Ionospheric prediction algorithm and its application in low-latitude regions based on the physically constrained polynomial model

With sufficient consideration of the ionospheric variation subject to solar activity and geomagnetic variation, a high-precision ionospheric prediction model, i.e., a physically constrained polynomial model (PCPM), was constructed by adding the Kp index, Dst index, F10.7P, R sunspot number, and other geomagnetic and solar variation indexes. Based on the data of Continuously Operating Reference System (CORS) stations in China Southern Power Grid, the ionospheric prediction results of PCPM were compared with the widely used and well-recognized seasonal enhanced product autoregressive integrated moving average (Arima) model without additional physical parameters. The prediction performance of the two models in different prediction time spans and the influence of model products on Precise Point Positioning-Real Time Kinematic (PPP-RTK) was emphatically analyzed. The experimental results demonstrated that the accuracy of PCPM is better than that of the seasonal Arima model on the first day of prediction, the prediction accuracy of the two models decreases with the increase of prediction time span, and the lowest accuracy of the seasonal Arima model on the first three days is 1.5 TECU. The Arima model outperforms the PCPM over a long-time span; in light of the overall accuracy, PCPM is more available in five southern provinces, outperforming the seasonal Arima model; the results of the PPP-RTK experiment showed that the PCPM could not only improve the velocity of ambiguity fixing but also improve the positioning accuracy after ambiguity fixing. Compared with the seasonal Arima model, the velocity of ambiguity fixing is increased by 16.7%, and the positioning accuracy after ambiguity fixing is improved by 61.0% in the E direction and 6.9% in the U direction compared with the results without ionospheric constraints.


Introduction
The ionosphere is an atmospheric ionization region with a height of 60-2000 km from the group and is an important part of the terrestrial and solar space environment.Radio signals are also subjected to refraction, reflection, scattering, and absorption during ionosphere transmission.This will delay global navigation satellite system (GNSS) signals and cause several hundred-meter errors, making it a rather thorny problem in GNSS data processing [1][2][3][4][5] .For dual-frequency receivers, the ionosphere can be eliminated by constructing ionospheric combined observations, which are inversely proportional to the square of signal frequency.Still, for single-frequency receivers, the ionospheric errors are mainly corrected and eliminated by ionospheric models.To this end, the ionospheric errors affect GNSS's service mode and application range to a certain extent [6][7][8] .Currently, the commonly used ionospheric models mainly include the Klobuchar model, Bent model, international reference ionosphere (IRI) model, and other empirical models, all of which are mathematical models formed by statistical results, and whose model correction effect can only reach about 60%.The accuracy of these models in ionospheric prediction cannot meet the needs of daily operations across the board.Moreover, due to the distribution of modeling data, these models are not ideal for regional applications in China, especially in low-latitude regions covered by five southern provinces (Guangdong, Yunnan, Hainan, Guangxi, and Guizhou), and there are time and space limitations in the models [9][10][11][12] .Therefore, domestic scholars have studied the time sequence prediction model, neural network prediction model, and other models.Still, the above models do not consider the external physical parameters that affect ionospheric variations, and the ionospheric variations cannot be interpreted only by mathematical modeling [13][14][15][16][17][18][19] .The physical parameters such as solar radiation index and geomagnetic index that affect ionospheric variations are not introduced in the modeling and prediction of models.CORS is a local GNSS reference network composed of continuous operating reference stations.At present, there are more than 1000 continuously operating reference stations in the five southern provinces, which constitute continuously operating reference station systems covering the whole country, serving as an important aspect of GNSS application and development in the five southern provinces.However, CORS stations often face failure problems such as power failure and network disconnection, which leads to CORS stations being unable to provide services for users.Maintenance often takes several days for remote base stations and base station failures often cause engineering delays and waste of service resources [20] .Therefore, an ionospheric prediction model based on CORS observation data is urgently needed.
In this paper, based on the ionospheric oblique delay calculated from CORS data, Kp index, Dst index, and F10.7 data provided by the National Aeronautics and Space Administration (NASA) were used to constrain physical parameters, polynomial epoch-by-epoch fitting prediction method was adopted, a PCPM is constructed to analyze the prediction performance and accuracy under different prediction time spans.In comparison with the seasonal product Arima model in five southern provinces, the improvement effect of the model prediction products in PPP-RTK was emphatically analyzed.

Physically constrained polynomial model
Ionospheric variations are closely related to solar activity and geomagnetic variations.This section mainly describes the correlation indexes of solar activity and geomagnetic variations and uses these physical parameters to model and predict the ionosphere.Details are described below.
1) Solar activity index Extreme Ultraviolet (EUV) solar radiation is the main ionization source of the Earth's ionosphere, so the solar activity index can well reflect the ionosphere variations.Solar activity indexes commonly include sunspot number (SSN) and 10.7 cm radio emission flux (F10.7).Due to the lack of long-term continuous observation of solar EUV radiation, the solar radiation flux (F10.7) with a wavelength of 10.7 cm observed on the ground is often used to express solar radiation variations.The determination of solar radio emission intensity in a 100 MHz bandwidth centered on 2800 MHz (wavelength of 10.7 cm) with an hour as the center is expressed in solar luminous flux units (sfu).F10.7P (P) is a parameter calculated using F10.7, which can better reflect the intensity of solar EUV radiation flux.Its calculation method is as follows: 10.7 10.7 10.7/2 (1) where F10.7A is the 81-day average of F10.7.
2) Geomagnetic index Geomagnetic indices mainly include Ks, Kp, Ap (unit 2nT), Cp, C9, Q-DAY, D-DAY, DST, AE index, etc.Among them, the Kp index is a 3-hour index derived from the Ks index (obtained from the data of 13 geomagnetic stations), which is the average of horizontal disturbance grades of two components in three components of geomagnetic observation.The geomagnetic indices of Ap, Cp, C9, Q-DAY, and D-DAY are derived from the Kp index.The kp index reflects the state of the earth's magnetic field according to its numerical value.A Kp index in the range of 0-2 indicates that the earth's magnetic field is calm, 2-3 indicates geomagnetic disturbance, 3-4 indicates that the geomagnetic field is active, 4-5 indicates that the geomagnetic field has small magnetic storms, 5-6.5 indicates that the geomagnetic field has generated large magnetic storms, and 6.5 or more has generated severe magnetic storms.The Dst index refers to the geomagnetic index used by stations in middle and low latitudes, which is the instantaneous average of global equatorial H component disturbance.Dst index and Kp index are the physical parameters representing geomagnetic variations used in this ionospheric modeling and prediction.
3) Physically constrained polynomial modeling There are a lot of physical parameters reflecting solar activity and geomagnetic variations to enumerate.This paper solves the correlation coefficient between physical parameters and ionospheric data.The physical parameters with the largest correlation coefficient, such as the Dst index, F10.7P, and Kp index, were selected to construct multivariate polynomial equations with ionospheric time sequence data.
⋯     ⋯     ⋯   (2) where  is the ionospheric delay value at time t;  is the value of F10.7P at time ;  is the value of the Dst index at time ;  is the value of the Kp index at time ; , , and  are the polynomial orders of F10.7P, Dst index, and Kp index respectively; , , and  are polynomial coefficients.According to the observation data of  (   1) days, the matrix equation is constructed: According to the principle of least squares: (7) The ionospheric physically constrained polynomial model at time  can be obtained by substituting Equation (7) to Equation ( 2), and the ionospheric value at the prediction time can be obtained by substituting F10.7P, Dst index, and Kp indices to the equation.The ionospheric prediction work can be completed for continuous time periods by modeling continuous radian epoch by epoch.

Experimental analysis
In this paper, the observation data of 10 stations in five southern provinces in 2018 (from January 1, 2018, to March 2, 2018) were selected, and the ionospheric data calculated by the non-combined PPP method was used as the data source for modeling.Considering that the solar rotation period is 25 days, this paper selected 27 days as a group of data for modeling.The data site distribution is shown in Figure 1.

Figure 1 Data site distribution diagram
In this paper, we obtain the ionospheric arc segment by the elevation angle alignment interception instead of time interception to acquire experimental data.By using the elevation angle and puncture point position, the position of satellite puncture points every day is judged, and the ionospheric delay close to the puncture point is aligned, so the arc segment alignment judgment interception operation is carried out, and the epoch time attribute is marked to form modeling ionospheric data.The 27-day ionospheric skew parameters after arc alignment and three physical parameters are selected to construct a polynomial matrix epoch by epoch, and the polynomial coefficients form the matrix to be solved.The polynomial coefficients are solved by Equations ( 2)- (7).The order of each physical parameter is traversed 0-8 in turn, and the process of model equation construction and model solving is repeated.The polynomial solution of each epoch needs to be traversed 279 times, the order with the highest internal coincidence accuracy is selected, and the polynomial parameters are output to complete the polynomial modeling of each epoch.
After the polynomial order of each epoch model is determined, the model is used for prediction analysis.Because the polynomial is prone to producing an edge effect, it will lead to spikes.In ionospheric applications, the smoothness of the ionosphere is of great significance.In order to ensure the smoothness of the predicted arc segment, polynomial fitting is fitted to the predicted arc segment to generate a smooth ionospheric arc segment.In this process, polynomial fitting can better weaken the influence of spikes compared with moving average models.This study obtained from preliminary experiments that the average accuracy of polynomial fitting ionospheric arc segment is 0.1 TECU, so the overall fitting of the ionospheric arc segment after polynomial prediction will not affect the prediction accuracy.
To explore the preliminary effect of the PCPM, the estimated ionospheric data for 27 days from January 3, 2018 to January 30, 2018 were selected to predict the ionospheric data on January 31, 2018 and compared with the ionospheric oblique delay estimated with the observed data on January 31, 2018, and with the results of directly replacing the day with the previous day (January 30, 2018), where the individual satellite results for station QXQB (27.02°N, 106.02°E,Beijing) are shown in Figure 2. From the above figures, it can be found that PCPM can well predict the ionospheric variation trend, and it is closer to the true value, and the predicted ionosphere is smoother and has a higher coincidence with the true value, which fully demonstrates the feasibility of PCPM for ionospheric modeling and prediction.

Accuracy analysis under different time spans
In order to verify the influence of PCPM under different time spans, this section selects five months of data for five sets of predictions, and the experimental data are described in the Table 1: Table 1 Description of the experience Using 27-day data to forecast one day, the prediction residual error of the above experimental prediction time results is classified and counted, and the percentage distributions of the prediction residual error of less than 1 TECU, 1-2 TECU, 2-3 TECU, and more than 3 TECU were counted.The statistical results are shown in Table 2.In Table 2, the residual classification of the seasonal Arima model and PCPM in Chapter 3 is compared.The data show that 82.85% of the prediction residuals of the PCPM are less than 1 TECU, which is 10.85% higher than the Arima model.95.14% of the data residuals of the PCPM are less than 2 TECU.At the same time, there is no data with residuals greater than 3 TECU in this model, while 0.63 of the data residuals of the seasonal Arima model are still more than 3 TECU.Through the above table, we can draw the conclusion that the PCPM is smaller than the seasonal Arima model on the whole, which shows that the prediction accuracy of the PCPM in five southern provinces is higher.In order to compare the prediction accuracy of the PCPM and the seasonal Arima model more intuitively, taking the QXQB station as an example, we selected the data of Doy 28, Doy 87, and Doy 148 for three days and drew the ionospheric series predicted by the two models as shown in Figure 3. From Figure 3, it can be noted that the satellite prediction results of the PCPM are relatively closer to the true value than the seasonal Arima model for some time periods, but it can be seen from the overall map that the accuracy of the two models is equal, the details of seasonal Arima model are described more finely, and the trend of PCPM is smoother.The main reason is that the details are lost after the PCPM is fitted again by the polynomial.Therefore, in practical application, if more attention is paid to the detailed changes of the ionosphere, the seasonal Arima model is selected for prediction; if importance is attached to the smoothness of the ionosphere, the PCPM is a better choice.In order to further compare the accuracy of the PCPM and the seasonal Arima model, this paper makes accuracy statistics on five groups of experimental results in different periods, and the statistical results are shown in Table 3  Prediction analysis of five sets of experiments for five months of data revealed that the accuracy of the seasonal Arima model of 0.97 TECU improved by 0.87 TECU compared to the combined result of five sets of experiments for Fdata of 1.84 TECU, a percentage improvement of 47.28%.The comprehensive accuracy of PCPM is 0.82 TECU, which is 15.5% higher than that of the seasonal Arima model.The average bias of the PCPM is -0.27TECU, and the absolute bias is less than that of the seasonal Arima model, which shows that the prediction accuracy of the PCPM is better than that of the Arima model.The accuracy of the seasonal Arima model in UT10-14 is about 0.6 TECU in the evening time, which is better than the other two time periods, and the accuracy amid the afternoon time is the lowest among the three time periods.The main reason for this phenomenon is the influence of the sun, which is the strongest in the afternoon, during which the ionosphere is active, while the ionosphere is calm at night.PCPM in UT4-10 that is, local time in the afternoon, the model prediction is the best, indicating that additional physical parameters can well describe the impact of solar activity on ionospheric variations.In order to better analyze the influence of time span on the PCPM, this paper used the above experimental data to analyze the accuracy of 27-day prediction for 7 days.The accuracy statistics of the PCPM are shown in Table 4, the accuracy statistics of the seasonal Arima model are shown in Table 5, and the comparison chart of the 7-day prediction is shown in Figure 3.It can be seen from Figure 4 that the RMS values of the prediction results show a fluctuating upward trend.That is, with the increase of the prediction time span, the prediction accuracy of the model becomes lower.The prediction accuracy of the seasonal Arima model from the 28th to the 30th day is good, which is less than 1.5 TECU.The prediction accuracy of the seasonal Arima model changes slowly in the first three days.In March and April, the data of the two groups of experiments after three days show the phenomenon that RMS is greater than 3 TECU, demonstrating that the stability of prediction accuracy of the model becomes worse after three days.The prediction accuracy of the PCPM decreases with the increase of the prediction time span.The prediction accuracy of the PCPM on the first day is higher than that of the seasonal Arima model, but in the experiment of April, the prediction accuracy of PCPM on the second day jumps to 3 TECU.Hence, for practical application, the continuous, usable days of PCPM are only one day.Compared with the seasonal Arima model, the model performs poorly in multi-day prediction.In this regard, when CORS stations are cut off from power and network in five southern provinces, the ionospheric prediction products using the seasonal Airma model can only provide high-precision and stable ionospheric services for three days, and the accuracy of ionospheric prediction products after three days is insufficient to provide users with operations.It is sufficient to say that PCPM can only be provided for one day.

Application analysis of PPP-RTK
In actual production, the statistical accuracy of products often cannot represent the effect of their practical application, so this paper will use the prediction products produced by PCPM to study and analyze the application of PPP-RTK.The observation data of QXQB station UT6:30-UT7:30 on Doy 28, that is, 14:30-15:30 local time, with a sampling rate of 30 s, are selected for experimental analysis, and the performance of ionospheric products predicted by the PCPM and Arima model in positioning for one day by using 27-day data is compared and analyzed.The residual sequence diagram of positioning results is shown in Figure 5, the statistical table of ambiguity fixed time is shown in Table 6, the statistical table of overall positioning results is shown in Table 7, and the statistical table of accuracy after ambiguity fixed is shown in Table 8.In terms of the velocity of ambiguity fixing, it can be seen from Figure 5 and Table 6 that the ambiguity fixing of the PCPM is the fastest, taking 10 minutes, which is 16.7% higher than that of the seasonal Arima model and 37.5% higher than that of the non-ionospheric-constrained experiment, that is, the ionospheric prediction products of PCPM can effectively improve the ambiguity.
Positioning accuracy: It can be seen from the overall positioning accuracy statistics in Table 7 that the positioning accuracy with ionospheric constraints is generally better than that without ionospheric constraints, among which the PCPM has the highest accuracy.Compared with the seasonal Arima model, the PCPM has improved the positioning accuracy by 55.5% in the E direction, 45.5% in the N direction, and 46.7% in the U direction.That is, PCPM has outstanding performance in improving positioning accuracy.
Accuracy of ambiguity fixing: According to the data in Table 8, after ambiguity fixing, the positioning results of the PCPM are significantly better than the other three groups of experimental results, and the accuracy of ambiguity fixing is improved by 61.0% in the E direction and 6.9% in the U direction compared with the results without ionospheric constraints.The seasonal Arima model and the results of the previous day's ionospheric product experiment have no effect on improving the accuracy of ambiguity fixing, mainly because the ionospheric product accuracy of PCPM is higher and smoother.In contrast, the seasonal Arima model has higher overall accuracy, but its maximum and minimum biases are larger and its smoothness is lower.So PCPM can effectively improve the velocity and accuracy of PPP-RTK ambiguity fixing, seasonal Arima model can only improve the velocity of ambiguity fixing.

Conclusion 1)
According to the accuracy comparison of model prediction in different time spans, the results show that the average bias of the seasonal Arima model is 72.01%within 1 TECU, 95.21% less than 2 TECU, and 0.63% more than 3 TECU.The prediction residual of PCPM is less than 1 TECU (82.85%), which is 10.85% higher than that of Arima model.The prediction residual of the PCPM is less than 2 TECU (95.14%), and there is no data with a residual greater than 3 TECU in this model.The prediction accuracy of the PCPM is better than that of the seasonal Arima model on the first day of prediction.The prediction accuracy of the two models decreases with the increase in prediction time span, and the lowest prediction accuracy of the seasonal Arima model on the first three days is 1.5 TECU.The Arima model is better than the PCPM in the long time span.
2) The results of the PPP-RTK experiment show that the seasonal Arima model prediction product can effectively enhance the velocity of ambiguity fixing.In comparison with the Total Electron Content (TEC) constraint of the same period of the previous day, the velocity of ambiguity fixing of the model prediction product is increased by 11.5%, and compared with the fixed velocity without ionospheric products, it is increased by 25%; compared with the seasonal Arima model, the PCPM can not only improve the velocity of ambiguity fixing but also improve the positioning accuracy after ambiguity fixing.The velocity of ambiguity fixing is increased by 16.7%, and the positioning accuracy after ambiguity fixing is improved by 61.0% in the E direction and 6.9% in the U direction compared with the result without ionosphere constraint.

About the author:
Weizhao Huang (1981.11-),male, Han nationality, born in Nan'an, Fujian Province, graduated from South China University of Technology and Shenzhen Power Supply Bureau with a master's degree.He is a senior engineer and has been engaged in primary equipment management, artificial intelligence technology research, digital transformation of power transmission and transformation, and other related aspects all year round.

Figure 2
Figure 2 STEC prediction value and the true value of each satellite in QXQB station on January 31, 2018

Figure 3
Figure 3 Comparison of prediction results of QXQB station

Figure 4
Figure 4 Comparison chart of precision prediction for 7 days of the experiment

Table 2
Statistical table of classification percentage of predicted residuals value Δ (%) below.Table 3 Statistical table of prediction accuracy

Table 4
Statistical table of seven-day prediction results of the PCPM

Table 5
Statistical table of seven-day prediction results of the seasonal Arima model

Table 6
The time of ambition fixing

Table 7
Statistical table of overall positioning accuracy