Geophysical Unsupervised Signal Processing...SPSC - Geophysical Unsupervised Signal Processing Reﬂectivity Function → Earth response to an ideal impulsive, seismic source Fuchs

SPSC - Geophysical Unsupervised Signal Processing

Geophysical Unsupervised Signal Processing

Fuchs Anna & Pessentheiner Hannes

Signal Processing and Speech Communication Laboratory

Advanced Signal Processing 2

Fuchs Anna & Pessentheiner Hannes Advanced Signal Processing 2 page 1/50


Outline

IntroductionModel Description and AssumptionsSummary

Approaches for Unsupervised ProcessingWiener Filter and Unsupervised ProcessingPredictive DeconvolutionBlind Source SeparationICA-Based Seismic DeconvolutionResultsSummary

Degenerate Unmixing Estimation Technique – DUET



Reflection Seismic



Reflectivity Function

→ Earth response to an ideal impulsive, seismic source



How?

Information on the subsurface of an area under analysis

Produce seismic waves (dynamite, air-guns,...)

Reflection on subsurface

Sensor grids located in the surface measure reflections

Why?

Exploration and monitoring of hydrocarbon reservoirs

Assessing sites for CO2 sequestration

Nuclear waste deposition

Information about structure of earth



Model DescriptionUnknown

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.

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.

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.

....

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.

Source Signals Mixing Matrix AObservedSignals

s1s1

s2s2

s3

s3

sm

x1

x2

xm

a11

a12

a1m

a21

a22

a2m

an1

an2

amm

. . .

. . .

. . .

Different Goals

Finding s1(n) – Reflectivity Function

Finding A – Wavelets

Separation of si(n)



Problems:

Seismic source and subsurface propagation not ideal

Output is mixture of different waves which must be identifiedand separated

Hypothesis:

Linear distortion and mixtures

Convolution relationship – source signal and propagationenvironment are not ideal

Mixing – linear combination



Summary

Geophysical applications include Estimation of different sources Estimation of reflectivity function

Problems due to the not ideal environment conditions

Prior assumption of model to be able to use known algorithms



Approaches for Unsupervised Processing [1]

Wiener Filter and Unsupervised Processing

Geophysical SP - prominent role to develop unsupervisedmethods

Wieners theory to seismology (Enders A. Robinson 1954 [2])

SOS HOS

Blind Deconvolution

Unsupervised Processing

Blind Source Separation



Question

Additional information to the measured data about inputsources and the convolution/mixing system

If one of them is known – supervised methods

Assumption

No additional information available



Source Signature – known

Deterministic supervised deconvolution in seismology

e.g. estimate reflectivity function through linear filtering,using the Wiener-Levinson minimization of the MSE

Source Signature – unknown

Unsupervised task

Exploiting prior knowledge about structure of source signatureand the reflectivity



Task

Get reflectivity function from measured data usingdeconvolution

Predictive Deconvolution

Wiener theory on prediction and filtering could be used forpredictive, unsupervised deconvolution [2]

Two Hypothesis

1. The seismic wavelet is the impulse response of an all-pole,minimum phase system.

2. The impulse response of the layered earth model behaves likea decorrelated (white) signal, so that it has a flat frequencyspectrum.



Predictive Deconvolution

Deconvolution using prediction-error filter uses only SOS –simplification of task BUT

Prediction-error filter acts as a whitening filter – ideallyrecovers uncorrelated signals

Problem if seismic wavelet cannot be modeled as an all-pole,minimum phase filter



Limitation of linear predictive deconvolution

Next step – from SOS to HOS







Unknown

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.

....

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.

.

Source Signals Mixing Matrix AObservedSignals

s1s1

s2s2

s3

s3

sm

x1

x2

xm

a11

a12

a1m

a21

a22

a2m

an1

an2

amm

. . .

. . .

. . .

x(n) = [x1(n) x2(n) ... xM (n)]T

s(n) = [s1(n) s2(n) ... sN (n)]T

x = A · s → s = W · x with W = A−1



Goal: Estimate all source signals

W can recover sources if mixing system is linear, memory-lessand time-invariant

SOS: W is adjusted so as to decorrelate the recovered signalss (using PCA)

HOS: W is adjusted so that recovered signals becomemutually independent



Blind Source Separation using SOS and HOS



ICA-Based seismic Deconvolution

ICA – Independent Component Analysis

Computational model for separating multiple sources

Produces additive subcomponents

Assumption:

- Mutual statistical independence- Source signals are non-Gaussian and i.i.d (– satisfied byreflectivity)



Modelx = As

x(0)x(1)

.

.

.x(N − 1)

︸︷︷︸

Trace

= A

ρ(0)ρ(1)

.

.

.ρ(N − 1)

︸︷︷︸

Reflectivity

A =

h(0) 0 . . . 0 0 . . . 0h(1) h(0) . . . 0 0 . . . 0

.

.

. h(1)...

.

.

.

.

.

.... 0

h(Nh − 1)

.

.

.... h(0) 0

... 0

.

.

. h(Nh − 1)... h(1) h(0)

...

.

.

.

0

.

.

....

.

.

.

.

.

.... 0

0 0 . . . h(Nh − 1) h(Nh − 2) . . . h(0)

→ Single Snapshot → delayed versions of x(n) and ρ(n) – N

snapshots



Step 1 – Rearrangement

Obtaining of M < N mixtures

Improvement of statistical properties of the mixture matrix

X =

0 0 . . . 0 x(0) . . . x(N −M)...

.... . .

. . ....

. . ....

0 x(0) . . . . . . x(M − 2) . . . x(N − 2)x(0) x(1) . . . . . . x(M − 1) . . . x(N − 1)



Step 2 – Data Whitening

Decorrelation of the signals

Equalize power

z(n) = WSOSx(n)

WSOS obtained with SOS

Step 3 – Dimension Reduction

Linear transformation – new set of Nh mixtures

x(n) = NTkW

TSOSz(n)



Step 4 – ICA

s(n) = WHOSx(n)

WHOS = QWSOS with Q given by ICA

Step 5 – Wavelet and Reflectivity Estimation

si = hTi x(n)

Optimal Pair (s∗(n),h∗) gives original reflectivity and wavelet

hi from WHOS



Results I – Synthetic Data

(a) Noiseless synthetic trace; (b) Synthetic reflectivity function; (c)Linear prediction error filter; (d) Estimated reflectivity based onestimated wavelet



Results II

(a) 35-points Berlage wavelet; (b) estimated wavelet by B-ICA; (c)zero-pole plot for the original wavelet; (d) zero-pole plot forestimated wavelet



Summary

Differences between supervised and unsupervised processing

Predictive deconvolution to estimate reflectivity function

Limitations of predictive deconvolution – go to HOS

Blind source separation to estimate sources

Combination of deconvolution and BSS – ICA-Based seismicDeconvolution



Volcano Hazards

many villages close to volcanos

predict eruptions, tremors, earthquakes to avoid casualties

Problem:lack of prediction knowledge

- number of sources?- types of sources?

Solution:use blind source separationfor data-mining

Landscape after ashfall.

How to improve knowledge of volcanic behaviour?



DUET: Degenerate Unmixing Estimation Technique

Features

for blind separation of multiple sources

- Volcano Tectonic (VT) events- Long Period (LP) events

based on pair of seismograph sensors (stations)

second sensor enables use of a time-frequency–lattice (τ, ω)

stations separated by less than half a wavelength of interest

performs best if there’s no signal overlap between sources

robust behaviour in echoic mixtures



DUET: Degenerate Unmixing Estimation Technique

Sensor 2

Sensor 1

δ

Seismic sources & absorbing sensors.

Are there any assumptions?Fuchs Anna & Pessentheiner Hannes Advanced Signal Processing 2 page 29/50


Assumptions

Anechoic Mixing Model

W-Disjoint Orthogonality

Sensor 2

Sensor 1

Anechoic Mixing Model.

Source 1

Source 2

t

f

1 Hz

5 Hz

W-Disjoint Orthogonality.



Assumptions: Anechoic Mixing Model

no realistic model

direct paths only (non-echoic signals)

stations provide attenuation & delay parameters of waves

Sensor 2

Sensor 1

Anechoic (Ideal) Mixing Model.

Sensor 2

Sensor 1

Echoic (Realistic) Mixing Model.



Assumptions: Anechoic Mixing Model cont’d

model

xk(t) :=

N∑

j=1

ak,j · sj(t− δk,j) k = 1, 2

where

xk(t) . . . mixture of k-th sensor

sj(t) . . . source signal

ak,j . . . attenuation coefficient

δk,j . . . time delay

k . . . sensor index

j . . . source index



Assumptions: W-Disjoint Orthogonality (WDO)

disjoint supports of signals’ windowed Fourier transforms (F)

every (τ, ω)-point in a mixture xj(t) dominated bycontribution of at most one source

Ω1 ∩ Ω2 = 0

Ω1

Ω2

Ω

Disjoint sets Ω1 & Ω2.

t

f

S1

! !"

S2

S1

S2 S1

S2

frame-size

WDO and non-WDO signals.



Assumptions: W-Disjoint Orthogonality (WDO) cont’d

given window function w(t) & source signals sj(t), sk(t)

given windowed F of sj(τ, ω), sk(τ, ω)

sj(τ, ω) :=1√2π

∫ +∞

−∞

w(t− τ)sj(t)e−iwtdt

where ω is the angular frequency and τ is the window shift

WDO if supports of sj(τ, ω), sk(τ, ω) are disjoint

sj(τ, ω) · sk(τ, ω) = 0 ∀τ, ω, ∀j 6= k



Assumptions: W-Disjoint Orthogonality (WDO) cont’d

if w(t) = 1- simply F of whole mixture- signals have to be frequency-disjoint to satisfy WDO-condition(frequency-division multiplexed signals)

if w(t) = δ(t)- signals have to be time-disjoint to satisfy WDO-condition(time-division multiplexed signals)

t

f

S2

S1 S1 S1

S2 S2

Frequency-Division Multiplexed Signals.

t

f

S2S1 S1

Time-Division Multiplexed Signals.

How does the algorithm work?



Signal Representation

mixture signals can be decomposed into weighted & shiftedsource signals in time & frequency domain

xk(t) :=

N∑

j=1

ak,j · sj(t− δk,j)

xk(τ, ω) :=

N∑

j=1

ak,j · sj(τ, ω) · e−iωδk,j

δ1

δ2

Sources, sensors, and delays.



Signal Representation cont’d

without loss of generality

a1,j = 1 δ1,j = 0 a2,j = aj δ2,j = δj

interested in the differences of both captured signals

vectorized mixture signals (k = 1, 2)

[x1(τ, ω)x2(τ, ω)

]

︸︷︷︸mixture signals

=

[1 . . . 1

a1e−iωδ1 . . . aNe−iωδN

]

︸︷︷︸relative propagation matrix

s1(τ, ω). . .

sN (τ, ω)

︸︷︷︸source signals

How to decompose the mixture signals?



Binary Mask

given mixture signal xk(t)

given F of xk(τ, ω)

xk(τ, ω) :=1√2π

∫ +∞

−∞

N∑

j=1

ak,j · sj(t− δk,j)e−iwtdt

xk(τ, ω) := a1 · s1(τ, ω) · e−iωδ1

+ a2 · s2(τ, ω) · e−iωδ2

+ . . .

+ aN · sN (τ, ω) · e−iωδN



Binary Mask cont’d

source separation in case of WDO using an ideal binary mask

sj(τ, ω) := Mj(τ, ω) · xk(τ, ω), ∀τ, ω

where

Mj(τ, ω) :=

1 sj(τ, ω) 6= 0

0 otherwise

and

Mj(τ, ω) ·Mk(τ, ω) = 0 ∀τ, ω ∀j 6= k



Binary Mask cont’d

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

2,200

Samples

Frequency (Hz) Binary Mask Source 2

1

2

3

Binary mask of a mixture signal for source 2.

How do we obtain coefficients aj & δj in case of WDO-signals?



Parameter Estimation

WDO- & Anechoic Source Signals

ratio of (τ, ω)-representations of mixtures depends on activesource component mixing parameters only

R(τ, ω) :=x2(τ, ω)

x1(τ, ω)

assume set of (τ, ω) on Ωj := (τ, ω) : Mj(τ, ω) = 1

R(τ, ω) := aj · e−iωδj

aj := |R(τ, ω)| δj := − 1

ωargR(τ, ω)

Do we have ideal conditions in real world?



Parameter Estimation cont’d

Approximated WDO- & Echoic Source Signals

in field scenarios perfect WDO never given

high degree of WDO approximation per frame if

- small overlap of source spectra in frequency- small overlap of source spectra in time

increase degree of WDO approximation by

- changing sensor position- changing window type/size (ao)

In case of non-WDO, why is the DUET still suitable?

Volcano Tectonic and Long Period events ...

- hardly occur simultaneously and, thus,- exhibit high degree of WDO approximation



Parameter Estimation cont’d

accurate estimation of true coefficients required

α = aj −1

aj

αj =

∑(τ,ω)∈Ωj

|x1(τ, ω) · x2(τ, ω)|2 · α∑(τ,ω)∈Ωj

|x1(τ, ω) · x2(τ, ω)|2

δj =

∑(τ,ω)∈Ωj

|x1(τ, ω) · x2(τ, ω)|2 · δ∑(τ,ω)∈Ωj

|x1(τ, ω) · x2(τ, ω)|2

How do we obtain mask Mj to separate source signals?



Parameter Estimation & Demixing cont’d

1. construct (τ, ω)-representations x1(τ, ω) & x2(τ, ω)

2. label each non-zero (τ, ω)-point with computed pair

(α , δ)

3. weight each pair with energy of corresponding (τ, ω)-point

4. generate histogram of these labels

5. find peaks in histogram for each source

6. construct masks Mj to determine Ωj for each source

7. apply masks to mixture

sj(τ, ω) = Mj(τ, ω) · x1(τ, ω)

8. convert each sj(τ, ω) back into time domain



Results (with both parameters α and δ)

(α, δ)-lattice for one (top left) and six (top right) sources. Original, mixed, andseparated signals (bottom).



Results (with single parameter δ)

0 1,000 2,000 3,000 4,000 5,000 6,000−0.02

−0.01

0

0.01

0.02

Samples

0 1,000 2,000 3,000 4,000 5,000 6,000−0.02

−0.01

0

0.01

0.02

Samples

(a)

(b)

0 1,000 2,000 3,000 4,000 5,000 6,000−0.01

−0.005

0

0.005

0.01

Samples

0 1,000 2,000 3,000 4,000 5,000 6,000−0.01

−0.005

0

0.005

0.01

Samples(a)

Mixtures of Long Period (LP) and Volcano Tectonic (VT) events at bothsensors (top/bottom left). Band-pass filtered mixtures (top/bottom right).



Results (with single parameter δ) cont’d

0 1,000 2,000 3,000 4,000 5,000 6,000−0.01

−0.005

0

0.005

0.01

Samples

0 1,000 2,000 3,000 4,000 5,000 6,000−0.02

−0.01

0

0.01

0.02

Samples

−30 −20 −10 0 10 20 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Delay in Samples

Energy

SamplesFrequency (Hz)

0 1,000 2,000 3,000 4,000 5,000 6,000

2

4

6

8

Samples

Frequency (Hz)

0 1,000 2,000 3,000 4,000 5,000 6,000

2

4

6

8

Amplitude (dB)

−5

−4

−3

−2

−1

Amplitude (dB)

−4

−3

−2

−1

(b)

Band-pass filtered mixtures (top left) and corresponding delay histogram of VTand LP events (top right). Separated signals of VT (bottom top) and LP

(bottom bottom).



Conclusion

DUET applied to geophysical data

for blind source separation of VT and LP events

results show that DUET ...

- is a powerful tool- for identifying, separating, and locating VT and LP events- to better understand the source mechanism behind tremors,earthquakes, and eruptions caused by volcanic activities



References

Andre K. Takahata and Everton Z. Nadalin and Rafael Ferrari and Leonardo Tomazeli Duarte and Ricardo

Suyama and Renato R. Lopes and Joao Marcos Travassos Romano and Martin Tygel,“Unsupervised Processing of Geophysical Signals: A Review of Some Key Aspects of Blind Deconvolutionand Blind Source Separation,”IEEE Signal Process. Mag., vol. 29, pp. 27–35, 2012.

E. A. Robinson,

“Predictive decomposition of time series with applications to seismic exploration”,Ph.D. dissertation,Dept. Geol. Geophys., MIT, Cambridge, 1954.

Yilmaz, O., Rickard, S.,

“Blind Separation of Speech Mixtures via Time-Frequency Masking”,IEEE Transactions on Signal Processing, Vol. 52, No. 7, July, 2004.

Moni, A., Bean, C. J., Lokmer, I., Rickard, S.,

“Source Separation on Seismic Data: Application in a geophysical setting”,IEEE Signal Processing Magazine, May, 2012.

Moni, A., Bean, C. J., Lokmer, I., Rickard, S.,

“Seismic Signal Source Separation”,ISSC 2011, Trinity College Dublin, June 2011.


Documents

Geophysical Unsupervised Signal Processing...SPSC - Geophysical Unsupervised Signal Processing Reﬂectivity Function → Earth response to an ideal impulsive, seismic source Fuchs