Extraction of common mode error based on SSA method and its impact analysis on GNSS vertical coordinate time series
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摘要: 本研究基于德国北部24个GNSS测站8 a的数据，引入奇异谱分析方法(singular spectrum analysis，SSA)，提出一种顾及不同残差子分量互相关性及子分量贡献率的共模误差(common mode error，CME)识别方法. 探讨了CME对GNSS坐标时间序列噪声和参数估计的影响. 通过与主成分分析(principal component analysis，PCA)方法的对比发现，提出的新方法与PCA方法提取的CME结果非常接近，证实了新方法的可行性. GNSS的CME序列主要包含白噪声(white noise，WN)、闪烁噪声(flicker noise，FN)和非整数谱指数幂律噪声(power law noise，PL). 在剔除CME后，各测站的WN和有色噪声量级分别平均下降了30.32%和52.61%，说明CME中有色噪声占主导地位. 同时，CME改正后，坐标的周年周期和半年周期振幅均有所减小，参数拟合的均方根误差(root mean squared error，RMSE)降低了16.7%. 综上所述，新方法在提高GNSS坐标时间序列质量方面具有重要实际意义.
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关键词:
- 共模误差(CME) /
- GNSS /
- 奇异谱分析 /
- 噪声 /
- 主成分分析(PCA)
Abstract: This study, based on eight years of data from 24 Global Navigation Satellite Systems (GNSS) stations in northern Germany, introduces the singular spectrum analysis method. It proposes a common mode error identification method that considers the inter-correlation of different residual subcomponents and their contribution rates. The impact of common mode errors on GNSS coordinate time series noise and parameter estimation is explored. Compared with the principal component analysis (PCA) method, it is found that the method proposed in this paper closely aligns with the common mode errors extracted by PCA, confirming the feasibility of the new method. The GNSS common mode error sequence mainly contains white noise, flicker noise, and power-law noise with non-integer spectral indices. After removing common mode errors, the magnitude of white noise and colored noise at each station decreased by an average of 30.32% and 52.61% respectively, indicating that colored noise dominates in common mode errors. Furthermore, after correcting common mode errors, the annual and semi-annual cycle amplitudes of coordinates are reduced, and the root mean square error of parameter fitting is decreased by 16.7%. In summary, the method described in this paper is of significant practical importance in improving the quality of GNSS coordinate time series. -
表 1 本文方法提取共模误差结果
测站 阶次 贡献率/% 测站 阶次 贡献率/% ESH5 8 13.94 TGWV 7 11.77 ESBC 10 14.23 FLDW 8 12.90 ESBH 8 12.91 TGEM 8 11.82 RANT 8 12.99 KNOC 8 13.44 TGDA 10 14.47 DZYL 9 13.22 TGME 11 12.54 DELZ 10 14.45 TGCU 10 14.37 TGD2 8 13.37 TGZU 7 12.67 TGDU 9 13.49 HELG 9 11.19 BORJ 10 14.15 HEL2 8 12.96 TGBF 8 13.96 TGBU 8 12.35 FYHA 7 12.15 TGBH 10 13.84 HOLT 7 11.84 
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表 2 PCA方法提取CME贡献率统计
测站 贡献率/% 测站 贡献率/% ESH5 14.36 TGWV 10.81 ESBC 14.65 FLDW 13.08 ESBH 11.36 TGEM 12.41 RANT 11.69 KNOC 15.24 TGDA 15.61 DZYL 13.69 TGME 13.74 DELZ 14.58 TGCU 12.54 TGD2 15.98 TGZU 11.87 TGDU 14.01 HELG 10.71 BORJ 14.39 HEL2 12.04 TGBF 13.14 TGBU 15.69 FYHA 13.49 TGBH 14.98 HOLT 12.07
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表 3 CME改正前后对坐标时间序列噪声特性的影响
测站 改正前 改正后 改正前后各分量改善的百分比/% 最佳噪声模型 各噪声分量值/mm 最佳噪声模型 各噪声分量值/mm ESH5 WN+PL 4.82+9.65 WN+PL 3.69+3.65 23.44+62.18 ESBC WN+PL 4.75+8.95 WN+PL 4.02+3.25 15.37+63.69 ESBH WN+PL 4.98+10.36 WN+PL 2.36+4.32 52.61+58.30 RANT WN+FN 5.32+12.03 WN+PL 3.12+5.1 41.35+57.61 TGDA WN+PL 4.69+11.25 WN+PL 2.02+6.25 56.93+44.44 TGME WN+PL 8.32+11.054 WN+PL 7.32+5.14 12.02+53.50 TGCU WN+FN 5.23+14.23 WN+FN 4.15+6.98 20.65+50.95 TGZU WN+PL 6.21+14.02 WN+PL 3.34+5.21 46.22+62.84 HELG WN+PL 5.65+11.36 WN+PL 2.36+5.43 58.23+52.20 HEL2 WN+PL 4.98+9.23 WN+PL 1.36 +3.69 72.69+60.2 TGBU WN+PL 6.24+10.67 WN+PL 2.36+5.36 62.18+49.77 TGBH WN+PL 5.54+11.05 WN+PL 4.32+5.87 22.02+46.88 ESH5 WN+PL 5.95+12.36 WN+PL 3.98+5.13 33.11+58.50 ESBC WN+FN 6.24+14.25 WN+FN 5.02+6.11 19.55+57.12 ESBH WN+PL 6.21+9.87 WN+PL 4.28+3.25 31.08+67.07 RANT WN+PL 5.65+8.26 WN+PL 4.98+3.64 11.86+55.93 TGDA WN+PL 4.87+13.91 WN+PL 3.65+6.53 25.05+53.06 TGME WN+PL 5.21+15.36 WN+PL 3.53+8.32 32.25+45.83 TGCU WN+PL 5.69+11.03 WN+PL 3.25+6.42 42.88+41.80 TGZU WN+FN 5.14+15.03 WN+PL 3.32+5.12 35.41+65.93 HELG WN+PL 6.32+12.36 WN+PL 3.36+7.25 46.84+41.34 HEL2 WN+FN 5.24+13.02 WN+FN 4.36+5.82 16.79+55.30 TGBU WN+PL 4.25+14.03 WN+PL 3.65+7.54 14.12+46.26 TGBH WN+PL 6.24+12.36 WN+PL 4.36+5.53 30.13+55.26 TGWV WN+PL 5.21+13.65 WN+PL 3.21+0.59 38.39+95.68 FLDW WN+PL 5.12+11.02 WN+PL 2.65+1.36 48.24+87.66 TGEM WN+PL 4.52+14.36 WN+PL 3.36+1.98 25.66+86.21 KNOC WN+FN 6.24+13.65 WN+PL 4.63+2.36 25.80+82.71 DZYL WN+PL 5.74+11.69 WN+PL 3.02+5.66 47.39+51.58 DELZ WN+PL 4.05+14.03 WN+PL 2.65+6.01 34.57+57.16 TGD2 WN+FN 5.06+13.25 WN+FN 2.39+5.98 52.77+54.87 TGDU WN+PL 6.3+14.14 WN+PL 4.36+7.01 30.79+50.42 BORJ WN+PL 2.87+15.02 WN+PL 2.01+7.65 29.97+49.07 TGBF WN+PL 5.24+12.36 WN+PL 3.98+8.06 24.05+34.79 FYHA WN+PL 4.93+5.69 WN+PL 3.68+3.25 25.35+42.88 HOLT WN+PL 5.07+9.54 WN+PL 3.98+5.36 21.50+43.82 TGWV WN+PL 7.09+6.68 WN+PL 5.36+2.36 24.40+64.67 FLDW WN+PL 5.91+9.25 WN+PL 5.36+5.87 9.31+36.54 TGEM WN+PL 6.45+6.45 WN+PL 4.69+6.35 27.29+1.55 KNOC WN+FN 6.81+10.36 WN+PL 3.98+6.98 41.56+32.63 DZYL WN+PL 5.67+13.21 WN+PL 5.02+4.5 11.46+65.93 DELZ WN+PL 5.48+10.98 WN+PL 5.36+6.97 2.19+36.52 TGD2 WN+FN 5.21+11.05 WN+FN 4.69+5.23 9.98+52.67 TGDU WN+PL 6.71+9.21 WN+PL 5.21+8.35 22.35+9.34 BORJ WN+PL 6.52+8.65 WN+PL 5.36+5.41 17.79+37.46 TGBF WN+PL 5.32+7.98 WN+n 4.69+5.69 11.84+28.70 FYHA WN+PL 4.96+10.95 WN+PL 3.69+4.36 25.60+60.18 HOLT WN+PL 4.87+12.32 WN+PL 3.69+5.36 24.23+56.49
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表 4 CME对坐标时间序列参数拟合的影响
改正状态 周年振幅/mm 周年相位/(°) 半年振幅/mm 半年相位/(°) 站速度/(mm·a−1) RMSE 改正前 5.89 26.5 1.36 152.3 0.58 1.02 改正后 5.61 25.2 1.25 151.6 0.58 0.85
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