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


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:


1000

5

m

Recovered cross section at 450 S Recovered cross section at 1380 W

“Keel”


Quartz Rhyolit

e

Massive Sulphide

Mafic Volcanics

Ele

vati

on (

m) 20

0016

00

-2000 -1100Easting (m)

Geologic cross section

Physical properties






- 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”


Quartz Rhyolit

e

Massive Sulphide

Mafic Volcanics

Ele

vati

on (

m) 20

0016

00

-2000 -1100Easting (m)

Geologic cross section

Physical properties






- 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.

eldad haber

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