Time-varying Wavelet Research Articles

Do oil and natural gas prices affect carbon efficiency? Daily evidence from China by wavelet transform-based approaches

This study analyzes the impacts of oil and natural gas prices on China's carbon efficiency using wavelet transform-based time series methods. To this end, the study applies the continuous wavelet transform (CWT), wavelet transform coherence (WTC), and a time-varying wavelet causality test (TVWCT) to daily data during the period from October 31, 2012 to November 3, 2022. The CWT results show that there are volatilities in oil prices due to the coronavirus 2019 (COVID-19) pandemic, in natural gas prices because of the Russia-Ukraine war, and in carbon efficiency resulting from the Paris Climate Agreement. The WTC results demonstrate that increasing oil prices promote carbon efficiency in the 2014–2016 and 2018–2020 periods, while natural gas prices have a weak impact on carbon efficiency in 2022. Moreover, the TVWCT results prove that there is a strong causality leading from oil and natural gas prices to carbon efficiency in the long run. These results highlight that China can use volatilities in oil and natural gas prices as a policy tool to enhance carbon efficiency.

Environmental reverberations of geopolitical risk and economic policy uncertainty resulting from the Russia-Ukraine conflict: A wavelet based approach for sectoral CO2 emissions

After the COVID-19 pandemic, Russia invaded Ukraine in February 2022, and a natural gas crisis between the European Union (EU) and Russia has begun. These events have negatively affected humanity and resulted in economic and environmental consequences. Against this background, this study examines the impact of geopolitical risk (GPR) and economic policy uncertainty (EPU) caused by the Russia-Ukraine conflict, on sectoral carbon dioxide (CO2) emissions. To this end, the study analyzes data from January 1997 to October 2022 by using wavelet transform coherence (WTC) and time-varying wavelet causality test (TVWCT) approaches. The WTC results show that GPR and EPU reduce CO2 emissions in the residential, commercial, industrial, and electricity sectors, while GPR increases CO2 emissions in the transportation sector during the period from January 2019 to October 2022, which includes Russia-Ukraine conflict. The WTC analysis also indicates that the reduction in CO2 emissions provided by the EPU is higher than that of the GPR for several periods. According to the TVWCT, there are causal impacts of the GPR and the EPU on sectoral CO2 emissions, but the timing of the causal impacts differs between the raw and decomposed data. The results suggest that the EPU has a larger impact on reducing sectoral CO2 emissions during the Ukraine-Russia crisis and that production disruptions due to uncertainty have the greatest impact on reducing CO2 emissions in the electric power and transportation sectors.

Nonstationary seismic–well tying with time-varying wavelets

Seismic–well tying is an important technique for correlating well-logging curves in depth with seismic traces in time. An appropriate seismic–well tying technique must account for two types of nonstationarity: the nonstationary time errors in the synthetic seismic trace caused by the inaccurate time–depth relationship established based on sonic-logging velocity and the nonstationary seismic signals due to the time-varying wavelets during wave propagation. The nonstationary problems related to the time–depth relationship and the time-varying wavelets are interrelated in seismic-well tying procedure. We implemented a nonstationary seismic–well tying method by iteratively updating the time-depth relationship and estimating the time-varying wavelets. From the estimated time-varying wavelets, we also estimated a Q-value by assuming that the subsurface medium has a constant Q at depth and used the constant Q to constrain the variation of the seismic wavelet during propagation. Then, we used the improved time–depth relationship and time-varying wavelets with the Q constraint for further iterations. In the iterative procedure, we quantified the accuracy of the seismic–well tying result using the correlation coefficient between the synthetic and the true seismic trace in each iteration and evaluated the reliability using the normalized mean-square errors among the wavelets estimated in different iterations.

An Improved Bat Algorithm With Time-Varying Wavelet Perturbations for Cloud Computing Resources Scheduling

Resources scheduling is a major challenge in cloud computing because of its ability to provide many on-demand information technology services according to needs of customers. In order to acquire the best balance between speed of operation, average response time, and integrated system utilization in the resource allocation process in cloud computing, an improved bat algorithm with time-varying wavelet perturbations was proposed. The algorithm provided a perturbation strategy of time-varying Morlet wavelet with the waving property to prevent from local optimum greatly and improve the converging speed and accuracy through the guide of individual distribution to control diversity and time-varying coefficient of wavelets. The experiments showed the proposed could significantly upgrade the overall performance and the capability of resource scheduling in cloud service compared to similar algorithms.

Spectral extrapolation principles and application: Mindoro Island, Philippines, seismic data

Spectral extrapolation is a bandwidth extension technique that we implement by combining spectral inversion with constraints, time-variant wavelet extraction, and targeted broadband filtering. We explain the principles of spectral extrapolation as a valid and effective bandwidth extension method and demonstrate its application to a 2D onshore Philippines legacy seismic data set using a time-variant wavelet extraction, resulting in a tripling of the frequency range of the spectrum. The results indicate significant potential for mapping complex stratigraphy and geomorphological features not evident on the input seismic data images, yielding information about reservoir distribution and connectivity that is often critical for optimal well placement.

Nonstationary seismic reflectivity inversion based on prior-engaged semisupervised deep learning method

Reflectivity inversion methods based on a stationary convolution model are essential for seismic data processing. They compress the seismic wavelet, and by broadening the bandwidth of seismic data, they assist the interpretation of seismic sections. Unfortunately, they do not apply to realistic nonstationary deconvolution cases in which the seismic wavelet varies as it propagates in the subsurface. Deep learning techniques have been proposed to solve inverse problems in which networks can behave as the regularizer of the inverse problem. Our goal is to adopt a semisupervised deep learning approach to invert reflectivity when the propagating wavelet is considered unknown and time-variant. To this end, we have designed a prior-engaged neural network by unrolling an alternating iterative optimization algorithm, in which convolutional neural networks are used to solve two subproblems. One is to invert the reflectivity, and the other is to estimate the time-varying wavelets. In general, it is well known that, when working with geophysical inverse problems such as ours, one has limited access to labeled data for training the network. We circumvent the problem by training the network via a data-consistency cost function in which seismic traces are honored. Reflectivity estimates also are honored at spatial coordinates in which true reflectivity series derived from borehole data are available. The cost function also penalizes time-varying wavelets from varying abruptly along the spatial direction. Experiments are conducted to find the effectiveness of our method. We also compare our approach to a nonstationary blind deconvolution algorithm based on regularized inversion. Our findings reveal that the developed method improves the vertical resolution of seismic sections with noticeable correlation coefficient improvements over the nonstationary blind deconvolution. In addition, our method is less sensitive to initial estimates of nonstationary wavelets. Moreover, it needs less human intervention when setting parameters than regularized inversion.

Examining the asymmetries between equity and commodity ETFs during COVID-19

Previous studies have neither examined the volatility co-movements across stock and commodity markets in terms of both time and frequency nor differentiated between bad and good volatility and the potential asymmetric effect. To address this gap, we computed 5-min price data, the positive and negative semivariances on five leading Exchange Traded Funds (ETFs) covering the US equity market, crude oil, natural gas, gold, and silver markets from January 2, 2019 to May 29, 2020, and then draw on the wavelet coherency methodology and the time-varying wavelet coherence measure. The results show that the negative realized volatility co-movements are stronger during the COVID-19 outbreak, especially at short and medium frequencies. The US stock market leads energy and precious metals in the short-run frequency. However, over the long-run, the lead-lag pattern mostly alternates over time for all cases. Notably, the realized volatilities of US equities and precious metals are shaped by the COVID-19 outbreak, reflecting the quest of investors for protection from the market volatilities by investing in gold and silver. This latest finding is confirmed by the wavelet coherence measure. Further results show asymmetric co-movements emanating especially from realized negative semivariance of equities and energy markets around the pandemic outbreak across short-time horizons. We also notice that the COVID-19 outbreak increases the procyclical movement of all ETFs in the short-term. The effect is more pronounced among the US equity and precious metals markets, whereas no significant countercyclical connectedness is observed among these markets. Our findings support previous evidence showing that gold and silver can serve as safe-haven assets due to their low coherence with risky assets.

Improved generalized S-transform deconvolution for non-stationary seismic data

Improving the vertical resolution is one of the significant tasks for seismic data processing. Most traditional resolution-enhancement techniques assume that the seismic wavelet is time-invariant. However, the seismic wavelet varies with seismic wave propagation in the subsurface. To solve this issue, a new spectral-modeling method is proposed to extract the time-varying wavelet using improved generalized S-transform (IGST) and higher-order Fourier series. The IGST based on modified time-window function can effectively improve the resolution of the time-frequency (t-f) spectrum. The high-order Fourier series is used to fit on the logarithm t-f spectrum and achieve the high-precision time-varying wavelet. The proposed method is composed of four steps in the implementation. Firstly, the seismic data is decomposed by the IGST and converted to the logarithm t-f domain. Secondly, the time-varying wavelet spectrum is modeled at each time sample using a higher-order Fourier series. Thirdly, the boxcar smoothing method is used to smooth the time-varying wavelet spectrum and extract the time-varying wavelet with Hilbert transform. Finally, using the time-varying wavelet spectrum to spectrally balance seismic data to flatten the seismic response. Synthetic and field data examples demonstrate the feasibility of the proposed method in improving the signal-to-noise ratio and enhancing the vertical resolution.

Machine learning-based vertical resolution enhancement considering the seismic attenuation

The vertical resolution of seismic data is important for the interpretation of geologic features at a fine scale. To improve the resolution of seismic data, spectral enhancement has progressed mainly due to one-dimensional (1D) convolutional models and deconvolution filters; however, recent studies have applied machine learning (ML) algorithms. To generate successful outcome using ML techniques, it is important to reflect the features of target data in the training stage of the ML model. One of main features of seismic field data is a non-stationary wavelet over time due to the change of frequency content as the wavelet passes through specific regions, such as a gas reservoir. However, since conventional methods are generally based on a 1D convolutional model, which assumes that the propagating wavelet is stationary, they show low performance for the parts where the frequency content of the wavelet is significantly changed. In this study, we developed a spectral enhancement algorithm using ML techniques, which can be applied to a seismic trace with different frequency contents over time. We showed that sparse spike inversion results of all target data, rather than just well log data, can be useful to obtain information on the reflectivity series for generating a training data set. To verify the performance of the new algorithm, we compared the spectrally enhanced results from the ML model trained by stationary wavelets and time-variant wavelets and confirmed that the latter wavelets provided better results. In addition, we proposed a quality control (QC) method for verifying the spectrally enhanced results. • We developed a spectral enhancement ML algorithm consider the seismic attenuation. • The ML model trained by time-variant wavelets yielded better results for the area with seismic attenuation. • The results obtained by the ML were verified the validity by our proposed QC method only using seismic data.

Multichannel adaptive deconvolution based on streaming prediction-error filter

Abstract Deconvolution mainly improves the resolution of seismic data by compressing seismic wavelets, which is of great significance in high-resolution processing of seismic data. Prediction-error filtering/least-square inverse filtering is widely used in seismic deconvolution and usually assumes that seismic data is stationary. Affected by factors such as earth filtering, actual seismic wavelets are time- and space-varying. Adaptive prediction-error filters are designed to effectively characterise the nonstationarity of seismic data by using iterative methods, however, it leads to problems such as slow calculation speed and high memory cost when dealing with large-scale data. We have proposed an adaptive deconvolution method based on a streaming prediction-error filter. Instead of using slow iterations, mathematical underdetermined problems with the new local smoothness constraints are analytically solved to predict time-varying seismic wavelets. To avoid the discontinuity of deconvolution results along the space axis, both time and space constraints are used to implement multichannel adaptive deconvolution. Meanwhile, we define the parameter of the time-varying prediction step that keeps the relative amplitude relationship among different reflections. The new deconvolution improves the resolution along the time direction while reducing the computational costs by a streaming computation, which is suitable for handling nonstationary large-scale data. Synthetic model and field data tests show that the proposed method can effectively improve the resolution of nonstationary seismic data, while maintaining the lateral continuity of seismic events. Furthermore, the relative amplitude relationship of different reflections is reasonably preserved.

Nonstationary seismic inversion: joint estimation for acoustic impedance, attenuation factor and source wavelet

Seismic signal can be expressed by nonstationary convolution model (NCM) which integrates acoustic impedance (AI), attenuation factor (AF) and source wavelet (SW) into a single formula. Although it provides attractive potential to invert AI, AF and SW, simultaneously, effective joint inversion algorithm has not been developed because of the extreme instability of this nonlinear inverse problem. In this paper, we propose an alternating optimization scheme to achieve this nonlinear joint inversion. Our algorithm repeatedly alternates among three subproblems corresponding to AI, AF and SW recovery until changes in inverted models become smaller than the user-defined tolerances. Also, when we optimize one parameter, other two parameters are fixed. NCM is an explicit linear formula for AI; therefore, AI recovery is accomplished by linear inversion which is regularized by low-frequency model and isotropy total variation domain sparse constraints. However, NCM is a complicated nonlinear formula for AF. To facilitate the AF inversion, we propose a centroid frequency-based attenuation tomography method whose forward operator and observations are acquired from the time-varying wavelet amplitude spectra which is estimated according to Gabor domain factorization of NCM. SW is decoupled from NCM based on Toeplitz structure constraint, and we obtain an orthogonal wavelet transform domain sparse regularized SW inverse subproblem. Split Bregman technique is adopted to optimize AI and SW inverse subproblems. Numerical test and field data application confirm that the proposed nonstationary seismic inversion algorithm can stably generate accurate estimates of AI, AF and SW, simultaneously.

A data-driven method for time-varying wavelet extraction based on the local frequency spectrum

Seismic wavelet extraction always plays a central role in high-resolution seismic processing. Conventional methods assume that seismic data are stationary when a constant wavelet is considered, which ignores the time-varying characteristics of seismic wavelets. In reality, seismic data are nonstationary because of attenuation, scattering, and other physical processes during propagation, which means that the frequency spectrum of seismic signal changes from shallow to deep formations. We have developed a time-varying wavelet extraction method by using a highly energy-concentrated time-frequency representation technique. Time-varying wavelets are generated according to the local frequency spectrum at every instant. In addition, because the estimations of parameters for wavelet extraction are fully data-driven, the results of the proposed method are more accurate and suitable for the nonstationary nature of actual seismic data. Synthetic tests indicate the reliability and robustness of the proposed method, even under noise contamination. By applying the time-varying wavelet extracted using the proposed method to seismic inversion on a field data example, we obtain the deconvolution result with improved resolution and a better fit to the well-log reflectivity compared to that by using conventional wavelet extraction methods.

Time-varying wavelet estimation and deconvolution for nonstationary data based on a FWE function

• We present a nonstationary convolution model constituted of the reflectivity series and a time-varying wavelet matrix. • We propose a novel wavelet estimation method toextract time-varying wavelet at every instant for nonstationary seismic data . • We achieve blind deconvolution without assumptions about factor Q value and the source wavelet.

Correlation between mars’ semidiurnal LOD amplitude variation and atmospheric dust cycle index

The Mars’ rotational rate, or equivalently its length-of-day (LOD), varies slightly due to global atmospheric changes. And the atmospheric dust cycle is considered to be one key process impacting the Mars’ atmosphere. Lewis and Barker (2005) demonstrated the close link of the semidiurnal tidal amplitude with the atmospheric dust content. In consideration of tidal effects in the Mars’ variable rotation, possible connection may exist between the Mars’ semidiurnal LOD amplitude variation and atmospheric dust cycles. In this study, the Mars’ LOD variations which are influenced by global atmosphere, and the atmospheric dust cycle index (ADCI) are computed based on eight realistic scenarios of the Martian Years (MY) 24–31 of the Martian Climate Database. The semidiurnal LOD amplitude change and the ADCI are found to be correlated at the 99% significance level, and they present similar time-varying wavelet spectral structures. In mature stages of global dust storms in MYs 25 and 28, relatively high LOD and ADCI values occurred synchronously across the time scales, ranging from a few months’ high frequency to seasonal and over one year’s lower frequency band. The close relation between the semidiurnal LOD amplitude variation and the ADCI reflects the strong coupling between the solid Mars and surficial atmosphere system, which relates primarily to the semidiurnal atmospheric tide.

Multi-trace nonstationary sparse inversion with structural constraints

The recorded seismic signals are attenuated and spatially correlated due to their propagation through an elastic earth and the sedimentary rule of strata. This attenuation phenomenon is quantified by means of the earth quality factor (Q) or the attenuation factor (1∕Q). Nowadays, the related Q-compensation and multi-trace inversion for the seismic data are two challenging problems when used for enhancing the temporal resolution and preserving the spatial continuity. Separately estimating Q and reflectivity are difficult and produce the uncertainty or ill-condition problems. To overcome these limitations, we have developed a multi-trace nonstationary sparse inversion with structural constraint. Using prior dipping-angle information and reflectivity sparsity property, the proposed method simultaneously estimates equivalent-Q and reflectivity with structural constraint. Constructed by the source wavelet and different scanned equivalent-Q, a series of time-varying (nonstationary) wavelet matrices are provided for the forward-modeling schemes and the corresponding inversions. When the Q-model is infinitely close to the true attenuation mechanism, the corresponding inverted reflectivity is comparatively sparse and quantified as maximum sparsity or minimum sparse representation. A sparse representation function, such as l0.1-norm, is used for sparsity measurement of the inverted reflectivity corresponding to each scanned Q. Through optimizing these sparse representation values, a suitable equivalent-Q, as well as the corresponding inverted reflectivity with structural preservation and Q-attenuation, is determined. The synthetic and field examples both confirmed a substantial improvement on seismic records, especially for Q-estimation, structure preservation and Q-compensation.

Combining CEEMD and recursive least square for the extraction of time-varying seismic wavelets

The vertical resolution of seismic data is greatly affected by time-varying seismic wavelets. To break through the shape restriction in amplitude estimation and the dependency upon the constant or piecewise stationary assumptions in phase estimation in the existing methods, we propose a time-varying wavelet extraction method combining CEEMD (Complementary Ensemble Empirical Mode Decomposition) and RLS (Recursive Least Square). According to the differences that the amplitude spectra of wavelets and reflection coefficient are smooth and oscillating respectively, the logarithm amplitude spectra of the seismogram at different times are decomposed into multi-layer components with different oscillation scales by CEEMD, and the amplitude spectra of time-varying wavelets can be estimated by filtering the oscillating components and reconstructing the smooth components. Hilbert transform is applied to the estimated amplitude spectra to build time-varying minimum-phase wavelets, and the system functions and pole-zeros in Z domain of minimum-phase wavelets can be estimated using RLS. Finally, time-varying mixed-phase wavelets are extracted by reconfiguring the pole-zeros and determining the optimal combinations under the constraint of local similarity. This method does not need to calculate the Q value. The numerical simulation and real seismic data processing results demonstrate that the proposed method can improve the accuracy of time-varying wavelet extraction compared to the conventional method.

Multi-Ricker Spectral Modeling in the S-transform Domain for Enhancing Vertical Resolution of Seismic Reflection Data

DOI: 10.17014/ijog.6.3.223-233We present a relatively straightforward methodology for extending seismic bandwidth and hence enhancing the seismic resolution by performing time-variant deconvolution. We use the generalized S-transform (GST) approach in order to properly compute the time-frequency components of the seismic reflection trace. In estimating the time-variant wavelet, we propose a spectral modeling method named multi-Ricker spectral approximation (MRA). After obtaining the estimated wavelet spectrum at each time sample, a deconvolution filter can then be built and applied in the S-transform domain. This proposed time-variant seismic enhancement method needs neither information on subsurface attenuation model nor assumption that the subsurface reflectivity is random. It is a data-driven methodology which is based on the seismic data only. We validate this proposed method on a synthetic and apply to a field data. Results show that, after enhancement, overall seismic bandwidth can be extended resulting in higher vertical resolution. Correlation with VSP corridor stack at well location ensures that the generated reflection details after enhancement is geologically plausible.

Multi-Ricker Spectral Modeling in the S-transform Domain for Enhancing Vertical Resolution of Seismic Reflection Data

DOI: 10.17014/ijog.6.3.223-233 We present a relatively straightforward methodology for extending seismic bandwidth and hence enhancing the seismic resolution by performing time-variant deconvolution. We use the generalized S-transform (GST) approach in order to properly compute the time-frequency components of the seismic reflection trace. In estimating the time-variant wavelet, we propose a spectral modeling method named multi-Ricker spectral approximation (MRA). After obtaining the estimated wavelet spectrum at each time sample, a deconvolution filter can then be built and applied in the S-transform domain. This proposed time-variant seismic enhancement method needs neither information on subsurface attenuation model nor assumption that the subsurface reflectivity is random. It is a data-driven methodology which is based on the seismic data only. We validate this proposed method on a synthetic and apply to a field data. Results show that, after enhancement, overall seismic bandwidth can be extended resulting in higher vertical resolution. Correlation with VSP corridor stack at well location ensures that the generated reflection details after enhancement is geologically plausible.

Time-varying wavelet estimation and its applications in deconvolution and seismic inversion

Wavelet holds an essential role in seismic data processing and characterization, for examples deconvolution and seismic inversion. Unfortunately, wavelet is an unknown data. Several existing methods attempt to estimate and extract the wavelet from seismic data. However, the methods give only a single wavelet from one seismic trace. When seismic data are non-stationer, single wavelet usage will cause a problem, that is raising the error. This paper proposes a time-varying wavelet estimation method to accommodate this problem. It uses matrix diagonalization to estimate a set of wavelets. Next, the time-varying wavelet is applied to deconvolution and seismic inversion. The experiment shows that time-varying wavelet improves the results in both deconvolution and seismic inversion. The errors decreased and spectrum bandwidth broadened.

Multitrace Semiblind Nonstationary Deconvolution

We proposed a multitrace semiblind nonstationary deconvolution method. The proposed method estimates reflectivity and source wavelet simultaneously for pursuing high-resolution seismic processing. The mathematical framework is derived based on convolution exchange law and Fourier transform property. In this framework, seismic records are treated as the convolution of a time-varying wavelet and nonattenuated reflectivity or the convolution of a constant wavelet and attenuated reflectivity. Using these two equivalence relations, we devise an objective function containing two variables, the reflectivity and wavelet. In addition, we add the 2-D total variation constraint to the cost function, which preserves lateral and vertical continuity of the estimated reflectivity. The cost function is solved by alternating iteration and proximal splitting methods, under the assumptions of a known attenuation model and sparse reflectivity. In addition, the mathematical framework is extended to implement semiblind deconvolution in an approximate layered earth model. To demonstrate the effectiveness of the proposed method, we apply the proposed method to synthetic data and field data and confirm that the proposed method can achieve better reflectivity and source wavelet.