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State Space Approach to Signal Extraction Problems in SeismologyPowerPoint Presentation

State Space Approach to Signal Extraction Problems in Seismology

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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)

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

- 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

Small Experimental, Survey Data

Parametric Models + AIC

Huge Observations, Complex Systems

- Flexible Modeling
- Smoothness priors
- Automatic Procedures

Unknown Parameter

Noise

Smoothness PriorSimple 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

Crucial

parameter

Bayesian Interpretation

Multiply by and exponentiate

Smoothness

Prior

Determination of by ABIC (Akaike 1980)

Applications of State Space Model Trend Estimation, Seasonal Adjustment Time-Varying Variance Models, Volatility Signal Extraction, Decomposition

- Modeling Nonstationarity
- in mean

- in variance

- in covariance
- Time-Varying Coefficient Models, TVAR model

State Space Models

Linear Gaussian

Nonlinear

Non-Gaussian

Nonlinear Non-Gaussian

Discrete state

Discrete obs.

General

Normal approx.

Piecewise

Linear

Step function

Normal mixture

Monte Carlo approx.

Recursive Filter/Smootherfor State Estimation0. 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

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

Time-varying parameter

Augmented State Vector

Non-Gaussian or Monte Carlo Smoother

Simultaneous Estimation of State and Parameter

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

- Detection of Micro Earthquakes
- Extraction of Coseismic Effects
- Analysis of OBS (Ocean Bottom
- Seismograph) Data

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

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

Difficulties

- Presence of many missing
and outlyingobservations

Outlier

Missing

- Strongly affected by barometric air pressure, earth tide and rain

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

Extraction of Coseismic Effect

Component Models

Observation

Trend

Air Pressure Effect

Earth Tide Effect

Observation Noise

Extraction of Coseismic Effect

Component Models

Observation

Trend

Air Pressure Effect

Earth Tide Effect

Precipitation Effect

Observation Noise

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

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 EffectOriginal

T+P+ET

T+P+ET+R

Signal

Air Pressure

Effect

Earth Tide

Effect

P & ET

Removed

Precipitation

Effect

P， ET & R

Removed

Min AIC model

m=25, l=2, k=5

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

> 4cm

>1cm

Rain Water level

Magnitude

Distance

Coseismic Effect

Earthquake Water level

Effect of EarthquakeFindings

- 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?

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

Objectives

Estimation of Underground Structure

Intermediate objectives

Detection of Reflection & Refraction Waves

Estimation of parameters （hj , vj）

Model for Decomposition

Self-Organizing Model

Decomposition of Ch-701 (D=4km)

Observed

Decomposition of Ch-721 (D=8km)

Observed

“Spatial” Filter/Smoother

k:Time-lag

Spatial Model(Ignoring time series structure)

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

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 the Differences of the Arrival Times Between Adjacent Channels

Epicentral Distance

Model for Decomposition Adjacent Channels

Spatial-Temporal Model Adjacent Channels

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

Spatial-Temporal Filtering Adjacent Channels

Decomposition Adjacent Channels

New findings, Adjacent Channels

Automatic procedure

SummarySignal extraction and knowledge discovery by statistical modeling

- Use of information from data and
- Prior knowledge
- State Space Modeling
- Filtering/smoothing & SOSS

Time-varying Spectrum Adjacent Channels

AR model Autocovariance Spectrum

Time-varying Nonstationary

Time-varying AR model

Time-varying spectrum

Estimation of Nonstationary AR Model Adjacent Channels

Model for Time-changes of Coefficients

State Space Representation

Kronecker product

State Space Representation Adjacent Channels

Case: k = 1

Time-varying Spectrum Adjacent Channels

Precipitation Effect Adjacent Channels

Estimation of Arrival Time Adjacent Channels

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 Time Adjacent Channels

Locally Stationary AR Model

Background Noise Seismic Signal

Background Noise Model

Seismic Signal Model

Estimation of Arrival Time Adjacent Channels

AIC of the Total Models

Min AIC Estimate

of Arrival Time

Model & Implementations Adjacent Channels

LSAR model:

Ozaki and Tong (1976)

Householder implementation:

Kitagawa and Akaike (1979)

Kalman filter implementation:

State Space Representation of AR Model Adjacent Channels

New datayn

Lower Order Models Adjacent Channels

Levinson recursion

Posterior Probabilities of Arrival Times Adjacent Channels

AIC: -2(Bias corrected log-likelihood)

Likelihood of the arrival time

Posterior probability of the arrival time

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