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Regression analysis and variable selection to determine the key subduction-zone parameters that determine the maximum earthquake magnitude


Large variations in the maximum earthquake magnitude (\(M_{{\text{max}}}\)) have been observed among the world’s subduction zones. There is still no universal relationship between \(M_{{\text{max}}}\) and a given subduction-zone parameter, such as plate age, plate dip angle, or plate velocity, which suggests that multiple parameters control \(M_{{\text{max}}}\). Here, we conduct exhaustive variable selections that are based on three evaluation criteria; leave-one-out cross-validation errors (LOOCVE), Akaike information criterion (AIC), and Bayesian information criterion (BIC) to determine the combination of subduction-zone parameters that best explains \(M_{{\text{max}}}\). Multiple linear regression analyses are applied using 18 subduction-zone parameters as potential candidates for the explanatory variables of \(M_{{\text{max}}}\). The minimum BIC is obtained when five variables (trench sediment thickness, existence of an accretionary prism, upper-plate crustal thickness, bending radius of the subducting oceanic plate, and trench depth) are selected as explanatory variables; each variable contributes positively to \(M_{{\text{max}}}\). Minimum LOOCVE and AIC values are obtained when eight variables (the five parameters for BIC, plus the along-strike plate convergence rate, age of the subducting plate, and maximum depth of the subducting plate) are selected. Our selection of the trench sediment thickness and plate bending radius contributing to \(M_{{\text{max}}}\) is consistent with previous studies. The results show that increasing upper-plate crustal thickness results in a large \(M_{{\text{max}}}\). In addition to smoothing the subducting-plate interface via subducted sediments, along-dip extension of the crustal area along the convergent plate boundary would be important for generating a large earthquake.

Graphic Abstract


Large earthquakes (magnitude \(M \ge 8\)) rarely occur, but are often detrimental to life and property. They have usually been observed along subduction zones and some continental-collision zones, with a significant variation in the maximum earthquake magnitude (\(M_{{\text{max}}}\)) detected among the world’s subduction zones (Fig. 1); for example, the 1960 M9.5 Chile earthquake along the South-Central Chilean subduction zone is the largest recorded earthquake to date, whereas a M-7+ event has not been observed along the South Kermadec subduction zone. Seismologists generally assume that large earthquakes are associated with certain subduction settings, with numerous relationships between \(M_{{\text{max}}}\) (or the Gutenberg–Richter b-values) and various parameters that characterize the tectonic features of subduction zones (hereafter referred to as “subduction-zone parameters”) proposed (e.g., Wirth et al. 2022; Marzocchi et al. 2016); for example, the age of the subducting plate (e.g. Ruff and Kanamori 1980; Nishikawa and Ide 2014), angle or curvature radius of the subducting plate (e.g., Ruff and Kanamori 1980; Bletery et al. 2016), seafloor sediment thickness at the subduction trench (e.g., Ruff 1989; Heuret et al. 2012; Scholl et al. 2015; Brizzi et al. 2018), subducted sediment thickness (Seno 2017), fore-arc structures (e.g., Song and Simons 2003; Wells et al. 2003), upper-plate strain (e.g., Heuret et al. 2012), trench migration velocity (e.g., Schellart and Rawlinson 2013), upper-plate motion (e.g., Scholz and Campos 1995), width of the subducting plate or trench length (Schellart and Rawlinson 2013; Brizzi et al. 2018), and topographic roughness or seafloor smoothness along the subducting plate (e.g., Wang and Bilek 2014; Lallemand et al. 2018) have all been analyzed to infer \(M_{{\text{max}}}\). Schellart and Rawlinson (2013) have investigated 24 physical parameters that characterize subduction zones, but were unable to find any parameters that had a large correlation with \(M_{{\text{max}}}\) (correlation coefficient less than 0.5). Thus, there is still no consistent relationship between \(M_{{\text{max}}}\) and these individual subduction-zone parameters, which suggests that multiple factors may be involved. Alternatively, observational errors in subduction zone parameters may make it difficult to relate such parameters to \(M_{{\text{max}}}\), and it is therefore necessary to select parameters with a low signal-to-noise ratio to correctly predict \(M_{{\text{max}}}\).

The ability to determine the key subduction-zone parameters that influence the occurrence of large earthquakes is limited by the ability to effectively derive a small number of essential elements from a desired phenomenon. An exhaustive variable selection procedure combined with regression or discriminant analysis, which is a primitive machine-learning-based method, is a powerful approach to derive a small number of essential variables from complex processes (e.g., Kuwatani et al. 2014; Igarashi et al. 2018; Ueki et al. 2020; Itano et al. 2020; Nakao et al. 2022). This method is suitable for determining a combination of subduction-zone parameters that can reasonably explain \(M_{{\text{max}}}\), and can be instrumental in gaining scientific insight into the origin of large earthquakes; however, this machine-learning approach has not been applied to derive a \(M_{{\text{max}}}\) relationship to date. Variable selection is a reasonable approach for addressing the present problem for two key reasons. First, the sample locations for subduction-zone parameters are limited. For example, the observed subduction styles, including the velocity and shape of the subducting plate, exhibit much smaller variations than those simulated in laboratory and numerical experiments (e.g., Schellart 2011; Nakao et al. 2016). Model selection with cross-validation would also be useful in enhancing the predictability of \(M_{{\text{max}}}\) with limited observations. Second, the observed subduction-zone parameters, including the velocity and geometry of the subducting oceanic plate, contain large uncertainties, such that a model may be overfit due to these uncertainties if a variable selection approach is not employed. Therefore, we employ an exhaustive variable selection approach in this study to infer which subduction-zone parameters may explain local variations in \(M_{{\text{max}}}\).

Data and methods


We investigate 18 types of subduction-zone parameters via regression analysis to determine the set of parameters that can effectively constrain \(M_{{\text{max}}}\). Subduction-zone parameters are sampled at 2-degree intervals following Heuret and Lallemand (2005), at which both \(M_{{\text{max}}}\) and its explanatory variables are sampled. We incorporate present-day observations in our analysis, and therefore evaluate potential \(M_{{\text{max}}}\) values under present-day tectonic conditions.

The objective variable, \(M_{{\text{max}}}\), is taken from the SubMap 4.3 database and is based on the rupture areas of large subduction earthquakes (\(M\ge 8\)) that occurred at depths less than 70 km during the 1900–2007 period (Heuret et al. 2011), as well as the 2011 Tohoku-oki Earthquake (M9.1, Northeast Japan; Yagi and Fukahata 2011). In addition, we included known historical earthquakes, including the 1700 Cascadia earthquake (M9.0, North America; Satake et al. 1996), the 1707 Hoei earthquake (M8.6, Southwest Japan; Fujiwara et al. 2020), and the 1833 Sumatra earthquake (M8.8, Indonesia; Zachariasen et al. 1999). The segmentation of \(M_{{\text{max}}}\) is defined based on three criteria of Heuret et al. (2011): (1) the rupture area inferred for M+8.0 earthquakes must be included in a single segment; (2) the transects with homogeneous activity in the seismogenic zone were grouped; and (3) the transects with homogeneous geometries in the seismogenic zone were grouped.

The subduction-zone parameters that we investigate in relation to \(M_{{\text{max}}}\) are listed in Table 1 and shown graphically in Fig. 2, with \(M_{{\text{max}}}\) values obtained at 169 locations along subduction zones worldwide (Fig. 1). We mainly employed the subduction-zone parameters from the SubMap 4.3 database (e.g., Heuret and Lallemand 2005) in our analysis. This data set lacks observations along some subduction zones (e.g., Mediterranean subduction zones); therefore, we excluded all sample locations with at least two missing variables. We conducted leave-one-out cross validation, a method to enhance the model predictability for unknown samples, to minimize the effects of this omission. We employed a neighboring value at the locations with only one missing variable. We consider that the effect to be small because the operation was applied to only 0.3% of the total data, and because the subduction-zone parameters generally vary continuously along a trench. Furthermore, we removed the subduction-zone parameters that are dependent on \(M_{{\text{max}}}\) by definition, such as the equivalent representative magnitude (\(M_\text{MRR}\); the earthquake magnitude calculated from MRR (moment release rate), where MRR is the integrated seismic moment during a century and along 1000 km of the trench), from the regression analysis. The trench length (or slab width), which is a potential controlling factor for \(M_{{\text{max}}}\) (Schellart and Rawlinson 2013; Brizzi et al. 2018), is not used as an explanatory variable of \(M_{{\text{max}}}\) in this study, because there can be a spurious correlation between \(M_{{\text{max}}}\) and the trench length; more precisely, a smaller \(M_{{\text{max}}}\) is generally expected to be observed within a limited timeframe as the trench length becomes smaller, even if the occurrence of a large earthquake is completely random.

The details of the 18 analyzed subduction-zone variables are as follows. A is the age of the subducting oceanic plate at the trench (Müller et al. 1997). \(a_{\text{s}}\) is the mean dip angle of the subducting oceanic plate over the 0–125 km depth range, which is measured using hypocenters of Engdahl et al. (1998) along Wadati–Benioff zones and plate boundaries (Lallemand et al. 2005). CMP and TNS (compression and tension, respectively) are dummy parameters, which take 0 or 1, to express the strain state of the upper plate (“UPS” in Heuret et al. 2011): \((\hbox {CMP}, \hbox {TNS}) = (1, 0)\) for compressible upper plates, (0, 1) for tensile upper plates, and (0, 0) for neutral upper plates. The upper-plate stress is originally classified using an “ordinal scale” into three types based on the focal mechanisms of shallow earthquakes occurring at depths less than 40 km (Heuret et al. 2011); the two dummy variables CMP and TNS are necessary to express the ordinal scale in the regression modeling. MT, or the margin type, is a dummy variable that expresses either the accretionary or erosional conditions of the upper-plate margin: \(\hbox {MT} = 1\) for accretionary margins and 0 for erosional margins. \(R_{{\text{c}}}\) is the bending radius of the subducting oceanic plate, which is measured such that a circle of radius \(R_{{\text{c}}}\) within a trench-normal vertical cross section fits hypocenters of Engdahl et al. (1998) along the upper limit of the Wadati–Benioff zone over the 0–150 km depth range (Heuret 2005; Wu et al. 2008). \(R_{\text{SW}}\), \(R_{\text{IW}}\), and \(R_{\text{LW}}\) are the seafloor roughnesses of the subducting oceanic plate for different bathymetric wavelengths, which have been defined by Lallemand et al. (2018): 12–20 km (short wavelengths), 20–80 km (intermediate wavelengths), and 80–100 km (long wavelengths), respectively. \(T_{{\text{c}}}\) is the thickness of the margin of the upper plate, which is taken from CRUST 1.0 (Laske et al. 2013). Here, \(T_{{\text{c}}}\) is defined as the maximum thickness of the crust from the trench to the volcanic arc. \(T_{\text{sed}}\) is the sediment thickness at the trench, which is taken from GlobSed 3 (Straume et al. 2019). \(v_{\text{sn}}\) and \(v_{\text{ss}}\) are the trench-normal and -parallel components, respectively, of the convergence rate at the trench (Lallemand et al. 2008). \(v_{\text{dn}}\) is the trench-normal component of the extension rate of the upper-plate margin (Lallemand et al. 2008). \(v_{\text{dn}}\) is positive for back-arc spreading and negative for back-arc shortening. \(v_{\text{tn}}\) is the trench-normal component of the trench migration rate, with a positive value for a retreating trench (oceanward motion) and negative value for an advancing trench (continent-ward motion). We referenced the absolute velocity \(v_{\text{tn}}\) from the SB04 model (Steinberger et al. 2004; Lallemand et al. 2008), which is adjusted using the Indo-Atlantic hotspots as a reference. We referenced SB04 because the Indo-Atlantic hotspot reference frame better explains the geometry of subducting slabs beneath global subduction zones, which is sensitive to trench motion (Schellart 2011; Schellart and Rawlinson 2013; Nakao et al. 2022). We additionally applied the \(v_{\text{tn}}\) based on the Pacific hotspot reference frame (HS3; Gripp and Gordon 2002) to confirm that the influence of the reference frame on the analytical results is small, as shown in Additional file 1: Fig. S7. \(Z_{\text{seis}}\) is the maximum depth of deep earthquakes. \(Z_{{\text{t}}}\) is the trench depth. \(Z_{\text{tomo}}\) is the maximum depth of the subducting plate, which has been constrained from high-velocity seismic anomalies (Heuret and Lallemand 2005).

Regression analysis

We relate \(M_{{\text{max}}}\) (objective variable) to the subduction-zone parameters (explanatory variables) via regression analysis. We evaluate the contribution of each explanatory variable to \(M_{{\text{max}}}\) by standardizing the i-th explanatory variables at location j as follows:

$$\begin{aligned} x_{ij}^\prime = \frac{ x_{ij} - m_i }{s_i}, \end{aligned}$$

where \(x_{ij}\) is an unstandardized explanatory variable (i.e., \(A, a_{\text{s}},\ldots , Z_{\text{tomo}}\)), \(m_i = \frac{1}{J} \sum _{j=1}^{J} x_{ij}\) is the empirical mean value of variable i, \(s_i = \{ \frac{1}{J-1} \sum _{j=1}^J(x_{ij} - m_i)^2 \}^\frac{1}{2}\) is the empirical standard deviation of variable i, and J is the number of locations used for training. We randomly selected 95% of the 169 locations in Fig. 1 as training data for the regression analysis, with the remaining 5% used as test data to validate the optimal models (i.e., \(J = 161\)). Hereafter, a standardized variable is expressed using a prime symbol.

We assume that \(M_{{\text{max}}}\) is a linear combination of the explanatory variables:

$$\begin{aligned} f^\prime _j ({{\textbf {a}}}; \,{{\textbf {c}}}) = a_0 + \sum _{i=1}^{I} c_i a_i x_{ij}^\prime + \varepsilon _j, \end{aligned}$$

where \(f^\prime _j ({{\textbf {a}}};\,{{\textbf {c}}})\) is the predicted maximum earthquake magnitude at location j, \({{\textbf {a}}} = (a_0, a_1,\ldots , a_I)\) is a vector of the coefficients, I is the number of explanatory variables (\(I=18\)), \({{\textbf {c}}} = (c_1,\ldots , c_I)\) is a vector of parameters that control whether \(x_{ij}^{\prime }\) is included in the regression analysis (i.e., \(c_i =0\) or 1), and \(\varepsilon _j\) is Gaussian observation noise. A linear model is determined when \({{\textbf {c}}}\) is fixed, with this linear model specified by the configuration of \({{\textbf {c}}}\). Although such a simple linear combination of subduction-zone parameters is often used to express \(M_{{\text{max}}}\) (e.g., Brizzi et al. 2018), this assumption is not physically derived. We therefore conduct an error analysis to check the validity of this assumption.

The coefficients are estimated via a least-squares method for a given \({{\textbf {c}}}\), such that:

$$\begin{aligned} {{\textbf {a}}}_{{\text {LS}}} = \mathop {{\mathrm{arg\,min}}}\limits _{{\textbf {a}}} \sum _{j=1}^J \left( f^\prime _j({{\textbf {a; c}}}) - M^\prime _{{\text {max}},j} \right) ^2, \end{aligned}$$

where \(M^\prime _{\text{max},j}\) is the maximum earthquake magnitude observed at location j.

Multicollinearity, which occurs when a pair of explanatory variables has a significantly large correlation coefficient, causes problems during multiple regression analysis. For example, strong multicollinearity may prohibit the selection of an explanatory variable, even if that explanatory variable has a significant relationship to the objective variable. We verified that each of the pairs among the 18 explanatory variables have variance inflation factors (VIFs; an index for multicollinearity) below \(\sim\) 3.5 (Additional file 1: Fig. S2), thereby indicating that multicollinearity should not largely impact our results (Hair et al. 2009).

Model selection

We assume that a limited number of subduction-zone parameters essentially characterizes \(M_{{\text{max}}}\), such that we can determine the best combination of the subduction-zone parameters from our analysis. We select the optimal models from \(2^I\, ({=}\,262{,}144)\) cases by calculating three criteria, the leave-one-out cross-validation error (LOOCVE), Akaike information criterion (AIC) (Akaike 1974), and Bayesian information criterion (BIC) (Schwarz 1978), for each case.

Cross-validation is a method that divides the samples into test and training data sets, with the model performance iteratively evaluated based on the data sizes. Leave-one-out cross-validation is a special case whereby one sample is left as test data to evaluate a model that is generated using the other samples, with this procedure repeated based on the number of samples:

$$\begin{aligned} \text{LOOCVE} ({\textbf {c}})& = \sqrt{ \frac{1}{J}\sum _{j=1}^J \left( f_{-j}^\prime ({\textbf {c}}) - M_{\text{max},j}^\prime \right) ^2}, \end{aligned}$$

where \(f_{-j}^\prime ({\textbf {c}})\) is the maximum earthquake magnitude at the j-th location that was predicted by the model using the other locations. Cross-validation can enhance the generalized performance of a prediction model using a small number of observations, which is common in subduction zones.

AIC and BIC are evaluation criteria that balance the misfit with the number of explanatory variables. We can avoid overfitting to noisy subduction-zone parameters by implementing either AIC or BIC. AIC and BIC are defined as

$$\begin{aligned} \text{AIC} ({\textbf {c}}) = -2 \ln {L ({\textbf {c}})} + 2 \left( \sum _{i=1}^I c_i +2 \right) \end{aligned}$$


$$\begin{aligned} \text{BIC} ({\textbf {c}}) = -2 \ln {L ({\textbf {c}})} + \left( \sum _{i=1}^I c_i +2 \right) \ln J, \end{aligned}$$

respectively, where \(L({\textbf {c}})\) is the maximum likelihood of the model. The term \(\left( \sum _{i=1}^I c_i +2 \right)\) in Eqs. (5) and (6) indicates the number of parameters; the number of \(a_i\) values used in model c, intercept \(a_0\), and observation noise are all counted as parameters in these evaluations. \(L({\textbf {c}})\) is calculated as

$$\begin{aligned} \ln L ({\textbf {c}}) = -\frac{J}{2} \ln (2 \pi {\hat{\sigma }}^2 ({\textbf {c}}) ) - \frac{J}{2}, \end{aligned}$$

where \({\hat{\sigma }} ({\textbf {c}})\) is the maximum likelihood estimate of the error variance, which can be written as

$$\begin{aligned} {\hat{\sigma }}^2 ({\textbf {c}}) = \frac{1}{J} \sum _{j=1}^{J} \left( f_j^\prime ({\textbf {a}}_{\text{LS}}; {\textbf {c}}) - M_{\text{max},j}^\prime \right) ^2. \end{aligned}$$

Exhaustive model evaluation was conducted to determine the smallest LOOCVE, AIC, and BIC values, which both enhanced the predictability and minimized the overfitting of the final model.


Optimal models

We obtained minimum LOOCVE and AIC values among the \(2^{18}\) cases by characterizing \(M_{{\text{max}}}\) using eight explanatory variables: A, MT, \(R_{{\text{c}}}\), \(T_{{\text{c}}}\), \(T_{\text{sed}}\), \(v_{\text{ss}}\), \(Z_{{\text{t}}}\), and \(Z_{\text{tomo}}\) (Fig. 3). The standardized and unstandardized forms of the optimal model are

$$\begin{aligned} f_{\text{LOOCVE}}^\prime& = f_{\text{AIC}}^\prime = 5.4 \times 10^{-16} +0.15 A ^{\prime } +0.30 \text{MT} ^{\prime } +0.30 R_{{\text{c}}} ^{\prime } \nonumber \\{} & \quad +0.46 T_{{\text{c}}} ^{\prime } +0.16 T_{\text{sed}} ^{\prime } +0.13 v_{\text{ss}} ^{\prime } +0.17 Z_{{\text{t}}} ^{\prime } -0.14 Z_{\text{tomo}} ^{\prime } \end{aligned}$$


$$\begin{aligned} f_{\text{LOOCVE}} = f_{\text{AIC}} = 6.7 +2.1 \times 10^{-3} A \nonumber \\{} \quad+0.37 \text{MT} +9.1 \times 10^{-4} R_{{\text{c}}}+0.018 T_{{\text{c}}} +0.13 T_{\text{sed}} \nonumber \\{} \quad+4.4 \times 10^{-3} v_{\text{ss}} +0.078 Z_{{\text{t}}} -2.7 \times 10^{-4} Z_{\text{tomo}} ^{\prime }, \end{aligned}$$

respectively. The minimum BIC is obtained when the model includes only five parameters, which are also used in the \(f_\text{LOOCVE}^\prime\) and \(f_{\text{AIC}}^\prime\) (Fig. 3). The standardized and unstandardized forms of the optimal model in terms of BIC are

$$\begin{aligned} f_{{\text{BIC}}}^\prime = 5.53 \times 10^{-16} +0.32 \text{MT} ^{\prime } +0.30 R_{{\text{c}}} ^{\prime } +0.40 T_{{\text{c}}} ^{\prime } +0.16 T_{\text{sed}} ^{\prime } +0.24 Z_{{\text{t}}} ^{\prime } \end{aligned}$$


$$\begin{aligned} f_{{\text{BIC}}} = 6.7 +0.40 \text{MT} +9.0 \times 10^{-4} R_{{\text{c}}} +0.015 T_{{\text{c}}} +0.12 T_{\text{sed}} +0.081 Z_{{\text{t}}}, \end{aligned}$$

respectively. Other parameters, including the angle of the subducting oceanic plate, seafloor roughness, upper-plate strain, and trench-normal plate velocities, were not selected as explanatory parameters in the both optimal models. Although the \(R_\text{IW}\)\(R_{\text{LW}}\) pair has a slightly large VIF (3.5; Additional file 1: Fig. S2), neither is selected; therefore, our regression analysis did yield very weak multicollinearity (Hair et al. 2009).

The upper-plate crustal thickness, \(T_{{\text{c}}}\), makes the largest contribution to each optimal model among the selected explanatory variables. In fact, all large earthquakes \((M>9)\) have occurred along trenches where the upper plate consists of continental lithosphere; e.g., the 1960 Chile M9.5, 1964 Alaska M9.2, 2004 Sumatra–Andaman M9.1, 2011 Northeast Japan 9.1, and 1952 Kamchatka M9.0 earthquakes. Conversely, no large earthquakes \((M>8.5)\) have been observed along the Mariana, Tonga–Kermadec, and South Sandwich subduction zones, where the upper plate consists of oceanic lithosphere, which has a homogeneous crustal thickness of \(\sim\) 7 km (White et al. 1992).

The margin type, MT, and the trench sediment thickness, \(T_\text{sed}\), also make large positive contributions to \(f_{\text{LOOCVE}}\), \(f_{\text{AIC}}\), and \(f_{{\text{BIC}}}\), followed by \(T_{{\text{c}}}\). An accretionary prism \((\hbox {MT} = 1)\) generally develops where there are thick oceanic sediments, such that both variables have a strong positive correlation (Additional file 1: Fig. S1). Accretionary prisms and large \(T_{\text{sed}}\) values are found along the Cascadia, Alaska, Antilles, Andaman, and Hikurangi subduction zones, and some great earthquakes, such as the 1964 Alaska M9.2 and 2004 Sumatra–Andaman M9.1 earthquakes, occurred in these regions.

Comparison between \(f_{{\text{BIC}}}\) and \(M_{{\text{max}}}\)

Hereafter, we compare the observed \(M_{{\text{max}}}\) with the optimal model that yields the minimum BIC (\(f_{{\text{BIC}}}\)) for simplicity. This is because BIC generally yields a simpler model than LOOCVE and AIC and better fits our purpose. \(f_{\text{LOOCVE}}\) (\(f_{\text{AIC}}\)) yields approximately the same values as \(f_{{\text{BIC}}}\) (Additional file 1: Figs. S3, S4, S6).

Figure 4a shows that the optimal model \(f_{{\text{BIC}}}\) can predict \(M_{{\text{max}}}\) from both the test and training data sets within the 95% prediction intervals, whereas some of the \(f_\text{BIC}\) values along the (A) South Kermadec and (B) South-Central Chile subduction zones are outside of the prediction intervals (Fig. 4a, c). Possible reasons for the limitations of our analysis along these three subduction zones will be discussed in the Discussion section.

Figure 4b shows that the error between the predicted and maximum earthquake magnitudes, \(f_{{\text{BIC}}}^\prime - M_{{\text{max}}}^\prime\), possesses a Gaussian-like distribution. A Q–Q plot, which can be used to quantify the distribution of the error (Additional file 1: Fig. S5b), shows that the error \(f_{{\text{BIC}}}^\prime - M_{{\text{max}}}^\prime\) is well aligned with the theoretical Gaussian distribution.


Possible effects of subduction-zone parameters on \(M_{{\text{max}}}\)

Our analysis indicates that the trench sediment thickness, \(T_\text{sed}\), is an essential factor for obtaining a large \(M_{{\text{max}}}\), which is consistent with previous studies (e.g., Heuret et al. 2012; Brizzi et al. 2018). There are several explanations of the effects of oceanic sediments on \(M_{{\text{max}}}\). One is that the subducted sediment layer creates structural coherence between two converging plates, establishing the potential for plate locking due to diagenesis (e.g., Ruff 1989). Another explanation focuses on the small permeability of the subducted sediments (Seno 2017). The subducted oceanic crust dehydrates as the temperature and pressure increase, and the released fluid migrates toward the overlying sediment layer. Seno (2017) proposed that a thick sediment layer will act as an impermeable layer, preventing the migration of fluid from the underlying crustal layer, and the pore-fluid pressure along the subducted plate interface above the sediment layer will remain small. This mechanism may account for the positive correlation between the stress drop due to megaquakes and sediment thickness (Seno 2017).

Our study reveals that the upper-plate crustal thickness, \(T_{{\text{c}}}\), is another essential factor in generating large \(M_{{\text{max}}}\). This result is consistent with Heuret et al. (2011), who showed that continental upper plates can host larger earthquakes than oceanic upper plates. A possible explanation for the positive relationship between \(T_{{\text{c}}}\) and \(M_{{\text{max}}}\) is that the rupture areas (i.e., exponential function of earthquake magnitude) of large earthquakes are roughly limited to the brittle crust of the upper plate. This is because the serpentine in the mantle component of the upper plate along the plate boundary, which generally forms via plate dehydration and has a low shearing strength, tends to inhibit shear stress accumulation and instead release it via ductile deformation (e.g., Katayama et al. 2012). A key exception is the 2011 Tohoku earthquake, which occurred where the upper continental crust is slightly thinner (\(\sim\) 30 km); however, the main rupture area is still estimated to be at shallow depths near the Japan Trench, with insignificant slip detected below the continental Moho (e.g., Yagi and Fukahata 2011).

The bending radius, \(R_{{\text{c}}}\), was selected as an explanatory variable for \(M_{{\text{max}}}\) in our analysis, but the dip angle, \(a_{\text{s}}\), was not. This result is consistent with a previous observational study (Bletery et al. 2016), in which the bending radius has the stronger correlation with \(M_{{\text{max}}}\) compared to the dip angle. This is probably due to the fact that \(a_{\text{s}}\) is the average angle for the entire slab depth (0–125 km), whereas large earthquakes generally occur at shallow depths (0–70 km). Therefore, \(R_{{\text{c}}}\) has a stronger relationship with \(M_{{\text{max}}}\). We propose that, where \(R_{{\text{c}}}\) is larger, more of the plate boundary is in contact with the crustal part of the upper plate, along which elastic stress would accumulate.

It is unclear why the trench depth, \(Z_{{\text{t}}}\), is selected in the optimal model and why its coefficient is positive, particularly since \(Z_{{\text{t}}}\) generally reflects negative slab buoyancy and is expected to have a negative contribution to the earthquake size (Nishikawa and Ide 2014). Although \(Z_{{\text{t}}}\) is negatively correlated with \(T_{\text{sed}}\) as sediment fills a trench, multicollinearity does not greatly affect our results due to the small correlation coefficient (\(\sim \,{-}\,0.4\); Additional file 1: Figs. S1 and S2), and \(Z_{{\text{t}}}\) would have a role independent of \(T_{\text{sed}}\). An explanation for the positive relationship between \(Z_{{\text{t}}}\) and \(M_{{\text{max}}}\) is that, when a subductinhg slab contains a large amount of water, both of \(Z_{{\text{t}}}\) and \(M_{{\text{max}}}\) become small. That is, subducting plates containing a large amount of water have a large positive buoyancy (Nakao et al. 2016, 2018), thereby yielding small \(Z_{{\text{t}}}\) values. Meanwhile, such a plate would cause significant dehydration and subsequently yield large pore-fluid pressures along the plate boundary, inhibiting stress accumulation along the plate boundary. However, this explanation is highly speculative; numerical simulations are required to reveal the physical mechanisms that may induce large earthquakes along deep trenches.

A smooth seafloor is often regarded as an essential factor in generating large earthquakes (e.g., Wang and Bilek 2014) because a smooth plate interface contributes to a coherent plate boundary. However, the seafloor roughness, at least at wavelengths greater than 12 km, is not selected as an explanatory variable in our analysis. A possible reason for this omission is that the subducted sediment layers are up to \(\sim\)1 .6 km thick (Seno 2017), which may cover and smooth a large percentage of the seafloor relief, whereas the typical seafloor roughness is \(\sim\) 0.5 km, with a standard deviation of \(\sim\) 0.5 km (Table 1). This may be the reason why the seafloor roughness is not selected as a universal explanatory variable in our analysis.

The upper-plate stress was not selected as an explanatory variable of \(M_{{\text{max}}}\) in our analysis, even though it had been selected in some previous studies (Heuret et al. 2011, 2012). Our results suggest that the present-day stress is not a good indicator for evaluating the potential \(M_{{\text{max}}}\). A possible interpretation of our result is that the stress patterns change temporally throughout earthquake cycles. For example, the upper-plate stress immediately changed from compressional to tensional, as observed before and after the 2011 Tohoku earthquake (Hasegawa et al. 2012).

Thus, we propose that multiple factors influence \(M_{{\text{max}}}\). For example, the 1868 Peru earthquake (Mw8.5\(-\)9.2; Lay and Nishenko 2022; McCaffrey 2008) has occurred where the trench sediment is thin (< 1 km), suggesting that it is difficult to attribute the magnitude of this earthquake to the sole role of the sediment. Rather, this historical earthquake may be related to the thick continental crust (\(T_{{\text{c}}}\sim\) 60 km), which is newly found as a factor for large \(M_{{\text{max}}}\) in our study. However, it is difficult to generate a \(M \ge 8\) earthquake considering a typical value of \(T_{{\text{c}}}\); therefore, the combined effects of other factors, including \(T_{\text{sed}}\) and \(R_{{\text{c}}}\), are necessary to generate a \(M \ge 8\) earthquake.

Previous studies have proposed numerous possible mechanisms for the genesis of large earthquakes as in Introduction. Our exhaustive model evaluation has detected the subduction-zone parameters that possess a strong relationship with \(M_{{\text{max}}}\), which enables us to evaluate the plausibility of the proposed mechanisms for generating large earthquakes. Our presented approach will therefore assist in clarifying problems that include complex processes.

Origin of the misfit between \(f_{{\text{BIC}}}\) and \(M_{{\text{max}}}\)

Here we discuss why there is high degree of misfit between \(f_\text{BIC}\) and \(M_{{\text{max}}}\) at some of the analyzed locations (Fig. 4a). There are large \(f_{{\text{BIC}}}\) values along the South Kermadec subduction zone (A in Fig. 4a, c) because the subduction-zone parameters are almost the same as those along the North Kermadec subduction zone, whereas \(M_{{\text{max}}}\) is significantly smaller. If our analysis accurately reflects a sufficient number of factors for constraining the earthquake magnitude, then a M-8 class earthquake will potentially occur along the South Kermadec subduction zone. Or, the North and South Kermadec regions should be integrated into the same group. The South-Central Chile subduction zone (B in Fig. 4a, c) is another outlier. These results suggest that the 1960 Chile earthquakes are linked to tectonic processes that are not captured by the subduction parameters considered in our analysis, which focuses on large-scale tectonic features. Ridge subduction, petit-spots, and hydrothermal circulation are potential candidates for the elevated earthquake magnitudes along the South-Central Chile subduction zone. However, we cannot identify the factor(s) here, but hope to resolve this in a future study.

The misfit between the predicted and observed maximum earthquake magnitudes, \(f_{{\text{BIC}}} - M_{{\text{max}}}\), yields a Gaussian-like distribution among the 169 analyzed locations. The least-squares method, which is used in our regression analysis, is based on the assumption that the error follows a Gaussian distribution. Therefore, it is suggested that our modeling, which is based on the assumption that \(f_{{\text{BIC}}}\) is a linear combination of the subduction-zone parameters, is not unreasonable. In addition, the Gaussian-like distribution of the misfit justifies applying \(M_{{\text{max}}}\) in Fig. 1 as the objective variable to some extent although \(M_{{\text{max}}}\) may lack some unknown historical earthquakes because of the observation duration shorter than megathrust earthquake cycles (\(10^2\)\(10^3\) years). This would be because large earthquakes have been observed in this \(10^2\) year correspondingly to potential (or ideal) \(M_{{\text{max}}}\) following the Gutenberg–Richter’s law.

Fig. 1
figure 1

Map of the observed maximum earthquake magnitude, \(M_{{\text{max}}}\) (unstandardized). The data is referred from SubMap 4.3 and includes the 169 locations. The white lines indicate plate boundaries

Table 1 Analyzed subduction-zone parameters in this study
Fig. 2
figure 2

Schematic cross-section of a subduction-zone, with the analyzed subduction-zone parameters labeled. The stars indicate earthquake hypocenters. The gray, pink, yellow, and blue regions indicate lithospheric rocks, the upper crust, trench sediments, and seawater, respectively. See Table 1 for details of each parameter

Fig. 3
figure 3

Results for determining the subduction-zone parameters that are related to \(M_{{\text{max}}}\). The graph shows the coefficients, \(a_i\), for each standardized explanatory variable that yields the smallest LOOCVE and AIC (gray bars; Eq. 9), and BIC (black bars; Eq. 11). An explanation of the symbols is provided in Table 1

Fig. 4
figure 4

Comparison between observed and modeled maximum earthquake magnitudes. a Unstandardized observed maximum earthquake magnitude, \(M_{{\text{max}}}\) (horizontal axis), versus the predicted value, \(f_{{\text{BIC}}}\) (vertical axis), at the 169 analyzed locations. The open gray circles and closed red circles indicate the training and test data sets, respectively. The error bars indicate the 95% prediction intervals. \(f_{{\text{BIC}}}=M_{{\text{max}}}\) along the black line. Samples A (South Kermadec) and B (South-Central Chile) are outliers. b Histogram of the error between the predicted and observed standardized maximum earthquake magnitudes (\(f_\text{BIC}^\prime - M_{{\text{max}}}^\prime\)). c Map of the error between the predicted and observed unstandardized maximum earthquake magnitudes (\(f_{{\text{BIC}}} - M_{{\text{max}}}\)) at the 169 analyzed locations. Warm colors indicate \(f_{{\text{BIC}}} > M_{{\text{max}}}\), and cool colors indicate \(f_{{\text{BIC}}}<M_{{\text{max}}}\). The white lines indicate plate boundaries


We conducted multiple regression analyses and exhaustive model evaluation to determine the subduction-zone parameters that control maximum earthquake magnitude, \(M_{{\text{max}}}\). The smallest LOOCVE and AIC evaluation criteria were obtained when eight parameters, the trench sediment thickness, \(T_{\text{sed}}\), existence of an accretionary prism, MT, upper-plate thickness, \(T_{{\text{c}}}\), bending radius of the subducting oceanic plate, \(R_{{\text{c}}}\), trench depth, \(Z_{{\text{t}}}\), age of the subducting plate, A, along-strike convergence rate along the trench, \(v_{\text{ss}}\), and maximum depth of the subducting plate, \(Z_{\text{tomo}}\), were selected as explanatory variables to express \(M_{{\text{max}}}\). Furthermore, the combination of only five variables, \(T_{\text{sed}}\), MT, \(T_{{\text{c}}}\), \(R_{{\text{c}}}\), and \(Z_{{\text{t}}}\), yields the smallest BIC. The seafloor roughness, trench-normal plate and trench velocities, upper-plate stress, and dip angle of the subducting oceanic plate are notable subduction-zone parameters that were not selected as explanatory variables. Our results are consistent with previous studies that have proposed \(T_{\text{sed}}\) as the primary factor controlling \(M_{{\text{max}}}\). We provided new insight that \(T_{{\text{c}}}\) also has a positive effect on producing large \(M_{{\text{max}}}\), which suggests that along-dip extension of crustal areas along a converging plate boundary are important in generating large earthquakes. We used five tectonic conditions, \(T_{\text{sed}}\), MT, \(T_{{\text{c}}}\), \(R_{{\text{c}}}\), and \(Z_{{\text{t}}}\), to demonstrate that our optimal model can explain almost all of the observed \(M_{{\text{max}}}\) values within the 95% confident interval, although our model fails to predict some samples, such as the 1960 M9.5 Chile earthquake. An evaluation of additional mechanisms that may cause these outliers will be conducted to understand the processes controlling the earthquakes that are not explained by our model. An investigation of the genesis of large earthquakes using numerical simulations that consider the essential subduction-zone factors for generating large \(M_{{\text{max}}}\) will also be undertaken, as our analyses do not explain the physical meanings of the detected parameters.

Availability of data and materials

All of the data used in this study are available from SubMap 4.3 (Heuret and Lallemand 2005,, CRUST 1.0 (Laske et al. 2013,, and GlobSed 3 (Straume et al. 2019,



Akaike information criterion


Bayesian information criterion


Leave-one-out cross-validation error

M max :

Maximum earthquake magnitude


Margine type






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We thank two anonymous reviewers for their constructive comments and Naoki Uchida and Aitaro Kato for editing this article. We used the R language (R Core Team 2022) and the MuMIn package (Barton and Barton 2015) for model selection. Some of the figures were produced using Generic Mapping Tools (Wessel et al. 2019).


This work was supported by Japan Science and Technology Agency CREST Grant No. JPMJCR1761; JSPS KAKENHI Grant Nos. JP19K04026, JP20K04119, JP22K14131, and JP22H03653; and the Joint Research Programs 2021-B-01 and 2022-B-06 of the Earthquake Research Institute, the University of Tokyo.

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AN designed the research, collected the data, conducted the analysis, and wrote the original draft under the supervision of TK. TK and KU developed the analytical method. TK, KU, KY, and TY reviewed and edited the original draft. HH and SA verified the analytical method and reviewed and edited the statistical discussions in the manuscript.

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AN (Corresponding author, Post-doctoral Researcher), TK (Deputy Group Leader, Senior Researcher), KU (Researcher), KY (Researcher), and TY (Research Assistant) belongs to Solid-Earth Data Science Group (SDG) of JAMSTEC. TK employed AN and TY under the CREST project until March 2023, and AN is currently an assistant professor at Akita University. TK and HH (Professor of ISM) are principal researchers of the CREST project. HH and SA (Chief Senior Researcher of AIST) work with SDG through Joint Research program of ERI.

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Correspondence to Atsushi Nakao.

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Nakao, A., Kuwatani, T., Ueki, K. et al. Regression analysis and variable selection to determine the key subduction-zone parameters that determine the maximum earthquake magnitude. Earth Planets Space 75, 78 (2023).

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  • Earthquake magnitude
  • Subduction zone
  • Multiple regression analysis
  • Exhaustive model evaluation
  • Plate tectonics