\documentclass[epub,pdf]{ebook}
\usepackage[cm]{sfmath}
\usepackage{amsmath}
\usepackage{color}
\usepackage[pdftoolbar=false, pdfpagemode=UseNone, linkbordercolor={0 0 0}]{hyperref}

\ident{ES}{19}{2}{2019}{2019ES000656}
\cpright{}{2019}
\pagestyle{empty}
\definecolor{brown}{RGB}{157,23,4}

\def\deg{$^{\circ}$}

\begin{document}

\vspace*{2\baselineskip}

\title{On the directions and structure  of the short-term magnetic variations}

\author{A. Khokhlov$^{1,2,3}$, R. Krasnoperov$^2$, B.~Nikolov$^2$, J. Nikolova$^2$, M. Dobrovolsky$^2$, V.~Petrov$^4$,D. Kudin$^{2,4}$, I.~Belov$^2$}

\affil{1}{Bauman Moscow State Technical University, Moscow, Russia}
\affil{2}{Geophysical Center RAS, Moscow, Russia}
\affil{3}{IEPT RAS, Moscow, Russia}
\affil{4}{IZMIRAN RAS, Troitsk, Moscow Region, Russia}

\abstract{We study the directional structure of the $1$-minute magnetic variations and demonstrate  their
polarization-like asymmetry: statistically these directions are close to certain 2D plane in 3D space
and the orientation of such a plane  depends on the observatory location. We describe
the method of the magnetic data processing  and some partial results, however in this publication we are avoiding the
reasonings concerning the  physical origins of the detected effect and its consequences for the observatory data analysis in general.}


\section{Introduction}
The geomagnetic field is continuously measured by a network of magnetic observatories, thus, the mathematical and computational technologies for data processing should take into account the actual
conditions in which observation of the Earth magnetic field
is performed at these ob\-ser\-va\-to\-ries. For instance the effective identification of noise
and elimination of its influence on final data is
an important part of the data processing.

A magnetic observatory should be constructed so that
only the natural magnetic field is present.
In particular, the location is selected so that  local magnetic anomalies, be it from
geological or artificial origin, must be eliminated and this in principle should lead to
the fact that the observable short-term effects expected to be homogeneous.
The spatial structure of the magnetic field is therefore well-tested (ty\-pi\-cal\-ly this requires
the spatial gradients to be of the order of magnitude 1~nT~m$^{-1}$),  temporal magnetic field variations must
be identical inside the entire observation space. However the study of the short-term variations was mainly
reduced to their absolute or component estimations without serious interest to their directional structure [\itc{Khomutov et al.,} \reflink{Khom17}{2017}].

What is important is that such a study of the directional structure may be extracted not necessarily from the observatory datasets but also from a variometer, which is a magnetometer designed to monitor the temporal variations of
magnetic field components relative to a fixed baseline.  The numerous magnetic stations therefore  give rise to more complete  research of the  local specific details  of the directional variations. Indeed the installation of the variometer presumes the
reference system (the
most popular orientations are $H$, $D$, $Z$ and $X$, $Y$, $Z$) and that the variometer orientation
is stable over time.

We address to the directions that result from magnetic variations of the magnetic vector using either  variometer or observatory  data. Such a  direction is a unit 3D-vector and the  data is in one-one correspondence with the points of the unit sphere. The plan is  to consider some statistical effects using the experimental probability density distribution. Generally speaking an arbitrary  collection of data does not allow application of the statistical methods: the core of any statistics assumes that the data population is the part of some stationary process. For this assumption of stationarity we need some preliminary  magnetic data processing since in real conditions the magnetic vector is subjected to daily and long-term secular variations, there are non-perturbed  days and magnetic  storm distortions, also the sufficient amount of data is necessary  to reveal the (initially hidden) statistical effects.

We may use for the $1$-minute variations
the term ``noise'' which is by no doubt relative. For instance, temporary signals in the magnetic field caused by
the sources in the ionosphere-magnetosphere are typically considered as noise (especially during gro\-und magnetic survey and
interpretation of  results),
the signals, which have
sources closer than a few tens of kilometres, typically  considered as noise  during the  observatory  measurements
[\itc{Santarelli et al.,} \reflink{Sant14}{2014}].
 Of course, there are also some sources of man-made noise such as DC railways, which in the case of appropriate conductivity of the  crust can
produce a significant effect in the magnetic data at large distances. It is important that these effects in principle differ in their durations, spectral shapes etc., the resolution of 1 s and the randomization of the observational periods may reduce the chance that  the directional distribution results from the  rare fact of the exceptional man-made or natural noise.\vspace*{.5ex}

With all this precautions it comes out that the experimental probability density of $1$-minute variations is not at all directionally isotropic.
This is not new, since the registered magnetic variations may result from the  induction effects of the electric  currents in the crustal  layers --  in the situation of homogeneity at  the local scale one  may expect that the  directions of  magnetic variations  share the horizontal plane.
This is not the case however: indeed the 2D plane concentration of directions exists but this plane is in general  neither horizontal nor perpendicular to the local magnetic field lines.  We checked also the orientation of these planes in various observatories and saw no clear latitudinal or longitudinal dependencies. We may speculate that this is purely local effect but it is not equally obvious  from station to station. Also, this effect depends on the strength of variation: the stronger the variation the closer its direction to the preferred 2D-plane.

At present we have no  clear physical interpretation. For this   we need  at least several time-scales to state that this  polarization effect is purely local. So we exhibit the approach now and plan  to proceed with further research  using various scales in  time and space.

\section{Methods}

Without the loss of generality we may assume the initial magnetic data to be the sequence $\left\{{\mathbf{B}}\right\}_n$ of the 3D magnetic vectors ${\mathbf{B}}_k={\mathbf{B}}(t_{k})$
where $\delta t=t_{k+1}-t_k$ equals to 1 minute. We aim to construct the population of 1~min magnetic variations ${\mathbf{b}}=\delta{\mathbf{B}}$ whatever this means.
The real magnetic signal has some memory at a scales larger than 1-minute but we want to exclude this type of correlations in the sequence of
 variations. Therefore we take into account some other time scale $\Delta t \gg\delta t$ (in our  implementation $\Delta t$ is equal to 10 minutes) and perform some preprocessing.

\begin{sidewaysfigure}               % Fig 1
\figurewidth{140mm}
\setimage{}{}{155mm}{}{2019es000656-f01}
\fcaplabel{Data preprocessing: magnetic data collected  from  13.02.2013 to  27.02.2013 for the time interval  12:00--13:00 UTC  at Arti 	 (ARS) 	 observatory (location 56.433${}^\circ$N, 58.567${}^\circ$E). Left: horizontal axes show the sequential number
of magnetic data, the intensity of $\lvert{\mathbf{B}}-\bar{{\mathbf{B}}}\rvert$ in nT, here $\bar{{\mathbf{B}}}$ is the mean field (top plot) and corresponding absolute values of variations $\lvert{\mathbf{b}}\rvert$ in nT (bottom plot). Right: histogram of $\lvert{\mathbf{b}}\rvert$ in the semilogscale.}
\end{sidewaysfigure}


The right plot at \figref{1}
shows the almost linear decay of the histogram, this corresponds to the absolute difference of the two  exponential distributions related each to the intensities
of $\lvert{\mathbf{B}}\rvert$ (the latter can be in turn tested directly).

\subsection{Data Preprocessing}

\enlargethispage{-.85pc}

Let us divide  the time into sequential segments $\Delta t$-periods $(t_k,t_k+\Delta t]$
and construct the second order approximation of $\left({\mathbf{B}}(t_k),\ldots,{\mathbf{B}}(t_k+\Delta t)\right)$  at each segment --
this mimic the $\Delta t$-scale trends of the initial sequence. Now  for an arbitrary $t_i\in (t_k,t_k+\Delta t]$ we use
the  vector value of this approximation $\tilde{{\mathbf{B}}}(t_i)$ and  set ${\mathbf{u}}_i=\delta{\mathbf{B}}_i={\mathbf{B}}(t_i)-\tilde{{\mathbf{B}}}(t_i)$.


It can be tested that the sequence $\left\{{\mathbf{b}}\right\}_n$ demonstrates much
more stationary behaviour  and less correlations  -- see for instance \figref{1} %%Fig.~\ref{ARSPreproc}
-- than those  in the sequence of the first
differences  $\{{\mathbf{B}}_{k+1}-{\mathbf{B}}_k\}$ (the latter mimic the sequence of time derivatives).  Nevertheless  at scales
 $\gg\Delta t$ there persist long-term changes of the absolute values $\lvert{\mathbf{b}}\rvert$ -- this typically reflect
the daily variation of the magnetic activity. The next transformation ${\mathbf{b}}\mapsto{\mathbf{u}}$ is
the normalization: ${\mathbf{u}}={\mathbf{b}}/\lvert{\mathbf{b}}\rvert$ therefore we get the set $U$ of unit 3D vectors ${\mathbf{u}}\in{\mathbb{R}}^3$. For  this we may use magnetic data collected  in the fixed UTC time interval (one or many).  To illustrate here both approaches  we choose the following examples:
\begin{enumerate}
\item ARS unperturbed magnetic field. Time interval 12:00--13:00 UTC, data collected from  13.02.2013 to  27.02.2013   at Arti (ARS) 	 observatory (location 56.433${}^\circ$N, 58.567${}^\circ$E),
\item EBR unperturbed magnetic field. Time interval 12:00--18:00 UTC, data collected from  23.05.2015 to  03.06.2015   at Ebro (EBR) 	 observatory (location 40.957${}^\circ$N, 0.333${}^\circ$E),
\item KIV  perturbed magnetic field. Data collected  00:00--23:59 01.06.2013 at Kiev (KIV) observatory	(location 50.720${}^\circ$N, 30.300${}^\circ$E),
\item SPG  perturbed magnetic field. Data collected 00:00--23:59  18.03.2015 at Saint Petersburg (SPG) observatory	 (location 60.542${}^\circ$N, 29.716${}^\circ$E)
[\itc{Sidorov et al.,} \reflink{Sidorov2017}{2017}].
\end{enumerate}

In fact we have considered also many other observatories and data sets. The selected examples are of more or less typical behavior.


\begin{sidewaysfigure}               % Fig 2
\figurewidth{140mm}
\setimage{}{}{155mm}{}{2019es000656-f02}
\fcaplabel{Polar diagrams for ARS, EBR, KIV, SPG data (see main text).}
\end{sidewaysfigure}

\subsection{Directional Data Selection and Plots}

Actual values for ${\mathbf{b}}$ that appear during the  preprocessing are very small in general and  sometimes give zero values for $\lvert{\mathbf{b}}\rvert$ (because of finite precision of magnetometer) and in that latter case directional unit vector is not even defined. Zeros may also appear in one or even two components of vector $\lvert{\mathbf{b}}\rvert$, the corresponding directions then will occupy coordinate planes or axes -- but this are exceptional rare situations and we exclude them from $U$  to concentrate our attention on visually obvious features of  a  distribution over unit sphere $S^2$.
The histogram (implemented as polar diagram see \figref{2}) of plane projection of vectors ${\mathbf{u}}$ onto OXY plane clarify the directional structure of the set $U$, however in incomplete way.

In order to consider the 3D structure we may either compute
approximative density  of points over sphere (see details below) or use 3D plot to show points at sphere $S^2$; both approaches are shown for the ARS magnetic data at \figref{3}.  %%Fig.~\ref{Ars2fig}.
It is important for the 3D plots to remember that
in virtue of the discrete nature of variations (remember the precision of the data!) we sometimes get several different directions as one point at the unit sphere $S^2$.


\begin{sidewaysfigure}               % Fig 3
\figurewidth{140mm}
\setimage{}{}{155mm}{}{2019es000656-f03}
\fcaplabel{The directions ${\mathbf{u}}$  for the  ARS magnetic data (see main text). Left: relative densities (see main text for the
details) for the different latitudes relative densities (see main text for the
details), middle:  3D-plot of all directions  ${\mathbf{u}}\in U$, right: shown are only those directions ${\mathbf{u}}\in U$ that
correspond to variations $\lvert{\mathbf{b}}\rvert> 0.7$~nT.}
\end{sidewaysfigure}

We may define the average global density of $N$ points over sphere as $\hat\mu=N/4\pi$ (where $N$ is the total number of points). For a given latitude interval  $(\theta_0,\theta_1)$ in a similar way we define
the corresponding  density  $\mu(\theta_0,\theta_1)=N(\theta_0,\theta_1)/S(\theta_0,\theta_1)$, here
$N(\theta_0,\theta_1)$ is the  number of points with latitude
$\theta\in(\theta_0,\theta_1)$, $S(\theta_0,\theta_1)$ -- the  corresponding area.
 Now we may choose specific latitudes $\{\theta_i\}$ to provide all equal areas $S(\theta_{k-1},\theta_k)$
 and to consider the relative densities $\mu(\theta_{k-1},\theta_k)/\hat{\mu}$; this may show density irregularities.
While the  3D-plot of all points ${\mathbf{u}}\in U$ may be too cumbersome, the plot of smaller subset may reveal the geometrical pattern of distribution.




Now address the KIV and SPG data: in general the typical variations $\lvert{\mathbf{b}}\rvert$ there are order of magnitude stronger then those of ARS data, see \figref{4},  %%Fig.~\ref{KIVSPG1},
we also obviously see the inhomogeneous nature of the directional distribution, the projections of the two clusters  are shown on \figref{2}.  %%Fig.~\ref{AllRoses}.

\begin{sidewaysfigure}              % Fig 4
\figurewidth{140mm}
\setimage{}{}{155mm}{}{2019es000656-f04}
\fcaplabel{Horizontal axes show the sequential number of the selected magnetic data,
vertical axe show the intensity in nT.
Left: intensity $\lvert{\mathbf{B}}-\bar{{\mathbf{B}}}\rvert$ in nT, $\bar{{\mathbf{B}}}$ is the mean field  (top plot) and corresponding absolute values of variations $\lvert{\mathbf{b}}\rvert$ (bottom plot) for the KIV data; right: the similar plots for the SPG data.}
\end{sidewaysfigure}

\begin{sidewaysfigure}               % Fig 5
\figurewidth{140mm}
\setimage{}{}{155mm}{}{2019es000656-f05}
\fcaplabel{Left: 3D plots of all directions  ${\mathbf{u}}\in U$ for the  KIV data; right: the similar plot for the SPG data.}
\end{sidewaysfigure}

For the corresponding 3D plots no selection of the strong variations ${\mathbf{b}}$ is needed: the  geometrical structure of density is obvious for both cases.
Taking into account   3D-plots of  \figref{5}
we  may speculate that the   clusters of directions ${\mathbf{u}}$  are located in the vicinity of  certain inclined \mbox{2D-plane}.
We may even algorithmically compute  the best 2D-plane $Ax+By+Cz=0$ that approximates the given  distribution of ${\mathbf{u}}$ in each case, indeed we may compute the orientation tensor for all ${\mathbf{u}}$ directions and derive its eigenvalues  and corresponding  eigenvectors. The smallest eigenvalue defines the 2D plane (that is perpendicular to the corresponding eigenvector $(A,B,C)$), the smaller this eigenvalue the more concentrated ${\mathbf{u}}$ directions in the  vicinity of this plane. These  inhomogeneous distributions seem to be  the important feature of the magnetometer location which in principle may affect the magnetic data. Let us further name  these planes as ``polarization planes'' for clarity and shortness.


To clarify the relation between the intensities of the va\-ri\-ations ${\mathbf{b}}$ and the densities of the corresponding points at sphere $S^2$ let us consider the mutual scatter plot of the two quantities: $\lvert{\mathbf{b}}\rvert$ and $\alpha$, where $\alpha$ is the angle (in degrees) between the polarization plane and corresponding ${\mathbf{u}}$. We see on \figref{6} %%Fig.~\ref{Anticor}
the obvious negative correlation between them: the stronger the variation ${\mathbf{b}}$ the closer its direction to the polarization plane and smaller the angle $\alpha$. This effect was already demonstrated in a different way by \figref{3}, %%.~\ref{Ars2fig},
however for KIV data the  polarization plane is inclined  $30^\circ$ therefore the   dependencies of the density  from the latitude are not that obvious.



The polarization plane of EBR data is inclined even more and here we may also use the alternative way to present the
relations between $\alpha$ angle and  $\lvert{\mathbf{b}}\rvert$, see \figref{7}. %%Fig.~\ref{EBR}.
We again have weak variations  and therefore polarization plane is less contrast on 3D plot.

\begin{sidewaysfigure}               % Fig 6
\figurewidth{140mm}
\setimage{}{}{155mm}{}{2019es000656-f06}
\fcaplabel{Left: shown are the pairs $(\alpha,\lvert{\mathbf{b}}\rvert)$ in coordinate axes (degrees, nT)  for all directions ${\mathbf{u}}\in U$ for the  KIV data (see main text for the definition of the angle $\alpha$); right: the similar plot for the SPG data.
Grid lines mark 5\% and 95\% quantiles}
\end{sidewaysfigure}


\begin{sidewaysfigure}               % Fig 7
\figurewidth{140mm}
\setimage{}{}{155mm}{}{2019es000656-f07}
\fcaplabel{Left: 3D plots of all directions  ${\mathbf{u}}\in U$ for the  EBR data; right: absolute values of variations $\lvert{\mathbf{b}}\rvert$ in nT  (top) and corresponding $\alpha$ angles in degrees (bottom) for the  EBR data (see main text).
Horizontal axes show the sequential number of the selected magnetic data, vertical axes
show the intensity in nT.}
\end{sidewaysfigure}


\subsection{Directions and  Data Characteristics}


By the nature of preprocessing algorithm the resulting unit vectors ${\mathbf{u}}_i$ are of different orientations, in real situation we get two big clusters that are oppositely directed. Obviously vectors ${\mathbf{u}}_k$ are linked to discrete derivatives (first
differences)  $\left\{{\mathbf{B}}_{k+1}-{\mathbf{B}}_k\right\}$, however, in virtue of the damping of local trends, we may postulate only qualitatively  ``the bigger the derivative $\lvert{\mathbf{B}}_{k+1}-{\mathbf{B}}_k\rvert$ the bigger the value of $\lvert{\mathbf{b}}\rvert$''.
The cluster structure can be revealed in two ways:  either make the orthogonal projection of all ${\mathbf{u}}_i$ onto the polarization plane and then consider plane distribution or  compute   the usual polar diagram at $OXY$ plane.

Another type of analysis may clarify the relations between $\lvert{\mathbf{b}}\rvert$-values and the  angles between the corresponding ${\mathbf{b}}$ and polarization plane (see \figref{6}). %%Fig.~\ref{Anticor} ).
As a general rule we see here the negative correlation between this two entities:
``the bigger the $\lvert{\mathbf{b}}_k\rvert$ the smaller the angle between ${\mathbf{b}}$ and the polarization plane''. Since the extreme values of field derivatives correspond to large values of ${\mathbf{b}}$ therefore  one may expect the strongest field derivative to be parallel to the polarization plane.

 What will happen if we change preprocessing to the simple first-differences algorithm and then normalize the discrete derivatives? Essentially the spherical distribution of directions will be much the same as shown above, however some regular patterns (because of local trends) may appear. \vspace*{-\baselineskip}




\section{Results}

After the routine cross-check of numerous magnetic ob\-ser\-va\-to\-ries we may state the following:
\begin{enumerate}
\item Almost any magnetometer reveals the existence of the polarization plane for the directions of $1$-minute variations.
However, especially if considered the data from only the undisturbed periods of magnetic activity,  the   polarization effect  at certain locations  can be very weak. In contrast the data from nearabout time of the  magnetic storms  show the polarization  effect in much more obvious way.\vspace*{-.33pc}
\item The orientation of the polarization plane does not depend on the particular choice of the seasonal data: such a  plane seems to be stable for a given magnetometer at least at the scale from one day to decades.\vspace*{-.33pc}
\item The orientation of the polarization plane differs from location to location: it need not to be horizontal, sometimes (see \figref{7} for instance) it is inclined  more than  $60^\circ$ to the horizontal plane. No clear dependencies between the inclination of the polarization plane and the inclination of the local main magnetic field lines were detected. Also we were yet unable to derive any functional dependency from the  geographical locations (latitude and longitude).\vspace*{-.33pc}
\item  The strong $1$-minute variations of the magnetic field  at each observatory  demonstrate the clear tendency to stay parallel to the polarization plane.  Moreover  the strong variations show the structure of two opposite clusters. In other words we may expect the stron\-gest variation to lie within rather limited solid angle of 3D-directions.
\end{enumerate}

\section{Discussion and Conclusions}

To stay within the traditional knowledge we may speculate that studied short-term magnetic  variations mainly appeared as local induction effects related to the electric currents in the crust. The structure of the conducting media need not  be necessarily homogeneous therefore there is no reason to expect that all such electric currents will provide only horizontal  magnetic variations. Nevertheless it seems amazing that there exist the stable geometrical structure of the short-term  magnetic variations
and it  is not dependent of the time-scale. We are not yet ready to compare the different time-scales: right now we have considered only the available most frequent magnetic variations [\itc{Gvishiani et al.,} \reflink{Gvis14}{2014}].

It is important to remark that this polarization effect appeared  obvious only after the non-linear transformation (normalization) of  variations. Without this the plane polarization  cannot be easily detected by  the separate component analysis  of the magnetic field.

On the other hand this polarization effect can be easily verified on a wide family of magnetic variometers since we do not need  any absolute calibration of the magnetic data.  Up to now we used mostly INTERMAGNET observatory data but the network of  variometers sometimes is much more dense; the  further research will  clarify the space scale of the polarization effect using  neighboring magnetic stations.

If only local conditions are responsible for the orientation of the polarization plane we then indeed may have a sort of statistical prediction of the local magnetic activity effects at least at the short time-scale.

\subsubsection{Acknowledgments.}
This work was suppor\-ted by the Russian Science Foundation grant (pro\-ject No. 17-77-20034 ``Creation of maps of geomagnetic activity characteristics zoning for the territory of the Russian Federation'').
The results presented in this paper rely on data collected at magnetic observatories [\itc{Love \& Chulliat,} \reflink{Love13}{2013}]. We thank the national institutes that support them and INTERMAGNET for promoting high standards of magnetic observatory practice (\url{http://www.intermagnet.org}.
The presented results are also based on the data collected at Saint Petersburg observatory (SPG). We thank the Geophysical Center of RAS (GC RAS) and Saint Petersburg branch of the Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radiowave Propagation of the Russian Academy of Sciences (IZMIRAN SPb Branch), for supporting its operation and Russian-Ukrainian Geomagnetic Data Center (RUGDC) for making data available on the Internet.
The authors wish to express gratitude to the scientific staff of the Shared Use Facility ``Analytical Geomagnetic Data Center'' (\url{http://ckp.gcras.ru/}).


\begin{references}

\reference{Gvis14} Gvishiani, A., R. Lukianova, A. Soloviev, A. Khokhlov (2014), Survey of Geomagnetic Observations Made in the Northern Sector of Russia and New Methods for Analysing Them, {\itshape Surveys in Geophysics}, {\itshape 35}, no. 5, p. 1123--1154,  \href{https://doi.org/10.1007/s10712-014-9297-8}{\textcolor{brown}{\space\bfseries Cross\-ref}}

\reference{Khom17} Khomutov, S., O. Mandrikova, E. Budilova, K. Arora, L. Manjula (2017), Noise in raw data from magnetic observatories, {\itshape Geosci. Instrum. Method. Data Syst}, {\itshape 35}, no. 6, p. 329--343,  \href{https://doi.org/10.5194/gi-6-329-2017}{\textcolor{brown}{\space\bfseries Cross\-ref}}

\reference{Love13} Love, J., A. Chulliat (2013), An international network of magnetic observatories, {\itshape Transactions American Geophysical Union}, {\itshape 94}, no. 42, p. 373--374,  \href{https://doi.org/10.1002/2013EO420001}{\textcolor{brown}{\space\bfseries Cross\-ref}}

\reference{Sant14} Santarelli, L., P. Palangio, M. De Lauretis (2014), Electromagnetic background noise at L'Aquila Geomagnetic Observatory, {\itshape Ann. Geophys.Italy}, {\itshape 57}, p. G0211,  \href{https://doi.org/10.4401/ag-6299}{\textcolor{brown}{\space\bfseries Cross\-ref}}

\reference{Sido17} Sidorov, S., A. Soloviev, R. Krasnoperov (2017), Saint Petersburg magnetic observatory: from Voeikovo subdivision to INTERMAGNET certification, {\itshape Geosci. Instrum. Method. Data Syst}, {\itshape 35}, no. 6, p. 473--485,  \href{https://doi.org/10.5194/gi-6-473-2017}{\textcolor{brown}{\space\bfseries Cross\-ref}}
\end{references}

\end{document}


