Power-law frequency distribution of H/V spectral ratio of seismic signals: Evidence for a critical crust
© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2012
Received: 4 March 2011
Accepted: 5 August 2011
Published: 2 March 2012
We analyse records from several seismic stations in Central Italy (including some with multiple recording sites) for the statistical properties of spectra of both noise and earthquake signals. The main result is that the power-law scaling of most of the spectral H/V ratio statistical distributions leads to a so-called heavy tail. This is interpreted as a statistical distribution with a fractal dimension of about 2, which is almost certainly caused by a porous percolating medium beneath the stations due to some universal property of the crustal rocks crossed by the elastic waves propagation suggesting that the uppermost crust is in a critical state.
Key wordsSeismic signal critical crust H/V spectral ratio fractal dimension
Amongst the statistical distributions of physical phenomena, we can distinguish between wild and mild distributions according to the character of their extreme values, i.e. their distribution tails. In particular, the terms wild or mild are here related to the presence of heavy or thin tails, respectively, in the frequency-size distributions. Current usage of these terms regards the idea that heavy-tailed distributions are those that have power-law frequency-size distributions, and where not all moments are finite; these distributions arise from scale-invariant processes (Ghil et al., 2011). What is interesting is that distributions with a wild character, such as the power-law distributions, in contrast to those with mild ones, such as Gaussian distributions, are ubiquitous in many natural phenomena characterised by some criticality process (Sornette, 2006, 2009). This can be explained by the presence of their heavy tail that emerges from the non-linear coupling properties, when the probability of values far from the statistical mode (such as the occurrence of disasters with respect to normal situations) has still some chance of occurring. Thus, the presence of this heavy tail could be a sign of non-linear characteristics of the dynamical system of the quantities under investigation.
Statistical distributions of some quantities related to seismicity are of considerable importance for scientific studies either per se, or in order to assess reasonable seismic hazard.
Another known statistical relationship is the length distribution of crustal faults, usually described by a fractal (power law) distribution (Scholz et al., 1993) with a fractal dimension in the range 1.4 to 1.7 (Barton, 1985).
Statistical studies of the seismic signal per se can be more difficult. They are mainly based on seismometer data, and may be equipment-dependent. We attempt this approach, but try to avoid equipment-dependence.
It is well-established that seismic site response is related to the characteristics of the uppermost surface layers. Several techniques have been developed to estimate the seismic site response. Amongst these, Nakamura (1989) proposed a simple method to deduce the seismic site response from the spectral ratio of the horizontal H and vertical V components of microtremors (seismic noise). In principle, the ratio between two components eliminates seismometer transfer-functions, and provides an objective measure of the mechanical characteristics of the medium through which the seismic waves propagate and/or other information related to physical phenomena. Nakamura’s spectral (H/V) ratio technique has been largely applied to study site effects (Lermo and Chavez-Garcia, 1994; Field and Jacob, 1995) with the main objective of estimating the fundamental frequency response of soft deposits. This technique is not wholly straightforward as it includes a filtering procedure that is not completely objective in that it requires an arbitrary choice of appropriate parameters. However, by analysing H/V spectral ratios: (i) we avoid any conventional filtering in both time and frequency domains; on the other hand, (ii) we make some restrictions on the analysed data. In this short note, we do not deal with the aspects related to the strict interpretation of the spectral H/V ratio (see, for instance, Fah et al., 2001; Cara et al., 2003; Castellaro and Mulargia, 2009; Surve and Mohan, 2010; Tuan et al., 2011) but we analyse only its statistical frequency distribution for many seismometer digital segments where there appears only seismic noise, mainly produced by anthropic noise such as traffic, industry, etc. We study the heavy tail of the statistical distribution because we are particularly interested in the non-linear properties characterising the seismic phenomenon and/or the medium involved. We then extend this approach to some seismic events. This latter analysis has not been completely developed so only some general considerations can be made.
This paper is organised as follows: we describe the data collection, the related sites of observation, and then we analyse the data in terms of their statistical distribution. Finally, we interpret and discuss the results in terms of fractal properties.
2. Data Collection
Details of sites of Italian seismic stations used in this study.
1. Fabriano (Borgo)
Holocenic-Pleistocenic fluvial-colluvial deposits.
Marly Tertiary bedrock at 10 meters depth.
2. Celano 1
Holocenic-Pleistocenic fluvial and fluvioglacial deposits
(gravels and sands). Carbonatic Bedrock at 10–100 m.
3. Celano 2
Holocenic-Pleistocenic lacustrine deposits (clays, silts and sands).
The bedrock is at several hundred metres depth.
4. Celano 3
Holocenic-Pleistocenic lacustrine deposits (clays, silts and sands). The bedrock is at several hundred metres depth.
5. Celano 4
Outcropping Mesozoic bedrock (stratified limestone).
6. Celano 5
Sub-outcropping Mesozoic bedrock (massive limestone).
7. Fabriano (Vallina)
Holocenic detrital deposits: rubbles, pebbles, gravel with clay.
The depth of the bedrock is at about twenty meters.
8. Fabriano (Cupo)
Holocenic detrital deposits: pebbles, gravel, sands with clays.
The depth of the bedrock is at about twenty meters.
9. Fabriano (Belvedere)
Outcropping Tertiary bedrock: marls, silty marls and calcarenites.
10. Fabriano (Spina)
Holocenic eluvial-colluvial deposits: clays, limes and sands.
The Miocenic-Tertiary bedrock constituted by marly clays is at 3 to 10 metres depth.
3. Data Analysis and Results
First, the three-component (E, N and V) digital data of the seismic noise from each seismic station listed in Table 1 have been fast-Fourier transformed (providing , and , respectively, where indicates the operation of the Fourier transformation over the X component). Second, we estimated the spectral H-component from E and N, such as and computed the spectral ratio. In the following, we will refer to simply as X, meaning that it is derived from a fastFourier transformed operation over the component X, and squaring its value. Then, we estimated the statistical distributions for H, V spectral components and the H/V spectral ratio. Let N (H/V) be the number of occurrences of a given ratio H/V within M samples; p(x) = N(H/V)/M is the estimate of the probability of occurrence of a certain value x of H/V (the actual p(x) would be with the limit of M to infinity). The behaviour of p(x) with x is the statistical probability distribution. However, since we will investigate the scaling properties of the statistical distribution, we can simply analyse the behaviour of N (H/V) with H/V.
This scaling property of the spectral H/V ratio distribution does not depend on the type of instrument used, nor on the kind of frequency distribution that has been analysed. Confirmation of this can be seen when the same analysis is applied to VBB seismometer data from Caltabellotta (Sicily) as shown by Fig. 2 (bottom). Again, a line of slope −2.9 in the log-log distribution diagram closely follows most of the distribution.
The statistical analysis here has been performed with the specific objective of analysing the heavy tail of the distribution, in particular excluding those H/V values smaller then or equal to 1 and greater than around 100, allowing us to focus the analysis over two orders of units. We also built the statistical distribution without considering data points after the first zero value. All this avoids considering the lowest values of both horizontal and vertical components, which could be critical being probably too close to the instrument resolution. The fact of having considered a band-limited H/V should also avoid any aliasing effects, because one could expect that aliasing is usually transferred to small values of H and V, so producing fictitious high or low values of H/V.
4. A Fractal Interpretation
Let us recall that a power-law distribution p(n) with a slope −α (i.e., p(x) ∼ x−α) corresponds to a power-law cumulative distribution with exponent −β = −α + 1 (the only exception is α = 1, when the cumulative distribution is logarithmic), in our case β = 1.9. It is usually assumed that, when β is between 0 and 3, the distribution can be considered fractal with a fractal dimension D = β (Schroeder, 1991); however, the relation between D and β is often considered valid even outside this interval of β (Turcotte, 1997, pages 16–17). In our case, this means that D is close to 2.
The H/V spectral ratio could be, in principle, indicative of the characteristics of the medium through which the seismic waves propagate. The fact that the scaling exponent of most of the spectral ratio distribution is practically the same, either in seismic or aseismic activity, and independent of the recording location, could be interpreted as a more general signature of the propagation medium. This general aspect cannot be extracted by simply analysing the spectral ratio behaviour with frequency.
The fact that the statistical distribution tail of the spectral ratio H/V has a power-law form also for some level of seismic activity (at least, this arises from the analysis of a few seismic events), would imply that the fractal properties are characterising not only the more superficial rocks but the whole crust. Morein et al. (1997) presented a simple 2D array of slider blocks connected by springs, interacting via static friction with a surface. Extending their findings to a 3D percolation model, would result in a fractal dimension similar to that found in our analysis, suggesting that the earthquakes form the foundation for a percolation cluster (Robertson et al., 1995).
Our results not only support this fact, but also add the important consideration that the 3D percolation model seems to persist even without earthquakes, that is without an explicit energy source, since our work concerns the whole seismic signal (noise and earthquake). Since the power-law distribution of percolation models is an indication of a critical state (e.g. Jensen, 1998), we argue that the H/V power-law distribution is evidence that the uppermost crust is in a critical state (Bak and Tang, 1989; Bak, 1996; also Crampin, 2006). The value (near 2) of the fractal dimension in our analysis is close to that attributed to the distributed seismicity of the Gutenberg-Richter relationship (1); it is also the same as that found for the spatial distribution of earthquakes (Turcotte, 1997, chap. 4). We do not believe that these coincidences are fortuitous, but suggest that they arise naturally from the critical state of the rocks composing the uppermost crust.
Analysing the statistical distribution of the horizontal-to-vertical spectral ratio of the seismic signal (both microtremors and seismic events), we find that the power-law cumulative distribution always has a scaling exponent of ∼2. This result is not in contradiction with the use of the H/V spectral ratio for isolating the resonance frequency of soft deposits: the latter application is just the interpretation of a point (or a small subset of points) of the whole statistical distribution. Where this particular distribution comes from is not an easy problem to solve. Power-law distributions are wild distributions that seem to characterise systems with very non-linear dynamics, or with stochastic multiplicative amplification effects (Sornette, 2006). Thus, seismic wave propagation, although apparently stochastic in each component of the seismic signal, is, in some way, organised in terms of distributions of the H/V spectral ratio values. This organisation is probably indicative of the strong non-linear properties of rocks forming the uppermost crust, likely resulting from a critical state (Crampin, 2006). Crampin demonstrates that the criticality is caused by the distributions of fluid-saturated microcracks in virtually all rocks, being so closely-spaced that they verge on failure in fracturing and hence are critical-systems. To better exploit the characteristics of the H/V statistical distributions with particular attention to the cases of increasing seismic activity, which have here been only marginally investigated, some further investigation is needed. This will possibly be the subject of a future work.
This work was initiated whilst ADS lectured as Contracted Professor of Geophysics at the University of Chieti. We would like to thank our colleagues of the University of Chieti (UC) and INGV. Especially: Bruno Di Sabatino and Mario Rain-one of UC, for their discussions and for providing a stimulating environment to play with this idea; Paolo Casale of INGV for providing the data from the VBB station of Caltabellotta (Sicily, Italy); Paolo Favali and Patrizia Tosi of INGV for reading a preliminary version of the work and providing some interesting suggestions. We thank Stuart Crampin and an anonymous referee for their useful comments in reviewing the manuscript. The Raggiano data was taken from a CD-ROM of the executive project GNDT-CNR 1998, 6a1 (prepared by Govoni, A., Spallarossa, P. Augliera, L. Trojani; coordinator: M. Cattaneo). Part of this work was carried out under the SAGA-4-EPR Project, funded by the Italian Foreign Office. We dedicate this work to our colleague and beloved friend Bruno Di Sabatino who passed away in September 2010.
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