Wave-fields separation for seismic recorders distributed at non-flat recording surfaces

A computing device for generating an image based on separating up-going and down-going wave fields (U, D) in seismic data related to a subsurface of a body of water or to a subsurface of a body of rock, the computing device comprising:

• an interface configured to receive seismic data (Po, Zo) recorded in the time-space domain with seismic recorders distributed on a first datum, wherein the first datum is non-flat;
• and
• a processor connected to the interface and configured to,
• transform the seismic data (P0, Z0) from the time-space domain to transformed seismic data for water pressure (P) and transformed seismic data for particle velocity (Z), in another domain,
• receive a mathematical relation that relates (1) the transformed seismic data for water pressure (P) to desired up-going and down-going wave fields (U, D), wherein the transformed seismic data for water pressure (P) is expressed as a first function of the desired up-going and down-going wave fields (U, D) on a second planar datum, and (2) the transformed seismic data for particle velocity (Z) to the desired up-going and down-going wave fields (U, D), wherein the transformed seismic data for particle velocity (Z) is expressed as a second function of the desired up-going and down-going wave fields (U, D) on the second planar datum,
• solve with an inversion procedure the first and second functions to obtain the up-going and down-going wave fields (U, D) for the second datum, and
• generate the image based on at least one of the desired up-going and down-going wave fields (U, D) for the second datum,
• wherein the second datum is different from the first datum, and
• wherein each of the first and second functions includes extrapolator operators for extrapolating the desired up-going and down-going wave fields (U, D), respectively, to the second datum.

A non-transitory computer readable medium including computer executable instructions, wherein the instructions, when executed by a processor, implement instructions for generating an image based on separating up-going and down-going wave fields (U, D) in seismic data related to a subsurface of a body of water or to a subsurface of a body of rock, the instructions comprising:

• receiving seismic data (Po, Zo) recorded in the time-space domain with seismic recorders distributed on a first datum, wherein the first datum is non-flat;
• transforming the seismic data (P0, Z0) from the time-space domain to transformed seismic data for water pressure (P) and transformed seismic data for particle velocity (Z), in another domain;
• establishing, in the another domain, a mathematical relation that relates (1) the transformed seismic data for water pressure (P) to desired up-going and down-going wave fields (U, D), wherein the transformed seismic data for water pressure (P) is expressed as a first function of the desired up-going and down-going wave fields (U, D) on a second planar datum, and (2) the transformed seismic data for particle velocity (Z) to the desired up-going and down-going wave fields (U, D), wherein the transformed seismic data for particle velocity (Z) is expressed as a second function of the desired up-going and down-going wave fields (U, D) on the second planar datum;
• solving with an inversion procedure, run on the processor, the first and second functions to obtain the up-going and down-going wave fields (U, D) for the second datum;
• and
• generating the image based on at least one of the desired up-going and down-going wave fields (U, D) for the second datum,
• wherein the second datum is different from the first datum, and
• wherein each of the first and second functions includes extrapolator operators for extrapolating the desired up-going and down-going wave fields (U, D), respectively, to the second datum.