147.

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where θ_s is solar zenith angle, θ_n is the angle of slope of terrain surface, Φ_s is the solar azimuth angle, and Φ_n is the surface aspect of the slope angle (see Figure 1.11). Slope is defined as a plane tangent to the surface containing two components: one is gradient, which specifies the rate of change in elevation, and the other is aspect, which measures the direction of the gradient. The values of several parameters are required to solve these equations, namely, solar zenith angle θ_s, solar azimuth angle Φ_s, the surface slope θ_n and aspect Φ_n. Both θ_s and Φ_s can be obtained from the image header file, while slope and aspect can be derived by co-registering the image with a digital elevation model (DEM). A variety of approaches can be employed to calculate the slope and aspect from a DEM. Skidmore (1989) provides a good review.

In the case of the non-Lambertian reflectance assumption, Smith et al. (1980) suggest the following function to correct for the topographic effect:

(1.20)

where θ_e is effective view angle (Figure 1.11), cos(θ_i) is defined in Equation (1.19), and k is known as the Minnaert constant (Minnaert, 1941) describing the bidirectional reflection distribution function of the surface, the type of scattering dependence, and surface roughness (Smith et al., 1980). A Lambertian surface is defined by a k value of 1.0, and Equation (1.20) then reduces to Equation (1.18).

In order to solve Equation (1.20), one needs to relate the effective view angle θ_e and k (the method for deriving cos(θ_i) is described in Equation (1.19)). If Landsat TM or MSS imagery is used, θ_e can be regarded as the same as θ_n (i.e. the slope of terrain surface) since Landsat has a narrow view angle. In the case of SPOT HRV data, which can acquire imagery through an angle of ±27°, θ_e should be set to , the satellite view angle. To estimate the Minnaert constant k, Equation (1.20) is converted into logarithmic form as:

(1.21)

The term k is then equal to the slope of regression line of the plot of log(cos(θ_i)cos(θ_e)) on the x-axis against log(Lcos(θ_e)) on the y-axis.

To calibrate for the topographic effect using a non-Lambertian assumption is more complicated than that based on a Lambertian reflectance assumption. As the computational cost and calibration accuracy is concerned, Smith et al. (1980) suggest that when surface slopes are less than 25° and effective illumination angles are less than 45° then the Lambertian assumption is more valid. Under such circumstances, one can use Equation (1.18) to carry out topographic effect correction, and the calibration accu

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