2-D MODELING AND DECON

This chapter shows many examples to give some feeling for multidimensional spectra, deconvolution, and modeling. In each of Figures 2-7 we see three panels. The left panel is a two-dimensional data set which is an image of something that might be familiar. Let this data d(x,y) have Fourier Transform D(k_x,k_y).

Define noise n(x,y) to be a plane of random numbers. In Fourier space, these numbers are N(k_x,k_y). Now compute an amplitude spectrum by smoothing $A=\sqrt{\overline{N(k_x,k_y)} N(k_x,k_y})$.Using this noise we compute the synthetic data in the center frame. It has the 2-D spectrum $N\times A$which is the same 2-D amplitude spectrum of the given data but a different phase spectrum.

For each illustration, notice the similarities and differences between the real data and the synthetic data.

The rightmost frame has Fourier Transform D/A, the original data with its spectrum divided out. What we see is called ``deconvolved data''. It is also called ``prediction error''. Theoretically, its output is ``white''. Actually its spectrum is only approximately white because the spectrum is smoothed before it is divided out. (The name prediction error derives from a simple model that can be applied to all data. This simple model is that all data can be modeled as white noise into a filter where the filter has a spectrum that is a smoothed version of that of the data.)

 

granite
Figure 1 Synthetic granite matches the training image quite well. The prediction error is large at crystal grain boundaries and almost seems to outline the grains.

 

wood
Figure 2 Synthetic wood grain has too little white. This is because of the nonsymmetric brightness histogram of natural wood. Again, the deconvolution looks random as expected.

 

herr
Figure 3 A banker's suit (left). A student's suit (center). My suit (right). The deconvolution is large where the weave changes direction (herring bone spine).

 

basket
Figure 4 Basket weave. The synthetic data fails to segregate the two dips into a checkerboard pattern. The deconvolution looks structured.

 

brick
Figure 5 Brick. Synthetic brick edges are everywhere and do not enclose blocks containing a fixed color. PEF output highlights the mortar.

 

ridges
Figure 6 Ridges. A spectacular failure of the stationarity assumption. All dips are present but in different locations. The ridges have been sharpened by the deconvolution.

 

WGstack
Figure 7 Gulf of Mexico seismic section, modeled, and deconvolved. Do you see any drilling prospects in the synthetic data? The deconvolution suppresses the strong horizontal layering giving a better view of the hyperbolas.