2015), which is a

model of the Earth’s topography based on a surface spherical

harmonic expansion up to degree 10,800. The abbreviation

RET stands for the rock-equivalent topography and means

that several mass layers with different densities (here, rock,

water and ice) were condensed into a single layer equivalent

to topographic rock, here with the density ρ=2670 kg m−3.

For our study, the topography was synthesized up to de-

gree nmax =2159, which approximately corresponds to the

5 arc-min resolution of global gravity ﬁeld models such as

EGM2008 (Pavlis et al, 2012). The implied gravity ﬁeld is

here modelled up to degree N=21,590 (cf. Eq. 24), being

the tenth multiple of the topography bandwidth, while em-

ploying up to 30 powers of the input topography (pmax =

30). This is necessary, because a band-limited topography

generates a full-banded gravity ﬁeld (e.g., Balmino, 1994;

Balmino et al, 2012; Hirt and Kuhn, 2014).

Cap integration in spectral gravity forward modelling up to the full gravity tensor 9

Fig. 3 Indirect check (in double precision) on the numerical accuracy

of Q2,2,j

np (r,ψ0)(Eqs. 27 and 109) for k=0 and varying harmonic de-

gree nand topography power p. Contrary to the tests from Section 4.1,

here the two terms on the left-hand and the entire right-hand side of

Eq. (109) were evaluated with 256 signiﬁcant digits (accurate up to 28

or more digits) but then each term was separately converted into dou-

ble precision. After the conversion, the near- and far-zone coefﬁcients

were summed in double precision and Eq. (27) was used to compute

δA. The base-10 logarithm of δAis shown in the plot. The values of

−16 represent the maximum 16-digit accuracy, while 0 and larger val-

ues indicate zero accurate digits

Importantly, the settings of the experiment allow us not

only to verify the correctness of the equations from Sec-

tion 3, but also to study the divergence effect of cap-modiﬁed

spherical harmonic series. While global spectral gravity for-

ward modelling from Section 2 has already been examined

for the divergence effect (Hirt et al, 2016; Hirt and Kuhn,

2017; Rexer, 2017), showing its presence for the Earth and

Moon when the maximum degree of the spherical harmonic

series is large enough, this is the ﬁrst experiment of its kind

for cap-modiﬁed spectral modelling.

Spatially, the validation is performed with near-global

coverage (within the [−80◦,80◦]latitude range) for near-

zone gravity effects and, out of necessity, only regionally

for the far-zone effects. The latter is done because compu-

tational demands associated with delivering reference val-

ues via spatial-domain Newtonian integration are enormous

when working on a global scale with such high resolutions

as in this study. Nevertheless, we evaluated the far-zone ef-

fects over two challenging computational areas of the Hi-

malayas (latitude: [20.05◦,44.96◦], longitude: [70.04◦,104.96◦])

and Kiribati ([−4.95◦,4.96◦],[185.04◦,199.96◦]), which can

reasonably well serve as a benchmark for the prediction over

the entire Earth’s surface as long as the values of nmax,pmax

and Nare similar to ours. These regions were selected, be-

cause they seem to be the most prone to the divergence effect

as shown in Hirt et al (2016).

The validation is performed at points arranged in a 5 arc-

min equiangular grid which approximately corresponds to

the spatial resolution of the degree-2159 topography. The

points are distributed globally, excluding 10◦polar caps (near-

global validation). The radial component of their position is

twofold,

•either 1 m above the RET2014 topography Hif H>0 m

or 1 m above the reference sphere R=6,378.137 km if

H≤0 m, brieﬂy referred to as 1 m above the Earth’s

surface (Sections 4.2.3 and 4.2.4), and

•on a Brillouin sphere with the radius R=6,378.137 km+

7 km (a sphere being completely outside the masses,

Sans`

o and Sideris, 2013) (Section 4.2.5).

The former set-up enables to identify the divergence effect if

present. The latter case, being free of this error by deﬁnition,

allows us to discriminate a possible divergence effect from

errors associated with the evaluation of the truncation coef-

ﬁcients (Section 4.1.1). Note that our spatial-domain New-

tonian integration software (Section 4.2.1) is free of the di-

vergence of spherical harmonics and is capable of delivering

gravity effects within our target accuracy (cf. Section 4.1) as

indirectly shown in Sections 4.2.3, 4.2.5 and 4.2.4. Beyond

doubts, the reference values can therefore serve as a bench-

mark for identifying the divergence effect, similarly as in

Hirt et al (2016) or Hirt and Kuhn (2017).

4.2.1 Spatial-domain Newtonian integration

To obtain the reference gravity values, we use spatial-domain

Newtonian integration software (cf. Bucha et al, 2016) that

combines i) the polyhedron-based routine developed by Tsoulis

(2012) and ii) tesseroids (Grombein et al, 2013). The inte-

gration radius ψ0was set to a spherical distance of 100 km/

6,378.137 km ≈0.90◦.

Near-zone gravity effects. For accurate gravity forward

modelling, the integration domain (0◦≤ψ≤ψ0) is subdi-

vided into an inner zone (ψ≤0.25◦), where we rely on poly-

hedral modelling, and an outer zone (0.25◦<ψ≤ψ0), be-

ing modelled by tesseroids (note that in our case the singu-

larity of the integral kernels of tesseroids does not cause any

deterioration that would be larger than our target accuracy

speciﬁed in Section 4.1, see also Section 4.3.4 of Bucha et al,

2016). Generally, polyhedral gravity forward modelling is

slower but more accurate than the tesseroid-based one (e.g.,

Bucha et al, 2016), so is here used to forward model the cru-

cial innermost masses, and vice versa. The radius of 0.25◦

that separates the two zones was empirically found to be

a reasonable compromise between the accuracy and com-

putational costs. The RET2014 topography was synthesized

globally at the spatial resolution of 10 arc-sec, thus with an

10 Bucha B., Hirt C. and Kuhn M.

oversampling factor of ∼30. This grid represents the shape

of the topographic masses that were subsequently forward

modelled using our Newtonian integration software.

The statistics of the 10 obtained forward modelled quan-

tities (VIn,Vx,In,Vy,In ,Vz,In,Vxx,In ,Vxy,In,Vxz,In ,Vyy,In,

Vyz,In,Vzz,In ) are reported in Table 1. For the sake of brevity,

shown in Fig. 4 is only Vz,In , which was chosen as represen-

tative for visualization purposes, given that it is closely re-

lated to the widely used gravity anomalies and disturbances.

Note that while the near-zone effects are shown in Fig. 4

over two areas only, the Himalayas and Kiribati, they were

computed near-globally within the [−80◦,80◦]latitude limit

as already discussed. A complete picture of the near-zone

reference gravity effects is provided in Figs. S1 – S4 of

ESM.

Far-zone gravity effects. In case of far-zone effects, tesseroids

are used over the entire integration domain (ψ0≤ψ≤180◦).

This is permissible, given the attenuation of gravity signal

with increasing distance from the evaluation point. To de-

crease computation time, the resolution of the forward mod-

elled RET2014 topography is here lowered from 10 arc-sec

to 30 arc-sec (oversampling factor of 10) and forward mod-

elling is restricted to two areas, the Himalayas and Kiribati

(see Fig. 5 and Table 1). Nevertheless these areas represent

a worst-case scenario and are certainly challenging for ac-

curate gravity forward modelling both in the spatial and the

spectral domain. The reference far-zone gravity effects are

shown for each functional in Figs. S5 – S8 of ESM.

4.2.2 Cap-modiﬁed spectral gravity forward modelling

Next, we performed cap-modiﬁed spectral gravity forward

modelling (Eqs. 6 – 23) on the Earth’s topography via the

gradient approach (Section 3.3). For the analytical continua-

tion in the gradient approach, we use the radius 6,378,137 m+

7000 m (a sphere outside of all masses) and the Taylor series

truncated at kmax =40 (cf. Eq. 9 of Bucha et al, 2019a). The

maximum topography integer power is gradually set up to

pmax =30. The maximum harmonic degree N(cf. Eq. 24)

of the gravity effects varies from nmax to 10nmax , where

nmax =2159. The truncation coefﬁcients were evaluated us-

ing 256 signiﬁcant digits, then they were converted to double

precision (cf. Section 4.1) and, ﬁnally, the entire harmonic

synthesis was conduced in double precision.

Before the validation itself, we show in Fig. 6 dimen-

sionless degree variances (cf. Eq. 15 of Bucha et al, 2019a)

of the total near- and far-zone gravity signal Vz,jas well as

of the individual gravity contributions generated by the pth

power of the topography. The quantity Vz,jwas selected as

an example, as it is frequently employed in practice in the

form of gravity disturbances. Brieﬂy, both near- and far-zone

effects possess an important portion of the signal even well

beyond the maximum degree of the topography (cf. Hirt and

Kuhn, 2014; Hirt et al, 2016; Hirt and Kuhn, 2017; Bucha

et al, 2019a). Similarly as in Bucha et al (2019a), the de-

gree variances of the far-zone gravity effects exhibit a strong

arch-like pattern, which is caused by the Molodensky’s trun-

cation coefﬁcients. It is seen that the power of the near-zone

gravity signal is stronger and decays more slowly than its

far-zone counterpart which is consistent with the decay of

gravitational signal with distance form the source (near-zone

effects show generally more power in higher frequencies

than far-zone effects).

Note that after a certain degree, the curves in Fig. 6 start

to oscillate around the same level, e.g., ∼10−46 for near-

zone effects and p≥6 (see also the spectrum of the far-

zone effects beyond degree ∼15,000 that is shown by the

thick black line). This is caused by numerical issues asso-

ciated with the growing range of the ¯

Hnmp coefﬁcients (cf.

Eq. 4) with increasing p. More speciﬁcally, Fig. 7 shows that

the algorithms for spherical harmonic synthesis and analysis

that we used (the Gauss–Legendre quadrature from Sneeuw

1994 combined with fully-normalized Legendre functions

evaluated after Fukushima 2012) were not able to capture

such a wide range of magnitudes in double precision. Fur-

ther examples with similar numerical issues can be seen in

Hirt et al (2016) or Bucha et al (2019a). As a potential rem-

edy, the whole computation process could be performed in

quadruple precision. This was successfully tested with a degree-

360 topography, but it is not used in the ﬁnal computations

with the degree-2159 topography due to its signiﬁcantly longer

computation time. We also tried to normalize the input sig-

nal by a single constant factor but without success. Never-

theless, these inferior coefﬁcients can be ignored when the

magnitudes they produce are of negligible strengths in terms

of gravity (as it is in Fig. 6) or excluded when the numeri-

cal inaccuracies are ampliﬁed too much as would happen

for ultra-high harmonic degrees. Fig. 7 therefore shows that,

sooner or later, some different strategy may be required to

accurately recover the ¯

Hnmp coefﬁcients for high values of

n,m,nmax and p.

4.2.3 Near-zone gravity effects: validation 1 m above the

topography

In this validation, the near-zone gravity effects are computed

at a 5 arc-min equiangular grid with the radial position of

the evaluation points being either 1 m above the RET2014

topography Hif H>0 m or 1 m above the reference sphere

R=6,378.137 km if H≤0 m. The latter is done to avoid

computations inside the masses. In that case, our software

for the spatial-domain Newtonian integration would yield

(in agreement with potential theory) non-harmonic gravita-

tional potential and its derivatives, while the cap-modiﬁed

spectral technique, being based on a ﬁnite linear combina-

tion of harmonic functions (cf. Sections 2 and 3), gives nec-

Cap integration in spectral gravity forward modelling up to the full gravity tensor 11

80˚ 90˚ 100˚

30˚

40˚

−600 −500 −400 −300 −200 −100 0 100

−170˚ −165˚

−4˚

0˚

4˚

150 200 250 300 350 400 450

Fig. 4 Near-zone gravitational effects (mGal) implied by the Earth’s degree-2159 RET2014 topography in terms of Vz,In (mGal) over the Hi-

malayas (left panel) and Kiribati (right panel). The computation points are deﬁned by a 5 arc-min equiangular grid and reside 1 m above the

Earth’s topography (cf. Section 4.2.3). The values were obtained by a divergence-free spatial-domain Newtonian integration and will later serve

as a reference for the validation of cap-modiﬁed spectral gravity forward modelling. Note that the element Vz,In represents the positive ﬁrst-order

radial derivative of VIn which is why its sign is opposite with respect to Bucha et al (2019a), who worked with the negative derivative known as

the gravity disturbance. The short-scale wavy-like features that can be seen in the left panel, especially around the Himalayas, are caused by the

oscillating nature of topography that is expanded in surface spherical harmonics

80˚ 90˚ 100˚

30˚

40˚

70 80 90 100 110

−170˚ −165˚

−4˚

0˚

4˚

208 209 210 211 212

Fig. 5 The same as Fig. 4, but with far-zone gravitational effects (mGal) on Vz,In over the Himalayas (left panel) and Kiribati (right panel)

essarily a harmonic potential even inside the masses (an ana-

lytically downward continued external potential; e.g., Moritz

2010; Freeden and Gerhards 2013), where the true gravita-

tional potential is non-harmonic.

In Table S1 of ESM, we provide statistics of the vali-

dation (RMS of the discrepancies and the maximum of their

absolute values, here denoted as MAX). For each functional,

the obtained discrepancies are plotted in Figs. S9 and S10 of

ESM. In Fig. 8, we show the RMS values as a function of

the maximum harmonic degree Nfor various pmax . Several

conclusions can be drawn based on the validation.

•VIn (requires Qj

np (r,ψ0)): Within our target accuracy

(∼0.001 m2s−2), the cap-modiﬁed harmonic series for

the potential converges rather fast when compared with

the other quantities. This is not surprising, given that

the most signiﬁcant portion of the gravitational poten-

tial signal is contained within its low harmonics, here

up to degree ∼2159. The best agreement achieved with

the least effort is RMS =0.0053 m2s−2and MAX =

0.035 m2s−2and is reached with pmax =15 and N=

8636. When translated into geoid undulations, a sub-

millimetre RMS accuracy could be achieved in geoid

computations. Note that a slightly worse but fairly com-

12 Bucha B., Hirt C. and Kuhn M.

Table 1 Characteristics of the near- and far-zone reference gravity effects obtained from the spatial-domain Newtonian integration of the degree-

2159 RET2014 topography. The computation points are placed 1 m above the Earth’s topography (Sections 4.2.3 and 4.2.4). Note that while the

statistics for the near-zone effects are based on near-global datasets (5 arc-min equiangular grid within the [−80◦,80◦]latitude limits; 1920×4320

nodes), the far-zone statistics are based on two much smaller areas, the Himalayas (300×420 nodes) plus Kiribati (120 ×180 nodes), representing

∼1.8 % of grid nodes of the former one. Therefore, the statistics for the far-zone effects likely do not provide a complete global picture of the

signals. The abbreviation STD stands for the standard deviation

Quantity Unit Near-zone effects Far-zone effects

Min Max Mean STD Min Max Mean STD

Vm−2s−2−490.898 589.679 −139.061 176.223 −26273.726 −9819.973 −13835.625 5259.778

VxmGal −362.166 425.608 −0.356 25.157 −250.559 553.289 100.048 188.022

VymGal −364.518 368.921 −4.201e−6 24.394 −364.448 272.190 −14.718 109.172

VzmGal −713.520 642.281 140.271 179.917 60.147 211.948 106.818 44.871

Vxx E−338.470 287.899 7.386 16.628 −18.442 23.440 6.592 11.424

Vxy E−183.657 160.344 3.569e−3 7.609 −9.174 9.586 −1.492e−2 2.010

Vxz E−364.062 286.933 7.212e−2 15.679 −2.161 1.020 −0.235 0.445

Vyy E−403.187 318.051 7.385 16.223 −18.765 29.584 6.680 12.637

Vyz E−322.009 315.731 2.020e−4 15.391 −0.985 1.170 3.208e−2 0.261

Vzz E−361.698 659.755 −14.771 28.776 −49.187 36.360 −13.271 23.692

Fig. 6 Dimensionless degree variances (cf. Eq. 15 of Bucha et al, 2019a) of near-zone (up to ψ0≈0.90◦) and far-zone (beyond ψ0≈0.90◦) gravity

effects (Vz,j) shown as a function of integer power pof the Earth’s degree-2159 topography. The degree variances refer to a Brillouin sphere that

is outside of all masses with the radius R=6,378.137 km +7 km (the maximum elevation from RET2014 is ∼6.7 km for nmax =2159), where

spherical harmonic series converge. No divergence effect can therefore be seen in this ﬁgure as opposed to, for instance, Fig. 3.14 of Rexer (2017),

who employed global spectral gravity forward modelling (nmax =2160, pmax =50, N=21,600) to provide degree variances referring to the

reference sphere that is partially inside the gravitating masses, where spherical harmonic series may converge or diverge

parable sub-millimetre RMS accuracy is seen already

with pmax =5 and N=4318 (RMS =0.0054 m2s−2),

but at the cost of a worse MAX value (0.280 m2s−2). A

careful inspection of Fig. S9 of ESM, where the differ-

ences are plotted, reveals that divergence effect emerges

over the Himalayas. However, its magnitude is too low

to allow us draw reliable conclusions on the divergence

of spherical harmonics on the Earth’s surface. As can

be seen from Figs. S9 and S10, the divergence effect

also appears to be present in all other studied quanti-

ties, so this observation will not be repeated below. Two

exceptions that will be discussed are Vz,In and Vzz,In, for

which the magnitude of the divergence effect is sufﬁ-

ciently large to formulate reliable conclusions.

•Vx,In,Vy,In (both require Q1,1,j

np (r,ψ0)): For the two quan-

tities, the lowest RMS errors obtained are 4.1 and 2.7 µGal

and were achieved with pmax =10, N=10,795 and N=

12,954, respectively. Beyond these values, a massive de-

terioration is seen, worsening the RMS errors gradually

up the order of ∼1018 mGal (cf. Table S1 of ESM). How-

ever, this is not caused by the divergence effect of spher-

ical harmonics, but instead by the numerical issues re-

lated to the evaluation of Q1,1,j

np (r,ψ0)(cf. Section 4.1.1).

This conclusion will be conﬁrmed in Section 4.2.5.

•Vz,In (requires Q1,0,j

np (r,ψ0)): Using pmax =15 and N=

15,113, we achieved a 2 µGal RMS agreement, which is

our best result for this quantity. When further increasing

the two parameters, especially pmax , the accuracy can

slightly be improved over some regions, but the spheri-

Cap integration in spectral gravity forward modelling up to the full gravity tensor 13

Fig. 7 Dimensionless degree variances of the topographic height func-

tion and its ﬁrst 30 integer powers. The almost horizontal tails of the

curves indicate parts of the spectra that were not recovered accurately

via spherical harmonic analysis and synthesis in double precision

cal harmonic series starts to produce invalid results over

other parts of the Earth’s surface. This effect is well vis-

ible over the Himalayas (Fig. 9), where increasing pmax

results in decreased accuracy with artiﬁcial fan-like struc-

tures covering large portions of the Earth’s surface, sim-

ilarly as in the studies by Hirt et al (2016), Hirt and

Kuhn (2017) and Rexer (2017). The same feature, but of

a smaller magnitude, can also be seen over Kiribati when

pmax =30. In that case, an area from which the fan-line

structure appears to emerge is seen around the latitude

−1◦and the longitude −168◦. The enlarged discrepan-

cies are clearly reﬂected also in Fig. 8 when pmax =30.

At ﬁrst, the RMS error decreases with Ngrowing up to

10,795 as could be expected, but then the agreement

exacerbates from 7.2µGal up to 11.1µGal for N=

19,431, following by a slight improvement of 10.5µGal

for N=21,590. Even more obvious impact of the diver-

gence effect can be seen in terms of the MAX criterion

(cf. Table S1 of ESM), where a sudden degradation starts

already with pmax =15 and N=15,113. Supported by

these observations as well as by the studies of Hirt et al

(2016), Hirt and Kuhn (2017) and Rexer (2017), this be-

haviour is here concluded to be caused by the divergence

effect of spherical harmonics. The numerical issues as-

sociated with the evaluation of truncation coefﬁcients

from Section 4.1.1 are here rejected as the main cause

of these discrepancies. This is justiﬁed by experiments

that will be presented in Section 4.2.5, where RMS and

MAX are shown to drop by 1 and 3 orders of magnitude,

respectively, when evaluating Vz,In on a Brillouin sphere,

on which the series converges by deﬁnition. Neverthe-

less, even on the Earth’s surface expanded up to degree

2159, a 2 µGal near-global RMS value was achieved that

could be considered as satisfactory for current applica-

tions.

•Vxx,In,Vyy,In (both require Q2,0,j

np (r,ψ0)and Q2,2,j

np (r,ψ0)):

For both Vxx,In and Vyy,In , we observe RMS errors slightly

below 1 E. As already indicated by Fig. 3, increased

discrepancies can be expected for large pmax values (cf.

Fig. 3). Here, these numerical issues produced RMS er-

rors at the order of ∼1018 E for pmax =30. As will be

shown with the Vzz,In element, the errors associated with

the Q2,0,j

np (r,ψ0)coefﬁcients are negligible in this case

and the bulk of the discrepancies is due to the inaccura-

cies related to the Q2,2,j

np (r,ψ0)coefﬁcients.

•Vxy,In (requires Q2,2,j

np (r,ψ0)): Similarly as in the previ-

ous case, sub-E RMS errors were achieved, of which the

best one is 0.2 E for pmax =10 and N=8636. Again,

the pmax parameter should be chosen carefully, as highly

inaccurate results may be obtained (RMS errors at the

order of 1018 E) when pmax =20 and 30.

•Vxz,In,Vyz,In (both require Q2,1,j

np (r,ψ0)): The best RMS

agreement with the reference values, 0.079 E and 0.069 E,

respectively, was achieved with pmax =10 and N=10,795.

Similarly as with the other horizontal derivatives of the

gravitational potential that involve differentiation of Qj

np (r,ψ0)

with respect to ψ, spurious artefacts start to be clearly

detectable as soon as pmax is high enough (here beyond

pmax =10).

•Vzz,In (requires Q2,0,j

np (r,ψ0)): For this quantity, we achieved

1.07 E RMS error with pmax =10 and N=6477. As

no differentiation of Qj

np (r,ψ0)with respect to ψis in-

volved, we do not observe deterioration with increasing

pmax and/or N. Similarly as with Vz,In, the divergence

effect starts to dominate over the signal when pmax =30

and Nis larger than, say, 15,113.

4.2.4 Far-zone gravity effects: validation 1 m above the

topography

Here, the experiment from the previous section is repeated,

but this time we evaluate far-zone gravity effects. The dis-

crepancies are shown in Figs. S13 – S16 of ESM and the

statistics are reported in Table S2 (ESM). Brieﬂy, similar ac-

curacy was achieved as with the near-zone effects (RMS of

0.03 m s−2for the potential, 0.8 – 20 µGal for the elements

of the gravitational vector and 0.1 – 3 mE for the elements

of the gravitational tensor). In all cases except for V,Vxand

Vy, the statistics are more favourable, in some cases almost

by three orders of magnitude (e.g., Vzz). The decreased ac-

curacy in V,Vxand Vyby about one order of magnitude

could be explained by the fact that far-zone masses generate

in this case signal of larger magnitudes than the near-zone

masses (cf. Table 1) which then causes an overall lower ac-

curacy in the absolute sense. However, the relative accuracy

remains comparable with the near-zone effects. Also, note

14 Bucha B., Hirt C. and Kuhn M.

10-3

10-2

10-1

m2 s-2

VIn

10-3

10-2

10-1

100

mGal

Vx,In

10-3

10-2

10-1

100

mGal

Vy,In

10-3

10-2

10-1

100

mGal

Vz,In

10-1

100

101

E

Vxx,In

10-1

100

E

Vxy,In

10-2

10-1

100

101

E

Vxz,In

10-1

100

101

E

Vyy,In

2159 6477 10795 15113 19431

Max. spherical harmonic degree N

10-2

10-1

100

101

E

Vyz,In

2159 6477 10795 15113 19431

Max. spherical harmonic degree N

1

1.1

1.2

2

E

Vzz,In

pmax = 5 pmax = 10 pmax = 15 pmax = 20 pmax = 30

Fig. 8 RMS of discrepancies between the cap-modiﬁed spectral technique and the spatial-domain Newtonian integration in terms of near-zone

gravity effects evaluated 1 m above the Earth’s topography as a function of the maximum degree N=2159,4318,...,21590 with varying pmax

(cf. Eqs. 24 and 25). For Vx,In,Vy,In,Vxx,In ,Vxy,In,Vxz,In ,Vyy,In and Vyz,In, the RMS values signiﬁcantly exceed the upper limits in the vertical

axes when pmax ≥20, so are not shown in the plots. These inferior results are caused by the numerical issues associated with the evaluation of the

truncation coefﬁcients (cf. Section 4.1.1). Detailed statistics are reported in Table S1 of ESM

Cap integration in spectral gravity forward modelling up to the full gravity tensor 15

80˚ 90˚ 100˚

30˚

40˚

pmax = 30

−0.010 −0.005 0.000 0.005 0.010

−170˚ −165˚

−4˚

0˚

4˚

pmax = 30

30˚

40˚

pmax = 20

−4˚

0˚

4˚

pmax = 20

30˚

40˚

pmax = 15

−4˚

0˚

4˚

pmax = 15

Fig. 9 Near-zone effect differences Vz,In between cap-modiﬁed spectral gravity forward modelling and reference values over the Himalayas (left

column) and Kiribati (right column). While the pmax value varies from 15 to 30 in the cap-modiﬁed spectral technique, the maximum degree Nis

ﬁxed to 21,590 (cf. Eq. 24). The computation points are placed 1 m above the topography. Statistics based on the near-global discrepancies can be

found in Table S1 of ESM. Unit in mGal

16 Bucha B., Hirt C. and Kuhn M.

that the far-zone effects are here evaluated only over com-

plex areas of the Himalayas and Kiribati which, as we ex-

pect, may worsen the RMS value when compared with the

near-global evaluation from the previous section, which in-

cluded also ﬂat areas of the Earth’s surface.

As for the convergence rate, no signiﬁcant improvement

is observed after pmax =5 and N=2159 (cf. Table S2 of

ESM), both of which are the lowest values that we study

here. Spherical harmonic series therefore converge signif-

icantly faster for far-zone effects than for near-zone ones,

provided that the integration radius is large enough (here

ψ0≈0.9◦). This is within the expectations because of the

attenuation of short-scale signals with distance. As a result,

far-zone effects from the cap-modiﬁed spectral technique

may require to employ only a ﬁrst few powers of the to-

pography and the maximum degree may not need to be ex-

tended beyond the resolution of the input topography (or at

least substantially less than with the near-zone effects). This

greatly simpliﬁes the computations and appears to be an

analogy to spatial-domain gravity forward modelling, where

coarser grid resolutions are used to improve the computa-

tional speed when evaluating far-zone effects.

In Fig. 10, we show the discrepancies for Vz,Out as an

example. Importantly, despite the high values of pmax and

N(30 and 21,590, respectively), no divergence effect is vis-

ible as compared to the near-zone effects (the bottom row

of Fig. 9), and a sub-µGal accuracy was achieved (cf. Ta-

ble S2 of ESM). As an explanation, near-zone effects con-

tain more signal power in high harmonics (cf. Fig. 6) which,

in turn, may readily cause the divergence effect when evalu-

ating the series on the topography and pmax and Nare high

enough. Opposed to this, the signal power of far-zone ef-

fects is strongest in low and medium harmonics and rather

negligible in high-degree harmonics as already discussed.

Similarly as in the Bucha et al (2019a) study, we ob-

serve a longitudinal stripe pattern both in Figs. 9 and 10.

For now, we still do not have a satisfactory explanation, but

we expect that this is most likely caused by numerical in-

accuracies associated with the cap-modiﬁed spectral tech-

nique. These may include the computation of i) the ¯

Hnmp

coefﬁcients (cf. Section 4.2.2), ii) the truncation coefﬁcients

(Section 4.1.1) or iii) the spherical harmonic synthesis. Nev-

ertheless, the achieved accuracy still seems to be sufﬁciently

high for many current practical applications.

Finally, when pmax ≥15, the results for quantities with at

least one horizontal derivative start to deteriorate, indicating

that it is difﬁcult to reach a high accuracy in this case. Again,

this is caused by the numerical issues related to Q1,1,j

np (r,ψ0),

Q2,1,j

np (r,ψ0)and Q2,2,j

np (r,ψ0)(cf. Section 4.1).

4.2.5 Near-zone gravity effects: validation on a Brillouin

sphere

Here, we provide the results of the same experiment as in

Section 4.2.3 but with the evaluation points placed on a Bril-

louin sphere having a constant radius of RB=6,378.137 km+

7 km (the maximum elevation from the RET2014 model

is ∼6.7 km for nmax =2159). This radius ensures that all

evaluation points are located in a space, where the series in

Eqs. (6) – (23) converge uniformly and absolutely by def-

inition. Therefore, we assume that if the large discrepan-

cies present in Section 4.2.3 diminish, then they stem from

the divergence effect. Otherwise, if the large errors persist,

speciﬁcally for any of the quantities that involve at least one

horizontal derivative, they are assign to the known numeri-

cal issues discussed in Section 4.1.1.

Brieﬂy, it is seen from Fig. 11 that the convergence is

now signiﬁcantly faster than in Fig. 8 and the discrepan-

cies dropped for many of the quantities (cf. Table S3 and

Figs. S11 and S12 of ESM). Taking Vz,In with pmax =30

and N=21,590 as an example, the RMS and MAX values

decreased from 0.011 mGal and 5.0 mGal to 0.001 mGal

and 0.005 mGal, respectively. An improvement is seen also

for VIn and Vzz,In. This supports our conclusion that the in-

creased discrepancies observed in Fig. 9 are indeed caused

by the divergence effect.

In case of Vx,In,Vy,In ,Vxx,In,Vxy,In ,Vxz,In,Vyy,In and

Vyz,In, the enlarged discrepancies remain present when pmax ≥

15 which shows that these quantities should be computed ei-

ther with lower values of pmax and N(but still allowing for

a high overall accuracy, cf. Section 4.2.3)—or with an ex-

tended number of signiﬁcant digits in the synthesis.

Not shown here, but the evaluation of far-zone effects

on a Brillouin sphere is in agreement with the conclusions

drawn in this section.

4.2.6 Concluding remarks on the validation

First, it is obvious that our experiments with truncated spher-

ical harmonic series (Eqs. 6 – 23) cannot in principle reveal

whether the original inﬁnite series converge or diverge. This

is because what we actually deal with in practice is in fact a

ﬁnite series of real numbers, which necessarily converges to

a real number. When we speak about the divergence effect,

we refer to a series behaviour yielding for a certain max-

imum degree N1a worse approximation of the true value

than for some N2<N1. Note that this, however, differs from

the deﬁnition of divergent inﬁnite series. As a consequence,

even a series suffering from the divergence effect may in-

deed converge to the true value when extended up to inﬁnity

(for instance, this appears to be the case of the Taylor se-

ries in Figs. 5 and 9 of Balmino et al 2012 and Bucha et al

2019b, respectively). Nevertheless, for some applications in

Cap integration in spectral gravity forward modelling up to the full gravity tensor 17

80˚ 90˚ 100˚

30˚

40˚

−0.004 −0.002 0.000 0.002 0.004

−170˚ −165˚

−4˚

0˚

4˚

Fig. 10 Far-zone effect differences Vz,Out between cap-modiﬁed spectral gravity forward modelling and reference values over the Himalayas (left

column) and Kiribati (right column). In both cases, the pmax =30 and N=21,590 were used in the synthesis (cf. Eq. 24). The computation points

are placed 1 m above the topography. Statistics of the discrepancies can be found in Table S2 of ESM. Unit in mGal

physical geodesy, the divergence effect is not a desired be-

haviour and should be identiﬁed if present. In case of, for

instance, RTM study, it could lead to grossly invalid results

with respect to the observed terrestrial gravity. If the series in

Eqs. (6) – (23) are divergent when extended up to inﬁnity but

are able to represent the true value with a [...] high degree

of accuracy [...] (Gradshteyn and Ryzhik, 2007) after a suit-

able truncation, then they can be considered as asymptotic

(e.g., Sj¨

oberg, 1977; Moritz, 2003; Gradshteyn and Ryzhik,

2007; Sj¨

oberg and Bagherbandi, 2017).

Second, our validation is performed at a 5 arc-min equian-

gular grid, while the gravity signals, here expanded up to de-

gree 21,590, possess spatial information down to the ∼30 arc-

sec resolution. Our experiments are thus not able to examine

whether the divergence effect is present, roughly speaking,

inside the cells of the 5 arc-min grid.

Third, our experiments are not capable of revealing the

divergence effect at magnitudes close to, say, the µGal-level

for gravity if present. This is because the ∼µGal accuracy of

the two techniques is not sufﬁcient to draw such conclusions

reliably.

Finally, our experiments were restricted to studying the

divergence effect for gravity ﬁeld expanded up to degree

21,590. As a consequence, they cannot be used to extrapo-

late the series behaviour beyond that degree. These conclu-

sions are therefore valid only for the degree-2159 RET2014

topography with the near- and far-zone gravity effects (ψ0≈

0.90◦) modelled up to degree 21,590. However, it seems to

be reasonable to extend this conclusion from one particular

degree-2159 model, here RET2014, to all topographic mod-

els expanded up to the same maximum degree. This is be-

cause they all share similar spectral properties, and therefore

similar behaviour of the divergence effect can be expected.

Importantly, this generalization is proposed exclusively for

the Earth’s topographic models and not generally also for

other bodies like, for instance, the Earth’s Moon. This re-

ﬂects the fact that a different planetary surface may produce

a substantially different behaviour of the divergence effect

(cf. Hirt et al, 2016; Hirt and Kuhn, 2017; Rexer, 2017).

As a general conclusion, based on the tests from Sec-

tions 4.2.3, 4.2.4 and 4.2.5, we are conﬁdent in saying that

the enlarged discrepancies in Vzand Vzz from Section 4.2.3

are caused by the divergence effect of spherical harmon-

ics, while in case of Vx,Vy,Vxx,Vxy,Vxz ,Vyy,Vyz and

pmax ≥20, they originate from numerical issues associated

with ultra-high degree cap-modiﬁed spectral modelling. Nev-

ertheless, using appropriate values for pmax and N, accuracy

that appears to be sufﬁcient for many practical applications

was achieved in this study over the entire Earth’s surface

in both cases (∼0.1m2s−2for the gravitational potential,

∼µGal for the gravitational vector and ∼E for the gravita-

tional tensor elements).

5 Conclusions

This paper extends cap-modiﬁed spectral gravity forward

modelling from the ability of delivering gravitational po-

tential and its radial derivatives (presented by Bucha et al,

2019a) to the full gravitational vector and tensor in the local

north-oriented reference frame. In addition, equations for all

radial derivatives of the horizontal components are provided.

Using advanced forward modelling methods, algorithms

and computational resources, the new technique has been

successfully validated in a numerical experiment for all 10

gravity ﬁeld quantities against an independent and divergence-

free spatial-domain forward modelling. The gravitating body,

18 Bucha B., Hirt C. and Kuhn M.

10-3

10-2

m2 s-2

VIn

10-3

10-2

10-1

mGal

Vx,In

10-3

10-2

10-1

mGal

Vy,In

10-4

10-3

10-2

10-1

mGal

Vz,In

10-2

100

E

Vxx,In

10-4

10-2

100

E

Vxy,In

10-3

10-2

10-1

100

E

Vxz,In

10-2

100

E

Vyy,In

2159 6477 10795 15113 19431

Max. spherical harmonic degree N

10-3

10-2

10-1

100

E

Vyz,In

2159 6477 10795 15113 19431

Max. spherical harmonic degree N

10-2

100

E

Vzz,In

pmax = 5 pmax = 10 pmax = 15 pmax = 20 pmax = 30

Fig. 11 RMS of discrepancies between the cap-modiﬁed spectral technique and the spatial-domain Newtonian integration in terms of near-zone

gravity effects evaluated on a Brillouin sphere (radius RB=6,378.137 km+7 km) as a function of the maximum degree N=2159,4318,...,21590

with varying pmax (cf. Eqs. 24 and 25). For Vx,In,Vy,In ,Vxx,In,Vxy,In ,Vxz,In,Vyy,In and Vyz,In, the RMS values signiﬁcantly exceed the upper limits

in the vertical axes when pmax ≥20, so are not shown in the plots. These inferior results are caused by the numerical issues associated with the

evaluation of the truncation coefﬁcients (cf. Section 4.1.1). Detailed statistics are reported in Table S3 of ESM

Cap integration in spectral gravity forward modelling up to the full gravity tensor 19

the Earth’s degree-2159 topography, was forward modelled,

yielding its implied gravity ﬁeld up to degree 21,590. One

of the most challenging steps was the accurate calculation of

truncation coefﬁcients up to ultra-high degrees (here 21,600),

high integer powers of the topography (30) and high-order

radial derivatives of the truncation coefﬁcients (40). To this

end, we had to extend the number of signiﬁcant digits from

16 in double precision to 256 digits. This rather huge num-

ber of digits ensured numerical evaluation of the coefﬁcients

with 24-digit or better accuracy. Despite this fairly sufﬁcient

number of common digits in terms of double precision, we

have found out that the loss of signiﬁcance may occur for

rather advanced but realistic conditions. To avoid these is-

sues, one can lower the maximum topography power pmax

and maximum harmonic degree Nfor the most problematic

coefﬁcients Q1,1,j

np ,Q2,1,j

np ,Q2,2,j

np and still obtain acceptable

accuracy. Using this strategy, we obtained RMS errors at

the level of 0.005 m2s−2(gravitational potential), 4 µGal

(gravitational vector) and 0.07 – 1 E (gravitational tensor)

for near-zone effects, and similar accuracy was achieved for

the far-zone effects (0.03 m2s−2, 0.8 – 20 µGal, 0.1 – 3 mE,

respectively). In the former case, we used i) pmax =10 (in

some cases 15) topography powers, ii) the maximum degree

of N=∼10,795 and iii) kmax =40 radial derivatives for the

continuation in the gradient approach. In the latter case, a

higher convergence rate was observed and thus signiﬁcantly

lower values of pmax and Nwere sufﬁcient while keeping a

comparable accuracy (10 and 2159, respectively; kmax might

also be lowered, but this was not studied). Together with

other indirect validations discussed in the manuscript, these

results demonstrate the correctness of the newly derived equa-

tions. It needs to be stressed, however, that avoiding the nu-

merical issues by lowering pmax and Nmay not be an ac-

ceptable strategy for more complex topographies than that

in our study (nmax >2159).

The demanding character of our numerical experiments

was drawn by the intention to study the convergence/divergence

behaviour of both global and cap-modiﬁed spectral gravity

forward modelling on the topography. We have shown that a

severe divergence effect can be observed when the spher-

ical harmonic series are evaluated on the Earth’s surface,

a region in which the series may no longer converge. We

have also demonstrated that the issue becomes more serious

when improving the completeness of the modelled gravity

ﬁeld, that is, when increasing the pmax and Nparameters.

Including ﬁndings from other recent studies (e.g., Garmier

and Barriot, 2001; Takahashi and Scheeres, 2014; Hu and

Jekeli, 2015; Hirt et al, 2016; Reimond and Baur, 2016; Se-

bera et al, 2016; Hirt and Kuhn, 2017; Rexer, 2017; Bucha

et al, 2019b; Chen et al, 2019), the divergence issue of spher-

ical harmonic series on planetary surfaces may soon become

a more urgent issue than perhaps expected before. To be

more speciﬁc, this study has shown that the divergence ef-

fect comes into play at least around degree 10,795 in terms

of RMS errors (see also the degree variances in Fig. 3.14

of Rexer 2017). We expect that the divergence effect may

be detectable at even lower degrees if pmax >30. Therefore,

as we believe, various spherical harmonic representations of

the true potential in the vicinity of the ﬁeld-generating body

(e.g., Sacerdote and Sans`

o, 2010; Sans`

o and Sideris, 2013;

Bucha et al, 2019b) should further be examined to enable re-

liable spherical harmonic gravity ﬁeld modelling close to the

Earth’s surface. In the space external to the smallest sphere

enclosing all gravitating masses, spherical harmonic series

are, however, convergent as known from potential theory

(e.g., Hotine, 1969).

As an outlook on further developments in cap-modiﬁed

spectral modelling, the probably most urgent issue is the ac-

curate and efﬁcient evaluation of Molodensky’s truncation

coefﬁcients for high values of n,pand k. As shown in this

paper, more than 200 signiﬁcant digits can easily be lost

when targeting at high values for n,pand k. In terms of the

computational speed, it took about one week to compute the

near- and far-zone coefﬁcients using a PC with Intel R

CoreTMi7-

6800K CPU, 128 GB of RAM and a 250 GB SSD drive (our

code is not parallelized because its most time-consuming

parts involve recurrence relations). Because of this, we de-

cided to release the ﬁnal sets of truncation coefﬁcients that

were used in this study (cf. Data availability after this sec-

tion), making it possible for others to reuse them and thus

avoid the somewhat cumbersome computations with 256 sig-

niﬁcant digits. The next issue, which will become relevant

for a further development of spectral gravity forward mod-

elling techniques, was illustrated in Fig. 7. The ﬁgure im-

plies that high powers of the topographic height function

may be difﬁcult to accurately evaluate, because they cover a

wide range of magnitudes which may be problematic when

standard algorithms for harmonic analysis and synthesis are

used in double precision. This was not recognized in pre-

vious works on global and cap-modiﬁed spectral forward

modelling. Next, the observed loss of signiﬁcance indicate

that our strategy may need to be modiﬁed if one intends to

extend the modelling beyond the levels of resolution and

completeness of the modelling reached in this paper. Besides

these computational and numerical challenges, a study on

the relation between the divergence effect and the integra-

tion radius could be beneﬁcial. It might reveal, for instance,

an integration radius (possibly multiple radii depending on

the location) for which the series for far-zone effects start to

suffer from the divergence effect on a detectable level in a

closed-loop environment.

Author contributions: BB, CH and MK designed the study; BB con-

ducted all the numerical experiments and drafted the manuscript; all

authors discussed and commented on the manuscript.

20 Bucha B., Hirt C. and Kuhn M.

Data availability: Input data: The input RET2014 topography (Hirt

and Rexer, 2015) is available at http://ddfe.curtin.edu.au/models/

Earth2014.

Output data: Gravity effects from spatial and spectral gravity forward

modelling and spherical harmonic coefﬁcients of 30 integer powers

of the degree-2159 RET2014 topography are available on the request

from BB (∼400 GB). The evaluated truncation coefﬁcients Qj

np ,Q1,0,j

np ,

Q1,1,j

np ,Q2,0,j

np ,Q2,1,j

np ,Q2,2,j

np are available for download at http://edisk.

cvt.stuba.sk/~xbuchab/.

Computer codes: Each of the following routines is written in Mat-

lab and was tested with Matlab R2015b and R2018b. The software

packages for spherical harmonic synthesis, GrafLab (Bucha and Jan´

ak,

2013) and isGrafLab (Bucha and Jan´

ak, 2014), routines to perform

ultra-high-degree surface spherical harmonic analysis and to compute

truncation coefﬁcients Qj

np ,Q1,0,j

np ,Q1,1,j

np ,Q2,0,j

np ,Q2,1,j

np ,Q2,2,j

np are avail-

able at http://edisk.cvt.stuba.sk/~xbuchab/.

Acknowledgements BB was supported by the project VEGA 1/0750/18.

The computations were performed at the HPC centre at the Slovak

University of Technology in Bratislava and at the Slovak Academy of

Sciences, which are parts of the Slovak Infrastructure of High Perfor-

mance Computing (SIVVP project, ITMS code 26230120002, funded

by the European region development funds, ERDF). The maps were

produced using the Generic Mapping Tools (Wessel and Smith, 1998).

A Derivation of the ﬁrst-order potential derivatives in

LNOF for cap-modiﬁed spectral gravity forward

modelling

In this appendix, cap-modiﬁed spectral gravity forward mod-

elling is presented for the ﬁrst-order potential derivatives in

LNOF, including the horizontal ones. For the sake of brevity,

the derivation is shown only for the near-zone gravitational

effects (inside-cap integration). For the far-zone effects (outside-

cap integration), we provide only the ﬁnal formulae, because

the derivation can easily be reproduced, simply by chang-

ing the integration domain. Our derivation is based on the

idea by Molodensky et al (1962) and follows the manner by,

for instance, Heiskanen and Moritz (1967) and ˇ

Sprl´

ak et al

(2015).

The starting point is the expression for the topographic

potential induced by topographic masses that have a con-

stant mass density ρand are located inside a spherical cap

centred at the evaluation point (Bucha et al, 2019a),

VIn(r,ϕ,λ) = GρR2

∞

∑

p=1

ψ0

Z

ψ=0

2π

Z

α=0

Hp(ψ,α)Kp(r,ψ)

×sinψdαdψ,

(28)

where ψand αare the spherical distance and azimuth, re-

spectively, ψ0∈[0,π]is the spherical distance deﬁning the

spherical cap and Hp(ψ,α)is the pth integer power of the

topographic height function (cf. Eq. 4). For r>R, the in-

tegral kernels Kp(r,ψ)can be express either via a spectral

relation

Kp(r,ψ) =

∞

∑

n=0R

rn+1

p

∏

r=1

(n+4−r)

p!(n+3)P

n,0(cosψ),(29)

where P

n,0is the un-normalized Legendre function of de-

gree n, or through closed spatial formulae

K1(r,ψ) = R

l(r,ψ),

K2(r,ψ) = 1

2K1(r,ψ)−r∂K1(r,ψ)

∂r,

Kp(r,ψ) = 1

p!

p−2

∑

s=1

aps rp−s∂p−sK1(r,ψ)

∂rp−s,p≥3,

(30)

with the Euclidean distance

l(r,ψ) = pr2−2R r cosψ+R2(31)

and the coefﬁcients

aps = (−1)p−1(p−1)!(p−3)!

(p−s)!(p−s−2)!(s−1)!.(32)

Spatial and spectral relations for the radial derivatives of

Kp(r,ψ),p≥1, are provided in Appendix A.4 (Eqs. 67 and

70, respectively).

Throughout all the derivations in Appendices A and B,

we assume that

r>max(R+ˆ

H(ϕ,λ)),(33)

ensuring that the order of summation and integration can be

interchanged whenever necessary. Then, the resulting inﬁ-

nite spherical harmonic series for the topographic potential

and its derivatives are absolutely and uniformly convergent.

Otherwise, the series may converge or diverge.

Next, we formally extend the cap integration from Eq. (28)

to the whole sphere. This can be achieved with discontinu-

ous integral kernels

KIn

p(r,ψ) = (Kp(r,ψ)for 0 ≤ψ≤ψ0,

0 for ψ0<ψ≤π,(34)

and leads to

VIn(r,ϕ,λ) = GρR2

∞

∑

p=1

π

Z

ψ=0

2π

Z

α=0

Hp(ψ,α)KIn

p(r,ψ)

×sinψdαdψ.

(35)

Note that the gravitational potential in Eq. (35) still remains

to be implied only by the inside-cap masses despite the global

integration (cf. the zero case for KIn

p(r,ψ)in Eq. 34).

Cap integration in spectral gravity forward modelling up to the full gravity tensor 21

To obtain the sought ﬁrst-order derivatives of VIn(r,ϕ,λ)

in LNOF, we introduce the following differential operators

(e.g., Heiskanen and Moritz, 1967),

Dx=1

r

∂

∂ ϕ =−cos αD1,1,

Dy=−1

rcosϕ

∂

∂ λ =sin αD1,1,

Dz=D1,0,

(36)

where we utilized the relations (ibid.)

∂

∂ ϕ =−cos α∂

∂ ψ ,

1

cosϕ

∂

∂ λ =−sin α∂

∂ ψ ,

(37)

and introduced the substitutions

D1,0=∂

∂r,

D1,1=1

r

∂

∂ ψ =−1

rsinψ∂

∂cosψ.

(38)

In Eq. (38), the ﬁrst superscript next to D, here being equal

to 1, implies that the differential operator is related to the

ﬁrst-order derivatives of the gravitational potential. The sec-

ond superscript denotes the order of the differentiation with

respect to ψ(here either 0 or 1). Importantly, the two ex-

pressions for D1,1are equal when considering that D1,1will

only be applied to isotropic kernels Kp(r,ψ)(cf. Eqs. 29 and

30), each of which can generally be expressed by a conver-

gent series

K(r,ψ) =

∞

∑

n=0R

rn+1

knP

n,0(cosψ),r>R.(39)

After applying the differential operators from Eq. (36)

to the gravitational potential from Eq. (35), the ﬁrst-order

derivatives of VIn in LNOF are obtained as

Vv,In(r,ϕ,λ) = GρR2

∞

∑

p=1

π

Z

ψ=0

2π

Z

α=0

Hp(ψ,α)Kv,In

p(r,ψ)

×sinψdαdψ,v={x,y,z},

(40)

where the integral kernels

Kv,In

p(r,ψ) = DvKIn

p(r,ψ),v={x,y,z},(41)

read

Kx,In

p(r,ψ) = −cos αK1,1,In

p(r,ψ),

Ky,In

p(r,ψ) = sinαK1,1,In

p(r,ψ),

Kz,In

p(r,ψ) = K1,0,In

p(r,ψ),

(42)

with

K1,i,In

p(r,ψ) = D1,iKIn

p(r,ψ),i=0,1.(43)

Eq. (42) reveals that the three integral kernels Kv,In

p(r,ψ),

v={x,y,z}, from Eq. (40) can be expressed in terms of two

kernels only, K1,i,In

p(r,ψ),i=0,1. As a result, only two sets

of Molodensky’s truncation coefﬁcients are now needed to

compute the three elements of the gravitational vector (see

also ˇ

Sprl´

ak et al, 2015).

Next, the kernels K1,i,In

p(r,ψ)are expanded in series of

un-normalized Legendre functions of the ﬁrst kind (e.g., de Witte,

1967; ˇ

Sprl´

ak et al, 2015),

K1,i,In

p(r,ψ) =

∞

∑

n=i

2n+1

2Q1,i,In

np (r,ψ0)P

n,i(cosψ),i=0,1.

(44)

The coefﬁcients Q1,i,In

np (r,ψ0)are called Molodensky’s trun-

cation coefﬁcients and are deﬁned as (ibid.)

Q1,i,In

np (r,ψ0) = (n−i)!

(n+i)!

π

Z

0

K1,i,In

p(r,ψ)P

n,i(cosψ)sin ψdψ

=(n−i)!

(n+i)!

ψ0

Z

0

D1,iKp(r,ψ)P

n,i(cosψ)sin ψdψ,

(45)

where i=0,1. Formulae suitable for practical evaluation of

these coefﬁcients are discussed in Appendices A.1 and A.2.

Substituting Eq. (44) into Eq. (42), with the help of Eq. (40)

and the relations (e.g., Hagiwara, 1972; Eshagh, 2009; ˇ

Sprl´

ak

et al, 2015)

H(p)

n(ϕ,λ) = 2n+1

4π

π

Z

ψ=0

2π

Z

λ=0

Hp(ψ,α)P

n,0(cosψ)

×sinψdαdψ,

(46)

∂H(p)

n(ϕ,λ)

∂ ϕ =2n+1

4π

π

Z

ψ=0

2π

Z

λ=0

Hp(ψ,α)P

n,1(cosψ)cos α

×sinψdαdψ

(47)

and

1

cosϕ

∂H(p)

n(ϕ,λ)

∂ λ =2n+1

4π

π

Z

ψ=0

2π

Z

λ=0

Hp(ψ,α)P

n,1(cosψ)

×sinαsin ψdαdψ,

(48)

22 Bucha B., Hirt C. and Kuhn M.

we get

Vx,In(r,ϕ,λ) = −2πGρR2

∞

∑

p=1

∞

∑

n=1

Q1,1,In

np (r,ψ0)

×∂H(p)

n(ϕ,λ)

∂ ϕ ,

(49)

Vy,In(r,ϕ,λ) = 2πGρR2

cosϕ

∞

∑

p=1

∞

∑

n=1

Q1,1,In

np (r,ψ0)

×∂H(p)

n(ϕ,λ)

∂ λ ,

(50)

Vz,In(r,ϕ,λ) = 2πGρR2

∞

∑

p=1

∞

∑

n=0

Q1,0,In

np (r,ψ0)

×H(p)

n(ϕ,λ).

(51)

The sought Eqs. (9) – (11) for j=‘In’ are obtained from

Eqs. (49) – (51) when truncating the series over pat some

ﬁnite pmax, then considering that

H(p)

n=

n

∑

m=−n

¯

Hnmp Ynm(ϕ,λ)(52)

and, ﬁnally, utilizing (cf. Lemma 4.1 of Freeden and Schnei-

der, 1998)

H(p)

n(ϕ,λ) = 0 for n>p×nmax .(53)

As discussed before, the far-zone effects (j=‘Out’) can

be derived by changing the integration domain to ψ∈[ψ0,π].

Also, we note that H(p)

n(ϕ,λ)stands for the nth-degree Laplace’s

surface spherical harmonic function of the pth power of the

topographic height function H. It must not be confused with

the pth power of the nth-degree Laplace’s surface spher-

ical harmonic function. The same applies to the notation

Hp

n(ϕ,λ)of Bucha et al (2019a), though they omitted the

brackets in the superscript.

For future reference, we also provide the formulae for

Molodensky’s truncation coefﬁcients related to far-zone ef-

fects,

Q1,i,Out

np (r,ψ0) = (n−i)!

(n+i)!

π

Z

0

K1,i,Out

p(r,ψ)P

n,i(cosψ)

×sinψdψ

=(n−i)!

(n+i)!

π

Z

ψ0

D1,iKp(r,ψ)P

n,i(cosψ)

×sinψdψ,i=0,1,

(54)

where we introduced the kernels

K1,i,Out

p(r,ψ) = D1,iKOut

p(r,ψ),i=0,1,(55)

with

KOut

p(r,ψ) = (0 for 0 ≤ψ<ψ0,

Kp(r,ψ)for ψ0≤ψ≤π.(56)

A.1 Spectral representation of truncation coefﬁcients for

the ﬁrst-order potential derivatives in LNOF

The spectral relations for the inner-zone truncation coefﬁ-

cients Q1,0,In

np (r,ψ0)and Q1,1,In

np (r,ψ0)can be obtained from

Eq. (45) and with the help of Eqs. (29) and (38). For Q1,0,In

np (r,ψ0),

we have (n≥0)

Q1,0,In

np (r,ψ0) = −1

R

∞

∑

l=0

(l+1)R

rl+2

p

∏

r=1

(l+4−r)

p!(l+3)

×

ψ0

Z

0

P

l,0(cosψ)P

n,0(cosψ)sin ψdψ,

(57)

and the coefﬁcients Q1,1,In

np (r,ψ0)read (n≥1)

Q1,1,In

np (r,ψ0) = −1

n(n+1)

1

R

∞

∑

l=1R

rl+2

p

∏

r=1

(l+4−r)

p!(l+3)

×

ψ0

Z

0

P

l,1(cosψ)P

n,1(cosψ)sin ψdψ.

(58)

Note that the integral in Eq. (57) can be evaluated analyti-

cally using recurrence relations (e.g., Paul, 1973; Moreaux

et al, 1999). The integral in Eq. (58) with un-normalized

Legendre functions can be computed analytically similarly

as shown, for instance, in Pail et al (2001) or Hwang (1991)

for fully normalized Legendre functions.

For the sake of brevity, the formulae for the far-zone

truncation coefﬁcients Q1,0,Out

np (r,ψ0)and Q1,1,Out

np (r,ψ0)are

omitted here, but can be derived from Eq. (54). This yields

formally similar relations as in Eqs. (57) and (58) but with

the integration domain ψ∈[ψ0,π].

A.2 Closed forms of truncation coefﬁcients for the

ﬁrst-order potential derivatives in LNOF

After generalizing Eqs. (45) and (54) to a single expres-

sion via the superscript j={‘In’,‘Out’}and considering

Eq. (38), the closed form for truncation coefﬁcients with

i=0 immediately reads

Q1,0,j

np (r,ψ0) = ∂

∂r

π

Z

0

Kj

p(r,ψ)P

n,0(cosψ)sin ψdψ

=∂

∂rQj

np (r,ψ0),n≥0.

(59)

The newly introduced coefﬁcients Qj

np (r,ψ0)are deﬁned in

Eqs. (28) and (32) of Bucha et al (2019a), wherein spectral

Cap integration in spectral gravity forward modelling up to the full gravity tensor 23

and recurrence relations can be found for an arbitrary n,p

and order of the radial derivative.

For i=1 in Eqs. (45) and (54), we have

Q1,1,j

np (r,ψ0) = 1

rcjsinψ0P

n,1(cosψ0)

n(n+1)Kp(r,ψ0)

−Qj

np (r,ψ0),n≥1,

(60)

with

cj=(1 for j=‘In’ ,

−1 for j=‘Out’ .(61)

Eq. (60) was obtained from Eqs. (45) and (54) using integra-

tion by parts and the relation (e.g., Hagiwara, 1972)

d

dψ(sinψP

n,1(cosψ)) = n(n+1)P

n,0(cosψ)sin ψ.(62)

Note that it must hold in Eq. (62) that ψ∈[0,π], a condition

which is fulﬁlled in our case (cf. Eqs. 45, 54).

A.3 Spectral relations for an arbitrary radial derivative of

truncation coefﬁcients related to the ﬁrst-order potential

derivatives in LNOF

Differentiation of Eqs. (57) and (58) with respect to rdi-

rectly leads to spectral relations for the kth radial derivative,

k≥1, of Q1,0,In

np (r,ψ0)and Q1,1,In

np (r,ψ0),

∂kQ1,0,In

np (r,ψ0)

∂rk= (−1)k+11

Rk+1

∞

∑

l=0

k+1

∏

e=1

(l+e)

×R

rl+k+2

p

∏

r=1

(l+4−r)

p!(l+3)

×

ψ0

Z

0

P

l,0(cosψ)P

n,0(cosψ)sin ψdψ,

(63)

∂kQ1,1,In

np (r,ψ0)

∂rk= (−1)k+11

n(n+1)

1

Rk+1

∞

∑

l=1

k+1

∏

e=2

(l+e)

×R

rl+k+2

p

∏

r=1

(l+4−r)

p!(l+3)

×

ψ0

Z

0

P

l,1(cosψ)P

n,1(cosψ)sin ψdψ,

(64)

where the former relation holds for n≥0 and latter one for

n≥1.

The spectral relations for the kth radial derivative of

Q1,0,Out

np (r,ψ0)and Q1,1,Out

np (r,ψ0)are formally similar to

Eqs. (63) and (64) but with the integration domain ψ∈[ψ0,π].

A.4 Closed forms for an arbitrary radial derivative of

truncation coefﬁcients related to the ﬁrst-order potential

derivatives in LNOF

Closed forms of the kth radial derivative of the truncation

coefﬁcients Q1,0,j

np (r,ψ0)and Q1,1,j

np (r,ψ0)can be obtained by

differentiating, respectively, Eqs. (59) and (60) with respect

to r. This leads to

∂k

∂rkQ1,0,j

np (r,ψ0) = ∂k+1

∂rk+1Qj

np (r,ψ0),n≥0,k≥0,(65)

and

∂k

∂rkQ1,1,j

np (r,ψ0) =

k

∑

q=0k

q(−1)k−q(k−q)!1

rk−q+1

×cjsinψ0P

n,1(cosψ0)

n(n+1)

∂q

∂rqKp(r,ψ0)

−∂q

∂rqQj

np (r,ψ0),n≥1,k≥0.

(66)

In Eq. (66), we employed the general Leibniz rule that

provides a formula for an nth derivative of a product of two

n-times differentiable functions. Similarly as in Appendix A.2,

the derivatives ∂q(Qj

np (r,ψ0))/∂rqcan be computed for all

q≥0 using the relations from Bucha et al (2019a). The

last missing expressions are those for the radial derivatives

of Kp(r,ψ0). The closed spatial relations for the kth radial

derivative, k≥0, can be obtained from Eq. (30),

∂k

∂rkK1(r,ψ) = R∂k

∂rk1

l(r,ψ),

∂k

∂rkK2(r,ψ) = 1

2−(k−1)∂k

∂rkK1(r,ψ)

−r∂k+1

∂rk+1K1(r,ψ),

∂k

∂rkKp(r,ψ) = 1

p!

p−2

∑

s=1

aps

k

∑

q=0k

qR(k−q)

p−s(r)

×∂p−s+q

∂rp−s+qK1(r,ψ),p≥3,

(67)

24 Bucha B., Hirt C. and Kuhn M.

where (Martinec, 1998)

∂k

∂rk1

l(r,ψ)=

1

l(r,ψ),k=0,

k

∑

(k+t)is even

t=0

(−1)k+t

2

×(k−t+1)!!(k+t−1)!!

(k−t+1)!

k!

t!

×(r−Rcosψ)t

lk+t+1(r,ψ),k≥1,

(68)

and

R(q)

w(r) = dq

drqrw=

rw,q=0,w≥1,

q

∏

j=1

(w−j+1)rw−q,q≥1,w≥1.

(69)

The spectral relations for ∂k(Kp(r,ψ0))/∂rkwere derived

by differentiating Eq. (29) with respect to r,

∂kKp(r,ψ)

∂rk=(−1)k

Rk

∞

∑

n=0

k

∏

l=1

(n+l)R

rn+k+1

×

p

∏

r=1

(n+4−r)

p!(n+3)P

n,0(cosψ),k≥1.

(70)

B Derivation of the second-order potential derivatives

in LNOF for cap-modiﬁed spectral gravity forward

modelling

In this appendix, we derive Eqs. (13) – (23) to compute the

second-order derivatives of the topographic gravitational po-

tential in LNOF. The derivation closely follows the ideas of

Appendix A as well as that of ˇ

Sprl´

ak et al (2015). Therefore,

they are limited to the most important steps for the sake of

brevity.

First, we rewrite the differential operators from Eqs. (10)

and (11) of ˇ

Sprl´

ak et al (2015) in terms of spherical polar

coordinates (r,ψ,α),

Dxx =−1

2D2,0+cos2αD2,2,

Dxy =−sin2αD2,2,

Dxz =cosαD2,1,

Dyy =−1

2D2,0−cos2αD2,2,

Dyz =−sinαD2,1,

Dzz =D2,0,

(71)

where

D2,0=∂2

∂r2,

D2,1=∂2

∂r∂ ψ −1

r·=−sinψ∂2

∂r∂cosψ−1

r·,

D2,2=1

2r2∂2

∂ ψ 2−cotψ∂

∂ ψ =1

2r2sin2ψ∂2

∂(cosψ)2.

(72)

The notation −1

r·stands for the multiplication of the term

−1

rand the integral kernels. Again, it is presupposed that

the differential operators from Eqs. (71) and (72) will only

be applied to isotropic kernels having the form of Eq. (39).

After applying the differential operators from Eq. (71) to

Eq. (35), we get

Vuv,In(r,ϕ,λ) = GρR2

∞

∑

p=1

π

Z

ψ=0

2π

Z

α=0

Hp(ψ,α)Kuv,In

p(r,ψ)

×sinψdαdψ,u,v={x,y,z},

(73)

where we introduced integral kernels

Kuv,In

p(r,ψ) = DuvKIn

p(r,ψ),u,v={x,y,z},(74)

which have the form

Kxx,In

p=−1

2K2,0,In

p(r,ψ) + cos2αK2,2,In

p(r,ψ),

Kxy,In

p=−sin2αK2,2,In

p(r,ψ),

Kxz,In

p=cosαK2,1,In

p(r,ψ),

Kyy,In

p=−1

2K2,0,In

p(r,ψ)−cos2αK2,2,In

p(r,ψ),

Kyz,In

p=−sinαK2,1,In

p(r,ψ),

Kzz,In

p=K2,0,In

p(r,ψ),

(75)

with

K2,i,In

p(r,ψ) = D2,iKIn

p(r,ψ),i=0,1,2.(76)

Next, the kernels from Eq. (76) are expanded in series of

un-normalized Legendre functions as

K2,i,In

p(r,ψ) =

∞

∑

n=i

2n+1

2Q2,i,In

np (r,ψ0)P

n,i(cosψ)(77)

with i=0,1,2 and Molodensky’s truncation coefﬁcients

Q2,i,In

np (r,ψ0)deﬁned as

Q2,i,In

np (r,ψ0) = (n−i)!

(n+i)!

π

Z

0

K2,i,In

p(r,ψ)P

n,i(cosψ)sin ψdψ

=(n−i)!

(n+i)!

ψ0

Z

0

D2,iKp(r,ψ)P

n,i(cosψ)sin ψdψ.

Cap integration in spectral gravity forward modelling up to the full gravity tensor 25

(78)

Numerical evaluation of these coefﬁcients via spectral and

closed spatial relations is discussed in Appendices B.1 and

B.2, respectively.

With the help of Eqs. (78), (77), (76), (75) and (73) of

this paper as well as using Eqs. (47) and (48) of ˇ

Sprl´

ak et al

(2015), we arrive at the expressions

Vxx,In(r,ϕ,λ) = 2πGρR2

×

∞

∑

p=1

∞

∑

n=0"−1

2Q2,0,In

np (r,ψ0)H(p)

n(ϕ,λ)

+Q2,2,In

np (r,ψ0) n(n+1)H(p)

n(ϕ,λ)

+2∂2H(p)

n(ϕ,λ)

∂ ϕ 2!#,

(79)

Vxy,In(r,ϕ,λ) = −2πGρR2

cosϕ

∞

∑

p=1

∞

∑

n=0

Q2,2,In

np (r,ψ0)

×2 tanϕ∂H(p)

n(ϕ,λ)

∂ λ +∂2H(p)

n(ϕ,λ)

∂ λ ∂ ϕ !,

(80)

Vxz,In(r,ϕ,λ) = 2πGρR2

∞

∑

p=1

∞

∑

n=0

Q2,1,In

np (r,ψ0)

×∂H(p)

n(ϕ,λ)

∂ ϕ ,

(81)

Vyy,In(r,ϕ,λ) = 2πGρR2

×

∞

∑

p=1

∞

∑

n=0"−1

2Q2,0,In

np (r,ψ0)H(p)

n(ϕ,λ)

−Q2,2,In

np (r,ψ0) n(n+1)H(p)

n(ϕ,λ)

+2∂2H(p)

n(ϕ,λ)

∂ ϕ 2!#,

(82)

Vyz,In(r,ϕ,λ) = −2πGρR2

cosϕ

∞

∑

p=1

∞

∑

n=0

Q2,1,In

np (r,ψ0)

×∂H(p)

n(ϕ,λ)

∂ λ ,

(83)

Vzz,In(r,ϕ,λ) = 2πGρR2

∞

∑

p=1

∞

∑

n=0

Q2,0,In

np (r,ψ0)

×H(p)

n(ϕ,λ).

(84)

Note that the non-existing coefﬁcients Q2,1,In

0p(r,ψ0),

Q2,2,In

0p(r,ψ0), and Q2,2,In

1p(r,ψ0)are set to zero in Eqs. (79) –

(84).

Finally, after truncating the series over pat some ﬁ-

nite pmax and considering Eqs. (52) and (53), we obtain

Eqs. (13) – (23) for j=‘In’.

The relations for j=‘Out’ can similarly be derived by

changing the integration domain to ψ∈[ψ0,π]which re-

quires to introduce the truncation coefﬁcients

Q2,i,Out

np (r,ψ0) = (n−i)!

(n+i)!

π

Z

0

K2,i,Out

p(r,ψ)P

n,i(cosψ)

×sinψdψ

=(n−i)!

(n+i)!

π

Z

ψ0

D2,iKp(r,ψ)P

n,i(cosψ)

×sinψdψ,i=0,1,2,

(85)

with

K2,i,Out

p(r,ψ) = D2,iKOut

p(r,ψ),i=0,1,2.(86)

B.1 Spectral representation of truncation coefﬁcients for

the second-order potential derivatives in LNOF

The inner-zone truncation coefﬁcients expressed in the spec-

tral form read

Q2,0,In

np (r,ψ0) = 1

R2

∞

∑

l=0

(l+1)(l+2)R

rl+3

p

∏

r=1

(l+4−r)

p!(l+3)

×

ψ0

Z

0

P

l,0(cosψ)P

n,0(cosψ)sin ψdψ,

(87)

Q2,1,In

np (r,ψ0) = −1

(n+1)n

1

R2

∞

∑

l=1

(l+2)R

rl+3

×

p

∏

r=1

(l+4−r)

p!(l+3)

×

ψ0

Z

0

P

l,1(cosψ)P

n,1(cosψ)sin ψdψ

(88)

26 Bucha B., Hirt C. and Kuhn M.

and

Q2,2,In

np (r,ψ0) = 1

(n+2)(n+1)n(n−1)

1

2R2

×

∞

∑

l=2R

rl+3

p

∏

r=1

(l+4−r)

p!(l+3)

×

ψ0

Z

0

P

l,2(cosψ)P

n,2(cosψ)sin ψdψ,

(89)

for n≥0, n≥1 and n≥2, respectively. These equations

were derived from Eq. (78) using Eqs. (29) and (72). The

far-zone coefﬁcients can be derived in a similar manner from

Eq. (85), see also Appendix A.1. Note that the integrals in

Eqs. (87) – (89) can be evaluated analytically (cf. Appendix A.1).

B.2 Closed forms of truncation coefﬁcients for the

second-order potential derivatives in LNOF

Considering Eq. (72), the closed form of truncation coefﬁ-

cients for the second-order potential derivatives (Eqs. 78 and

85) reads for i=0 (n≥0)

Q2,0,j

np (r,ψ0) = ∂Q1,0,j

np (r,ψ0)

∂r=∂2Qj

np (r,ψ0)

∂r2.(90)

For i=1, we have the relation (n≥1)

Q2,1,j

np (r,ψ0) = 1

rcjsinψ0P

n,1(cosψ0)

n(n+1)

×1

rKp(r,ψ0)−∂

∂rKp(r,ψ0)

−1

rQj

np (r,ψ0)−∂

∂rQj

np (r,ψ0)

=1

rQ1,1,j

np (r,ψ0)−∂

∂rr Q1,1,j

np (r,ψ0),

(91)

where we employed integration by parts together with Eq. (62).

Finally, the formula for i=2 reads (n≥2)

Q2,2,j

np (r,ψ0) = 1

2r2cjsinψ0P

n,2(cosψ0)

(n+2)(n+1)n(n−1)

∂Kp(r,ψ0)

∂ ψ

−cjsinψ0P

n,1(cosψ0)

n(n+1)Kp(r,ψ0)

+Qj

np (r,ψ0).

(92)

This equation was derived by applying integration by parts

twice, utilizing Eq. (62) and the recurrence relation (e.g.,

Freeden and Schreiner, 2009)

sin2ψdP

n,m(cosψ)

dcos ψ−mcos ψP

n,m(cosψ