DERIVATION OF ROTATED McCLELLAN FILTERS

The McClellan filters proposed by Hale 1991 to approximate the $\cos\mid\vec k\mid$ filter have the important advantage of being compact, and consequently inexpensive to apply as convolutions in space domain. The lower-order filter is applied by convolving the data with a 9-point stencil, while the higher-order one by convolving the data with a 17-point stencil. However, both filters have an anisotropic impulse response; that is, their spectra vary with azimuth. They fit very well the desired circular spectrum of $\cos\mid\vec k\mid$ along the axes, but differ from it at high wavenumbers. The error is dependent on the azimuthal direction and is maximum for azimuths rotated 45 degrees with respect to the axes. In this section we derive a filter that is more isotropic than either of the McClellan filters and has a computational cost comparable to the 17-point one. The new filter is obtained by averaging the 9-point McClellan filter with a 45 degrees rotated version of itself.

The representation of the 9-point McClellan filter in the wavenumber domain is  

$$\cos\sqrt{k_x^2 + k_y^2}\approx MC_0\left(k_x,k_y\right)=-1+{1\over{2}}\left(1+\cos k_x\right)\left(1+\cos k_y\right).$$ (1)

In the wavenumber domain, this filter can be easily rotated by an angle $\theta$by applying the change of variables,

$$\begin{eqnarray} & k_x^{\theta}= k_x\cos \theta- k_y\sin \theta \nonumber \\ & k_y^{\theta}= k_x\sin \theta+ k_y\cos \theta.\end{eqnarray}$$ (2)

Applying this rotation to the McClellan filter we obtain the general expression for a filter aligned with an azimuth rotated by $\theta$with respect to the axes,

$$\begin{eqnarray} \lefteqn{ MC_{\theta}\left(k_x,k_y; \theta \right)= } \nonumber... ...ht]\left[1+\cos\left(k_x\sin \theta+ k_y\cos \theta\right)\right].\end{eqnarray}$$ (3)

For an arbitrary $\theta$ this filter cannot be approximated in the space domain with a compact convolutional operator, but in the special case of $\theta$ being equal to 45 degrees, the general expression of equation (3) simplifies to  

$$MC_{45}\left(k_x,k_y; \theta=45^{\circ}\right)= -1+{1\over{2... ...}}}\right)+ \cos\left({k_y\over{\sqrt{2}}}\right)\right] }^{2}.$$ (4)

The spectrum of the the rotated operator is shown in Figure . As expected, it is the rotation of the spectrum of the 9-point McClellan filter (dotted line). The spectrum of the rotated filter matches the spectrum of the desired filter (solid line) along the diagonals, but not along the axes. Before averaging the two filters we scale the rotated filter such that it matches $\cos\mid\vec k\mid$ along the axes. This scaling is a non-linear stretch of the wavenumbers axes, that is defined by the relationships

$$\begin{eqnarray} & \cos \overline{k_x}= -1+{1\over{2}}{\left[ \cos\left({k_x\ove... ...2}}{\left[ \cos\left({k_y\over{\sqrt{2}}}\right)+ 1 \right] }^{2},\end{eqnarray}$$ (5)

where the bars indicate that the wavenumber axes have been remapped. Solving for $\cos\left({k_x/{\sqrt{2}}}\right)$ and $\cos\left({k_y/{\sqrt{2}}}\right)$ in equation (5) leads to the relationships

$$\begin{eqnarray} & \cos\left({k_x\over{\sqrt{2}}}\right)= -1+2\cos\left({\overli... ...{\sqrt{2}}}\right)= -1+2\cos\left({\overline{k_y}\over{2}}\right),\end{eqnarray}$$ (6)

which upon substitution in equation (4) yield the desired filter  

$$\overline{{MC}_{45}}\left(\overline{k_x},\overline{k_y}\righ... ...ht)+ \cos\left({\overline{k_y}\over{2}}\right)-1 \right] }^{2}.$$ (7)

 

rot
Figure 1 Contours of constant amplitude for the spectra of the 9-point McClellan transformation filter (double dots), the 45 degree rotate 9-point McClellan filter (dash) and the ideal circularly symmetric filter (solid).

The spectrum of the new operator is shown in Figure with a dashed line. As expected, it matches the spectrum of $\cos\mid\vec k\mid$along the axes, but not along the diagonals. However, the arithmetic and the geometric averages of the original 9-point McClellan filter $MC_0\left(k_x,k_y\right)$ with the newly derived rotated filter $\overline{{MC}_{45}}\left(k_x,k_y\right)$are filters with almost isotropic spectrum. Figure shows the spectrum of the averaged filters (dot dash for the arithmetic average and double dash for the geometric average) compared with the ideal filter (solid line) and the 17-point McClellan filter proposed by Hale (fine dash). The spectra of the averaged filters are similar and both match exactly the ideal spectrum along the axis and deviate from it considerably less than the spectrum of the 17-point McClellan filter across the whole range of azimuths.

 

rotscal
Figure 2 Contours of constant amplitude for the spectra of the scaled McClellan ($\overline{{MC}_{45}}$transformation filter (dash) and the ideal circularly symmetric filter (solid).

The filter $\overline{{MC}_{45}}\left(k_x,k_y\right)$ can be efficiently applied to the data because the operator
${\left[ \cos\left({k_x/2}\right)+ \cos\left({k_y/2}\right)-1 \right) }]$is separable; its space-domain representation is a cross-shaped convolutional operator. The arms of the cross are the space-domain representation of the $\cos\left({\k/2}\right)$ operator. This filter can be approximated in different ways; for computing the migration examples shown in the next section we windowed its ideal impulse response with a 7-point Kaiser window Oppenheim and Shafer (1975). Consequently, to apply $\overline{{MC}_{45}}\left(k_x,k_y\right)$ we convolved the data twice with a 13-point cross; longer filters can be used if more accuracy is required.

 

avg
Figure 3 Contours of constant amplitude for spectra of the 17-point modified McClellan filter (double dots), the arithmetic averaged filter (dot dash), the geometric averaged filter (double dash) and the ideal circularly symmetric filter (solid).

The application of the arithmetic averaged filter requires a convolution with the 9-point McClellan filter followed by two convolutions with a 13-point cross. The total cost of this operation is higher than convolving the data with the 17-point McClellan filter; though it yields more accurate results. However, the geometric average is as close to the ideal filter as the arithmetic average, and thus a very good approximation is also achieved by applying $MC_0\left(k_x,k_y\right)$ and $\overline{{MC}_{45}}\left(k_x,k_y\right)$ alternately between depth steps. The total computational cost of this procedure is about the same as applying the 17-point McClellan filter.