Sparsity-Enhanced Constrained Least-Squares Spectral Analysis with Greedy-FISTA

Abstract

The utilization of the inversion-based algorithm for spectral decomposition using constrained least-squares spectral analysis (CLSSA) facilitates a time–frequency spectrum with higher temporal and frequency resolution. The conventional CLSSA algorithm is solved by optimizing an L2-norm regularized least-squares misfit function using Gaussian elimination, which suffers from intensive computational cost. Instead of solving an L2-norm regularized misfit function, we propose to use an L1-norm regularized objective function to enhance the sparsity of the resulting time–frequency spectra. Then, we utilize a faster, smarter, and greedier algorithm named greedy-FISTA to enhance the computational efficiency. Compared to the short-time Fourier transform, continuous wavelet transform, and the conventional CLSSA method, the sparsity-enhanced CLSSA with the greedy-FISTA is capable of achieving time–frequency spectra with higher resolution but with much less computational cost. The applicability of this sparsity-enhanced CLSSA method is demonstrated through synthetic and real data examples.

1. Introduction

It has long been recognized that frequency anomaly may serve as an indicator of hydrocarbon reservoirs since Taner et al. [1] applied the complex trace analysis to seismic signal analysis. Various methods have been proposed to extract frequency-related attributes for interpreting seismic data. Notably, the spectral decomposition method introduced by Partyka et al. [2] has attracted significant attention for reservoir characterization. By applying the discrete Fourier transform (DFT) to the seismic data, this method images and maps stratigraphic details, facilitating the analysis of fine structures or attenuation anomalies typically associated with hydrocarbon reservoirs [3].

In general, spectral decomposition involves converting the seismic data from the t-x domain to the t-x-f domain. This conversion significantly increases data redundancy with an increasing number of frequency samplings, thus demanding large storage and high computational efficiency. Several algorithms are employed for spectral decomposition, each with distinct features. For example, spectral decomposition using the short-time Fourier transform (STFT) offers the lowest computational cost but suffers from a trade-off between the temporal and spectral resolution due to the windowing effects. Another extensively employed spectral decomposition method is the continuous wavelet transform (CWT), which decomposes seismic traces into separate frequency components by using mother wavelets with varying degrees of stretching, being equivalent to narrowband filtering in the time domain [4,5]. Compared to the STFT, the CWT method is less affected by the time window thanks to the implicitly implemented frequency-dependent window. However, it should be noted that the comparable time-frequency spectra can be achieved by either the CWT or the STFT method with the careful selection of the window function [6]. Another group of spectral decomposition methods involves user-defined wavelet dictionaries, such as matching pursuit and basis pursuit [7]. This group of methods usually produces results with high temporal and spectral resolution, but at the expense of intensive computational cost. Because of their high resolutions, many studies focus on improving the efficiency of these user-defined wavelet dictionary-based approaches [8,9,10].

Another class of methods for increasing the resolution is by using the compressive sensing-based spectral decomposition. For example, Flandrin and Borgnat [11] proposed a time-frequency analysis method for obtaining sharp time-frequency decomposition by making use of the compressive sensing techniques, however, with the price of a heavy computational load. A recent review by Sejdic et al. [12] concluded that compressive sensing-based time-frequency representations could be very helpful, but it is still in its early stage and requires further effort.

The least-squares spectral analysis (LSSA), originally designed by Vaníček [13], is a spectral analysis approach based on least-squares that can also be used for spectral decomposition. It estimates a frequency spectrum based on the least-squares fit of sinusoids to the time series by iteratively reducing the misfit. The constrained least-squares spectral analysis (CLSSA) proposed by Puryear et al. [14] is an improved LSSA algorithm by introducing additional constraints to the LSSA. This CLSSA algorithm is then capable of providing spectral analysis with high accuracy even with a limited number of sampling points, which makes it a superior alternative to Fourier spectral analysis. Traditionally, the CLSSA method directly solves an L2-regularized least-squares misfit function for the Fourier series coefficients, which effectively reduces the windowing effects and makes it suitable for detecting thin layers or estimating attenuation effects. However, the CLSSA is known to be computationally intensive, particularly with large data volumes [8,10,15].

Motivated by the need for efficiency, we introduce a simple but cost-effective approach to solving the inverse problem formulated in the CLSSA method by using the greedy fast iterative shrinkage-thresholding algorithm (greedy-FISTA). The greedy-FISTA is a newly developed variant of the FISTA, inheriting its merits but with reduced computational cost and increased efficiency [16]. Additionally, the sparsity of results can be enhanced since the misfit function with L1-regularization can be solved using greedy-FISTA.

The paper is organized as follows: First, we briefly introduce the theory of least-squares spectral analysis and the greedy-FISTA and define all the relevant equations. Next, we evaluate the method using synthetic seismic traces to analyze the feature of the least-squares spectral analysis by using the greedy-FISTA, and compare it to other commonly used spectral decomposition methods. Finally, we test the proposed method on the PSTM dataset, demonstrating that high-resolution, reliable t-f representations can be obtained at a lower cost.

2. Methods

2.1. Least-Squares Spectral Analysis

Following the classical theory of Fourier transform and the complex trace analysis introduced by [1], the windowed complex seismic trace d = d_r + id_i, obtained by Hilbert transform, can be formulated in a matrix form as

$$Fm=d$$

(1)

The elements in vector m are complex frequency-dependent coefficients, equivalent to the Fourier coefficient. The matrix F is the inverse Fourier transform operator composed of sinusoidal functions within a truncated frequency and time range.

$$F\left(t,f\right)=e^{i\theta \left(t,f\right)}=\cos \left(2\pi ft\right)+i\sin \left(2\pi ft\right)$$

(2)

Thus, relation 1 can be regarded as a matrix notation of the inverse discrete Fourier transform. The timing $t\in \left[-T,T\right]$ denotes the deviation of the sampling point from the window center t₀. The time interval between two adjacent sampling points is denoted by Δt, and the parameter f is the frequency in Hz. Thus, in a discrete form, the dimension of the matrix F^N×K is determined by the number of time samples N = 2^x + 1 ($x\in Z^{+}$) and the number of frequencies K that one wants to evaluate. The model vector m^K×1 contains the K coefficient of frequencies ranging from 0 to f_max = (K − 1)Δf, with Δf being the frequency interval. Therefore, the elements in matrix F can be denoted by $F_{n,k}=W^{nk}$, with $n\in \left[-{\displaystyle \frac{N-1}{2}},{\displaystyle \frac{N-1}{2}}\right]$, $k\in \left[0,K-1\right]$, and $W=e^{i2\pi \Delta t\Delta f}$.

For the STFT algorithm, $\Delta f\ge {\displaystyle \frac{1}{N\cdot 2T}}$ is usually required. Then, for a very short window, the frequency resolution is significantly reduced and is severely distorted by the short window. Therefore, to increase the frequency resolution, the frequency interval ∆f should be allowed to be smaller than ${\displaystyle \frac{1}{N\cdot 2T}}$. With such selection, where K > N, the problem of solving for m from Equation (1) becomes under-determined. To address this problem and mitigate the impact of the window function, the constraint least-squares spectral analysis method is introduced by essentially applying the scaled diagonalized Hann taper function $W_d^{N\times N}=\beta \cdot \mathrm{Diag}\left(0.5+0.5\cos \left({\displaystyle \frac{2\pi n\Delta t}{N}}\right)\right)$ to constrain the data and by using the diagonal matrix $W_m^{N\times N}$ to constrain the model space [14]. Here, the scaling factor β is obtained by taking the data envelope value at the center of the window. Thus, Equation (1) can be modified as

$$F_{w}m=W_{d}d \mathrm{with} F_w=W_{d}F{W}_m{\left(W_m\right)}^{-1}$$

(3)

Here, the model-weighting matrix $W_m^{N\times N}$ can be defined $W_m^{N\times N}=I$ or, $W_m^{N\times N}=\mathrm{Diag}\left(\left|m\right|\right)$ for obtaining more compact spectra. The coefficient m is inverted by first formulating Equation (3) into an L2-norm weighted least-squares objective function and then solving it by using Gaussian elimination [14]. This approach exhibits improved time and frequency resolution relative to the conventional STFT method, but may have high computational costs and be nonintuitive [15]. This disadvantage hinders the wide application of the CLSSA method.

2.2. The Greedy Fast Iterative Shrinkage-Threshold Algorithm

The main objective of this paper is to reduce the computational cost by employing an L1-norm weighted objective function, commonly referred to as the Lasso problem.

$$\min \left({‖F_{w}m-W_{d}d‖}_2^2+\lambda {‖m‖}_1\right)=\min \left(F\left(m\right)+\lambda R\left(m\right)\right)$$

(4)

The first term $F\left(m\right)={‖F_{w}m-W_{d}d‖}_2^2$, which represents the misfit, is used to measure the differences between the observed and the modeled data. The second term $R\left(m\right)={‖m‖}_1$ is used to penalize nonsparse solutions. Note that in the conventional CLSSA by [14], an L2 norm $R\left(m\right)={‖m‖}_2$ is used. The main difference between ${‖m‖}_2$ and ${‖m‖}_1$ is that the first one leads to an overall smoothed result, while the latter promotes a sparse outcome. There are also other strategies which can help achieve more focused and blocky results, such as gravity inversion using minimum gradient support [17] and the L0 norm for sparse constraints [18]. A tunable sparsity approach based on minimum gradient support has also been proposed by Vignoli et al. [19], which can help achieve sharper results at different levels. In this presentation, we will focus on the L1 norm. The trade-off parameter λ is used to balance the weight of the two terms. The selection of λ depends on the purpose of the spectral decomposition. For example, if one intends to use CLSSA with L1 norm to detect seismic reflectors, a large value of λ should be chosen to promote aggressive sparsity. An adaptive scheme for λ, as demonstrated by [18], can also be applied. Equation (4) is usually solved by a modified version of the gradient descent method since the conventional ones are not applicable. For example, the iterative shrinkage-thresholding algorithm (ISTA) solves the Lasso problem using a modified gradient descent method that incorporates near-end change [20]. In each iteration, the algorithm uses the minimum value point x_j of the approximate function as the starting point x_j+1 for the next iteration. The fast-iterative shrinkage-thresholding algorithm (FISTA) is an improved version of ISTA that uses a linear combination of the two previous iteration points x_j and x_j+1 to accelerate the convergence [21]. Theoretical proofs demonstrate that the worst-case convergence rate of ISTA is O(1/j), while FISTA boasts a faster rate of O(1/j²) [21].

The original FISTA by [21] usually suffers from local oscillation that usually damps its efficiency. To solve this problem, ref. [16] proposes a greedy version of the original FISTA (greedy-FISTA) by simply keeping the inertial parameter a_j = 1, and choosing as large as possible step size γ at each iteration, and implementing two more steps before the loop into another iteration. These two steps are named “restarting” and “safeguard”, which help to achieve monotonic performance and speedup without leading to numerical divergence due to large step size γ.

The workflow of the greedy-FISTA scheme can be summarized as follows:

• Set ${\displaystyle \frac{1}{L}}\le \gamma <{\displaystyle \frac{2}{L}}$ and $\xi <1,S>1$, and m₋₁ = m₀, j = 1; the parameter L is the Lipschitz constant. The parameter j counts the number of iterations.
• Update the model m_j+1 by

  $$m_{j+1}={\mathrm{prox}}_{\gamma R}\left(y_j-\gamma \nabla F\left(y_j\right)\right)$$

  (5a)

  with

  $$y_j=m_j+a_j\left(m_j-m_{j-1}\right) \mathrm{with} a_j\equiv 1$$

  (5b)
  The proximity operator of R(m) is defined as

  $${\mathrm{prox}}_{\gamma R}\left(•\right)=\arg \min \gamma R\left(m\right)+{\displaystyle \frac{1}{2}}{‖m-•‖}^2$$

  (5c)
• If ${\left(y_j-m_{j+1}\right)}^T\left(m_{j+1}-m_j\right)\ge 0$, then $y_j=m_j$. This is the step called “restarting” that helps to speed up.
• $‖m_{j+1}-m_j‖\ge S‖m_1-m_0‖$, then $\gamma =\max \left(\xi \gamma ,L^{-1}\right)$. This is the “safeguard” that helps to avoid numerical divergence.
• The algorithm stops when the maximum iterations or the tolerance is reached.

Compared to the original FISTA, steps 3 and 4 are required by the greedy-FISTA and a_j is a constant at all the iterations. These updates can help to alleviate the computational cost remarkably, as shown by a random numerical test on the converge rate of the original and greedy-FISTA in Figure 1. The development of the greedy-FISTA and its comparison to the original FISTA can be found in [16]. Here, below, we mainly focus on the application of greedy-FISTA to the seismic time-frequency decomposition.

3. Examples

3.1. Synthetic Tests

By using the convolution theory, we constructed a noise-free synthetic record as shown in Figure 2a, which is composed of four identical reflections, each with a dominant frequency of 30 Hz. By applying the conventional Gabor transform, continuous wavelet transform (CWT), the conventional CLSSA method proposed by [14], and the greedy-FISTA-assisted CLSSA, we obtained the time-frequency spectra depicted in Figure 2b–e, respectively. The results in Figure 2d, e indicate that the results obtained by two CLSSAs have higher resolution compared to those by using the Gabor transform and the CWT (Figure 2b,c). This test also shows that for noise-free seismic data, the inverted time-frequency spectra can be equally acceptable. However, for this test, the greedy-FISTA-assisted CLSSA is more efficient as it only costs 0.021 s but the conventional CLSSA requires 0.175 s.

In the test shown in Figure 2, a trade-off parameter λ = 0.5 is employed, and the total number of iterations N are, respectively, 1 and 20 for the conventional CLSSA and greedy-FISTA-assisted CLSSA.

For a test on the noise-contaminated signal, a 10 dB white Gaussian noise is added, but the same setting is applied as in Figure 2 for a fair comparison. The results are shown in Figure 3. The outcome indicates that the greedy-FISTA-assisted CLSSA exhibits greater resilience in handling noisy signals, although not as good as the Gabor transform and the CWT. This superiority can be attributed to the fact that greedy-FISTA prioritizes the sparsity of signal and is consequently less susceptible to noise interference. However, it should be noted that an acceptable result can be obtained by applying the conventional CLSSA with an adjusted number of iterations or modified λ for the L2 norm, but with increased computational cost.

We further compare the results between the conventional and the greedy-FISTA-assisted CLSSA by varying the trade-off parameter λ and the number of iterations Ni. The results presented in Figure 4 and Figure 5 reveal that the conventional CLSSA is more sensitive to the number of iterations, whereas the greedy-FISTA-assisted CLSSA is more sensitive to the trade-off parameter λ. This observation suggests that the conventional CLSSA incurs higher computational costs, as evidenced by the recorded running times in

Table 1. Comparisons of the computational cost of the tests in Figure 4 and Figure 5 with different CLSSA algorithms.

Ni	λ	Conventional (ms)	Greedy-FISTA (ms)
2	5	97.3	3.6
2	15	96.8	3.3
4	5	166.7	10.1
4	15	149.2	9.3

. Specifically, with Ni = 2, the greedy-FISTA-assisted CLSSA demonstrates approximately 30 times greater efficiency compared to the conventional CLSSA algorithm. Even with Ni = 4, the greedy-FISTA-assisted CLSSA remains less computationally expensive and more viable. This comparison indicates that the CLSSA can be more practical in daily work with the greedy-FISTA.

3.2. Field Data Test

We employ the greedy-FISTA-assisted CLSSA to analyze a real seismic profile (Figure 6) obtained from the field, aiming to aid in stratigraphic characterization. The strongest reflection in Figure 6 is caused by the existence of a coal layer as shown by the impedance well log. Adjacent to the coal layer, there are many thin layers with mild impedance contrast which can be easily identified from the well log but difficult to be distinguished from the stacked seismic image. This is mainly due to the fact that reflections from those thin layers are masked by the strong reflection caused by the coal layer. Consequently, one of the primary tasks is to try to unveil the thin layers (indicated by the arrows in Figure 6) that may be obscured by this strong reflection. While the conventional strategies often involve removing this coal layer-related strong reflection, we opt for an alternative approach by utilizing the time-frequency analysis. The essence of this approach lies in seeking additional information from various frequency slices to facilitate the characterization of the thin layers.

Figure 7 presents the spectral decomposition results obtained using Gabor (Figure 7a,b) and CWT (Figure 7c,d). It is evident that neither method reveals the thin-layer structures.

Figure 8a, Figure 8b, Figure 8c and Figure 8d are the spectral decomposition results at varying frequencies obtained by, respectively, the conventional (Figure 8a,b) and greedy-FISTA-assisted CLSSA method (Figure 8c,d). It is clear that with appropriately selected parameters, the results obtained by both algorithms are equally well. However, the running time of the conventional one is about 10 times more than the new one proposed in the current paper. Such notably improved computational efficiency makes the high-resolution spectral decomposition practical for daily processes. Also, the greedy-FISTA-assisted CLSSA provides sharper and clearer time-frequency spectra.

The reflections from the thin layers adjacent to the coal layer are difficult to be directly distinguished from the input seismic profile in Figure 6 or be detected from the time–frequency spectra obtained by the Gabor and CWT methods (Figure 7). However, both the conventional and greedy-FISTA-assisted CLSSA are capable of identifying the thin layers (Figure 8). Particularly, the results in Figure 8c,d demonstrate that the greedy-FISTA-assisted CLSSA can help uncover more fine structures than the conventional CLSSA (see the events indicated by the arrows). Comparing the resulting spectral decomposition at 10 Hz to that at 60 Hz suggests that the reflection related to the coal layer has a wider bandwidth than the thin layers indicated by the arrows. Thus, by scanning the frequency slices of the seismic profile and combining them with the well log data, we should be able to characterize the thin layers without going through a complicated reflection removal workflow.

4. Conclusions

Our study highlights the effectiveness of the greedy-FISTA-assisted constrained least-squares spectral analysis (CLSSA) method in extracting the time–frequency features and interpreting seismic data for understanding subsurface structures. We have shown that greedy-FISTA outperforms the traditional CLSSA approaches, especially in handling noise and speeding up computations. By using greedy-FISTA, we successfully identified thin layers near prominent reflections like coal layers with high efficiency. This emphasizes the importance of looking at frequency details for better geological insights. The numerical and field tests suggest that greedy-FISTA-assisted CLSSA is a valuable tool for advancing seismic interpretation and exploration efforts.