Dictionary:Kernel function

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1. For any linear transform,

${\displaystyle F(\xi )=\int _{-\infty }^{\infty }\!\!K(x,\xi )\,f(x)\,\mathrm {d} x}$

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:

${\displaystyle V={\frac {C}{r}}+2C\int \!\!K(\lambda )J_{o}(\lambda r)\,\mathrm {d} \lambda }$

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.

References

1. Koefoed, O., 1965, Direct methods of interpreting resistivity observations: Geophys. Prosp., 13, 568–591.
2. Paul, M. K., 1968, Notes on ‘‘Direct interpretation of resistivity profiles for Wenner electrode configuration’’ by O. Koefoed: Geophys. Prosp., 16, 159–162.
3. Pekeris, C. L., 1940, Direct method of interpretation in resistivity prospecting: Geophysics, 5, 31–42.
4. Vozoff, K., 1956, Numerical resistivity analysis: Horizontal layers: Geophysics, 23, 536–556.