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\title{Mercury's DEM and FAG fractal structure -- indicator for meteorite bombardment by different density space bodies}

\author{Boyko Ranguelov$^{1}$, Rosen Iliev$^{2}$}

\affil{1}{University of Mining and Geology ``St. Ivan Rilski'' Faculty of Geology Exploration Department of Applied Geophysics, Sofia,  Bulgaria}
\affil{2}{Institute for Space Research and Technology, Bulgarian Academy of Sciences, Sofia,  Bulgaria}

\abstract{Over the past few decades Messenger spacecraft missions have provided to the scientific community a huge amount of new data on the geology and physics of the planet closest to the Sun -- Mercury. The collected data became the starting material for the building of the gravity field model of the Mercury -- HgM008. Based on it, a very recent NASA scientific team has released a high-quality ``free-air'' gravity map for the topography of the small planet. This enables new analyzes and interpretations of Mercury's physics and geology. The present study presents the results of Mercury's free-air gravity field (FAG) and digital elevation model (DEM) analysis using the (multi)fractal approach. The obtained results shed new light on the natural processes that have taken place during the geological evolution of Mercury. The results confirmed clear differences between the two hemispheres of the \linebreak\newpage}\noindent\centerline{\parbox{100mm}{  planet. Within the northern hemisphere fractal dimensions of FAG and DEM have variations ($R^2$) 0.908 and 0.942, while within the southern hemisphere $R^2$ of FAG and DEM have values 0.975 and 0.857. The results obtained determine the different intensity and density characteristics of space objects colliding with Mercury's two hemispheres, which necessitates additional interpretations.}}


\section{Methods and Data}

\subsection{Fractal Dimension Estimation}

      In the present study the fractal analysis is performed using fractal surface approach.
Based on the variogram
[\itc{Mark and Aronson,} \reflink{Mark84}{1984}],
 the fractal calculator (FocalID) generates
an image through a window around each raster pixel. In this way the fractal calculator
initially estimates a variogram, and
\begin{eqnarray*} %\begin{equation}\label{1}
  y(h)= \mathrm{Var} (Z_i - Z_j)
\end{eqnarray*} % \end{equation}                             % (1)
where $i$,  $j$ are spaced by the distance vector $h$.
      Then derived by regressing the logarithm of the distance vector with the logarithm of
the variance
[\itc{Zhou and Lam,} \reflink{Zhou05}{2005}]
is calculated the slope of regression. Finally, the fractal
dimension ($D$) is estimated through the following formula:

\begin{eqnarray*} % \begin{equation}\label{2}
D=3-(B/2)                                                       \end{eqnarray*} %  \end{equation}                             % (2)
where $D$ is fractal dimension and $B$ is the slope of the regression.

      The fractal value of each pixel reflects the variation complexity
[\itc{Pentland,} \reflink{Pent84}{1984}]
of
the gravity field or topography. The fractal signal value is much higher, when elevation or
gravity values have a more complex variation in regard to their neighboring pixel cells.

\section{Data and Software}

\begin{sloppypar}
      The analysis of the Mercury's ``free-air'' gravity anomalies was performed using data
(in GMT format) from the recently released Hermean (HgM008) gravity field model
[\itc{Genova et al.,} \reflink{Geno19}{2019}].
Input data derived from MESSENGER (Mercury Surface, Space Environment,
Geochemistry and Ranging) spacecraft missions.
\end{sloppypar}


\begin{sloppypar}
     Mercury's topography analysis is based on data from Mercury Global Digital
Elevation Model (DEM)~v2
[\itc{Becker et al.,} \reflink{Beck16}{2016}]
derived from MESSENGER spacecraft
missions
[\itc{Solomon et al.,} \reflink{Solo01}{2001}].
The DEM is created at $665 \times 665$~m spatial resolution.
\end{sloppypar}


\begin{sloppypar}
      The gravity and DEM data have been processed and explored using Geographic
Information System (GIS) -- SAGA-GIS
[\itc{Conrad et al.,} \reflink{Conr15}{2015}],
QGIS
[\itc{Thiede et al.,} \reflink{Thie14}{2014}]
and
LandSurf
(Wood, J., The LandSurf Manual,
\href{http://www.staff.city.ac.uk/~jwo/landserf/landserf230/doc/landserfManual.pdf}{http://www.staff.city.ac.uk/~jw o/landserf/landserf230/doc/landserfManual.pdf})
% [\itc{Wood,} \reflink{Wood09}{2009}]
free software.
\end{sloppypar}


\section{Free-Air Gravity Anomalies and Digital Elevation Model (DEM) of the Mercury}

\begin{sidewaysfigure}                      % Fig 1
\figurewidth{150mm}
\setimage{}{}{140mm}{}{2019es000678-f01}
\fcaplabel{Spatial and frequency distribution of the Mercury's FAG anomalies.}
\end{sidewaysfigure}


\begin{sloppypar}
      In gravimetry, ``free-air'' or Faye's gravity anomaly is an anomaly in free atmosphere
calculated from the observed value of the power of the normal gravity field, reduced to the
height at the point of observation. On Mercury's topography the ``free-air'' gravity field
reflects the elevation differences on the surface
(\figref{1}).           % (Fig.1).
The free-air gravity values are best
expressed by interaction between
negative (craters) and positive (mountains, ridges)
landforms. For its part, the digital elevation model % (DEM)
describes the spatial distribution
and interconnections of landforms.
\end{sloppypar}
\vspace*{-.5ex}


      Mercury's free-air gravity field % (FAG)
varies from $-227$ to $+141$~mGals. The negative
gravity anomalies dominated over the positive ones
(\figref{1}).           % (Fig.1).
Almost 55\%
of the total area of
the gravity field is occupied by negative gravity anomalies, 40.5\%
of positive gravity
anomalies and only 4.5\%
is for ``normal'' gravity field. As a whole the FAG is more variable
within the northern hemisphere
(\figref{2}),           % (Fig.2),
than within the southern hemisphere
(\figref{3})           % (Fig.3)
of the
planet. The southern hemisphere of the planet (especially within latitudes above
30\deg--35\deg) is
entirely dominated by a negative gravity field. Within the equatorial regions of the planet
(from $0\deg$ to $25\deg-30\deg$ in both hemispheres), the FAG is represented by values ranging from the
highest to the lowest ones.

\begin{sidewaysfigure}                      % Fig 2
\figurewidth{150mm}
\setimage{}{}{140mm}{}{2019es000678-f02}
\fcaplabel{Spatial and frequency distribution of the % Free-air gravity
FAG anomalies within the
northern hemisphere of Mercury.}
\end{sidewaysfigure}

\begin{sidewaysfigure}                      % Fig 3
\figurewidth{150mm}
\setimage{}{}{140mm}{}{2019es000678-f03}
\fcaplabel{Spatial and frequency distribution of the
FAG anomalies within the southern hemisphere of Mercury.}
\end{sidewaysfigure}

\begin{sidewaysfigure}                      % Fig 4
\figurewidth{150mm}
\setimage{}{}{140mm}{}{2019es000678-f04}
\fcaplabel{Spatial and frequency distribution of the Mercury DEM.}
\end{sidewaysfigure}

\begin{sidewaysfigure}                      % Fig 5
\figurewidth{150mm}
\setimage{}{}{140mm}{}{2019es000678-f05}
\fcaplabel{Spatial and frequency distribution of the DEM within the northern hemisphere of Mercury.}
\end{sidewaysfigure}

\begin{sidewaysfigure}                      % Fig 6
\figurewidth{150mm}
\setimage{}{}{140mm}{}{2019es000678-f06}
\fcaplabel{Spatial and frequency distribution of the DEM within the southern hemisphere of Mercury.}
\end{sidewaysfigure}

	The digital elevation model % (DEM)
of Mercury's topography varies from $-5272$~m to
4273~m
(\figref{4}).           % (Fig.4).
Negative values occupy about 60\%,
while the positive ones occupy about
40\%
of the planet's surface. Negative landforms are represented by impact craters, while the
positive ones by mountains and ridges. In the northern hemisphere, the negative landforms
prevail
(\figref{5}),           % (Fig.5),
while within the southern hemisphere the positive landforms are more
dominant
(\figref{6}).           % (Fig.6).
 Most of mountains and ridges are localized up to $30\deg -35\deg$ northern and
southern latitude, while the northern latitudes of the planet are predominantly represented by
negative shapes and lowest values of gravity field.



              In the course of the study, significant differences between Mercury's northern and
southern hemispheres were found, both in terms of gravity and hypsometry. As we will see
from the next section, these differences are well described by the variance of their fractal dimensions.

\begin{sidewaystable}                                  % Table 1
\tablewidth{140mm}
\ftablabel{Fractal Analysis of Mercury's FAG and DEM (Key: FD -- Fractal Dimension; $R^2$ -- Coefficient of Determination)}
\vspace{5pt}
%%%\small
\begin{tabular}
{@{}l@{\hspace{15pt}}
c@{\hspace{12pt}}
c@{\hspace{12pt}}
c@{\hspace{12pt}}
c@{\hspace{12pt}}
c@{\hspace{12pt}}
c@{}}
\hline
\\ [-7pt]
& \multicolumn{6}{c}{Free-Air Gravity (FAG)}  \\
& \multicolumn{3}{c}{Northern hemisphere} & \multicolumn{3}{c}{Southern hemisphere} \\
&   All values&  Positive&  Negative & All values & Positive & Negative \\
  [7pt]  \hline   \\ [-4pt]
FD min & 2.10  &      2.01  &    2.01   &    2.13    &    2.01   &   2.01  \\
FD max & 2.56  &      2.38  &    2.41   &    2.64    &    2.40   &   2.44   \\
$R^2$     & 0.908 &      0.968 &    0.967  &     0.975  &    0.994  &    0.990  \\
[7pt]  \hline   \\ [-4pt]
  & \multicolumn{6}{c}{Digital Elevation Model (DEM)}  \\
& \multicolumn{3}{c}{Northern hemisphere} & \multicolumn{3}{c}{Southern hemisphere} \\
&   All values&  Positive&  Negative & All values & Positive & Negative \\
[7pt]  \hline   \\ [-4pt]
 FD min &2.01    &    2.0   &    2.01   &      2.01   &   2.0    &    2.01 \\
FD max &2.64    &    2.45  &    2.58   &     2.58    &  2.47    &   2.54  \\
$R^2$     & 0.942  &     0.903&    0.940  &      0.857  &    0.863 &     0.910  \\
[4pt] \hline
\end{tabular}
\end{sidewaystable}


\section{Results and Discussion}

      The results obtained in the course of the study of Mercury's FAG and DEM fractal
structure are summarized in
\tabref{1}.  % Table 1.
The main conclusions and interpretations are discussed
further.

      As we can see from the table, the FD min for both FAG and DEM in northern
hemisphere is approximately equal. Vice versa -- FD~max are relatively different, but the
tendencies are kept. In southern hemisphere FD~max for DEM are larger (both for positive
and negative values), but all values are similar in FD~min. The $R^2$ value is lower for the DEM
of southern hemisphere. This indicated higher non-linearity in spatial distribution of
landforms to the south of Mercury. Within the southern hemisphere the transition from high
to low landforms is clearly expressed. To the north, the positive and negative areas of the
terrain are more compact and larger.




\begin{sloppypar}
      The comparison between the northern and southern hemisphere of FAG (positive and
negative) anomalies
(\figref{7} and \figref{8})                % (Fig.7, 8)
shows clearly that the areas of both types of FAG anomalies
dominated in their sizes and numbers. For example, the negative values in northern
hemisphere have higher areas within range of FD between 2.36 and 2.40. For the southern
hemisphere the same is true for the range between 2.37 and 2.40. Same tendency is clearly
visual for the positive values.
\end{sloppypar}\vspace{.5ex}


      The analysis of the FD for both hemispheres shows the following peculiarities: The
fractal dimensions for both cases starts as values of 2 and finished at levels 2.4. Up to 2.27
(for northern hemisphere -- NH) and 2.30 (to the southern one -- SH) the fractal dimensions
reflects both -- positive and negative anomalies. These values show that in both cases the
nonlinearity is clearly expressed but at almost the same values of areas (positive and
negative). The appearance of some spikes in the distributions is not very clear and will need
additional investigations. After 2.27 (NH) and 2.30 (SH) the dynamics in the changes of FD's
is sharper and rather specific for NH and SH. The FD of the positive anomalies in NH
demonstrates a maximum between 2.37 and 2.38, which means specific fractal structure (i.e.
fragmentation). The FD distribution of the positive anomalies in SH is unimodal, as well as
both distributions of the FD's related to the negative anomalies. These specifics also need
some additional investigations, but clearly confirm that the northern hemisphere is much
more frequently attacked by the meteorites, asteroid and/or comets.


      Generally within the northern hemisphere of Mercury $R^2$ of fractal dimensions of both
FAG and DEM are almost identical
(\figref{7} and \figref{8}),           %(Fig.7, 8),
but this does not apply to the planet's southern
hemisphere.

\begin{figure}                      % Fig 7
\figurewidth{90mm}
\setimage{}{}{115mm}{}{2019es000678-f07}
\fcaplabel{Fractal dimensions frequency of FAG positive and negative values within the northern hemisphere of Mercury.}
\end{figure}


\begin{figure}                      % Fig 8
\figurewidth{90mm}
\setimage{}{}{115mm}{}{2019es000678-f08}
\fcaplabel{Fractal dimensions frequency of FAG positive and negative values within the southern hemisphere of Mercury.}
\end{figure}


\section{Conclusion}

      The distributions of the DEM and FAG of northern and southern hemispheres of
Mercury are studied. The results show clear fractal properties of the positive and negative
areas of DEM and FAG. The correlation between them is confirmed. Sometimes there are
some discrepancies in the correlation which probably means lack of coincidence in the FAG
and DEM due to the different densities.

      The northern hemisphere is much more bombarded by meteorites and asteroids then
the southern one. The fractal analysis confirmed this fact clearly. This means that the free
space flying objects bombarding the Solar system dominated the North direction. If this is a
space specifics, it must be proved for other planetary objects in the Solar system too. The
more dense free flying objects in the Solar system are about 1.5--2.0 times rarer then the less
dense. It could be related to the physical properties of the meteorites and asteroids. Usually
the less dense objects are identified as rocky objects, and the denser -- as richer of metallic
components space objects. This means that the ``iron'' objects (probably) are rarer within the
Solar system. Probably it can be proved by the gravimetric studies to other terrestrial planets
and satellites in the Solar system.


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\end{document}
