New 2D-Variational Mode Decomposition-Based Techniques for Seismic Attribute Enhancement

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New 2D-VMD-based seismic attributes have been suggested. They provide significant advantages over traditional approaches and that combining complementary methods can further improve seismic interpretation outcomes.

Abstract

Seismic attributes are widely used to enhance the interpretation of structural, stratigraphic, and lithologic features in subsurface data. Their effectiveness, however, can be limited by noise, resolution constraints, and processing artifacts. This study suggests new seismic attributes computed using 2D-Variational Mode Decomposition (2D-VMD), which are specifically Mode-Weighted Spectral Discontinuity (MWSD) (in Module and Phase modes), VMD-Directionality Coherence (VDC), Instantaneous Frequency Concentration (IFC-VMD), and Instantaneous Bandwidth Dispersion (IBD-VMD). The proposed 2D-VMD-based attributes are compared with seven key conventional seismic attributes: dip, azimuth, chaos, coherence (semblance), curvature (mean curvature), instantaneous frequency, and instantaneous bandwidth (Hilbert transform). Through applications on simulated and real seismic data, each method is compared in terms of its ability to enhance attribute stability, resolution, and interpretability while mitigating limitations such as noise sensitivity and loss of detail. Results indicate that MWSD (Module) is optimal for amplitude stability, MWSD (Phase) for phase-sensitive applications, VDC for high-resolution structural delineation, IFC-VMD for complex geological settings, and IBD-VMD for abrupt feature changes. The findings demonstrate that these new 2D-VMD-based techniques provide significant advantages over traditional approaches and that combining complementary methods can further improve seismic interpretation outcomes.

1. Introduction

Seismic attributes have become indispensable tools in modern geophysical interpretation, enabling the extraction of structural, stratigraphic, and lithologic information that may be difficult to discern from raw seismic data alone. Attributes such as dip, azimuth, coherence, curvature, instantaneous frequency, and bandwidth provide interpreters with quantitative measures that can reveal subtle geological features, improve fault and fracture mapping, and aid in reservoir characterization [1,2,3,4,5,6]. However, the reliability and resolution of these attributes are often challenged by factors such as seismic noise, attenuation, limited bandwidth, and interference effects, which can obscure or distort subsurface features.

Over the past two decades, various signal decomposition and denoising approaches have been developed to address these challenges. More recently, Variational Mode Decomposition (VMD) has emerged as a robust alternative, offering adaptive and noise-resilient separation of seismic signals into intrinsic mode functions. It has been applied in geosciences to process seismic, induced polarization, ground penetrating radar, and volcanic signals. Yu and Ma (2018) reported that complex VMD delivers higher denoising quality than f-x deconvolution and Empirical Mode Decomposition (EMD) in seismic noise attenuation [7]. Naveed et al. showed that VMD enhances the reconstruction of induced polarization decay and separates low-frequency signals from noise [8,9]. Xue et al. (2016) and Liu et al. (2015, 2016) described VMD’s robustness to noise and its ability to provide clearer time-frequency representations [10,11,12], while Zhang et al. (2018) noted that standard VMD yields sparser decompositions for improved noise separation in ground penetrating radar data [13]. For seismic noise attenuation, Zhou and Chi (2020) improved VMD by combining Kurtosis method, a waveform matching extension algorithm, and the Singular Spectrum Analysis (SSA)) to improve de-noising precision [14]. In addition, Wang et al. (2019) reported that coupling VMD with the Wigner–Ville distribution suppresses cross-term interference and improves spectral resolution [15]. Proaño et al. (2018) documented a 99.26% accuracy in event onset and ending detection for volcano monitoring [16]. These studies agree that VMD’s adaptive, optimization-based framework often outperforms EMD, Complete Ensemble EMD (CEEMD), Short-Time Fourier Transform (STFT), and Wavelet Transform (WT) for noise attenuation and time-frequency analysis in geoscience applications.

Moreover, a two-dimensional version of VMD (2D-VMD) has been introduced. Schmitt et al. (2015) suggested an improved variational mode decomposition method for internal waves separation, and the reduction in artifacts and improved separation using the suggested method compared with the discussed approaches [17]. Horne et al. (2021) reported effective wave mode separation and SNR optimization using 2D-VMD with respect to preexisting internal wave decomposition methods [18]. Liu et al. (2024) demonstrated that 2D-VMD is superior in denoising of seismic data compared with the other methods (EMD, VMD) [19].

Based on this concept, new two-dimensional VMD (2D-VMD)-based techniques have been introduced, specifically tailored for seismic attribute enhancement. The proposed attributes are Mode-Weighted Spectral Discontinuity (MWSD) (Module and phase), VMD-Directionality Coherence (VDC), Instantaneous Frequency Concentration (IFC-VMD), Instantaneous Bandwidth Dispersion (IBD-VMD).

This paper presents a comparative evaluation of these advanced attributes with seven commonly used seismic attributes, which are specifically: azimuth, dip, chaos, coherence, curvature, instantaneous frequency, instantaneous bandwidth (Hilbert), with these advanced methods. The study focuses on identifying each method’s strengths, limitations, and suitability for specific interpretation objectives, with the ultimate goal of providing practical guidelines for their application in seismic interpretation workflows. Applications on simulated and real seismic data confirm that these approaches offer a powerful toolkit for optimizing attribute computation in diverse geological settings.

2. Theory of Variational Mode Decomposition (1D and 2D)

2.1. Introduction to VMD

Variational Mode Decomposition (VMD) is a modern adaptive and non-recursive signal decomposition method introduced by Dragomiretskiy and Zosso (2014) to overcome limitations of earlier techniques such as Empirical Mode Decomposition (EMD) [20]. Unlike the heuristic EMD, VMD is found in a solid mathematical framework. It frames the decomposition process as a variational optimization problem, guaranteeing well-posedness, improved stability, and superior performance in noisy environments.

The objective of VMD is to decompose a signal into a fixed number of band-limited intrinsic mode functions (IMFs), each characterized by a narrow spectral bandwidth and a center frequency identified through optimization. This enables an efficient separation of different oscillatory modes within a signal.

2.2. One-Dimensional VMD Formulation

In the one-dimensional case, the input signal $f(t)$ is assumed to be the superposition of $K$ modes:

$$f\left(t\right)={\sum }_{k=1}^K u_k(t)$$

(1)

Each mode $u_k(t)$ is assumed to have a compact frequency support around a center frequency ${\omega }_k$. The decomposition is completed by solving the following constrained optimization problem:

$$\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{min} \left\{{\sum }_{k=1}^K {‖{\partial }_t\left[\left(\delta (t)+{\displaystyle \frac{j}{\pi t}}\right) * u_k(t)\right]\cdot e^{-j{\omega }_{k}t}‖}_2^2\right\}$$

(2)

subject to:

$${\sum }_{k=1}^K u_k(t)=f(t)$$

(3)

where $\delta$ is the Dirac distribution, $j$ is the imaginary unit ($j^2=-1)$, $t$ is time script, ${‖.‖}_p^2$ is the squared $L^p$—norm of gradient (where p = 2), and * denotes convolution.

Here, the analytic signal is derived via the Hilbert transform and subsequently shifted to baseband for bandwidth estimation. The bandwidth for each mode is quantified by the squared L²-norm of its temporal gradient. The constrained optimization problem is resolved using the Augmented Lagrangian framework in conjunction with the Alternating Direction Method of Multipliers (ADMM), which iteratively updates $u_k,{\omega }_k$, and the Lagrange multiplier until convergence is achieved [20].

2.3. Extension to 2D VMD

For two-dimensional data, such as seismic time slices, horizon slices, or images, the principle is extended to two spatial dimensions ($x,y$). The signal $f(x,y)$ is decomposed into K 2D modes $u_k(x,y)$ with compact support in the 2D spatial frequency domain.

The optimization problem becomes:

$$\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{min} \left\{{\sum }_{k=1}^K {‖\mathit{\nabla }\left[u_k(x,y)\cdot e^{-j{\omega }_k\cdot r}\right]‖}_2^2\right\}$$

(4)

Subject to:

$${\sum }_{k=1}^K u_k(x,y)=f(x,y)$$

(5)

Here, $\mathit{\nabla }$ is the 2D gradient operator, measuring spatial bandwidth; ${\omega }_k$ denotes the 2D center frequency vector of mode k; and $r = (x,y)$ represents the spatial position vector. The 2D VMD decomposes the input into spatially localized components, each characterized by a specific orientation and spatial scale, making it particularly suited for seismic attribute enhancement [17,18,19].

3. Theoretical Basis of the 2D-VMD-Based Seismic Attribute

3.1. Mode-Weighted Spectral Discontinuity (MWSD)

The procedure to compute MWSD attribute is detailed as follows:

• Apply 2D-VMD: Decompose a seismic slice (inline, crossline, or time slice) into $K$ intrinsic mode functions (IMFs), each capturing distinct frequency bands.

$$\mathrm{S}(x,z)\to {\left\{u_k(x,z)\right\}}_{k=1}^K$$

(6)

where $S(x,z)$ denotes the input seismic section, and $u_k(x,z)$ represent the 2D VMD modes. These notations will be used hereafter.

• Compute Local Gradient of Each Mode: For each IMF, compute the magnitude of the spatial gradient:

  $$G_k(x,z)=\sqrt{{\left({\displaystyle \frac{\partial u_k}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial u_k}{\partial z}}\right)}^2}$$

  (7)

This captures local edge strength or structural discontinuity per mode.

• Compute Frequency-Weighted Combination: Assign higher weights to higher-frequency modes (since they are more sensitive to small-scale discontinuities):

$$MWSD(x,z)={\sum }_{k=1}^K w_k\cdot G_k(x,z)$$

(8)

where $w_k={\displaystyle \frac{w_k}{\sum {\omega }_k}}$, and ${\omega }_k$ is the central frequency of the $k$-th mode (estimated during VMD).

The obtained attribute map MWSD ($x,z$) is used for highlighting edges (faults, stratigraphic breaks), especially those embedded in noisy or complex signal zones.

To enhance structural discontinuities (e.g., faults, fractures) and stratigraphic terminations by analyzing the spectral content and spatial gradients of intrinsic mode functions (IMFs) obtained via 2D-VMD.

3.2. VMD-Directionality Coherence (VDC)

The VDC attribute is calculated as follows:

• Compute Local Gradient Orientation: For each mode, compute:

$$\mathrm{Gradient} \mathrm{in} \mathrm{x} \mathrm{and} \mathrm{z}:\mathit{\nabla }{u}_k=\left({\partial }_x{u}_k,{\partial }_z{u}_k\right)$$

(9)

$$\mathrm{Orientation} \mathrm{angle}: {\mathrm{\theta }}_{\mathrm{k}}(\mathrm{x},\mathrm{z})=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n} 2\left({\partial }_{\mathrm{z}}{\mathrm{u}}_{\mathrm{k}},{\partial }_{\mathrm{x}}{\mathrm{u}}_{\mathrm{k}}\right)$$

(10)

• Calculate Angular Consistency Across Modes: Measure the circular variance of orientations across the $K$ modes:

$$VDC(x,z)=1-{\displaystyle \frac{1}{K}}\left|{\sum }_{k=1}^K e^{j{\theta }_k(x,z)}\right|$$

(11)

The VDC attribute captures the directional alignment of local features across multiple frequency scales, extracted by 2D-VMD, enhancing detection of bedding orientation, fracture alignment, and chaotic versus structured zones. Low VDC values reflects orientations are aligned (coherent reflectors), whereas high values correspond to scattered orientations (chaotic or fractured zones).

3.3. Instantaneous Frequency Concentration (IFC-VMD)

The IFC-VMD attribute is calculated as follows:

• Compute Local Energy Contribution

For each mode, compute the normalized energy at each location:

$$E_k(x,z)={\displaystyle \frac{u_k(x,z{)}^2}{{\sum }_{j=1}^K u_j(x,z{)}^2+ϵ}}$$

(12)

where $ϵ$ is a small constant to avoid division by zero.

• Compute Weighted Instantaneous Frequency Map

The instantaneous frequency concentration is calculated as:

$$IFC(x,z)={\sum }_{k=1}^K {\omega }_k\cdot E_k(x,z)$$

(13)

This attribute provides a spatial map of where the signal energy is concentrated in frequency. Higher values correspond to higher frequency energy (thin beds, high-impedance contrasts, gas presence, fractures), while lower values are associated with low-frequency events (Broad reflectors, shale, compact lithology).

In contrast with the conventional Hilbert IF attribute, based on the analytic signal, the IFC-VMD attribute is based on the 2D-VMD method. It uses mode energy weighting combined with mode-specific center frequencies ${\mathrm{\omega }}_{\mathrm{k}}$ to construct a physically meaningful spectral attribute. It is analogous to instantaneous frequency, but stable, multi-scale, and interpretable.

3.4. Instantaneous Bandwidth Dispersion (IBD-VMD)

The IBD-VMD is computed using the following approach:

• Compute Normalized Local Energy of Each Mode

$$E_k(x,z)={\displaystyle \frac{u_k(x,z{)}^2}{{\sum }_j u_j(x,z{)}^2+ϵ}}$$

(14)

• Compute Local Bandwidth Dispersion

Use the weighted variance of the center frequencies at each pixel:

$$IBD(x,z)=\sqrt{{\sum }_k E_k(x,z)\cdot {\left({\omega }_k-\bar{\omega }(x,z)\right)}^2}$$

(15)

where

$$\bar{\omega }(x,z)={\sum }_k E_k(x,z)\cdot {\omega }_k$$

(16)

This is similar to the standard deviation of frequency locally—a measure of spectral spread.

Unlike classical instantaneous bandwidth (derived from the analytic signal and second derivatives of phase), this new attribute captures local variation in spectral content across VMD modes. It reflects how spread out or focused the energy is in the frequency domain at each location, and highlights zones where spectral content is locally dispersed across VMD modes (indicative of thin beds, diffractions, interference).

Low IBD values are associated with spectrally focused, single events (simple reflections), whereas High values corresponds to spectral interference (thin layers, pinch-outs, stratigraphic variation).

For recall, Instantaneous Bandwidth Dispersion (IBD) describes the variability of the instantaneous bandwidth, which represents the local spread of spectral energy around the instantaneous frequency [21,22,23].

High IBD values occur when the spectrum broadens, which typically reflects complex seismic responses such as interference between closely spaced reflectors (thin-bed tuning) [24], abrupt vertical impedance changes, or scattering/diffractions caused by small-scale heterogeneities [25].

In contrast, low IBD values reflect a narrow local frequency band and are commonly associated with laterally continuous, single-event reflectors. Because different physical processes can produce similar spectral broadening, IBD should not be interpreted in isolation.

Rather, it is most effective when integrated with other seismic attributes such as instantaneous frequency, amplitude envelope, and coherence, and calibrated with geological and well data [26].

4. Theoretical Background of the Traditional Seismic Attributes Used

To highlight the benefit of the proposed 2D-VMD-based attributes, several traditional seismic attributes were employed in this study for structural and stratigraphic analysis.

Table 1. Summary of the used traditional seismic attributes with their descriptions, mathematical formulations, and key references.

Attribute (References)	Description	Formula
Azimuth [27,28]	Measures the orientation (angle) of seismic reflectors in the horizontal plane.	$\mathrm{Azimuth} =\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{\partial t/\partial x}{\partial t/\partial y}}\right)$, where $t(x,y)$ is the two-way travel time surface.
Dip Magnitude [27]	Measures the steepness of seismic reflectors in the vertical direction.	$\mathrm{D}\mathrm{i}\mathrm{p}=\sqrt{{\left({\displaystyle \frac{\partial t}{\partial x}}\right)}^2+{\left({\displaystyle \frac{\partial t}{\partial y}}\right)}^2}$
Chaos Attribute [29,30]	Quantifies local discontinuity or randomness in seismic amplitudes; highlights structural complexity	Formula (variance-based): $\mathrm{Chaos} =1-{\displaystyle \frac{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\sum _i {\lambda }_i}}$ where ${\lambda }_i$ are eigenvalues of the covariance matrix of local seismic gradients.
Coherence (Semblance) [22,23]	Measures trace-to-trace similarity; detects discontinuities like faults and channels.	$C={\displaystyle \frac{{\left({\sum }_{i=1}^N s_i(t)\right)}^2}{N\cdot {\sum }_{i=1}^N s_i^2(t)}}$ where $s_i(t)$ are seismic samples across $N$ traces.
Curvature (Mean Curvature) [28,31]	Measures bending of reflectors; sensitive to folds and fractures.	$k_m={\displaystyle \frac{k_1+k_2}{2}}$ where $k_1,k_2$ are the maximum and minimum curvatures from the reflector surface geometry.
Instantaneous Frequency [21]	Derivative of the instantaneous phase; indicates local frequency content of the signal.	$f(t)={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{d\varphi (t)}{dt}}$ where $\varphi \left(t\right)=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{H[s(t)]}{s(t)}}\right)$ and $H[\cdot ]$ is the Hilbert transform.
Instantaneous Bandwidth (Hilbert) [21,22,23,24,25,26]	Measures spectral spread around the instantaneous frequency; relates to absorption and heterogeneity.	$B(t)={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{d}{dt}}\mathrm{l}\mathrm{n}A(t)$ where $A(t)=\sqrt{s^2(t)+H[s(t){]}^2}$ is the instantaneous amplitude.

summarizes the main attributes considered, outlining their theoretical basis, mathematical formulation, and key references. These attributes provide critical insights into reflector geometry, continuity, and spectral characteristics, and serve as benchmarks for comparison with the advanced 2D-VMD methods.

5. Application to Simulated Data

In order to demonstrate the potential of the suggested VMD- based attributes, a synthetic seismic section was generated. It represents a synthetic or simplified original seismic section. The section contains several distinct patterns. There is a large, irregularly shaped body in the upper left, characterized by concentric rings or arcs of reflections. A series of parallel, dipping reflections occupy the central part of the section. Below this, there is a relatively uniform area, with a small, rectangular feature at the bottom. This figure serves as the input for the attribute analysis shown in the following.

Then, the newly proposed attributes—Mode-Weighted Spectral Discontinuity (MWSD) (Module and phase), VMD-Directionality Coherence (VDC), Instantaneous Frequency Concentration (IFC-VMD), Instantaneous Bandwidth Dispersion (IBD-VMD))—together with conventional seismic attributes such as azimuth, dip, chaos, coherence (semblance), curvature (mean curvature), instantaneous frequency, instantaneous bandwidth (Hilbert) were applied to the simulated seismic line, and their results were systematically compared.

Figure 1 presents the simulated seismic line and its corresponding modes obtained using the 2D-VMD method and Figure 2 presents the obtained seismic attribute maps.

5.1. Azimuth Attribute (In Radians)

This Azimuth attribute map illustrates the azimuthal orientation of the seismic reflectors. The attribute effectively maps the directional changes across the model, showing the angular variation in reflectors. Linear patterns aligned with the true reflector orientations are well captured, especially in regions with consistent dip directions. However, azimuth estimation becomes less reliable in structurally complex or low-coherence zones, where chaotic patterns may appear due to reflector termination interferences.

5.2. Dip Magnitude Attribute

The dip magnitude attribute represents the computed dip magnitude of the reflectors. It highlights zones with high-angle dips, such as fault planes and steeply inclined bedding. The attribute performs well in outlining steep features and transitions between horizontal and dipping strata. However, its resolution may degrade near discontinuities or terminations, where dip estimation becomes challenging due to reflector disruption or poor signal-to-noise ratio.

5.3. Chaos Attribute

The chaos attribute quantifies the degree of structural disorder in the seismic data. In this image, it successfully highlights regions of high structural complexity, such as faults and discontinuities, by assigning higher chaos values. These areas appear as bright or irregular patches, indicating a lack of local coherence. While chaos is effective in identifying broken or disturbed reflectors, it does not provide directional information and may overestimate disorder in areas affected by noise or small-scale heterogeneities.

5.4. Coherence Attribute (Semblance)

The coherence attribute measures the similarity of seismic traces. High indicates a high degree of similarity and continuity, often associated with a continuous geological layer or a strong reflector. Low coherence indicates a lack of similarity, which can be caused by faults, fractures, channels, or other discontinuities. The image shows a large, roughly pear-shaped region of high coherence (yellow/orange) with a few small, separate areas of high coherence. This suggests a continuous and well-defined geological feature. There is a distinct, sharp boundary around this feature where coherence drops to very low values (black). This black boundary likely represents a significant geological discontinuity. The interior of the feature is also not perfectly uniform, showing some subtle variations in coherence (darker orange/red) that could indicate internal variations in the rock properties or subtle fracturing. A small, rectangular area of high coherence is also visible at the bottom of the main feature.

5.5. Curvature Attribute (Mean Curvature)

The curvature attribute measures the geometric bending of a seismic horizon. Mean curvature is a specific type that averages the two principal curvatures at each point. High curvature values often highlight small-scale features like faults, fractures, folds, and channels.

Areas with low curvature (values near zero, represented by darker red) are flat or smoothly curved. The curvature map displays a texture-like pattern of high and low curvature values. Unlike the coherence map, which shows large, distinct bodies, this map is more finely detailed.

There are numerous small, linear, and curvilinear features with high curvature (bright yellow). The overall region seems to be a composite of these fine-grained patterns. The attribute values are normalized, ranging from 0 to 1, with a high concentration of intermediate to high values.

5.6. Instantaneous Frequency Attribute (Hz)

Instantaneous frequency is a measure of the frequency of the seismic signal at each point in time. It is derived from the complex seismic trace using the Hilbert transform. Variations in instantaneous frequency can be caused by changes in rock properties (e.g., lithology, porosity, fluid content) or bed thickness. High frequencies (bright yellow/white) can be associated with thin beds or hard rock, while low frequencies (dark orange/red, or negative values as shown on the scale) can be associated with thick beds or softer rock.

This map shows a pattern of frequency variations, with distinct “patches” of different frequency values. The color scale ranges from negative values (−0.4) to positive values (0.4), indicating a wide range of frequency responses. The prominent feature is a large, irregularly shaped body with a mix of high (yellow) and low (orange) frequencies, but with distinct, blocky sections. There are also linear and curvilinear features where the frequency changes abruptly, suggesting geological boundaries.

5.7. Instantaneous Bandwidth (Hilbert) Attribute

Instantaneous bandwidth is another attribute derived from the Hilbert transform. It measures the rate of change in instantaneous frequency. High bandwidth (bright yellow/white) indicates a rapidly changing frequency, which can be caused by thin-bed tuning, unconformities, or pinch-outs. Low bandwidth (dark red/black) indicates a more stable, constant frequency, typical of thick, uniform layers.

This figure shows a map dominated by very low bandwidth values (dark black/red). There are small, scattered “hot spots” or small, localized zones of high bandwidth (yellow/white). These high-bandwidth zones are often linear or point-like.

5.8. MWSD (Module) Attribute

The figure shows two distinct, elongated, elliptical bodies. The body on the top left is a solid, uniform red, suggesting a very strong and continuous seismic response. The body on the bottom right has a striped or hatched pattern of varying red and orange colors, which also indicates strong reflections but with a periodic internal variation. The rest of the map is dark blue, representing an area with very low seismic energy.

This attribute clearly delineates two separate geological features. The uniform red body could represent a thick, homogeneous geological layer or a feature with a consistent acoustic impedance contrast. The striped body suggests a feature with internal variations, possibly representing a series of thin beds, a channel fill with internal stratification, or a set of parallel faults or fractures. This attribute is a fundamental tool for outlining the geometry of geological bodies.

5.9. MWSD (Phase) Attribute

This figure shows a striking contrast. The upper-left region and the area around the two elliptical bodies are primarily dark blue, indicating a consistent or uniform phase response. The interior of the two elliptical bodies, however, shows a chaotic mix of colors, primarily red and green/yellow. This indicates a high degree of phase variation within these features. The rest of the map is a mosaic of different phase values, with a large, irregularly shaped red-and-green zone in the upper right.

The chaotic phase response within the elliptical bodies suggests internal complexities. This could be due to variations in bed thickness, changes in lithology, or the presence of fluids that are causing phase shifts in the seismic signal. The uniform phase response outside of these bodies indicates a more consistent geological background. The large, irregular red-and-green zone could represent another geological feature with a distinct phase signature, possibly indicating a different stratigraphic unit or a region of structural deformation. This attribute is useful for identifying subtle stratigraphic variations and potential fluid contacts.

5.10. Instantaneous Frequency Concentration (IFC-VMD) Attribute

Instantaneous Frequency Concentration (IFC-VMD) measures how concentrated the frequency of the seismic signal is at each point. It is derived from a VMD analysis. High concentration (bright red) indicates a very pure, single-frequency signal, which can be associated with thick, uniform layers or specific resonant frequencies. Low concentration (dark blue) indicates a signal with a broad frequency spectrum, which can be caused by thin-bed tuning, discontinuities, or complex lithology.

This figure clearly highlights the two elliptical bodies from the other panels. The top-left body shows a high frequency concentration (red), especially towards its center. The bottom-right body shows a lower frequency concentration (blue/green) with a distinct striated or hatched pattern, similar to the module map. The areas outside these bodies are a mixture of high and low concentrations. Circular features with high concentration are also visible at the top-left and bottom of the image.

This attribute is a valuable tool for distinguishing between different lithological units and for identifying zones of uniform versus complex deposition or deformation.

5.11. VMD-Directionality Coherence (VDC) Attribute

This attribute combines the concepts of and directional coherence. It measures the coherence of seismic traces along a specific direction. High coherence (bright red) indicates continuous, well-aligned reflections, while low coherence (dark blue) indicates discontinuities or random reflections. The “directionality” component is key, as it enhances features oriented in a particular direction.

This figure shows a busy, complex pattern. The elliptical body on the bottom right is characterized by very high coherence (bright red) with a clear, striated pattern, suggesting a strong, continuous geological feature with a preferred orientation. The top-left elliptical body has a lower coherence (blue/green), with a different internal pattern. The area outside the main bodies is a mixture of low and high coherence zones, with some linear features and patches of varying coherence. A large, complex, and irregular feature is also visible in the upper part of the image, with low coherence (dark blue) and internal striations.

This attribute is highly effective at delineating features with a specific directional trend. The bright red, striated pattern in the bottom-right body confirms that it is a highly coherent feature with a distinct internal structure, likely sedimentary layers or fractures. The lower coherence in the top-left body suggests it is less coherent or that its internal features do not align with the directionality of the analysis. This attribute is particularly useful for mapping fractures, faults, and sedimentary channels.

5.12. Instantaneous Bandwidth Dispersion (IBD-VMD) Attribute

Instantaneous Bandwidth Dispersion (IBD) is a measure of the variability or spread of instantaneous frequencies within a seismic signal, as derived from a VMD analysis. High dispersion (bright red/yellow) indicates a wide range of frequencies and a rapidly changing frequency content. This can be caused by geological discontinuities, thin-bed tuning, or complex layering. Low dispersion (dark blue/black) indicates a more stable, narrow frequency range, characteristic of thick, uniform layers.

This figure shows a striking and detailed pattern. There are two main elliptical bodies. The top-left body is outlined by a bright red/yellow halo, while its interior is a darker blue. The bottom-right body is also outlined in red/yellow, and its internal striped pattern is also highlighted in red/yellow, with darker blue lines in between. The rest of the section is dark blue, with the exception of a few other features. There is a smaller elliptical feature and a rectangular feature, both with high-dispersion outlines. A large, complex, and irregularly shaped zone of high dispersion is also visible in the upper part of the image, possibly representing a structural or stratigraphic boundary.

The high dispersion values (red/yellow) are concentrated along the boundaries and internal structures of the geological features. This indicates that these are zones where the frequency content of the seismic signal changes rapidly. The sharp red outlines around the elliptical and rectangular bodies are excellent indicators of their boundaries, suggesting they might be faults, unconformities, or other sharp contacts. The internal striping of high dispersion in the bottom-right body suggests internal stratification, thin-bed sequences, or a series of closely spaced discontinuities. This attribute is a powerful edge-detection tool for highlighting discontinuities and internal structures that might be subtle on other attribute maps.

6. Application to Real Seismic Data

To demonstrate the potential of the suggested 2D-VMD attributes, two seismic sections taken from Algerian fields have been used [32,33].

The first seismic profile is located in the Ahnet Basin (Southwestern Algeria), part of the larger Saharan Platform. The Ahnet Basin is a Paleozoic intracratonic basin characterized by thick successions of Cambrian to Carboniferous sedimentary rocks. It is mainly composed of alternating sandstones, shales, and carbonates, with important Silurian and Devonian source rocks. The basin has been subject to limited tectonic deformation, preserving relatively simple structures such as gentle folds and faults, making it a favorable area for hydrocarbon accumulation [34,35,36,37,38,39,40,41,42,43,44].

The second seismic profile crosses the Illizi Basin (Southeastern Algeria), one of the most prolific hydrocarbon provinces in Algeria. The Illizi Basin is filled with Paleozoic sedimentary successions, dominated by Cambro-Ordovician sandstones overlain by Silurian shales (notably the “Hot Shale”) and Devonian to Carboniferous sequences. The basin’s structural style is marked by a combination of gentle folds and fault-related traps, with localized tectonic reactivation along Paleozoic and Hercynian lineaments. This region has been extensively explored and is well known for its giant gas and oil fields [45,46,47,48,49,50,51,52,53].

6.1. Application to Seismic Section 1

The section shows a layered stratigraphy. The upper part (roughly from sample number 1 to 600 on the y-axis) is characterized by a series of high-amplitude, sub-horizontal reflections (bright yellow and red stripes) (Figure 3). These reflections are relatively continuous but show some lateral variations. Below this, from about sample number 600 to 1200, the reflections are much weaker, with a generally lower amplitude and more uniform. A prominent, high-amplitude reflection is visible around 600, separating the two zones. There are some signs of subtle faulting or folding, particularly in the upper section.

This section shows a clear change in geological character with depth. The upper part likely represents a highly stratified package of sedimentary layers, possibly with significant lithological variations causing the strong reflections. The lower part could represent a more uniform geological unit, such as a thick shale or basement rock, with fewer significant impedance contrasts. The strong reflector around 600 is a key stratigraphic boundary, possibly an unconformity.

The obtained seismic attributes computed from the seismic line 1 are discussed in the following (Figure 4).

6.1.1. Azimuth Attribute (In Radians)

The figure shows a highly textured and complex pattern. There is no single dominant color, indicating a high degree of variation in the local azimuth. The upper part of the section is particularly chaotic, with a mix of black, orange, and yellow speckles and lines. The lower part is slightly more uniform but still shows significant variation. There are also distinct, roughly horizontal bands of a more consistent azimuth. The boundaries of the main geological units from the original section are not as clearly defined here.

The chaotic and rapidly changing azimuth values suggest a high degree of structural complexity. This could be caused by numerous small-scale faults, fractures, or a highly folded or deformed stratigraphic package. The attribute is effectively highlighting the localized changes in orientation that might be subtle on the original section. The more uniform bands could represent zones of more consistent dip.

6.1.2. Dip Magnitude Attribute

This dip magnitude attribute map clearly correlates with the original seismic section. The upper part of the section, with its horizontal to sub-horizontal reflections, shows a very low dip magnitude (dark red/black). These lines correspond to the high-amplitude reflections in the original section. The lower part of the section is almost entirely black, indicating a very low dip magnitude.

This attribute is excellent for highlighting key reflectors and identifying structural dips. The bright yellow lines correspond to the most significant reflectors, indicating that they are well-defined horizons. The low dip magnitude overall confirms that the stratigraphy is generally flat-lying. This attribute can be used to trace specific horizons and to identify subtle dips or folds that might be hard to see on the original seismic section.

6.1.3. Chaos Attribute

The map shows a very clear pattern. The upper part of the section, from roughly sample number 1 to 600, has a high chaos value (bright yellow/white). This zone is highly chaotic, with a complex texture. The lower part of the section, sample number from 600 to 1200, has a much lower chaos value (darker red), indicating a more coherent response. A distinct horizontal band of very high chaos is visible around sample number 600, separating the two zones. There are also horizontal bands of high chaos within the upper zone.

This attribute is a powerful tool for mapping discontinuities and disrupted zones. The high chaos in the upper zone suggests it is a highly deformed or fractured stratigraphic package. The prominent high-chaos band around 600 likely corresponds to the major stratigraphic boundary seen in the original section, indicating it is an unconformity or a major erosional surface. The low chaos in the lower section suggests it is a more uniform and undisturbed geological unit. This attribute effectively highlights areas of geological complexity and disruption.

6.1.4. Coherence Attribute (Semblance)

This figure displays a coherence attribute map, which measures the similarity of seismic traces. High coherence (values approaching 1, represented by bright yellow/white) indicates a high degree of similarity and continuity, often associated with a continuous geological layer. Low coherence (values approaching 0, represented by dark black/red) indicates a lack of similarity, which can be caused by faults, fractures, channels, or other discontinuities.

The map is dominated by low coherence values (dark red and black). However, there are numerous thin, bright yellow/orange lines that appear to be sub-horizontal. These lines are not perfectly continuous and show lateral variations and breaks. There is a distinct, dark black band from roughly 0 to 100 on the y-axis, indicating a zone of very low coherence. Below this, the bright lines become more prominent, especially around the 600–800 y-axis range.

The low overall coherence suggests a complex and potentially discontinuous stratigraphy, with numerous small-scale faults, fractures, or lateral facies changes. The bright lines of high coherence likely correspond to the most significant and continuous reflectors, which are still not perfectly continuous. This attribute is effectively highlighting the major stratigraphic horizons and their lateral disruptions. The very low coherence zone at the top could represent a chaotic zone, an unconformity, or a region of very poor data quality.

6.1.5. Curvature Attribute (Mean Curvature)

The map is almost entirely a uniform orange/yellow color, with very little variation. There are no prominent lineaments, features, or significant color contrasts visible. The color scale ranges from 0 to 1, and the entire map seems to be at a high, consistent value. There are some very subtle, scattered dark red pixels, but they do not form any discernible pattern.

The uniform, high curvature value across the entire map is unusual. It suggests either that the seismic data is uniformly and highly curved, which is unlikely for a large section, or, more plausibly, that the analysis is highlighting a dominant high-frequency component or texture rather than individual geological features. It is also possible that the specific processing parameters or the nature of the input data are causing this uniform result, making the attribute less effective for identifying subtle geological features in this particular instance. Without further context, the interpretation is difficult, but it strongly suggests a limitation in how this attribute is revealing structural information from this specific dataset.

6.1.6. Instantaneous Frequency Attribute (Hz)

The map shows a clear layered pattern of varying frequencies. There are prominent, sub-horizontal bands of different colors, indicating distinct frequency zones. The top part of the map (roughly 0–600 on the y-axis) is characterized by a mix of yellow and orange horizontal bands, suggesting a layered stratigraphy with varying frequency responses. Below this, the frequency becomes more uniform and lower (more orange/red) towards the bottom of the section. The color scale ranges from −100 to 100 Hz.

The horizontal banding indicates a layered geological environment where the rock properties are changing vertically. The higher frequency zones (yellow) in the upper part of the section could represent thinner beds or specific lithologies, while the lower frequency zones (orange/red) could represent thicker beds or different lithologies. The change to a more uniform, lower frequency in the bottom half of the section suggests a change to a different, more homogeneous geological unit, possibly a thick shale or basement rock. This attribute is a powerful tool for lithological and stratigraphic interpretation.

6.1.7. Instantaneous Bandwidth (Hilbert) (Hz)

The map is overwhelmingly black, with very few and very small scattered white pixels. The color scale goes up to 20,000, but the vast majority of the data is at or near the lowest value, represented by black. There is no discernible pattern or geological feature highlighted by this attribute.

The consistently low bandwidth values suggest that the frequency content of the seismic signal is very stable across the entire section. This is a very unusual result for a typical seismic dataset. It could indicate that the data has been processed in a way that removes frequency variations, or that the attribute is not well-suited to this specific dataset. Alternatively, if the data is a synthetic model, it may have been generated with a very narrow frequency spectrum. The lack of any significant features makes this attribute uninformative for geological analysis in this specific case.

6.1.8. MWSD Attribute (Module)

The map is overwhelmingly dark blue, indicating a very low-energy seismic response for most of the section. However, there are several distinct horizontal bands of higher amplitude (light blue and green). These bands are most prominent in the upper half of the section (approximately 0 to 600 on the y-axis). The bands are not perfectly continuous, showing some lateral variations and breaks. The values on the color scale range up to 5000, but the colors on the map suggest most values are much lower.

This attribute highlights a few key geological horizons. The horizontal bands of higher amplitude correspond to significant seismic reflectors, which are likely representing continuous geological layers with a strong acoustic impedance contrast. The lack of high-amplitude features in the rest of the section suggests either a very uniform geological unit or a zone of poor seismic data quality. This map is effectively acting as a filter, showing only the most energetic parts of the seismic signal.

6.1.9. MWSD Attribute (Phase)

The “Phase” map represents the phase of the seismic signal in a specific frequency band. Phase information is often sensitive to changes in bed thickness or fluid content and can reveal subtle stratigraphic details that are not visible in the magnitude (module) data. The color scale, ranging from −0.5 to −0.75, suggests values in a range, representing phase angles in radians.

The map shows a highly striated, horizontal pattern. The entire section is a complex mosaic of bright red/orange and dark blue lines. These lines are sub-horizontal and show some lateral continuity but also frequent changes in color. There are no large, distinct features or bodies. The texture appears quite uniform across the entire section, with subtle changes in the horizontal banding.

The phase attribute is revealing a layered stratigraphy. The alternating red and blue horizontal lines indicate changes in the phase of the seismic signal, which can be caused by variations in rock properties, bed thickness, or fluid content. The complexity and high-frequency nature of these phase changes suggest a finely stratified geological section with numerous thin beds. This attribute is particularly useful for identifying subtle stratigraphic variations that may not be apparent on other attribute maps.

6.1.10. VMD-Directionality Coherence (VDC)

This map is a complex mix of colors, with a predominance of yellow and green/blue. The overall impression is one of a highly textured and chaotic pattern. There are numerous small, horizontal and sub-horizontal bands of high coherence (yellow), but they are not perfectly continuous. The upper half of the section appears to have slightly higher overall coherence than the lower half. There are no large, distinct fault lines or other major discontinuities clearly visible.

The high degree of coherence variation suggests a complex geological environment. The yellow bands indicate areas of good reflection continuity, corresponding to the layered stratigraphy. The blue/green patches and breaks within these bands suggest numerous small-scale discontinuities, such as subtle faults, fractures, or lateral facies changes. The attribute is effectively highlighting the patchy and discontinuous nature of the geological layers, making it a good tool for mapping zones of geological disruption.

6.1.11. Instantaneous Frequency Concentration (IFC-VMD)

This map shows a clear layered pattern. The upper part of the section (approximately 0 to 600 on the y-axis) is characterized by prominent, high-concentration (red/yellow) horizontal bands. These bands are separated by zones of lower concentration (green/blue). The lower part of the section (below 600) is dominated by lower frequency concentration (blue and dark red), with fewer prominent high-concentration bands.

The high-concentration bands in the upper section correspond to zones with a dominant, single frequency. This could indicate thick, uniform geological layers. The lower-concentration zones in between them could be thinner layers or zones where the frequency content is more complex. The transition to a predominantly low-concentration zone in the lower half of the section suggests a change to a more complex stratigraphy with a broader frequency spectrum, possibly indicating a series of thin beds, a disturbed zone, or a change in lithology. This attribute is a valuable tool for distinguishing between different lithological units and for identifying zones of uniform versus complex deposition.

6.1.12. Instantaneous Bandwidth Dispersion (IBD-VMD)

This figure shows a highly textured and complex pattern across the entire section. The colors range from dark blue to bright red, indicating a wide range of dispersion values. The upper part of the section (0 to 600) is a mosaic of different colors, with a higher concentration of red/yellow zones. These high-dispersion zones appear as horizontal, wavy bands, corresponding to the reflections in the original seismic section. The lower part of the section (below 600) is also highly textured but with a slightly higher concentration of blue, indicating lower dispersion overall.

The high dispersion values (red/yellow) in the upper section are concentrated along the seismic reflectors. This indicates that these are zones where the frequency content of the seismic signal changes rapidly, likely due to thin-bed tuning or stratigraphic changes. The wavy, horizontal bands of high dispersion are excellent indicators of the layered stratigraphy and any folding or deformation. The lower dispersion values (blue) in the lower section suggest a more uniform geological unit with a more stable frequency response. However, the presence of many smaller, high-dispersion patches suggests that even this deeper zone has some internal complexities or discontinuities. This attribute is a powerful edge-detection tool for highlighting stratigraphic and structural features that might be subtle on the original seismic section.

6.2. Application to Seismic Section 2

This seismic line presents seismic reflections appearing continuous or discontinuous horizontal to dipping patterns. The upper part of the section (approximately 0 to 500 on the y-axis) shows a series of sub-horizontal reflections that are relatively continuous but show some lateral variations. These reflections appear to have a higher amplitude than the lower part of the section (Figure 5). Below 500 on the y-axis, the seismic signal is more uniform and lower in amplitude, with fewer distinct, high-energy reflections. A distinct boundary or a zone with different reflection character is visible around 500. There are some signs of subtle faulting or folding, particularly in the upper section.

This section shows a clear change in geological character with depth. The upper part likely represents a stratified package of sedimentary layers, with the strong reflections indicating significant lithological variations. The lower part could represent a more uniform geological unit, with fewer significant impedance contrasts.

In the following, the seismic attributes obtained from seismic line 2 are discussed (Figure 6).

6.2.1. Azimuth Attribute (In Radians)

The figure shows a highly textured and complex pattern. The upper part of the section (approximately 0 to 500 on the y-axis) is particularly chaotic, with a mix of black, white, and yellow speckles and lines. The lower part is slightly more uniform but still shows significant variation. There are also distinct, roughly horizontal bands of a more consistent azimuth.

The chaotic and rapidly changing azimuth values suggest a high degree of structural complexity. This could be caused by numerous small-scale faults, fractures, or a highly folded or deformed stratigraphic package. The attribute is effectively highlighting the localized changes in orientation that might be subtle on the original section. The more uniform bands could represent zones of more consistent dip.

6.2.2. Dip Magnitude Attribute

This map clearly correlates with the features in the original seismic section. The upper part of the section (approximately 0 to 500 on the y-axis) shows a very low dip magnitude (dark red/black), but there are numerous thin, bright yellow/white horizontal lines. These lines correspond to the high-amplitude reflections in the original section. The lower part of the section is almost entirely black, indicating a very low dip magnitude.

This attribute is excellent for highlighting key reflectors and identifying structural dips. The bright yellow lines correspond to the most significant reflectors, indicating that they are well-defined horizons. The low dip magnitude overall confirms that the stratigraphy is generally flat-lying.

6.2.3. Chaos Attribute

The map shows a very clear pattern. The upper part of the section, from roughly 0 to 500 on the y-axis, has a high chaos value (bright yellow/white). This zone is highly chaotic, with a complex texture. The lower part of the section, from 500 to 2000, has a much lower chaos value (darker red), indicating a more coherent response. A distinct horizontal band of very high chaos is visible around 500, separating the two zones.

This attribute is a powerful tool for mapping discontinuities and disrupted zones. The high chaos in the upper zone suggests it is a highly deformed or fractured stratigraphic package. The prominent high-chaos band around 500 likely corresponds to the major stratigraphic boundary seen in the original section, indicating it is an unconformity or a major erosional surface. The low chaos in the lower section suggests it is a more uniform and undisturbed geological unit.

6.2.4. Coherence Attribute (Semblance)

The map is dominated by a reddish-orange color with horizontal bands of brighter yellow and black. The upper part of the section (approximately 0 to 800 on the y-axis) shows continuous, high-coherence (yellow) bands that are fairly uniform. Below this, from about 800 to 2000, the map becomes much more chaotic, with a mix of high and low coherence speckles and short, discontinuous lineaments. There are also distinct, thin, dark black lines that appear to be horizontal, indicating zones of very low coherence.

The high-coherence bands in the upper section suggest a package of well-layered and continuous sedimentary rocks. The transition to a more chaotic pattern in the lower section suggests a significant change in geology. This could be a zone of intense fracturing, faulting, or a change to a different rock type with more complex and discontinuous reflections, like a chaotic basement or a highly deformed zone. The thin, low-coherence lines could be subtle faults or stratigraphic discontinuities.

6.2.5. Curvature Attribute (Mean Curvature)

The map is almost entirely a uniform orange/yellow color, with a scale ranging from 0 to 1. There are no prominent lineaments, features, or significant color contrasts visible. It is a solid block of color with a few very subtle, scattered pixels that do not form any discernible pattern.

The uniform, high curvature value across the entire map is highly unusual for a geological section. This result suggests that the attribute is not revealing any meaningful structural information from this dataset. It could be a result of the specific processing parameters, a characteristic of the input data, or a limitation in how this attribute is applied to the data. It is not providing useful information for geological analysis in this case.

6.2.6. Instantaneous Frequency Attribute (Hz)

The map is dominated by orange and red, with a color scale ranging from −200 to 200 Hz. The overall impression is a highly textured and granular pattern with no clear horizontal banding or distinct features. There are no prominent zones of high or low frequency. The values seem to be concentrated around zero, with a high degree of pixel-to-pixel variation.

The lack of distinct horizontal bands or large-scale features suggests a complex stratigraphy with rapid vertical and lateral changes in frequency. This could indicate a finely layered or chaotic geological environment. The overall uniform, speckled appearance makes it difficult to distinguish specific geological units, suggesting that this attribute may not be the most effective for broad lithological or stratigraphic interpretation in this dataset.

6.2.7. Instantaneous Bandwidth (Hilbert) Attribute (Hz)

The map is almost entirely black, with very few and very small scattered white pixels. The color scale goes up to 500 on the y-axis, but the vast majority of the data is at or near the lowest value, represented by black. There is no discernible pattern or geological feature highlighted by this attribute.

The consistently low bandwidth values suggest that the frequency content of the seismic signal is very stable across the entire section. This is a very unusual result for a typical seismic dataset. It could indicate that the data has been processed in a way that removes frequency variations, or that the attribute is not well-suited to this specific dataset. Alternatively, it could simply be uninformative for this specific data, as it shows no significant variations.

6.2.8. MWSD (Module) Attribute

The most prominent feature is a continuous, high-amplitude layer visible between 500 and 1000 on the vertical axis. This reflector appears laterally homogeneous, showing a strong and consistent response across the profile, while the zones above and below are dominated by much lower amplitudes, illustrated by the dark blue coloring.

The high-amplitude event most likely corresponds to a significant geological boundary or layer characterized by a strong impedance contrast. Such a feature could represent a hard lithology like limestone, a fluid-saturated reservoir sand, or even a volcanic intrusion. Its lateral consistency suggests that this reflector is continuous and horizontally uniform within the surveyed area. In contrast, the low-amplitude zones above and below the event are indicative of more homogeneous or weakly reflective materials, such as shales or sands lacking fluid contacts.

6.2.9. MWSD (Phase) Attribute

Unlike the Module map, no clear continuous layer is observed. Instead, the phase attribute exhibits erratic shifts with scattered patches of varying colors, reflecting a high degree of variability and suggesting a complex or heterogeneous subsurface at this scale. The absence of a coherent pattern at the location of the strong reflector identified in the Module map indicates that the phase response is emphasizing smaller-scale variations. This behavior may result from scattering caused by small-scale heterogeneities, multiple reflections, or interference effects.

6.2.10. VMD-Directionality Coherence (VDC) Attribute

The overall map attribute is characterized by a mix of blue tones, indicating low coherence, and yellow to green shades, reflecting moderate coherence. A distinct zone of relatively high coherence, displayed in yellow, green, and light blue, is observed between 500 and 1000 on the vertical axis, approximately corresponding to the high-amplitude reflector seen in the Module map. However, this high-coherence band is not entirely continuous, as it is interrupted by scattered low-coherence blue patches. In contrast, the zones above and below are generally dominated by very low coherence values.

The presence of a moderate-to-high coherence band confirms that the feature identified in the Module map corresponds to a geologically continuous layer, suggesting that the reflector is largely intact and not significantly affected by faulting or fracturing within this interval. The scattered low-coherence patches within the band, however, point to localized disruptions, which may represent small-scale faults, fractures, or lateral facies variations. In contrast, the very low coherence observed in the upper and lower sections of the profile implies a disorganized subsurface, potentially reflecting chaotic sedimentary deposits or intervals lacking well-defined, continuous reflectors.

6.2.11. Instantaneous Frequency Concentration (IFC-VMD) Attribute

On this map attribute, high values (red) indicate zones where the signal is dominated by a single frequency, whereas low values (blue) reflect broadband or noisy responses.

The chaotic, speckled distribution observed suggests that the frequency content varies rapidly across the profile, likely reflecting strong subsurface heterogeneity in elastic properties that induces scattering and abrupt spectral changes. This attribute appears particularly sensitive to small-scale heterogeneities, such as variations in rock type, fluid content, or fracture density, which may not be evident in the other attribute maps. Although the absence of a coherent large-scale pattern limits its utility for mapping major geological structures, it remains valuable for detecting subtle localized features or for seismic data quality control.

6.2.12. Instantaneous Bandwidth Dispersion (IBD-VMD) Attribute

As mentioned earlier, high dispersion values (red) indicate zones where a wide frequency range is present, often caused by scattering from small-scale heterogeneities, thin layering, or noise. Conversely, low dispersion values (blue) represent more monochromatic or spectrally pure signals, typically associated with thicker and more homogeneous geological units where scattering and interference are minimal.

The chaotic appearance of the IBD map suggests that the seismic frequency content is highly variable at small scales, making this attribute particularly useful for identifying areas affected by fracturing, gas saturation, or subtle stratigraphic variations that may not be visible in the original amplitude data. By emphasizing frequency variability, IBD provides complementary insights that enhance interpretation when combined with conventional seismic attributes.

7. Discussion

Seismic attributes are crucial for enhancing subsurface interpretation by extracting specific features from seismic data. Each attribute has a primary function but also limitations that must be considered when choosing the appropriate analysis method. Advanced decomposition and denoising techniques such as MWSD (Module and Phase), VDC, IFC-VMD, and IBD-VMD provide complementary advantages for improving attribute reliability, resolution, and interpretability (

Table 2. Comparative analysis of the proposed 2D VMD-based and conventional seismic attributes.

Attribute	Primary Function	Limitation	MWSD (Module) Advantage	MWSD (Phase) Advantage	VDC Advantage	IFC-VMD Advantage	IBD-VMD Advantage
Dip	Estimates reflector slopes	Sensitive to noise in steep dips	Good lateral continuity detection	Improved phase stability	High-resolution structure delineation	Better noise suppression in complex geology	Handles abrupt dip changes well
Azimuth	Measures reflector orientation	Affected by poor signal-to-noise ratio	Stable azimuth estimation in noisy data	Better orientation continuity	Enhanced azimuth resolution	Accurate under varying illumination	Good for small-scale azimuthal variations
Chaos	Detects structural discontinuities	Prone to highlight noise as features	Suppresses random noise while preserving faults	Improved phase for discontinuity mapping	Good edge preservation	Enhanced chaotic zone definition	Accurate in small chaotic zones
Coherence (semblance)	Measures similarity between traces	Resolution decreases with increasing window size	Improved similarity detection	Phase-insensitive similarity mapping	High vertical resolution	Better detection in noisy areas	Sensitive to subtle stratigraphic features
Curvature (Mean Curvature)	Measures reflector curvature	Sensitive to noise; may produce artifacts	Good noise suppression while keeping curvature detail	Better curvature continuity	Highlights subtle features	Improved detection of small-scale folds	Handles mixed curvature types well
instantaneous frequency	Estimates temporal frequency variation	Affected by attenuation and tuning	Stable frequency estimation	Better phase consistency	Enhanced thin-bed detection	Accurate under complex wavelets	Good in low SNR environments
instantaneous bandwidth (Hilbert)	Estimates bandwidth variations	Sensitive to noise	Better bandwidth stability	Improved phase estimation for bandwidth	Good for lithology changes	Enhanced detection of attenuation zones	Handles abrupt bandwidth changes

).

7.1. Structural Attributes

Dip and azimuth estimation are fundamental for characterizing reflector geometry. While dip highlights slope variations and azimuth measures reflector orientation, both are sensitive to noise, particularly in steep dips or areas with poor signal-to-noise ratio. MWSD (Module) excels at maintaining lateral continuity and stable estimation, while MWSD (Phase) enhances phase stability and orientation continuity. VDC offers high-resolution structural delineation, and IFC-VMD provides robustness in complex geological settings. IBD-VMD is especially useful for handling abrupt changes in dip or small-scale azimuthal variations.

7.2. Discontinuity and Similarity Attributes

Chaos and coherence (semblance) help in identifying faults, fractures, and stratigraphic discontinuities. Chaos detects structural complexity but may misinterpret noise as geologic features. Coherence measures trace similarity, but resolution decreases with larger windows. MWSD-based approaches effectively suppress random noise while preserving edges, with the Phase mode offering improved phase handling. VDC ensures edge preservation and high vertical resolution, while IFC-VMD enhances definition in chaotic zones and noisy environments. IBD-VMD improves sensitivity to subtle features and small-scale discontinuities.

7.3. Geometric Attributes

Curvature (mean curvature) is essential for mapping folds, fractures, and subtle structural trends, but it is highly sensitive to noise, potentially producing artifacts. MWSD (Module) reduces noise while retaining curvature detail, whereas MWSD (Phase) ensures continuity in curvature interpretation. VDC is adept at highlighting subtle features, and IFC-VMD improves detection of small-scale folds. IBD-VMD performs well with mixed curvature types, maintaining interpretative clarity even in heterogeneous geology.

7.4. Spectral Attributes

Instantaneous frequency and instantaneous bandwidth (Hilbert transform) reveal variations in temporal frequency and bandwidth, which are critical for stratigraphic and lithologic analysis. Both are noise-sensitive and can be influenced by attenuation or tuning effects. MWSD (Module) ensures stable frequency and bandwidth estimates, while MWSD (Phase) provides consistent phase information. VDC enhances thin-bed detection and lithologic discrimination, IFC-VMD improves accuracy under complex wavelets or attenuation zones, and IBD-VMD excels in low signal-to-noise conditions and abrupt spectral changes.

7.5. Overall Assessment

The synthesis shows that no single method universally outperforms the others; rather, each has unique strengths for specific scenarios. MWSD (Module) is generally superior for stability and noise suppression, MWSD (Phase) for phase-sensitive applications, VDC for high-resolution and edge preservation, IFC-VMD for complex geology and noise-affected areas, and IBD-VMD for abrupt feature changes and fine-scale resolution. Optimal seismic interpretation often involves combining these approaches to leverage their complementary advantages.

8. Conclusions

The proposed attributes (MWSD, VDC, IFC-VMD, and IBD-VMD) build upon the limitations of traditional seismic attributes by integrating multi-scale, frequency-aware analysis through Variational Mode Decomposition (VMD). Traditional attributes such as azimuth, dip, chaos, and coherence are effective for basic structural interpretation but lack frequency selectivity and often struggle in noisy or complex geological environments.

By contrast, the VMD-based methods allow for decomposition into well-defined frequency modes, enabling targeted analysis of structural, spectral, and directional features. This results in improved noise robustness, better fault continuity mapping, and more reliable detection of subtle boundaries and attenuation zones. Consequently, these new attributes offer enhanced interpretability and precision in complex subsurface settings, making them particularly valuable for advanced seismic interpretation workflows.

While the current work focuses on structural interpretation using 2D-VMD-derived attributes, it is important to recognize that VMD’s capacity for robust signal decomposition and denoising has implications for shear-wave velocity (Vs) estimation. By isolating cleaner, mode-specific seismic components, the proposed attributes may facilitate more accurate vs. inversion in future studies, a promising direction for expanding the applicability of VMD in quantitative seismic analysis.