Today modeling and analysis of the Earth's magnetic eld is a key to understand how geomagnetic eld parameters are distributed on the Earth's surface, its subsoil and in circumterrestrial space. Geomagnetic data collected by the sets of geomagnetic stations and observatories, various scienti c organizations, space vehicles and individual researchers requires its systematization, interpretation and analysis with special approach, which is based on methods of mathematical and spatial modeling and allows representing the cuts of geomagnetic data in easily accessible web-oriented way. However, the analysis proved that despite of wide variety and dynamic evolution of modern information technologies, special program complexes and tools for data processing, analysis and visualization, the approach of modeling and analysis of geomagnetic eld data is not enough developed. In the paper the authors suggest web-based solution, which provides the mechanisms to calculate, analyze and visualize parameters of geomagnetic eld and its variations and one of the services called "Geomagnetic Calculator".
One of the most important components of near-Earth space is the Earth's magnetosphere, where natural and anthropogenic factors cause a number of variations and anomalies, which are dangerous for the bio- and technosphere. It is well known that geomagnetic variations and magnetic anomalies with a high degree of probability can provoke short-term rearrangements of the vegetative-humoral and cardiovascular systems of man, which is fraught with serious disastrous consequences. Extreme geomagnetic phenomena lead to complete failure of power supply networks, the emergence of strong currents in pipelines, almost complete cessation of radio communications at all frequencies, the so-called "magnetic braking" of the Earth's arti cial satellite.
Such an impact on the technosphere often causes irreparable economic damage. And this is far from a complete list of the consequences to which the changes in the normal state of the geomagnetosphere can lead. The main way to prevent adverse e ects of geomagnetic factors on the bio- and technosphere is the forecast and monitoring of the geomagnetic situation. The importance of solving the problem of model-ing and analysis of geomagnetic eld and its variations is di cult to overestimate.
On the one hand, it is a system for observing and evaluating the current geomagnetic situation, and on the other hand, a means of informational support for the pro-cess of preparing and making managerial decisions in the relevant applied eld (biology, medicine, geophysics, geology, meteorology, etc.). Registration of the parameters of the geomagnetic eld is traditionally conducted both on the surface of the Earth and from outer space. So, among the ground-based instruments, the most important are magnetic observatories and stations recording both the total vector of the magnetic eld and its variations. The maximum density of magnetic observatories falls on the territory of the European part of the continent, while there is a complete absence in most countries of Asia, Africa and South America, whose population is many times larger than the countries of Western Europe. Measurements of the Earth's magnetic eld from outer space are currently carried out mainly by the European system of SWARM satellites. A large number of sources of geomagnetic data, both global and local, as well as a huge number of their consumers require an integrated and easily access approach to solving this problem. It is obvious that the solution is to be based on modern information technologies, which will provide a signi cant in-crease in the level of information and intellectual support for the ecological monitor-ing of geomagnetic variations and anomalies of the natural and anthropogenic nature of origin.
The analysis of the problem solutions proved, that information technologies providing calculation, analysis and visualization of geomagnetic data are poorly developed. One of the known solutions is the information service, which is provided by NOAA and available at http://www.ngdc.noaa.gov/geomag-web [NOAA Magnetic Field Calculators]. Although the calculation results are acceptable enough, the solution has a lot of disadvantages, such as no visualization tools, low ergonomics and e ciency. The same disadvantages are inherent to other similar services, which are provided by BGS, CIRES (http://geomag.org/models/igrfplus-declina-tion.html) and other leading organizations [Thomp14].
According to geomagnetic data analysis there is another well-known solution. It is the Interregional Geomagnetic Data Center of the Russian-Ukrainian INTERMAGNET segment, which is operated by the Geophysical Center of the Russian Academy of Sciences [Russian-Ukrainian Geomagnetic Data Center]. Geomagnetic data are transmitted from observatories located in Russia and Ukraine [Gv15, 8]. The solution is based on fuzzy logic approach and is intended to real-time recognition of arti cial (anthropogenic) disturbances in incoming data [Sol13].
One more key element of the mentioned problem is concerned with a number of heterogeneous data sources, which collect and share historical and current values of parameters of space weather, geomagnetic eld and its variations. These are powerful web services provided by INTERMAGNET, BGS and other organizations [Ham13][Kol17][Love13][Mac13]. The main thing here is that all these data sources are not integrated. There are no any tools, which provide to user a single access to space weather and geomagnetic data. In the paper the authors suggest a web-based solution of the problem, which provides modeling, monitoring, analytical control, two- and three-dimensional visualizations of geomagnetic eld and its variations parameters. With this solution a user can get access to data, automatically analyze them and use the results for solving applied problems.
Analysis of the known research results in relative areas, such as modeling and analysis of a whole-space transient electromagnetic eld [Jiang17], also proved, that they can not be applied to the geomagnetic eld studies, which special features are concerned with three-component intensity vector as well as a restrict mathematical model based on 5-year calculated common coe cients (IGRF model). The known results on 2.5-dimensional FDTD geoelectric modeling do not take into account, that geomagnetic state depends on various natural and technogenic factors and strongly determined by the spatiotemporal characteristics of the geographical point, where the parameters are to be calculated.
Thus, the proposed originality is concerned with a technical solution of geomagnetic eld analysis, modeling and graphical interpretation, based on the modern information technologies with a greater accent on using web technologies. So the proposed results can be available to a wide range of users with di erent scienti c interests and competency.
The full vector of the Earth's magnetic eld intensity in any geographical point with spatiotemporal coordinates (latitude, longitude, altitude, year) is de ned as a sum of three components:
Bge = B1 + B2 + B3: Pnm(cos ) = 1 3 5 ::: r (n + m)!(n m)! sinm [cosn m (n m)(n 2(2n m 1) 1) cosn m 2 + :::] (5) where m is a normalization factor ( m = 2 for m 1 and m = 1 for m = 0); n is a degree of spherical harmonics; m is an order of spherical harmonics.
In a number of scienti c problems, some geospatial data (for example, positions of satellites in space) is represented in coordinates ; ; h, which are based on the Earth's surface approximation by spheroid. In these problems, the Earth's ellipticity is neglected with no di erence between spherical and geodesic coordinates. But in accurate calculations, the Earth's pole compression should be taken into account [Thomp07].
So, the components X0; Y 0; Z0 of induction vector of intraterrestrial sources geomagnetic eld in nT are de ned as follows:
N U = RE X n
X (gnm cos m n=1 m=0 + hnm sin m )( RE )n+1Pnm cos : r where r is the distance from the Earth's center to the observation point (geocentric distance), [km]; is the longitude from Greenwich meridian, [degrees]; is the polar angle (colatitude, = ( =2) 0, [degrees], where 0 is the latitude in spherical coordinates, [degrees]); RE is an average radius of the Earth, RE = 6371:03 km; gnm(t); hnm(t) are spherical harmonic coe cients, [nT], which depend on time period; Pnm are Schmidt seminormalized associated Legendre functions of degree n and order m.
Due to the main eld temporal variations, the coe cients of harmonic series (spherical harmonic coe cients) are periodically (once in 5 years) recalculated with the new experimental data [Mer96].
The change of the main eld for one year (or secular variation) is also represented by spherical harmonics series.
Schmidt seminormalized associated Legendre functions Pnm in general can be de ned as an orthogonal polynomial, which is represented as:
The main eld model is based on spherical harmonic series, which are depending on geographical coordinates.
The scalar potential of intraterrestrial sources geomagnetic eld induction U [nT km] at any point with spherical coordinates r, , as follows:
X0 =
N = X n
X (gnm cos m n=1 m=0 + hnm sin m )( RE )n+2 @Pnm cos r @ where B1 is an intensity vector of geomagnetic eld of intraterrestrial sources; B2 is a regular component of intensity vector of geomagnetic eld of magnetosphere cur-rents, which is calculated in solar-magnetosphere coordinate system; B3 is a geomagnetic eld intensity vector component with technogenic origin.
The summand B3 represents the magnetic eld component of technogenic origin, which occurs due to the human life activity and usually has wide and unpredictable amplitude and frequency range and progress intention.
Normal geomagnetic eld is supposed as a value of B1 vector with excluding a component, which is caused by rocks magnetic properties (including magnetic anomalies). This component is excluded as a geomagnetic variation:
B0 = B1 where B0 is undisturbed geomagnetic eld intensity in the point with spatiotemporal coordinates; B10 is component of intraterrestrial sources of geomagnetic eld, which represents magnetic properties of the rocks [Gv15].
So, geomagnetic variations in the point with spatiotemporal coordinates can be de ned as follows: Bgmv =
B10 + B2 + B3: (1) (2) (3) (4) (6) Y 0 =
N = X n
X (gnm sin m n=1 m=0 (7) Z0 = @U = XN Xn (n + 1)(gnm cos m + hnm sin m )( RE )n+2@Pnm cos : (8) @r n=1 m=0 r
Thus, the geomagnetic eld induction vector distribution in space is described by a number of geomagnetic elements: orthogonal components of geomagnetic eld; modulus of geomagnetic eld induction vector; angle elements of geomagnetic eld (geomagnetic dipole).
The described model of geomagnetic eld is the basis of web-oriented solution for analysis and modeling of geomagnetic eld and its variations. 3
The most common problem in analyzing the Earth's geomagnetic eld concerns with calculating of the parameters of normal geomagnetic eld in a speci c point with given geographical coordinates. Its solution is based on applying some special types that are known as geomagnetic calculators.
The "Geomagnetic Calculator" is a web-based application providing calculation parameters of geomagnetic eld and its secular variations according to the set of coordinates and dates (Fig. 1) [Vorob15b]. A user of any level can calculate and analyze parameters of geomagnetic eld at his current location or at any other point on the Earth.
The "Geomagnetic Calculator" is a responsive application, which does not depend on device type and parameters but is transferable and works similarly in any device, i.e. phones, tablets, desktops. This platform- and device-independency is realized on the basis of the special framework "Bootstrap" (http://getbootstrap.com) that is a set of programming libraries for HTML, CSS and JavaScript. Flexibility and performance of this application tool are also increased by supporting HTML5 and CSS3 standards.
To calculate parameters of geomagnetic eld, a user has to de ne spatiotemporal coordinates of the point of interest on the Earth's surface. In "Geomagnetic Calculator" geodetic latitude and longitude can be de ned in various ways.
The simplest way to de ne the current geographical position of the user is a geolocation function. This function automatically takes the IP address of the device, which a user operates to access the Internet. This functionality allows the user to get on his location without searching for it on the map or lling the appropriate input elds.
Another way to de ne a point is to pick it on the map. A user can move over the map (using keyboard or mouse) and click on the speci c point. All necessary spatiotemporal parameters of the point are calculated automatically and are immediately displayed on the screen.
Both the aforementioned ways of location de nition do not require the exact coordinates of location as all necessary geospatial data are obtained automatically. A good, even though more complicated, way to calculate parameters of geomagnetic eld and present the speci c point on the map with high accuracy (up to a few meters) is by entering manually the coordinates into the input elds and hereafter the application displays the point on the map and the parameters of normal geomagnetic eld.
In addition, the "Geomagnetic Calculator" provides a geocoding functionality to de ne a speci c point. Geocoding is the transformation from of an address of full form, i.e. "postal code, country, city, street, building number" or of short form, i.e. just one item or their combination, into the coordinates set "north latitude, east longitude". Thus, in order to nd the point to be calculated, one can just enter in the "Search" eld the address of the location he is interested in.
The main functionality of the "Geomagnetic Calculator" provides e ective and re-liable calculation and analysis of parameters of normal (undisturbed) geomagnetic eld by the spatiotemporal coordinates with an error value of no more than 0.1% . "Geomagnetic Calculator" provides the calculation of the following parameters of geomagnetic eld: north component of geomagnetic eld induction vector; vertical component of geomagnetic eld induction vector; magnetic declination and dip; scalar potential of geomagnetic eld induction vector. 4
The authors suggested solution uses classic mechanisms of digital signal processing (DSP) to provide a set of tools for digital processing of data about space weather, GMF and GMV parameters, such as following [Vorob15b]: linear, nonlinear and adaptive ltering of information signal, which provides noise reduction and signal bandpassing in frequency domain as well as analysis of correlation of adjacent information signals parameters (Fig. 2, a); signal analysis in time domain, which provides a calculation of maximal, minimal and average values, variance and standard deviation of information signal; spectral and time-frequency analysis, which is implemented by two ways. First way is an analysis of periodogram as an estimation of power spectral density based on calculation of square of data sequence Fourier transformation module with statistical averaging if information signal. Second way is an estimation of wavelet scalogram of information signal (Fig. 2, b).
Main methodic of DSP, which is used in authors solution, includes the following steps.
Reading the discrete information signal (IS) as JSON/CSV/XML array Detection, analysis and exclusion of missing measurements Calculation and exclusion of IS constant component Caclulation of IS variance Calculation of numerical value of IS frequency threshold detector Calculation of direct Discrete Fourier Transformation (DFT) and periodogram based on Fast Fourier Transform (FFT) Calculation of the biggest IS frequency with power spectral density less than IS frequency threshold detector Calculation of low frequencies digital lter cuto frequency Preparation of discrete IS (original array) for digital ltering Calculation of direct DFT of IS, which is prepared to ltering Multiplication of Fourier image and transfer function Calculation of reverse DFT of IS on the basis of FFT Preparation of the array to save / output
Save / output of the ltered array in JSON/CSV/XML format 5
The proposed software solution [Vorob15][Vorob15b][Vorob17] for visualizing the parameters of the main geomagnetic eld is based on a combination of modern information technologies for computer visualization of Cesium and WebGL data, which in turn are based on an innovative web-oriented HTML5 platform that e ectively implements the presentation of multimedia data (in particular, 3D images) in web applications. Through these technologies, a three-dimensional graphical interpretation of the geospatial binding of the visualized parameters of the main geomagnetic eld is realized at the level of a specialized component - a virtual globe designed to represent both the Earth's surface and the nature of the distribution of the geomagnetic eld parameters on it. When working with a web application, the end user interacts directly with the interactive virtual globe, managing it with the mouse cursor to update the data in a timely manner, display individual geographical points or areas, and change the scale of their presentation.
In general, the solution of the task of visualizing the parameters of the main geo-magnetic eld can be represented in the form of the following generalized stages: 1. Create a virtual globe instance. 2. Con guring a virtual globe instance (scaling constraint, camera control, widget display, basic graphic layer). 3. Select the type of data source (KML, JSON, etc.) to display on the virtual globe. 4. Con guring the data source (boot method, refresh rate, color gamut, etc.). 5. Bind the data source to the virtual globe instance. 6. Cleaning the existing layers (except for the base layer). 7. Display a new layer (layers) on the virtual globe based on the speci ed data source.
At the software level, all of the above steps are implemented by the authors through the JavaScript programming language, as well as functions and methods of the Cesium library. A feature of Cesium technology is the ability to switch between di erent rendering modes (two- and threedimensional or pseudo-3D image) through standard tool widgets implemented in one API and not requiring additional program code. In this connection, all graphic solutions for visualizing the parameters of the main geomagnetic eld at the software level are realized once and do not recon gure under the change of the presentation. In this case, the level lines automatically adapt to the mode of graphical interpretation chosen by the user without any distortion or loss of functionality, which is also an essential advantage of the chosen visualization technology.
Another advantage of the Cesium software library and technology is the default geo-location and reverse geocoding tools, which allows to take full advantage of the capabilities of modern geoinformation technologies when solving geospatial data visualization problems. With this functionality, the end user can nd their location on a virtual globe at the touch of a button or move to a point corresponding to the text address they have entered (in Russian or English). At the same time, the implementation of such functionality does not require additional program code from the application developer.
The approach presented in this paper to visualize the parameters of the main geo-magnetic eld provides an increase in the e ciency and e ciency of the analytical control of the corresponding geomagnetic data by presenting it to the end user in the form most adapted for visual analysis and making appropriate decisions in the subject area. Thus, the comparative analysis of the values of the parameters of the main geo-magnetic eld for the period 2010-2015 made it possible to conclude that during this period the northern magnetic pole shifted from the point with coordinates (-80.015 N, 72.219 E) to the point with coordinates (-80.312 N, {72.621 E), which is about 33.93 km in the southern (geographical) direction. In the paper the authors suggested a number of solutions, which are based on a paradigm of heterogeneous data sources (satellite, air, marine, ground observations, and magnetic observatories) combination into integrated information space with platform-independent mechanisms of DSP and data interpretation. Also the authors proposed an approach whose goal was to replace the traditional textual and tabular representation of geomagnetic data with the results of their graphical two- and three-dimensional interpretation for subsequent visual analysis by specialists in the relevant applied eld of knowledge.
On the basis of the solution formalized by the authors, a specialized web-based service was developed and implemented that provides interactive visualization of the parameters of the main geomagnetic eld and its variations through innovative plug-in independent graphics acceleration technologies with the ability to vary image modes (2D, 3D, 2.5D), control the granular representation of geomagnetic Data and support for geolocation and geocoding functions. 7
The reported study was partly funded by RFBR according to the research project No.18-07-00193. [Gv15]
A.D. Gvishiani, R.Yu. Luk'janova. Geoinformatics and the Earth's magnetic eld observations: Russian segment. Fizika Zemli | Physics of the Earth, 2:3{20, 2015.