Reverse Time Migration of up and down going signal for ocean bottom data |
The derivation for decomposing over/under pressure waves into up-going and down-going signals is best done in the Fourier domain. For a thorough review of this method, please refer to Sonneland et al. (1986). Denote and to be the Fourier-transformed measurements of compressional waves at depths (over) and (under). Theoretically, is a sum of the up-going and down-going components. Likewise for :
Down-going waves arrive at the under array () before the over () array. Therefore, shifting forward in time would match the signal . Similarly, up-going waves visit the over array first. Therefore, shifting forward in time would match the signal . This relationship is equivalent to a phase-shift in the Fourier domain:
where , and is the usual dispersion relation. Finally, substituting equation 6 into equation 5 yields the formula for the up-going and down-going waves at the receivers:
Over/under acquisition is used to eliminate receiver ghosting and water reverberation. Although over/under arrays are rarely placed on the sea floor in real seismic surveys, this technique allows easy generation of up-going and down-going waves at the sea bottom for synthetic examples or modeling. For the remaining of this paper, we will denote the operation that separates over/under data into up-down data in equation 7 as .
Reverse Time Migration of up and down going signal for ocean bottom data |