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PATTERN IDENTIFICATION IN THE DYNAMIC FINGERPRINT OF SEISMICALLY ACTIVE ZONES

The final version of this paper was published in: Quaternary International, 171-172, 45-51

http://www.sciencedirect.com/science/article/pii/S1040618207000298

Cristian Suteanua*, Cristian Ioanab

a Department of Geography and Environmental Studies Program, Saint Mary's University, 923 Robie

St., Halifax, Nova Scotia, B3H 3C3 Canada. E-mail: cristian.suteanu@smu.ca

b Datasystem, 214 r. Lecourbe, 75015 Paris, France. E-mail: cioana@datasystem.fr Abstract. The aim of this paper is to present a methodology designed to characterize seismically active areas from the point of view of their dynamic fingerprints, which rely on space-time aspects of seismicity patterns. According to this methodology, earthquake events are represented as points in phase space, which leads to an "events thread" connecting the points in their temporal sequence. A multiscale analysis procedure is applied to the events thread, and scaling exponents are identified. The proposed framework considers projections of the events thread corresponding to different spatial orientations. For each orientation, the events thread is characterized from the point of view of signal persistence. The resulting persistence diagrams portray anisotropy aspects of the dynamic fingerprint. The identification of anisotropic properties of the dynamic fingerprint provides a new perspective on seismically active zones, complementing the information from geological and geophysical studies. The paper describes the proposed methodological framework, and presents applications focusing on two different regions: the Aleutian Arc and Hawaii. Keywords: seismicity, anisotropy, multiscale analysis, persistence. 1. Introduction

The quantitative characterization of earthquake patterns in a given area may significantly

contribute to a better understanding of underlying processes and their evolution in time. The study of

earthquake events from the point of view of their space-time distribution represents an important but

challenging task (Stock and Smith, 2000; Fuji and Matsu’ura, 2000; Carbone et al., 2005). This is

particularly so if one is interested in more than a single aspect of seismicity – such as the spatial

distribution of epicenters, or the distribution of event magnitude.

The problem regarding a rigorous and relevant description of multi-parameter earthquake

patterns has been addressed in different ways. From the point of view of the practical characterization

of multi-dimensional dynamic fingerprints, the Events Thread Analysis (ETA) represents a useful

approach: it is based on the idea that spatial, temporal, and energy aspects of earthquake occurrence

should be considered together in a unified way (Suteanu, 1998).

In this paper we present an expansion of the ETA method, which is designed to address not

only the earthquake occurrence pattern as a whole, but also the spatial orientation related to

correlations in seismicity patterns. The proposed approach leads to the identification and

* Corresponding author. E-mail: cristian.suteanu@smu.ca. Tel: 001-902-420-5731. Fax: 001-902-496-8213.

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characterization of pattern anisotropy in 3-dimensional space, highlighting its relationship with scaling

aspects.

In the next section we introduce the principle of ETA and explain the proposed new

methodological framework, including the smoothing dimensions method used for the multiscale

analysis. Section 3 describes applications of this analysis framework, focusing on seismicity patterns

in two regions – the Aleutian Arc and Hawaii. Finally, in section 4 we discuss the results and

synthesize the conclusions of this study.

2. The dynamic fingerprint and the multiscale approach to anisotropic aspects

2.1. The Events Thread Analysis (ETA) method

In order to characterize seismicity patterns, different aspects of earthquake occurrence have been

studied in detail, and various scaling laws have been identified. Most of these studies focus separately

on distinct categories of parameters, such as space, time, or event size, eventually investigating

correlations between scaling exponents (Hirata, 1989; Bodri, 1994; Wang and Ou, 1998; Lei and

Kusunose, 1999; Nanjo and Nagahama, 2000; Nicholson et al., 2000; Lapenna et al., 2000; Roberts et

al., 2004). The problem of earthquakes succession has received special attention: Godano et al.

(1999) apply a multifractal approach to the distribution of time intervals between events, to produce

earthquake catalogs of independent events; while Abe and Suzuki (2003, 2005) study scaling aspects

of distances between successive earthquakes, Carbone et al. (2005) consider the waiting

times between seismic events, for which they find a general scaling law.

ETA starts from the premise that an earthquake may change the context in which future events

will occur, thus recognizing the importance of irreversibility aspects in processes involved in seismic

events (Suteanu, 1998). This approach emphasizes the meaning of information regarding the

succession of earthquakes and considers therefore both temporal and spatial aspects at different

scales. According to the principles of ETA, a seismically active area can be viewed as a dissipative

system: earthquake events contribute to changes in such a system, which may in their turn influence

the way the system evolves in the future, at different scales in space and in time. According to this

model, space-time correlations should be identifiable: their quantitative characterization could thus be

used for a comprehensive description of seismicity (Suteanu, 1998).

The ETA method has provided a useful approach to seismicity studies, supporting a

quantitative and comparative evaluation of earthquake patterns. Moreover, this method has proven to

be characterized by particular robustness regarding earthquake catalog completeness and accuracy

(Suteanu et al., 2001).

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Figure 1. Examples of event threads: segments corresponding to the Aleutian Arc (a) and Hawaii (b).

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According to the ETA method, each earthquake event is represented by a point in phase

space, which can be based on spatial, spatio-temporal, or space-time-energy coordinates (Suteanu,

1998). The resulting "events thread" connecting the points in their temporal sequence (Fig. 1) is then

analyzed in a multiscale perspective. To this end, one can use a variety of methods. We briefly

describe below the smoothing dimensions approach, which is applied in our studies due to its

accuracy and reliability.

2.2. Multiscale analysis: smoothing dimensions

The fractal analysis of real-world time series, which can often be strongly irregular, represents

a powerful tool for pattern characterization (Falconer, 1990). Among the practical methods developed

for this purpose, the smoothing dimensions approach (Munteanu et al., 1995) enjoys particular

advantages, such as high precision and accuracy and a robust behaviour in applications involving a

large variety of signals (Ioana et al., 1997).

The principle of the group of methods leading to smoothing dimensions relies on the

identification of scaling properties of parameters attached to time series, evaluated after successive

steps of signal “smoothing”. Due to its effectiveness regarding real time series analysis (Ioana et al.,

1997) we have used the di smoothing dimensions method for our ETA. In this case, signal smoothing

occurs in the frequency domain and is given by the cutoff frequency fc.

Given a signal u(t), we filter it using a series of cutoff frequency values fc. The result consists

of the series of filtered signals ufc(t), for which we compute the Fourier transform:

dte)t(u)(U tj

fcfc

(1)

and the parameter

0

d|)(U|)u(I fcfc (2)

For discrete time series, Eq. (2) becomes:

2

1

/M

k

kfc .|U|)U(I (3)

We check the following relation:

id

cfc )f()U(I (4)

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Figure 2. Definition of the angles kappa and gamma for the evaluation of pattern anisotropy.

If Eq. (4) holds, then di represents the smoothing dimension for the analyzed time series. The

following equation relating di to the fractal dimension df of the signal has been proven (Munteanu et

al., 1995):

51.dd if (5)

which, given the relationship between the fractal dimension df of the signal and its Hurst exponent H

(Feder, 1988), leads to:

id.H 50 (6)

The Hurst exponent is particularly useful for the characterization of scaling properties of

patterns, because H is directly related to persistence (Feder, 1988). Hurst values larger than 0.5

indicate a persistent signal, which means that fluctuations in a certain direction have the tendency to

be continued in the same direction. On the contrary, when fluctuations in one direction have the

tendency to be promptly followed by changes in the opposite direction, Hurst values are lower than

0.5, which is a sign of antipersistence. A Hurst value of 0.5 corresponds to uncorrelated, random

fluctuations.

The application of this method to the analysis of events threads is thus expected to provide

information regarding the seismicity patterns and to identify pattern change.

2.3 Events thread analysis and dynamic fingerprint anisotropy

The events thread corresponding to a seismically active zone – generated as explained in

section 2.1 – can be analyzed directly in phase space, which leads to the identification of the overall

dynamic fingerprint of the zone (Suteanu et al., 2001).

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Using the same approach, we can go farther, and analyze the direction dependence of

correlations and reveal anisotropy aspects related to scaling properties. To this end, we project the

events thread on a line which is positioned in such a way as to correspond to different successive

spatial directions.

Figure 2 presents the principle of the events thread anisotropy evaluation and the angles used

for this purpose. The events thread is projected on the line D, defined by two angles: gamma, the

angle between D and the vertical Z-axis, and kappa, the angle between the X-axis and Dp, which is the

projection of D on the XOY plane. X-axis and Y-axis values increase eastwards, and northwards,

respectively. We rotate D stepwise on a cone, with a step delta kappa; the angle gamma of the cone is

changed stepwise – with a step delta gamma – after each completed rotation. At every step, the 3-

dimensional events thread is projected on the line D. Each events thread projection is then

characterized by the Hurst exponent H.

In view of the multiscale analysis, the resulting projected time series ai is normalized. First, we

construct bi, as a zero mean vector of the same length with ai:

N

a

ab

N

i

ii

1

(7)

Then we generate bsi, by dividing all the elements of the vector by the standard deviation:

)b(std

bb

i

i

si (8)

Thereby, we obtain time series that are comparable from the point of view of their pattern,

independent of the actual values involved.

Finally, the cumulative summation ci of all the elements of the events thread

N

sii bc1

(9)

produces the “landscape” of the events thread, which is effectively approachable with the described

multiscale analysis methods. The time series ci is then evaluated with the help of the smoothing

dimensions method presented above.

Figure 3 presents a projection landscape example. Its visual appearance suggests that it has

fractal properties: looking at smaller parts of the profile at different scales, one can notice that their

patterns are similar to those of larger parts of the signal, as well as to the pattern of the whole

segment. However, it is by quantitative pattern analysis – such as the smoothing dimensions analysis

presented above – that one should rigorously identify and characterize such self-similarity properties.

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Figure 3. A fragment from an events thread projection landscape for Hawaii (profile height in arbitrary units).

The resulting persistence values obtained for each spatial direction are represented in 3-

dimensional diagrams: they lead to surfaces defined by the coordinates kappa and gamma, with

heights representing the values of the Hurst exponent (Fig. 4).

At this point, an isotropic behaviour would lead to a flat landscape in gamma-kappa

coordinates: the overall height of the flat surface would reflect the degree of persistence in the

seismicity pattern. A shape that would be different from a flat surface would indicate the presence of

anisotropic properties. One has thus a tool to check whether variations are present in the persistence

“relief” – they would correspond to changes in persistence for certain spatial directions.

3. Applications

3.1. Regions and analyzed data

We illustrate the applicability of the proposed framework by considering two different types of

context: along the Aleutian Arc and in Hawaii. In both cases more than a single category of

earthquakes are involved: while most earthquakes along the Aleutian Arc are of tectonic origin, events

related to volcanic activity are also present (Levin et al., 2005); on the other hand, tectonic

earthquakes also occur in Hawaii (Wolfe et al., 2004), where volcanic processes are dominant.

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Figure 4. Persistence diagram for earthquake patterns corresponding to the Aleutian Arc. The angles gamma

and kappa are given in radians. H is the Hurst exponent.

Figure 5. Persistence diagram for the Aleutian Arc, shuffled catalog. Same coordinates as in Fig. 4;

however, the scale range for H is different from the one in Fig.4.

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Forming a boundary between the North American and the Pacific plates, the Aleutian Arc is

characterized by different types of relative motion including normal convergence, highly oblique

convergence, and pure transform motion (Levin et al., 2005). It represents a particularly active area,

where large earthquakes have occurred in the past and are expected in the future (Conrad et al.,

2004).

Volcanism in Hawaii is related to hot spot activity: the island of Hawaii contains five major

volcanoes; among them, Kilauea has been the most active one during the last decades (Wolfe et al.,

2004). Earthquakes are caused by magma movement, by stress change due to volcano loading and

lithospheric flexure, as well as by thermal cooling stress (Wolfe et a., 2004).

For the present study we analyze data from the Advanced National Seismic System (ANSS)

catalog. For the Aleutian Arc, data are collected from a rectangular area defined by 45° to 60° northern

latitude and 60° to 180° western longitude. This segment corresponds mainly to subduction processes

(Levin et al., 2005). To ensure catalog homogeneity regarding the recorded minimum magnitude

threshold, we consider the time interval from 1997 to 2003, which leads to a number of 4096 analyzed

events (the total number of events is adjusted to a number equal to a power of 2, which is imposed by

the Fourier analysis involved in the applied procedure, as described in section 2.2).

For earthquakes in Hawaii, we choose the spatial limits corresponding to 18.5° to 20.5°

northern latitude and 154.5° to 156.5° western longitude. For the reasons explained above, we

analyze the time interval from 1989 to 1992, considering again 4096 events.

Hypocentre data are first converted to distance units and then processed according to the

methodology described in chapter 2.

3.2. Results: anisotropy in persistence diagrams

Figure 4 shows the surface corresponding to the direction dependence of persistence for the

Aleutian Arc. The surface is far from being flat: a strong anisotropic character is visible. There is an

obvious drop in persistence approaching values of uncorrelated noise (H = 0.5), corresponding to the

direction perpendicular to the Arc. However, for values of the angle kappa that are different from this

orientation, there is a clearly persistent behaviour. The values of gamma have a lower impact on the

persistence diagram; however, persistence increases as the angle gamma grows. The lowest values

of persistence – and therefore the closest position to uncorrelated noise – occurs thus for the vertical

direction and for the direction perpendicular to the Arc.

The shape of the surface in the diagram may depend on the spatial distribution of the events,

as well as on their succession in time. It is difficult to distinguish the contribution of the different factors

from such a diagram alone. In order to find out whether the sequence of events plays a significant

role, we perform the analysis for the same data also considering a shuffled catalog. The results of this

analysis are shown in Figure 5. They highlight a different pattern: the values on the surface are close

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Figure 6. Persistence diagram for Hawaii. Same coordinates as in Fig. 4.

Figure 7. Persistence diagram for Hawaii, shuffled catalog. Same coordinates as in Fig. 4 however, the scale

range for H is different from the one in Fig.6.

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to H=0.5 (uncorrelated noise). Even if there is still a visible change in persistence for the direction

along the Arc, the pattern is much closer to isotropic random behaviour. The differences between the

two diagrams emphasize the key role played by the succession of events in time. The persistent

character of the dynamic fingerprint for angles that are different from the vertical direction and from the

orientation perpendicular to the arc indicates that there are indeed correlations between events, which

refer both to their spatial location, and to their time of occurrence.

When applied to the earthquake events in Hawaii, the methodology leads to the persistence

landscape in Figure 6. The diagram is different from the case of the Aleutian Arc: there is no sharp

drop in persistence. The spatial behaviour of the seismic activity is moderately orientation dependent.

However, the orientation in the horizontal plane defines together with the vertical component a set of

angle combinations for which persistence is significant and higher than for the other angles. Again, in

order to identify possible implications for earthquake studies related to the underlying volcanic and

tectonic processes, more information is required: one has to find out to what extent this anisotropy is

also a consequence of the succession of events in time, as opposed to the mere spatial location of the

earthquakes. The same type of analysis applied to the shuffled catalog reveals a persistence surface

that is again very different from the case of the original data (Fig. 7). The resulting flat surface is in fact

very close to uncorrelated noise, which emphasizes the importance of the temporal sequence of

events. The Hurst exponent scale is the same in Figures 4 and 6, and in Figures 5 and 7, respectively,

to make a comparison between the two cases possible. A more narrow scale interval is used in

Figures 4 and 6, to provide details regarding the persistence surface.

4. Discussion and conclusions

The proposed methodology is based on an integrative approach to seismicity patterns. While

the initially developed ETA was applied to assess the dynamic fingerprint of a seismically active zone

as a whole, the presented approach focuses on the identification of spatial anisotropy characterizing

patterns of earthquake occurrence. To accomplish this task, the spatial information is taken explicitly

in consideration, while the temporal dimension is included indirectly, as the succession of events in

time. However, by analyzing the resulting events threads both for the original data and for shuffled

catalogs, the effect of the temporal sequence becomes visible. The described smoothing dimensions

method applied to events threads represents thus a multiscale approach referring to space and time.

The proposed ETA approach reveals features of seismicity patterns that are not identifiable

using conventional methods. It offers a rigorous description of the spatial anisotropy of seismicity,

which is not only an expression of the location of hypocenters, but also a consequence of the

succession of events in time. Since these different effects can be distinguished from each other with

the help of this framework, the proposed approach provides a quantitative link between spatial

seismicity patterns and their temporal aspects.

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The methodology is particularly useful in the case of comparative studies concerning different

seismically active zones. If applied to the same zone for successive time intervals, it supports studies

regarding the temporal evolution of seismicity. Another possibly fruitful application concerns the

assessment of earthquake models and their critical comparison to earthquake catalogs.

Information from a new perspective can thus be obtained and interpreted together with results

from other studies, i.e. geological and geophysical investigation. Persistence aspects offer indications

that can be used in risk assessment studies: if a seismicity pattern is monitored and investigated

continuously with the proposed method, the resulting persistence values add information about the

series of events expected in the future. This adds a fine-tuning tool to risk evaluations based on

overall event patterns: beyond the probability of event occurrence corresponding to a certain time

interval, one can gain insights regarding temporary tendencies in relation to certain spatial directions.

In a wider context, the events thread approach can be applied to a variety of geosystems,

especially in situations where the significance of events succession is studied. Depending on the

concrete application, the method can be complemented by spatial analysis methods as well as by

statistical analysis of time series: the required phase space is selected both in terms of its dimension

and of its variables, which can lead to a flexible and powerful framework for the analysis of complex

natural systems.

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