On signal contribution functions and sensitivity for frequency-domain electromagnetic and direct current electrical resistivity methods

Signal contribution functions can be integrated over space to calculate the response of an electrical or electromagnetic technique to a given resistivity distribution. On the other hand, sensitivity functions show how the measured signal changes with a change in resistivity in a region of the ground. Signal contribution functions and sensitivity have been previously presented for the direct current resistivity technique. While useful forms of both the signal contribution function and sensitivity are proportional to the correlation of the current densities from the normal and reciprocal configurations, it has not previously been shown how sensitivity can be derived directly from the signal contribution. For frequency-domain electromagnetic techniques, there are existing expressions for sensitivity but not for the signal contribution. In this contribution, I show how the direct current signal contribution can be differentiated to obtain the sensitivity. I also derive an expression for the signal contribution function for frequency-domain electromagnetics and show how it can be differentiated to obtain the sensitivity. The new signal contribution function has a term that, like the sensitivity, is proportional to the electrical current densities from the normal and reciprocal configurations and an additional term that is proportional to the correlation of the magnetic fields from the normal and reciprocal configurations. I show plots of these two terms and investigate their magnitudes as a function of the induction number. This new expression will be useful for testing numerical models and aids in understanding the measured results in frequency-domain electromagnetics.