The largest ERT inversion ever conducted was searching for subsurface contamination at Hanford's B-tank farm, site of underground tanks used to contain hazardous nuclear wastes. This challenge involved approximately 5,000 electrodes, 220,000 measurements, and an imaging computational model of about 3 million elements. Unlike leading commercial software, E4D-RT was able to analyze that enormous data set all at once using supercomputer processing. The following demonstrates the visualization of that contamination E4D-RT provided.
E4D computes the subsurface potential distribution arising from a known current source using the weak form finite element solution to the Poison equation on an unstructured tetrahedral mesh. Mesh generation capabilities are provided by the 2D triangular mesh generator Triangle , and the 3D tetrahedral mesh generator TetGen . E4D constructs the input files for Triangle and TetGen b ased on a user generated mesh configuration fileto produce a customized tetrahedral mesh. The interface provided by E4D enabl es efficient modeling of surface topography subsurface features with known location and dimension (i.e. geologic boundaries, wellbores, buried tanks and pipes, water table boundaries, non-point electrodes etc.). These features can be used to constrain the inverse problem using a flexible set of conditional imaging constraints based the method of Iteratively Reweighted Least Squares, thereby enabling users to incorporate known information into the inversion problem to improve imaging resolution.
Forward modeling is referred to herein as the process whereby E4D simulates the subsurface potential distribution arising from a current injection event, and/or to the process whereby E4D simulates an Electrical Resistivity Tomography or Spectral Induced Polarization survey. Forward modeling is a critical component of the imaging algorithm, and is also essential for assessing the performance of a given electrode configuration and measurement sequence under likely field conditions. E4D was designed to facilitate forward modeling studies by enabling visualization of potential distributions (both real and complex), by generating the files by which to compare simulated data with observed data, and by simulating ERT and SIP surveys. When executed in forward modes, E4D automatically generates the survey files necessary to invert the synthetic data in order to investigate, for example, the influence of electrode placement, measurement sequence, model constraints, mesh refinement, and data noise on imaging resolution. E4D also computes the analytic ERT solution for a homogeneous half-space, which facilitates analysis of numeric solution accuracy for a given mesh refinement. Forward modeling examples are provided and used extensively in the tutorial sections of the E4D User Guide contained at the end of each chapter.
E4D is implemented with a flexible set of tools enabling users to supply the inversion with prior information. These tools include customized mesh capabilities, which enable users to place known boundaries in the mesh, and to use those boundaries to formulate constraints on the inversion. The constraints themselves are user chosen according to one or more structural metrics coupled with a corresponding weighting function, and are applied using the method of Iteratively Reweighted Least Squares. The weighting functions determine the conditions under which the structural metrics (e.g. similarity constraints, inequality constraints, known value constraints etc.) should be applied to the inversion. Constraint settings can be adjusted to inform the inversion of numerous conditions, for example, the upper and lower conductivity limits of a particular mesh zone, the sign of the change in conductivity across a known boundary or time, the conductivity gradient at which a similarity constraint should be removed, or the conductivity values that the inversion is allowed to use to fit the data (i.e. a piecewise constant inversion). Some examples of these capabilities are provided in the tutorial sections of the E4D User Guide. Details concerning the inverse solution constraint capabilities implemented in E4D are provided in the E4D Theory Guide.
E4D provides the capability to model and invert Spectral Induced Polarization (SIP) data using an approach that partially decouples the amplitude and phase inversions so that they can be executed separately. By so doing, all computations may be executed in the real number domain, thereby removing the necessity to store complex numbers, which reduces inversion memory requirements by 50%. The decoupling also facilitates the capability to specify different noise levels and impose different inversion constraints on the amplitude and phase data. Furthermore, all of the solution constraint options available for ERT inversions are also available for SIP inversions.
"Tank-scale" imaging refers to ERT or SIP imaging where current flow within the system is constrained to a closed set of zero-flux current boundary conditions. E4D provides the capability to model and invert tank scale data using a computational mesh that conforms to tank dimension. All of the capabilities available in standard ERT and SIP modes are also available for tank-scale imaging.
Time lapse ERT image of fluid transport within dual saturated-unsaturated tank scale system
E4D links field data collection hardware with supercomputing systems via wireless internet, thereby enabling real-time subsurface imaging. The example below demonstrates real-time imaging of uranium contamination remediation within the vadose zone (i.e. the subsurface region between the groundwater table and the surface) at the Hanford Site. Here a phosphate solution is infiltrated from 2 meters below the surface. The solution binds uranium to the sediments, thereby preventing it from migrating downward to the groundwater. E4D was used to monitor phosphate infiltration in real time. Images were provided every 12 minutes for three weeks, enabling site remediation operators to ‘see’ the distribution of phosphate in the subsurface. Results were delivered in real-time by secured internet link.
E4D was designed specifically to address the computational demands of inverting large ERT and SIP data sets by leveraging distributed-memory computing resources. E4D is highly scalable, and can be used on systems ranging from laptop computers to state-of-the-art supercomputing systems. The largest inversion executed to date involved a data set with approximately 5000 electrodes, 220,000 measurements, and a mesh with approximately 3 million elements, executed on NERSC supercomputing resources using 10,000 processors.
E4D is written in Fortran95 and uses Message Passing Interface (MPI) libraries for communication between processors. Forward solutions are computed using start-of-the-art solver routines from the Portable, Extensible Toolkit for Scientific Computing (PETSc), and a customized parallel Conjugate Gradient Least Squares algorithm designed for the inverse problem. E4D uses a master-slave configuration, where a master process communicates with user inputs and orchestrates computations, and slave processes accommodate the primary computational burdens (i.e. forward modeling, Jacobian matrix construction, and inversion) in terms of both CPU cycles and memory requirements.
Spectral Induced Polarization tomography has been recognized as having potential for high content root phenotyping diagnosis. Roots of many plants have been observed to exhibit anomalous electrical properties in comparison to host soils. This provides the opportunity for non-invasive root phenotyping through electrical tomographic imaging. Below is shown an in silico feasibility assessment of the capability to image root zone electrical properties using a vertical mini-electrode array, conducted using E4D. The left image shows the true root distribution, which includes a series of roots of approximately 2 mm in diameter. The right image shows the corresponding 3D tomographic imaging results. The boundaries and depth of the root zone are well recovered in the tomographic image. The root density is manifest in the tomographic image in terms of the electrical conductivity magnitude. Such relationships between root zones and their corresponding electrical properties can provide critical phenotypic metrics for identifying plants with desirable root traits.