Separation algorithms for marine simultaneous-source data generally require encoded sources. Proposed encoding schemes include random time delays (time dithers), periodic time sequences (such as those referred to as seismic apparition), and periodic phase sequences (for sources with fully controlled phase like a marine vibrator). At a given frequency, time dithers spread energy at a given wavenumber over all wavenumbers, phase sequences shift the energy by a fixed wavenumber (independent of frequency), and time sequences split energy over multiple wavenumbers in a frequency-dependent way. The way the encoding scheme distributes energy in the wavenumber domain is important because separation algorithms generally assume that, in the absence of encoding, all energy falls into the signal cone. Time dithering allows separation by inversion. At low frequencies, the inverse problem is overdetermined and easily solved. At higher frequencies, sparse inversion works well, provided the data exhibit a sufficiently sparse representation (consistent with compressive sensing theory). Phase sequencing naturally separates the sources in the wavenumber domain at low frequencies. At higher frequencies, ambiguities must be resolved using assumptions such as limited dispersion and limited complexity. Time sequencing allows a simple separation at low frequencies based on a scaling and subtraction process in the wavenumber domain. However, the scaling becomes unstable near notch frequencies, including DC. At higher frequencies, a similar problem to that for phase sequencing must be solved. The encoding schemes, therefore, have similar overall properties and require similar assumptions, but differ in some potentially important details. Phase sequencing is clearly only applicable to phase-controllable sources, and the different encoding schemes have other implications for data acquisition, for example, with respect to operational complexity, efficiency, spatial sampling, and tolerance to errors.
Deep S3PR: Simultaneous Source Separation and Phase Retrieval Using Deep Generative Models