Marked spatio-temporal point patterns, periodicity analysis and earthquakes: an analytical extension including hypocenter depth

Abstract

The longitude, latitude and depth of the hypocenter in 3-D space and the date and time of rupture makes an earthquake a “point” in a spatio-temporal point pattern, observed over a region and months, years or decades. The magnitude of earthquakes marks the point pattern, as would hypocenter depth do if only the longitude and latitude of epicenters were used for location in 2-D space. Stochastic declustering, based on a space–time ETAS model (ETAS: epidemic-type aftershock sequence), is a procedure that can be applied in the preliminary stage of an earthquake catalog data analysis. Stochastic declustering procedures have underlying assumptions, such as the time independence of the background intensity function whether the spatial framework is 2-D or 3-D, and a separate treatment of hypocenter depth from longitude and latitude when the spatial framework is 3-D. Cyclical processes in the Earth, including tides and seasonal surface water loads, can introduce periodic behavior in earthquake occurrence and related variables. The effects of ETAS-based 2-D and 3-D declustering on the outcome of periodicity analyses performed from the resulting earthquake data catalogs are studied. The research objectives and statistical challenges include the detection of periodicities for hypocenter depth in addition to monthly earthquake number, and the risk of missing observations for hypocenter depth when the monthly earthquake number after declustering is zero. A version of the method of multi-frequential periodogram analysis (MFPA) that allows for missing observations in the input temporal series is presented in detail, and applied to hypocenter depth (monthly mean and median) for central and northern California from January 2006 to December 2014. The results obtained for the 2-D and 3-D declustered catalogs are compared with those for the original catalog for this region. A semiannual periodicity in hypocenter depth is detected for the original and 2-D declustered catalogs, and fitted with the goal of relating it to periodicities found in the time series of monthly earthquake numbers. Using these results for central and northern California earthquakes, some of the assumptions on the intensity function of the spatio-temporal point process in stochastic declustering are discussed and future research perspectives are proposed.

Appendices

Appendix 1

1.1 Equations and statistical assumptions in the 2-D ETAS-based declustering procedure

In Zhuang et al. (2002), the conditional intensity function of the spatio-temporal point process proposed to model clustering in seismic activity, given the space–time magnitude occurrence history ($\mathscr{H}_{t}$) of the earthquakes up to time t, is written

$$\lambda (t,x,y,M|{\mathscr{H}}_{t} ) = \mu \left( {x,y,M} \right) + \sum\nolimits_{{\{ a,t_{a} < t\} }} {\kappa (M_{a} )g(t{-}t_{a} )f(x{-}\xi_{a} ,y{-}\eta_{a} |{M^{*}}_{a} )j(M|{M^{*}}_{a} )} ,$$

(1)

where subscript a is used for an ancestor and $\kappa (M_{a} )g(t{-}t_{a} )f(x{-}\xi_{a} ,y{-}\eta_{a} |{M^{*}}_{a} )j(M|{M^{*}}_{a} )$ represents the intensity function of the nonstationary Poisson process associated with the ath ancestor, under the following four “assumptions (a)–(d)”:

• a. The occurrence rate of background events is assumed to be a function µ(·) of spatial location of the epicenter and magnitude, but not of time.

• b. Each “ancestor” event produces “offspring” independently and the expected number of direct offspring from an individual ancestor is assumed to depend on its magnitude M and is denoted κ(M).

• c. The probability distribution of the amount of time separating the occurrence of an offspring event from its direct ancestor is a function of the time lag $\Delta t$, and is independent of magnitude, so its density function has the form g($\Delta t$). It is also independent of what happens between the two occurrence times.

• d. The probability distributions of the epicenter location (x, y) and magnitude M of an offspring event are dependent on the magnitude M* and the epicenter location (ξ, η) of its direct ancestor, so their density functions are written $f\left( {x{-}\xi ,y{-}\eta |M^{*} } \right)$ and $j\left( {M|M^{*} } \right)$, respectively.

Under a threefold “assumption (e)” which specifies that the magnitude distribution of a background event is independent of its location, the magnitude distribution of a direct offspring event is independent of the size of its ancestor, and the magnitude distributions for the background events and their offspring are identical (Zhuang et al. 2002), it follows the decomposition

$$\lambda (t,x,y,M|{\mathscr{H}}_{t} ) \, = j\left( M \right)\lambda (t,x,y|{\mathscr{H}}_{t} ),$$

(2)

where $\lambda (t,x,y|{\mathscr{H}}_{t} ) = \mu \left( {x,y} \right) + \sum\nolimits_{{\{ a,t_{a} < t\} }} {\kappa (M_{a} )g(t{-}t_{a} )f(x{-}\xi_{a} ,y{-}\eta_{a} |{M^{*}}_{a} )} .$

Appendix 2

2.1 Equations and statistical assumptions in the 3-D ETAS-based declustering procedure

The background seismicity rate (µ) is important in statistical models because it is related to tectonic loading. In the hypocentral extension of Guo et al. (2015), it continues to be assumed constant in time and variable in space, while the formulation for the cluster seismicity rate of an earthquake with magnitude M is similar to the second term on the right-hand side of (1) (Zhuang et al. 2002), with the same assumptions (a)–(d) but a main difference and further assumptions. The main difference is that in Guo et al. (2015), Zhuang et al.’s (2002) cluster seismicity rate is multiplied by a function h(z, zʹ), where z denotes the focal (hypocenter) depth of an aftershock (offspring event) and zʹ, the depth of the main shock (parent event). Further assumptions include: there is separability between depth and (longitude, latitude) and the corresponding features of the point-process model; the hypocenter depths (z, zʹ) follow a beta distribution with the shape parameter determined by the depth of the parent event, which provides suitable characteristics for the probability density function of depth in the formulation of Guo et al. (2015).

In the resulting 3-D ETAS model, the time-varying seismicity rate function is written

$$\lambda (t,x,y,z|{\mathscr{H}}_{t} ) = \mu \left( {x,y,z} \right) \, + \sum\nolimits_{{\{ a,t_{a} < t\} }} { \tau (t - t_{a} ,x - x_{a} ,y - y_{a} ,z{-}z_{a} ;M_{a} ,z_{a} )} ,$$

(3)

where z_a is the hypocenter depth of a parent event, and the cluster seismicity rate is

$$\tau \left( {t,x,y,z;M,z^{\prime}} \right) \, = \kappa \left( M \right)g\left( t \right)f\left( {x,y;M} \right)h\left( {z,z^{\prime}} \right),$$

(4)

with the same functions κ(·), g(·) and f(·) as in (1) (Zhuang et al. 2002), and h(z, zʹ), the probability density function of the beta distribution used for the hypocenter depths (z, zʹ).

Appendix 3

3.1 MFPA variant for temporal series and MFP-derived coherency analysis: Matrix notations and basic elements

Consider a temporal series of length n > 2K + 1, arranged in a column vector y = $\left( {Y_{{t_{1} }} , \ldots ,Y_{{t_{n} }} } \right)^{{\text{T}}}$ with ^T the transpose operator), and a vector ${{\varvec{\upomega}}} = \,\left( {\omega_{1} , \ldots ,\omega_{K} } \right)^{{\text{T}}}$ of K frequencies $\omega_{{1}} \, < \, \ldots \, < \,\omega_{K}$, for which the MFPA statistic has to be calculated. Let X(${{\varvec{\upomega}}}$) denote the n × (2K + 1) matrix made of a first column of ones, followed by $\cos \left( {\omega_{k} \cdot t_{i} } \right)$, i = 1, …, n and k = 1, …, K (even columns) and $\sin \left( {\omega_{k} \cdot t_{i} } \right)$, i = 1, …, n and k = 1, …, K (next odd columns). With ⁻¹ the inverse operator, the univariate MFPA statistic I^M(${{\varvec{\upomega}}}$) is written in matrix notation (Dutilleul 2001, 2011):

$$I^{M} ({{\varvec{\upomega}}}) = {\mathbf{y}}^{{\text{T}}} {\mathbf{X}}({{\varvec{\upomega}}}) \, ({\mathbf{X}}({{\varvec{\upomega}}})^{{\text{T}}} {\mathbf{X}}({{\varvec{\upomega}}}))^{{{-}{1}}} {\mathbf{X}}({{\varvec{\upomega}}})^{{\text{T}}} {\mathbf{y}}.$$

(5)

Let y₁ and y₂ be two time series or temporal series made of observations at the same n times, equally spaced or not. With ${I_{1}}^{M} ({{\varvec{\upomega}}})$ and ${I_{2}}^{M} ({{\varvec{\upomega}}})$, the two univariate multi-frequential periodograms, and ${I_{12}}^{M} ({{\varvec{\upomega}}})\, = \,{{\mathbf{y}}_{1}}^{{\text{T}}} {\mathbf{X}}({{\varvec{\upomega}}}) \, ({\mathbf{X}}({{\varvec{\upomega}}})^{{\text{T}}} {\mathbf{X}}({{\varvec{\upomega}}}))^{{{-}{1}}} {\mathbf{X}}({{\varvec{\upomega}}})^{{\text{T}}} {\mathbf{y}}_{{2}}$, the corresponding cross periodogram at frequencies ${{\varvec{\upomega}}}$ (based on a scalar product instead of a squared Euclidean norm), the MFP-derived coherency statistic is given by

$$\frac{{{I_{12}}^{M} \left( {{\varvec{\upomega}}} \right)}}{{\sqrt {{ I_{1}}^{M} \left( {{\varvec{\upomega}}} \right)\,{I_{2}}^{M} \left( {{\varvec{\upomega}}} \right)} }},$$

(6)

and can be interpreted as a correlation coefficient ($- \,{1}\, \le \, \cdots \le \,\, + \,{1}$) in the frequency domain (Dutilleul 2001, 2011), for a vector of frequencies in the general case or a scalar frequency in the uni-frequential case, K = 1. The classical t-test for Pearson’s r sample correlation coefficient can be applied, with or without Bonferroni correction for the repetition of tests when the significance of more than one MFP-derived coherency statistic is assessed. The frequencies $\omega_{k}$ (k = 1, … K) are not constrained to be Fourier frequencies, i.e., they do not have to correspond to an integer number of cycles over the series.

In closing, it must be noted that the vector of frequencies, $\varvec{\upomega}$, is usually unknown in (5) and estimated in an iterative procedure in the univariate version of the MFPA, whereas it can be replaced with a known scalar, $\omega$, set at the frequency corresponding to a periodicity of particular interest (e.g., 6 or 12 months) in (6) for the computation of the MFP-derived coherency statistic in the bivariate version.