Multiple seismic attribute analyses for determination of bottom simulating reflector of gas hydrate seismic data in the Ulleung Basin of Korea

Abstract

The bottom simulating reflector (BSR), the boundary between the gas hydrate and the free gas zone, is considered to be the most common evidence in seismic data analysis for gas hydrate exploration. Multiple seismic attribute analyses of reflectivity and acoustic impedance from the post-stack deconvolution and complex analysis of instantaneous attribute properties including the amplitude envelope, instantaneous frequency, phase, and first derivative of the amplitude of seismic data have been used to effectively confirm the existence of a BSR as the base of gas hydrate stability zone. In this paper, we consider individual seismic attribute analysis and integrate the results of those attributes to locate the position of the BSR. The outputs from conventional seismic data processing of the gas hydrate data set in the Ulleung Basin were used as inputs for multiple analyses. Applying multiple attribute analyses to the individual seismic traces showed that the identical anomalies found in two-way travel time (TWT) between 3.1 and 3.2 s from the results of complex analyses and l ₁ norm deconvolution indicated the location of the BSR.

Appendices

Appendix A: Seismic complex analyses

The analytic seismic trace is given as follows:

$$ F(t) = f(t) + ig(t) $$

(A-1)

where f(t) is the real part corresponding to the recorded seismic data and g(t) the imaginary part of the complex trace, is the Hilbert transform of f(t) that is defined as follows:

$$ g(t) = {\frac{1}{\pi }}\int\limits_{ - \infty }^{ + \infty } {{\frac{f(\tau - t)}{\tau - t}}d\tau } , $$

(A-2)

where τ is delay time.

Furthermore, we define E(t), θ(t), and ω(t) as the amplitude envelope, instantaneous phase, and frequency, respectively. The total instantaneous energy or amplitude envelope is computed as follows:

$$ E(t) = \sqrt {f^{2} (t) + g^{2} (t)} . $$

(A-3)

As given in Eq. A-3, the envelope is dependent on the phase, and relates to the acoustic impedance contrast or reflectivity. This attribute can be used as an effective discriminator for several characteristics such as possible gas accumulation, major changes of lithology and depositional environments.

The first derivative of the amplitude envelope represents the variation of the energy of the reflected events. It also represents the energy loss. The lower ratio indicates greater energy loss effects. The first derivative of the amplitude envelope is calculated as follows:

$$ {\frac{dE(t)}{dt}} = E(t)*dif\left( {f(t)} \right) , $$

(A-4)

where * denotes the convolution operator and dif(f(t)) is the differentiation operator.

The second derivative of amplitude envelope ${\frac{{d^{2} E(t)}}{{dt^{2} }}}$ gives the measure of the sharpness of the envelope peak, and could be used as the principle attribute to indicate sharp changes of lithology, as well as a change in the depositional environment, even when the corresponding envelope amplitude is low.

The instantaneous phase is determined from the complex form of the seismic trace given in Eq. A-1 as follows:

$$ \theta (t) = \arctan \,\left( {{\frac{g(x,t)}{f(x,t)}}} \right). $$

(A-5)

The information in the instantaneous phase is dependent on the amplitude, and relates to the propagation phase of seismic wave front. The phase attribute can be effectively used as discriminator of geometrical shape classification.

Instantaneous frequency is defined as

$$ \omega (t) = {\frac{d\theta }{dt}} = {\frac{d}{dt}}\arctan \,\left( {{\frac{g(x,t)}{f(x,t)}}} \right) = {\frac{{g(x,t){\frac{df(x,t)}{dt}} - f(x,t){\frac{dg(x,t)}{dt}}}}{{f^{2} (x,t) + g^{2} (x,t)}}} . $$

(A-6)

This equation shows that the nature of instantaneous frequency is phase variation with time, which is different from the frequency in a Fourier transform. As seen in Eq. A-6 the instantaneous frequency is negative whenever $g(x,t){\frac{df(x,t)}{dt}} - f(x,t){\frac{dg(x,t)}{dt}}$ is negative. The instantaneous frequency can be related to wave propagation and results from changes in the depositional environment, as well as being a direct hydrocarbon indicator because of a low frequency anomaly.

Appendix B: l ₁ norm deconvolution

The seismic trace x(t) is created by convolving the reflectivity r(t) and source wavelet s(t), and the additive noise term n(t) as given

$$ x(t) = s(t)*r(t) + n(t) , $$

(B-1)

where * is denoted for the convolution operator.

If the subsurface is assumed to be flat, the reflectivity function will be zero everywhere except at those times τ _k that correspond to the two-way travel time to the kth layer. The reflectivity function could be expressed as Delta function as given in Eq. B-2:

$$ r(t) = \sum\limits_{k = 1}^{NL} {r_{k} \delta (t - \tau_{k} ),} $$

(B-2)

where NL is the total number of layers and r _k is the reflectivity coefficient at the interface between the kth and the k + 1th layers. The acoustic impedance in the kth layer is defined as follows:

$$ \xi_{k} = v_{k} \rho_{k} $$

(B-3)

where v _k and ρ _k is the velocity and density of the kth layer, respectively. The reflectivity coefficient at the base of the kth layer depends upon the acoustic impedance contrast and is given by the following:

$$ r_{k} = {\frac{{\xi_{k + 1} - \xi_{k} }}{{\xi_{k + 1} + \xi_{k} }}} $$

(B-4)

Rearrangement of Eq. B-4 yields

$$ \xi_{k + 1} = \xi_{k} {\frac{{1 + r_{k} }}{{1 - r_{k} }}} = \xi_{1} \prod\limits_{i = 1}^{k} {{\frac{{1 + r_{i} }}{{1 - r_{i} }}}} . $$

(B-5)

The relationship between acoustic impedance and reflectivity function is approximated as (Peterson et al. 1955; Waters 1978)

$$ r(t) = {\frac{1}{2}}\,{\frac{{d\left[ {\ln \xi (t)} \right]}}{dt}}. $$

(B-6)

The solution of ordinary derivative Eq. B-6 is given as

$$ \xi (t) = \xi (0)\exp \left[ {2\int\limits_{0}^{t} {r(u)du} } \right], $$

(B-7)

where ξ(0) = ξ ₁ is the surface impedance. We define

$$ \eta (t) = {\frac{\xi (t)}{\xi (0)}}. $$

(B-8)

So that, from Eq. B-7 we have

$$ \eta (t) = {\frac{\xi (t)}{\xi (0)}} = \exp \left[ {2\int\limits_{0}^{t} {r(u)du} } \right]. $$

(B-9)

The linear program approach is used to extract information about r(t) is used only in “appraisal” methods whose goals were to recover unique information about the reflectivity in the form of averages < r(t) > = r(t) * a(t), where a(t) is the averaging function (Oldenburg et al. 1983). The alternative way of extracting information about the reflectivity function is to write the continuous form of Eq. B-1.

$$ x(t_{j} ) = \int {r(t)w(t_{j} - t)dt,\quad \, j = 1,2, \ldots ,N} . $$

(B-10)

Equation (B-10) can be used to construct acceptable models, that is, function r(t), which reproduces the data to within an error justified by its accuracy. The primary obstacle for this approach is non-uniqueness. There are a finite number of acceptable modes, and they will not necessarily be similar. This non-uniqueness must be reduced if construction methods are to be useful, and in order to do this, more information must be supplied. One piece of information comes from the assumption that the Earth structure is adequately represented by a sequence of homogeneous layers. With that hypothesis, the reflectivity function could be expressed in mathematical form, as given in Eq. B-2. The usefulness of Eq. B-2 was discussed in more detail by Oldenburg et al. (1983). There are many ways to achieve the reasonable reflectivity in the form of Eq. B-10, and one of very important fact for this issue is to assume the source wavelet is known or well predicted. Because the deconvolution is considered as inversion technique, the corresponding solution is to iteratively minimize the objective function, that is defined by some norms in which the l ₂ norm is mostly used. The comparison between two algorithms based on l ₁ and l ₂ norm given by Barrodole et al. (1984) states out that the l ₁ norm procedures are shown to exhibit different characteristics which are often desirable, and the results are generally superior to those of the l ₂ norm procedures for one at a time spike extraction.

The extraction of reflectivity function could use the l ₁ norm, consequently minimizing as

$$ \phi_{1} = \int {\left| {{\frac{d\xi (t)}{dt}}} \right|dt} $$

(B-11)

or

$$ \phi_{2} = \int {\left| {{\frac{d\eta (t)}{dt}}} \right|dt} . $$

(B-12)

Either ϕ ₁ or ϕ ₂ can be minimized, but the minimization of ϕ ₂ leads to particularly simple results. Using the definition of η(t) and Eq. B-9, we obtain

$$ 2r(t) = {\frac{d\eta (t)}{dt}}. $$

(B-13)

Thus, minimizing ϕ ₂ is equivalent to minimizing

$$ \phi = \int {r(t)dt} . $$

(B-14)

The time discrete form of reflectivity function, r(t) can be expressed as

$$ r(t) = \sum\limits_{n = 0}^{N - 1} {r_{n} \delta (t - n\Updelta ),} $$

(B-15)

where N is the number of time samples and Δ is the time interval. The substitution of Eq. B-15 into Eq. B-14 yields

$$ \phi = \sum\limits_{n = 0}^{N - 1} {\left| {r_{n} } \right|} , $$

(B-16)

as an objective function to be minimized. The minimization of Eq. B-16 is precisely the problem dealt with by Levy and Fullagar (1981). Their solution is briefly outlined here. Taking the Fourier transform of Eq. B-10 and separating it into real and imaginary parts, we have

$$ \text{Re} \{ R_{j} \} = \sum\limits_{n = 0}^{N - 1} {r_{n} \cos \left( {{\frac{2\pi jn}{N}}} \right)} $$

(B-17)

$$ \text{Im} \{ R_{j} \} = \sum\limits_{n = 0}^{N - 1} {r_{n} \sin \left( {{\frac{2\pi jn}{N}}} \right)} . $$

(B-18)

The most straightforward estimate of the Fourier transform of reflectivity R _j is from the Fourier transform of seismic trace (X) and source wavelet (W), which is:

$$ R_{j} \cong {\frac{{X_{j} }}{{W_{j} }}} $$

(B-19)

In fact the seismic trace is contaminated by noise, and, in practice, the source wavelet can only be approximated. Nevertheless, errors for the real and imaginary portions of R _j can often be estimated, and, hence, $\text{Re} \{ R_{j} \}$and $\text{Im} \{ R_{j} \}$ can be admitted as a constraint in the linear programming algorithm by requiring

$$ \text{Re} \{ R_{j} \} - \alpha \delta_{j} \le \sum\limits_{n = 0}^{N - 1} {r_{n} \cos \left( {{\frac{2\pi jn}{N}}} \right)} \le \text{Re} \{ R_{j} \} + \alpha \delta_{j} $$

(B-20)

$$ \text{Im} \{ R_{j} \} - \alpha \delta_{j} \le \sum\limits_{n = 0}^{N - 1} {r_{n} \sin \left( {{\frac{2\pi jn}{N}}} \right)} \le \text{Im} \{ R_{j} \} + \alpha \delta_{j} . $$

(B-21)

where δ _j is the standard deviation of $\text{Re} \{ R_{j} \}$ or $\text{Im} \{ R_{j} \}$ and α is the factor determining the goodness of fit. The detailed linear programming is discussed by Oldenburg et al. (1983).