The Bulgarian Geomagnetic Reference Field (BulGRF) for 2015.0
epoch and its secular variation model prediction up to 2020.0 is produced and
presented in this paper. The main field model is based on the well-known
polynomial approximation in latitude and longitude of the geomagnetic field
elements. The challenge in our modelling strategy was to update the absolute
field geomagnetic data from 1980.0 up to 2015.0 using secular measurements
unevenly distributed in time and space. As a result, our model gives a set of
six coefficients for the horizontal

In the era of satellite observations, a variety of global models have been developed to represent the Earth's magnetic field. Because of the high altitude, they have spatial resolutions of hundreds of kilometres and focus mostly on the core field and large-scale lithospheric signals (e.g. CHAOS model series: Olsen et al., 2006; Finlay et al., 2016; GRIMM: Lesur et al., 2008). Study of intermediate-wavelength anomalies requires a combination of satellite, and/or airborne and ground measurements, as, for example, applied on the global scale for the World Digital Magnetic Anomaly Map (WDMAM) (Korhonen et al., 2007; Dyment et al., 2015) and of course the most recent comprehensive model (CM5) of the geomagnetic field (Sabaka et al., 2015) which co-estimate the major field sources using many different datasets. It uses joint inversion to describe field contributions from core, lithospheric, and external fields, along with associated Earth-induced signals. Regional models, on the other hand, are produced from data measured inside the area of interest thus being able to represent more accurately the field behaviour over that area. Unlike the global models, they are capable of modelling wavelengths in the kilometric range, and thus providing a better spatial resolution.

There are different techniques suitable for modelling the field in regions covering from a few squared geographical degrees to continental scales. Models for some countries are developed by means of revised spherical cap harmonic analysis (R-SCHA) (Thébault et al., 2006; Korte and Thébault, 2007; Qamili et al., 2010). Others successfully use simpler procedures such as second-degree polynomial fitting (e.g ITGRF: De Santis et al., 2003; Dominici et al., 2007; Žagar and Radovan, 2012) to represent the space and temporal behaviour of the field better than the global models. Kovács et al. (2015) compared the advantages and disadvantages of polynomial and adjusted spherical harmonic analysis (ASHA) (De Santis, 1992) models over the territories of Croatia and Hungary. They concluded that because of its physical justification the obtained ASHA model is superior to the polynomial models, although the latter resulted in lower residuals.

In the case of Bulgaria we chose to apply the procedure of second-degree polynomial fitting for three reasons: (1) we aim at developing a simple reference model up to 2020 which will be used when carrying out magnetic surveys, (2) our repeat stations are too limited in space and too scattered in time in order to obtain a good fit for a SCHA model, and (3) the territory of Bulgaria is small enough to neglect the curvature of the Earth in the investigated area. In this paper we present an up-to-date version of the Bulgarian geomagnetic reference model that is valid for epoch 2015 with a secular variation model until 2020. The model, characterized by a set of six coefficients for each element of the field was produced by least-squares fitting of second-degree polynomial of geographical coordinates to 473 points over the territory of Bulgaria.

Comparison of the modelled values with real measurements is a method for model validation. Of course, a full match cannot be expected because some contribution also comes from the magnetized rocks of the Earth's crust which turns the modelling into a tool for crustal anomaly investigation. We use our model also to evaluate the magnitude of the well-delineated magnetic anomaly in Panagyurishte (PAG) observatory region which was the main obstacle when in 1935–1936 experts discussed the location for building a magnetic observatory in Bulgaria. The anomaly is caused by two andesitic veins located inside the crust south of Panagyurishte which contain high levels of ferromagnetic minerals. Nevertheless, after consultation with geologists who discussed the possible viscous nature of the rock magnetization and/or shifting of the masses relative to the felsic granite above which the observatory will be built it was decided that this anomaly could be accepted as a constant.

PAG observatory (red square) and repeat stations of Bulgaria (green triangles) used for geomagnetic model compilation.

The repeat station network of Bulgaria was established in 1934 (Kostov and
Nozharov, 1987). The eight points selected were then supplemented by seven
more points in 1964 (Fig. 1). Up to 1980 repeat stations were measured every
3 years and then, because of the small secular variation, every 5 years. The last absolute geomagnetic survey was performed in the period
1978–1980. The geomagnetic elements

The big challenge in our modelling strategy was to update the absolute field
geomagnetic data from 1980.0 to 2015.0 using measurements that are sparse
and unevenly distributed in time and space. The main problem was the data
extrapolation from the last measured value to the aimed 2015.0 epoch. To
solve this problem we investigated the correlation between the secular
variation trends of every single component in each repeat station and the
secular variation trends in Panagyurishte (PAG) observatory for the same
period. Thus, we obtained coefficients allowing us to calculate the secular
variations of the geomagnetic field elements in repeat stations for the
extended time intervals without real measurements, using data from
Panagyurishte observatory and the equation:

Extrapolated element secular variation for four repeat stations
(Slavotin, Bekleme, Popovo, and Michurin):

Using the above relation, secular variation values for every year (between 1980
and 2015) for each component for all repeat stations were calculated. Example
plots of predicted

As a result of all calculations, for each geomagnetic element (

Isoporic maps of the accumulated change in the main field
elements from 1980 to 2015 for

Confidence intervals of the empirically determined statistical
estimate of the coefficients

Although the surface polynomials were the first analytical method used to produce regional models (Haines, 1990) they are still used because of the relatively simple procedure of model generation and subsequent ease of use for calculations.

In the present research we fit a second degree polynomial of the form
(Buchvarov and Kostov, 1981)

We use the well-known technique of least squares (Wilks, 1967) according to which if we
have a function

Then, the empirically determined statistical estimate of the coefficients
having minimum variance is (Wilks, 1967)

Other important parameters of our analysis (see for example Buchvarov,
1977) are as follows:

the statistical estimation variance:

the limits of the confidence intervals of the coefficients:

where

the confidence intervals of the calculated variables:

where

the correlation coefficient (for assessing the significance of the regression):

During the calculations the following procedure was applied to remove
anomalous points or possible errors in measurements (Buchvarov and Kostov,
1981). Using data from all observational points, a statistical estimate of
the regression coefficients

As mentioned above, the input data for producing the 2015.0 model are
the last absolute geomagnetic field measurements (1980) updated with the
prepared isoporic maps from secular measurements and neighbouring
observatories. Annual mean values from 2015.0 to 2020.0 are computed by
extrapolation of Panagyurishte observatory data, assuming an autoregressive
secular variation to define the coefficients

Calculation of the four AR models (for geomagnetic field elements

Empirically determined statistical estimate of the coefficients

Values of the BulGRF 2015.0 coefficients

Maps of the modelled geomagnetic field elements of BulGRF epoch
2015.0:

Values of the BulGRF model epoch 2015.0 together with the
calculated confidence interval (dashed line), IGRF-12 and real data of

The

Since the 2015.0 BulGRF model was developed using independent polynomials
for each element, it was necessary to check the self-consistency of the
model. We checked if the geometrical constraint (Haines, 1990) is satisfied
using the equation

To test the validity of the model we compared its values with data not used
for model generation. These are the PAG observatory (42.52

Very well expressed however is the difference between the three models and
the real data in PAG observatory. It is clear that if someone measures the
magnetic field at a point on the Earth's surface, they cannot expect to get
the value predicted by our model or the IGRF model. The reason for this is well
known: there is a significant contribution which comes from the magnetized
rocks of the Earth's crust – typically 200–300 nT, but it can reach
bigger values in the anomalous regions. They are “shared” among the
components but mostly affect the

Concerning the case with PAG observatory difference, it can be clearly seen
in Fig. 6 that calculated annual means are strongly below the values of
BulGRF model and both global models (IGRF-12 and CHAOS-6) as well. It was
expected because of the already mentioned anomaly found in the region before
the observatory construction. This anomaly is also observed on the detailed
exploratory magnetic anomaly map of Bulgaria prepared by Pchelarov (2000),
where this region falls in a well-delineated crustal anomaly with 150 nT
magnitude of

The main point in the future will be to check whether this anomaly has a constant residual value in time or whether it varies along with the main geomagnetic field thus influencing the secular variation.

The advantages of the proposed 2015.0 BulGRF model with extension to 2020 can be summarized in three points: (1) a second-degree polynomial approximation of the geomagnetic field elements of Bulgaria is elaborated which is closer to the base level of the geomagnetic field over a limited region like the Bulgarian territory; (2) because of the long periods without secular measurements in Bulgaria there was a need of recent model compilation which now is fulfilled; and (3) BulGRF is an accurate model of the regional geomagnetic field which together with its simplicity makes it a useful tool for reducing to a common epoch the magnetic surveys carried out over the Bulgarian territory up to 2020.

Panagyurishte observatory data and secular measurements data are available from the World Data Center Edinburg
(

The authors declare that they have no conflict of interest.

This article is part of the special issue “The Earth's magnetic field: measurements, data, and applications from ground observations (ANGEO/GI inter-journal SI)”. It is a result of the XVIIth IAGA Workshop on Geomagnetic Observatory Instruments, Data Acquisition and Processing, Dourbes, Belgium, 4–10 September 2016.

We would like to thank Ivan Buchvarov for providing his knowledge and expertise in geomagnetic data processing. We thank IAGA for the young researcher financial support for participating in the XVIIth IAGA Workshop in Dourbes, where this research was first presented and discussed.

We thank the editor and the two reviewers for the constructive criticism and comments that helped us to improve the manuscript.

This research is supported by contract no: 8/20.04.2016 from the Program for career development of young scientists, BAS. Edited by: Georgios Balasis Reviewed by: two anonymous referees