3-D decoupled inversion of complex conductivity data in the real number domain

Complex conductivity imaging (also called induced polarization imaging or spectral induced polarization imaging when conducted at multiple frequencies) involves estimating the frequency-dependent complex electrical conductivity distribution of the subsurface. The superior diagnostic capabilities provided by complex conductivity spectra have driven advancements in mechanistic understanding of complex conductivity as well as modelling and inversion approaches over the past several decades. In this work, we demonstrate the theory and application for an approach to 3-D modelling and inversion of complex conductivity data in the real number domain. Beginning from first principles, we demonstrate how the equations for the real and imaginary components of the complex potential may be decoupled. This leads to a description of the real and imaginary source current terms, and a corresponding assessment of error arising from an assumption necessary to complete the decoupled modelling. We show that for most earth materials, which exhibit relatively small phases (e.g. less than 0.2 radians) in complex conductivity, these errors become insignificant. For higher phase materials, the errors may be quantified and corrected through an iterative procedure. We demonstrate the accuracy of numerical forward solutions by direct comparison to corresponding analytic solutions. We demonstrate the inversion using both synthetic and field examples with data collected over a waste infiltration trench, at frequencies ranging from 0.5 to 7.5 Hz. In addition to insight provided by the decoupled equations, we see two primary advantages of the decoupled inversion in comparison to fully coupled inversion in complex number domain. First, memory requirements are reduced by a factor of two because it is unnecessary to simultaneously store both the real and complex components and associated Jacobian matrices. Second, and most importantly, data noise is specified separately for the real and imaginary components of potential (or magnitude and phase), which removes the ambiguity in data misfit residuals that exists in the complex number domain, thereby enabling the inversion to appropriately fit the data. Additionally, the modelling and inversion algorithms are equivalent to the direct current electrical resistivity tomography counterpart. This enables direct application of advancements and approaches developed originally for electrical resistivity tomography.