SOURCE SEPARATION METHOD

1 . A method, comprising:

obtaining wave field recordings generated by least one receiver based on simultaneous or near-simultaneous activation of at least two sources having at least one parameter, the at least one parameter being one of a source signal amplitude, a source signal spectrum, a source activation time, and a source location at activation time, the at least one parameter varying non-periodically between the at least two sources from one activation to a following activation along an activation line describing a trajectory of the at least two sources in space or a temporal sequence of activations of a line of the at least two sources;

modelling the obtained wave field recordings using a model that incorporates a non-periodic variation in the at least one parameter as a sum of wave fields, generated by the at least two sources individually, the obtained wave filed recordings having bounded support in a transform domain;

using the model, inverting the obtained wave field recordings to separate a contribution of at least one of the at least two sources to the obtained wave field recordings;

generating a sub-surface representation of structures or Earth media properties from the contribution of the at least one of the at least two sources; and

outputting the generated sub-surface representation.

2 . The method of claim 1 , wherein the bounded support comprises cone-shaped regions in a frequency wavenumber domain.

3 . The method of claim 2 , wherein the modelling step comprises modelling the obtained wave field recordings using unequally spaced discrete Fourier transform operators.

4 . The method of claim 1 , wherein the inverting step further comprises solving a system of equations.

5 . The method of claim 4 , wherein the inverting step further comprises solving the system of equations iteratively.

6 . The method of claim 5 , wherein the inverting step further comprises solving the system of equations iteratively including use of a preconditioner.

7 . The method of claim 6 , where the preconditioner includes at least one of a least-squares solution for periodic timing, signal amplitude, or signal spectrum variations, and regular activation locations.

8 . The method of claim 1 , wherein the variations include at least one of the source activation time, the source signal amplitude, and the source signal spectrum, and the model includes application of a cyclic convolution matrix and a corresponding inverse of the cyclic convolution matrix using a transform of a modulation function associated with the variations.

9 . The method of claim 8 , wherein the inverting step further comprises computing a least-squares solution of a system of equations.

10 . The method of claim 1 , wherein at least one source of the at least two sources is not part of the same survey, and wherein the activation time of the at least one source is approximately known.

11 . The method of claim 10 , wherein activation times are determined from the obtained wave field recordings.

12 . The method of claim 1 , wherein the at least one parameter is varied in a non-periodic controlled or uncontrolled manner.

13 . The method of claim 1 , wherein the at least one parameter is varied in a non-periodic manner and is composed of a periodic part overlaid with non-periodic fluctuations.

14 . The method of claim 1 , wherein the inverting step further comprises regularizing data onto a regular sampling grid in time and or space.

15 . The method of claim 1 , wherein the transform domain is one of a Fourier transform domain, a Radon transform domain, a Gabor time-frequency domain, and a tau-p transform domain.

16 . The method of claim 15 , wherein each corresponding transform is replaced by its respective representation or mathematical equivalent in corresponding transform domains in a space or time domain.

17 . The method of claim 1 , wherein computations in the modelling and inverting steps are performing using fast solvers to reduce the time complexity of the modelling and inverting steps.

18 . The method of claim 1 , wherein the obtained wave field recordings comprise multiple components, and one or more of the multiple components is modeled and inverted to separate the contribution using information derived from one or more of the other multiple components.

19 . The method of claim 18 , where the information derived from the one or more of the other multiple components comprises local directionality information.

20 . The method of claim 18 , further comprising determining local directionality information jointly from two or more of the multiple components.

21 . An apparatus, comprising:

processing circuitry configured to

obtain wave field recordings generated by least one receiver based on simultaneous or near-simultaneous activation of at least two sources having at least one parameter, the at least one parameter being one of a source signal amplitude, a source signal spectrum, a source activation time, and a source location at activation time, the at least one parameter varying non-periodically between the at least two sources from one activation to a following activation along an activation line describing a trajectory of the at least two sources in space or a temporal sequence of activations of a line of the at least two sources;

model the obtained wave field recordings using a model that incorporates a non-periodic variation in the at least one parameter as a sum of wave fields, generated by the at least two sources individually, each of the wave fields having bounded support in a transform domain;

using the model, invert the obtained wave field recordings to separate a contribution of at least one of the at least two sources to the obtained wave field recordings;

generate a sub-surface representation of structures or Earth media properties from the contribution of the at least one of the at least two sources; and

output the generated sub-surface representation.