The International Sunspot Number, provided by the SIDC in Belgium [6] with worldwide contributing observatories, represents a good indicator for the solar activity. The cycles shown in Figure 1 of approximately 11 years can easily be recognized. During the solar maxima, the number of CMEs (Coronal Mass Ejections) increases. Those CMEs consist mainly of Electrons and Protons, but also heavier elements, originating in solar spots [7,8].

Some of these ejections are directed towards the earth, called magnetic storms that cause e.g. the Northern Lights but also drastically compress the magnetosphere on the day-side of the earth and stretch it on the nightside [9].

Regarding the seismic activity, the worldwide ANSS catalogue [10], the Centennial catalogue, as well as regional data are used to create a representative earthquake index. The data is restricted to events with a maximum depth of 40 km, assuming the postulated effect from the magnetic field to be limited to crustal depths. Furthermore only earthquakes of magnitude 5 and larger are used for the test statistics.

The worldwide ANSS catalogue is created by merging the master earthquake catalogues from contributing ANSS member institutions and then removing duplicate events, or non-unique solutions for the same event. For the study period the catalogue contains 39,343 events with the mentioned characteristics.

For comparison concerning large events also the Centennial earthquake catalogue, which is regarded complete down to magnitude 6.5 between the 1930’s and today, is used [11]. The mentioned catalogue is a composition of different contributing catalogues giving a picture of worldwide significant earthquake events (Figure 2). There are 2081 earthquake registrations with magnitude greater 6.5 in the period of interest.

As Tavares and Azevedo [1] report regionally different effects and for better comparison with the magnetic field strength, the events (from the ANSS catalogue) around the 3 observatories (see following section) are separately taken into consideration. The regions can roughly be described as, Northern South America, Central Europe and Alaska.

Finally, in the attempt to prove Tavares and Azevedo [1] thesis, total field strength values obtained from the geomagnetic observatories in Fúquene (Colombia), Niemegk (Germany) and College (Fairbanks, Alaska, USA) are used. The Observatories represent not only different latitudes (Figure 3), but also different tectonic regimes. Alaska and Northern South America are mostly characterized by subduction earthquakes, whereas Central Europe represents in large parts stable continent with increasing activity towards the Mediterranean.

Especially the data from Fúquene at the beginning of the observation period (starting 1955) shows errors or missing values. In all three time series the data is revised and corrected for such flaws. However, as the goal of this study is to find the 11-year cycles of the solar activity, minor errors in the range of days in the appearance of the curves are not expected to influence the results.

The standard cross-correlation method is adopted to find patterns in the occurrence of Earthquakes that resemble the 11 year solar cycles. Thus, the results have to show significantly higher correlation coefficients than the null-hypothesis (cross-correlation of the Sunspot Index with a random time series). The standard numerical cross-correlation function of two vectors x and y of length N is given by:

For comparison, the series x and y are normalized before using cross-correlation, so that their autocorrelations at zero lag (m = 0) is identically to 1.0. Moreover, possible trends were removed before applying the algorithm. The cross-correlation has also the characteristic of showing common periodicities in the correlated series, for example in the three magnetic field strength variations.

The cointegration method is a statistical method mostly used in economics to find dependencies in two or more time series. The basic concept can be expressed in the following manner (Definition from Sørensen [12]):

The two time series y t and x t are cointegrated if there exists a parameter α such that

is a stationary process.

This definition applies to I(1) processes for and, which means they are integrated of the order 1, thus becoming stationary after differentiation. One also says the process has a unit root, i.e.

for a = 1 (3)

This is the equation of a random walk, so that at every time point the value of depends on the value before and a random term. A standard unit root test would be e.g. testing the Null-Hypothesis of a = 1 (unit-root) versus a < 1 (stationary). For a = 1, the time series is nonstationary and will be drifting from its initial value with increasing standard deviation. The series and are cointegrated if they are both following such a random walk, but having a common trend, i.e. they don’t drift away too far from each other, such that there is always a relationship between the variables as in Equation (2).

The cointegration relationship is also valid for processes of higher order I(p) and there are methods to test cointegration for more than two variables, for example the Engle-Granger Cointegration Test, used in this article [13]. The basic concept of this test is a regression on the following relation (being the n-th time series of same length as):

After calculating the regression, the residuals are tested for a unit-root.

The big advantage of cointegration testing is that normal regression for uncorrelated traces can, in the case of random walks, yield results showing a correlation between the time series where in fact there is none. This phenomenon is called spurious regression.

Figure 4 shows the results of cross-correlation between the Sunspot Number and the Earthquake and Magnetic Field Strength Data, respectively. A graphical comparison of the Centennial Data is also given in Figure 2. The maximum time lag in Figure 4(a) is restricted to ±2 years, assuming that the effects from high solar activity should be found in that time window. For comparison also the mean-cross-correlation-coefficient between 500 random series and the Sunspot Index is shown as a limiting line.

It becomes quite clear, that none of the series used shows a significant positive correlation. However, it seems like there is a negative correlation especially pronounced for the worldwide ANSS catalogue, which plots

at exactly zero time lag with a coefficient of almost –0.4 (mean random + one standard deviation ≈ ±0.3). It is quite noticeable, that there seems to be stronger negative correlation. The picture changes only slightly when considering time lags of up to five years (Figure 4(b)). However, a time lag of five to six years means, as well as negative values, anti-cyclicality, i.e. more earthquakes in solar minimum phases. The maximum positive correlation coefficient can be found in the comparison of the Sunspot Number with the Earthquake Data from Central Europe, with a value slightly above 0.3 and a time lag of four years. A more detailed picture of this particular case is given in Figure 5(b).

Furthermore, common periodicity in the magnetic field strength data is found according to Figure 6. The most prominent feature is an approx. 20 year cycle that seems to be a stable feature of all three correlations.

Yet another approach is the comparison between the magnetic field strength and the regional earthquake data.

It shows the highest correlation for the case of Niemegk and Central Europe. The correlation coefficient is about 0.4, whereas correlation with a random earthquake profile plus one standard deviation is about 0.3. However, Figure 7 shows that the correlation almost exactly resembles the input values of field strength at Niemegk. This also means that there is equally high correlation not only for zero, but also for about 18 years time lag, which is difficult to explain.

As mentioned in Section 3.2, in order to adopt the cointegration method it is important, that the analysed time series have a unit root. Intuitively one already suspects the geomagnetic records to meet the criteria for

cointegrated time series. The series follow a random walk (secular variations) with some degree of individual, random freedom (magnetic storms, currents in the ionosphere...), but undoubtedly their general behaviour is linked to each other.

On the other hand, the Sunspot Number does not show those random walk characteristics, resembling a stationary process with a rather strict periodicity. Also the earthquake occurrence is stationary, following (in a basic model) a Poisson Distribution.

The test statistics show indeed a cointegration relation between the three magnetic field strength records. The graph for the Residuals u_t in Figure 8 is given by

where are the records from the observatories, constant factors and c is a constant that shifts the curve to zero mean.

Applying the cointegration with the Sunspot data, although it contains no unit root, results either in an associated factor of α almost zero, or, in the case of SSN = y_t in Equation (4), in the residuals being identically to the Index. Although this is not exactly a valid test, this could mean that Sunspot Number and magnetic field strength (in a relation of year-long cycles) are independent. If there was a common trend or cyclicality the curve form of the Sunspot Index wouldn’t appear in the Residuals.

As the earthquake data is linked to the same problem, further cointegration testing is omitted.