An Ensemble Approach for Improved Short-to-Intermediate-Term Seismic Potential Evaluation

Abstract

Pattern informatics (PI), load/unload response ratio (LURR), state vector (SV), and accelerating moment release (AMR) are four previously unrelated subjects, which are sensitive, in varying ways, to the earthquake’s source. Previous studies have indicated that the spatial extent of the stress perturbation caused by an earthquake scales with the moment of the event, allowing us to combine these methods for seismic hazard evaluation. The long-range earthquake forecasting method PI is applied to search for the seismic hotspots and identify the areas where large earthquake could be expected. And the LURR and SV methods are adopted to assess short-to-intermediate-term seismic potential in each of the critical regions derived from the PI hotspots, while the AMR method is used to provide us with asymptotic estimates of time and magnitude of the potential earthquakes. This new approach, by combining the LURR, SV and AMR methods with the choice of identified area of PI hotspots, is devised to augment current techniques for seismic hazard estimation. Using the approach, we tested the strong earthquakes occurred in Yunnan–Sichuan region, China between January 1, 2013 and December 31, 2014. We found that most of the large earthquakes, especially the earthquakes with magnitude greater than 6.0 occurred in the seismic hazard regions predicted. Similar results have been obtained in the prediction of annual earthquake tendency in Chinese mainland in 2014 and 2015. The studies evidenced that the ensemble approach could be a useful tool to detect short-to-intermediate-term precursory information of future large earthquakes.

Appendix

1.1 Pattern Informatics

The pattern informatics (PI) method, which is an example of a phase dynamical measure, was proposed to detect the characteristic precursory patterns before large earthquakes (Rundle et al. 2000, 2003). Detailed procedures of this method can be outlined as follows:

• 1. To divide the study region into square bins with side length of Δx.

• 2. To define average rate of occurrence of earthquakes in the i-th bin over the period t _b to t

  $$I_{i} (t_{b} ,t) = \frac{1}{{t - t_{b} }}\sum\limits_{{t^{\prime} = t_{b} }}^{t} {N_{i} (t^{\prime})} ,$$

  (4)

  where t _b varies between t ₀ and t ₁ at a time step of Δt, and t ₀ is the initial time. The time interval t _b–t ₁ is the reference period. N _i (t) is the number of earthquakes with magnitude greater than M _c in the i-th bin. M _c is the magnitude cutoff.

• 3. To normalize the activity rate function,

  $$\widehat{I}_{i} (t_{b} ,t) = \frac{{I_{i} (t_{b} ,t) - \langle I_{i} (t_{b} ,t)\rangle }}{{\sigma (t_{b} ,t)}},$$

  (5)

  where  $\langle I_{{\text{i}}} (t_{b} ,t) \rangle$ and σ(t _b,t) are the average activity rate function and its spatial standard deviation over all the bins at time t.

• 4. To assess the change of normalized activity rate function for the time period t ₁–t ₂,

  $$\Delta I_{i} (t_{b} ,t_{1} ,t_{2} ) = \widehat{I}_{i} (t_{b} ,t_{2} ) - \widehat{I}_{i} (t_{b} ,t_{1} )$$

  (6)

• 5. To calculate the probability of change of activity in the i-th bin,

  $$P_{i} (t_{0} ,t_{1} ,t_{2} ) = \overline{{\Delta I_{i} (t_{0} ,t_{1} ,t_{2} )}}^{2} ,$$

  (7)

  where $\overline{{\Delta I_{i} (t_{0} ,t_{1} ,t_{2} )}} = \frac{1}{{t_{1} - t_{0} }}\sum\nolimits_{{t_{b} = t_{0} }}^{{t_{1} }} {\Delta I_{i} (t_{b} ,t_{1} ,t_{2} )}$. In phase dynamical systems, probabilities are related to the square of the associated vector phase function (Rundle et al. 2000).

• 6. To evaluate the difference between the P _i(t ₀,t ₁,t ₂) and its spatial mean ${\langle P_i(t_0,t_1,t_2)\rangle},$ representing the probability of change in activity relative to the background,

$$\Delta P_{i} (t_{0} ,t_{1} ,t_{2} ) = P_{i} (t_{0} ,t_{1} ,t_{2} ) - \langle P_{i} (t_{0} ,t_{1} ,t_{2} )\rangle .$$

(8)

The hotspots are defined to be the bins (or the regions) where ΔP _i(t ₀, t ₁, t ₂) is positive.

1.2 Load/Unload Response Ratio

Over the past decade, an earthquake prediction method named the load/unload response ratio (LURR) has been developed by Yin and others (Yin et al. 2000; Zhang et al. 2006). In earthquake prediction practice using the method, the seismic energy release within certain temporal and spatial windows is usually used as data input. Loading and unloading periods are determined by calculating the earth tide-induced Coulomb failure stress change along a tectonically favored rupture direction on a specified fault plane (Hainzl et al. 2010; Harris 1998). The Coulomb failure stress is defined as:

$$CFS = \tau_{n} + f\sigma_{n} ,$$

(9)

where f, τ _n and σ _n stand for inner frictional coefficient, shear stress and normal stress (positive in tension), respectively, and n is the normal of the fault plane on which the CFS reaches its maximum. When the change of Coulomb failure stress (ΔCFS) >0, it is in a loading state; and when ΔCFS <0, it is in an unloading state. The LURR is, thus, expressed as a ratio between energy released during loading and that released during unloading periods:

$$Y_{m} = \frac{{\left( {\sum\nolimits_{i = 1}^{N + } {E_{i}^{m} } } \right)_{ + } }}{{\left( {\sum\nolimits_{i = 1}^{N - } {E_{i}^{m} } } \right)_{ - } }},$$

(10)

where E _i is seismic energy released by the i-th event, and N+ or N− represents the numbers of events that occurred during the loading and unloading stages, respectively. When m = 1/2, E ^m denotes the Benioff strain. Note that the focal mechanisms of the small earthquakes are assumed in agreement with that of the main shock to contribute positively to ΔCFS for the main shock. This assumption is supported by studies of Hauksson (1994), Hauksson et al. (2002), and Hardebeck and Hauksson (2001), which demonstrated that the focal mechanisms of regional small earthquakes prior to the Landers and Hector Mine earthquakes were quite consistent with that of the ensuing main shocks. To avoid volatile fluctuations due to poor statistics, the loading and unloading periods are usually summed over many load–unload cycles within the time window. Circular region is usually adopted as the spatial window, and the optimal critical region scale for LURR evaluation is determined by computing the LURR anomaly within differently sized regions centered at epicenter of the upcoming large event to reach the maximum LURR precursory anomaly (Yin et al. 2000). The forecasting time window, from months to years, is determined by the magnitude of the ensuing earthquake: the larger the earthquake, the longer the time.

1.3 State Vector

The idea of SV is adopted from statistical physics into seismology to characterize the spatial and temporal evolution of seismic activities (Yu et al. 2006). This method is defined by dividing the target region into n uniform square subregions, and the sum of seismic magnitudes in each subregion within certain temporal window is computed as the component of an n-dimensional vector. If a series of vectors at different times are known, the temporal and spatial evolution of seismicity may be obtained. Previous studies show that anomalously high SV peaks have often been observed months to years before large earthquakes (Yu et al. 2006). Generally, four scalars are defined to measure evolution of the vectors.

• 1. Modulus of the vectors,

  $$M = \left| {\vec{V}_{k} } \right|,$$

  (11)

  where $\vec{V}_{k}$ is the state vector at time t _k (k = 1, 2…n), whose temporal window is T at a sliding step of Δt.

• 2. Angle between two consecutive vectors,

  $$A_{\text{s}} = \arccos \left( {\frac{{\vec{V}_{k + 1} \vec{V}_{k} }}{{\left| {\vec{V}_{k + 1} } \right|\left| {\vec{V}_{k} } \right|}}} \right).$$

  (12)

• 3. Modulus of the increment vector:

  $${\text{IM}} = \left| {\vec{V}_{k + 1} - \vec{V}_{k} } \right|.$$

  (13)

• 4. Angle between vector $\vec{V}_{k}$ and equalized vector $\vec{V}_{e}$,

  $$A_{\text{c}} = \arccos \left( {\frac{{\vec{V}_{e} \vec{V}_{k} }}{{\left| {\vec{V}_{e} } \right|\left| {\vec{V}_{k} } \right|}}} \right),$$

  (14)

  where the equalized vector $\vec{V}_{e}$ consists of equal components.

1.4 Accelerating Moment Release

Prior to the occurrence of large or great earthquakes, the accelerating moment release (AMR) is usually observed (Jaume and Sykes 1999). Bufe and Varnes (1993) suggested that a simple power-law time-to-failure equation derived from damage mechanics could be used to model the observed seismicity. This hypothesis is an outgrowth of efforts to characterize large earthquakes as a critical phenomenon (Rundle 1989). The function has the following form:

$$\varepsilon_{\text{p}} (t) = A + B(t_{\text{c}} - t)^{z} ,$$

(15)

where t _c is the time of the large event, B is negative and z is the exponent. A is the value of ε(t) when t = t _c (i.e., the final Benioff strain up to and including the largest event). The cumulative Benioff strain at time t is defined as:

$$\varepsilon (t) = \sum\limits_{i = 1}^{N(t)} {E_{i} (t)}^{{\frac{1}{2}}} ,$$

(16)

where E _i is the energy of the i-th event and N(t) is the number of events at time t.