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 .
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 (x_{x},x_{y}). In these equations, t_{1} is the input time after PS-NMO, t_{2} 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, B_{1} and B_{2} 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 (t_{1}), the P velocity (v_{p}), and . Therefore, PS-AMO presents a non-linear dependency on the traveltime after normal moveout (t_{1}), 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.