Stochastic variability in the atmospheric activity of Jupiter - Springer Link

S T O C H A S T I C VARIABILITY IN T H E A T M O S P H E R I C A C T I V I T Y O F JUPITER I. XANTHAKIS,* I. LIRITZIS and B. PETROPOULOS Research Center for Astronomy and Applied Mathematics, Academy of Athens, 14 Anagnostopoulou Str., Athens 106 73 and C. B A N O S and E. SARRIS Astronomical Institute, National Observatory of Athens (Received 22 August 1994)

Abstract. The atmospheric activity of Jupiter exhibits quasi-periodic character attributed to the stochastic nature of the involved mechanisms. Periodicities between 4-33 years are obtained employing four spectrum analysis methods, (power spectrum employing the Blackman-Tukey window, maximum entropy, Fourier, autocorrelation), whilst, their significance and their stationarity has been established with the application of general statistical tests (Kolmogorov-Smimov, one sample and two sample test, randomness test, chi-square, various orders of autoregressive process, analysis of truncated records).

1. Introduction Activity in the atmosphere of Jupiter is manifested by the apparition of dark matter which develops into the form of belts, of periodically varying intensity and width parallel to the equator of the planet. Peek (1958) and Beebe et al. (1989) in review works present the variation of atmospheric activity as a function of time, while Sanchez-Lavega et al. (1985, 1991) and Kuhen et al. (1993) give a more recent detailed descriptions of the atmospheric activity changes in particular regions. The activity is preceded by the appearance of dark modes and strips, according to Focas and Banos (1964), Banos (1966) which spread out and form the dark belts. A method measuring the activity on Jupiter was presented by Focas and Banos (1964) based on photometric measurements on photographic plates which lead to the establishment of a coefficient of activity R. The coefficient of activity in a spherical zone between +45 ° on Jupiter is given by the following expression. 1 f +45°

R(0 = C J-45o [(1 - I(~)]d~,

(1)

When R is the photometric coefficient of activity, f is the planetometric latitude, C is the constant reference area with limits +45 °, and [ ( ~ ) = B s ( ~ ) / B is the ratio between the intensities B s of a point of the central S(c2) meridian and B of * Deceased. Earth, Moon and Planets 66: 189-212, 1994. @ 1994 Kluwer Academic Publishers. Printed in the Netherlands.

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J. XANTHAKIS ET AL.

the brightest point near the center of the disc, if the center is not the brightest, with limits 0 to 1. Xanthakis et al. (1991) have given a statistical analysis of the coefficient of activity R for the time period 1915-1985 and determined periodicities of 22, 8 and 6 years. The period of 22 years has been obtained by fitting the measured values with trigonometric functions using the method of successive approximations. In order to perform a more extensive study of the search for possible (quasi-) periodicities in the coefficient of atmospheric activity R for the above period, more sophisticated statistical methods of spectrum analysis, have been applied. In this work we have used the same values of the coefficient of atmospheric activity of Jupiter given by Xanthakis et al. (1991), obtained by Focas (1971), Focas and Banos (1964) and Banos and Sarris (1985). In the present work we have investigated the variation of the coefficient of atmospheric activity of Jupiter, that is, if this variation inheres a periodic or stochastic process for the time interval 1916-1985. For this reason, the time-series analysis of Jupiter's atmospheric activity (abbreviated JAA) was approached by three methods of spectrum analysis; Fourier (FFT), Maximum entropy (MESA) and power spectrum (smoothed spectrum using the Blackman-Tukey window). Various tests of significance and stationarity were applied to examine the reliability of the obtained periodicities. Moreover, in our endeavour to interpret the obtained periodic cycles a correlation study was made between JAA and sunspot numbers. Finally, an attempt was made to apply prediction models to forecast the JAA variation in the next few years. 2. Selection of Data

The plates used in this work have been taken: 1. From the archives of Lowell Observatory for the period 1916-1955. 2. From the archives of the National Observatory of Athens for the period 19551985 which include plates taken with the 16'r refractor of Athens, the 25 ~r refractor of Pedele Athens and the 48 Hreflector of Kryonerion Station (northern Peloponesse, Greece). The selection of the plates has been made with special care in order to avoid overexposed and underexposed plates. With this criterion we can have plates giving a contrast as true as possible between dark, semitone and bright areas on the image of the planet. Each plate had a sensitometric step wedge for photometric purposes. Sharpness of the details, visible on the negative, was examined also with care. Taking into account all the criteria for the selection of plates, we obtained measurements in the linear part of the log E - D curve of each plate. Details of the method for the calculation of the coefficient of activity are given by Focas and Banos (1964).

STOCHASTIC VARIABILITY IN THE ATMOSPHERIC ACTIVITY OF JUPITER

191

TABLEI Summarystatisticsof the JAA time series annualvariation. Variable:

Datm

Sample size Average Median Mode Geometric mean Variance Standard deviation Standard error Minimum Maximum Range Lower quartile Upper quartile Interquartile range Skewness Standardizedskewness Kurtosis Standardizedkurtosis

70.0 0.2434 0.243 0.24 0.242212 5.78736E-4 0.0240569 2.875335E-3 0.181 0.307 0.126 0.229 0.258 0.029 4.20071E-3 0.0143482 0.532379 0.90921

A standard error of 3% is estimated on the measurements of the microphotometric tracings. In our present work two separate statistical analyses for two different time series one for the time period 1915-1955, Focas (1971), and another for the time period 1955-1985 have been made.

3. Statistics of JAA Time.Series The statistics of JAA time-series annual variation is shown in Table I (JAA = Datm). The distribution histogram (Figure 1) shows that the number of the annual JAA can be fit quite well by a normal (Gaussian) distribution (chi-square = 6.8, confidence = 34%, median value = 0.243, standard deviation = 0.024 and maximum range in frequency -4-0.216). The distribution function (histogram) is almost symmetric. Furthermore, from the application of one-sample test analysis, the Ho hypothesis about constancy in the variation of JAA is rejected. The smoothing of time-series of JAA (Figure 2) by 3-term and 5-term moving average exhibits low and high frequency components. Also, this JAA can be

192

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represented by a polynomial fit of 8th order (Figure 3), which smooths out highfrequency terms and gives insight to possible long-term modulation. Indeed, the low-frequency component shows that the period of annual JAA values is modulated by one long oscillation of about 33 years.

193

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The high-frequency component superimposed on this modulation appears visually to be of a stochastic nature. To examine the possible random variation in the JAA time-series the "randomness test" was applied, on four different JAA time-series which refer to two runs tests.

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195

TABLE II Test of randomness which refers to two runs test i.e. the number that the time-series arises or falls and this number equals the number of points that change phase plus one. Data: datm Median = 0.244 based on 70 observations. Number of runs above and below median = 18 Expected number = 35.9714 Large sample test statistic Z = -4.21056 Two-tailed probability of equaling or exceeding Z = 2.54903E-5 Number of runs up and down = 27 Expected number = 44.333 Large sample test statistic Z = -4.94481 Two-tailed probability of equaling or exceeding Z = 7.6334E-7 NOTE: 3 adjacent values ignored.

As the low probability (which is larger or equal of Z ) indicates, the n u m b e r of runs a b o v e and below the median is significantly b e l o w what would be expected, if the respective time-series were random. The results for the runs test up and d o w n show a total o f 27 runs up and d o w n respectively (Table II), which is significantly less than would be expected in a r a n d o m sequence. In three cases 1-3 adjacent values in the sequence were equal and dropped f r o m the computations. The test statistic follows approximately a normal distribution for large samples as has been shown earlier. Since the confidence level for the a b o v e mentioned four J A A time-series is quite below 0.01 ( Z = 7.6 x 10 -7, Z = 0, Z = 3.7 x 10 -14, Z = 6.34 x 10 -14 and Z = 6.34 × 10 -8 respectively) we conclude that these time-series are not random. It is worth noticing that apart o f the original J A A series, the smoothed series also preserve the property of periodic variation. The autocorrelation function was, also, taken for J A A series (Figure 4). It is noticed that below 90% confidence level (doted lines) the time-series is firstly o f semi-stochastic nature, and secondly, is bound by white noise.

4. Spectrum Analysis The spectrum analysis offers a c o m p a c t description o f the J A A time-series in the frequency domain.

196

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In order to examine the accuracy and stability of the inherent periodicities, the frequency and power spectra density (PSD) were estimated in several complementary ways.


197

(a) Fourier analysis (FFT), (b) Maximum entropy (MESA) and (c) BlackmanTukey power spectrum analysis (PSA), which includes a z2-test too (Liritzis, 1990, Liritzis and Tsapanos, 1993). The results are almost similar, but some details in the computed sideband structure of power spectrum density (PSD) with frequency (i.e. the resolution) in MESA depends upon the order of autoregressive process or the number of lags respectively. Analytically, the spectrum analysis per method is described below. a) FOURIER (FFT) Initially, from the JAA time-series, which had been already smoothed with moving average terms of the kind R~t ) = ( / ~ t - 1 + 2Rt + R~+1)/4, by Xanthakis et al. (1991), the mean trend represented by a polynomial of 2nd order, was subtracted (DATM2 file), (Figure 5). It was noticed that applying a 1st order (mean) or a 3rd order polynomial the result does not differ from this smoothing. Following the 2nd order polynomial subtraction the residuals (DATM2) were analysed by FFT. Figure 6 shows the PSD and several periodic terms between 4-33 years. In fact, these quasiperiods or cycles are: 33 + 5, 14 + 1, 10 -4- 0.5, 7 ± 0.5, 6 ± 0.3 and 4.3 ± 0.2 years. The ± sign is the estimated error in the respective PSD peak calculated from the FWHM of each peak. In order to examine the effect that the subtraction of the trend has on PSD distribution, as well as, on fitting problems due to time-series sideband effects, the mean value of the JAA series was subtracted. The FFT results on the new residuals gave similar PSD structure with respective significance level (5') (KolmogorovSmimov test) in the obtained quasiperiodicities of 4.3 (S > 95%), 6 (S > 95%), 7(S > 95%), 10(S > 95%), 14(75 < S < 95%), and 33(75% < 5" < 95%), years (Figure 7). A short cosinus bell was also applied to 10% of the data in both ends of the JAA time-series and the DATM2 time-series was reanalysed. This tapering gave similar results to the previous spectrum analysis. Finally, an 8th order polynomial (Figure 3) was subtracted, that is, the modulated low-frequency component of ~33 years was subtracted. The analysed residuals gave the PSD of higher-frequencies with quasiperiodic terms ranging between 4.3-14 years (Figure 8). The quasiperiods and the respective significance levels applying again the Kolmogorov-Smirnov test of cumulative frequencies of PSD against frequency, are 4.3 (S > 95%), 6(S > 95%), 7(75% < S < 95%), 10(S < 75%) and 14.3 ( 99%), 13.3 ± 0.5 (S = 95%), 10 4- 0.5 (S = 90%), 6.7 ± 0.2 (S = 95%), 6 ± 0.3 (S > 95%), for L = 60. The Lag = 30 gives a broad PSD spectrum with three prominent components. A long one of 90 years (60-180 years), one at 36 ± 3 (S > 90%) and another one of 20 ± 2 years (S = 97%), as well as, two others of low variance, at 12.8 ± 0.2 (S < 90%) and 9.5 ± 0.5 (S < 90%). It is apparent that Lag = 30 provides a lower resolution and the peaks are superimposed on a high background. 5. Future Trend of JAA Finally, an attempt was made to forecast JAA in the next few years employing two models: (a) the Holt's linear exponential smoothing and (b) Winter's seasonal smoothing. The (a) procedure uses two different smoothing constants. The first constant (alpha) is used to smooth the estimate of the level. The second constant (beta) is used for smoothing the estimate of the trend (a > 0, b > 1).


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