1. For any linear transform,
K(x,ξ) is the 'kernel'. 2. A mathematical function of resistivity and depth that can be calculated from apparent resistivity data, from which one tries to derive the resistivity stratification. Koefoed (1965, 568–591) derives the kernel function for Schlumberger-configuration data and Paul (1968, 159–162) for Wenner-configuration data. The derivation of resistivity stratification from the kernel is shown by Pekeris (1940) and Vozoff (1956). The electric potential V at the surface of a horizontally layered earth because of a dc point-source that is also located at the surface was expressed by Stefanesco in 1930 as a Hankel integral:
where r is the distance from the point source to the observation point, Jo(λ r) is a Bessel function, λ is a phantom variable of integration, C is a constant, and K(λ) is the kernel function. Also called Stefanesco function.
- Koefoed, O., 1965, Direct methods of interpreting resistivity observations: Geophys. Prosp., 13, 568–591.
- Paul, M. K., 1968, Notes on ‘‘Direct interpretation of resistivity profiles for Wenner electrode configuration’’ by O. Koefoed: Geophys. Prosp., 16, 159–162.
- Pekeris, C. L., 1940, Direct method of interpretation in resistivity prospecting: Geophysics, 5, 31–42.
- Vozoff, K., 1956, Numerical resistivity analysis: Horizontal layers: Geophysics, 23, 536–556.