基于级别建模的灰色GM(1,1)模型在卫星钟差中长期预报中的应用
Application of the Grey GM(1,1) Based on Stepwise-Ratio Modelling to Medium and Long-Term Prediction of Satellite Clock Bias

针对常规灰色GM(1,1)模型无法反映卫星钟差级比动态变化的问题，提出以钟差相邻历元的级比为建模序列，建立能够体现钟差级比序列变化趋势的级比离散灰色GM(1,1) (stepwise-ratio discrete GM(1,1)，SDGM(1,1))模型。首先，在卫星钟差相邻历元之间作级比生成级比序列；其次，以钟差级比序列为建模对象，建立SDGM(1,1))模型对钟差级比进行外推；最后，根据钟差序列与级比序列之间的关系，将级比外推值还原得到卫星钟差预报值。采用国际GNSS服务组织(International GNSS Service, IGS)提供的最终精密GPS卫星钟差数据进行1 d短期和30 d中长期预报试验，并和常规GM(1,1)模型、二次多项式模型的预报结果进行对比。结果表明，SDGM(1,1))模型可以较好地预报出卫星钟差级比的变化趋势，1 d钟差平均预报精度相比于GM(1,1)和二次多项式模型分别提升62.38%和37.96%，平均预报稳定度分别提升26.6%和10.87%；对于30 d钟差预报，与二次多项式模型相比，SDGM(1,1))模型的平均预报精度和预报稳定度分别提升75.18%和76.74%，比GM(1,1)模型的平均预报精度和预报稳定度高2个数量级。
Aiming at the problem that the conventional grey GM(1,1) model cannot reflect the dynamic change of satellite clock bias stepwise-ratio, this paper proposes to take the stepwise-ratio of adjacent epochs of clock bias as the modeling sequence, and establishes the stepwise-ratio discrete grey GM(1,1) model, denoted as SDGM(1,1), which can reflect the change trend of clock bias stepwise-ratio sequence. The stepwise-ratio sequence is first generated by making stepwise-ratio calculation between adjacent epochs of satellite clock bias. Secondly, taking the clock bias stepwise-ratio sequence as the modeling object, a SDGM(1,1) model is established to extrapolate the clock bias stepwise-ratio. Finally, according to the relationship between the clock bias and stepwise-ratio sequence, the extrapolated stepwise-ratio values are restored to obtain the satellite clock bias predictions. The final products of precise GPS satellite clock bias provide by the International GNSS Service (IGS) are taken as data basis to carry out satellite clock bias prediction up into one day and 30 days in future. The results show that the SDGM(1,1) model can better predict the change trend of satellite clock bias stepwise-ratio. Compared with the GM(1,1) and quadratic polynomial model, the average prediction accuracy of one-day clock bias is improved by 62.38% and 37.96% respectively, and the average prediction stability is improved by 26.6% and 10.87% respectively. For the 30-day clock bias prediction, in contrast to the quadratic polynomial model, the average prediction accuracy and prediction stability of the SDGM(1,1) model are improved by 75.18% and 76.74% respectively, and the average accuracy and stability are two orders of magnitude higher than those of the conventional GM(1,1) model.