Turning noise into signal: a paradox? Kees Wapenaar, Delft University of Technology Roel Snieder, Colorado School of Mines Presented at: Making Waves about Seismics: a Tribute to Peter Hubral’s achievements, not only in Geophysics Karlsruhe, February 28, 2007

The Green’s function emerges from the cross-correlation of the diffuse wave field at two points of observation:

Weaver and Lobkis

Campillo and Paul

Seismic interferometry at global scale: US-Array

‘Turning noise into signal’ works in practice and we have a theory that explains it …. ……so what is the paradox? Extraction of signal is fairly robust, despite: Assumptions about source distribution are never fulfilled in practical situations. Signal extraction relies for a large part on multiple scattering. Is it stable?

Laplace, 1814: The physical world is a deterministic clockwork

Laplace, 1814: The physical world is a deterministic clockwork Poincaré, 1903: OK Pierre, but uncertainties in initial conditions lead to chaos at later times

Laplace, 1814: The physical world is a deterministic clockwork Poincaré, 1903: OK Pierre, but uncertainties in initial conditions lead to chaos at later times Heisenberg, True Henri, but at atomic 1927: scale only probabilities are determined

Laplace, 1814: The physical world is a deterministic clockwork Poincaré, 1903: OK Pierre, but uncertainties in initial conditions lead to chaos at later times Heisenberg, True Henri, but at atomic 1927: scale only probabilities are determined Astonishing Werner! But let’s go back to macroscopic physics and now look at waves

Particle scattering: chaotic after 8 scatterers Wave scattering: still stable after 30+ scatterers

F Einstein, 1905, Brownian motion Kubo, 1966, fluctuation-dissipation theorem

Conclusion: Robustness of ‘turning noise into signal’ is explained by stability of wave propagation Finally, Let’s see how this can be generalized

Dissipating media (no time-reversal invariance) Applications for EM waves in conducting media, diffusion, acoustic waves in viscous media, etc. Systems with higher order DV’s Applications for e.g. bending waves

Flowing media (non-reciprocal Green’s functions)

Schroedinger’s equation ‘Zero offset’

General vectorial formulation (for example, electroseismic)