The 1/6-th trick

Given the filter D₂ (k), defined in formula (10), we can construct an accurate approximation to the second derivative operator -k² by considering a filter ratio (another Padé-type approximation) of the form  

$$-k^2 \approx \frac{D_2(k)}{1 + \beta D_2 (k)}\;,$$ (25)

where $\beta$ is an adjustable constant Claerbout (1985). The actual Padé coefficient is $\beta=1/12$. As pointed out by Francis Muir, the value of $\beta = 1/4 - 1/\pi^2 \approx 1/6.726$gives an exact fit at the Nyquist frequency $k = \pi$. Fitting the derivative operator in the L₁ norm yields the value of $\beta \approx 1/8.13$. All these approximations are shown in Figure 10.

 

sixth Figure 10 The second-derivative operator in the wavenumber domain and its approximations.

B