1 Spatial Modelling of the Terrestrial Environment - Georeferencial

186 Spatial Modelling of the Terrestrial Environment
At wavelengths around 4 µm in the MIR terrestrial atmospheric window (3–5 µm), the
exponent b has a value equal to 4, whilst the multiplier a has a value of 3 × 10 −9 . Expanding
this relationship to represent a fire ‘hotspot’ pixel containing n subpixel thermal components
of fractional area A n and temperature T n (which may be the different smouldering and
flaming parts of the fire), provides:
L h,MIR = aε MIR
n∑
i=1
A n T 4
n (3)
where L MIR,h and ε MIR are the hot ‘fire’ pixel spectral radiance and surface spectral emissivity
in the appropriate MIR spectral band.
For the same modelled fire activity within a pixel, the energy emitted over all wavelengths
is given by Stefan’sLaw:
FRE TRUE = A sampl · εσ
n∑
i=1
A n T 4
n (4)
where FRE TRUE is the Fire Radiative Energy (J s −1 ) emitted over all wavelength, A sampl is
the ground-pixel area (m 2 ), ε is the surface emissivity averaged over all wavelengths, σ is
the Stefan–Boltzmann constant (5.67 × 10 −8 Js −1 m −2 K −4 ).
Equating (3) and (4) provides the following equation relating FRE to the MIR spectral
radiance recorded at the fire pixel:
FRE MIR = A sampl · σ.ε
L MIR,h (5)
a · ε MIR
Since L MIR,h represents the MIR radiance from the fire only, when analysing real remotely
sensed data this should be calculated by subtracting the background MIR radiance L MIR,bg
(estimated from neighbouring non-fire pixels) from that of the fire pixel (L MIR,h ). Langaas
(1995) suggests that for larger wildfires (the type most likely to be analysed via moderateresolution
satellite imagery) the flames can be assumed to radiate as black or grey bodies
(i.e., ε = ε MIR ), which we suggest is a reasonable assumption for our purposes, especially
so since, when viewing from above, the hot surface material will form the background to
any observation of the actual flames themselves. Thus, the emissivity terms are removed
from the equation, and setting A sampl equal to the 1 km 2 pixel area of the MODIS 3.9 µm
channel provides the algorithm for estimate FRE from MIR radiances recorded at MODIS
fire pixels:
FRE = 1.89 × 10 7 (L MIR − L MIR,bg ) (6)
where spectral radiance has units of W m −2 sr −1 µm −1 and FRE units of J s −1 or Watts.
Wooster et al. (2003) have compared the FRE estimates derived via equation (6) to
those from the empirically derived equation provided by Kaufman et al. (1998a), which is
used in the MODIS fire products (Kaufman et al., 1998b), and the results show excellent
agreement (r 2 = 0.99). However, the physically based nature of equation (6) allows it to
be easily adapted for use with other satellite-based sensor working in the 3–5 µm atmospheric
window. The Hot Spot Recognition System (HSRS), carried by the Bi-spectral
InfraRed Detection (BIRD) small experimental satellite (Briess et al., 2003), is one such
sensor and Figure 9.4 shows a comparison of MODIS and HSRS MIR images for one of