PSPM as a spreading operator

The PSPM operator defined in equations (1) can be written as function of the half-offset h, half-velocity V, and source-receiver travel time t_h:  

$$\left \{ \begin{array} {l} \displaystyle{t_0 = { { {(t^2_h... ...}{{sin^2{\alpha}} \over {V^2}}} }} }}\\ \end{array} \right.$$ (2)

Defining the variables $\tau = { t^2_h-{h^2 \over V^2}},\; p=\displaystyle{{ {sin{\alpha}} \over {V} }}\$,we can rewrite equations (2) as

 

$$\left \{ \begin{array} {l} t_0 = \displaystyle{{ {\tau}^2 ... ...^2p} \over {\sqrt { {\tau}^2 + h^2p^2}} }}\end{array} \right.$$ (3)

Equations (3) are parametric representations of the PSPM operator, function of two new variables: $\tau$ and $\bf{p}$.The variable $\tau$ is the NMO corrected constant-offset travel time, while the variable $\bf{p}/2$ is the p_x=dt / dx parameter.

In Figure 1 the PSPM operator is mapped using equations (3). The elliptic curves represent curves of constant $\tau$parameter, while the vertical curves are curves of constant $\bf{p}$ parameter. The PSPM operator is represented by $\bf{t_0}$ and $\bf{x_0}$ coordinates, which is a velocity independent characterization. The velocity cutoff is contained in the parameter $\bf{p}$, which is increasing toward the edges of the plot.

The cutoff is determined by the parameter $\bf{p}$. Higher velocities introduce the restriction in the parameter $\bf{p}$ which cannot be higher than 1 / V, where V is half-velocity. A medium with velocity V₁ will have a tighter cutoff than a medium with lower velocity V₂, (V₁ > V₂). In the first case the $\bf{p}$ parameter takes values from to 1 / V₁, (${p} \in [0,{1 \over {V_1}}]$) while in the second case the $\bf{p}$ parameter ranges from to 1 / V₂ (${p} \in [0,{1 \over {V_2}}]$). Two different media will share the same PSPM operator for the same values of $\bf{p}$. On the same DMO ellipse two media with velocities V₁,V₂ migrate an impulse to the same place if the equality $p={\sin{{\alpha}_1} / {V_1}}={\sin{{\alpha}_2} / {V_2}}$is satisfied. For a higher velocity medium, a high dip is migrated in the same place as a lower dip in a lower velocity medium.

Equations (3) map the PSPM operator as a function of the variables $\bf{p}$ and $\tau$. In a constant velocity medium, for each point in the ($\bf{x_0}$, $\bf{t_0}$)space there corresponds a pair of values ($\bf{p}$, $\tau$) and vice-versa. For a medium with variable velocity with depth V(z), it was shown that for certain velocity models the DMO impulse response presents triplications (Popovici and Biondi, 1989). Supposing the PSPM operator is represented in a system of coordinates ($\bf{p}$, $\tau$), for certain velocity models the transformation $(\bf{x_0},\bf{t_0}) \longrightarrow (\bf{p},\mbox{\boldmath$\tau$})$is not unique, thus generating the triplications in the PSPM curve. In a medium with variable velocity, this can be determined by following the variation of the $\bf{p}$ parameter along the equal travel time reflector. When the value of $\bf{p}$ is repeated along the same reflector with a given $\tau$,the PSPM operator will have multiple values of the pair of parameters (x₀,t₀) for the same value of $\bf{p}$.According to this scheme it can be predicted for what types of media the PSPM operator will be multivalued in the parameter $\bf{p}$.