Approximations in seismic interferometry and their effects on surface waves

P>We investigate common approximations and assumptions in seismic interferometry. The interferometric equation, valid for the full elastic wavefield, gives the Green's function between two arbitrary points by cross-correlating signals recorded at each point. The relation is exact, even for complicated lossless media, provided the signals are generated on a closed surface surrounding the two points and are from both unidirectional point-forces and deformation-rate-tensor sources. A necessary approximation to the exact interferometric equation is the use of signals from point-force sources only. Even in simple layered media, the Green's function retrieval can then be imperfect, especially for waves other than fundamental mode surface waves. We show that this is due to cross terms between different modes that occur even if a full source boundary is present. When sources are located at the free surface only, a realistic scenario for ambient noise, the cross terms can overwhelm the higher mode surface waves. Sources then need to be very far away, or organized in a band rather than a surrounding surface to overcome this cross-term problem. If sources are correlated, convergence of higher modes is very hard to achieve. In our examples of simultaneously acting sources, the phase of the higher modes only converges correctly towards the true solution if sources are acting in the stationary phase regions. This offers an explanation for some recent body wave observations, where only interstation paths in-line with the prevailing source direction were considered. The phase error resulting from incomplete distributions around the stationary phase region generally leads to an error smaller than 1 per cent for realistic applications.