Bayesian approach to uncertainty assessment in seismic · PDF fileBayesian approach to...

Bayesian approach to uncertainty assessment in

seismic imaging

S. Dossou-Gbété

(Joint work with L. Bordes, E. Landa, T. Johng'Ay)

Laboratoire de Mathématiques et de leurs Applications-Pau, UMR CNRS 5142Université de Pau et des Pays de l'Adour, Pau (France)

Workshop on Mathematics for IndustryCFD & Probabilistic Analysis

December 5th 2012 at BCAM, Bilbao (Spain)

S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 1 /

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Outline

1 Introduction

2 Residual moveout analysis

3 Bayesian approach to residual moveout correction

4 Results of experiments on synthetic data


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1 Introduction

Seismic imaging

Seismic experiment data subsets

Depth migration


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1 Introduction

Seismic imaging


Depth migration


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Seismic imaging

The main goal of seismic imaging is to provide accurate knowledge of

some target earth subsurface, as locations of layers and cracks inside

the earth. This knownlesge is used to determine geologic features

such as oil or gaz location.


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Seismic imaging

Main features of a seismic experiment undertaken for data acquistion

are as follows:

• Controlled source of acoustic energy is used to generate

acoustic waves that will travel inside a target subsurface of the

Earth;

• As the wave�eld propagates it is re�ected, an up-going

wave�eld, by subsurface heterogeneities,

• Amplitude and time of backscattered seismic energy are

recorded by receivers spread out along a linear or areal array at

the Earth surface.


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Seismic imaging

Figure: Seismic experiment


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Seismic imaging

Data parameters are:

• time t,

• sources location xs ,

• receivers location xr ,

• half o�set h = 12 ‖xs − xr‖

Data processing turns recorded signals into images of the geologic

structure of the target subsurface.


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1 Introduction

Seismic imaging


Depth migration


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Gathers: special seismic experiment data subsets

One feature of seismic data is redundancy.

Before data processing recorded signals are sorted and gathered

according to common value of an acquisition parameter

• shot location: common shot gather (CSG)

• half o�set (distance betwen shot location and receiver

location): common o�set gather (COG)


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Gathers: special seismic experiment data subsets ...

• midpoint between shot location an receiver location: common

midpoint gathers (CMP).

The

plot of amplitude versus arrival

times and o�set usually shows

that seismic re�ection data

exhibits space-time structures.

This is one of

the most striking visual feature

of seismic re�ection data.

This feature is characterized by a hyperbolic shape which is called

normal move out.


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Gathers: special seismic experiment data subsets ...

To create an

2D image (vertical section) of

the subsurface, the acquisition

is done along a linear

array of receiver locations.

The traces constituting

the 2D seismic data

are organized into a cube:

• one dimension of this cube

represents the time axis,

• a second dimension for the

CMP's positions

• a third dimension for the

O�set axis


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1 Introduction

Seismic imaging


Depth migration


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Depth migration

A underground location can be described using either two di�erent

coordinates systems:

1 depth coordinates system: an underground location is

described by a pair (x ,d) where x is the corresponding position

at the earth surface and d is depth below the position x ;

2 time coordinates system: an undergound location is described

by a pair (x ,t) such that a wave starting from that location hit

the surface of the earth at position x after a traveltime t.


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Depth migration

Depth migration aims to produce image of some target subsurface

of the earth where locations are described in depth coordinates using

data made up amplitudes and traveltime of signals recorded at the

surface of the earth.


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Depth migration

Achievement of this goal require an accurate knowledge of the seismic

velocity in depth coordinates is needed, but unfortunatly this is never

available and one should guess and try.


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Residual moveout


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What is residual moveout?

When depth migration is

applied with a correct

seismic velocity, one ob-

tains �at events on com-

mon depth point gathers.

Stacking is done to im-

prove the signal to noise

ratio and get a single im-

age


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In case where depth

migration is done with

a wrong velocity, struc-

tured event appear on

common depth point

gathers, characterized

by an hyperbolic shape

which is called residual

moveout.

This hyperbolic shape is

modeled using the equa-

tion zhj =√

z20 + (γ2 − 1) h2j where γ = vcis the residual move out

parameter, v the migration velocity, c the medium velocity , z0 the

depth of event at the zero-o�set and hj the jth half o�set.


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Residual moveout analysis is an important building block in the depth

migration work �ow.

Since the actual seismic velocity is not available, depth migration is

achieved through an iterative process where the analysis of residual

moveout is used at each step to update the velocity.


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Statistical model

Bayesian approach to residual moveout correction


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Statistical model



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Statistical model

Let A = (aij)j=1:Mi=1:N be a matrix whose elements are values gathered

in a CIG, i bieng a depth index and j an o�set index.

One assumes that:

• aij is a realisation on a random variable Aij such that the

conditionaly to γ A | γ ∼ N((

aγ (zi ,hj) ,σ2I))

• aγ (zi ,hj) = a0 (z0 (zi ,hj ,γ)) is the amplitude of the �stack�

trace along the event zi =√

z20 + (γ2 − 1) h2j at depth z0.

• γ is generate by a probability distribution Π.

Model parameters are: the functionals aγ (zi ,hj), γ and σ2.


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Marginal distribution of A

This is obtain by integrated the conditional distribution with

respect to the distribution Π

f (a) =1

(2πσ2)NM2

ˆexp

− 1

2σ2

N∑i=1

M∑j=1

(aij − aγ (zi ,hj))2

dΠ (γ)


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Statistical model



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Estimation of the functionals aγ (z ,hj)

Since we haven't the amplitude at any depth, we resort to kernel

regression method to estimate the amplitude at any depth.

Lets write aj (u) a functional estimation of the amplitude at depth

u and o�set of index j , then

aγ (z ,hj) =1

M

M∑j=1

a(√

z20 + (γ2 − 1) h2j

)= a0 (z0 (z ,hj ,γ))


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Bayesian estimation of γ

Since aγ (zi ,hj) are not known the posterior distribution of γ cannot

be computed.

Using the estimates aγ (zi ,hj) of aγ (zi ,hj), we resort to the

following estimation of the density of this distribution as

f (γ | a) =

1

(2πσ2)NM2

exp

− 12σ2

N∑i=1

M∑j=1

(aij − aγ (zi ,hj))2

Π (γ)

f (a)

Metropolis-Hastings algorithm is used to compute the various

descriptive parameters of the posterior distribution of γ (mean,

standard deviation, quantiles: : :) and so describe the uncertainty

on the parameter .


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Synthetic data with constant velocity

Synthetic data with constant velocity and with AVO


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