﻿ CORS站高程非线性速度场及方差波动模型构建方法
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CORS站高程非线性速度场及方差波动模型构建方法

1. 辽宁工程技术大学测绘与地理科学学院, 辽宁 阜新 123000;
2. 中国测绘科学研究院, 北京 100830

Height nonlinear velocity field and variance fluctuation model construction method for CORS stations
ZHANG Hengjing1,2, CUI Dongdong1,2, CHENG Pengfei2
1. School of Geomatics, Liaoning Technical University, Fuxin 123000, China;
2. Chinese Academy of Surveying and Mapping, Beijing 100830, China
Abstract: The basic concept of establishing linear velocity field with fixed period term in CORS stations height time series is described, and the problem of deviation between given period and actual period is pointed out. Therefore, this paper proposes a nonlinear velocity field modeling method for CORS stations height time series. This method takes the linear least squares solution as the initial value of iteration, and uses the gauss-newtoni-teration algorithm to solve the unknown parameters of the nonlinear velocity field model, realizing the nonlinear fitting of height time series data of CORS stations. The test method for the heteroscedasticity of the residual square sequence of the fitting model is given, and the basic criteria for the establishment of GARCH (p, q) model to reflect the fluctuations of non-stationary sequence are expounded. The six CORS stations at home and abroad more than 20 years as the research object, the height of time series nonlinear velocity motion model is set up, the results show that the CORS stations height movement does not exist strict year or half year cycle, the approximate periodic motion is most obvious, the approximate period of two years than the minimum, cycle in deviation of 12%, half year cycle is 18%, two year period is 6%, nonlinear modeling accuracy and effect is better than the linear model as a whole. The ARCH test method is used to obtain the heteroscedasticity of the residual square sequence of the nonlinear model of CORS stations height, that is, the residual square sequence has non-stationary characteristics. GARCH (p, q) model is introduced to model the non-stationary residual square sequence of the height component of CORS stations, which reflects the non-stationary fluctuation of the residual square sequence. The feasibility of GARCH (p, q) model in modeling non-stationary residual square series of CORS stations height is verified, which provides an idea for future modeling of non-stationary noise series of CORS stations height and reconstruction of nonlinear velocity field with GARCH (p, q) model.
Key words: height time series of CORS stations    nonlinear velocity field    heteroskedasticity    GARCH model

CORS站坐标运动特征预测分析是维持2000中国大地坐标系统(China geodetic coordinate system 2000，CGCS2000)框架准确性和现势性的重要基础[1]。通过对国内多个CORS站高程坐标分量序列特征分析发现，CORS站在垂直方向表现出明显的非线性和周期性特征，并且不同的CORS站对应周期性变化模式和运动规律也不尽相同[2]。影响全球定位系统(global positioning system，GPS)高程分量周期性变化的因素很多，主要包括地壳运动、冰雪、土壤水分、大气压负载以及CORS站测量技术的高低等[3-4]。在分析CORS站高程坐标分量运动特征时，只考虑线性运动，直接在线性最小二乘拟合模型中对周期项赋予固定数值[5]，而忽略其非线性运动特征，导致建立的CORS站高程线性速度场模型对于垂向速率估计和误差估计存在很大的偏差[6-8]。同时CORS站周围存在物理、地理因素及测量仪器等产生的噪声，也会影响CORS站高程运动的规律。在对我国CORS站高程时间序列数据噪声模型研究时，文献[9]利用功率谱分析CORS高程序列的噪声性质，并采用极大似然法定量地估计噪声序列中有色噪声的分量。文献[10]利用不同组合的噪声模型对CORS站坐标时间序列的噪声进行分析，并计算了积雪深度、大气压负载、土壤湿度负载等对CORS站位移的影响，得到中国区域CORS站坐标序列的噪声特性主要表现为白噪声+闪烁噪声和白噪声+带通幂律噪声。文献[11]利用AR模型对CORS站高程噪声序列建模，并与CORS站高程非线性拟合模型结合进行高程运动的预报。

1 CORS站高程时间序列非线性建模方法

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LS求解非线性模型时，给定未知参数初值的精度不同，一次平差结果可能不满足参数的精度要求，采用高斯-牛顿迭代算法，计算流程如图 1所示，其中

 图 1 CORS站高程非线性建模流程 Fig. 1 The nonlinear modeling process of CORS stations height

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2 CORS站高程非线性建模试验

 参数 BJFS KUNM SHAO BOGO YAR2 AUCK a/m 0.001 90 0.005 90 0.005 70 0.000 65 -0.000 99 -0.000 22 b/ma－1 0.002 20 -0.001 00 -0.001 10 -0.000 28 0.000 43 -0.000 28 A1/m 0.005 20 0.005 80 0.005 70 0.004 30 0.005 40 0.005 90 A2/m 0.001 20 0.001 60 0.001 80 0.000 90 0.000 80 0.000 94 A3/m 0.000 25 0.001 30 0.001 20 0.000 64 — — f1/a－1 1.000 00 1.000 00 1.000 00 1.000 00 1.000 00 1.000 00 f2/a－1 2.000 00 2.000 00 2.000 00 2.000 00 2.000 00 2.000 00 f3/a－1 0.500 00 0.500 00 0.500 00 0.500 00 — — φ1/rad 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 φ2/rad 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 0.000 00 φ3/rad 0.000 00 0.000 00 0.000 00 0.000 00 — —

 参数 BJFS KUNM SHAO BOGO YAR2 AUCK a/m 0.001 80 0.006 00 0.006 80 0.000 66 -0.001 00 -0.000 23 b/ma－1 0.002 20 -0.001 10 -0.001 00 -0.000 28 0.000 42 -0.000 28 A1/m 0.006 90 0.007 50 0.007 60 0.005 00 0.006 30 0.006 80 f1/a－1 0.988 70 1.01 250 1.012 80 0.998 00 1.01000 1.120 40 φ1/rad -1.674 40 0.073 50 0.082 90 -1.470 40 -0.408 70 -0.146 00 A2/m 0.001 00 0.001 30 0.001 50 0.000 70 0.000 75 0.001 00 f2/a－1 1.990 50 1.870 70 1.816 80 1.981 80 2.020 00 2.042 80 φ2/rad 3.353 70 1.501 50 1.438 40 2.940 80 2.448 90 1.068 50 A3/m 0.000 23 0.001 10 0.001 30 0.000 60 — — f3/a－1 0.470 90 0.486 00 0.48700 0.437 60 — — φ3/rad 2.279 40 0.979 90 0.550 20 2.105 24 — — (nl)rmse/m 0.006 00 0.007 90 0.007 88 0.004 90 0.006 03 0.005 31 (l)rmse/m 0.008 10 0.009 22 0.009 30 0.006 41 0.008 14 0.007 71 rnew 0.976 64 0.952 31 0.952 46 0.965 23 0.953 64 0.964 85 R2 0.852 30 0.823 45 0.819 95 0.886 52 0.852 32 0.886 58

 图 2 6个CORS站高程数据建模结果 Fig. 2 The height data modeling results of 6 CORS stations

3 ARCH效应检验与GARCH(p, q)波动模型

3.1 ARCH效应检验方法

 图 3 CORS站高程残差平方序列非平稳性检验流程 Fig. 3 The nonstationarity test procedure of CORS stations height residual square sequence

 图 4 自相关性Q检验流程 Fig. 4 The autocorrelation of Q test procedure

 图 5 异方差性ARCH检验流程 Fig. 5 The ARCH test procedure of heteroscedasticity

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3.1.1 残差平方序列自相关性检验

 图 6 两个CORS站高程的残差平方序列 Fig. 6 The residual squared sequence of two CORS stations height

 图 7 两个CORS站高程残差平方序列的自相关图 Fig. 7 The autocorrelation graph of two CORS stations height residual square sequences

3.1.2 残差平方序列异方差性检验

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3.2 残差平方序列GARCH(p, q)波动模型

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 图 8 两个CORS站高程残差平方序列的偏相关图 Fig. 8 The partial correlogram graph of two CORS stations height residual square sequences

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 parameter value standard error t statistic constant 0.000 000 20 0.000 006 15 0.132 911 00 GARCH(1) 0.688 887 00 0.375 217 00 1.835 970 00 ARCH(1) 0.167 274 00 0.906 637 00 0.184 500 00 offset 0.000 048 96 0.000 138 28 0.354 015 00

 parameter value standard error t statistic constant 0.000 000 20 0.000 001 70 0.117 767 00 GARCH(1) 0.521 826 00 0.648 599 00 0.804 54200 ARCH(1) 0.214 017 00 0.012 011 80 17.817 200 0 offset 0.000 026 29 0.000 118 60 0.221 713 00

 图 9 残差平方序列与条件方差序列趋势变化比较 Fig. 9 Comparison of the trend between the residual squared sequence and the conditional variance sequence

4 结论

(1) 建立了CORS站高程分量非线性速度场周期拟合模型。CORS站高程分量周期性运动以近似年周期项为主，近似半年周期项次之，近似两年变化的比重几乎可以忽略不计。周期项并不是严格的年、半年或两年，固定周期项的线性建模估计结果存在偏差，采用非线性模型对CORS站高程数据建模精度明显优于传统固定周期项的线性模型。

(2) 建立了描述CORS站高程残差平方序列非平稳波动的GARCH(1, 1)模型。非线性建模后的CORS站高程残差平方序列自相关性和异方差性检验结果表明，残差平方序列是非平稳的随机过程，即非线性建模后的CORS站高程残差序列具有非平稳性。GARCH(pq)模型克服了AR模型只能对平稳序列建模的不足，为下一步基于GARCH(pq)模型对CORS站高程非平稳残差序列建模和非线性速度场重构提供了思路。

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http://dx.doi.org/10.11947/j.AGCS.2019.20190017

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#### 文章信息

ZHANG Hengjing, CUI Dongdong, CHENG Pengfei
CORS站高程非线性速度场及方差波动模型构建方法
Height nonlinear velocity field and variance fluctuation model construction method for CORS stations

Acta Geodaetica et Cartographica Sinica, 2019, 48(9): 1096-1106
http://dx.doi.org/10.11947/j.AGCS.2019.20190017