3D pyramid interpolation

3D pyramid Transform between pyramid domain and $f$-$\bf x$ domain

I discussed the 2D pyramid transform in Shen (2008), and the 3D version is almost the same, except that some scalars become vectors. In the 3D pyramid transform, spatial grid spacing is calculated for each frequency $f$ using the equation

$$\Delta{\bf x}(f)=\frac{\Delta{\bf {x_0}}{\bf v}}{f{\bf {n_{sf}}}},$$ (1)

where $\Delta{\bf x}(f)$, $\Delta{\bf {x_0}}$, $\bf v$ and $\bf {n_{sf}}$ are all 2D vectors. $\Delta{\bf x_0}$ is the uniform spatial grid spacing in the original $f$-$\bf x$ data, $\bf v$ is the velocity that controls the slope of the pyramid and $\bf n_{sf}$ is the sampling factor in pyramid domain. By changing this factor we can control how densely the pyramid domain is sampled. In situations where events to be interpolated are not perfectly stationary, dense sampling is preferrable since the information in the low frequencies cannot be represented well by only a few points. In 3D, the inversion scheme that transforms data in $f$-$\bf x$ space to the pyramid domain is as follows:

$$\bf {Lm - d \approx 0,}$$ (2)

where $\bf {m}$ is the data in the pyramid domain, $\bf {d}$ is the known data in $f$-$\bf x$ space, and $\bf L$ is the 2D linear interpolation operator in 3D pyramid transform. The 3D pyramid transform from pyramid domain to $f$-$\bf x$ domina uses the following equation :

$$\bf { d = Lm.}$$ (3)

Where now $\bf m$ is known and $\bf d$ is unknown data in $f$-$\bf x$ space.

3D pyramid interpolation