Multidimensional autoregression

Simple dip filters

Convolution in two dimensions is just like convolution in one dimension except that convolution is done on two axes. The input and output data are planes of numbers and the filter is also a plane. A two-dimensional filter is a small plane of numbers that is convolved over a big data plane of numbers.

Suppose the data set is a collection of seismograms uniformly sampled in space. In other words, the data is numbers in a $(t,x)$ -plane. For example, the following filter destroys any wavefront aligned along the direction of a line containing both the ``+1'' and the ``$-1$ ''.

$$\begin{array}{cc} -1 &\cdot \ \cdot &\cdot \ \cdot &1 \end{array}$$ (22)

The next filter destroys a wave with a slope in the opposite direction:

$$\begin{array}{cc} \cdot &1 \ -1 &\cdot \end{array}$$ (23)

To convolve the above two filters, we can reverse either on (on both axes) and correlate them, so that you can get

$$\begin{array}{ccc} \cdot & -1 &\cdot \ 1 &\cdot &\cdot \ \cdot &\cdot & 1 \ \cdot & -1 &\cdot \end{array}$$ (24)

which destroys waves of both slopes.

A two-dimensional filter that can be a dip-rejection filter like (22) or (23) is

$$\begin{array}{cc} a & \cdot \ b & \cdot \ c & 1 \ d & \cdot \ e & \cdot \end{array}$$ (25)

where the coefficients $(a,b,c,d,e)$ are to be estimated by least squares in order to minimize the power out of the filter. (In the filter table, the time axis runs vertically.)

Fitting the filter to two neighboring traces that are identical but for a time shift, we see that the filter coefficients $(a,b,c,d,e)$ should turn out to be something like $(-1,0,0,0,0)$ or $(0,0,-.5,-.5, 0)$ , depending on the dip (stepout) of the data. But if the two channels are not fully coherent, we expect to see something like $(-.9,0,0,0,0)$ or $(0,0,-.4,-.4,0)$ . To find filters such as (24), we adjust coefficients to minimize the power out of filter shapes, as in

$$\begin{array}{ccccc} v & a & \cdot \ w & b & \cdot \ x & c & 1 \ y & d & \cdot \ z & e & \cdot \end{array}$$ (26)

With 1-dimensional filters, we think mainly of power spectra, and with 2-dimensional filters we can think of temporal spectra and spatial spectra. What is new, however, is that in two dimensions we can think of dip spectra (which is when a 2-dimensional spectrum has a particularly common form, namely when energy organizes on radial lines in the $(\omega,k_x)$ -plane). As a short (three-term) 1-dimensional filter can devour a sinusoid, we have seen that simple 2-dimensional filters can devour a small number of dips.

Multidimensional autoregression