基於卡爾曼濾波器之GPS時間序列分析

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楊晟右(Cheng-Yu Yang) 查詢紙本館藏

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地球科學學系

論文名稱

基於卡爾曼濾波器之GPS時間序列分析
(GPS Time Series Analysis via Kalman Filter)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ GPS時間序列
★ 卡爾曼濾波器

關鍵字(英)

論文目次

摘要 iv
Abstract v
目錄 vi
圖目錄 vii
第一章緒論 1
1.1 研究動機 1
1.2 論文架構 5
第二章研究方法 6
2.1 GPS時間序列分析 6
2.2 狀態空間模型（state-space model, SSM） 7
2.3 卡爾曼濾波器（Kalman filter, KF） 9
2.4 RTS平滑器（Rauch-Tung-Striebel smoother, RTS smoother） 12
2.5 超參數估計（hyperparameter estimation） 13
第三章雜訊分析 17
3.1 增廣狀態（Augmented state） 17
3.2 頻譜指數及雜訊振幅估計 18
3.3 結果 19
3.4 討論 36
第四章震後變形 38
4.1 前人研究 38
4.2 引入加速度 40
4.3 合成資料測試（Synthetic data test） 43
4.4 結果 49
4.4.1 時間序列 49
4.4.2 美濃震前速度場 50
4.4.3 美濃同震位移 50
4.4.4 美濃震後位移 51
4.5 應變分析 66
4.5.1 美濃震前應變率 67
4.5.2 美濃同震應變 67
4.5.3 美濃震後應變 68
第五章結論 79
參考文獻 81

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指導教授

張午龍(Wu-Lung Chang)

審核日期

2024-1-26

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