# Seismic Reflectivity and Fractal (Random) Behavior

*One of the fundamental assumptions of the convolutional model (and subsequently a lot of deconvolution methods) is that reflectivity is a purely random (or "white") process. Overall, the only predictable part of a seismogram is that of the source wavelet. Now, reflectivity is NOT purely random. Despite this, the convolutional model is still a very useful tool.

-My question(s) is this: what if reflectivity is somehow predictable? What would that (for lack of a better word/phrase) look like? How would that jeopardize the usefulness of different deconvolution methods...? Thanks in advance to anyone who can help clarify this.

• Out of curiosity, in what way do you mean reflectivity is not purely random?
– Z W
Jul 30 '17 at 8:07

## 1 Answer

The assumption that the earth's reflectivity is random (i.e. has a white spectrum) is only necessary for the statistical extraction of a wavelet from the data itself via the amplitude spectrum of the autocorrelation function. The convolutional model still works regardless of the reflectivity pattern, although getting a wavelet may be more difficult if the "white reflectivity" assumption does not hold.

On the other hand, in most sedimentary basins, earth's reflectivity pattern tends to not be random. Sedimentary deposition happens in nested cycles forming smaller cyclic layers within large cycle layers. This fractal nature is observed in the amplitude spectrum of real (i.e. measured) reflectivity series. Instead of a having a flat (i.e. white) amplitude spectrum, real reflectivity series tend to have low amplitudes in the lower frequencies and high amplitudes in the high frequencies (i.e. blue) while generally conforming to a consistent positive slope line when ploted in octave vs dB. The slope of this line is a characteristic of the local geology and contains information about the fractal nature of the layering.

This concept is used in a technique called spectral blueing (and also similarily in coloured inversion), where seismic data is forced to match the characteristic reflectivity amplitude spectrum line causing an increase in apparent bandwidth.