Eldad Haber
description
Transcript of Eldad Haber
Eldad Haber
Inversion of 3D Electromagnetics: A maturing technique in applied geophysics
Collaborators: Doug Oldenburg, Roman Shekhtman,Scott Napier, and Rob Eso
Outline:
• Introduction: Example problems– Environmental, geotechnical, resource exploration
• Geophysical surveys
• Inversion
• Mineral exploration example
• Summary/discussion
Environmental: UXO• Military proving grounds
• Regions of conflict
• Avalanche control
http://www.nohowinc.com/
http://www.dma.state.mn.us/
http://www.centennialofflight.gov
Environmental: How do we find UXO?
?
Geotechnical: A Canadian potash mining
Geotechnical problem
• Water gushing into the mine
Mineral exploration
What do we have?
Map of surface geology
Solutions … Geophysics
Energy from sourceEnergy from source
Physical propertiesand contrasts
Physical propertiesand contrasts
MeasurementsMeasurements
- Density- Magnetic susceptibility- Electrical resistivity- Chargeability- others …
- Gravity- Magnetic field anomalies- DC or electromagnetics- Induced polarization- etc. …
physical properties
Physical properties• UXO:
– Electrical conductivity and magnetic susceptibility
• Water (at potash mine): – Electrical conductivity: high if it has dissolved salt
• Minerals: – magnetic susceptibility
– electrical conductivity
– chargeability
– density
(Loop or grounded electrode)
Source
Waveform
Time
I (half sine, step…)
Measurements
(E, H, dB/dt)
Borehole
Depth
(E, H, dB/dt)
3D TEM Setup
Outline:
• Introduction: Example problems– Environmental, geotechnical, resource exploration
• TEM forward modelling
• Inversion
• Mineral exploration example
• Summary/discussion
Mathematical Setup
Maxwell’s Equations
Boundary conditions
Initial conditions
This must be solved in both space and time.
time [0, tf]
Implicit Method
• Backward Differentiation Formula (BDF)
• E n depends upon E n-1, H n-1 (BDF1)
where
Need initial conditions to start time stepping.
Initial Conditions
Time stepping starts with fields at time t = 0
0 t
I(t)Case 1:
0t
Case 2: E0, H0 fields are from DC currents
Solve
DC Resistivity
MMR
Formulation as potentials
(1)
(2)
where and
Helmholtz decomposition
Coulomb Gauge
Yields
Discretization with FV on staggered grid
-- A, J are defined on the faces
-- is in the centers
-- H is on the edges
Finite Volume discretization:
Each equation is integrated over a volume.
Yields:
Formulating the forward problem
Source term:
Maxwell’s equations:
Write:
Maxwell’s Equations:
Solve: Aj (m) uj = qj using BICGStab with ILU preconditioner.
Solving the system
Let nc = number of cells
number of unknowns ~ 7(nc)3
if nc = 64 matrix size = (2x106)2
Solve
Solve An, Φ system using BICGstab with block ILU
preconditioner (same as EH3D)
The Cominco example
SurfaceSurface targe
tHost resistivity = 200 Ohm-mHost resistivity = 200 Ohm-m
X
Y
Z
(0,0,0)
Loop source, 1km squareLoop source, 1km square
625 m
Target (x, y, z) = (250, 500, 100)Conductivity = 1.0 S/m
The “Cominco” model: loop source, conductive target in a host
Forward modelling of fields
Velocity of smoke ring
Nabighian (1979)
Currents decay
Induction finished t~0.01 sec
Sampling time for movies: 62 frames [10^-6 – 10^-2]
Examples: H field
Examples: J field
Examples: Js field
Examples: Surface (Ex,Hx, dB/dt)
Examples: Profiles of (E,H)
Conclusion
(Loop or grounded electrode)Source
Waveform
Time
I (half sine, step…)
Surface Measurement
(E, H, dB/dt)
Borehole Measurement
Depth
(E, H, dB/dt)
Next Step: Inversion
What is Inversion?
Goal: Estimate the Earth modelInversion
??
Airborne, surface or borehole
measurements
Inversionprocessing
Inversion procedure:
• Divide earth into cells (each with fixed size and unknown value).
• Inversion: find values for cells
• Use mathematical optimization theory.
• Difficulties:
– Solution is non-unique.
– Computationally demanding.
MeasurementsMeasurements Pre-processingPre-processing InversionInversion
Physical property distributions = Inversion
Physical property distributions = Inversion
Prior informationPrior information
3D, and ~ 105 cells
1
2
Inversion as optimization: 3 parts
Inversion as optimization:
= d + m. 0 < < is a constant
Choose such that d < Tolerance
A priori information: reference model, structural detail...
...)(
)()(2
020
dvx
mmdvmmm
S
x
S
sm
Model objective function:
• s, x … constants
• m0 : reference model
Misfit: 2
1
][
N
i i
obsii
d
dmF
• i : standard deviation
Inverse problem
Minimize = d + m
2)( refm mmW
)(][,)][(2
QumdmWd fFF obsd where
: Regularization parameterQ: Projection matrix u: Potentials : Observations : Model and Reference model
obsd
refmm,Wd, W : Measurement error, model weighting
Solving the inverse problem
)(]][[)( refTobsT
dT F mmWWdmWJmg
m
Differentiating the objective function with model m
where sensitivity matrix
and
),( HES
f
Gauss-Newton method
)()( mgmWWJJ δTT
The sensitivity matrix J has been normalized by dW
and the gradient is
Solve g(m) = 0, and let F[m+m] = F[m] + J m
Matrices Wd, W, S, Q, , G(m,u) are SPARSE!
Solution of the matrix system
IPCG solver with preconditioner
WWIM T 1.0
)()( mgmWWJJ δTT
Computations:
wf Forward modelling: Solve
(1) J v = -Wd S Q G v w
w
(2) JT v = - GT QT ST WdT v
wf Adjoint modelling: Solve
mmm δkk 1Update the model
So each CG iteration has two forward modellings:
Choose 0, mref
Evaluate (mref), g(mref), matrices Wd, W...
22)()][( ref
obsd
md
F mmWdmW
Recall we are solving …
End
For k = 1 max iterations
• Line search for step length
tolk
kdd
)(
)(or or 1*
mg
mg• Exit if
• IPCG to solve )()( mgmWWJJ δTT
• Update model mmm δkk 1
Reduce
For cooling loop
End
Flow chart
Two-prism example
• Loop size 130 x 130 m
• Step off current
• Receivers: Hx, Hy, Hz, Ex, Ey inside the loop
• Times: 32 logarithmically spaced (10-6 – 10-3)
• Gaussian noise (1%) added (N=16,000)
• Inversion model: 423
• Starting and reference model equals true halfspace
Two-prism inversion
-1.75
-2.02
-2.30
Misfit for Two-Prism Example
0 5 10 15 20 25 30 35 404
4.5
5
5.5
6
iteration
log 1
0(m
isfit
)
“Keel”
Tertiary BrecciaMafic Volcanics
Quartz Rhyolit
e
Massive Sulphide
Mafic Volcanics
Ele
vati
on (
m) 20
0016
00
-2000 -1100Easting (m)
LocationGeologic cross section
Physical properties
Field Example: San Nicolas Deposit
Unit Density Susceptibility Resistivity Chargeability(g/cc) (S.I. x10) (ohmm) (msec)
Qal 2 0 50 5Tv 2.3 0 20-30 10Mst./Lst. 2.4 0 150 20Mafic Vol. 2.7 0 80 30Mafic/IntVol. 2.7 0 80 30Sulphide 3.5 10 20 200Qtz. Rhyolite 2.4 0 100 10Graphitic Mst. 2.4 0 100+ 30
Unit Density Susceptibility Resistivity Chargeability(g/cc) (S.I. x103 ) (ohm- m) (msec)
Qal 2 0 50 5Tv 2.3 0 20 -30 10Mst./Lst. 2.4 0 150 20Mafic Vol. 2.7 0 80 30Mafic/IntVol. 2.7 0 80 30Sulphide 3.5 10 20 200Qtz. Rhyolite 2.4 0 100 10Graphitic Mst. 2.4 0 100+ 30
- 10(20)
- 5- 5
- 10- 5
- 30- 40- 50- 50
- 20- 70
Introduction to UTEM Geophysics Survey at San Nicolas
• 3 large loop transmitters– 2 km by 1.5 km
• dB/dt receivers– mainly z component
• transmitter waveform
– 30 Hz sawtooth wave
– dI/dt constant over half cycle
I(t)
dIdt
dBdt
15 ms
Loop 1Loop 2
Loop 9
San Nicolas UTEM Geophysics Survey
dBz/dt
nT/s
UTEM channel 4 (1.513ms)
easting
1075
-1300-3000 -220
nort
hin
g
A simplified procedure for inverting time-domain electromagnetic (TEM) surveys
forward model
a priori
information
discretize
background model
understanding the data
error assignment
inversions
evaluate results
validate
UTEM geophysics survey at San Nicolas
• 10 time channels (0.024 – 12.1 ms )
• Number of data inverted: 3523
• Error assignment: percentage + floor
• Reduced volume: 3.3 × 2.3 × 2.3 km
• Number of cells: 241,920
• Reference model: 90 m layer (10 ohm-m)
• 100 ohm-m halfspace
• Sensitivity weighting for the source loop
• Model objective function: (10^-4, 1,1,1)
Fitting the Observations
View from SW
Observed 15 m iso-surface 1000.0
31.0
1.0
observedpredicteddBz/dt
nT/s
log10(t)
One decay curve: Observed and predicted
Observed
Predicted
San Nicolas inversion results:
easting northing-1000 -500-2500 -500
1000
5
m
Recovered cross section at 450 S Recovered cross section at 1380 W
easting northing-1000 -500-2500 -500
1000
5
m
Resistivity from drilling at 1380 WResistivity from drilling at 450 S
Question: In this case we have extensive drilling and a rock model
to compare. How about the other surveys?
•3 transmitter loops
•3000 (?)
•240,000 cells
•First 3D inversion of TD electromagnetics
for mineral exploration
Stopping Point:
easting northing-1000 -500-2500 -500
1000
5
m
Recovered cross section at 450 S Recovered cross section at 1380 W
“Keel”
Tertiary BrecciaMafic Volcanics
Quartz Rhyolit
e
Massive Sulphide
Mafic Volcanics
Ele
vati
on (
m) 20
0016
00
-2000 -1100Easting (m)
Geologic cross section
Physical properties
Field Example: San Nicolas Deposit
Unit Density Susceptibility Resistivity Chargeability(g/cc) (S.I. x10) (ohmm) (msec)
Qal 2 0 50 5Tv 2.3 0 20-30 10Mst./Lst. 2.4 0 150 20Mafic Vol. 2.7 0 80 30Mafic/IntVol. 2.7 0 80 30Sulphide 3.5 10 20 200Qtz. Rhyolite 2.4 0 100 10Graphitic Mst. 2.4 0 100+ 30
Unit Density Susceptibility Resistivity Chargeability(g/cc) (S.I. x103 ) (ohm- m) (msec)
Qal 2 0 50 5Tv 2.3 0 20 -30 10Mst./Lst. 2.4 0 150 20Mafic Vol. 2.7 0 80 30Mafic/IntVol. 2.7 0 80 30Sulphide 3.5 10 20 200Qtz. Rhyolite 2.4 0 100 10Graphitic Mst. 2.4 0 100+ 30
- 10(20)
- 5- 5
- 10- 5
- 30- 40- 50- 50
- 20- 70
DC resistivity
CSAMT
Other conductivity surveys
Gravity
Magnetics
Induced Polarization
Other Surveys
Outcrop geology
1.7km
3.7k
mTransmitter: 15 frequencies (0.5 - 8192 Hz)
- 3 receiver lines spaced 200m apart;- 60 stations per line @ 25m spacing.
Grid North
Surface projection of the San Nicolas ore body.
San Nicolás: CSAMT survey layout
10
55
300
ohm-m3D model from many 1D column-models
Isosurface view of the same 3D conductivity model
San Nicolás: CSAMT 1D inversion results
10
55
300
ohm-m
Isosurface view of the same 3D conductivity model
San Nicolás: CSAMT 3D inversion results
3D inversion results: Frequencies ..
DC Resistivity
1D CSAMT
3D CSAMT
3D UTEM
-950-2150 -1550
-950-2150 -1550
-950-2150 -1550
-950-2150 -1550
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
0
200
400
600
800
“Keel”
Tertiary BrecciaMafic Volcanics
Quartz Rhyolit
e
Massive Sulphide
Mafic Volcanics
Ele
vati
on (
m) 20
0016
00
-2000 -1100Easting (m)
Geologic cross section
Physical properties
Field Example: San Nicolas Deposit
Unit Density Susceptibility Resistivity Chargeability(g/cc) (S.I. x10) (ohmm) (msec)
Qal 2 0 50 5Tv 2.3 0 20-30 10Mst./Lst. 2.4 0 150 20Mafic Vol. 2.7 0 80 30Mafic/IntVol. 2.7 0 80 30Sulphide 3.5 10 20 200Qtz. Rhyolite 2.4 0 100 10Graphitic Mst. 2.4 0 100+ 30
Unit Density Susceptibility Resistivity Chargeability(g/cc) (S.I. x103 ) (ohm- m) (msec)
Qal 2 0 50 5Tv 2.3 0 20 -30 10Mst./Lst. 2.4 0 150 20Mafic Vol. 2.7 0 80 30Mafic/IntVol. 2.7 0 80 30Sulphide 3.5 10 20 200Qtz. Rhyolite 2.4 0 100 10Graphitic Mst. 2.4 0 100+ 30
- 10(20)
- 5- 5
- 10- 5
- 30- 40- 50- 50
- 20- 70
DC resistivity
CSAMT
Other conductivity surveys
Gravity
Magnetics
Induced Polarization
Other Surveys
density-contrast
conductivitychargeability
magnetic susceptibility
San Nicolás: local scale inversion results (north facing)
Summary
• Developed a practical 3D inversion.
• Surface and borehole applications
• Has worked well in a field example.
• Future work:
– Applications and workflow development
– Extension to multi-source
– Unstructured grids
• Global comment: We can now invert most types of non-seismic surveys to recover a 3D physical property model.