Jökull - 01.12.1957, Blaðsíða 10

n'y.trttj = JrfrtJGfar.y.zJtfz (20)
The surface gradient due to this part is there-
fore
Fy(0,(t1+At))=gCeil(t1+At) (21)
The surface gradient at the end of the second
period is therefore underestimated if the fol-
lowing expression is applied
g (1 + Ce, i (tx + At) + Ce> 2(At)) (22)
Accordingly, by the proper choice of the
second correction factor the following ineqality
holds for any time t > t^
(Ty)y=0>g(l+Ce,i (t) + C0i2 (t-tj)) (23)
which furnishes the lower limit to the surface
gradient.
The expressions (17) and (23) give upper and
lower limits to the surface gradient in the case
of two periods of erosion. The method is easily
extended to the case of more than two periods.
The surface gradient will in the following be
approximated by the average of the two limits.
In general, the correction factors are small com-
pared to unity and the difference between the
limits is, therefore, relatively small.
Approximations in the case of a short period
of no erosion following a period of erosion.
In the case where a period of erosion is fol-
lowed by a much shorter period of no erosion,
the changes of the surface gradient in the
second period will be relatively small and can
be regarded as a small correction to the surface
gradient at the end of the first period. A use-
able approximation to this correction can be
derived in the following way.
It will be assumed again that f (t) = 0 and
that the initial temperature distribution at the
beginning of the erosional period is T = gy
where g is a constant gradient. The tempera-
ture distribution at the end of this period is
therefore given by equation (8). In order to
obtain the temperature distribution at the encl
of the second period, this expression has to
be used as the initial temperature distribution
for the second period.
An approximation in the present case is
found by assuming that the total erosion dur-
ing the first period occurred instantaneously
at the time t0 before the end of this period.
The time t0 can be found by the condition
that the approximation should give the exact
surface gradient at the end of the period.
Expression (9) can in most practical cases be
written
(Ty)y = 0=g(l+br/Vit) (24)
where t is the time since the beginning of the
erosion and b a constant. On the other hand,
the surface gradient at the time t in the case
of an instantaneous erosion of the same amount
at the time t — t0 is given by expression (14)
(Ty)y=o =g(l +r/V"ato) (25)
By equating (24) and (25) we find
t0 = t/ab2 (26)
The surface gradient at the end of the
second period of the length At where At << t
can therefore be approximated by
<t,),=.=S(> + ^==7^) (27)
The surface gradient at the end of the second
period is therefore obtained by the multiplica-
tion of the correction factor at the end of the
first period by the factor
,/ -------------- (28
V t + jtb2 At
CORRECTION OF THE OBSERVED
WELL TEMPERATURES
In order to attempt a thermal dating of the
geological events three working hypotheses (A),
(B) and (C) will be applied.
The first hypothesis or hypothesis (A) will
be based on the assumption that the uplift
took place at the end of the Tertiary period
and that the present landscape forms, that is,
the fiords and valleys are the work of the
entire Pleistocene period. It will furthermore
be assumed that the present remnants of the
peneplaine represent the true height at the
beginning of the Pleistocene glaciation.
Hypothesis (B) will be based on the as-
sumption that the uplift took place in the
middle of the Pleistocene period and that the
8