联合半参数规则学习的电离层TEC预报

doi:10.11947/j.AGCS.2025.20250209

测绘学报 ›› 2025, Vol. 54 ›› Issue (10): 1741-1756.doi: 10.11947/j.AGCS.2025.20250209

联合半参数规则学习的电离层TEC预报

潘雄¹(), 赵子瑄¹, 平常¹, 金丽宏²(), 刘立龙³

^1.武汉纺织大学计算机与人工智能学院，湖北　武汉　430200
^2.武汉纺织大学数学与统计学院，湖北　武汉　430200
^3.桂林理工大学测绘地理信息学院，广西　桂林　541004

收稿日期:2025-05-19 修回日期:2025-09-12 出版日期:2025-11-14 发布日期:2025-11-14
通讯作者: 金丽宏 E-mail:pxjlh@163.com;33384351@qq.com
作者简介:潘雄（1973—），男，博士，教授，研究方向为深度学习、卫星导航定位。E-mail：pxjlh@163.com
基金资助:
广西自然科学基金(2024GXNSFDA010041);国家自然科学基金(42174010)

Ionospheric TEC prediction incorporating semi-parametric and rule-learning

Xiong PAN¹(), Zixuan ZHAO¹, Chang PING¹, Lihong JIN²(), Lilong LIU³

^1.School of Computer Science and Artificial Intelligence, Wuhan Textile University, Wuhan 430200, China
^2.School of Mathematics and Statistics, Wuhan Textile University, Wuhan 430200, China
^3.College of Geomatics and Geoinformation, Guilin University of Technology, Guilin 541004, China

Received:2025-05-19 Revised:2025-09-12 Online:2025-11-14 Published:2025-11-14
Contact: Lihong JIN E-mail:pxjlh@163.com;33384351@qq.com
About author:PAN Xiong (1973—), male, PhD, professor, majors in deep learning and satellite navigation positioning. E-mail: pxjlh@163.com
Supported by:
The Guangxi Natural Science Foundation of China(2024GXNSFDA010041);The National Natural Science Foundation of China(42174010)

摘要/Abstract

摘要：

针对电离层球谐函数周期特征选取困难、影响因素众多的难题，本文引入规则学习，基于半参数模型建立了半参数与规则学习球谐函数组合模型（Semi-SH-RL）。首先，引入规则学习，基于电离层先验知识构建规则集和约束集，精准提取电离层的周期；其次，在规则学习的规则层中引入自注意力机制与剪枝层，优化特征权重并剔除冗余周期，得到电离层球谐函数的优化周期项；然后，为了减弱模型计算误差与窗宽参数的影响，将周期项估计值与窗宽参数综合考虑，建立半参数变系数球谐函数新模型；最后，利用欧洲定轨中心6 a的数据验证本文模型的适用性。结果表明：得到剪枝的特征后再通过自注意力机制赋权，能够有效改进规则学习神经网络模型，优化周期的选择，改进规则学习神经网络能捕捉到8层月周期和5层日周期特征，有效分解周期。Semi-SH-RL模型与QP、C1PG、LSTM、Semi-SH、Semi-SH-AR、Semi-SH-LSTM 6种模型对比，纬度方面小于0.5 TECU的残差均值改进率分别为41.01%、28.01%、22.40%、11.02%、12.63%和8.30%；单天预报平均误差在1、3和5 TECU以内占比分别为62.61%、94.95%、98.97%，均优于其他对比模型；滑动预报5 d较Semi-SH和Semi-SH-AR模型提升了4.80%和4.54%。本文模型的周期项特征对于不同阶次的球谐函数具有一致性，周期值经一次规则学习后就能精度收敛，预报效率显著提高。

关键词: 电离层TEC预报, 半参数模型, 规则学习, 自注意力机制, 剪枝

Abstract:

Aiming at the challenges of selecting periodic features of ionospheric spherical harmonic functions and the multitude of influencing factors, this paper establishes an ionospheric total electron content (TEC) forecasting mode—semiparametric-spherical harmonic-rule learning (Semi-SH-RL), which integrates semi-parametric methods with rule learning. Firstly, rule learning is introduced to construct rule sets and constraint sets based on prior knowledge, enabling precise extraction of ionospheric periods. Secondly, a self-attention mechanism and a pruning layer are incorporated to optimize feature weights and eliminate redundant periods, thereby obtaining the periodic terms of ionospheric spherical harmonic functions. Then, to mitigate the impact of model computational errors and window width parameters, a semi-parametric varying-coefficient spherical harmonic function model is established by considering both the estimated periodic terms and window width parameters. Finally, the applicability of the proposed theory is validated using six years of data from the Center for Orbit Determination in Europe (CODE). The results show that the improved rule-learning neural network can capture eight-layer monthly and five-layer daily periodic features, effectively decompose periods and reduce noise, and improve the rule-learning neural network model and the selection of periods by assigning weights through the self-attention mechanism after obtaining pruned features. Semi-SH-RL is compared with quadratic programming (QP), CODE'S 1-day predicted GIM (C1PG), long short-term memory (LSTM), semiparametric-spherical harmonic (Semi-SH), semiparametric-spherical harmonic-auto-regressive model (Semi-SH-AR), and Semi-SH-LSTM. In terms of latitude, the mean improvement rates of residuals less than 0.5 TECU are 41.01%, 28.01%, 22.40%, 11.02%, 12.63%, and 8.30%, respectively. Semi-SH-RL model achieves average single-day forecasting errors within 1, 3, and 5 TECU for 62.61%, 94.95%, and 98.97% respectively, outperforming other comparative models. For a five-day sliding forecast, the model shows improvements of 4.80% and 4.54% over the Semi-SH and Semi-SH-AR models. The periodic features of the model exhibit consistency across spherical harmonic functions of different orders, and the periodic values converge in accuracy after a single round of rule learning, significantly enhancing forecasting efficiency.

Key words: ionospheric TEC prediction, semi-parametric model, rule-learning, self-attention mechanism, pruning

中图分类号:

P228

潘雄, 赵子瑄, 平常, 金丽宏, 刘立龙. 联合半参数规则学习的电离层TEC预报[J]. 测绘学报, 2025, 54(10): 1741-1756.

Xiong PAN, Zixuan ZHAO, Chang PING, Lihong JIN, Lilong LIU. Ionospheric TEC prediction incorporating semi-parametric and rule-learning[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(10): 1741-1756.

图/表 15

图1

图2

图3

图4

图5

表1

图6

表2

不同数据段单天预报时的精度稳定度"

数据段	模型	残差绝对值（\|Δ\|）范围/（%）			精度评定/TECU
数据段	模型	\|Δ\|<1TECU	\|Δ\|<3TECU	\|Δ\|<5TECU	E（\|Δ\|）	var（Δ）	RMSE
1	QP	39.07	83.23	96.13	1.734 6	2.324 3	2.309 4
	C1PG	48.30	89.67	98.49	1.389 7	1.513 7	1.856 1
	LSTM	48.16	92.90	98.86	1.294 4	1.224 5	1.702 9
	Semi-SH	56.32	94.39	98.87	1.136 5	1.125 2	1.554 6
	Semi-SH-AR	55.40	94.22	98.88	1.153 8	1.137 7	1.571 3
	Semi-SH-LSTM	59.01	94.48	98.88	1.093 2	1.148 3	1.530 9
	Semi-SH-RL	63.01	95.18	99.00	1.014 7	1.055 9	1.444 1
2	QP	56.67	92.92	98.22	1.181 6	1.393 7	1.670 3
	C1PG	61.11	95.14	99.29	1.052 8	0.945 5	1.433 2
	LSTM	57.77	96.15	99.44	1.058 9	0.882 5	1.415 6
	Semi-SH	68.83	97.64	99.62	0.847 9	0.676 1	1.181 1
	Semi-SH-AR	69.52	97.65	99.62	0.837 1	0.676 0	1.173 3
	Semi-SH-LSTM	67.68	97.76	99.67	0.870 3	0.658 4	1.189 9
	Semi-SH-RL	72.68	98.43	99.77	0.760 1	0.545 3	1.059 7
3	QP	57.88	94.60	99.19	1.099 9	1.024 9	1.494 9
	C1PG	74.87	96.70	99.68	0.801 2	0.718 1	1.166 2
	LSTM	64.67	98.54	99.96	0.876 2	0.491 1	1.122 0
	Semi-SH	72.66	98.63	99.97	0.765 9	0.459 5	1.022 8
	Semi-SH-AR	72.62	98.62	99.97	0.766 3	0.460 0	1.023 3
	Semi-SH-LSTM	75.09	98.99	99.98	0.727 3	0.398 8	0.963 2
	Semi-SH-RL	76.27	98.87	99.97	0.702 4	0.432 2	0.962 1
4	QP	43.95	86.54	95.78	1.626 3	2.827 0	2.339 2
	C1PG	55.00	92.64	98.26	1.235 8	1.454 6	1.726 8
	LSTM	40.10	87.52	97.22	1.621 1	2.294 6	2.218 7
	Semi-SH	47.60	91.21	98.22	1.373 8	1.764 9	1.911 1
	Semi-SH-AR	49.80	91.68	98.23	1.331 2	1.765 8	1.880 9
	Semi-SH-LSTM	47.41	91.66	98.24	1.357 6	1.611 1	1.858 6
	Semi-SH-RL	54.93	93.06	98.44	1.206 0	1.502 5	1.719 6
5	QP	39.90	83.38	94.73	1.755 2	2.722 4	2.408 9
	C1PG	41.01	82.85	95.80	1.732 9	2.287 8	2.300 1
	LSTM	40.10	83.34	95.16	1.736 2	2.604 3	2.370 4
	Semi-SH	46.15	86.62	96.37	1.534 6	2.228 2	2.140 9
	Semi-SH-AR	43.26	85.18	95.86	1.628 1	2.387 2	2.244 5
	Semi-SH-LSTM	37.54	87.04	97.36	1.629 7	1.784 4	2.107 2
	Semi-SH-RL	50.31	89.89	97.45	1.369 3	1.783 3	1.912 7
6	QP	43.38	86.61	96.23	1.599 7	2.370 9	2.220 3
	C1PG	48.53	85.58	97.39	1.513 3	1.906 8	2.048 6
	LSTM	42.17	89.22	97.38	1.514 1	1.780 1	2.018 0
	Semi-SH	51.92	91.60	98.31	1.287 3	1.494 2	1.775 2
	Semi-SH-AR	53.25	92.21	98.32	1.250 8	1.447 8	1.735 6
	Semi-SH-LSTM	49.68	92.42	98.18	1.293 7	1.412 9	1.756 9
	Semi-SH-RL	58.44	94.26	99.21	1.092 6	1.061 7	1.501 9
均值	QP	46.81	87.88	96.71	1.499 6	2.110 5	2.073 8
	C1PG	54.80	90.43	98.15	1.287 6	1.471 1	1.755 2
	LSTM	48.83	91.28	98.00	1.350 2	1.546 2	1.807 9
	Semi-SH	57.25	93.35	98.56	1.157 7	1.291 4	1.597 6
	Semi-SH-AR	57.31	93.26	98.48	1.161 2	1.312 4	1.604 8
	Semi-SH-LSTM	56.07	93.73	98.72	1.162 0	1.169 0	1.567 8
	Semi-SH-RL	62.61	94.95	98.97	1.024 2	1.063 5	1.433 3

表2

图7

表3

图8

表4

表5

图9

表6

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试验	模型	各数据段的RMSE/TECU							耗时/s
试验	模型	1	2	3	4	5	6	均值	耗时/s
1	Semi-SH	1.554 6	1.181 1	1.022 8	1.911 1	2.140 9	1.775 2	1.597 6	9.62
2	Semi-SH-RLv0	1.725 0	1.418 3	1.304 6	2.158 1	2.656 3	2.006 4	1.878 1	10.44
3	Semi-SH-RLv0-SA	2.344 4	1.290 2	1.192 5	2.210 9	2.595 5	2.313 3	1.991 1	12.72
4	Semi-SH-RLv0-Prun	1.467 8	1.264 0	1.231 7	2.072 5	1.936 2	1.679 3	1.608 6	11.79
5	Semi-SH-RL（RLv0-SA-Prun	）1.444 1	1.059 7	0.962 1	1.719 6	1.912 7	1.501 9	1.433 3	11.07

步长/h	残差绝对值（\|Δ\|）范围/（%）			精度评定/TECU
步长/h	\|Δ\|<1 TECU	\|Δ\|<3 TECU	\|Δ\|<5 TECU	E（\|Δ\|）	var（Δ）	RMSE
12	59.04	94.63	99.00	1.090 1	1.100 1	1.512 7
15	59.05	94.66	99.00	1.089 4	1.096 0	1.510 9
20	59.11	94.62	99.00	1.089 5	1.099 4	1.512 0
24	59.51	94.77	99.01	1.079 3	1.084 2	1.499 7
30	59.01	94.49	98.99	1.094 1	1.102 9	1.516 6
40	58.48	94.13	98.91	1.114 6	1.160 8	1.550 2
60	54.66	92.99	98.77	1.200 5	1.269 2	1.646 3
120	46.31	83.84	95.88	1.612 8	2.522 5	2.263 6

模型	第1天	第2天	第3天	第4天	第5天	均值
QP	2.309 4	2.183 1	2.020 5	1.883 7	2.077 6	2.095 7
LSTM	1.702 9	1.841 7	1.486 4	1.433 2	1.673 2	1.628 6
Semi-SH	1.554 6	1.767 6	1.470 4	1.427 1	1.641 4	1.572 5
Semi-SH-AR	1.571 3	1.763 8	1.455 2	1.411 1	1.635 5	1.568 2
Semi-SH-LSTM	1.530 9	1.601 9	1.344 4	1.338 4	1.606 6	1.485 7
Semi-SH-RL	1.445 4	1.576 7	1.407 8	1.402 6	1.649 5	1.497 0

模型	单天预报耗时	多天预报耗时
QP	296.32	1 533.10
LSTM	845.77	4 761.60
Semi-SH	9.62	52.11
Semi-SH-AR	9.66	53.35
Semi-SH-LSTM	994.25	5 534.66
Semi-SH-RL	11.07	56.17

联合半参数规则学习的电离层TEC预报

Ionospheric TEC prediction incorporating semi-parametric and rule-learning

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模型	[0，0.5）TECU	[0.5，1.0）TECU	[1.0，1.5）TECU	[1.5，2.0）TECU	[2.0，2.5）TECU	[2.5，3.0）TECU	>3.0 TECU
QP	25	189	350	321	212	119	80
C1PG	15	387	393	247	210	42	2
LSTM	19	360	530	281	84	17	5
Semi-SH	39	541	469	196	40	9	2
Semi-SH-AR	34	515	490	199	47	8	3
Semi-SH-LSTM	38	592	444	178	37	7	0
Semi-SH-RL	73	648	438	114	20	3	0