Introduction
Conductive seawater moving through the ambient Earth's main magnetic field
Bmain induces a secondary magnetic field
Boc. Due to their periodic nature, magnetic signals
generated by ocean tides are particularly easy to detect and have been
studied in observatory data as early as, e.g., . However, the
extraction of global models for magnetic fields induced by the dominating M2
tide from satellite data has only recently become possible (e.g.,
). Although the extraction procedures used
are solely based on the temporal periodicity of the tidal signal (and not on
further information on the spatial localization of the sources), they seem,
by visual inspection, to coincide very well with results obtained by forward
models such as in . A more extensive comparison of
forward models of electromagnetic ocean tidal signals based on different
ocean tide models has recently been published in . In that
work, it was shown that the residuals between the different models can exceed
the nominal noise level of the Swarm satellite mission. The ability to
extract M2 tidal magnetic field signals in satellite data more precisely can
therefore help in constraining ocean tide models. In and
it has been shown that an M2 tidal magnetic field model can
also be used to constrain 1-D models of the Earth's mantle conductivity, and
forward studies in have shown that lateral variations in
the conductivity of the ocean water itself should have a detectable influence
on the measured magnetic field (although the latter study was performed for
general ocean circulation and not for tidal current systems).
In this short paper, we want to illustrate the effects that different
(spatially localized) sets of trial functions can have on the approximation
of the magnetic field induced by the M2 tide in the first place.
In particular, we describe a possible setup for the inclusion of spatial
localization constraints (in addition to the constraint of temporal
periodicity) for the approximation of ocean-tide-induced magnetic fields.
Clearly, the velocity field u that is responsible for the
generation of the corresponding secondary magnetic field
Boc vanishes over the continents. The precise connection
between u and Boc is given by the time-harmonic
Maxwell equations:
∇×Boc=μ0σ(E+u×Bmain),∇×E=iωBoc,∇⋅Boc=0,
where we assume to have knowledge of Bmain, the
underlying conductivity σ, and the frequency ω. Furthermore,
E denotes the electric field and μ0 is the vacuum permeability.
Instead of using a fixed velocity field u, we substitute it by a
set of functions {uℓ}ℓ=1,…,L (e.g., vectorial
Slepian functions as in that are
localized over the oceans) to obtain a set of corresponding trial functions
{Bℓ}ℓ=1,…,L that each solve
Eq. (). The latter is suitable for the approximation of
Boc and reflects the spatial localization of the sources
of the induced magnetic signal in the oceans. Thus, a magnetic field model
that is based on an expansion of the signal in terms of the function system
{Bℓ}ℓ=1,…,L automatically reflects the spatial
origin of the signal as well as its temporal periodicity (described by the
frequency ω). Additionally, due to the linear connection between
u and Boc, an approximation of
Boc directly yields an approximation of the underlying
tidal current velocity u in terms of the functions
{uℓ}ℓ=1,…,L. However, a model of the underlying
conductivity σ has to be assumed for the construction of the
Bℓ. Throughout this paper, we fix the underlying conductivity,
meaning that we do not test the influence of a variation of the conductivity
model on the approximation of Boc. The goal of the paper
is rather the illustration of the effect of the general constraint that the
(unknown) underlying u is restricted to the oceans. In a
forthcoming study, the simultaneous reconstruction of u and
approximation of Boc, and a comparison with existing
models, shall be investigated more thoroughly. Since the connection between
σ and Boc is nonlinear, a simultaneous
determination of σ and Boc (assuming a fixed
velocity field model for u) is not as straightforward. A detailed
description of the trial functions is provided in Sect. .
In Sect. , we illustrate our approach with input data
derived from the (satellite- and observatory-data-based) CM5 model of
and from data derived from a forward model based on the X3DG
solver from . We approximate these input data sets
separately in terms of time-periodic vector spherical harmonics, a system of
spatially localized trial functions that contains no particular information
on the underlying sources (in our case, Abel–Poisson kernels), and the new
set of trial functions indicated in the previous paragraph, respectively. We
also include an example with artificial continental noise. The residuals with
respect to the input data show that the use of the function system
{Bℓ}ℓ=1,…,L can filter out undesired contributions
to the M2 tidal magnetic field over the continents, without neglecting data
over the continents. These residuals can reach up to 15% of the maximal
signal strength and have a magnitude that should be detectable at satellite
altitude.
The kernel K(r⋅,aη1) for ar=0.91 (a)
and ar=0.67 (b). The fixed nodal point η1∈S is marked by a white cross.
Method and function systems
Given a so-called dictionary D of trial functions, we use the
Regularized (Orthogonal) Functional Matching Pursuit (cf.
for details) for the
approximation of Boc. Briefly speaking, this is a
greedy-type algorithm that yields an approximation
B‾N=∑i=1Nαidi
of Boc by iteratively choosing coefficients
αi∈R and dictionary elements di∈D via
argminα,d‖Ri-1-αFd‖RM2+λ‖B‾i-1+αd‖H2.
Ri-1=b-FB‾i-1 denotes the
residual between the data b∈RM and the approximation
after i-1 iterations. In this particular setup, F represents
the linear operator that evaluates a function at the M locations where data
are provided, and H is a suitable Hilbert space for the
regularization of the problem. The parameter λ controls
the trade-off between the data misfit ‖Ri‖RM2 and
the regularizing term ‖B‾i‖H2, which
imposes a certain property to B‾i such as smoothness (as
in our case). The Regularized (Orthogonal) Functional Matching Pursuit, in
general, has the advantage that it can easily deal with different
dictionaries D (or combinations of such) from which the
approximant B‾N is built. However, any other
approximation method could be used as well with the proposed function
systems. In this paper, we use the term “dictionary” simply to describe a
set of arbitrary functions that we consider suitable for our purposes. These
functions do not necessarily have to satisfy particular mathematical
properties such as orthogonality or completeness. Therefore, we call such
functions “trial functions” rather than, e.g., “basis functions”.
In the following, we briefly introduce some function systems that can be used
for the constitution of D. In particular, Sect.
describes in more detail the aforementioned trial functions
{Bℓ}ℓ=1,…,L that contain temporal and spatial
constraints tailored for ocean-tide-induced magnetic fields.
Absolute value of the vectorial Slepian function
g50(3) with 50th best localization over the oceans
(a) and g1630(3) with the 50th worst
localization over the oceans (b), for bandlimit
N=40.
Vector spherical harmonics
We briefly recapitulate the notion of classical vector spherical harmonics in
a form that we need at a few occasions later on. By
Sr={x∈R3:|x|=r}, we denote the sphere of
radius r, while S=S1 stands for the unit
sphere. Every unit vector ξ=ξ(t,φ)∈S can be
expressed in spherical coordinates with longitude φ and polar
distance t=cos(ϑ), where ϑ is the corresponding
co-latitude. By Yn,k, we denote fully normalized spherical harmonics
of degree n and order k: for every n∈N0 and
k=-n,…,n,
Yn,k(ξ)=2n+14π(n-|k|)!(n+|k|)!Pn|k|(t)2cos(kφ),k<0,1,k=0,2sin(kφ),k>0.
The involved associated Legendre functions are, for t∈[-1,1],
defined as
Pnk(t)=(-1)k2nn!1-t2k/2ddtn+k(t2-1)n.
If f is a scalar-valued square-integrable function on S, then,
for every degree n∈N0 and order k=-n,…,n, the values
f^(n,k)=∫Sf(η)Yn,k(η)dS(η)
are called the Fourier coefficients of the function f.
Going over to the vectorial setting, it is well known that every
square-integrable vector field f on the unit sphere can be uniquely
decomposed into its radial and two tangential components such that
f=erf1+∇Sf2+LSf3,
with scalar-valued functions f1,f2,f3 and the radial unit vector
er=1-t2cos(φ),1-t2sin(φ),tT.
By the surface gradient ∇S, we denote the tangential
component of the usual Euclidean gradient ∇, i.e.,
∇S=eφ11-t2∂∂φ+et1-t2∂∂t,
with unit vectors
et=-tcos(φ),-tsin(φ),1-t2T and
eφ=-sin(φ),cos(φ),0T. Moreover,
the surface curl gradient LS is defined by
LSf(ξ)=ξ×∇Sf(ξ),
where × is the usual cross product in R3. In other
words,
LS=-eφ1-t2∂∂t+et11-t2∂∂φ.
Hence, we define three types of vector spherical harmonics: the radial
yn,k(1)=erYn,k,
for degrees n≥0 and orders k=-n,…,n, as well as the tangential
yn,k(2)=1n(n+1)∇SYn,k,yn,k(3)=1n(n+1)LSYn,k,
for degrees n≥1 and orders k=-n,…,n. Note that, for convenience,
we set y0,0(2)=y0,0(3)=0. It should
further be noted that the vector spherical harmonics in Eq. ()
are surface-curl-free while those in Eq. () are
surface-divergence-free. In analogy to Eq. (), we can
now define the Fourier coefficients
f^(n,k)(i)=∫Sf(η)⋅yn,k(i)(η)dS(η)
of square-integrable vector fields f on the unit sphere.
The vector spherical harmonics from above are defined solely on the unit
sphere and can, therefore, only be used for the expansion of vector-valued
functions on S. However, for the approximation of satellite
potential field data it is necessary to have related functions that are also
defined in the exterior of a sphere. For that purpose, we define the
following gradients of harmonic extensions of (scalar) spherical harmonics:
hn,k(x)=1r2arn∇SYn,k(ξ)-ξ(n+1)Yn,k(ξ),
for r=|x|>a and ξ=x|x|∈S, where a is the radius
of a reference sphere, e.g., Earth's mean radius.
Vectorial Abel–Poisson kernel
While the set of functions
{hn,k}n∈N0,k=-n,…,n from Eq. ()
is suitable for the global approximation of potential field data, we are also
interested in localized functions. One possible choice is the Abel–Poisson
kernel (see, e.g., ). For
x,y∈R3, |x|>|y|, it is defined by
K(x,y)=14π|x|2-|y|2|x-y|3.
That is, with unit vectors ξ,η∈S and radii r>a>0, we
have
K(rξ,aη)=14πr2-a2a2+r2-2ar(ξ⋅η)3/2,
which shows that K only depends on the spherical distance between ξ and
η, since |ξ-η|2=2(1-ξ⋅η). Therefore, the kernel is
radially symmetric if we keep one of the variables fixed (we strictly keep
the second argument fixed, here aη). The degree of localization is
determined by the ratio ar. The closer it is to 1, i.e., the
smaller the difference between the radii a and r, the better the
spatial localization of K(r⋅,aη) around η. In our case, we
choose a to be the Earth's mean radius, and r is the radius of the sphere
at which we evaluate the kernel. An illustration of the kernel is provided in
Fig. .
Absolute value of the trial function B50re
corresponding to u=g50(3) at time t=0 (a)
and the trial function B1630re corresponding to
u=g1630(3) at time t=0 (b). Models of
surface shell conductance on Sa and absolute value of
Bmain on Sa used for the generation of the
trial functions (c, d).
The corresponding vectorial Abel–Poisson kernel is simply defined by
k(x,y)=∇xK(x,y)=14π∑n=0∞∑k=-nnYn,ky|y|hn,k(x).
Further calculations show
k(x,y)=14π2|x-y|3x-3|x|2-|y|2|x-y|5(x-y).
A spatially localized alternative to
{hn,k}n∈N0,k=-n,…,n could then be defined by
the set of functions {k(⋅,aηi)}i=1,…,M, where
η1,…,ηM∈S is a fixed set of adequately
distributed nodal points.
Spherical Slepian functions
While the localization of Abel–Poisson kernels is of radially symmetric
nature, one is often interested in regions of more complex geometry, e.g.,
continents or oceans. Spherical Slepian functions, for instance, provide an
orthonormal system of functions that can reflect localization in such general
predefined regions Γ⊂S (see, e.g.,
for
details).
Specifically, the function f showing the best localization in Γ,
is the one that maximizes the energy ratio
λΓ(f)=∫Γ|f(η)|2dS(η)∫S|f(η)|2dS(η),
i.e., the one with an energy ratio closest to 1. Let us now assume that
g(i) is a bandlimited vectorial function of type i with
bandlimit N; i.e., it can be expanded as
g(i)=∑n=0N∑k=-nng^(n,k)(i)yn,k(i).
Further, the matrix
P=(P(n,k),(m,j))∈R(N+1)2×(N+1)2 contains (properly sorted) all of the appearing inner
products
P(n,k),(m,j)=∫Γyn,k(i)(η)⋅ym,j(i)(η)dS(η)
and
g^=(g^(n,k)(i))T∈R(N+1)2,
with n=0,…,N and k=-n,…,n. If we now restrict ourselves to
normalized functions g(i) (i.e.,
∫S|g(i)(η)|2dS(η)=g^Tg^=1), one obtains the simple expression
λΓ(g(i))=g^TPg^. Eventually, the maximization of the energy ratio
Eq. () leads to the eigenvalue problem
Pg^=λg^.
The eigenvalues λℓ are the possible energy ratios and the
corresponding eigenvectors g^ℓ contain the Fourier
coefficients of bandlimited functions gℓ(i) attaining the
energy ratio
λℓ=λΓgℓ(i). The set of
functions {gℓ(i)}ℓ=1,…,(N+1)2 is ordered
such that 1≥λ1≥λ2≥…≥λ(N+1)2≥0.
Accumulated energy for {uℓ}ℓ=1,…,L
(a) and
{Bℓre,Bℓim}ℓ=1,…,L
(b), with L=1200.
In typical scenarios, it turns out that the eigenvalues are clustered close
to 1 and close to zero. Those eigenvalues λ1,…,λL which
are closer to 1 determine the subset
{gℓ(i)}ℓ=1,…,L of well-localized Slepian
functions that should be used for approximation in Γ. The code for the
generation of vectorial Slepian functions has been kindly supplied in
. For our situation, where Γ denotes the
region of (a spherical) Earth which is covered by oceans, an illustration is
provided in Fig. .
Absolute value of the radial part of the tidal model
BocCM5 as well as the forward model
BocX3DG at an altitude of 300 km above
the Earth's surface.
Physics-based trial functions
We start with the time-harmonic Maxwell equations as already indicated in
Eq. (). For simplicity, we assume a 1-D (only radially
varying) conductivity model for σ within the ball Ba, and
at the surface Sa we allow a laterally varying conductivity
(cf. the bottom left image in Fig. for an illustration).
Further, the magnetic field Bmain is taken from the CHAOS-5
model (see ) and u is supposed to denote a
depth-integrated velocity field that is restricted to Sa (in
fact, within the numerical framework of the X3DG solver, we assume a constant
ocean depth of 1km, with u being tangential to the sphere and
independent of the depth). Since we are mainly interested in tidal velocity
fields, it is a reasonable assumption that u is
surface-divergence-free for most parts of the oceans. The latter means that
u can be expanded in terms of vector spherical harmonics or vectorial
Slepian functions of type 3, i.e., yn,k(3) or
gℓ(3), respectively.
Absolute value of the radial part of approximations of
BocCM5 based on dictionary D1
(a), dictionary D2 (b), and dictionary
D3 (c), as well as the corresponding residuals with
respect to BocCM5 (d, e, f). Note
the different scales in the bottom row which are chosen in order to emphasize
the spatial distribution of the residuals.
For the generation of the tailored trial functions
{Bℓ}ℓ=1,…,L, we therefore substitute u by a
set of surface-divergence-free functions {uℓ}ℓ=1,…,L
that reflect spatial localization within the oceans. More precisely, we
choose
uℓ=gℓ(3),
where gℓ(3) is the ℓth best localized vectorial
Slepian function of type 3. The corresponding solution
Boc of Eq. () within this setup then
provides an auxiliary function B̃ℓ. It should be noted
that in order to obtain Maxwell's equation in the time-harmonic form
Eq. (), one has to apply a Fourier transform in time.
Therefore, for the actual trial function Bℓ, we have to invert
the Fourier transform and get
Bℓ(x,t)=e-iωtB̃ℓ(x),x∈R3,t∈R.
For technical reasons, we choose to work in a real-valued framework, so that
the real and imaginary part of Bℓ each yield a trial function
Bℓre(x,t)=cos(ωt)B̃ℓre(x)+sin(ωt)B̃ℓim(x),Bℓim(x,t)=sin(ωt)B̃ℓre(x)-cos(ωt)B̃ℓim(x).
Thus, each choice of uℓ yields two functions
Bℓre and Bℓim that reflect
the temporal periodicity of the tidal magnetic field as well as the spatial
localization of the sources within the oceans. An illustration for the M2
tide with ω=2π12.42h can be found in Fig. . For
the computation of the B̃ℓ as solutions of
Eq. (), we have used the X3DG solver from
.
Absolute value of the radial part of approximations of
BocX3DG based on dictionary D1
(a), dictionary D2 (b), and dictionary
D3 (c), as well as the corresponding residuals with
respect to BocX3DG (d, e, f).
Figure shows the accumulated energy
∑ℓ=1L|uℓ(ξ)|2, for ξ∈S, of the
underlying functions uℓ that describe the velocity field and the
accumulated energy
∑ℓ=1L|Bℓre(x,t)|2+|Bℓim(x,t)|2,
for x∈Sr with r=a+300 km and time t=0, of the
corresponding trial functions. In both cases, one can clearly see the spatial
localization over the oceans. However, the accumulated energy of the trial
functions additionally reflects the influence of the conductivity σ
and the main/core magnetic field Bmain indicated in
Fig. .
Absolute value of the radial part of
BocX3DG (a) and model of
continental “noise” e (b), as well as superposition
Boce of both of
these (c).
Examples
For our experiments we rely on the CM5 geomagnetic field model (cf.
) and a forward model based on the M2 depth-integrated
tidal velocity field from TPXO8-ATLAS (cf. ) that
has also been used in . The contribution of CM5 that is due
to the oceanic M2 tide is given as an expansion in terms of spherical
harmonics up to degree 18; we denote it as
BocCM5 for the remainder of this section
and sample it at M=250000 points which are taken from actual Swarm
satellite tracks. The forward model has been computed via the X3DG solver
based on the surface conductance and the main/core magnetic field model
indicated in the bottom row of Fig. and a depth-integrated M2
tidal velocity field from TPXO8-ATLAS. We denote it by
BocX3DG and evaluate it on the same point
grid as before. These samples are used as input data
b∈RM for the Regularized (Orthogonal) Functional
Matching Pursuit, which works iteratively as indicated in
Eq. (). In the following, we want to illustrate the influence
of the choice of different function systems (i.e., the choice of different
dictionaries D) on the approximation of
BocCM5 and
BocX3DG. For that purpose, we choose three
different dictionaries: the spherical-harmonic-based
D1={cos(ωt)hn,k(x),sin(ωt)hn,k(x)}n=0,…,20,k=-n,…,n,
with ω=2π12.42h and the functions hn,k from
Eq. (); the Abel–Poisson-kernel-based
D2={cos(ωt)k(x,aηi),sin(ωt)k(x,aηi)}i=1,…,Mp,
where a=6371.2 km, {ηi}i=1,…,Mp is a Reuter grid on
S with Mp=6201 nearly equally distributed points (see, e.g.,
, p. 137), and k given as in Eq. (); and
D3={Bℓre(x,t),Bℓim(x,t)}ℓ=1,…,1200,
with Bℓre and Bℓim, the
physics-based trial functions from Eqs. () and
().
Absolute value of the radial part of approximations of superposition
Boce based on dictionary D1
(a), dictionary D2 (b), and dictionary
D3 (c), as well as the corresponding residuals with
respect to original BocX3DG without the
continental “noise” e (d, e, f).
The actual signals that we want to approximate are indicated in
Fig. . The approximations B‾N of
BocCM5 together with the residuals
|B‾N-BocCM5| for each of
the three dictionaries above are shown in Fig. , whereas the
respective approximations of BocX3DG and
corresponding residuals
|B‾N-BocX3DG| are
displayed in Fig. .
The difference between the approximations B‾N
of the undisturbed BocX3DG (as shown in
Fig. a, b, and c) and the approximations
B‾Ne of the noisy
Boce (as shown in Fig. a, b,
and c).
Root mean square errors (RMSEs) corresponding to the approximations
of the undisturbed forward model BocX3DG as
well as the noisy model
Boce=BocX3DG+e
compared to the “ground truth” BocX3DG
with the three different dictionaries. Left-hand columns show errors for the
approximation of BocX3DG (compare
Fig. ), while center columns show errors for the
approximation of Boce (compare
Fig. ). The right-hand columns display the RMSEs of the
difference between the respective approximations (as shown in
Fig. ).
B‾N-BocX3DG
B‾Ne-BocX3DG
B‾N-B‾Ne
RMSE
D1
D2
D3
D1
D2
D3
D1
D2
D3
Overall
0.060533
0.029952
0.064133
0.107972
0.128185
0.086386
0.089407
0.126170
0.057929
Continents
0.045495
0.021553
0.055487
0.160529
0.237245
0.085791
0.153771
0.236059
0.067152
Oceans
0.064366
0.032047
0.086563
0.086145
0.066827
0.086563
0.057396
0.062991
0.054858
In the case of BocCM5 as the underlying
signal, it can be seen in Fig. that the dictionary
D1 yields the overall best approximation, which, however, is not
surprising since we try to fit a spherical-harmonic-based model with a
spherical-harmonic-based dictionary. The result for dictionary
D2 shows a more localized pattern in the residual, as is
expected for the use of Abel–Poisson kernels. However, the maxima in the
residual are not correlated to specific continental or oceanic structures but
they mainly coincide with the maxima of the original signal
BocCM5. The situation for dictionary
D3 of physics-based trial functions is different. The agreement
of the approximation with BocCM5 is good
over the oceans but significant deviations exist over the continents. The
latter could be an indication that the original model
BocCM5 contains contributions over the
continents whose physical origin is not due to induction by oceanic tides.
Some smaller deviations over oceanic areas exist around southern Africa and
the east of Australia. Since we are dealing with the approximation of a
low-degree (up to degree 18) spherical-harmonic-based M2 tidal magnetic field
model by localized trial functions, one cannot reliably say if the latter
deviations are artifacts from the approximation procedure or if they have a
physical origin. However, those are areas with a shallower ocean topography,
so the assumption of surface-divergence-free depth-integrated tidal
velocities (which we made for our choice of the underlying uℓ)
and the assumption of a constant ocean depth (we chose a depth of 1 km for
the generation of the B̃ℓ via the X3DG solver) might not
be accurate in these areas. Nonetheless, the residuals over the continents
show that the use of the adapted trial functions might eventually deliver
improved tidal magnetic field models that correct unrealistic continental
contributions without disregarding continental areas entirely.
The residuals of the approximations of the forward model
BocX3DG in Fig. , on the
other hand, indicate that the quality of the approximations does not vary too
much (at least on scales that are relevant for satellite data approximation)
among the three tested function systems. This is mainly due to the fact that
the input model BocX3DG already reflects
certain spatial localization properties over the oceans. In such a scenario
(if additionally solely interested in the approximation of the signal and not
the underlying velocity fields) it would, therefore, not be necessary to use
the adapted trial functions that we have introduced. The crucial point,
however, is that satellite data typically contain undesired contributions
over the continents that are not due to ocean-tide-generated magnetic fields.
In order to illustrate this behavior, we use the following additional
example. We take randomly distributed Fourier coefficients to construct a
“noise” function e with a bandlimit of degree 40, i.e.,
e=∑n=140∑k=-nne^(n,k)yn,k(3),
where the Fourier coefficients e^(n,k) are normally distributed
with zero mean and a variance such that the amplitude of e is in the
range of the oceanic signal BocX3DG. This
function e is then restricted to the continents and eventually
superposed with the forward model (cf. Fig. ). For the sake
of clarity, we denote the approximations of the noisy data
Boce=BocX3DG+e
by B‾Ne instead of B‾N.
The latter still represents the approximation of
BocX3DG without extra continental noise e.
In Fig. one can directly see the influence which the
continental noise has on the approximation depending on the various
dictionaries (Fig. shows the same quantities for the
approximations in the undisturbed setup). In the case of dictionary
D1, the spherical harmonics also approximate a part of the
continental data which in turn also has some impact on the approximation in
oceanic areas. Due to the localization of the kernels contained in dictionary
D2, the (undesired) reconstruction of the continental noise is
even more accurate, while the reconstruction over the oceans only changes
very slightly. With the proposed physics-based functions in dictionary
D3, however, the influence of the continental noise is much less
apparent. The maxima occur very close to the coastline, which is most likely
due to numerical issues stemming from discontinuities of the data
Boce in coastal areas. A closer look at the
differences |B‾N-B‾Ne|
between the approximations of noisy and undisturbed data is given in
Fig. . This shows again that the inclusion of continental
noise has a smaller effect on the approximation via physics-based trial
functions than on the approximations via the other tested trial functions.
Moreover, the corresponding root mean square errors of the approximations
B‾N and B‾Ne,
respectively, can be found in Table . In both cases, we compared
the approximations to the undisturbed data
BocX3DG in order to emphasize the impact of
continental noise on the overall approximation. The errors over continental
and oceanic regions are provided separately.