1 INTRODUCTION

The task of determining the position of the North and South magnetic poles has always been in the focus of attention of both travelers and magnetologists, and it has not lost its importance and relevance to date. To solve this problem, which is difficult in theory, methodology, and organization, expeditionary studies, observations, and theoretical research were carried out. Their goal was and remains to develop methods and algorithms for estimating the position of the Earth’s magnetic poles on the basis of direct measurements, to construct regional models, and to use global models of the Earth’s magnetic field.

It is of high significance to carry out geomagnetic studies aimed at determining the position of the South magnetic pole (SMP), since the knowledge of current positions of the magnetic poles and their dynamics in time is necessary to solve a number of fundamental geophysical problems, which brings us closer to understanding the nature of the geomagnetic field and its generation mechanism (Merrill et al., 1998), as well as how and why its inversions occur.

At present, there is no generally accepted hypothesis answering the question on what processes are indicated by changes in the position and the travel speed of the poles (Witze, 2019). As such, the following were considered: local processes in the core and associated jerks (Newitt et. al., 2002; Campbell, 2003) and/or global changes leading to the field inversion (Kuznetsov, 2006).

To know the actual position of the magnetic poles is also of great practical importance, since it can be used to construct correctly global models of the geomagnetic field and verify their accuracy in polar regions, where the observed geomagnetic field most substantially diverges from the field predicted by global models. In particular, a noticeable change in the position of the North magnetic pole makes it necessary to update the so-called World Magnetic Model (WMM), on which, among other things, the functioning of all navigational systems is based.

In addition, an important issue is to estimate the impact of true shifts of the magnetic poles on the upper atmosphere (Lyakhov et al., 2006), terrestrial thermosphere and ionosphere (Namgaladze et al., 2018), climate (Belikov et al. 2018), biosphere (Reshetnyak and Pavlov, 2016) and, ultimately, humans (Valet and Valladas, 2010).

1.1 Historical Background and Definitions

The magnetic pole is a wandering point on the surface of the Northern (Southern) hemisphere of the Earth, where the geomagnetic field vector T is directed vertically, i.e., the horizontal component H is zero. Since all lines of equal magnetic declination D converge at the magnetic pole, the declination is not defined at the pole itself. Moreover, since the geomagnetic field has a nondipole component, this convergence is not radial even in the polar regions. The fact that a compass does not point to the geographic north was noticed as early as the 11th century in China and the late 15th century in the Old World. This served as an impetus for intensifying regular measurements of the declination. It became obvious that the availability of reliable maps of the declination values for water areas was a prerequisite for ensuring navigation. Values of D were regularly measured during voyages of the British Admiralty ships.

The beginning of magnetic measurements in the polar regions dates back to the 16th century and is associated with searching for the northeastern passage to China and India (Hakluyt, 1589; de Veer, 1598). In the 19th century, several unsuccessful attempts were made to reach the SMP, among which there are expeditions undertaken by J.S.C. Dumont-d’Urville (1837−1840), C. Wilkes (1838−1842), and J.C. Ross (1839−1843) (Ross, 1847)—explorers from France, America, and Great Britain, respectively.

To determine the SMP location, the magnetic declination was calculated for the first time on January 23, 1838, by Clément Adrien Vincendon-Dumoulin (Voyage au Pôle …, 1842−1846), a hydrographer, who participated the 1837−1840 expedition of Dumont-d’Urville on the corvettes Astrolabe and Zélée in Antarctica and Oceania, during which Adélie Land was discovered.

In the 20th century: on January 16, 1909, three members of the expedition of Sir Ernest Henry Shackleton (Douglas Mawson, Edgeworth David, and Alistair McKay) declared that, according to the inclination measurements, they found the SMP (Shackleton, 2014). At that time, the SMP was on land, as the historical model, called gufm, shows (Jackson et al., 2000). The magnetic pole coordinates calculated from this model are available, for example, on the website of the National Geophysical Data Center (NGDC). The explorers set up a flagpole at the site of the pole and declared this territory the possessions of the British Empire. To reach the SMP, they walked 1260 miles in total, dragging a sled and supplies that weighed approximately 670 pounds. The pole position, which they determined on January 16, 1909, was 72°25′ S, 155°16′ E (Magidovich and Magidovich, 1985).

The research of the motion of the magnetic and geomagnetic poles and the evolution of the Earth’s magnetic field in both polar regions during the last millennia is described by Korte and Mandea (2008). Historical geomagnetic data obtained from 1510 to 1930 were collected during the last two decades of the 20th century, and their comprehensive review is given by Jonkers et al. (2003). The dynamics of the motion of both poles in 2015−2019 was analyzed using a large volume of data collected by three satellites of the Swarm mission (Regi et al., 2021).

1.2 Geomagnetic Research of Antarctica from J. Cook to Admiral Vladimirskii

The first geomagnetic measurements in the Antarctic region were made during the second round-the-world expedition of J. Cook (1772−1775) (Cook, 1777; The Journals of Captain James Cook, 1961) (see Fig. 1a). The declination measurements were performed by an astronomer of the expedition W. Beyley; and their results were published and used, in particular, for constructing the historical gufm model of the geomagnetic field. However, the astronomer apparently did not estimate the SMP position from the data obtained, which was beyond the scope of objectives of the expedition.

Fig. 1.
figure 1

A scheme for the distribution of points of the declination measurements performed during the expeditions of Cook (a) and Bellingshausen (b). The arrows show the results of measuring the declination, and the star symbol shows the SMP position in the epoch of the expeditions according to the gufm model.

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For the first time, the SMP position was experimentally determined in the course of the round-the-world Antarctic expedition undertaken by Russian navigators F. Bellingshausen and M. Lazarev in 1819−1821. To solve this problem, the sloops Vostok and Mirnyi passed round Antarctica and made 146 measurements of the magnetic declination along the way (Fig. 1b). Partially, where possible, the expedition members repeated the route of the Cook expedition. This allowed Raspopov et al. (2014) to recently estimate the secular trend in the declination during the 48 years between the two expeditions. The results of these measurements were used by Bellingshausen to calculate the SMP position. According to his calculations, the pole was located at coordinates of 76° S, 142°30′ E. However, it is unknown which particular points with the measured declination were used by Bellingshausen to calculate these coordinates. In his report, Bellingshausen (1949) listed only 52 points with measured values of the declination, 46 of which were near the Antarctica region. Later, at the request of J.C.F. Gauss, Bellingshausen gave him 203 points with the declination values specified. The method, by which Bellingshausen determined the SMP coordinates, is not described in his reports and correspondence with Gauss. We may only suppose that he used the technique, which is described in the paper by Duperrey (1841) as one of the possible approaches to determining the position of the pole. According to this technique, straight lines in the direction of the measured declination are mapped in the polar projection; and the intersection of these lines should give the position of the pole.

The program of the anniversary round-the-world expedition of the vessel Admiral Vladimirskii (2019 ̶ 2020) was worked out so that its course would basically repeat the round-the-world expedition of Bellingshausen. Along with the associated geophysical and oceanographic studies, special research on magnetic measurements was planned to determine experimentally the SMP coordinates. Within the frames of this paper, we estimated the SMP position from the magnetic measurements performed during the expeditions of Cook and Bellingshausen (PART 1) and the vessel Admiral Vladimirskii (PART 2).

2 DETERMINING THE SMP POSITION FROM THE DATA OF THE COOK AND BELLINGSHAUSEN EXPEDITIONS

2.1 Methods to Determine the Magnetic Pole Position

The position of the magnetic pole may be determined in three ways: by direct measurements of the inclination (an experimental method), by processing the magnetic data surveyed in circumpolar regions (an algorithmic method), or by determining the pole from global models of the geomagnetic field (an analytical method) (Newitt, et al., 2009). To determine the magnetic pole position by experiment, one should carry out a magnetic survey directly in the area of the pole. Note that, regardless of the method, it is a challenge to find the magnetic pole position by instruments due to several reasons: first, due to the measurement errors, the area, within which the horizontal component is very small (i.e., the inclination is close to the maximal value, 90°), is rather extensive; second, the magnetic pole position is not at a fixed point, i.e., the poles may move around during a day by distances from first tens to hundreds of kilometers from their mean position, which depends on geomagnetic disturbances in the ionosphere and magnetosphere (Wasserfall, 1938; Dawson and Newitt, 1982); and, finally, the polar regions are relatively inaccessible for expeditionary activity.

Nevertheless, if a set of points with measured values of any component or several components of the geomagnetic field is available, one can calculate the pole position from these data. To solve this problem, several approaches applicable to certain data types were proposed. The best known of them are the methods of local spherical harmonic analysis, virtual magnetic pole, polynomials, etc. (Newitt, et al., 2009). The capabilities of each of these approaches are limited by the effects that appear at edges; consequently, it is important that the searched pole is located within the region formed by the points of measurements.

The analytical technique for determining the pole position is based on the use of global models representing the geomagnetic field in the form of a series expansion in spherical harmonic functions. In contrast to a local model, the coefficients of global models can be used to calculate the values of all components of the field at any point on the Earth’s surface, in particular, in the polar region. The results of calculations depend on the model chosen: for example, the model may take into account or ignore the anomalous magnetic field. In the former case, the result should be closer to that of the experimental methods. In practice, the geomagnetic field is nearly vertical within an oval-shaped region; and the size and position of this region may change every day (Dawson and Newitt, 1982).

2.2 Input Data

As initial data for determining the SMP position, we used the results of the declination measurements performed during the second round-the-world expedition of J. Cook (1772−1775) and the round-the-world Antarctic expedition undertaken by Russian navigators F. Bellingshausen and M. Lazarev (1819−1821). The Cook expedition was not tasked with determining the position of the SMP. Its goal was to establish the fact that there is a continent in the Southern Ocean. For the present study, we used the data obtained by Cook only at the latitudes higher than 45° S.

Prof. I.M. Simonov, an astronomer of the Bellingshausen expedition, measured the declination D at 203 points during the expedition. Exactly these D values were given to Gauss, who used them to check his proposed representation of the geomagnetic field as expansion in spherical functions and published these data (Gauss, 1877). From these data, we chose 146 points, the latitudes of which are higher than 45° S, as in processing the Cook data. Figure 1b shows the geographic location of these points. The SMP location was determined by several methods, including the analytical one that uses the historical gufm model. The applicability of all of the methods was tested on models: at points with geographic coordinates of the real measurements of D, the declination values were calculated using the gufm model at the time of measurements. From the modeled declination values, the position of the pole was determined by each of the methods. After that, the obtained coordinates of the poles were compared to the SMP coordinates calculated by the gufm model. Actually, for the modeled declination values, the error of a particular method is defined by the difference in the coordinates. It is rather difficult to reduce this error only by measuring the declination, since all of the methods use the dipole approximation in varying degrees.

2.3 Two-Circle Method

We conventionally called the first method, which we used to determine the coordinates of the magnetic pole from the declination data obtained in the Cook and Bellingshausen expeditions, the “two-circle method” because we did not encounter in literature a description of such a method for solving the problem considered. Naturally, we do not claim to be the authors of this method, since its intuitive idea follows from an inherent property of declination. In fact, since all isogonic lines converge to the point of the magnetic pole, the lines of large circles constructed in the direction prescribed by the declination in the dipole approximation, if they do not lie on the same isogon, intersect at the magnetic pole.

Thus, the two-circle method is as follows: a large circle, passing along a specified azimuth (the declination at this point) is drawn through a point with known declination. The same is done for a second point with its azimuth (declination). These two circles intersect at two points, and the SMP position may be estimated by the southern intersection point. For the purely dipole field, this estimate will be accurate, since each of the large circles passes through the dipole pole. Let us call the point obtained by this method the virtual magnetic pole (VMP) analogously to the paper by Kuznetsov (1998) and in contrast to the concept of the virtual geomagnetic pole accepted in paleomagnetology.

The error of this method includes two components: a random error of measurements (an instrumental error) and a systematic error arisen from the nondipole component of the field. The first error is illustrated in Fig. 2: it is seen that when the points are close to each other small changes in declination lead to significant shifts of the intersection point.

Fig. 2.
figure 2

The error in determining the SMP position for two values of the angular distance between the points of measurements: 10° (a) and 90° (b). Random errors in determining D are assumed to be ±2°.

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When applying the method to a purely dipole model, we may show that for a distance of 250 km (corresponding to 5° in longitude at a latitude of about 60° S) between neighboring points and a measurement error of 2°, the error in determining the VMP position is nearly 2000 km. To find the optimal distance between the points, which should be chosen when determining the VMF with this method, we performed calculations for two variants, as if the measurements were carried out in the geocentric dipole field and the gufm model field. The coordinates of measurement sites of the Bellingshausen expedition were used as points of measuring the declinations, while the declination values were calculated in accordance with the field considered. A small perturbation was added to the declination, and two estimates of the pole position, depending on the field type, were obtained with this method. In the first case, the error is the distance to the estimated position to the geographic pole, while in the second case it is the distance to the pole of the gufm model. Since the route of a ship around Antarctica may be considered as sublongitudinal in the first approximation, the optimal longitudinal distance between the points forming a pair was determined. For the first model, the optimal longitudinal distance between the points in a pair was estimated at 90°. For the second model, this longitudinal distance is 65°.

Figure 3 shows the systematic error of the method. Two large circles are drawn through two points according to the azimuths built by the gufm model for the epoch of 1820. The points are taken from the Bellingshausen expedition data and their longitudes differ approximately by 90°. The VMP and the modeled pole do not coincide because the model contains the nondipole component of the field.

Fig. 3.
figure 3

The systematic error in determining the magnetic pole position by the method of large circles. The points, for which the declination was calculated using the gufm model, are marked by filled circles. The solid line shows the great circle arcs. The VMP position is marked by a square. The star symbol shows the SMP position in the epoch of 1820 according to the gufm model.

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To estimate the SMP position from the data of the Cook and Bellingshausen expeditions, we first took the pairs of points spaced by 40°−120° for each of the expeditions and calculated the VMP position with the described method; then, to find the SMP position, we used the bootstrapping method, which is a method for analyzing the probability distributions, which relies on the Monte-Carlo multiple generation of samples from a sample available (Efron and Tibshirani, 1986). This method allows one to estimate rather accurately the mean value and the variance for the data containing systematic errors. The coordinates of the SMP position estimated by this method from the experimental data of the Cook and Bellingshausen expeditions and the corresponding modeled data are presented in Table 1.

Table 1.   The SMP coordinates obtained by different methods
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2.4 The Equivalent-Dipole Method

This method also uses the dipole approximation to describe the field in the SMP region. Let us consider a dipole of the magnetic moment M located at the center of the Earth. The dipole field is described by the well-known formula

$${\mathbf{B}} = - \frac{{{{\mu }_{0}}}}{{4\pi }}\left( {\frac{{{\mathbf{M}}~}}{{{{R}^{3}}}} - \frac{{3\left( {{\mathbf{MR}}} \right){\mathbf{R}}}}{{{{R}^{5}}}}} \right),$$

where R is the radius-vector of the observation point in the geocentric coordinate system. Let us write this expression in the matrix form B(R) = F(R) M, where F(R) is the matrix

$${\text{F}}\left( {\mathbf{R}} \right) = \frac{{{{\mu }_{0}}}}{{4\pi {{R}^{5}}}}\left( {3{\mathbf{R}}{{{\mathbf{R}}}^{T}} - {{R}^{2}}E} \right),$$

while E is a unit matrix of 3 × 3. When moving to the topocentric coordinate system with the origin at the observation point (φ, λ), we obtain the magnetic field vector B' = S(φ, λ) F(R) M. Let us denote A (φ, λ, R) = S(φ, λ)F(R), where S(φ, λ) is the (3 × 3) orthogonal matrix for transforming the coordinates of the vector when passing from the geocentric coordinate system to the topocentric one. Figure 4 shows the coordinate systems used and the dipole axis.

Fig. 4.
figure 4

The coordinate systems in the equivalent-dipole method. The axes of the geocentric and topocentric coordinate systems are designated by X, Y, and Z and by x ′, y ′, and z ′, respectively. The dipole location is shown by a diamond. Nm is the point of intersection of the dipole axis with the Earth’s surface; φ and λ are the geographic coordinates; and D is the magnetic declination.

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The horizontal components of the vector B' are linked through the declination as follows: \(B_{y}^{{\text{'}}}\) = tan(D)\(B_{x}^{{\text{'}}}\). Denote by aj the rows of the matrix A and express the components of B through aj. Then, we have

$$\begin{gathered} B_{y}^{'} - {\text{tan}}\left( D \right)M_{x}^{'} = \left( {{{{\mathbf{a}}}_{2}},~{\mathbf{M}}} \right) - {\text{tan}}\left( D \right)\left( {{{{\mathbf{a}}}_{1}},{\mathbf{M}}} \right) \\ = \left( {{{{\mathbf{a}}}_{2}} - {\text{tan}}\left( D \right){{{\mathbf{a}}}_{1}},{\mathbf{M}}} \right) = 0. \\ \end{gathered} $$

Since the magnitude of the magnetic momentum is of no importance for determining the SMP position, we assume Mz = 1. Then, the equation is no longer homogeneous, and we obtain a system of linear equations with respect to Mx and My, which can be solved by the least-squares method (LSM). As a result, we derive the momentum vector M, which allows us to determine the position of the magnetic pole for the equivalent dipole found. Note that geometrically the equation defines a plane with the normal a2 − tan(D)a1, in which the magnetic moment lies. The method was tested on models; the obtained values of the SMP coordinates were compared to those resulted from the gufm model for the epoch of 1820. The error approximately amounted to 150 km. The values for the SMP coordinates obtained by this method from the modeled data and the Cook and Bellingshausen data are given in Table 1.

2.5 Field Approximation by a Sum of Spherical Harmonics

To construct a spherical harmonic model from a number of points, for which the declination were measured, two methods were proposed. The first of them, analogously to the equivalent-dipole method described above, is based on the definition of the declination D = arctan(Y/X). In this case, the relation X sin D = Y cos D is true. An overview of this method may be found in a paper by Baraclow (1974). The components X and Y can be represented in the form of a series expansion in terms of spherical functions:

$$\begin{gathered} X = \sum\limits_{n{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {\frac{R}{r}} \right)}}^{{n{\kern 1pt} + {\kern 1pt} 2}}}} \mathop \sum \limits_{m{\kern 1pt} = {\kern 1pt} 0}^n \frac{{dP_{n}^{n}}}{{d\theta }}(g_{n}^{m}{\kern 1pt} {\text{cos}}{\kern 1pt} (m\lambda ) + h_{n}^{m}{\text{sin}}\left. {\left( {m\lambda } \right)} \right), \\ Y = \sum\limits_{n{\kern 1pt} = {\kern 1pt} 1}^N {{{{\left( {\frac{R}{r}} \right)}}^{{n{\kern 1pt} + {\kern 1pt} 2}}}} \mathop \sum \limits_{m{\kern 1pt} = {\kern 1pt} 0}^n mP_{n}^{m}(g_{n}^{m}{\kern 1pt} {\text{sin}}(m\lambda ) - h_{n}^{m}{\kern 1pt} {\text{cos}}\left. {\left( {m\lambda } \right)} \right). \\ \end{gathered} $$

Since we are interested only in the magnetic pole position rather than in the absolute magnitudes of the field, we may put the first coefficient \(g_{1}^{0}\) = 1 and search for all other coefficients in the form of ratios \({{g_{n}^{m}} \mathord{\left/ {\vphantom {{g_{n}^{m}} {g_{1}^{0}}}} \right. \kern-0em} {g_{1}^{0}}}\) and \({{h_{n}^{m}} \mathord{\left/ {\vphantom {{h_{n}^{m}} {g_{1}^{0}}}} \right. \kern-0em} {g_{1}^{0}}}\). In this case, confining ourselves to the nth order of the polynomial, we derive a system of linear equations with respect to the coefficients, which can be solved by the LSM. With the obtained coefficients, we calculate the horizontal component H and determine the pole position. The approximation of the declination D for the polynomial orders n = 1 and n = 2 is shown in Fig. 5а. In our case, the points with the measured values of D are located within a ring encompassing the pole. Since the field components outside the ring, bounding the measurement points, are calculated by extrapolation, the increase in the order of the polynomials may result in growing the error of determining the pole position, even when this may improve the accuracy of the approximation for D. For example, if the model is limited to the order of polynomials n = 3, the extrapolation error is small and does not influence the resulting SMP position. However, for high-order models with n = 12−14, such as the gufm model, the extrapolation error appears already in the approximation by polynomials with n = 2. When n grows the spatial distribution of the component H calculated with the obtained coefficients begins to differ noticeably from that based on the gufm model, and additional minima appear. Although the solution approximates the specified declination values with better accuracy, the extrapolation error grows with increasing n and impedes determination of the SMP position. To determine the pole position with this method by approximating the Cook and Bellingshausen data, we limited ourselves to n = 1. The coordinates of the SMP are given in Table 1.

Fig. 5.
figure 5

The approximations of D by spherical functions obtained with the approximation method (Section 2.5) (a) and the modified linearization method (Section 2.6) (b). The measured values of D and the results of approximation with n = 1 (for both methods) are shown by a thick line and a line with diamonds, respectively. The dotted lines present the results of approximation with n = 2 and 3 in panels (a) and (b), respectively.

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2.6 The Linearization Method

The problem of constructing a spherical harmonic model of the geomagnetic field only from the data on declination D may be solved by linearization (Baraclow, 1974), which is performed by differentiating the expression for D.

Let us consider the expression for small increments of the declination

$$\delta D = \delta {\text{arctan}}\frac{X}{Y} = \frac{1}{{{{H}^{2}}}}\left( {X\delta Y - Y\delta X} \right).$$

The derived formula is linear relative to the increments. Let us consider the following iteration algorithm.

We set some initial values of X0 and Y0, calculate H0 and D0, and compare them to the measured values of D. The linearized formula allows us to calculate δD. Let us represent δX and δY as sums of spherical functions with the coefficients \(\delta g_{n}^{m}\) and \(\delta h_{n}^{m}\) and substitute them into the expression for δD. We obtain a system of linear equations with respect to \(\delta g_{n}^{m}\) and \(\delta h_{n}^{m}\), by which we calculate the corrections δX and δY for X0 and Y0 and then, using these corrections, calculate X1 = X1 + δX, Y1 = Y1 + δY, H1, D1, and a new value for δD. In this way, the iteration process is constructed. A peculiarity of this approach is that the eigenfunctions of the Laplace operator should be constructed for the “spherical cap,” which is a difficult task. However, it is the geometry of points that suggested how to modify this approach; and exactly this modification was applied to the present analysis. To construct the zero approximation, we assumed that the pole is located at the center of the circular region bounded by the points of measurements. In this case, we may assume the modulus of the horizontal component H in all of the points to be the same (a dipole model) and equal to H0. Then, we may calculate X0 = H0 cos D and Y0 = H0 sin D. For determining the pole position, the magnitude of H0 does not matter and may be assumed of any value. By expressing X0 and Y0 through the coefficients \(g_{n}^{m}~\) and \(h_{n}^{m}\), we derive a system of equations relative to these coefficients. The solution of the system yields the zero approximation of the spherical coefficients, and the value of H1 is calculated from them. By substituting the quantities H0 by H1 in the system of equations with respect to \(g_{n}^{m}\) and \(h_{n}^{m}\) and by using the measured values of D, we obtain a new system of equations; by solving this system, we find new estimates for \(g_{n}^{m}\) and \(h_{n}^{m}\). By repeating the above procedure, we construct the iteration process. At each of the steps, along with Hi, the synthesized values of Di are calculated from the obtained coefficients, which may be compared to the measured values. The dispersion in the obtained values of the difference serves as a criterion for terminating the iteration process.

When the models constructed from the Bellingshausen data were used, four or five steps were required for the described iterative process to converge, depending on the order of polynomials. At the same time, the matrix of the system of linear equations became ill-conditioned at n = 3. Application of the method to the real data yielded similar results, but required some points to be additionally rejected. Figure 5b shows the results of approximation by this method for the modeled and measured values of D and the polynomial order n = 2. Since the described method is ultimately based on the approximation of D, the calculation of H for the area may include the extrapolation error outside “the data ring”, analogously to the result of the previous method. The estimates of the SMP coordinates obtained at n = 3 are given in Table 1.

The described method is not universal. Its applicability strongly depends on the spatial structure of the distribution of measurement points and their proximity to the pole. Thus, for the Cook data, the spatial structure of the measurement points does not meet the main requirement of the method; consequently, we could not achieve the method converging even with the modeled data. For this reason, the method was not applied to the Cook data.

3 DISCUSSION OF THE RESULTS

To determine the SMP position from the declination values measured during the Cook and Bellingshausen expeditions, different methods were used. These methods are based on the computer technology capabilities, well-developed mathematical apparatus, and models of the Earth’s magnetic field, which were not available at the time of these expeditions. The estimates of the SMP coordinates obtained by the four methods are summarized in Table 1. In addition, Table 1 contains the coordinates of the pole calculated by the same methods from the modeled data, i.e., the declination values taken from the gufm model for the epochs corresponding to the time of the expeditions. The pole coordinates calculated by Bellingshausen are also added. As one would expect, the obtained estimates of the SMP position are significantly dispersed in both latitude and longitude.

In Figure 6, all of the SMP estimates obtained from both the measured and modeled data are shown by filled and open circles, respectively, for the Cook (panel (a)) and Bellingshausen (panel (b)) expeditions. The azimuthal projection, which is commonly accepted for depicting polar regions, is used in the maps. The mean values of the SMP coordinates for both the expedition and modeled data are shown in the same figure by star symbols (filled and open ones, respectively). For each averaged position of the pole, the 50% confidence ellipses were calculated; they are shown in Fig. 6 with solid and dashed lines for the measured and modeled data, respectively.

Fig. 6.
figure 6

Estimates of the SMP position obtained by using the proposed methods from the modeled data and from the Cook and Bellingshausen observations. The open and filled circles show the results for the modeled and observed data, respectively; and the open and filled star symbols indicate the corresponding averaged coordinates. The SMP positions resulted from the gufm model for the epochs of 1771 (a) and 1820 (b) are shown by filled diamonds. The solid and dashed ellipses correspond to the 50% confidence interval for the observed and modeled data, respectively.

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As is seen in Fig. 6, the size of the confidence ellipse for the Bellingshausen expedition is significantly smaller than that for the Cook expedition, which is apparently due to the greater accuracy of measurements performed during the expedition undertaken later by Bellingshausen. For each of the expeditions, the axes of the confidence ellipses for the modeled and experimental data are approximately the same in size. However, they are oriented differently. For the Cook expedition data, the both confidence ellipses are coaxial, while the major axes of the ellipses corresponding to the Bellingshausen expedition make a significant angle.

To numerically characterize the scattering of the VMP location, the distances from the pole position of the gufm model to those found from the measured and modeled data by different methods were calculated. The results are presented in Tables 2 and 3 for the data of the Bellingshausen and Cook expeditions, respectively. The distance to the pole position determined by Bellingshausen was added to Table 2. As is seen from Table 2, the pole position calculated by Bellingshausen is the most remote from the gufm model pole, which may be caused by limited tools and methods of the data processing and analysis available to Bellingshausen. At the same time, it is worth noting that the distances from the gufm model pole to the poles obtained by averaging the SMP positions calculated from the experimental data of Bellingshausen and the corresponding modeled data are close (30 km). The difference could appear due to a variable component of the geomagnetic field contained in the experimental data.

Table 2.   The distances of the SMP position obtained from the Bellingshausen expedition data to the pole position determined by Bellingshausen himself and to the pole according to the gumf model
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Table 3.   The distances of the SMP position obtained from the Cook expedition data to the pole according to the gumf model
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For the data of the Cook expedition, the distances from the average pole positions obtained from the experimental and modeled data differ by 100 km from that calculated with the gufm model, which is most likely caused by the accuracy of determining the declination and the navigation errors in the Cook expedition. At the same time, for the both expeditions, the average SMP positions obtained from the modeled data differ by 106 and 130 km from those calculated from the gufm model. Since this quantity characterizes the error of the methods, we may conclude that the dipole model is not sufficient to describe the geomagnetic field in the SMP region. At the same time, the SMP positions calculated for the epoch of 1820 by the truncated gufm model for n = 1 (a dipole approximation) and n = 2 (a quadrupole approximation) are (78.65° S, 118.06° E) and (75.72° S, 145.42° E), respectively. From the comparison to the data in Table 1, it is clear that our estimates of the SMP coordinates are much closer to the pole location resulted from a complete version of the gufm model than to the dipole and quadrupole approximations. In other words, despite the drawbacks, the proposed methods allowed us to take into account, at least partly, the influence of the nondipole component. On the other hand, our results indirectly confirm that one may use the gufm model to describe the geomagnetic field in the SMP region at the time of the Bellingshausen and Cook expeditions.

4 CONCLUSIONS

In this paper, the SMP positions in the late 18th century and the early 19th century were estimated. As experimental data, we used the results of the declination measurements carried out in the Antarctic region during the expeditions of Cook (1772) and Bellingshausen (1820).

To solve the problem of determining the coordinates of the pole only from the data on declination, several methods were used, which were preliminary tested on models. Although the estimates of the SMP coordinates obtained by each of these methods form a cloud of points, the average value is only 100−130 km lower than that obtained from the gufm model. Since all of the methods stem from the assumption that the magnetic field near the pole is close to the dipole field, the above error characterizes the degree of influence of the nondipole component of the geomagnetic field on the declination magnitude. The significantly greater distance to the pole of the gufm model obtained from the declination values measured in the Cook expedition is most likely caused by a larger error both in the declination measurements themselves and in determining the coordinates of points, at which they were carried out.

Thus, this study gives the experience of determining the SMP coordinates from the experimental data with the use of only the declination values. The main conclusion is that for the epochs of the Cook and Bellingshausen expeditions, the nondipole component of the geomagnetic field produced a significant impact on the position of the SMP.