# Dictionary:Kernel function

**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^{[1]}) derives the kernel function for Schlumberger-configuration data and Paul (1968, 159–162^{[2]}) for Wenner-configuration data. The derivation of resistivity stratification from the kernel is shown by Pekeris (1940^{[3]}) and Vozoff (1956^{[4]}). 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, *J*_{o}(*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**.

## References

- ↑ 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.