TWO 1-D PEFS VERSUS ONE 2-D PEF

Here we look at the difference between using two 1-D PEFs, and one 2-D PEF. Figure shows an example of sparse tracks; it is not realistic in the upper-left corner (where it will be used for testing), in a quarter-circular disk where the data covers the model densely. Such a dense region is ideal for determining the 2-D PEF. Indeed, we cannot determine a 2-D PEF from the sparse data lines, because at any place you put the filter (unless there are enough adjacent data lines), unknown filter coefficients will multiply missing data. So every fitting goal is nonlinear and hence abandoned by the algorithm.

 

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Figure 6 Synthetic wavefield (left) and as observed over survey lines (right). The wavefield is a superposition of waves from three directions.

The set of test data shown in Figure is a superposition of three functions like plane waves. One plane wave looks like low-frequency horizontal layers. Notice that the various layers vary in strength with depth. The second wave is dipping about $30^\circ$ down to the right and its waveform is perfectly sinusoidal. The third wave dips down $45^\circ$ to the left and its waveform is bandpassed random noise like the horizontal beds. These waves will be handled differently by different processing schemes, so I hope you can identify all three. If you have difficulty, view the figure at a grazing angle from various directions.

Later we will make use of the dense data region, but first let $\bold U$ be the east-west PE operator and $\bold V$ be the north-south operator and let the signal or image be $\bold h = h(x,y)$.The fitting residuals are  

$$\begin{array} {lll} \bold 0 &\approx& (\bold I - \bold J) (... ... \ \bold h \\ \bold 0 &\approx& \bold V \ \bold h \end{array}$$ (19)

where $\bold d$ is data (or binned data) and $(\bold I-\bold J)$masks the map onto the data.

Figure shows the result of using a single one-dimensional PEF along either the vertical or the horizontal axis.

 

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Figure 7 Interpolation by 1-D PEF along the vertical axis (left) and along the horizontal axis (right).

Figure compares the use of a pair of 1-D PEFs versus a single 2-D PEF (which needs the ``cheat'' corner in Figure .

 

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Figure 8 Data infilled by a pair of 1-D PEFs (left) and by a single 2-D PEF (right).

Studying Figure we conclude (what theory predicts) that

• These waves are predictable with a pair of 1-D filters:
  • Horizontal (or vertical) plane-wave with random waveform
  • Dipping plane-wave with a sinusoidal waveform
• These waves are predictable with a single 2-D filter:
  • both of the above
  • Dipping plane-wave with a random waveform