Topographic Corrections

(4.8) The raster layers slope, aspect, elevation, satellite image, and vegetation raster were combined together using the procedure described in section 4.2.1.2. This created a dataset that could be used for creating polar plots (in Matlab) to represent the relationship between aspect, slope, elevation, reflectance value and vegetation class. Hence, the dataset created here, was set up such that each row represented a unique combination of elevation, aspect, slope and reflectance values for each band with its number of occurrences in the count column. The vegetation raster contained a numerical code for each of the vegetation 1 classes, and the aspect and slope values were converted to integer values before the combination.

4.2.3 Atmospheric Corrections

It was not considered necessary to conduct an atmospheric correction on any im-ages. Most analyses were conducted with only a single date image, see results section 5.3.2.3 for further description.

4.2.4 Topographic Corrections

An ideal slope-aspect correction removes all topographically induced illumination variation so that two objects having the same reflectance properties show the same

digital number despite their different orientation to the sun’s position. The c-correction and the cosine topographic c-correction methods were the two used. The following closely follows Teillet (1986) and Meyer et al. (1993). The calculations for both methods were performed in matlab. The slope and aspect raster layers were derived in Envi4.2 using a 25m DEM over the area (see section 4.2.2.2 for procedure). The suns elevation angle and azimuth were given in the satellite image header information. The sun’s zenith angle is calculated by subtracting the sun’s elevation angle from 90. Topographic corrections were performed on one summer image dated 24.07.1994.

Cosine Correction Method

The amount of irradiance (radiation from the sun) reaching a sloping pixel is pro-portional to the cosine of the incidence angle i, where i is defined as the angle between the normal of the pixel in question and the zenith direction (the sun in this case) see figure 4.5. The cosine law takes the sun’s position into account in the form of the sun’s zenith angle(sz), but assumes the solar constant and the distance between the sun and the earth to be constant for all scenes. In this thesis an image from only one time period was used so the assumptions were correct.

Figure 4.5: Diagram illustrating how the solar zenith angle(sz)and incidence angle(i)are measured. The inclined slope can be thought of as the earth’s surface, with the satellite directly in the zenith and the sun at some other angle. Copied and modified from Teillet (1986)

The cosine correction is a strongly trigonometric approach based on a basic phys-ical law assuming Lambertian reflection characteristics of objects and neglecting

4.2 Image Variation 51

the presence of an atmosphere. The cosine correction is represented by the fol-lowing equations:

Lh=Ltcos(sz)

cos(i) (4.9)

cos(i) = cos(sz) cos(slope) + sin(sz) sin(slope) cos(azimuth−aspect) (4.10) where,Lh= the radiance observed for a horizontal surface (i.e. the new corrected value), Lt = radiance observed over a sloped terrain (i.e. original value,,) sz = sun’s zenith angle, and i = sun’s incident angle in relation to the normal on a pixel.

The cosine correction method only models the direct part of the irradiance. In real-ity regions which are weakly illuminated by direct sunlight, receive a considerable amount of diffuse irradiance. In such areas, the cosine correction has a dispropor-tional brightening effect (because the diffuse radiation is not taken into account).

The smaller thecos(i)(Eq. 4.10), the stronger this over-correction is. For pixels in complete self-shadow (i.e. whencos(i) = 0), and in faintly illuminated areas, the digital numbers (DNs) saturate and lead to artifacts in the corrected image.

A linear regression was calculated for the original and cosine correction method to understand the changes the topographic method had made with the data. A linear regression describes the relationship between 2 variables. A function in matlab polyfit, finds the coefficients of a polynomialp(x)of degreen(in this case 1 -linear) that fits the data,p(x(i))toy(i), in a least squares sense. The resultpis a row vector of lengthn+ 1containing the polynomial coefficients in descending powers (Matlab, 2005). These coefficients can then be plugged into the linear equation described in 4.11 and plotted.

C-Correction Method

The c-correction method is very similar to the cosine correction method but brings the original data into the form:

Lt=mcos(i) +b (4.11)

This corresponds to a regression line (also used in the statistical-empirical ap-proach) with the original DN for a spectral band on theY axis andcos(i)on the

X axis. This method is based on a significant correlation between a dependent and one or several independent variables. With the help of a regression function the influence of the independent variables can be corrected. The quality of such a correlation depends on the degree of explanation of the regression function. The c-correction method introduces a parameter c which is the quotient of b and m of the regression line. The c parameter was derived by calculating the regression line in matlab for data from the satellite image that represented the Good, Very Good, and Less Good grazing classes. All other areas were masked out to give the best possible regression function. These areas were then the areas that were topographically corrected and no other. The c parameter is built in to the cosine law as a additive term:

Lh=Ltcos(sz) +c

cos(i) +c (4.12)

c= b

m (4.13)

where, Lh = radiance observed on a horizontal surface (i.e. the new value), Lt

= radiance observed over a sloped terrain (i.e. original value),sz = sun’s zenith angle,i= sun’s inclination angle in relation to the normal on a pixel,c= correction parameter, m = inclination of regression line (i.e. gradient),b = intercept of the regression line, andcos(i)is defined in equation 4.10.

According to Teillet (1986) the parameter c creates the effect of path radiance on the slope-aspect correction, however the physical analogies are not exact. Math-ematically, the effect of c is similar to that of the minnaert constant i.e. that it increases the denominator and weakens the over-correction of faintly illuminated pixels as a consequence.