This report presents a description and several examples of application of a new program for the estimation of effective seismic Q. The program is named qest and employs an algorithm based on a method of spectral ratios wherein the ratios are computed one sample at a time using a continuously time-variant DFT. The marine data examples show that the values computed by the program are consistent, agree well overall with expected values for marine sediments, and show anomalously low Q values associated with gas reservoirs. These low Q values are also shown to be associated with anomalies in the peak, or dominant, Fourier frequency in the data.
The dissipation of the energy of propagating seismic waves is a physical reality. The mechanisms for this dissipation are generally understood and are attributed to many effects, among which is specific attenuation, which is proportional to Q-1. Typical values of attenuation for sediments are shown in Figure 1. The reader should understand that in the discussion that follows, the use of Q will refer to the inverse of attenuation. The effects of Q on seismic data are to reduce the contribution of higher frequencies to the wave form, resulting in a decrease in amplitude, and to induce a temporal delay in the arrival time of the wave form (Berkhout,1987). Procedures for compensating recorded data for other dissipative effects such as spherical divergence and transmission losses are a common part of any seismic data processing flow. Compensation for attenuation (cf. Luh, 1993; McGlynn and Crider, 1986; Bickel and Natarjan, 1985; Hale, 1981, 1982) is, however, generally ignored, due at least in part to the difficulties involved in measuring the value of Q. Meissner and Theilen (1983) point out that while it is unlikely that all of the effects can be separated, as long as the measured Q-values are consistent and really characterize a specific layer, one need not worry too much about the individual contributions to energy dissipation. If measurements are made on this premise, it is unlikely that the value measured will be the true Q value, but measurement of a meaningful relative value for Q should indeed be possible.
The usefulness of information obtained from measurements of Q is dependent upon the understanding of the source(s) of the dissipative Q effects. These sources are numerous and are related to the sedimentological parameters of the propagation medium. Meissner and Theilen (1983) and Klimentos (1995), suggest that two principal sources of Q effects in porous sediments are fluid flow and electro-chemical processes and that the Q effects are enhanced by increased hydrocarbon saturation. Others (e.g. Madatov and Helle, 1994) attribute the Q effects to pore pressure. While both of these positions seem well founded, reconciliation between them seems to suggest that any measurement of Q should be able to identify both a localized Q effect, due at least in part to increased hydrocarbon saturation in sediments, and a more general, or ``low frequency'', effect due to overpressure. Measurements of consistent Q values, then, particularly when considered with confirming measurements such as AVO, should provide an additional aid in the identification of significant accumulations of hydrocarbons.
This report presents examples of measurements of Q from seismic data collected in areas of hydrocarbon discoveries for the purpose of verifying that consistent Q values can be measured from seismic data and that anomalous values of Q may be associated with zones of significant hydrocarbon saturation. The theory and methodology of the method for measuring Q is presented. The measured Q values for several marine data examples are presented as color contour displays, along with displays of the dominant Fourier frequency in the data
An expression for an attenuated propagating plane wave as a function of time and frequency, is given by (Luh (1993), Bickel and Natarajan (1985))
U(w,t) = U(w,0)exp[-wt/2Q + i wt/(piQ) ln(w/w0)] (1)
where w is frequency in radians/sec and Q is the effective Q (the integration of the interval Q's).
A measure of the energy density of the waveform is computed as the autocorrelation, or power spectrum, of the plane wave, given by
U*(w,t)U(w,t) = U*(w,0)U(w,0)exp[-wt/Q], (2)
where the ``*'' signifies complex conjugate.
Now, if we assume that U(w,t) is the frequency domain representation of a spike ( U(w,0) = 1.) which has propagated one time unit through the attenuating medium and we take a linear regression of the natural logarithm of the ratio of the output spectrum to the input spectrum, we find that the specific attenuation, in dB per wavelength, is given by
where L is the slope of the log of the power spectrum at time t, Q is the quality factor, and the constant is 10 log10 e, which is derived from the conversion of the power spectrum to dB.
In the general application of the method of spectral ratios (Butler, 1979; Jacobson et al., 1981; Spencer et al, 1982; Stainsby and Worthington, 1985), the ratio is computed for windows separated in time by about one second and slope of the ratio is computed by linear regression over all frequencies. However, this procedure is adversely impacted by changes in the shape of and local maxima in the power spectra due to changes in the local stratigraphy (cf. Spencer et al, 1982; Okaya, 1995; Partyka, 1995) and noise, and the large temporal separations introduce unknown variables into the ratio computations. Also, since the Q effects are expected to be most pronounced for the high frequencies, including low frequency information in the regression may tend to increase the measurement error. In the current procedure, the uncertainties introduced by the large separation are minimized through a continuously time-variant spectral estimation. The stratigraphic effects are minimized by identifying the local dominant, or peak, spectral component and measuring the slope relative to that maximum over at least an octave above that component. Additionally, distortions of the low frequency retion of the spectrum due to acquisition and processing are avoided by relying on the higher frequencies. Tests have shown that this method gives much more consistent results than those obtained using the entire spectrum for windows widely separated in time.
The spectra and, therefore the spectral ratios, are computed for each time sample by computing the modulus of a continuously time-variant Discrete Fourier Transform, or DFT. This DFT is derived by writing the DFT over the first n-length temporal window as a Z-transform
F1(w) = a0 + a1 Z1 + a2 Z2 + ..........+ an Zn,
where the coefficients (a's) are the data sample values, w is angular frequency, and Z is exp[-iwk], where k=0,1,...,(N-1) and w = 2pinu/M, nu=0,1,...,(M-1) . The transform of the second window, relative to the first, is given by
F2(w) = a1 + a2 Z1 + a3 Z2 +.............+ anZn-1 + an+1 Zn,
F2(w) = (F1(w)-a0) Z-1 + an+1 Zn
This recursive operation requires few mathematical operations to compute each successive transform, and is therefore fast and efficient. A similar procedure is used to remove the average from the data windows prior to computing each transform.
Program qest is simple to use. The data to be analyzed requires no special processing and the program itself requires few parameters. The data should be processed in the same manner as for AVO analysis, with care taken to ensure that the scaling is proper, that normal moveout is correctly applied, and that multiples have been removed as effectively as possible. Deconvolution can be applied (apparently) without adversely affecting the analysis. The data should not be Q compensated, not even for a constant Q! The data can be stacked and migrated [note that complete analysis in the (non-zero) offset domain requires the introduction of an additional, incident angle-dependent term for which the application has not been worked out yet]. It is advisable to apply a mix to the data prior to the analysis to improve statistics. If the tool is to be used to estimate Q for applying compensation in the processing, a plausible plan is to perform the analysis on shot data with spherical divergence (and transmission) compensation, with and without (spiking) deconvolution, and to use the estimate that seems most reasonable.
Program qest may be executed on any of the computing platforms normally used for seismic data processing. The program has only two key parameters that you might want to examine. These parameters are the window length and the bandwidth over which to perform reqression to find the slope of the log of the power spectrum. The program defaults these parameters to 500 ms and 1.5 octaves, respectively. In the tests to date, these seem to be good parameters, but you might want to test them. Tests indicate the the analysis is sensitive to both of these parameters. The sensitivity to window length is not unexpected, since we expect to see attenuation effects due to lithology changes, pore fluid changes, and pore pressure changes which may have shorter or longer effective wavelengths, and you will want to use the window length which best suits the nature of your analysis (shorter windows for lithologic/fluid analysis, longer windows for pressure analysis). The sensitivity to bandwidth is a function of the statistics available for the linear regression and, in general, the more statistics available (broader bandwidth), the better the regression will be and the better the Q estimate will be. Check the amplitude spectra in your data before specifying bandwidths much greater than about two octaves, though.
Figure 1. Compilation of in-situ attenuation values for P-waves in various geologic age rocks at depth (From notes for Amoco's Seismic Rock Properties course). Note that an attenuation value of 0.1 corresponds to a Q value of about 270, while an attenuation value of 2 corresponds to a Q value of about 14 (see equation (3)).
In lieu of application to synthetic data and to hasten attempts to calibrate these measurements against known sedimentological environments, program qest has been applied to marine data from three different areas. For marine sediments, Q values in the range of about 60 to 200, corresponding to attenuation values from about .15 to about .5 in Figure 1, are considered normal (Miessner and Theilen, 1983). In the first example presented, hydrocarbons have been discovered. In the second example, AVO analysis strongly suggests the presence of hydrocarbons. The third example is presented as a case in which hydrocarbons have been discovered but no anomalous Q can be detected, possibly due to thin reservoirs and insufficient frequency content in the data but most likely due to improper data perparation.
The first case, illustrated in Figures 2a-c, illustrates application of program qest to data over a recent gas discovery within the structural high highlighted at about 3.1 sec. Clearly associated with the high amplitudes at the level of the reservoir is a zone of low Q as well as zone of low dominant frequency. Note that both the low Q zone and the low frequency zone are consistent with structure and with each other.
Figure 2a. Seismic data showing reflections from gas discovery at about 3.1 seconds. The bright amplitudes correspond to the producing horizons. In the color scale to the right of the seismic data, the numbers in the cells are only indices and have no meaning. The numbers outside and to the right of the cells give the data magnitudes to which the associated colors correspond on the data display.
Figure 2b. qest output for the data in Figure 2a. Note the zone of low Q values in the range of the mid 20's to low 30's in the center of the figure at about 3.1 seconds. This zone corresponds to the gas reservoir indicated in Figure 2a. Note that over the rest of the display the Q values are primarily in the range of 60 to 90. The spurious high values are considered noise.
Figure 2c. Dominant Fourier spectral components for the data in Figure 2a. A zone of low frequencies in the range of mid 10's to low 20's is associated with the zone of low Q indicated in Figure 2b.
The second case is illustrated in Figures 3a-c. These data show a zone containing bright amplitudes related to a structural high and its (left) flank in the highlighted zone. AVO analysis has indicated strong positive anomalies associated with each of these amplitudes. The Q estimation data also clearly indicate a zone of low Q and low dominant frequency in the highlighted zone. Another high amplitude event, outside the highlighted zone, on the right flank at about 2.1 sec, is evident in Figure 3a. The estimated Q and dominant frequency for this zone are also anamolous relative to the surrounding sediments. However, while the estimated Q is anomalously low, the dominant frequency is anomalously high. These anomalies are unexplained at the present time.
Figure 3a. Portion of seismic section showing a zone of bright amplitudes on the right side of the figure, between 1.9 sec. (top of structure) and 2.1 sec. (left flank of structure). In this figure, black corresponds to positive amplitude and white to negative amplitude. These bright amplitudes have been associated with AVO anomalies.
Figure 3b. Output from program qest for the data in Figure 3a. Note the zone of low Q values in the range of the mid to low 20's associated with the high amplitudes on the left flank of the structure, whereas the values elsewhere are in the range of the upper 50's to mid 100's.
Figure 3c. Dominant Fourier spectral components for the data in Figure 3a. Note the zone of very low frequency (in the range 5-10 Hz) associated with the zone of low Q in Figure 3b while elsewhere the dominant frequencies are generally greater than 25 Hz.
The third case, in Figures 4a-c, illustrates data from an area of known condensate production from thin (about 15 feet) reservoirs. Several stacked reservoirs have been discovered in zone corresponding to the highlighted area. These data have been processed with spectral balancing and time-variant scaling as well as with time-variant filtering, so it is unlikely that Q estimation will be successful on this data. Accordingly, the Q estimation and peak Fourier frequency displays in Figures 4b and 4c show no distinguishable zones of either low Q or low dominant frequency in the area around the reservoirs, nor anywhere else, for that matter. These results are entirely inconclusive because of the inappropriate processing and are presented simply to show that processing of the data to be analyzed should be carefully considered.
Figure 4a. Portion of a seismic line showing a zone (highlighted), between about 1.3 and 1.5 seconds on the right portion of the figure, of thin (about 15 feet) reservoirs. Unlike the data in the previous examples, these data have been not been properly processed for Q estimation analysis.
Figure 4b. Output from program qest for the data in Figure 4a. Note that while there does appear to be a zone of lower Q in the highlighted zone, there is little differentiation from the surrounding values. The general trend to lower Q with increasing depth is attributed to the time-variant filter and scaling applied to the data.
Figure 4c. Dominant spectral components output from program qest for the data in Figure 4a. Note that, as for the Q estimate, while there appear to be low frequencies associated with the highlighted reservoir zone, there is little differentiation from the surrounding areas. The trend to lower frequencies imposed by the time-variant filtering applied to the data during processing is evident.
The theory and methodology for an algorithm to estimate seismic Q have been presented. This algorithm has been implemented in program qest . Examples of the use of program qest to estimate Q for marine data show that reasonable values of effective seismic Q may be derived from seismic data. These examples also show that when reservoir thickness and freqency content of the data are sufficient, low values of effective Q may be observed from seismic data and associated with hydrocarbon-bearing sediments and anomalies in Q are associated with anomalies in AVO. Observation of the peak, or dominant, Fourier spectral component suggests that low Q zones may be associated with zones of low frequency. All of these observations are subject to further calibration.
Berkhout, A.J. 1987, Applied Seismic Wave Theory, Elsevier Publishing Co., New York, pp 359-365.
Bickel, S.H. and Natarajan, R.R., 1985, Plane-wave Q Deconvolution, Geophysics, Vol. 50, No. 9, pp 1426-1439.
Butler, T. M., 1979, Seismic Attenuation Studies Using Frequency Domain Synthetic Seismograms, M.S. Thesis, Texas A&M University, 154 pp.
Hale, D., 1981, An Inverse-Q Filter, Stanford Exploration Project Report Number 26, pp 231-244.
Hale, D., 1982, Q-Adaptive Deconvolution, Stanford Exploration Project Report Number 30, pp 133-158.
Luh,P.C., 1993, Wavelet Attenuation and Bright-Spot Detection, in Offset-Dependent Reflectivity - Theory and Practice of AVO Analysis, J.P. Castagna and M.M. Backus ,eds, Society of Exploration Geophysicists, pp 190-198.
Jacobson, R.S., Shor, G.G. Jr., and Dorman, L.M., 1981, Linear Inversion of Body Wave Data - Part II: Attenuation Versus Depth Using Spectral Ratios, Geophysics, Vol. 46, No. 2, pp 152-162.
Madatov, A.G, and Helle, H.B., 1994, Pore Pressure From Seismic Attenuation.
McGlynn, J.D and Crider, R.L., 1986, Inverse Q Filtering, Presented at the Frebruary, 1986 RGTS Meeting, Tulsa, OK.
Meissner, R., and Theilen, F., 1983, Attenuation of Seismic Waves in Sediments, Proceedings of the Eleventh World Petroleum Congress, Volume 2, SP3, pp 363-379.
Okaya, D.A., 1995, Spectral Properties of the Earth's Contribution to Seismic Resolution, Geophysics, vol. 60., No. 1, pp 241-251.
Partyka, G, 1995, Seismic Attribute Techniques, Notes and Reference Material, Amoco Tulsa Technology Center.
Spencer, T.W., Sonnad, J.R., and Butler, T.M., 1982, Seismic Q - Stratigraphy or Dissipation, Geophysics, Vol. 47, No. 1, pp 16-24.
Stainsby,S. D. and Worthington, M.H., 1985, Q Estimation from Vertical Seismic Profile Data and Anomalous Variations in the Central North Sea, Geophysics, Vol. 50, No. 4, pp 615-626.
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