ArticlePDF Available

Abstract and Figures

Cap-modified spectral gravity forward modelling is extended in this paper to the full gravity vector and tensor 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 arbitrary 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 surfaces. In a numerical study, we exemplarily apply the new technique by forward modelling Earth's degree-2159 topography up to degree 21, 590, employing 30 topography powers. The experiment shows that near- and far-zone gravity effects can be synthesized on the topography with an accuracy (RMS) of 0.005 -- 0.03 m^2 s^−2 (potential), 0.8 -- 20 µGal (gravity vector) and 0.1 mE -- 1 E (gravity tensor). The numerical experiment also shows that the divergence effect of spherical harmonics comes into play around degree 10, 795 when evaluating the series on the Earth's surface. The difficult-to-compute truncation coefficients that are employed in the study are made freely available at http://edisk.cvt.stuba.sk/~xbuchab/ and are accompanied 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 importance in spectral gravity forward modelling techniques.
Content may be subject to copyright.
Citation:
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Note: This is a preprint (author’s own manuscript that has not been peer reviewed) of an article accepted for publication in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 m2s2(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 having 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+4i)
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 m2s2,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/R0.90(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=
AAreference
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 p2, 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 p12.
Beyond p12, 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 p12 (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 m3.
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 10polar 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
H0 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 (0ψψ0) is subdi-
vided into an inner zone (ψ0.25), where we rely on poly-
hedral modelling, and an outer zone (0.25<ψψ0), 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 (ψ0ψ180).
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., 1046 for near-
zone effects and p6 (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 H0 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˚
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˚
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 m2s2), 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 m2s2and MAX =
0.035 m2s2and 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
Vm2s2490.898 589.679 139.061 176.223 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 4.201e6 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 E338.470 287.899 7.386 16.628 18.442 23.440 6.592 11.424
Vxy E183.657 160.344 3.569e3 7.609 9.174 9.586 1.492e2 2.010
Vxz E364.062 286.933 7.212e2 15.679 2.161 1.020 0.235 0.445
Vyy E403.187 318.051 7.385 16.223 18.765 29.584 6.680 12.637
Vyz E322.009 315.731 2.020e4 15.391 0.985 1.170 3.208e2 0.261
Vzz E361.698 659.755 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 ψ00.90) and far-zone (beyond ψ00.90) 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 m2s2),
but at the cost of a worse MAX value (0.280 m2s2). 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
1and 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 s2for 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˚
pmax = 30
30˚
40˚
pmax = 20
−4˚
pmax = 20
30˚
40˚
pmax = 15
−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
ψ00.9). 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˚
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 (0.1m2s2for 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 m2s2(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 m2s2, 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+4r)
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)rK1(r,ψ)
r,
Kp(r,ψ) = 1
p!
p2
s=1
aps rpspsK1(r,ψ)
rps,p3,
(30)
with the Euclidean distance
l(r,ψ) = pr22R r cosψ+R2(31)
and the coefficients
aps = (1)p1(p1)!(p3)!
(ps)!(ps2)!(s1)!.(32)
Spatial and spectral relations for the radial derivatives of
Kp(r,ψ),p1, 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 ψψ0,
0 for ψ0<ψπ,(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) = (ni)!
(n+i)!
π
Z
0
K1,i,In
p(r,ψ)P
n,i(cosψ)sin ψdψ
=(ni)!
(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) = (ni)!
(n+i)!
π
Z
0
K1,i,Out
p(r,ψ)P
n,i(cosψ)
×sinψdψ
=(ni)!
(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 ψ0ψπ.(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 (n0)
Q1,0,In
np (r,ψ0) = 1
R
l=0
(l+1)R
rl+2
p
r=1
(l+4r)
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 (n1)
Q1,1,In
np (r,ψ0) = 1
n(n+1)
1
R
l=1R
rl+2
p
r=1
(l+4r)
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),n0.
(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),n1,
(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,
k1, 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+4r)
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+4r)
p!(l+3)
×
ψ0
Z
0
P
l,1(cosψ)P
n,1(cosψ)sin ψdψ,
(64)
where the former relation holds for n0 and latter one for
n1.
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),n0,k0,(65)
and
k
rkQ1,1,j
np (r,ψ0) =
k
q=0k
q(1)kq(kq)!1
rkq+1
×cjsinψ0P
n,1(cosψ0)
n(n+1)
q
rqKp(r,ψ0)
q
rqQj
np (r,ψ0),n1,k0.
(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
q0 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, k0, can be obtained from Eq. (30),
k
rkK1(r,ψ) = Rk
rk1
l(r,ψ),
k
rkK2(r,ψ) = 1
2(k1)k
rkK1(r,ψ)
rk+1
rk+1K1(r,ψ),
k
rkKp(r,ψ) = 1
p!
p2
s=1
aps
k
q=0k
qR(kq)
ps(r)
×ps+q
rps+qK1(r,ψ),p3,
(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
×(kt+1)!!(k+t1)!!
(kt+1)!
k!
t!
×(rRcosψ)t
lk+t+1(r,ψ),k1,
(68)
and
R(q)
w(r) = dq
drqrw=
rw,q=0,w1,
q
j=1
(wj+1)rwq,q1,w1.
(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+4r)
p!(n+3)P
n,0(cosψ),k1.
(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
2D2,0cos2αD2,2,
Dyz =sinαD2,1,
Dzz =D2,0,
(71)
where
D2,0=2
r2,
D2,1=2
r∂ ψ 1
r·=sinψ2
rcosψ1
r·,
D2,2=1
2r22
∂ ψ 2cotψ
∂ ψ =1
2r2sin2ψ2
(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) = (ni)!
(n+i)!
π
Z
0
K2,i,In
p(r,ψ)P
n,i(cosψ)sin ψdψ
=(ni)!
(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)
+22H(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 tanϕH(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)
+22H(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) = (ni)!
(n+i)!
π
Z
0
K2,i,Out
p(r,ψ)P
n,i(cosψ)
×sinψdψ
=(ni)!
(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+4r)
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+4r)
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(n1)
1
2R2
×
l=2R
rl+3
p
r=1
(l+4r)
p!(l+3)
×
ψ0
Z
0
P
l,2(cosψ)P
n,2(cosψ)sin ψdψ,
(89)
for n0, n1 and n2, 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 (n0)
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 (n1)
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 (n2)
Q2,2,j
np (r,ψ0) = 1
2r2cjsinψ0P
n,2(cosψ0)
(n+2)(n+1)n(n1)
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 ψmcos ψP
n,m(cosψ