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# PS Azimuth Moveout

Azimuth moveout for converted waves (PS-AMO) is especially designed to process PS data, since it handles the asymmetry of the raypaths. PS-AMO moves events across a common reflection point according to their geological dips Rosales and Biondi (2006).

Theoretically, the cascade of any imaging operator with its corresponding forward-modeling operator generates a partial-prestack operator Biondi et al. (1998). A cascade operation of PS-DMO and its inverse (PS-DMO-1) is the basic procedure that I follow to derive the PS-AMO operator. First, I present a Kirchhoff integral derivation of the PS-AMO operator using a 3-D extension of the 2-D PS-DMO operator.

Following the derivation of the AMO operator Biondi et al. (1998), I collapse the PS-DMO operator with its inverse. Figure  schematically illustrates the PS-AMO transformation. The axes are the x and y CMP coordinates. Figure  shows four important vectors. The vectors and are transformation vectors, extensions of the offset vectors and respectively, according to the equations that will follow. These transformation vectors ( and ) are responsible for the lateral shift needed for transforming a trace from the CMP domain into the CRP domain and vice versa. Figure  shows the surface representation of a trace with input offset vector , reflection point at the origin, and azimuth . This trace is 1) translated to its corresponding CRP position using the transformation vector ; 2) transformed into zero offset () by a time shift with the PS-DMO operator (an intermediate step); 3) converted into equivalent data in the CRP domain with; and finally, 4) translated to its corresponding CMP position, using the transformation vector , with output offset vector , midpoint , and azimuth .

plane_new
Figure 1
Schematic of the PS-AMO transformation. An input trace with offset vector and reflection point at the origin is transformed into equivalent data with offset vector and midpoint position after a transformation in and out of the CRP domain with the transformation vectors and .

Appendix D presents the derivation of the PS-AMO operator, which is as follows:

 (35)

where

 (36) (37)

and

 (38) (39)

Equation , combined with equations -to-, represents an asymmetrical saddle on the CMP coordinates (xx,xy). In these equations, t1 is the input time after PS-NMO, t2 is the time after PS-AMO and before inverse PS-NMO, and are the input and output offset vectors, respectively, is a scaling factor (), is the P-to-S velocity ratio, and are the input and output azimuth positions, respectively, B1 and B2 are the scalar quantities that relate the transformation vectors, and , with the final position () for the input trace, and is the final CMP trace position. Similarly, is the transformation vector from the original trace position to the intermediate zero-offset position, and is the transformation vector from the intermediate zero-offset position to the final trace position.

The PS-AMO operator transforms the input trace, with offset vector and reflection point at the origin, into an equivalent trace with offset vector and a reflection point shifted by the vector , as shown in Figure .

The PS-AMO operator depends on the P-to-S velocity ratio (). Equation depends on the transformation vectors ( and ), and the transformation vectors depend on the traveltime after normal moveout (t1), the P velocity (vp), and . Therefore, PS-AMO presents a non-linear dependency on the traveltime after normal moveout (t1), the P velocity and . Because of this, PS-AMO varies with respect to traveltime even in constant-velocity media.

It is important to note that for a value of , both transformation vectors and become zero, and equation reduces to the known expression for AMO. Also, the PS-DMO operator, used in this section, assumes constant velocity; therefore, the PS-AMO operator of equation is based on a constant velocity assumption. Next, we discuss a computationally efficient implementation of the PS-AMO operator in the frequency-wavenumber log-stretch domain.