Hi there! Today’s post is a pretty much different than usual. I will present you a mathematical analysis of planetary features. More precisely, a paper investigating the fractal structure of the surface of Mars. This is a paper entitled Mars topography investigated through the wavelet method: A multidimensional study of its fractal structure, by Adrien Deliège, Thomas Kleyntssens and Samuel Nicolay, which has been recently published in Planetary and Space Science. This study has been conducted at the University of Liège (Belgium).

The surface of Mars

The Mars Orbiter Laser Altimeter (MOLA), as instrument of Mars Global Surveyor, provided us a very accurate map of the whole surface of Mars, which is far from boring. It has for instance an hemispheric asymmetry, the Northern hemispheric being composed of pretty flat, new terrains, which the Southern one is very cratered (several thousands of craters). The northern new terrains are made of lava, which is a fingerprint of past geophysical activity. Moreover, Mars has two icy polar caps.

Among the remarkable features are:

• Olympus Mons, which is the highest known mountain in the Solar System. This is a former volcano, which rises 22 km above the surrounding volcanic plains.
• The Tharsis region, which contains many volcanoes.
• Hellas Planitia, which is a huge impact basin (diameter: 2300 km, depth: 7 km), located in the Southern hemisphere.

You can find below an annotated map, please click!

The mission Mars Global Surveyor

The missions Mars Global Surveyor (MGS) is a NASA mission, which has been launched in November 1996, and has been inserted into orbit around Mars 10 months later, i.e. September 1997. It became silent in November 2006 after 3 extensions of the nominal mission, and gave us invaluable data during almost 10 years. It embarked 5 scientific instruments:

• the Mars Orbiter Camera (MOC), a wide angle camera which gave us images of the surface and of the two satellites of Mars Phobos and Deimos,
• the Mars Orbiter Laser Altimeter (MOLA), which gave us the most accurate topographic measurements of Mars. The study I present today uses its data,
• the Thermal Emission Spectrometer (TES), which studied the atmosphere of Mars, and the thermal emission of the surface. This instrument observed in the infrared band,
• the magnetometer, which studied the magnetic field of Mars,
• and the radio-science, which measured the gravity field of the planet.

Mars Global Surveyor was of great help to prepare the further missions. It allowed in particular to identify landing sites for rovers.

The rich topography of Mars has encouraged many scientists to characterize it with a fractal structure.

Fractals and multifractals

A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale, see the following figure, which shows the well-known Mandelbrot set.

The Mandelbrot set, plotted by myself after an inspiration from Rosetta Code. The zoom on the right shows the same structure than on the left, with a larger scale.

It is tempting to quantify the “fractality” of such a set. A convenient indicator is the Hausdorff dimension, which is an extension of the dimension of a space. A line is a space of dimension 1, a plane is of dimension 2, and a volume of dimension 3. Now, if you look at the Mandelbrot set, for instance, its contour is a line of infinite length (actually depending on the resolution of the plot), which tends to fill the plane, but does not fill it entirely. So, it makes sense that its dimension should be a real number larger than 1 and smaller than 2. The Hausdorff dimension quantifies how a fractal set fills the space. The Hausdorff dimension of the Mandelbrot set is 2, the one of the coastline of Great-Britain is 1.25, and the one of the coastline of Norway is 1.52.

For a natural object, things are not necessarily that easy, in the sense that some parts of the objects could look like a fractal, and some not, or look like another fractal. Then the object is said multifractal.

The Hausdorff dimension is not the only possible measure of a fractal object. In the paper I present today, the authors use the Hölder exponent, which represents how continuous the function is. Here, the function is the height of a terrain, it depends on its coordinates, i.e. longitude and latitude, on the surface of Mars. The Hölder exponent is usually more appropriate for sets of numerical data.

The wavelet transforms

The wavelet transform is a mathematical transform, which aims at measuring the periodicity of a phenomenon, and gives the amplitude of a periodic contribution, at a given period. In our case, the idea is to measure periodic patterns in the spatial evolution of the height of the surface of Mars.
For that, the authors use more specifically the wavelet leaders methods, which will in particular give them the Hölder exponent, and tell them how (mono)fractal / multifractal the topography of Mars is.

Results

The “fractality” actually depends on the scale you are considering. The authors disposed of MOLA data, with a resolution of 0.463 km. They analyzed them twice, once in performing 1-D analyses, in considering the longitude and the latitude independently, and once in a 2-dimensional analysis, which is probably new in this context. And here are their results:

• The surface of Mars is monofractal if you look at it at scales smaller than 15 km.
• It is multifractal for scales larger than 60 km (the authors considered that the range 15-60 km is a transtition zone).
• The “monofractality” is better in longitude than in latitude. This could be due to the hemispherical asymmetry of Mars, to the polar caps, and / or to the fact that the representation surface is just a planar projection, which necessarily alters it.
• Some features can be detected from the variations of the Hölder exponent, especially the plains. However, this technique seems to fail for the volcanoes.

Some links

• The study. You can also find a poster version here.
• The profile of Adrien Deliège on ResearchGate
• The webpage of Thomas Kleyntssens.
• The webpage of Samuel Nicolay.
• The mission Mars Global Surveyor

That’s it for today! I hope you enjoyed this post. I particularly like the idea to give a mathematical representation of a natural object. Please feel free to comment! You can also subscribe to the Twitter @planetmechanix and to the RSS feed.