Joint Inversion Using Multi-Objective Global ...

Joint Inversion Using Multi-Objective Global Optimization Methods Peter G. Leli`evre1 , Rodrigo Bijani2 and Colin G. Farquharson*1 [email protected] http://www.esd.mun.ca/~peter/Home.html http://www.esd.mun.ca/~farq

1 Memorial

University, Department of Earth Sciences, St. John’s, Canada 2 Observat´ orio Nacional, Rio de Janeiro, Brasil

EAGE Conference, Vienna 2016, WS10

Introduction

Sulphide example

Results from typical strategy

Results from PMOGO strategy

Conclusion

Typical numerical approach for mesh-based inversion (underdetermined) • Local descent-based minimization • Weighted aggregate objective function

min fa (m) = m

X

wi fi (m)

i

Leli` evre1 , Bijani2 and Farquharson1 , [email protected] Joint Inversion Using Multi-Objective Global Optimization Methods

1

Memorial University, 2 Observat´ orio Nacional (1 / 18)

Introduction

Sulphide example



Conclusion


min fa (m) = m

X

wi fi (m)

i

N 1 X fd = N j=1

djpred (m) − djobs

!2

σj2


1


Introduction

Sulphide example



Conclusion


min fa (m) = m

X

wi fi (m)

i

N 1 X fd = N j=1


!2

σj2

min fa (m) = λfd (m) + fm (m) m


1


Introduction

Sulphide example



Conclusion


min fa (m) = m

X

wi fi (m)

i

N 1 X fd = N j=1


!2

σj2

min fa (m) = λfd (m) + fm (m) m


1


Introduction

Sulphide example



Conclusion

Disadvantages of the typical inversion approach

1

Appropriate weights must be determined for the aggregate objective function

minm fa (m) = λfd (m) + fm (m)

fa (m) =

X

+

X

λi fd,i (mi ) +

i

X

αj fm,j (mj )

j

ρk fc,k (mk,1 , mk,2 )

k


1


Introduction

Sulphide example



Conclusion


1


2

All objective functions must be differentiable


1


Introduction

Sulphide example



Conclusion


1


2

All objective functions must be differentiable

3

Local minima entrapment may occur, providing suboptimal solutions


1


Introduction

Sulphide example



Conclusion

Pareto Multi-Objective Global Optimization (PMOGO) • Multi-Objective Problem (MOP):

h i min f(m) = f1 (m), f2 (m), . . . , fL (m) m


1


Introduction

Sulphide example



Conclusion



• Concept of dominance:

fi (ma ) ≤ fi (mb )

for i = 1, . . . , L

fi (ma ) < fi (mb )

for at least one i


1


Introduction

Sulphide example



Conclusion





for i = 1, . . . , L

fi (ma ) < fi (mb )

for at least one i


1


Introduction

Sulphide example



Conclusion





for i = 1, . . . , L

fi (ma ) < fi (mb )

for at least one i

• We want to converge to the Pareto-optimal

curve/surface (related to Tikhonov curve)


1


Introduction

Sulphide example



Conclusion





for i = 1, . . . , L

fi (ma ) < fi (mb )

for at least one i

• We want to converge to the Pareto-optimal

curve/surface (related to Tikhonov curve) • Non-dominated Sorting Genetic Algorithm

(NSGA-II) of Deb et al. (2002) Leli` evre1 , Bijani2 and Farquharson1 , [email protected] Joint Inversion Using Multi-Objective Global Optimization Methods

1


Introduction

Sulphide example



Conclusion

Massive sulphide deposit scenario from Carter-McAuslan et al. (2015) 0

0

0

(b)

(c)

50

50

100

100

100

150

150

150

200

Depth (m)

50

Depth (m)

Depth (m)

(a)

200

250

250

250

300

300

300

350

350

350

400

400 0

50

0

100 x (m) 1

150

200

2

2

background (gneiss) intrusive (troctolite) lens (sulphide)

400 0

50

−2

Density (g/cc)

1

Rock types:

200

−1

100 x (m) 0

150

1

200

2

Slowness (s/km)

1

Leli` evre , Bijani and Farquharson , [email protected] Joint Inversion Using Multi-Objective Global Optimization Methods

0

50

1

100 x (m) 2

150

200

3

Rock unit ID

1


Introduction

Sulphide example



Conclusion

Joint inversion of gravity data and traveltimes Aggregate objective function: Data misfits Model roughness Coupling h i h i fa (m1 , m2 ) = λ1 fd1 (m1 ) + λ2 fd2 (m2 ) + fm1 (m1 ) + fm2 (m2 ) + ρfc (m1 , m2 )


1


Introduction

Sulphide example



Conclusion

Joint inversion of gravity data and traveltimes Aggregate objective function: Data misfits Model roughness Coupling h i h i fa (m1 , m2 ) = λ1 fd1 (m1 ) + λ2 fd2 (m2 ) + fm1 (m1 ) + fm2 (m2 ) + ρfc (m1 , m2 )

zik2

= (m1,k −

wik2 zik2

i=1 k=1 u1,i )2 +

2

2

1

1

Slowness (s/km)

fc (m1 , m2 ) =

C X M X

Slowness (s/km)

Joint coupling term (fuzzy c-means):

0

−1

(m2,k − u2,i )

−1

2 −2

−2 0

1

2

1


0

1 Density (g/cc)

2

0

1

1 Density (g/cc)

2

2

Memorial University, Observat´ orio Nacional (5 / 18)

Introduction

Sulphide example



Conclusion

Independent inversion results (aggregate & local minimization) 0

0

0

(a)

(b)

(c)

50

50

50

100

100

100

150

150

150

200

Slowness (s/km)

200

Depth (m)

Depth (m)

Depth (m)

(d) 2

200

250

250

250

300

300

300

350

350

350

400

400

400

1

0

−1

−2 0

50

0

100 x (m) 1

150

200

0

2

−2

Density (g/cc)

1

2

50

−1

100 x (m) 0

150

1

200

2

Slowness (s/km)

1


0

50

1

100 x (m) 2

150

0

200

1 Density (g/cc)

2

3

Rock unit ID

1


Introduction

Sulphide example



Conclusion

Joint inversion with careful local minimization (hand holding) 0

0

0

(a)

(b)

(c)

50

50

50

100

100

100

150

150

150

200

Slowness (s/km)

200

Depth (m)

Depth (m)

Depth (m)

(d) 2

200

250

250

250

300

300

300

350

350

350

400

400

400

1

0

−1

−2 0

50

0

100 x (m) 1

150

200

0

2

−2

Density (g/cc)

1

2

50

−1

100 x (m) 0

150

1

200

2

Slowness (s/km)

1


0

50

1

100 x (m) 2

150

0

200

1 Density (g/cc)

2

3

Rock unit ID

1


Introduction

Sulphide example



Conclusion

Joint inversion with careless local minimization 0

0

0

(a)

(b)

(c)

50

50

50

100

100

100

150

150

150

200

Slowness (s/km)

200

Depth (m)

Depth (m)

Depth (m)

(d) 2

200

250

250

250

300

300

300

350

350

350

400

400

400

1

0

−1

−2 0

50

0

100 x (m) 1

150

200

0

2

−2

Density (g/cc)

1

2

50

−1

100 x (m) 0

150

1

200

2

Slowness (s/km)

1


0

50

1

100 x (m) 2

150

0

200

1 Density (g/cc)

2

3

Rock unit ID

1


Introduction

Sulphide example



Conclusion

Lithological inversion 0

50

• Define several a priori rock types (3 here) 100

• Define physical property distributions for each rock type

Depth (m)

150

(homogeneous here) 200

• PMOGO inversion assigns a rock type to each mesh cell

250

• Simple to add numerically complicated topological

300

constraints, e.g. the model must contain: 350

400 0

50

1

100 x (m)

150

2

200

• 4 contiguous regions • all 3 rock types • 2 regions corresponding to the background

3

Rock unit ID

1

2

1


1


Sulphide example


Regularization 300 400 500 600

3

fit

.2

is

1.0 1.2 Gravity Misfit

1

.0 M 1 ty vi ra

G

0.8

2

1.2 0.8 1.0 Gravity Misfit

8 1 0.

2

1

Seismic Misfit

1

2

3

3

Pareto surface vs curve

Conclusion

3D Pareto front SeiRegularization smic Mis 300 fit400 500 600

Lithological inversion:


Regularization 300 400 500 600 S e is m ic M is fit

Introduction

3

3

1

1

1 0.8

2

2

2

2

1

Regularization Seism ic Misfit 30 40 50 60

Regularization Seismic Misfi t 300 400 500 600

Regularization Seismic Misfi 300 400 500t 600

3


1


0.8

0.8

1.0 1.2 Gravity Misfit 1



Introduction

Sulphide example


Conclusion

2

3

3

Regularization 300 400 500 600 S e is m ic M is fit

3D Pareto front

8 1 0.

2

G

ra

.0 M 1 ty vi

SeiRegularization smic Mis 300 fit400 500 600

Lithological inversion:


Joint Inversion Using Multi-Objective Global Optimization Methods

.2

Leli` evre1 , Bijani2 and Farquharson1 , [email protected]

1


it

0.8

f is

1

1


Introduction

Sulphide example


Lithological inversion: 0

0

(c)

50

50

100

100

100

150

150

150 Depth (m)

50

Depth (m)

Depth (m)

(b)

200

250

250

300

300

300

350

350

350

400 0

50

100 x (m)

150

200

Misfits: (a) expected targets (b) high gravity (c) high seismic

200

250

400

Conclusion

three models in the Pareto front

0

(a)

200


400 0

50

100 x (m)

150


200

0

50

100 x (m)

150

200

1


Introduction

Sulphide example


Lithological inversion: 0

0

(c)

50

50

100

100

100

150

150

150 Depth (m)

50

Depth (m)

Depth (m)

(b)

200

Rock types: 200

250

250

250

300

300

300

350

350

350

400

400 0

50

100 x (m)

0

50

150

200

2

100

(a) background (b) intrusive (c) sulphide

400 0

0

Percent

1

Conclusion

some rudimentary statistics

0

(a)

200


50

100 x (m)

150

50 Percent

1


200

0

100

0

50

100 x (m) 50

150

200

100

Percent

1


Introduction

Sulphide example



Conclusion

Surface-based inversion • Parameterization defines interfaces between rock units • No mesh required (very fast forward problem) • PMOGO inversion moves vertices


1


Sulphide example


Curvature 0.1 0.2 .0

M is m ic is S e

2.5

.0 1 .0 G

.0 2 it f is

.5 M 1 ity v ra

2 .5

2.5

2.0

2.0 2.5 1.0 1.5 Gravity Misfit

1


1.5

1.0

.0

.5 1

2.0

1.5

1.0

3.0

.5

Seismic MiCurvature sfit 0.1 0.2

3 2

Seismic Misfit

2

3.0

2.5

1.0

Conclusion

3D Pareto front Curvature 0.1 0.2

Surface-based inversion:


fit

Introduction

3.0

Seismic Misfi Curvature t 0.1 0.2

2.5

Se ismic Misfit Curvature 0.1 0.2

3.0 2.5

2.0

2.0

1.0

2


2.5

1


2.0

Curvature 0.1 0.2 1.0

1.0

1.0

2.5

1.5

1.5

1.5

1

Seismic Misfi

t

3.0

1.0


2.5

1.0

1.5 2.0 Gravity Misfit 1

2.5


Introduction

Sulphide example


Conclusion

3D Pareto front

M is fit

.0 .5

2

is m ic

2

Seismic MCurvature isfit 0.1 0.2

3

Curvature 0.1 0.2

Surface-based inversion:


S e

.0

3.0

1 .5 .0 1

2.5

ra

.5 M 1 ity v

.0 G 1

2.0

1.5

.0 2 it f is

1.0 2.5


.5


2

1.0

1


Introduction

Sulphide example

Surface-based inversion: 50



Conclusion

three models in the Pareto front

0

50

100

100

Legend: - - feasible region — true interfaces — expected target misfits — both misfits high — high seismic misfit

150

200

Depth (m)

Depth (m)

150

250

300

350

200

250

400 0

50

100 x (m)

150

200

Leli` evre1 , Bijani2 and Farquharson1 , [email protected] 300 Joint Inversion Using Multi-Objective Global Optimization Methods

1


Introduction

Sulphide example

Surface-based inversion: 50



Conclusion

statistics for surrounding solutions

0

50

100

100

Legend: - - feasible region — true interfaces — mean model • 2-st. dev. ellipses

150

200

Depth (m)

Depth (m)

150

250

300

350

200

250

400 0

50

100 x (m)

150

200

Leli` evre1 , Bijani2 and Farquharson1 , [email protected] 300 Joint Inversion Using Multi-Objective Global Optimization Methods

1


Introduction

Sulphide example



Conclusion

Advantages and disadvantages of PMOGO Advantages: • Obviate the requirement of having to deal with trade-off parameters

→ Joint inversion greatly simplified


1


Introduction

Sulphide example



Conclusion


→ Joint inversion greatly simplified • Automatically provide a suite of solutions across the Pareto front

→ Opportunities to calculate statistics


1


Introduction

Sulphide example



Conclusion



→ Opportunities to calculate statistics • Any numerically complicated objective functions or constraints can be used • Avoid local minima entrapment

→ Fundamentally different model parameterizations


1


Introduction

Sulphide example



Conclusion




→ Fundamentally different model parameterizations • Easy to parallelize


1


Introduction

Sulphide example



Conclusion




→ Fundamentally different model parameterizations • Easy to parallelize

Disadvantages: • Increased computing time (100-300x for mesh-based inversions, WITH CAVEATS) Leli` evre1 , Bijani2 and Farquharson1 , [email protected] Joint Inversion Using Multi-Objective Global Optimization Methods

1


Introduction

Sulphide example



Conclusion

References

• Bijani et al., 2015, Three-dimensional gravity inversion using graph theory to delineate the

skeleton of homogeneous sources. Geophysics, 80, G53–G66 • Carter-McAuslan, Lelievre & Farquharson, 2015, A study of fuzzy c-means coupling for

joint inversion, using seismic tomography and gravity data test scenarios. Geophysics, 80, W1–W15 • Deb et al., 2002, A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans.

on Evol. Comp., 6, 182–197


1


Joint Inversion Using Multi-Objective Global ...

Joint Inversion Using Multi-Objective Global ...

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