Eur. Phys. J. Special Topics 143, 217–222 (2007) c EDP Sciences, Springer-Verlag 2007 DOI: 10.1140/epjst/e2007-00090-2
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Power spectrum crossover in sediments of a paleolake disturbed by volcanism G. Mart´ınez-Mekler1 , E. Ugalde2 , and G. Vilaclara3 1
2 3
Instituto de Ciencias F´ısicas, Universidad Nacional Aut´ onoma de M´exico, Apdo. Postal 48-3, 62251 Cuernavaca, Morelos, M´exico e-mail:
[email protected] ´ Instituto de F´ısica, Universidad Aut´ onoma de San Luis Potos´ı, Alvaro Obreg´ on 64, 7800 San Luis Potos´ı, SLP, M´exico Limnolog´ıa Tropical, Divisi´ on de Investigaci´ on y Posgrado, FES-Iztacala, Universidad Nacional Aut´ onoma de M´exico, Apdo. Postal 314, Tlanepantla, Edo. M´exico, M´exico
Abstract. We study density fluctuations from sediments of a paleolake in central Mexico that was subjected to volcanic perturbations by means of computed tomography (CT) measurements on blocks chiselled out of mines at the lake’s bed. The mine walls show laminations corresponding to the alternation of low density diatom sediments and high density volcanic ash depositions. We have previously shown that there is a range of scales where these fluctuations present a self-similar behavior [1]. Here we relate density correlation calculations to the power spectrum of the fluctuations. We show that a scaling region in the power spectrum coincides with the scaling region in the correlations produced by relaxation from intense volcanic perturbations to steady state fluctuations. There appears to be a kinklike crossover in the power spectrum from mid range scaling to a shorter range scale invariance. This, together with the density probability distribution of the fluctuations, draws attention to the dominant role of rare events. We believe that our analysis may be useful for the understanding of other phenomena with similar power spectrum properties, in which a scale invariance in the unperturbed system is altered by external perturbations that induce an additional scaling behavior.
1 Paleolake sediments Mexico has numerous diatomite beds from paleolakes in regions subjected to intense volcanic activity [2]. Laminae are common in these beds (see Fig. 1.(a)) when diatomite blocks are exposed to X-Rays, a rich and complex structure becomes manifest which may be related to short-period environmental changes (see tomography plate in Fig. 1.(b)). For the case of diatomite sediments, X-ray attenuation measured with computed tomography (CT) is particularly suited for the fine scale determination of this structure. We have performed such measurements for blocks dug out of the walls of the Santa Barbara and El Lucero mines which lie at the bed of an extinct Pliocene-Pleistocene lake in the state of Tlaxcala, Mexico (N19◦ 24’, W98◦ 18). Fig. 2 shows the CT X-ray attenuation series ρ(n) expressed in Hounsfield units (based on a relative scale, 0 for water and −1000 for air) of concatenated blocks with data taken every 0.25 mm, along transects perpendicular to the lamination pattern, measuring a total length of 1.4 meters, corresponding to the range 0 < n < 5607. Since we have verified that these X-ray attenuation values are proportional to sediment density values [3], we shall hereafter interchange both terms indiscriminately. Particle induced X-ray emission (PIXE) measurements which can determine chemical composition at the spatial resolution of the CT, as well as our microscope
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(b)
(a) Fig. 1. a) Laminated sediment from Santa Barbara mine, Tlaxcala, M´exico. b) Computed tomography plate, the arrows show the X ray image of two intense volcanic event registers in the mine.
observations, show that high density material is mainly of volcanic origin while low density is rich in diatom remains [4]. This is mostly due to the empty space left in between the diatom valves (shells). Furthermore, the surface roughness of the material of volcanic origin indicates that the pronounced peaks in the density series correspond to direct ash falls from the stronger volcanic events and not to volcanic material dragged in from the catchment stores (by climatic phenomena). The CT series can hence register the occurrence of the higher intensity volcanic events in the present record, and can be used to discriminate between diatomite and layers rich in volcanic materials. The unusual presence of strong events corresponding to the bigger volcanic eruptions can be appreciated from the right hand tail of the probability distribution histogram of the density fluctuations shown in Fig. 3. Our interest in the study of these sediments is manifold, on one hand they provide information on volcanic events of geophysical interest which occurred about two million years ago, during a span of around 6000 years; on the other hand they may contribute to an understanding of the paleolake’s evolution subject to volcanism. Paleolimnological studies report sediments mainly as records of past climate changes [5–7], without giving due attention to geological changes. Although the effect of tephras (layers of volcanic cinders) in diatom composition has been explored [8, 9], it has not been considered as a main driver for lake evolution. If we regard geological activity, associated to frequent volcanic emissions, as a forcing that can be as important as climate for lake changes in central Mexico, a study of the scaling properties of the CT
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ρ(n) (H.U.)
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Fig. 2. Computer tomography absorption radiation measurements ρ(n), expressed in Hounsfield units. corresponding to 1.4 meters of the mine walls.
Relative Frequenci es
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Fig. 3. Relative frequencies with which the CT density fluctuations shown in Fig. 2 occur as a function of their magnitude expressed in Hounsfield units (H.U.). Bars are the normalized histogram of the fluctuations.
registers may be revealing. In the following, we extend our previous scaling analysis [1], leaving the connection with our diatom microscopy studies to a future report.
2 Computed tomography scaling analysis 2.1 Correlation scaling In order to explore the presence of correlations in the data shown in figure 2, following the procedure outlined in [1], we determine for each integer value s in the interval [0,500] the differences ∆s ρ(n) = ρ(n + s) − ρ(n) for 0 < n < 5607− s. We then retain the negative values which correspond to density drops spaced out sunits and delete the lower 3% of the data in order to diminish the effect of excessively high peaks. From the remaining set we select the minimum value which we denote as min∆s ρ(n). This quantity measures the most pronounced density drop at a given scale s. Fig. 4 is the log-log plot of | min ∆s ρ(n)| as a function of s. Three distinct regions delimited by the vertical dotted lines, at s = 10 and s = 90 , can be identified. The left region gives information related to density drops along the slopes of high fluctuation peaks. The middle region shows a power law behavior indicative of scaling behavior which we associate to density correlations present during relaxation processes from the stronger
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perturbations to “steady state” conditions. The region to the right we attribute to the steady state fluctuation behavior.
|min∆sρ(n)| (H.U.)
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Fig. 4. Log-log plot of | min ∆s ρ(n)| defined in the text, as a function of scale s. The bold line is a least square fit with a 0.297 ± 0.006 slope and parametric correlation coefficient of 0.98. It is an indication of a power law behavior in the range of s values between 10 and 90 which we associate to a scale invariant regime in the relaxation processes, for which the data fluctuations appear to be correlated.
The middle region is dominated by the occurrence of the relatively high fluctuations and provides a statistical measure of the lake’s response to the strong external perturbations; the power law behavior evidenced by the bold line best fit with slope 0.297±0.006 is an indication of the interrelation of this response at different scales within this range. For distances beyond the second dotted vertical line, the small and intermediate fluctuations are independent from the big perturbations; this line determines a scale above which there is a “loss of memory” of the occurrence of the strong perturbations. This value of s (90), typifies a resilience duration related to the relaxation processes. In an equivalent analysis for the more perturbed shaded region of the mine walls shown in the lower part of Fig. 1.(a)., the middle region of Fig. 4 is not present [1]; the higher frequency of strong external volcanic perturbations appears to impede the recovery scaling behavior by interrupting the relaxation processes during which the correlations in the fluctuations settle in. Additionally, if instead of looking into the negative density differences, we consider positive differences [1], the middle region of Fig. 4 is again not present, both for the strongly perturbed case as well as for the less perturbed data of Fig. 2. This is an indication that the scaling is a manifestation of relaxation processes. 2.2 Power spectrum scaling If we perform the Fourier transform of ρ(n) in Fig. 2, we obtain the Fourier coefficients ρ(k) N given by: ρ(k) = ρ(n) exp(i2πkn/N ), where N is the total amount of data. n=1
In Fig. 5 we show the positive square root of the power spectrum of ρ i. i.e. the Fourier amplitude magnitude |ρ(k)|, in a log-log plot against the wave number k expressed in inverse spacing units (1/0.25 mm). The vertical lines indicate the wave numbers corresponding to the s values of the vertical lines shown in Fig. 4, taken in inverse order, namely k = 5607/90 = 62.3 and k = 5607/10 = 560.7. The white line shows a least square fit to the data in between the vertical lines with slope −1.07 ± 0.05. A point that should be stressed from the comparison of Figs 4 and 5 is that the stronger fluctuations which give rise to the to statistical scale invariance of the maximum density drops |min∆ s ρ(n)| plotted in Fig. 4 for the intermediate scale range, roughly in-between s = 10 and s = 90, also regulate the power law trend coming from all the fluctuations shown in Fig. 5 for the same range. Our interpretation is that the external (allocthonous)
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|ρ(k)| (H.U)
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Fig. 5. Square root of the power spectrum of the density fluctuations shown in Fig. 2. The dashed vertical lines delimit kvalues corresponding to the scaling region suggested in Fig. 4. The white line is a least square fit, determined for these kvalues, with slope −1.07±0.05 and parametric correlation coefficient of −0.69.
volcanic perturbations appear to have triggered correlations in the internal (authoctonous) lake fluctuations which might have been generated by diatom population self-organizing processes leading to “steady state” conditions. Finally, we relate the apparent power law behavior beyond the second vertical line to internal steady state like fluctuations present at smaller scales, not associated to the transient relaxation dynamics mentioned above. This type of crossover with a kink in the log-log plot of the power spectrum has been encountered in other natural phenomena which share the feature of having a scale invariant regime perturbed externally. Such is the case of the magnetic field fluctuations in the poles perturbed by solar wind [10]. Furthermore, power spectrum crossover kinks such as the above have also been reported in sandpile models [11]. A general comment is appropriate at this stage. Usually a power law requires a power functional form for a range of several decades. Though in figures 3 and 4 this behavior is found for only one decade our findings are indicative of a power law trend within a reduced range and provide a means for distinguishing different data correlation regimes. With regard to the slope values, they are not expected to fulfill exponent scaling relations. If the straight lines shown in Figs. 4 and 5 would have held for a range of values covering most of the data and if the bulk and tails of the data probability distribution would have presented the same scale invariance, then the exponent scaling relation 2(α + β) = −1 should have been approximately satisfied, where α and β are the slopes coming respectively from Figs. 4 and 5.
3 Summary Our data is a unique source for the exploration of the effects that the stronger external perturbations have on internal fluctuations. Statistical correlations are present, which generate scale invariance in the relaxation processes from big perturbations to steady state conditions. The large rare events appear to have triggered the onset of correlations in the system’s response to these disturbances. This scaling might be related to self-organizing processes in the lake’s population dynamics during recovery periods. The absence of the above in more perturbed regions, where the higher frequency of the disturbances interrupts the relaxation processes, suggests that scaling is present if the recovery time (resilience time) is shorter or comparable to the characteristic time between the strong perturbations. We acknowledge partial financial from grants: CONACyT-34512E, CONACyT-47836F, ECOSANUIES-M04-M01, and DGAPA-UNAM-IN116198 for financial support. G.M.-M. and E.U. thank the hospitality of the Centre de Physique Th´eorique-CNRS Luminy.
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