Electrical resistivity tomography (ERT) is a geophysical technique for imaging sub-surfaces structures from electrical measurements made at the surface, or by electrodes in bore holes. It is closely related to the medical imaging technique electrical impedance tomography, and mathematically is the same inverse problem. The technique evolved from techniques of electrical prospecting that predate digital computers, where layers or anomalies were sought rather than images. Early work on the mathematical problem in the 1930s assumed a layered medium (see for example Langer, Slichter). Tikhonov who is best known for his work on regularization of inverse problems also worked on this problem. He explains in detail how to solve the ERT problem in a simple case of 2-layered medium. During the 1940s he collaborated with geophyicists and without the aid of computers they discovered large deposits of copper. As a result they were awarded a State Prize of Soviet Union. Geophysics, the study of the earth by quantitative physical methods, especially by seismic reflection and refraction, geodesy, gravity, magnetic, electrical, electromagnetic, and radioactivity methods. ... Electrical Impedance Tomograpy (EIT), is a medical imaging technique in which an image of the conductivity or permittivity of part of the body is inferred from surface electrical measurements. ... The inverse problem is the task that often occurs in many branches of science and mathematics where the values of some model parameter(s) must be obtained via manipulation of observed data. ... This article or section does not cite its references or sources. ... Andrey Nikolayevich Tychonoff (Андрей Николаевич Тихонов: October 30, 1906–1993) was a Russian mathematician. ... // Events and trends World War II was a truly global conflict with many facets: immense human suffering, fierce indoctrination, and the use of new, extremely devastating weapons such as the atomic bomb. ...

When adequate computers became widely available the inverse problem of ERT could be solved numerically, and the work of Loke and Barker at Birmingham University was among the first such solution, and their approach is still widely used. The University of Birmingham is the oldest of three universities in the English city of Birmingham. ...

Applications of ERT include mineral prospecting, monitoring of ground water flow and archeology Archaeology or sometimes in American English archeology (from the Greek words αρχαίος = ancient and λόγος = word/speech) is the study of human cultures through the recovery, documentation and analysis of material remains, including architecture, artefacts, biofacts, human remains, and landscapes. ...

References

• R.E. Langer, On an inverse problem in differential equations, Bull Am Math Soc , 39, pp814--820, 1933.
• L.B. Slichter, The interpretation of the resistivity prospecting method for horizontal structures, J Appl Phys, v4, pp307--322, 1933.
• R.E. Langer, On determination of earth conductivity from observed surface potentials, Bull Am Math Soc, 10, pp747--754, 1936.
• AN Tikhonov, On the Uniqueness of the problem of electrical prospecting,

Dokl. Acad. Nauk. SSSR, 69, 797-800, 1949. (in Russian)

• A P Calderón On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro. 1980. Scanned copy of paper
• M. H. Loke, and R. D. Barker , Rapid least-squares inversion of apparent resistivity pseudo-sections using quasi-Newton method: Geophysical Prospecting, 48, 181-152, 1996.
• M. H. Loke, and R.D. Barker , , Practical techniques for 3D resistivity surveys and data inversion: Geophysical prospecting, 44, 499-523, 1996.