Cauchy Wavelet Research Articles

Phase retrieval of bandlimited functions for the wavelet transform

We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.

Limits of Detection for EXAFS Characterization of Heterogeneous Single-Atom Catalysts

Single-atom catalysts (SACs), consisting of individual metal atoms dispersed on a support, attract attention due to their unique reactivity, efficient use of precious metals, and precise chemical tunability. Characterization of the metal species is crucial to substantiate structure–function relationships. Authors often use─and referees often require─X-ray absorption spectroscopy (XAS) data to prove the absence of clustered metal (or metal oxide) structures after pre-treatment and under in situ or operando conditions. However, there has been no critical assessment of the limitations of XAS in substantiating such conclusive statements, which is particularly important given the potential outsized influence of minority catalyst structures in dictating catalytic activity. In this article, we quantitatively assess the detection limits of XAS to identify metal (or metal oxide) clusters in samples containing predominantly single atoms by modeling the extended X-ray absorption fine structure (EXAFS) of mixtures of structures. We identified that a significant fraction of clusters can coexist with SAC active sites (e.g., ∼10% metallic Pt or ∼40% oxidized Pt clusters in Pt/CeO2 SACs), while eluding detection via EXAFS with any statistical significance. To generalize these conclusions, a descriptor-based screening of bulk metal oxides using a continuous Cauchy wavelet transform was proposed that suggests certain materials for which differentiating atomically dispersed metal species and metal oxide clusters would be infeasible by EXAFS (e.g., ReOx). Based on this analysis, we suggest best practices for the study of SACs using EXAFS and provide recommendations to ensure that conclusions do not outpace the evidence used to support them. In this rapidly expanding research area, rigorous characterization will lead to greater understanding of the behavior of SACs and ultimately improved catalytic materials.

Strategy to Probe the Local Atomic Structure of Luminescent Rare Earth Complexes by X-ray Absorption Near-Edge Spectroscopy Simulation Using a Machine Learning-Based PyFitIt Approach

Rare earth(III) β-diketonates are highly remarkable luminophores in the visible spectral region among the rare earth compounds, owing to the efficient contribution from the 4f-4f intraconfigurational transitions. To get detailed structural insight into the RE3+ sites (RE = Eu, Gd, and Sm), X-ray absorption near-edge spectroscopy (XANES) can be very potent in probing the local chemical environment around the RE3+ ion. In this work, a PyFitIt machine learning approach was employed as a new strategy to simulate the Eu, Gd, and Sm L3-edge XANES and thereby determine the local atomic structure of the luminescence RE3+ β-diketonate complexes, [Eu(tta)3(H2O)2], [C4mim][Eu(dbm)4], [Gd(tta)3(H2O)2], and [Sm(dbm)3(phen)] (tta, 3-thenoyltrifluoroacetonate; dbm, dibenzoylmethane; phen, phenanthroline; and C4mim, 1-butyl-3-methylimidazolium bromide). Continuous Cauchy wavelet transform validated the PyFitIt calculated XANES by visualizing very efficiently the coordination geometries, composed of O and O/N backscatterers around the RE3+ (RE = Eu and Gd) and Sm3+ ions, respectively, as a pinkish-red color map in the two-dimensional images of the corresponding complexes. Extended X-ray absorption fine structure fit in Artemis also corroborated the three-dimensional structures generated by PyFitIt XANES simulation for all the compounds. Though, relatively slightly higher bond distance values for the Sm3+ complex are due to the higher atomic radius of the Sm3+ ion when compared to the Eu3+ and Gd3+ complexes. Meanwhile, higher Debye-Waller factor (σ2) values for the [C4mim][Eu(dbm)4] when compared to the [Eu(tta)3(H2O)2] indicated the structure disorder, owing to the distortion in the local geometry. It is noteworthy that the optical properties, described mainly by the Ωλ (λ = 2 and 4) 4f-4f intensity parameters, are very sensitive to the local coordination environment around the Eu3+ ion. Thus, a close agreement between the experimental and theoretically calculated Ωλ parameter values confirmed that the PyFitIt calculated square antiprismatic structures are precisely similar to the real structures of the Eu3+ complexes.

A fractal uncertainty principle for Bergman spaces and analytic wavelets

Motivated by results of Dyatlov on Fourier uncertainty principles for Cantor sets and by similar results of Knutsen for joint time-frequency representations (i.e., the short-time Fourier transform (STFT) with a Gaussian window, equivalent to Fock spaces), we suggest a general setting relating localization and uncertainty and prove, within this context, an uncertainty principle for Cantor sets in Bergman spaces on the unit disk, where the Cantor set is defined as a union of annuli that are equidistributed in the hyperbolic measure. The result can be written in terms of analytic Cauchy wavelets. As in the case of the STFT considered by Knutsen, our result consists of a two-sided bound for the norm of a localization operator involving the fractal dimension log⁡2/log⁡3 in the exponent. As in the STFT case and in Dyatlov's fractal uncertainty principle, the (hyperbolic) measure of the dilated iterates of the Cantor set in the disk tends to infinity, while the corresponding norm of the localization operator tends to zero.

CWT-Based Method for Extracting Seismic Velocity Dispersion

Time–frequency (T–F) analysis has powerful applications in various fields such as geoscience and engineering. Seismograms are distorted by dispersion and attenuation in such a way that they alter their amplitude and phase spectra. Velocity dispersion is usually neglected in the conventional seismic data processing. However, it has a severe effect on seismic data processing if dispersion is intense in high-attenuation media. Most of the studies on dispersion are theoretical models and laboratory measurements, whereas velocity dispersion analysis directly from field data is rare. In this letter, a method is proposed to estimate the dispersion of vertical seismic profile (VSP) data based on continuous wavelet transform (CWT) in time–frequency domain. The Morlet wavelet, three-parameter wavelet (TPW), and Cauchy wavelet are chosen and compared as mother wavelets in CWT to determine the variations of velocity with frequency. Then, the detailed process of extracting velocity dispersion of propagating waves in a dissipative medium is proposed by using their T–F spectra. Finally, the synthetic and field VSP data are used to demonstrate the validity of the proposed method. The results indicate that the velocity dispersion in synthetic data estimated by TPW and Cauchy wavelet generally match well with the theoretical values while it is not the case for the Morlet wavelet. The velocity dispersion of field data is weak and the estimated velocity basically matches well with the logging data.

Wavelet‐based approach of time series model for modal identification of a bridge with incomplete input

AbstractIdentification of modal parameters of a bridge from its earthquake responses is crucial for performing damage assessment of the structure. However, all the input base excitations of the bridge may not be measured because of economic concerns and sensor malfunctions. Consequently, evaluating the modal parameters of a bridge under the consideration of incomplete input measurements is a challenging and important task. An approach that combines the continuous Cauchy wavelet transform with an autoregressive time‐varying moving average with exogenous input (AR‐TVMA‐X) model is proposed in this study to identify the modal parameters of a multispan bridge under multiple support earthquake excitations with incomplete measurements. The efficiency and efficacy of the proposed approach are first validated using numerically simulated responses of a three‐span continuous beam subjected to multiple support nonstationary excitations. A standard procedure of using the proposed approach to identify the modal parameters is established according to comprehensive studies on the effects of noise in the data, the number of supports whose excitations are used in the AR‐TVMA‐X model, and the orders of the AR‐TVMA‐X model on the accuracy of identifying the modal parameters. This procedure is further applied to process the earthquake responses of a two‐span cable‐stayed 510‐m‐long bridge to demonstrate the engineering applicability of the proposed approach.

In-situ studies of oxidation/reduction of copper in Cu-CHA SCR catalysts: Comparison of fresh and SO2-poisoned catalysts

SO2-poisoning results in deactivation of Cu-CHA SCR under standard SCR conditions; however regeneration at 700 °C completely restores the SCR performance. To understand the nature of these effects, Cu-species in the fresh and poisoned catalysts were characterized by in-situ temperature-dependent time-resolved Cu K-edge X-ray absorption spectroscopy using the multivariate curve resolution alternating least squares (MCR-ALS) approach and continuous Cauchy wavelet transforms. The extracted chemically-meaningful reference spectra of Cu-species were analyzed by DFT-assisted XANES calculations. Cu-bisulfates were found as the most energetically favorable poisoned Cu-species. The response of Cu-species to a reducing environment differs in the fresh and SO2-poisoned catalysts. Differences in reducibility are related to the formation of quasi-linear Cu-complexes in the SO2-poisoned catalyst formed during heating in H2/He. Heating in H2/He leads to partial desulfurization of the poisoned catalyst. Cooling in H2/He after heating results in more facile formation of Cu-metal clusters in fresh catalyst than in SO2-poisoned.

Characterization of Analytic Wavelet Transforms and a New Phaseless Reconstruction Algorithm

We obtain a characterization of all wavelets leading to analytic wavelet transforms (WT). The characterization is obtained as a by-product of the theoretical foundations of a new method for wavelet phase reconstruction from magnitude-only coefficients. The cornerstone of our analysis is an expression of the partial derivatives of the continuous WT, which results in phase-magnitude relationships similar to the short-time Fourier transform (STFT) setting and valid for the generalized family of Cauchy wavelets. We show that the existence of such relations is equivalent to analyticity of the WT up to a multiplicative weight and a scaling of the mother wavelet. The implementation of the new phaseless reconstruction method is considered in detail and compared to previous methods. It is shown that the proposed method provides significant performance gains and a great flexibility regarding accuracy versus complexity. Additionally, we discuss the relation between scalogram reassignment operators and the wavelet transform phase gradient and present an observation on the phase around zeros of the WT.

Investigation of the Relationship Between Electrical Stimulation Frequency and Muscle Frequency Response Under Submaximal Contractions.

Neuromuscular electrical stimulation (NMES) is a common tool that is used in clinical and laboratory experiments and can be combined with mechanomyography (MMG) for biofeedback in neuroprostheses. However, it is not clear if the electrical current applied to neuromuscular tissues influences the MMG signal in submaximal contractions. The objective of this study is to investigate whether the electrical stimulation frequency influences the mechanomyographic frequency response of the rectus femoris muscle during submaximal contractions. Thirteen male participants performed three maximal voluntary isometric contractions (MVIC) recorded in isometric conditions to determine the maximal force of knee extensors. This was followed by the application of nine modulated NMES frequencies (20, 25, 30, 35, 40, 45, 50, 75, and 100 Hz) to evoke 5% MVIC. Muscle behavior was monitored by the analysis of MMG signals, which were decomposed into frequency bands by using a Cauchy wavelet transform. For each applied electrical stimulus frequency, the mean MMG spectral/frequency response was estimated for each axis (X, Y, and Z axes) of the MMG sensor with the values of the frequency bands used as weights (weighted mean). Only with respect to the Z (perpendicular) axis of the MMG signal, the stimulus frequency of 20 Hz did not exhibit any difference with the weighted mean (P = 0.666). For the frequencies of 20 and 25 Hz, the MMG signal displayed the bands between 12 and 16 Hz in the three axes (P < 0.050). In the frequencies from 30 to 100 Hz, the muscle presented a higher concentration of the MMG signal between the 22 and 29 Hz bands for the X and Z axes, and between 16 and 34 Hz bands for the Y axis (P < 0.050 for all cases). We observed that MMG signals are not dependent on the applied NMES frequency, because their frequency contents tend to mainly remain between the 20- and 25-Hz bands. Hence, NMES does not interfere with the use of MMG in neuroprosthesis.

Mechanomyography-Based Wearable Monitor of Quasi-Isometric Muscle Fatigue for Motor Neural Prostheses.

A motor neural prosthesis based on surface functional electrical stimulation (sFES) can restore functional movement (e.g., standing, walking) in patients with a spinal cord injury (SCI). sFES generates muscle contractions in antigravity muscles and allows balance-assisted standing. This induced standing has several benefits, such as improved cardiovascular function, decreased incidence of urinary infections, reduced joint contractures, and muscle atrophy. The duration of sFES assisted standing is limited due to the quick onset of muscle fatigue. Currently, there is no method available to reliably estimate real-time muscle fatigue during sFES. Simply monitoring the M-wave changes is not suitable due to the high signal disturbances that arise during multi-channel electrical stimulation. Mechanomyography (MMG) is immune to electrical stimulation artifacts and can be used to detect subtle vibrations on the surface of the skin related to activation of the underlying muscle's motor units (MU). The aim of this study was to develop a method for detecting muscle fatigue brought on by sFES. The method was tested in three different heads of the quadriceps muscle in SCI patients during electrically elicited quasi-isometric contraction. Six spinal cord-injured male volunteers, with no voluntary control of the quadriceps muscle participated in the study. Electrical bursts of voltage-controlled monophasic square pulses at frequencies of 1 kHz (50% duty cycle) at 50 Hz (15% duty cycle) were used to generate thigh muscle contractions that controlled the knee joint in the sagittal plane. The pulse amplitudes were set to position the knee joint at a 5° angle from the horizontal plane and when the knee angle dropped to 20° (e.g., the quadriceps were unable to hold the lower leg in the desired position), the test was terminated. Two data segments lasting 10 s each, at the beginning and end of each test, were analyzed. The muscle contraction was assessed by MMG sensors positioned on the rectus femoris, vastus lateralis, and vastus medialis muscles. Data segments were decomposed into 11 frequency bands using a Cauchy wavelet transform. In the initial time interval (non-fatigued muscle), the power peak was concentrated in the 11.31 Hz frequency band. In the final interval (muscle fatigued) this peak shifted to lower frequencies (2 and 6 Hz frequency bands). The decreased frequency was most prominent during the last 4 s of the recordings. It was shown that MMG could be used as a real-time indicator of muscle fatigue during FES-induced isometric contraction of quadriceps; hence, MMG could be used in closed-loop control as a fatigue detector. Subsequent studies for non-isometric contractions could possibly lead to prediction of muscle fatigue before contractile failure during functional use of the muscle.

Locating damaged storeys in a structure based on its identified modal parameters in Cauchy wavelet domain

• A flexibility-based procedure for accurately locating damaged storeys of a structure is developed. • The modal parameters that identified in continuous Cauchy wavelet domain are employed. • The accuracy and effectiveness of the proposed approach are confirmed by numerical simulations. • Acceleration responses of three-story and eight-story steel frames in shaking table test are also processed. Structural damage detection based on the changes of dynamic properties is a major topic for structural health monitoring. In this paper, efforts are made to extend the flexibility-based damage localization methods, especially the damage locating vectors (DLVs) method, to the case of earthquake vibration, where the finite element model and mass matrices are not available. First, a new method using continuous Cauchy wavelet transform (CCWT) and state-variable time series model is proposed to identify the modal parameters of a structure. Then the flexibility matrix can be constructed from the identified modal parameters. Second, a modified DLVs damage assessment approach is also proposed to locate damage positions in the structure through a weighted relative displacement index (WRDI). This index is calculated by using DLVs vectors determined from the change of flexibility matrix before and after damage of the structure. Numerical analyses demonstrate that the proposed process can indeed monitor the variation of stiffness for each storey. These two approaches are further applied to process the dynamic responses of three-storey and eight-storey steel frames in shaking table tests. The proposed scheme is also proved to be superior to mode shape based methods (CMS, COMAC) in monitoring the variation of stiffness for each storey.

Operando X-ray Absorption Spectroscopy Study of Atomic Phase Reversibility with Wavelet Transform in the Lithium-Rich Manganese Based Oxide Cathode

Lithium-rich, Mn-based composites, such as Li(Li0.2Mn0.54Ni0.13Co0.13)O2, retain immense interest for researchers and practitioners as one of the most promising candidates for electric vehicle applications (EV/HEV/PHEV). Enabling the commercialization of this battery system requires reducing intrinsic drawbacks, that is, large capacity fading during first cycle or low-rate capability; it is of great importance to optimize the composite design of the battery cathode. Information about the electronic structure and atomic environment of the transition metals (Mn, Co, and Ni) provides crucial insight into the structure of high-energy density cathodes. In this study, we present precise characterization of the atomic environments of each transition metal by operando X-ray absorption spectroscopy using continuous Cauchy wavelet transform (CCWT) analysis during the first electrochemical cycle at the charge rate of 0.5 C. The CCWT interpretation provides continuous decomposition of extended X-ray absorption fine s...

Application of Cauchy wavelet transformation to identify time-variant modal parameters of structures

This work proposes a procedure for accurately identifying instantaneous modal parameters of a linear time-varying system using a time-varying autoregressive with exogenous input (TVARX) model with the continuous Cauchy wavelet transform (CCWT). An appropriate TVARX model is established using the velocity and displacement responses of the system under consideration. The time-varying coefficients of the TVARX are expanded as piecewise polynomial functions. CCWTs with various scale parameters are then applied to the TVARX model to evaluate the instantaneous modal parameters of different modes. The CCWTs of the velocity and displacement responses are analytically obtained from the CCWT of the measured acceleration responses. The effectiveness and accuracy of the proposed procedure are validated by numerical simulations of single and multiple degrees of freedom systems that have periodically varying and sharply varying stiffness and damping coefficients. The effects of noise, the Cauchy wavelet function and the order of the polynomial on the evaluation of the modal parameters are explored in processing the numerically simulated acceleration responses of systems with a single degree of freedom subjected to base excitation. Finally, the proposed procedure is adopted to determine the modal parameters of a five-story symmetric steel frame from its measured acceleration responses in a shaking table test. The measured strains reveal the yielding of columns in the first story. The variations of the identified instantaneous natural frequencies and modal damping ratios with time are consistent with the physical phenomena that are observed from the measured strains and base excitation acceleration.

Application of damage locating vectors approach on monitoring shock isolation system

Structural damage detection based on the changes of dynamic properties is a major topic for structural health monitoring. In this paper, efforts are made to extend the flexibility-based damage localization methods, especially the damage locating vectors (DLVs) method, to the case of earthquake vibration, where the finite element model and mass matrices are not available. First, a new method using continuous Cauchy wavelet transform (CCWT) and ARX (autoregressive with exogenous input) model is applied to identify the modal parameters of a five-storey steel frame with seismic base isolation system LRB from its simulated acceleration responses under 10% and 100% of Chi-Chi Earthquake excitation (Taiwan, 1999). The DLVs, which determined from the change of flexibility matrix between two cases, are then used to monitor the shock isolation device in the structure through a weighted relative displacement index (WRDI). The proposed scheme is also proved to be superior to mode shape based methods (MAC, COMAC) in monitoring shock isolation system.

Cauchy wavelet transform of ultra-distributions in tube domains

In this paper, Cauchy wavelet transform of ultra-distributions in tube domains is defined and its various properties are studied using the theory of Cauchy integrals and Poisson integrals of ultra-distributions.

Phase Retrieval for the Cauchy Wavelet Transform

We consider the phase retrieval problem in which one tries to reconstruct a function from the modulus of its wavelet transform. We study the uniqueness and stability of the reconstruction. In the case where the wavelets are Cauchy wavelets, we prove that the modulus of the wavelet transform uniquely determines the function up to a global phase. We show that the reconstruction operator is continuous but not uniformly continuous. We describe how to construct pairs of functions which are far away in \(L^2\)-norm but whose wavelet transforms are very close, in modulus. The principle is to modulate the wavelet transform of a fixed initial function by a phase which varies slowly in both time and frequency. This construction seems to cover all the instabilities that we observe in practice; we give a partial formal justification to this fact. Finally, we describe an exact reconstruction algorithm and use it to numerically confirm our analysis of the stability question.

Characterization of permeability anisotropy using wavelet analysis

Wavelet analysis is an image analysis technique that can extract local information at multiple scales. Because of this capability, wavelet analysis can be used to identify dominant scales in statistically heterogeneous and anisotropic random fields. We develop and test a wavelet analysis method for identifying dominant scales and orientations in permeability fields and for identifying boundaries between regions with different dominant orientations. We evaluate three different wavelets (fully anisotropic Morlet wavelet, Mexican hat wavelet, and Cauchy wavelet) and show that the Morlet wavelet is the most effective of these three wavelets in identifying dominant orientations. We also investigate the use of several different quantitative wavelet measures in identifying dominant scales and orientations in permeability fields. The technique is demonstrated using both a synthetic data set with known characteristics and a laboratory‐collected permeability data set from Massillon sandstone.

Quantitative Adsorption and Local Structures of Gallium(III) at the Water-α-FeOOH Interface

The adsorption of Ga(III) at the water-alpha-FeOOH (goethite) interface has been investigated by means of quantitative adsorption experiments, extended X-ray absorption fine structure (EXAFS) spectroscopy, and surface complexation modeling. Under the conditions studied, pH range 3-11 and surface coverages of 0.9-3.2 micromol/m2, Ga(III) was found to adsorb strongly to alpha-FeOOH, and the surface species were more resistant toward hydrolysis and formation of soluble Ga(OH)4- than either solid gallium hydroxides or soluble polynuclear complexes. The EXAFS measurements revealed the presence of octahedral Ga(III) complexes at the water-alpha-FeOOH interface, with practically no structural variations as a function of pH or total gallium concentration. Analysis of the first coordination shell required an anharmonic model indicating a distorted geometry of the GaO6 octahedra, with mean Ga-O distances at 1.96-1.98 angstroms. A method based on the continuous Cauchy wavelet transforms (CCWT) was used to identify backscattering atoms in the higher coordination shells. This analysis indicated predominately Fe backscattering, and the quantitative data fitting resulted in three Ga-Fe paths at 3.05, 3.2, and 3.55 angstroms, which correspond to two edge-sharing and one corner-sharing linkage, respectively. The collective results from EXAFS spectroscopy showed that Ga(III) adsorbs to Fe equivalent sites at the surface alpha-FeOOH as an extension of the rows of Fe octahedra in the bulk structure. This interpretation was further corroborated by a Ga-Fe-Fe multiple scattering path at 6.13 angstroms. The quantitative adsorption and proton data were modeled using a surface complexation formalism based on a 1 pK(a) constant capacitance model. In agreement with the EXAFS results, the model obtained included one predominating surface complex with the stoichiometry [triple bond]FeOGa(OH)2(-0.5) and the stability constant log beta(intr.) = -2.55 +/- 0.04 ([triple bond]FeOH(-0.5) + Ga3+ + 2H2O <--> [triple bond]FeOGa(OH)2(-0.5) + 3H+).

Continuous Cauchy wavelet transform of XAFS spectra

The continuous Cauchy wavelet transform (CCWT) is applied to the analysis of XAFS spectra. Thanks to that method, XANES and EXAFS signals can be visualized in three-dimensions: the wavevector (k), the interatomic distance uncorrected for phase-shifts (R′) and the CCWT modulus (corresponding to the continuous decomposition of the amplitude terms). Applied to EXAFS spectra, the CCWT analysis provides straightforward qualitative information related to the k-range of each “R′-EXAFS” contribution. Such information is particularly useful to perform next nearest-neighbors identification, despite the presence of spectral artifacts such as multiple-scattering features, multi-electronic excitations or noise. When applied to XANES spectra, the CCWT analysis helps highly to measure the “spectral limit” between XANES and EXAFS regions, as well as the energy range required to model properly next-nearest neighbors. To further illustrate the potential of CCWT analyses applied to XAFS spectra, we present examples related to: (1) a XANES spectrum collected at the Ti K-edge for titanite (CaTiSiO5); (2) an experimental Au LIII-edge EXAFS spectrum for gold sorbed on goethite (FeO(OH)).

Continuous Cauchy wavelet transform analyses of EXAFS spectra: A qualitative approach

To better understand the extended X-ray absorption fine structure (EXAFS) spectroscopic information obtained for complex materials such as those encountered in Earth materials, we propose to use the Continuous Cauchy Wavelet Transform (CCWT). Thanks to this method, EXAFS spectra can be visualized in three dimensions: the wavevector (k), the interatomic distance uncorrected for phase-shifts (R’), and the CCWT modulus (corresponding to the continuous decomposition of the EXAFS amplitude terms). Consequently, more straightforward qualitative interpretations of EXAFS spectra can be performed, even when spectral artifacts are present, such as multiple-scattering features, multi-electronic excitations, or noise. More particularly, this method provides important information concerning the k range of each EXAFS contribution, such as next nearest-neighbors identification. To illustrate the potential of CCWT analyses applied to EXAFS spectra, we present experimental and theoretical spectra obtained for thorite and zircon at the Th LII and Zr K edges, respectively. Then, we present CCWT analyses of EXAFS spectra collected for amorphous materials of geochemical and environmental interest, including sodium trisilicate glass and an aqueous chloride solution, at the Mo K and Au LIII edges, respectively. Further studies based on CCWT phase terms are underway, in order to quantitatively characterize anharmonic information from EXAFS contributions.