SIGNAL ENHANCEMENT BY PREDICTION

In historic exploration-industry use, prediction-error filtering provides temporal predictions that are immediately subtracted from the data itself. In recent years, Luis Canales proposed and developed a process of looking at the spatial predictions themselves and this process has become quite popular. The idea is that because noise is unpredictable, better-looking seismic data can result from looking at the spatial predictions than looking at the data itself. Although Canales' process is done in the temporal-frequency domain, we can also do it in the time domain, where we can maintain tighter control over nonstationarity and statistical fluctuations. The form of the prediction-error filter is

$$\begin{array} {ccccccc} a &a &a &\cdot &\cdot &\cdot &\cdot ... ...&\cdot &\cdot \\ a &a &a &\cdot &\cdot &\cdot &\cdot\end{array}$$ (7)

and the prediction is the same without the ``1''. It is perplexing that the spatial prediction has a horizontal direction. Some people average the left and the right, but here I have not. An alternative is to use interpolation,

$$\begin{array} {ccccccc} a &a &a &\cdot &a &a &a \\ a &a &a &... ... a &a &a &\cdot &a &a &a \\ a &a &a &\cdot &a &a &a \end{array}$$ (8)

but here I have not. In either case, it is important to realize that after the filter coefficients are determined by minimizing output power, the ``1'' in the filter is replaced by zero before it is used. Thus the methods are prediction or interpolation and they should not be called ``deconvolution''.

 

idapred
Figure 12 Stack of Shearer's IDA data (left). Prediction (right). Notice that the time scale is minutes and the offset is degrees of angle on the earth's surface.

To compare the spatial predictions to the data itself, I selected the interesting data set shown in Figure 12. The data plane is a stack of earthquakes. At early times, before 93 minutes travel time, the data resembles a common-midpoint gather. At later times, the strong surface waves travel round the earth and past the antipodes and come back towards the source. Otherwise, there are remarkable similarities to conventional exploration seismic data. Many fewer earthquakes are observed near 180 degrees than near 90 degrees for the simple geometrical reason that the 10 degrees surrounding the equator is a much bigger area than the 10 degrees surrounding the pole. Thus the quality of the stacks degrades rapidly toward the poles. Although data quality is poor at the poles themselves, notice that waves going beyond the antipodes come back toward the source. The data has a large dynamic range that I compressed by various range- and time-dependent gain multipliers and in the last step before display, I took the signed square roots of the values of the stack.

Professor Peter Shearer gathered the earthquakes from the IDA network, an array of about 25 widely distributed gravimeters, donated by Cecil Green, and Shearer selected most of the shallow-depth earthquakes of magnitude greater than about 6 over the 1981-91 time interval, and sorted them by epicentral distance into bins $1^\circ$ wide and stacked them. He generously shared his edited data with me and I have been restacking it, compensating for amplitude in various ways, and planning time and filtering compensations.