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电离层延迟是GNSS定位的主要误差来源，会对定位产生几米到几十米的定位误差[1]，严重制约了单频接收机的定位精度。如何合理、准确地模拟和预报电离层延迟，尤其在低纬度地区，已成为研究电离层的一个重要课题[2-3]。目前，单频接收机通常采用Klobuchar模型[4]、IRI模型[5]、NeQuick模型[6]等对电离层延迟进行改正。其中，Klobuchar模型因计算简单方便在单频接收机中得到广泛应用，但电离层延迟改正率仅能达到50%~60%且无法反映夜间电离层变化规律[7]。许多学者从不同方面对Klobuchar模型进行改进[8-10]。文献[8]将模型8参数扩展为14参数并对模型初始相位引入纬度改正，在中国地区取得较好效果。文献[9]通过采用非线性迭代引入振幅改正参数的方法建立适用于小区域的模型，使模型值更接近“真实值”。国内外大部分研究成果虽然从各方面对模型进行改进并取得较好的改正效果，但仍存在对电离层整体改正率不高，不能很好地反映夜间电离层变化特性等不足。
指数平滑法是预测时间序列的常用方法之一，能对趋势数据直接进行平滑处理，对原时间序列进行预测，已广泛应用于各领域[11-13]，并取得较好的效果。本文利用Holt指数平滑模型[13]的特点，对Klobuchar模型进行改进，并采用双频观测模型对改进效果进行评估。
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Holt指数平滑模型[13]由Holt于1957年提出，其基本思想是：假定所有已知数据对预测数据均有影响，但近期数据对预测值影响较大，远期数据对预测值影响较小，且影响力随时间的推移呈几何级数变化。无季节模型表达式为：
$$ \left\{ \begin{array}{l} {S_t} = \alpha {X_t} + \left( {1-\alpha } \right)\left( {{S_{t-1}}-{b_{t - 1}}} \right)\\ {b_t} = \beta \left( {{S_t} - {S_{t - 1}}} \right) + \left( {1 - \beta } \right){b_{t - 1}}\\ {F_{t + m}} = {S_t} + m{b_t} \end{array} \right. $$ (1) 式中，St为t时刻的稳定成分；Xt为t时刻的观测值；bt为t时刻的趋势成分；α、β∈[0, 1]为平滑参数；Ft+m为m期的预测值；m为预测期数。
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Klobuchar模型[14]自1987年发布以后，因计算简单方便而广泛应用于单频接收机中：
$$ {I_2}\left( t \right) = \left\{ \begin{array}{l} {A_1} + {A_2}\cos \left[{\frac{{2{\rm{ \mathit{ π} }}}}{{{A_4}}}\left( {t-{A_3}} \right)} \right], \left| {t -{A_3}} \right| < \frac{{{A_4}}}{4}\\ {A_1}, t{\rm{为其他值}} \end{array} \right. $$ (2) 式中各参数意义参见文献[14]。
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本文利用IGS中心提供的数据通过Klobuchar模型与双频观测模型[15]分别解算前6 d电离层总电子含量(total electron content, TEC)值。采用Holt无季节模型对各历元前6 d两种模型差值进行1 d预测，利用预测所得差值对Klobuchar模型第7 d的TEC值进行改进：
$$ \begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;I{'_2}\left( t \right) = \\ \left\{ \begin{array}{l} {A_1} + {A_2}\cos \left[{\frac{{2{\rm{ \mathit{ π} }}}}{{{A_4}}}\left( {t-{A_3}} \right)} \right] + {\mathit{\Delta }_r}, \left| {t -{A_3}} \right| < \frac{{{A_4}}}{4}\\ {A_1} + {\mathit{\Delta }_r}, t{\rm{为其他值}} \end{array} \right. \end{array} $$ (3) 式中，Δr为采用无季节模型预测所得第7 d差值; 其他各参数含义参照Klobuchar模型。
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Klobuchar模型主要应用于中、低纬度地区，在中纬度地区改正效果优于低纬度地区，分别选取IGS中心提供的2013年年积日63~69、2010年年积日147~153低纬度(Pimo站、Guam站)和中纬度(Crao站、Chan站)作为活跃期和平静期数据。根据本文§1.3建立改进模型，利用双频观测模型对电离层延迟改正效果能够达到95%的特点[15]，将双频观测模型解算所得的TEC值作为真值，把改进模型值与双频观测模型值进行比较。采用基本模型与改进模型相对于双频观测模型的改正率(以下简称改正率)P和均方根误差RMSE来评定模型效果：
$$ \left\{ \begin{array}{l} {P_{{i_j}}} = 1-\frac{{\left| {{K_{ij}}-{D_j}} \right|}}{{{D_j}}} \times 100\%, i = 1\;{\rm{或}}\;{\rm{2}}\\ {\rm{RMS}}{{\rm{E}}_{{i_t}}} = \sqrt {\frac{{\sum\limits_{j = n}^m {{{\left( {{K_{ij}}-{D_j}} \right)}^2}} }}{{m - n}}}, i = 1\;{\rm{或}}\;{\rm{2}} \end{array} \right. $$ (4) 式中，i=1表示基本模型；i=2表示改进模型；Kij表示第j个历元的Klobuchar模型值；Dj表示第j个历元的双频观测模型值；|Ki-Dj |表示第j个历元的两种模型差值的绝对值；n表示时段的起始历元；m表示时段的结束历元。
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图 1为活跃期不同纬度基本模型、改进模型同双频观测模型的对比图，图 2为平静期不同纬度基本模型、改进模型同双频观测模型的对比图。图 1~2中，UTC表示世界时；VTEC表示天顶电离层总电子含量。
图 1 活跃期低纬度与中纬度3种模型对比
Figure 1. Comparison of Three Model in Ionospheric Active Period over Low-Latitude and Mid-Latitude Areas
图 2 平静期低纬度与中纬度3种模型对比
Figure 2. Comparison of Three Model in Ionospheric Quiet Period over Low-Latitude and Mid-Latitude Areas
由图 1和图 2可以看出，在电离层活跃期和平静期，改进模型与双频观测模型在低、中纬度均符合较好，说明改进模型能较好地反映电离层变化特性，特别是夜间电离层变化特性。在大多数情况下，改进模型与双频观测模型的差值在3 TECu以内，与双频观测模型精度相当，且两种模型差值随纬度的增加而减小。特别在低纬度地区，两种模型差值要明显高于中纬度地区。平静期改进模型的改正效果略低于活跃期，但总体上仍有较大的改善。
表 1、表 2分别给出了1 d 6个时段(每个时段4 h)活跃期、平静期基本模型与改进模型精度统计情况。表中改正率在前，均方根误差(RMSE)在后, 以“改正率/RMSE”形式表现。
表 1 活跃期不同时段基本模型与改进模型改正率和RMSE统计表
Table 1. Statistics of Relative Accuracy and RMSE Between Basic Model and Improved Model in Ionospheric Active Period
站名 模型 [0, 4] [4, 8] [8, 12] [12, 16] [16, 20] [20, 24] Pimo 基本 81.74%/7.18 82.34%/8.59 84.19%/7.39 64.02%/7.56 59.84%/4.83 50.55%/4.70 改进 97.52%/0.93 96.32%/1.83 90.43%/3.74 88.25%/2.46 79.55%/2.28 93.62%/0.33 Guam 基本 73.98%/12.53 77.87%/11.63 78.47%/7.80 46.96%/11.86 57.71%/4.65 78.22%/5.33 改进 98.37%/0.87 97.45%/1.36 82.87%/5.67 92.01%/2.37 94.87%/0.41 89.37%/1.94 Crao 基本 -63.50%/5.69 66.31%/2.55 42.29%/9.56 43.71%/9.02 31.37%/5.14 -83.73%/5.95 改进 87.14%/0.55 89.62%/1.14 91.19%/1.45 96.23%/0.72 91.95%/0.61 82.79%/0.57 Chan 基本 62.83%/9.24 95.43%/1.62 84.32%/2.36 92.71%/0.76 92.26%/0.81 69.56%/3.48 改进 91.80%/2.05 93.43%/1.88 91.16%/1.37 92.51%/0.74 86.97%/1.23 88.66%/1.81 表 2 平静期不同时段基本模型与改进模型改正率和RMSE统计表
Table 2. Statistics of Relative Accuracy and RMSE Between Basic Model and Improved Model in Ionospheric Quiet Period
站名 模型 [0, 4] [4, 8] [8, 12] [12, 16] [16, 20] [20, 24] Pimo 基本 86.30%/2.36 72.07%/4.95 69.22%/3.97 -160.64%/4.26 21.33%/6.58 -19.97%/4.70 改进 76.90%/3.34 83.78%/2.98 83.83%/2.09 46.57%/2.92 61.54%/1.20 78.05%/1.90 Guam 基本 92.98%/1.52 94.06%/1.79 84.38%/0.95 43.99%/0.83 -37.09% /0.47 73.73%/1.29 改进 98.59%/0.55 92.02%/1.14 92.30%/1.45 89.76%/0.72 88.57%/0.62 89.28%/0.57 Crao 基本 -221.59%/6.94 -35.77%/5.64 -87.48%/9.83 0.51%/7.95 77.96%/6.61 -287.77%/7.31 改进 93.39%/0.16 92.24% /0.44 93.82%/0.38 88.59%/0.93 86.07%/0.72 84.96%/0.34 Chan 基本 77.70%/2.38 80.09%/2.51 46.95%/4.44 92.24%/0.79 23.46%/3.94 73.61%/2.94 改进 90.08%/1.03 95.45%/0.63 69.25%/2.52 90.08%/1.02 84.69%/0.96 86.19%/1.81 由表 1~2可以看出，活跃期与平静期的改进模型改正率比基本模型均有显著提升。在活跃期，低纬度地区改进模型日间的改正率比基本模型提高约15%，RMSE提高约6 TECu，夜间的改正率提高约30%，RMSE提高约5 TECu；中纬度地区改进模型日间的改正率比基本模型提高约20%，RMSE提高约3 TECu，夜间的改正率提高约70%，RMSE提高约4 TECu。在平静期，低纬度地区改进模型日间的改正率比基本模型提高约2%，RMSE提高约1 TECu，夜间的改正率提高约65%，RMSE提高约3.5 TECu；中纬度地区改进模型日间的改正率比基本模型提高约70%，RMSE提高约4 TECu，夜间的改正率提高约80%，RMSE提高约4 TECu。
结合表 1和表 2可知，无论在活跃期还是平静期，大多数时段低纬度地区改进模型的改正率优于中纬度地区，但RMSE比中纬度地区高，出现这种情况是由于低纬度地区受太阳直射影响，测站VTEC值较大，导致即使两种模型差值较大，在低纬度地区模型改正率较高，RMSE较大。基本模型改正率在夜间普遍出现负值，是由于基本模型将夜间电离层延迟量设为常量，无法较好地反映电离层夜间变化特性。此外，低纬度地区Pimo站平静期0~4 h统计精度显著低于活跃期均值，可能是由于受2010年06月16日03:16:27 UTC巴布亚岛地震影响，在震前第14 d该时段电离层出现异常[16]。
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本文利用IGS中心提供的中、低纬度地区活跃期、平静期数据，采用Holt指数平滑模型对每个历元前6 d Klobuchar模型和双频观测模型的TEC差值进行1 d预测，利用预测所得差值对第7 d Klobuchar模型TEC值进行改进，经实验分析发现，在电离层平静期和活跃期，改进模型在中、低纬度地区与双频观测模型均符合得较好，能很好地反映电离层变化特性，特别是夜间电离层变化特性。改进模型的改正率比基本模型有显著提升，中纬度地区改进模型的改正效果优于低纬度地区。日间改进模型的改正效果优于基本模型，夜间改进模型的改正效果比基本模型有显著提升。
由于本文采用的数据有限，无法确定对其他区域的适用性，仍需采用更多的数据验证模型的适用性。
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摘要: 利用IGS（International GNSS Service）中心提供的中、低纬度地区平静期、活跃期观测数据，通过Klobuchar模型与双频观测模型解算电离层总电子含量（total electron content，TEC）值。采用Holt指数平滑模型对每个历元前6 d两种模型差值进行1 d预测，利用预测所得差值对Klobuchar模型第7 d的TEC值进行改进。实验结果表明，无论在电离层活跃期还是平静期，改进模型改正效果比基本模型有显著提升，改进模型能更好地反映电离层变化特性，尤其是夜间电离层变化特性。Abstract: We use Klobuchar model and Dual-frequency observation model to calculate TEC values with the data provided by IGS center over low-latitude and mid-latitude areas in ionospheric quiet period and active period. Then improve 7th day's TEC values calculate by Klobuchar model with the difference values come from the forecast difference values in each epoch of the preceeding 6 days between Klobuchar model and Dual-frequency observation model by the Holt exponential smoothing model. The experiment results show that the correction effect of improved model has a significant improvement than the basic model no matter whether it is in in ionospheric quiet period or active period. In addition, the improved model can better reflect the ionospheric changing characteristics, especially the night ionospheric changing characteristics.
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Key words:
- Klobuchar /
- Holt exponential smoothing model /
- VTEC /
- ionosphere /
- time series
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图 1 活跃期低纬度与中纬度3种模型对比
Figure 1. Comparison of Three Model in Ionospheric Active Period over Low-Latitude and Mid-Latitude Areas
图 2 平静期低纬度与中纬度3种模型对比
Figure 2. Comparison of Three Model in Ionospheric Quiet Period over Low-Latitude and Mid-Latitude Areas
表 1 活跃期不同时段基本模型与改进模型改正率和RMSE统计表
Table 1. Statistics of Relative Accuracy and RMSE Between Basic Model and Improved Model in Ionospheric Active Period
站名 模型 [0, 4] [4, 8] [8, 12] [12, 16] [16, 20] [20, 24] Pimo 基本 81.74%/7.18 82.34%/8.59 84.19%/7.39 64.02%/7.56 59.84%/4.83 50.55%/4.70 改进 97.52%/0.93 96.32%/1.83 90.43%/3.74 88.25%/2.46 79.55%/2.28 93.62%/0.33 Guam 基本 73.98%/12.53 77.87%/11.63 78.47%/7.80 46.96%/11.86 57.71%/4.65 78.22%/5.33 改进 98.37%/0.87 97.45%/1.36 82.87%/5.67 92.01%/2.37 94.87%/0.41 89.37%/1.94 Crao 基本 -63.50%/5.69 66.31%/2.55 42.29%/9.56 43.71%/9.02 31.37%/5.14 -83.73%/5.95 改进 87.14%/0.55 89.62%/1.14 91.19%/1.45 96.23%/0.72 91.95%/0.61 82.79%/0.57 Chan 基本 62.83%/9.24 95.43%/1.62 84.32%/2.36 92.71%/0.76 92.26%/0.81 69.56%/3.48 改进 91.80%/2.05 93.43%/1.88 91.16%/1.37 92.51%/0.74 86.97%/1.23 88.66%/1.81 
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表 2 平静期不同时段基本模型与改进模型改正率和RMSE统计表
Table 2. Statistics of Relative Accuracy and RMSE Between Basic Model and Improved Model in Ionospheric Quiet Period
站名 模型 [0, 4] [4, 8] [8, 12] [12, 16] [16, 20] [20, 24] Pimo 基本 86.30%/2.36 72.07%/4.95 69.22%/3.97 -160.64%/4.26 21.33%/6.58 -19.97%/4.70 改进 76.90%/3.34 83.78%/2.98 83.83%/2.09 46.57%/2.92 61.54%/1.20 78.05%/1.90 Guam 基本 92.98%/1.52 94.06%/1.79 84.38%/0.95 43.99%/0.83 -37.09% /0.47 73.73%/1.29 改进 98.59%/0.55 92.02%/1.14 92.30%/1.45 89.76%/0.72 88.57%/0.62 89.28%/0.57 Crao 基本 -221.59%/6.94 -35.77%/5.64 -87.48%/9.83 0.51%/7.95 77.96%/6.61 -287.77%/7.31 改进 93.39%/0.16 92.24% /0.44 93.82%/0.38 88.59%/0.93 86.07%/0.72 84.96%/0.34 Chan 基本 77.70%/2.38 80.09%/2.51 46.95%/4.44 92.24%/0.79 23.46%/3.94 73.61%/2.94 改进 90.08%/1.03 95.45%/0.63 69.25%/2.52 90.08%/1.02 84.69%/0.96 86.19%/1.81
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