Ionosphere is the ionized part of the upper atmosphere containing free electrons and ions. The critical frequency (foF2) is one of the most important parameters of the ionospheric electron density, which affects the functionality of radio communication and navigation equipment. This frequency is a very variable parameter, which depends on various geophysical parameters. Therefore, improving the accuracy of predicting the critical frequency foF2 of the ionosphere is of great importance for the proper operation of radio communication and navigation equipment. Various physical and empirical models are used to predict this parameter. The large amount of observational data and development of modern machine learning algorithms make it possible to use new approaches for predicting ionospheric parameters. In the paper we analyzed a neural network created for predicting foF2 measured by Irkutsk Digisonde, as a function of various geophysical parameters. We studied the contribution of these parameters to the predicted foF2 frequency. It was shown that the critical frequency of foF2 most strongly depends on 10.7 cm solar radiation and on the local solar time.
The characteristics of the ionosphere are of signi cant practical importance for
radio communications. Prediction of these characteristics allows one to work
more reliably with radio communication and navigation equipment. Many
methods have been developed for predicting the ionospheric characteristics [
The results were obtained using the equipment of Center for Common Use \Angara" http://ckp-rf.ru/ckp/3056. available measurement data and the growth of computational capabilities allow one to use very complex empirical prediction models - arti cial neural networks.
One of the key ionospheric parameters is the foF2 frequency. The critical frequency foF2 is the maximal frequency of a radio wave propagating vertically and re ecting from the ionosphere [2]. If the frequency of the radio wave exceeds the critical frequency the wave penetrates the ionosphere. The foF2 critical frequency is frequently used for predicting the radio wave propagation characteristics. So the knowledge of this frequency is of practical importance for radio communications.
The foF2 dependence on the indices of solar and geomagnetic activity is well known [3]. The dependence of foF2 on the solar radiation intensity at the wavelength of 10.7 cm is especially signi cant. The physical reason of such a dependence is an increase in atmospheric ionization with an increase in solar radiation intensity, which can be detected by an increase in solar radio emission ux. The dependence of foF2 on the geomagnetic activity (for example on apindex) is also known. In this paper, we use Dst, ap and Kp geomagnetic activity indices for predicting foF2. The planetary index Kp describes the geomagnetic disturbance in the three-hour interval. The initial data for the calculation of the Kp index is the K-index data of the twelve observatories located between 63 and 48 north and south of the geomagnetic latitudes. The K-index takes values from 0 to 9, where 0 corresponds to undisturbed conditions and 9 corresponds to a very strong geomagnetic disturbance [4].
The index ap is calculated from the Kp-index data. It represents the change in the most disturbed geomagnetic eld component D or H over the 3-hour time interval at mid-latitude stations and is calculated in units of 2 . The ap index is refered as the planetary amplitude in the 3-hour interval. The Dstindex characterizes the intensity of the symmetric ring current, which is typically observed in the recovery phase of a geomagnetic storm. Dst-index is the average value of the geomagnetic disturbance over the hourly interval, calculated over the network of low-latitude geomagnetic stations separated by longitude. The unit of measurement for the Dst-index is . To calculate the Dst-index, the data from 4 geomagnetic stations are used [4].
For analysis we use measurements of the foF2 critical frequency near Irkutsk, made with the DPS-4 digisonde from 2009 to 2016. The DPS-4 digisonde has two 150W transmitters, four receiving antennas and a \crossed vertical rhombs" transmitting antenna system. The digitization module has the ability to simultaneously register in the automatic mode the following radio signal parameters: amplitude, frequency, height (range), angles of arrival, phase, polarization, and Doppler frequency shift of the radio waves re ected by the ionosphere [5]. Observations of the digisonde are obtained from the ISTP database [6]. Solar and geomagnetic indices for the same period are obtained from publically available databases (OMNI database [7]).
To take into account the periodicity of foF2 changes, in addition to the solar and geomagnetic indices mentioned above, a periodic function is used that depends on the day number in a year. The periodic functions that depend on the hour of the day have also been used.
cosDOY (LST ) = cos cosHOD(LST ) = cos sinHOD(LST ) = sin 2 2 2
DOY 365:25
HOD 24 HOD 24 (1) (2) (3) where HOD is the ordinal hour number of the day (from 0 to 23, we use local solar time), DOY is the ordinal day number of the year (from 1 to 365 or 366).
We train our foF2 prediction algorithms based on the following dataset: cos DOY (LST), cos HOD (LST) and sin HOD (LST), f10.7 solar index, Dst, Kp, ap geomagnetic indicies, foF2.
The resulting dataset is divided into two subsets: the training dataset and the test dataset in a ratio of 80 to 20%, respectively. The whole experimental data set consists of 61675 elements.
An arti cial neural network is a collection of interconnected perceptrons (neurons) aggregated into layers. Each perceptron in the neural network converts its input to its output by making a linear combination of input values and using it as an argument for some ('activation') function:
OutN (w; x) = (w1 x1 + w2 x2 + w3 x3 + : : : + wn xn + b) (4) where fxijx2;x3;x4;x5; : : : ; xng - is a set of input attributes, wi; b - weights, and (x) is an activation function.
For prediction of foF2 we trained a neural network with two hidden layers with 11 and 5 neurons in each, and a sigmoid activation function. The model was built on the basis of the Python scikit-learn software library and the Perceptron Multi-layer class [8], with \adam" solution algorithm, and a regularization coe cient of 0.001. We used an algorithm with back propagation of an error for training the neural network [9]. 3
An example of the forecast over test dataset is shown in Figure 1. The prediction over test dataset provides a Pearson correlation coe cient of 0.89, a root mean square error of 0.84MHz and a mean absolute percent error of 14.7%. A high correlation coe cient and a rather small root-mean-square error indicate that the model is of su cient quality to predict the ionospheric critical frequency. To estimate the degree of in uence of various input parameters on the nal result we made an analysis of the weights inside each neuron. last column of the Table 1 and Table 2 corresponds to the sum of the absolute values of the weight coe
cients in each neuron. The rows of the table are sorted in descending order by this parameter, allowing us to nd the neurons of higher importance.
As one can see from Table 1, the
rst three most in uential neurons (marked by color) of the
rst layer have signi cant prevalence of weights corresponding to local time and date (cos HOD (LST), sin HOD (LST), cos DOY (LST)), showing the importance of regular daily and seasonal variations in foF2 dynamics.
As one can see from Table 2 the rst most in uential neuron (marked by color) of the second layer contains large coe cients (-5.139 and 3.106) at the outputs of the tenth and eleventh neurons of the rst layer (which are the most in uential in the rst layer). At the same time, the coe cients at the output of the sixth, seven and eighth neurons of the rst layer are also signi cant (1.944, 1.508 and 1.437). These neurons of the rst layer have the greatest weights corresponding to the index f10.7 (solar radiation ux at a wavelength of 10.7 cm).
The output layer (see Table 3) has coe cients of approximately the same order. This allows us to ignore the output neuron and to limit the analysis of the most in uential neurons to analysis of the rst two layers. From this we can conclude that the main contribution to the neural network output is produced by solar radiation index f10.7 and solar local time and date. 4
In this paper we described our experience in creating and training a neural network for predicting the critical frequency foF2 using data obtained at Irkutsk with the DPS4 Digisonde for the period 2009-2016, as well as solar and geomagnetic indices for the same period. It is shown that average model accuracy is about 0.84MHz and Pearson correlation coe cient is 0.89.
We analyzed the structure of the network and showed that variations in the ionospheric critical frequency foF2 are mostly caused by f10.7 index and periodical daily and seasonal variations. The dependence of foF2 on the geomagnetic indices Dst, ap, and Kp is much weaker. This result corresponds well with physical mechanisms of formation F2 layer in the ionosphere and can be explained by an increase in the ionospheric ionization intensity with the solar radiation intensi cation and the daily and seasonal dynamics of the solar zenith angle. 2. J.K. Hargreaves. The Upper Atmosphere and Solar-Terrestrial Relations. An introduction to the aerospace environment. 1982 (Dzh. K. Hargrivs. Verhnyaya atmosfera i solnechno-zemnye svyazi. Vvedenie v ziku okolozemnoj kosmicheskoj sredy. L.: Gidrometeoizdat, 1982. In Russia) 3. M.G. Deminov, G. S. Ivanov-Holodnyj, E.V. Nepomnyashchaya. Dependence of the quasi-biennial variations of the critical frequency of the F2 layer on the indices of solar and geomagnetic activity. // Geomagnetism and aeronomy. 2002. (M.G. Deminov, G. S. Ivanov-Holodnyj, E.V. Nepomnyashchaya. Zavisimost' kvazidvuhletnih variacij kriticheskoj chastoty F2-sloya ot indeksov solnechnoj i geomagnitnoj aktivnosti. // Geomagnetizm i aeronomiya. 2002. T. 42, 1, s. 112-115 In Russia) 4. Zabolotnaya N. A. Indices of geomagnetic activity. Reference manual. 2007. (Zabolotnaya N. A. Indeksy geomagnitnoj aktivnosti. Spravochnoe posobie. M.: LKI, 2007. In Russia) 5. V.F. Smirnov, A.E. Stepanov. New opportunities in high-latitude ionosphere studies: Dps-4 digisonde - rst results on localization measurements and dynamics of largescale ionospheric structures in Yakutsk. // Journal Solar-Terrestrial Physics. 2004. (V.F. Smirnov, A.E. Stepanov. Novye vozmozhnosti v issledovaniyah vysokoshirotnoj ionosfery: digizond Dps-4 { pervye rezul'taty po izmereniyam lokalizacii i dinamiki krupnomasshtabnyh struktur ionosfery v Yakutske. // Solnechno-Zemnaya Fizika. 2004. 5 (118), str.: 105-106 In Russia) 6. Ionospheric database, http://hawk.iszf.irk.ru/dps data/ 7. Interface to produce plots, listings or output les from OMNI 2, https://omniweb.gsfc.nasa.gov/form/dx1.html 8. Neural network models (supervised), https://scikitlearn.org/stable/modules/neural networks supervised.html#multi-layerperceptron 9. Backpropagation Algorithm, http://u dl.stanford.edu/wiki/index.php/Backpropagation Algorithm