Math 2.0

Geophysics Definition

Electrical resistivity tomography

Builds on Geophysical inversionCalderón problemGreen's function

Definition

Let Ω\Omega be the subsurface, with conductivity σ(x)>0\sigma(x) > 0 and resistivity ρ=1/σ\rho = 1/\sigma. A current II injected at electrode AA and withdrawn at electrode BB produces a potential φ\varphi. It solves the following equation, with no normal current through the ground surface: ∇⋅(σ∇φ)=−I δA+I δBin Ω.\nabla \cdot (\sigma \nabla \varphi) = -I\, \delta_A + I\, \delta_B \quad \text{in } \Omega. Here δA\delta_A and δB\delta_B are point sources at the electrodes. A measurement records the voltage ΔV=φ(M)−φ(N)\Delta V = \varphi(M) - \varphi(N) between two other electrodes MM and NN. The apparent resistivity of the configuration is ρa=K ΔV/I\rho_a = K\, \Delta V / I, where the geometric factor KK is chosen so that ρa=ρ\rho_a = \rho for a homogeneous half-space. Electrical resistivity tomography is the geophysical inversion that recovers σ\sigma on Ω\Omega from the values ΔV/I\Delta V / I of many electrode configurations (A,B,M,N)(A, B, M, N).

Remarks

In a homogeneous half-space a single surface electrode with current II gives φ(x)=ρI/(2π∣x−A∣)\varphi(x) = \rho I / (2 \pi \abs{x - A}), a Green's function of the problem with the image of the source in the surface. Example: the Wenner array has A,M,N,BA, M, N, B on a line with equal spacings aa; then ΔV=ρI2π(1a−12a−12a+1a)=ρI2πa\Delta V = \frac{\rho I}{2\pi}\bigl(\frac{1}{a} - \frac{1}{2a} - \frac{1}{2a} + \frac{1}{a}\bigr) = \frac{\rho I}{2\pi a}, so K=2πaK = 2 \pi a. Non-example: a single resistance measurement with one fixed configuration gives one number, which cannot separate depth from lateral changes, so it is not a tomography. With all current and voltage pairs on the boundary of a bounded body, the data form the Neumann-to-Dirichlet map of the Calderón problem. This map carries the same information as the Dirichlet-to-Neumann map. Field surveys collect a finite subset and use regularised inversion.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Boundary, concept, no evidence levelBoundaryconceptContinuous function, concept, no evidence levelContinuous functionconceptDirichlet-to-Neumann operator, concept, no evidence levelDirichlet-to-Neumann …conceptDomain in the complex plane, concept, no evidence levelDomain in the complex…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptGeophysical inversion, concept, no evidence levelGeophysical inversionconceptGreen's function, concept, no evidence levelGreen's functionconceptInverse problem, concept, no evidence levelInverse problemconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptHarmonic function, concept, no evidence levelHarmonic functionconceptDirichlet problem, problem, no evidence levelDirichlet problemproblemCalderón problem, problem, no evidence levelCalderón problemproblemElectrical resistivity tomography, definition, no evidence levelElectrical resistivit…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Boundary concept
  2. Continuous function concept
  3. Dirichlet-to-Neumann operator concept
  4. Domain in the complex plane concept
  5. Euclidean space object
  6. Function concept
  7. Geophysical inversion concept
  8. Green's function concept
  9. Inverse problem concept
  10. Total derivative concept
  11. Function of several variables concept
  12. Vector-valued function concept
  13. Differentiable function of several variables concept
  14. Partial derivative concept
  15. Vector field concept
  16. Gradient concept
  17. Divergence concept
  18. Laplacian concept
  19. Harmonic function concept
  20. Dirichlet problem problem
  21. Calderón problem problem
  22. Electrical resistivity tomography definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Electrical_resistivity_tomography
About this page
TagXZNARSEBZ1
URIm2://XZNARSEBZ1
Kinddefinition
WritingW0 stub
MSC86A22, 35R30, 35J25
DefinesElectrical resistivity tomography
UsesGeophysical inversion, Calderón problem, Green's function
Changed3 November 2026

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