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- About
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ANGEO | Articles | Volume 36, issue 5

Ann. Geophys., 36, 1393-1402, 2018

https://doi.org/10.5194/angeo-36-1393-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/angeo-36-1393-2018

© Author(s) 2018. This work is distributed under

the Creative Commons Attribution 4.0 License.

Special issue: Dynamics and interaction of processes in the Earth and its...

**Regular paper**
18 Oct 2018

**Regular paper** | 18 Oct 2018

Approximation of spatial structures of tidal magnetic fields

^{1}Computational Science Center, University of Vienna, 1090 Vienna, Austria^{2}Helmholtz Centre Potsdam German Research Centre for Geosciences – GFZ, Section 2.3 Geomagnetism, 14467 Potsdam, Germany

Abstract

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The extraction of the magnetic signal induced by the oceanic M2 tide is typically based solely on the temporal periodicity of the signal. Here, we propose a system of tailored trial functions that additionally takes the spatial constraint into account that the sources of the signal are localized within the oceans. This construction requires knowledge of the underlying conductivity model but not of the inducing tidal current velocity. Approximations of existing tidal magnetic field models with these trial functions and comparisons with approximations based on other localized and nonlocalized trial functions are illustrated.

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How to cite.

Telschow, R., Gerhards, C., and Rother, M.: On the approximation of spatial structures of global tidal magnetic field models, Ann. Geophys., 36, 1393-1402, https://doi.org/10.5194/angeo-36-1393-2018, 2018.

1 Introduction

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Conductive seawater moving through the ambient Earth's main magnetic field
*B*_{main} induces a secondary magnetic field
*B*_{oc}. 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., Malin (1970). 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.,
Sabaka et al., 2015, 2016; Tyler et al., 2003). 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 Kuvshinov and Olsen (2005). A more extensive comparison of
forward models of electromagnetic ocean tidal signals based on different
ocean tide models has recently been published in Saynisch et al. (2018). 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 Grayver et al. (2016) and
Schnepf et al. (2015) 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 Irrgang et al. (2016) 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

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathrm{\nabla}\times {\mathit{B}}_{\mathrm{oc}}={\mathit{\mu}}_{\mathrm{0}}\mathit{\sigma}(\mathit{E}+\mathit{u}\times {\mathit{B}}_{\mathrm{main}}),\\ \text{(1)}& {\displaystyle}& {\displaystyle}\mathrm{\nabla}\times \mathit{E}=i\mathit{\omega}{\mathit{B}}_{\mathrm{oc}},{\displaystyle}& {\displaystyle}\mathrm{\nabla}\cdot {\mathit{B}}_{\mathrm{oc}}=\mathrm{0},\end{array}$$

where we assume to have knowledge of *B*_{main}, the
underlying conductivity *σ*, and the frequency *ω*. Furthermore,
** E** denotes the electric field and

In Sect. 3, we illustrate our approach with input data derived from the (satellite- and observatory-data-based) CM5 model of Sabaka et al. (2015) and from data derived from a forward model based on the X3DG solver from Kuvshinov (2008). 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 $\mathit{\{}{\mathit{B}}_{\mathrm{\ell}}{\mathit{\}}}_{\mathrm{\ell}=\mathrm{1},\mathrm{\dots},L}$ 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.

2 Method and function systems

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Given a so-called dictionary 𝒟 of trial functions, we use the
Regularized (Orthogonal) Functional Matching Pursuit (cf.
Fischer and Michel, 2013; Michel and Telschow, 2014, 2016 for details) for the
approximation of *B*_{oc}. Briefly speaking, this is a
greedy-type algorithm that yields an approximation

$$}{\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}=\sum _{i=\mathrm{1}}^{N}{\mathit{\alpha}}_{i}{\mathit{d}}_{i$$

of *B*_{oc} by iteratively choosing coefficients
*α*_{i}∈ℝ and dictionary elements *d*_{i}∈𝒟 via

$$\begin{array}{}\text{(2)}& {\displaystyle}{\mathrm{argmin}}_{\mathit{\alpha},\mathit{d}}\left(\Vert {\mathit{R}}_{i-\mathrm{1}}-\mathit{\alpha}\mathcal{F}\mathit{d}{\Vert}_{{\mathbb{R}}^{M}}^{\mathrm{2}}+\mathit{\lambda}\Vert {\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{i-\mathrm{1}}+\mathit{\alpha}\mathit{d}{\Vert}_{\mathcal{H}}^{\mathrm{2}}\right).\end{array}$$

${\mathit{R}}_{i-\mathrm{1}}=\mathit{b}-\mathcal{F}{\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{i-\mathrm{1}}$ denotes the
residual between the data ** b**∈ℝ

In the following, we briefly introduce some function systems that can be used for the constitution of 𝒟. In particular, Sect. 2.4 describes in more detail the aforementioned trial functions $\mathit{\{}{\mathit{B}}_{\mathrm{\ell}}{\mathit{\}}}_{\mathrm{\ell}=\mathrm{1},\mathrm{\dots},L}$ that contain temporal and spatial constraints tailored for ocean-tide-induced magnetic fields.

We briefly recapitulate the notion of classical vector spherical harmonics in
a form that we need at a few occasions later on. By
${\mathbb{S}}_{r}=\mathit{\{}x\in {\mathbb{R}}^{\mathrm{3}}:|x|=r\mathit{\}}$, we denote the sphere of
radius *r*, while 𝕊=𝕊_{1} stands for the unit
sphere. Every unit vector $\mathit{\xi}=\mathit{\xi}(t,\mathit{\phi})\in \mathbb{S}$ can be
expressed in spherical coordinates with longitude *φ* and polar
distance *t*=cos(*ϑ*), where *ϑ* is the corresponding
co-latitude. By *Y*_{n,k}, we denote fully normalized spherical harmonics
of degree *n* and order *k*: for every *n*∈ℕ_{0} and
$k=-n,\mathrm{\dots},n$,

$$}{Y}_{n,k}\left(\mathit{\xi}\right)=\sqrt{{\displaystyle \frac{\mathrm{2}n+\mathrm{1}}{\mathrm{4}\mathit{\pi}}}{\displaystyle \frac{(n-|k\left|\right)\mathrm{!}}{(n+|k\left|\right)\mathrm{!}}}}\phantom{\rule{0.25em}{0ex}}{P}_{n}^{\left|k\right|}\left(t\right)\left\{\begin{array}{ll}\sqrt{\mathrm{2}}\mathrm{cos}\left(k\mathit{\phi}\right),& k<\mathrm{0},\\ \mathrm{1},& k=\mathrm{0},\\ \sqrt{\mathrm{2}}\mathrm{sin}\left(k\mathit{\phi}\right),& k>\mathrm{0}.\end{array}\right.$$

The involved associated Legendre functions are, for $t\in [-\mathrm{1},\mathrm{1}]$, defined as

$$}{P}_{n}^{k}\left(t\right)={\displaystyle \frac{(-\mathrm{1}{)}^{k}}{{\mathrm{2}}^{n}n\mathrm{!}}}{\left(\mathrm{1}-{t}^{\mathrm{2}}\right)}^{k/\mathrm{2}}{\left({\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\right)}^{n+k}({t}^{\mathrm{2}}-\mathrm{1}{)}^{n}.$$

If *f* is a scalar-valued square-integrable function on 𝕊, then,
for every degree *n*∈ℕ_{0} and order $k=-n,\mathrm{\dots},n$, the values

$$\begin{array}{}\text{(3)}& {\displaystyle}{\widehat{f}}_{(n,k)}=\underset{\mathbb{S}}{\int}f\left(\mathit{\eta}\right){Y}_{n,k}\left(\mathit{\eta}\right)\mathrm{d}S\left(\mathit{\eta}\right)\end{array}$$

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

$$}\mathit{f}={\mathit{e}}_{r}{f}_{\mathrm{1}}+{\mathrm{\nabla}}_{\mathbb{S}}{f}_{\mathrm{2}}+{\mathit{L}}_{\mathbb{S}}{f}_{\mathrm{3}},$$

with scalar-valued functions ${f}_{\mathrm{1}},{f}_{\mathrm{2}},{f}_{\mathrm{3}}$ and the radial unit vector
${\mathit{e}}_{r}={\left(\sqrt{\mathrm{1}-{t}^{\mathrm{2}}}\mathrm{cos}\left(\mathit{\phi}\right),\sqrt{\mathrm{1}-{t}^{\mathrm{2}}}\mathrm{sin}\left(\mathit{\phi}\right),t\right)}^{T}$.
By the surface gradient ∇_{𝕊}, we denote the tangential
component of the usual Euclidean gradient ∇, i.e.,

$$}{\mathrm{\nabla}}_{\mathbb{S}}={\mathit{e}}_{\mathit{\phi}}{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{1}-{t}^{\mathrm{2}}}}}{\displaystyle \frac{\partial}{\partial \mathit{\phi}}}+{\mathit{e}}_{t}\sqrt{\mathrm{1}-{t}^{\mathrm{2}}}{\displaystyle \frac{\partial}{\partial t}},$$

with unit vectors
${\mathit{e}}_{t}={\left(-t\mathrm{cos}\left(\mathit{\phi}\right),-t\mathrm{sin}\left(\mathit{\phi}\right),\sqrt{\mathrm{1}-{t}^{\mathrm{2}}}\right)}^{T}$ and
${\mathit{e}}_{\mathit{\phi}}={\left(-\mathrm{sin}\left(\mathit{\phi}\right),\mathrm{cos}\left(\mathit{\phi}\right),\mathrm{0}\right)}^{T}$. Moreover,
the surface curl gradient *L*_{𝕊} is defined by
${\mathit{L}}_{\mathbb{S}}f\left(\mathit{\xi}\right)=\mathit{\xi}\times {\mathrm{\nabla}}_{\mathbb{S}}f\left(\mathit{\xi}\right)$,
where × is the usual cross product in ℝ^{3}. In other
words,

$$}{\mathit{L}}_{\mathbb{S}}=-{\mathit{e}}_{\mathit{\phi}}\sqrt{\mathrm{1}-{t}^{\mathrm{2}}}{\displaystyle \frac{\partial}{\partial t}}+{\mathit{e}}_{t}{\displaystyle \frac{\mathrm{1}}{\sqrt{\mathrm{1}-{t}^{\mathrm{2}}}}}{\displaystyle \frac{\partial}{\partial \mathit{\phi}}}.$$

Hence, we define three types of vector spherical harmonics: the radial

$$}{\mathit{y}}_{n,k}^{\left(\mathrm{1}\right)}{\displaystyle}={\mathit{e}}_{r}\phantom{\rule{0.125em}{0ex}}{Y}_{n,k},$$

for degrees *n*≥0 and orders $k=-n,\mathrm{\dots},n$, as well as the tangential

$$\begin{array}{}\text{(4)}& {\displaystyle}& {\displaystyle}{\mathit{y}}_{n,k}^{\left(\mathrm{2}\right)}=\sqrt{{\displaystyle \frac{\mathrm{1}}{n(n+\mathrm{1})}}}{\mathrm{\nabla}}_{\mathbb{S}}{Y}_{n,k},\text{(5)}& {\displaystyle}& {\displaystyle}{\mathit{y}}_{n,k}^{\left(\mathrm{3}\right)}=\sqrt{{\displaystyle \frac{\mathrm{1}}{n(n+\mathrm{1})}}}{\mathit{L}}_{\mathbb{S}}{Y}_{n,k},\end{array}$$

for degrees *n*≥1 and orders $k=-n,\mathrm{\dots},n$. Note that, for convenience,
we set ${\mathit{y}}_{\mathrm{0},\mathrm{0}}^{\left(\mathrm{2}\right)}={\mathit{y}}_{\mathrm{0},\mathrm{0}}^{\left(\mathrm{3}\right)}=\mathrm{0}$. It should
further be noted that the vector spherical harmonics in Eq. (4)
are surface-curl-free while those in Eq. (5) are
surface-divergence-free. In analogy to Eq. (3), we can
now define the Fourier coefficients

$$}{\widehat{f}}_{(n,k)}^{\left(i\right)}=\underset{\mathbb{S}}{\int}\mathit{f}\left(\mathit{\eta}\right)\cdot {\mathit{y}}_{n,k}^{\left(i\right)}\left(\mathit{\eta}\right)\mathrm{d}S\left(\mathit{\eta}\right)$$

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:

$$\begin{array}{}\text{(6)}& {\displaystyle}{\mathit{h}}_{n,k}\left(x\right)={\displaystyle \frac{\mathrm{1}}{{r}^{\mathrm{2}}}}{\left({\displaystyle \frac{a}{r}}\right)}^{n}\left({\mathrm{\nabla}}_{\mathbb{S}}{Y}_{n,k}\left(\mathit{\xi}\right)-\mathit{\xi}(n+\mathrm{1}){Y}_{n,k}\left(\mathit{\xi}\right)\right),\end{array}$$

for $r=\left|x\right|>a$ and $\mathit{\xi}=\frac{x}{\left|x\right|}\in \mathbb{S}$, where *a* is the radius
of a reference sphere, e.g., Earth's mean radius.

While the set of functions $\mathit{\{}{\mathit{h}}_{n,k}{\mathit{\}}}_{n\in {\mathbb{N}}_{\mathrm{0}},k=-n,\mathrm{\dots},n}$ from Eq. (6) 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., Freeden and Gerhards, 2012; Freeden et al., 1998). For $x,y\in {\mathbb{R}}^{\mathrm{3}}$, $\left|x\right|>\left|y\right|$, it is defined by

$$}K(x,y)={\displaystyle \frac{\mathrm{1}}{\mathrm{4}\mathit{\pi}}}{\displaystyle \frac{|x{|}^{\mathrm{2}}-|y{|}^{\mathrm{2}}}{|x-y{|}^{\mathrm{3}}}}\phantom{\rule{0.125em}{0ex}}.$$

That is, with unit vectors $\mathit{\xi},\mathit{\eta}\in \mathbb{S}$ and radii $r>a>\mathrm{0}$, we have

$$}K(r\mathit{\xi},a\mathit{\eta})={\displaystyle \frac{\mathrm{1}}{\mathrm{4}\mathit{\pi}}}{\displaystyle \frac{{r}^{\mathrm{2}}-{a}^{\mathrm{2}}}{{\left({a}^{\mathrm{2}}+{r}^{\mathrm{2}}-\mathrm{2}ar(\mathit{\xi}\cdot \mathit{\eta})\right)}^{\mathrm{3}/\mathrm{2}}}},$$

which shows that *K* only depends on the spherical distance between *ξ* and
*η*, since $|\mathit{\xi}-\mathit{\eta}{|}^{\mathrm{2}}=\mathrm{2}(\mathrm{1}-\mathit{\xi}\cdot \mathit{\eta})$. Therefore, the kernel is
radially symmetric if we keep one of the variables fixed (we strictly keep
the second argument fixed, here *a**η*). The degree of localization is
determined by the ratio $\frac{a}{r}$. The closer it is to 1, i.e., the
smaller the difference between the radii *a* and *r*, the better the
spatial localization of $K(r\cdot ,a\mathit{\eta})$ 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. 1.

The corresponding vectorial Abel–Poisson kernel is simply defined by

$$}\mathit{k}(x,y){\displaystyle}={\mathrm{\nabla}}_{x}K(x,y)={\displaystyle \frac{\mathrm{1}}{\mathrm{4}\mathit{\pi}}}\sum _{n=\mathrm{0}}^{\mathrm{\infty}}\sum _{k=-n}^{n}{Y}_{n,k}\left({\displaystyle \frac{y}{\left|y\right|}}\right){h}_{n,k}\left(x\right).$$

Further calculations show

$$\begin{array}{}\text{(7)}& {\displaystyle}\mathit{k}(x,y)={\displaystyle \frac{\mathrm{1}}{\mathrm{4}\mathit{\pi}}}\left({\displaystyle \frac{\mathrm{2}}{|x-y{|}^{\mathrm{3}}}}\phantom{\rule{0.125em}{0ex}}x\phantom{\rule{0.33em}{0ex}}-\phantom{\rule{0.33em}{0ex}}\mathrm{3}\phantom{\rule{0.125em}{0ex}}{\displaystyle \frac{|x{|}^{\mathrm{2}}-|y{|}^{\mathrm{2}}}{|x-y{|}^{\mathrm{5}}}}(x-y)\right).\end{array}$$

A spatially localized alternative to $\mathit{\{}{\mathit{h}}_{n,k}{\mathit{\}}}_{n\in {\mathbb{N}}_{\mathrm{0}},k=-n,\mathrm{\dots},n}$ could then be defined by the set of functions $\mathit{\left\{}\mathit{k}\right(\cdot ,a{\mathit{\eta}}_{i}){\mathit{\}}}_{i=\mathrm{1},\mathrm{\dots},M}$, where ${\mathit{\eta}}_{\mathrm{1}},\mathrm{\dots},{\mathit{\eta}}_{M}\in \mathbb{S}$ is a fixed set of adequately distributed nodal points.

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., Plattner and Simons, 2015, 2017a; Simons and Plattner, 2015; Simons et al., 2006 for details).

Specifically, the function *f* showing the best localization in Γ,
is the one that maximizes the energy ratio

$$\begin{array}{}\text{(8)}& {\displaystyle}{\mathit{\lambda}}_{\mathrm{\Gamma}}\left(\mathit{f}\right)={\displaystyle \frac{{\int}_{\mathrm{\Gamma}}\left|\mathit{f}\right(\mathit{\eta}\left){|}^{\mathrm{2}}\mathrm{d}S\right(\mathit{\eta})}{{\int}_{\mathbb{S}}\left|\mathit{f}\right(\mathit{\eta}\left){|}^{\mathrm{2}}\mathrm{d}S\right(\mathit{\eta})}}\phantom{\rule{0.125em}{0ex}},\end{array}$$

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

$$}{\mathit{g}}^{\left(i\right)}=\sum _{n=\mathrm{0}}^{N}\sum _{k=-n}^{n}{\widehat{g}}_{(n,k)}^{\left(i\right)}{\mathit{y}}_{n,k}^{\left(i\right)}\phantom{\rule{0.125em}{0ex}}.$$

Further, the matrix
$\mathbf{P}=\left({\mathbf{P}}_{(n,k),(m,j)}\right)\in {\mathbb{R}}^{(N+\mathrm{1}{)}^{\mathrm{2}}\times (N+\mathrm{1}{)}^{\mathrm{2}}}$ contains (properly sorted^{2}) all of the appearing inner
products

$$}{\mathbf{P}}_{(n,k),(m,j)}=\underset{\mathrm{\Gamma}}{\int}{\mathit{y}}_{n,k}^{\left(i\right)}\left(\mathit{\eta}\right)\cdot {\mathit{y}}_{m,j}^{\left(i\right)}\left(\mathit{\eta}\right)\mathrm{d}S\left(\mathit{\eta}\right)$$

and
$\widehat{\mathit{g}}=({\widehat{g}}_{(n,k)}^{\left(i\right)}{)}^{\mathrm{T}}\in {\mathbb{R}}^{(N+\mathrm{1}{)}^{\mathrm{2}}}$,
with $n=\mathrm{0},\mathrm{\dots},N$ and $k=-n,\mathrm{\dots},n$. If we now restrict ourselves to
normalized functions *g*^{(i)} (i.e.,
${\int}_{\mathbb{S}}\left|{\mathit{g}}^{\left(i\right)}\right(\mathit{\eta}\left){|}^{\mathrm{2}}\mathrm{d}S\right(\mathit{\eta})={\widehat{\mathit{g}}}^{\mathrm{T}}\widehat{\mathit{g}}=\mathrm{1}$), one obtains the simple expression
${\mathit{\lambda}}_{\mathrm{\Gamma}}\left({\mathit{g}}^{\left(i\right)}\right)={\widehat{\mathit{g}}}^{\mathrm{T}}\mathbf{P}\widehat{\mathit{g}}$. Eventually, the maximization of the energy ratio
Eq. (8) leads to the eigenvalue problem

$$\mathbf{P}\widehat{\mathit{g}}=\mathit{\lambda}\widehat{\mathit{g}}\phantom{\rule{0.125em}{0ex}}.$$

The eigenvalues *λ*_{ℓ} are the possible energy ratios and the
corresponding eigenvectors ${\widehat{\mathit{g}}}_{\mathrm{\ell}}$ contain the Fourier
coefficients of bandlimited functions ${\mathit{g}}_{\mathrm{\ell}}^{\left(i\right)}$ attaining the
energy ratio
${\mathit{\lambda}}_{\mathrm{\ell}}={\mathit{\lambda}}_{\mathrm{\Gamma}}\left({\mathit{g}}_{\mathrm{\ell}}^{\left(i\right)}\right)$. The set of
functions $\mathit{\{}{\mathit{g}}_{\mathrm{\ell}}^{\left(i\right)}{\mathit{\}}}_{\mathrm{\ell}=\mathrm{1},\mathrm{\dots},(N+\mathrm{1}{)}^{\mathrm{2}}}$ is ordered
such that $\mathrm{1}\ge {\mathit{\lambda}}_{\mathrm{1}}\ge {\mathit{\lambda}}_{\mathrm{2}}\ge \mathrm{\dots}\ge {\mathit{\lambda}}_{(N+\mathrm{1}{)}^{\mathrm{2}}}\ge \mathrm{0}$.

In typical scenarios, it turns out that the eigenvalues are clustered close to 1 and close to zero. Those eigenvalues ${\mathit{\lambda}}_{\mathrm{1}},\mathrm{\dots},{\mathit{\lambda}}_{L}$ which are closer to 1 determine the subset $\mathit{\{}{\mathit{g}}_{\mathrm{\ell}}^{\left(i\right)}{\mathit{\}}}_{\mathrm{\ell}=\mathrm{1},\mathrm{\dots},L}$ 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 Plattner and Simons (2017b). For our situation, where Γ denotes the region of (a spherical) Earth which is covered by oceans, an illustration is provided in Fig. 2.

We start with the time-harmonic Maxwell equations as already indicated in
Eq. (1). For simplicity, we assume a 1-D (only radially
varying) conductivity model for *σ* within the ball 𝔹_{a}, and
at the surface 𝕊_{a} we allow a laterally varying conductivity
(cf. the bottom left image in Fig. 3 for an illustration).
Further, the magnetic field *B*_{main} is taken from the CHAOS-5
model (see Finlay et al., 2015) and ** u** is supposed to denote a
depth-integrated velocity field that is restricted to 𝕊

For the generation of the tailored trial functions
$\mathit{\{}{\mathit{B}}_{\mathrm{\ell}}{\mathit{\}}}_{\mathrm{\ell}=\mathrm{1},\mathrm{\dots},L}$, we therefore substitute ** u** by a
set of surface-divergence-free functions $\mathit{\{}{\mathit{u}}_{\mathrm{\ell}}{\mathit{\}}}_{\mathrm{\ell}=\mathrm{1},\mathrm{\dots},L}$
that reflect spatial localization within the oceans. More precisely, we
choose

$${\mathit{u}}_{\mathrm{\ell}}={\mathit{g}}_{\mathrm{\ell}}^{\left(\mathrm{3}\right)},$$

where ${\mathit{g}}_{\mathrm{\ell}}^{\left(\mathrm{3}\right)}$ is the ℓth best localized vectorial
Slepian function of type 3. The corresponding solution
*B*_{oc} of Eq. (1) within this setup then
provides an auxiliary function ${\stackrel{\mathrm{\u0303}}{\mathit{B}}}_{\mathrm{\ell}}$. It should be noted
that in order to obtain Maxwell's equation in the time-harmonic form
Eq. (1), one has to apply a Fourier transform in time.
Therefore, for the actual trial function *B*_{ℓ}, we have to invert
the Fourier transform and get

$$}{\mathit{B}}_{\mathrm{\ell}}(x,t)={e}^{-i\mathit{\omega}t}{\stackrel{\mathrm{\u0303}}{\mathit{B}}}_{\mathrm{\ell}}\left(x\right),\phantom{\rule{1em}{0ex}}x\in {\mathbb{R}}^{\mathrm{3}},t\in \mathbb{R}.$$

For technical reasons, we choose to work in a real-valued framework, so that
the real and imaginary part of *B*_{ℓ} each yield a trial function

$$\begin{array}{}\text{(9)}& {\displaystyle}& {\displaystyle}{\mathit{B}}_{\mathrm{\ell}}^{\mathrm{re}}(x,t)=\mathrm{cos}\left(\mathit{\omega}t\right)\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{\u0303}}{\mathit{B}}}_{\mathrm{\ell}}^{\mathrm{re}}\left(x\right)+\mathrm{sin}\left(\mathit{\omega}t\right)\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{\u0303}}{\mathit{B}}}_{\mathrm{\ell}}^{\mathrm{im}}\left(x\right),\text{(10)}& {\displaystyle}& {\displaystyle}{\mathit{B}}_{\mathrm{\ell}}^{\mathrm{im}}(x,t)=\mathrm{sin}\left(\mathit{\omega}t\right)\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{\u0303}}{\mathit{B}}}_{\mathrm{\ell}}^{\mathrm{re}}\left(x\right)-\mathrm{cos}\left(\mathit{\omega}t\right)\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{\u0303}}{\mathit{B}}}_{\mathrm{\ell}}^{\mathrm{im}}\left(x\right).\end{array}$$

Thus, each choice of *u*_{ℓ} yields two functions
${\mathit{B}}_{\mathrm{\ell}}^{\mathrm{re}}$ and ${\mathit{B}}_{\mathrm{\ell}}^{\mathrm{im}}$ 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 $\mathit{\omega}=\frac{\mathrm{2}\mathit{\pi}}{\mathrm{12.42}h}$ can be found in Fig. 3. For
the computation of the ${\stackrel{\mathrm{\u0303}}{\mathit{B}}}_{\mathrm{\ell}}$ as solutions of
Eq. (1), we have used the X3DG solver from
Kuvshinov (2008).

Figure 4 shows the accumulated energy
${\sum}_{\mathrm{\ell}=\mathrm{1}}^{L}\left|{\mathit{u}}_{\mathrm{\ell}}\right(\mathit{\xi}){|}^{\mathrm{2}}$, for *ξ*∈𝕊, of the
underlying functions *u*_{ℓ} that describe the velocity field and the
accumulated energy
${\sum}_{\mathrm{\ell}=\mathrm{1}}^{L}\left|{\mathit{B}}_{\mathrm{\ell}}^{\mathrm{re}}\right(x,t){|}^{\mathrm{2}}+|{\mathit{B}}_{\mathrm{\ell}}^{\mathrm{im}}(x,t){|}^{\mathrm{2}}$,
for *x*∈𝕊_{r} with $r=a+\mathrm{300}$ 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 *B*_{main} indicated in
Fig. 3.

3 Examples

Back to toptop
For our experiments we rely on the CM5 geomagnetic field model (cf.
Sabaka et al., 2015) and a forward model based on the M2 depth-integrated
tidal velocity field from TPXO8-ATLAS (cf. Egbert and Erofeeva, 2002) that
has also been used in Kuvshinov (2008). 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
${\mathit{B}}_{\mathrm{oc}}^{\mathrm{CM}\mathrm{5}}$ for the remainder of this section
and sample it at *M*=250 000 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. 3 and a depth-integrated M2
tidal velocity field from TPXO8-ATLAS. We denote it by
${\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}$ and evaluate it on the same point
grid as before. These samples are used as input data
** b**∈ℝ

$$}{\mathcal{D}}_{\mathrm{1}}=\mathit{\left\{}\mathrm{cos}\right(\mathit{\omega}t\left){\mathit{h}}_{n,k}\right(x),\mathrm{sin}(\mathit{\omega}t\left){\mathit{h}}_{n,k}\right(x){\mathit{\}}}_{n=\mathrm{0},\mathrm{\dots},\mathrm{20},\phantom{\rule{0.125em}{0ex}}k=-n,\mathrm{\dots},n},$$

with $\mathit{\omega}=\frac{\mathrm{2}\mathit{\pi}}{\mathrm{12.42}h}$ and the functions *h*_{n,k} from
Eq. (6); the Abel–Poisson-kernel-based

$$}{\mathcal{D}}_{\mathrm{2}}=\mathit{\left\{}\mathrm{cos}\right(\mathit{\omega}t\left)\mathit{k}\right(x,a{\mathit{\eta}}_{i}),\mathrm{sin}(\mathit{\omega}t\left)\mathit{k}\right(x,a{\mathit{\eta}}_{i}){\mathit{\}}}_{i=\mathrm{1},\mathrm{\dots},{M}_{p}},$$

where *a*=6371.2 km, $\mathit{\{}{\mathit{\eta}}_{i}{\mathit{\}}}_{i=\mathrm{1},\mathrm{\dots},{M}_{p}}$ is a Reuter grid on
𝕊 with *M*_{p}=6201 nearly equally distributed points (see, e.g.,
Michel (2013), p. 137), and ** k** given as in Eq. (7); and

$$}{\mathcal{D}}_{\mathrm{3}}=\mathit{\left\{}{\mathit{B}}_{\mathrm{\ell}}^{\mathrm{re}}\right(x,t),{\mathit{B}}_{\mathrm{\ell}}^{\mathrm{im}}(x,t){\mathit{\}}}_{\mathrm{\ell}=\mathrm{1},\mathrm{\dots},\mathrm{1200}},$$

with ${\mathit{B}}_{\mathrm{\ell}}^{\mathrm{re}}$ and ${\mathit{B}}_{\mathrm{\ell}}^{\mathrm{im}}$, the physics-based trial functions from Eqs. (9) and (10).

The actual signals that we want to approximate are indicated in Fig. 5. The approximations ${\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}$ of ${\mathit{B}}_{\mathrm{oc}}^{\mathrm{CM}\mathrm{5}}$ together with the residuals $|{\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}-{\mathit{B}}_{\mathrm{oc}}^{\mathrm{CM}\mathrm{5}}|$ for each of the three dictionaries above are shown in Fig. 6, whereas the respective approximations of ${\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}$ and corresponding residuals $|{\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}-{\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}|$ are displayed in Fig. 7.

In the case of ${\mathit{B}}_{\mathrm{oc}}^{\mathrm{CM}\mathrm{5}}$ as the underlying
signal, it can be seen in Fig. 6 that the dictionary
𝒟_{1} 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
𝒟_{2} 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
${\mathit{B}}_{\mathrm{oc}}^{\mathrm{CM}\mathrm{5}}$. The situation for dictionary
𝒟_{3} of physics-based trial functions is different. The agreement
of the approximation with ${\mathit{B}}_{\mathrm{oc}}^{\mathrm{CM}\mathrm{5}}$ is good
over the oceans but significant deviations exist over the continents. The
latter could be an indication that the original model
${\mathit{B}}_{\mathrm{oc}}^{\mathrm{CM}\mathrm{5}}$ 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 *u*_{ℓ})
and the assumption of a constant ocean depth (we chose a depth of 1 km for
the generation of the ${\stackrel{\mathrm{\u0303}}{\mathit{B}}}_{\mathrm{\ell}}$ 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
${\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}$ in Fig. 7, 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 ${\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}$ 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.,

$$\mathit{e}=\sum _{n=\mathrm{1}}^{\mathrm{40}}\sum _{k=-n}^{n}{\widehat{e}}_{(n,k)}{\mathit{y}}_{n,k}^{\left(\mathrm{3}\right)},$$

where the Fourier coefficients ${\widehat{e}}_{(n,k)}$ are normally distributed
with zero mean and a variance such that the amplitude of ** e** is in the
range of the oceanic signal ${\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}$. This
function

$${\mathit{B}}_{\mathrm{oc}}^{\mathit{e}}={\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}+\mathit{e}$$

by ${\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}^{\mathit{e}}$ instead of ${\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}$.
The latter still represents the approximation of
${\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}$ without extra continental noise ** e**.

In Fig. 9 one can directly see the influence which the
continental noise has on the approximation depending on the various
dictionaries (Fig. 7 shows the same quantities for the
approximations in the undisturbed setup). In the case of dictionary
𝒟_{1}, 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
𝒟_{2}, 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
𝒟_{3}, 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
${\mathit{B}}_{\mathrm{oc}}^{\mathit{e}}$ in coastal areas. A closer look at the
differences $|{\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}-{\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}^{\mathit{e}}|$
between the approximations of noisy and undisturbed data is given in
Fig. 10. 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
${\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}$ and ${\stackrel{\mathrm{\u203e}}{\mathit{B}}}_{N}^{\mathit{e}}$,
respectively, can be found in Table 1. In both cases, we compared
the approximations to the undisturbed data
${\mathit{B}}_{\mathrm{oc}}^{\mathrm{X}\mathrm{3}\mathrm{DG}}$ in order to emphasize the impact of
continental noise on the overall approximation. The errors over continental
and oceanic regions are provided separately.

4 Conclusions

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The main goal of this paper is to study the errors that are made by the approximation of tidal magnetic fields by use of different sets of trial functions. While, e.g., Saynisch et al. (2018) compared forward models for the M2 tidal magnetic field based on different tidal models, we aim at illustrating the effect of the involved trial functions on the possible extraction of the tidal magnetic field from satellite data in the first place. The indicated residuals for the synthetic examples show that the use of the presented adapted physics-based trial functions could have a detectable effect for the extraction of such signals in satellite data. These trial functions reflect the underlying physics in the sense that they satisfy the time-harmonic Maxwell equations and that they include knowledge of the ambient core magnetic field and the Earth's conductivity, but they are not designed to rely on detailed oceanographic models. The latter can be advantageous since the extracted magnetic field induced by ocean tides might eventually be used to make inferences on such models.

Data availability

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Data availability.

The authors would like to thank the following providers of software and models: Alexey Kuvshinov for the code of the X3DG solver and a processed version of the depth-integrated tidal ocean velocities from TPXO8-ATLAS, Nils Olsen for the coefficients of the M2 tidal contribution in the CM5 model, Alain Plattner for the code for the generation of vectorial Slepian functions.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement

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Special issue statement.

This article is part of the special issue “Dynamics and interaction of processes in the Earth and its space environment: the perspective from low Earth orbiting satellites and beyond”. It is not associated with a conference.

Acknowledgements

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Acknowledgements.

This work was supported by DFG grant GE 2781/1-1 within the Research Priority
Program SPP 1788 “DynamicEarth”.

Edited by:
Juergen Kusche

Reviewed by: two anonymous referees

References

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In our case, we use the norm
$\Vert f{\Vert}_{\mathcal{H}}^{\mathrm{2}}={\sum}_{n,k}(n+\frac{\mathrm{1}}{\mathrm{2}}{)}^{\mathrm{4}}{\widehat{f}}_{(n,k)}$, but
other norms can be used as well depending on the property one wants to impose
on *f*. By ${\widehat{f}}_{(n,k)}$ we denote the Fourier coefficients of *f* as
indicated in Eq. (3).

Typically, the order is
$(\mathrm{0},\mathrm{0}),(\mathrm{1},-\mathrm{1}),(\mathrm{1},\mathrm{0}),(\mathrm{1},\mathrm{1}),(\mathrm{2},-\mathrm{2}),\mathrm{\dots}$ such that the pair (*n*,*k*) is at
position ${n}^{\mathrm{2}}+n+k+\mathrm{1}$ in a row or column.