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Nonlinear response of radon and its progeny in spring emission
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  Nonlinear response of radon and its progeny in spring emission Nisith K. Das a,  , Prasanta Sen b,  , Rakesh K. Bhandari a , Bikash Sinha a,b a Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata 700 064, India b Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700 064, India a r t i c l e i n f o  Article history: Received 26 March 2008Accepted 25 September 2008 Keywords: Radon-222Polonium-218Thermal springsNonlinear quantifiers a b s t r a c t Temporal fluctuations of Rn-222 and Po-218 emanated from a thermal spring have been investigated.Nonlinear statistical approach has been employed to the time sequences as to bring out the ingrainedstructure of the experimental data and the underlying mechanism therein. It is observed that theirregular time series are nonrandom and consistent with the nonlinear process. In addition, our findingsreveal that the experimental time dependent data bears the signature of chaotic traits. &  2008 Elsevier Ltd. All rights reserved. 1. Introduction Thermal spring provides the deep earth volatiles a majorpathway to the surface. The amount of volatile release is largelygoverned by the nature of the fluid reservoir and permeability of the conduits, however the radiogenic gases such as radon (Rn-222) and its progeny (Po-218) owe to the spread of   a -emitters inhost rocks and the flow rate of the carrier gases (N 2 , CO 2 , CH 4 ).Continuous temporal measurement of radon and its progeny arebeing carried out for a considerably long time in thermal springgases. Radon emerges as an important component of two phasefluid flow from the thermal spring through intricate network of fluid conducting channels to the surface.Natural and geophysical observations are not always regular innature, however, in many occasions follow Gaussian distribution(Turcotte,1992). Accordingly, many a times it is impossible to geta trend in these observations. Fractal statistics turns to be usefulto describe the natural irregularity of temporal data and helps todiagnose the observed irregularities which are not truly randombut having temporal correlation. It has been amply demonstratedthat many seemingly random events in nature, such as earthquake(Papadopoulos and Vassilis, 1992 ) , fault system, function of heartand brain besides several other phenomena have inbuilt config-uration that can be expressed by fractional dimension (Cox andWang, 1993; Feder, 1998). Fractal approaches are also being used extensively to characterize real data extracted from broadspectrum of physics, (Takayasu, 1990) atmospheric sciences,geophysics and even biological sciences (Nonnenmacher et al.,1994; Mandelbrot, 1983). In an attempt to explore the underlying structure and the governing mechanism of manifestly aberrantappearance of the time series, fractal statistics have been pursuedwith time dependent experimental data.Since the time series is obtained from natural thermal springemission the data sets invariably contain noise. The presence of noise in the time series under study could mask the actualdynamical information of the system as also affects the precisionof computation of the nonlinear quantifiers. To do away with thenoise present in the signals singular value decomposition (SVD)technique is used. Since the small singular values primarilyrepresent noise, filtered time series, after removal of small SVDs’,with less noise is derived (Das et al., 2006a) for nonlinear analysis.Because surrogate data provides fairly reasonable test foridentifying nonlinearity in real time data set, we employed themethod of surrogate data to test whether the experimental timeseries is in the state exhibiting nonlinear characteristics orconsistent with the linear correlated noise (Theiler et al., 1992).The statistical quantifiers namely correlation dimension andLyapunov exponent are computed for Rn-222 and Po-218 timeseries and those are compared with the corresponding valuesevaluated for surrogates which are conveniently designated asRnS and PoS, respectively. In the present case, the ensemble of stochastic surrogates are generated by phase randomizationkeeping the mean, variance as well as the power spectrum sameas that of the srcinal experimental time series.The macroscopic interactions of the upwelling thermal fluidswith the physical parameters of the adjoining crust likely to bringabout microscopic alterations in the compositions of the escapegases and temporal properties. So as to explore the nonlinear ARTICLE IN PRESS Contents lists available at ScienceDirectjournal homepage: www.elsevier.com/locate/apradiso Applied Radiation and Isotopes 0969-8043/$-see front matter  &  2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.apradiso.2008.09.016  Corresponding authors. Department of Atomic Energy, Variable EnergyCyclotron Centre, 1/AF, Bidhannagar, Kolkata 700 064, West Bengal, India. Tel.:+913323214983; fax: +913323374637. E-mail addresses:  nkdas@veccal.ernet.in, helium_vec@yahoo.co.uk (N.K. Das), prasanta.sen@saha.ac.in (P. Sen).Applied Radiation and Isotopes 67 (2009) 313–318  characteristics of the time series, this paper deals with theanalysis of the experimental data through power spectrum,Hurst’s exponent ( H  ), correlation dimension ( D 2) and Lyapunovexponent ( l ). This would enable us to understand the physicalcauses that lead to irregular yet non-random traits of the timeseries and helps to unravel the underlying attributes of concen-tration profiles of Rn-222 and Po-218. 2. Experimental The monitoring thermal spring is located at Bakreswar(23 1 52 0 30 00 N; 87 1 02 0 30 00 E) in the district of Birbhum in WestBengal, India and situated about 250km away from Kolkata. It islocated in a region of considerable structural complexity with ahighly heterogeneous geological structure of Archean age andwithin a seismo-tectonic range. The computed heat flow throughthis geothermal region is close to 200mW/m 2 , which is more thantwice the global average (Nagar et al., 1996).Adequate precautionary measures have been taken to ensurethat the observed changes in the concentrations are not caused bymeteorological factors but indeed be the manifestations of genuine alterations taking place at depths. Bubbling gases emergefrom fractured rocks at a pressure close to 1.6bar with a flow-rateof about 200Nm 3 /h and at a temperature of 69 1 C. Changes inatmospheric pressure and temperature, therefore, have littleeffect, if any, on the gas concentration variations or on the gasflow rate. The thermal spring gas is composed of nitrogen(  92.2vol%) in addition to helium (  1.4vol%), argon (  2.1vol%), oxygen (  0.9vol %) and methane (  3.4vol%) and is saturatedwith water vapor. The escaping gas bubbles from the spring aretrapped under an inverted SS funnel and transferred by a SS tubeinto the adjacent monitoring laboratory. In order to prevent loss of counting efficiency of radon progenies through moisture, thespring gas is completely dehydrated by passing it through a seriesof drying columns filled with anhydrous CaCl 2 . A buffer volume atthe entrance of the drying columns ensures a steady gas flow sothat the rate or volume of gas taken as sample is uniform duringthe course of the measurements (Das et al., 2006b).An electronic radon monitor type Sarad DOSEman (SARADGmbH, Germany) determines the radon and progeny concentra-tions. The measurement range of this instrument is from 10Bq/m 3 to 4MBq/m 3 for  222 Rn and the minimum detection limit for theprogeny is 0.38counts/min. A 200mm 2 semiconductor detector isplaced inside the measurement chamber to convert the detecteddecays of   222 Rn and the short-lived progeny into energyequivalent voltage pulses. An internal multi channel analyzer(MCA) creates count sums assigned to the different nuclides within a pre-defined sample interval. A chronological sequence of thecounts recorded with in a previously assigned time interval arestored in a non-volatile memory and is subsequently transferredto a computer directory compatible to EXCEL through an infraredinterface module for analysis and data processing. The timewindow set for the above instrument was 120min. All datacollected over an interval of 24h at the spring site are off loadedby the local computer and transferred once a day through theinternet VSAT facility to our main laboratory at Kolkata foranalysis and storage. 3. Result and discussion The observed time dependent variations of radon and itsprogeny from the spring emanations are given in Figs. 1a and b,respectively. The variations look to be quite random. To make outthe statistical characteristics of the data, the frequency distribu-tions of the two time series are plotted in Figs. 2a and b,respectively. A Gaussian distribution is fitted to the observedfrequency distribution and plotted over the histogram for bettercomparison. Even though it shows that the observed dataresembles somewhat Gaussian trend, however, it does not accordwell with the Gaussian distribution. Instead, the probabilities of random white noise and Gaussian noise processes turn to be zero.  3.1. Power spectra Spectral analysis has been used extensively in medicine(Goldberger, 1992) and geology (Dolan et al., 1998). Figs. 3a and b show the power spectral density of radon and its progeny,respectively. For the data set  x ( t  ) the power spectrum is defined as S  (  f  )  ¼  C  .(1/  f  b ), where  S  (  f  ) is the system’s power spectral density,frequency is  f   and  b  is the spectral exponent while  C   stands forproportionality constant (Sakata et al., 1999). The exponent  b  isestimated bylinear regression analysis of the plotof log S  (  f  ) versuslog(  f  ). In practice,  b  varying between 0.5–2.0 is defined as the 1/  f  scaling (Esen et al., 2001). The broad band power spectra of theobserved time series displaying exponential decay towards thehigher frequency with 1/  f   scaling are highly suggestive of underlying fractal structure of the time series (Dolan et al.,1998). It is observed that multiple scaling regions appeared in the ARTICLE IN PRESS Fig. 1.  (a) Time series presentation of Rn-222 fluctuations in spring gases and (b)time series presentation of Po-218 fluctuations in spring gases. N.K. Das et al. / Applied Radiation and Isotopes 67 (2009) 313–318 314  spectrum, while the lowest frequency regions share a commontrend and at high frequencies prominent bumps are present. Therespective spectral exponents are 1.79 for Rn-222 and 1.65 for Po-218. It is rather evident that the temporalvariations of Rn-222 andits immediate progeny Po-218 are typical of 1/  f   fluctuations.Generically the systems undergoing advective motion andunder dynamic equilibrium having thermal noise, in all prob-ability, capture the signature of uncorrelated or white noise. Theresult shows 1/  f   frequency scaling is present in the recorded timeseries having a self similar pattern and it may be considered as asort of temporal fractal. The power law distribution of variablessuch as concentrations and emission rate of progeny may beconsidered as the fingerprint of a self organized critical (SOC)system.  3.2. Rescaled range (R/S) analysis The rescaled range ( R / S  ) analysis is a simple yet robust non-parametric method for fast fractal analysis. This is performed onthe discrete time series data set {  x t  } of dimension  N   by calculatingthe mean,  ¯  x ð N  Þ , standard deviation,  S  ( N  ) and the cumulativedeparture,  X  ( n , N  ), where, ¯  x ð N  Þ ¼  1 N  X N t  ¼ 1  x t   and  S  ð N  Þ ¼  1 N  X N t  ¼ 1 ð  x t     ¯  x ð N  ÞÞ 2 " # 1 = 2 (1)Range of cumulative departure of the data is given by R ð N  Þ ¼  max f  X  ð n ; N  Þg  min f  X  ð n ; N  Þg  (2)Where cumulative departure is defined as follows  X  ð n ; N  Þ ¼ X nt  ¼ 1 ð  xt     x ð N  ÞÞ  0 o n p N   (3)The associated  R / S   analysis for the two time series are presentedin Figs. 4a and b, respectively. It is computed as / R / S  S ffi ( n ) H  . Theslope of the plot of   / R / S  S  versus the time lag  n  on a doublylogarithmic plot gives rise to  H  . The magnitude of   H   indicateswhether a time series is random or successive increments in timeseries are not independent.The fractal dimension,  D , is determined from  H   as  D  ¼  2  H  (Hurst, 1951).The correlation between two successive steps or incrementswas represented by (Feder, 1998):  r  ¼  2 2 H   1  1.For  H   equals to 0.5 ( r  ¼  0) the time series represent a randomwalk or uncorrelated white noise and each observation is fullyindependent of all prior observations.  H   lying between 0.5 and 1( r  positive) implies persistent time series characterized by longmemory effects. The successive increments are positively corre-lated with the preceding observations. For 0 o H  o 0.5, ( r  negative)the exponents indicate anti-persistent and each datavalue is morelikely to have a negative correlation with preceding values. Thefractal dimensions turn to be 1.25 ( H   ¼  0.75) and 1.33 ( H   ¼  0.67)for Rn-222 and Po-218 towards the smaller time scales and thatfor the higher time scales become 1.67 ( H   ¼  0.23) and 1.79( H   ¼  0.21), respectively.Even if the data appears to be randomly distributed in time,  R / S  analysis affirms the time series to be nonrandom. Like the powerspectrum, the  R / S   plots reveal the signature of two distinct scalingregions in each time series. Incidentally, the two time series dataarefound tobe combination of twokinds of signals, persistent andnon-persistent. The two time series express that the data set are ARTICLE IN PRESS Fig. 2.  (a) Frequency distribution of Rn-222 concentration and (b) frequencydistribution of Po-218counts/sec. Fig. 3.  (a) Power Spectrum of Rn-222 time series and (b) power spectra of Po-218time series. N.K. Das et al. / Applied Radiation and Isotopes 67 (2009) 313–318  315  fairly persistent towards the lower time scale while these becomeanti-persistent towards the higher time scale. The analysis, thus,implies two different type of underlying mechanisms and areresponsible for producing the random like fluctuations of the timeseries data. This may be pointed to the fact that the emission of radon and its progeny from their parent element having differenthalf lives constitute a process while their propagation throughfluid conducting channels accompanying other carrier gases suchas N 2 , CH 4 , He and thermal water to the surface involves anothersystem. Mutual interplay between these two processes may leadto the observed variations giving rise to two scaling regions.  3.3. Correlation dimension The dimension of attractor provides important informationabout the nature of the systems giving off temporal signal and theeffective degrees of freedom present in the physical system underconsideration. The measure of correlation dimension ( D 2) helpsus to determine whether a signal behaves like a random or chaoticdistribution. The correlation dimension ( D 2) is computed usingthe sphere-counting algorithm set forth by Grassberger–Procaccia(Grassberger and Procaccia, 1983). The idea is to construct afunction  C  ( r  ) which is the probability of finding those pairs withina circle of radius  r  , the separation of two arbitrary points  x i  and  x  j of the reconstructed phase space. This is done by calculating theseparation between every pair of   N   data points and sorting theminto bins of width  dr   proportional to  r  . A correlation dimensioncan be evaluated using the distances between each pair of pointsin the set of   N   number of points, s ð i ;  j Þ ¼ j  x i    x  j j  (4)A correlation function,  C  ( r  ) is then calculated as C  ð r  Þ ¼  1 N  2   ½ number of pairs  ð i ;  j Þ  such that  s ð i ;  j Þ o r    (5)Alternatively, it may be represented by C  ð r  Þ ¼  1 N  2   X N i ¼ 1 X N  j ¼ i þ 1 Y ½ r     s ð i ;  j Þ  (6)where  Y (  f  ) is the heavyside function. In the limit of an infinitenumber of data and for vanishingly small  r  , the correlation sumshould scale like a power law,  C  ( r  ) p r  D and the correlationdimension is then represented as, D 2  ¼  lim r  ! 0 log  C  ð r  Þ log ð r  Þ  (7)The  D 2 for each time series was computed using lag reconstructedsignals with the pre-determined embedding dimension by thefalse nearest neighbor method. We plotted log C  ( r  ) as a function of log( r  ) and computed  D 2 from the slope of a linear fit.It is seen from the Fig. 5 that the computed  D 2 for the giventime series tend to be constant as the embedding dimensionincreases and the processes thus turn to be deterministic ones. Inthe case where  D 2 increases with the embedding dimension theprocess will happen to be stochastic one. Figs. 6a and 7a reveal that the values of   D 2 for Rn-222 appears to be 2.78 while that forthe Po-218 it becomes 2.54 and those for the correspondingsurrogates come out to be 2.59 and 2.75, respectively, as shown inFigs. 6b and 7b, respectively. The values of   D 2 calculated from thecorresponding surrogates and their deviations are given in Table 1below. It may be noted that the values ( D 2) obtained from thesrcinal time series are at variance with the surrogates. Thesevalues substantiates the Kaplan–Yorke  ( Kaplan and Yorke, 1979)conjecture as to figure out the deterministic traits of the attractor.  3.4. Lyapunov exponent  The Lyapunov exponent ( l ) represents the nature of timeevolution of close trajectories in the phase space and is known tobe the reliable indicator to distinguish between a chaotic from anon-chaotic process. It quantifies chaos by means of estimatingthe sensitivity to perturbations in initial conditions. If the initialseparation of two close trajectories grow exponentially, the value ARTICLE IN PRESS Fig. 4.  (a) Plot of   / R / S  S  vs. ( n ) for Rn-222 time series displaying two scalingregions and (b) plot of   / R / S  S  vs. ( n ) for Po-218 time series displaying two scalingregions. Fig. 5.  Correlation dimension vs. embedding dimension for Rn-222 and Po-218,respectively. N.K. Das et al. / Applied Radiation and Isotopes 67 (2009) 313–318 316  of   l  becomes positive and implies that the dynamics underlyingthe measured signals capture the signature of chaos. A zeroexponent indicates that the system is in a steady state mode(conserves information) as also a negative  l  implies a dissipativeor non-conservative system that approaches equilibrium quickly.To evaluate the Lyapunov exponents it is necessary to monitorthe behavior of two closely neighboring points in phase space as afunction of time. The initial separation of two points in theEuclidian sense is determined and then the separation isexamined over several orbits. Had the two near by trajectoriesstarted off with an initial separation of   d 0 ( t   ¼  0) then thetrajectories diverge with time satisfying the relation,  d t   ¼  d 0 e l t  ,where  d t   is the separation at time  t  . As a matter of fact,  l  is used tocalculate the logarithm of time,  l  ¼  log d 0 /log d t  . For periodicmotion  l max  is equal to zero, indicating that the limit cycleconserves its information in time. The algorithm used in thepresent study to calculate the largest Lyapunov exponent is due toWolf (Wolf et al., 1985). This algorithm, involves embedding thetime series in  m -dimensional space and monitoring the diver-gence of trajectories initially close to each other in that phasespace. The largest Lyapunov exponent can then be estimated from l max  ¼  1 D t  X M k ¼ 1 log 2 d ð t  k Þ d 0 ð t  k  1 Þ  (8)where  d 0 ( t  k  1 ) is the separation distance of two initially closepoints in the trajectory. After a fixed evolution time  D t  , the finaldistance between the two points evolves to  d ( t  k ).In the present study, we have estimated the largest Lyapunovexponent for the experimental and corresponding surrogates asgiven in Table 2 hereunder. It may be noted that there aresignificant variations of the  l  values between the srcinal andsurrogate of the respective data. 4. Conclusion The time dependent variation of radon and its progeny appearto be random in nature, however, through power spectral andHurst’s rescaled analysis non-random pattern of variation isestablished. Estimation of invariants unravels the inherent traitsof the time sequences affirming the nonlinear attributes of theprocess causing radon emission. ARTICLE IN PRESS Fig. 6.  (a) Plot of log  C  ( r  ) vs. log( r  ) to estimate correlation dimension for Rn-222time series and (b) plot of log  C  ( r  ) vs. log( r  ) to estimate correlation dimension forsurrogates corresponding to Rn-222 time series. Fig. 7.  (a) Plot of log  C  ( r  ) vs. log( r  ) to estimate correlation dimension for Po-218time series and (b) plot of log  C  ( r  ) vs. log( r  ) to estimate correlation dimension forsurrogates corresponding to Po-218 time series.  Table 1 Estimated values of correlation dimensions of the experimental time series andcorresponding surrogatesD2_exp D2_surro Variation (%)Rn-222 2.78 2.59 7.33Po-218 2.54 2.75 8.26  Table 2 Estimated values of Lyapunov exponents and corresponding surrogates l _exp  l _surro Variation (%)Rn-222 0.319 7 0.017 0.447 7 0.019 40.12Po-218 0.286 7 0.019 0.414 7 0.017 44.75 N.K. Das et al. / Applied Radiation and Isotopes 67 (2009) 313–318  317
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