A step-by-step separation scheme

The first order operator can be further approximated as a cascade of three operators.

$$A3\ =\ \left[ \begin{array} {ccccc} 1 & & 0 & & 0 \\ & & &... ...}.p_y \\ & & & & \\ 0 & & y_{32}.p_y & & 1\end{array}\right]$$

pre-multiplying,

$$A2\ =\ \left[ \begin{array} {ccccc} 1 & & 0 & & x_{13}.p_x ... ... & & 0 \\ & & & & \\ x_{31}.p_x & & 0 & & 1\end{array}\right]$$

pre-multiplying,

$$A1\ =\ \left[ \begin{array} {ccccc} \cos(\phi)/2 & & \sin(\... ...\phi)/2 & & 0 \\ & & & & \\ 0 & & 0 & & 1/2\end{array}\right]$$

$$P_{decomposed} = A3\,A2\,A1\ P_{raw}\ A1^T\,a2^T\,A3^T$$

The separation procedure has three stages.

A1

A rotation about the z-axis to separate the two S-wavetypes.

A2

An operator to separate P from S₁. This operator has two free parameters that need to be estimated.

A3

An operator to separate P from S₂. This operator also has two free parameters that need to be estimated.