Parametric Beam Formation in Rock

P. A. Johnson and T. M. Hopson
Geological Engineering Group MS D443
Los Alamos National Laboratory
Los Alamos, NM 87545

B. P. Bonner
Earth Sciences Department
Lawrence Livermore National Laboratory
Livermore, CA 94550

T. J. Shankland
Geological Engineering Group MS D443
Los Alamos National Laboratory
Los Alamos, NM 87545


Abstract

The highly nonlinear elastic properties of rock may enable a new means of imaging Earth structure through parametric formation of a difference frequency beam from the interaction of two collimated primary drastic waves. Because the difference frequency beam can have the narrow collimation of the higher frequency primaries, it could be used as a directional wave source. Such a low frequency wave propagates farther than the primary signals to locate features not currently detectable by waves generated from conventional sources. The concept of a nonlinearly-derived source arose from research in underwater acoustics [1] where the idea of beating collinear high frequency beams to produce a collimated beam at the lower, and less attenuated difference frequency originated; this work led to the development of nonlinear, directional sources in water [2].

The large elastic nonlinearity of rocks is due to the network of microfractures they generally contyain; as these cracks close with applied stress, there are correspondingly large changes in leastic moduli [3]. The change in a modulus M with pressure P, dM/dP, can be nearly two orders of magnitude larger in rocks than in an uncracked material such as a liquid or crystal. Nonlinear conversion efficiency from primary to difference frequency signals increases with dM/dP and is therefore higher for rocks than for uncracked materials [4]. However, the enhanced nonlinear conversion is partially diminished by the large elastic wave attenuation also characteristic of rocks [3].

The purpose of this paper is to show how nonlinear elastic waves generated within a material can be made to interfere and produce a strong difference frequency signal at distances where the primary signals have disappeared due to attenuation. In addition, we show results from low frequency attenuation studies using a torsional oscillator [5,6] aimed at determining the microstructural properties that control the nonlinear response in rock, with the goal of improving parametric array generation and understanding where the departure between linear and nonlinear elasticity takes place in rock. Lastly, we demonstrate a sensitive, low noise frequency domain travel time (FDTT) method for application with the parametrically-generated difference signals. The FDTT method is of general applicability for accurate measurements of travel time in the presence of noise [7,8,9].