On unified dual fields and Einstein deconvolution

Abstract

In many physical phenomena, the laws governing motion can be looked at as the relationship between unified dual fields which are continuous in time and space. Both fields are activated by a single source. The most notable example of such phenomena is electromagnetism, in which the dual fields are the electric field and the magnetic field. Another example is acoustics, in which the dual fields are the particle‐velocity field and the pressure field. The two fields are activated by the same source and satisfy two first‐order partial differential equations, such as those obtained by Newton’s laws or Maxwell’s equations. These equations are symmetrical in time and space, i.e., they obey the same wave equation, which differs only in the interface condition changing sign. The generalization of the Einstein velocity addition equation to a layered system explains how multiple reflections are generated. This result shows how dual sensors at a receiver point at depth provide the information required for a new deconvolution method. This method is called Einstein deconvolution in honor of Albert Einstein. Einstein deconvolution requires measurements of the pressure signal, the particle velocity signal, and the rock impedance, all at the receiver point. From these measurements, the downgoing and upgoing waves at the receiver are computed. Einstein deconvolution is the process of deconvolving the upgoing wave by the downgoing wave. Knowledge of the source signature is not required. Einstein deconvolution removes the unknown source signature and strips off the effects of all the layers above the receiver point. Specifically, the output of Einstein deconvolution is the unit‐impulse reflection response of the layers below the receiver point. Compared with the field data, the unit‐impulse reflection response gives a much clearer picture of the deep horizons, a desirable result in all remote detection problems. In addition, the unit‐impulse reflection response is precisely the input required to perform dynamic deconvolution. Dynamic deconvolution yields the reflectivity (i.e., reflection‐ coefficient series) of the interfaces below the receiver point. Alternatively, predictive deconvolution can be used instead of dynamic deconvolution.