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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
5. MATHEMATICAL PREDICTION OF GLACIER
DYNAMIC
Some years after from the beginning of the dynamic study of
the rock glacier of Argualas, it was seen that there was a
relation between the climatic change and the movements of the
glacier. For instance, this relation is more important in 1994,
when the media temperatures were 3° C higher than the rest of
the 90's in the Pyrenees. Obviously, there was an important
glacier dynamic that year (Figure 3).
This correspondence gives us to a study line which try to
foresee in a short period of time the movement of the glacier,
through the climatic data and the geodesic measure.
1992 1993 194 1995 19% 197 198 199 200
| Rod 15
| 0400
E e MÀ
| 0,100
| 0,000
|
Figure 3. Glacier dynamic of rod number 15.
Through the employ of predictive processes, the future behavior
of some targets (geodesic rods, presignaled control points) can
be estimated.
The predictive study can be focused on “temporary sequences”
and “dynamic systems”. Concretely, a dynamic system has
been developed because there are little data to be applied to
temporary sequences.
The dynamic system shows the change of a system through the
passing of time. This change can be described through a
mathematical model, which can be a system of differential
equations.
The needful information for the development of dynamic
systems is: |
* Geodesic coordinates (x, yz) of the rods in different
studies.
* Time of observations. This fact shows us of the period
between campaigns.
e Weather information (rainfalls and temperature)
between the studies.
The mathematical program “Mathematica” has been used to
solve the dynamic system. With the weather and geodesic
values of previous studies we can establish a second order
polynomial for predicting weather conditions in future studies.
The coordinates (x, yz) are used in six geodesic studies, the
first of them have not enough weather data and the sixth is used
as test (López et al., 2002).
With these facts, 12 equations can be creates, and the system
has 12 unknown (a,b, c, d,e, j, gh k,L,o, p).
F(pet) [a£ & bp cep? «d, et * jtp* gp ^h, P ipso! «q)
With the value of these unknowns, the weather conditions for
the sixth observation can be solved:
(p:232.69 dmm, 1:9.29 ^C)
The different positions of a rod through the effect of a
deformation can be considered as the succession of a function
about its initial function:
L chi A «Bae n +
x =F.x pO or FE". ry
907
X qp d(x)
Vila eh h(y)
z n k ! 9 (p.t) q(z) n-l
For example, the results of P1 rod coordinates obtained with the
dynamic system are:
X:1216713 y:1063,780 z: 847,909
The coordinates of the same rod P1 through the geodesic
observation are:
+. 1216753 y 1063418 = 847,938
It can be seen that there is a difference of + 4 cm among all the
coordinates of the dynamic calculation system and the geodesic
measures. So, the method is acceptable for the predictive
determination of “control points”. To the predictive system it
can be imposed some conditions to make it better for future
applications:
e Establishment of a minimal and maximal period of
time among different observations.
* Minimal number of campaigns so that, the predictive
system is acceptable.
e See if the given coordinates through the
photogrammetric method are the ideal ones to
develop this predictive technique. It has been shown
than the geodesic method is acceptable.
6. CLOSE RANGE PHOTOGRAMMETRY IN THE
GLACIER DYNAMIC
6.1 Justification of close range photogrammetry technique
Different techniques can be employed to study any kind of rock
glacier: Interferometry radar, geodesy (angle, distance),
levelling (geometric, trigonometric), global positioning system,
photogrammetry (aerial, terrestrial, close range).
Obviously, surveying around to Argualas glacier (walls of 300
metres) makes the employment of these techniques rather
difficult. In the case of Argualas glacier, the geodesic
techniques, G.P.S. and close range photogrammetry have been
employed.
The geodesic techniques do not let us follow in a detail the
movement of the glacier, because only some points can be
measured. On the other hand, G.P.S. has given problems
because of the multipath effect near glacier walls (Sanjosé,
2003). In addition, to obtain the geodesic dynamic the same
points (rods) of the last campaign must be looked for in the land
and these points can be hidden by of the movement of stones.
Aerial photogrammetry is very expensive just for analysing
only one glacier; its employ could be more efficient to study all
the rock glaciers of the Pyrenees (26 glaciers). Apart from that,
the precision of the flights must be studied in relation with close
range photogrammetry because the plane must not touch the
walls of the glacier, so the plane should fly over them. This fact
makes the photographic quality different. In any case, whatever
kind of photogrammetric technique which is employed in the
Argualas rock glacier must let smaller precisions than 6 cm in
the displacement.
The advantage of photogrammetry (aerial, terrestrial) over the
geodesy is than the photogrammetry can show the position of a
lot of points (photographic information), and it can collect
information in the country much faster than geodesy. The close
range photogrammetry program “C.D.W.” has shown excellent
results in the study of the Argualas glacier.
