Non-Dipole and Regional Effects on the Geomagnetic Dipole Moment Estimation

Abstract

The study of the temporal evolution of the dipole moment variations is a forefront research topic in Earth sciences. It constrains geodynamo simulations and is used to correct cosmogenic isotope production, which is evidence of past solar activity, and it is used to study possible correlations between the geomagnetic field and the climate. In this work, we have analysed the main error sources in the geomagnetic dipole moment computation from palaeomagnetic data: the influence of the non-dipole terms in the average approach, the inhomogeneous distribution of the current palaeomagnetic database, and the averaging procedure used to obtain the evolution of the dipole moment. To evaluate and quantify these effects, we have used synthetic data from a global model based on instrumental and satellite data, the International Geomagnetic Reference Field: 11th generation. Results indicate that the non-dipole terms contribute on a global scale of <6 % in the averaged dipole moment, whereas the regional non-dipole contribution can show deviations of up to 35 % in some regions such as Oceania, and different temporal trends with respect to the global dipole moment evolution in other ones, such as Europe and Asia. A regional weighting scheme seems the best option to mitigate these effects in the dipole moment average approach. But when directional and intensity palaeomagnetic information is available on a global scale, and in spite of the inhomogeneity of the database, global modelling presents more reliable values of the geomagnetic dipole moment.

1 Introduction

The dipole field is the major contributor to the geomagnetic field at the earth’s surface. Its time evolution plays a significant role in constraining geodynamo models (e.g. Glatzmaier and Roberts, 1995; Christensen et al., 2010). In addition, accurate determinations of the past dipole moment are needed for appropriately correcting the production rate of cosmogenic isotopes (¹⁴C, ¹⁰Be) used for reconstructing scenarios of past solar activity (e.g. Muscheler et al., 2007; Vieira et al., 2011; Roth and Joos, 2013). Finally, geomagnetic dipole moment evolution at decadal and centennial time scales is necessary to address debated questions as the possible link between geomagnetic field variations and earth’s climate (e.g. Gallet et al., 2005; Usoskin et al., 2008; Genevey et al., 2013).

The dipole moment (DM) can be estimated when a global geomagnetic model is available. Spherical harmonic analysis (SHA) is the methodology usually used in the generation of global models of the earth’s magnetic field (Whaler and Gubbins, 1981). This technique is based on SH expansion developed by Gauss in 1838, being the potential of the internal geomagnetic field established at any point (r, θ, λ) over the earth’s surface:

$$V\left( {r,\theta ,\lambda ,t} \right) = a\mathop \sum \limits_{n = 1}^{N} \mathop \sum \limits_{m = 0}^{n} \left( \frac{a}{r} \right)^{n + 1} P_{n}^{m} \left( {\cos \theta } \right)\left( {g_{n}^{m} \left( t \right) \cdot \cos m\lambda + h_{n}^{m} \left( t \right) \cdot \sin m\lambda } \right)$$

(1)

where a is the mean radius of the earth (a = 6,371.2 km), $P_{n}^{m}$ are the associated Legendre functions with integer degree n and integer order m, $\cos m\lambda$ and $\sin m\lambda$ the Fourier functions, and the N is the maximum degree of the spatial expansion. $g_{n}^{m} \left( t \right)$ and $h_{n}^{m} \left( t \right)$ are the spherical harmonic coefficients, also denoted as Gauss coefficients.

The DM is easily calculated from the three first Gauss coefficients. These coefficients (g ⁰₁ , g ¹₁ and h ¹₁ ) provide the contribution of an inclined geocentric dipole, and the DM can be obtained as (see Jacobs, 1991):

$${\text{DM}} = \frac{4\pi }{{\mu_{0} }}a^{3} \sqrt {\left( {g_{1}^{0} } \right)^{2} + \left( {g_{1}^{1} } \right)^{2} + \left( {h_{1}^{1} } \right)^{2} }$$

(2)

where μ ₀ is the magnetic permeability of the free space (μ ₀ = 4π × 10⁷ VsA⁻¹m⁻¹).

When only the axial geocentric dipole is considered, i.e. it is aligned with the earth’s rotational axis, the DM is derived in the axial dipole moment (ADM):

$${\text{ADM}} = \frac{4\pi }{{\mu_{0} }}a^{3} \left| {g_{1}^{0} } \right|$$

(3)

Nowadays, a dipole tilted by approximately 11° accounts for more than 98 % of the geomagnetic field observed on the earth’s surface. The international geomagnetic reference field (IGRF) models describe the evolution of the field during the last century. Their last generation, the IGRF-11 (Finlay et al., 2010), covers the time span from 1900 to 2010 and is developed using instrumental data and satellite data (for the last few decades). During the last century, both DM and ADM are decreasing with rates around 50 × 10⁻³ Am²/year.

To extend the knowledge of field variations to the past, historical directional data (Jonkers et al., 2003), which came from shipboard observations for navigational purposes, have been used in global modelling. The GUFM1 model (Jackson et al., 2000) is the model based on historical and instrumental data collected from 1590 to 1990 AD. But, due to the lack of historical intensity data before 1832 (Gauss, 1833), when Gauss developed a method for its measurement, this model had to assume an estimation of the temporal evolution of the first Gaussian coefficient (g ⁰₁ ) prior to this epoch. Jackson et al. (2000) extrapolated linearly the value of this coefficient in the year 1840, and they assumed a constant rate of temporal evolution of 15 nT/year, which corresponds to the average value of the time derivative of g ⁰₁ from 1850 to 1990.

Prior to 1590 AD, there are no direct measurements of the geomagnetic field elements (declination D, inclination I, and intensity F), and the description of the field is based on indirect measurements of magnetized materials, such as sediments, lava flows, or heated archaeological artefacts. Each provides different types of palaeomagnetic information due to the different processes involved in its remanence acquisition.

The archaeomagnetic and lava flow data acquire their magnetization by a thermoremanence (TRM) mechanism. Archaeomagnetic data come from heated archaeological structures as pottery, tiles, or bricks. They recorded the geomagnetic field acting during their last heating–cooling process. In the case of lava flows, the magnetization was recorded during their natural cooling after eruption. If the age of these cooling events is well-controlled, these data provide spot records of the ancient geomagnetic field. For this reason, detailed reconstructions of the geomagnetic field variations generally use this kind of information (e.g. Kovacheva et al., 2009; Genevey et al., 2013).

In contrast, sediments acquire a magnetization throughout depositional and/or post-depositional remanent magnetization processes (DRM and/or pDRM, respectively). This magnetization mechanism is delayed due to the compaction time required to lock in the magnetization. Consequently, geomagnetic field variations recorded by sediments are smoothed and global models derived from this kind of information present smaller variations of the geomagnetic field elements (Korte et al., 2009). In addition, from sedimentary data, only relative intensities can be determined (e.g. Tauxe, 1993) in contrast to archaeomagnetic and volcanic data, which provide absolute palaeointensities.

In terms of data distribution, the present spatial and temporal distribution of the archaeomagnetic and volcanic data is very inhomogeneous (Fig. 1): for the last 14,000 years, the spatial distribution presents a clear lack of data in the Southern Hemisphere and a high concentration in the European region. In time, 83 % of the data are concentrated into the last 3,000 years, whereas the remaining 17 % is distributed between 12000 and 1000 BC (Fig. 1). The sedimentary data present a slightly better distribution in both space and time (Donadini et al., 2009), and some authors (e.g. Korte et al., 2009; Korte and Constable, 2011; Licht et al., 2013) have preferred to include them in the geomagnetic field reconstructions.

Fig. 1

Spatial (a–c) and temporal (d–f) distribution of the archaeomagnetic and lava flow data for the last 14,000 years. Adapted from Pavón-Carrasco et al. (2014)

Following the above mentioned, the time evolution of the DM given by the global models depends on the data used. On one hand, the archaeomagnetic and volcanic data provide higher temporal variability, but an overfitting of the available data could produce artificially high frequency in the temporal variability. On the other hand, the inclusion of sediment data increases the smoothness of the DM variability. If we take into account the behaviour of the geomagnetic field during the last 170 years (available time span for the GUFM1 and IGRF models), the DM presents a slow temporal evolution. This could be in agreement with the use of sediment data, but a problem arises in that the true frequency content of the DM changes is still not well-known for the last millennia. For this reason, we prefer to use only archaeomagnetic and volcanic data in our study, avoiding the different problems related to the use of sediment data.

A habitual practice to estimate the DM from palaeomagnetic data is by using the geomagnetic field elements F and I and calculating the so-called virtual dipole moment (VDM _i ) as follows (e.g. Genevey et al., 2008):

$${\text{VDM}}_{i} = \frac{{4\pi F_{i} a^{3} \sqrt {1 + 3\cos^{2} I_{i} } }}{{2\mu_{0} }}$$

(4)

where the sub-index i indicates the value of the VDM for an individual point on the earth’s surface. However, not all the intensity data available in the current palaeomagnetic database corresponds to inclination data. This is a common problem in archaeomagnetic studies, for example, when archaeointensities are determined from ceramic fragments where the orientation at the time of cooling is not known. In these cases, the VDM _i cannot be estimated and it is commonly substituted by the virtual axial dipole moment (VADM _i ), obtained as

$${\text{VADM}}_{i} = \frac{4\pi }{{\mu_{0} }}\frac{{F_{i} a^{3} }}{{\sqrt {1 + 3\sin^{2} \phi_{i} } }}$$

(5)

where $\phi_{i}$ is the latitude of the studied site. In palaeomagnetism, the common procedure to analyse the evolution of the V(A)DM on a millennial timescale is by averaging local values of V(A)DM_i to obtain a mean V(A)DM at regional or global scales (e.g. Yang et al., 2000; Macouin et al., 2004; Genevey et al., 2008). The best averaging procedure is still an open question and different authors have followed distinct approaches (see Genevey et al., 2008 for a deeper discussion).

The main objective of this work is to assess the reliability of the different ways proposed in the literature for magnetic DM determinations from palaeomagnetic data. First (Sect. 2), we carry out a quantitative determination of the non-dipole effect when an averaging process is considered. This effect is produced because the geomagnetic field at the earth’s surface, which is recorded by a palaeomagnetic material, contains information about the whole field, not only from the dipole field. The main assumption of palaeomagnetic studies is that the non-dipole contribution of the geomagnetic field is cancelled by averaging.

The next section (Sect. 3) is focused on the impact of the sparse palaeomagnetic data distribution (see Fig. 1) on the computation of the global averages of the V(A)DM. The strong geographical bias of the intensity database toward Eurasia might likely produce erroneous estimations of the global V(A)DM. Korte and Constable (2005b) noticed that some care had to be taken to weight the data properly as a function of their location. Genevey et al. (2008) proposed a simple first-order weighting scheme. We study the reliability of this kind of averaging procedure. We denote this analysis as a regional effect.

In the last section (Sect. 4), we study the limitations of the current palaeomagnetic database to generate geomagnetic field models. During the last few decades, global geomagnetic models based on palaeomagnetic data have been developed (Korte and Constable, 2003, 2005a, 2011; Korte et al., 2009; Korte and Constable, 2011; Licht et al., 2013; Pavón-Carrasco et al., 2014). These models allow a direct estimation of the (A)DM (Eqs. 2 and 3). However, they are based on strongly biased databases (see Fig. 1). The objective of this section is to evaluate the effect of the use of a sparse database as input data in the models and to determine how the Gaussian coefficients are affected indirectly for the database used. Henceforth, this effect will be known as the regional indirect effect.

2 Non-Dipole Effect

Global averages of the V(A)DM _i are commonly developed in palaeomagnetism in order to determine the long-scale temporal evolution of the geomagnetic dipole moment. The main assumption is that the non-dipole contribution of the field is cancelled when these averages are calculated. To check the reliability of this assertion, we propose to work with the IGRF-11 model, which spans between 1900 and 2010.

The IGRF-11 model was generated using instrumental data collected from geomagnetic observatories and satellites (CHAMP, Ørsted, and SAC-C missions). This model is based on a spherical harmonic expansion whose maximum degree, N, is chosen so that the coefficients of the model are reliably determined given the available coverage and quality of observations. For IGRF-11, N was chosen to be 10 up to 1995; thereafter it is extended to N = 13 to take advantage of the accurate data provided by the Ørsted and CHAMP satellites.

To evaluate the non-dipole effect in the global computation of the V(A)DM, we used synthetic data (F and I) from the IGRF-11 model. Synthetic data computed are defined in a geocentric framework. The high accuracy and the good worldwide coverage of this model assure that our results will not be affected by the regional indirect effect. The data were synthesized in a total of 2,561 points distributed homogeneously over the earth’s surface by using all the coefficients of the harmonic expansion. We have computed, every 5 years, the individual VDM _i and VADM _i using Eqs. 4 and 5, respectively. Then, we have calculated the average values, i.e. the global averages, denoted as V(A)DM, which have been compared with the theoretical values DM and ADM (Eqs. 2 and 3) provided by the three first Gaussian coefficients of the IGRF-11 model. The comparison has been quantified by the difference between the V(A)DM and the (A)DM as follows:

$$\sigma_{\text{VADM}} = \frac{{{\text{VADM}}\text{ - }{\text{ADM}}}}{\text{ADM}} \times 100\,\%$$

(6)

$$\sigma_{\text{VDM}} = \frac{{{\text{VDM}}\text{ - }{\text{DM}}}}{\text{DM}} \times 100\,\%$$

(7)

The relative differences are plotted in Fig. 2a and summarized in Table 1S (Supplementary Material). Average values for the whole temporal interval gives differences of 5.4 % between the VADM estimation and the ADM, and 1.7 % between VDM and DM. This result confirms that the non-dipole terms are not completely cancelled after the averaging procedure. However, their contributions are lower (always below 6 % for the last 110 years) than the common errors on palaeointensity estimations: around 10 % (see Donadini et al., 2009). It is also interesting to point out that all σ _V(A)DM are positive, which reflects that the V(A)DM is always higher than the (A)DM in the time span from 1900 to 2010.

Fig. 2

a Relative errors between VADM and ADM ($\sigma_{VADM}$) and VDM and DM ($\sigma_{VDM}$) calculated from Eqs. 6 and 7. Temporal evolution of the b VADM and c VDM curves obtained with synthetic data of the IGRF-11 model, for N = 1 (solid blue), N = 2 (solid green), N = 3 (solid red), and N = 13 (solid black), together with the (A)DM curves represented with crosses, by comparison

To investigate which are the most important non-dipole terms affecting the V(A)DM, we have computed these magnitudes varying the maximum degree N (from the dipole, N = 1, up to the total field, N = 13) of the harmonic expansion of the IGRF-11 model. In Fig. 2b, c the temporal evolution of the V(A)DM computed from the first three field contributions (dipole, N = 1; dipole + quadrupole, N = 2; and dipole + quadrupole + octupole, N = 3) and the total field (N = 13) are shown, together with the theoretical (A)DM. We can observe the well-known decrease of the dipole moment, (A)DM, during the last century. This trend is also presented in the computed V(A)DM, with a decrease of 5.9 % for the VADM and 6.4 % for the VDM, calculated using all the harmonic contributions (N = 13). This decreasing tendency is observed in the entire time interval and does not depend on the degree N considered for the analysis. The first three harmonic terms (N from 1 to 3) present the highest contributions to the V(A)DM estimations. When including N = 4 and higher terms, no significant differences are observed (see Fig. 1S of the Supplementary Material).

3 Regional Effect

In the previous section, we analysed the influence of the non-dipole contributions to the global V(A)DM estimation with synthetic data from the IGRF-11 model and a dense grid homogeneously distributed all around the world. This kind of homogeneous database is not realistic when we are dealing with palaeomagnetic data. In this section, the objective is to study how the inhomogeneous spatial and temporal distribution of the palaeomagnetic database (Fig. 1) affects the regional averages of the V(A)DM _i .

First, we have calculated different regional averages of the V(A)DM _i on a continental scale using a homogeneous grid for each continent (denoted as V(A)DM_continent). Secondly, we used the original locations of the intensity palaeomagnetic database of Genevey et al. (2008) for the last 3,000 years and computed global estimations of the V(A)DM _i directly [V(A)DM] or by using a regional weighting scheme [V(A)DM_W]. In both cases, the data were synthesized using the IGRF-11.

We have called this procedure the regional effect (RE), to distinguish it from our previous study. However, we have to remark that the regional effect is also due to non-dipole contributions, which are highlighted by the regional average computation approach.

3.1 Regional Average of the V(A)DM _i on a Continental Scale Using a Homogeneous Database

We selected six different spherical cap areas of 30° of radius, centred in the star points of Fig. 3, corresponding to the continental regions of North America, Europe and Northern Africa, Asia, South America, Central and South Africa, and Oceania. In each selected area, the synthetic data were generated considering a homogeneous distribution with a density of 173 points in each spherical cap. The quantification of the RE [Axial Regional Effect (ARE), and RE] was calculated by the relative difference between the V(A)DM_continent for each continent (with a sub-index indicating the name of the continent) and the theoretical values of the (A)DM as follows:

Fig. 3

Map showing the considered continental areas in the study of regional effect. The spherical caps (green circles) are of 30° of radius, and are centred in the yellow stars. The geographical distribution of the ArcheoInt database (Genevey et al., 2008) for the last 3,000 years are also shown (black and grey points). Definition of the eight regions (each 30° width, both in latitude and longitude) chosen for the next VADM and VDM computations as in Genevey et al. (2008): 1 Western Europe (latitudes between 30N and 60N, longitudes between 10W and 20E), 2 Central Europe and near East (latitudes between 30N and 60N, longitudes between 20E and 50E), 3 Central Asia (latitudes between 12N and 42N, longitudes between 55E and 85E), 4 Eastern Eurasia (China; latitudes between 20N and 50N, longitudes between 95E and 125E), 5 Far East (Japan; latitudes between 20N and 50N, longitudes between 127E and 157E), 6 Pacific (Hawaii; latitudes between 5N and 35N, longitudes between 190E and 220E), 7 Southwest part of North America (latitudes between 17N and 47N, longitudes between 235E and 265E), 8 Northwest part of South America (Peru; latitudes < 0, longitudes between 270E and 300E)

$${\text{ARE}} = \frac{{{\text{VADM}}_{\text{continent}} - {\text{ADM}}}}{\text{ADM}} \times 100 \%$$

(8)

$${\text{RE}} = \frac{{{\text{VDM}}_{\text{continent}} - {\text{DM}}}}{\text{DM}} \times 100 \%$$

(9)

To investigate the origin of the differences between the V(A)DM_continent and (A)DM estimations we have carried out a more detailed study of the different multipolar contributions affecting the selected regions. Apart from the V(A)DM_continent calculated considering the total field (N = 13), we have also computed the V(A)DM_continent for the total field without the quadrupole contribution, the total field without the quadrupole and octupole contributions, and using only the dipole field (N = 1). The ARE/RE values for all the above mentioned contributions are given in Table 1 and along the text, and plotted in Fig. 4 along with the theoretical (A)DM.

Table 1 Errors (rms) of V(A)DM_continental estimations for the period 1900–2010. The (axial) regional effects, (A)RE, are computed from Eqs. 8 and 9

Fig. 4

Regional averaged of VADM (left column) and VDM (right column) [V(A)DM_continent] curves, synthesized from IGRF-11 model to N = 13 (solid black), N = 13 minus quadrupole term (solid green), N = 13 minus quadrupole and octupole terms (solid red), and N = 1 (solid blue) in a North America, b Europe and Northern Africa, c Asia, d South America, e Central and South Africa, f Oceania. The ADM and DM curves (crosses) are shown for comparison

In North America (Fig. 4a), we observe higher V(A)DM _{North America} estimations than real (A)DM values, but they present a similar temporal trend. The difference between the V(A)DM_{North America} calculated from N = 1 and N = 13 accounts for the importance of the higher non-dipole terms (N > 3) in this region. The small difference observed between the VDM_{North America} and the DM is due to the octupole field that contributes around 3 % to the VDM_{North America}.

In Europe and Northern Africa (Fig. 4b), the main difference between regional and theoretical DM estimations is the temporal evolution of these magnitudes. In contrast to the global decreasing trend of the (A)DM, the V(A)DM_Europe curves present an increasing trend with a minimum around the year 1930. As can be observed in Fig. 4b, these anomalous values are related to the local effect of the quadrupole (with contributions around −9.8 % for the VADM_Europe and −2.3 % for the VDM_Europe) and octupole terms (4.4 % for the VADM_Europe, 5.5 % for the VDM_Europe), because the increasing trend disappears when removing these contributions.

In Asia (Fig. 4c), the V(A)DM_Asia estimations are higher than the (A)DM values. A nearly constant or slightly increasing temporal trend of the VDM is suggested by the regional averages, with a small relative maximum around the year 1960. The quadrupole is the main source of differences between regional and theoretical values with a percentage of contribution of 14.2 % for the VADM_Asia and 11.1 % for the VDM_Asia.

The continent with lower V(A)DM_continent values than the (A)DM is South America (Fig. 4d). Here, deviations between regional averages and (A)DM estimations are greater than 19 % for the VADM_{South America} and 15 % for the VDM_{South America}. This area is under the influence of the South Atlantic Anomaly (SAA) with intensity values lower than expected for that region. The difference between the V(A)DM_{South America} and the (A)DM is mainly due to the quadrupole term (−16.2 % for the VADM_{South America}, −14.8 % for the VDM_{South America}). The contribution of the octupole term affects around the 5.5 % to the VADM_{South America} and 3.5 % to the VDM_{South America}). In this case, the quadrupole and octupole terms act in opposite directions. The first one decreases the value of V(A)DM_{South America}, whereas the second one increases it, being the most powerful the quadrupole term.

In Africa (Fig. 4e), the most important non-dipole term is the quadrupole, with a contribution of the −8.0 and −9.5 % for VADM_Africa and VDM_Africa respectively. However, the V(A)DM_Africa and (A)DM estimations are similar, consequently the non-dipole contribution in the RE is not so strong.

Finally, Oceania is the region where the geomagnetic field is more affected by the non-dipole terms (Fig. 4f). Here the V(A)DM_Oceania reaches the highest values (up to 10.5 × 10²² Am² for the VADM_Oceania and 9.5 × 10²² Am² for the VDM_Oceania), associated with the quadrupole (18.2 % for the VADM_Oceania, 11.8 % and for the VDM_Oceania) and with the octupole terms (7.3 % for the VADM_Oceania, 8.2 % for the VDM_Oceania). Differences between V(A)DM_Oceania and (A)DM estimations are about 35 % for the VADM_Oceania (ARE) and 19 % for the VDM_Oceania (RE).

The values contained in Table 1 show that, in general, ARE is greater than RE, and that these errors can be locally very high. The high errors and the differences observed between VADM_continent and VDM_continent suggests that the use of mixed VADM/VDM curves, commonly combined in palaeomagnetism due to the lack of inclination values (e.g. Genevey et al., 2008), introduces an additional source of errors. Then, it is not an appropriate approach.

On the other hand, the palaeomagnetic database for the last 14,000 years is clearly biased (Fig. 1): for the last 8,000 years the archaeomagnetic data are concentrated in Eurasia, while for the earlier times, from 12000 to 6000 BC, the available data are mainly lava flows from Hawaii and North America. This means that if V(A)DM estimations are not adequately averaged, they might be clearly influenced by the regional effect. However, we must point out that the RE depends on the geomagnetic field structure and then it is time-dependent, i.e. our values cannot be directly extrapolated for the past, but provide a reliable idea about the order of magnitude of the regional effect.

3.2 Regional Average of the V(A)DM _i Using Simulations of the Palaeointensity Data Distribution

One of most important problems in the ancient DM estimation, in both V(A)DM and (A)DM, is the inhomogeneous palaeomagnetic database. Most of the palaeomagnetic data are concentrated in the Northern Hemisphere (around 95 % of the archaeomagnetic and lava flow data for the last 3,000 years (Donadini et al., 2009)). This heterogeneous spatial distribution generates problems in the V(A)DM (global average) such as an overestimation of the regions with more available data, as is the case of Eurasia (Genevey et al., 2008). In order to correct this RE in the V(A)DM estimation, Genevey et al. (2008) proposed a simple, first-order regional weighting scheme based on the definition of eight regions (rectangle regions in Fig. 3). These regions were selected taking into account the locations of the palaeointensity data compiled in the database ArcheoInt (Genevey et al., 2008) for the last 3,000 years. They considered that each selected region contains enough palaeointensity information. Here, in order to check the reliability of the regional weighting scheme of Genevey et al. ( 2008 ), we simulate their procedure but using synthetic data from the IGRF-11 model.

The data were synthesized at the locations of the ArcheoInt database (Fig. 3). Since the database contains palaeomagnetic data for the last 3,000 years, whereas the IGRF-11 model only spans from 1900 to 2010, we had to adapt linearly the time interval covered by the database to the last century. That is to say, we simulated a synthetic database with field information given by the IGRF-11 model at the locations of the ArcheoInt database (sites represented in Fig. 3) and we attributed to each data point a fictitious age (linearly adapted) within the 1900–2010 time interval. That is to say, the assigned age has been estimated as follows: t ₂ = m × (1,000 + t ₁) + 1900, where m = 110/2,900, t ₂ the time adapted in the new synthetic database, and t ₁ the time given by the ArcheoInt database.

Two important points to remark: (1) we used all the locations of the ArcheoInt database. That is, we did not introduce the selection criteria used by Genevey et al. (2008) to consider only high quality palaeointensity data. (2) We have synthesized both inclination and intensity data at all locations. However, some of the data of ArcheoInt provide only intensity values without inclination data (58 % of the intensity data) and, therefore, the VDM _i could not be always calculated. This is the reason why the authors used mixed VADM/VDM curves. Consequently, we are considering the best case scenario (i.e. lower errors are expected) for the regional averaging procedure proposed by Genevey et al. (2008).

The regional weighting scheme of Genevey et al. (2008), consists of calculating eight regional VADM and VADM/VDM curves for each selected region, by using the classical sliding overlapping windows technique, and then computing the averaged global VADM and VADM/VDM curves (assuming equal weight for each region).

To estimate the temporal evolution of the V(A)DM_regional we have transformed the original 500-year window shifted by 250 years and a 200-year window shifted by 100 years, into a 20-year window shifted by 10 years and 10-year window shifted by 5 years, respectively. We calculated the regional average V(A)DM_regional from each region and time window, and an estimation of the global weighted averaged V(A)DM, denoted as V(A)DM_W, was obtained. The different V(A)DM_regional for each region are plotted in the supplementary Fig. 2S and the global V(A)DM_W is plotted in the Fig. 5. For comparison, we have also added the global V(A)DM directly calculated from all data, without the regional weighting procedure. The theoretical (A)DM curves are represented as well.

Fig. 5

Effect of the geographic bias in the distribution of the synthetic data on the estimates of the (left) global VADM and (right) global VDM variation curves. Computations are performed using the selected data (see Fig. 3) smoothed over overlapping sliding windows of a 20 years shifted by 10 years, b 10 years shifted by 5 years. Solid blue V(A)DM_W computed with the weighting scheme of Genevey et al., (2008), solid red V(A)DM calculated from all data without the regional weighting scheme, dashed lines show the error band with a level of confidence of 65 %, computed as the involving of 500 perturbed databases (see text for more details), crosses (A)DM calculated from the first three Gaussian coefficients

In order to provide a more realistic result, we have perturbed our synthetic database using a set of 500 random perturbations obtained from Gaussian distributions with mean values equal to zero and standard deviations equal to the standard deviation of the archaeomagnetic data for the last 3,000 years (4.2° for inclinations and 8.6 µT for intensities, Donadini et al., 2009). We have repeated the previous process using the new datasets of perturbed data. The results provide the bands at ~65 % confidence level (dashed lines in Fig. 5) for the V(A)DM_W and for the global V(A)DM (without the regional weighting scheme).

Our results indicate that the variability reported is related to the spatial and temporal distribution of the data. The data distribution is more different among shorter windows, leading to differences in influence of regional bias from one window to the next. The higher variability is an artefact of the RE varying with the data distribution.

The VADM presents an increasing temporal trend, with a maximum value around the year 1970 which is a clear artefact. An increasing trend was also observed in the VADM_Europe curve for the European continent in our previous study (see Fig. 4b). This means that when VADM is obtained from global averaging, the European zone is overestimated because it is the region with more available data (up to 55 %, regions 1 and 2 in Fig. 3). So, when the regional weighting scheme is applied, the influence of European data is weakened and the VADM evolution is more similar to the ADM trend. Although still higher VADM_W values than ADM are obtained, which means that the RE has not been completely cancelled. Deviations between the global V(A)DM_W/V(A)DM and the (A)DM are outlined in Table 2. Lower relative errors between V(A)DM and (A)DM than between V(A)DM_W and (A)DM are obtained. However, this result does not mean that the use of the regional weighting scheme is inappropriate. As we discussed previously, the results are more consistent with the theoretical trend when a regional weighting is considered.

Table 2 Errors (rms) of regionally weighting averaged V(A)DM [V(A)DM_W] estimations for the period 1900–2010 and the V(A)DM without regional weighting scheme, together with the error band (confidence level of ~65 %). The comparisons are developed as a function of (A)DM for the same temporal interval

The lower differences between VDM_W and VDM trends and the lower errors in relation with DM values (see Table 2), are related to the use of more field information: the inclination in addition to the intensity. With this additional information changes in the tilt of the dipole are also considered and, therefore, a more accurate description of the DM is expected. It is important to note that the original ArcheoInt database contains inclination information of around 48 % of the sites. Consequently the errors that we have obtained are the lowest that could be reached.

Finally, we would like to point out that the error bands of the V(A)DM are narrower than those of the V(A)DM_W. The reason is the average procedure: the V(A)DM is obtained with all the data and this high number of data gives lower standard deviations. This is not the case for the V(A)DM_W, where the lower number of regions to be averaged (eight regions) increases the standard deviations.

4 Regional Indirect Effect

In this section, we want to analyse the influence of a sparse database in the models generated from palaeomagnetic/archaeomagnetic data (e.g. Korte et al., 2009, 2011; Korte and Constable, 2011; Licht et al., 2013; Pavón-Carrasco et al., 2014) and especially, its effects on the (A)DM estimation.

We have developed a geomagnetic global model by using the same synthetic database of the previous Sect. 3b, including a new set of synthetic data for the declination, which is necessary to develop the global model. The global model, called IGRF-11_S, was obtained by using the classical approach from palaeomagnetic data (Korte and Constable, 2003): the SHA technique in space and the penalized cubic B-splines (De Boor, 2001) in time. In terms of the SHA, the potential of the internal geomagnetic field can be established at any point (r, θ, λ) over the earth’s surface as (1). The usual time-dependent Gaussian coefficients [g ^m _n (t) and h ^m _n (t)] may be developed using penalized cubic B-splines defined by the matrix B _q (t), as follows:

$$\begin{gathered} g_{n}^{m} \left( t \right) = \mathop \sum \limits_{q = 1}^{Q} g_{n,q}^{m} B_{q} \left( t \right) \hfill \\ h_{n}^{m} \left( t \right) = \mathop \sum \limits_{q = 1}^{Q} h_{n,q}^{m} B_{q} (t) \hfill \\ \end{gathered}$$

(10)

where Q is the maximum degree of the temporal expansion and $g_{n,q}^{m} \left( t \right)$ and $h_{n,q}^{m} \left( t \right)$ are the time-dependent spherical harmonic coefficients.

In palaeomagnetic studies, the measures of the geomagnetic field are D, I, and F. These components cannot be expressed as a linear combination of the Gaussian coefficients. For this reason, any scalar element of the geomagnetic field d (declination, inclination, or intensity) must be given as a non-linear function f, related to Eq.1 and depending on the time-dependent Gaussian coefficients:

$$d = f\left( {\vec{m}} \right) + \varepsilon$$

(11)

where $\vec{m}$ contains all the Gaussian coefficients and $\varepsilon$ is the error. To find the optimal set of time-dependent Gaussian coefficients, we chose the regularized least-squares inversion applying the Newton–Raphson iterative approach (Gubbins and Bloxham, 1985):

$$\vec{m}_{i + 1} = \vec{m}_{i} + \left( {\hat{A}^{\prime }_{i} \times \hat{A}_{i} + \alpha \times \hat{S} + \tau \times \hat{T}} \right)^{ - 1} \times \left( {\hat{A}^{\prime }_{i} \times \vec{\gamma }_{i} - \alpha \times \hat{S} \times \vec{m}_{i} - \tau \times \hat{T} \times \vec{m}_{i} } \right)$$

(12)

where $\hat{A}$ is the matrix of parameters which depends on the SH functions in space and time (the so-called Frechet matrix) and $\hat{A}^{{\prime }}$ is the transpose of $\hat{A}$. $\vec{\gamma }$ is the vector of differences between the input data and modelled data for the ith iteration. The $\hat{S}$ and $\hat{T}$ matrices are the spatial and temporal regularization matrices, respectively, with damping parameters α and τ. The index i indicates the number of the iteration, which requires a first initial solution $\vec{m}_{0}$. To create the B-spline base we have selected knot points between 1899 and 2011 every 4 years.

The spatial regularization minimizes the Ohmic dissipation at the core-mantle boundary (CMB) (Gubbins, 1975), which can be written as:

$$\hat{S} = \frac{4\pi }{{t_{e} - t_{s} }}\mathop \int \limits_{{t_{s} }}^{{t_{e} }} \mathop \sum \limits_{n = 1}^{N} \frac{{\left( {n + 1} \right)\left( {2n + 1} \right)\left( {2n + 3} \right)}}{n}\left( \frac{a}{c} \right)^{2n + 3} \mathop \sum \limits_{m = 0}^{n} \left[ {\left( {g_{n}^{m} (t)} \right)^{2} + \left( {h_{n}^{m} (t)} \right)^{2} } \right]dt$$

(13)

where t _s and t _e are the initial and final epoch respectively and c is the mean radius of the CMB. The temporal regularization minimizes the second time derivative of the radial field at the CMB (Bloxham and Jackson, 1992), as follows:

$$\hat{T} = \frac{1}{{t_{e} - t_{s} }}\mathop \int \limits_{{t_{s} }}^{{t_{e} }} \oint\limits_{\varOmega } {\left( {\partial_{t}^{2} B_{r} } \right)_{r = c}^{2} {\text{d}}\varOmega {\text{d}}t}$$

(14)

where dΩ is the differential solid angle over the sphere Ω. The choice of the best regularization is applied to obtain a model with the minimal complexity and a reasonable fit to the data. After carrying out several tests with different values of damping parameters (e.g. Licht et al., 2013; Pavón-Carrasco et al., 2014), we have chosen α and τ equal to 5 × 10⁻⁹ μT⁻² and 10⁻³ μT⁻² year⁴, respectively. Again, in order to provide a more realistic result, we have also used the 500 perturbations from the dataset of the previous section. In this case, the new element (the declination) was perturbed by a Gaussian distribution with mean 0° and standard deviation equal to 6.1° (from Donadini et al., 2009). A total of 500 models were developed providing the error modelling as the standard deviation of the Gaussian coefficients.

The IGRF-11_S model is compared with the original IGRF-11. In Fig. 6 the maps of D, I, F at 1955 (central epoch of the considered time interval: 1900–2010) from the IGRF-11_S and IGRF-11 models are represented. The differences between these geomagnetic elements are also represented, together with the locations of the data in the time span from 1950 to 1960 (the three knot points considered for 1955). These data represent less than 10 % of the total data. We can observe that the IGRF-11_S reproduces very well the main characteristics of the geomagnetic field (see also the maps provided in the Supplementary Material). The good representation of the Southern Atlantic Anomaly by the IGRF-11_S model is highlighted, in spite of a lack information available from this region (and from the Southern Hemisphere in general).

Fig. 6

Declination, D, inclination, I, and intensity, F, maps at 1955 for (left column) global model generated using synthetic data from ArcheoInt IGRF-11_S. (center column) IGRF-11 model. Maps for different years are given as Supplementary Material, (right column). Residual between both models is also shown

The major differences between both models are located in the regions with absence of data: Africa and Antarctica. The highest differences in declination are found in Antarctica and the Southern Indian Ocean. Discrepancies in inclination are low, with the exception of a small dipole in central Southern Africa that produces inclination differences up to 8° and −10°. The major disagreement in intensity is observed in Southern Africa and the Southern Atlantic Ocean, with higher intensities than the IGRF model (around 12 μT). These artefacts are due to the absence of information to reproduce adequately the SAA, i.e. the main differences between the IGRF-11_S and the IGRF-11 are located in the region affected by the SAA from where there is not enough available palaeomagnetic information.

Finally, and using the new set of Gaussian coefficients provided by the IGRF-11_S model, we have calculated the (A)DM curves, denoted as (A)DM_S. The coefficients’ error is used to obtain the error bands at ~65 % confidence level. Figure 7 shows the (A)DM_S curves together with the (A)DM curves of the original IGRF-11 model, and Table 3 summarizes the relative errors between them. We can observe a similar temporal trend between all (A)DM, with lower values at the beginning of the time interval, and higher values at the end, likely due to the inhomogeneous data distribution. In spite of this, we can observe that when we consider the error band, the theoretical (A)DM curves lie in the error band.

Fig. 7

For IGRF-11_S a ADM_S and b DM_S curves together with the error band at ~65 % confidence level are shown. The theoretical (A)DM is also plotted for comparison

Table 3 Errors (rms) for the deviation between ADM and DM of the geomagnetic field calculated with the Gauss coefficients of IGRF-11_S and IGRF-11 models from 1900 to 2010, every 5 years, together with the error band (confidence level of ~65 %)

In contrast to the regional weighting scheme where artificial variations of the DM were obtained for small sliding windows, dipole variations obtained by modelling reproduces much better the theoretical dipole moment. But we have to take into account that we have used synthetic data, i.e. we have considered the best situation. And more importantly, in this last case we have increased the amount of palaeomagnetic information (including declinations) with respect to the previous section. From this analysis, we could conclude that when directional palaeomagnetic information is available, the best method to compute the geomagnetic dipole moment evolution is from global modelling.

5 Conclusions

In this work, we have evaluated and quantified different sources of error introduced in the geomagnetic dipole moment estimation. The principal errors considered in this study come from (1) the averaged procedure of the non-dipole contribution, because palaeomagnetic data record all the contributions of the geomagnetic field and not only the dipole field (non-dipole effect); and (2) the effect of the current palaeomagnetic database distribution in the averaged procedure (regional effect) and in the generation of the global geomagnetic field models (regional indirect effect). To evaluate these errors we have used the IGRF-11 model.

Firstly, we have estimated the non-dipole effect in the global and regional averages for the last century. Although the main assumption is that the non-dipole terms are cancelled in the averaged procedure, we have reported that this contribution is not cancelled completely. In the global averages, it can reach 5.8 % for the VADM and 2.3 % for the VDM. The most important terms are the quadrupole and octupole. For the last 110 years, the non-dipole effect is small (never greater than 6 %) if we compare with the values of the palaeointensity errors (around 10 %).

In the regional averages, the non-dipole effect can give rise to deviations between the V(A)DM and (A)DM higher than 35 % in some continental regions, such as Oceania. Again, the quadrupole and octupole terms are the most important non-dipolarity sources, the quadrupole effect associated with the Southern Atlantic Anomaly being especially important. This term produces a decrease of about 15–16 % in the V(A)DM_continent over the Southern American region. Another interesting artefact is the anomalous evolution trend of V(A)DM_continent observed in Europe and Asia. The V(A)DM_Europe and V(A)DM_Asia are increasing whereas (A)DM is decreasing for the time span from 1900 to 2010.

Because of the sparse palaeomagnetic database, which is clearly biased towards the European region (with more available data), the regional evolution of the V(A)DM_Europe could affect the global averages. One of the methods proposed to avoid this overestimation is the first-order regional weighting scheme proposed by Genevey et al. (2008). Our results confirm the improvement of the V(A)DM when the regional weighting scheme is considered, with respect to the global average on the total database. However, mathematical artefacts are created with this procedure depending on the size of the temporal sliding windows used and the number of data available. The smaller the temporal sliding window is, the more artefacts appear. This varying regional bias effect might also affect the power distribution between dipole and non-dipole contributions in palaeomagnetic SHA models.

Finally, we have analyzed the effect of the current data distribution for the last 3,000 years on the generation of the global geomagnetic field models. We have generated a synthetic model (IGRF-11_S) with the D, I, F synthesized in the locations of the ArcheoInt database with the temporal interval adapted linearly to the last century. The results confirm that the main differences between the model created and the IGRF-11 are located in regions with lack of data (e.g. Africa). Moreover, we can observe a good agreement between the (A)DM calculated from the IGRF-11_S and the IGRF-11 models, lying in the error band with a confidence level of ~65 %.

From this analysis, we might conclude that when directional and intensity palaeomagnetic information is available, in spite of the inhomogeneity of the database, the best method to compute the geomagnetic dipole moment evolution is from global modelling. The (A)DM_model seems to be the most appropriate parameter to correct the cosmogenic isotopes production or to study the possible correlations between the geomagnetic field and the climate.