粒子フィルタ法を利用した日本沿岸部に おける潮位の長期変動解析 長尾大道 樋口知之（統計数理研究所） 三浦 哲 稲津大祐（東北大学理学研究科） 第 1 回 データ同化ワークショップ Apr. 22, 2011 Outline  Time-series analysis of tide gauge records using the PF Univariate analysis Multivariate analysis Event detection using a non-Gaussian distribution  Future plans  Summary

Distribution of tidal observatories Continuous observation since 1884 ~ 150 observatories Monthly means from 1966 to 2008 (i.e., 43 years) corrected to the 1000hPa constant-pressure surface by using atmospheric pressure data Tidal Data along the Coastline of Japan 第 1 回 データ同化ワークショップ Apr. 22, 2011

(blue) (black) (red) Example of Application ±std. err 第 1 回 データ同化ワークショップ Apr. 22, 2011

monthly means at Aburatsubo observatory Crust uplift ~2m when Great Kanto EQ Crustal deformation (GSI website) Long-Term Trend in Tide Gauge Data 第 1 回 データ同化ワークショップ Apr. 22, 2011

Oceanic Variations Kato and Tsumura (1979) Several Years to Decadal Variations in Tide Gauge Data 第 1 回 データ同化ワークショップ Apr. 22, 2011 Residual obtained by subtract of trend & seasonal variations from original data

Clustering of AR components (Ward’s method) 第 1 回 データ同化ワークショップ Apr. 22, 2011 Kobayashi (2008)

Observation model State Space Model for Univariate Analysis datatrend (long-term variation) seasonal (annual variation) AR (several-years variation) observation noise cf. Kato & Tsumura method (1979) We are going to improve this to detect a sudden baseline jump such as due to an earthquake take time-varying annual variations into consideration deal with missing values as easy as possible 第 1 回 データ同化ワークショップ Apr. 22, 2011

(long-term variation) System noise v 1 expresses slight changes from a linear trend (annual variation) System noise v 2 expresses small temporal changes of amplitude and phase (several-years variation) AR model extracts several- years variation System model State Space Model for Univariate Analysis Trend component Seasonal componentAR component Observation noise component (follows a Gaussian distribution) 第 1 回 データ同化ワークショップ Apr. 22, 2011

Unknown Parameters to be Optimized in Univariate Analysis AR coefficients variances of system noises variance of observation noise initial state vector 第 1 回 データ同化ワークショップ Apr. 22, 2011

State Space Model Linear formcf. Non-linear form State vector 第 1 回 データ同化ワークショップ Apr. 22, 2011

Step 1: One-step ahead prediction Each distribution is approximated as an ensemble of particles Step 2: Filtering Successive Estimation of the States : state at time t when that at time at t-1 is given: observation data at times 1 to t 第 1 回 データ同化ワークショップ Apr. 22, 2011

Sample a parameter vector from an appropriate prior Calculate likelihood of the time series Optimum parameters Sample N initial state vectors from an appropriate prior Calculate one-step ahead prediction by Resample N particles on the basis of likelihood End of time series? Enough number of parameter vectors? No Flowchart of Model Parameter Estimation No Yes 第 1 回 データ同化ワークショップ Apr. 22, 2011

DELL Precision T5400 x 24nodes (192 cores in total) for data assimilation (theoretical performance >2TFlops) ・ CPU: Intel Xeon 2.83GHz Quad core×2 ・ Memory: 32GB/node ・ Intel Fortran + MPI PC Cluster of Data Assimilation Group 第 1 回 データ同化ワークショップ Apr. 22, 2011

Land Sinking & Sea Level Changes along the coastline of Japan Sea of Japan Pacific Ocean View from South Sea of Japan Pacific Ocean View from North Oceanic Current “Kuroshio” 50cm 第 1 回 データ同化ワークショップ Apr. 22, 2011

Observation model State Space Model for Univariate AnalysisState Space Model for Multivariate Analysis vector form Observatory # 第 1 回 データ同化ワークショップ Apr. 22, 2011

(linear trend) System noise v 1 expresses slight changes from a linear trend (annual variation) System noise v 2 expresses small temporal changes of amplitude and phase (several years variation) Multivariate AR model extracts spatial correlation between observatories System model State Space Model for Multivariate Analysis Trend component Seasonal componentAR component Observation noise component 第 1 回 データ同化ワークショップ Apr. 22, 2011

Multivariate AR model AR coefficient matrix indicates cross-correlation between each observation degree All roots of are enforced to be outside the unit circle in the complex plane using the Lehman-Schur method. University of London, December 10, /21 1 1

Unknown Parameters to be Optimized in Multivariate Analysis AR coefficients variances of system noises variance of observation noise initial state vector 第 1 回 データ同化ワークショップ Apr. 22, 2011

Comparison between Multivariate & Univariate Analyses Noise level is drastically reduced !! Multivariate Analysis Univariate Analysis 第 1 回 データ同化ワークショップ Apr. 22, 2011

monthly means at Aburatsubo observatory Crust uplift ~2m when Great Kanto EQ Crustal deformation (GSI website) Long-Term Trend in Tide Gauge Data 第 1 回 データ同化ワークショップ Apr. 22, 2011

(linear trend) System noise v 1 expresses slight changes from a linear trend (annual variation) System noise v 2 expresses small temporal changes of amplitude and phase (several years variation) Multivariate AR model extracts spatial correlation between observatories System model State Space Model for Multivariate Analysis Trend component Seasonal componentAR component Observation noise component with event detection (follows Cauchy distribution) 第 1 回 データ同化ワークショップ Apr. 22, 2011

Unknown Parameters to be Optimized in Multivariate Analysis with Event Detection AR coefficients variances of system noises variance of observation noise initial state vector Scale factor of Cauchy distribution 第 1 回 データ同化ワークショップ Apr. 22, 2011

Sea Level Change due to the 1923 Great Kanto Earthquake Cauchy distribution: Gauss distribution: 第 1 回 データ同化ワークショップ Apr. 22, 2011

Cauchy 分布 Sea Level Changes due to Off-Miyagi Earthquakes 第 1 回 データ同化ワークショップ Apr. 22, 2011

SSH Anomalies Estimated by MRI.COM Yasuda and Sakurai (2006) temporal & spatial resolution?

Summary We develop a particle filter code of univariate/multivariate time series analysis, which is applicable to any time series data in various field of science. The particle filter algorithm is effective such as a sudden event detection, i.e., situations that non-Gaussian distributions are required in the model. Does a modeling of several-years oceanic variations by using MRI.COM help to detect coseismic/post-seismic deformations? 第 1 回 データ同化ワークショップ Apr. 22, 2011