Appendix – Detailed descriptions for the metrics tested in this study. Metric names, symbols, and descriptions are taken from the SPIP software programdocumentation (SPIPTM)as well as descriptions provided by McGarigal et al. (2009) and Gadelmawla et al. (2002). Classification of metrics as well as discussions on the relationships between surface metrics and patch-based metrics are fromMcGarigal et al. (2009). Additional details on the surface metrics described below can be found by accessing the SPIP user manual (available from All of the metrics described below and used in the study can be calculated with and without correcting for the overall mean height of the surface – or plane fit to the surface. The descriptions note which metrics were based on a correction for the overall mean.
Amplitude Metrics
Roughness Average (Sa): The average absolute deviation of the surface heights from the mean. Sa is a general measure of overall surface variability and is therefore a measure of landscape diversity. Since it is a non-spatial measure, it is analogous to patch-based diversity metrics. Sa does not differentiate between different shapes in the surface height profile, only their absolute heights. The equation for Sa is:
whereMxN is a rectangular image, z is the height value, and x and y are individual pixel locations. The default unit for Sa is nm when computed in SPIP.
Root Mean Square Roughness (Sq): The standard deviation of the distribution of surface heights. It is a measure of overall surface variability, like Sa, but is more sensitive to large deviations from the surface mean than Sa. The equation for Sq is:
whereMxN is a rectangular image, z is the height value, and x and y are individual pixel locations. The default unit is nm when computed in SPIP.
Surface Skewness (Ssk): describes the asymmetry of the surface height distribution above the mean. Ssk is sensitive to occasional deep valleys or high peaks. A surface with an equal number of peaks and valleys will have a surface skewness of zero indicating symmetric height distributions. Similarly, surfaces with no peaks but occasional deep valleys will have negative skewSsk< 0, and flat surfaces with no valley but with peaks will have positive skew,Ssk> 0. Values numerically greater than 1.0 may indicate extreme holes or peaks on the surface. High skewness values, either positive or negative, indicate a landscape with a dominating surface height. This can be compared to the patch-mosaic model in terms of having a dominant matrix underlying the landscape. As such, this metric can be interpreted as a measure of landscape dominance similar to patch-based evenness metrics. The equation for Sskis:
Surface Kurtosis (Sku): describes the “peakedness” of the surface topography. Since Sku is relative to the mean value, it is sensitive to occasional deep valleys or high peaks. For Gaussian height distributions,Skuapproaches 3.0 when increasing the number of pixels. A surface with relatively little area in high peaks above or deep valleys below the mean has a high surface kurtosis (Sku>3). A surface with relatively even distribution of heights above and below the mean has low surface kurtosis (Sku<3). Smaller values indicate broader height distributions andvisa versafor values greater than 3.0. In terms of comparison with patch-based metrics, a landscape with high surface kurtosis has a dominant surface height, which is akin to the matrix for a patch mosaic model. Therefore, Sku can also be interpreted as a measure of landscape dominance. The equation for Skuis:
Ten Point Height (S10z): defined as the average height above the mean surface of the five highest local maximums plus the average height of the five lowest local minimums. When there are less than five valid maximums or five valid minimums, the parameter is not defined.S10z is a general measure of overall surface variability and is sensitive to occasional high peaks or deep valleys. Interpretation of this metric is similar to Sa and Sq, and the three are likely to be correlated for real-world applications. The equation for S10zis:
wherezpiand zviare the height of the ithhighest local maximums (peaks) and the ithlowest local minimums (valleys) respectively. The default unit is nm when computed in SPIP.
Root Mean Square Gradient (Sdq): is the RMS-value of the surface slope within the sampling area, and is defined as:
Surface Bearing Metrics
Surface bearing metrics are based on the surface bearing area ratio curve (also known as the BAC or Abbott Curve). The Abbott curve is calculated by inverting the cumulative height distribution histogram, and the bearing area curve represents the cumulative form of the surface height distribution used in the amplitude metrics (see McGarigal et al. 2009; Gadelmawla et al. 2002 for examples of the BAC/Abbott Curve). The BAC is generally divided into three zones known as the “peak”, “core” and “valley” zones. The peak zone corresponds to the top 5% of the surface height range. The core zone corresponds to the 5%-80% surface height range. The valley zone corresponds to the bottom 20% of the surface height range. Surface bearing metrics are sensitive to the overall height distribution and variability but are insensitive to the spatial arrangement of that variability. Surface bearing metrics are measures of landscape composition, not configuration.
Surface Bearing Index (Sbi): defined as the ratio of the root mean square roughness (Sq) to the height from the top of the surface to the height at 5% bearing area. It is an overall measure of surface variability and is sensitive to occasional high peaks. However it is insensitive to occasional deep valleys. Sbi approaches 0.608 for Gaussian height distributions, so surfaces with relatively few high peaks will have a low Sbi(less than 0.608). Similarly, surfaces with relatively many high peaks, or surfaces without any high peaks, will have a high surface bearing index. In terms of patch-based concepts, Sbican be interpreted as a measure of landscape dominance. The equation for Sbiis:
where z.05 is the surface height at 5% bearing area. For a Gaussian height distribution, Sbiapproaches 0.608 for an increasing number of pixels. Large Sbiindicates a good bearing property. Note that negative peak artifacts may cause this parameter to be underestimated. In such cases, it may be appropriate to perform noise filtering.
Core Fluid Retention Index (Sci): measures the area above the Abbott Curve in the core zone. Sci is a measure of the shape of the surface height profile like Ssk and Sku, and it is sensitive to both occasional high peaks and deep valleys. Sci approaches 1.56 for Gaussian height distributions for increasing number of pixels. As such, surfaces with relatively few high peaks and/or low valleys – indicating the area under the Abbott curve in the core zone is large - will have a high Sci (>1.56). Surfaces with many high peaks and/or low valleys will have low Sci values (< 1.56). The equation for Sci is:
whereVv(Zx) is the void area over the bearing area ratio curve and under the horizontal lineZx. Large values ofSciindicate that the void volume in the core zone is large. For all surfaces,Sciranges between 0 and 0.75(Z0.05 -Z0.80)/Sq.
Spatial Metrics
Summit Density (Sds):defined asthe number of local peaks, or maximums, per area. Sds is a simple measure of overall spatial variability in surface height. Larger Sds values indicate an increase in the spatial heterogeneity of the surface attribute and lower values indicate an increase in homogeneity. Sds is sensitive to noisy peaks and should be interpreted with care. In terms of patch-based metrics, it is analogous to patch density. The equation for Sds is:
Note: Sds is sensitive to noisy peaks and should therefore be interpreted carefully
Surface Area Ratio (Sdr): expresses the increment of the interfacial surface area relative to the area of the projected (flat) x,y plane. Stated otherwise, it is the ratio between the surface area of the flat plane with the same x,y dimensions. For completely flat surfaces, the surface area and the area of the reference x,y plane are exactly the same, and Sdr = 0%. Sdr increases as local slope variability increases. In terms of patch-based metrics, Sdr is somewhat analogous to contrast-weighted edge density.
The equation for Sdr is:
whereAklis defined as:
Texture Direction (Std): defined as the angle of the dominating texture in the image. For images consisting of parallel ridges, the texture direction is parallel to the direction of the ridges. If the ridges are perpendicular to the X-scan directionStd= 0. If the angle of the ridges is turned clockwise, the angle is positive and if the angle is turned counter-clockwise, the angle is negative. This parameter is only meaningful if there is a dominating direction on the sample. Std is calculatedfrom the Fourier spectrum. The relative amplitudes for the different angles are found by summation of the amplitudes alongMequiangularly separated radial lines.The result is called the angular spectrum. Note that the Fourier spectrum is translated so that the DC component is at (M/2,M/2). The anglea of the i-th line isa=p/M, where i=0, 1, ..,M-1. The amplitude sum, A(α) , at a line with the angle,α, is defined as:
Theangular spectrum is calculated by the following formula:
For non-integer values ofand, the value ofF(u(p),v(q)) is found by linear interpolation between the values ofF(u(p),v(q)) in the 2x2 neighboring pixels. The line having the angle,α, with the highest amplitude sum,Amax, is the dominating direction in the Fourier transformed image and is perpendicular to the texture direction on the image.
Note that due to 1/f noise, a dominating direction parallel to the x-axis is often found.
The Texture Direction Index(Stdi), is a measure of how dominant the dominating direction is. It is defined as the average amplitude sum divided by the amplitude sum of the dominating direction. Stdi ranges between 0 and 1 where surfaces with very dominant directions will have Stdi values close to zero. Stdi will be close to one if the amplitude sum for all directions are similar. The equation for Stdi is:
Radial Wave Index (Srwi)is a measure of how dominant the dominating radial wavelength is. It is defined as the average amplitude sum divided by the amplitude sum of the dominating wavelength. Srwi ranges between 0 and 1. Swri will be close to zero if there is a very dominating wavelength. It will be close to 1 if there is no dominating wavelength. The equation for Swri is:
Fractal Dimension (Sfd)is calculated for different angles by analyzing the Fourier amplitude spectrum. For different angles, the amplitude Fourier profile is extracted and the logarithm of the frequency and amplitude coordinates is calculated. The fractal dimension, D, for each direction is then calculated as:
wheres is the (negative) slope of the log - log curves. The reported fractal dimension is the average for all directions.
Texture Aspect Ratio (Str20, Str37) are used to identify texture strength (uniformity of texture aspect). It is defined as the ratio of the fastest to slowest decay to correlation 20% and 37% of the autocorrelation function respectively. In principle, the texture aspect ratio value ranges between 0 and 1. For a surface with a dominant lay, the parameters will tend towards 0, whereas a spatially isotropic texture will result in a Str value of 1. If the autocorrelation for some direction does not decay below 20% or 37%, the associated parameters will not be reported. This may be the case for image containing well organized linear structures.