Jökull - 01.01.2019, Síða 79

Pálsson et al.
to conclude that there may be a general relationship,
that maximum magnitude is different for inflation and
deflation of a caldera. In this light we may take the
magnitude of the large earthquakes that took place at
Bárðarbunga during the period 1974–1996 as indica-
tion of deflation of the volcano during that time (Ein-
arsson, 1991a) rather than inflation as suggested by
several authors (Zobin, 1999: Nettles and Ekström,
1998; Bjarnason, 2014).
The seismic efficiency during the collapse period
is low. If the potential seismic moment, sometimes
called geodetic moment, is estimated from the def-
inition MG = µuA and assumed all the faulting (u)
takes place on the caldera boundary fault, a total of
65 m, we get MG equal to 1.4 x 1020 Nm. The area
of the caldera fault (A) is estimated from the diameter
of the caldera block of 6 km and depth to the brittle-
ductile transition of 8 km, both values derived from
Guðmundsson et al. (2016). A value of 13 GPa is used
for the shear modulus (µ) of the brittle crust, follow-
ing Grapenthin et al. (2006) and assuming the shear
modulus is one third of the Young’s modulus. The re-
leased seismic moment of all earthquakes during the
collapse (M0), on the other hand, is 9.3 x 1018 Nm,
a factor of 15 lower. Here we use M = 2/3 log M0
– 6.0 to convert magnitude to seismic moment. The
seismic efficiency or moment ratio M0/MG is 0.066
and falls within the range of values for magmatically
controlled events (e.g. Pedersen et al., 2007). It is
thus clear that a substantial part of the fault displace-
ment takes place by aseismic creep. The same con-
clusion may be drawn from the fact that the hypocen-
ters are unevenly distributed along the ring fault. The
earthquakes line up along the northern and southern
sections of the fault, whereas the eastern and western
parts are almost devoid of earthquake sources (Fig-
ure 1). It is therefore conceivable that different sec-
tions of the fault react very differently to the high
strain rate at the boundaries of the caldera block.
The deviation from the linear relationship of the
Gutenberg-Richter equation becomes extreme in the
case of the caldera earthquakes during the collapse
period. As we show, the magnitude-frequency dis-
tribution of these earthquakes can be modeled as the
sum of two populations, population A of small earth-
quakes, which behaves like the Gutenberg-Richter
equation prescribes, and population B of larger events
which can be modeled with a normal distribution.
This is by no means a unique model of the over-all
distribution, but calls for some speculation on possi-
ble explanations. We note that these events take place
during very unusual circumstances, in a limited vol-
ume of rock, bounded by a circular fault, and un-
der very high strain rate. Most of the earthquakes
take place on the boundary fault, as seen in Figure 1.
The block was subsiding at a fairly steady rate as
high as half a meter per day. One may argue that
the dimension of the source faults of the population
A earthquakes was well within the dimension of the
caldera block and to them the faulted volume was like
any other infinite, seismically active volume of rock,
hence in accord with the Gutenberg-Richter relation.
The earthquakes of population B were larger, how-
ever, and involved a good part of the circular caldera
fault where different size limitations are in effect.
There may be a characteristic earthquake, the mag-
nitude of which may be determined by the circum-
ference of the caldera and the subsidence rate of the
caldera floor. We note, for example, that the magni-
tude of these earthquakes appears to decrease as the
collapse continues and the subsidence rate decreases
(Figure 5). The source dimension of the events of pop-
ulation B may be estimated, for example, by apply-
ing the source scaling relationships of Abercrombie
(1995). An earthquake of magnitude 4 would have a
source dimension of 160 m -1600 m, depending on the
assumed stress drop, varying between 100 MPa and
0.1 MPa, respectively. The corresponding values for
a magnitude 5 event would be 500 m – 5000 m. Con-
sidering the volcanic environment and the low seismic
efficiency one may argue that the stress drop is more
likely to be on the low side, and therefore the source
dimensions on the high side.
It may be of some importance for monitoring pur-
poses to determine when the magnitude distribution
in a remote caldera becomes bi-modal as in the case
of Bárðarbunga. The statistical problem of detection
is worth a special study and is outside of the scope
of this paper. We point out, however, that the earth-
quakes exceeding magnitude 5 began four days into
78 JÖKULL No. 69, 2019