sfspectra computes spectrum, the absolute value of the 1-D Fourier transform along the fast axis.

The following example from rsf/su/rsfkasper shows a seismic shot gather before and after high-pass filtering and the corresponding spectra computed with sfspectra.

sfspectra has only one option: all=. If all=n (the default behavior), the spectrum is computed separately for each trace. If all=y, the output is the spectrum averaged across all traces.

For a two-dimensional version, try sfspectra2. The following figures from jsg/seislet/lena show the infamous Lena image and the “seismic Lena” (after Chris Liner and David Monk)

and the corresponding 2-D spectra computed with sfspectra2

Previous programs of the month

• sfnoise
• sfgraph
• sfclip
• sfagc
• sfenvelope
• sfcontour
• sfsmooth

tkvpconvert provides a Tkinter GUI (graphical user interface) for vpconvert. It is a silly exercise, which simply demonstrates an ability to generate specialized GUIs for Madagascar scripts and programs.

sfsmooth, one of the most useful Madagascar programs, implements smoothing by triangle filtering.

The idea of triangle filtering is explained in Jon Claerbout’s books Processing versus Inversion and Image Estimation by Example. You can find the explanation in the chapter Smoothing with box and triangle. The triangle filter of radius is a correlation of two box filters. In the Z-transform notation,
$$T_k(Z) = B_k(1/Z)\,B_k(Z)$$
where
$$B_k(Z) = \displaystyle \frac{1}{k}\,\left(1+Z+Z^2+\cdots+Z^k\right) = \frac{1}{k}\,\frac{1-Z^{k+1}}{1-Z}$$
and $k$ is the smoothing radius. Triangle filtering is a very efficient operation, because it requires only $N$ multiplications for the input of size $N$.

Triangle filtering serves as a good approximation to more expensive Gaussian filtering. Repeated triangle filtering rapidly approaches Gaussian filtering (a consequence of the central limit theorem). The following example from jsg/shape/smoo demonstrates how repeated application of triangle smoothing turns a triangle filter into a Gaussian filter

The following example from gee/ajt/triangle demonstrated how sfsmooth handles boundary conditions: the energy is reflected from the side in order to preserve the zero-frequency (DC) component of the input signal. Repeated smoothing or smoothing with a very large radius turns every signal into a constant.

In multiple dimensions, triangle smoothing is applied sequentially in different directions, possible with different radii (controlled by rect#= ) parameter. The following example from gee/ajt/mound shows shapes created by 2-D triangle smoothing after one pass and two passes (repeat=2).

Triangle smoothing is theoretically a self-adjoint operator. Numerically, the adjoint and forward operators are different. The action is controlled by the adj= parameter.

For smoothing with a box instead of a triangle, use sfboxsmooth.

For a different approach to smoothing, try sfgaussmooth, which implements smoothing by recursive Gaussian filtering.

Previous programs of the month

• sfnoise
• sfgraph
• sfclip
• sfagc
• sfenvelope
• sfcontour

sfcontour is a program for generating contour plots of 2-D data. It shares many of its parameters with other 2-D plotting programs, such as sfgrey and sfgraph. You can access these parameters by running sfdoc stdplot.
Let us discuss some of the parameters specific to sfcontour.
The number of contours, their origin and increment are given by nc=, c0=, dc=. If these parameters are not specified, they are determined from the data values. The contour lines in the following picture from jsg/agath/radon were created with sfcontour c0=10 dc=10 nc=10

Alternatively, the contours can be specified in a list with c= or in a file with cfile=.
By default, sfcontour plots only contours for the positive values of the input. If you want both positive and negative values to be represented, use allpos=n.
A 3-D version of sfcontour is sfcontour3.
The contour lines in the following picture from jsg/flat/comaz were created with sfcontour3 cfile=