1. State Space Approach to Signal Extraction Problems in Seismology Genshiro Kitagawa The Institute of Statistical Mathematics IMA, Minneapolis Nov. 15, 2001 Collaborators: Will Gersch (Univ. Hawaii) Tetsuo Takanami (Univ. Hokkaido) Norio Matsumoto (Geological Survey of Japan)

2. Roles of Statistical Models Data Information Model as a “tool” for extracting information Modeling based on the characteristics of the object and the objective of the analysis. Unify information supplied by data and prior knowledge. Bayes models, state space models etc.

3. Outline Method –Flexible Statistical Modeling –State Space Modeling Applications –Extraction of Signal from Noisy Data –Automatic Data Cleaning –Detection of Coseismic Effect in Groundwater Level –Analysis of OBS (Ocean Bottom Seismograph) Data JASA(1996) + ISR(2001) + some new

4. Change of Statistical Problems Flexible Modeling Smoothness priors Automatic Procedures Huge Observations, Complex Systems Small Experimental, Survey Data Parametric Models + AIC

5. Smoothness Prior Simple Smoothing Problem Observation Unknown Parameter Noise Penalized Least Squares Whittaker (1923), Shiller (1973), Akaike(1980), Kitagawa-Gersch(1996) Infidelity to the data Infidelity to smoothness

6. Automatic Parameter Determination via Bayesian Interpretation Bayesian Interpretation Multiply by and exponentiate Determination of by ABIC (Akaike 1980) Crucial parameter Smoothness Prior

7. Time Series Interpretation and State Space Modeling State Space Model Equivalent Model

8. Applications of State Space Model Modeling Nonstationarity in mean Trend Estimation, Seasonal Adjustment in variance Time-Varying Variance Models, Volatility in covariance Time-Varying Coefficient Models, TVAR model Signal Extraction, Decomposition

9. State Space Models Nonlinear Non-Gaussian General Linear Gaussian Nonlinear Non-Gaussian Discrete state Discrete obs.

10. Kalman Filter Prediction Filter Smoothing Initial Prediction Filter

11. Non-Gaussian Filter/Smoother Prediction Filter Smoother

12. Recursive Filter/Smoother for State Estimation 0. Gaussian Approximation Kalman filter/smoother 1. Piecewise-linear or Step Approx. Non-Gaussian filter/smoother 2. Gaussian Mixture Approx. Gaussian-sum filter/smoother 3. Monte Carlo Based Method Sequential Monte Carlo filter/smoother True Normal approx. Piecewise Linear Step function Normal mixture Monte Carlo approx.

13. Sequential Monte Carlo Filter System Noise Importance Weight (Bayes factor) Predictive Distribution Filter Distribution Resampling Gordon et al. (1993), Kitagawa (1996) Doucet, de Freitas and Gordon (2001) “Sequential Monte Carlo Methods in Practice”

14. Self-Tuned State Space Model Augmented State Vector Non-Gaussian or Monte Carlo Smoother Simultaneous Estimation of State and Parameter Time-varying parameter

15. Tools for Time Series Modeling Model Representaion – Generic: State Space Models – Specific: Smoothness Priors Estimation – State: Sequential Filters – Parameter: MLE, Bayes, SOSS Evaluation – AIC

16. Examples 1. Detection of Micro Earthquakes 2. Extraction of Coseismic Effects 3. Analysis of OBS (Ocean Bottom Seismograph) Data

17. Extraction of Signal From Noisy Data Basic Model Component Models Observed

18. State Space Model

19. Extraction of Micro Earthquake Observed Seismic Signal Background Noise 0 400 800 1200 1600 2000 2400 2800 15 0 -15 15 0 -15 15 0 -15 4 2 0 -2 -4 -6 Time-varying Variance (in log 10 )

20. Extraction of Micro Earthquake Background Noise Earthquake Signal Observed

21. Extraction of Earthquake Signal Observed S-wave P-wave Background Noise

22. 3D-Modeling P-wave S-wave E-W N-S U-D P-wave E-W N-S U-D S-wave

23. Detection of Coseismic Effects Observation Well Geological Survey of Japan Precipitation Groundwater Level Air Pressure Earth Tide dT = 2min., 20years Japan Tokai Area 5M observations

24. Detection of Coseismic Effect in Groundwater Level Difficulties Presence of many missing and outlying observations Outlier Missing Strongly affected by barometric air pressure, earth tide and rain

25. Automatic Data Cleaning State Space Model Observation Noise Model

26. Model for Outliers -5 -4 -3 -2 -1 0 1 2 3 4 5 Mixture -5 -4 -3 -2 -1 0 1 2 3 4 5

27. Missing and Outlying Observations Gaussian Mixture OriginalCleaned

28. Detection of Coseismic Effects 1981 1982 19831984 19851986 19871988 1989 1990 Strongly affected the covariates such as barometric air pressure, earth tide and rain Difficult to find out Coseismic Effect

29. Pressure Effect Air Pressure Pressure Effect

30. Extraction of Coseismic Effect Component Models Observation Trend Air Pressure Effect Earth Tide Effect Observation Noise

31. State Space Representation

32. AIC Values

33. Precipitation Effect Original Pressure, Earth-Tide removed

34. Extraction of Coseismic Effect Component Models Observation Trend Air Pressure Effect Earth Tide Effect Precipitation Effect Observation Noise

35. State Space Model

36. Groundwater Level Air Pressure Effect Earth Tide Effect Precipitation Effect min AIC model m=25, l=2, k=5 M=4.8, D=48km Extraction of Coseismic Effects Corrected Water Level

37. Detected Coseismic Effect Original T+P+ET+R M=4.8 D=48km M=6.8 D=128km M=7.0 D=375km M=5.7 D=66km M=7.7 D=622km M=6.0 D=113km M=6.2 D=150km M=5.0 D=57km M=7.9 D=742km T+P+ET Signal

38. Min AIC model m=25, l=2, k=5 Original Air Pressure Effect Earth Tide Effect P & ET Removed Precipitation Effect P ， ET & R Removed

39. Coseismic Effect 19811982 19831984 19851986 19871988 19891990 M=7.0 D=375km M=4.8 D=48km M=5.7 D=66km M=7.7 D=622km M=6.0 D=113km M=6.2 D=150km M=5.0 D=57km M=7.9 D=742km M=6.8 D=128km M=6.0 D=126km M=6.7 D=226km M=5.7 D=122km M=6.5 D=96km 1981 1982 19831984 19851986 19871988 1989 1990

40. Effect of Earthquake Earthquake Water level Rain Water level Distance MagnitudeCoseismic Effect > 16cm > 4cm > 1cm

41. Findings Drop of level Detected for earthquakes with M > 2.62 log D + 0.2 Amount of drop ~ f( M  2.62 log D ) Without coseismic effect water level increases 6cm/year increase of stress in this area?

42. Exploring Underground Structure by OBS (Ocean Bottom Seismogram) Data Bottom OBS Sea Surface

43. Observations by an Experiment Off Norway （ Depth 1500-2000m ） 39 OBS, (Distance: about 10km ） Air-gun Signal from a Ship （ 982 times: Interval 70sec., 200m ） Observation （ dT=1/256sec., T =60sec., 4- Ch ） 4 Channel Time Series N=15360, 982 39 series Hokkaido University + University of Bergen

44. Time-Adjusted (Shifted) Time Series

45. An Example of the Observations OBS-4 N=7500 M=1560 OBS-31 N=15360 M=982 High S/N Low S/N

46. Direct wave, Reflection, Refraction Direct Wave Refraction Wave Reflection Wave

47. Objectives Estimation of Underground Structure Detection of Reflection & Refraction Waves Estimation of parameters （ h j, v j ） Intermediate objectives

48. Time series at hypocenter (D=0) Wave(0)Wave(000)Wave(00000) Wave(011)Wave(00011)

49. Model for Decomposition Self-Organizing Model

50. Decomposition of Ch-701 (D=4km) Observed

51. Decomposition of Ch-721 (D=8km) Observed

52. A Small Portion of Data

53. “Spatial” Filter/Smoother k: Time-lag

54. Spatial Model (Ignoring time series structure) Series j-1 Series j : Time-lag=k

55. Local Cross-Correlation Function Time Location 08 730 630

56. Spatial-Temporal Model

57. Model of Propagation Path Parallel Structure Width Velocity Water

58. Examples of Wave Path Wave(0)Wave(000)Wave(01) Wave(011)Wave(0121)Wave(000121) Wave(01221)Wave(012321)Wave(00012321)

59. Path Models and Arrival Times

60. Path models and arrival times(OBS4) Distance (km) Arrival Time (sec.)

61. Local Time Lag -10 -8 -6 -4 -2 0 2 4 6 8 10 D: Distance (km) 876543210876543210 Arrival Time (sec.)

62. Path Models and the Differences of the Arrival Times Between Adjacent Channels Epicentral Distance

63. Model for Decomposition

64. Spatial-Temporal Model Time-lag (Channel j-1 Channel j ) = k

65. Spatial-Temporal Filtering

66. Spatial-Temporal Decomposition Reflection waveDirect wave

67. Mt. Usu Eruption Data Hokkaido, Japan March 31, 2000 13:07-

68. Volatility and component models Hokkaido, Japan March 31, 2000 13:07-

69. Decomposition

70. Summary Signal extraction and knowledge discovery by statistical modeling Use of information from data and Prior knowledge State Space Modeling Filtering/smoothing & SOSS New findings, Automatic procedure

72. Time-varying Spectrum AR model Autocovariance Spectrum Time-varying Nonstationary Time-varying AR model Time-varying spectrum

73. Estimation of Nonstationary AR Model State Space Representation Model for Time-changes of Coefficients Kronecker product

74. State Space Representation For k = 1 For k = 2 Kronecker Product

75. State Space Representation Case: k = 1

76. Time-varying Coefficients Gauss model Cauchy model

77. Time-varying Spectrum

78. Precipitation Effect

79. Estimation of Arrival Time PS Estimation of Arrival Times Estimation of Hypocenter Locally Stationary AR Model Automatic Modeling by Information Criterion AIC Automatic & Fast Algorithm Prediction of Tsunami

80. Estimation of Arrival Time Locally Stationary AR Model Seismic Signal Model Background Noise Seismic Signal Background Noise Model

81. Estimation of Arrival Time AIC of the Total Models Min AIC Estimate of Arrival Time

82. Model & Implementations LSAR model: Ozaki and Tong (1976) Householder implementation: Kitagawa and Akaike (1979) Kalman filter implementation:

83. State Space Representation of AR Model New data y n

84. Lower Order Models Levinson recursion

85. Arrival Times of P-waves AIC 2000

86. Arrival Times of S-waves AIC 100

87. Posterior Probabilities of Arrival Times AIC: - 2(Bias corrected log-likelihood) Likelihood of the arrival time Posterior probability of the arrival time