Max A. Meju
PETRONAS Exploration,
Petronas Twin Towers, Kuala Lumpur
Workshop on Uncertainty Analysis in Geophysical
Inverse Problems, November 7 - 11, 2011. Leiden, The
Netherlands
» Introduction
- Sources of uncertainty: nature of
geophysical data, modelling error
» Methods of uncertainty estimation
» Extremal bounds analysis : the nonlinear mostsquares approach
- Examples from electromagnetic inversion
» Challenges & way forward
- uncertainty reduction
- simple metrics of model uncertainty
» Conclusion
• There are various sources of uncertainty - from
field surveying to model interpretation
• Quantifying and reducing the impact of
uncertainty in regularized inversion of incomplete,
insufficient and inconsistent observational data is
a difficult proposition
• Finding a simple composite measure or index of
uncertainty in model interpretation remains a
major challenge
Main causes of uncertainty in
geophysical data inversion
•Band-limited field measurements (due
to technical limitations)
•Measurement errors
•Geological heterogeneity
•Nonlinearity of physical (forward)
process. Simulation/prediction errors.
Nature of field data: incomplete, insufficient & inconsistent
MT amplitude and phase data for different periods (sec.).
The nature of EM field data and
model equivalency lead to
uncertainty in interpretation
Meju & Sakkas, JGR 2007
3 equivalent 2D resistivity models for
the same EM data set from Kenya.
» Formally, model uncertainty and non-uniqueness can
be reduced by combining measurements of
fundamentally different physical attributes of a
subsurface target (i.e., coupled-field analysis) or by
using available a priori information about the target,
and accounting for 3D geological heterogeneity.
» Several approaches to quantifying the uncertainty
arising from band-limited data, data errors, and
nonlinearity.
» Bayesian probability
density function
estimation (mostly
applied to small-size
geophysical inverse
problems)
» Monte Carlo covariance
method for large-scale
geophysical inverse
problems
» Deterministic (mostly
applied to small-size
inverse problems) :
- linear sensitivity analysis
(Backus-Gilbert; funnelfunction appraisal
methods etc),
- minimax,
- nonlinear extremal
bounds analysis (Mostsquares method)
» Problem statement:
‘Given an optimal model for a regularised parameter
estimation problem, mopt, find, on account of data
errors, solution equivalency and non-uniqueness,
other models that satisfy a specified threshold
misfit, qms and are consistent with the constraints
imposed by the data uncertainties’
Problem formulation:
Extremize the function mTb, subject to the constraint
q = (Wd-Wf(m))T(Wd-Wf(m)) + mTLTLm + (m-mo)T(m-mo)
≤ qms,
(1)
where b is the model projection operator, and L and  are
respectively the Laplacian operator and regularisation
parameter used for generating mls.
The expected value of qms is n-l, where there are l active
constraints in the problem.
NB: The above problem formulation is appropriate for model
construction algorithms that use the additional stabilisation
measure, (m-mo)T(m-mo), to place a bound on the absolute
size of model perturbation.
An equivalent mathematical
statement to Equation (1) is:
Extremize:
mTb + 1/2µ{(Wd-Wf(m))T(Wd-Wf(m)) +
mTLTLm + (m-mo)T(m-mo) - qms}
(2)
Next, we need to linearize the expression so that we
can solve it using established linear methods!
mTb + 1/2µ{(Wd-Wf(m))T(Wd-Wf(m)) +
mTLTLm + (m-mo)T(m-mo) - qms}
ℓ=
NB: If we linearize d=f(m), using Taylor
series expansion, we obtain y=Ax.
So,
ℓ = (m0+ x)Tb + 1/2µ{(Wy-WAx)T(Wy-WAx) +
(m0+ x)TLTL(m0+ x) + xTx - qms} (3)
Next, find dℓ/dx and equate to zero for
minima/maxima!
ℓ = m0Tb + xTb + 1/2µ {yTWTWy - WxTATWy -
WyTWAx + xTATWTWAx + (m0TLTLm0) +
2(m0TLTLx) + (xTLTLx) + xTx - qms}
So,
dℓ/dx = b + 1/2µ { -2ATEy + 2ATEAx + 2LTLm0 +
2LTLx + 2Ix} = 0
(4)
where E = WTW for notational simplicity!
Next, assemble like-terms and form the Most-Squares
Normal Equations!
From the previous slide,
b + 1/µ [ATEAx + LTLx + Ix] - 1/µ[ATEy - LTLm0] = 0.
Therefore,
[ATEA + LTL + I]x = [ATEy - LTLm0 -µb], (5)
which are the Most-squares Normal Equations (akin to
the Least-squares normal equations) and I is the
identity matrix.
Finally, solve the Normal Equations for x!
From the Normal Equations, we obtain
x
= [ATEA + LTL + I]-1 [ATEy - LTLm0 -µb].
(6)
Nonlinearity is dealt with using an iterative formula
of the form
mk+1 = mk + x
= mk + [ATEA + LTL + I]-1 [ATEy - LTLm0 -µb]
(7)
where A and y are evaluated at mk and the iteration
is begun at k = 0.
Alternatively, obtain the direct estimate of m!
Recall that m=mo+x. Replacing x with m-mo in the
Normal Equations yields
m = [ATEA + LTL + I]-1 {ATEd* + Im0 -µb}
(8)
where d* = [y + Am0], and
 = ± {[qms - qls]/(bT [ATEA + LTL + I]-1 b)}0.5.
There are two (plus and minus) solutions for  and may
be regarded as the upper and lower resistivity bounds
for the specified qms if the elements of the projection
operator b are set to unity.
» The result of regularised 2D MT inversion served as
the input. This is the best model obtained in a
Least-squares sense. It is then subdivided into
blocks for appraisal.
» Most-squares model bounds are calculated for each
block of the model for a specified threshold misfit
of 10%, i.e., qms = qls 1.1, with b values set to unity.
Example of a simple application by Meju & Sakkas, JGR 2007
qms = qls 1.1, with b values set to unity
A. 2D Least-squares inversion model
B. 2D Most-squares
inversion model.
» Appraisal of layered-earth models: Mapping regions
inaccessible to a given data set (to prevent the overinterpretation of field data)
- Synthetic MT amplitude and phase data
(smooth-versus-rough-models)
- COPROD MT field data (standard test) and effective
depth of inference
- Time-domain CSEM models and effective depth of
inference
lower
envelope
Model response covers the region
spanned by error bars of the
amplitude and phase data
Upper
envelope
For high quality data, narrow
solution envelopes are obtained
by setting the elements of the
projection vector ‘b’ to unity. That
is, only two models are
generated which are maximally
consistent with the data.
Model response covers the
region spanned by error
bars of the amplitude data
For high quality data, narrow solution
envelopes are obtained by setting the
elements of the projection vector ‘b’ to
unity. That is, only two extreme models
are generated.
Benchmark: Effective depth of inference (zone of
structural concordance in pooled models) for
COPROD MT data
inaccessible
region
Region
constrained
by field data
inaccessible
region
Smooth models are generated for a range of regularisation factors (0.05 to
0.8). The resulting pool of models share common features only when they
are constrained by the field data. The models diverge (splay) at depths
where the data fail to constrain them (Meju, 1988).
Region not
constrained
by field data
Region not
constrained
by field data
Note that the Most-squares model diverge or splay
where the models are poorly-constrained by the noisy
field data.
Region not
constrained
by field data
Conventional least-squares inversion
of high-quality data leads to a single
smooth- or rough-model
Region not
constrained
by field data
Region
constrained
by field data
Region not
constrained
by field data
Models from inversion using
two different regularisation
factors, 0.01 and 0.5
Models from skin-depth
of investigation
High quality field
data; narrow
solution envelopes
obtained by setting
the elements of the
projection vector
‘b’ to unity. That is,
only two extreme
models are
generated.
Note that model responses
cover the region spanned by
data error bars
Extreme parameter
sets are obtained
by setting only one
element of the
projection vector
‘b’ to unity at a
time, yielding 2M
extreme models for
the M model
parameters.
Note that model responses
cover the region spanned
by data error bars
Uncertainty visualisation in multiple dimensions:
The search for a simple index of common structure!
What are the common features in these 3 statistically equivalent
2D resistivity models? How can we measure ‘common structure’?
» Is there a simple metric for presenting or visualizing
uncertainty in inversion?
» The geometrical attributes of extreme models from
Most-squares inversion, and especially their degree of
structural similarity, may be used as one quantitative
measure of uncertainty (Meju,2009, GRL).
» Let mp and mm be the property fields for the Mostsquares plus and minus solutions.
» A criterion to apply is that the cross-products of the
gradients of the mp and mm property fields must be
zero at an important boundary or target zone.
» If mp(x,y,z) denotes the vector field of the gradient of
the model derived using the plus solution and
mm(x,y,z) denotes the gradient of the model from the
minus solution, we define the cross-gradients function
t (x,y,z) = mp(x,y,z)  mm(x,y,z).
» The vector field of the cross-gradients function is
p
m
p
m
p
m
p
m
p
m

 m p m m
m m
m m
m m
m m
m m
t ( x , y , z )  

,

,

z
z
y
z
x
x
z
x y
y
x
 y
» and t (x,y,z) = 0 implies structural similarity while
departure from zero at any given location in the model
(m(x,y,z)) may be taken as an interpretative measure of
structural uncertainty at that location (Meju 2009).




» Using mathematical methods to combine (disparate or
correlated) measurements taken from different
platforms so as to reconcile geoscientific conflicts or
reduce interpretational ambiguities.
» ‘Mathematics of conflict resolution’ requires realistic
physical models!
» Uncertainty quantification in large-scale geophysical
inversion is a major challenge
» Simple robust interpretation or visualization of
uncertainty is not available!
Example: Joint
cross-gradient
inversion of
electromagnetic
resistivity and
seismic traveltime data
Gallardo & Meju, 2003 GRL;
2004 JGR; 2007 GJI; 2011 RoG.
Gallardo et al., 2005 Inv. Prob.
Basis of structure-coupled joint inversion
. True model
Test model and
2D grid design.
Seismic
ray
coverage for the
above
synthetic
test model. The
inverted triangles
mark the locations
of the seismic
shot-points.
Separate
Joint inversion
a) MT-resistivity model obtained by separate inversion. b) Seismic model obtained by
separate inversion of the seismic data. c) MT-resistivity and d) seismic velocity
models obtained by joint inversion using cross-gradients constraints. The thick lines
outline the test structures that are the exploration targets (units A-F).
Model appraisal: cross-gradient map of
interpretational uncertainty
Simple metric of uncertainty: Cross-gradient
values are zero in areas well-constrained zones
having common structure. It is non-zero in zones
of conflicting subsurface structures.
5. Concluding comments:
The outstanding challenges in
uncertainty quantification are
• Geological complexity and nonlinearity
• Uncertainties in forward modelling of
physical reality! Computational limitation
• Fusing physical and non-physical data
• Data homogenization (how to give equalimportance to various data sources!)
That’s all folks !
Thanks for your kind attention !
Email: maxwell_meju@petronas.com.my
Access pdfs of my publications at: http://www.es.lancs.ac.uk/giei/publications.htm
Useful references