3d potential field inversion for wireframe surface geometryfarq/pdfs/peter_seg_2015.pdf · 3d...

3D Potential Field Inversion forWireframe Surface Geometry

Peter G. Lelievre1, Colin G. Farquharson1 and Rodrigo Bijani2

[email protected]

http://www.esd.mun.ca/~peter/Home.html

http://www.esd.mun.ca/~farq

1Memorial University, Department of Earth Sciences, St. John’s, Canada2Observatorio Nacional, Rio de Janeiro, Brasil

SEG 85th Annual Meeting, Oct. 2015, New Orleans, GM2

Introduction Mesh-based inversion Surface-based inversion (2D) Surface-based inversion (3D) Conclusion

Geophysical inversion primer

Earth model (e.g. density) Survey data (e.g. gravity)

Forward problem

Inverse problem

Lelievre1, Farquharson1 and Bijani2, [email protected] 1Memorial University, 2Observatorio Nacional

3D potential field inversion for wireframe surface geometry (2 / 27)


Motivation

(a) (b)

• Geophysical numerical methods typically work with mesh-baseddistributions of physical properties (a)

• Geologists’ interpretations about the Earth typically involve wireframecontacts between distinct rock units (b)

• There is a disconnect here!




Types of geophysical inversion

1 Discrete body inversion

2 Mesh-based inversion

3 Surface-based inversion




1. Discrete body inversion

Simplified representation of the Earth:• Simple shapes for one or more causative

target bodies

• Homogeneous background

Inversion:

• Few parameters (e.g. shape, location)

• Data best-fit problem

• Low computational requirements

• Stochastic investigations feasible




2. Mesh-based inversion

General representation of the Earth:• Mesh of tightly packed cells

• Piecewise (pixellated) distributionof physical properties

Inversion:

• Many parameters (many cells)

• High computational requirements

• Stochastic investigations not veryfeasible

Rectilinear Quadtree

Unstructured




2. Mesh-based inversion

General representation of the Earth:• Mesh of tightly packed cells

• Piecewise (pixellated) distributionof physical properties

Inversion:

• Many parameters (many cells)

• High computational requirements

• Stochastic investigations not veryfeasible




3. Surface-based inversion

Flexible representation of the Earth:• Wireframe of nodes and

connecting facets representingcontacts between rock units

• How geological models are built

Inversion:

• Surface geometry defined bymoderate number of parameters

• Moderate computationalrequirements

• Stochastic investigationssomewhat feasible

Richardson & MacInnes, 1989, The inversion of gravity datainto three-dimensional polyhedral models, JGR




3. Surface-based inversion

Flexible representation of the Earth:• Wireframe of nodes and

connecting facets representingcontacts between rock units

• How geological models are built

Inversion:

• Surface geometry defined bymoderate number of parameters

• Moderate computationalrequirements

• Stochastic investigationssomewhat feasible




Mesh-based inversion for smooth distributions

• Objective function

Φ = Φd + βΦm

• Data misfit

Φd =∑i

(F (m)i − di

σi

)2

• Model structure (regularization)

Φm =∑j

wj(mj − pj)2 +

∑j

∑k

wj ,k(mj −mk)2

[smallness term] + [smoothness term]

• Deterministic local optimization approach: one “best” solution




Mesh-based inversion for sharper features


Φ = Φd + βΦm

• Data misfit

Φd =∑i

(F (m)i − di

σi

)2


Φm =∑j

wj(mj − pj)2 +

∑j

∑k

wj ,k |mj −mk |p + Ψ


• Different norm, measures or re-weighted iterative procedure can help






Φ = Φd + βΦm

• Data misfit

Φd =∑i

(F (m)i − di

σi

)2


Φm =∑j

wj(mj − pj)2 +

∑j

∑k

wj ,k(mj −mk)2


• The safest and most effective approach is to hardwire the surfaces





Hardwiring surfaces ⇒ unstructured meshes become important

Inversion of gravity gradiometry data:

Unconstrained Hardwired surface Additional regularization







3 Surface-based inversion (2D)




2D Surface-based inversion for geometry

• Inversion seeks positions of nodes inwireframe model

• Only data misfit is required

Φd =∑i

(F (m)i − di

σi

)2

• Regularization not required: work on coarserepresentation, refine e.g. splines

• Global optimization strategies (PSO, GA, MCMC) provide statistics:

⇒ many solution samples




Cocagne Subbasin: gravity survey data

CALEDONIAUPLIF

T

MONCTONSUBBASIN

HASTINGS UPLIFT

WESTMORLAND UPLIFT

SACKVILLESUBBASIN

NEWBRUNSW

ICK

PLATFORM

BelleisleFault

COCAG

NE

SUBBASIN

Smith

Creek

Fault Nor

th River

Fault

ZONE

DEFORMED

INDIA

N

MO

UNTAIN

Sussex

Chi

gnec

toB

ay

NovaScotia

Northumberland

Strait

Alma

Bayof

Fundy

Shediac

Sackville

CUM

BERLA

ND

SUBBASIN

0 10 km

0 10 km

66°

U.S.A.

NewBrunswick

Nova

Scotia

P.E.I.

Gulfof

St. Lawrence Newfoundlandand Labrador

Qu becé

0 100 km

Area of Figure 1

MARITIMESBASIN

Buctouche

Richibucto

BerryMillsFault

Moncton

Area ofFigure 2

Late Devonian–Carboniferousdeep basins

Pre-Late Devonian crystallinebasement

Carboniferous shallow basins

Stoney Creek Field (oil/gas)

McCully Field (gas)

Penobsquis mine (potash/salt)

Fault

Figure 1. Distribution of Late Devonian–Carboniferous uplifts and subbasins in theMaritimes Basin of eastern New Brunswick(modified after St. Peter 2006; St. Peter andJohnson 2009), as well as the location ofpresent-day oil, natural gas, and potash/saltproducers. Inset map shows the area of theMaritimes Basin in eastern Canada.

101

(previously interpreted trace)

113

65°10’ 65°00’ 64°50’ 64°40’ 64°30’

46°40’

46°30’

46°20’

46°10’

Figure 8. New terrain-corrected Bouguer anomaly map of the report area, with fBlue lines E–E' and W–W' mark the location of

section profiles in Figures 11 and 12, respectively.

ault boundaries(modified after Smith 2008) shown as thick grey lines.

Gravity stations are as identified in Figure 6. Figure 1shows the location of this report area in eastern New Brunswick.

0 5 10 km

W

W'

E'

E

BelleisleFault

CormiervilleFault

Smith CreekFault

Berry MillsFault

J. Evangelatos & K. E. Butler, 2010. New gravity data in Eastern New Brunswick: implications for structural delineation ofthe Cocagne Subbasin. In Geological Investigations in New Brunswick for 2009. Edited by G. L. Martin. New BrunswickDepartment of Natural Resources; Lands, Minerals and Petroleum Division, Mineral Resource Report 2010–1, p. 98–122.




Cocagne Subbasin: rudimentary geological model

Figure 11. A two-dimensional geological model, simulating the observed gravity response across theeastern part of the Cocagne Subbasin along profile E–E' in Figure 8. The magnetic profile is extractedfrom the regional aeromagnetic dataset (Canadian Belleisle,Cormierville, and Smith Creek fault traces in Figure 8 are shown here as red ticks on the horizontal axis.

Aeromagnetic Data Base 2010). The

Belleisle Fault4

0 5 10 15

Smith Creek Fault

Basement rock(2700 kg/m )3

Basement rock(2710 kg/m )3

Sedimentaryrock (2550 kg/m )3

Cormierville Fault

3

2

1

Depth(km)

0

E (North) E' (South)

-5

0

5

10

600

400

200

0

Gravity(mGal)

Magnetics(nT)

Distance (km)

VerticalExaggeration = 2

Modelled

Observed

Observed

118

constraints on subsurface densities might be obtained by consulting well logs for the fewmoderately deep boreholes in the region, although none penetrates deeply into the Cocagnegravity low itself.

Structural constraints might also be obtained by analyzing old seismic reflection data from thearea. Indeed, Durling and Marillier (1990a) discuss a structural interpretation of marineseismic reflection data acquired from the nearby offshore area in the Northumberland Straitand Gulf of St. Lawrence. Further information about correlations between the onshore and

J. Evangelatos & K. E. Butler, 2010. New gravity data in Eastern New Brunswick: implications for structural delineation ofthe Cocagne Subbasin. In Geological Investigations in New Brunswick for 2009. Edited by G. L. Martin. New BrunswickDepartment of Natural Resources; Lands, Minerals and Petroleum Division, Mineral Resource Report 2010–1, p. 98–122.




Cocagne Subbasin: mesh-based inversion

−15

−10

−5

0

5

10

Gra

vity (

mG

al)

0 5000 10000 15000

Distance (m)

ObservedModelled

−15

−10

−5

0

5

10

Gra

vity (

mG

al)

0 5000 10000 15000

Distance (m)

ObservedPredicted

Reference model Recovered model0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

−0.2 −0.1 0.0 0.1

Anomalous density (kg/m3)

0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

−0.2 −0.1 0.0 0.1

Anomalous density (kg/m3)




Cocagne Subbasin: mesh-based inversion

−15

−10

−5

0

5

10

Gra

vity (

mG

al)

0 5000 10000 15000

Distance (m)

ObservedModelledPredicted

Absolute difference Percent difference0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

0.00 0.01 0.02 0.03 0.04 0.05

Absolute density difference (kg/m3)

0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

0 10 20 30 40 50

Percent density difference




Cocagne Subbasin: surface-based inversion

−15

−10

−5

0

5

10

Gra

vity (

mG

al)

0 5000 10000 15000

Distance (m)


−15

−10

−5

0

5

10

Gra

vity (

mG

al)

0 5000 10000 15000

Distance (m)


2D wireframe model Splined model0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

0.04 g/cc

0.03 g/cc

−0.12 g/cc

0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

0.04 g/cc

0.03 g/cc

−0.12 g/cc




Options for visualizing uncertaintyEnsemble of solutions Error bars (1 st. dev.)

0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

0.04 g/cc

0.03 g/cc

−0.12 g/cc

0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

0.04 g/cc

0.03 g/cc

−0.12 g/cc

Sized by st. dev. Coloured by st. dev.0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

0.04 g/cc

0.03 g/cc

−0.12 g/cc

0

1

2

3

4

De

pth

(km

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Distance (km)

0.04 g/cc

0.03 g/cc

−0.12 g/cc

0 500 1000

Standard deviation







3 Surface-based inversion (3D)




Surface-based inversion for geometry

• Inversion seeks positions of nodes inwireframe model

• Only data misfit is required

Φd =∑i

(F (m)i − di

σi

)2

• Regularization not required: work on coarserepresentation, refine e.g. surface subdivision

• Global optimization strategies (PSO, GA, MCMC) provide statistics:

⇒ many solution samples




Surface-based inversion for geometry

Use coarse control surfaces rather than the surfaces themselves!

Subdivision with cubicB-spline interpolation

Subdiv., Dyn-Levin-Gregory interp.

Subdiv., cubic B-spline

Bezier surface

Wojciech mula at pl.wikipedia. Licensed under CCBY-SA 3.0 via Wikimedia Commons




Synthetic example #1: Isolated bodyBlack wireframe = true modelRed wireframe = control surface for true modelGreen wireframe = control surface for recovered modelWhite box = bounds on control nodes during inversionColoured surface = recovered model (standard deviations, red high)Coloured points = gravity data

Overhead view Side view




Synthetic example #2: Sheet-like contact surfaceBlack wireframe = true modelRed wireframe = control surface for true modelGreen wireframe = control surface for recovered modelWhite box = bounds on control nodes during inversionLight grey surface = recovered model (standard deviations not plotted here)Coloured points = gravity data





Real data example: IOCG deposit, gravity data


Surface-based inversion result consistent with understanding of geology,significant differences to mesh-based result require further study




Summary

• Mesh-based “distribution inversion” is standard and numericallywell-behaved until you try to recover sharp features (best way is tohardwire them)

• Surface-based “geometry inversion” is challenging but:• can model sharp contacts and provide statistical information• geological and geophysical models can be, in essence, the same Earth

model: there is no longer a disconnect!




Future work

• Joint surface-based inversion• Multi-objective optimization methods for joint inverse problems ...

• Alternate global optimization approaches• Particle swarm, genetic algorithms, ant colony, ...

• Work directly with complicated 3D common Earth models ...

• Hybrid approach ...



(additional slides follow)

Future work

Multi-objective genetic algorithms for joint inversion

Single-objective GA:• Aggregate of objectives:

min(f ) f = f1 + λf2 + ...

• One single best solution

• Difficult to find best λ value(s)

Multi-objective GA:

• Objectives treated separately:min(f1, f2)

• Several solutions along the Paretofront (nondominated)



Future work

Building and manipulating complicated 3D models

FacetModeller



Future work

Can we extract the control surfaces, rather than thesurfaces themselves, from the modelling software?

Subdiv., cubic B-spline Bezier surface

Wojciech mula at pl.wikipedia. Licensed under CCBY-SA 3.0 via Wikimedia Commons



Future work

A third approach?

1 Mesh-based inversion with sharp features


3 Hybrid approach?Lelievre1, Farquharson1 and Bijani2, [email protected] 1Memorial University, 2Observatorio Nacional


3d potential field inversion for wireframe surface geometryfarq/pdfs/peter_seg_2015.pdf · 3d...

Documents