surface. Both, direct and isotropically diffuse solar radiation in shadow illuminate the simulated soil surface. 
The model assumes that reflectance from any point on a sphere as well as on the horizontal surface is 
Lambertian. The models mentioned above was validated by ground measurements of soil reflectance 
demonstrating a close similarity to the model-generated data. 
Relations found between remotely sensed data and parameters of the illumination and 
viewing geometry of interpreted surfaces have been used to correct their images before classification. The 
dependence of the data upon the sensor view angle is especially important for airborne and satellite scanners 
viewing the surfaces at wide scan angles (Barnsley, 1984; Foody, 1988; Kowalik et al., 1982; Royer at al., 
1985). The directional reflectance of soil surfaces as non-Lambertian reflectors has been explained by 
interactions of the directional component of solar irradiation with irregularities of the surfaces, i.e., soil 
aggregates, clods and soil microrelief configurations. These rough elements produce shadowing effects which 
change the level and angular distribution of solar energy leaving the soil surfaces (Ciemiewski, 1987; Cooper 
and Smith, 1985; Graetz and Gentle, 1982; Huete, 1987; Milton and Webb, 1987; Norman et al., 1985; Pech et 
al., 1986). 
The aim of this paper is to present a mathematical model of the influence of soil surface 
roughness, solar illumination and viewing of a soil surface on soil reflectance in the visible and reflective near- 
infrared range. It is a further improvement of the previous model (Ciemiewski, 1989) and the derived version 
of the model developed by Verbrugghe and Ciemiewski (1993). 
2. METHODS 
2.1. The model 
The model assumes that wave energy in the visible and near-infrared range reflected from anisotropic soil 
surfaces is strongly correlated with the area of sunlit soil surface fragments and significantly reduced by the 
area of shaded soil fragments. Furthermore, the energy leaving the sunlit soil fragments is directly proportional 
to the energy coming to them, that is, it also depends on the angle of incidence of the sunbeams on these 
directly illuminated parts. 
The model predicts the reflectance distribution of an horizontal soil surface along the solar 
principal plane in the wavelength range mentioned above. Equal-sized spheroids of horizontal (a) and vertical 
(b) radii, lying on a horizontal plane simulate the soil surface (Fig. 1). They are arranged on the horizontal 
surface so their centers in the horizontal projection are at the distance d, independently of the azimuthal 
position of the solar principal plane. This regularity in the spacing of the spheroids expresses isotropic 
features of the simulated soil surface geometry. The shadowed and sunlit fragments of the structure are 
observed by a sensor inside of the (r f ) and the (r b ) radii of the basic view area of the sensor which changes with 
the view zenith angle (0v) as: 
r f = r b = 1/2 d cos0v. (1) 
Along these radii the model calculates segments of the sunlit (I) and shadowed fragments (S) of the given 
spheroid (Is, Ss), the adjoining spheroid (la, Sa), and the ground surface between the spheroids (Ig, Sg). The 
model divides curvilinear slopes of the calculated sunlit soil surface segments into many (j) simple linear sub 
slopes of the angle pi. The position of border points between the sunlit and shadowed fragments, and also the 
sub-slope angles were found analytically by solving trigonometrical equations. The pi angles in relation to the 
azimuth position of the soil slopes (<t>r), and angles of the sunbeam direction, 0s and <J>s, decide about wave 
energy reaching these sunlit fragments. This energy is determined using the factor Ep, as: 
Epj = COS0S cospi + sinpi sin0s (sin<J>s smj>r +cos<|>s COS<t>r), (2) 
where 4>r is 90° for the forward pi angles and 270° for the backward pi; 4>s equals 90° for all the solar azimuth 
angles. The factor Ep; is 1 if the sunbeams reach the analyzed slope perpendicularly. It equals 0 when the 
sunbeams are tangential to a given slope. Negative values of this factor mean that the sunbeams do not reach 
the slope directly, i.e., that it slope is shaded. 
Assuming that the total energy leaving the sunlit soil fragments is directly proportional to Epj 
and the length of sunlit soil sub-segments Ii, and that the energy leaving the shaded fragments of an isotropic 
distribution is proportional to the length of shaded segments, relative luminance of the analyzed soil surface 
(L) visible to the sensor from the given direction (0v) can be formulated as: