利用加權最小二乘法作為估計器在GPS的時間序列分析中是最常見的做法，雖其原理簡單易實現、耗時極低，但在實際的應用中會發現擬合結果往往不如預期，這是因為GPS時間序列中含有許多我們無從得知的暫態訊號及具有時變性的訊號存在，為了克服上述問題，基於隨機過程的最小二乘估計，也就是卡爾曼濾波器便被應用在此類問題上。
而無論在卡爾曼濾波器抑或是最小二乘法當中，具有時間關聯的色雜訊都是不可忽視的問題，在Didova et al. (2016)中應用Bode and Shannon (1950)的shaping filter的方式，利用自回歸模型建構色雜訊模型，上述研究經測試認為AR(3)能夠有效模擬GPS時間序列中之色雜訊，因此，此研究基於此方法，並以地調所全台96座測站座為範例，濾出時間序列之色雜訊及白雜訊以進行雜訊分析，並與Bos et al. (2013)傳統Hector之方法比較其在雜訊分析上之異同。
在卡爾曼濾波器中，由於積分隨機遊走(integrated random walk)能夠有效的模擬長周期的趨勢變化，因此在狀態空間模型(state space model)中常利用此一模型來表示GPS時間序列中的趨勢訊號，但此種趨勢模型是將斜率視為一隨機遊走(random walk)，因此若是斜率在短時間內有強時間關聯性的變化，此一趨勢模型便不能有效的對此種訊號進行估計，而震後變形便是符合此一特性中最常見的訊號之一。由於震後變形的初期，斜率的變化具有強關聯性的，因此，常會發現利用積分隨機遊走作為趨勢模型會造成震後變形的過度平滑，此一問題會導致震後變形初期的低估，並間接的影響到同震位移的估計。因此，不同於積分隨機遊走模型將加速度視為白噪音，此研究中提出在震後利用一階自回歸模型AR(1)將加速度引入狀態空間模型中，以解決震後變形初期過度平滑的問題，提升震後及同震變形估計上的準確性。此外，本研究將卡爾曼濾波器估計之瞬時速度場用於應變率之計算，以台灣西南部地區75座GPS測站座為範例進行分析，以提供於2016年之美濃地震前後3年之地表變形的另一種觀測數據。

Weighted least-squares is commonly used in GPS time series analysis due to its simplicity and efficiency. However, practical applications often face fitting issues because of unknown transient and time-varying signals within GPS data. To solve this, this research adopts the Kalman filter, a least squares estimation method for stochastic process.
Temporal correlation of colored noise poses a challenge in both Kalman filtering and least squares methods. A study by Didova et al. (2016) addresses this by applying the shaping filter proposed by Bode and Shannon (1950), using autoregressive models to create a colored noise model. They found that AR(3) models effectively simulate colored noise in GPS time series. As a result, this research uses this approach, taking 96 stations across Taiwan from the Geological Survey Institute as examples. It filters out colored and white noise from the time series for noise analysis, comparing it with the traditional Hector method by Bos et al. (2013) to examine the differences in noise analysis.
In the Kalman filter, using the integrated random walk to model trends variation tends to oversmooth signals like post-seismic deformation due to its inability to handle time-correlated velocity changes effectively. To address this, instead of treating acceleration as white noise, this research introduces an AR(1) model for acceleration in the state space model during post-seismic period, improving accuracy in estimating both post-seismic and co-seismic deformations. Furthermore, the study utilizes the Kalman filter-estimated velocity/displacement field for strain rate/strain calculations, exemplifying its application using 75 GPS stations in southwestern Taiwan before and after the 2016 Meinong earthquake, offering additional data for surface deformation analysis three years after and before Meinong earthquake.

Agnew, D.C., 1992. The time-domain behavior of power-law noises. Geophys. Res. Lett. 19, 333–336. https://doi.org/10.1029/91GL02832
Béon, M.L., Huang, M.-H., Suppe, J., Huang, S.-T., Pathier, E., Huang, W.-J., Chen, C.-L., Fruneau, B., Baize, S., Ching, K.-E., Hu, J.-C., 2017. Shallow geological structures triggered during the Mw 6.4 Meinong earthquake, southwestern Taiwan. Terr. Atmos. Ocean. Sci. 28, 663–681. https://doi.org/10.3319/TAO.2017.03.20.02
Bode, H.W., Shannon, C.E., 1950. A Simplified Derivation of Linear Least Square Smoothing and Prediction Theory. Proc. IRE 38, 417–425. https://doi.org/10.1109/JRPROC.1950.231821
Bos, M.S., Fernandes, R.M.S., Williams, S.D.P., Bastos, L., 2013. Fast error analysis of continuous GNSS observations with missing data. J Geod 87, 351–360. https://doi.org/10.1007/s00190-012-0605-0
Box, G.E.P., Jenkins, G.M., Reinsel, G.C., Ljung, G.M., 2016. Time series analysis: forecasting and control, Fifth edition. ed, Wiley series in probability and statistics. John Wiley & Sons, Inc, Hoboken, New Jersey.
Ching, K.-E., Rau, R.-J., Lee, J.-C., Hu, J.-C., 2007. Contemporary deformation of tectonic escape in SW Taiwan from GPS observations, 1995–2005. Earth and Planetary Science Letters 262, 601–619. https://doi.org/10.1016/j.epsl.2007.08.017
Davis, J.A., Greenhall, C.A., Stacey, P.W., 2005. A Kalman filter clock algorithm for use in the presence of flicker frequency modulation noise. Metrologia 42, 1–10. https://doi.org/10.1088/0026-1394/42/1/001
Davis, J.L., Wernicke, B.P., Tamisiea, M.E., 2012. On seasonal signals in geodetic time series. J. Geophys. Res. 117, 2011JB008690. https://doi.org/10.1029/2011JB008690
Dempster, A.P., Laird, N.M., Rubin, D.B., 1977. Maximum Likelihood from Incomplete Data Via the EM Algorithm. Journal of the Royal Statistical Society: Series B (Methodological) 39, 1–22. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x
Didova, O., Gunter, B., Riva, R., Klees, R., Roese-Koerner, L., 2016. An approach for estimating time-variable rates from geodetic time series. J Geod 90, 1207–1221. https://doi.org/10.1007/s00190-016-0918-5
Dmitrieva, K., Segall, P., DeMets, C., 2015. Network-based estimation of time-dependent noise in GPS position time series. J Geod 89, 591–606. https://doi.org/10.1007/s00190-015-0801-9
Dong, D., Fang, P., Bock, Y., Cheng, M.K., Miyazaki, S., 2002. Anatomy of apparent seasonal variations from GPS-derived site position time series: SEASONAL VARIATIONS FROM GPS SITE TIME SERIES. J. Geophys. Res. 107, ETG 9-1-ETG 9-16. https://doi.org/10.1029/2001JB000573
Fukuda, J., Higuchi, T., Miyazaki, S., Kato, T., 2004. A new approach to time-dependent inversion of geodetic data using a Monte Carlo mixture Kalman filter. Geophysical Journal International 159, 17–39. https://doi.org/10.1111/j.1365-246X.2004.02383.x
Hines, T.T., Hetland, E.A., 2016. Rheologic constraints on the upper mantle from 5 years of postseismic deformation following the El Mayor‐Cucapah earthquake. JGR Solid Earth 121, 6809–6827. https://doi.org/10.1002/2016JB013114
Holmes, E.E., 2013. Derivation of an EM algorithm for constrained and unconstrained multivariate autoregressive state-space (MARSS) models.
Hosking, J.R.M., 1981. Fractional differencing. Biometrika 68, 165–176. https://doi.org/10.1093/biomet/68.1.165
Huang, M.-H., Tung, H., Fielding, E.J., Huang, H.-H., Liang, C., Huang, C., Hu, J.-C., 2016. Multiple fault slip triggered above the 2016 M w 6.4 MeiNong earthquake in Taiwan: Coseismic Slip Model of MeiNong Earthquake. Geophys. Res. Lett. 43, 7459–7467. https://doi.org/10.1002/2016GL069351
Ji, K.H., Herring, T.A., 2013. A method for detecting transient signals in GPS position time-series: smoothing and principal component analysis. Geophysical Journal International 193, 171–186. https://doi.org/10.1093/gji/ggt003
Kailath, T., 1968. An innovations approach to least-squares estimation--Part I: Linear filtering in additive white noise. IEEE Trans. Automat. Contr. 13, 646–655. https://doi.org/10.1109/TAC.1968.1099025
Kalman, R.E., 1960. A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering 82, 35–45. https://doi.org/10.1115/1.3662552
Kennedy, J., Eberhart, R., 1995. Particle swarm optimization, in: Proceedings of ICNN’95 - International Conference on Neural Networks. Presented at the ICNN’95 - International Conference on Neural Networks, IEEE, Perth, WA, Australia, pp. 1942–1948. https://doi.org/10.1109/ICNN.1995.488968
Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., 1983. Optimization by Simulated Annealing. Science 220, 671–680. https://doi.org/10.1126/science.220.4598.671
Klos, A., Bos, M.S., Bogusz, J., 2018. Detecting time-varying seasonal signal in GPS position time series with different noise levels. GPS Solut 22, 21. https://doi.org/10.1007/s10291-017-0686-6
Kolmogorov, A., 1941. Stationary sequences in Hilbert space. Bull. Math. Univ. Moscow 2.
Kolmogorov, A., 1939. Sur l’interpolation et extrapolation des suites stationnaires. Comptes Rendus de l’Académie des Sciences 208, 2043–2045.
Langbein, J., Johnson, H., 1997. Correlated errors in geodetic time series: Implications for time-dependent deformation. J. Geophys. Res. 102, 591–603. https://doi.org/10.1029/96JB02945
Mai, H.A., Lee, J.-C., Chen, K.H., Wen, K.-L., 2021. Coulomb stress changes triggering surface pop-up during the 2016 Mw 6.4 Meinong earthquake with implications for earthquake-induced mud diapiring in SW Taiwan. Journal of Asian Earth Sciences 218, 104847. https://doi.org/10.1016/j.jseaes.2021.104847
Mao, A., Harrison, C.G.A., Dixon, T.H., 1999. Noise in GPS coordinate time series. J. Geophys. Res. 104, 2797–2816. https://doi.org/10.1029/1998JB900033
Ming, F., Yang, Y., Zeng, A., Zhao, B., 2019. Decomposition of geodetic time series: A combined simulated annealing algorithm and Kalman filter approach. Advances in Space Research 64, 1130–1147. https://doi.org/10.1016/j.asr.2019.05.049
Mohamed, A.H., Schwarz, K.P., 1999. Adaptive Kalman Filtering for INS/GPS. Journal of Geodesy 73, 193–203. https://doi.org/10.1007/s001900050236
Nikolaidis, R., 2002. Observation of geodetic and seismic deformation with the Global Positioning System. Ph.D. Thesis.
Rau, R.-J., Wen, Y.-Y., Ching, K.-E., Hsieh, M.-C., Lo, Y.-T., Chiu, C.-Y., Hashimoto, M., 2022. Origin of coseismic anelastic deformation during the 2016 Mw 6.4 Meinong Earthquake, Taiwan. Tectonophysics 836, 229428. https://doi.org/10.1016/j.tecto.2022.229428
Rauch, H.E., Tung, F., Striebel, C.T., 1965. Maximum likelihood estimates of linear dynamic systems. AIAA Journal 3, 1445–1450. https://doi.org/10.2514/3.3166
Sage, A.P., Husa, G.W., 1969. Adaptive Filtering with Unknown Prior Statistics. Proc. Joint Autom. Control Conf 760–769.
Segall, P., Matthews, M., 1997. Time dependent inversion of geodetic data. J. Geophys. Res. 102, 22391–22409. https://doi.org/10.1029/97JB01795
Shan, C., Zhou, W., Yang, Y., Jiang, Z., 2020. Multi-Fading Factor and Updated Monitoring Strategy Adaptive Kalman Filter-Based Variational Bayesian. Sensors 21, 198. https://doi.org/10.3390/s21010198
Shen, Z., Wang, M., Zeng, Y., Wang, F., 2015. Optimal Interpolation of Spatially Discretized Geodetic Data. Bulletin of the Seismological Society of America 105, 2117–2127. https://doi.org/10.1785/0120140247
Shen, Z.-K., Jackson, D.D., Ge, B.X., 1996. Crustal deformation across and beyond the Los Angeles basin from geodetic measurements. J. Geophys. Res. 101, 27957–27980. https://doi.org/10.1029/96JB02544
Shi, Y., Eberhart, R., 1998. A modified particle swarm optimizer, in: 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98TH8360). Presented at the 1998 IEEE International Conference on Evolutionary Computation Proceedings. IEEE World Congress on Computational Intelligence, IEEE, Anchorage, AK, USA, pp. 69–73. https://doi.org/10.1109/ICEC.1998.699146
Shumway, R.H., Stoffer, D.S., 1982. AN APPROACH TO TIME SERIES SMOOTHING AND FORECASTING USING THE EM ALGORITHM. J Time Series Analysis 3, 253–264. https://doi.org/10.1111/j.1467-9892.1982.tb00349.x
Thrun, S., Burgard, W., Fox, D., 2005. Probabilistic robotics, Intelligent robotics and autonomous agents. MIT Press, Cambridge, Mass.
Wiener, N., Masani, P., 1957. The prediction theory of multivariate stochastic processes: I. The regularity condition. Acta Math. 98, 111–150. https://doi.org/10.1007/BF02404472
Williams, S.D.P., 2004. Error analysis of continuous GPS position time series. J. Geophys. Res. 109, B03412. https://doi.org/10.1029/2003JB002741
Wold, H., 1938. A Study in The Analysis of Stationary Time Series.
Wornell, G.W., 1993. Wavelet-based representations for the 1/f family of fractal processes. Proc. IEEE 81, 1428–1450. https://doi.org/10.1109/5.241506
Zhi-Hui Zhan, Jun Zhang, Yun Li, Chung, H.S.-H., 2009. Adaptive Particle Swarm Optimization. IEEE Trans. Syst., Man, Cybern. B 39, 1362–1381. https://doi.org/10.1109/TSMCB.2009.2015956
李宥葭, 2017. 台灣G P S時間序列的雜訊分析. 國立中央大學, 桃園.
蔡佩京, 2018. Abnormally Large Postseismic Deformation Caused by Reactivated Mud Diapirism on the Accretionary Wedge: Constrained by the 2016 Meinong Earthquake (フェローシップ事業成果報告書). 公益財団法人日本台湾交流協会.
賴力嘉, 2023. 2016 年Mw 6.4 美濃地震觸發褶皺逆衝帶的淺層構造滑動所產生的反向變形. 國立成功大學.