Premonitory patterns of seismicity months before a large earthquake: Five case histories in Southern California

Abstract

This article explores the problem of short-term earthquake prediction based on spatio-temporal variations of seismicity. Previous approaches to this problem have used precursory seismicity patterns that precede large earthquakes with “intermediate” lead times of years. Examples include increases of earthquake correlation range and increases of seismic activity. Here, we look for a renormalization of these patterns that would reduce the predictive lead time from years to months. We demonstrate a combination of renormalized patterns that preceded within 1–7 months five large (M ≥ 6.4) strike-slip earthquakes in southeastern California since 1960. An algorithm for short-term prediction is formulated. The algorithm is self-adapting to the level of seismicity: it can be transferred without readaptation from earthquake to earthquake and from area to area. Exhaustive retrospective tests show that the algorithm is stable to variations of its adjustable elements. This finding encourages further tests in other regions. The final test, as always, should be advance prediction. The suggested algorithm has a simple qualitative interpretation in terms of deformations around a soon-to-break fault: the blocks surrounding that fault began to move as a whole. A more general interpretation comes from the phenomenon of self-similarity since our premonitory patterns retain their predictive power after renormalization to smaller spatial and temporal scales. The suggested algorithm is designed to provide a short-term approximation to an intermediate-term prediction. It remains unclear whether it could be used independently. It seems worthwhile to explore similar renormalizations for other premonitory seismicity patterns.

We consider here a particular approach to earthquake prediction: prediction of large earthquakes based on changes of seismicity in the intermediate magnitude range. The approach we follow here is reviewed in ref. 1. It is focused on well defined prediction algorithms that can be validated by advance prediction. Such a focus is essential both for a fundamental understanding of crustal dynamics and enhancing earthquake preparedness.

Previous studies of observed and modeled seismicity have found a variety of spatio-temporal patterns of medium-magnitude seismicity that emerge as a large earthquake approaches. Well established are “intermediate-term” patterns, which emerge years before a large earthquake and within hundreds of kilometers from the incipient source (2–10, 12–19).¶ Here, we introduce short-term patterns that emerge with a lead time of months. Our working hypothesis is that some short-term patterns may be defined by renormalization of intermediate-term ones, so that they differ only in the values of adjustable numerical parameters. We consider patterns that reflect two specific premonitory changes of seismicity in the intermediate magnitude range: increases of earthquake correlation range and increases of seismic activity.

Premonitory increases of earthquake correlation range were recognized during the last decade. Several patterns reflecting the rise of correlation range have been suggested (6, 14, 20–24). Here we consider two of them, ROC (range of correlation) and Accord, which were found first in a model (20, 21) and then in observations (22, 23). The pattern ROC was observed months before large earthquakes in the Lesser Antilles (22), which makes it especially relevant to the present study.

Increases of seismic activity years to decades before a large earthquake have been well documented during the last century for seismic regions worldwide (2, 5, 12, 13, 15, 17, 18, 25–28). Here, we consider a specific pattern U reflecting the rise of seismic activity; it was introduced into earthquake prediction research by Schreider (29).

Data

We use the standard Southern California Seismographic Network earthquake catalog produced by the California Institute of Technology (30). The analysis is carried out within the area shown in Fig. 1 during 1960–2001. Targets for prediction are the five large (M ≥ 6.4) strike-slip earthquakes listed in Table 1 and mapped on Fig. 1.

Fig. 1.

Area considered for analysis. Circles mark large earthquakes. The area S is considered for earthquakes with 6.4 ≤ M < 7 (the Borrego Mountain, 1968; Imperial Valley, 1979; and Superstition Hills, 1987). The union of areas S and N (SN) is considered for the earthquakes with M ≥ 7 (Landers, 1992 and Hector Mine, 1999). Fault map after Faults of Southern California, Southern California Earthquake Data Center and ref. 31.

Table 1.

Large earthquakes targeted for prediction

Location	Date	°N, °W	ML(SCSN)	MS(PDE)
Borrego Mountain	April 9, 1968	33.19, 116.12	6.5
Imperial Valley	Oct. 15, 1979	32.61, 115.31	6.4	6.9
Superstition Hills	Nov. 24, 1987	33.01, 115.85	6.6	6.6
Landers	June 28, 1992	34.20, 116.43	7.3	7.6
Hector Mine	Oct. 16, 1999	34.59, 116.27	7.1	7.4

^*Ref. 30; SCSN, Southern California Seismographic Network.

^† PDE, Preliminary Determination of Epicenters, ftp://ghtftp.cr.usgs.gov/pub/pde.

Evolution of seismicity before the large earthquakes is analyzed within the two areas shown in Fig. 1. For the earthquakes with magnitude below 7 (Borrego Mountain, 1968; Imperial Valley, 1979; and Superstition Hills, 1987) we consider the area S, which includes the southern reaches of the San Andreas fault network. The union of the areas S and N (denoted by SN) is considered for the larger Landers (1992) and Hector Mine (1999) earthquakes. The area SN is the principal plate-boundary fault network of southern California and northwestern Mexico, south of the Garlock fault. The northeastern margin of this area roughly corresponds to the Landers–Mojave earthquake line (32).

The areas considered are chosen in one of many possible ways. This nonuniqueness is also the case with other elements of the subsequent analysis: subdivision of the areas (Fig. 2), the choice of premonitory seismicity patterns, the formalization of these patterns (next section), choosing the values of numerical parameters (Table 2), etc. The influence of these free choices on the prediction results is analyzed in Stability Test: The Error Diagram.

Fig. 2.

Divisions into sub-areas by a coarse formal grid (a) and seismicity and fault-based grid (b).

Table 2.

Adjustable parameters of the prediction algorithm

n0, yr−1	TR, days	CR	TA, days	QA, %	κ	CU, yr−1
20 (40)	10	5	15	99	15	2
15–30 (30–60)	5–30	3–8	7–30	98–99	5–30	0.8–10

Values used in Patterns ROC, Accord, and U vs. Large Earthquakes (Figs. 3 and 4) are given in bold. The variation ranges for the stability tests (Fig. 5) are given in plain font. Values of n₀ in brackets refer to the patterns Accord and ROC within the area SN.

Premonitory Seismicity Patterns: Definitions

Previous studies have found a family of premonitory seismicity patterns with common scaling and similar definitions (1). Each of those patterns is captured by a specific functional F(t) defined on a sequence (t_j, M_j, x_j), j = 1, 2, …  of main shocks within a given area and magnitude range M_j > m₀. Here t_j is the occurrence time of the jth main shock, M_j its magnitude, and x_j coordinates of epicenter. The lower magnitude threshold m₀ is determined by the condition n(m₀) = n₀ yr⁻¹, where n(M) is the average annual number of main shocks with magnitude M or larger, n₀ is an adjustable parameter. The same value of n₀ may correspond to different lower magnitude thresholds due to spatial and temporal variations on the seismic activity; this dependence introduces self-normalization of premonitory patterns (8, 9). Emergence of the pattern is defined by the condition F(t) ≥ C_F. The patterns are defined on the earthquake sequence with aftershocks eliminated. The identification of aftershocks is made by the coarse windowing introduced in refs. 33 and 34. For brevity, remaining earthquakes are called the main shocks.

Prediction is targeted at large earthquakes defined by the condition M_j ≥ M₀. Previous studies suggest that the lower magnitude m₀ of the earthquakes that form premonitory patterns is 3–4 units less than the lower magnitude M₀ of the earthquakes to be predicted (1).

The pattern ROC reflects the nearly simultaneous occurrence of medium-magnitude main shocks at large distances (20–22). The functional R(t), which captures this pattern, is defined within a narrow sliding time window (t − T_R, t). It counts the number of pairs (i, j) of main shocks such that r_ij ≥ r₀; here r_ij is the distance between the epicenters of the main shocks i and j. The threshold r₀ is normalized by the minimal magnitude M₀ of the earthquakes targeted for prediction: r₀ = A⋅10^bM₀. Such normalization is used in the well established prediction algorithms CN, M8, and SSE (8, 9).¶ The coefficients are A = 0.03, B = 0.5.¶ We take M₀ = 6.4 in the area S and M₀ = 7.0 in SN. Emergence of the pattern ROC is defined by the condition R(t) ≥ C_R, C_R being an adjustable parameter.

The pattern Accord reflects a nearly simultaneous rise of seismic activity in several parts of the area considered (20, 21, 23), hence its name. The area is divided into K sub-areas, and the functional A(t), which captures the pattern, is defined as the number of sub-areas, which have at least one earthquake within a sliding time window (t − T_A, t). This definition is a marginal case of the following general one (20): for each sub-area k = 1, … , K its seismic activity is estimated by a chosen measure in a sliding time window. High activity is defined by the condition that this measure exceeds a certain threshold. The functional A(t) is defined as the number of the sub-areas where activity is high at the moment t.

Emergence of the pattern Accord is defined by the condition A(t) ≥ C_A. The threshold C_A is adapted to the level of seismicity by a binomial model. Let p be the probability to have at least one main shock within a given sub-area during a time interval T_A. With a uniform distribution of earthquakes, the probability that at least A out of K sub-areas have one or more main shocks is P(A) = Σ_a≥A p^a(1 − p)^K−a with p = n₀⋅T_A⋅K⁻¹. We chose the threshold C_A as the Q_A percentile of the distribution P.

Seismicity in this model is assumed to be (i) uniformly distributed and (ii) independent in each sub-area. Both of these assumptions lead to overestimation of the probability P(A) for large A, so for high Q_A the functional A(t) exceeds the threshold C_A more rarely than suggested by the model. Thus, we use a reliable lower estimation. Note that the level of seismic activity is used in many earthquake prediction algorithms (1, 2, 4, 5, 7–9, 12–15, 17–19),¶ although for much wider windows than considered here: years instead of weeks.

We consider two alternative divisions into sub-areas. One is a coarse grid oriented along the southern San Andreas fault (Fig. 2a). An advantage of this division is its simplicity and independence of nonunique interpretations of fault maps and seismicity. This simplicity is achieved at the cost of disregarding this potentially relevant information. To use this information we introduce an alternative division (Fig. 2b) where each sub-area roughly corresponds to a relatively large fault. The borders between sub-areas are chosen to avoid dense clouds of epicenters. This division takes into account the fault network and territorial distribution of seismicity at the cost of nonuniqueness in transforming these data into boundaries of sub-areas.

The pattern U reflects the increase of the earthquake occurrence rate. It is captured by the functional U(t_j) = 1/(t_j− t_j−κ+1), j = κ, κ + 1, … defined at the occurrence times t_j of main shocks used for analysis. Here the denominator is the length of the minimal time interval, which covers the last κ earthquakes. Emergence of the pattern U is defined by the condition U(t_j) ≥ C_U, C_U being an adjustable parameter.

Patterns ROC, Accord, and U vs. Large Earthquakes

In this section we demonstrate the emergence of the above patterns before the five large earthquakes we consider (see Table 1, Fig. 1). We consider 5-year intervals, each ending at one of the large earthquakes (5 years is a typical lead time for the intermediate-term premonitory seismicity patterns). Analyzing seismicity before the Hector Mine earthquake, we eliminated sub-areas N1, N2, and N3 (Fig. 2b) where activity rose after the Landers earthquake. Thus we avoided mixing post-Landers activity, decaying but still high, with pre-Hector Mine activity.

Functionals R(t), A(t), and U(t) have been computed with the values of the adjustable parameters indicated in bold in Table 2 and sub-areas shown in Fig. 2b. The corresponding values of the self-adapted thresholds m₀, r₀, and C_A calculated from the adjustable parameters are listed in bold in Table 3. They are determined within the time intervals (t_f − 5 yr, t_f − 1 yr), t_f being the moment of a large earthquake. The last year is eliminated from the statistics, since our goal is to find what changed there compared with previous years. For area Swe take n₀ = 20 yr⁻¹, as in the widely used algorithm M8 (8). In the area SN, which contains twice as many sub-areas as S, we take n₀ = 40 yr⁻¹ for the patterns ROC and Accord. This procedure ensures that both areas have the same average number of main shocks per sub-area.

Table 3.

Self-adapted thresholds calculated from adjustable parameters

Pattern	Threshold	Large earthquakes
		Borrego Mountain	Imperial Valley	Superstition Hills	Landers	Hector Mine
ROC	m0	2.5	3.0	2.9	2.9	3.0
		2.5–2.8	2.9–3.1	2.7–3.0	2.7–3.0	2.7–3.1
	r0, km	47	47	47	95	95
		16–67	16–90	16–80	32–140	32–135
Accord	m0	2.5	3.0	2.9	2.9	3.0
		2.5–2.8	2.9–3.1	2.7–3.0	2.7–3.0	2.7–3.1
	CA	4	4	4	6	6
		3–6	3–6	3–6	4–8	4–8
U	m0	2.5	3.0	2.9	3.2	3.3
		2.5–2.8	2.9–3.1	2.7–3.0	3.1–3.3	3.1–3.4

Fig. 3 shows the behavior of each functional before the five large earthquakes considered. By definition, a pattern emerges when the functional is equal or larger than the respective threshold (shown by horizontal dotted line in Fig. 3). Patterns emerge sporadically within isolated narrow time intervals. Duration of emergence intervals is up to weeks for patterns ROC and Accord and up to months for pattern U.

Fig. 3.

Individual performance of premonitory patterns. Functionals R(t), A(t), and U(t) during 5-year intervals ending at a large earthquake. The five lines in each panel correspond, from top to bottom, to the large earthquakes in the order of occurrence: Borrego Mountain, 1968; Imperial Valley, 1979; Superstition Hills, 1987; Landers, 1992; and Hector Mine, 1999. Functionals R(t) and A(t), by definition take nonnegative integer values; functional U(t) takes real positive values. Dashed horizontal lines show the thresholds C_U, C_A, and C_R. See details in PatternsROC, Accord, and U vs. Large Earthquakes.

The pattern U precedes each large earthquake by 2 years or less. Besides, it emerges 4.5 years before the Superstition Hills earthquake. Thus, this pattern still remains an intermediate-term one. At the same time, its lead time is much shorter than 5 years, which is typical for most of the patterns reflecting premonitory rise of activity (1).

The patterns ROC and Accord precede four large earthquakes (Borrego Mountain, 1968; Imperial Valley, 1979; Landers, 1992; and Hector Mine, 1999) by 3 months or less and the Superstition Hills earthquake by 7 months. These patterns also emerge with a larger (up to 58 months) lead time before the Borrego Mountain, Superstition Hills, and Landers earthquakes. In the previous applications the lead time was also months for ROC (22) but it was years for Accord (23).

The patterns ROC and Accord do not seem promising as individual short-term precursors since they would produce many false alarms. Next, we demonstrate that these patterns give better performance when they emerge in the wake of the pattern U. To formalize this observation, we formulate the following prediction algorithm:

(i) Whenever the pattern U emerges [U(t) ≥ C_U], a waiting period is declared for the subsequent time interval τ_U.

(ii) When both the patterns ROC and Accordemerge [R(t) ≥ C_R and A(t) ≥ C_A] during a waiting period, an alarm is declared for the subsequent time interval τ_A.

The performance of this algorithm is shown in Fig. 4, which consists of five panels, one for each target earthquake. The top line in each panel shows alarms produced by the algorithm. The dark bars in three bottom lines show emergence intervals of the patterns ROC, Accord, and U. Notable is the temporal correlation between the patterns, particularly ROC and Accord. Waiting periods declared after the emergence of the pattern U are marked by light shadowing. Parameters of the functionals R(t), A(t), and U(t) are the same as in Fig. 3, τ_U = 24 months and τ_A = 7 months. The resulting lead times between the declaration of an alarm and the subsequent large earthquake vary from 49 to 205 days.

Fig. 4.

Short-term alarms. Each panel shows a 5-year interval ending with a large earthquake. The top line shows (by checkered bars) the alarms retrospectively determined by the algorithm. The three lower lines correspond to premonitory patterns indicated at the left. Dark bars show the time intervals when a pattern emerges. Light bars in the bottom line mark waiting periods.

Fig. 4 might suggest an alternative algorithm, based on the fact that all three patterns emerge in arbitrary order within 7 months before a large earthquake. What would be the outcome of this alternative algorithm? Generally, what would be the outcome of prediction if we vary each of its adjustable elements? We answer this question in the next section by exploring a wide range of possible alternatives.

Stability Test: The Error Diagram

Here, we evaluate the stability of our algorithm: vary its adjustable elements, repeat prediction, and compare the results on an error diagram. The error diagram was introduced into seismological studies by G. Molchan (35); its definition follows.

Consider a prediction algorithm applied on a certain territory during the period T years. Suppose that A alarms are declared and A_f of them are false. N strong earthquakes occurred and N_f of them are not covered by alarms (unpredicted). The alarms cover altogether the time D year. Performance of the algorithm is characterized by three dimensionless errors: the relative duration of alarms, τ = D/T; the fraction of failures to predict, n = N_f/N; and the fraction of false alarms, f = A_f/A. The error diagram shows the values of τ, n, and f for different versions of a prediction algorithm.

We have considerably varied the adjustable elements of the algorithm. In particular, we have tested four alternative rules for alarm declaration based on interplay of the three individual patterns; four versions of sub-areas (used to define the pattern Accord) of different size and orientation; and a wide range of adjustable numerical parameters varied uniformly within the ranges indicated in the second line of Table 2. The corresponding ranges of self-adapted thresholds m₀, r₀, and C_A are indicated in Table 3. Note that the wide ranges of variation for the parameter n₀ (say, from 15 to 30 in area S) correspond to modest ranges of variation for the magnitude threshold m₀ (typically, 0.3 units of magnitude).

Altogether we have considered >1,000 sets of algorithm's adjustable elements. Prediction was done with each of them; the results are juxtaposed in the error diagram of Fig. 5. The error diagram shows that prediction performance is reasonably stable: 23% of the versions considered predict all five large earthquakes with no false alarms (corresponding points lie at the origin of the error diagram), and 60% give the error score A_f + N_f ≤ 1 (no more than one false alarm or one unpredicted earthquake). Parameters corresponding to these predictions fill a reasonably wide domain in the parameter space. The average duration of alarms is shorter than 5 months per earthquake. The limits where the good performance fails (say, correspond to A_f + N_f ≥ 3) are reached in 20% of the versions. Other results may be summed up as follows:

Fig. 5.

Error diagram. The diagram compares performance of the prediction algorithm for 1,008 sets of its adjustable elements (see definition in Stability Test: The Error Diagram). Values n = 0.2, 0.4, etc. correspond to N_f = 1, 2, etc. failures to predict. Interval 0.1 on the τ scale is 30 months (6 months per large earthquake). Numbers on the right show how many sets correspond to a given (f, n) combination. Numbers on the left show how many sets correspond to a given value of n and interval of τ. The line n + τ = 1 shown (Left) corresponds to a random binomial prediction: alarm is declared at each time step with probability τ. Note the stability of performance: 235 (23%) variants predict all five earthquakes with no false alarms (an example marked by the large circle is shown in Fig. 4); 270 (26%) variants predict all five earthquakes with one false alarm; 92 (9%) variants predict four earthquakes with no false alarm.

(i) The most stable is prediction of the Landers earthquake, which is predicted by 94% of the variants considered; the least stable is prediction of the Borrego Mountain earthquake (79%).

(ii) Important for our prediction is the spatial scale of the sub-areas (20–50 km) and not a particular position of borders. This notion is supported by the fact that formal sub-areas (Fig. 2a) and those based on the fault map and seismicity (Fig. 2b) give similar performance.

(iii) The rise of activity (pattern U) and rise of correlation range (patterns ROC and Accord) indeed play different roles in the transition to a large earthquake. This finding is supported by the fact that the prediction rule suggested earlier (uniform use of the individual patterns) gives more errors than the rules that use ROC and Accorddifferently from U.

More details are given in Table 4 and Supporting Text, which are published as supporting information on the PNAS web site, www.pnas.org.

Discussion

We renormalized in time and space three previously known premonitory seismicity patterns, which reflect the increase of seismic activity and increase of earthquake correlation range. A combination of the renormalized patterns provides, in retrospect, a short-term prediction (with lead time of months) of the five largest earthquakes in Southeastern California, including the Landers and Hector Mine earthquakes.

What Is a Possible Physical Interpretation of a Premonitory Earthquake Sequence Introduced Here?

Rise of earthquake correlation range indicates that the blocks surrounding a soon-to-break fault start to aggregate, tending to move as a whole. The pattern Accord captures this phenomenon on a relatively high level of averaging. The pattern ROC captures this phenomenon without any spatial averaging, by the emergence of individual earthquakes. The displacement rate should be sufficiently large, which is ensured by the general rise of seismic activity (pattern U).

These processes are not permanent; they are realized in sporadic episodes reflected by distinct peaks of the functionals. Each peak lasts for weeks (patterns ROC and Accord) or months (pattern U). Similar irregularity on a larger scale is exhibited by the intermediate-term patterns.

Why Renormalization?

It can be naturally explained by the phenomenon of self-similarity, a general feature of many nonlinear systems. Such systems exhibit well defined behavior patterns (36–38), which emerge in similar form on different scales. Generally, the renormalization to short-term scales is well explored in the modeling of critical transitions, although not in connection with prediction problem (39–44). If our results reflect reality, the premonitory patterns we analyzed exhibit a specific kind of similarity: they retain their predictive power in different spatio-temporal scales.

Further Tests.

Exhaustive retrospective tests show that the suggested algorithm is reasonably stable to variations of its adjustable elements (numerical parameters being only part of them). This finding encourages its further test in other regions. Such tests are facilitated by the fact that the algorithm is self-adapting to the level of background seismicity. Results of such tests for other parts of California, the Eastern Mediterranean, and Japan will be published elsewhere.

Transition from Intermediate- to Short-Term Prediction?

By definition, our algorithm can be used as a second approximation to an intermediate-term one, which would provide initial alarms within several hundred kilometers and several years. It remains unclear whether our algorithm can be used independently.

A consecutive approximation approach was pioneered with singular success in prediction of the Haicheng earthquake in China in 1975 (28). Kossobokov et al. (45) describe a premonitory short-term rise of activity in areas of intermediate-term alarms of the algorithm MSc in the Pacific Rim and California.

What Do We Rely On?

The quest for short-term earthquake precursors has been less than successful so far. A multitude of such precursors have been suggested over the decades, but hardly any were validated by systematic applications, with a reproducible count of errors and successes. Many current efforts are focused on electromagnetic precursors (11). Our hope rests on taking advantage of the following previous findings: extents of averaging of seismicity in time and space; normalized self-adapting definitions; the recently introduced patterns, which reflect increase of earthquake correlation range; and a consecutive approximation approach.

Abbreviations

• ROC, range of correlation