The seismic source characterization defines the earthquake sources, their geometry, and the rate that earthquakes of various magnitudes are expected to occur on each source (magnitude recurrence relation). The first seismic source model (SSM1) is based on areal source zones from Bungum et al. [
6] with additions from Musson and Sergeant [
23] for locations near the UK, and the second seismic source model (SSM2) is based on the areal source model of the SHARE project [
16]. In addition, four seismic source models, using smoothed gridded seismicity, were developed using an earthquake catalogue merged from the International Seismological Centre on-line bulletin [
30] and the British Geological Survey [
31]. All six seismic source models were then combined in a logic tree framework, with weights of 0.2 to SSM1, 0.4 to SSM2, and 0.1 to each of the four models developed using smoothed gridded seismicity.
4.1. Magnitude Recurrence Relation
A magnitude recurrence relation describes the rate at which earthquakes with magnitudes greater than or equal to
M occur on a source
N(M). The recurrence relation is calculated by integrating the magnitude probability density function (
fm) from
M to the maximum magnitude considered (
Mmax) and multiplying by the activity rate (
Nmin):
where the activity rate is the rate of earthquakes above a minimum magnitude (
Mmin) and the magnitude probability density function describes the relative number of earthquakes of various magnitudes that are expected to occur. All sources in this study used a truncated exponential model, which is a modified version of the Gutenburg and Richter model [
32], and a minimum moment magnitude of
Mmin = 4.0. The minimum magnitude was set at 4.0 because this is likely the minimum magnitude to cause damage to infrastructure [
33]. In addition to the minimum magnitude, maximum magnitude, and the activity rate, the truncated exponential model also requires a b-value. The b-value is the slope of the rate of earthquakes in log space and represents the ratio between large and small magnitude earthquakes for a source.
4.2. Seismic Source Model 1
Bungum et al. (2000) [
6] was the first regional seismic hazard study for the entire North Sea and was a joint effort by Norwegian and UK researchers. Bungum et al. [
5] described the model for Norway and the Norwegian sector of the North Sea, and the report by EQE [
4] describes the model for the UK and the UK sector of the North Sea. Bungum et al. [
5] used a coarse seismic source model consisting of 24 areal source zones and a fine seismic source model consisting of 37 areal source zones. The coarse model is the same as that used for Norway in the GSHAP project [
18]. All source zones used a truncated exponential model with three sets of activity rate and b-value pairs implemented with different weights in a logic tree. EQE [
4] extended both seismic source models to the UK and defined 38 areal source zones for the fine model and 26 for the coarse model. However, the values, weights and distributions of the activity rates of the different seismic source zones of the model used by EQE [
4] are not publicly available.
As a result, this study used the areal source zones of Musson and Sergeant [
23], who developed a seismic source model for the UK that includes some zones in the southern part of the North Sea. Their model was developed from the model of Chadwick et al. [
24], discussed in
Section 3.2, which is based mainly on tectonics and kinematics. Musson and Sergeant [
23] modelled 20 different activity rate and b-value pairs for each areal source zone.
Seismic source model 1 (SSM1) is a combination of the fine models of Musson and Sergeant [
23] and Bungum et al. [
5]. Only the mean activity rate and b-values for each zone are used instead of the entire logic tree of different weighted values.
Figure 1 shows the geometry and location of each of the areal source zones and
Table 1 lists their activity rates (N) and b-values. Zones that begin with NOR are from Bungum et al. [
5] and the rest are from Musson and Sergeant [
23], except NOR0. Areal source zone NOR0 was added to fill the gap between the two studies, using similar values as for Zone NOR2. This is a conservative assumption because the Musson and Sergeant [
23] model has no seismic source for this area.
Table 2 lists the maximum magnitudes used, which are the same as in Musson and Sergeant [
23] and Bungum et al. [
5].
4.4. Smooth Gridded Seismicity Source Models
Four seismic source models were developed based on an earthquake catalogue compiled from the International Seismological Centre on-line bulletin [
30] and the British Geological Survey [
31] for earthquakes occurring within a 500 km radius of the site. The ISC catalogue is based on instrument recordings and for this area contains data from 1927 to 2020. The BGS catalogue contains both instrumental and historical earthquakes from 1382 to 2020. The two catalogues were merged and duplicates were removed, with precedence given to the BGS catalogue.
Figure 3 shows the merged earthquake catalogue. The largest earthquake in the database is the 1931 Dogger Bank earthquake (blue pentagon in
Figure 3). According to the BGS catalogue, it occurred 111 km to the southwest of the site and had a local magnitude
ML = 6.1, whereas the ISC catalogue places the earthquake at 126 km from the site with a surface wave magnitude
MS = 5.6. The more conservative interpretation from the BGS was used in the final catalogue.
The earthquake magnitudes were all converted to moment magnitude (
Mw) using the midcontinent region magnitude conversion equations given in NRC [
35] and those proposed by Grünthal et al. [
17], with equal weight given to both methods in the logic tree framework. The NRC [
35] conversion equations were developed for the Central and Eastern United States, which is a similar tectonic region to the North Sea, and the Grünthal et al. [
17] equations were developed for central, northern and north-western Europe. Several other studies [
36,
37,
38] also used the method proposed by Grünthal et al. [
17] to convert magnitudes for sites in the UK.
PSHA assumes that all earthquake events are independent; therefore, dependent events such as foreshocks and aftershocks must be removed. The declustering models of Grünthal [
39] and Reasenburg [
40] were used. The Grünthal [
39] method uses a magnitude-dependent space and time window to define foreshocks and aftershocks. It is based on regression analyses of earthquakes from central Europe. The Reasenburg [
40] model uses a time window based on Omori’s law, which models the expected rate of aftershocks, and a time-dependent Poisson process. It allows for higher order aftershocks (i.e., aftershocks of aftershocks). The space window is calculated based on the estimated source dimensions of the most recent previous event in the cluster and the source dimensions of the largest event in the cluster.
The method of Stepp [
41] was used to estimate completeness of the earthquake catalogue. A catalogue is complete when all earthquakes with
M >
Mmin that occurred in a given area and time period are included in the catalogue. Evaluating catalogue completeness is important to prevent under-prediction of the activity rate, which can lead to unconservative results. Normally, most catalogues are incomplete for smaller magnitude earthquakes because they are harder to detect without a strong ground motion station nearby.
The method of Stepp [
41] evaluates earthquake catalogues for completeness for individual magnitude bins. This allows information for larger earthquakes from historical records to be used with information from instrument records that have a much shorter time period. The Stepp [
41] method assumes the earthquake sequence can be modelled as a Poisson distribution, and uses the statistical property that the variance of the estimate of a sample mean is inversely proportional to the number of observations in the sample. Therefore, the rate of occurrence of earthquakes should be approximately constant and the catalogue is incomplete when it starts to decrease. However, the mean rate of occurrence and the standard deviation will only be stable and constant in the subinterval that is not only complete, but also long enough to give a good estimate (i.e., the sample size is statistically large enough). As a result, the data might fluctuate for the first few years due to the small sample size.
Table 4 lists the completeness years for different minimum magnitudes as well as the completeness years calculated by other studies for nearby regions. The completeness years in this study are similar to those of [
38] for the UK sector of the North Sea and the completeness years used by the SHARE project [
16] for northern Europe. The others have significantly earlier completeness years (longer time periods) because they include large portions of onshore UK, which have better records of earthquake occurrence than offshore areas.
After the earthquake catalogues were corrected for earthquake magnitude, dependent events and completeness, the activity rate, b-value and uncertainty bounds were calculated using the maximum likelihood method [
42]. The activity rates and b-values were calculated for four alternative source models (
Table 5 and
Figure 4). Models SS-NRC12-GR and SS-NRC12-RE use the NRC [
35] magnitude conversion equations, and SS-G09-GR and SS-G09-RE use the magnitude conversion equations proposed by Grünthal et al. [
17]. SS-NRC12-GR and SS-G09-GR use the Grünthal [
39] declustering method, and models SS-NRC12-RE and SS-G09-RE use the Reasenberg [
40] declustering method.
For each of the four source models based on earthquake catalogues, one areal source zone of 700 × 700 km centered on the site was modelled using smoothed gridded seismicity. Smoothed gridded seismicity is a grid of very small areal sources with different activity rates but the same magnitude probability density function and b-value. The different activity rates represent the spatial variability of earthquake occurrence. The relative rates of each cell are based not just on the earthquakes that occurred in that cell, but a weighted average of the rates of the cell and the cells around it. A Gaussian distribution with a 30 km radius and 0.1 × 0.1 degree grid cells was used to calculate the smoothed gridded seismicity.
Figure 5 shows the smoothed gridded seismicity models.
Maximum magnitudes of 6.5, 6.7, 6.9 and 7.0 were used with weights of 0.5, 0.2, 0.2 and 0.1 in the logic tree framework, which was similar to several previous studies [
36,
37,
38]. The largest earthquake in the database is the 1931 Dogger Bank earthquake that had a
Mw = 5.8 (
ML = 6.1), which was less than these values.