Sviluppo di modelli numerici per la simulazione WP2: Monitoraggio di inquinanti nel sottosuolo con i

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Sviluppo di modelli numerici per la simulazione
WP2 Monitoraggio di inquinanti nel sottosuolo
con inversione di dati

• Cristina Manzi, Ernesto Bonomi
• and Enrico Pieroni
• Environmental and Imaging Sciences, CRS4
• www.crs4.it

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WP2 reconstruction and imaging

• Applied mathematical approach
• Developement of imaging strategies
• Inversion technique ?
  Reconstruction

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Three strategies

• Linear inversion in frequency domain
• Non-linear inversion ? Multifrequency
  approach
• 2D Time domain technique

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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 conjugate gradient algorithm performs as a
  regularizing strategy without tuning of any
  parameters

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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 (II)
Apparent conductivity mS/m
Height m
Depth m
Conductivity profile mS/m
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The Inverse Model (II)
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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Inverse Problem Solution
Conductivity mS/m
Depth m
Depth m
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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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Projected Conjugate Gradient
Projection strategy
Convergence of the algorithm
Conjugate gradient performs as a regularization
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Borchers data set
Conductivity mS/m
Depth m
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A Field Data Example the Poetto Beach

• 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
• 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

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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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Multifrequency analysis

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

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Two strategies
Construction of intermediate solution supplied
by the conjugate gradient

• Average Solution (AS)

• Average Gradient (AG)

Cost function
Global gradient
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An example
Conductivity mS/m
Depth m
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TDEM technique

• Maxwells equations in time domain
• Finite element discretization in spatial domain
• Crank-Nicolson scheme for time discretization
• Secondary electric field

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TDEM technique
Spatial discretization
Temporal discretization

• Mitsuhata boundary conditions

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TDEM - Numerical examples

• Two layered soil with different resistivity
• Time discretization

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TDEM Numerical examples
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TDEM Numerical examples
Variation of the secondary electric field 15m
above the current line on the surface
In late time the slope of the curve is
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Conclusions I Linear model

• Regularization Tikhonov or Conjugate Gradient
• Ill-conditioned problem ? Stability

Projected Conjugate Gradient
Experimental data provide credibility to our
results on the EM38 linear inversion strategy
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Conclusions II Non Linear model

• Local minima
• Constraints ? projection
• Data and instruments?
• Ottimization strategy

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Conclusions III

• In the multi-frequency model, the inversion
  combines an iterative scheme implementing a
  constrained non-linear conjugate gradient
• The AS method gives a good accuracy for shallow
  and deeper layers while the AG method performs
  correctly only in the near zone surface
• TDEM direct algorithm provides a reliable
  reconstruction of the secondary electric field
  for a 2D medium excited by a infinite line source
  
• ? TDEM inversion

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• COLLABORATION WITH
• Gian Piero Deidda, Departement of Territorial
  Engineering, UNICA
• Brian T. Borchers, Department of
  Mathematics, New Mexico Tech, Socorro
• Eiichi Arai, Metal Mining Agency of Japan
• Yuji Mitsuhata, National Institute of Advanced
  Industrial Science and Technology, Japan