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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).

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state space approach to signal extraction problems in seismology

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)

roles of statistical models
Roles of Statistical Models

Model as a “tool”

for extracting

information

Data

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.

outline
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

change of statistical problems

Small Experimental, Survey Data

Parametric Models + AIC

Huge Observations, Complex Systems

  • Flexible Modeling
  • Smoothness priors
  • Automatic Procedures
Change of Statistical Problems
smoothness prior

Observation

Unknown Parameter

Noise

Smoothness Prior

Simple Smoothing Problem

Infidelity to

smoothness

Infidelity

to the data

Penalized Least Squares

Whittaker (1923), Shiller (1973), Akaike(1980), Kitagawa-Gersch(1996)

automatic parameter determination via bayesian interpretation
Automatic Parameter Determination via Bayesian Interpretation

Crucial

parameter

Bayesian Interpretation

Multiply by and exponentiate

Smoothness

Prior

Determination of by ABIC (Akaike 1980)

applications of state space model
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
state space models
State Space Models

Linear Gaussian

Nonlinear

Non-Gaussian

Nonlinear Non-Gaussian

Discrete state

Discrete obs.

General

kalman filter
Kalman Filter

Initial

Prediction

Prediction

Filter

Filter

Smoothing

non gaussian filter smoother
Non-Gaussian Filter/Smoother

Prediction

Filter

Smoother

recursive filter smoother for state estimation

True

Normal approx.

Piecewise

Linear

Step function

Normal mixture

Monte Carlo approx.

Recursive Filter/Smootherfor 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

sequential monte carlo filter
Sequential Monte Carlo Filter

System Noise

Predictive Distribution

Importance Weight (Bayes factor)

Filter Distribution Resampling

Gordon et al. (1993), Kitagawa (1996)

Doucet, de Freitas and Gordon (2001)

“Sequential Monte Carlo Methods in Practice”

self tuned state space model
Self-Tuned State Space Model

Time-varying parameter

Augmented State Vector

Non-Gaussian or Monte Carlo Smoother

Simultaneous Estimation of State and Parameter

tools for time series modeling
Tools for Time Series Modeling
  • Model Representaion
    • Generic: State Space Models
    • Specific: Smoothness Priors
  • Estimation
    • State: Sequential Filters
    • Parameter: MLE, Bayes, SOSS
  • Evaluation
    • AIC
examples
Examples
  • Detection of Micro Earthquakes
  • Extraction of Coseismic Effects
  • Analysis of OBS (Ocean Bottom
  • Seismograph) Data
extraction of micro earthquake
Extraction of Micro Earthquake

15

0

-15

15

0

-15

15

0

-15

4

2

0

-2

-4

-6

Observed

Background Noise

Seismic Signal

Time-varying Variance

(in log10)

0 400 800 1200 1600 2000 2400 2800

extraction of micro earthquake20
Extraction of Micro Earthquake

Observed

Earthquake Signal

Background Noise

extraction of earthquake signal
Extraction of Earthquake Signal

Observed

S-wave

P-wave

Background Noise

3d modeling

U-D

N-S

E-W

P-wave

3D-Modeling

S-wave

P-wave

U-D

N-S

E-W

S-wave

slide23

Detection of Coseismic Effects

Groundwater Level

Precipitation

Air Pressure

Earth Tide

dT = 2min., 20years

Japan

Tokai Area

Observation Well

Geological Survey of Japan

5M observations

detection of coseismic effect in groundwater level
Detection of Coseismic Effect in Groundwater Level

Difficulties

  • Presence of many missing

and outlyingobservations

Outlier

Missing

  • Strongly affected by barometric air pressure, earth tide and rain
automatic data cleaning
Automatic Data Cleaning

State Space Model

Observation Noise Model

model for outliers

Mixture

-5 -4 -3 -2 -1 0 1 2 3 4 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

Model for Outliers
missing and outlying observations
Missing and Outlying Observations

Gaussian

Mixture

Original

Cleaned

slide28

Detection of Coseismic Effects

1981

1982

1983

1984

1985

1986

1987

1988

1989

1990

Strongly affected the

covariates such as

barometric air

pressure, earth tide

and rain

Difficult to find out Coseismic Effect

slide29

Pressure Effect

Air Pressure

Pressure Effect

extraction of coseismic effect
Extraction of Coseismic Effect

Component Models

Observation

Trend

Air Pressure Effect

Earth Tide Effect

Observation Noise

slide33

Precipitation Effect

Pressure, Earth-Tide removed

Original

extraction of coseismic effect34
Extraction of Coseismic Effect

Component Models

Observation

Trend

Air Pressure Effect

Earth Tide Effect

Precipitation Effect

Observation Noise

slide36

Air Pressure Effect

Earth Tide Effect

Precipitation Effect

Extraction of Coseismic Effects

Groundwater Level

M=4.8, D=48km

min AIC model

m=25, l=2, k=5

Corrected Water Level

detected coseismic effect

M=4.8

D=48km

M=6.0

D=113km

M=6.8

D=128km

M=7.7

D=622km

M=7.9

D=742km

M=5.7

D=66km

M=6.2

D=150km

M=5.0

D=57km

M=7.0

D=375km

Detected Coseismic Effect

Original

T+P+ET

T+P+ET+R

Signal

slide38

Original

Air Pressure

Effect

Earth Tide

Effect

P & ET

Removed

Precipitation

Effect

P, ET & R

Removed

Min AIC model

m=25, l=2, k=5

slide39

1981

1982

1981

1982

M=7.0

D=375km

M=4.8

D=48km

1983

1984

M=6.0

D=113km

M=6.8

D=128km

1983

1984

M=7.7

D=622km

M=7.9

D=742km

M=5.7

D=66km

M=6.2

D=150km

M=5.0

D=57km

1985

1986

1985

1986

M=6.0

D=126km

1987

1988

1987

1988

M=6.7

D=226km

1989

1990

1989

1990

M=5.7

D=122km

M=6.5

D=96km

Coseismic Effect

effect of earthquake

> 16cm

> 4cm

>1cm

Rain Water level

Magnitude

Distance

Coseismic Effect

Earthquake Water level

Effect of Earthquake
findings
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?
observations by an experiment

4 Channel Time Series

N=15360, 98239 series

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)

Hokkaido University + University of Bergen

an example of the observations
An Example of the Observations

OBS-4

N=7500

M=1560

OBS-31

N=15360

M=982

Low S/N

High S/N

direct wave reflection refraction
Direct wave, Reflection, Refraction

Refraction Wave

Direct Wave

Reflection Wave

objectives
Objectives

Estimation of Underground Structure

Intermediate objectives

Detection of Reflection & Refraction Waves

Estimation of parameters (hj , vj)

time series at hypocenter d 0
Time series at hypocenter (D=0)

Wave(011)

Wave(00011)

Wave(0)

Wave(000)

Wave(00000)

model for decomposition
Model for Decomposition

Self-Organizing Model

spatial model ignoring time series structure
Spatial Model(Ignoring time series structure)

Series j-1 Series j : Time-lag=k

model of propagation path
Model of Propagation Path

Parallel Structure

Water

Width

Velocity

examples of wave path
Examples of Wave Path

Wave(0)

Wave(000)

Wave(01)

Wave(011)

Wave(0121)

Wave(000121)

Wave(01221)

Wave(012321)

Wave(00012321)

path models and arrival times obs4
Path models and arrival times(OBS4)

Arrival Time (sec.)

Distance (km)

local time lag

8

7

6

5

4

3

2

1

0

Arrival Time (sec.)

-10 -8 -6 -4 -2 0 2 4 6 8 10

D: Distance (km)

Local Time Lag
spatial temporal model64
Spatial-Temporal Model

Time-lag (Channel j-1 Channel j ) =k

spatial temporal decomposition
Spatial-Temporal Decomposition

Reflection wave

Direct wave

mt usu eruption data
Mt. Usu Eruption Data

Hokkaido, Japan

March 31, 2000 13:07-

volatility and component models
Volatility and component models

Hokkaido, Japan

March 31, 2000 13:07-

summary

New findings,

Automatic procedure

Summary

Signal extraction and knowledge discovery by statistical modeling

  • Use of information from data and
  • Prior knowledge
  • State Space Modeling
  • Filtering/smoothing & SOSS
time varying spectrum
Time-varying Spectrum

AR model Autocovariance Spectrum

Time-varying Nonstationary

Time-varying AR model

Time-varying spectrum

estimation of nonstationary ar model
Estimation of Nonstationary AR Model

Model for Time-changes of Coefficients

State Space Representation

Kronecker product

state space representation74
State Space Representation

For k = 1

For k = 2

Kronecker Product

time varying coefficients
Time-varying Coefficients

Gauss model

Cauchy model

estimation of arrival time
Estimation of Arrival Time

P

S

Prediction of Tsunami

Estimation of Arrival Times

Automatic & Fast Algorithm

Estimation of Hypocenter

Locally Stationary AR Model

Automatic Modeling by

Information Criterion AIC

estimation of arrival time80
Estimation of Arrival Time

Locally Stationary AR Model

Background Noise Seismic Signal

Background Noise Model

Seismic Signal Model

estimation of arrival time81
Estimation of Arrival Time

AIC of the Total Models

Min AIC Estimate

of Arrival Time

model implementations
Model & Implementations

LSAR model:

Ozaki and Tong (1976)

Householder implementation:

Kitagawa and Akaike (1979)

Kalman filter implementation:

lower order models
Lower Order Models

Levinson recursion

posterior probabilities of arrival times
Posterior Probabilities of Arrival Times

AIC: -2(Bias corrected log-likelihood)

Likelihood of the arrival time

Posterior probability of the arrival time