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Detection in seismic data using curvelet transform and tensor-based elliptical adaptive structuring elements
Journal Pre-proof Detection in seismic data using curvelet transform and tensorbased elliptical adaptive structuring elements
Bahareh Boustani, Abdolrahim Javaherian, Majid Nabi-Bidhendi, Siyavash Torabi, Hamid Reza Amindavar PII:
S0926-9851(18)30889-9
DOI:
https://doi.org/10.1016/j.jappgeo.2019.103881
Reference:
APPGEO 103881
To appear in:
Journal of Applied Geophysics
Received date:
12 October 2018
Revised date:
24 September 2019
Accepted date:
31 October 2019
Please cite this article as: B. Boustani, A. Javaherian, M. Nabi-Bidhendi, et al., Detection in seismic data using curvelet transform and tensor-based elliptical adaptive structuring elements, Journal of Applied Geophysics(2019), https://doi.org/10.1016/ j.jappgeo.2019.103881
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© 2019 Published by Elsevier.
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Detection in seismic data using curvelet transform and tensorbased elliptical adaptive structuring elements Bahareh Boustani a, Abdolrahim Javaheriana,b,*, Majid Nabi-Bidhendib, Siyavash Torabic, Hamid Reza Amindavard Department of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran
b
DMT Petrlogic, GMbH & Co. KG
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c
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Institute of Geophysics, University of Tehran, Tehran, Iran
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a
d
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Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran
ABSTRACT
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*Corresponding author: [email protected], [email protected]
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Channels are one of the vital issues in the exploration of oil and gas. They can be
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considered as reservoirs if they are filled with porous and permeable material and placed in a suitable position. A detailed study of the channels can help identify the sedimentation processes of the area, the intensity, and direction of the sea currents. Manual interpretation of the channels is difficult and requires skill. Therefore, in this paper, the adaptive curvelet and morphological gradient algorithm (ACMG) is used for the automatic interpretation of the channels. First, the morphological gradient is applied to extract the edges of the channels; then, the curvelet transform 1
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is used to increase the signal-to-noise ratio. The morphological top-hat operator extracts the local maxima of curvelet sub-bands. In this workflow, we applied an elliptical adaptive structuring element (EASE) based on Gradient structure tensor (GST). For the construction of the 2D GST, the horizontal and vertical gradients of the image were calculated by the first-order Gaussian derivative. The eigenvalue
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decomposition of each structure tensor can provide an estimate of the direction and
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anisotropy rate of the image objects. Hence, parameters of the elliptical structuring
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element are obtained by eigenvalue decomposition of the GST. To evaluate
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ACMG, we compared the final output with the results of the non-adaptive curvelet
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and morphological gradient algorithm (CMG). Also, we compared it with the
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common edge-detectors such as Canny, Sobel, Laplacian of Gaussian, and similarity attribute. The comparison shows that both CMG and ACMG well
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extracted the edge of the channels with the higher signal-to-noise ratio.
Keywords: channel edge; elliptical adaptive structuring element; gradient structure tensor; morphological gradient; curvelet transform; top-hat
1. Introduction
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Channel detection is essential in the petroleum exploration industry. Since, if the channels are filled with the porous and permeable material, they can be a good place to accumulate hydrocarbon. In addition, a detailed study of the channels can help to better know of sedimentation process in the area (Alsouki et al., 2008). Also, channels in stratigraphic studies are important for the development and
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production of oil reservoirs (Mirkamali et al., 2014). A channel in a seismic section
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is considered as an interruption of reflection continuity, but their recognition is
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easier in time or horizon slices of the 3D seismic data set. However, manual
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interpretation of the channels is a tedious task which requires skill and training. So,
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several attributes have been introduced for smoother and more accurate channel
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interpretation (Chopra and Marfurt, 2007). Coherence is an edge-sensitive attribute which is used to mapping channel edges
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(Bahorich and Farmer, 1995). Spectral decomposition complements the coherence
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and edge-based attributes since it is sensitive to channel thickness rather than its width (Partyka et al., 1999). Liu and Marfurt (2007) combined peak frequency and peak amplitude with the coherence to define channel thickness in addition to its width. Most-positive and most-negative curvature are also helpful in channel identification (Chopra and Marfurt, 2008; Lozano and Marfurt, 2008). Kadlec et al. (2008)
used
Gaussian-smoothed
first-order structure tensor and
eigen-
decomposition of the second-order structure tensor to segment channel features in 3
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3D seismic volumes. Al-Dossary (2015) compared running average, wavelet transform, and polynomial fitting noise reduction algorithms before Sobel filter to enhance channel patterns in seismic data. Ghazanfari and Javaherian (2016) combined seismic attributes using the artificial neural network to detect buried channels. Karbalaali et al. (2017a, b and 2018) detected channel edges in 2D and
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3D seismic data using cone-adapted and pyramid-adapted shearlet transform. They
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first maximize shearlet coefficients in each pixel at the finest scale of
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decomposition, then histogram-based thresholding and morphological thinning are
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applied.
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The curvelet transform as a multi-scale and multi-directional transform was used
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in many different seismic application such as ground roll attenuation (Yarham et al., 2004 and 2006; Naghizadeh and Sacchi, 2011; Boustani et al., 2013; Hejazi et
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al., 2013; Yuan et al., 2018), random noise attenuation (Kumar and Herrmann,
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2009; Wang et al., 2010; Cao et al., 2015; Zhang et al., 2018) and multiple attenuation (Herrmann et al., 2008; Yu and Yan, 2011; Nguyen and Dyer, 2016). Mathematical morphological filtering (MMF) in seismic data processing first introduced by Wang et al. (2005). Li et al. (2005) used MMF, i.e., an average of the open-close and close-open filters in random noise attenuation of the seismic trace. They examined the effect of shape, width, and amplitude of the structuring element (SE) on the filter output. Wang et al. (2008) introduced multi-scaled 4
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morphology using self-dilation of a semicircle structuring element. Each morphology scale is obtained by computing the MMF by structuring element of that scale and subtracting the result from the previous stage. He then attenuated the ground roll from seismic data by removing the related morphological scales and reconstruction of others. Duan et al. (2010) and Yu et al. (2014) also utilized the
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multi-scaled morphology for ground roll noise attenuation of seismic data. The
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semicircle SE was used in the temporal direction of every single trace to suppress
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low-frequency noise by the compound top-hat filtering (CTF) (Li et al., 2016).
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MMF with straight-line structuring element was employed for coherent noise
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attenuation by sliding the SE along the noise (spatial) direction (Huang et al.,
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2017). Huang et al. (2018) used multi-scaled MMF for weak signal detection of microseismic data. They used a weighted average of morphological scales to
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reconstruct data. The weights are chosen in the previous work manually (Wang et
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al., 2008; Chen et al., 2015) or adaptively using a varimax norm (Yuan et al. 2016). But they proposed a non-stationary weighting operator which can be obtained by solving an inverse problem in a shaping regularization approach. Huang and Wang (2018) for random noise attenuation generalized MMF into 2D in which SE is a matrix, not a vector. Planar MMF has been more successful in random noise reduction since morphological operators on a single trace do not have enough ability to detect signal from random noise. 5
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In this paper, we applied the adaptive curvelet transform and morphological gradient (ACMG) algorithm to enhance data and detect channel edges to the synthetic and real data sets containing channels. For this purpose, we used the elliptical adaptive structuring element. We compared the results of ACMG with the common edge detection filters in image processing such as Canny, Sobel,
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Laplacian of Gaussian (LoG), and similarity attribute.
Curvelet transform
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2.1.
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2. Methodology
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One of the multi-scale and multi-directional transforms which is proposed to
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compensate shortcoming of the wavelet transform in two and higher dimensions was curvelet transform (Candes and Donoho, 1999). It has high directional and
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anisotropic properties due to the decomposition of the data along with radial
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wedges in the 2D Fourier domain. It is the second generation curvelet transform which is introduced by Candes et al. (2006). To digitize the curvelet transform, they proposed two algorithms, i.e., unequally-spaced fast Fourier transform (USFFT), and wrapping. Because the latter is easier and faster than the first one, we used wrapping-based fast discrete curvelet transform in this study. The source code of the algorithm is available in CurveLab package. For the detailed
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description of the theory and equations of the fast discrete curvelet transform (FDCT) is referred to (Candes et al., 2006).
2.2.
Mathematical morphology
The mathematical morphology was first proposed by Matheron and Serra around
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1964 to investigate the geometric structures of geological samples (Serra, 1982). It
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was introduced by Serra (1982) to image processing. The language of
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mathematical morphology is set theory, since it uses the simple mathematical
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concepts such as intersection and union of sets (Soille, 2004). As a powerful tool, it
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is applicable in many image processing issues such as texture analysis, edge
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detection and noise reduction. Morphological operators analyze the image structures based on their shape and do this with a small template called structuring
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element. A 2D structuring element is a zero matrix whose elements inside a given
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shape are 1. The shape of SE should be selected according to the desired processing. The basic morphological operator is “erosion” and “dilation” and all other operators are expressed in terms of these two. Once probing the image by the structuring element, if structuring element fits the image, the central pixel is a pixel of eroded image, If the structuring element hits the image, the central pixel is a pixel of dilated image and if the structuring element neither fits nor hits the image, the central pixel does not belong to any eroded and dilated images. This operation 7
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is shown schematically in Fig. 1. Indeed, dilation increases and erosion decreases the border pixels. In other words, dilation gives the maximum and erosion gives the minimum of an image intensity values in the neighborhood of the SE. Two basic morphological filters are “opening”, i.e., erosion followed by dilation and “closing”, i.e., dilation followed by erosion. In the grayscale image, opening
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removes the objects which are brighter than the background and smaller than the
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SE. On the contrary, closing removes the objects which are darker than the
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background and smaller than the SE. Sometimes the objects which are removed by
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the opening and closing are not noises, so they can be recovered using “top -hat”
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and “bottom-hat” operators. Top-hat is the difference between the original image
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and its opening and bottom-hat is the difference between the closing and the original image. The edges in the image are areas with high grey level variations.
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Gradient operators can enhance these variations. Morphological gradients are
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operators which can strengthen variations of pixel intensity in SE’s neighborhood. The simplest morphological gradient algorithms are defined in terms of the difference between the results of the dilation, erosion and the original image. Fig. 2 shows some aforementioned morphological operators on a simple binary test image. The success of the mathematical morphology is due to its simple mathematical description and many powerful tools in image analysis which it provides (Soille, 2004). 8
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2.2.1. Adaptive mathematical morphology Traditional mathematical morphology uses a fixed structuring element for the whole image. This method is useful for detecting image features of a certain size and shape. But selecting a proper structuring element for images whose features
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varying in size and shape have difficulty. One solution is to filter these images
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iteratively with SEs in different sizes, and finally, for each point, use the result of
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an SE. But this method takes a lot of time because the whole image should be
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processed many times; however, a small fraction of calculation is required for the
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final result. Thus, the tendency towards adaptive morphology has increased, which
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its origin is the theory introduced by the Serra (1988). One method of defining adaptive morphology is structure-based, i.e., the structuring element aligned to
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edges and contours. This can be done only using considering the orientation, or the
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anisotropy rate, or the distance to the edges. Relation to this approach, Landstrom and Thurley (2013) introduced an elliptical adaptive structuring element based on the gradient structure tensor (GST). The eigenvectors and eigenvalues of the GST include information about the orientation of the image structures (edges) and the anisotropy rate (Landstrom and Thurley, 2013; Curic et al., 2014).
2.2.2. 2D gradient structure tensor 9
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Gradient structure tensor is a well-known characteristic in image processing which is constructed from the smoothed product of partial derivatives of the image. Bakker et al. (1999) introduced an edge-preserving and orientation adaptive filter using gradient structure tensor. In seismology, Bakker (2002) also presented the edge sensitive attribute such as coherency based on gradient structure tensor. If
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(x,y) is the coordinate of a pixel of the image, and f(x,y) is corresponding gray-
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level value. 2D gradient structure tensor (GST), T(x,y), which represents the local
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directional structures in the image is defined by the following 2˟2 matrix
)
[
](
)
(
(
)
(
))
(1)
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(
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(Landstrom and Thuley, 2013; Wang et al., 2017):
and
demonstrate partial derivative of the image along the x-axis and y-
axis,
(
) , and
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where
is a Gaussian kernel with standard deviation σ. This
smoothing regularizes the matrix and is important to obtain a stable estimation of feature orientations. Eigenvalue decomposition of GST can estimate the orientation and anisotropy of image structures. So, in the next step for each pixel (x,y), the eigenvalues l1(x,y) and l2(x,y) (l1 > l2), and the corresponding eigenvectors e1(x,y) and e2(x,y) are calculated from the symmetric matrix of GST. These eigenvalues contain information about the image edges (Landstrom and Thuley, 2013): 10
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l1 ≈ l2 >> 0: No dominant direction (edge crossing or point), l1 >> l2 ≈ 0: Strong dominant direction (edge), l1 ≈ l2 ≈ 0: No dominant direction (no edge).
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The eigenvector e1 shows the dominant local gradient in the image, while the
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eigenvector e2, which is orthogonal to e1, indicates the direction of the smallest
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variations.
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2.2.3. Elliptical Adaptive structuring element
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Now, we can define a flat elliptical structuring element SE(a,b,φ) for each pixel
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(x,y), so that a and b are the semi-major axis and semi-minor axis of the ellipse and
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φ is the angle between the semi-major axis and x-axis. Fig. 3 shows the ellipse parameters and their relation to the eigenvectors of GST. Hence, the ellipse parameters for each pixel are obtained from the following equations (Landstrom and Thuley, 2013; Wang et al., 2017): (
)
(
)
( (
)
(
)
) (
(
)
) (
(2)
)
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where M is the maximum allowable semi-major axis length which is defined by the user.
is a small value, i.e. machine epsilon. For the all values of the l1(x,y) and
l2(x,y) we have 0 ≤ b(x,y) ≤ a(x,y) ≤ M and a(x,y) + b(x,y) = M. the orientation is obtained from the corresponding eigenvector (Landstrom and Thuley, 2013; Curic
(
(
)
(
)
( ) and
) )
}
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{
)
(3)
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where
)
(
(
) are the components of the eigenvector
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(
(
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et al., 2014):
(
).
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This structuring element changes dynamically from a line of length M, near the
Curvelet and morphological gradient
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2.3.
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dominant edges, to a circle of radius M/2, in isotropic regions in the image.
Gradient operators called edge-detectors because they enhance high-frequency events of the image, which coincide with the edges. In this study, we used a median filter before morphological gradient to smooth the seismic image. Then, we extract the channel edges by the morphological gradient, i.e., the difference between dilation and erosion, using EASE. Finally, the curvelet transform is used in post-processing to increase the signal-to-noise ratio. Channels in time slices of 12
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the 3D seismic data are considered as curvilinear structures with different thicknesses and widths. Therefore, the multi-scale and multi-directional curvelet transform can be a good choice for separating them from noise. The curvelet coefficients which are parallel to the edges have magnitudes much greater than the zero. Hence, we extract the maximum curvelet coefficients in all sub-bands using
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the modified morphological top-hat algorithm (Miri and Mahloojifar, 2011). By
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definition, the top-hat operator results in local maxima of the data. In the curvelet
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domain, we first calculated the absolute value of all curvelet sub-bands. Then, a
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morphological close-open filter is applied, and the minimum of the filter result and
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original sub-band is calculated. To compute modified top-hat, this minimum is
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subtracted from original sub-band for all curvelet sub-bands. Finally, the inverse curvelet transform is applied and the channel edge map is generated. Fig. 4 depicts
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the flowchart of the ACMG edge-detection algorithm. In generating EASE, we
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used Gaussian first-order derivative of the standard deviation σ 1 to calculate the gradient of the image in vertical and horizontal directions. Then, we constructed the components of GST matrix according to Eq (1). A Gaussian filter of standard deviation σ2, the same as σ in Eq (1), is used for smoothing of the GST components. Then, the eigenvalue decomposition is applied to the GST. Finally, the ellipse parameters are calculated according to the Eqs. (2) and (3). So, the only
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parameters that should be defined by the user are σ 1, σ2 and M (maximum allowable semi-major axis length). The step by step proposed method is as follows: a) Applying the median filter. b) Generating EASE based on 2D gradient structure tensor. c) Applying the morphological gradient by the EASE.
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d) Applying wrapping based fast discrete curvelet transform.
e) Applying the modified top-hat algorithm using EASE to the absolute value of all
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curvelet sub-bands.
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f) Applying the inverse fast discrete curvelet transform.
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3. Results and discussion
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To verify the quality of the EASE, in Fig. 5, the results of the closing on a portion of the real data containing channels are presented. Fig. 5a is the original data containing the linear horizontal noise which is the effect of the acquisition footprint. Figs. 5b, c, and d show the results of the closing of 5a by disk SE, multilength and multi-direction linear SE (used in the proposed CMG algorithm), and EASE, respectively. As can be seen, all three structuring elements have removed the noise, but the disk SE also eliminates parts of the channels. The linear SE has not reconstructed the edges of the channels as smoothly as the EASE. 14
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The proposed methodology is developed in the form of a MATLAB code and evaluated on synthetic and real seismic time slices containing channels. The parameters of the ACMG algorithm which is used for the synthetic and real data are presented in Table 1. We determined the optimum values for these parameters by the trial and error. The median filter with a 5*5 window size usually gives the
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best result. The default value is considered for the number of scales and directions
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of curvelet transform because of the negligible effect they have on the filter output.
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The desired output resulted from the M (maximum allowable semi-major axis
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length of the ellipse) value between 6 and 10, which 8 is considered appropriate in
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this study. Outside this range, if a much smaller value is selected for M, the edges will not be extracted, and the noise is stronger. If a much larger value is selected,
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although more noise will be reduced, the edges will be destroyed. If the σ1 value is
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smaller than 1, then the filter does not respond well. The values between 4 and 6
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are proper for σ1. Outside this range, if it is much smaller, the edges will not be well extracted and if it is much larger, the edges will be destroyed. The values between 3 and 5 for σ2 can lead to an acceptable output. Again, a much smaller or larger value of σ2 has the same results as σ1 in edge detection.
3.1.
Synthetic data
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The synthetic data from the convolution of a reflectivity cube with a Ricker wavelet with the peak frequency 15 Hz with the sampling interval of 2 ms and spatial interval of 10 m. The earth model used in the construction of the reflectivity cube consists of a horizontal and dip layering in which a sinusoidal channel is embedded. Fig. 6a shows a time slice of this data set with the signal-to-noise ratio
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(S/N) of 1. To the preprocessed data, we applied a 2D median filter in a 5*5
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window to smooth image before edge detection (Fig. 6b). Figs 6c and d show the
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edge detection results using adaptive morphological gradient and after applying
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modified top-hat on all curvelet subbands. The edge detection results by
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morphological gradient without curvelet transform is very noisy. But after applying
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the curvelet filter, the channel edges could be more appropriately extracted. Since the top-hat algorithm extracts local maxima of an image, using the absolute value
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of curvelet subbands in Figs 6e and 6f affect the edge detection results compared
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with Fig 6d. Then, as Figs 6g-h, we used the edge detection algorithms (Sobel, and LoG) on it. All three ACMG, CMG and Sobel methods have well extracted the channel edges, while the ACMG has a higher signal-to-noise ratio.
3.2.
Real data
The real data is a time slice of a 3D seismic volume of southern Iran (Fig. 7a). Fig. 7b depicts the result of preprocessing by the median filter with 5*5 windows. Fig 16
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7c shows the result of edge detection using the adaptive morphological gradient which is noisy. After applying curvelet transform in Fig 7d, channel edges have separated from noise. Figs 7e shows the result of channel edge detection using CMG algorithm. It can be seen that the adaptive SE has been able to better extract the thin part of the channels. Compared to the Sobel, LoG, and similarity methods
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in Figs. 7f-h, it is concluded that the ACMG and CMG methods have been able to
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detect the edges of the channels, while the noise level is lower than that of the
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others.
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4. Quantitative assessment
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We used four criteria to evaluate the proposed method quantitatively. They are the true positive rate (TPR), false positive rate (FPR), true negative rate (TNR) and
,
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,
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accuracy (ACC):
(4)
, (
)
,
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where TP and TN indicate the correctly detected edge and background pixels, respectively. FP illustrates background pixels, mistakenly regarded as edge pixels. FN shows edge pixels, mistakenly recognized as background pixels. To understand how well the proposed algorithm correctly detects channel edges from the background, we converted the image to binary. To do this, global thresholding via
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Otsu method and then morphological thinning was used (Karbalaali et al., 2017a,
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b, and 2018). Figs. 8 and 9 show the binary results of the synthetic and real data,
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respectively. Figs 8a and 9a are the ground truth binary edge images obtained by
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the manual pick. Figs 8b and 9b and Figs 8c and 9c show the result of binarization
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and thinning of ACMG and CMG edge detection algorithm, respectively. Figs. 8d-
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f demonstrate the results of Canny edge detection with threshold [0.06, 0.4], Sobel edge detection with threshold 0.004, and LoG edge detection with threshold 0.0005
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for synthetic data. For Canny, Sobel, and LoG, we used MATLAB edge detection
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function and chose the optimal thresholds by trial and error. Figs. 9d-f show the result of the Canny edge detection with threshold [0.04, 0.22], Sobel, and LoG for real data. Fig. 10 depicts the results of the quantitative measurements between methods. For the synthetic data, in ACMG and CMG, TPR is much more, FPR is less, and TNR and ACC is more than other methods. For the real data, in ACMG and CMG, TPR is slightly more, FPR is less, and TNR and ACC is more than other methods. TPR of all methods is low because our ground truth is not sufficiently 18
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precise and used only to compare the methods. In general, compared with other methods, the ACMG method has yielded acceptable results.
5. Conclusions Mathematical morphology is a powerful tool in image processing which is useful
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in many purposes such as extraction of the object’s boundaries. The curvelet
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transform, as a multi-scale and a multi-directional transformation due to the
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directional properties and high anisotropic characteristics, can be adapted to the
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curved discontinuities in an image. Hence, a combination of the morphological
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filters and the curvelet transform can be used to detect channel edges of seismic
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data. By applying the method to the synthetic and real data it is concluded that the curvelet and morphological gradient algorithm with multi-directional linear
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structuring element can well extract the channel edges, but the adaptive curvelet
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and morphological gradient algorithm with elliptical adaptive structuring element can better detect the thin part of the channels. Besides, in the non-adaptive CMG algorithm, the entire operation must be repeated with each structuring element, i.e., 60 times in this study. The qualitative and quantitative comparisons show that the proposed method can identify channel edges with a higher signal-to-noise ratio than the Canny, Sobel, and LoG edge-detectors can.
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Acknowledgements The authors gratefully thank the constructors of CurveLab for making the code of the Fast Discrete Curvelet Transform (FDCT) available.
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Karbalaali, H., Javaherian, A., Dahlke, S., Torabi, S., 2017b, Channel edge detection using 2D complex shearlet transform: a case study from the South Caspian Sea, Exploration Geophysics, doi.org/10.1071/EG17057, A-I. Karbalaali, H., Javaherian, A., Dahlke, S., Reisenhofer, R., Torabi, S., 2018, Seismic channel edge detection using 3D shearlets—a study on synthetic and real channelised 3D seismic data, Geophysical Prospecting, 66, 1272-1289.
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spectral decomposition in reservoir characterization, The Leading Edge, 18, 353-
Serra, J., 1982, Image Analysis and Mathematical Morphology, Academic Press, London. Serra, J., 1988, Image Analysis and Mathematical Morphology, Vol. 2, Academic Press, New York, NY.
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Wang, Y., Lu, W., Wang, B., Chen, J., and Ding, J., 2017, Extraction of strong
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Wang, R. Q., Zheng, G. J., Fu, H. Z., and Li, Q., 2005, Noise-eliminated method
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by morphologic filtering in seismic data processing, Oil Geophysical Prospecting,
Yarham, C., Boeniger, U., and Herrmann, F.J., 2006, Curvelet-based ground roll removal, Annual International Meeting: SEG, New Orleans, 2777-2782. Yarham, C., Trad, D., and Herrmann, F.J., 2004, Curvelet processing and imaging: adaptive ground roll removal, CSEG National Convention, Great ExplorationCanada and Beyond, 1-4. 26
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Yu, J., Wang, R., Liu, T., Zhang, Zh., Wu, J., Jiang, Y., Sun, L., and Xia, P., 2014, Seismic energy dispersion compensation by multi-scale morphology, Petroleum Science, 11, 376-384. Yu, M., and Yan, Zh., 2011, Flexible surface multiple attenuation using the
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Yuan, H., Hu, Z. D., Liu, Zh., and Ma, J. W., 2018, Ground roll attenuation based
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adaptive multi-scaled morphology for weak signal detection of thin interbedded
Abstracts.
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reservoir, 78th Annual International Conference and Exhibition, EAGE, Extended
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Zhang, H., Yang, H., Li, H., Huang, G., and Ding, Zh., 2018, Random noise
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attenuation of non-uniformly sampled 3D seismic data along two spatial coordinates using non-equispaced curvelet transform, Applied Geophysics, 151, 221-233.
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Fig 1. (a) Probing of a binary image using a 3˟3 square structuring element (shown in blue). The gray and white pixels of the image have 0 and 1 values, respectively. In structuring elements,
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Journal Pre-proof pixels inside of dash line are 0 and pixels inside of the solid line are 1. The square SE in pixel 1 fits the image, so this pixel is part of (b) eroded image. In pixel 2, the square SE hits the image, so this pixel is part of (c) dilated image. In pixel 3, the square SE neither fits nor hits the image, so it does not belong to any eroded and dilated images. (d) Eroded image and (e) dilated image using a linear SE (shown in red) with a length of 10 and angle of 30 degrees. (f), and (g) the opened (erosion followed by dilation) image using square SE and linear SE, respectively. As can
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Fig 2. (a) A simple binary image and its (b) dilation, (c) erosion, (d) opening, (e) top-hat, morphological gradient which has resulted from (f) dilated and original image subtraction, and
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(g) original and eroded image subtraction. The square structuring element has a length of 5
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pixels. The red lines show the original object that superimposed on the results.
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Fig 3. Ellipse parameters a, b and φ, and their relation to eigenvectors e1 and e2 of GST
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Computing morphological gradient using dilation and erosion subtraction
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Applying wrapping based forward curvelet transform
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Calculating absolute value of all subbands
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Computing minimum of each subband and its close-open
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Subtraction of each subband and its previous step minimum
Applying the inverse curvelet transform
2D channel edge map
Fig 4. The flowchart of the adaptive curvelet and morphological gradient algorithm.
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Fig 7. (a) A time slice of a 3D seismic data set containing channels. (b) The result of a median filter. The result of edge detection (c) using the morphological gradient with EASE and (d) after applying curvelet transform. (e, f, and g) The edge detection by the CMG algorithm, the Sobel, and the LoG filter, respectively. (h) The similarity attribute of (a). The boxes in the upper left of figures are missing data. 36
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Fig 9. (a) Ground truth binary image of the real data set resulted from Fig. 7a using the manual
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picking for quantitative assessment. (b, and c) Binarization and thinning of ACMG, and CMG
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edge detection, (d) Canny edge detection with threshold [0.04, 0.22], (e, and f) binarization and
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Journal Pre-proof Highlights
Channels can have reservoir potential; also can be a drilling hazard. We used the curvelet transform and the morphological gradient in channel edge detection. Morphological gradient algorithm detects channel edges. In post processing, curvelet transform is used to improve the signal -to-noise ratio. We used elliptical adaptive structuring element based on local structure tensor.
Channel edge Table
1. Input parameters of the ACMG algorithm for the synthetic and real
Synthetic
Window size of the median filter
5˟5
5˟5
Number of the curvelet scales
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16
16
4
1
σ2 (standard deviation of the Gaussian filter for smoothing LST components)
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3
M (maximum allowable semimajor axis length of the ellipse)
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Number of the curvelet directions in the 2nd scale
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Parameters
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data.
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σ1 (standard deviation of the Gaussian derivative kernel for computing image gradient)
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Real
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Conflicts of Interest
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The authors declare this paper and the data included whether in the form presented or its base format present no conflicts of interest.
44
Bahareh Boustani, Abdolrahim Javaherian, Majid Nabi-Bidhendi, Siyavash Torabi, Hamid Reza Amindavar PII:
S0926-9851(18)30889-9
DOI:
https://doi.org/10.1016/j.jappgeo.2019.103881
Reference:
APPGEO 103881
To appear in:
Journal of Applied Geophysics
Received date:
12 October 2018
Revised date:
24 September 2019
Accepted date:
31 October 2019
Please cite this article as: B. Boustani, A. Javaherian, M. Nabi-Bidhendi, et al., Detection in seismic data using curvelet transform and tensor-based elliptical adaptive structuring elements, Journal of Applied Geophysics(2019), https://doi.org/10.1016/ j.jappgeo.2019.103881
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© 2019 Published by Elsevier.
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Detection in seismic data using curvelet transform and tensorbased elliptical adaptive structuring elements Bahareh Boustani a, Abdolrahim Javaheriana,b,*, Majid Nabi-Bidhendib, Siyavash Torabic, Hamid Reza Amindavard Department of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran
b
DMT Petrlogic, GMbH & Co. KG
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Institute of Geophysics, University of Tehran, Tehran, Iran
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Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran
ABSTRACT
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*Corresponding author: [email protected], [email protected]
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Channels are one of the vital issues in the exploration of oil and gas. They can be
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considered as reservoirs if they are filled with porous and permeable material and placed in a suitable position. A detailed study of the channels can help identify the sedimentation processes of the area, the intensity, and direction of the sea currents. Manual interpretation of the channels is difficult and requires skill. Therefore, in this paper, the adaptive curvelet and morphological gradient algorithm (ACMG) is used for the automatic interpretation of the channels. First, the morphological gradient is applied to extract the edges of the channels; then, the curvelet transform 1
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is used to increase the signal-to-noise ratio. The morphological top-hat operator extracts the local maxima of curvelet sub-bands. In this workflow, we applied an elliptical adaptive structuring element (EASE) based on Gradient structure tensor (GST). For the construction of the 2D GST, the horizontal and vertical gradients of the image were calculated by the first-order Gaussian derivative. The eigenvalue
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decomposition of each structure tensor can provide an estimate of the direction and
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anisotropy rate of the image objects. Hence, parameters of the elliptical structuring
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element are obtained by eigenvalue decomposition of the GST. To evaluate
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ACMG, we compared the final output with the results of the non-adaptive curvelet
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and morphological gradient algorithm (CMG). Also, we compared it with the
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common edge-detectors such as Canny, Sobel, Laplacian of Gaussian, and similarity attribute. The comparison shows that both CMG and ACMG well
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extracted the edge of the channels with the higher signal-to-noise ratio.
Keywords: channel edge; elliptical adaptive structuring element; gradient structure tensor; morphological gradient; curvelet transform; top-hat
1. Introduction
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Channel detection is essential in the petroleum exploration industry. Since, if the channels are filled with the porous and permeable material, they can be a good place to accumulate hydrocarbon. In addition, a detailed study of the channels can help to better know of sedimentation process in the area (Alsouki et al., 2008). Also, channels in stratigraphic studies are important for the development and
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production of oil reservoirs (Mirkamali et al., 2014). A channel in a seismic section
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is considered as an interruption of reflection continuity, but their recognition is
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easier in time or horizon slices of the 3D seismic data set. However, manual
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interpretation of the channels is a tedious task which requires skill and training. So,
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several attributes have been introduced for smoother and more accurate channel
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interpretation (Chopra and Marfurt, 2007). Coherence is an edge-sensitive attribute which is used to mapping channel edges
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(Bahorich and Farmer, 1995). Spectral decomposition complements the coherence
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and edge-based attributes since it is sensitive to channel thickness rather than its width (Partyka et al., 1999). Liu and Marfurt (2007) combined peak frequency and peak amplitude with the coherence to define channel thickness in addition to its width. Most-positive and most-negative curvature are also helpful in channel identification (Chopra and Marfurt, 2008; Lozano and Marfurt, 2008). Kadlec et al. (2008)
used
Gaussian-smoothed
first-order structure tensor and
eigen-
decomposition of the second-order structure tensor to segment channel features in 3
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3D seismic volumes. Al-Dossary (2015) compared running average, wavelet transform, and polynomial fitting noise reduction algorithms before Sobel filter to enhance channel patterns in seismic data. Ghazanfari and Javaherian (2016) combined seismic attributes using the artificial neural network to detect buried channels. Karbalaali et al. (2017a, b and 2018) detected channel edges in 2D and
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3D seismic data using cone-adapted and pyramid-adapted shearlet transform. They
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first maximize shearlet coefficients in each pixel at the finest scale of
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decomposition, then histogram-based thresholding and morphological thinning are
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applied.
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The curvelet transform as a multi-scale and multi-directional transform was used
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in many different seismic application such as ground roll attenuation (Yarham et al., 2004 and 2006; Naghizadeh and Sacchi, 2011; Boustani et al., 2013; Hejazi et
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al., 2013; Yuan et al., 2018), random noise attenuation (Kumar and Herrmann,
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2009; Wang et al., 2010; Cao et al., 2015; Zhang et al., 2018) and multiple attenuation (Herrmann et al., 2008; Yu and Yan, 2011; Nguyen and Dyer, 2016). Mathematical morphological filtering (MMF) in seismic data processing first introduced by Wang et al. (2005). Li et al. (2005) used MMF, i.e., an average of the open-close and close-open filters in random noise attenuation of the seismic trace. They examined the effect of shape, width, and amplitude of the structuring element (SE) on the filter output. Wang et al. (2008) introduced multi-scaled 4
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morphology using self-dilation of a semicircle structuring element. Each morphology scale is obtained by computing the MMF by structuring element of that scale and subtracting the result from the previous stage. He then attenuated the ground roll from seismic data by removing the related morphological scales and reconstruction of others. Duan et al. (2010) and Yu et al. (2014) also utilized the
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multi-scaled morphology for ground roll noise attenuation of seismic data. The
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semicircle SE was used in the temporal direction of every single trace to suppress
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low-frequency noise by the compound top-hat filtering (CTF) (Li et al., 2016).
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MMF with straight-line structuring element was employed for coherent noise
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attenuation by sliding the SE along the noise (spatial) direction (Huang et al.,
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2017). Huang et al. (2018) used multi-scaled MMF for weak signal detection of microseismic data. They used a weighted average of morphological scales to
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reconstruct data. The weights are chosen in the previous work manually (Wang et
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al., 2008; Chen et al., 2015) or adaptively using a varimax norm (Yuan et al. 2016). But they proposed a non-stationary weighting operator which can be obtained by solving an inverse problem in a shaping regularization approach. Huang and Wang (2018) for random noise attenuation generalized MMF into 2D in which SE is a matrix, not a vector. Planar MMF has been more successful in random noise reduction since morphological operators on a single trace do not have enough ability to detect signal from random noise. 5
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In this paper, we applied the adaptive curvelet transform and morphological gradient (ACMG) algorithm to enhance data and detect channel edges to the synthetic and real data sets containing channels. For this purpose, we used the elliptical adaptive structuring element. We compared the results of ACMG with the common edge detection filters in image processing such as Canny, Sobel,
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Laplacian of Gaussian (LoG), and similarity attribute.
Curvelet transform
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2.1.
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2. Methodology
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One of the multi-scale and multi-directional transforms which is proposed to
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compensate shortcoming of the wavelet transform in two and higher dimensions was curvelet transform (Candes and Donoho, 1999). It has high directional and
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anisotropic properties due to the decomposition of the data along with radial
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wedges in the 2D Fourier domain. It is the second generation curvelet transform which is introduced by Candes et al. (2006). To digitize the curvelet transform, they proposed two algorithms, i.e., unequally-spaced fast Fourier transform (USFFT), and wrapping. Because the latter is easier and faster than the first one, we used wrapping-based fast discrete curvelet transform in this study. The source code of the algorithm is available in CurveLab package. For the detailed
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description of the theory and equations of the fast discrete curvelet transform (FDCT) is referred to (Candes et al., 2006).
2.2.
Mathematical morphology
The mathematical morphology was first proposed by Matheron and Serra around
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1964 to investigate the geometric structures of geological samples (Serra, 1982). It
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was introduced by Serra (1982) to image processing. The language of
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mathematical morphology is set theory, since it uses the simple mathematical
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concepts such as intersection and union of sets (Soille, 2004). As a powerful tool, it
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is applicable in many image processing issues such as texture analysis, edge
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detection and noise reduction. Morphological operators analyze the image structures based on their shape and do this with a small template called structuring
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element. A 2D structuring element is a zero matrix whose elements inside a given
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shape are 1. The shape of SE should be selected according to the desired processing. The basic morphological operator is “erosion” and “dilation” and all other operators are expressed in terms of these two. Once probing the image by the structuring element, if structuring element fits the image, the central pixel is a pixel of eroded image, If the structuring element hits the image, the central pixel is a pixel of dilated image and if the structuring element neither fits nor hits the image, the central pixel does not belong to any eroded and dilated images. This operation 7
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is shown schematically in Fig. 1. Indeed, dilation increases and erosion decreases the border pixels. In other words, dilation gives the maximum and erosion gives the minimum of an image intensity values in the neighborhood of the SE. Two basic morphological filters are “opening”, i.e., erosion followed by dilation and “closing”, i.e., dilation followed by erosion. In the grayscale image, opening
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removes the objects which are brighter than the background and smaller than the
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SE. On the contrary, closing removes the objects which are darker than the
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background and smaller than the SE. Sometimes the objects which are removed by
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the opening and closing are not noises, so they can be recovered using “top -hat”
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and “bottom-hat” operators. Top-hat is the difference between the original image
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and its opening and bottom-hat is the difference between the closing and the original image. The edges in the image are areas with high grey level variations.
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Gradient operators can enhance these variations. Morphological gradients are
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operators which can strengthen variations of pixel intensity in SE’s neighborhood. The simplest morphological gradient algorithms are defined in terms of the difference between the results of the dilation, erosion and the original image. Fig. 2 shows some aforementioned morphological operators on a simple binary test image. The success of the mathematical morphology is due to its simple mathematical description and many powerful tools in image analysis which it provides (Soille, 2004). 8
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2.2.1. Adaptive mathematical morphology Traditional mathematical morphology uses a fixed structuring element for the whole image. This method is useful for detecting image features of a certain size and shape. But selecting a proper structuring element for images whose features
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varying in size and shape have difficulty. One solution is to filter these images
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iteratively with SEs in different sizes, and finally, for each point, use the result of
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an SE. But this method takes a lot of time because the whole image should be
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processed many times; however, a small fraction of calculation is required for the
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final result. Thus, the tendency towards adaptive morphology has increased, which
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its origin is the theory introduced by the Serra (1988). One method of defining adaptive morphology is structure-based, i.e., the structuring element aligned to
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edges and contours. This can be done only using considering the orientation, or the
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anisotropy rate, or the distance to the edges. Relation to this approach, Landstrom and Thurley (2013) introduced an elliptical adaptive structuring element based on the gradient structure tensor (GST). The eigenvectors and eigenvalues of the GST include information about the orientation of the image structures (edges) and the anisotropy rate (Landstrom and Thurley, 2013; Curic et al., 2014).
2.2.2. 2D gradient structure tensor 9
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Gradient structure tensor is a well-known characteristic in image processing which is constructed from the smoothed product of partial derivatives of the image. Bakker et al. (1999) introduced an edge-preserving and orientation adaptive filter using gradient structure tensor. In seismology, Bakker (2002) also presented the edge sensitive attribute such as coherency based on gradient structure tensor. If
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(x,y) is the coordinate of a pixel of the image, and f(x,y) is corresponding gray-
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level value. 2D gradient structure tensor (GST), T(x,y), which represents the local
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directional structures in the image is defined by the following 2˟2 matrix
)
[
](
)
(
(
)
(
))
(1)
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(
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(Landstrom and Thuley, 2013; Wang et al., 2017):
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demonstrate partial derivative of the image along the x-axis and y-
axis,
(
) , and
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where
is a Gaussian kernel with standard deviation σ. This
smoothing regularizes the matrix and is important to obtain a stable estimation of feature orientations. Eigenvalue decomposition of GST can estimate the orientation and anisotropy of image structures. So, in the next step for each pixel (x,y), the eigenvalues l1(x,y) and l2(x,y) (l1 > l2), and the corresponding eigenvectors e1(x,y) and e2(x,y) are calculated from the symmetric matrix of GST. These eigenvalues contain information about the image edges (Landstrom and Thuley, 2013): 10
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l1 ≈ l2 >> 0: No dominant direction (edge crossing or point), l1 >> l2 ≈ 0: Strong dominant direction (edge), l1 ≈ l2 ≈ 0: No dominant direction (no edge).
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2.2.3. Elliptical Adaptive structuring element
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Now, we can define a flat elliptical structuring element SE(a,b,φ) for each pixel
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(x,y), so that a and b are the semi-major axis and semi-minor axis of the ellipse and
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φ is the angle between the semi-major axis and x-axis. Fig. 3 shows the ellipse parameters and their relation to the eigenvectors of GST. Hence, the ellipse parameters for each pixel are obtained from the following equations (Landstrom and Thuley, 2013; Wang et al., 2017): (
)
(
)
( (
)
(
)
) (
(
)
) (
(2)
)
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where M is the maximum allowable semi-major axis length which is defined by the user.
is a small value, i.e. machine epsilon. For the all values of the l1(x,y) and
l2(x,y) we have 0 ≤ b(x,y) ≤ a(x,y) ≤ M and a(x,y) + b(x,y) = M. the orientation is obtained from the corresponding eigenvector (Landstrom and Thuley, 2013; Curic
(
(
)
(
)
( ) and
) )
}
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{
)
(3)
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where
)
(
(
) are the components of the eigenvector
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(
(
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et al., 2014):
(
).
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This structuring element changes dynamically from a line of length M, near the
Curvelet and morphological gradient
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2.3.
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dominant edges, to a circle of radius M/2, in isotropic regions in the image.
Gradient operators called edge-detectors because they enhance high-frequency events of the image, which coincide with the edges. In this study, we used a median filter before morphological gradient to smooth the seismic image. Then, we extract the channel edges by the morphological gradient, i.e., the difference between dilation and erosion, using EASE. Finally, the curvelet transform is used in post-processing to increase the signal-to-noise ratio. Channels in time slices of 12
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the 3D seismic data are considered as curvilinear structures with different thicknesses and widths. Therefore, the multi-scale and multi-directional curvelet transform can be a good choice for separating them from noise. The curvelet coefficients which are parallel to the edges have magnitudes much greater than the zero. Hence, we extract the maximum curvelet coefficients in all sub-bands using
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the modified morphological top-hat algorithm (Miri and Mahloojifar, 2011). By
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definition, the top-hat operator results in local maxima of the data. In the curvelet
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domain, we first calculated the absolute value of all curvelet sub-bands. Then, a
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morphological close-open filter is applied, and the minimum of the filter result and
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original sub-band is calculated. To compute modified top-hat, this minimum is
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subtracted from original sub-band for all curvelet sub-bands. Finally, the inverse curvelet transform is applied and the channel edge map is generated. Fig. 4 depicts
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the flowchart of the ACMG edge-detection algorithm. In generating EASE, we
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used Gaussian first-order derivative of the standard deviation σ 1 to calculate the gradient of the image in vertical and horizontal directions. Then, we constructed the components of GST matrix according to Eq (1). A Gaussian filter of standard deviation σ2, the same as σ in Eq (1), is used for smoothing of the GST components. Then, the eigenvalue decomposition is applied to the GST. Finally, the ellipse parameters are calculated according to the Eqs. (2) and (3). So, the only
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parameters that should be defined by the user are σ 1, σ2 and M (maximum allowable semi-major axis length). The step by step proposed method is as follows: a) Applying the median filter. b) Generating EASE based on 2D gradient structure tensor. c) Applying the morphological gradient by the EASE.
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d) Applying wrapping based fast discrete curvelet transform.
e) Applying the modified top-hat algorithm using EASE to the absolute value of all
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curvelet sub-bands.
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f) Applying the inverse fast discrete curvelet transform.
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3. Results and discussion
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To verify the quality of the EASE, in Fig. 5, the results of the closing on a portion of the real data containing channels are presented. Fig. 5a is the original data containing the linear horizontal noise which is the effect of the acquisition footprint. Figs. 5b, c, and d show the results of the closing of 5a by disk SE, multilength and multi-direction linear SE (used in the proposed CMG algorithm), and EASE, respectively. As can be seen, all three structuring elements have removed the noise, but the disk SE also eliminates parts of the channels. The linear SE has not reconstructed the edges of the channels as smoothly as the EASE. 14
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The proposed methodology is developed in the form of a MATLAB code and evaluated on synthetic and real seismic time slices containing channels. The parameters of the ACMG algorithm which is used for the synthetic and real data are presented in Table 1. We determined the optimum values for these parameters by the trial and error. The median filter with a 5*5 window size usually gives the
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best result. The default value is considered for the number of scales and directions
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of curvelet transform because of the negligible effect they have on the filter output.
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The desired output resulted from the M (maximum allowable semi-major axis
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length of the ellipse) value between 6 and 10, which 8 is considered appropriate in
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this study. Outside this range, if a much smaller value is selected for M, the edges will not be extracted, and the noise is stronger. If a much larger value is selected,
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although more noise will be reduced, the edges will be destroyed. If the σ1 value is
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smaller than 1, then the filter does not respond well. The values between 4 and 6
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are proper for σ1. Outside this range, if it is much smaller, the edges will not be well extracted and if it is much larger, the edges will be destroyed. The values between 3 and 5 for σ2 can lead to an acceptable output. Again, a much smaller or larger value of σ2 has the same results as σ1 in edge detection.
3.1.
Synthetic data
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The synthetic data from the convolution of a reflectivity cube with a Ricker wavelet with the peak frequency 15 Hz with the sampling interval of 2 ms and spatial interval of 10 m. The earth model used in the construction of the reflectivity cube consists of a horizontal and dip layering in which a sinusoidal channel is embedded. Fig. 6a shows a time slice of this data set with the signal-to-noise ratio
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(S/N) of 1. To the preprocessed data, we applied a 2D median filter in a 5*5
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window to smooth image before edge detection (Fig. 6b). Figs 6c and d show the
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edge detection results using adaptive morphological gradient and after applying
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modified top-hat on all curvelet subbands. The edge detection results by
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morphological gradient without curvelet transform is very noisy. But after applying
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the curvelet filter, the channel edges could be more appropriately extracted. Since the top-hat algorithm extracts local maxima of an image, using the absolute value
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of curvelet subbands in Figs 6e and 6f affect the edge detection results compared
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with Fig 6d. Then, as Figs 6g-h, we used the edge detection algorithms (Sobel, and LoG) on it. All three ACMG, CMG and Sobel methods have well extracted the channel edges, while the ACMG has a higher signal-to-noise ratio.
3.2.
Real data
The real data is a time slice of a 3D seismic volume of southern Iran (Fig. 7a). Fig. 7b depicts the result of preprocessing by the median filter with 5*5 windows. Fig 16
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7c shows the result of edge detection using the adaptive morphological gradient which is noisy. After applying curvelet transform in Fig 7d, channel edges have separated from noise. Figs 7e shows the result of channel edge detection using CMG algorithm. It can be seen that the adaptive SE has been able to better extract the thin part of the channels. Compared to the Sobel, LoG, and similarity methods
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in Figs. 7f-h, it is concluded that the ACMG and CMG methods have been able to
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detect the edges of the channels, while the noise level is lower than that of the
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others.
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4. Quantitative assessment
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We used four criteria to evaluate the proposed method quantitatively. They are the true positive rate (TPR), false positive rate (FPR), true negative rate (TNR) and
,
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,
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accuracy (ACC):
(4)
, (
)
,
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where TP and TN indicate the correctly detected edge and background pixels, respectively. FP illustrates background pixels, mistakenly regarded as edge pixels. FN shows edge pixels, mistakenly recognized as background pixels. To understand how well the proposed algorithm correctly detects channel edges from the background, we converted the image to binary. To do this, global thresholding via
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Otsu method and then morphological thinning was used (Karbalaali et al., 2017a,
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b, and 2018). Figs. 8 and 9 show the binary results of the synthetic and real data,
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respectively. Figs 8a and 9a are the ground truth binary edge images obtained by
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the manual pick. Figs 8b and 9b and Figs 8c and 9c show the result of binarization
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and thinning of ACMG and CMG edge detection algorithm, respectively. Figs. 8d-
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f demonstrate the results of Canny edge detection with threshold [0.06, 0.4], Sobel edge detection with threshold 0.004, and LoG edge detection with threshold 0.0005
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for synthetic data. For Canny, Sobel, and LoG, we used MATLAB edge detection
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function and chose the optimal thresholds by trial and error. Figs. 9d-f show the result of the Canny edge detection with threshold [0.04, 0.22], Sobel, and LoG for real data. Fig. 10 depicts the results of the quantitative measurements between methods. For the synthetic data, in ACMG and CMG, TPR is much more, FPR is less, and TNR and ACC is more than other methods. For the real data, in ACMG and CMG, TPR is slightly more, FPR is less, and TNR and ACC is more than other methods. TPR of all methods is low because our ground truth is not sufficiently 18
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precise and used only to compare the methods. In general, compared with other methods, the ACMG method has yielded acceptable results.
5. Conclusions Mathematical morphology is a powerful tool in image processing which is useful
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in many purposes such as extraction of the object’s boundaries. The curvelet
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transform, as a multi-scale and a multi-directional transformation due to the
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directional properties and high anisotropic characteristics, can be adapted to the
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curved discontinuities in an image. Hence, a combination of the morphological
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filters and the curvelet transform can be used to detect channel edges of seismic
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data. By applying the method to the synthetic and real data it is concluded that the curvelet and morphological gradient algorithm with multi-directional linear
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structuring element can well extract the channel edges, but the adaptive curvelet
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and morphological gradient algorithm with elliptical adaptive structuring element can better detect the thin part of the channels. Besides, in the non-adaptive CMG algorithm, the entire operation must be repeated with each structuring element, i.e., 60 times in this study. The qualitative and quantitative comparisons show that the proposed method can identify channel edges with a higher signal-to-noise ratio than the Canny, Sobel, and LoG edge-detectors can.
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Acknowledgements The authors gratefully thank the constructors of CurveLab for making the code of the Fast Discrete Curvelet Transform (FDCT) available.
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Fig 1. (a) Probing of a binary image using a 3˟3 square structuring element (shown in blue). The gray and white pixels of the image have 0 and 1 values, respectively. In structuring elements,
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Fig 2. (a) A simple binary image and its (b) dilation, (c) erosion, (d) opening, (e) top-hat, morphological gradient which has resulted from (f) dilated and original image subtraction, and
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(g) original and eroded image subtraction. The square structuring element has a length of 5
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pixels. The red lines show the original object that superimposed on the results.
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Fig 3. Ellipse parameters a, b and φ, and their relation to eigenvectors e1 and e2 of GST
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Calculating 2D structure tensor
Eigenvalue decomposition: l1, l2, e1, e2
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Applying wrapping based forward curvelet transform
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Calculating absolute value of all subbands
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Computing minimum of each subband and its close-open
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Subtraction of each subband and its previous step minimum
Applying the inverse curvelet transform
2D channel edge map
Fig 4. The flowchart of the adaptive curvelet and morphological gradient algorithm.
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(c) multi-length and multi-direction linear structuring element, and (d) elliptical adaptive
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Fig 7. (a) A time slice of a 3D seismic data set containing channels. (b) The result of a median filter. The result of edge detection (c) using the morphological gradient with EASE and (d) after applying curvelet transform. (e, f, and g) The edge detection by the CMG algorithm, the Sobel, and the LoG filter, respectively. (h) The similarity attribute of (a). The boxes in the upper left of figures are missing data. 36
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edge detection, (d) Canny edge detection with threshold [0.04, 0.22], (e, and f) binarization and
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Channels can have reservoir potential; also can be a drilling hazard. We used the curvelet transform and the morphological gradient in channel edge detection. Morphological gradient algorithm detects channel edges. In post processing, curvelet transform is used to improve the signal -to-noise ratio. We used elliptical adaptive structuring element based on local structure tensor.
Channel edge Table
1. Input parameters of the ACMG algorithm for the synthetic and real
Synthetic
Window size of the median filter
5˟5
5˟5
Number of the curvelet scales
6
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7
16
16
4
1
σ2 (standard deviation of the Gaussian filter for smoothing LST components)
3
3
M (maximum allowable semimajor axis length of the ellipse)
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Number of the curvelet directions in the 2nd scale
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Parameters
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σ1 (standard deviation of the Gaussian derivative kernel for computing image gradient)
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Conflicts of Interest
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The authors declare this paper and the data included whether in the form presented or its base format present no conflicts of interest.
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