Jökull - 01.01.2019, Side 75

Pálsson et al.
Figure 2. log N as a function of M for two differ-
ent periods. The red curve shows the period imme-
diately following the eruption and the caldera col-
lapse, April 1 – December 23, 2015. The b-value
is 0.87. The blue curve is for the period October
29, 2016 – June 4, 2017 and shows an anomalously
low b-value, 0.48. Both data sets contain 1000 earth-
quakes. Lines with a slope of -0.48 and -0.87 are
shown for comparison. The 95% confidence inter-
val of these values is ±0.03 and ±0.05, respectively.
– Stærðardreifing jarðskjálfta á tveimur mismunandi
tímabilum. Rauða línuritið gildir fyrir tímabilið rétt
á eftir að gosinu lauk í Holuhrauni og hrun Bárðar-
bunguöskjunnar hætti, 1. apríl – 23. desember 2015.
Gildið á b er 0,87. Bláa línuritið sýnir dreifinguna á
tímabilinu 29. október 2016 – 4. júní 2017. Gildið
á b er 0,48. Á báðum tímabilunum mældust 1000
skjálftar. Línur með hallatölunum -0,87 og -0,48 eru
sýndar til viðmiðunar. 95% öryggismörk fyrir halla-
tölurnar eru ±0.05 og ±0.03.
METHODS
The earthquake data were taken from the online
records of the Icelandic Meteorological Office, avail-
able on their website. Two representative samples of
magnitude-frequency distributions are shown in Fig-
ure 2. Both contain 1000 events. The curves show
a familiar pattern of a linear relationship in the mid-
dle section and significant deviations from linearity at
the low and high magnitude end of the curves. At the
lower end the deviation is clearly caused by the in-
completeness of the earthquake catalog. Small events
are missing. The knee in the curve shows the limit of
detection for the seismic network. At the high magni-
tude end the curve is determined by only few events,
ten or less. These points have only a limited effect
on the slope of the curve. We try to circumvent these
irregularities by fitting the distribution curve by three
line-segments. The b-value is taken to be given by the
slope of the middle segment, that represents a large
proportion of the data.
The R package “segmented” was used to fit a
segmented linear regression model to the log of the
cumulative distribution of earthquake magnitudes as
suggested by the Gutenberg-Richter relation. The
package implements an iterative procedure described
in Muggeo (2003). Three segments were used, with
the slope of the central one giving the b-value. The
package also supplies estimates for a 95% confidence
interval for the slope. It should be noted that this esti-
mate does not take into account the varying frequency
of earthquakes, and with it the varying time period
used for estimating the b-value. Further research is
needed to determine how this affects the uncertainty.
When calculating a time series of b-values, a mov-
ing window with a fixed number of earthquakes was
used, with the b-value being calculated at every stop.
The amount of smoothing was controlled by the win-
dow size and by how far the window was moved after
each iteration. The window size was chosen in an ad
hoc way as being large enough to provide adequate
smoothing to reveal underlying trends but no larger.
RESULTS
A preliminary inspection of the data set reveals large
variations of the b-value with time. This is clearly
seen in Figure 2, where the events of two different
time slots are compared. Magnitude distribution is
74 JÖKULL No. 69, 2019