Finite-volume modelling of geophysicalelectromagnetic data using potentials on
unstructured staggered grids
Hormoz Jahandari and Colin G. Farquharson
Department of Earth SciencesSt. John’s, Newfoundland, Canada
SEG 85th Annual Meeting, New Orleans
October 20, 2015
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Contents
1 Unstructured grids
2 Finite-volume discretization of Maxwell’s equations (directEM-field and potential formulation)
3 Example for a grounded wire source
4 Example for a helicopter EM survey
5 Conclusions
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Unstructured grids
Model irregular structures
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Unstructured grids
Topographical features
Geological interfaces
Local refinement (at observation points, sources, interfaces)
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Dual tetrahedral-Voronoı grids
Grid generator: TetGen (Si, 2004)
tetrahedral grid Voronoı grid
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Staggered finite-volume schemes
Magnetic field divergence free
Easy for implementingboundary conditions
Satisfies the continuity oftangential E
Physically meaningful
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Staggered finite-volume schemes
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Staggered finite-volume schemes
Direct EM-field method
Unknowns are E and/or H
Simpler
Smaller system of equations
Ill-conditioned
EM Potential (A� φ) method
Unknowns are A and φ
Larger system of equations
Well-conditioned
Allows studying the galvanic and inductive parts
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Direct EM-field formulation of Maxwell’s equations
Maxwell’s equations:
∇� E � �iωµ0H� iωµ0Mp
∇�H � σ E� Jp
Helmholtz equation for electric field
∇�∇� E� iωµ0σE � �iωµ0Jp � iωµ0p∇�Mpq
Homogeneous Dirichlet boundary condition:
E � 0 at8
orE � τ � 0 on Γ
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EM-field formulation: Discretization
Integral form of Maxwell’s equations:
¾
BSD
E � d lD � �iµ0ω
¼
SD
H � dSD � iµ0ω
¼
SD
Mp � dSD
¾
BSV
H � d lV � σ
¼
SV
E � dSV �
¼
SV
Jp � dSV
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EM-field formulation: Discretization
Discretized form of Maxwell’s equations:
WDj
q�1
Eipj,qq lDipj,qq � �iµ0ωHj S
Dj � iµ0ωMpj S
Dj
WVi
k�1
Hjpi,kqlVjpi,kq � σEi S
Vi � Jp i S
Vi .
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EM-field formulation: Discretization
Discretized form of Helmholtz equation:
WVi
k�1
����
WDj
q�1
Eipj,qq lDipj,qq
� lVjpi,kq
SDjpi,kq
� � iωµ0σEi S
Vi
� �iωµ0
WVi
k�1
Mpjpi,kq
lVjpi,kq
SDjpi,kq
� iωµ0Jp i
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EM potential (A-φ) formulation of Maxwell’s equations
Magnetic vector and electric scalar potentials:
E � �iωA�∇φµ0H � ∇� A
Helmholtz equation in terms of potentials with Coulomb gauge
∇�∇� A�∇ p∇ � Aq � iωµ0σA� σµ0∇φ � µ0Jp � µ0∇�Mp
Conservation of charge
�∇ � J � ∇ � Jp
iω∇ � σA�∇ � σ∇φ � ∇ � Jp
Homogeneous Dirichlet boundary condition:
pA, φq � 0 at8
orpA � τ, φq � 0 on Γ
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EM potential (A-φ) formulation of Maxwell’s equations
Relations used for the finite-volume discretization
∇�H� µ�10 ∇ψ � iωσA� σ∇φ � Jp �∇�Mp (1)
µ0H � ∇� A (2)
ψ � ∇ � A (3)
iω∇ � σA�∇ � σ∇φ � ∇ � Jp (4)
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The system of equations for the A-φ method
Decompose A and φ into real and imaginary parts:
A � Are � iAim ; φ � φre � iφim
Resulting block matrix equation:����
A �ωB C 0ωB A 0 �C0 �ωD E 0ωD 0 0 E
���
����
Are
Aim
Φre
Φim
��� �
����
S1
0S2
0
���
0
500
1000
1500
2000
2500
0 500 1000 1500 2000 2500
Sparse coefficient matrix, A−φ method
(a) refinement at
observation points
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Solution of the finite-volume schemes
Direct solution: MUMPS sparse direct solver (Amestoy et. al, 2006)
Iterative solution: BiCGSTAB and GMRES solvers from SPARSKIT(Saad, 1990)
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Example 1: grounded wire
100 m wire along the x axis operating at 3 Hz
Dimensions of the prism: 120 � 200 � 400 m
σground � 0.02 S{m ; σprism � 0.2 S{m
Observation points along the x axis
−400
0
z (m)
0 500 1000
x (m)
100 m
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Example 1: grounded wire
Dimensions of the domain: 40 � 40 � 40 km
Number of tetrahedra: 162,689
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Example 1: grounded wire
(on Apple Mac Pro; 2.26 GHz Quad-Core Intel Xeon processor)
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Example 1: grounded wire
Scattered electric field
10−11
10−10
10−9
10−8
10−7
10−6
Sec
ondar
y E
x (
V/m
)
800 1200
x (m)
In phase
IE
FE ; potential method (Α−φ)
FV ; direct E−field method
FV ; potential method (Α−φ)10−11
10−10
10−9
10−8
10−7
10−6
800 1200
x (m)
Quadrature
IE
FE ; potential method (Α−φ)
FV ; direct E−field method
FV ; potential method (Α−φ)
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Example 1: grounded wire
Galvanic part (�∇φ)
Inductive part (�iωA)
Total electric field:E � �∇φ� iωA
−100
0
100
y (m)
900 1000 1100
In phase
−∇φ
−100
0
100
900 1000 1100
Quadrature
−∇φ
−100
0
100
y (m)
900 1000 1100
−iωA
−100
0
100
900 1000 1100
−iωA
−100
0
100
y (m)
900 1000 1100
x (m)
E
−100
0
100
900 1000 1100
x (m)
E10−9
10−8
10−7
10−6
E (V/m)
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Example 1: grounded wire
Galvanic part (�σ∇φ)
Inductive part (�iωσA)
Total current density:J � �σ∇φ� iωσA
−100
0
100
y (m)
900 1000 1100
In phase
−σ∇φ
−100
0
100
900 1000 1100
Quadrature
−σ∇φ
−100
0
100
y (m)
900 1000 1100
−iωσA
−100
0
100
900 1000 1100
−iωσA
−100
0
100
y (m)
900 1000 1100
x (m)
J
−100
0
100
900 1000 1100
x (m)
J
10−10
10−9
10−8
10−7
J (A/m2)
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Example 1: grounded wire
Cumulative error versus the changing cell size at the observation points
1e−08
1e−07
1e−06
1e−05
Cum
ula
tive
erro
r (V
/m)
10
Cell size (m)
in phase, no interpolation (E−field method)
quadrature, no interpolation (E−field method)
in phase, no interpolation (A− φ method)
quadrature, no interpolation (A− φ method)
full line: 1st order
dashed line: 0.5th order
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Example 1: grounded wire
10−16
10−12
10−8
10−4
100
Resid
ual
no
rm
0 500 1000 1500 2000
BCGSTAB
a without Coulomb gauge
1
3
10−16
10−12
10−8
10−4
100
Resid
ual
no
rm
0 500 1000 1500 2000
Iteration
c with Coulomb gauge
1
3
5
10−16
10−12
10−8
10−4
100
0 500 1000 1500 2000
GMRES
b without Coulomb gauge
0x50
0x200
0x500
3x50
3x200
3x500
10−16
10−12
10−8
10−4
100
0 500 1000 1500 2000
Iteration
d with Coulomb gauge
0x50
0x500
3x200
3x500
5x500
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Example 2: Ovoid, HEM survey
Ovoid: massive sulfide ore body, Voisey’s Bay, Labrador, Canada
HEM survey of the region has been simulated
Transmitter and receiver towed below the helicopter 30 m above ground
Transmitter-receiver separation was 8 m and the frequencies were 900and 7200 Hz
σground � 0.001 and σovoid � 100 S/m were chosen by try-and-error
Number of tetrahedra: 240, 692
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Example 2: Ovoid, HEM survey
Topography of the region
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Example 2: Ovoid, HEM survey
White dots show the observation points
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Example 2: Ovoid, HEM survey
The plan view
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Example 2: Ovoid, HEM survey
Grid refined at the sources and observation points
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Example 2: Ovoid, HEM survey
FV results (circles) vs real HEM data (lines)
0
200
400
Hz(p
pm
)
6242800 6243000 6243200 6243400
Northing (m)
blue: 900 Hz
red: 7200 Hz
dashed line: HEM ; in phase
full line: HEM ; quadrature
FV ; in phase
FV ; quadrature
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Example 2: Ovoid, HEM survey
Amplitude of the horizontal component of total current density at avertical section along the observation profile
−100
−50
0
50
100
z (m)
In phase
900 Hz
Quadrature
900 Hz
−100
−50
0
50
100
z (m)
6243000 6243200
y (m)
7200 Hz
6243000 6243200
y (m)
7200 Hz
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
J (A/m2)
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Example 2: Ovoid, HEM survey
Galvanic part (�σ∇φ)
Inductive part (�iωσA)
Total current density:J � �σ∇φ� iωσA
6243000
6243200
6243400
y (m)
In phase
−σ∇φ
Quadrature
−σ∇φ
6243000
6243200
6243400
y (m)
−iωσA −iωσA
6243000
6243200
6243400
y (m)
555600 555800 556000
x (m)
J
555600 555800 556000
x (m)
J
10−16
10−15
10−14
10−13
10−12
10−11
10−10
10−9
10−8
10−7
J (A/m2)
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Conclusions
A finite-volume approach was used for modelling total field EM data. Itused staggered tetrahedral-Voronoı grids.
The potential formulation of Maxwell’s equation was discretized andsolved and compared to the solution of the EM-field formulation.
Accuracy and versatility were tested using two examples: one with agrounded wire source and a small conductivity contrast; another one witha realistic body, magnetic sources and a large conductivity contrast.
Unlike the EM-field scheme, the A� φ scheme could be solved usinggeneric iterative solvers.
The gauged problems were harder to solve than the ungauged problems.
Solutions were decomposed into galvanic and inductive parts. The resultswere in good agreement with the type of sources that were used.
While both EM-field and potential schemes possessed the same trends ofaccuracy, the potential approach showed lower cumulative errors.
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Acknowledgements
ACOA(Atlantic Canada Opportunities Agency)
NSERC(Natural Sciences and Engineering Research Council ofCanada)
Vale
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References
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Ansari, S. M., and C. G. Farquharson, 2013, Three-dimensional modeling of controlled-sourceelectromagnetic response for inductive and galvanic components: Presented at the 5th InternationalSymposium on Three-Dimensional Electromagnetics.
Farquharson, C. G., and D. W. Oldenburg, 2002, An integral-equation solution to the geophysicalelectromagnetic forward-modelling problem, in Three-Dimensional Electromagnetics: Proceedings of theSecond International Symposium: Elsevier, 319.
Farquharson, C. G., Duckworth, K. and D. W. Oldenburg, 2006. Comparison of integral equation andphysical scale modeling of the electromagnetic responses of models with large conductivity contrasts,Geophysics, 71, G169-G177.
Madsen, N. K., and R. W. Ziolkowski, 1990, A three-dimensional modified finite volume technique forMaxwell’s equations: Electromagnetics, 10, 147-161.
Si, H., 2004. TetGen, a quality tetrahedral mesh generator and three-dimensional delaunay triangulator,v1.3 (Technical Report No. 9). Weierstrass Institute for Applied Analysis and Stochastics.
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