0 -> \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.}}
1 -> \section{Methods and Data}
2 -> 
3 -> \subsection{Fractal Dimension Estimation}
4 -> 
5 ->       In the present study the fractal analysis is performed using fractal surface approach.
6 -> Based on the variogram
7 -> [\itc{Mark and Aronson,} \reflink{Mark84}{1984}],
8 ->  the fractal calculator (FocalID) generates
9 -> an image through a window around each raster pixel. In this way the fractal calculator
10 -> initially estimates a variogram, and
11 -> \begin{eqnarray*} %\begin{equation}\label{1}
12 ->   y(h)= \mathrm{Var} (Z_i - Z_j)
13 -> \end{eqnarray*} % \end{equation}                             % (1)
14 -> where $i$,  $j$ are spaced by the distance vector $h$.
15 ->       Then derived by regressing the logarithm of the distance vector with the logarithm of
16 -> the variance
17 -> [\itc{Zhou and Lam,} \reflink{Zhou05}{2005}]
18 -> is calculated the slope of regression. Finally, the fractal
19 -> dimension ($D$) is estimated through the following formula:
20 -> 
21 -> \begin{eqnarray*} % \begin{equation}\label{2}
22 -> D=3-(B/2)                                                       \end{eqnarray*} %  \end{equation}                             % (2)
23 -> where $D$ is fractal dimension and $B$ is the slope of the regression.
24 -> 
25 ->       The fractal value of each pixel reflects the variation complexity
26 -> [\itc{Pentland,} \reflink{Pent84}{1984}]
27 -> of
28 -> the gravity field or topography. The fractal signal value is much higher, when elevation or
29 -> gravity values have a more complex variation in regard to their neighboring pixel cells.
30 -> 
31 -> \section{Data and Software}
32 -> 
33 -> \begin{sloppypar}
34 ->       The analysis of the Mercury's ``free-air'' gravity anomalies was performed using data
35 -> (in GMT format) from the recently released Hermean (HgM008) gravity field model
36 -> [\itc{Genova et al.,} \reflink{Geno19}{2019}].
37 -> Input data derived from MESSENGER (Mercury Surface, Space Environment,
38 -> Geochemistry and Ranging) spacecraft missions.
39 -> \end{sloppypar}
40 -> 
41 -> \begin{figure*}[t]                     % Fig 1
42 -> \figurewidth{35pc}
43 -> \setimage{}{}{35pc}{}{2019es000678-f01}
44 -> \shortcaption{Spatial and frequency distribution of the Mercury's FAG anomalies.}
45 -> \end{figure*}
46 -> 
47 -> 
48 -> \def\bottomfraction{.8}
49 -> \def\textfraction{.05}
50 -> 
51 -> \begin{figure}[b]                     % Fig 2
52 -> \figurewidth{35pc}
53 -> \setimage{10.5pc}{}{35pc}{}{2019es000678-f02}
54 -> \crosscaption{Spatial and frequency distribution of the % Free-air gravity
55 -> FAG anomalies within the
56 -> northern hemisphere of Mercury.}
57 -> \end{figure}
58 -> 
59 -> \begin{sloppypar}
60 ->      Mercury's topography analysis is based on data from Mercury Global Digital
61 -> Elevation Model (DEM)~v2
62 -> [\itc{Becker et al.,} \reflink{Beck16}{2016}]
63 -> derived from MESSENGER spacecraft
64 -> missions
65 -> [\itc{Solomon et al.,} \reflink{Solo01}{2001}].
66 -> The DEM is created at $665 \times 665$~m spatial resolution.
67 -> \end{sloppypar}
68 -> 
69 -> 
70 -> \begin{sloppypar}
71 ->       The gravity and DEM data have been processed and explored using Geographic
72 -> Information System (GIS) -- SAGA-GIS
73 -> [\itc{Conrad et al.,} \reflink{Conr15}{2015}],
74 -> QGIS
75 -> [\itc{Thiede et al.,} \reflink{Thie14}{2014}]
76 -> and
77 -> LandSurf
78 -> (Wood, J., The LandSurf Manual,
79 -> \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})
80 -> % [\itc{Wood,} \reflink{Wood09}{2009}]
81 -> free software.
82 -> \end{sloppypar}
83 -> 
84 -> 
85 -> \section{Free-Air Gravity Anomalies and Digital Elevation Model (DEM) of the Mercury}
86 -> 
87 -> \begin{figure*}[t]                     % Fig 3
88 -> \figurewidth{35pc}
89 -> \setimage{}{}{35pc}{}{2019es000678-f03}
90 -> \caption{Spatial and frequency distribution of the
91 -> FAG anomalies within the southern hemisphere of Mercury.}
92 -> \end{figure*}
93 -> 
94 -> \vspace*{.1ex}
95 -> \begin{sloppypar}
96 ->       In gravimetry, ``free-air'' or Faye's gravity anomaly is an anomaly in free atmosphere
97 -> calculated from the observed value of the power of the normal gravity field, reduced to the
98 -> height at the point of observation. On Mercury's topography the ``free-air'' gravity field
99 -> reflects the elevation differences on the surface
100 -> (\figref{1}).           % (Fig.1).
101 -> The free-air gravity values are best
102 -> expressed by interaction between
103 -> negative (craters) and positive (mountains, ridges)
104 -> landforms. For its part, the digital elevation model % (DEM)
105 -> describes the spatial distribution
106 -> and interconnections of landforms.
107 -> \end{sloppypar}
108 -> \vspace*{-.5ex}
109 -> 
110 -> \begin{figure}[b]                     % Fig 4
111 -> \figurewidth{35pc}
112 -> \setimage{10.5pc}{}{35pc}{}{2019es000678-f04}
113 -> \shortcaption{Spatial and frequency distribution of the Mercury DEM.}
114 -> \end{figure}
115 -> 
116 ->       Mercury's free-air gravity field % (FAG)
117 -> varies from $-227$ to $+141$~mGals. The negative
118 -> gravity anomalies dominated over the positive ones
119 -> (\figref{1}).           % (Fig.1).
120 -> Almost 55\%
121 -> of the total area of
122 -> the gravity field is occupied by negative gravity anomalies, 40.5\%
123 -> of positive gravity
124 -> anomalies and only 4.5\%
125 -> is for ``normal'' gravity field. As a whole the FAG is more variable
126 -> within the northern hemisphere
127 -> (\figref{2}),           % (Fig.2),
128 -> than within the southern hemisphere
129 -> (\figref{3})           % (Fig.3)
130 -> of the
131 -> planet. The southern hemisphere of the planet (especially within latitudes above
132 -> 30\deg--35\deg) is
133 -> entirely dominated by a negative gravity field. Within the equatorial regions of the planet
134 -> (from $0\deg$ to $25\deg-30\deg$ in both hemispheres), the FAG is represented by values ranging from the
135 -> highest to the lowest ones.
136 -> 
137 -> \begin{figure*}[t]                     % Fig 5
138 -> \figurewidth{35pc}
139 -> \setimage{}{}{35pc}{}{2019es000678-f05}
140 -> \caption{Spatial and frequency distribution of the DEM within the northern hemisphere of Mercury.}
141 -> \end{figure*}
142 -> 
143 -> 	The digital elevation model % (DEM)
144 -> of Mercury's topography varies from $-5272$~m to
145 -> 4273~m
146 -> (\figref{4}).           % (Fig.4).
147 -> \enlargethispage{-24\baselineskip}
148 -> Negative values occupy about 60\%,
149 -> while the positive ones occupy about
150 -> 40\%
151 -> of the planet's surface. Negative landforms are represented by impact craters, while the
152 -> positive ones by mountains and ridges. In the northern hemisphere, the negative landforms
153 -> prevail
154 -> (\figref{5}),           % (Fig.5),
155 -> while within the southern hemisphere the positive landforms are more
156 -> dominant
157 -> (\figref{6}).           % (Fig.6).
158 ->  Most of mountains and ridges are localized up to $30\deg -35\deg$ northern and
159 -> southern latitude, while the northern latitudes of the planet are predominantly represented by
160 -> negative shapes and lowest values of gravity field.
161 -> 
162 -> \begin{figure}[b]                     % Fig 6
163 -> \figurewidth{35pc}
164 -> \setimage{10.5pc}{}{35pc}{}{2019es000678-f06}
165 -> \crosscaption{Spatial and frequency distribution of the DEM within the southern hemisphere of Mercury.}
166 -> \end{figure}
167 -> 
168 -> 
169 ->               In the course of the study, significant differences between Mercury's northern and
170 -> southern hemispheres were found, both in terms of gravity and hypsometry. As we will see
171 -> from the next section, these differences are well described by the variance of their fractal dimensions.
172 -> 
173 -> 
174 -> 
175 -> 
176 -> \section{Results and Discussion}
177 -> 
178 ->       The results obtained in the course of the study of Mercury's FAG and DEM fractal
179 -> structure are summarized in
180 -> \tabref{1}.  % Table 1.
181 -> The main conclusions and interpretations are discussed
182 -> further.
183 -> 
184 ->       As we can see from the table, the FD min for both FAG and DEM in northern
185 -> hemisphere is approximately equal. Vice versa -- FD~max are relatively different, but the
186 -> tendencies are kept. In southern hemisphere FD~max for DEM are larger (both for positive
187 -> and negative values), but all values are similar in FD~min. The $R^2$ value is lower for the DEM
188 -> of southern hemisphere. This indicated higher non-linearity in spatial distribution of
189 -> landforms to the south of Mercury. Within the southern hemisphere the transition from high
190 -> to low landforms is clearly expressed. To the north, the positive and negative areas of the
191 -> terrain are more compact and larger.
192 -> 
193 -> 
194 -> \begin{table*}[t]                                   % Table 1
195 -> \tablewidth{41pc}
196 -> \caption{Fractal Analysis of Mercury's FAG and DEM (Key: FD -- Fractal Dimension; $R^2$ -- Coefficient of Determination)}
197 -> \vspace{5pt}
198 -> \begin{tabular}
199 -> {@{}l@{\hspace{43pt}}
200 -> c@{\hspace{31pt}}
201 -> c@{\hspace{31pt}}
202 -> c@{\hspace{31pt}}
203 -> c@{\hspace{31pt}}
204 -> c@{\hspace{31pt}}
205 -> c@{}}
206 -> \hline
207 -> \\ [-7pt]
208 -> & \multicolumn{6}{c}{Free-Air Gravity (FAG)}  \\
209 -> & \multicolumn{3}{c}{Northern hemisphere} & \multicolumn{3}{c}{Southern hemisphere} \\
210 -> &   All values&  Positive&  Negative & All values & Positive & Negative \\
211 ->   [7pt]  \hline   \\ [-4pt]
212 -> FD min & 2.10  &      2.01  &    2.01   &    2.13    &    2.01   &   2.01  \\
213 -> FD max & 2.56  &      2.38  &    2.41   &    2.64    &    2.40   &   2.44   \\
214 -> $R^2$     & 0.908 &      0.968 &    0.967  &     0.975  &    0.994  &    0.990  \\
215 -> [7pt]  \hline   \\ [-4pt]
216 ->   & \multicolumn{6}{c}{Digital Elevation Model (DEM)}  \\
217 -> & \multicolumn{3}{c}{Northern hemisphere} & \multicolumn{3}{c}{Southern hemisphere} \\
218 -> &   All values&  Positive&  Negative & All values & Positive & Negative \\
219 -> [7pt]  \hline   \\ [-4pt]
220 ->  FD min &2.01    &    2.0   &    2.01   &      2.01   &   2.0    &    2.01 \\
221 -> FD max &2.64    &    2.45  &    2.58   &     2.58    &  2.47    &   2.54  \\
222 -> $R^2$     & 0.942  &     0.903&    0.940  &      0.857  &    0.863 &     0.910  \\
223 -> [4pt] \hline
224 -> \end{tabular}
225 -> \end{table*}
226 -> 
227 -> 
228 -> \begin{sloppypar}
229 ->       The comparison between the northern and southern hemisphere of FAG (positive and
230 -> negative) anomalies
231 -> (\figref{7} and \figref{8})                % (Fig.7, 8)
232 -> shows clearly that the areas of both types of FAG anomalies
233 -> dominated in their sizes and numbers. For example, the negative values in northern
234 -> hemisphere have higher areas within range of FD between 2.36 and 2.40. For the southern
235 -> hemisphere the same is true for the range between 2.37 and 2.40. Same tendency is clearly
236 -> visual for the positive values.
237 -> \end{sloppypar}\vspace{.5ex}
238 -> 
239 -> 
240 ->       The analysis of the FD for both hemispheres shows the following peculiarities: The
241 -> fractal dimensions for both cases starts as values of 2 and finished at levels 2.4. Up to 2.27
242 -> (for northern hemisphere -- NH) and 2.30 (to the southern one -- SH) the fractal dimensions
243 -> reflects both -- positive and negative anomalies. These values show that in both cases the
244 -> nonlinearity is clearly expressed but at almost the same values of areas (positive and
245 -> negative). The appearance of some spikes in the distributions is not very clear and will need
246 -> additional investigations. After 2.27 (NH) and 2.30 (SH) the dynamics in the changes of FD's
247 -> is sharper and rather specific for NH and SH. The FD of the positive anomalies in NH
248 -> demonstrates a maximum between 2.37 and 2.38, which means specific fractal structure (i.e.
249 -> fragmentation). The FD distribution of the positive anomalies in SH is unimodal, as well as
250 -> both distributions of the FD's related to the negative anomalies. These specifics also need
251 -> some additional investigations, but clearly confirm that the northern hemisphere is much
252 -> more frequently attacked by the meteorites, asteroid and/or comets.
253 -> 
254 -> 
255 ->       Generally within the northern hemisphere of Mercury $R^2$ of fractal dimensions of both
256 -> FAG and DEM are almost identical
257 -> (\figref{7} and \figref{8}),           %(Fig.7, 8),
258 -> but this does not apply to the planet's southern
259 -> hemisphere.
260 -> 
261 -> \begin{figure*}                     % Fig 7
262 -> \figurewidth{35pc}
263 -> \setimage{}{}{35pc}{}{2019es000678-f07}
264 -> \caption{Fractal dimensions frequency of FAG positive and negative values within the northern hemisphere of Mercury.}
265 -> \end{figure*}
266 -> 
267 -> \begin{figure*}                    % Fig 8
268 -> \figurewidth{35pc}
269 -> \setimage{}{}{35pc}{}{2019es000678-f08}
270 -> \caption{Fractal dimensions frequency of FAG positive and negative values within the southern hemisphere of Mercury.}
271 -> \end{figure*}
272 -> 
273 -> \section{Conclusion}
274 -> 
275 ->       The distributions of the DEM and FAG of northern and southern hemispheres of
276 -> Mercury are studied. The results show clear fractal properties of the positive and negative
277 -> areas of DEM and FAG. The correlation between them is confirmed. Sometimes there are
278 -> some discrepancies in the correlation which probably means lack of coincidence in the FAG
279 -> and DEM due to the different densities.
280 -> 
281 ->       The northern hemisphere is much more bombarded by meteorites and asteroids then
282 -> the southern one. The fractal analysis confirmed this fact clearly. This means that the free
283 -> space flying objects bombarding the Solar system dominated the North direction. If this is a
284 -> space specifics, it must be proved for other planetary objects in the Solar system too. The
285 -> more dense free flying objects in the Solar system are about 1.5--2.0 times rarer then the less
286 -> dense. It could be related to the physical properties of the meteorites and asteroids. Usually
287 -> the less dense objects are identified as rocky objects, and the denser -- as richer of metallic
288 -> components space objects. This means that the ``iron'' objects (probably) are rarer within the
289 -> Solar system. Probably it can be proved by the gravimetric studies to other terrestrial planets
290 -> and satellites in the Solar system.
291 -> 
292 -> 

Total strings in array = 293
