Human vision system-based structural similarity model for evaluating seismic image quality

The quality of a seismic section is generally evaluated based on relevant qualitative and quantitative criteria. We have developed a new quantitative model to evaluate seismic sections, i.e., a full-reference human visual system based on seismic image structural similarity. With this model, we compared and analyzed two seismic sections before and after data processing in terms of energy intensity, contrast, and seismic reflection configuration similarity measures. We evaluated the quality of the seismic section in combination with the three measures, and their changing trends before and after data processing are represented with relevant quantitative indicators. The numerical simulation used algorithms and actual data, and the results indicate that compared with conventional evaluation methods, our method is easier to understand and simpler to calculate. Our method can also highlight the differences between changing seismic sections and improve the consistency of the objective evaluation results of seismic images with the corresponding human subjective perception.


INTRODUCTION
Vision is an important human ability.The human vision system (HVS) is a complicated information processing system comprising many connected neurons.The HVS can be divided into two parts: (1) the optical imaging system (i.e., eyes) and (2) the visual nervous system (i.e., retina, lateral geniculate body, and visual cortex).The fundamental goal of visual image quality evaluation (Marr, 1982) is to develop computational models that can accurately and automatically perceive image quality, whereas its ultimate goal is to perceive images by substituting the HVS with computers.In simulating the human eyes for evaluation, image-quality evaluation should be based on the consistency of the quantitative measurements with the corresponding human subjective observations.Image-quality evaluation is performed using various techniques; however, HVS models (Ismail et al., 2002) based on the function of the eye have always been research hotspots.The picture quality scale (Makoto et al., 1998) of such models is consistent with visual perception to a great extent, and it sufficiently matches subjective experiment data.Noise quality and distortion measures (Niranjan et al., 2000) are objective quality-evaluation methods for image degradation models that analyze the effects of frequency distortion and additive noise on vision systems.Fuzzy models (Weken et al., 2004) allow the comparison of similarity and consistency between images before and after processing under fuzzy theory and evaluation of image quality using similarity measures.The singular-value decomposition model (Aleksandr et al., 2006), which is highly consistent with subjective perception, can be applied to images of different distortion types and can measure the quality of images of mixed distortion types.The visual signal-to-noise model (Chandler et al., 2007) is a quality-evaluation method for natural images based on wavelet-domain visual signal-to-noise ratio (S/N).This method, which is based on physical brightness and visual angle, can adapt to different visual conditions.The visual information fidelity model (Sheikh et al., 2006) solves image quality evaluation problems in terms of information communication and sharing.The structural similarity model (Wang et al., 2004), from the perspective of image formation, combines structural information in the scene, image brightness, and contrast, with full reference evaluation of image quality.
Peer-reviewed code related to this article can be found at http://software.seg.org/2018/0007.Manuscript received by the Editor 16 April 2016; revised manuscript received 11 January 2018; published ahead of production 09 May 2018; published online 02 August 2018.Li (1994) presents and defines two important concepts: visual resolution and visual S/N.The term "vision" is interpreted as allowing relevant personnel to obtain all useful information on seismic sections.For a high visual resolution, the theory of seismic image enhancement is introduced to enlarge the dynamic range of a selected feature without increasing the inherent information content of the data, thereby allowing easy detection.Accordingly, the enlarged range can also enhance the interpretation, analysis, recognition, and measurement accuracy of subsequent seismic data and avoid multiple solutions for any geophysics or geology interpretation.The current quality of seismic sections after processing is generally evaluated with relevant qualitative and quantitative criteria.Qualitative indicators are often provided for a data processing personnel's subjective evaluation of a section with active seismic waves, significant reflection characteristics, and high-resolution seismic data.Quantitative indicators are often provided to evaluate the quality of a seismic section after processing in terms of bandwidth, S/N, peak S/N, and differential analysis before and after denoising.Frequency bandwidth (Xu et al., 2015) is an important indicator to evaluate whether data processing was successful, but it is easily influenced by aliasing and wavelet sidelobes.The S/N allows the evaluation of seismic signal quality, but many problems and difficulties emerge in its statistical algorithm and scope of application (Zhang, 2011).Differential analysis before and after denoising cannot directly reveal the detailed differences before and after data processing.
We propose a full reference-quality evaluation method based on seismic data structural similarity (SDSS) in this work.Under this method, we evaluated changes in S/N in combination with wave intensity, local amplitude variation, seismic reflection configuration similarity, and consistency of objective evaluation results with the corresponding human subjective perception.Chopra et al. (2006) introduce the texture property of a seismic image.The energy property of a seismic image is a measure of the texture uniformity of the image.Its formula is written as follows:

METHODS AND PRINCIPLES
(1) where P k;q denotes the amplitude and n and m are the number of vertical and horizontal points of the window.The mean energy in the time window is defined by μ x ¼ ðð1∕NÞEnergyÞ 1 2 , where N denotes the total number of sampling points in the time window, reflecting intensity (i.e., brightness), x denotes the seismic image before being processed (the original seismic image), y denotes the seismic image after being processed (the reference seismic image), and x i and y i denote the amplitude at the sampling point.The term μ y is defined similarly to μ x .The variance σ x of amplitude in the time window corresponding to the seismic section x is defined as follows (the definition of σ y is the same as σ x ): (2) where σ x and σ y represent the seismic image contrast with the local variation measure in the time window.
The term σ xy denotes the correlation coefficient in the time window corresponding to the section x and y, which indicates the reflection configuration similarity (3) Then, the seismic energy, contrast, and configuration similarity measures are defined as follows: The function lðx; yÞ is the seismic energy measure in the time window, lðx; yÞ The function cðx; yÞ is the contrast measure in the time window, cðx; yÞ (5) The function sðx; yÞ is the configuration similarity measure in the time window, Figure 1.Flowchart of structural similarity.
where C 1 , C 2 , and C 3 are all small positive numbers that avoid the singularities caused when the denominator is close to or equal to zero.
Mean SDSS (MSDSS) is a measure for evaluating the overall quality of a seismic image: where M denotes the total number of sampling points in the SDSS section.
The flowchart of this algorithm is shown in Figure 1.

THEORETICAL MODEL
The evaluation results of the seismic section obtained with SDSS and MSDSS were verified using the convolution model with stepped faults in addition to random noise, which we will gradually increase.We adopted 20 sets of seismic data with different S/Ns for the SDSS computational analysis, and we drew MSDSS curves (Figure 3) with the S/N defined as S∕N ¼ 10 log 10 ðkx orig k 2 2 ∕kx orig − y ref k 2 2 Þ, where x orig denotes the original seismic data and y ref denotes the reference seismic data.We compared and analyzed 4 of the 20 sets of typical seismic data (Figure 2) in the original, SDSS, and reference sections.The selected wavelet is zero phase with a dominant frequency of 50 Hz, 128 sampling points in the time window, and a time window size of 5 traces × 11 sampling points, where α ¼ 1, β ¼ 1, and γ ¼ 1.
Figure 2a represents the original noise-free seismic data, the S/N of which is an infinite value.The corresponding MSDSS is one, which indicates that the reference and original sections are identical.Figure 2b, 2d, 2f, and 2h depicts noise-added seismic data with S/Ns of 34.4, 16.34, 0.26, and −10.27 dB, respectively.The Figure 2c right figure presents the difference between the reference and original sections.Clearly, the noise source is mainly a randomly interfering noise (the Figure 2e, 2g, and 2i right figures are very similar).From Figure 2c, 2e, 2g, and 2i (MSDSS values of 0.904, 0.434, 0.215, and 0.108, respectively), when S/N is high, the SDSS section shows minor effectsincluding similar waveforms and high SDSS valuesnear the seismic reflection event because the noise energy is lower than that of the seismic reflection event.However, for the other parts affected by noise, the structural similarity significantly changes and is manifested in a small SDSS value.With reduced S/N and enhanced random noise energy, the structural similarity near the seismic reflection event also declines, primarily because of the significant differences between the original and reference sections shown by distortions of the reflected waveform caused by random noise.Figure 3a and 3b shows the S/N curve and MSDSS value curve based on the 20 sets of data, respectively.The S/N decreases exponentially with the gradual increase in random noise energy, and the changing trend of the MSDSS value is consistent with S/N, although its rate of change is slightly lower.Given that the computation complexity of MSDSS (Oðn 2 Þ) is similar to that of the S∕NðOðn 2 ÞÞ and that the intersection between them corresponds linearly (Figure 3c), both can be used as a reference for statistics related to the S/N.

PRACTICAL APPLICATIONS
Example 1 shows a time-domain superimposed section in western China (Figure 4a) with 1296 traces consisting of 3500 sampling points, a 2 ms sampling interval, and an S/N of 2.06 dB.As shown in the noise analysis, the present experiment focused on the coherent noise produced by the near-surface low-frequency surface wave.The "black box" test was performed without considering any specific data processing algorithm.We analyzed the difference between the original and reference sections (Figure 4b; S∕N ¼ 25.55 dB), investigated the denoising effects, and evaluated the quality of the seismic section after processing.Figure 4c shows the difference between the original section and the reference section, and it illustrates that the coherent noise produced by the near-surface low-frequency surface wave within the 0.4-2.0s interval is properly suppressed.The energy of a strong seismic reflection event on the plane of unconformity within the 4.5-4.8s interval is attenuated and the vertical coherent noise is suppressed.Figure 4d presents the SDSS section with an MSDSS of 0.683.Compared with those in Figure 4c, the seismic section exhibits considerable changes before and after processing.In Figure 4c, the SDSS value in the blue and light blue areas is relatively high, and the waveforms before and after denoising only show slight changes.The SDSS value in the red to bright-pink areas gradually decreases, but the waveforms before and after denoising show significant changes.The area with considerable changes within the 0.4-2.0s interval is consistent with the coherent noise produced by the surface waveform.The 2.0-5.0 s interval mainly involves the energy attenuation of a strong lateral seismic reflection event.The 5.0-7.0 s interval mainly involves the deep area affected by coherent noise.The seismic section before and after denoising shows only slight changes and retains useful information on weak deep reflection signals.
Example 2 represents the multioffset seismic section of a region before superposition (Figure 5a), with 2000 traces, 3500 sampling points per trace, and 2 ms sampling.The focus was on coherent noise, in particular from surface waves and multiples.Figure 5c shows the difference between the original and resultant sections: The coherent noise produced by the near-surface low-frequency surface wave within the interval of 0.4-2.0s is properly suppressed, and the energy of the strong seismic reflection event on the plane of unconformity within the interval of 4.5-4.8s is attenuated, whereas the vertical coherent noise is suppressed.Figure 5d represents the SDSS section with MSDSS of 0.837.The SDSS values in the blue and light blue areas are relatively high, and slight changes in the waveform are observed.The SDSS values in the red to bright-pink areas gradually decrease, and the waveform shows significant changes.The areas with significant changes exhibit the morphological characteristics of properly characterized coherent noise.

CONCLUSION
Based on HVS, a full-reference SDSS is proposed in this work.Energy intensity, contrast, and seismic reflection configuration similarity measures are used between two seismic sections before and after data processing.An SDSS model is established, and mean measures are calculated as quantitative indicators for evaluating the quality of seismic section.In the theoretical model, the computation complexity of the MSDSS is similar to S/N, and the intersection between them corresponds linearly; hence, both can be used as a reference for statistics related to the S/N and for evaluating seismic data processing results.The SDSS seismic section shows changing trends before and after processing with quantitative indicators, thus highlighting the differences in the changes in the seismic section and verifying the feasibility of the model.In this study, typical seismic sections before and after superposition are selected for filtering, enhancement, etc. SDSS and MSDSS are also analyzed quantitatively.The results reveal that the consistency of the objective evaluation results with the corresponding human subjective perception can be improved.The algorithm is also applicable in high-dimensional data and in the evaluation of the quality of seismic section to meet actual production needs.