Choice of mother wavelets in CWT spectral decomposition
TL;DR: In this article, the relative value and use of Morlet, Mexican Hat, Derivative of Gaussian (DOG), and Shannon wavelets in the analysis of a fluvial-deltaic system were evaluated.
Abstract: Summary Spectral decomposition carried out with the use of the continuous wavelet transform requires the choice of a mother wavelet, which in turn is used to derive a family of wavelet functions. These wavelet functions are scaled and shifted to ‘fit’ them to the input seismic data traces. Unlike the fixedlength discrete Fourier transform method, the continuous wavelet transform (CWT) window varies with frequency, resulting in better temporal resolution at high frequencies and better frequency resolution. We evaluate the relative value and use of Morlet, Mexican Hat, Derivative of Gaussian (DOG), and the Shannon wavelets in the analysis of a fluvial-deltaic system. Spectral decomposition carried out on two seismic data volumes shows that the Morlet wavelet is more robust and yields better results than the others. While we do not suggest that this conclusion be generalized, we do recommend that this exercise be carried out on a test volume to select the best mother wavelet to be used in the spectral decomposition.
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TL;DR: The interpretation of discrete stratigraphic features on seismic data is limited by its bandwidth and its signal-to-noise ratio as mentioned in this paper, and thus, well-resolved reflections from the top and base of subtle stratigraphy geologic boundaries occur only for thick features imaged by broadband data.
Abstract: The interpretation of discrete stratigraphic features on seismic data is limited by its bandwidth and its signal-to-noise ratio. Unfortunately, well-resolved reflections from the top and base of subtle stratigraphic geologic boundaries occur only for thick features imaged by broadband data. Seismically thin stratigraphic features approaching a quarter-wavelength in thickness give rise to composite, or “tuned,” seismic reflections. Different spectral-decomposition methods provide an effective way of examining the seismic response of stratigraphic geologic features in terms of spectral components and thus help in interpretation. Phase components help with interpretation of the discontinuity features as well as stratigraphic features such as onlap, offlap, and erosional unconformities. Applications of an often overlooked attribute derived during spectral decomposition, called the voice components, can be illustrated in terms of more accurate interpretation of the subsurface features. An “amplitude-f...
45 citations
Cites methods from "Choice of mother wavelets in CWT sp..."
...After choosing an appropriate mother wavelet (Chopra and Marfurt, 2015), the scaled members of the wavelet family are defined by simple scaling and shifting of the mother wavelet....
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TL;DR: In this article, the authors assess the relationship between climatic variability events (El Nino Southern Oscillation, ENSO), land use and shallow lakes' areal variations in the temperate plains of Argentina for the period 2002-2011.
Abstract: Areal variation in lakes is closely associated with other factors such as intensity of land use of the hydrological basins, climatology, geomorphology and depth. The main objective of this study is to assess the relationship between climatic variability events (El Nino Southern Oscillation, ENSO), land use and shallow lakes’ areal variations in the temperate plains of Argentina for the period 2002–2011. The standard precipitation evaporation index (SPEI) was employed in this study in conjunction with the oceanic Nino index (ONI) and in situ annual precipitation data to determine wet, dry and normal years. Shallow lakes’ areal variation was analyzed by satellite imagery (MODIS) processing and the variability of the land use intensity was determined based on agricultural estimations. Frequency in the maximum of shallow lake areas was relatively more dependent on any changes in the rainfall regime (ENSO events) than in the land use (decrease/increase of the cultivated area). The El Nino (2002/3) and La Nina (2007/8) events clearly affected the precipitations of the total studied lake regions, with maximum areas in 2002 and minimum in 2008. The results obtained may be an important contribution to mitigate flooding and droughts in the studied area.
37 citations
TL;DR: The proposed hybrid denoising scheme of wavelet-based higher order correlative stacking (HOCS) in the curvelet domain improves noisy seismic data significantly with respect to both signal-to-noise ratio and fidelity.
Abstract: To whiten random noise and identify coherent noise while preserving the features of seismic events, a hybrid denoising scheme of wavelet-based higher order correlative stacking (HOCS) in the curvelet domain is proposed. The proposed algorithm uses HOCS to isolate the coefficients of seismic events in the curvelet domain. It then removes the noises and recovers signals recorded in noisy environment, without the need to choose an arbitrary threshold; the HOCS method selects a threshold automatically in the curvelet domain. Therefore, with the HOCS, it is possible to capture the features of useful signals with good correlations at all scales and all angles, then to remove the features of coherent noise with disordered correlations. Using interpretive seismic records of karst cavities and hidden sinkhole detections after artificial backfill, we show that the proposed scheme improves noisy seismic data significantly with respect to both signal-to-noise ratio and fidelity. To demonstrate the advantages of this hybrid denoising scheme, a comparison of the performances between different individual denoising methods is investigated for complex seismic records contaminated with different types of noise. Numerical case studies and three field data examples validate the effectiveness of the hybrid denoising scheme proposed in this paper.
28 citations
Cites background from "Choice of mother wavelets in CWT sp..."
...can be expressed in terms of a tensor integral of discrete 1-D wavelets [18], [19]; thus, this kind of 2-D representation cannot be utilized to reconstruct line-shaped edge features [20], [21]....
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TL;DR: The number of volumetric seismic attributes has increased to the point at which interpreters are overwhelmed and cannot analyze all of the information that is a... as mentioned in this paper The number of VOLUME 7, 2019
Abstract: During the past two decades, the number of volumetric seismic attributes has increased to the point at which interpreters are overwhelmed and cannot analyze all of the information that is a...
23 citations
TL;DR: Melbourne, Australia, a city that recently overcame a 12 year "Millennium Drought", is used to evaluate the relative importance of climatic and anthropogenic drivers of urban water demand (using wavelet-based approaches), and the relative contribution of various water saving strategies to demand reduction during the Millennium Drought.
Abstract: Cities in drought prone regions of the world such as South East Australia are faced with escalating water scarcity and security challenges. Here we use 72 years of urban water consumption data from Melbourne, Australia, a city that recently overcame a 12 year "Millennium Drought", to evaluate (1) the relative importance of climatic and anthropogenic drivers of urban water demand (using wavelet-based approaches) and (2) the relative contribution of various water saving strategies to demand reduction during the Millennium Drought. Our analysis points to conservation as a dominant driver of urban water savings (69%), followed by nonrevenue water reduction (e.g., reduced meter error and leaks in the potable distribution system; 29%), and potable substitution with alternative sources like rain or recycled water (3%). Per-capita consumption exhibited both climatic and anthropogenic signatures, with rainfall and temperature explaining approximately 55% of the variance. Anthropogenic controls were also strong (up to 45% variance explained). These controls were nonstationary and frequency-specific, with conservation measures like outdoor water restrictions impacting seasonal water use and technological innovation/changing social norms impacting lower frequency (baseline) use. The above-noted nonstationarity implies that wavelets, which do not assume stationarity, show promise for use in future predictive models of demand.
17 citations
References
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TL;DR: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions, chosen in order to best match the signal structures.
Abstract: The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of Gabor functions a matching pursuit defines an adaptive time-frequency transform. They derive a signal energy distribution in the time-frequency plane, which does not include interference terms, unlike Wigner and Cohen class distributions. A matching pursuit isolates the signal structures that are coherent with respect to a given dictionary. An application to pattern extraction from noisy signals is described. They compare a matching pursuit decomposition with a signal expansion over an optimized wavepacket orthonormal basis, selected with the algorithm of Coifman and Wickerhauser see (IEEE Trans. Informat. Theory, vol. 38, Mar. 1992). >
9,380 citations
"Choice of mother wavelets in CWT sp..." refers background or methods in this paper
...Since then other methods have been introduced, including the continuous wavelet transform (CWT) (Sinha et al., 2005), the S-transform (Stockwell et al., 1996), or the matching pursuit decomposition (Mallat and Zhang, 1993)....
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..., 1996), or the matching pursuit decomposition (Mallat and Zhang, 1993)....
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TL;DR: The S transform is shown to have some desirable characteristics that are absent in the continuous wavelet transform, and provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum.
Abstract: The S transform, which is introduced in the present correspondence, is an extension of the ideas of the continuous wavelet transform (CWT) and is based on a moving and scalable localizing Gaussian window. It is shown to have some desirable characteristics that are absent in the continuous wavelet transform. The S transform is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates.
2,752 citations
"Choice of mother wavelets in CWT sp..." refers background or methods in this paper
...The S-transform method is better than the continuous wavelet transform method, as it yields good temporal and spectral resolution....
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...Since then other methods have been introduced, including the continuous wavelet transform (CWT) (Sinha et al., 2005), the S-transform (Stockwell et al., 1996), or the matching pursuit decomposition (Mallat and Zhang, 1993)....
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..., 2005), the S-transform (Stockwell et al., 1996), or the matching pursuit decomposition (Mallat and Zhang, 1993)....
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Journal Article•
TL;DR: The S transform as discussed by the authors is an extension to the ideas of the Gabor transform and the Wavelet transform, based on a moving and scalable localising Gaussian window and is shown here to have characteristics that are superior to either of the transforms.
Abstract: The S transform, an extension to the ideas of the Gabor transform and the Wavelet transform, is based on a moving and scalable localising Gaussian window and is shown here to have characteristics that are superior to either of the transforms. The S transform is fully convertible both forward and inverse from the time domain to the 2-D frequency translation (time) domain and to the familiar Fourier frequency domain. Parallel to the translation (time) axis, the S transform collapses as the Fourier transform. The amplitude frequency-time spectrum and the phase frequency-time spectrum are both useful in defining local spectral characteristics. The superior properties of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis while the localising scalable Gaussian window dilates and translates. As a result, the phase spectrum is absolute in the sense that it is always referred to the origin of the time axis, the fixed reference point. The real and imaginary spectrum can be localised independently with a resolution in time corresponding to the period of the basis functions in question. Changes in the absolute phase ofa constituent frequency can be followed along the time axis and useful information can be extracted. An analysis of a sum of two oppositely progressing chirp signals provides a spectacular example of the power of the S transform. Other examples of the applications of the Stransform to synthetic as well as real data are provided.
2,323 citations
TL;DR: Spectral decomposition provides a novel means of utilizing seismic data and the discrete Fourier transform (DFT) for imaging and mapping temporal bed thickness and geologic discontinuities over large 3D seismic surveys.
Abstract: Spectral decomposition provides a novel means of utilizing seismic data and the discrete Fourier transform (DFT) for imaging and mapping temporal bed thickness and geologic discontinuities over large 3-D seismic surveys. By transforming the seismic data into the frequency domain via the DFT, the amplitude spectra delineate temporal bed thickness variability while the phase spectra indicate lateral geologic discontinuities. This technology has delineated stratigraphic settings (such as channel sands and structural settings involving complex fault systems) in 3-D surveys.
792 citations
"Choice of mother wavelets in CWT sp..." refers methods in this paper
...(Partyka et al. (1999) and Marfurt and Kirlin (2001) used a fixed length short window discrete Fourier transform (SWDFT)....
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TL;DR: In this paper, the authors presented a new methodology for computing a time-frequency map for nonstationary signals using the continuous-wavelet transform (CWT), which does not require preselecting a window length and does not have a fixed timefrequency resolution over the timefrequency space.
Abstract: This paper presents a new methodology for computing a time-frequency map for nonstationary signals using the continuous-wavelet transform (CWT). The conventional method of producing a time-frequency map using the short time Fourier transform (STFT) limits time-frequency resolution by a predefined window length. In contrast, the CWT method does not require preselecting a window length and does not have a fixed time-frequency resolution over the timefrequency space. CWT uses dilation and translation of a wavelet to produce a time-scale map. A single scale encompasses a frequency band and is inversely proportional to the time support of the dilated wavelet. Previous workers have converted a time-scale map into a time-frequency map by taking the center frequencies of each scale. We transform the time-scale map by taking the Fourier transform of the inverse CWT to produce a time-frequency map. Thus, a time-scale map is converted into a time-frequency map in which the amplitudes of individual frequencies rather than frequency bands are represented. We refer to such a map as the time-frequency CWT (TFCWT). We validate our approach with a nonstationary synthetic example and compare the results with the STFT and a typical CWT spectrum. Two field examples illustrate that the TFCWT potentially can be used to detect frequency shadows caused by hydrocarbons and to identify subtle stratigraphic features for reservoir characterization.
446 citations
"Choice of mother wavelets in CWT sp..." refers methods in this paper
...Since then other methods have been introduced, including the continuous wavelet transform (CWT) (Sinha et al., 2005), the S-transform (Stockwell et al., 1996), or the matching pursuit decomposition (Mallat and Zhang, 1993)....
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...R ed is tr ib ut io n su bj ec t t o SE G li ce ns e or c op yr ig ht ; s ee T er m s of U se a t h ttp :// lib ra ry .s eg .o rg / Figure 3: Stratal slices from 40 Hz frequency volumes for a horizon close to 1200ms showing CWT spectral decomposition carried out using (a) Morlet, (b) Mexican Hat, (c) DOG, and (d) Shannon mother wavelets....
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...One can compute SWDFT and CWT either by convolving a time domain seismic trace with a kernel function in the time domain, or by multiplying the spectrum of the seismic trace with a suite of filter banks (the Fourier transform of the kernel functions) and converting back to time....
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...Our results show that the Morlet wavelet exhibits more robust results in the CWT approach as compared with the other mother wavelets....
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...The choice of mother wavelets in the CWT method for spectral decomposition is important....
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