During interpretation of layered sequences using band limited seismic data, it is useful to have some idea regarding the resolution of thin beds. In the past, limits of resolution and detection were linked to attributes of the bandlimited temporal response. This usually involved:
(1) an unrealistically simplified view of the subsurface (i.e. a single layer embedded in a quiet background), and
(2) a simple filter design (e.g. Ormsby, Ricker, etc...) that makes no attempt to decouple spectral colour from the spectral wavelet/filter shape.
Simplifying in this manner was convenient both computationally and for visualization, but it does not provide a realistic view of the interplay between the more complex earth and the seismic signal.
As the seismic interpretation toolkit continues to expand and improve, it is getting easier to model and investigate seismic response from more complex layer-stacking patterns. In addition, such probing can now be done in both time and frequency domains. Resolution and detection can now be investigated with respect to traditional attributes such as:
- peak-to-trough time separation
- dominant frequency
- upper frequency
It can now also be approached as a function of:
- available signal bandwidth
- signal to noise at each frequency
- complexity of the layer stacking pattern
- gross thickness of the layer-stacking pattern
- impedance contrasts
(Partyka G., 2005 SEG Distinguished Lecture)
One concept that is now well accepted, is that reflectivity and impedance profiles do not exhibit a white spectrum (Velzeboer 1982). This concept of spectral colour was further developed by Walden & Hoskens (1985) and Lancaster & Whitcombe (2000). Their work on reflectivity and impedance shows:
Reflectivity exhibits a "blue" spectrum (i.e. less energy at lower frequencies, more energy at higher frequencies).
Impedance exhibits a "red" spectrum (i.e. more energy at lower frequencies, less energy at higher frequencies).
We can represent a layer-stacking pattern (i.e. the geology) as reflectivity or impedance. The geological content is exactly the same in both cases, but the spectral response (i.e. spectral colour) of reflectivity is dramatically different from the spectral response (i.e. spectral colour) of impedance. The dominant frequency is significantly higher in the reflectivity case than in the impedance case. If we filter both cases with exactly the same bandpass filter, we still end up with a significantly different spectral responses (even though the geological content is exactly the same).
If we define resolution according to the traditional view, (e.g. using peak-to-trough time separation, dominant frequency, etc...), the reflectivity version provides more resolution than the impedance version. But how can this be? The geological content is identical. The bandpass filter parameters are identical. It is just the representation (spectral colour and phase) that is different. The problem is that the traditional approach to resolution and detection makes no attempt to decouple spectral colour from the wavelet shape.
By incorporating spectral colour as part of the modeling workflow, and decoupling it from the seismic wavelet/filter shape, we can better understand the impact of each component on the combined response. These individual components are:
- the geological content (i.e. local interference pattern) controlled by the reflective target material
- the spectral colour, controlled by choice of reflectivity (blue) or impedance (red) to represent the reflective material.
- the seismic wavelet (e.g. Ormsby, Ricker, etc....)
The spectral colour can be characterized by the best least-squares fit log/log straight lines that fit the impedance and reflectivity spectra of the input model. This is fundamentally what is done during the accepted industry techniques of blueing and coloured inversion (Lancaster & Whitcombe 2000). The spectral colour can then be removed by removing the log-log linear trend of the spectral response.
By being able to model the interplay between the individual components, we gain a better understanding of how each one impacts the ability to resolve or detect reflective material. The purpose of this modeling tool is to:
(1) construct a geological scenario (impedance/reflectivity profile),
(2) specify a filter shape, and
(3) view the bandlimited response (interference pattern) in both time and frequency, with and without the presence of spectral colour.
This simple program shows the effects of spectral colour and filter parameters on a sequence of differing acoustic impedance blocks. The blocks themselves are defined by the times to reflecting horizons with the acoustic impedance corresponding to the layer above, similar to the way simple velocity models are used for time to depth conversion.
The layering model and filter parameters are defined in the form that is displayed when SpectralColourAndResolution, below is selected. Initially, this form is populated with an example set of values demonstrating the format for describing the model.
The user has a choice of filtering with either a trapezoidal Ormsby filter, defined by its four corner frequencies, or a Ricker wavelet, defined by its peak frequency. The filter and trace parameters are entered in the appropriate boxes. The layering model is defined as a set of horizon time / impedance of the layer above pairs in the lower text box.
The trapezoidal Ormsby filter is defined by its corner points such that 0 ≤ f1 ≤ f2 ≤ f3 ≤ f4.
Frequencies below f1 and above f4 are not passed. Between frequencies f2 and f3 there is no attenuation. The filter strength between f1 to f2 and f3 to f4 varies linearly.
The Ricker wavelet based filter is defined by its peak frequency.
The output sample interval is used to define the filtered traces. The trace samples can be optionally displayed.
The impedance model is defined in the text area below the parameter input boxes. The symbol '#' is used to indicate that the rest of the line, in which it occurs, is a comment.
The model comprises a series of horizon time / impedance of the layer above pairs. The times are in milliseconds and represent the end of the block with the associated impedance. Thus the first pair represents the background impedance and the time of the start of the first layer. The last time pair represents the end time and impedance of the last block. Times later than this in the model have the background impedance value assigned. The background can be considered to extend to infinity in both directions, thereby eliminating artifacts created by a finite length model. Thus a typical input would be:
200 1 # first horizon at 200 ms with a background impedance of 1 unit extending infinitely upwards
210 2 # Second horizon at 210ms, with an impedance of 2 units in the layer above.
215 1.5 # Third horizon at 210ms, with an impedance of 1.5 units in the layer above.
225 2 # Time ofthe last horizon, which has an impedance of 2 units in the layer above. Below this, extending to infinity, the model reverts to the background value of 1 unit as defined in the first pair.
The 'Calculate and Display' button selects calculation and display of the filtered traces and their associated spectra.
When a valid model has been generated the associated spectra and traces can be downloaded as comma delimited, '*.csv', files using the 'Save Spectra' and 'Save Traces' buttons.