The multifractality of the internal geomagnetic field

It is well established that the external geomagnetic field exhibits distinct multifractal behaviour with multiple characteristic timescales that are a manifest of the complex solar-driven dynamics of the magnetosphere or ionosphere. The internal magnetic field on the other hand is characterized by much longer timescales. Consequently, in order to observe any multifractal behaviour, there is needed a time series of magnetic dipolar moment spanning thousands or even millions of years of the past geomagnetic field of the core. Here, we analyse two series that describe the dipolar ingredient of the geomagnetic field for the last 4 and 2 million years, respectively. The first series is constructed from a continuous streak of sedimentary samples while the other series is a composite of a myriad of shorter time series contained within the 2 million years frame. We analysed the Generalized Hurst Exponent through the Multi Fractal-Detrended Fluctuation Analysis method and in each case, we observed typical multifractal structures. These results may be the manifestation of different phenomena evolving in the liquid outer core, possibly providing hints and insights into the details of the corresponding mechanisms. Analysis of the time evolution of the Generalized Hurst Exponent for each series displays a complicated behaviour. Typically the anti-persistent character of the series becomes more evident when close to e dipole reversal. Special efforts need to be dedicated in identifying connections between geomagnetic reversals and time dependence of the Generalized Hurst Exponent.


Introduction
The geomagnetic field of the Earth is distinctly complex having a rich variety of temporal and spatial features [1,2,3].The field observed on the Earth's surface is in reality a superposition of several magnetic fields with sources of varied nature [1].This combination gives rise to an intricate system characterized by multiple temporal and spatial scales [2].The dominant contribution in the measured geomagnetic field pertains to the dipolar magnetic whose origins lie in the complicated interaction between magnetic field and fluid flow in the liquid outer core of Earth [1,3,4,5].Another great contributor is the external magnetic field which originates in the terrestrial magnetospheres and ionospheres and are heavily driven by the solar radiation [1,2].
The dipolar field can be studies on different temporal scales, focusing on recent geological epochs, by constructing series of the so-called virtual axial dipolar moment (VADM).Obviously the spatial variation are difficult to be resolved when working with paleo-data and the VADM seems a reasonable quantity to work with [1,2,4,5].Typically, the samples can be collected from multiple sources, like: sedimentary rocks, igneous rocks or archaeological artefacts, or others sources of relevance [4,5].The details are explained in the Data section.In this paper we focus on two VADM time series.The first series is compiled from a single streak of sample measurements covering the last 4 Myr (read million years) [4].The second series is a composition of a myriad of existing shorter time series constructed from samples of various origins, spanning a cumulative period of 2 Myr [5].
Many geophysical series, he same being observed in many complex systems, are known to possess some degree of persistence or memory and many display multifractal behaviour.In plain language style, a multifractal series has the property that its parts, with different lengths, have different statistical features when compared to the series itself [6].Such property is widely observed in the external geomagnetic field signals [7,8,9,10].In this paper we pose the question: Is the internally generated dipolar magnetic field multifractal as well?

Data
In the present study we focus on three series of magnetic field data.The first one is the 4 Myr long series of paleo-magnetic dipolar moment magnitude constructed by analysing samples drilled from the floor of the Indian Ocean [4].The series spans approximately 4 Myr starting nearly three decades ago stretching back in time.During this time interval, have occurred several inversions of the dipolar geomagnetic field that are visible in the time series (Fig. 1).Some of them are well-known and extensively studied like the Brunhes-Matuyama [11], Jaramillo and Olduvai inversions [1], etc.The measurements are used initially to construct the series of relative paleo-intensity (RPI) [4].In order to gauge the absolute paleo-intensity (API), there can be employed measurements carried over igneous rocks samples or other quantities of geophysical interest [4,5,12].The gauging typically consists in linearly scaling the RPI using appropriate coefficients [4].

Figure 1.
The palaeomagnetic time series constructed from data extracted by the collected samples on the Indian Ocean floor.This series gives the magnetic field for the last 4 Myr.This figure is drawn with data from [4].
The second series, spanning 2 Myr, is a composite of shorter time series within the time span [5].It has been constructed on several collections of series obtained from paleo-magnetic measurements on samples gathered from lava formations, samples from lake sediments, measurements on pottery or other materials sampled from archaeological sites, etc.The authors tried to diversify the stack as much as possible in terms of spatial and temporal coverage, despite there being some bias due to the nature of each series type.The whole stack consists of 76 time series that were merged by employing a penalized maximum likelihood model.The authors claimed that this composite series is comparably close to the time series already published elsewhere [5].

Figure 2.
The series of the PADM2M series which describes the VADM for the last 2 Myr.This figure is drawn with data from [5].The Virtual Axial Dipolar Moment (VADM) is normalized.

Data and Theoretical methods
The generalized Multi Fractal Detrended Fluctuation Analysis (MF-DFA) procedure can be casted into a five steps procedure.Kantelhardt et al., 2002 [5], represent a pivotal work in formulating and initially exploring such method, whose work we follow here.
The first (non-mandatory) step determines the "profile" of a given compact series, which contains very few zero entries (known as a compact series), as follows: In this formula, N is the length of the original series.It must be mentioned that the first step is not mandatory because the profile can be calculated indirectly and then removed in step 3.
The next step consists of dividing the profile Y (i) (assuming step 1 is completed) into Ns nonoverlapping segments, each of length s.Generally N is not a multiple of the time scale s.Thus, there lies the possibility of excluding some of the elements of the series.In order to avoid such problem, the procedure is repeated twice, starting from both ends producing altogether 2Ns segments.
In the third step we calculate by a simple least-squares method the local trend for each of the 2Ns segments.There is no limitation for the order of the polynomial employed in the fitting process.Consequently, the MF-DFA method is labelled MF-DFAn where n is the degree of the polynomial used in this step.In this paper we use show the results for the linear case because the use of quadratic polynomials leads to similar results.Afterwards, it becomes possible to compute the variances: Each variance is calculated in the forward and backward direction for the series.
In the next step there is calculated the average of the variances for all segments such to obtain the q th order fluctuation function: Definition (4) provides a global measure for the series and it changes with the order q.Such change provides insights in the fractal nature of the series.
In principle the number q can take any real value, but the integer values are found to provide a measure of persistence and are easier to interpret regarding the multifractal structure of the analysed series [5,6,7].In the final step, it is determined the scaling behaviour of the fluctuation function by analysing the log-log plots of Fq(s) versus s, where q takes values in a given range of interest.If the algorithm converges, Fq(s) is scaled with s as described by a power-law [5,6,7,8], that reads: Here H(q) is known as the Generalized Hurst Exponent (GHE) [5][6][7][8][9].The multifractal properties of a series are evident by calculating the scaling exponent 1.
Then, through a Legendre transformation encoded into: one can calculate the singularity spectrum .
This spectrum is flat for a mono-fractal series (in which case each portion of the series has the same properties as the whole series) and not flat for a multifractal series.Typically, the shape of this spectrum is considered a solid evidence regarding the fractality of the series [5,9].Mathematically, if 0 < H (q) < 0.5, the series is considered anti-persistent and indicates that the following values in a series are usually different to the preceding values.Conversely if 0.5 < H (q) < 1, the series is said to be persistent indicating that the following values of the series are usually likely close the preceding values.Finally, if H (q) = 0.5, the series is statistically the same as a random walk.In practice, this assertion is true when H (q) is close to 0.5 [5][6][7][8][9].
The value of H(q) give indication about the fractal nature of the signal where any kind of dependency on q indicates multi-fractality and time-correlations.It is known that the GHE for negative q describes the scaling of small fluctuations because the segments with small variance dominate for in this regime.In contrast, the segments with large variance have a stronger influence for positive q.While the range of physical significance of GHE lies between 0 and 1, it is possible for MF-DFA to produce values greater than 1.In such cases there are present non-stationarities in the time series and the method is considered to be unsuccessful [7,8].
The MF-DFA can be applied successfully in gaining insights regarding the temporal evolution of GHE.We have applied MF-DFA as described above on a moving window with different widths, sliding along the series.Basically this is a combination of the approaches described in [6] and [7].This method yields the temporal evolution of the GHE for a given series with reasonable length and provides crucial information on the temporal dynamics of the fractal structure of the series.
Multifractal structures may arise usually for two main reasons: fat tail distribution or long-term correlations present in systems with memory [5].The latter can be destroyed by a simple reshuffle of the original series exposing a rather artificial multifractality.Conversely, if the multifractality is still present after the reshuffling, this indicates an inherent feature of the time series related to the fat tail distribution.

Results
The 4 Myr time series of dipolar magnetic field is characterized by a robustly determined Generalized Hurst exponent.Here we show (Figure 3, left panel) the H(q = 2) value which coincides with the Hurst exponent of a stationary series [6].Usually it is considered as a reference value for characterising the presence of persistence in the given series.We observe that the value of H is very close to the critical value of 0.5 strongly suggesting for a random walk-like series.Such property is also observed for the external field studied elsewhere [10].The reshuffling of the original series has minor effects whatsoever manifested as a mild decrease in the value of H. the series is inherently multifractal.
The 2 Myr time series (Figure 3, right panel) is characterized by a lower value of H which indicates a mild anti-persistence.There are observed some mild non-stationarities because relation (5) is not robustly satisfied.The reshuffling process sheds light into such observation because the algorithm of determination of the Generalized Hurst exponents fails.Apparently, the time series is constructed in a problematic fashion that may be due to the algorithm used to compile the stacks of paleo-magnetic data.Also, there may be present parasitical long-term correlations that may be induced by the algorithm employed by the authors.The singularity spectrum shows undoubtfully the multifractal nature of each time series (Figure 4).Furthermore, the reshuffling of each series has no effect at all.This fact suggests that the spectrum is more reliable tool in identifying multifractal properties that just the determination of the Generalized Hurst exponent.The temporal evolution of the Generalized Hurst exponent is quite intriguing.It is calculated following the algorithm described in [7] for various window widths.Here (Figure 5) is analysed the width of 400 units.Interestingly there is observed a mild anti-persistence in which is directed to random walk-like values when the magnitude of the moment reaches near zero.Such occurrences pertain to reversals [1,2,3,13,14,15] indicating a possible connection which begs further investigations.
Figure 5.The temporal dynamics of H is quite complex and deserves future attention.Here (Figure 6) is shown the case with window width of 400 units.

Discussions and Conclusions
What can be the causes of multifractality manifested in the time series of the dipolar magnetic field?Quite possibly this is the result of the combination of many mechanisms characterized by multiple timescales, like: the different circulation time scales of the fluid in the outer core [1,2,3]; non-isotropic turbulence [12]; local features of the interface between the outer core and the mantle [16,17], heterogeneous heat flux through the core [1], or other factors not mentioned here.
Under first-hand analysis, the main geomagnetic field for the last 4 Myr, as provided from the paleo/magnetic measurements, is multifractal.The features of the series are unaffected by the reshuffling, indicating inherent multifractality.Regarding the composite series spanning 2 Myr we conclude that it has strong non-stationarity because the generalized Hurst exponent is poorly determined when the series is reshuffled.The Hurst exponent H (for q = 2) is very close to 0.5 for both unshuffled series.Mathematically, the VADM series have characteristics of a random walk.
The multifractality of the 4 Myr series limits the use of the existing data to predict the future evolution of the magnetic field of the core.Therefore, different paradigms for different time scales are needed.Fundamentally this is a manifestation of the complexity already characterising the processes that govern the fluid flow in the outer core.In this paper we have analysed only the time series of magnitudes of the dipolar field.In a forthcoming paper we will analyse the properties of the polarity series, which contain the well-known polarity reversals of the dipolar field.The main objective will be related to identify possible connections between the Generalized Hurst exponent and polarity reversals.
It is possible to analyse the time evolution of GHE for a given series.The time window width plays a crucial role where the greater the width, the smaller the variation of GHE.However, it is observed that the value of the Generalized Hurst exponent gets closer to 0.5 when the magnitude of the dipolar moment drops to small values and becomes considerably lower than 0.5 when approaching a reversal.The latter seems obvious because in such a case the adjacent magnetic moment values tend to be different.Can this property be used to signal incoming dipolar field reversals need to be investigated in future works.

Figure 3 .
Figure 3.The GHE for q = 2: a) for the 4 Myr time series [4] (left panel), b) 2 Myr time series [5] (right panel).The reshuffling process has virtually no effect on the 4 Myr series, but exposes the strong nonstationarities that characterize the 2 Myr time series.

Figure 4 .
Figure 4.The singularity spectrum of the 4 Myr time series (left panel) and of the 2 Myr time series (right panel).The reshuffling has no effect in the spectrum.