The soil roughness of the SMAPVEX08 data set is described using three parameters; root mean square height (*rms height* or *sigma*), correlation length *(L)* and the correlation function *f(L)*. The root mean square height describes the random surface characteristics, while the correlation length and correlation function describe the periodicity of the soil surface. The correlation function is characterized by a power coefficient n ranging from 1 to 2, where 1 represents a Gaussian height distribution and 2 represents an exponential height distribution. The periodicity and random components of the soil surface roughness are schematically shown in Figures 1a and 1b.

In terms of the mean surface height and the second moment , the *rms* height is represented by:

**(Equation 1)**

where *z* is the surface height in cm.

To determine the correlation length and the correlation function, the surface autocorrelation curve needs to be computed. The surface autocorrelation is a measure of the degree of correlation between the height *z(x)* at point *x* and the height *z(x+d)* at point *x + d*. The following equation can be used to calculate the autocorrelation curve:

**(Equation 2)**

Once the autocorrelation curve has been computed, the correlation length can be determined. The correlation length is defined as the distance *(d)* at which the autocorrelation is less than *e-1 (ï‚» 0.3678)*. The computed correlation length can be used to fit the theoretical correlation function to the measured autocorrelation curve by optimizing the power coefficient *(n)*. The correlation function is mathematically represented by:

**(Equation 3)**

Where *L* is the correlation length (cm) and *n* is the power coefficient describing the correlation function, which is dimensionless.

#### Sampling Strategy

At several sites, several representative locations were selected for roughness sampling. At each sampling location, one roughness picture along the row direction and one in the cross-row direction was taken. In the absence of row structure, the two pictures were taken at perpendicular angles. At some roughness sampling sites, one picture was taken. In other cases, pictures were missing.

#### Digitizing the Pictures

The commercial program Didger 3 was used to digitize the roughness pictures. Before scanning, the dimensions of the board were identified in Didger 3 using reference points on the board. The soil surface was digitized by taking a height measurement at every 1/2 cm (grid scanning). This scanning method provides a random (or normal) distribution of the surface height, which is required for a correct computation of the *rms* height. However, with this method of scanning some variation in the surface height is neglected, which could influence the computation of the correlation length.

#### Calculation of the Roughness Parameters

The roughness parameters were calculated using a simple spreadsheet program. Because of the variability in x increment of the digitized surface, the surface was resampled to the nearest 1 mm. The root mean square error was then calculated. Correlation length was calculated as the length at which the autocorrelation function is equal to *e-1*. The power coefficient was determined by visual comparison of the autocorrelation curves and idealized power curves with some guidance by the root mean square error between the curves.