Inversion of EM38 Electrical Conductivity Data The Least Squares Minimization with Tikhonov Regulari

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Inversion of EM38 Electrical Conductivity
DataThe Least Squares Minimization with
Tikhonov Regularization a Case Study

• Ernesto Bonomi and Cristina Manzi
• Environmental and Imaging Sciences, CRS4
• Gian Piero Deidda
• Department of Territorial Engineering, UNICA

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Introduction (I)

• Geophysical EM surveys aim to provide information
  about conductivity of the Earth
• Vadose zone characterization
• Ground water and salinity monitoring
• Detection of contaminants in soils and acquifers
• Detection of metallic debris
• From FEM measurements of the ground apparent
  electrical conductivity, the problem is to supply
  the conductivity profile of the subsurface

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Introduction (II)

• Quantitative inference about subsurface
  conductivity is an ill-posed problem
• Least squares inverse problem
• Tikhonov regularization
• The aim of this work
• illustrate how Tikhonov regularization may be
• the cause of misleading results

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EM38 Instrument

• Fixed frequency f14.6 kHz
• Fixed coil spacing s1 m
• Apparent conductivity (NBs/d ltlt 1)
• Hp primary field
• Hs secondary field
• d skin depth
• Horizontal and vertical configurations

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EM38 Linear Response Model

• McNeills model for a stratified medium
• - s(z) conductivity at depth z
• - FH,V sensitivity of the
  instrument

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The Forward Model (I)

• Apparent conductivity data
• (2NxM)-linear system
• - m constant

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The Forward Model (II)
Apparent conductivity mS/m
Height m
Depth m
Conductivity profile mS/m
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The Inverse Problem
Apparent conductivity mS/m
Height m
Depth m
Conductivity profile mS/m
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Least Squares Problem

• Cost function
• The minimum of e reached for the conductivity
  profile
• Ill-conditioning

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Tikhonov Regularization

• Enhance stability
• trade-off between and
• Ln a discrete differential operator
• New least squares problem
• Solution

Condition number
a
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Tikhonov Regularization

• Enhance stability
• trade-off between and
• Ln a discrete differential operator
• New least squares problem
• Solution

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L-curve Construction

• Tuning a, achieve an acceptable balance among
  stability, accuracy and regularity
• Recipe Optimal value aopt determined by the
  point on the corner of the L-curve

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L-curve Two Examples
Number of data N11 Number of layers M32
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Inverse Problem Solution
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The solver

• Constrain the optimal solution within the
  feasible set
• Projected conjugate gradient
• The problem is extremely ill-conditioned
• However best solution for a0, in the sense of
  proximity to the true conductivity profile

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Eigenvalues of

• Most of the eigenvalues are clustered in a small
  interval, the remaining lie to the right

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Convergence

• CG algorithm converges faster if most of the
  eigenvalues are clustered in a small interval

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A Field Data Example the Poetto Beach

• Near surface material
• medium- to fine-grained sand (gt 60 of quartz)
  4-5 m
• Sea water table depth, varying during the day
  about 2 m
• Five soundings, every 10 m, along a profile
  orthogonal to the shore, starting 65 m before
• EM38 height from 0 to 1.5 m, with a 0.1 m step,
  N16 for each coil-mode configuration

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Apparent Conductivity mS/m
Top layer less conductive than the underlying
ones
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Subsurface Conductivity mS/m
Sandair with a 30-40 porosity low conductivity
Sand fully saturated by salt water high
conductivity
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Conclusion

• Computer experiments provide credibility to our
  results on the EM38 inversion data
• The problem is highly ill-conditioned
• Tikhonov approach and the L-curve criterion is
  the cause of misleading conductivity profiles
• Eigenvalues of the initial least squares problem
  are clustered
• Using the CG, no regularization!!!

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Future Activities

• Multifrequency analysis non-linear inversion of
  the magneto-telluric equation
• Forward problem
• Adjoint problem
• Minimization projected conjugate gradient

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An example