Synonyms

Microrelief; Microtopography

Definition

Surface roughness is usually defined at the human scales of centimeter to a few meter; larger scales are usually considered as topography. Relief at these scales is familiar to field geologists working at the outcrop scale and those interested in interpretation of landforms and earth-surface processes that form and modify them.

Scientific usefulness

One important surficial geologic process is aeolian erosion, transport, and deposition of sediments. The shear stress wind produces at the earth's surface is strongly affected by the surface roughness. The aerodynamic roughness parameter, z0, depends on the wind speed profile as a function of height about the ground (Greeley et al., 1997). This parameter is used by geologists interested in aeolian processes as well as climatologists seeking to quantify atmospheric coupling with the solid earth.

Windblown dust and sand can also modify surface roughness by mantling and attenuating surface roughness (Farr, 1992; Arvidson et al., 1993). This can lead to estimates of relative age for surfaces such as lava flows or alluvial fans exposed to the same rate of aeolian deposition (Farr, 1992; Farr and Chadwick, 1996).

Streambed and ocean-bottom roughness also affect the flow and transport capabilities of water in those environments (e.g., Butler et al., 2001).

Other geologic processes produce or modify surface roughness, in particular volcanic eruptions which may mantle surfaces with ash or produce new roughness elements through extrusion of lava flows which can be relatively smooth pahoehoe or extremely rough aa. Roughness of lava flows can provide information on their eruption characteristics, such as rate and temperature (e.g., Lescinsky et al., 2007).

Land surface roughness strongly affects many remote sensing techniques. Observations of reflected visible-near-infrared wavelengths are affected by sub-resolution self-shadowing of roughness elements. Thus, rougher surfaces are darker, and the shadows are illuminated by sky light or reflections from adjacent land, shifting the spectral signature of the surface (Adams and Gillespie, 2006). At thermal infrared and microwave wavelengths, which are dominated by emission from solar-heated surfaces, roughness as well as larger-scale topography affects the initial heating of the surface while roughness also affects the efficiency of emission (Ulaby et al., 1982). Active microwave (radar) systems image surfaces through scattering of a transmitted wave from the surface. Smooth surfaces at the scale of the wavelengths, which are typically centimeter-meter, reflect energy away from the receiving antenna and are imaged as dark surfaces, while rough surfaces scatter the incident energy in all directions and show up in bright tones on radar images (Henderson and Lewis, 1998).

Much work has gone into quantitative models which seek to remove the effects of roughness on sub-resolution shadowing and thermal heating and emission (Tsang et al., 2000; Adams and Gillespie, 2006). In the radar area, inversion models have been developed which estimate the surface roughness from radar observations at different angles, polarizations, and wavelengths (Ulaby et al., 1982; Van Zyl et al., 1991; Evans et al., 1992; Dubois et al., 1995; Tsang et al., 2000).

Quantifying surface roughness

Good reviews of techniques for describing quantitatively surface roughness can be found in Dierking (1999), Thomas (1999), Shepard et al. (2001), and Campbell (2002), Chap. 3. The simplest description of surface roughness is an estimate of the standard deviation (or root-mean square: RMS) of the surface heights (Table 1).

Table 1 Measures of surface roughness for two natural surfaces. aa is a rough lava flow surface at Pisgah lava field in the Mojave Desert. Playa is a smooth dry lake surface at Lunar Crater volcanic field in central Nevada. Profiles were measured at 1 cm spacing
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Describing the roughness of a surface by its RMS height leaves out any description of the scales of the roughness. One way to describe the scale of the roughness is to calculate the correlation length of profiles. Correlation length is a measure of how quickly heights change when moving along a profile. The autocorrelation function for a surface profile is calculated by sequentially stepping the profile across a stationary copy, multiplying, and normalizing. The autocorrelation is unity for 0 steps, or lags, and then drops as the number of lags increases (Figure 1). The rate of the drop-off, measured by the lag at which the autocorrelation value drops to 1/e, is called the correlation length, l. Smoother surfaces tend to have larger correlation lengths (Ulaby et al., 1982; p. 822).

Figure 1
figure 4

Profiles and surface roughness measures of two natural surfaces: an aa lava surface (rough) and a playa surface (smooth). (a) Profiles: aa profile has been offset 20 cm for clarity. (b) Autocorrelation functions for the two profiles. Note rapid drop-off of aa autocorrelation. Horizontal line is at 1/e. (c) Power spectra of the two profiles. Note they are nearly linear (in log-log plot) and parallel. Playa has much less power at all spatial frequencies.

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Another way to describe quantitatively both the amplitude and scale of surface roughness is through the power spectrum, or power spectral density, usually of profiles. The power spectrum is basically the Fourier transform of the profile (or two-dimensional microtopography) (e.g., Bendat and Piersol, 1986; Brown and Scholz, 1985; Austin et al., 1994). This produces a plot showing power, or variance, as a function of spatial frequency or scale (Figure 1). When plotted in log-log coordinates, the functions are found to be approximately linear (e.g., Berry and Hannay, 1978; Farr, 1992; Shepard et al., 2001), indicating a power-law relationship between roughness and scale. This relationship simplifies the quantitative description of the power spectrum of a profile to two numbers: the slope and offset (Table 1). Power spectrum slope is a measure of self-similarity and is related to fractal dimension of a profile, D, by (Brown, 1985):

$$ {\mathrm{ D}=\frac{{5+\mathrm{ slope}}}{2}} $$

Power spectrum offset is a measure of overall roughness, sometimes called “roughness amplitude” (Huang and Turcotte, 1989, 1990; Goff, 1990).

Measurement of surface roughness

Measurement of surface roughness at scales of centimeter to several meters is difficult, especially as the area covered must be large enough to make statistically significant calculations of quantities described above for natural surfaces. This usually means measuring an area or profiles 10–20 m or more in size. Techniques used to date include templates (Shepard et al., 2001); stereophotogrammetry from handheld and balloon-borne cameras and from a helicopter (Wall et al., 1991; Farr, 1992); and more recently ground-based lidar systems (Morris et al., 2008). All of these techniques have provided cm or better resolution, but helicopter stereophotogrammetry and ground-based lidar have provided the best coverage.

Summary

Land surface roughness at scales of centimeter to several meters is important in several areas of earth science as well as in the interpretation of remote sensing data. Roughness can be quantified in a variety of ways, but power spectral analysis is best at describing roughness and its scaling properties.

Cross-references

Geomorphology

Geophysical Retrieval, Inverse Problems in Remote Sensing

Land Surface Emissivity

Lidar Systems

Microwave Radiometers

Microwave Surface Scattering and Emission

Radars

Radar, Scatterometers

Radar, Synthetic Aperture

Surface Truth

Trafficability of Desert Terrains