 Research Letter
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Statistical characterization of fullmargin rupture recurrence for Cascadia subduction zone using event time resampling and Gaussian mixture model
Geoscience Letters volumeÂ 10, ArticleÂ number:Â 52 (2023)
Abstract
Earthquake occurrence modeling of large subduction events involves significant uncertainty, stemming from the scarcity of geological data and inaccuracy of dating techniques. The previous research on statistical modeling of fullmargin ruptures of the Cascadia subduction zone attempted to address these issues. However, the adopted resampling method to account for the uncertain marine turbidite age data from the Cascadia subduction zone was not sufficient in the sample size. This study presents a statistical approach based on the Gaussian mixture model applied to significantly larger resampled Cascadia age data. The results suggest that the 3component Gaussian mixture model outperforms the 2component Gaussian mixture model and the 1component renewal models by capturing the long gap and shortterm clustering. The developed Gaussian mixture model is well suited to apply to probabilistic seismic and tsunami hazard analysis and the calculation of longterm probability of the future fullmargin Cascadia events by considering the elapsed time since the last event.
Introduction
Earthquake occurrence modeling of large magnitude events from major subduction zones and faults is a critical element in probabilistic seismic and tsunami hazard analysis, but involves significant uncertainty (Davies et al. 2018; Baker et al. 2021; Behrens et al. 2021; Mori et al. 2022). The main challenges of this modeling stem from the scarcity and the uncertainty/ambiguity of the data (Rhoades et al. 1994; Headquarters for Earthquake Research Promotion 2019). For the former, despite various efforts for identifying large paleoseismic events from geological data in major subduction zones (Philibosian and Meltzner 2020), the number of large earthquakes is rather small compared to the number of smalltomoderate earthquakes. For the latter, event times are often estimated based on radiocarbon dating of sediment samples and geological cores, and estimated event times are uncertain with a possible margin of error of several decades to a few hundred years.
By recognizing nonPoissonian and quasiperiodic recurrence characteristics of major subduction zones and matured faults, various earthquake occurrence models have been proposed in the literature (e.g., Ogata 1999; Sykes and Menke 2006). Both physicsinspired occurrence models (e.g., Shimazaki and Nakata 1980; Console et al. 2008) and statistical renewal models (e.g., Cornell and Winterstein 1988; Matthews et al. 2002; Abaimov et al. 2008) have been adopted in seismic and tsunami hazard assessments. A renewal process can characterize the evolution of occurrence probability with time by specifying an interarrival time (IAT) distribution of earthquakes. Popular IAT distributions include the lognormal distribution, Brownian Passage Time distribution (Matthews et al. 2002), and Weibull distribution (Abaimov et al. 2008), noting that the homogeneous Poisson model corresponds to the exponential distribution. Typically, the IAT distribution is characterized by three parameters: mean recurrence time, coefficient of variation (CoV), and elapsed time since the previous event. Although the concept of renewaltype earthquake occurrence models is appealing, when the number of earthquake occurrence time data is small, the superiority of the timedependent renewal model to the timeindependent Poisson model is not always obvious (Williams et al. 2019; Griffin et al. 2020).
The Cascadia subduction zone (CSZ) is located at the convergent boundary where the oceanic Juan de Fuca Plate, Gorda Plate, and Explorer Plate and the continental North American Plate interact (Fig.Â 1). The CSZ extends 1100Â km from British Columbia to Northern California and has a convergence rate of 30 to 45Â mm/year (DeMets et al. 2010). In the past, the CSZ ruptured in fullmargin earthquakes of moment magnitude (M_{w}) greater than 8.5 with an average recurrence period of 530Â years (Goldfinger et al. 2012) and the last event in 1700 (Satake et al. 2003). The sedimentary records of the CSZ events can be observed in the form of marine turbidite deposits (Goldfinger et al. 2012) and buried soil/marsh data (Atwater et al. 2015). Using the Cascadia turbidite records produced by Goldfinger et al. (2012), Kulkarni et al. (2013) carried out statistical modeling and seismic gap analysis of fullmargin CSZ ruptures, whose magnitude ranges were between 8.49 and 9.13, by using a hierarchical clustering method. They pointed out that there may be a relationship between the earthquake size and the interarrival time, noting that the earthquake size can be correlated with the fault rupture length and the thickness of the turbidite layers or turbidite volumes. Recognizing the significant uncertainty associated with the radiocarbon dating of deepsea geological cores and synchronous turbidite event analysis, Monte Carlo resampling of the CSZ age data was conducted; however, the size of the resampled CSZ age data was small (20 simulated catalogs; each contains 18 events). They proposed that the Cascadia fullrupture records can be divided into several clusters, punctuated by longer gaps. Their investigation was motivated to present statistical evidence of clustering and seismic gap as plausible earthquake occurrence patterns in the CSZ.
This study conducts statistical modeling of fullrupture CSZ recurrence using event time resampling and Gaussian mixture method. The same information of the past CSZ age data suggested by Goldfinger et al. (2012) is used, and the same resampling approach as in Kulkarni et al. (2013) is adopted to dealing with the uncertainty of the CSZ age data. Each event can be represented by multiple possible CSZ age data, and the uncertainty of each age data, associated with radiocarbon dating, can be approximated by a triangular distribution based on the best, lower, and upper estimates. The statistical modeling carried out in this study differs from that of Kulkarni et al. (2013) in two major ways. Firstly, the size of the resampled CSZ age data is significantly larger (5000 vs. 20 simulated catalogs). By treating the simulated catalogs as possible realizations of the underlying original age data, more stable characteristics of the CSZ age data can be examined. Secondly, the Gaussian mixture model is fitted to the IAT distribution based on the simulated CSZ age data, which exhibit multicomponent features. A major advantage of the proposed method is that when the elapsed time since the last event is considered in probabilistic seismic and tsunami hazard analysis, the mixing proportions of the multiple normal distributions can be updated with this information. The important contributions of this research include the development of a viable statistical method for the fullmargin CSZ ruptures by capturing the multicomponent characteristics of the IAT data and the implementation of the developed method in the context of probabilistic seismic and tsunami hazard analysis.
Data
Event times of fullmargin ruptures in the Cascadia subduction zone
The CSZ extends from Vancouver Island to Cape Mendocino (Fig.Â 1), and multiple submarine channelcanyon systems are distributed along the continental margin of the CSZ. During major storms, earthquakes, and tsunamis, turbidity currents can be triggered by rivercarried sandy and silty sediments, sliding down the continental shelf. From branching tributaries, turbidity currents merge into the main channel and form a large turbidite. By contrast, smalltomoderate storms and farfield tsunamis are unlikely to induce synchronized turbidity currents along the entire continental margin of the CSZ.
To develop a catalog of megathrust subduction earthquakes in the CSZ, Goldfinger et al. (2012) carried out extensive coring surveys spanning the entire margin of the CSZ and analyzed collected samples using marine radiocarbon dating and stratigraphic correlation techniques. Due to the synchronous occurrence of turbidite currents along the 500km coast, the northern CSZ events are best explained by the paleoseismic events that can trigger a CSZwide event. For the southern CSZ, most of the turbidite samples are wellcorresponded and correlated with the spatial extent of onshore paleoseismic records, while there are uncorrelated turbidites that were likely to be depositional products after smaller earthquakes, local storms, or farfield tsunamis. The northern portion of the CSZ ruptured less frequently with mean recurrence periods of 500 to 530Â years and has a strong spatial correlation with the southern half, thus leading to a synchronized full or nearfull rupture which could result in M_{w}9class megathrust events. The southern portion of the CSZ, in addition to the wholeregion ruptures, experienced additional smaller earthquakes (M_{w}8class events) according to the turbidite records with mean recurrence periods of 240 to 320Â years. According to Goldfinger et al. (2012), 19 welldated turbidites that ruptured the northern CSZ were identified and denoted by T1 to T18, noting that there is T17a which was considered as a separate fullmargin rupture from T17. T1 corresponds to the most recent 1700 event, whereas T18 corresponds to the oldest event in 9795 calibrated years before the present. Among the 19 turbidite events, T2 was eventually excluded from the list of fullmargin events because this event was not consistently recorded on buried soil/marsh data on land, implying that T2 might have been caused by nonseismic sources.
Kulkarni et al. (2013) adopted the fullmargin rupture data by Goldfinger et al. (2012) to develop an earthquake clustering model for the CSZ (see Table 1 of Kulkarni et al. (2013)). The data consist of 83 turbidite ages for the 19 events (i.e., T1 to T18, including T2 and T17a); different events have different numbers of age data (for instance, T3 has 8 age data). Each of the turbidite data comes with three age estimates, i.e., best,â€‰+â€‰2 sigma bound, andâ€‰âˆ’â€‰2 sigma bound. Kulkarni et al. (2013) considered that the probability distribution of the individual age data can be represented by the triangular distribution with the best estimate as the mode and the twosigma bounds as the upper and lower limits. The same interpretations of the turbidite data are adopted in this study. The data used by Kulkarni et al. (2013) are shown in Fig.Â 2a. Kulkarni et al. (2013) observed several long gaps after largemagnitude events that produced large turbidite masses (see Fig.Â 2 of Kulkarni et al. (2013)) and hypothesized that a large earthquake magnitude is associated with a longer interarrival time. They were motivated to develop an earthquake occurrence model that distinguishes intercluster data and gap data using a hierarchical clustering method. After excluding T2 from the dataset, they obtained the 17 IAT data from the 18 age data. They indicated that a renewal model with the normal IAT distribution performs well with the central portion of the IAT data but is not suitable for characterizing the lower and upper tails. This can be seen in Fig.Â 2b. More details of the seismic gap analysis for the CSZ can be found in Kulkarni et al. (2013).
Monte Carlo resampling of the Cascadia age data
Due to nonnegligible uncertainty of the CSZ age data from Goldfinger et al. (2012), Kulkarni et al. (2013) conducted 20 Monte Carlo resampling of the CSZ age data and used for their hierarchical clustering analysis. The procedure of the resampling is summarized as:

1.
Set the number of Monte Carlo resampling.

2.
For each turbidite event, choose one of the data randomly with equal chance (i.e., all listed data in Table 1 of Kulkarni et al. (2013) are regarded as equally reliable).

3.
Sample the age of the turbidite data chosen in Step 2 from the triangle distribution, which is determined by the best andâ€‰+â€‰/â€‰âˆ’â€‰2 sigma bounds.

4.
Repeat Steps 2 and 3 for all turbidite data. The IAT data can be obtained for each catalog.

5.
Repeat Steps 2 to 4 for the resampling number specified in Step 1.
Because the radiocarbon dated ages involve large uncertainty and some of the adjacent turbidite events are only separated by a few hundred years, the abovementioned resampling procedure can result in the reversed order of the simulated events. In this study, when the simulated CSZ age catalog has negative IAT data (i.e., reversed order), the simulated trial is discarded; the rejection rate is approximately 20%. Note that Kulkarni et al. (2013) considered the simulated CSZ age catalogs that have IAT data longer than 100Â years only because the shortest IAT in the original CSZ age catalog was about 100Â years. This resampling method results in a high rejection rate of the simulated catalogs (approximately, 62% of the simulated catalogs are abandoned). Since the threshold of 100Â years is arbitrary and this high rejection rate could cause bias in statistical modeling, all IAT data are required to be positive (i.e., data do not have to be greater than 100Â years). It is noteworthy that the analyses are also carried out by allowing negative IAT data (i.e., no rejection); the results are not significantly different from the baseline results. Although the simulation procedure adopted in this study is the same as Kulkarni et al. (2013), considering a larger resampling size and adopting a less subjective rejection criterion of the simulated turbidite age data will lead to robust characteristics of the simulated CSZ event time data.
FigureÂ 3 shows a histogram of the Cascadia IAT data distribution from 5000 resampling simulations. Each simulated catalog consists of 18 events and 17 IAT data can be calculated (note: all IAT data are positive). Therefore, the histogram is based on 85,000 IAT data. The simulated IAT data are right skewed and exhibit heavy tails on both upper and lower sides, compared with the normal distribution (see Fig.Â 2b). The longer IAT data are associated with long gaps, whereas the shorter IAT data are related to shorttime clustering. The mean and standard deviation of the simulated IAT data are 561Â years and 272Â years, respectively, and thus the CoV is calculated as 0.485.
Methods
A Gaussian mixture model comprised multiple Gaussian components, each captures clustering in the data. For onedimensional data, such as the IAT data for the fullmargin Cascadia rupture case, the model for K components can be expressed by:
where Ï€_{k} is the mixing proportion of the kth component and its summation over all K components equals 1, and Âµ_{k} and Ïƒ_{k} are the mean and standard deviation of the kth component, respectively. The kth mixing proportions represents the probability of observing data that come from the kth Gaussian component. With the increase of K, the number of model parameters increases; for instance, for Kâ€‰=â€‰1, 2, and 3, the number of parameters is 2, 5, and 8, respectively.
The model parameters of the Gaussian mixture model can be estimated using the Expectationâ€“Maximization (EM) algorithm (McLachlan and Peel 2000). The algorithm attempts to maximize the log likelihood function of the Gaussian mixture model for given data in two steps iteratively. For the specified value of K, initial values for component means, covariance matrices, and mixing proportions are generated through a kmeansâ€‰+â€‰â€‰+â€‰technique. In the Expectation step, the algorithm computes posterior probabilities of component memberships for each data point. Subsequently, in the Maximization step, with the component membership posterior probabilities, component means, covariance matrices, and mixing proportions are estimated based on the maximum likelihood method. The EM steps are iterated until the convergence is achieved or the iteration number reaches the specified trial value. Since the success of the EM algorithm depends on the data complexity and initial values and the solution may converge to a local minimum, multiple runs of the EM algorithm are made. The results with the highest log likelihood value is adopted as the final estimate. In this study, the parameter estimation is performed using the fitgmdist function in MATLAB, and the number of EM initializations is set to 50.
The Gaussian mixture model is fitted to the simulated IAT data by considering Kâ€‰=â€‰1, 2, and 3. The inspection of the histogram shown in Fig.Â 3 indicates that the central, lower, and upper portions of the IAT data can be represented by different components. When Kâ€‰=â€‰1, the Gaussian mixture model is equivalent to a renewal model with the normal IAT distribution. In this sense, the goodnessoffit of renewal models with other IAT distributions (e.g., lognormal and Weibull) can be compared with the 1component Gaussian mixture model. To facilitate the comparison of the Gaussian mixture models with different numbers of model parameters, the Akaike Information Criterion (AIC) can be used:
where N_{p} is the number of model parameters and lnL is the log likelihood value of the model. For the same data, a model with a smaller AIC value is superior. The AIC values shall be used as model performance indicator in a relative sense. Therefore, the difference of the AIC values for the two competing models (e.g., 2component versus 3component Gaussian mixture models) should be evaluated. Burnham and Anderson (2004) suggest that a model with the AIC value difference greater than 2 with respect to the competing model is considered to have weak support.
Results and discussion
Model performances of the Gaussian mixture models
To begin with the performance evaluations of different statistical models for the simulated IAT data of the fullmargin CSZ ruptures, the 1component Gaussian mixture model (i.e., renewal model with the normal IAT distribution) and other renewal models are considered. The AIC values with the Gaussian, exponential, gamma, lognormal, extreme value, Weibull, Rayleigh, and Brownian Passage Time distributions are obtained as 1,194,100, 1,246,239, 1,197,931, 1,222,593, 1,220,980, 1,190,912, 1,191,436, and 1,271,206, respectively. The Weibull distribution (i.e., AICâ€‰=â€‰1,190,912) is superior to other models, including the Gaussian model (i.e., AICâ€‰=â€‰1,194,100). The fitted 1component model based on the Weibull distribution is also shown in Fig.Â 3. Although the Weibull distribution captures the unsymmetrical feature of the simulated IAT data (i.e., longer upper tail), the fitted model reveals significant mismatches on both sides of the dominant mode of the simulated data. To improve the fit, more complex statistical models can be considered.
Next, the 2component and 3component Gaussian mixture models are used to characterize the simulated data. The results of the model fit are shown in Table 1 and Fig.Â 4. The visual inspection and AIC values of the fitted Gaussian mixture models indicate that the multicomponent Gaussian models outperform all singlecomponent models that are examined previously and that the 3component Gaussian mixture model is superior to the 2component Gaussian model (AICâ€‰=â€‰1,185,057 versus 1,186,767). When the 3component model is implemented, three components correspond to clusters with mean recurrence periods of 503, 905, and 167Â years; the second component corresponds to the long gaps with the mixing proportion of 0.240, whereas the third component corresponds to the shortterm clustering with the mixing proportion of 0.114. When the number of components is restricted to Kâ€‰=â€‰2, the longgap component is retained in the model, while the less obvious shortterm clustering component is merged with the most dominant central component. In short, the datadriven statistical investigations carried out in this section indicate strong support for the 3component cluster model for the fullmargin CSZ ruptures, outperforming the 1 and 2component clustering models.
Simulations of earthquake event times by considering the elapsed time
The fitted 3component Gaussian mixture model can be used in probabilistic seismic and tsunami hazard analysis for the CSZ by considering the current elapsed time since the last 1700 event, i.e., T_{E}â€‰=â€‰323Â years. The adjustment to the 3component Gaussian mixture model (Fig.Â 4b) is straightforward by modifying the mixing proportions of the 3 components. First, for each component, calculate the probability that the IAT is longer than T_{E}, and then multiply this probability by the original mixing proportion of the component. Once the same calculations are performed for all three components, these quantities are normalized to 1.0, and can be used as the updated mixing proportions. This calculation step is demonstrated in Table 2.
The simulation steps of stochastic event catalogs for nonzero elapsed times are as follows:

1.
Considering the fitted 3component Gaussian mixture model and the elapsed time T_{E}, calculate the modified mixing proportions for all components (Table 2).

2.
Set the number of stochastic event catalogs to be simulated. Set the duration for the hazard analysis T_{D}.

3.
For simulating the occurrence time of the first event in the catalog, T_{E} must be considered. Simulate a uniform random number between 0 and 1 to select a suitable mixture component by using the modified mixing proportions. Subsequently, simulate t_{IAT} from the conditional probability distribution of the selected component.

4.
If t_{IAT} is less than T_{D}, the simulated event should be registered as t_{1}â€‰=â€‰t_{IAT} in a stochastic event catalog and proceed to the second event; otherwise, the simulation process for this catalog realization is stopped (i.e., no event occurs over a period of T_{D}).

5.
For the second event, T_{E} is reset to 0 and generate t_{IAT} using the original proportions for the Gaussian mixture model (similar to Step 3). When t_{2}â€‰=â€‰t_{1}â€‰+â€‰t_{IAT} is less than T_{D}, the second event is registered in the stochastic event catalog; otherwise, the simulation for this catalog realization is stopped.

6.
Continue Step 5 until the updated time of the most recent event exceeds T_{D}.

7.
Repeat Steps 3 to 6 as many times as required to generate stochastic event sets that reflect the multicomponent earthquake occurrence characteristics and the data uncertainty of the fullmargin CSZ ruptures.
The abovementioned simulation procedure is implemented to develop a histogram of the 3component Gaussian mixture model conditioned on the elapsed time of 323Â years (Fig.Â 5). When the elapsed time is 323Â years, the modified proportion for the shortterm clustering is significantly decreased from 0.114 to 0.007. Consequently, the proportions for the dominant central component and the longgap component are increased. The probability density function shown in Fig.Â 5 is renormalized (or conditioned) by considering no occurrence of the fullmargin CSZ event in the last 323Â years. Therefore, its height is taller than the probability distribution shown in Fig.Â 4b. It is noted that the longterm probability of the future fullmargin CSZ events can be computed using the results shown in Fig.Â 5. For instance, by considering T_{D}â€‰=â€‰50Â years, the probability can be calculated as 6.6%.
Magnitude distribution of the Cascadia fullmargin ruptures
The magnitude frequency distribution is an important element for the earthquake occurrence modeling. In many cases, the temporal earthquake occurrence process and the earthquake size distribution are related. For this reason, it is useful to mention how the magnitude distribution of the fullmargin CSZ ruptures can be characterized. Goldfinger et al. (2012) estimated the earthquake magnitudes of the fullmargin CSZ events (see Table 8 of Goldfinger et al. (2012)). A direct approach is to adopt this empirical magnitude distribution. Alternatively, other magnitude distributions can be adopted. For instance, the (characteristic) uniform distribution may be used by specifying the lower and upper magnitude limits of the fullmargin rupture scenarios. A truncated normal distribution with the lower and upper magnitude limits can also be adopted by specifying the mean and standard deviation based on the empirical magnitude data. The abovementioned approaches essentially decouple the temporal occurrence model and the magnitude frequency model. If a coupled characterization of the temporal occurrence and earthquake size is desirable, a stronger hypothesis, such as a hierarchical earthquake occurrence model by Kulkarni et al. (2013), which can capture the dependency between the waiting time and the earthquake magnitude, could be adopted in probabilistic seismic and tsunami hazard analyses. A discussion as to which of the hypothesized models is more suitable is beyond the scope of this study.
Conclusions
The earthquake occurrence modeling for major subduction events involves significant data uncertainty due to the scarcity of available geological data and the adopted dating techniques. Due to the complex nature of major subduction zones, the data may exhibit the clustering characteristics, separated by long gaps. The marine turbidite age data for the fullmargin CSZ ruptures, which were compiled by Goldfinger et al. (2012), revealed such data features. Building upon the previous work by Kulkarni et al. (2013) which attempted to characterize such clustering behavior of the Cascadia fullmargin events using a hierarchical clustering analysis, this study adopted the Gaussian mixture model as an alternative method. To fully capture the uncertainty of the underlying marine turbiditebased Cascadia age data, the number of Monte Carlo resampling was increased substantially (from 20 to 5000).
The statistical modeling results clearly showed that the 3component Gaussian mixture model outperforms the 2component Gaussian mixture model and the 1component renewal models. In addition to the most dominant central component with the mean recurrence period of 500Â years, the longgap component, and the shortterm clustering component were identified. The developed Gaussian mixture model can be used to generate stochastic event sets for probabilistic seismic and tsunami hazard analysis and to evaluate the longterm probability of the future fullmargin CSZ events. The computer algorithm and code are provided to promote the use of the proposed Gaussian mixture method.
As a final remark, it is important to mention the limitations of the results presented in this study. Although the marine turbidite age data for the fullmargin CSZ ruptures by Goldfinger et al. (2012) and Kulkarni et al. (2013) were taken as a valid dataset for statistical modeling of the fullrupture CSZ recurrence, a subset of the identified fullmargin CSZ ruptures can be adopted by considering the age data to certain calibrated years before the present. For instance, onshorebased paleotsunami records are generally shorter than offshorebased paleotsunami records. The reduced data size will result in different recurrence models for the fullmargin CSZ ruptures. To reflect epistemic uncertainty associated with the data selection of the earthquake occurrence modeling, such different models can be implemented in a logic tree for probabilistic seismic and tsunami hazard analysis.
Abbreviations
 AIC:

Akaike Information Criterion
 CoV:

Coefficient of variation
 CSZ:

Cascadia subduction zone
 EM:

Expectation maximization
 IAT:

Interarrival time
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Funding
The work is funded by the Canada Research Chair program (950232015) and the NSERC Discovery Grant (RGPIN201905898).
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Supplementary Information
Additional file 1.
A MATLAB code to fit the Gaussian mixture model to the interarrival time data of the Cascadia subduction zone. The Cascadiaturbidite age data suggested by Kulkarni et al. 2013) are included in the MATLAB code.
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Goda, K. Statistical characterization of fullmargin rupture recurrence for Cascadia subduction zone using event time resampling and Gaussian mixture model. Geosci. Lett. 10, 52 (2023). https://doi.org/10.1186/s40562023003066
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DOI: https://doi.org/10.1186/s40562023003066