Eldad HaberInversion of 3D Electromagnetics: A maturing technique in applied geophysicsCollaborators: 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: UXOMilitary proving groundsRegions of conflictAvalanche control

Environmental: How do we find UXO?

?

Geotechnical: A Canadian potash mining

Geotechnical problemWater gushing into the mine

Mineral explorationWhat do we have?Map of surface geology

Solutions GeophysicsEnergy from sourcePhysical propertiesand contrastsMeasurements- Density- Magnetic susceptibility- Electrical resistivity- Chargeability- others - Gravity- Magnetic field anomalies- DC or electromagnetics- Induced polarization- etc. physical properties

Physical propertiesUXO:Electrical conductivity and magnetic susceptibility

Water (at potash mine): Electrical conductivity: high if it has dissolved salt

Minerals: magnetic susceptibility electrical conductivity chargeabilitydensity

3D TEM Setup

Outline: Introduction: Example problems Environmental, geotechnical, resource exploration

TEM forward modelling

Inversion

Mineral exploration example

Summary/discussion

Mathematical SetupThis must be solved in both space and time.time [0, tf]

Implicit MethodBackward Differentiation Formula (BDF)E n depends upon E n-1, H n-1 (BDF1)whereNeed initial conditions to start time stepping.

Initial ConditionsTime stepping starts with fields at time t = 0

Formulation as potentials(1)(2)whereandHelmholtz decompositionCoulomb Gauge Yields

Discretization with FV on staggered grid-- A, J are defined on the faces-- is in the centers -- H is on the edgesFinite Volume discretization: Each equation is integrated over a volume.Yields:

Formulating the forward problemSource term:Maxwells equations:Write:Maxwells Equations:Solve: Aj (m) uj = qj using BICGStab with ILU preconditioner.

Equations:Solve...Forward problem continued

Solving the systemLet nc = number of cellsnumber of unknowns ~ 7(nc)3if nc = 64 matrix size = (2x106)2Solve

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

Forward modelling of fieldsVelocity of smoke ringNabighian (1979)Currents decayInduction finished t~0.01 secSampling 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

What is Inversion?Goal: Estimate the Earth model

Inversion procedure:Divide earth into cells (each with fixed size and unknown value).Inversion: find values for cellsUse mathematical optimization theory.Difficulties:Solution is non-unique.Computationally demanding.

Inversion as optimization: 3 parts

Inverse problem Minimize = d + m where : Regularization parameterQ: Projection matrix u: Potentials : Observations : Model and Reference modelWd, W : Measurement error, model weighting

Solving the inverse problem Differentiating the objective function with model m where sensitivity matrix and

Gauss-Newton method The sensitivity matrix J has been normalized by 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 Computations: So each CG iteration has two forward modellings:

Flow chart

Two-prism example Loop size 130 x 130 mStep off currentReceivers: Hx, Hy, Hz, Ex, Ey inside the loopTimes: 32 logarithmically spaced (10-6 10-3)Gaussian noise (1%) added (N=16,000)

Inversion model: 423Starting and reference model equals true halfspace

Two-prism inversion

Misfit for Two-Prism Example

LocationGeologic cross section Physical propertiesField Example: San Nicolas Deposit

Introduction to UTEM Geophysics Survey at San Nicolas3 large loop transmitters2 km by 1.5 kmdB/dt receiversmainly z componenttransmitter waveform30 Hz sawtooth wavedI/dt constant over half cycle

San Nicolas UTEM Geophysics SurveydBz/dtnT/sUTEM channel 4 (1.513ms)easting1075-1300-3000-220northing

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

UTEM geophysics survey at San Nicolas 10 time channels (0.024 12.1 ms )Number of data inverted: 3523Error assignment: percentage + floorReduced volume: 3.3 2.3 2.3 kmNumber of cells: 241,920Reference model: 90 m layer (10 ohm-m) 100 ohm-m halfspace Sensitivity weighting for the source loopModel objective function: (10^-4, 1,1,1)

Fitting the Observations

San Nicolas inversion results:eastingnorthing-1000-500-2500-50010005mRecovered cross section at 450 SRecovered cross section at 1380 W

Question: In this case we have extensive drilling and a rock modelto compare. How about the other surveys? 3 transmitter loops3000 (?) 240,000 cellsFirst 3D inversion of TD electromagneticsfor mineral exploration Stopping Point:eastingnorthing-1000-500-2500-50010005mRecovered cross section at 450 SRecovered cross section at 1380 W

Geologic cross section Physical propertiesField Example: San Nicolas Deposit

1.7km3.7kmTransmitter: 15 frequencies (0.5 - 8192 Hz)- 3 receiver lines spaced 200m apart;- 60 stations per line @ 25m spacing.Grid NorthSurface projection of the San Nicolas ore body.San Nicols: CSAMT survey layout

3D model from many 1D column-modelsIsosurface view of the same 3D conductivity modelSan Nicols: CSAMT 1D inversion results

Isosurface view of the same 3D conductivity modelSan Nicols: CSAMT 3D inversion results3D inversion results: Frequencies ..

DC Resistivity1D CSAMT3D CSAMT3D UTEM-950-2150-1550-950-2150-1550-950-2150-1550-950-2150-15500200400600800020040060080002004006008000200400600800

Geologic cross section Physical propertiesField Example: San Nicolas Deposit

density-contrastconductivitychargeabilitymagnetic susceptibilitySan Nicols: local scale inversion results (north facing)

Summary Developed a practical 3D inversion.Surface and borehole applicationsHas worked well in a field example.Future work:Applications and workflow developmentExtension to multi-source Unstructured gridsGlobal comment: We can now invert most types of non-seismic surveys to recover a 3D physical property model.

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This is pretty complicated at first glance but all of the elements are important. (Unfortuanately flowcharts tend to be that way)Also, it depends critically upon what you plan to say and how particular parts of the slide are brought up.

Possibly: omit top part on survey design. Important elements:Data acquisitionModelling: check discretization errors and make sure mesh, time steps etc are ok. Understanding : normalizations, orientations (this could flash in red!!!!)(generate a crude model: do responses look like the observations (ie are we in the right ball-park) is the target likely to be detectable ( this should have been done)

Error assignmentInversionsEvaluate results (possible iterate) (a prior information about the model, altering details about errors)

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