Retrieving the potential field response from cross-correlations

We show that the two-point cross-correlation of potential-field recordings is equal to the Green’s function between the two points. This holds under the condition that spatially and temporally uncorrelated noise sources exist throughout the volume. They should have a known amplitude spectrum and their correlated strengths should be equal to the real part of the dissipative medium property function. Natural fluctuations, such as thermal noise, may occur that satisfy the necessary conditions. When these fluctuations are random deviations from a state of thermal equilibrium the fluctuation-dissipation theorem determines these external sources. This allows for Green’s function retrieval for all types of fields that satisfy a similar quasi-static field equation. Under equilibrium conditions, possible downhole reservoir applications include virtual source DC electric resistivity measurements, fluid flow measurements and local temperature estimations.