A method for estimating a far field seismic energy source signature, comprising:
- entering into a computer detected near field seismic signals corresponding to actuation of each one of a plurality of seismic energy sources in an array of seismic energy sources, the near field seismic signals detected at two spaced apart locations in the near field of each seismic energy source, the at least two spaced apart locations being arranged such that a direction of propagation of the detected near field seismic signals is determinable from the detected near field signals;
- in the computer, determining a notional source signature for each seismic energy source and a notional ghost for each seismic energy source using the detected near field seismic signals, wherein the determining a notional source signature and determining a notional ghost comprises inversion of the expression,
- h
- i
-
- (
- t
- )
- =
- ∑
- j
- =
- 1
- n
-
- p
- j
-
- (
- t
- )
- *
- δ
-
- (
- t
- -
- τ
- ji
- )
- /
- r
- ji
- +
- ∑
- j
- =
- 1
- n
-
- a
- j
-
- (
- t
- )
- *
- δ
-
- (
- t
- -
- τ
- gji
- )
- /
- r
- gji
- in which hi(t) denotes the ith seismic sensor data recording with respect to time t, pj(t) denotes the jth notional source waveform, rji denotes a possibly time variant distance from a moving seismic energy source bubble to a moving seismic sensor, aj(t) is a notional ghost waveform, rgji is a distance between a moving virtual source to a moving seismic sensor and τji is a time for seismic energy to travel from an airgun discharge bubble center to the moving seismic sensor, the expression denoted in block matrix form as,
- (
- D
- 11
- D
- 1
-
- j
- ⋯
- D
- 1
-
- n
- D
- i
-
-
- 1
- D
- ii
- ⋯
- D
- in
- ⋮
- ⋮
- ⋮
- ⋮
- ⋮
- ⋮
- D
- m
-
-
- 1
- D
- mj
- ⋯
- D
- mn
- |
- G
- 11
- G
- 1
-
- j
- …
- G
- 1
-
- n
- G
- i
-
-
- 1
- G
- ii
- ⋯
- G
- in
- ⋮
- ⋮
- ⋮
- ⋮
- ⋮
- ⋮
- G
- m
-
-
- 1
- G
- mj
- ⋯
- G
- mn
- )
- ·
- (
- p
- 1
- p
- j
- ⋮
- p
- n
- a
- 1
- a
- j
- ⋮
- a
- n
- )
- =
- (
- h
- 1
- h
- i
- ⋮
- ⋮
- h
- m
- )
- wherein Dij is formed by taking the identity matrix and convolving each column with δ(t−τji)/rji, and Gij is formed by taking the identity matrix and convolving each column with δ(t−τgji)/rgji;
- in the computer, determining a far field signature for each of the plurality of seismic energy sources using the determined notional source signature and notional ghost source signature of each seismic energy source;
- and
- using the far field signature for each of the plurality of seismic energy sources to correct distortion in detected reflected seismic signals.