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Inversion of 3D Electromagnetics: A maturing technique in applied geophysics. Eldad Haber. Collaborators: Doug Oldenburg, Roman Shekhtman,Scott Napier, and Rob Eso. Outline:. Introduction: Example problems Environmental, geotechnical, resource exploration Geophysical surveys Inversion

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Inversion of 3D Electromagnetics:

A maturing technique in applied geophysics

Eldad Haber

Collaborators: Doug Oldenburg, Roman Shekhtman,Scott Napier, and Rob Eso


Outline
Outline:

  • Introduction: Example problems

    • Environmental, geotechnical, resource exploration

  • Geophysical surveys

  • Inversion

  • Mineral exploration example

  • Summary/discussion


Environmental uxo

http://www.nohowinc.com/

http://www.dma.state.mn.us/

http://www.centennialofflight.gov

Environmental: UXO

  • Military proving grounds

  • Regions of conflict

  • Avalanche control



Geotechnical a canadian potash mining
Geotechnical: A Canadian potash mining


Geotechnical problem
Geotechnical problem

  • Water gushing into the mine


Mineral exploration
Mineral exploration

What do we have?

Map of surface geology


Solutions geophysics
Solutions … Geophysics

Energy from source

physical properties

- Density

- Magnetic susceptibility

- Electrical resistivity

- Chargeability

- others …

Physical properties

and contrasts

- Gravity

- Magnetic field anomalies

- DC or electromagnetics

- Induced polarization

- etc. …

Measurements


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


3d tem setup

Waveform

(half sine, step…)

I

Time

Source

(Loop or grounded electrode)

Measurements

(E, H, dB/dt)

Borehole

(E, H, dB/dt)

Depth

3D TEM Setup


Outline1
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

Mathematical Setup

time [0, tf]

This must be solved in both space and time.


Implicit method
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

I(t)

Case 1:

t

0

Case 2:

E0, H0 fields are from DC currents

t

0

Solve

DC Resistivity

MMR

Initial Conditions

Time stepping starts with fields at time t = 0


Formulation as potentials
Formulation as potentials

(1)

(2)

where

and

Helmholtz decomposition

Coulomb Gauge

Yields


Discretization with fv on staggered grid
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
Formulating the forward problem

Maxwell’s Equations:

Source term:

Maxwell’s equations:

Write:

Solve: Aj (m)uj = qj using BICGStab with ILU preconditioner.


Solving the system

Solve An, Φ system using BICGstab with block ILU preconditioner (same as EH3D)

Solving the system

Solve

Let nc = number of cells

number of unknowns ~ 7(nc)3

if nc = 64 matrix size = (2x106)2


The cominco example

Y

Z

Loop source, 1km square

625 m

X

(0,0,0)

target

Surface

Target (x, y, z) = (250, 500, 100)

Conductivity = 1.0 S/m

Host resistivity = 200 Ohm-m

The Cominco example

The “Cominco” model: loop source, conductive target in a host


Forward modelling of fields
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]







Waveform

Source

(half sine, step…)

I

(Loop or grounded electrode)

Time

Surface Measurement

(E, H, dB/dt)

Borehole Measurement

(E, H, dB/dt)

Depth

Next Step: Inversion

Conclusion


What is inversion

Airborne, surface or borehole measurements

Inversion

processing

?

Inversion

What is Inversion?

Goal: Estimate the Earth model


Inversion procedure

Prior information

Inversion

Measurements

Pre-processing

Physical property

distributions = Inversion

3D, and ~ 105 cells

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.


Inversion as optimization 3 parts

A priori information: reference model, structural detail...

Model objective function:

  • s, x … constants

  • m0 : reference model

Misfit:

  •  i: standard deviation

Inversion as optimization:

 = d +  m.0 <  <  is a constant

Choose  such that d < Tolerance

Inversion as optimization: 3 parts


Inverse problem

Minimize  = d +  m

where

 : Regularization parameter

Q: Projection matrix

u: Potentials

: Observations

: Model and Reference model

Wd,W : Measurement error, model weighting


Solving the inverse problem

Differentiating the objective function with model m

where sensitivity matrix

and


Gauss-Newton method

Solve g(m) = 0, and let F[m+m] = F[m] + J m

The sensitivity matrix J has been normalized by

and the gradient is

Matrices Wd, W, S, Q, , G(m,u) are SPARSE!


IPCG solver with preconditioner

(1) J v = -Wd S QG v

Forward modelling:

Solve

(2) JT v = - GT QT ST WdT v

Adjoint modelling:

Solve

Update the model

Solution of the matrix system

Computations:

So each CG iteration has two forward modellings:


Flow chart

Recall we are solving …

Choose0, mref

Evaluate (mref), g(mref), matrices Wd, W...

For  cooling loop

For k = 1  max iterations

  • IPCG to solve

  • Line search for step length 

  • Update model

  • Exit if

End

Reduce 

End

Flow chart


Two prism example
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

Two-prism inversion



Field example san nicolas deposit

Tertiary Breccia

Mafic Volcanics

2000

Quartz Rhyolite

Massive Sulphide

Elevation (m)

Mafic Volcanics

“Keel”

Unit

Unit

Density Susceptibility Resistivity

Density Susceptibility Resistivity

Chargeability

Chargeability

(g/cc) (S.I. x10

(g/cc) (S.I. x103

) (ohm

) (ohm

- m) (msec)

m) (msec)

1600

- 10

(20)

- 5

- 5

- 10

- 5

- 30

- 40

- 50

- 50

- 20

- 70

Qal

Qal

2 0

2 0

50 5

50 5

-2000

Easting (m)

-1100

Tv

Tv

2.3 0

2.3 0

20

20

-

-

30

30

10

10

Mst./Lst

Mst./Lst

.

.

2.4 0

2.4 0

150 20

150 20

Mafic Vol.

Mafic Vol.

2.7 0

2.7 0

80 30

80 30

Mafic/Int

Mafic/Int

Vol.

Vol.

2.7 0

2.7 0

80 30

80 30

3.5 10

3.5 10

20 200

20 200

Sulphide

Sulphide

Qtz

Qtz

. Rhyolite

. Rhyolite

2.4 0

2.4 0

100 10

100 10

Graphitic

Graphitic

Mst

Mst

.

.

2.4 0

2.4 0

100+ 30

100+ 30

Field Example: San Nicolas Deposit

Geologic cross section

Location

Physical properties


Introduction to utem geophysics survey at san nicolas

3 large loop transmitters

2 km by 1.5 km

dB/dt receivers

mainly z component

I(t)

15 ms

dI

dt

dB

dt

Introduction to UTEM Geophysics Survey at San Nicolas

  • transmitter waveform

    • 30 Hz sawtooth wave

    • dI/dt constant over half cycle


San nicolas utem geophysics survey

Loop 2

Loop 1

Loop 9

San Nicolas UTEM Geophysics Survey

UTEM channel 4 (1.513ms)

1075

dBz/dt

northing

nT/s

-1300

-220

-3000

easting


A simplified procedure for inverting time domain electromagnetic tem surveys

understanding the data

background model

discretize

validate

forward model

error assignment

a priori

information

inversions

evaluate results

A simplified procedure for inverting time-domain electromagnetic (TEM) surveys


Utem geophysics survey at san nicolas
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

Observed 15 m iso-surface

1000.0

31.0

1.0

View from SW

One decay curve: Observed and predicted

Observed

Predicted

observed

dBz/dt

nT/s

predicted

log10(t)

Fitting the Observations


San nicolas inversion results

1000

Resistivity from drilling at 450 S

Resistivity from drilling at 1380 W

m

-500

-1000

-500

-2500

5

northing

easting

San Nicolas inversion results:

1000

Recovered cross section at 450 S

Recovered cross section at 1380 W

m

-500

-1000

-500

-2500

5

northing

easting


Stopping point
Stopping Point:

1000

Recovered cross section at 450 S

Recovered cross section at 1380 W

m

  • 3 transmitter loops

  • 3000 (?)

  • 240,000 cells

  • First 3D inversion of TD electromagnetics

    for mineral exploration

-500

-1000

-500

-2500

5

northing

easting

Question: In this case we have extensive drilling and a rock model

to compare. How about the other surveys?


Field example san nicolas deposit1

Tertiary Breccia

Mafic Volcanics

2000

Quartz Rhyolite

Massive Sulphide

Elevation (m)

Other conductivity surveys

DC resistivity

CSAMT

Mafic Volcanics

Other Surveys

“Keel”

Unit

Unit

Density Susceptibility Resistivity

Density Susceptibility Resistivity

Chargeability

Chargeability

(g/cc) (S.I. x103

(g/cc) (S.I. x10

) (ohm

) (ohm

- m) (msec)

m) (msec)

1600

Gravity

Magnetics

Induced Polarization

- 10

(20)

- 5

- 5

- 10

- 5

- 30

- 40

- 50

- 50

- 20

- 70

Qal

Qal

2 0

2 0

50 5

50 5

-2000

Easting (m)

-1100

Tv

Tv

2.3 0

2.3 0

20

20

-

-

30

30

10

10

Mst./Lst

Mst./Lst

.

.

2.4 0

2.4 0

150 20

150 20

Mafic Vol.

Mafic Vol.

2.7 0

2.7 0

80 30

80 30

Mafic/Int

Mafic/Int

Vol.

Vol.

2.7 0

2.7 0

80 30

80 30

3.5 10

3.5 10

20 200

20 200

Sulphide

Sulphide

Qtz

Qtz

. Rhyolite

. Rhyolite

2.4 0

2.4 0

100 10

100 10

Graphitic

Graphitic

Mst

Mst

.

.

2.4 0

2.4 0

100+ 30

100+ 30

Field Example: San Nicolas Deposit

Geologic cross section

Physical properties


San nicol s csamt survey layout
San Nicolás: CSAMT survey layout

Outcrop geology

Grid

North

Transmitter: 15 frequencies (0.5 - 8192 Hz)

1.7km

3.7km

Surface projection of the San Nicolas ore body.

- 3 receiver lines spaced 200m apart;

- 60 stations per line @ 25m spacing.


San nicol s csamt 1d inversion results
San Nicolás: CSAMT 1D inversion results

300

ohm-m

55

10

3D model from many 1D column-models

Isosurface view of the same 3D conductivity model


San nicol s csamt 3d inversion results
San Nicolás: CSAMT 3D inversion results

300

ohm-m

55

10

3D inversion results: Frequencies ..

Isosurface view of the same 3D conductivity model


DC Resistivity

3D CSAMT

0

0

200

200

400

400

600

600

800

800

-950

-950

-2150

-1550

-2150

-1550

3D UTEM

1D CSAMT

0

0

200

200

400

400

600

600

800

800

-950

-2150

-1550

-950

-2150

-1550


Field example san nicolas deposit2

Tertiary Breccia

Mafic Volcanics

2000

Quartz Rhyolite

Massive Sulphide

Elevation (m)

Other conductivity surveys

DC resistivity

CSAMT

Mafic Volcanics

Other Surveys

“Keel”

Unit

Unit

Density Susceptibility Resistivity

Density Susceptibility Resistivity

Chargeability

Chargeability

(g/cc) (S.I. x103

(g/cc) (S.I. x10

) (ohm

) (ohm

- m) (msec)

m) (msec)

1600

Gravity

Magnetics

Induced Polarization

- 10

(20)

- 5

- 5

- 10

- 5

- 30

- 40

- 50

- 50

- 20

- 70

Qal

Qal

2 0

2 0

50 5

50 5

-2000

Easting (m)

-1100

Tv

Tv

2.3 0

2.3 0

20

20

-

-

30

30

10

10

Mst./Lst

Mst./Lst

.

.

2.4 0

2.4 0

150 20

150 20

Mafic Vol.

Mafic Vol.

2.7 0

2.7 0

80 30

80 30

Mafic/Int

Mafic/Int

Vol.

Vol.

2.7 0

2.7 0

80 30

80 30

3.5 10

3.5 10

20 200

20 200

Sulphide

Sulphide

Qtz

Qtz

. Rhyolite

. Rhyolite

2.4 0

2.4 0

100 10

100 10

Graphitic

Graphitic

Mst

Mst

.

.

2.4 0

2.4 0

100+ 30

100+ 30

Field Example: San Nicolas Deposit

Geologic cross section

Physical properties


San nicol s local scale inversion results north facing
San Nicolás: local scale inversion results (north facing)

magnetic susceptibility

density-contrast

conductivity

chargeability


Summary
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.


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