## Lunar slopes and Hurst exponent

Michael Kenneth Barker

There are many different statistical measures of topographic “surface roughness.” The dataset on this page contains maps of two such measures: Slope and Hurst exponent.

The slope maps give the median absolute (unsigned) slope in the along-track direction based on altimetry data acquired by the LRO-LOLA instrument. The point-to-point slopes have been calculated along track at baselines of 57m, 225m, and 560m, and binned in 0.25-degree overlapping windows, from which the median was reported. These values were then interpolated at a resolution of 16 pix/deg by 16 pix/deg and projected using a simple cylindrical projection with the Generic Mapping Tool's programs 'surface' and 'grdimage'. Map values are in units of degrees. The LOLA Laser 1 and 2 data through mission phase LRO_NO_13 are the source for this data set.

The Hurst exponent represents the slope of the best-fit line to a plot of log(v(dx)) vs. log(dx), where v(dx) is the RMS deviation and dx is the baseline (the deviogram). The value of the Hurst exponent has been calculated from LOLA altimetry data in 1-degree windows spaced 0.25 degrees apart along track, from RMS slopes over the baseline range from 57 m to 2.7 km. Slopes at the 1-degree (30 km) baseline have been removed to eliminate bias on the order of the window size. Many deviograms experience a breakover point marking a transition from one Hurst exponent to another. For each data point, the linear fit to the deviogram spans the baseline range from 57 m to the breakover point, if it exists. Although the baseline range used in this method varies over the surface, it avoids including fits to nonlinear sections of each deviogram and thus presents a more accurate estimate of the Hurst exponent at the smallest available scales. Hurst exponent values were then interpolated at a resolution of 8 pix/deg by 8 pix/deg and projected using a simple cylindrical projection with the Generic Mapping Tool's programs 'surface' and 'grdimage'. The LOLA Laser 1 and 2 data through mission phase LRO_NO_13 are the source for this data set.

The various products provided here are described in further detail in the accompanying detached PDS labels with the corresponding file names.

Details on the derivation and application of these quantities is elaborated upon in Rosenburg et al. (2011).

Please acknowledge Rosenburg et al. (2011) if you use these data products.

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**References**

Rosenburg, M. A., et al., "Global surface slopes and roughness of the Moon from the Lunar Orbiter Laser Altimeter", 2011, JGR, 116, E2.