Transcript Document

Topological Derivatives and other Embeddings
for
Ocean Floor Tsunami data
Prabhakar G. Vaidya, Nithin Nagaraj, Sajini Anand P S
Mathematical Modelling Unit
National Institute of Advanced Studies
IISc. Campus, Bangalore-12
(research partly supported by ISRO and DST)
International Conference on Non-linear Waves and Tsunami
March 6-10, Kolkatta
Role of Embedding in non-linear signal processing
1. Topological embedding
2. Modelling
3. Detection
4. Noise Removal
5. Prediction
What is Embedding?
…
RD
no inverse
Object of
Dimension
D
P
Injection
Q’
Q
RN
…
Topologically
conjugate
Object
Takens Embedding
Scalar variables measured at periodic interval
Y( t)
Y( t  h)
 X( t) 
 X( t  h) 
 X( t  2h) 


 X( t  h) 
 X( t  2h) 
 X( t  3h) 


Linear Systems:
For D dimensional dynamics, almost always
embedding dimension N = D, where N
is the dimension of the data vector
Non-linear systems:
Takens theorem(1981): N>2D
Topological Conjugacy
1. Embedding ensures that there is a continuous smooth map
from embedded variables to the original variables.
2. All the bifurcations and other properties are mimicked in the
embedded picture
Derivative embedding
Y1
X
Y2
d
X
dt
Y3
d2
2
X
dt
1. Close to physical Intuition
2. Prone to serious errors due to noise
Topological Derivatives
Topological derivatives are the same as exact derivatives for
i) No Noise
ii) Low dimensional dynamics
iii) Very high sampling rate
For noisy data:
• Noise is reduced
• For low sampling rate, it is diffeomorphic with actual
derivatives.
• Yields low dimensional or the best low dimensional
dynamics.
Ref: P. G. Vaidya, Monitoring and speeding up chaotic synchronization, Chaos,
Solitons and Fractals, Volume 17, Number 2, July 2003, pp. 433-439(7)
De-tiding of Tide-gauge data
High-Pass filter output for de-tiding (cut-off freq. =
0.314 Hz)
Tide gauge data
Power spectrum of data
Low-Pass filter output for de-spiking (cut-off
freq. = 0.628 Hz)
Period=24 hours
(Earth rotation)
Period=12 hours
(Moon revolution)
 De-tiding involves calculating the period contributed by the Earth’s rotation
and the Moon’s revolution which results in tides and removing them from signal.
We have done this in the Fourier domain using a High-Pass filter.
*Source: West Coast/Alaska Tsunami Warning Center: http://wcatwc.arh.noaa.gov/IndianOSite/IndianO12-26-04.htm
Ocean Floor Data Analysis
Bottom Pressure Recorded Data*
(sampling rate: 56.25 secs)
High pass filter output
(cut-off frequency is
lower 6% of the
spectrum )
*Source: Data obtained from National geophysical data center (http://www.ngdc.noaa.gov/seg/hazard/tsu.shtml)
Topological Derivative Embedding
Evolution of the signal
*Source: Data obtained from National geophysical data center (http://www.ngdc.noaa.gov/seg/hazard/tsu.shtml)
Time evolution of the position and velocity vectors
( H X H Y H T)
( H X H Y H T)
Disscussion and conclusions
1. Embedding started as a way to get Pictures from data
(Farmer ,Packard 1980)
2. We have shown how from a single variable using
topological derivatives and other methods we can get state
space plots
3. These have a potential for helping detecting Tsunami’s in
deepwater even in the presence of large noise