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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 /
26
Outline
1 Introduction
2 Residual moveout analysis
3 Bayesian approach to residual moveout correction
4 Results of experiments on synthetic data
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 2 /
26
Outline
1 Introduction
2 Residual moveout analysis
3 Bayesian approach to residual moveout correction
4 Results of experiments on synthetic data
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 2 /
26
Outline
1 Introduction
2 Residual moveout analysis
3 Bayesian approach to residual moveout correction
4 Results of experiments on synthetic data
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 2 /
26
Outline
1 Introduction
2 Residual moveout analysis
3 Bayesian approach to residual moveout correction
4 Results of experiments on synthetic data
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 2 /
26
1 Introduction
Seismic imaging
Seismic experiment data subsets
Depth migration
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 3 /
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1 Introduction
Seismic imaging
Seismic experiment data subsets
Depth migration
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 4 /
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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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 5 /
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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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 5 /
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Seismic imaging
Figure: Seismic experiment
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 5 /
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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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 5 /
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1 Introduction
Seismic imaging
Seismic experiment data subsets
Depth migration
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 6 /
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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)
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 7 /
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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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 8 /
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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
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 9 /
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1 Introduction
Seismic imaging
Seismic experiment data subsets
Depth migration
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 10 /
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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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 11 /
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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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 11 /
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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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 11 /
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2 Residual moveout analysis
Residual moveout
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 12 /
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2 Residual moveout analysis
Residual moveout
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 13 /
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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
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 14 /
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What is residual moveout?
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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 14 /
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What is residual moveout?
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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 14 /
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3 Bayesian approach to residual moveout correction
Statistical model
Bayesian approach to residual moveout correction
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 15 /
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3 Bayesian approach to residual moveout correction
Statistical model
Bayesian approach to residual moveout correction
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 16 /
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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.
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 17 /
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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)
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 18 /
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3 Bayesian approach to residual moveout correction
Statistical model
Bayesian approach to residual moveout correction
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 19 /
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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))
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 20 /
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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 .
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 21 /
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4 Results of experiments on synthetic data
Synthetic data with constant velocity
Synthetic data with constant velocity and with AVO
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 22 /
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4 Results of experiments on synthetic data
Synthetic data with constant velocity
Synthetic data with constant velocity and with AVO
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 23 /
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Synthetic data with constant velocity
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 24 /
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4 Results of experiments on synthetic data
Synthetic data with constant velocity
Synthetic data with constant velocity and with AVO
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 25 /
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Synthetic data with constant velocity and with AVO
S. Dossou-Gbété & al. (LMAP) Uncertainty assesment in seismic imagingBCAM-012-11-05 26 /
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