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where a, b and r are as shown in fig. 1. T is the temperature at
distance r from the centre, T, the temperature at distance a, and T,
the temperature at distance b.
Figure l. Basis of radial heat flow model
These results can be used in the following way. Take one of the
measured temperatures and assume it is ‘correct’: i.e., it would fit
equations 2 and 3 exactly if all the other quantities were known.
Then, for a range of values of fire temperature (at radius a),
position and radius, calculate the implied temperature at the other
grid points. (If the fire position is defined relative to one grid
point the distances to all the other points follow.) The set of
values of fire position, radius and temperature which produces the
best match between calculated and measured temperatures is the
solution to the problem.
A FORTRAN program was written to perform these calculations.
The following ‘space’ was searched for solutions in the case of
fire 141
X co-ordinate : -100 to 100 m with resolution 2m
y co-ordinate : -100 to 100 m with resolution 2m
z co-ordinate : -100 to 100 m with resolution 2m
temperature: 750 to 2500 K with resolution 50 K
fire radius: 2 to 30 m with resolution 2 m
and for fire 143 solutions in the intervals
X co-ordinate : -250 to 250 m with resolution 10 m
y co-ordinate: -250 to 250 m with resolution 10 m
z co-ordinate: -200 to 200 m with resolution 10 m
temperature: 500 to 2500 K with resolution 100 K
fire radius: 2 to 30 m with resolution 2 m
The *top left" corner of the measurement grid was assigned the
co-ordinates (0, 0, 0) in both cases. Positive z co-ordinates (above
the level of the origin) were tried as solutions because of the hilly
relief in the test areas. It would be quite possible to have an
underground fire above the level of the measurement grid.
2.5 Radial Model Results
The results of applying this model were rather better for fire 141
than for fire 143. The best solution found in the case of fire 141
was for a fire with a temperature of 1050 K and radius 2 m
725
centred at co-ordinates (22, 26, 0). The average difference
between calculated and observed temperatures was then 7.65 K
per grid point. The important point is that all the other ‘good’
solutions had very similar ‘co-ordinates’. No other region of
solutions was found even with the quite fine intervals used. The
solution was, therefore, well-defined. With a five-dimensional
space to search, the intervals used were about the finest practical.
The best solutions for fire 143 did not match the observed
temperatures as closely; the best solution had an average error of
11.84 K. The problem was that according to this solution, the
temperatures at 30 cm would then be higher than those at 50 cm,
the opposite of what was observed. It could be that the solar heat
flux penetrated to greater depths at this test site because of a
difference in soil properties and that this affected the results.
The results for fire 141, which at first sight look encouraging,
have to be carefully interpreted. What the solutions actually
imply is that an object of a particular radius with a particular
surface temperature would produce certain temperature patterns at
the surface - in some cases very similar to the observed
temperatures. This is not the same as saying that the fires
definitely have these radii and temperatures even if the model is
correct, because this model does not apply to the region in which
the heat is generated. It might be possible to model the
temperature distribution within this region if an ‘edge of fire’
temperature could be defined, but this is difficult. The main
usefulness of the model, in fact, is in determining the fire position.
Equation 2 could also be used to estimate the total heat flux form
the fire if the thermal conductivity of the rocks were known. This
could then be converted to a rate of coal burning, which is another
thing we wish to know.
The other main limitation of this model is that it can only really
deal with situations of constant thermal conductivity. In the field,
there could be many different rock types present between the fire
and the area where temperature measurements are made. It would
not be practical to apply analytical techniques in this situation. A
numerical technique, such as the finite element method, is
required for this. The ANSYS? finite element software has
recently been acquired and is now being used for this purpose.
3. CONCLUSIONS AND FUTURE WORK
A coal fire detection system based on Landsat-TM and airborne
data, at least, seems feasible. The TM data can be used to detect
larger fires and to map areas worthy of more detailed
investigation using airborne data. Care has to be taken to remove
solar heating effects from day-time data. It would be preferable to
always use night-time ‘pre-dawn’ data for detection but these can
be difficult to acquire, especially in the case of TM.
The radial heat flow model could be useful for determining fire
locations. It and the anticipated numerical model require further
testing. This should be possible after a further field campaign this
summer. It is planned then to repeat the soil temperature
measurements at sites in Ningxia province and the data gathered
will be used as inputs to the models. The difference is that, for
the Ningxia fires, extensive borehole temperature data down to
the fires exist; everything we want to model is in effect already
known. The models’ usefulness will be assessed by comparing
the new results with these temperature data.
The other major piece of work still to be carried out is to find a
way of ‘correcting’ the surface temperatures determined from the
remote sensing data for the influence of the temperature variations
described earlier. This would remove the need for sub-surface
temperature measurements. The solar heating maps are not
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996
