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By: Alireza Aghasi, Eric L. Miller, Andrew Ramsburg and Linda Abriola

Parametric Shape-based Inversion in Electrical Impedance Tomography for the Characterization of Subsurface Contaminant Distribution. By: Alireza Aghasi, Eric L. Miller, Andrew Ramsburg and Linda Abriola Tufts University, Medford, MA. Introduction. Motivation

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By: Alireza Aghasi, Eric L. Miller, Andrew Ramsburg and Linda Abriola

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  1. Parametric Shape-based Inversion in Electrical Impedance Tomography for the Characterization of Subsurface Contaminant Distribution By: Alireza Aghasi, Eric L. Miller, Andrew Ramsburg and Linda Abriola Tufts University, Medford, MA

  2. Introduction • Motivation • Down-gradient mass flux and concentration signals have been linked to source zone distribution of DNAPL • Knowledge of distribution (e.g., location, configuration) can aid in remediation planning • Goal of this work • Develop new inverse methods for identification of DNAPL configuration • Evaluate techniques in the context of electrical impedance tomography

  3. Inverse Problems • Mathematically Complicated and hard to analyze • Ill-Posed (Obtaining a certain and unique answer is not possible) • Computationally very expensive Measurement Object Physics Object Descriptor Measurement Medical Imaging Geophysics Astronomy and Remote Sensing Ocean Tomography

  4. Electrical Impedance Tomography • Electrical Impedance Tomography (EIT) is a technique of imaging subsurface structures through some measurements made on the surface or through boreholes. • Poisson’s Equation: Electric Potential Electrical Conductivity Electrical Permittivity Current Source Distribution Forward Problem: Knowing electrical conductivity and permittivity of the media and the current s(x,y,z), we then find the electric potential. Inverse Problem: Knowing the Electric Potential at some points (nodes) and the current s(x,y,z), we then estimate the conductivity and permittivity distribution throughout the media.

  5. Example of Pixel Based Reconstruction • Simulations made for zero frequency (DC current) • Parameter of interest is the resistivity: • Simulation details: • Saturation distribution from UTChem • Resistivity values from Newmark et al., JEEG., 3, 7–13. • More intense near sensor regularization • Imposing positivity • Better reconstruction • Relative Error:87% • Only Tikhonov Regularization • Poor Reconstruction • Near sensor distortions • Relative Error:191% • 40x40 Grid: 10m x 8m • 30 sensors • 40 experiments • SNR=60 dB

  6. Shape-Based Reconstruction • Problems with pixel based reconstruction: • Too many unknowns • Low resolution • Very ill-posed A low order representation of resistivity distribution • Level-Set Idea: The footprint or basically the shape, an important quantity of interest Level-set Curve

  7. Reconstruction Using Basis Expansions • Parametric Shape Based Reconstruction (Level-Sets approach) • Assuming the resistivity to be piecewise constant in the background ( ) and foreground ( ), we estimate source zone boundaries, background resistivity and smooth (low order, constant in this talk) foreground resistivity. • In some sense, an adaptive thresholding approach aimed at recovering “average” resistivity of source zone along with better delineation of boundary. Unknown

  8. What Are the Appropriate Basis Functions • Representing every basis function as a monotonic radial basis function • For every radial basis function (e.g. a Gaussian function), the center, width and the height can vary and are the unknowns in the inversion process. • The unknown shape can be represented as some level-set of this basis expansion. Height Width Center

  9. Choice of Compactly Supported Radial Basis Functions (CSRBF) • They have the bell-shaped property of Gaussian functions, and become strictly zero after a certain radius and causes a huge degree of sparsity in matrices. • Unlike other non-orthogonal basis functions, representation of any continuous function in terms of the summation of CSRBFs (collocation) is proved to be always stable. Gaussian CSRBF

  10. Shape Based Approach Initial level-set curve Final level-set curve • SNR=60 dB • 40x40 Grid: 10m x 8m • 30 sensors • 40 experiments • SNR=60 dB

  11. Evolution Process (Movie)

  12. Performing the Parametric Reconstruction (3D) • 30x30x30 Grid: 10m x 10m x 10m • 100 sensors • 120 experiments • SNR=60 dB

  13. Shape Based Approach (3D) • SNR=60 dB Original Shape Initial Level-set Final Reconstruction

  14. Evolution Process(Movie)

  15. Using different frequencies • SNR=60 dB • The foreground and background conductivity and permittivity assumed to be constant and a priori known • 4 different frequencies used to generate more data: (0, 10, 1K, 10k) Hz Original Reconstructed

  16. Conclusion • A new representation of electrical conductivity-permittivity is discussed, which is closer to the nature of unknowns. • The use of adaptive CSRBFs shows to be • Better in reconstruction • Faster in convergence • More Reliable • Bypassing the regularizations This project is funded and supported by US National Science Foundation (NSF) under Grant EAR 0838313, and we thank them for their support.

  17. Thank You!

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