Theory

 

        The imaging of ERT data depends on the development of a sophisticated inverse algorithm to reconstruct three-dimensional (3-D) conductivity distributions from electrical resistance measurements.  The results presented here used the 3-D anisotropic inverse algorithm described by LaBrecque and Casale⁸.  The algorithm uses the so-called Occam’s inversion method and seeks to find the smoothest possible solution that still fits the data within a specified a-priori value^3,8.  An iterative approach is used in which a series of linear approximations are used to search for a solution to the nonlinear inverse problem. Each iteration starts by comparing the data with a forward solution, the numerical approximation of the equation:

 

                                                                                ,                                                 (1)

 

where V is the electrical potential in the earth,    is the anisotropic electrical conductivity tensor and I is the distribution of electric source currents within the earth.  An anisotropic version of the finite-difference formulation of Dey and Morrison⁹ is used to solve for the potentials at a pair of receiver electrodes within a 3-D anisotropic earth. This formulation uses the simplified form of anisotropy that assumes that the axes of the conductivity ellipse are aligned with the coordinate directions so that the conductivity tensor simplifies to

 

                                                                 .                             (2)

 

A conjugate-gradient routine with a symmetric successive-over-relaxation (SSOR) preconditioner is used to solve the linear system of equations for the forward problem¹⁰. 

 

        To start the inversion, the user must provide an initial guess for the earth conductivity structure.  Usually, this is a homogeneous half-space with a best-guess estimate of the conductivity.  The forward model is used to create a local, linear approximation to the nonlinear relation between the conductivities and the data.  Using this linear approximation, the algorithm then seeks to improve the estimate of the conductivity.  We define the optimal conductivity structure, and thus determine how it is “improved” by the objective function,

 

                                                                                                     ,         (3)

 

where d is the vector of data values, m is the vector of parameters, g(m) is the forward solution,   W_D is the diagonal data weight matrix, R is the regularization operator which is discussed below, d_obs  is the vector of observed data, and a is an empirical factor that controls the amount of regularization versus the fit of the forward model to the data.  Minimizing the objective function for a large value of a results in a smooth solution but a poor data fit.  The optimal solution corresponds to the largest possible value of a which still fits the data to some a-priori value.  LaBrecque et al.³ discuss a method of iteratively determining a for 3-D inversion.  In the new algorithm, the parameters, m, are the natural logarithms of the three components of conductivity (X, Y, and Z) of each cell in the finite-difference mesh.

 

The nonlinear iterations can be expressed as m_n+1 = m_n + Dm_n. The parameter change vector at the n^th iteration, Dm_n , is obtained by solving the linear system

 

                                                                ,      (4)

 

where the elements of the sensitivity matrix,  G^T, are given by 

 

                                .                                                                           (5)

 

The system of equations given by Equation (4) is positive-definite and is solved using the conjugate-gradient method with a diagonal preconditioner³.

 

        The regularization operator is discussed in more detail by LaBrecque and Casale⁸ but has the general form

  

                                                                ,      (6)

               

where  , , and  matrices are used to control the roughness in the X, Y and Z directions respectively and  is used to minimize the anisotropy. 

                The code also implements a differencing inversion scheme similar to that described by LaBrecque and Yang⁶ to allow effective imaging of small changes in background resistivity. Our difference inversion algorithm is a modified version of an Occam's inverse method ^3,8  in which we invert on the difference in data in terms of the difference in parameters using a system of the form: 

 

                                                                                 ,               (7)

 

where

d⁰_obs is a prior  “background” data vector and

 m⁰ is a model derived by Occam’s inversion of the  background data.

 

The objective function given by Equation (4) becomes:

 

 

                   .    (8)

 

LaBrecque and Yang⁶ showed that the scheme improves the ability of ERT to monitor small changes over time.  

 

References

 

1.     Daily, W. D., Ramirez, A., LaBrecque, D. J., and Nitao, J., 1992, Electrical resistivity tomography of vadose water movement: Water Resources Research, 28, 1429-1442.

 

2.     Ramirez, A. L., Daily, W. D., and Newmark, R. L., 1995, Electrical resistance tomography for steam injection monitoring and process control; Journal of Environmental and Engineering Geophysics, 0, 39-52.

 

3.    LaBrecque, D. J., Morelli, G., Daily W., Ramirez, A., and Lundegard, P., 1999, Occam's Inversion of 3D ERT data: in: Spies, B., (Ed.), Three-Dimensional Electromagnetics, SEG, Tulsa, 575-590.

 

4.     Stubben, M. A., and LaBrecque, D. J., 1998, 3-D ERT inversion to monitor and injection experiment: Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems(SAGEEP)’98, 603-612.

 

5.     Slater, L., A. Binley, D. Brown, 1997, "Electrical Imaging of the Response of Fractures to Ground Water Salinity Change", Ground Water, 35(3), 436-442.

 

6.     LaBrecque, D. J. and Yang, X., 2001, Difference inversion of ERT data:  a fast inversion method for 3-D in situ monitoring: Journal of Environmental and Engineering Geophysics, 5, 83-90.

 

 

7.     Ramirez, A., Daily, W., Buettner, M.,  and LaBrecque, D., 1997, Electrical Resistivity Monitoring of the Thermomechanical Heater Test inYucca Mountain:  Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems (SAGEEP) ’97, 11-20.

 

8. LaBrecque and Casale, 2002, Experience with Anisotropic Inversion for Electrical Resistivity Tomography, submitted to Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems (SAGEEP) ‘02.

 

9.     Dey, A., and Morrison, H. F., 1979, Resistivity modeling for arbitrarily shaped three-dimensional structures; Geophysics, 44, 753-780.

 

 

10.   Spitzer, K., 1995, A 3-D finite difference algorithm for DC resistivity modeling using conjugate gradient methods: Geophysical J. Int'l, 123, 903-914.