State space approach to signal extraction problems in seismology
Download
1 / 87

State Space Approach to Signal Extraction Problems in Seismology - PowerPoint PPT Presentation


  • 146 Views
  • Uploaded on

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).

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'State Space Approach to Signal Extraction Problems in Seismology' - kacia


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
State space approach to signal extraction problems in seismology l.jpg

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 l.jpg
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 l.jpg
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 l.jpg

Small Experimental, Survey Data

Parametric Models + AIC

Huge Observations, Complex Systems

  • Flexible Modeling

  • Smoothness priors

  • Automatic Procedures

Change of Statistical Problems


Smoothness prior l.jpg

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 l.jpg
Automatic Parameter Determination via Bayesian Interpretation

Crucial

parameter

Bayesian Interpretation

Multiply by and exponentiate

Smoothness

Prior

Determination of by ABIC (Akaike 1980)


Time series interpretation and state space modeling l.jpg
Time Series Interpretationand State Space Modeling

Equivalent Model

State Space Model


Applications of state space model l.jpg
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 l.jpg
    State Space Models

    Linear Gaussian

    Nonlinear

    Non-Gaussian

    Nonlinear Non-Gaussian

    Discrete state

    Discrete obs.

    General


    Kalman filter l.jpg
    Kalman Filter

    Initial

    Prediction

    Prediction

    Filter

    Filter

    Smoothing


    Non gaussian filter smoother l.jpg
    Non-Gaussian Filter/Smoother

    Prediction

    Filter

    Smoother


    Recursive filter smoother for state estimation l.jpg

    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 l.jpg
    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 l.jpg
    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 l.jpg
    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 l.jpg
    Examples

    • Detection of Micro Earthquakes

    • Extraction of Coseismic Effects

    • Analysis of OBS (Ocean Bottom

    • Seismograph) Data


    Extraction of signal from noisy data l.jpg

    Observed

    Extraction of Signal From Noisy Data

    Component Models

    Basic Model



    Extraction of micro earthquake l.jpg
    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 l.jpg
    Extraction of Micro Earthquake

    Observed

    Earthquake Signal

    Background Noise


    Extraction of earthquake signal l.jpg
    Extraction of Earthquake Signal

    Observed

    S-wave

    P-wave

    Background Noise


    3d modeling l.jpg

    U-D

    N-S

    E-W

    P-wave

    3D-Modeling

    S-wave

    P-wave

    U-D

    N-S

    E-W

    S-wave


    Slide23 l.jpg

    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 l.jpg
    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 l.jpg
    Automatic Data Cleaning

    State Space Model

    Observation Noise Model


    Model for outliers l.jpg

    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 l.jpg
    Missing and Outlying Observations

    Gaussian

    Mixture

    Original

    Cleaned


    Slide28 l.jpg

    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 l.jpg

    Pressure Effect

    Air Pressure

    Pressure Effect


    Extraction of coseismic effect l.jpg
    Extraction of Coseismic Effect

    Component Models

    Observation

    Trend

    Air Pressure Effect

    Earth Tide Effect

    Observation Noise




    Slide33 l.jpg

    Precipitation Effect

    Pressure, Earth-Tide removed

    Original


    Extraction of coseismic effect34 l.jpg
    Extraction of Coseismic Effect

    Component Models

    Observation

    Trend

    Air Pressure Effect

    Earth Tide Effect

    Precipitation Effect

    Observation Noise



    Slide36 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    > 16cm

    > 4cm

    >1cm

    Rain Water level

    Magnitude

    Distance

    Coseismic Effect

    Earthquake Water level

    Effect of Earthquake


    Findings l.jpg
    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?


    Exploring underground structure by obs ocean bottom seismogram data l.jpg

    Sea Surface

    OBS

    Exploring Underground Structure by OBS(Ocean Bottom Seismogram) Data

    Bottom


    Observations by an experiment l.jpg

    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 l.jpg
    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 l.jpg
    Direct wave, Reflection, Refraction

    Refraction Wave

    Direct Wave

    Reflection Wave


    Objectives l.jpg
    Objectives

    Estimation of Underground Structure

    Intermediate objectives

    Detection of Reflection & Refraction Waves

    Estimation of parameters (hj , vj)


    Time series at hypocenter d 0 l.jpg
    Time series at hypocenter (D=0)

    Wave(011)

    Wave(00011)

    Wave(0)

    Wave(000)

    Wave(00000)


    Model for decomposition l.jpg
    Model for Decomposition

    Self-Organizing Model


    Decomposition of ch 701 d 4km l.jpg
    Decomposition of Ch-701 (D=4km)

    Observed


    Decomposition of ch 721 d 8km l.jpg
    Decomposition of Ch-721 (D=8km)

    Observed




    Spatial model ignoring time series structure l.jpg
    Spatial Model(Ignoring time series structure)

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


    Local cross correlation function l.jpg
    Local Cross-Correlation Function

    630

    Location

    730

    0

    Time

    8



    Model of propagation path l.jpg
    Model of Propagation Path

    Parallel Structure

    Water

    Width

    Velocity


    Examples of wave path l.jpg
    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 l.jpg
    Path models and arrival times(OBS4)

    Arrival Time (sec.)

    Distance (km)


    Local time lag l.jpg

    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



    Model for decomposition63 l.jpg
    Model for Decomposition Adjacent Channels


    Spatial temporal model64 l.jpg
    Spatial-Temporal Model Adjacent Channels

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


    Spatial temporal filtering l.jpg
    Spatial-Temporal Filtering Adjacent Channels


    Spatial temporal decomposition l.jpg
    Spatial-Temporal Decomposition Adjacent Channels

    Reflection wave

    Direct wave


    Mt usu eruption data l.jpg
    Mt. Usu Adjacent Channels Eruption Data

    Hokkaido, Japan

    March 31, 2000 13:07-


    Volatility and component models l.jpg
    Volatility and component models Adjacent Channels

    Hokkaido, Japan

    March 31, 2000 13:07-


    Decomposition l.jpg
    Decomposition Adjacent Channels


    Summary l.jpg

    New findings, Adjacent Channels

    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 l.jpg
    Time-varying Spectrum Adjacent Channels

    AR model Autocovariance Spectrum

    Time-varying Nonstationary

    Time-varying AR model

    Time-varying spectrum


    Estimation of nonstationary ar model l.jpg
    Estimation of Nonstationary AR Model Adjacent Channels

    Model for Time-changes of Coefficients

    State Space Representation

    Kronecker product


    State space representation74 l.jpg
    State Space Representation Adjacent Channels

    For k = 1

    For k = 2

    Kronecker Product


    State space representation75 l.jpg
    State Space Representation Adjacent Channels

    Case: k = 1


    Time varying coefficients l.jpg
    Time-varying Coefficients Adjacent Channels

    Gauss model

    Cauchy model


    Time varying spectrum77 l.jpg
    Time-varying Spectrum Adjacent Channels


    Slide78 l.jpg

    Precipitation Effect Adjacent Channels


    Estimation of arrival time l.jpg
    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 time80 l.jpg
    Estimation of Arrival Time Adjacent Channels

    Locally Stationary AR Model

    Background Noise Seismic Signal

    Background Noise Model

    Seismic Signal Model


    Estimation of arrival time81 l.jpg
    Estimation of Arrival Time Adjacent Channels

    AIC of the Total Models

    Min AIC Estimate

    of Arrival Time


    Model implementations l.jpg
    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 l.jpg
    State Space Representation of AR Model Adjacent Channels

    New datayn


    Lower order models l.jpg
    Lower Order Models Adjacent Channels

    Levinson recursion


    Arrival times of p waves l.jpg
    Arrival Times of P-waves Adjacent Channels

    AIC

    AIC

    AIC

    2000


    Arrival times of s waves l.jpg
    Arrival Times of S-waves Adjacent Channels

    AIC

    AIC

    AIC

    100


    Posterior probabilities of arrival times l.jpg
    Posterior Probabilities of Arrival Times Adjacent Channels

    AIC: -2(Bias corrected log-likelihood)

    Likelihood of the arrival time

    Posterior probability of the arrival time


    ad