HOW TO HANDLE CORRELATED NOISE IN SEISMIC
INVERSION ?
Franc¸ois Renard and Patrick Lailly
´
Institut Franc¸ais du Petrole
KIM consortium project
Annual meeting, Rueil-Malmaison
, 2000
December
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Introduction
Earth model that “best” fits the data
Seismic inverse problems
Minimization of a least squares function
In general, use of Euclidian/
norms
Lack of realism when the seismic data involve important
correlations
Objective : a first step towards a solution
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Outline
How to solve an inverse problem ?
Bayesian approach
determinist approach
Waveform inversion
Reflection traveltime tomography
Conclusion
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Bayesian approach (1)
Formulation of the inverse problem in terms of states of information
We have at our disposal a state of information a priori
on the data
on the unknowns
on the theory
What is the state of information a posteriori on the unknowns ?
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Bayesian approach (2)
States of information
probability density functions (pdf)
Use of Gaussian pdf
good approximation of the behaviour of many phenomena in
nature
the associated mathematics are tractable
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Bayesian approach (3)
Dimension N : x=vector
is the mean of the distribution
is the covariance matrix
specifies the uncertainties (diagonal elements)
specifies the correlations between these uncertainties
(off-diagonal elements)
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Bayesian approach (4)
The Bayesian solution of the inverse problem is the maximum of the
a posteriori pdf :
, where
are the a priori pdf on the data and on the model
and
parameters
Important assumption : independence between these two pieces
of information
Under the Gaussian pdf assumption
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Bayesian approach (5)
Solution
function
of the inverse problem is the minimum of the objective
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is Gaussian
is the solution of a linear system
Linear case :
generalized least squares criterion
Determinist approach (1)
Minimization of a cost function
in model space
chosen so as to provide a “relevant” quantification of the
deviations from the a priori model
requires an a priori knowledge of the confidence we can put
on the a priori model
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in data space
chosen so as to provide a “relevant” quantification of the
misfits
requires an a priori knowledge of the main features of the
noise
Determinist approach (2)
These norms must reflect the confidence we have in the seismic
data and in the a priori model, respectively
acceptable trade off between contradictory
If properly chosen
terms
The pieces of information on the data and on the a priori model
may be dependent
Bayesian approach
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Waveform inversion
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Waveform inversion - classical formulation (1)
Inversion of seismograms
Earth model (impedance,...)
minimizing
is the linear/non linear forward modeling operator
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In general, use of a
norm in the data space
Waveform inversion - classical formulation (2)
! Seismograms often corrupted by (high amplitude) correlated
noises
surface waves
multiples
tube waves (for VSP records)
Inversion of signal + correlated noise
unsatisfactory results of inversion
norm is not suited for such applications
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Waveform inversion - classical formulation (3)
“Classical” solution
filter the noise out
inversion using a
norm
! There is no perfect filter
An inversion approach with a dedicated (semi-) norm in the data
space is likely to take the most of the information redundancy
correlated noise must not have any effect on the cost function
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The VSP 1D inverse problem (1)
Given VSP records
observed at receivers , find the
and the source function
acoustic impedance distribution
that minimize (Mace´ and Lailly, 1986)
depth of the first receiver)
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(
solution of the system
The VSP 1D inverse problem (2)
Experiment on synthetic data
Use of an actual well log
impedance
12000
11000
10000
9000
8000
7000
6000
5000
4000
3000
0
200
400
600
depth (ms)
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The VSP 1D inverse problem (3)
Modeling of the associated VSP
Corruption by a high amplitude correlated noise
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The VSP 1D inverse problem (4)
impedance
x 10
5
2
1.5
1
0.5
0
100
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120
140
160
180
200
220
240
260
depth (ms)
Inversion of the noise-corrupted seismogram with the
norm
The VSP 1D inverse problem (5)
New (semi-) norm adapted to the correlated noise
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b
: correlation vector
The VSP 1D inverse problem (6)
Inversion of the noise-corrupted seismogram with the new (semi-)
norm
impedance
11000
10000
9000
8000
7000
6000
5000
4000
3000
100
120
140
160
180
200
220
240
260
280
depth (ms)
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The VSP 1D inverse problem (7)
D
✕
d
obs
✕
✕ ✕ f(m exact)
f(m2)
f(m1)
f(M)
m1 : Solution with the
new formulation
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m2 : Solution with the
classical formulation
The VSP 1D inverse problem (8)
Conclusions
New formulation of waveform inversion in case of seismograms
corrupted by correlated noise
Demonstration of the efficiency of this new approach on a simple
synthetic example
Future work
case of multiple correlation directions
case of dispersive noise
real data
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Reflection Traveltime Tomography (RTT)
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Reflection tomography - classical formulation (1)
find
Non linear problem : inversion of traveltimes
model
velocity/depth
that minimizes (Delprat-Jannaud and Lailly, 1993)
Solved by a Gauss-Newton algorithm
given a current model
, minimize
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Reflection tomography - classical formulation (2)
Choice of a (semi-) norm in the model space
in
If the number of data is sufficient, the linearized inverse problem is
well-posed
it has a unique solution
the solution is all the more stable as is high
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Reflection tomography - classical formulation (3)
is chosen diagonal
In practice,
diagonal
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not realistic
Very strong assumption
: uncertainty associated with data
data are considered as mutually independent
Reflection tomography - classical formulation (4)
A typical survey for a RTT study
dense near-offset grid
interface geometry
few multi-offset lines
velocity model
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Reflection tomography - classical formulation (5)
Quite an anisotropic discretization
directions along which kinematic data are finely sampled but the
correlation of information are not taken into account
other directions along which the kinematic data are very sparse
enough information ?
footprint of the discretization on the a posteriori pdf ...
... and on all derived quantities (solution model, evaluated
numerical artifact
uncertainties...)
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Reflection tomography - classical formulation (6)
Solution : formalize the kinematic information in an intrinsic way
need for a formulation of the continuum problem taking
correlations of information into account
need for a discretization strategy that allows an accurate
approximation of the continuum problem
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goal : to obtain an intrinsic solution
Correlation of information in RTT (1)
Choice of a “realistic” covariance operator on the data
uncertainties between two points are all the more related as the
points are close
correlation between two distant points is negligible
A classical model : exponential covariance operator
)
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= standard deviation
= correlation length (
Correlation of information in RTT (2)
Some realizations of Gaussian random functions with exponential
covariance and zero mean
Effects of variance changes
Effects of correlation length changes
variance=0.005 − lambda=0.01
0.4
0.2
0
−0.2
−0.4
0
2
4
6
8
10
variance=0.005 − lambda=0.1
variance=0.005 − lambda=0.1
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
0
2
4
6
8
10
−0.4
0
2
0.2
1
0.1
0.5
0
0
−0.1
−0.5
−0.2
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−1
0
2
4
6
4
6
8
10
variance=0.05 − lambda=0.1
variance=0.005 − lambda=1.0
8
10
0
1
2
3
4
5
6
7
8
9
10
Correlation of information in RTT (3)
exponential covariance operator is invertible !
)
Associated norm (
Too complex in practice : use of
and
Too restrictive :
must be constant
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Non independent pieces of information
determinist approach
Correlation of information in RTT (4)
We want to minimize the new cost function
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Correlation of information in RTT (5)
: offset
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and
: CMP coordinates
Correlation of information in RTT (6)
Main change : solution of the linearized inverse problem
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“Classical” formulation :
New formulation :
Correlation of information in RTT (7)
Some important results
need for a calculation of ray perturbations
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calculation of impact point perturbations
verifies
Bending equation
velocities
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explicit interfaces
(1)
and
,
Computation of
Model
and
,
Computation of
(2)
Model perturbation
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First order Taylor expression
satisfies Fermat’s principle in
Trajectory
and
Calculation of impact points perturbations
solution of
Solution of a linear system
each ray
each model parameter
for
Computation of second derivatives of time w.r.t.
impact point coordinates (
)
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)
model parameters (
,
Computation of
(3)
Summary of the new formulation of RTT (1)
Data
A priori information on the data :
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A priori information on the model
choose of adequate regularization weights
Summary of the new formulation of RTT (2)
Ray tracing and calculation of the non linear cost function
and
Computation of
Computation of the “classical” Jacobian
Formulation of the linearized inverse problem based on the use of
the Hessian matrix
avoids the storage of 4 huge Jacobians
Minimization of the new linearized cost function thanks to a
conjugate gradient algorithm
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Experiments on synthetic data (1)
Btert (CASSIS) :
parameters
Raised by
to avoid triplications in the data
1.307
0.000
1.044
13.000
7.000
0.109
X
0.509
2.089
0.908
3.133
1.307
4.178
6.267
0.109
7.311
0.509
8.356
0.908
9.400
1.307
Z
9.400
6.267
5.222
3.133
Z
Y
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1.000
-5.000
11.000
9.000
7.000
5.000
3.000
1.000
-1.000
13.000
X
0.000
0.109
0.172
0.235
0.299
0.362
0.425
0.488
0.551
0.614
0.677
0.740
0.803
0.866
0.929
0.992
1.055
1.118
1.181
1.244
-3.000
-5.000
Z (km)
Y
Experiments on synthetic data (2)
velocity :
Constant
parameters
X
0.000
0.632
0.947
1.263
1.579
0.429
0.429
0.857
0.857
1.286
1.286
1.714
1.714
2.143
2.143
2.571
2.571
3.000
3.000
Z
Z
1.895
2.211
2.526
2.842
3.158
3.474
3.789
4.105
4.421
4.737
5.053
5.368
5.684
6.000
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9.400
8.057
6.714
5.371
4.029
2.686
1.343
0.000
13.000
10.429
7.857
5.286
2.714
0.143
-2.429
-5.000
0.000
0.000
0.316
V e loc ity (km/s)
Y
Experiments on synthetic data (3)
2 types of data
2400 zero-offset data (regular grid)
8800 multi-offset data, modeled by CMPs in the
maximum offset =
direction
Constant uncertainties for all the data
and
directions for all the data
Progressive decrease of the regularization weights (KIM 99)
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)
Use of constant arbitrary initial model (
Constant correlation lengths in
Experiments on synthetic data (4)
Map migration
given the velocity, inversion of zero-offset traveltimes to recover
the depth of the interface
very low misfits
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Experiments on synthetic data (5)
Reflection tomography using multi-offset data
very low misfits
Validation of
the computation of
the calculation of the gradient of the cost function
How to discretize the kinematic information ?
see you next year for constructive answers !
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Conclusion (1)
Classical formulation of two geophysical inverse problems
standard reflection tomography
the kinematic information depends on the discretization of
the data space
the solution of the inverse problem is a numerical artifact
waveform inversion
the norm in the data space is not adapted to the inversion of
seismograms corrupted by very strong correlated noise
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Conclusion (2)
Solution
determinist approach
waveform inversion
use of a (semi-) norm in the data space, adapted to the
correlations involved in the noise
illustration of the potential of this approach on VSP data
reflection tomography
Objective : an intrinsic representation of the kinematic
information
how to discretize this information ?
use of a more realistic norm in the data space
correlation of information taken into account
encouraging experiments on synthetic data
more experiments will come soon !
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