Green's function retrieval from the CCF of random waves and energy conservation for an obstacle of arbitrary shape: noise source distribution on a large surrounding shell

Abstract

SUMMARY For imaging the earth structure, the cross-correlation function (CCF) of random waves as ambient noise or coda waves has been widely used for the estimation of the Green’s function. We precisely study the condition for the Green’s function retrieval in relation to the energy conservation for a single obstacle of arbitrary shape. When an obstacle is placed in a 2-D homogeneous medium, the Green’s function is written by a double series expansion using Hankel functions of the first kind which represent outgoing waves. When two receivers and the scattering obstacle are illuminated by uncorrelated noise sources randomly and uniformly distributed on a closed circle of a large radius surrounding them, the lag-time derivative of the CCF of random waves at the two receivers can be written by a convolution of the antisymmetrized Green’s function and the autocorrelation function of the noise source time function. We explicitly derive the constraint for the Hankel function expansion coefficients as the sufficient condition for the Green’s function retrieval. We show that the constraint is equal to the generalized optical theorem derived from the energy conservation principle. Physical meaningofthegeneralizedopticaltheorembecomesclearwhentheHankelfunctionexpansion coefficients are transformed into scattering amplitudes in the framework of the conventional scattering theory. In the 3-D case, the Green’s function is written by a double series expansion using spherical Hankel functions of the first kind and spherical harmonic functions. When two receivers and the scattering obstacle are illuminated by noise sources randomly and uniformly distributed on a closed spherical shell of a large radius surrounding them, we explicitly derive the constraint for the spherical Hankel function expansion coefficients for the Green’s function retrieval and the energy conservation. We note that the derivation of the constraint does not assume that two receivers are in the far field of the scattering obstacle.