


 Predicting rugged waterbottom multiples through wavefield extrapolation with rejection and injection  

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Theoretically, wavefield rejection and injection can be carried out for any of the wavefield extrapolation algorithms. However, the computational efficiency can differ greatly. Recursive Kirchhoff wavefield extrapolation in both spacefrequency and spacetime domain is the most suitable algorithm.
Because the basic theory and extrapolation operator based on the Kirchhoff integral are widely known (Schneider, 1978; Margrave and Daley, 2001; Berkhout, 1981), I derive only the formulae corresponding to upward and downward wavefield extrapolation of both the upgoing and downgoing waves.
In general, the wavefield extrapolation from depth
to
can be written as follows:

(1) 

(2) 
where
and
are the Fourier transform over
,
and
of the downgong and upgoing waves at position (
,
,
), respectively. The terms
,
and
are the three components of the wavenumber vector, and
is the angular frequency. The sign
before the depth interval
relates to the upward and downward wavefield extrapolations. Thus, there are four types of wavefield extrapolations in total.
On the other hand, the Kirchhoff integral in the spacefrequency domain is

(3) 
where
and
are the shorthand notations of (
,
,
) and (
,
,
) . The term
is the outward normal of the surface
. The wavefield
and Green function
satisfy the following Helmholtz equations:

(4) 

(5) 
where
is wave propagation velocity. Obviously, it is not easy to relate equation 3 to the upward and downward wavefield extrapolation of both the upgoing and downgoing waves.
Suppose
consists of a horizontal
surface at
and a hemisphere
which contains point A and satisfies the Sommerfeild radiation condition. Transforming the Kirchhoff integral from the (
,
,
,
) domain into (
,
,
,
) domain, equation 3 becomes (Berkhout, 1989)

(6) 
where
and
represent
and
.
Furthermore, equation 6 can be mathematically expressed by

(7) 
where
and
are the downgoing wave and Green’s function respectively, and
and
are the upgoing wave and Green’s function respectively.
Using the definitions of the downgoing and upgoing waves,

(8) 

(9) 
Equation 7 can be simplified to

(10) 
and furthermore to

(11) 
Thus the four types of wavefield extrapolations in terms of the Kirchhoff integral can be easily defined based on equation 11, as shown in figure 3.
In practice, generally we transform the equation 11 from the (
,
,
,
) domain back into the(
,
,
,
) or (
,
,
,
) domain In this case, the downgoing and upgoing Green’s functions will change accordingly, as shown in figure 3. The terms
and
are the causal and anticausal Green’s functions , or the “outgoing ” and “ingoing ” waves respectively.
After the above derivation, we have all types of wavefield extrapolations in terms of the Kirchhoff integral. In waterbottom multiple prediction, we use only types (a) and (c), that is, the downward wavefield extrapolation of the downgoing wave and the upward wavefield extrapolation of the upgoing wave.


vel
Figure 4. The structure and interval velocity of the geological model. [NR]






shotmult1
Figure 5. The shot gather at the leftmost location(left) and predicted multiples (right). [NR]






shotmult2
Figure 6. The shot gather at the central location (left) and predicted multiples (right). [NR]







 Predicting rugged waterbottom multiples through wavefield extrapolation with rejection and injection  

Next: Synthetic example
Up: Predicting rugged waterbottom multiples
Previous: Wavefield extrapolation with rejection
20100519