Non-stationary ETAS to model earthquake occurrences affected by episodic aseismic transients

Abstract

We present a non-stationary epidemic-type aftershock sequence (ETAS) model in which the usual assumption of stationary background rate is relaxed. Such a model could be used for modeling seismic sequences affected by aseismic transients such as fluid/magma intrusion, slow slip earthquakes (SSEs), etc. The non-stationary background rate is expressed as a linear combination of B-splines, and a method is proposed that allows for simultaneous estimation of background rate as well as other ETAS model parameters. We also present an extension to this non-stationary ETAS model where an adaptive roughness penalty function is used and consequently provides better estimates of rapidly varying background rate functions. The performance of the proposed methods is demonstrated on synthetic catalogs and an application to detect earthquake swarms (possibly associated with SSEs) in Hikurangi margin (North Island, New Zealand) is presented.

Introduction

Earthquakes are generated by a complex system—the Earth. Lithospheric plates gliding past one another causes stress buildup in the crustal rocks which is released in short, sudden bursts causing earthquakes. While this plate motion is the major source of earthquake activity, several other factors like the heterogeneity of the earth’s crust, the local stress conditions in the seismogenic area, fluid content, the disposition of existing faults and their stress histories, etc., also determine the size, location and time of occurrence of an earthquake. Since it is difficult to represent such a complex system physically, stochastic models are increasingly being used to model earthquake occurrences (Console et al. 2010; Helmstetter and Sornette 2002; Kagan and Knopoff 1981; Marsan and Lengline 2008; Ogata 1988, 1999). Well-established empirical laws like the Gutenberg-Richter law and the Omori’s law facilitate the formulation of stochastic models of earthquake occurrences such as the epidemic-type aftershock sequence (ETAS) model (Console 2003; Helmstetter 2003; Ogata 1988, 1998).

The ETAS model (Ogata 1988, 1998) is currently the most popular model to describe seismicity in a region and to test various hypotheses related to it. It is based on the premise that every earthquake has a magnitude-dependent ability to trigger aftershocks. The model considers that the earthquake sequence comprises aftershocks (events triggered by other earthquakes) and background earthquakes (events that occur independent of other earthquakes). The aftershocks are triggered because of internal stress adjustments in the seismogenic system initiated by the occurrence of an earthquake, while the background earthquakes are generally caused by forces related to plate tectonics. In addition, aseismic transient forces related to fluid/magma intrusion, slow slip etc., can also trigger (swarms of) earthquakes. Because such earthquakes do not follow typical mainshock–aftershock patterns, they are included in the model as background events. To analyze earthquake sequences that include such swarms which are short-lived, the usual assumption of stationary background activity needs to be relaxed. This leads to an ETAS model with non-stationary background rate, referred to as non-stationary ETAS model in this study.

The traditional epidemic-type aftershock sequence (ETAS) model (Ogata 1988, 1998) assumes the background activity to be stationary and hence a single (constant) parameter ($\mu _0$) is enough to represent it. On the contrary, a non-stationary background rate warrants representation by a continuous function $\mu (t)$ which needs to be estimated from the observed data. Some previous studies (Hainzl and Ogata 2005; Lombardi et al. 2006, 2010; Marsan et al. 2013a; Reverso et al. 2015) have presented analysis of earthquake occurrences (sequences) using ETAS models with non-stationary background rate. Initial studies such as Hainzl and Ogata (2005) and Lombardi et al. (2006, 2010) approximated the non-stationary background rate by fitting stationary ETAS model to data in moving windows. Later, Marsan et al. (2013a) adapted the iterative procedure of Zhuang et al. (2002) to the case of non-stationary ETAS model. However, these methods do not always produce a smooth solution and often result in an estimate of $\mu (t)$ that is wiggly, indicating overfit to the data. Marsan et al. (2013) described a procedure based on ETAS model to detect earthquake swarms triggered by aseismic transients. This was further extended in Reverso et al. (2015). However, the results from this method are sensitive to the choice of width of regular grid used to represent background rate. In a different approach, Llenos and McGuire (2011) estimated the background rate by modeling the aftershock activity using ETAS, and subtracting this from the total observed seismicity rate. An assumption of zero background rate is made in estimating the aftershock activity by fitting ETAS model to the complete sequence of earthquakes, and thus the estimate of $\mu (t)$ obtained by this method could be biased.

Recently, Kumazawa and Ogata (2014) adapted the hierarchical modeling approach of Ogata (2004, 2011) to non-stationary ETAS model. In this approach, the background rate function $\mu (t)$ is expressed as a piece-wise linear function made up of linear pieces between each pair of consecutive earthquake occurrence times. A two-step iterative procedure involving penalized maximum likelihood estimation (penalized MLE) and Type-II maximum likelihood method (Type-II MLE) was proposed for estimation of the piece-wise linear $\mu (t)$. In their method, however, only $\mu (t)$ is estimated while employing a priori ETAS model parameters. In their study, the ETAS model parameters were fixed to those obtained by fitting a stationary ETAS model to the earthquake catalog corresponding to a wider region around the study area. Often, such pre-determination of model parameters may not be feasible because the seismicity is localized (e.g., reservoir associated seismicity) or the model parameters are spatially inhomogeneous. Aside, owing to the strong dependence between $\mu (t)$ and other ETAS model parameters (Harte 2013; Marsan et al. 2013a), input of erroneous model parameters to the algorithm would result in biased estimates of the background rate. Therefore, a method that can simultaneously estimate both, the model parameters and the background rate, becomes desirable. Our study attempts to address this issue.

The method proposed in this study is largely based on Kumazawa and Ogata (2014). We express the non-stationary background rate function $\mu (t)$ as a spline function. Instead of using all earthquake occurrence times as knots to represent this spline function, we express it as a linear combination of finite number of basis splines (B-splines), akin to the P-splines method of Eilers and Marx (1996). Additionally, we employ the L-curve procedure (Frasso and Eilers 2015) to choose the optimal smoothing parameter in place of Type-II MLE. The penalized MLE method is then used to simultaneously invert for both the background rate $\mu (t)$ and other ETAS model parameters related to aftershock activity. Such simultaneous estimation of all unknowns in the model produces more reliable results.

Further, except for the method of Kumazawa and Ogata (2014), none of the above cited methods employ an explicit roughness penalty function to constrain the wiggliness of the estimated $\mu (t)$, and therefore, could lead to estimates that overfit the data. Even the method of Kumazawa and Ogata (2014) uses a global smoothness parameter to weigh the roughness penalty function in the expression for penalized log-likelihood. Thus, these methods are less effective to estimate the background rate function in situations where significant non-uniformity in the roughness of $\mu (t)$ exists. This situation often arises while analyzing earthquake sequences of long duration that are affected by occasional aseismic transients. To model such sequences, we therefore extend the method of Kumazawa and Ogata (2014) by representing the smoothness parameter as yet another spline function.

We explore the performance of these proposed methods on synthetic catalogs simulated using two different types of background rate functions. Further, an application of these methods to earthquake catalog from Gisborne area near Hikurangi subduction zone (New Zealand) is presented. The estimated background rate exhibits a few peaks indicating brief periods of increased seismicity. Examination of continuous global positioning system (cGPS) data recorded at nearby stations suggests that these peaks coincide with periods of reversals in recorded displacements that indicate slow slip.

The ETAS model

The epidemic-type aftershock sequence (ETAS) model was introduced by Ogata (1988) and, since then, is widely used in the analysis of earthquake sequences. It is a point process model which assumes that the earthquake sequence is made up of aftershocks and background events. Aftershocks include those events that are triggered by other earthquakes while background events are those that occur independent of other earthquakes. So, according to the model, the sequence of observed earthquakes is generated by a non-homogeneous Poisson process with rate $\lambda (t)$ expressed as

$$\begin{aligned} \lambda (t) = \mu (t) + \nu (t) \end{aligned}$$

(1)

where $\mu (t)$ and $\nu (t)$ are the rates of background and aftershock events, respectively.

Aftershock rate can be expressed in parametric form using well-known empirical characteristics of aftershocks—(a) the rate of aftershock activity triggered by an earthquake decays according to the Omori-Utsu law (Utsu et al. 1995) and (b) the total number of aftershocks triggered by an earthquake depends exponentially on its magnitude. Thus, the rate of aftershock activity triggered by an earthquake i occurring at time $t_i$ with magnitude $M_i$ is

$$\begin{aligned} \xi _i(t) = K e^{\alpha (M_i-M_0)} (t-t_i+c)^{-p} \end{aligned}$$

(2)

where $\{K, \alpha , c, p\}$ are unknown parameters and $M_0$ is the cutoff magnitude of the catalog. Note that only the earthquakes with magnitudes above the cutoff magnitude are considered for analysis using the ETAS model.

The total aftershock rate at any time t can thus be obtained as

$$\begin{aligned} \nu (t) = \sum _{i:t_i<t} K e^{\alpha (M_i-M_0)} (t-t_i+c)^{-p} \end{aligned}$$

(3)

Background rate is generally considered to be constant $\mu (t)=\mu _0$ in time. This is because, for the short time spans of observed catalogs, the effect of long-term plate tectonic forces, which are primarily responsible for background activity, can be considered to be constant. However, when aseismic transient forcings associated with slow slip earthquakes or fluid intrusion, etc., are believed to affect the observed earthquake sequence, a time-varying background rate function $\mu (t)$ is required for better modeling of data. A non-stationary ETAS model with such a time-varying background rate is considered in this study. We assume that the other model parameters are stationary over the observed time period.

Unlike aftershock rate, the background rate function $\mu (t)$ cannot be expressed in a general parametric form. Various authors have employed different ways to represent $\mu (t)$ and adopted different procedures for inversion/estimation of the unknowns as enunciated in the earlier section. For example, Kumazawa and Ogata (2014) represented the background rate as piece-wise linear function and performed estimation using an iterative procedure involving penalized maximum likelihood and Type-II maximum likelihood method. On the other hand, Hainzl et al. (2013) and Marsan et al. (2013a) used a different iterative procedure where they (a) begin with a homogeneous ETAS model and estimate its model parameters; (b) compute the probabilities of each event to be a background event (given by $\mu (t_i)/\lambda (t_i)$); (c) estimate the background rate by smoothing these probabilities; (d) use this estimated $\mu (t)$ as the new background rate and re-estimate the other model parameters. The steps (b)–(d) are repeated till convergence to obtain final estimates.

In the present study, we express the background rate $\mu (t)$ as a spline function. Unlike Kumazawa and Ogata (2014) where all the earthquake occurrence times are used as internal knots of the spline function, we employ only a fixed number M of B-spline basis functions as in P-splines methodology (Eilers and Marx 1996). That is, $\mu (t)$ is represented by

$$\begin{aligned} \mu (t) = \sum _{i=1} ^M {\phi _iB_i(t,d,\kappa _t)} \end{aligned}$$

(4)

where M is the total number of B-splines, $\phi _i$ are spline coefficients, $B_i(t,d,\kappa _t), i=1,2,3,{\ldots }M,$ are the B-spline basis functions of degree d computed over the knot vector $\kappa _t$. B-spline basis functions for any given knot vector and spline degree can be computed using the deBoor’s algorithm (De Boor 1978). The degree and knot vector of B-splines determine the flexibility of the spline function in fitting the observations. A spline function of degree d would have continuous derivatives of up to $d-1$. So, the larger the degree of spline function, the higher would be the smoothness. On the other hand, the knot vector determines the local flexibility of the spline function. The closer the knots, the more flexible would be the spline function in fitting the observations. In this study, we employ quantile spaced knots (Ruppert et al. 2003). That is, knots are chosen such that any two consecutive knots contain the equal number of events. This kind of knot placement is better suited for modeling any sudden surges in background activity compared to uniform knot placement.

Thus, the non-stationary ETAS model can be described as a non-homogeneous Poisson process with rate

$$\begin{aligned} \lambda (t) = \sum _{i=1} ^M {\phi _iB_i(t,d,\kappa _t)} + \sum _{i:t_i<t} K e^{\alpha (M_i-M_0)} (t-t_i+c)^{-p} \end{aligned}$$

(5)

The unknowns in this model are $\Phi = \{\phi _i, i=1,2,3,{\ldots },M\}$ corresponding to background activity, and other model parameters $\Theta =\{K, \alpha , c, p\}$ related to aftershock activity. Estimating these unknowns and plugging them in the above expression would give us the non-stationary ETAS model that best describes the observed earthquake sequence.

Estimation

In the simpler case of homogeneous ETAS model, where $\mu (t)=\mu _0$, the unknowns are only $\{\mu _0, K, \alpha , c, p\}$. They can be estimated using maximum likelihood estimation (MLE) method by maximizing the log-likelihood function $\log {L}$ (Ogata 1998)

$$\begin{aligned} \log {L} = \sum _{\{i:S<t_i<T\}} \log {\lambda (t_i)} -\int _S^T{\lambda (t)}\hbox {d}t \end{aligned}$$

(6)

where [S, T] is the time domain containing the observations.

However, simple maximum likelihood method does not provide good estimates in case of a non-stationary ETAS model. The MLE estimates would be such that the background rate overfits the data. To avoid this, a roughness penalty has to be applied to constrain the wiggliness of the estimated $\mu (t)$. Thus, the model unknowns $\Phi$ and $\Theta$ are estimated by penalized maximum likelihood estimation (penalized MLE) method. Estimated $\hat{\Phi }$ and $\hat{\Theta }$ are those that maximize the penalized log-likelihood objective (Kumazawa and Ogata 2014)

$$\begin{aligned} R(\Phi , \Theta , \tau ) = \log {L(\Phi ,\Theta )}-\tau \times Q(\Phi ) \end{aligned}$$

(7)

where $\tau$ is the regularization or smoothing parameter and $Q(\Phi )$ is the roughness penalty function. The roughness penalty is generally considered in the form of integrated squared mth-order derivative of the desired function

$$\begin{aligned} Q(\Phi ) = \int _S ^T[{\mu ^{(m)}(t)}]^2\hbox {d}t \end{aligned}$$

(8)

When $\mu (t)$ is expressed in terms of B-spline basis, the penalty function can be conveniently expressed as

$$\begin{aligned} Q(\Phi ) = \Phi ^{\prime }P\Phi \end{aligned}$$

(9)

where the penalty matrix P is a symmetric matrix with elements

$$\begin{aligned} P_{ij} = \int _S ^T {B_i^{(m)}(t,d,\kappa _t)B_j^{(m)}(t,d,\kappa _t)}\hbox {d}t \end{aligned}$$

(10)

The smoothing parameter $\tau$ in Eq. (7) controls the relative contribution of goodness-of-fit criterion (here log-likelihood) and the roughness penalty function in determining the values of estimated parameters. Large $\tau$ values in a penalized MLE lead to over smoothed estimates of $\mu (t)$, while a small τ results in a $\mu (t)$ which is under smoothed. It is therefore important to employ a $\tau$ that provides optimal smoothing.

Choosing optimal smoothness parameter

Consider the penalized log-likelihood objective function given in Eq. (7). The penalty function is used to regularize the inversion and the smoothness parameter $\tau$ plays the role of a regularization parameter. The purpose of the penalty function is to provide additional constraints to the inversion to impart stability.

As the penalty function depends only on $\Phi$, it thus provides constraints for the parameters in $\Phi$ alone. Depending on the choice of the order m of roughness penalty, the penalty matrix P could be rank deficient. For a ridge penalty ($m=0$), the penalty matrix would be full rank ($r= M$); for a penalty on first-order derivatives ($m=1$), the penalty matrix would have a rank $M-1$, and so on. In general, for a penalty on the mth-order derivative, the penalty matrix would be rank deficient by m and would have a rank of $r=M-m$. This rank deficiency implies that the penalty matrix cannot simultaneously constrain all the parameters $\phi _i$ in $\Phi$; only $r = M-m$ of them would be constrained. In brief, of the $M+4$ number of unknown parameters ($\Phi$ of dimension M and $\Theta$ of dimension 4), there exist $M-m$ constrained parameters, say $\Phi _f=\{\phi _{i}, i=1,2,3,{\ldots }M-m\}$, and $4+m$ unconstrained parameters $\Theta \cup \Phi _d$ where $\Phi _{d}=\{\phi _{i}, i=M-m+1,{\ldots },M\}$. Apart from these, the optimal smoothing parameter $\tau$ also has to be estimated. Kumazawa et al. (2016) assumed a priori $\Theta$ and were hence left only with constrained parameters $\Phi _f$ and the other unknowns $\eta = \{\Phi _d, \tau \}$. They employed the penalized MLE step to estimate $\Phi _f$ and the Type-II MLE to estimate $\eta$.

Type-II maximum likelihood estimation

Consider the expression for penalized log-likelihood given in Eq. (7). From the Bayesian perspective, applying a roughness penalty to the log-likelihood function is equivalent to putting a prior on the variables. To understand this better, let us exponentiate both sides of Eq. (7)

$$\begin{aligned} e^{R(\Phi , \Theta , \tau )} = e^{\log {L(\Phi ,\Theta )}}\times e^{-\tau \times Q(\Phi ,\tau )}. \end{aligned}$$

(11)

In the above equation, $e^{-\tau \times Q(\Phi ,\tau )}$ corresponds to prior, $e^{\log {L(\Phi ,\Theta )}}$ is the likelihood and $e^{R(\Phi , \Theta , \tau )}$ is proportional to the posterior. Note that estimating the parameters that maximizes the penalized log-likelihood objective is same as finding the mode of the above posterior. Thus, penalized maximum likelihood estimation is equivalent to maximum a posteriori (MAP) estimation.

The prior in expression (11) is improper because of the mentioned rank deficiency of the penalty matrix. However, the penalty matrix has rank $M-m$ and thus the prior is proper on the $M-m$ parameters $\Phi _f$. So, we can normalize this part of the prior as $e^{-\tau \times Q(\Phi _f)}/\int {e^{-\tau \times Q(\Phi _f)}}$ to obtain a prior, which is a proper probability density function. Using this, the posterior $T(\Phi _f, \eta )$ can be written as

$$\begin{aligned} T(\Phi _f, \eta ) = L(\Phi ,\Theta )\times \frac{ e^{-Q'(\Phi _f, \tau )}}{\int {e^{-Q'(\Phi _f, \tau )}}} \end{aligned}$$

(12)

where $Q'(\Phi _f,\tau )= \tau \times Q(\Phi _f)$ is the roughness penalty scaled by the smoothing parameter $\tau$.

Note that there is no prior on the parameters $\eta$. They are treated as hyperparameters and are estimated by maximizing the posterior that is marginalized over $\Phi _{f}$ as

$$\begin{aligned} {\Lambda (\eta )} = {\int {T(\Phi _f, \eta )}\hbox {d}\Phi _f} \end{aligned}$$

(13)

This procedure of estimating hyperparameters by maximizing the marginalized likelihood is called the Type-II maximum likelihood procedure (Kumazawa and Ogata 2014; Ogata 2011). This procedure is also known as empirical Bayes (Bishop 2006). The Akaike Bayesian Information Criterion (ABIC) is equal to $-\,2 \times \max \log {\Lambda (\eta )} + \dim (\eta )$. Thus, estimation using Type-II MLE is equivalent to choosing a model that minimizes the Akaike Bayesian Information Criterion (ABIC).

Computing the marginalized posterior involves integration over $\Phi _f=\{\phi _{i}, i=1,2,3,{\ldots }M-m\}$ which has large dimensionality. This integration is difficult to compute in practice; so, Laplace approximation is used to approximate the posterior by a Gaussian distribution and thereby simplify this integration. Using Laplace approximation, the logarithm of marginal likelihood can be written as (Ogata 2011)

$$\begin{aligned} log{\Lambda (\eta )} = R\left( \hat{\Phi }_f|\eta \right) - \frac{1}{2} \log {\det {H_{R}\left( \hat{\Phi }_f|\eta \right) }} + \frac{1}{2} \log {\det {H_{Q'}\left( \hat{\Phi }_f|\eta \right) }} \end{aligned}$$

(14)

where $H_{R}(\hat{\Phi }_f|\eta )$ and $H_{Q'}(\hat{\Phi }_f|\eta )$ denote the Hessians of penalized log-likelihood and roughness penalty, respectively, evaluated at $\hat{\Phi }_f$ corresponding to the peak of penalized log-likelihood function. These $\hat{\Phi }_f$ are generally unknown; in fact, it is our aim to estimate these. However, both $\Phi _f$ and $\eta$ can be estimated by an iterative procedure consisting of two steps: (a) given $\eta$, $\Phi _f$ is estimated by penalized MLE and (b) using this estimate $\hat{\Phi }_f$, $\eta$ is estimated using Type-II MLE by maximizing the approximated marginal likelihood objective given in Eq. (14). $\Phi _f$ and $\eta$ are re-estimated in each iteration, until convergence. This iterative procedure was adopted by Kumazawa and Ogata (2014) to estimate $\Phi _f$ and $\eta = \{\Phi _d, \tau \}$ given a priori known $\Theta$. Even in the case when $\Theta$ are unknown, this iterative procedure can be used to estimate $\Phi _f$ (step (a)) and redefined $\eta = \{\Phi _d, \Theta , \tau \}$ (step (b)) as was done by Ogata (2011) in the context of spatially inhomogeneous ETAS. However, in practice, the results obtained using such an approach remain unsatisfactory, because the strongly dependent parameters $\Phi$ and $\Theta$ are estimated in two different steps. Thus, we propose to use the L-curve procedure of Frasso and Eilers (2015) to choose the optimal $\tau$ and then use penalized MLE to simultaneously estimate all model unknowns $\Theta$ and $\Phi$.

L-curve method

It is well known that a small smoothing parameter $\tau$ results in a $\mu (t)$ which is wiggly (under smoothed), while a large $\tau$ gives an estimate of $\mu (t)$ that is over smoothed. Therefore, a good strategy is to perform penalized MLE over a range of $\tau$, from small to large values, and then pick the optimal $\tau$. When the roughness penalty values corresponding to the estimates of $\Phi$ and $\Theta$ at each $\tau$ are plotted against the respective negative log-likelihood values, an approximate L-shaped curve is obtained. The vertical part of the curve is associated with $\tau$ values that provide under smoothed solutions and the horizontal part with over smoothed solutions. Hence, the $\tau$ value corresponding to the corner of the L-curve can be chosen as the optimal smoothness parameter $\hat{\tau }$ (Frasso and Eilers 2015). The estimates from penalized MLE corresponding to this $\hat{\tau }$ are the optimal estimates of $\hat{\Phi }$ and $\hat{\Theta }$. This procedure is similar to the L-curve method for choosing the optimal regularization parameter in ill-posed inverse problems (Hansen 1999; Sen and Stoffa 2013).

Adaptive roughness penalty function

The above-described combination of penalized MLE and L-curve method may work well to effectively describe earthquake occurrences in most cases. However, in situations where the background rate function has significant non-uniform roughness over its time domain, use of a global smoothness parameter as indicated in the proposed method could be insufficient to model the earthquake sequence effectively. Quantile knots can take care of such variable smoothness to some extent. But, this is not always sufficient. To obtain better results in such cases, an adaptive penalty function is needed that allows local variations in roughness. In this study, such an adaptive penalty function is obtained by expressing the smoothness parameter as another spline function (Baladandayuthapani et al. 2005) defined over $M_\tau (<< M)$ B-spline functions as

$$\begin{aligned} \tau (t) = \sum _{i=1} ^{M_\tau } {\tau _i}B_i(t,\kappa _\tau ,d_\tau ) \end{aligned}$$

(15)

where $\tau _i$, $d_\tau$ and $\kappa _\tau$ are the spline coefficients, degree and sub-knots corresponding to the smoothness parameter function $\tau (t)$. A small subset of background rate knots ($\kappa _t$) are chosen as sub-knots ($\kappa _\tau$) for $\tau$. Thus, the penalized log-likelihood function in Eq. (7) would now be

$$\begin{aligned} R(\Phi , \Theta , \tau ) = \log {L(\Phi ,\Theta )}-\int _S ^T\tau (t)[{\mu ^{(m)}(t)}]^2\hbox {d}t \end{aligned}$$

(16)

where $\tau (t)$ is given by Eq. (15).

Since there are now more than one smoothness parameters (the spline coefficients $\tau _i$), the L-curve methodology cannot be applied. However, the iterative procedure involving penalized MLE and Type-II MLE procedure (Kumazawa and Ogata 2014) described previously can be used to estimate the spline coefficients $\{\tau _i, i=1,2,{\ldots },M_\tau \}$ by including all of them in the hyper parameters vector $\eta$. Note that this method involving Type-II MLE gives reliable results only in the case when $\Phi$ is estimated given known $\Theta$. Since model parameters are not always known beforehand, estimates are first obtained using the penalized MLE and L-curve approach described above. As we shall see in the section on synthetic tests, these are reasonably close to the true values. Thus, for estimation using the adaptive penalty approach, we first estimate (a) $\hat{\Theta }_L$, $\hat{\Phi }_L$ and $\hat{\tau }_L$ using penalized MLE and L-curve approach. (b) Set $\Theta =\hat{\Theta }_L$ (c) with $\Phi = \hat{\Phi }_L$ and $\tau _i = \hat{\tau }_L$, estimate $\hat{\tau }(i)$ and $\hat{\Phi }_A$ using the iterative procedure involving Type-II MLE (d) on convergence of the iterative procedure, use the newly estimated $\hat{\tau }(t)$ and perform a final penalized MLE step to re-estimate both $\Phi$ and $\Theta$. In the case of a rapidly varying background rate function, as demonstrated in the following section, this adaptive procedure provides estimates of background rate that are much better than those obtained using a global smoothness parameter.

Synthetic tests

In this study, the purpose of testing is to learn if the combination of penalized MLE and the L-curve procedure is capable of determining all the model unknowns with reasonable accuracy. In addition, it is necessary to check whether all the model unknowns, $\mu (t)$ and $\Theta$, are identifiable. That is, can all the unknowns be determined uniquely from a given combination of model and data? Non-identifiability would arise, say, if certain amount of background activity can be described as aftershock activity and vice versa. It is therefore imperative to determine existence of non-identifiability and asses its impact on the results. In order to achieve this, we additionally perform estimation where we invert only for $\mu (t)$ (or equivalently $\Phi$) while constraining $\Theta$ to their true values. Large discrepancy in the estimated background rates obtained from the proposed simultaneous inversion scheme and that through constraining $\Theta$ is a reasonable indicator of non-identifiability. In such a case, the model parameters $\Theta$ and $\Phi$ cannot both be determined from the data and further constraints become necessary to resolve the model parameters. Additionally, results from the proposed methods are compared with those obtained from Type-II MLE approach described above. Various approaches tested using synthetic catalogs are listed in Table 1 for ease of reference.

Table 1 List of different types of models considered in the current study

Synthetic catalogs are simulated using non-stationary ETAS model with (a) a Gaussian-type background rate and (b) an Omori’s law-type background rate (Kumazawa and Ogata 2013; Marsan et al. 2013a). Both the background rate functions have an expected value of 500 over the time period of simulation. For each of these background rate functions and with a typical set of model parameters $\Theta _{\mathrm{true}}=\{K=0.008\,\hbox {events/day}, \alpha =2,\, c=0.01\,\hbox {day},\, p=1.1 \}$, 100 synthetic datasets are simulated over a time period of [0, 500] days. The synthetic catalogs are simulated as a branching process (Zhuang and Touati 2015). First, the background events are simulated using the given non-stationary $\mu (t)$ function by thinning method. Then, for each background event, aftershock sequences are simulated. Time sorted sequence of all these events combined together forms a simulated catalog. Magnitudes of the events are generated to follow Gutenberg-Richter law with b-value one and their values between 2 and 8, using the inverse transform method (Felzer et al. 2002; Zhuang and Touati 2015). The number of earthquakes in each synthetic catalog simulated using Gaussian-type background rate function ranges from 664 to 3597 among which $93\%$ of the catalogs contain less than 1200 events. On the other hand, the number of earthquakes in the catalogs simulated using Omori’s law-type background rate lies between 651 to 5142 with $92\%$ of the catalogs contain less than 1200 events. The number of background events in the catalogs range between [453,556] and [427,554] for the Gaussian-type and Omori’s law-type synthetic datasets, respectively.

For all the non-stationary ETAS models tested in this study, we employ 100 linear B-splines to represent background rate and a roughness penalty on the first-order derivatives $(m=1)$. For the adaptive penalty method, we express the smoothness parameter $\tau$ as a piece-wise constant spline function on 10 sub-knots. Wherever necessary, L-curves are computed using a range of log-spaced smoothing parameters $\{10^{-4}, 10^{-3.5},{\ldots },10^{7.5}, 10^{8}\}$. Since L-curve procedure involves performing penalized MLE for a large number of $\tau$ values, we did not estimate L-curves for all the 100 synthetic catalogs as that would be computationally expensive. Because all of 100 catalogs (of each type) are simulated using the same background rate function, optimal $\tau$ estimated for one typical catalog would be approximately valid for all of them. Thus, we estimated optimal $\tau$ by computing L-curve only for one catalog of each type (see Figs. 1a, 2a) and used this $\tau$ value as the approximate optimal $\tau$ for all the 100 synthetic catalogs of the corresponding type of background rate function. We provide this optimal $\tau$ value as the initial $\tau$ value even for the models employing Type-II MLE (NS_TII_MP and NS_TII). Note that, for all the algorithms, we provide $\Theta =\Theta _{true}$ and $\phi _i = 1 \forall i$ as the initial values to the algorithm except for the model with adaptive penalty. For the adaptive penalty model, we provide the final estimates of $\Phi$, $\Theta$ and $\tau$ from NS_L_MP model as the initial values.

Fig. 1

Results for the synthetic datasets simulated with a Gaussian background rate function. The L-curves computed for one typical synthetic dataset along with the point corresponding to the chosen optimal smoothness parameter are shown in (a). The estimated background rate function for all 100 synthetic catalogs for each type of model listed in Table 1 is presented in (b-f). The true background rate function and the 10, 90 percentile bounds of estimates for each model are also shown

Fig. 2

Results for the synthetic datasets simulated with a Omori’s law-type background rate function. The L-curves computed for one typical synthetic dataset along with the point corresponding to the chosen optimal smoothness parameter are shown in (a). The estimated background rate function for all 100 synthetic catalogs for each type of model listed in Table 1 is presented in (b–f). The true background rate function and the 10, 90 percentile bounds of estimates for each model are also shown

Figure 1b–f shows the background rates for all the 100 Gaussian-type synthetic catalogs, estimated using the non-stationary ETAS models listed in Table 1. It can be seen that for the simple case of Gaussian background rate, all the models are capable of providing good estimates of $\mu (t)$. Background rates estimated with models using a priori known model parameters (see Fig. 1c, e) and the models that allow simultaneous estimation along with $\mu (t)$ (see Fig. 1b, d) are nearly the same. This indicates that no significant non-identifiability exists and thus simultaneous estimation of both the model parameters and the background rate function $\mu (t)$ is feasible. Note that a few of the estimates obtained via adaptive penalty method (see Fig. 1f) show small block like distortions. This is caused by the usage of piece-wise constant $\tau$ function; over smoothing seems to occur in some windows. So, the adaptive penalty method is not very helpful in the case where the background rate is not rapidly varying (like the Gaussian background rate function).

Fig. 3

Box plots of estimated model parameters for the Gaussian synthetic datasets. The box plots of model parameters estimated for the 100 synthetic catalogs using each of the models NS_L_MP, NS_TII_MP and NS_Adapt are shown. The true values of the respective parameters are plotted as horizontal lines

Fig. 4

Box plots of estimated model parameters for the Omori’s law-type synthetic datasets. The box plots of model parameters estimated for the 100 synthetic catalogs using each of the models NS_L_MP, NS_TII_MP and NS_Adapt are shown. The true values of the respective parameters are plotted as horizontal lines

In contrast, the results in the case of Omori’s law-type background rate function indicate that the adaptive penalty method provides considerably better estimates of background rate. Estimates using the method employing L-curve approach with a global $\tau$ exhibit undesired wiggles (see Fig. 2b, c), whereas the estimates employing Type-II MLE approach seem to over smooth (see Fig. 2d, e), especially near the region with sudden jump. But using an adaptive penalty seems to damp the undesired wiggles (see Fig. 2f) and provide smoother estimates even while preserving the jump. So, the adaptive penalty method seems to be very helpful in this case of Omori’s law-type background rate function whose roughness is much more rapidly varying than the Gaussian background rate function.

Table 2 Estimates of the ETAS model aftershock parameters ($\hat{\Theta }$) for the earthquake occurrences in Gisborne region for the period 2012/01–2015/05

Box plots of the estimated model parameters $\Theta$ using Models NS_L_MP, NS_TII_MP and NS_Adapt are shown in Figs. 3 and 4, respectively, for catalogs simulated using Gaussian-type and Omori’s law-type background rates. Overall, both the proposed models NS_L_MP are NS_Adapt provide considerably better estimates than the model NS_TII_MP that employs Type-II MLE. Between these, the NS_Adapt model provides slightly better estimates as is evident by the closeness of the median values of the estimates to the true values, and lower interquartile ranges of the estimates. The difference is particularly prominent for the Omori’s law-type synthetic catalogs. Observe that the estimates of parameter p obtained with the model NS_TII_MP are close to the true value and have a lower interquartile range as well, even while the estimates of other model parameters are bad. This is possibly due to the inability of Type-II MLE step in exploring values away from the initial value (= true value) provided.

Fig. 5

Background rates of each kind of synthetic catalogs estimated using different number of splines ${{{M}}}$ and ${{{M}}}_{{\tau }}$. The estimated background rate function using all possible combinations of $M=\{50,100,150,200,250,300\}$ and $M_{\tau }=\{10,20,30,40,50,60\}$ such that $M_{\tau }<M$, estimated using the models NS_L_MP and NS_Adapt for a typical Gaussian synthetic catalog are, respectively, shown in (a, c). These plots for a typical Omori’s law-type synthetic catalog are presented in (b, d). The true background rate functions (dotted line) are also shown

Fig. 6

Seismicity Map of North Island, New Zealand and the target study area. Epicenters of earthquakes with magnitudes $M\ge 2.5$ in the North Island (New Zealand) region with depths shallower than 65 km for the period 2012/01 to 2017/5 selected from the GeoNet catalog. The contours (in mm/year) show the cumulative slip of all detected SSEs on the Hikurangi subduction interface since 2002 until 2012, b taken from Wallace et al. (2012). Only the earthquakes with epicenters in the spatial window shown are considered for analysis in this study. Also displayed in the figure are the locations of cGPS station whose data are examined in the present study to investigate the association of slow slip earthquakes with increased seismicity

The results reported above were computed using $M=100$ B-splines to represent the background rate function $\mu (t)$ and $M_{\tau } = 10$ B-splines for the smoothness parameter $\tau (t)$. It is well known that the estimates obtained using the P-splines methodology are nearly independent of the number of splines used, provided they are sufficiently large in number to produce an overfitting estimate in the absence of any roughness penalty (e.g., Baladandayuthapani et al. 2005; Ruppert 2002). To test if this remains valid even for the methods described in the current study, we analyze a typical synthetic catalog of each type, using all possible combinations of $M=\{ 50,100,150,200,250,300\}$ and $M_{\tau }=\{10,20,30,40,50,60\}$ such that $M_{\tau } < M$. The estimates of the background rate function $\mu (t)$ thus obtained for each type of synthetic catalog using the proposed models M_L_MP and M_Adapt are shown in Fig. 5. It can be seen that all these combinations of M and $M_{\tau }$ yield similar estimates (see Fig. 5; Additional file 1: Tables S1, S2, S3 and S4 **in supporting information). Thus, these results suggest that the estimates obtained from the proposed methods are not very sensitive to the choice of the number of splines used to represent the background rate and smoothness parameter. Although use of large number of splines M and $M_{\tau }$ may produce better results, the attendant computational cost increases steeply, especially with $M_{\tau }$. Therefore, the choice of M and $M_{\tau }$ must be guided by the prior knowledge/expectation of the underlying background rate function $\mu (t)$. In any case, it would be prudent to examine the estimates using few other values of M and $M_{\tau }$ to confirm their stability.

In the current study, we chose quantile based knot vector, for spline representation of $\mu (t)$, using all the earthquakes in the catalog. Therefore, more knots are inadvertently chosen even in the places where background rate is low but where aftershock activity is large. This can, in some cases, cause a small part of aftershock activity to be seen as background activity in the estimated model, owing to greater flexibility accorded by closely space knots. This is the cause of the occasional spurious bumps seen in the estimated background rate of synthetic catalogs. We observed that this issue chiefly arises when the analyzed catalogs contain large aftershock sequences and the background activity is such that a low smoothness parameter is needed. Using a low smoothness parameter $\tau$ further adds to the flexibility allowed by closely spaced knots and therefore can cause local overfitting which results in spurious bumps. Adaptive penalties can alleviate this problem to a major extent. But, we speculate that devising and using a better knot selection strategy would also produce better and improved results.

Application to New Zealand data

We examine the GeoNet earthquake catalog of northern Hikurangi margin (New Zealand) and the cGPS data available for the region. The main objective is to demonstrate that the non-stationary ETAS model developed in this study is capable of identifying the transient increases in seismicity associated with slow slip earthquakes.

Slow slip earthquakes (SSEs) have been observed at many subduction zones around the world. They have been found to accompany with tremors in most cases, and thus together they have been called Episodic Tremor and Slip (Rogers and Dragert 2003; Schwartz and Rokosky 2007). Some of the slow slip earthquakes have also been found to trigger earthquake swarms (Delahaye et al. 2009; Hirose et al. 2014). On the other hand, some large earthquakes have also been believed to affect slow slip (Wallace et al. 2014; Zigone et al. 2012). It is necessary to understand such interactions between normal earthquakes and slow slip earthquakes in order to better appraise the seismic hazard potential of such regions.

At the Hikurangi margin, the Pacific plate is obliquely subducting beneath the Australian plate. Slow slip here occurs at a shallow depth similar to Boso peninsula, Japan. At both these subduction zones, the slow slip earthquakes have been found to trigger earthquake swarms. Previous studies have identified earthquake swarms triggered by a few slow slip earthquakes in Hikurangi margin e.g., Gisborne 2004 SSE (Delahaye et al. 2009), Cape Turnagain 2011 SSE (Wallace et al. 2012), Puketiti 2010 SSE (Todd and Schwartz 2016). Here, we analyze the observed earthquakes near Gisborne (see Fig. 6), where multiple SSE have been documented, using the non-stationary ETAS model. From the estimated background rate, we would try to see if any anomalous seismicity is associated with slow slip earthquakes in the region.

Fig. 7

Results for the GeoNet catalog analyzed using non-stationary ETAS model. The magnitudes of earthquakes in the catalog examined in the study are plotted versus time in (a). The computed L-curve along with the point corresponding to the chosen smoothness parameter is shown in (b). The background rate functions estimated using the models NS_L_MP and NS_Adapt are presented in (c, d) respectively, along with the error bounds computed using Hessian matrix of $\Phi$ at the solution

Fig. 8

Association of slow slip earthquakes and the peaks in the estimated background rate. The east component of the displacement recorded by the LEYL, WAHU, MAHI, MAKO and ANAU cGPS stations are shown (a–e) and compared with the background rate (f) estimated using the non-stationary ETAS model with adaptive roughness penalty using $M=150$ and $M_{\tau }=30$. Note that the peaks in the estimated background rate correspond well with the reversals in the displacement that indicate slow slip earthquakes. Also shown in (g) are the estimates obtained using different values of M and $M_{\tau }$, similar to Fig. 5

The location algorithm and the reported magnitudes in GeoNet catalog changed since the beginning of 2012 (www.geonet.org.nz/data/supplementary/earthquake_location). Thus, for the analysis in this study we consider only the earthquake catalog since January 2012 until May 2017. The magnitude of completeness for the events in the target spatial window and the time period is estimated to be 1.9 M using maximum curvature method (Wiemer 2001). Thus, the non-stationary ETAS model is applied to 5007 events with magnitudes $M \ge 1.9$ (see Fig. 7a) and with depths less than 65 km, and whose epicenters are confined to the spatial window shown in Fig. 6. Given the large number of events in the catalog and the frequent occurrence of SSEs that could affect the background seismicity in the region, we chose to use ($M=$) 150 linear B-splines to express the background rate function $\mu (t)$. A roughness penalty in terms of first-order derivative $(m=1)$ is used. For modeling with adaptive penalty, the smoothness parameter $\tau (t)$ is expressed as a piece-wise constant spline made up of 30 B-splines.

Fig. 9

Spatial locations of earthquakes associated with peaks in the estimated background rate. Epicenters of earthquakes corresponding to each seismicity peak identified in Fig. 8e are plotted. These scatter plots are visually enhanced by adding color based on smoothed histogram as described in Eilers and Goeman (2004). Also plotted, where available, are the slip regions of the associated slow slip earthquakes. Distinct spatial zones where prominently identifiable seismicity clusters seem to occur are shown as colored rectangles (red, blue and green)

The computed L-curve is shown in Fig. 7b, where the point corresponding to the chosen optimal smoothness parameters is marked. Estimated background rate function $\hat{\mu }(t)$, corresponding to this optimal ${\hat{\tau}}$, is wiggly (see Fig. 7c) and thus the adaptive penalty approach as described above is applied. The background rate (shown in Fig. 7d) thus estimated looks reasonably better than the one estimated without adaptive penalty. The estimated model parameters $\Theta$ corresponding to both the non-stationary ETAS models, with and without adaptive penalty, along with the standard errors are presented in Table 2. Note that the standard errors are computed as square root of diagonal elements of covariance matrix. The covariance matrix is estimated by taking inverse of the Hessian matrix at the solution. For goodness-of-fit tests, see supplementary material.

To see if slow slip earthquakes are associated with increased seismicity that manifests as peaks in the estimated background activity, we look at cGPS data recorded at the nearby cGPS stations LEYL, WAHU, MAHI, MAKO and ANAU (see Fig. 8). Slip caused by slow earthquakes is recorded by continuous GPS (cGPS) stations as reversals in the direction of slip over time periods ranging from a few days to years. Comparing the estimated background rate function with the east component of cGPS data recorded at the said cGPS stations, it is apparent that the peaks in estimated background rate are associated with deviations in cGPS east-west displacement time series indicative of slow slip earthquakes (see Fig. 8).

To check the sensitivity of the estimated background rate (shown in Fig. 8f) on the number of splines employed, we repeat the analysis described for the synthetic catalogs in the previous section, using different combinations of M and $M_{\tau }$. The estimates of $\mu (t)$ thus obtained using the non-stationary ETAS model with adaptive penalty are shown in Fig. 8g and the corresponding model parameters in Table S5. These suggest that the considered combinations of M and $M_{\tau }$ produced similar estimates and reconfirm the presence of peaks in the background rate observed earlier (see Fig. 8).

It is pertinent to note that not all SSEs are associated with peaks in the background rate. For example, SSE in October 2014 do not seem to produce any pronounced increase in seismic activity, although there seems to be a faint peak in $\mu (t)$ (Fig. 8g) associated with this event. Lack of such recognizable increase in seismicity associated with this particular SSE was also observed by Todd and Schwartz (2016). They explain that as the SSE occurred far away from the shore, the associated increase in seismicity, if any, was not recorded by the onshore network. Employing a better catalog with a lower magnitude cutoff could possibly help identify more earthquake swarms associated with SSEs.

We plot the epicenters of the earthquakes that occurred during the time windows (shaded regions) corresponding to each of the peaks shown in Fig. 8. These scatter plots of earthquake locations (shown in Fig. 9) are visually enhanced adding color to individual plots based on smoothed histogram as described in Eilers and Goeman (2004). Such visual enhancement leads to better identification of spatial clusters of earthquakes. In each of the plots (Fig. 9), corresponding to peaks 2, 3, 4, 5, 7, 9, 11 and 12, distinct spatial clustering of earthquakes is visible. Approximate regions of slow slip available for a few SSEs (Koulali et al. 2017; Wallace and Eberhart-Phillips 2013; Wallace et al. 2017) are shown in the corresponding plots. It can be noticed that the seismicity clusters related to peaks 5 and 11 are located near the down dip edges of the corresponding slip patches (Fig. 9e, k). However, the earthquake cluster in Fig. 9k seems to be located within the slip region while that in Fig. 9e lies even outside the slip region. While the former lends support to the hypothesis that slow slip triggers earthquakes on locked asperities within the slip zone, the latter suggests triggering by increased static stress. Examination of the subplots in Fig. 9 suggests that the distinct spatial clusters of earthquakes associated with SSEs occupies three narrow spatial zones marked by colored rectangles (R, G and B). Spatial zone R (Fig. 9) coincides with a region where slow slip earthquakes have been detected in earlier studies (e.g., Koulali et al. 2017; see also contours in Fig. 6). Hence, it is possible that the earthquake swarms in this region are mainly caused by failure of locked asperities within the slip region. In contrast, earthquake clusters in spatial zone G are located inland, while slow slip occurrence region is situated almost exclusively offshore (Todd and Schwartz 2016). Thus, the earthquake swarms falling in zone G, if triggered by SSEs, must have been caused by the associated increase in static stress. Previous studies on SSEs near Gisborne, that is spatial zone B, found that the associated swarms of earthquakes occur close to the down dip edge of the slip area (Bartlow et al. 2014; Delahaye et al. 2009). Thus, these swarms in spatial zone B are most likely trigged by static stress increase (Delahaye et al. 2009) akin to region G. Irrespective of the triggering mechanisms, it is important to understand if these spatial zones are particularly susceptible to SSE-associated seismicity. Detailed modeling of slow slip events and estimation of associated Coulomb stress changes might shed more light in this direction. In addition, if epicentral locations too are modeled along with the earthquake occurrence times, using a non-stationary spatio-temporal ETAS model, spatial clusters of earthquakes associated with SSEs could be better identified.

Conclusions

In this study, we propose a P-splines-based non-stationary ETAS model and an estimation procedure involving (a) penalized maximum likelihood estimation and (b) L-curve method for choosing optimal smoothness parameter. This procedure allows for simultaneous estimation of both background rate function and the other ETAS model parameters. Such a non-stationary ETAS model is useful in modeling earthquake sequences affected by time varying processes such as fluid/magma intrusion. For example, modeling the earthquake sequence associated with a particular swarm would help in understanding the time evolution of that swarm and could provide useful insights into the underlying causative process. We also present a non-stationary ETAS model that employs adaptive roughness penalty function. Such a model provides superior results when the background rate has significant non-uniform smoothness over its domain. This adaptive penalty method is particularly useful when analyzing long duration earthquake catalogs belonging to regions affected by occasional aseismic transients. The performance of both the proposed methods was demonstrated on synthetic datasets. An application to data from Hikurangi margin (New Zealand) is presented where the observed earthquake sequence near Gisborne is analyzed to find instances of increased background rate (earthquake swarms). These episodes of increased seismicity are then compared with cGPS data to better understand their association with slow slip earthquakes. The non-stationary ETAS model and the estimation procedures described in this study allow us to model earthquake activity affected by transient aseismic processes and thus allow us to obtain meaningful insights into these processes.