In the present study, we investigate the scaling properties of the topography of Mars. Planetary topographic fields are well known to roughly exhibit (mono)fractal behavior. Indeed, the fractal formalism reproduces much of the variability observed in topography. Still, a single fractal dimension is not enough to explain the huge variability and intermittency. Previous studies have claimed that fractal dimensions might be different from one region to another, excluding a general description at the planetary scale. In this article, we analyze the Martian topographic data with a multifractal formalism to study the scaling intermittency. In the multifractal paradigm, the apparent local variation of the fractal dimension is interpreted as a statistical property of multifractal fields. We analyze the topography measured with the Mars Orbiter Laser altimeter (MOLA) at 300 m horizontal resolution, 1 m vertical resolution. We adapted the Haar fluctuation method to the irregularly sampled signal. The results suggest a multifractal behavior from the planetary scale down to 10 km. From 10 to 300 m, the topography seems to be simple monofractal. This transition indicates a significant change in the geological processes governing the Red Planet's surface.

The acquisition of altimetric data from the Mars Orbiter Laser altimeter
(MOLA) has motivated numerous analyses of the Martian topography, each one
aiming to properly characterize the surface roughness. A possible approach is
to assume that topography can be statistically described with quantitative
parameters able to characterize the geological units. Many statistical
indicators have been proposed and widely explored in order to study the
surface of Mars: root mean square (RMS) height, RMS slope, median slope

On Mars, different authors

This section first describes the Martian topography data set used in this study and the MOLA instrument. The second part contains some elements of the theory of universal multifractality. The third part contains the description of the Haar fluctuation tool we adapted in the case of the irregularly sampled Martian topography.

MOLA (Mars Orbiter Laser altimeter

The topography is a two-dimensional field providing, for each pair of
latitude–longitude, an elevation value. In order to investigate the scaling
properties of such fields, we must study the distribution of slopes at
different scales or, equivalently, the fluctuations of elevation from one
point to the other. The simplest way to define fluctuations is to compute the
first difference for each couple of elevation data. The formalism used in
this study is largely based on

Once fluctuations are defined, a common way to explore the scaling properties
of any geophysical field is to compute statistical moments of several orders
and at different scales. If the field is scaling and if the fluctuations are
defined properly, statistical moments will appear as straight lines on
log–log plots, meaning they follow a power-law dependency on scales. This
property is expressed by Eq. (

In the

Although three moments of distinct orders are in principle sufficient to
establish the curvature of

The

The

The

Mars is well known for presenting huge fluctuations of altitude on its
surface despite its relatively small radius in comparison to Earth
(

The definition is given by Eq. (

Statistical moments of several orders (from 0.1 to 2) as a function of scale for simulated series (left plot) and for actual MOLA data (right plot). In both plots, red diamonds (or red triangles) stand for moment of order 1 (or order 2). Integer-ordered moments are also computed in order to test the multifractal formalism. They are plotted in shades of red (left plot) and blue (right plot).

For each scale, we compute moments of several orders

Before going any further, we discuss in this section possible artifacts that
may cause the observed transition around 10

Uncertainty due to the accuracy of measurement at small scales:
the characteristics presented in Sect.

Due to the quasi-polar orbit of MOLA, the density of measurements
depends on latitudes. The minimum is obtained near the Equator, whereas the
maximum is observed near the poles. This could introduce a bias into our
results. Indeed, a global analysis might only reflect the statistics of
high-latitude regions where the signal is oversampled in comparison to low
latitudes. To check that hypothesis, we performed the following: for each
contributing fluctuation, we apply a factor

By construction, all fluctuations have a privileged orientation (north–south) due to the orientation of along-track series. We can not exclude the possibility that a similar analysis would produce a different result with a different along-track direction. This would occur in the case of a strong anisotropy of topography at the planetary scale, outside the scope of this analysis.

Over nearly 4 orders of magnitude, beginning from around 10

Linear fit on the two different scaling regimes (below and above
10

The multifractal formalism may be tested by adjusting the experimental
structure function

Over the range of scales covering only 1 order of magnitude, the behavior is
clearly different. Still, topography seems to exhibit scaling behavior over
that range. Figure

Our goal was to validate the accuracy of the universal multifractal formalism
to describe the global scaling properties of the Martian surface in order to
test whether the roughness intermittency is scaling. From our results,
multiscaling seems to occur over a large but restricted range of scale (above
10

Theoretical structure function

Estimates of the fractal and multifractal parameters for scales
below and above 10

This result is consistent with

We demonstrate that a change in processes governing the Martian topography
occurs at 10

The above definition for Haar fluctuations implicitly requires that
along-track points are regularly spaced. However, the actual MOLA data sets
exhibit different kinds of irregularities, mainly due to the occurrence of
clouds of various lengths and random dysfunction of the instrument that might
introduce a bias into our analysis. Figure

A few examples of irregularly spaced points in the MOLA database.

Comparison between effective irregularities in the MOLA database and artificial irregularities in simulations. Irregularities are expressed as a proportion of occurrences from the maximum possible in log10 space (0 means all fluctuations are observed). The blue line represents the effective irregularity of actual data. Red points are obtained from the synthetic irregularities in multifractal simulations. The good agreement between maxima of the blue curve and red points indicates that effective irregularity is accurately reproduced by simulations.

Statistical moments of order

Distributions of Haar fluctuations depending on the scale.

In order to take this into account, we define an adjustable quality criterion
for the Haar fluctuations (see

The next step is to define a threshold to be applied to the ratio so that irregular fluctuations can be excluded from the global analysis. The threshold has to be chosen carefully: a very restrictive threshold might be damageable by excluding a lot of fluctuations. Indeed, a huge amount of fluctuations is needed for a statistical purpose. On the other hand, a nonrestrictive threshold might involve irrelevant fluctuations in the calculation. That may also introduce a bias.

In this paragraph, we try to quantify how the choice of a threshold might
affect the analysis. For that purpose, we have analyzed multifractal
simulations after having manually introduced holes of different kinds into
the data. To produce relevant statistics, 1000 universal multifractal series
have been generated using the simulation technique called the fractionally
integrated flux (FIF) developed by

We can now compute Haar fluctuations on both biased and unbiased data and see
how the choice of a threshold impacts the analysis of the biased series. We
evaluate statistical moments of order

We acknowledge support from the Institut National des Sciences de l'Univers (INSU), the Centre National de la Recherche Scientifique (CNRS), the Centre National d'Etude Spatiale (CNES) and the Programme National de Planetologie. Edited by: L. Telesca Reviewed by: two anonymous referees