PS-DMO in the frequency-wavenumber log-stretch domain

The previous section shows a kinematic PS-DMO operator. Like PP-DMO, kinematic PS-DMO can not properly handle phase and amplitude (). (, ) discuss a PS-DMO operator on the f-k domain by using a truncation of power series for the moveout of reflections from dipping reflectors in a constant velocity media.

() discuss that the PP-DMO operator in the f-k domain is computationally expensive because the operator is temporarily non-stationary. He uses the idea of () to express a more accurate PP-DMO operator by a logarithmic time stretching.

() exploit the idea of computational efficiency of the logarithmic time stretching for the PS-DMO operator, however their formulation is not clearly expressed. Therefore, I reformulate their work using the PS-DMO smile presented in the previous section and following a procedure similar to () and (). This operation accounts for constant velocity case.

Starting from the PS wave DMO smile,  

 54#54 (23)

By following Hale's assumption that p₀(t₀,x,H) = p_n(t_n,x,H), we obtain the 2D PS-DMO operator in the f-k domain:  

 55#55 (24)

Equation () implies a change of variable from t₀ to t_n. Therefore, equation () becomes  

 56#56 (25)

where

57#57 (26)

Equation () is the base for PS-DMO in the f-k domain. By using a time log-stretch transform pair,

   58#58 (27)

The DMO operator in the f-k log-stretch domain becomes  

 59#59 (28)

where  

 60#60 (29)

When either kh or 61#61 gets the values of , the filter goes to as well.

The previous expression is equivalent to the one presented by (). Note that equation () is based on the assumption that p₀(t₀,x,H) = p_n(t_n,x,H). This doesn't include changes in midpoint position and/or common reflection point position. This allows for a correct kinematic operator but one with a poor amplitude distribution along the reflectors.

() refers to this problem in the log-stretch frequency wavenumber domain by reformulating Black's f-k DMO operator. This operator is based on the assumption that p₀(t₀,x₀,H) = p_n(t_n,x_n,H). The midpoint location also changes, leading to a more accurate distribution of amplitudes. After implementing the PS-DMO operator (), extending this operator is a feasible task. Following the derivation used by (), I state a more accurate PS-DMO operator in the log-stretch frequency wavenumber domain. This new operator differs from the previous one in the filter 62#62. The new filter is  

 63#63 (30)

This filter reduces to if kh=0 or kH if 64#64.

Note that for a value of 65#65, the filter reduces to the known expression for P waves data ().

Figure  shows a series of impulse responses for this operator in the frequency domain.

 

imps
Figure 2 Impulse responses for the DMO operator, PP case (left), PS case (right)

Figure  shows the same series of impulse responses as Figure  but with the new filter [equation ()]. Both operators create the same kinematic response. However, Figure  shows that the filter in equation () gives a more accurate amplitude distribution along the impulse response.

We can trust the PS results since the PP impulse response, obtained with the filter in equation () and 65#65, is the same as that obtained by (). Moreover, the amplitude distribution follows Jaramillo's result.

 

imps-new
Figure 3 Impulse responses for the new PS-DMO operator, PP case (left), PS case (right)