[1]Q analysis refers to the procedure for estimating Q directly from a reflection seismic trace. Conventional Q analysis method compares two seismic wavelets selected from different depth (or time) levels, but picking “clean” wavelets without interferences from other wavelet and noise from a reflection seismic trace is really a problem. Therefore, instead of analysing individual wavelets, I perform Q analysis using the Gabor transform spectrum which reveals the frequency content changing with time in a seismic trace. I propose two Q analysis methods based on the attenuation function and compensation function, respectively, each of which may produce a series of average values of Q^{−1} (inverse Q), averaging between the recording surface (or the water bottom) and the subsurface time samples. But the latter is much more stable than the former one. I then calculate the interval or layered values of Q^{−1} by a constrained linear inversion, which produces a stable estimation of the interval-Q series.

[2] In this research letter, I propose procedures and methods for estimating seismic Q values directly from a reflection seismic trace. The procedures are akin to the velocity analysis and thus are referred to as the Q analysis.

[3] Conventional Q estimation methods directly compare two seismic wavelets, selected at different depth (or time) levels from, for instance, a VSP downgoing wavefield [Wang, 2003]. When using reflection seismic data recorded at surface, however, it is difficult if not impossible, to pick “clean” wavelets from a seismic trace without interferences from other wavelet and noise [White, 1992; Dasgupta and Clark, 1998]. I present here the Q analysis methods that are based on the Gabor transform spectrum of a seismic trace, instead of analysing individual wavelets. Gabor transform reveals the frequency content changing with time, by modeling localized time and frequency characteristics of a signal simultaneously. It is in contrast to the Fourier transform which “considers phenomena in an infinite interval and this is very far from our everyday point of view” [Gabor, 1946]. Thus, the Gabor transform spectrum is an appropriate measurement for the seismic attenuation analysis.

[4] I propose two Q-analysis methods based on the amplitude attenuation and compensation functions, respectively. The primary difference between these two methods is the stability. The attenuation-based method, fitting a theoretical attenuation function to the data attenuation measurement in the least-squares sense, seems straightforward in implementation. However, when a plane wave travels beyond a certain distance, its amplitude is attenuated to a level weaker than the ambient noise and including it in the Q analysis may cause large errors in Q estimate. Therefore, I further propose a stable Q analysis method, which is based on the stabilized amplitude compensation function.

[5] Each of these two Q analysis methods may be used to produce a series of average values of Q^{−1} (inverse Q), averaging between the recording surface (or the water bottom) and the subsurface time samples. Once the average values of Q^{−1} are obtained, I then calculate the interval or layered values of Q^{−1} using a linear inversion approach.

2. Attenuation-Based Q Analysis

[6] For a given seismic trace, u(t), we can use Gabor transform to generate a time-variant frequency spectrum, U(τ, ω), where τ is the travel time and ω is the angular frequency. In U(τ, ω), considering only attenuation, we may express the amplitude of a plane wave explicitly as

where A_{0} is the amplitude at τ = 0, and constant Q^{−1} (inverse Q) is an average between τ = 0 and the current time τ. We may also rewrite equation (1) as a linear equation

Considering real data from reflection seismic which is usually band-limited, we may rewrite equation (2) as

where χ ≡ ωτ, and y(χ) ≡ ln[A^{2}(χ)/A^{2}(χ_{a})]. In practice, we may set A^{2}(χ_{a}) to be the maximum power, at the coordinate χ_{a} and fit data samples for χ ≥ χ_{a} using the linear equation (3).

[7] To fit data with equation (3), i.e., to estimate the slope Q^{−1}, we may set up a least-squares problem as follows:

where x is the digitized variable χ and y is the discrete data set y(χ). Taking derivative dJ/dQ = 0, it leads to

[8] Data fitting (equation (5)) is performed only within the range [χ_{a}, χ_{b}], where the upper limit χ_{b} corresponds to the threshold for cutting off the small values of the 1-D spectrum y(χ). Only the numerically significant part of y(χ) should be considered in y. Including excessively small numbers of y(χ) which are smaller than the ambient noise would cause large errors in Q estimation. The threshold is given as

where G is a specified threshold, a negative value in dB (say, −50 dB). The threshold is set naturally by the stabilization factor σ^{2} used in inverse Q filtering and is linked to the signal-to-noise ratio (S/N) of the data set.

[9] The physical meaning of such a threshold is that, in the minimization problem (4), the support region for χ ≡ ωτ is finite, as seismic signals are band-limited and have finite duration, due to the attenuation effect. The support region can be defined by

Note that the support region is not only strictly the function of Q, but also the function of the data S/N ratio.

[10]Figure 1 demonstrates the implementation of the constant-Q^{−1} analysis, which consists of the following three steps:

[11] (1) Performing Gabor transform to a seismic trace (Figure 1a) and producing a time-variant spectrum A^{2}(τ, ω) (Figure 1b);

[12] (2) Transforming the 2-D spectrum A^{2}(τ, ω) into the 1-D spectrum A^{2}(χ) (Figure 1c);

[13] (3) Estimating Q^{−1} (Figure 1d), using spectral data y(χ) within the support region.

[14] In the 2-D Gador transform spectrum (Figure 1b), seismic wavelets appears as localized energy envelops along the time (τ) direction vertically. Such a localization feature can be used for stratigraphic visualization and even for gas-shadow indication as by Castagna et al. [2003], in which they used wavelet transform to generate the time-frequency spectrum. It may also be used to estimate Q values directly, based on the shift of the centre frequency of the pulse to a lower value during anelastic wave propagation, as shown by Matheney and Nowack [1995] for crustal-scale seismic refraction data and Dasios et al. [2001] for sonic logging data.

[15] In the Q analysis here, I transform the Gabor spectrum from 2-D to 1-D first (Figure 1c). The advantage is that, after the transformation, the spectrum decreases monotonically along the axis χ, so that one can use a monotonic attenuation function to fit the data attenuation measurement. In this way, one conducts the Q analysis using information from the whole seismic trace, rather than comparing individual wavelets.

3. Compensation-Based Q Analysis

[16] Compensation-based Q analysis is an alternative to the previous attenuation-based method and should be more stable.

[17] Given the amplitude attenuation measurement, we can use it directly to design a gain curve for compensating the amplitude spectrum. We may then use such a data-driven gain curve to estimate Q, by fitting it with a theoretical compensation function. The gain curve is designed assuming we concern about only amplitude effect. However, once we have obtained the Q values explicitly, we may apply them in inverse Q filter to compensate the amplitude and correct the phase simultaneously.

[18] The samples of 1-D attenuation measurement A(χ) are smoothed by applying a median filter and then normalized by (χ) = A(χ)/A_{a}. A data-driven gain curve is then designed as

where Λ_{d} with subscript _{d} indicates that the gain function is derived directly from “data”. In contrast, a theoretical compensation function is expressed as

where β(χ, Q) = exp [−χ/2Q]. Finally, Q estimation becomes a minimization problem:

I here first perform a (five-point) median filtering to mitigate the outliers and then minimize the absolute deviation between the two gain functions, to make the minimization procedure robust in finding the Q value.

[19] The Q analysis procedure based on amplitude compensation is shown in Figure 2, in which (c) is the amplitude attenuation curve, (d) is the associated amplitude compensation curve and the synthetic (gray) curve corresponding to the estimated Q value (Q = 87.2) and, for comparison, (e) is the amplitude spectrum after Q compensation.

4. Interval-Q Calculation by Inversion

[20] Two Q-analysis methods presented in the previous sections can be used to generate an average-Q function, by simply repeating the constant-Q analysis for a series of T samples, {T_{1}, T_{2}, ⋯, T_{n}, ⋯}. This procedure is akin to the conventional velocity analysis in seismic data processing. Once a series of average Q values is obtained, we need to convert it to a series of layered Q values. I now show an interval-Q calculation method using constrained linear inversion approach.

[21] Suppose the earth is divided into N layers with interval-Q values, {Q_{n}^{−1}, n = 1, 2, ⋯, N}. If given the interval-Q values, the average-Q value may be calculated by

where (Q_{a m}^{−1})_{cal} indicates that it is a calculated average-Q at time T_{m} = Δt_{i}. If assuming the layer thickness be constant, Δt, then

where Δt_{n} = cΔt, which is a fraction of the constant thickness Δt, and c is within the range (0, 1]. Equation (11) becomes

Then, the calculation for interval-Q values can be defined as a minimization problem:

where Q_{a m}^{−1} is the observed average-Q values obtained from the preceding Q analysis.

[22] The minimization problem, together with a constraint dQ^{−1}/dt = 0, may be formed as the following linear equation,

where d is the known “data” vector, d = [Q_{a1}^{−1}, ⋯, Q_{a M}^{−1}]^{T}, consisted of M average-Q^{−1} values, q is the unknown “model” vector, q = [Q_{1}^{−1}, ⋯, Q_{N}^{−1}]^{T}, N of interval-Q^{−1} values, and λ is a tuning parameter controlling the trade-off between the minimization and its constraint. In the linear equation (15), Jacobian A is a M × N, lower triangular matrix and constraint B is a (N − 1) × N, dual-diagonal matrix with (1, −1) on the two main diagonals. The trade-off parameter λ is set as 0.01 in the following examples.

5. Application Example

[23] To demonstrate Q analysis (average-Q analysis, followed by interval-Q inversion), I use a real seismic section shown in Figure 3, which is the brute stack of the P-P wave traces from an ocean-bottom cable survey and has been corrected for the spherical divergence effect, before it may be used for estimating the earth Q model.

[24]Figure 4 depicts the details of Q analysis step by step. Figure 4a is the Gabor transform spectrum, averaging over all traces in the seismic section. This Gabor transform spectrum is used first to compute the attenuation measurement and then to derive the compensation function with respect to the variable of frequency-time product. Such data-driven compensation function may be used for the average-Q value estimation, as shown in Figures 4b–4d. The average-Q analyses are conducted at different times with an increment 500 ms, but Figures 4b–4d display the diagnoses only at three different times selectively. Figure 4e shows the result of average-Q estimates and the final interval-Q values, where the interval is set as 250 ms in the linear inversion.

[25] This time-variant Q function is used to design an inverse Q filter, which is then applied to the seismic section in Figure 3. The resultant seismic section, after stabilized inverse Q filtering [Wang, 2002; Guo and Wang, 2004], is plotted in Figure 5, which shows true amplitude variation of the seismic reflection wave, and provides a reliable information for further geological and lithological interpretation.

6. Conclusions

[26] This paper presents a novel method for Q analysis, performed in a similar way to what we do routine velocity analysis on reflection seismic section. For the Q analysis, stability is the key word:

[27] (1) Each of the two Q analysis methods, based on the attenuation function and compensation function, respectively, may be used to produce a series of average values of Q^{−1} (inverse Q), but the compensation-based Q analysis method is more stable than the attenuation-based method.

[28] (2) Once a series of average values of Q^{−1} is produced, stable calculation of layered or interval values of Q^{−1} is obtained by using a constrained linear inversion approach.