N. M. Rotanova, T. N. Bondar, and E. V. Kovalevskaya
Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences, Troitsk
The magnetic field and its secular variations ( SV ) observed on the Earth's surface are a complex function of space and time. The identification of regular patterns in the geomagnetic field relies not only on the acquisition of new experimental data but also on the application of more effective methods of analysis.
As was shown in [Astafyeva, 1996; Dremin et al., 2001; Dyakonov, 2002], the Fourier representation of a process and the application of the maximum entropy method [Cay and Marple, 1981] are ineffective for the analysis of processes with local features. This is due to the fact that the basis function inherent in these methods, being defined in the entire space, is a smooth and strictly periodic function. Such a function can provide constraints only on characteristic periods and intensity of their variations but not on the temporal localization and characteristic scales. The wavelet analysis [Daubechies, 1992; Mallat, 1989; Meyer, 1992] is free from these disadvantages. The wavelet transformation is based on the use of an essentially new basis (a wavelet) represented by a short wave with a zero mean and localized on the temporal (spatial) axis. Wavelets are constructed with the use of special basis functions enabling the representation of a 1-D process in the time-frequency (scale) plane and the analysis of its spectral properties as a function of time or scale.
Abroad, the use of wavelets dates back to the early 1990s. However, recently the interest in wavelets has sharply increased in Russia as well. Presently, a number of reviews have been published in Russian journals [Astafyeva, 1996; Dremin et al., 2001] and monographs devoted to wavelets have been translated [Chui, 2001; Daubechies, 2001]. Wavelets have also been widely used in geophysics [Alexandrescu et al., 1995; Alperovich and Zheludev, 1998; Gilbert et al., 1998; Ippolitov et al., 2001; Ivanov and Rotanova, 2000; Rotanova et al., 2002]. The goal of our work is to apply the wavelet analysis to the study of the temporal structure of the geomagnetic secular variations.
The wavelet transformation aims at the identification of the fine structure of a temporal or spatial series. In the general case, the family of analyzing wavelets is constructed in the form
(1) |
where a is the scale parameter, b is the shift parameter and n is the normalizing coefficient. Then, the continuous wavelet transformation of a discrete series f_{n}(t) is defined as the convolution of this series with the wavelet y(a, b):
(2) |
where y (a, b) is the basis wavelet function and the asterisk means a complex conjugation.
In our numerical calculations, we utilized the MHAT-wavelet [Rotanova et al., 2002], which has a narrow energy spectrum and two equal zero moments. The analytical expression for such a moment is
(3) |
The wavelet transformation of the temporal series f_{n}(t) yields a 2-D set of the coefficients W (a, b) providing constraints on the contributions of components of various scales and their variations with time. Based on the known values of W (a, b), it is easy to estimate some energy characteristics. Among the latter, the simplest and useful parameter is the energy density
(4) |
which characterizes the energy levels of the study process f (t) in the space (a, b).
Since an analogue of the Parseval equality exists for the wavelet transformation, the total energy of the process f (t) can be written, for real functions, through the amplitudes of this transformation as
(5) |
where C^{-1}_{y} is the normalizing coefficient similar to the coefficient in the Fourier transformation. The confidence interval of the energy spectrum of the wavelet transformation is determined as the probability that, at given parameters a and b, the true spectrum lies within the interval of the estimated wavelet power spectrum,
(6) |
where |W (a, b)|^{2} is the true wavelet spectrum, n = N-1, N is the number of observation points, p is the significance level, and x^{2}_{n} (p/2) is the chi-square distribution with n degrees of freedom.
A series of estimates averaged along the shear parameter b or along the scale coefficient a. In the first case, we have the time-averaged power spectrum
(7) |
where Db = b_{2}-b_{1}+1 is the number of averaging points on the horizontal axis b. In the second case, the summation is performed over the scale axis a, and the scale-averaged spectrum has the form
(8) |
Since we examine the series varying with time, the wavelet coefficients are also time dependent. A natural measure for fluctuations of the coefficients is the variance distributions over various scales. The variance at each scale a is calculated by the formula
(9) |
where N is the number of wavelet coefficients on a scale a in a given time interval.
The wavelet analysis is most suitable for studying the fractal behavior of a time series. This is due to the fact that the wavelet transformation provides a signal on various scales obtained by calculating the scalar product of the analyzing wavelet by the signal analyzed. As regards the wavelet coefficients, this implies the power-law behavior of their higher-order moments as a function of scale. The wavelet transformation allows one to calculated the higher-order moments Z_{q} on various scales a:
(10) |
where maximum values of |W (a, b)| are summed. Dremin et al. [[2001] showed that, in the case of a fractal process, this sum should vary as
(11) |
or
(12) |
As follows from (12), the necessary condition for the self-similarity of the process is a linear dependence of log Z_{q} on the scale a. If this condition is met, the dependence of the function t on the rank of the moment q defines the given process as mono- or multifractal. Monofractal processes have a single dimension, whereas multifractal processes are characterized by a set of such dimensions.
Finally, we dwell on a method for constructing the phase space portrayal and the attractor structure for the process studied. For this purpose, we make use of a procedure known in literature as the Takens procedure (in the theory of embedding of nonlinear dynamic systems [Takens, 1981]). This procedure reduces to the following.
Component vectors of the system state are constructed from the original temporal series SV(t):
(13) |
where g_{k} (t_{i}) = g[t_{i} + (k + 1)l] and l is the retarding time. The spatial distribution of the vectors g_{m}(t_{i}) forms the reconstructed phase space of the dimension m.
Figure 1 |
Figure 2 |
Figure 3 |
Figure 4 |
On these scales, fluctuations in the wavelet coefficients and therefore in the variance are highest, with maximum values of D^{} being observed in the 1970s and 1990s. These years are of particular significance from the standpoint of development of singularities. The distribution of the variance of the initial SV field D, shown in Figure 4e, does not exhibit any anomalous behavior in the years mentioned above, thereby indicating a dynamic origin of these phenomena. Thus, the temporal behavior of the variance of the distribution of wavelet coefficients shows that the scales 8-10 years are most sensitive to the development of a singularity in the SV^{y} field. It is possible that precisely these scales disturb the linear behavior of the sum Z_{q} as a function of the scale a.
Figure 5 |
Figure 6 |
The plots presented in Figure 5 indicate that the a dependence of log Z_{q} is nonlinear, i.e., on the whole, the observed process SV^{y} does not possess the property of self-similarity (fractality). However, this dependence is linear for all values of q at large scales ( a2830 ). Therefore, one can state that the process SV^{y} is fractal at large scales a. If this condition is met, the dependence of the function on the rank of moment q (12) determines the mono- or multifractality of the signal analyzed. We constructed the dependences t(q) for large scales a, and their examples for a=30 and 40 are given in Figures 6a and 6b. Noteworthy are two features of these plots: first, the function t(q) is not strictly linear, which is evidence of multifractality of the process, and second, the functions t(q) constructed for different scale values virtually coincide. For this reason, the function t(q) can be considered as a scale-independent measure of the fractal process.
Returning to the results shown in Figure 5, we should note that all plots of the a dependence of log Z_{q} reveal three different structures on the scales a_{1}110, a_{2}1030 and a_{3}>30. These structures appear to be due to different sources. The smaller scales can be naturally related to an external source, singularities of the jerk type from internal sources, and noise. As regards the larger scales, they are evidently related to internal sources.
Figure 7 |
Figure 8 |
Thus, the results of wavelet analysis show that the SV temporal series are described by a process possessing a wide range of time scales. Apparently, this process is a superposition of several components: a stochastic low-frequency component, several regular variations with possible superimposed singularities, and irregular variations of the noise type.
Figure 9 |
Presently, no theory has been developed for elucidating the origin of singularities on the Earth's surface. However, some hypotheses have long been proposed. Thus, in the opinion of Courtillot and Le Mouel [1988], a sharp variation is due to an abrupt change in the generation mode and is associated with additional movements in the upper part of the liquid core that arise in relation to a change in the rotation velocity of the inner part of the core. Possibly existing vertical movements in the liquid core known as upwellings [Voorhies, 1986] can either accelerate or decelerate the Earth's rotation, which should naturally have an effect on the SV generation. Runcorn [1985] believes that singularities in the geomagnetic field can arise due to instabilities in the toroidal field, whereas Hide [1985] regards such variations as a result of the interaction between Alfven waves in the liquid core.
Golovkov et al. [1996] obtained interesting numerical results for the interpretation of jerks in terms of the field of material velocities on the core surface. They showed that the leading mechanism of motions in the liquid core is provided by the rise and descent of material that are described the toroidal component of the velocity field. Vortex motions associated with the toroidal component should be regarded as a secondary effect of the vertical movements.
Let the source of a sharp change in the SV field be the following model of a local disturbance:
(14) |
where q is a step function. In this case, the surface field variation should yield a peak the length and shape of which are determined by the conductive mantle.
We assume that F (j, l) describes a compensated source ( _{S} B_{r} dS=0 ) of a small spatial size. After a signal of the chosen geometry having a steplike time dependence passes through the conductive mantle, the field B (R_{0}, t) on the Earth's surface assumes the form consistent with the observed field variation pattern [Braginsky and Fishman, 1977]. Such a theoretical dependence is sharply asymmetric and has a maximum. Near the epicenter, the maximum is attained over the time 0.3t. As seen, the role of the conductive mantle does not reduce to the delay of the signal by a certain time (retarding time) and to the Gaussian smoothing of a certain width (smoothing time). The temporal dependence is strongly non-Gaussian.
Characteristic slopes of the temporal dependence before and after the maximum are different. Note that the observation data possess the same properties. It is natural to estimate the mantle conductivity from the characteristic increase time because the post-maximum drop is masked by other smooth variations of the field. The characteristic increase time estimated from observation data is t5 years. Hence, the screening parameter is estimated at t17 years, which differs little from previous estimates of the screening and, accordingly, conductivity [Braginsky and Fishman, 1977] obtained from the analysis of local sources of 60- and 30-year variations amounting to ~10 ^{3} S/m at the core-mantle boundary [Papitashvili et al., 1982].
The main results of our study reduce to the following.
1. The wavelet analysis applied to SV temporal series of the last 100 years revealed singularities ~10 years long that were most intense in 1970 and 1990.
2. The field SV^{y} was subjected to the dispersion analysis based on the results of wavelet transformations. The analysis detected scales ( a = 8 10) that are most sensitive to singularities in the SV^{y} temporal series and resolve best the singularities of 1970 and 1990.
3. The results of the SV^{y} wavelet transformation were used for calculating higher moments Z_{q}; their estimates showed that the process analyzed does not possess the property of self-similarity (fractality). The process can be regarded as fractal only on large scales a.
4. The scale diagrams and the higher moments Z_{q} obtained from the SV^{y} wavelet analysis indicate the presence of three different structures depending on scale and most likely due to different sources.
5. The wavelet analysis results allowed us not only to identify singularities in the SV^{y} field but also to trace variations in its energy parameters throughout the time interval studied.
6. The phase portrayal (attractor) constructed for
SV confirmed the complexity of the
SV structure.
The range of the dimension of the
SV dynamic system is estimated at
2
8. Based on analysis of the jerks, the characteristic time of screening of the mantle was estimated at t17 years, which yields s_{H.M.} l.m. 10 ^{3} S/m.
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