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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
Data InformationModel 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.
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
• Flexible ModelingSmoothness priors
• Automatic Procedures
Huge Observations, Complex Systems
Small Experimental, Survey Data
Parametric Models + AIC
Smoothness Prior Simple Smoothing Problem
N
nn
kN
nnn
fffy
1
22
1
2min
Nnfy nnn ,,1,
ObservationUnknown ParameterNoise n
n
n
fy
Penalized Least Squares
Whittaker (1923), Shiller (1973), Akaike(1980), Kitagawa-Gersch(1996)
Infidelityto the data
Infidelity to smoothness
Automatic Parameter Determination via Bayesian Interpretation
Bayesian Interpretation
N
nn
kN
nnn ffy
1
22
1
2
N
nn
kN
nnn ffy
1
2
2
2
1
2
2 2exp
2
1exp
)|(),|(),|( ffypyf
Multiply by and exponentiate )2/(1 2
),( 22
Determination of by ABIC (Akaike 1980)
Crucial parameter
Smoothness Prior
Time Series Interpretation and State Space Modeling
x Fx Gvy Hx w
n n n
n n n
1
State Space Model
21
2
22
1
)()(
nn
N
nnn
N
n
ttty
nnn
nnn
wty
vtt
1
),0(~
),0(~2
2
Nw
Nv
n
n
2
22
Equivalent 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
Nonlinear Non-Gaussian
nnn
nnn
wHxy
GvFxx
1
General
x f x v
y h x wn n n
n n n
( , )
( , )1
)| (~
)| (~ 1
nn
nn
xHy
xFx
Linear Gaussian
NonlinearNon-Gaussian
Discrete stateDiscrete obs.
Kalman FilterPrediction
Filter
x F x
V F V F G Q Gn n n n n
n n n n n nT
n n nT
| |
| |
1 1 1
1 1 1
K V H H V H R
x x K y H x
V I K H V
n n n nT
n n n nT
n
n n n n n n n n n
n n n n n n
| |
| | |
| |
( )
( )
( )
1 11
1 1
1
SmoothingA V F V
x x A x x
V V A V V A
n n n nT
n n
n N n n n n N n n
n N n n n n N n n nT
| |
| | | |
| | | |
( )
( )
11
1 1
1 1
Initial
Prediction
Filter
yn
n n 1
n 1
Non-Gaussian Filter/Smoother
Prediction
Filter
Smoother
p x Y p x Yp x x p x Y
p x Ydxn N n n
n n n N
n nn( | ) ( | )
( | ) ( | )
( | )
1 1
11
p x Yp y x p x Y
p y Yn nn n n n
n n
( | )( | ) ( | )
( | )
1
1
p x Y p x x p x Y dxn n n n n n n( | ) ( | ) ( | )
1 1 1 1 1
Recursive Filter/Smootherfor State Estimation
0. Gaussian ApproximationKalman filter/smoother
1. Piecewise-linear or Step Approx. Non-Gaussian filter/smoother
2. Gaussian Mixture Approx.Gaussian-sum filter/smoother
3. Monte Carlo Based MethodSequential Monte Carlo filter/smoot
her
True
Normal approx.
PiecewiseLinear
Step function
Normal mixture
Monte Carlo approx.
Sequential Monte Carlo Filter
System Noisev p v j mn
j( ) ~ ( ) , ,1
Importance Weight (Bayes factor)
p F f vnj
nj
nj( ) ( ) ( )( , ) 1
Predictive Distribution
nj
n njp y p( ) ( )( | )
Filter Distribution Resampling pn
j( ) fnj( )
Gordon et al. (1993), Kitagawa (1996)Doucet, de Freitas and Gordon (2001)
“Sequential Monte Carlo Methods in Practice”
Self-Tuned State Space ModelAugmented State Vector
Non-Gaussian or Monte Carlo Smoother
Simultaneous Estimation of State and Parameter
nnn v 1
nnn xz and
,
n
nn
xz
n Time-varying parameter
Tools for Time Series Modeling
• Model Representaion– Generic: State Space Models– Specific: Smoothness Priors
• Estimation – State: Sequential Filters– Parameter: MLE, Bayes, SO
SS• Evaluation
– AIC
Examples
1. Detection of Micro Earthquakes
2. Extraction of Coseismic Effects
3. Analysis of OBS (Ocean Bottom Seismograph) Data
Extraction of Signal From Noisy Data
Basic Model
nnnn wsry r
s
w
n
n
n
Background Noise
Seismic Signal
Observation Noise ),0(~
),0(~
),0(~
2
22
21
1
1
Nw
Nv
Nu
vsbs
urar
n
n
n
njn
l
jjn
njn
m
jjn
Component ModelsObserved
State Space Modelx Fx Gw
y Hxn n n
n n n
1
x Fx Gw
y Hxn n n
n n n
1
2
22
21
21
21
1
1
1
1
,0
0,]001001[
00
00
10
00
00
01
,
01
10
0
01
1
,
nn
l
m
ln
n
n
mn
n
n
n
RQH
Gbbb
aaa
F
s
s
s
r
r
r
x
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
420
-2-4-6
Time-varying Variance(in log10)
Extraction of Micro Earthquake
Background Noise
Earthquake Signal
Observed
Extraction of Earthquake SignalObserved
S-wave
P-wave
Background Noise
3D-Modeling
zn
yn
xn
n
n
n
n
n
n
w
w
w
r
q
p
z
y
x
333
2122
111
P-wave S-wave
E-W
N-SU-D
P-wave
E-W
N-SU-D
S-wave
2
1
1 2221
1211
n
n
jn
jn
j jj
jj
n
n
v
v
r
q
bb
bb
r
q
n
m
jjnjn upap
1
jjj ,,PCA
Detection of Coseismic Effects
Observation WellGeological Survey of Japan
Precipitation
Groundwater Level
Air Pressure
Earth Tide
dT = 2min., 20years
Japan
Tokai Area
5M observations
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
Automatic Data Cleaning
State Space Model
),(),0()1(~)( Mixture)(
)(Cauchy
),0(~)( Gauss
22
22
2
NNwrw
wr
Nwr
Observation Noise Model
t t v
y t wn n n
n n n
1
Noisen Observatio :
Noise System :
Signal :
nObservatio :
n
n
n
n
w
v
t
y
Model for Outliers
0
0.2
0.4
0.6
0.8
1 41 81 121 161 201 241 281 321 361 401
GaussCauchy
-5 -4 -3 -2 -1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1 41 81 121 161 201 241 281 321 361 401
Mixture
-5 -4 -3 -2 -1 0 1 2 3 4 5
Missing and Outlying Observations
Gaussian
Mixture
Original Cleaned
Model AICGauss -8741Cauchy -8655Mixture -8936
Detection of Coseismic Effects1981 1982
1983 1984
1985 1986
1987 1988
1989 1990
Strongly affected thecovariates such asbarometric airpressure, earth tideand rain
Difficult to find out Coseismic Effect
Pressure Effect0t
min110
min180
in
m
iin paP
0
n
n
P
p Air PressurePressure Effect
Extraction of Coseismic Effect
Component Models
nnnnn EPty
n
n
n
P
t
y
n
nE
ObservationTrendAir Pressure Effect
Earth Tide EffectObservation Noise
,
,
0
0
in
l
iin
in
m
iinnn
k
etbE
paPwt
State Space Representation
nnn
nnn
wHxyGvFxx
1
,1
0
0
0
0
1
,
1
1
1
1
1
,
11
0
0
nnmnn
m
n
n
etetppH
GF
b
b
a
a
t
x
AIC Values
5955459566593795783227
5956359575593865783626
5956959580593935783025
5952659536593745781524
5948859498593685781923
3210
m
Precipitation Effect
nin
k
iiin
k
iin vrdRcR
11
Original
Pressure, Earth-Tide removed
Extraction of Coseismic Effect
Component Models
y t P E Rn n n n n n
n
n
n
P
t
y
n
n
n
R
E
ObservationTrendAir Pressure Effect
Earth Tide EffectPrecipitation EffectObservation Noise
nin
k
iiin
k
iinin
l
iin
in
m
iinnn
k
vrdRcRetbE
paPwt
110
0
,
,
State Space Model
nnn
nnn
wHxyGvFxx
1
,0011
00
0000
000
00
01
,
1
11
11
1
1
,
2
121
0
0
1
1
nnmnn
k
k
m
kn
n
n
n
n
etetppH
d
dd
G
ccc
F
b
ba
aR
RRt
x
Groundwater Level Air Pressure Effect
Earth Tide Effect
Precipitation Effect
min AIC modelm=25, l=2, k=5
M=4.8, D=48km
618096
618105
618004
618033
617342
616751
AIC
k
Extraction of Coseismic Effects
Corrected Water Level
Detected Coseismic Effect
Original
T+P+ET+RM=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
Min AIC modelm=25, l=2, k=5
Original
Air PressureEffect
Earth TideEffect
P & ET Removed
PrecipitationEffect
P ， ET & RRemoved
Coseismic Effect1981 1982
1983 1984
1985 1986
1987 1988
1989 1990
M=7.0 D=375kmM=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
1983 1984
1985 1986
1987 1988
1989 1990
Effect of Earthquake
Earthquake Water level
Rain Water level
Distance
Mag
nitu
de
DM log45.2
Cos
eism
ic E
ffec
t
> 16cm> 4cm
> 1cm
1log45.2 DM
0log45.2 DM
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
Bottom
OBS
Sea Surface
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-C
h ）4 Channel Time Series
N=15360, 982 39 series
Hokkaido University + University of Bergen
Time-Adjusted (Shifted) Time Series
An Example of the Observations
OBS-4 N=7500 M=1560
OBS-31 N=15360 M=982
-10000
0
10000
20000
30000
40000
50000
1 501 1001 1501 2001 2501 3001 3501 4001 4501 5001 5501
High S/N Low S/N
Direct wave, Reflection, Refraction
Direct Wave Refraction Wave
Reflection Wave
Objectives
Estimation of Underground Structure
Detection of Reflection & Refraction Waves
Estimation of parameters （ hj , vj ）
Intermediate objectives
-30000
0
30000
1 501 1001 1501 2001 2501
Time series at hypocenter (D=0)
221100
1100
00
/2/2/1221)Wave(0
/2/11)Wave(0
/)Wave(0
Time Arrivalpath
vhvhvkh
vhvkh
vkh
k
k
k
,5,3,1k
Wave(0) Wave(000) Wave(00000)
Wave(011) Wave(00011)
Model for Decomposition
nnnn wsry Noisen Observatio
WaveReflection
eDirect Wav
n
n
n
w
s
r
n
m
jjnjn
nj
jnjn
vsbs
urar
1
1
),0(~),0(~),0(~
2
22
21
NwNvNu
n
n
n
, of Estimation 22
22
21
21 nn Self-Organizing Model
Decomposition of Ch-701 (D=4km)
Observed
- 15000
- 10000
- 5000
0
5000
10000
1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401
R eflection W ave
D irect W ave- 1 0000
- 5000
0
5000
1 0000
1 201 401 6 01 801 10 01 1201 1401 1 601 180 1 2001 2 201 24 01
- 1 0 0 0 0
- 5 0 0 0
0
5 0 0 0
1 0 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
R eflection W ave
D irect W ave- 1 0000
- 5000
0
5000
1 0000
1 201 401 6 01 801 10 01 1201 1401 1 601 180 1 2001 2 201 24 01
- 1 0 0 0 0
- 5 0 0 0
0
5 0 0 0
1 0 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
- 1 0000
- 5000
0
5000
1 0000
1 201 401 6 01 801 10 01 1201 1401 1 601 180 1 2001 2 201 24 01
- 1 0 0 0 0
- 5 0 0 0
0
5 0 0 0
1 0 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
0
2
4
6
8
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
- 4
- 2
0
2
4
6
8
1 2 01 40 1 601 801 10 01 120 1 1401 1601 1 801 20 01 22 01 240 1
T au 1
T au 2
0
2
4
6
8
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
- 4
- 2
0
2
4
6
8
1 2 01 40 1 601 801 10 01 120 1 1401 1601 1 801 20 01 22 01 240 1
0
2
4
6
8
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
- 4
- 2
0
2
4
6
8
1 2 01 40 1 601 801 10 01 120 1 1401 1601 1 801 20 01 22 01 240 1
T au 1
T au 2
- 4000
- 2000
0
2000
4000
1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401
Decomposition of Ch-721 (D=8km)Observed
R e f l e c t i o n W a v e
D i r e c t W a v e
- 4 0 0 0
- 2 0 0 0
0
2 0 0 0
4 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
- 4 0 0 0
- 2 0 0 0
0
2 0 0 0
4 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
R e f l e c t i o n W a v e
D i r e c t W a v e
- 4 0 0 0
- 2 0 0 0
0
2 0 0 0
4 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
- 4 0 0 0
- 2 0 0 0
0
2 0 0 0
4 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
- 4 0 0 0
- 2 0 0 0
0
2 0 0 0
4 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
- 4 0 0 0
- 2 0 0 0
0
2 0 0 0
4 0 0 0
1 2 0 1 4 0 1 6 0 1 8 0 1 1 0 0 1 1 2 0 1 1 4 0 1 1 6 0 1 1 8 0 1 2 0 0 1 2 2 0 1 2 4 0 1
0
40000
80000
120000
160000
200000
1 501 1001 1501 2001 2501
A Small Portion of Data
“Spatial” Filter/Smoother
njsnj time, channel :
1, jkns
, jns
2,2 jkns
,1 jns
1,1 jkns
2,12 jkns 2,12 jkns
1,1 jkns
,1 jns
k: Time-lag
Spatial Model (Ignoring time series structure)
njnjnj
njjknjknnj
wsy
vsss
2 2,21,
jnjnjn
jnjknjn
wHxy
GvFxx
,,,
,1,,
Series j-1 Series j : Time-lag=k
1,
,,
jkn
jnjn s
sx
Local Cross-Correlation Function
Time
Loc
atio
n
0 8
730
630
1,1,11,1,11,
,,1,,1,
1,1,11,1,11,
jknjnjnjnjkn
jknjnjnjnjkn
jknjnjnjnjkn
sssss
sssss
sssss
Spatial-Temporal Model
njsnj time, channel :
Model of Propagation Path
0h
0d1h
2h
D
1d
Parallel Structure
Width km,,, 3210 hhhh
Velocity km/sec,,, 3210 vvvv
Water
Examples of Wave PathWave(0) Wave(000) Wave(01)
Wave(011) Wave(0121) Wave(000121)
Wave(01221) Wave(012321) Wave(00012321)
Path Models and Arrival Times
31
3223
22
12
213
21
11
203
20
10
12021
22
122
11
1202
20
10
12021
22
122
11
1202
20
10
011
1201
20
10
011
1201
20
10
220
10
220
10
220
10
221)Wave(01232
)23(231)Wave(00012
)2(2Wave(0121)
)3(3Wave(0001)
)(Wave(01)
25)Wave(00000
9Wave(000)
Wave(0)
dvdhvdhvdhv
ddDvdhvdhv
ddDvdhvdhv
dDvdhv
dDvdhv
Dhv
Dhv
Dhv
231303322 22,/ dddDdvvhvd ijiiij
Path models and arrival times(OBS4)
Distance (km)
Arr
ival
Tim
e (s
ec.)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
1 11 21 31 41 51 61 71 81 91 101
W0W000W00000W0000000W01W011W0121W000121W01221W012321W00012321W0123321W0111221W0122221W01111W0001W00011W0001221
Local Time Lag
1 5 9 13 17 21 25 29 33 37S1
S5
S9
S13
S17
S21
S25
S29
30-4020-3010-200-10-10-0
-10 -8 -6 -4 -2 0 2 4 6 8 10
D: Distance (km)
8
7
6
5
4
3
2
1
0
Arr
ival
Tim
e (s
ec.)
Path Models and the Differences of the Arrival Times Between Adjacent Channels
3.73.73.71)Wave(01232
6.146.146.146.14 Wave(0121)
5.205.205.205.20 Wave(01)
4.332.305.237.142.0 )Wave(00000
9.336.329.282.214.0 Wave(000)
1.349.334.335.312.1 Wave(0)
50201050 ModelPath
(km)
Epicentral Distance
Model for Decomposition
jnjnjnjn wsry ,,,,
sjnjhnjn
rjnjknjn
uss
urr
j
j
,1,,
,1,,
X
00
X0
Wof timeArrival:)W(
Wof timeArrival:)W(
)W(),W(
Xj
j
jjjj
T
T
ThTk
snmnmnn
rnnnn
vsbsbs
vrarar
,11
,11
waveReflection
eDirect wav
,
,
jn
jn
s
r
Spatial-Temporal Model
)|(
),|( )|( ),|(
,11,
,11,,11,,1
jnjkn
jnnjjknjnnjjknjnnj ssp
ssspsspsssp
)|( ),|( ,1,,1,1, jnjknjnjnjkn sspsssp
)|(
)|( )|( ),|(
,11,
1,,11,,1
jnjkn
njjknjnnjjknjnnj ssp
sspsspsssp
Time-lag (Channel j-1 Channel j ) = k
Spatial-Temporal Filtering
filtering recursive similar to n"observatio" an as Consider 1, njjkn ys
)|(
)|( )|( ),|(
,11,
1,,11,,1
jnjkn
njjknjnnjjknjnnj ssp
sspsspsssp
Spatial-Temporal Decomposition
Reflection wave Direct wave
Mt. Usu Eruption Data Hokkaido, Japan March 31, 2000 13:07-
Volatility and component models
Hokkaido, Japan March 31, 2000 13:07-
Decomposition
- 50000
0
50000
1 1501 3001 4501 6001 7501 9001 10501 12001 13501 15001 16501 18001 19501 21001 22501 24001 25501 27001 28501
1系列
- 50000
0
50000
1 1501 3001 4501 6001 7501 9001 10501 12001 13501 15001 16501 18001 19501 21001 22501 24001 25501 27001 28501
1系列
- 50000
0
50000
1 1501 3001 4501 6001 7501 9001 10501 12001 13501 15001 16501 18001 19501 21001 22501 24001 25501 27001 28501
1系列
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
Time-varying Spectrum
AR model Autocovariance Spectrum
Time-varying Nonstationary
),0(~, 2
1
Nwwyay nnjn
m
jjnn
2
1
2
2
1)(
m
j
ijfjn
nn
eafp
Time-varying AR model
Time-varying spectrum
Estimation of Nonstationary AR Model
njn
m
jjnn wyay
1
),0(~, 2Nvva jnjnjnk
Tknmknmnnn
mnnk
mk
mk
aaaax
yyHH
IGGIFF
),,,,,,(
),,(
,
1,1,11
1)(
)()(
State Space RepresentationState Space Representation
Model for Time-changes of Coefficients
Kronecker product
State Space Representation
1)1()1()1( HGF
For k = 1
For k = 2
01 ,0
1 ,
01
12 )1()1()2(
HGF
Kronecker Product
BaBa
BaBa
bb
bb
aa
aa
BA
mmpqp
q
mm
1
111
1
111
1
111
State Space Representation
Case: k = 1
n
mn
n
mnnn
nm
n
nm
n
mn
n
wa
ayyy
v
v
a
a
a
a
1
1
,
,1
1,
1,11
,,
1
1
1
1
Time-varying Coefficients
Gauss model Cauchy model
Time-varying Spectrum
Precipitation Effect
nin
k
iiin
k
iin vrdRcR
11
Estimation of Arrival TimeP S
Estimation of Arrival Times
Estimation of Hypocenter Locally Stationary AR Model
Automatic Modeling by Information Criterion AIC
Automatic & Fast Algorithmsec. 01.0T
Prediction of Tsunami
Estimation of Arrival TimeLocally Stationary AR Model
y a y v v Nn ii
m
n i n n
1
20, ~ ( , )
y b y w w Nn ii
l
n i n n
1
20, ~ ( , )
Seismic Signal Model
Background Noise Seismic Signal
Background Noise Model
Estimation of Arrival Time
AIC of the Total Models
L
L1-
L2
L1
L0
AICAIC
AICAICAIC
R
R1-
R2
R1
R0
AICAIC
AICAICAIC
LjAIC R
jAIC
Rj
Ljj AICAICAIC
Min AIC Estimate
of Arrival Time
Model & Implementations
LSAR model: Ozaki and Tong (1976)
Householder implementation: Kitagawa and Akaike (1979)
Kalman filter implementation:
State Space Representation of AR Model
njn
m
jjn wyay
1
mn
n
n
a
a
x 1
n
mn
n
mnnn
mn
n
mn
n
w
a
a
yyy
a
a
a
a
1
1
1
111
],,[,
New data yn
nnm
nn
nn
a
a
x
|,
|,1
| )(
11
11
221
2
mnmnnn
nnn
yayay
nn
n
)1(2logAIC 2 mn nn
Lower Order Models
njn
m
j
mjn wyay
1
12221
21
)(1
)(1
mmmm
mm
jjm
mm
mjm
j
a
a
aaaa
Levinson recursion
Arrival Times of P-waves
- 40
- 20
0
20
40
1 51 101 151 201 251 301 351 401
9800
10300
10800
1 51 101 151 201 251 301 351 401
AIC
- 30
0
30
1 51 101 151 201 251 301 351 401
4000
5000
6000
7000
8000
9000
1 51 101 151 201 251 301 351 401
AIC
- 20
0
20
1 51 101 151 201 251 301 351 401
8000
9000
10000
1 51 101 151 201 251 301 351 401
AIC
2000
Arrival Times of S-waves
-30
-15
0
15
30
1 51 101 151 201 251 301 351 401
1940
1950
1960
1970
1980
1 51 101 151 201 251 301 351 401
AIC
- 60
- 30
0
30
60
1 51 101 151 201 251 301 351 401
2090
2110
2130
2150
2170
1 51 101 151 201 251 301 351 401
AIC
- 80
- 40
0
40
80
1 51 101 151 201 251 301 351 401
2350
2450
2550
1 51 101 151 201 251 301 351 401
AIC
100
Posterior Probabilities of Arrival TimesAIC: -2(Bias corrected log-likelihood)
Likelihood of the arrival time
Posterior probability of the arrival time
kAIC
2
1exp
kkkp AIC
2
1exp)()(
1940
1950
1960
1970
1980
1 51 101 151 201 251 301 351 401
0
0.05
0.1
1 51 101 151 201 251 301 351 401
4000
5000
6000
7000
8000
9000
1 51 101 151 201 251 301 351 401
0
0.25
0.5
1 51 101 151 201 251 301 351 401
4800
4850
4900
4950
5000
1 6 11 16 21
0
0.25
0.5
1 6 11 16 21
