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- the illumination direction s, 
- the viewing direction v and 
- the orientation n of the local surface normal. 
These directions enclose 
- the incidence angle i; 
- the emittance angle e; 
- the phase angle g (see figure 1). 
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surface element 
Figure 1: Imaging geometry 
Besides these geometrical relations, the image grey 
values are influenced by 
- illumination (radiance and wavelength of the 
incident radiation; distance and extension of the 
Source); 
- atmosphere (absorption, transmission, refraction); 
- sensor (radiometric sensitivity of the opto-electronic 
components; interior and exterior orientation); 
- surface (light reflectance properties of the surface 
layer). 
In the investigated approach, the light source is 
introduced as a distant point light source with known 
radiometric characteristics. Atmospheric influences are 
considered to be neglectible, while the sensor 
parameters are assumed to be known by radiometric 
and geometric calibration. 
Next two radiometric models are presented which 
serve to describe the reflectance properties of 
planetary surfaces. Besides the well-known Lambert 
law the Lommel-Seeliger law is derived in more detail. 
In the field of planetary photometry a series of models 
for the description of light scattering on planetary 
surfaces were developed [Minnaert 1941; Hapke 1981; 
Lumme, Bowell 1981, McEwen 1991]. While 
photometry aims at the derivation of parameters 
describing geological state and situation of planetary 
regoliths, SFS tries to relate observed brightnesses to 
the angles i, e and g and thus surface topography, 
along with surface reflectance (or albedo). 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
   
   
In both models, light with irradiance E, which is 
assumed to be collimated, falls on a surface layer, 
enclosing the incidence angle i between local surface 
normal and the direction to the light source. The 
irradiance is partly absorbed and partly scattered back 
into the upper hemisphere. A sensor lying in direction 
v, which encloses the emittance angle e between 
viewing direction and local surface normal registers 
the incoming radiance L (i,e,g). 
To describe this direction-dependent reflectance the 
so-called bidirectional reflectance (BDR) r(i,eg) is 
defined as the ratio between radiance L.(ieg) 
scattered towards the sensor, and the incoming surface 
irradiance E,; 
r(i,e,g) = L (i,e,g)/E, (1) 
The relation between the BDR and other quantities 
related to reflectance (e.g. the bidirectional reflectance 
distribution function (BRDF)) is given by [Hapke 
1993]. 
The Lambert law of reflectance is based on the 
assumption that the brightness of a surface depends 
only on the incidence angle i and is independent of 
the emittance angle e, i.e. the surface looks equally 
bright from every viewing direction. Consequently, the 
scattered radiance L (i,e,g) has to be proportional to 
E, per unit surface area. The BDR for a Lambert 
surface is 
r(i) » A, : cosi (2) 
A, (Lambert albedo) is a constant which describes the 
ratio between reflected radiance and incoming 
irradiance per unit surface area. The Lambert law is 
widely used in SFS algorithms for its simplicity, 
though no natural surface strictly obeys it. Especially 
for low-albedo surfaces, such as rocky planetary 
bodies, the assumption of Lambertian reflectance is 
not valid. 
In order to derive a more general photometric 
function, a model for the reflection of electromagnetic 
radiation is presented which describes light scattering 
within a semiinfinite, particulate medium which 
scatters light only once. This model is known as the 
Lommel-Seeliger law and was first described by 
Seeliger in 1887. It extends the assumption that light 
reflection occurs at the boundary surface between two 
media only. Instead, light scattering is assumed to be 
a phenomenon which takes place at individual 
particles within a layer of infinite thickness below the 
apparent surface; the radiance observed at a sensor 
comes from light scattered by all particles in the 
medium that lie within the field of view of the sensor. 
an exhaustive treatment of this topic can be found in 
[Hapke 1993]; a short derivation of the Lommel- 
Seeliger law follows.