Declination annual mean time series longer than 1 century provided by 24 geomagnetic observatories worldwide, together with 5 Western European reconstructed declination series over the last 4 centuries, have been analyzed in terms of the frequency constituents of the secular variation at inter-decadal and sub-centennial timescales of 20–35 and 70–90 years. Observatory and reconstructed time series have been processed by several types of filtering, namely Hodrick–Prescott, running averages, and Butterworth. The Hodrick–Prescott filtering allows us to separate a quasi-oscillation at a decadal timescale, which is assumed to be related to external variations and called the “11-year constituent”, from a long-term trend. The latter has been decomposed into two other oscillations called “inter-decadal” and “sub-centennial” constituents by applying a Butterworth filtering with cutoffs at 30 and 73 years, respectively. The analysis shows that the generally accepted geomagnetic jerks occur around extrema in the time derivative of the trend and coincide with extrema in the time derivative of the 11-year constituent. The sub-centennial constituent is traced back to 1600 in the five 400-year-long time series and seems to be a major constituent of the secular variation, geomagnetic jerks included.

The temporal variation of the geomagnetic field has been monitored for decades, mainly by continuous recordings in geomagnetic observatories. In spite of their growing number, their geographical coverage is highly uneven. Since the longest recorded series of observations at geomagnetic observatories do not exceed some 150 years, research has seen increasing interest in historical spot measurements to construct time series of geomagnetic elements (declination and inclination) going back over centuries as long as possible (Malin and Bullard, 1981, for the London area; Cafarella et al., 1992a, b, for Rome; Barraclough, 1995, for Edinburgh; Alexandrescu et al., 1996, 1997; Mandea and Le Mouël, 2016, for the Paris area; Korte et al., 2009, for the Munich area). Korte et al. (2009) also included archeomagnetic data in order to infer information going back to AD 1400. Such an interest has also been present in Eastern Europe (Bucha, 1959, for the Czech and Slovak territories; Valach et al., 2004, for Slovakia; Atanasiu, 1968; Constantinescu, 1979; Soare et al., 1998, for Romania); reconstructions go back to 1850 and include intensity elements of the geomagnetic field. Also note the collection of declination and inclination data from sea voyages over the 15–18th centuries by Jonkers et al. (2003) and the spherical harmonics (SHs) model by Jackson et al. (2000). This model includes these data and describes the geomagnetic field evolution since 1590 for all geomagnetic elements (the intensity values are based on an assumed uniform dipole decay rate before 1850).

The last 4 decades – since the finding by Courtillot et al. (1978) of
“geomagnetic jerks” – have seen a research focus on the features of the
time evolution of the geomagnetic field originating in the Earth's core
(main field). Geomagnetic jerks are viewed phenomenologically as sharp
changes, within 1 year or so, in the temporal variation of the main field
(secular variation) expressed as the first time derivative of the
geomagnetic field time series or steps in the secular acceleration
expressed as the second time derivative. The definition of geomagnetic jerks
is usually illustrated by declination (

Periodicities have been found in the evolution of the geomagnetic field.
Fourier spectral analysis (e.g., Currie, 1973; Alldredge, 1977; Langel et
al., 1986), the maximum entropy method (Jin and Thomas, 1977),
empirical mode decomposition (Roberts et al., 2007; Jackson and Mound,
2010), and calculations of torsional waves in the Earth's core
(Zatman and Bloxham, 1997; Dickey and de Viron, 2009; Buffett et al., 2009)
have
pointed to periodicities at several timescales, such as

By analyzing the frequency content of the geomagnetic field variability, De
Santis et al. (2003) and Lesur et al. (2018) have been able to reveal the
behavior of the geomagnetic field as either chaotic or stochastic. The
former showed that the temporal spectra of the geomagnetic field at the
Earth's surface could be approximated for the 1871–2000 time span by a
power law with a negative slope of about

Demetrescu and Dobrica (2014) demonstrate the presence, in 24 observatory
time series (annual means of the

In the present paper we focus on declination data and revisit 24 time series
of observatory data by updating the available measurements with additional
ones since 2007; the last year is included in a previous analysis (Demetrescu
and Dobrica, 2014). Additional data allow us to better constrain the 1999
(Mandea et al., 2000) and 2007 (Chulliat et al., 2010) events and to infer a
possible external contribution to the 2003 geomagnetic jerk (Olsen and
Mandea, 2007). Here, new methods are applied in filtering the time series
and novel approaches regarding the quasi-periodicities of the constituents
at longer timescales are considered (Hodrick and Prescott, 1997).
Additionally, special attention is given to the 11-year
solar-cycle-related constituent present in the declination annual means.
Finally, we elaborate on our previous analysis of three very long
declination time series by (i) including two more for Paris
(Alexandrescu et al., 1996, 1997; Mandea and Le Mouël, 2016) and
Edinburgh (Barraclough, 1995), (ii) discussing the first time derivative of the
five time series, (iii) comparing in detail our analysis on jerk occurrence to
the Alexandrescu et al. (1997) and Korte et al. (2009) ones, and (iv) comparing
with time series provided by the

Annual means of declination as given by

Declination and its first time-derivative time series. Example for a high-standards geomagnetic observatory (Niemegk, NGK).

The five long time series referenced above for Edinburgh, London, Paris,
Munich, and Rome have been considered in the present study. Data used, in
the order of decreasing latitude of the location, are as follows.

Edinburgh: raw data published by Barraclough (1995), adjusted to Eskdalemuir Observatory (ESK);

London: raw data published by Malin and Bullard (1981), adjusted to Hartland Observatory (HAD);

Paris: raw data published by Alexandrescu et al. (1996, 1997) and reviewed recently by Mandea and Le Mouël (2016), adjusted to Chambon-la-Forêt Observatory (CLF);

Munich: 11-year filtered smoothing spline fitted to raw data, as published by Korte et al. (2009), adjusted to Fürstenfeldbruck Observatory (FUR);

Rome: assembled time series using data published by Cafarella et al. (1992b) for the Rome area and for three successively operating Italian observatories (Pola, 1881–1922; Castellaccio, 1933–1962; L'Aquila, 1960–2011), adjusted to L'Aquila Observatory (AQU).

FFT power spectrum: observatory declination time series

Fourier spectral analyses (FFT) of the 24 declination time series (Fig. 2a)
and of their time derivative (Fig. 2b), done on detrended time series by
removing a straight line fit from data, show a broad spectral peak at around
73 years (60–100) that dominates by far other (broad) peaks at

At this stage we also note differences between observatories regarding frequencies corresponding to these lines, which are commented upon below. Some of these differences could arise from the different lengths of the time series, as some tests (not shown here) of repeating the FFT for the same time series truncated to different lengths seem to indicate. Demetrescu and Dobrica (2014) notice that dominating powerful signals at larger timescales in the data tend to contaminate the filtered time series meant to show quasi-periodic variations at smaller timescales. That is why in the present paper we apply a Hodrick and Prescott (1997; HP) analysis, which is able to separate oscillatory features at smaller (e.g., decadal) timescales from trends representing variations at larger (e.g., centennial) timescales.

The HP filter separates a time series

Variations at larger timescales seen in the trend given by HP filtering
have been further decomposed into two other oscillations by applying a
Butterworth (1930) filtering with certain cutoffs corresponding to periods
of

Constituents of the first time derivative of the declination at
Niemegk Observatory.

We have applied the described methods on observatory and historical data.
With respect to observatory data, firstly we show in Fig. 3, as an example
of data processing, results for Niemegk. The first time derivative of
declination (the first differences in annual means) is shown in Fig. 3a, together with the trend given by the HP filter. The
cyclic component is also plotted (panel b). No difference exists when
the latter is compared to the superimposed time series obtained as residuals
of filtering the original time series with an 11-year running average window
or with a high-pass 11-year cutoff Butterworth (1930) filter. Verbanac et
al. (2007) show that the residual signal after removing the CM4 core field model
from the annual averages of European observatories has a clear solar cycle
signature and can be modeled down to

Constituents of the first time-derivative trend of declination at
Niemegk Observatory (the red curve in Fig. 3a).

Since the HP filtering applied to the trend is not able to further separate
it into constituents, we apply (a) running averages (Demetrescu and
Dobrica, 2014) and (b) Butterworth (1930) filtering to get the time series
corresponding to the

Let us discuss the differences between observatories seen in frequencies
corresponding to the broad spectral lines singled out above (Figs. 2 and S3).
The information regarding the actual
periodicities at various observatories would not be lost when adopting a
certain average value (e.g., 73 or 80 years) in the data processing. Indeed,
unless the window in the running average or the cutoff value in the
Butterworth filtering is a multiple of the hidden period in the data, the
filtered cyclical component is itself a cyclical component of the same
period as the original component (Appendix in Demetrescu and Dobrica, 2014).
This can be seen in Fig. 4, in which time series obtained using values of 22
and 30 years or 80 and 73 years for filtering data are
compared. In the case of

We remark here that, at odds with the internal inter-decadal and
sub-centennial constituents, the 11-year constituent is very noisy,
on the one
hand because errors in the annual means (measurement noise, baseline
definition, changes in pillars, etc.) are retained almost entirely in this
time series and on the other hand as a result of the time-derivative
operator that enhances noise and brings forward the harmonics of the 11-year
constituent that are not significant in the data (compare also Fig. 2a and b).
The solar cycle length variability (between 8 and 14 years over the
past 10 cycles) also contributes to the noise in the high-passed 11-year
time series. By superimposing spectra of the cyclic component for the 24
declination time series (Fig. S4), the noisiness is
evident. However, in spite of that, some specific lines can be
distinguished:

lines in the 15–25-year interval corresponding to the

lines in the 8–14-year domain corresponding to the 11-year constituent; and

lines in the 4–7- and 2–3-year domains corresponding to the first two harmonics of the 11-year constituent (we note that the 4–7-year signal covers the 6-year signal detected in variations of length of day (Holme and de Viron, 2013) and in wave processes discussed by Gillet et al. (2010, 2015), pointing to a possible core contribution to the observed variation. We also note the presence of stronger peaks in the 2–3-year period domain than those in the 4–7-year one. This observation is in line with the study by Ou et al., 2017).

The historical data have been processed as has been done for observatory
time series. Since data are sparser and sparser before

The first time derivative of the Paris declination time series
(black) and the HP trend (red)

Figure 5 shows the time derivative of the Paris time series together with the
superimposed HP trend; the inter-decadal and sub-centennial constituents of
the trend are also shown. The noise problem becomes stronger when the time
derivative of the declination series is considered. The time derivative
enhances, as expected, short time variations presented at decadal or
shorter timescales by less accurate historical data before

Constituents of the first time derivative of the declination at
all analyzed observatories (European time series in black and non-European in gray).
From

The two constituents of the secular variation for the 24 observatories considered in this study as obtained by an HP filtering, namely the trend and the decadal cyclic variations, are shown in Fig. 6. The trends are referred to as the average value for the time interval in which they are defined. The two constituents of the trend, the inter-decadal and the sub-centennial variations as obtained by a Butterworth filtering, are also plotted. We superimpose the time series from the 24 observatories corresponding to each of the timescales in order to emphasize common features and differences. We also indicate in Fig. 6 the accepted occurrence time of geomagnetic jerks (e.g., Mandea et al., 2010; Brown et al., 2013).

The cyclic constituent obtained from an HP filtering
(Fig. 6b)
is assumed to show the effects of external sources in the data; it is very
noisy and prevents in this form any interpretation regarding this
constituent of the recorded secular variation. However, plotting only data
from the considered European observatories (Fig. 7) emphasizes the strong
presence of the harmonics of the 11-year cycle superimposed on the 11-year
oscillations in the first part of the time series and on the significant
22-year oscillation in the last

The 2003 jerk shown in Fig. 6 has been evidenced for limited areas only
(

In Figs. 6 and 7 the vertical lines mark epochs of accepted geomagnetic
jerks; they occurred around extrema in the time derivative of the trend
variation, produced by a combination of the two constituents at

Demetrescu and Dobrica (2014) have previously analyzed three of the long time series of historical magnetic declination data (London, Munich, Rome) and showed that the sub-centennial variation is present back in time to the 15th century. Here, we define and characterize the sub-centennial variation in the case of the secular variation of declination for five available time series.

The declination evolution since AD 1400 at five locations in
Europe. Annual values as obtained by a cubic B-spline interpolation on
historical observations and observatory data (full black curve) and as
obtained from the

The time series showing the declination at the five locations are plotted in
Fig. 8. We also superimpose time series obtained from

The first time derivative of the HP trend in the data (red) and in the

In the following, only the sub-centennial constituent, which is less affected by
noise than the decadal and inter-decadal constituents, is discussed. The
five curves in Fig. 10, showing the sub-centennial constituent of the trend
plotted in Fig. 9, demonstrate that the latter is not restricted to the last
150 years. So does the sub-centennial constituent derived from

The first time derivative of the sub-centennial constituent at
the five locations from the trend of observed data (blue) and of

Several maxima and minima are evident in the sub-centennial constituent time-derivative plots (Fig. 10) before 1900, in addition to the known ones over the
20th century. By comparing the five time series of the sub-centennial
constituent time derivatives with each other and with the corresponding

Firstly, all curves as derived from the data and/or model show the same maxima
and minima of the sub-centennial constituent after 1850, namely maxima at
1850–1880 and 1920–1930 and minima at around 1900 and 1960; however, the
sub-centennial signal is noisier at the beginning of the five time series
and mismatches to

Secondly, the amplitude of the sub-centennial constituent before and after 1900 seems to be comparable, in spite of the lower quality of the data for the first 300 years of the time series. Larger-amplitude variations at the start of the time series probably stem from the poorly constrained data in that time interval.

Finally, before 1850 the noise in the data is more evident in the case of the London
time series. However, a synchronous maximum around 1800 and a minimum around
1820 can be seen in the London, Paris, and Rome curves, but not in the Munich
and in the

In terms of geomagnetic jerks, Alexandrescu et al. (1997), based on a synthetic declination curve for Paris inferred from the Paris and London series, recognized one event around 1870 in annual and monthly means time series, also detected in Helsinki, Fürstenfeldbruck, and Oslo data. Other possible events are noted around 1700, 1730, 1750, 1770, and 1785. These dates correspond within a decade to the maxima and minima of the sub-centennial variation time derivative plotted in Fig. 10 (1690, 1740, 1780, 1800, 1850–1860, 1900), which is satisfactory given the uncertainty on both sides of this comparison. Data prior to 1870 were considered by Alexandrescu et al. (1997) as too noisy or unreliable to clearly reveal geomagnetic jerks at least of comparable amplitude with the 1870, 1901, 1925, 1969, and 1978 jerks. However, as mentioned above, the amplitude of the sub-centennial variation before and after 1900 is comparable. Korte et al. (2009) compared the Munich smoothed secular variation time series with a time series of the same length (1400–2000) for Paris that contains archeomagnetic and other measured data prior to 1700, is adjusted to CLF, and is spline-smoothed in the same way as the Munich time series. A good agreement characterizes the time interval 1770–2000 and, as the authors stated, surprisingly consistent secular variation and acceleration between the smoothed curves from the two locations is found for the time span 1400–1580, in spite of the rather low quality of the data over this time span. Significant differences between the two locations exist, however, between 1580 and 1770. Korte et al. (2009) also note that in both curves the time interval 1765–1865 seems to be devoid of strong rapid secular changes.

When comparing the possible geomagnetic jerks (called “events” by Korte et
al., 2009) in the Munich curve with the maxima and minima of the first
time derivative of the sub-centennial constituent for the same location
plotted in Fig. 10, we find that most of them (1448, 1508, 1558, 1693, 1741,
1861, 1889, 1932) coincide within 0–3 years with the maxima and minima of our
sub-centennial constituent (1460, 1510, 1560, 1690, 1740, 1780, 1800,
1850–1860, 1900). In the time interval 1765–1865, considered to be devoid of
strong rapid secular changes by Korte et al. (2009), our analysis detects,
as mentioned above, inflections at 1790–1800 and at

We note that the similar variability shown by the field at the five European
locations is not surprising, as the on one hand a core source with Earth-surface effects on a large area was active the whole timespan of the model
(Stefan et al., 2017). On the other hand, the spatial resolution of the

Considering these results we can suggest that geomagnetic jerks are only part of a variation at a much longer timescale: the sub-centennial constituent. A certain contribution, most visible over the last 40 years, also comes from the 20–30-year inter-decadal constituent. The larger noise in the data before 1900 prevents a possible identification of the latter at earlier times, and the only variation that could be observed is sub-centennial.

Our results underline the importance of the time perspective one has on geomagnetic data: besides the contribution of the sub-centennial constituent in defining geomagnetic jerks, what we called “steady variation” based on 150 years of observatory data (Demetrescu and Dobrica, 2014) proves to be only part of a larger timescale variation when 400 years of data are available.

Declination annual means time series longer than 1 century provided by 24
geomagnetic observatories worldwide, together with 5 reconstructed
declination series over the last 4 centuries in Western Europe, have
been analyzed in terms of the frequency constituents of the secular variation at
inter-decadal and sub-centennial timescales of 20–35 and
70–80 years. Observatory time series until 2015 have been processed by
several types of filtering, namely Hodrick–Prescott, running averages, and
Butterworth. Average windows and cutoffs at 11, 30, and
73 years have been used to account for broad lines in the FFT spectra
corresponding to (a) the external solar-cycle-related contamination in the
annual averages, the so-called 11-year or

The accepted geomagnetic jerks occur around more pronounced extrema in the time derivative of the inter-decadal constituent and coincide with extrema in the time derivative of the 11-year constituent (except the 1999 and the 2007 events). Around 1925, 1969, and 2006, the extrema in the sub-centennial constituent coincide in time or are close to the extrema in the inter-decadal constituent, leading to more pronounced geomagnetic jerks. Phase differences between individual time series explain differences in the occurrence time, geographical distribution, and magnitude of geomagnetic jerks. Once the external contributions to the first differences in the observatory annual means – of comparable amplitude with the inter-decadal and sub-centennial constituents – are minimized, the observed secular variation no longer exhibits a clear V-shape at the time of geomagnetic jerks. We are aware, however, that in doing so, some possibly important internal signal could have also been removed, and further work is necessary to elucidate this problem.

The detected extrema in the historical data have been compared with events interpreted in terms of geomagnetic jerk occurrence dates proposed by other authors (Alexandrescu et al., 1997; Korte et al., 2009). It appears that possible “events” in jerk terms, at 1700, 1730, 1750, 1770, and 1785, considered with a question mark by Alexandrescu et al. (1997) because of too-noisy or unreliable data, are occurring close to the maxima and minima of the sub-centennial constituent. The sub-centennial constituent has comparable amplitudes before and after 1900, in spite of the lower quality of the data in the first 300 years of the analyzed time series, making it a reliable tracer of geomagnetic jerks in the past. Unfortunately, because of noise in the reconstructed time series, the inter-decadal variation, a constituent of the secular variation, could not be recovered and complete information on these phenomena is limited.

According to our results, epochs of geomagnetic jerks may vary as much as a couple of years from one series to another. Over the investigated period, some very clear long periods exist between two successive and well-defined jerks. Over this long-term tendency, less well-defined events can be observed, such as the many noted since magnetic satellite data have been available (e.g., Torta et al., 2015). We suggest that the geomagnetic jerk concept should be considered as a more general notion, namely the evolution of the secular variation as a result of the superposition of two (or more) constituents describing the effects of processes in the Earth's core at two (or more) timescales. Revealing the causes of these variations from the point of view of mechanisms in the core is beyond the scope of this work.

The data used in this paper are freely available at World Data Center of Geomagnetism,
Edinburgh,

The authors declare that they have no conflict of interest.

The historical geomagnetic declination has been compiled from papers by the following authors: Stuart R. C. Malin and Edward C. Bullard; Lili Cafarella, Angelo De Santis, and Antonio Meloni; David R. Barraclough, Mioara Alexandrescu, Vincent Courtillot, and Jean-Louis Le Mouël; and Monika Korte, Mioara Mandea, and Jurgen Matzka (op cit). The study has been carried out in the framework of the project IDEI-UEFISCDI 93/2011. Partial results were presented at MagNetE5 (Rome, 2011) and IUGG (Melbourne, 2011) meetings. We are indebted to Lili Cafarella (INGV-Rome), who kindly provided historical geomagnetic data for Italy, and to the anonymous geomagnetic observatory staff and the World Data Centers on Geomagnetism for obtaining and keeping the data used in this study. We also thank Susan Macmillan, Jon Mound, and the topical editor, Nicolas Gillet, for constructive remarks that improved the paper. Edited by: Nicolas Gillet Reviewed by: Jon Mound and Susan Macmillan