Estimating a time variant signal representing a seismic source

1. A method for estimating a time variant signal representing a seismic source, the method comprising:

obtaining seismic data recorded by receivers detecting seismic excitations generated by the seismic source, the recorded seismic data comprising direct arrivals;

deriving the time variant signal using an operator that relates the time variant signal to the acquired seismic data, wherein the operator is constrained such that the time variant signal is sparse in time, wherein the operator is obtained by inverting one or more equations representing propagation of one or more notional signatures of the seismic source directly to the receivers thus yielding the direct arrivals; and

using the time variant signal to generate an image of the subsurface from the seismic data.

2. The method of claim 1 , wherein the time variant signal comprises a notional source or a farfield signature.

3. The method of claim 1 , wherein the direct arrivals comprise a ghost reflection at the air-water interface.

4. The method of claim 1 , wherein the seismic source is a source array comprising a plurality of seismic sources.

5. The method of claim 4 , wherein:

the plurality of seismic sources are located at a plurality of positions; and

the operator comprises a model of the seismic source array comprising at least one notional source located at each one of the plurality of positions.

6. The method of claim 5 , wherein:

each seismic source comprises an air gun having a gun volume; and

the model of the seismic source array further comprises at least one notional source having an amplitude scaled by a factor proportional to a cube root of the gun volume at each one of the plurality of positions.

7. The method of claim 4 , wherein the method further comprises correcting the time variant signal to reduce seismic source array effects.

8. The method of claim 1 , wherein the at least one receiver comprises a receiver array comprising a plurality of individual receivers.

9. The method of claim 8 , wherein

the seismic data is obtained from the receiver array; and

the operator comprises a sum of signals from the plurality of individual receivers and a spatial distribution of the individual receivers.

10. The method of claim 1 , wherein the operator defines a linear problem that is constrained using sparseness weights.

11. The method of claim 10 , wherein the sparseness weights restrict the time variant signal to a time window.

12. The method of claim 10 , wherein the sparseness weights are derived based on an estimate of the time variant signal.

13. The method of claim 1 , wherein the operator is defined so as to describe the obtained seismic data by a series of spikes.

14. The method of claim 1 , wherein the method further comprises deriving low frequency and high frequency components of the time variant signal separately.

15. A computing system for estimating a time variant signal representing a seismic source, the computing system comprising:

a storage device comprising seismic data recorded by receivers and generated by the seismic source, the recorded seismic data comprising direct arrivals; and

a processor in communication with the storage device and configured to derive the time variant signal using an operator that relates the time variant signal to the recorded seismic data, wherein the operator is constrained such that the time variant signal is sparse in time, wherein the operator is obtained by inverting one or more equations representing propagation of one or more notional signatures of the seismic source directly to the receivers thus yielding the direct arrivals, and to use the time variant signal to generate an image of the subsurface from the seismic data.

16. The computing system of claim 15 , wherein the time variant signal comprises a notional source or a farfield signature.