In: Wagner W., Székely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
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method as implemented in this work requires spectral radiance 
measurements of the target and a reference panel and the 
temperature and reflectance of the panel surface. The panel is 
assumed to be perfectly diffuse (or, more precisely, the 
directional properties of the panel and target are assumed to be 
the same), its reflectance is the same as measured in the 
laboratory, and atmospheric transmission between the target 
and the sensor is assumed to be neglectable. It also assumes that 
there is no temporal variation of atmospheric transmission. 
The spectral radiance at the sensor can be written as 
L S (A,T) = e(A)L BB (A,T) + (1- £(A))L>(A) (1) 
where L s is the spectral radiance at the sensor, A is wavelength, 
T is the material surface temperature, s is the emissivity of the 
surface, L BB is the blackbody spectral radiance at the surface 
(described by Planck’s function) and L* is the downwelling 
radiance. Solving for e(X) gives 
e(A) = (L S (A,T) - L*(A)) / (L BB (A,T) - L l (A)). (2) 
L*(A) is comprised of all the radiance on the surface from the 
surroundings. Surroundings include the atmosphere and any 
clouds, buildings, sensors, and people in the vicinity. Variable 
in space and time, downwelling radiance can be complex 
quantity, and it is often the largest error in this approach to 
emissivity retrieval. Assuming the panel is a diffuse reflector, 
we can estimate L^(A) with 
I>(A) = (Lp(A) - £ p L BB (A,T p )) / (1- £p ) (3) 
> 
E 
UJ 
Silicate Emissivity Spectra by Size Class 
Wavelength (urn) 
Carbonate Emissivity Spectra by Size Class 
Wavelength (pm) 
where the subscript p refers to the panel, T p is invariant with 
wavelength and, for the panel and spectral range used, £ p is 
constant with a value of 0.040 Neither T nor T p could be 
measured well so they must be estimated from the data. For T, 
Planck’s functions were draped on L S (A,T) to define the T that 
best fit the spectrum, assuming the emissivity was 1.0 
somewhere in the spectrum. To determine the temperature of 
the panel, we draped the Planck’s function over the observed 
panel spectrum where the atmosphere is opaque. 
Two gravel materials at three size classes were measured at 
three view angles, and on three different days. One gravel was 
a calcite with a sharp spectral feature and the other was a 
silicate with broad features. The three size classes had mean 
sizes of 0.8 cm, 1.5 cm, and 5 cm, and view nadir angles were 
7°, 30 °, and 60°. Figure 3 shows that the expected patterns 
were not clearly observed for a 30° nadir angle. At any angle, 
the weakest features occurred for the largest size class, as 
expected, but the medium size class had the strongest features. 
This seems partly due to the sensor view to the south, with a 
large portion of gravel surfaces being shaded and cool and the 
small gravel could not maintain as large a temperature gradient 
across a smaller distance. The large gravel did have the biggest 
changes with view nadir angle. In any case, the expectation that 
spectral feature depths are inversely proportional to roughness 
seems to be an over-simplification: it can be modulated by other 
factors. It is useful to note that the repeatability of the retrievals 
was generally in the 1-2% range, with exceptions occurring 
when the downwelling radiance was large. 
Figure 3. Retrieved emissivity spectra for gravels of 
different size classes for two rock types viewed at 30° from 
nadir. 
2.3 Roughness Modelling and Validation 
Radiosity models explicitly describe the radiant interactions 
between surface facets or points. Radiosity describes the 
radiative interactions between surface elements, and in the 
thermal IR domain includes both emitted and reflected energy. 
While these models can be scaled to any dimension, as a 
practical matter, areas of 1 m 2 are appropriate for high- 
resolution remote sensing simulations. Input digital elevation 
surfaces at 1 cm resolution can be easily measured with laser 
profilometers. Eq. 4 defines the radiosity for a number of facets 
or points: 
B i= R i+p- J.Bj-F'ij, i,y = 1,2...« (4) 
where B t is the radiosity of a surface element i, R, is thermal 
energy released from a surface element, p is reflectivity of the 
surface, Fy is form factor from surface element j to surface 
element i, and n is the number of surface elements. The second 
term in equation (4) describes multiple scattering components— 
energy bounced one or more times among surface elements. 
The key step of the radiosity model is determining the form- 
factor matrix F. The basic form of a form factor is given by