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The Methodology of the MaximumLikelihood Approach

Estimation, detection, and exploration of seismicevents

Anna Maly and Patrick Pircher

SPSCTU Graz

21.01.2013

Introduction Data Model Parameter Estimation Signal Detection Calculation of Test Treshold tm Summary

Inhalt

1 Introduction

2 Data Model

3 Parameter Estimation

4 Signal Detection

5 Calculation of Test Treshold tm

6 Summary

Anna Maly and Patrick Pircher TU Graz The Methodology of the Maximum Likelihood Approach


Introduction

Seismic Signals are transient (duration shorterthan observation)

Explosions and Earthquakes give rise to anumber of different waves. (e.g. pressurewaves, shear waves, or surface waves)

Waves follow different paths.

At an array of sensors one can observe theso-called phases of the different types of waves.



Goal

Wave Parameter Estimation: Find the mostlikely parameters for the incoming data.(Parameters: Slowness, Velocity, Azimuth,Elevation)

Compute Test Statistic: Probability of beinga signal.

Signal Detection: Comparing the teststatistic with a threshold.



Maximum Likelihood Estimation

The MLE for a scalar parameter is defined tobe the value of ϑ that maximizes p(x;ϑ) for xfixed.

The maximization produces a ϑ that is afunction of x.

Log-Likelihood:

l(ϑ|A) = lnL(ϑ|A) (1)

Excellent statistical performance androbustness.



Data Model

Plane Wave Model

Array outputs sampled and STFT

Nonlinear regression model:

Xl(ω) = H(ω, ϑ)Sl(ω) + Ul(ω) (2)



Xl(ω) = H(ω, ϑ)Sl(ω) + Ul(ω) (3)

Sl(ω) . . . Fourier-transformed signal vectorUl(ω) . . . Noise vectorH(ω, ϑ) = [d1(ω) . . .dM(ω)] . . . Transfer functiondi(ω) . . . Steering vectorϑ = [ξ1, . . . , ξM ] . . . Nonlinear wave parametersξi . . . Slowness Vector



The little lMulti-Taping

l . . . from STFT:

Xl(ω) =1√T

T−1∑t=0

wl(t)x(t)e−jωt (4)

l = 1,. . . ,L, where wl(t)s are orthonormal windowfunctions. We use the multi taping technique.x(t) is divided into K snapshots of duration T each.These are Fourier transformed using L orthogonalwindows depending on snapshot duration T and theselected analysis bandwidth.



Parameter Estimation

Properties: normal distribution, independency

Minimize U:

1

L

L∑l=1

J∑j=1

|Xl(ωj))−H(ωj , ϑ)Sl(ωj)|2 (5)



Leads to the broadband log-likelihood function:

L(ϑ,S, ν) = −LL∑

l=1

J∑j=1

[Nlog(ν(ωj)) +1

ν(ωj)

(Xl(ωj))−H(ωj , ϑ)Sl(ωj))H

(Xl(ωj)−H(ωj , ϑ)Sl(ωj))]

ν(ωj). . . Noise level



Signal and noise parameters are separable

Dependence on Sl(ωj) and ν(ωj) can beremoved ⇒ Replacing unknown signal andnoise parameters by their ML estimates at fixedand unknown wave parameters

L(ϑ) = −J∑

j=1

logtr [(I− P(ωj , ϑ))Cx(ωj)] (6)

P(ωj , ϑ) . . . Projection Matrix onto the columnspace of the transfer matrix H(ωj , ϑ)

Cx(ωj) = 1L

∑Lj=1X

l(ωj)Xl(ωj)

H . . . Sample spectralmatrix



Signal DetectionNeyman-Pearson Theorem

N(µ,σ2) . . . Gaussian PDF

Example: Two hypotheseses:H0 : µ = 0. . . Null HypothesisH1 : µ = 1. . . Alternative Hypothesis

Determine if µ = 0 or µ = 1 based on a singleobservation x [0].



It’s all about the threshold.

−4 −3 −2 −1 0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

PDFs

x[0]



Type 1 Error:Decide H1, but H0 is true. ⇒ P(H1;H0)

Type 2 Error:Decide H0, but H1 is true.⇒ P(H0;H1)

Probability of Detection:PD = P(H1;H1)

Probability of False Alarm:PFA = P(H1;H0) = α



Neyman-Pearson Theorem / Likelihood Ratio

TestTo maximize PD for a given PFA = α decide H1 if

L(x) =p(x;H1)

p(x;H0)> γ (7)

where the threshold γ is found from

PFA =

∫{x :L(x)>γ}

p(x;H0)dx = α (8)



In our case: Testing the null hypothesis Hm againstalternative hypothesis Am. (m = 1 . . .Mmax →number of signals/sources emphasizes matrixdimension and number of parameters associatedwith the assumed model)For m = 1,

H1: Data contains only noise.A1: Data contains at least one signal.

For m = 2, . . . ,Mmax ,Hm: Data contains at most (m − 1)

signals.Am: Data contains at least m signals.



Starting from H1, the test decides if a signal ispresent.

No Signal → Procedure stops

Signal detected → Procedure goes to next step

Continues until Mmax is reached



p(x;Am)

p(x;Hm)⇒ Tm = supL(ϑA)− supL(ϑH) (9)

Tm ≥ tm ⇒ Signal detected, reject Hm,Tm < tm ⇒ retain Hm

Tm . . . Central F-distributed for Null HypothesisFrequencies independent, so sum over Tm possible



Calculation of Test Threshold

The threshold tm is chosen to keep aprespecified false alarm level at αm.

Fm(·) . . . Null distribution of Tm

tm = F−1m (αm)

Broadband case → No closed-form expression



Broadband case: Several ways to determine tm:

Normal Approximation

Cornish Fisher Expansion

Sequentially rejectiveBonferroni-Holmprocedure

False discovery rate (FDR) through theBenjamini-Hochberg procedure



Normal Approximation:

tm ≈ µm +σm√J

Φ−1(αm) (10)

Φ−1(αm). . . inverse of standard normal distributionat αm

Calculation of mean and variance can be found in“Hypothesis testing for geoacoustic environmentalmodels using likelihood ratio,” from C. F.Mecklenbrauker, P. Gerstoft, J. F. Bohme, and P.-J.Chung.



SummaryINPUT: Fourier transformed data, maximal numberof signals Mmax , initial value for estimated numberof signals M = 0FOR m = 1, . . . ,Mmax

wave parameter estimation

compute test statistic

signal detection

ENDOUTPUT: estimated wave parameters and detectednumber of signals.


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