As our focus is on Germany and France, we work in areas of primarily low to moderate seismicity and low tectonic deformation. Although active faults have been mapped in certain areas, most notably the Lower Rhine Graben (Vanneste et al., 2013), not all the assessments have aimed to represent these explicitly in the seismic source models or they have only chosen to do so in some branches. An active fault in the current context can be defined as a fault that is capable of generating earthquakes with M≥5.5 and with a non-zero minimum bound on the slip rate (e.g. Danciu et al., 2021). As such, the distribution of active faults is limited only to the fastest-slipping and most well constrained structures in and around the Rhine Graben, and each set of seismogenic source models principally comprises area source zones and/or gridded seismicity zoneless sources. These types of sources are known as distributed seismicity sources, and earthquake recurrence is modelled mostly by a double-truncated Gutenberg–Richter model whose parameters, a, b, COV(a,b), and maximum magnitude (MMAX), are constrained by fit to observed seismicity in each zone. The area zonations of the three models can be found in the Supplement Part A Sect. S1.

DE2016 adopts five alternative area source zonations (Models A, B, C, D, and E) alongside two zoneless smoothed seismicity models. For the area sources, Grünthal et al. (2018) explicitly formulate their logic tree as a combination of large-scale area source zones (LASZs) and small-scale area source zones (SASZs). Models A and B are LASZs, which assume that the regional-scale tectonics are the main factors delineating the seismic sources and that seismicity may be uniform across large areas when viewed at longer timescales than those captured by the observed seismicity. Models C to E are SASZs, which regard local-scale seismicity and geological features as the primary guide to the seismogenic sources and therefore delineate smaller-scale zones. The smoothed seismicity branches differ in approach from those found in both FR2020 and ESHM20, which use a smoothing kernel with an adaptive bandwidth but for which the bandwidth is calibrated on the local density of seismicity (e.g. Helmstetter and Werner, 2012). Instead, DE2016 uses an adaptive kernel with a magnitude-dependent bandwidth based on the method of Woo (1996). The two branches are equally weighted and consider the two cases in which the bandwidth is capped at 25 km (Hm≤25 km) and one in which it is unconstrained (Hm≤∞). One feature of note among the SASZ models is that Model C adds explicit active fault sources in the Lower Rhine Graben (LRG). These adopt the fault geometry proposed by Vanneste et al. (2013) but use observed seismicity with M≥5.3 across two catchments (area sources) to constrain long-term seismic activity rates for the faults. The seismic activity rates for M≥5.3 within the two catchments are distributed among the faults within the catchments according to their respective fault length, while for M<5.3 the catchments are treated as area sources. This combined area and fault source model receives the highest weighting of the five source models.

FR2020 adopts three area source zonations, which assimilate those implemented in previous studies by different organisations: Geoter (now Fugro) (GTR), Électricité de France (EDF), and the Institut de Radioprotection et de Sûreté Nucléaire (IRSN). In addition, a single zoneless source model branch is included, which is developed using smoothed seismicity with an adaptive kernel bandwidth applied to the observed seismicity in France from 1960 to 2017. The smoothed seismicity produces seismic sources in the form of 10 km × 10 km cells, with the activity rate (a value) varying cell by cell but the b value and MMAX calculated based on the location of the cell with respect to a set of superzones, i.e. large-scale area zones delineating tectonically based domains (Grands Domaines).

The seismogenic source model of ESHM20 follows a different approach to that of either FR2020 or DE2016. In terms of the number of different source models considered, the source model branch set is simpler. It contains one branch of exclusively area source zones and another branch for a combined smoothed seismicity and active fault model. As described in Danciu et al. (2021), the area source model aims to unify existing area source zonations from different national PSHA models across Europe, modifying the source geometries at the boundaries of models to ensure a seamless transition from one region to another. In the France–Germany region, the unified area source model adopts DE2016's Model C as its basis in Germany and the IRSN source model branch of FR2020 for France, alongside existing models from Belgium, Switzerland (Wiemer et al., 2016), and the UK (Mosca et al., 2022). The active fault and smoothed seismicity model includes explicit characterisation of faults in both the Upper and the Lower Rhine Graben regions, as well as numerous faults in France adapted from the dataset of Jomard et al. (2017). Information regarding the dataset of active faults can be found in Basili et al. (2023). Smoothed seismicity is characterised using an isotropic power law kernel with adaptive bandwidth, whose parameters are optimised using log-likelihood scoring (Nandan et al., 2022). To combine the smoothed seismicity with the active faults, a buffer zone is defined for each fault, within which magnitudes lower than a fault-size-dependent threshold are kept as smoothed seismicity, while magnitudes larger than the threshold are associated with the fault surface. For regions away from the fault, the b value and MMAX are based on values determined from area sources in a large-scale zonation, reflecting regional-scale tectonics (named TECTO). More information on the relevance of this is seen in Sect. 2.2.

In this first component we can see that the three PSHA models display both similarities and differences in their approach to characterising epistemic uncertainty in the seismogenic source model. FR2020 and DE2016 aim to represent uncertainty in the sources predominantly through multiple uniform area zonations, while ESHM20 divides its weights more evenly between two different source typology definitions. Though only DE2016 explicitly adopts the LASZ/SASZ characterisation, this same philosophy is present in FR2020's Grands Domaines model and ESHM20's TECTO model. In the FR2020 model the distinction between large- and small-scale zone models within the three zonations considered (GTR, EDF, and IRSN) is not as clear and intentional as it had been for DE2016. Where the contrast exists, it manifests mostly in the difference between the IRSN and EDF models (46 and 49 zones respectively) and the zonation provided by GTR (92 zones). Each of these three models could be described as delineating zones accounting for both geology and seismology, albeit in proportions that are difficult to define. Only DE2016 models the LASZ explicitly in its A and B source zonations; however, all three models will come to adopt similar approaches toward earthquake recurrence by using their LASZ as a basis for fitting their earthquake recurrence models, which may then inform (either by direct calibration or as a prior distribution) the MFRs for the small-scale area sources with few events. In that sense, their philosophies toward area zonation are similar, but their implementation differs.

Adaptive kernel smoothed seismicity source models are present in all respective logic trees, though each PSHA model has taken a different approach to characterisation and implementation. Both FR2020 and ESHM20 have used approaches similar to that of, for example, Helmstetter and Werner (2012), optimising the parameters controlling the adaptive kernel's bandwidth using log-likelihood analysis applied to a pseudo-prospective seismicity forecast. However, the models arrive at significantly different outcomes in terms of the spatial distribution of activity rate. DE2016 adopts a different approach by using magnitude-dependent adaptive kernels, which increase the bandwidth for larger magnitudes, meaning that the rate in many low-seismicity regions is dominated by activity from the most extreme events. This contrasts with the adaptive bandwidth methods used in FR2020 and ESHM20 for which the bandwidth is based on the density of seismicity. For FR2020 and DE2016 the total weight assigned to the smoothed seismicity branches is the same (0.25), while for ESHM20 the combined seismic seismicity and active fault branch receives half the total weight.

For the majority of the seismic sources found within the three source model logic trees (DE2016, FR2020, ESHM20), a truncated form of the Gutenberg–Richter model (Gutenberg and Richter, 1944) is adopted as the magnitude frequency relation. The only exceptions to this are the DE2016 smoothed seismicity models (which may be considered non-parametric recurrence models) and the branches of ESHM20 for which a tapered Pareto model is used. In all three regional seismic hazard models, epistemic uncertainty in the recurrence model is included, both in terms of its a and b values and in terms of its MMAX.

The first issue to address in comparing the derivation and representation of the magnitude frequency distribution is that of declustering, as all three models choose to remove foreshocks and aftershocks from their respective catalogue prior to fitting the MFR. This means that the distributions of activity rates shown subsequently refer to the rates of the mainshocks and not of the total seismicity. Both FR2020 and DE2016 claim to apply the declustering process described in Burkhard and Grünthal (2009), which is based upon earlier studies by Grünthal (1985). It is unclear whether the same code for implementation was adopted by both studies, so it is difficult to assess the extent to which the same seismic clusters are identified. ESHM20 explored the impact that the choice of declustering algorithm has on the resulting activity rate models, noting a contrast in the proportions of the catalogue removed by different algorithms when applied to more seismically active or stable regions (Danciu et al., 2021). Despite the different outputs of declustering, however, ESHM20 also opts to adopt the same algorithm as FR2020 and DE2016 to remove non-Poissonian events from the catalogue prior to calculation of activity rate in the final model. At present, the use of declustering remains common practice across many seismic hazard models, in both Europe and worldwide. Whether this will remain the case for future models remains an open question, particularly when emerging practice has shifted toward calculating activity rates using the complete catalogue in recent state-of-the-art PSHA models in the United States (Field et al., 2024) and New Zealand (Rollins et al., 2024).

The general form of the truncated Gutenberg–Richter model to determine the rate ν(M) of earthquakes with a magnitude greater than or equal to M is 1νM=ν0∫MMMAXβe-βme-βMMIN-e-βMMAXdm, where β=bln⁡(10), and ν0 is the rate of earthquakes greater than or equal to minimum magnitude MMIN, which can be retrieved from the a value by ν0=eαβ(e-βMMIN-e-βMMAX) where α=aln⁡(10). As both France and Germany are regions that would be characterised as having low to moderate seismicity, the number of events per individual source zone is often too small to determine a and b. All three models address this issue in a similar way by invoking the concept of large-scale superzones that span a sufficiently large region from which to define estimates of the recurrence parameters using a maximum likelihood estimator accounting for the temporal variation in catalogue completeness (Weichert, 1980). The a and b values from these superzones then act as prior distributions for estimates of each source zone in the respective seismogenic source models within a penalised maximum likelihood estimation (MLE) approach (FR2020) or alternatively by maximising a likelihood function assuming a common b value across multiple zones but with the seismicity rate varying for each zone (described in Appendix B of Stromeyer and Grünthal, 2015). For specific details of how the two approaches perform the MLE and how they account for uncertainties in the catalogue and its completeness, the reader is referred to the original publications. The relevant point here is that either approach will define for each source zone an expected a^ and b^ value (or similarly α^ and β^) and corresponding covariance matrix COV(α,β) from which we retrieve the uncertainties σα and σβ and their correlation ρσα,σβ. Where individual source zones contain very few events or span an insufficiently wide magnitude range, the distributions of the recurrence parameters may be informed by, or be fit according to, the superzone to which the source zone is assigned.

The superzone concept is critical for each of the models, not only in defining estimates of a and b values, but also for characterising MMAX. Here, both FR2020 and DE2016 adopt the EPRI methodology to characterise the distribution MMAX (Johnston et al., 1994; EPRI, 2012). This invokes a Bayesian approach in which a global prior Gaussian distribution of MMAX is defined based on the observed maximum magnitudes in analogous tectonically stable regions across the Earth, which is then updated for each superzone such that f(MMAX)=0 for MMAX<MMAXobs in any given region, and the posterior distribution combines the shape of the prior and corresponding likelihood function L(M|βNEQ). L is dependent on both the b value of the zone and the number of earthquakes observed during the corresponding period. The resulting posterior distribution is either sampled (in the case of FR2020) or approximated by a discrete set of weighted values using the approach of Miller and Rice (1983) (in the case of DE2016). ESHM20 updates an earlier work of Meletti et al. (2013) to define the MMAX distribution, which yields the three branches MMAXobs, MMAXobs+0.3, and MMAXobs+0.6 with assigned weights of 0.5, 0.4, and 0.1 respectively. Though not explicitly applying the EPRI methodology, the weights assigned to each of the three branches reflect an interpretation of a posterior distribution for f(MMAX) that is broadly consistent with that of the EPRI approach.

As the superzones act as larger-scale constraints on the parameters of the MFR (a, b, and MMAX) for regions of tectonic similarity, it is inevitable that their definition is based almost exclusively on tectonic and geological criteria rather than local-scale seismicity. This is applied consistently across all three models: the Grands Domaines for FR2020, the LASZ Model A for DE2016, and the TECTO model for ESHM20. The three superzonations are compared in Fig. 5. In the regions where these models overlap, there is a considerable degree of divergence in the tectonic zonations, with different models providing strongly contrasting interpretations of the extent of the larger-scale tectonic structures that influence the spatial distribution of seismicity. ESHM20 and DE2016 are perhaps more consistent with one another in defining three zones of similar extent that delineate the Paris Basin, the Upper Rhine Graben, and the South German Block. However, in the Lower Rhine Graben and continuing through Belgium and the Netherlands and into the North Sea, all three models diverge. Though far from the only factor that will eventually contribute toward the differences between the three models in terms of seismic hazard, this divergence in the tectonic interpretations in the superzone models will inevitably propagate into the recurrence models, particularly in regions of low seismicity where the superzones act to fix parameters of or provide strong priors for the resulting MFRs.

Though we have so far focused our attention on the definition of the superzones and their influence in constraining the MFRs themselves, equally important in terms of the impact on PSHA is how the resulting distributions of a^, b^, and COVa,b (or COV(α,β)) are evaluated within the logic tree. Here, there is yet again significant divergence between the models, with each model constructing the logic tree for MFR epistemic uncertainty using an entirely different approach.

DE2016 follows the methodology set out by Stromeyer and Grünthal (2015), who describe the uncertainty in cumulative activity rate ν at each magnitude m from the covariance matrix such that 2σ2m=1-mTCOVa,b1-m=σa2-2mσaσb+m2σb2. The cumulative rate of events greater than or equal to magnitude νc(m) then becomes 3νc,im=∫MMMAX10a‾-b‾m+σmzidm, where zi is the number of standard deviations of a standard normal distribution. The incremental activity rate in any given bin of width dm then simply becomes 10a‾-b‾m+σmzi. The uncertainty in each magnitude is now represented by a marginal distribution of N0,σm. This epistemic uncertainty can thus be mapped onto a discrete set of i=1,2,…,k branches such that zi and its corresponding weight, wi, are discrete approximations to the standard normal distribution according to the Gaussian quadrature approach of Miller and Rice (1983). As Eq. (1) is dependent on MMAX, the posterior distribution f(MMAX) returned by the EPRI approach for each zone is first approximated into five discrete branches using the same Gaussian quadrature method. Each of the five MMAX values is then input into Eq. (3), which is then discretised into four branches to approximate N0,σm. The epistemic uncertainty in the MFR for each area source is therefore represented by 20 logic tree branches (shown in Fig. 2).

ESHM20 starts from a similar point to DE2016, as it defines a^, b^, and COVa,b according to Stromeyer and Grünthal (2015) but then approximates the distribution differently. Monte Carlo sampling is used to generate 1 million realisations of a and b from the multivariate normal distribution, and from these samples the pairs corresponding to the 16th, 50th, and 84th percentile values are taken to define the lower, middle, and upper branches respectively, with weights of 0.2, 0.6, and 0.2 respectively. MMAX is defined independently of a and b using the three branches described previously. Though ESHM20 evaluates the multivariate distribution of a, b, and COVa,b in a slightly less formally correct manner compared to DE2016, one would still expect the distribution of resulting hazard curves to be similar. However, ESHM20 diverges further from both the DE2016 and the FR2020 approaches by introducing as an alternative set of MFR branches a tapered Gutenberg–Richter recurrence model (Kagan, 2002): 4νM0=MtM0βexp⁡Mt-M0McmforMt≤M0<∞, where M0 is the seismic moment of an event with magnitude m, Mt is the threshold moment, β=bln⁡(10), and Mcm is the corner moment. Unlike the truncated Gutenberg–Richter model, the tapered Gutenberg–Richter distribution is theoretically unbounded at large moments; however, the exponential decay in the functional form above Mcm effectively tapers the rate of events to triviality for magnitudes larger than the corresponding Mcm, so truncation can be safely applied within 0.2–0.3 magnitude units above Mcm with only minimal impact on the hazard calculation. For the set of branches corresponding to this distribution, the rate and b value are fixed according to the a^ and b^ values defined previously, while the three MMAX branches are applied as epistemic uncertainty in Mcm. In total, for area sources the source model logic tree contains 12 branches to represent the uncertainty in the MFR: for the truncated Gutenberg–Richter model three branches of a and b and another three of MMAX and for the tapered Gutenberg–Richter model only three branches for Mcm.

For both DE2016 and ESHM20 it is also necessary to define activity rates for both the smoothed seismicity sources and the active fault sources. Because of its implicitly non-parametric approach to defining activity rates, no MFR uncertainty is considered for the zoneless smoothed seismicity model of DE2016. Similarly, for ESHM20 the smoothed seismicity model is optimised through an iterative forecast testing approach, which yields a single preferred smoothed seismicity model without epistemic uncertainty in the MFR. Both models do define epistemic uncertainty in the activity rates for the fault-based models. In the case of DE2016 the maximum magnitudes on the composite fault sources are characterised according to their fault dimension using a normal distribution of NMMAX,0.3 (Vanneste et al., 2013). These distributions are mapped onto five branches using the Miller and Rice (1983) methodology. On-fault recurrence is modelled using a truncated Gutenberg–Richter relation, but as the authors could not constrain the proportion of aseismic slip, they opted to assign the seismicity for MW≥5.3 to the fault sources and the rest to their respective catchment zone (Model C, zones C15 and C22), with the proportion of seismicity rate for each fault assigned according to the relative length of the fault. This results in a total of 20 MFR branches on the fault sources, comprising five MMAX branches and the four branches of recurrence uncertainty from the catchment zones. In ESHM20 the recurrence models for the active fault sources also use a truncated Gutenberg–Richter model, albeit moment balanced from the geological coseismic slip rate. As the slip rates are themselves uncertain, three branches for alternative coseismic slip rates are considered along with three branches for MMAX.

FR2020 takes a different approach to characterising epistemic uncertainty compared to either ESHM20 or DE2016. For each area source and for each larger-scale superzone the seismicity is represented by a truncated Gutenberg–Richter model represented by a^, b^, and C(a,b), in addition to the posterior density function f(MMAX) that is defined for each superzone. Rather than discretise the distributions of ν(m) (as DE2016) or of a, b, and MMAX into a small set of branches according to Miller and Rice (1983), Drouet et al. (2020) instead use Monte Carlo sampling, drawing 100 samples from each distribution, with each sample then represented as an equally weighted MFR branch (weight =1/100) in the logic tree. Samples are drawn independently from f(MMAX) and the multivariate normal distribution representing the a and b value MVNa^b^,Ca,b. This results in a total of 400 source model branches from four source models (GTR, EDF, IRSN, and zoneless), each with 100 MFR samples. Implementation of the model revealed that the original authors had adopted a stratified sampling strategy for a and b, which is illustrated in more detail in the Supplement Part A Sect. S2.

To illustrate how the different approaches to characterisation and implementation of the MFRs in a logic tree can yield quite different distributions of activity rates for a given source, even where many inputs to the source model are similar, we consider the case of the Upper Rhine Graben (URG). Among the different source zonations within the different logic trees there are some differences to the exact shape of the source(s) in the Upper Rhine, though most models describe a source that follows the main outline of the graben starting just north of the Basel earthquake sequence in the south and terminating close to Karlsruhe in the northwest. We select the DEAS107 zone from the ESHM20 unified area source model branch, the Fosse-Rhenan Sud (FRS) zone from the FR2020 GTR source zonation, and the D051 zone from the DE2016 model to look at in detail as they depict similar geometries with respect to the spatial seismicity distribution. These sources are shown with seismicity from their respective earthquake catalogues in the top row of Fig. 6. Here we observe a first point of divergence, as the catalogues show remarkably different patterns of seismicity for the same zone. This is somewhat surprising as ESHM20 adopts the same earthquake catalogue as FR2020 within the French territory, namely FCAT-17 (Manchuel et al., 2018), and the same catalogue within the German territory as DE2016 for the post-1900 seismicity. Differences emerge in the pre-1900 earthquake catalogues as ESHM20 adopts the European Preinstrumental Earthquake Catalogue (EPICA) (Rovida et al., 2022), which is compiled independently from the other catalogues.

The next point of divergence can be seen in the estimate of completeness magnitude and its variation in time, which can be seen in the middle row of Fig. 6 and is given in Table 2. FR2020 and DE2016 estimate completeness using the method of Hakimhashemi and Grünthal (2012), although adopting different spatial zones to apply the method, while ESHM20 estimates completeness using an inversion method based on forecast testing (Nandan et al., 2022). Drouet et al. (2020) provide the uncertainty range for the completeness estimates, and although the preferred values are different for many magnitude bins, the earliest years of completeness for magnitudes in the range 4.0≤MW≤6.5 for the DE2016 and ESHM20 models are consistent with the uncertainty range shown in Table 2 for FR2020. Taking the best estimates, however, and contrasting these against the catalogues (shown in the middle row of Fig. 6), it is obvious that both the catalogues and the completeness estimates are dissimilar.

The bottom row of Fig. 6 shows the distributions of activity rate with magnitude for all the MFR branches assumed by the respective logic trees. Although each model uses some form of maximum likelihood estimate (Johnston et al., 1994; Stromeyer and Grünthal, 2015) to determine the Gutenberg–Richter parameters for the zone, the results are significantly different. ESHM20 has an expected a and b value of 1.9565 and 0.7334 respectively, which are mapped onto three branches of a-b pairs: (1.886, 0.685), (1.9565, 0.7443), and (2.0278, 0.803). By contrast, FR2020 yields a and b values of 2.3711±0.182 and 0.8696±0.0918 respectively, with ρab=0.8991, and while DE2016 is dependent on MMAX, the a and b values range from 3.89 to 2.86 and from 1.08 to 0.95 respectively. Not only do the MFR parameters themselves then vary significantly, but the different mappings onto logic tree branches also yield significantly different activity rate distributions, as illustrated in Fig. 6. ESHM20 places more weight on the middle branches, and in this case the MFR logic tree mixes both the truncated Gutenberg–Richter distribution and the tapered Pareto distribution. FR2020 clearly shows the largest spread of MFRs, which arises in part from the independence of a and b from MMAX and in part because of the large number of evenly weighted sample values. DE2016 is something of a middle point, with a narrower range of values and notably higher weights on a specific subset of branches.

The comparison here is not an exhaustive description of all the reasons for the differences in seismic hazard that we will eventually see between the three models, but it is illustrative of how they can diverge significantly in the critical information for PSHA (namely activity rate per magnitude bin) despite adopting theoretically similar approaches. Particularly insightful is the contrast in the way in which the distribution of a and b is mapped onto the epistemic uncertainty, which would potentially suggest that even if the three models produced a similar fit in their recurrence models, they could still diverge significantly in the resulting activity rate distributions inside the PSHA calculation. We discuss in the “Discussion and conclusion” section the implications for future harmonisation of the seismic hazard, but a key point to take from this brief analysis is that each step of the process, from the basic earthquake data through to the distribution of activity rates, requires both transparency and scrutiny. Though the models considered here are arguably better documented than many, there are still many steps in the processes that are not completely described, or if they are described it may be difficult to perceive how this can influence the hazard. These factors contribute to the differences in hazard model components and hazard model outputs shown in Sects. 4 and 5.

For the ground motion model (GMM) logic tree it is not necessarily the technical process itself and the decisions made therein that differ significantly between the three PSHA models but rather the general philosophy of how to characterise epistemic uncertainty. Specifically, between the three models we see an example of a multi-model (or “weight-on-model”) GMM logic tree (FR2020), a hybrid multi-model logic tree with backbone scaling factors (DE2016), and a fully regionalised scaled-backbone logic tree (ESHM20). All three models explicitly invoke the same objective of “capturing epistemic uncertainty in terms of the centre, body and range of the technically defensible interpretations of available data” (U. S. Nuclear Regulatory Commission, 2012). To contrast distributions of GMMs from different PSHA models, we created a set of trellis plots in which the GMM selections from two different models are plotted side by side for the same set of predictor variables. The range of GMM median or standard deviation values for the contrasting model is described by a shaded region beneath the GMMs for the model in question.

The GMM logic tree for DE2016 was initially based on a multi-model approach, with five models identified as suitable for application in Germany (Akkar et al., 2014a; Bindi et al., 2014; Derras et al., 2014; Cauzzi et al., 2015; Bindi et al., 2017), but it adds to each of these models a set of scaling factors to the median ground motions (0.7, 1.0, 1.25, and 1.5) to account for epistemic uncertainty in regional stress drop. Of the five models selected, Akkar et al. (2014a), Bindi et al. (2014), and Derras et al. (2014) are fit to data from the pan-European RESORCE dataset, while Cauzzi et al. (2015) and Bindi et al. (2017) are fit using data predominantly from Japan (supplemented by some records from other regions of the globe). The latter is fit using records from the NGA-West2 dataset but using a simpler functional form than the NGA-West2 GMMs. This makes Bindi et al. (2017) better suited for the level of parameterisation commonly found in moderate- to low-seismicity regions where seismogenic sources are predominantly based on distributed seismicity rather than directly on active faults. The DE2016 GMM logic tree combines a standard multi-model approach with elements of a scaled-backbone approach to capture some of the uncertainty in the underlying seismological properties of the target region; hence, we refer to it as a hybrid multi-model and backbone GMM logic tree.

Grünthal et al. (2018) outline several key factors that influence their decision-making process: (i) varying strengths of the different databases of ground motion, e.g. tectonic similarity for Europe (RESORCE), a wealth of short-distance records (NGA-West2), and detailed site parameterisation (Japan – Cauzzi et al., 2014); (ii) variation in functional form and how this influences ground motion prediction for small- to moderate-magnitude events; and (iii) the observation of several earthquakes with higher-than-average stress drop in stable regions of France, Germany, and the UK. The multi-model approach and the choice of models selected largely address the first two of these issues. Three different datasets (RESORCE, NGA-West2, and Japan) are represented, which also implicitly incorporate GMM source-region-to-source-region variability (i.e. Europe, western United States, and Japan). The highest weight assigned to the three GMMs derived from RESORCE and then split evenly between the three models therein is 0.5, while the Cauzzi et al. (2015) and Bindi et al. (2017) models receive equal weights of 0.25. Functional form variation and parameterisation are accounted for by mixing classical random effect models (each with slight differences in functional form) with purely data-driven neural network models (Derras et al., 2014).

The GMM logic tree adopted for FR2020 is the simplest of the three, using four ground motion models each assigned an equal weight of 0.25 (Ameri, 2014

The original paper of Drouet et al. (2020) indicated that the Ameri et al. (2017) model was adopted here; however, discussions with the authors revealed it was in fact the earlier Ameri (2014) model that was used.

In practice, the DE2016 and FR2020 approaches yield similar outcomes, with the same three source regions represented: local/Europe, Japan, and the western United States, with the local region receiving a weight of 0.5 and the other two a weight of 0.25 each. The two sets of GMMs for the DE2016 and FR2020 models are compared in terms of their range of median ground motions (Fig. 7) and their aleatory uncertainty σT (Fig. 8). For these comparisons and those in the subsequent figures (Figs. 9 to 12), the three scenario magnitudes (MW4.0,5.5, and 6.5) are chosen because they represent the minimum magnitude considered by the models (MW4.0), the typical controlling magnitude of the hazard for return periods of engineering interest (MW5.5), and a feasible but low-probability extreme scenario (MW6.5) that is close to the MMAX in stable zones but lower than the MMAX of the active fault sources found in the Rhine Graben and Alpine front.

The uncertainty in stress drop is the motivation behind adding the additional scaling factors, which capture both the possibility that stress drop may be lower in Germany than in the respective source regions of the models (0.75) and the possibility that it is higher (1.25 and 1.5). Weights of 0.36 are assigned to each of the 1.0 and 1.25 scaling factors, while the outer branches (for lower-than-average or much-higher-than-average stress drop) are assigned smaller weights of 0.14 each. This pushes the balance of the weight toward higher stress drop in Germany.

Compared to the strategies adopted for FR2020 and DE2016, the ESHM20 model has taken a different approach to defining a GMM logic tree that captures the centre, body, and range of the technically defensible interpretations of available data, and it does so by abandoning the multi-model concept entirely in favour of a regionalised scaled-backbone logic tree. The full explanation of the logic tree, including both its motivation and its calibration, is given in Weatherill et al. (2020). This change in approach is motivated in large part by the development of the Engineering Strong Motion (ESM) database and flatfile (Lanzano et al., 2019), which increases the number of ground motion records available in Europe by nearly an order of magnitude, particularly those of small- to moderate-magnitude earthquakes, including many more from France and Switzerland compared to RESORCE. The backbone GMM is fit to this dataset (Kotha et al., 2020), but with such a large volume of data additional random effects are included to capture region-to-region variability in the stress parameter scaling of the model (δL2Ll) and in the attenuation (δc3, where c3 is the coefficient of the anelastic attenuation term of the model). These two random effects are both normally distributed variables with means of 0 and standard deviations of τL2L and τc3 respectively, and individually they quantify the total regional variability in the stress parameter and residual attenuation within Europe. For regions with little or no ground motion data, the distributions of N(0,τL2L) and N(0,τc3) are mapped onto sets of discrete branches using the method of Miller and Rice (1983), making the model a scaled-backbone model. Where data are available the distributions can be adjusted to reflect the local stress parameter or attenuation properties implied by the data; thus the model is also regionalisable.

Even in the larger ESM flatfile there are few events from Germany, and those that are present are almost all located in the Upper Rhine Graben and Alpine Foreland. In France the majority of earthquake and records come from the Alpine and Pyrenees regions. Observations were available for the regions where δc3 could be calibrated, so regions of similar δc3 were grouped together to differentiate between regions of slower, average, or faster attenuation. These differences are reflected in the model, where the attenuation parameters of the backbone GMM for sites in these regions are adjusted to incorporate these differences. Altogether, the regionalised scaled-backbone logic tree maps the unadjusted (un-regionalised) δL2Ll term onto five branches and the regionalised δc3 term onto three branches, resulting in 15 GMM branches altogether. The median accelerations predicted by ESHM20 GMMs are compared against those of FR2020 and DE2016 in Figs. 9 and 10 respectively and the aleatory uncertainties in Figs. 11 and 12.

Among the most important trends to be seen in the plots in Figs. 7 to 12 are the general tendencies toward higher median ground motions at short distances and small magnitudes for the GMM logic trees of the DE2016 and ESHM20 models compared to those of FR2020. For larger magnitudes the trends reverse, and it is the ESHM20 GMM logic tree that provides a lower central tendency in the ground motions. At intermediate magnitudes and distances, where we are best constrained by data, ESHM20's GMM logic tree tends toward lower short-period motions at most magnitudes and distances, while longer-period motions are comparable. However, we note that the very high and very low stress parameter branches of the ESHM20 GMM logic tree that envelope the range of values in the plots have very little weight associated with them, and it is the three more central branches that have the greatest influence on the mean hazard.

For the DE2016 and FR2020 comparisons, the DE2016 GMMs tend to skew higher. This reflects the influence of the stress drop scaling, where more weight is put on toward the scaling factors greater than or equal to 1.0. Without these adjustments the GMM selections would likely have returned a similar centre and range of ground motions, except at near-source distances (RJB<10 km) where the Derras et al. (2014) GMM with the point-source to finite rupture distance correction seems to extrapolate toward much higher motion than the other models.

For the aleatory variability the ESHM20 is based on a scaled-backbone model with no branches for epistemic uncertainty in this term; thus the range of σT collapses to a simple line in Figs. 11 and 12. What is evident, however, is the heteroscedastic variability that is present in the ESHM20 model and in the Abrahamson et al. (2014) model. This results in lower σT at high magnitudes, which in turn lowers the aleatory uncertainty in the ESHM20 model compared to the other GMM logic trees. We also observed that the ESHM20 model shows a lower aleatory variability in general compared to the spread found in other GMM logic trees. Two factors play a role here: the first is that the Kotha et al. (2020) model was derived using robust linear mixed-effect regression that down-weights outlier values and the second is that the ESHM20 GMM implementation adopts different site-to-site variability (ϕS2S) for the cases when the site condition (VS30) is measured and when it is inferred from a proxy (Danciu et al., 2021; Crowley et al., 2021). For the measured VS30 case, which is the one being considered in the ESHM20 application, ϕS2S is reduced compared to most other GMMs shown here because it is fit to the site-to-site variability of the subset of stations with measured VS30, while most other models have calibrated this variability based on records from stations that mix measured and inferred VS30.

Although PSHA models have developed in sophistication over the decades, the fundamental framework for PSHA has remained largely unchanged since its establishment by Cornell (1968) and McGuire (1976). Arguably the most notable evolutions in practice emerge with the “grand inversion” methodology for modelling fault systems (e.g. Field et al., 2015, 2024) and more widespread usage of Monte Carlo techniques (e.g. Ebel and Kafka, 1999; Musson, 2000; Weatherill and Burton, 2010; Assatourians and Atkinson, 2014). These later adaptations do not alter this core probabilistic framework but rather evaluate it in a manner that may be flexible or better suited to incorporating new modelling developments or providing input into a broader range of applications. Yet, despite the robustness of the conceptual probabilistic seismic hazard integral, different PSHA software can be remarkably divergent in the way the input source and ground motion models are processed and translated into the PSHA framework.

Differences between PSHA software can be broadly grouped into three categories:

Irreconcilable discrepancies owing to fundamental differences in software operation. These can include characterisation of the seismic source and/or magnitude frequency relation and their discretisations within the hazard integral; treatment of rupture finiteness in distributed seismicity sources; and its scaling with earthquake magnitude, calculation of fault rupture to site distances, and evaluation and/or approximation of the statistical density functions to retrieve probabilities of exceedance of ground motion. Such differences can be identified but not necessarily replicated from one software to another without significant changes to the code.

The way that different software packages characterise common elements of a PSHA calculation and the corresponding impacts on the resulting hazard curves have been evaluated as part of the PEER Probabilistic Seismic Hazard Code Verifications (PEER Tests hereafter) (Thomas et al., 2010; Hale et al., 2018). These are elemental PSHA calculations usually comprising a single source, a ground motion model, and a limited number of target sites with fixed properties, which are designed specifically to assess how the different software programmes approach a particular modelling issue. The results are compared against either “exact” solutions calculated by hand, where possible, or the range of curves determined from the participating PSHA codes when the problem cannot be evaluated by hand.

The PEER Tests have been particularly insightful in identifying how and why PSHA codes diverge, which is especially important given that many codes participated in them (both proprietary and open source) that are widely used in commercial application. As they are elemental in nature, however, they cannot necessarily predict the extent to which different codes will yield different outputs for seismic hazard at a given location, where many modelling differences come into play. The importance of this type of application and the benefits of multi-software implementations of a seismic hazard model as part of a quality assurance (QA) process for the design of critical facilities have been emphasised by Bommer et al. (2015) and Tromans et al. (2019), among others, and both are becoming more widely used in practice. The QA application is only one context, however, and arguably a favourable one in which multiple parties are involved, and resources are often made available to document and debate the implementations and to resolve discrepancies as and when they emerge.

More relevant for the case at hand is migration of an existing or established hazard model from one software to another. Here the challenges are different, as the existing model forms the reference, and the new software may need to replicate the behaviour of the previous one in order to ensure consistency in the outputs. In some cases, if the new software user is different from or does not have the support of the original software developer, and the source code of the software is closed, then there can often be critical elements of the PSHA calculation process that are hidden from the user. In this instance complete agreement between the existing and migrated models may not be possible, primarily due to the irreconcilable differences between the software highlighted above. Instead, a target level of “acceptable agreement” between the previous and new implementation needs to be defined (e.g. Abbot et al., 2020; Allen et al., 2020).

In the migration processes described in this section, we set a target level of agreement in terms of the OpenQuake-calculated seismic hazard curves at given target sites agreeing with those produced from the original PSHA software code to within a ±10 % annual probability of exceedance (APoE) for the corresponding range of ground motion intensity measure levels (IMLs) for APoEs greater than 10⁻⁴ (corresponding to a return period of approximately 10 000 years). Although in many cases agreement can be achieved for lower APoEs, the irreconcilable differences due to issues of discretisation, rounding, numerical instability etc. may begin to influence the extreme tails of the distributions that assume greater importance at these longer return periods. An APoE of 10⁻⁴ is sufficient to span the range of return periods considered for conventional design building codes, which reflect the applications for which these specific hazard models are intended. As both the FR2020 model and the DE2016 model have logic trees, we undertake comparisons in two steps, the first comparing specific branches of the logic tree to ensure broad agreement over source and ground motion model combinations and the second comparing the curves in terms of the respective means and quantiles. We note that from the seismic hazard curves similar agreement targets could be set in terms of the IMLs for a fixed range of APoEs, which may be slightly more intuitive. Both options were explored, and no cases were found in which the agreement in curves for the IMLs failed to reach the set ±10 % target when the agreement in terms of APoEs did. As all three software programmes considered return seismic hazard curves in terms of PoE for a user-input set of IMLs, and statistics of means and quantiles were calculated based on PoE, we opted to use APoE as the variable for the comparisons to avoid introducing potential discrepancies from different interpolation approaches. Summaries of the migration issues for both FR2020 and DE2016 can be found in the Supplement Part A Sects. S3, S4, and S5, and further details of the issues encountered in the migration of FR2020 to OpenQuake can be found in Weatherill et al. (2022). Illustrative comparison plots of the two software implementations for both national seismic hazard maps and seismic hazard curves at selected locations can be seen in the Supplement Part B.

In OpenQuake the mean is calculated as the weighted arithmetic mean of the probabilities of exceedance (PoEs) for each given intensity measure level (IML). Similarly, quantiles are determined based on the probabilities of exceedance for each intensity measure level by sorting the PoEs from the lowest to the highest at each IML and interpolating the corresponding cumulative density function to the desired quantile values (typically 0.05, 0.16, 0.5 (median), 0.84, and 0.95). As OpenQuake adopts the earthquake rupture forecast (ERF) formulation for the PSHA calculation (Field et al., 2003), all hazard statistics are extracted from the probabilities of exceedance rather than the rates of exceedance. This formulation of the mean and quantiles represents one of several different ways of retrieving this term. Other PSHA software programmes may apply the statistics to the IMLs for a given PoE and/or work with the geometric means rather than arithmetic means, and each approach yields different results. From communication with the model developers, we verified that FR2020 defines the mean hazard as the arithmetic mean of the probabilities of exceedance, while for DE2016 the means are based on the arithmetic mean of the annual rates of exceedance. For consistency with OpenQuake, in the comparisons of means and quantiles, we retrieved these values from the complete suite of hazard curves and processed them identically rather than taking the mean or quantiles from the software itself.

We see in Sect. 2 how the three different models attempt to translate the uncertainty in a, b, and COV(a,b) into the logic tree and how this yields quite different distributions of activity rates. A factor that is not discussed is the issue of source-to-source correlation in the MFRs. To summarise, consider an idealised model with just four area sources, each with their own truncated Gutenberg–Richter MFR, and a corresponding logic tree with three branches for uncertainty in a and b (e.g. -Δ⋅a,b,a,b‾,+Δ⋅a,b) and three for uncertainty in MMAX (e.g. MMAXLOW,MMAX‾+MMAXHIGH). If the MFRs are fit independently for each zone, then the resulting logic tree would need to permute every combination of the MFR parameters for each source, which would in this simple case result in 94=6561 logic tree end branches, i.e. NBRANCHESNSRCS. Applying this same logic to the area source zonations for DE2016, for example, we have between 31 and 107 sources per model and 20 MFR branches, which would result in between 2031 and 20107 logic tree branches for each source model. This is clearly intractable for any PSHA calculation software, and OpenQuake cannot even construct such a logic tree from which to sample. A common alternative is to assume perfect correlation between the sources, which in the idealised case would be to apply the same branches (e.g. -a,b|MMAXlow,a,b‾|MMAXLOW,+a,b|MMAXLOW,-a,b| MMAX‾,…,+(a,b)|MMAXHIGH) to all of the sources at the same time. This results in a more manageable logic tree of just 9 branches in the simple idealised case and 20 MFR branches per source model in the DE2016 case.

Both DE2016 and ESHM20 adopt discrete MFRs for each of the sources, meaning that in order to execute the calculation, perfect correlation between sources had to be assumed in both cases. However, by sampling the MFRs for each source separately in the 100 branches, FR2020 preserves independence in the source model MFRs. This issue of correlation can impact on the outcomes of the hazard as the assumption of perfect source-to-source correlation in MFRs could conceivably assign disproportionately large weights to the extreme cases such that all sources may have higher or lower activity rates. This inflates the uncertainty, meaning that the resulting hazard distributions may be larger than intended, potentially skewing the mean toward higher values compared to the case in which MFR epistemic uncertainties are characterised independently for each source. Work is currently ongoing to explore this issue and its impacts on seismic risk assessment for a country in further detail.

A final issue of PSHA implementation relates to the scale of calculation, by which we refer to the volume of data and, by extension, the CPU time and RAM needed to execute the PSHA for logic trees of this size. Each of the three software programmes address this differently, and as two of the software programmes are proprietary, we have not been able to benchmark the calculations. For OpenQuake, however, this type of logic tree with many source and MFR branches has not been efficiently handled at the time of writing. The main reason for this is that for each source model and MFR branch a new earthquake rupture forecast is constructed. This requires re-calculation of distances and ground motions for each logic tree branch and MFR branch, increasing both the CPU requirements and the RAM requirements. Calculations here were run on a 192 CPU server with 760 GB RAM, and this was insufficient to execute the calculations in a single run. Instead, the models for FR2020 and DE2016 were split into subsets of branches, and the resulting hazard curves were later recombined and post-processed to retrieve the mean and quantiles. It is hoped that future efforts will be undertaken to improve the efficiency of the calculations for this type of epistemic uncertainty, which is commonly applied in regions of low to moderate seismicity.

Section 2 explains how all three models share some similarities in the source types that they use, as well as their differences. As each model adopts a logic tree with epistemic uncertainty in both the source types and the recurrences, how can one quantitatively compare not just the sources but also their respective distributions? The starting point is to render each source into a common representation that allows for quantitative comparisons of the models and their respective distributions. Each source branch of each model is translated into a three-dimensional array λ(ϕ,θ,M) of longitude, latitude, and magnitude, with each cell containing the incremental rate of activity for the corresponding longitude, latitude and magnitude bin λijm, where i=1,2,…,Nϕ corresponds to the longitude bin, j=1,2,…,Nθ to the latitude bin, and m=1,2,…,Nm to the magnitude bin. For area sources, the rate of the uniform area source is partitioned into each grid cell according to the proportion of total area overlapping with each cell. In the case of gridded seismicity, the rate assigned to each target grid cell corresponds to that of the original source cell's centroid falls (which can result in latitudinally dependent striations of “empty” cells depending on the different map projections used). Finally, for the fault sources the seismicity rate per cell is partitioned according to the proportion of the fault's surface projection that intersects the cell. All seismogenic sources here are shallow crustal sources, so although hypocentral depth is relevant to the seismic hazard, for the current purposes, rates are not distributed across different depth layers.

Each source model logic tree branch k of Nk total branches is rendered into the three-dimensional rate grid λk(ϕ,θ,M), and each grid is associated with its respective logic tree branch weight. This relatively simple translation of the respective source models into a common grid representation facilitates quantitative comparisons by virtue of simple descriptive statistics. For example, Fig. 13 shows the spatial variation in the mean cumulative rate of seismicity above M4.5 for each of the three models, which is weighted by the logic tree branch weight for each source branch: 5λϕ,θ|M≥4.5‾=∑k=1Nkwk⋅∑m=1NMλkϕ,θ,Mm⋅HMm≥4.5, where H[⋅] is the heaviside step function. Similarly, weighted percentiles can be extracted for each spatial bin, which we show in Fig. 13 as the 16th and 84th percentiles. The minimum magnitude mmin=M4.5 is used in these comparisons as this is the common minimum magnitude in the PSHA calculations for all three models. Other values of mmin could be compared depending on the relevant context; however, mmin=4.5 is sufficient to illustrate the application here. From these descriptive statistics we can extract a measure of the centre and body of the activity rate distributions, with the latter illustrated in terms of the weighted interquartile range in Fig. 14. Note that the striations in the maps for the FR2020 model emerge from the gridded seismicity branches being regularly Cartesian spaced every 10 km, while the reference grid is in a geodetic system (latitude and longitude).

It is not our intention to provide a complete interpretation of all the features visible in these maps, though for the comparisons of hazard in the France–Germany border region, noteworthy differences include the relative activity of the Albstadt Shear Zone (SE Germany) and the Upper and Lower Rhine Graben. The Albstadt Shear Zone is a particularly complex feature where the smoothed-seismicity-driven branches of DE2016 and ESHM20 produce a very localised zone of high activity, while several area zonations (particularly those based on regional tectonics) do not isolate this region from the larger regional seismicity. Therefore, higher quantiles tend to reflect the smoothed seismicity branches in which the ASZ is highly visible, and lower quantiles reflect the larger-scale zonations where the ASZ is not present.

Relative differences between the models can be quantified from this same characterisation via the use of difference maps, both for the mean activity rates (Fig. 15, top) and for the relative differences in the model range shown by the ratio of the interquartile ranges (Fig. 15, bottom). The difference maps present a somewhat incoherent picture, which is not unexpected given the complexities and variations in the constituent source models. We note that in the presentation of the relative comparisons in Fig. 15 (and in subsequent figures), we do not identify any specific model as a reference and instead show all combinations.

Comparisons of the mean and quantiles of the rate distributions such as those shown in Figs. 13 and 15 are certainly important as they highlight regions where the underlying source models have a general tendency toward increased or reduced activity. However, these metrics alone do not necessarily provide insight into the complete similarity or dissimilarity of the full distributions of activity rate divergence or how this divergence varies geographically. To visualise this sort of information we can instead adopt metrics from information theory to help quantify dissimilarity between distributions: weighted Kolmogorov–Smirnov statistic (DKS) (e.g. Monohan, 2001) and Wasserstein distance (DWS) (Vaserstein, 1969). If λk(ϕ,θ,mmin) is the rate grid for source branch k, with weight wk we can then define for each complete source model logic tree a probability distribution fMODEL(λk|ϕ,θ,mmin) at each location, where λk|ϕ,θ,mmin is the total activity rate in the spatial domain (ϕ,θ) greater than or equal to a specified minimum magnitude mmin. In the simplest case the spatial domain refers to each grid cell; however, this same process applies to any spatial subdomain of the region enclosed by the original rate grid and could be applied to larger regions or somehow coarsened with respect to the grid. If fMODELA(λk|ϕ,θ,mmin) and gMODELB(λk|ϕ,θ,mmin) are the respective empirical probability density functions for the two full seismic source models A and B implied by their logic trees, then 6DKS=supλkFMODELAλk|ϕ,θ,mmin-GMODELBλk|ϕ,θ,mmin and 7DWS=∫-∞∞FMODELAλk|ϕ,θ,mmin-GMODELBλk|ϕ,θ,mmindλk, where FMODELA and GMODELB are the respective empirical cumulative density functions of models A and B. The conceptual definitions of these terms are illustrated in Fig. 16, where we can see DKS as the maximum absolute distance between the empirical cumulative density functions (CDFs) and DWS as the total area enclosed between the CDFs. DKS is constrained to the domain [0, 1], with 0 indicating perfect agreement in the CDFs and 1 indicating no overlap in the respective ranges of λk, while DWS is constrained only by a lower bound of 0 (total agreement). By working on the cumulative density functions, both terms account for the distribution of weights in each of the logic trees.

With DKS and DWS we have metrics that allow us to assess spatial variation in the similarity between the effective rate distributions predicted by two different models, which is shown for the combinations of FR2020/DE2016, ESHM20/DE2016, and ESHM20/FR2020 in Fig. 17. The most immediate contrast between the maps using the two different metrics is the apparent “noisiness” of the DKS metric compared to that of DWS. The Kolmogorov–Smirnov statistic can appear to change significantly over short distances, often highlighting boundaries of source zones, while Wasserstein distances show a smoother transition, particularly in regions of higher seismicity. The sharp spatial contrasts and variability appear to be particularly exacerbated for comparisons involving ESHM20. This behaviour may be anticipated by the conceptual definitions of the metrics shown in Fig. 16, in which the largest absolute distance between empirical CDFs can change significantly even with relatively small changes in the shape of the CDF. In the empirical CDFs for FMODELA and GMODELB, notable changes in shape may appear from one source zone to another due to changes in the MFRs for each of the zones, while in the case of ESHM20 the comparatively few MFR branches result in empirical CDFs that are more step-like, which results from having gaps in the probability density function (PDF) that can arise due to coarse discretisation of the continuous distributions and/or transitions from one type of source or recurrence model to another. In this respect, DWS appears to be a better-suited metric for interpretation as it is less sensitive to small changes in the empirical CDFs than DKS. Focusing on this metric, in the France–Germany border regions we can see more coherent trends, such as greater divergence in the Lower Rhine Graben than along the Upper Rhine for all models, with the ESHM20 providing a notably divergent distribution here. Similarly, the Albstadt Shear Zone emerges as a point of divergence between FR2020 and the other two models.

The rendering of each model into regular rate grids allows us to make comparisons of the source models in a common framework and to interpret differences using simple descriptive statistics as well as through more non-parametric measures that are based on information theory. Here, we contrast the source models from the three different PSHA models (FR2020, DE2016, and ESHM20), though similar comparisons could be made for successive generations of models; however, one need not go back more than one or two generations of regional-scale models before concepts such as the logic tree are no longer found. From the comparisons of the source models shown here, a recommendation would be to compare models firstly via difference maps of mean rates and potentially a small number of selected quantiles and then to apply DWS to be able to interpret quantitatively how and where the distributions diverge.