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

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

Inverse Problems

Geophysics Medical Imaging

Astronomy and Remote Sensing Ocean Tomography

Physics

Object Measurement

MeasurementObject

Descriptor

Mathematically Complicated and hard to analyze

Ill-Posed (Obtaining a certain and unique answer is not possible)

Computationally very expensive

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:

:)(

:)(

:)(

:)(

rs

r

r

rv

Electric Potential

Electrical Conductivity

Electrical Permittivity

Current Source Distribution

)()(),( rsrvr

svK )(

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.

),,(

),(),(),(

zyxr

rirr

Example of Pixel Based Reconstruction

40x40 Grid: 10m x 8m

30 sensors 40 experiments SNR=60 dB

Only Tikhonov Regularization

Poor Reconstruction Near sensor distortions Relative Error:191%

More intense near sensor regularization

Imposing positivity Better reconstruction Relative Error:87%

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.

(x, y) 1(x, y)

Shape-Based Reconstruction

Problems with pixel based reconstruction:

• Too many unknowns• Low resolution• Very ill-posed

Level-Set Idea:

A low order representation of resistivity distribution

The footprint or basically the shape, an important quantity of interest

Level-set Curve

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

)()( xUxF bfb )(F

Nc

iiibfb rcUr

1

)()()(

bf

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.

Center

Width

Height

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

Nc

iiiibfb rracUr

1

)()()(

icupdate

irupdate

iaupdate

fbupdate

Shape Based Approach

SNR=60 dB

6.38b 4.194f 40x40 Grid: 10m x 8m 30 sensors 40 experiments SNR=60 dB

Initial level-set curve Final level-set curve

Evolution Process (Movie)

Performing the Parametric Reconstruction (3D)

30x30x30 Grid: 10m x 10m x 10m 100 sensors 120 experiments SNR=60 dB

Shape Based Approach (3D) SNR=60 dB

Original Shape Initial Level-set Final Reconstruction

Evolution Process(Movie)

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

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.

Thank You!