Seismic data represent an interface property wherein reflection events are seen due to relative changes in acoustic impedance of adjacent rock layers. Such observed amplitude changes may not indicate if the amplitude changes relate to lithology variations above or below an interface.

Acoustic impedance is a physical rock property, given as the product of density and velocity. Interestingly, both density and velocity are measured by well logs and impedance can be derived by multiplying the density log with the well velocity. Thus, while acoustic impedance is a layer property, seismic amplitudes are attributes of layer boundaries.

Seismic inversion for acoustic impedance is widely used in our industry today, mainly due to the ease and accuracy of interpretation of impedance data, but also because it allows an integrated approach to geological interpretation.

In a series of three articles of Geophysical Corner in the May, June and July 2015 issues of EXPLORER, the application of the different methods for transformation of stacked, prestack and multicomponent seismic data into impedance data were described. In this month’s column we revisit one of the methods, namely colored inversion, to describe in detail the methodology entailed and its application to a seismic dataset from Denmark.

Due to the band-limited nature of the source wavelet, seismic data are devoid of the low-and high-frequency information. Typically, the bandwidth of seismic data is roughly in the range of 8 to 80 hertz, unless the more recent acquisition of broadband seismic data is considered that may have the lower end of the frequency range lowered to 3 or 4 hertz.

The earliest method adopted for transforming seismic data into impedance is the recursive inversion (see the May 2015 article) yielding “relative impedance” data. As the low-frequency component (0 – 7 hertz or so) is usually missing in seismic data, it is sought from well-log data and added to the transformed relative impedance data to obtain “absolute impedance.” Such data correlate well with impedance logs, which are typically overlaid on the impedance sections to evaluate the correlations between the inverted and measured data at different well locations. Absolute impedance data are also required for derivation of rock physics properties and performing quantitative interpretation. Later, other methods were developed for impedance inversion with different levels of sophistication. All these methods are considered to require expert skills and are time consuming.

## Colored Inversion

The colored inversion method emerged out of an observation that when sparse-spike inversion (see the May 2015 article) generated impedance data were cross-correlated with the input seismic data, a single matching filter was derived. On convolving this filter with the input seismic data, the result was very similar to the impedance data derived using a more sophisticated inversion method, such as the sparse spike inversion. What this exercise suggested was that inversion over a specific time window could be approximated with a single filter. Another observation that came out was that the matching filter had a constant negative 90-degree phase, which is consistent with the observation that when a horizon is picked on input seismic data and overlaid on the equivalent impedance data, it is found to be off by the same phase. Also, the process of integration in recursive inversion (see the May 2015 article) results in a similar phase difference and is understood as resulting from the transformation from seismic reflections to geologic intervals.

As a digital filter may be specified by both its amplitude spectrum and its phase, the next question that arose was how to determine the amplitude spectrum. This was resolved by recalling that the Earth’s reflection coefficient series exhibits a predictable power law function of the form fβ, where f is the frequency and β is a positive constant that may vary from one field to another but may remain constant within a field. The frequently available acoustic impedance logs are found to depict a similar behavior, except the exponent is found to be negative, and may be taken as -α. Thus by assuming that the spectral trend of the inverted input seismic data will also be exhibited by the acoustic impedance in that area, the value of α could be determined by curve-fitting to acoustic impedance logs.

It is assumed in such filter applications that the phase of the input seismic data is zero.

## Application to a Danish Dataset

We demonstrate the application of colored inversion on a 3-D seismic data volume from Denmark, which was acquired near a place called Stenlille over an area that houses an underground natural gas storage facility to serve as a buffer for a supply of gas from the North Sea. The formation that stores the gas is a deep aquifer at a depth of 1,500 meters. Natural gas has been injected and stored at Stenlille since 1990, where the reservoir occurs within a domal subsurface structure and is covered by a tight caprock. The Upper Triassic Gassum Formation consisting of interbedded sandstones and mudstones is the reservoir where natural gas is stored by displacing formation water. Below the Gassum Formation are the impermeable mudstones of the Vinding, and other older formations. To keep the nomenclature simple, we simply refer to the top of the formation of interest as the “Upper target” and its base as the “Base target.”

Figure 1a shows the frequency spectrum of the detrended acoustic impedance logs from two wells in the area being considered. More wells tend to make the computation more robust. The equivalent frequency spectrum of the seismic data for traces at the well locations is shown in red. The thresholds in terms of frequency range to be used in the computation of relative impedance are shown by the two vertical red lines.

The same figure plotted on a log-log scale is shown in figure 1b, where the data cluster distribution is linearized and a best-fit line in cyan has been shown drawn through the data points, which mathematically approximates the well log spectrum. Notice in the frequency domain the thresholds for the seismic amplitude spectrum are shown with the dark red vertical lines restricting the frequency range between 16 and 100 hertz. The choice of such thresholds can influence the final results appreciably and need to be considered carefully. The two spectra shown now need to be compared. Thus, as the next step the seismic spectrum is simply subtracted from the modeled well spectrum, and the difference spectrum after a phase rotation of negative 90 degrees is transformed back to the time domain with the use of a Fourier transform. Such a computed filter is shown in figure 1c.

Finally, this filter is convolved with the input seismic data in a trace-by-trace fashion, and the colored inversion generated as the output. This colored inversion output is relative impedance and thus contains both positive and negative values of impedance.

An arbitrary line passing through two available wells (S-15 and S-19) extracted from the seismic volume is shown in figure 2a, and an equivalent line extracted from the computed colored inversion volume is shown in figure 2b. The colored inversion result shows reasonable correlation with the two overlaid logs and the low impedance zone in the middle of the target zone corresponding to the gas reservoir.

In figure 3a we show a stratal slice at a level through the target zone (gas reservoir) extracted from colored inversion volume, and an equivalent slice from the absolute volume is shown in figure 3b. Though the comparison might be considered as one of apples to oranges, but the reservoir seen around well S-19 is clearly imaged as low impedance in addition to a few other pockets. The overall distribution of relative impedance is not very different from the absolute impedance display.

## Conclusions

In the above analysis we have tried to explain in a simple way the steps that are utilized for the generation of colored inversion, which yields relative acoustic impedance from seismic data by utilizing well data. We have demonstrated the application of colored inversion to a dataset from Denmark acquired over a reservoir being used for storage of natural gas. A comparison of stratal display from colored inversion with the absolute impedance volume seems encouraging.

The advantages of the use of colored inversion are that it is an objective technique (other deterministic methods including prestack impedance inversion are very subjective), is fast, easy to implement, does not require any prior model and is reasonably accurate.

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