Reflection seismology has always been a valuable exploration method for the petroleum industry. Seismology works because the Earth is made up of layers of varying composition. Each of these layers possesses a unique value of acoustical impedance, which is a measure of the inherent resistance the material of the layer has to penetration by a sound wave. When an acoustic wave passes from one composition layer to another, a portion of the wave is reflected back and a portion continues in the original direction. This is exactly what is happening when people can hear their voices echo off of a distant object. Physically speaking, the acoustic impedance of the distant object is much greater than the air, so the majority of the sound wave is reflected back at the person; however, a small portion of the wave does continue through the object.

With respect to the Earth, every one of the compositional layers generate a reflection, some more prominent than others. Typically, the only attribute of the reflected seismic data that is utilized is the trace displacement versus time as measured at the location of the geophone.

One fundamental shortcoming with seismic-based exploration is that a subsurface layer, as depicted by the reflected waveform, can be shaped appropriately but still not contain hydrocarbons. For years geophysicists have been searching for ways to extract more information from seismic data to help eliminate these uncertainties. One specific area of interest has been to interpret the data in the frequency versus time domain as opposed to the displacement versus time domain. This requires transforming the data to a spectral domain. Although frequency analysis techniques have existed for more than a century, it has not been until recently that a suitable technique for seismic data has been presented. This new technique is called the Wavelet Transform (WT) and exists in several different forms, depending on the application.

Mathematically, when the WT is performed on a data set, the data is projected onto a discretized family of functions, called wavelets, by means of an inner product. In the classical sense this is the technique of breaking a signal up into its basic functional parts. In other words, the result from a projection of the data onto one of the wavelets is the component of the data that matches that wavelet. As an example, imagine listening to a song which contains vocals, a guitar and drums. Using the WT, project the song onto a wavelet that matches the harmonics of the guitar and you can isolate this portion. The result is a new song that only contains the guitar from the original. The same can be done for the vocals and the drums, and then all three can be recombined to recover the original score.

What becomes interesting is the ability to analyze or modify these portions once they are isolated. For instance, one listener may prefer a louder guitar. Once isolated via the WT, the guitar portion can be multiplied by some value to increase the magnitude and then recombined with the vocals and drums for a modified version of the song. This is not unlike the effects of a stereo system equalizer in the realm of musical entertainment, except that the WT has the ability to zero in on very localized events in the overall signal. This is equivalent to an equalizer that can isolate and amplify the occasional callous-laden pluck of the index finger of the 1st chair harp in the National Symphony. The equivalent bandwidth of the WT can be made as small as desired.

With respect to seismic data, the first step in computing the inner product between the data and a wavelet is to multiply the wavelet into a single seismic trace at a specific location in time. Next, the sum of all of these multiplied values is calculated, resulting in a single value whose magnitude represents the similarity between the trace and the wavelet at that location in time. This single value is called a wavelet coefficient. If this routine is repeated for wavelets of different frequencies at different points in time, the result is a 2-D array of wavelet coefficients that represents the frequency content of the trace versus time. Notice that performing the WT on seismic data adds a new dimension, so that a 1-D trace becomes a 2-D array, a 2-D section becomes a 3-D volume, and so forth. Once these new arrays or volumes are calculated, the subsequent operations can be performed.

One typical spectral analysis (SA) technique is to view a 2-D array of the wavelet coefficients for a single wavelet or frequency, or for a combination of frequencies. This section is arranged just like a 2-D seismic section, where the x axis is the trace number and the y axis is time. Figure 1 shows a low-frequency wavelet coefficient combination plot for a section from 0 to 1,000 milliseconds ranging over 35 traces. The large red spot located near trace number 15 at 900 milliseconds is a solution gas-drive oil well in the Mississippian Limestone formation in North Texas. The theory supporting this SA interpretation states that a gas reserve of sufficient size will create a frequency anomaly lower than the average frequency of the data. There are many factors responsible for this, but the most significant one is the difference in density between a gas and a solid. Realizing that the frequency of the wave is an observed property, any reflections that contain below-average frequencies were likely generated when the wave was propagating at a slower-than-average rate, such as it would when it passes through a gas-filled cavity. Because these low-frequency components travel more slowly, these anomalies are often observed below the actual structure on the seismic section, or at a later travel time. These anomalies are what geophysicists have been referring to when they speak of gas shadows in the frequency domain.

Using the same logic, a liquid should also be distinguishable in the frequency domain. The problem here is that the differences between the physical properties of oil and water are not nearly as great as the differences between oil and rock or water and rock. Though it is sometimes possible to identify only oil, simply identifying a liquid at the top of a typically productive structure can confirm a decent level of porosity. Figure 2 shows a combined wavelet coefficient plot for a frequency range above gas but below the average frequency of the data. At this frequency range, the spectrum can become cluttered with energy from the dominant reflectors since these frequencies are very close to the average frequency of the data. In this figure the anomaly located at trace number 27 at 800 milliseconds corresponds to another solution gas-drive oil well in the Mississippian Limestone. However, this particular well produced oil at a higher rate and produced a very small amount of gas compared to the previous example. When the liquid-indicating frequency range is analyzed for the first reservoir, a very questionable anomaly is observed at the reservoir location. Likewise, when the gas-indicating frequency range is analyzed for the second reservoir, a very weak anomaly is present. These SA results suggest that the first reservoir should be a stronger gas producer but a weaker oil producer than the second. The actual production records for these reservoirs indicate that the second reservoir accumulated just over 40,000 bbl of oil, whereas the reservoir depicted in Figure 1, though still a productive well, will be lucky to accumulate 20,000 bbl.

The technique has some limitations. With respect to a WT-based SA, low frequency wavelets give poor time resolution with good frequency resolution, whereas higher frequency wavelets give good time resolution with poor frequency resolution. To better understand this WT property, realize that the data was recorded at a fixed sampling rate, typically around one data point per millisecond. Since the wavelength of a low-frequency wave is longer in time than a high-frequency wave, there are fewer data points to describe one complete oscillation of a high-frequency event. With respect to time resolution, the time resolution becomes poor for low frequencies because one complete oscillation requires more time to evolve.

WT-based spectral whitening

Another WT-based analysis technique is to enhance certain frequency levels from the data. Using this technique, the uncertainty in time and frequency is not problematic. Remembering the musical example, it can be beneficial for the interpreter to view a typical seismic section that has had a specific frequency range amplified. Figure 3 shows a seismic section that contains the second reservoir from the previous section of this article. Though it is difficult to interpret, the center of the reef is located at trace number 27. Using the same family of wavelets as before, the wavelet coefficients resulting from the WT are amplified for a frequency range that typically indicates liquids; then the data is reconstructed using the inverse of the transform. Figure 4 shows this reconstructed section, with the shape of the reef now clearly visible.

Conclusions

Overall, the WT-based SA looks promising. With improvements to the algorithms as well as the discovery of more effective wavelets occurring as time progresses, geophysicists will continue to discover correlations between the spectral content of the data and the reserves. It should only be a matter of time before the technique finds its way into seismic processing programs for coefficient analysis, spectral whitening and even noise removal.

For more information, contact the author at phillips@digitalpassage.com.

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