1 Spatial Modelling of the Terrestrial Environment - Georeferencial

Modelling Ice Sheet Dynamics by Satellite-Derived Topography 15
surface slope. When τ is incorporated into the flow law of ice (known as Glen’s flow law)
to determine the surface velocity, U s ,wefind that it is proportional to the third power
of α:
U S − U bed = Aτ n h
(2)
(n + 1)
where n is the flow law exponent, usually taken to be 3, A is a variable in the flow law of
ice that determines its viscosity (Paterson, 1994) and U bed is the velocity at the bed. A has
an exponential relationship with temperature:
( q
)
A = A 0 exp − . (3)
RT
Equation (3) is an Arrhenius-type relationship, where A 0 is a temperature-independent
‘constant’, q is the activation energy for creep, R is the universal gas constant and T is
temperature. The relationships between velocity and τ and between viscosity and T result
in a highly non-linear system, with important consequences and challenges for numerical
modelling.
Using Glen’s flow law, the ice thickness, h, of an ice sheet can be shown to be related to
its length, L, and position, x (see Figure 2.1) as follows:
where
h 2+2/n = K (L 1+1/n − x 1+1/n ) (4)
K =
2(n + 2)1/n
ρg
( c
) 1/n
(5)
2A
and c is the net accumulation at the surface (Paterson, 1994). Various assumptions have been
used to derive equation (4) including a flat, horizontal bedrock and a uniform accumulation
rate. From equations (4) and (5), it can be seen that, assuming a value of 3 for n, the ice
thickness is insensitive to c (proportional to the eighth power), and that it varies as the
eighth root of the flow law parameter A. The significance of this will be discussed later.
2.2 Remote Sensing of Topography: Methodology
Probably the most frequently used method for obtaining topography from remote sensing
data is stereo-photogrammetry of airborne or satellite imagery. However, this approach,
using visible sensors such as SPOT, has a number of limitations over ice sheets. First,
there is rarely sufficient contrast to carry out stereo matching in the visible part of the EM
spectrum. Second, cloud is ubiquitous in the polar regions and difficult to discriminate from
snow. Third, to obtain absolute height measurements, ground control points are required,
which are rarely available. Consequently, other approaches have proved more valuable. Microwave
sensors are particularly useful in the polar regions as they are generally unaffected
by clouds and can operate day or night. The two instruments that have been used most successfully
for deriving topography are radar altimeters and synthetic aperture radars (SAR).
The methodologies for deriving topography from these two instruments are outlined.