We show that the two-point cross-correlation of potential-field recordings is proportional to the Green’s function between the two points. This holds under the condition that spatially and temporally uncorrelated quasi-static noise sources exist throughout the volume. 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.
The term interferometry is now well-known to mean the extraction of new responses from correlations of recordings. During the past eight years many interferometric methods have been developed for random fields and for controlled-source data. Many of the underlying theories have in common that the medium is assumed to be lossless. The main reason for this underlying assumption is that the wave equation in lossless media is invariant for time-reversal. For an overview of the theory of seismic interferometry or Greens function retrieval and its applications to passive as well as controlled-source data, we refer to a reprint book ofWapenaar et al. (2008), which contains a large number of papers on this subject and to the book of Schuster (2009).
It has been shown (Snieder, 2006, 2007; Weaver, 2008) that a volume distribution of uncorrelated noise sources, with source strengths proportional to the dissipation parameters of the medium, precisely compensates for the energy losses. This approach holds both for waves in dissipative media and for pure diffusion processes. Recently Wapenaar et al. (2006) and Snieder et al. (2007) showed that interferometry by cross-correlation, including its extensions for wave fields and diffusive fields in dissipative media, can be represented in a unified form. This naturally leads to the question of whether potential fields can also be retrieved by cross-correlation of noise measurements. Physical mechanisms for the noise sources were not identified in the previous studies on field correlations in dissipative media. Here we find that when these noise sources are caused by thermal fluctuations, the cross-correlation functions of these thermal noise sources are described by the fluctuation-dissipation theorem (FDT) (Callen & Welton, 1951). We first describe the principle using the quasi-static electric-field approximation to Maxwell’s equations. At the macroscopic level thermal fluctuations lead to fluctuations in the electric field, which then act as the source (Landau&Lifshitz, 1960). We derive an identity for the real part and the imaginary part of the Green’s function between two points from cross-correlations of field fluctuations at the same two points that are caused by sources distributed throughout the volume. We then identify thermal noise sources as possible sources satisfying all necessary requirements and show the relation with Brownian motion and the Johnson-Nyquist electric circuit noise. We then extend these findings to measurements both in a macroscopic piecewise-continuous open medium of infinite extent and a partially closed medium, such as when the surface of the earth is included.