from genomics to geology: hidden markov models for seismic data analysis samuel brown february 5,...

From Genomics to Geology:From Genomics to Geology:Hidden Markov Models for Hidden Markov Models for

Seismic Data AnalysisSeismic Data Analysis

Samuel BrownSamuel Brown

February 5, 2009February 5, 2009

Objective

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• Create a pattern recognition tool that can locate,

isolate, and characterize events in seismic data.

• Adapt hidden Markov model (HMM) search

algorithms from genomics applications and extend

them to work on multi-dimensional seismic data.

Outline

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• HMM Applications

• HMM Theory

• Generating Sequences

• Simple Markov Models

• Hidden Markov Models

• From Sequence Generation to Pattern Recognition

• Application to Seismic Data

Application Areas

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HMMs can be used to build powerful, fast pattern

recognition tools for noisy, incomplete data.

• Medical Informatics/Genomics

• hmmer – Sean Eddy, Howard Hughes Medical

Institute

• Speech Recognition

• Intelligence

Prerequisites

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• We will describe our data as a sequence of symbols

from a predefined alphabet

• For DNA, the alphabet consists of nucleic acids:

{A, C, G, T}

• A sample DNA sequence: CGATATGCG

• We will reference the symbols in a sequence by

their position, starting with 1; ie symbol 1 in the

previous sequence is ‘C’

Sequence Generation

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• Goal: build a probabilistic state machine (model)

that can generate any DNA sequence and

characterize quantitatively the probability that the

model generates any given sequence.

• Enter the Markov model (chain)

Markov Models

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• Primary Characteristic: The probability of any

symbol in a sequence depends solely on the

probability of the previous symbol.

• tAC = P(xi = A | xi-1 = C)

Markov Model State Machine

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• Each state with straight edges is an emitting state.

• B and E are special non-emitting states for

beginning and ending sequences.


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•A transition probability is assigned to each arrow:

• tAC = P(xi = A | xi-1 = C)

• The probability of sequence x of length L is:

• P(x) = P(E | xL)P(xL | xL-1)…P(x1 | B)


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•Given the sequence, CGAGTC, and a table of

transition probabilities, we can trace a path through

the state machine to get the probability of the

sequence.

Markov Model Example

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•Assume all transitions are equiprobable, ie,

.25 = tAC = tAG = tAT = tCA= …

P(CGAGTC) = (tEC)(tCG)(tGA)(tAG)(tGT)(tTC)(tCB)

= .00097

CpG Islands

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•The dinucleotide subsequence CG is relatively rare

in the human genome and is usually associated with

the beginning of coding DNA regions

• CpG islands are subsequences in which the CG

pattern is common, and there are more C and G

nucleotides in general.

CpG Island Model

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•We can define a new Markov model for CpG

islands, in which the transition probabilities are

adjusted to reflect the higher frequency of C and G

nucleotides.

Combined Model

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•What we really want is a model that can emit

normal DNA sequences and CpG islands, with low

probability transitions between the two regions.

Hidden Markov Model

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•We call this type of Markov model ‘hidden’

because one cannot immediately determine which

state emitted a given symbol.

Outline

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• HMM Theory






HMM Search

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•We can calculate the path through the model which

generates a given sequence with the highest

probability using the Viterbi algorithm.

• This allows us to identify portions in our sequence

that have a high

probability of being

CpG islands.

HMM Generalizations

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•Remove direct link between states and symbols and

allow any state to emit any symbol with a defined

probability distribution.

• ek(b) = P(xi= b | πi = k)

• Sequence probability is now a joint probability of

transition and emission probabilities:

• P(x, π) = Π eπi(xi) tπi,πi+1

HMM Generalizations

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•Create HMM based on specific sequences, with

(M)atch, (I)nsert and (D)elete states.

• Add B self-transition to be able to skip symbols.

• Allow for feedback from E to B to link recognized

sequence portions together.

Outline

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• HMM Theory






Searching a Trace

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•When searching a trace, what is our alphabet?

• Trace amplitudes + noise, which is assumed to be

normally distributed.

• What are our models?

• Library of scaled wavelets – one set of (M)atch,

(I)nsert, and (D)elete states for each sample.

Searching a Trace

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• Emitting states emit a trace sample with a high

probability when the trace sample amplitude is

within one standard deviation of the scaled model

sample amplitude.

• HMM search program will return a list of wavelet

types, central times, and amplitudes.

Searching a Trace

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•HMM search correctly identified all wavelet components in the

left trace, allowing us to synthesize the spiked trace.

Central Time Frequency Coefficient

.23s 18hz -1

.25s 20hz .5

.48s 20hz -.4

.498s 18hz 1.5

Searching a Trace

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What about noise?

Searching Across Traces

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•Taking the output wavelets from a 1D HMM, what

is the alphabet when we search across traces?

• Moveouts.

• What are our models?

• Library of target trajectories.

First Arrival 1D HMM

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To Be Continued…

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from genomics to geology: hidden markov models for seismic data analysis samuel brown february 5,...

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