Article

A Practical Guide to Wavelet Analysis

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Abstract

A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El NiñoSouthem Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmöller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of changes in ENSO variance on interdecadal timescales. Using new datasets that extend back to 1871, the Niño3 sea surface temperature and the Southern Oscillation index show significantly higher power during 1880-1920 and 1960-90, and lower power during 1920-60, as well as a possible 15-yr modulation of variance. The power Hovmöller of sea level pressure shows significant variations in 2-8-yr wavelet power in both longitude and time.

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... The proof of the previous equation is given in Appendix C.2. In this study, the implementation of the CWT and its inverse for discrete sequences by Torrence & Compo (1998) is used. An example of an admissible mother wavelet is the Morlet wavelet, which is a Gaussian-modulated sinusoidal pulse (Farge, 1992;Torrence & Compo, 1998): where ω 0 is a nondimensional frequency and is set to 6 (Farge, 1992;Torrence & Compo, 1998) and i denotes the imaginary unit. ...
... In this study, the implementation of the CWT and its inverse for discrete sequences by Torrence & Compo (1998) is used. An example of an admissible mother wavelet is the Morlet wavelet, which is a Gaussian-modulated sinusoidal pulse (Farge, 1992;Torrence & Compo, 1998): where ω 0 is a nondimensional frequency and is set to 6 (Farge, 1992;Torrence & Compo, 1998) and i denotes the imaginary unit. The real and imaginary parts of the wavelet are shown in Figure 4.2. ...
... In this study, the implementation of the CWT and its inverse for discrete sequences by Torrence & Compo (1998) is used. An example of an admissible mother wavelet is the Morlet wavelet, which is a Gaussian-modulated sinusoidal pulse (Farge, 1992;Torrence & Compo, 1998): where ω 0 is a nondimensional frequency and is set to 6 (Farge, 1992;Torrence & Compo, 1998) and i denotes the imaginary unit. The real and imaginary parts of the wavelet are shown in Figure 4.2. ...
Thesis
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Seismologists found a significant deterioration in station quality of seismological stations after the installation of wind turbines in the vicinity, which led to conflicts between wind turbine operators and seismological data services. Since the frequency bands of wind turbine noise and earthquakes overlap, spectral filtering, which is usually used in seismic data processing, is not effective. Here, two different techniques are presented to suppress wind turbine noise from seismological data: nonlinear thresholding and a denoising autoencder.Nonlinear thresholding works well when the seismic event is visible in the raw data but it fails when wind turbine noise dominates and the event is hidden. Once the denoising autoencoder is trained with a large dataset, it is able to remove wind turbine noise even when the event is completely masked by noise.
... Following Dowling (2021b) and Vidal-Tomás (2022a), we shed some light on the connections between NFT features and Web3 fungible tokens by means of the wavelet coherence approach, in order to analyse the co-movement and causality between the two-time series in terms of both time and frequency domain. According to Torrence and Compo (1998), the cross wavelet transform of two time-series x t and y t is defined by means of the continuous wavelet transform W x n (u, s) and W y n (u, s), as follows: ...
... As a consequence, it is not possible to identify positive and negative co-movements properly. To overcome this issue, we employed the phase difference proposed by Torrence and Compo (1998) that allowed us not only to distinguish between positive and negative co-movements but also to shed some light on the causal relationships between time series. Wavelet coherence phase difference is defined as: ...
... One of the main properties of this methodology is that it "can be used to analyse time series that contain nonstationary power at many different frequencies" (Torrence and Compo, 1998) (see also Daubechies, 1990). This fact was also supported by Cazelles et al. (2008) when contending that the wavelet coherence approach is "especially relevant to the analysis of nonstationary systems, like those observed in ecological systems." ...
Article
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This paper provides (i) a review of the existing literature on the metaverse and (ii) an empirical assessment of the current state of the Web3 meta-economy, with the focus on economic governance and metaverse commerce. We have analysed the entire Web3 metaverse niche, i.e. both the 196 available metaverse fungible tokens and all the non-fungible token (NFT) transactions belonging to the metaverse marketplace. Our results showed that economic governance is based on metaverse tokens that cannot be defined as reliable virtual currencies due to their explosive behaviour, negative performance, and higher volatility compared to traditional alternatives. Paradoxically, fiat currencies and stablecoins could be more appropriate candidates for the payment infrastructure. Moreover, we also observed that NFT prices are affected by the cryptocurrency market, which highlights the risk of metaverse commerce. For future research, developers and scholars must assess the different alternatives and infrastructures that can make the metaverse a persistent reality with a proper virtual economy. However, at present, it seems that the hype has run far ahead of reality.
... We use the power spectrum analysis based on a fast Fourier transform to compute the frequency spectrum of the time series 75 . However, because the time series shows a non-stationary signal, we also use the Morlet wavelet 76 and wavelet coherence spectrum to identify the nonstationary relationship between two time series 47 . The wavelet analysis is a useful method for the identification of periodic signals in the timefrequency space 76 , whereas the Wavelet coherence spectrum can well be used to study the time-varying relationship between two time series and their evolution over a continuous time-frequency space by considering the cross-wavelet transform, wavelet power spectrum and phase difference. ...
... However, because the time series shows a non-stationary signal, we also use the Morlet wavelet 76 and wavelet coherence spectrum to identify the nonstationary relationship between two time series 47 . The wavelet analysis is a useful method for the identification of periodic signals in the timefrequency space 76 , whereas the Wavelet coherence spectrum can well be used to study the time-varying relationship between two time series and their evolution over a continuous time-frequency space by considering the cross-wavelet transform, wavelet power spectrum and phase difference. By calculating the continuous wavelet transform one can well use the wavelet coherence spectrum to identify the relation between two time series (A and B) and their lead-lag relationship. ...
... In this wavelet coherence spectrum map, the direction of arrows can reveal the causal relationship between A and B 42 . The significance testing of power spectrum, wavelet spectrum and wavelet coherence spectrum can be found 55,56,76 . ...
Article
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Winter Arctic sea-ice concentration (SIC) decline plays an important role in Arctic amplification which, in turn, influences Arctic ecosystems, midlatitude weather and climate. SIC over the Barents-Kara Seas (BKS) shows large inter-annual variations, whose origin is still unclear. Here we find that interannual variations in winter BKS SIC have significantly strengthened in recent decades likely due to increased amplitudes of the El Niño-Southern Oscillation (ENSO) in a warming climate. La Niña leads to enhanced Atlantic Hadley cell and a positive phase North Atlantic Oscillation-like anomaly pattern, together with concurring Ural blocking, that transports Atlantic ocean heat and atmospheric moisture toward the BKS and promotes sea-ice melting via intensified surface warming. The reverse is seen during El Niño which leads to weakened Atlantic poleward transport and an increase in the BKS SIC. Thus, interannual variability of the BKS SIC partly originates from ENSO via the Atlantic pathway.
... A significance level of 0.05 was chosen to test the null hypotheses that no significant differences exist among aggregation types and that no significant differences exist in the number of predator dives among aggregation types. Wavelet analysis (Torrence and Compo 1998) was used to identify dominant temporal scales in krill aggregation mean acoustic backscatter (i.e., S v ) data from E2 and E4 datasets. Wavelet power was calculated using the R package WaveletComp (Rösch and Schmidbauer 2018) with a continuous Morlet mother wavelet function (Torrence and Compo 1998). ...
... Wavelet analysis (Torrence and Compo 1998) was used to identify dominant temporal scales in krill aggregation mean acoustic backscatter (i.e., S v ) data from E2 and E4 datasets. Wavelet power was calculated using the R package WaveletComp (Rösch and Schmidbauer 2018) with a continuous Morlet mother wavelet function (Torrence and Compo 1998). Peaks in the wavelet spectra indicate periods that contribute the most to the variance of the series (Cazelles et al. 2008). ...
... Peaks in the wavelet spectra indicate periods that contribute the most to the variance of the series (Cazelles et al. 2008). Statistical significance in each localized wavelet power was evaluated through comparison to a white noise background spectrum (i.e., a flat Fourier spectrum) defined using 100 simulations to establish a 95% confidence level for the significance of a peak in the wavelet power spectrum (Torrence and Compo 1998). ...
Article
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The Antarctic krill, Euphausia superba, is a major component of the Southern Ocean’s ecosystem. Limited high-resolution data on the relative importance of oceanographic processes on the behavioral responses of krill at traditional predator foraging grounds constitutes a major obstacle in the understanding of krill-environment coupling and ecosystem-based management of this resource. Aggregation structures of krill and predator interactions were investigated using active acoustic data collected by WBAT echosounders deployed on moorings in two hydrographically different sites in Bransfield Strait. Near Nelson Island, water flows from the northwest to southeast while Deception Island is influenced by stronger net current velocities from the southwest to northeast. Krill aggregations were identified and then classified in three clusters using a swarm-identification algorithm and hierarchical clustering using aggregation morphological characteristics: acoustic density, mean depth, center of mass, inertia, equivalent area, aggregation index, and proportion occupied. A total of 693 and 736 aggregations were detected at the mooring sites close to Nelson and Deception Islands. The three aggregation categories ranged from high to low densities, evenness, and dispersion and were distributed throughout the water column. Krill aggregation density distribution and mean thickness are influenced by krill mean depth, current velocities and direction. The majority of observed predator dive profiles occurred over the aggregation type with highest krill densities at both Nelson and Deception Islands, and within the first 25 m of the water column. The heterogeneity of krill aggregations potentially impacts predator foraging strategies and predator–krill interactions in the hydrodynamically active Bransfield Strait.
... Step 1a -apply wavelet transform to observations First, we apply the continuous wavelet transform (WT) to the observed time series. The main steps and equations for the WT are provided here, though the reader is referred to Torrence and Compo (1998) and Liu et al. (2011) for more details. ...
... Before applying the WT, a mother wavelet needs to be selected. In Torrence and Compo (1998), they discuss the key factors that should be considered when choosing the mother wavelet. There are four main considerations, including (i) orthogonal or nonorthogonal, (ii) complex or real, (iii) width, and (iv) shape. ...
... The wavelet transform (WT) expands the dimensionality of the original time series by introducing the timescale (or period) dimension. Wavelet power is also a function of both time and timescale (e.g., Torrence and Compo, 1998). This is illustrated in Fig. 2. The streamflow time series (Fig. 2a) is expanded into a 2-dimensional (2-D) wavelet power spectrum (Fig. 2b). ...
Article
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Streamflow timing errors (in the units of time) are rarely explicitly evaluated but are useful for model evaluation and development. Wavelet-based approaches have been shown to reliably quantify timing errors in streamflow simulations but have not been applied in a systematic way that is suitable for model evaluation. This paper provides a step-by-step methodology that objectively identifies events, and then estimates timing errors for those events, in a way that can be applied to large-sample, high-resolution predictions. Step 1 applies the wavelet transform to the observations and uses statistical significance to identify observed events. Step 2 utilizes the cross-wavelet transform to calculate the timing errors for the events identified in step 1; this includes the diagnostic of model event hits, and timing errors are only assessed for hits. The methodology is illustrated using real and simulated stream discharge data from several locations to highlight key method features. The method groups event timing errors by dominant timescales, which can be used to identify the potential processes contributing to the timing errors and the associated model development needs. For instance, timing errors that are associated with the diurnal melt cycle are identified. The method is also useful for documenting and evaluating model performance in terms of defined standards. This is illustrated by showing the version-over-version performance of the National Water Model (NWM) in terms of timing errors.
... Wavelet analysis. Wavelet analysis is a mathematical tool for analysing localised variations in time series 42 . ...
... Throughout this study, a complex Morlet wavelet function is used with ω 0 = 6 43 . The scales are inversely related to frequencies and for the chosen Morlet wavelet, the relation between the frequency f and the wavelet scale s is f ≈ 1 1.033s 42 . The amplitude squared of the wavelet coefficients ( |W x (s, n)| 2 ) can be interpreted as the power of the corresponding frequency at timepoint n . ...
... S represents a smoothing function in both time and scale. In this study, smoothing in time is done by convolution with a filter given by the absolute value of the complex wavelet function (a Gaussian for the Morlet wavelet) at each scale 42,43 . Smoothing in scale is done by computing a moving average in windows equal to the number of scales in one octave (5 in this study) 43 . ...
Article
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In neonates with hypoxic ischemic encephalopathy, the computation of wavelet coherence between electroencephalogram (EEG) power and regional cerebral oxygen saturation (rSO2) is a promising method for the assessment of neurovascular coupling (NVC), which in turn is a promising marker for brain injury. However, instabilities in arterial oxygen saturation (SpO2) limit the robustness of previously proposed methods. Therefore, we propose the use of partial wavelet coherence, which can eliminate the influence of SpO2. Furthermore, we study the added value of the novel NVC biomarkers for identification of brain injury compared to traditional EEG and NIRS biomarkers. 18 neonates with HIE were monitored for 72 h and classified into three groups based on short-term MRI outcome. Partial wavelet coherence was used to quantify the coupling between C3–C4 EEG bandpower (2–16 Hz) and rSO2, eliminating confounding effects of SpO2. NVC was defined as the amount of significant coherence in a frequency range of 0.25–1 mHz. Partial wavelet coherence successfully removed confounding influences of SpO2 when studying the coupling between EEG and rSO2. Decreased NVC was related to worse MRI outcome. Furthermore, the combination of NVC and EEG spectral edge frequency (SEF) improved the identification of neonates with mild vs moderate and severe MRI outcome compared to using EEG SEF alone. Partial wavelet coherence is an effective method for removing confounding effects of SpO2, improving the robustness of automated assessment of NVC in long-term EEG-NIRS recordings. The obtained NVC biomarkers are more sensitive to MRI outcome than traditional rSO2 biomarkers and provide complementary information to EEG biomarkers.
... Existen varias funciones de este tipo, sin embargo para determinar cual emplear, es necesario definir este parámetro según la naturaleza de los datos y las característica que queremos extraer (Moreno & García, 2012). En la figura 5 se pueden ver algunas de las formas de onda de las wavelets madre empleadas en Torrence & Compo (1998). Si bien existen diferentes tipos de WT, nosotros utilizamos la Transformada de Wavelet Continua (CWT, por sus sigla en ingles), la cual está definida por la suma de la multiplicación de una señal continua y la wavelet madre en su forma desplazada y escalada, teniendo la forma: Torrence & Compo (1998). ...
... En la figura 5 se pueden ver algunas de las formas de onda de las wavelets madre empleadas en Torrence & Compo (1998). Si bien existen diferentes tipos de WT, nosotros utilizamos la Transformada de Wavelet Continua (CWT, por sus sigla en ingles), la cual está definida por la suma de la multiplicación de una señal continua y la wavelet madre en su forma desplazada y escalada, teniendo la forma: Torrence & Compo (1998). ...
... Se incluyen módulos operacionales y en desarrollo.................................................................................17 Figura 3: Distribución espacial de las mediciones utilizadas para la evaluación de MOSA-CROCO.................................................................................................................................22 Figura 4: Mapa con estaciones meteorológicas utilizadas para la validación del modelo operacional MOSA-WRF......................................................................................................24 Figura 5: Cuatro bases de wavelet diferentes.Las gráficas muestran la parte real (linea continua) y la parte imaginaria (linea discontinua) para las wavelet en el dominio del tiempo y las wavelets correspondientes en el dominio de la frecuencia. Figura editada deTorrence & Compo (1998)....................................................................................................27 Figura 6: Esquema del modelo biogeoquímico Fennel.........................................................29 Figura 7: Dominio y batimetría utilizada en los 3 modelos biogeoquímicos (NPZD, PISCES y Fennel)................................................................................................................................32 Figura 8: Ubicación de las estaciones de los muestreos de IFOP (puntos rojos) y transecta a lo largo del MIC para evaluar los tres modelos biogeoquímicos (puntos negros)................34 Figura 9: (a) Dominio y batimetría utilizada del modelo MOSA-MAG. (b) Batimetría empleada en los Senos Skyring y Ottway (c) Batimetría empleada en el Golfo Almirante Montt.....................................................................................................................................35 Figura 10: Ubicación y series mensuales de caudales de agua dulce en MOSA-MAG........38Figura 11: Ubicación de las estaciones de CTD utilizadas en la evaluación de MOSA-MAG. ...
Technical Report
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En esta etapa del proyecto se evaluó del modelo oceanográfico operacional MOSACROCO mediante una validación espacio-temporal de las variables de temperatura, salinidad, altura del nivel del mar y corrientes pronosticadas entre los años 2017 y 2022. Del mismo modo, la evaluación de los vientos del modelo atmosférico operacional MOSA-WRF fue realizado con la información de tres estaciones meteorológicas disponibles a la fecha. A su vez, implementamos y evaluamos tres modelos biogeoquímicos: NPZD, PISCES y FENNEL. Finalmente, se implementó un modelo físico de alta resolución (~1.5 km) para la región de Magallanes.
... This study focuses primarily on emphasizing the importance of taking the COI into account in the design of ERP/EEG machine learning classification models which use features derived from the CWT. The COI is a boundary that is superimposed on the CWT scalogram to delineate the accurately computed wavelet coefficients from those that are inaccurate (artifact coefficients) due to the wavelet extending beyond the observation interval of the signal [1][2][3][4]. These edge-effect artifacts are due to the mechanics of convolving the signal and the wavelet near the beginning and near the end of the signal. ...
... Methods have also been proposed to approximate the COI, which include (a) using the time constant 1/ to delineate the borders of the cone of influence at each scale [2,36,37], (b) defining the extent of the COI at each scale as the point where the wavelet transform magnitude decays to 2% of its peak value [38,39], (c) delineating the COI by adding and subtracting (1/2) the wavelet footprint at the beginning and end of the observation interval at each scale [37,40,41], and (d) using the wavelet time interval that encompasses 95% of the wavelet's energy for Morse wavelets [42][43][44]. ...
... Over the years, many different CWTs have been proposed, which include, among many others, Morlet [63,64], Morse [44], Gaussian [2], and Mexican Hat [65,66]. The choice of the wavelet is application dependent, and in this study, the analytic Morlet wavelet is selected because it is frequently used to analyze the oscillatory behavior of ERPs and EEGs. ...
Article
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Features extracted from the wavelet transform coefficient matrix are widely used in the design of machine learning models to classify event-related potential (ERP) and electroencephalography (EEG) signals in a wide range of brain activity research and clinical studies. This novel study is aimed at dramatically improving the performance of such wavelet-based classifiers by exploiting information offered by the cone of influence (COI) of the continuous wavelet transform (CWT). The COI is a boundary that is superimposed on the wavelet scalogram to delineate the coefficients that are accurate from those that are inaccurate due to edge effects. The features derived from the inaccurate coefficients are, therefore, unreliable. In this study, it is hypothesized that the classifier performance would improve if unreliable features, which are outside the COI, are zeroed out, and the performance would improve even further if those features are cropped out completely. The entire, zeroed out, and cropped scalograms are referred to as the “same” (S)-scalogram, “zeroed out” (Z)-scalogram, and the “valid” (V)-scalogram, respectively. The strategy to validate the hypotheses is to formulate three classification approaches in which the feature vectors are extracted from the (a) S-scalogram in the standard manner, (b) Z-scalogram, and (c) V-scalogram. A subsampling strategy is developed to generate small-sample ERP ensembles to enable customized classifier design for single subjects, and a strategy is developed to select a subset of channels from multiple ERP channels. The three scalogram approaches are implemented using support vector machines, random forests, k-nearest neighbor, multilayer perceptron neural networks, and deep learning convolution neural networks. In order to validate the performance hypotheses, experiments are designed to classify the multi-channel ERPs of five subjects engaged in distinguishing between synonymous and non-synonymous word pairs. The results confirm that the classifiers using the Z-scalogram features outperform those using the S-scalogram features, and the classifiers using the V-scalogram features outperform those using the Z-scalogram features. Most importantly, the relative improvement of the V-scalogram classifiers over the standard S-scalogram classifiers is dramatic. Additionally, enabling the design of customized classifiers for individual subjects is an important contribution to ERP/EEG-based studies and diagnoses of patient-specific disorders.
... For the confidence levels, the coherence values should be tested against the null hypothesis of zero population coherence, i.e., whether the coherence exceeds expected values from arbitrary colored (e.g., white or red) noise backgrounds. While various methods have been employed for this statistical test, one common approach is to estimate the confidence levels by means of Monte Carlo simulations (Torrence and Compo, 1998;Grinsted et al., 2004;BjörgÓlafsdóttir et al., 2016). ...
... As a result, the application of Fourier analyses to solar time series often displaying quasi-periodic wave motion (e.g., spicules, fibrils, rapid blueshift excursions (RBEs), etc.; Beckers, 1968;De Pontieu et al., 2004, 2007aZaqarashvili and Erdélyi, 2009;Sekse et al., 2013a,b;Kuridze et al., 2015) may not be the most appropriate as a result of the limited lifetimes associated with these features. Wavelet techniques, pioneered by Torrence and Compo (1998), employ a time-localized oscillatory function that is continuous in both time and frequency (Bloomfield et al., 2004b), which allows them to be applied in the search for dynamic transient oscillations. The time resolution of the input dataset is preserved through the modulation of a simple sinusoid (synonymous with standard FFT approaches) with a Gaussian envelope, providing the Morlet wavelet commonly used in studies of waves in the solar atmosphere (Bloomfield et al., 2004a;Jess et al., 2007;Stangalini et al., 2012;Kobanov et al., 2013Kobanov et al., , 2015Jafarzadeh et al., 2017d). ...
... amplitude periodicity between times of 0 − 2200 s at a period of ≈ 210 s (≈ 4.7 mHz). This wave activity is highlighted in the wavelet transform by being bounded by the 95% confidence level isocontours across these times and periods, which is equivalent to the oscillatory behavior being significant at the 5% level (Torrence and Compo, 1998). To calculate the wavelet power thresholds corresponding to the 95% confidence isocontours, the wavelet background spectrum (i.e., the output theoretical background spectrum that has been smoothed by the wavelet function) is multiplied by the 95 th percentile value Figure 6 (top). ...
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Waves and oscillations have been observed in the Sun's atmosphere for over half a century. While such phenomena have readily been observed across the entire electromagnetic spectrum, spanning radio to gamma-ray sources, the underlying role of waves in the supply of energy to the outermost extremities of the Sun's corona has yet to be uncovered. Of particular interest is the lower solar atmosphere, including the photosphere and chromosphere, since these regions harbor the footpoints of powerful magnetic flux bundles that are able to guide oscillatory motion upwards from the solar surface. As a result, many of the current- and next-generation ground-based and space-borne observing facilities are focusing their attention on these tenuous layers of the lower solar atmosphere in an attempt to study, at the highest spatial and temporal scales possible, the mechanisms responsible for the generation, propagation, and ultimate dissipation of energetic wave phenomena. Here, we present a two-fold review that is designed to overview both the wave analyses techniques the solar physics community currently have at their disposal, as well as highlight scientific advancements made over the last decade. Importantly, while many ground-breaking studies will address and answer key problems in solar physics, the cutting-edge nature of their investigations will naturally pose yet more outstanding observational and/or theoretical questions that require subsequent follow-up work. This is not only to be expected, but should be embraced as a reminder of the era of rapid discovery we currently find ourselves in. We will highlight these open questions and suggest ways in which the solar physics community can address these in the years and decades to come.
... However, various researchers have implemented the wavelet approach to examine economic and financial time series. For instance, [83][84][85][86] suggested detailed and formal descriptions of wavelet analysis. ...
... To test the cross-correlation action among the frequency and time period, we employ the square coherence wavelet by presenting first transform of cross-wavelet following [84] subscribe the wavelet squared coherence as the next (7) and (8) models, respectively. ...
Article
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This paper investigates the association between CO2 emissions and a range of factors, including electricity consumption, economic growth, urbanization, and trade openness for six Gulf Cooperation Council (GCC) countries using data covering the 1965–2019 period. Namely, Oman, Saudi Arabia, the UAE, Kuwait, Bahrain, and Qatar. Contrasting with the standard literature, our empirical strategy uses the wavelet coherence approach on the frequency domain, thought to complement the time series econometric procedures reported earlier on this topic. Supplied at the country level, associated evidence presents far-reaching policy recommendations whose applications may directly benefit environmental planning and bring high information value for the sake of sustainable energies in the Gulf region.
... This allows better identification of the prevailing periodicities of variability and their temporal changes. The theory behind the wavelet transform is well discussed in the literature (Torrence and Compo 1998;Grinsted et al. 2004;Partal 2012). ...
... In this study, we adopted the CWT approach for identifying temporal variability and nonstationary trends in the long-term monthly and seasonal precipitation series. Since the wavelet of CWT is not completely localised, it has some edge artefacts called the "cone of influence" (COI) by Torrence and Compo (1998), which also provides a detailed description of the CWT methodology. ...
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This study aimed to provide broad insight into the long-term trends and periodicity of monthly and seasonal precipitation in Turkey and to evaluate their implications for sectoral water availability. Overall, Turkey’s monthly precipitation declined over the last five decades. While decreasing precipitation trends were dominant in the wet season, consistently increasing trends prevailed in the dry season. The monthly monotonic trends were marked by larger downward trends, especially in early winter and spring. The trend magnitudes in annual and wet season precipitation decreased from north to south and west to east, reaching a maximum value of 7.5 mm/year increase in the eastern Black Sea. The magnitude of dry season trends was much smaller but consistent across the country, varying by 1–5 mm/year. Monthly changes in the trend magnitudes varied between − 2.2 and 3.4 mm, reflecting both decrease and increase, respectively. The magnitude of the downward precipitation trends was higher in inland regions than in other regions. Spatial patterns in the trends evidenced that wet season precipitation variability largely governs annual precipitation variability. The wavelet spectrum indicated a strong annual signal for the monthly precipitation. The inland regions experienced the periodicity of wet years at much longer durations. An average 8-year periodicity was dominant for the annual precipitation, underlining its inter-annual variability and coincided well with the NAO spectrum. Assuming the identified trends persist in the future, a further increase in the magnitude of precipitation trends in coastal areas can enhance the flood risk. However, the precipitation decreases during the wet season are bound to have adverse consequences on sectoral-dependent water availability.
... Hydroclimatic time-series are intrinsically non-stationary, and they integrate a broad set of transient patterns varying within the temporal record. The wavelet transform allows to localize in both time and periodicity the (Torrence & Compo, 1998). In this research, we make use of the Morlet wavelet, which was successfully used in the past to analyze precipitation and discharge time-series (Carey et al., 2013;Pérez-Ciria et al., 2019). ...
... where W n (s) is the wavelet transform coefficients, the normalized wavelet (the Morlet wavelet in our case), (*) the complex conjugate, s the wavelet scale, n the localized time index, and n' the translated time index of the time ordinate x. The CWT enables to plot a global picture showing the varying amplitude at each time-scale and along the time-line (Torrence & Compo, 1998). ...
Article
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River discharge has experienced diverse changes in the last decades due to modification of hydrological patterns, anthropogenic intervention, re‐vegetation or annual and interannual climatic and atmospheric fluctuations. Assessing the recent changes in river discharge and understanding the main drivers of these changes is thus extremely important from theoretical and applied points of view. More specifically, here we want to draw attention toward the impacts of streamflow changes on reservoir storage and operation. We describe the hydrological dynamics of the Yesa reservoir draining catchment, located in the central Spanish Pyrenees, and characterize the reservoir operation modes over the last 60 years (1956–2020). We analyze concurrent climatic (precipitation, air temperature, drought index), atmospheric mechanisms, land cover (Normalized Different Vegetation Index) and discharge (inlet and outlet of Yesa reservoir) time‐series. By using the wavelet transform methodology, we detect historical breakpoints in the hydrological dynamics at different time‐scales. Distinctive periods are thus identified. More regular seasonal flows characterized the catchment's dynamics during the first decades of the study period, while the last decades were characterized by a high inter‐annual variability. These changes are primarily attributed to the natural re‐vegetation process that the catchment experienced. Furthermore, we related changes in atmospheric circulation with a decline of the long‐term discharge temporal features. This research contributes to the understanding of long‐term river discharge changes and helps to improve the reservoir management practices.
... The analytic Morlet wavelet transform was utilized to analyze the time series data comprising nonstationary power at diverse frequencies [25,26]. Cross wavelet power of paired head and eye velocities during vHIT was defined as wavelet transformation of the autocorrelation function and showed area with high coherent power in time-frequency JTEHM-00098-2022 domain. ...
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italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Objective: This study aimed to determine the impact on hearing prognosis of the coherent frequency with high magnitude-squared wavelet coherence (MSWC) in video head impulse test (vHIT) among patients with sudden sensorineural hearing loss with vertigo (SSNHLV) undergoing high-dose steroid treatment. Design: This study was a retrospective cohort study. Setting: SSNHLV patients treated at our referral center from December 2016 to December 2020 were examined. Participants: The cohort comprised 64 patients with SSNHLV undergoing high-dose steroid treatment. Main outcome measures: MSWC was measured by calculating the wavelet coherence analysis (WCA) at various frequencies from a vHIT. The hearing prognosis were analyzed using a multivariable Cox regression model and convolution neural network (CNN) of WCA. Results: There were 64 patients with a male-to-female ratio of 1:1.67. The greater highest coherent frequency of the posterior semicircular canal (SCC) was associated with the complete recovery (CR) of hearing. After adjustment for other factors, the result remained robust (hazard ratio [HR] 2.11, 95% confidence interval [CI] 1.86-2.35). In the feature extraction with Resnet-50 and proceeding SVM in the horizontal image cropping style, the classification accuracy [STD] for (CR vs. partial + no recovery [PR + NR]), (over-sampling of CR vs. PR + NR), (extensive data extraction of CR vs. PR + NR), and (interpolation of time series of CR vs. PR + NR) were 83.6% [7.4], 92.1% [6.8], 88.9% [7.5], and 91.6% [6.4], respectively. Conclusions: The high coherent frequency of the posterior SCC was a significantly independent factor that was associated with good hearing prognosis in the patients who have SSNHLV. WCA may be provided with comprehensive ability in vestibulo-ocular reflex (VOR) evaluation. CNN could be utilized to classify WCA, predict treatment outcomes, and facilitate vHIT interpretation. Feature extraction in CNN with proceeding SVM and horizontal cropping style of wavelet coherence plot performed better accuracy and offered more stable model for hearing outcomes in patients with SSNHLV than pure CNN classification.
... It is an integral transformation that allows obtaining the time and frequency information about a signal. Wavelet analysis helps analyze the harmonics of periods from hours to months in the time series data (Collineau & Brunet, 1993;Torrence & Compo, 1998). By using WT, we can decompose a time series into time-frequency space and determine both dominant modes of variability and how those modes vary with time. ...
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The variability of PM2.5 concentrations obtained from the air quality monitoring stations (AQMS) established at six different environments of the Pune Metropolitan Region (PMR), situated in the western part of India, is analyzed for the period 2014–2018. The PM2.5 concentrations showed an increasing trend at almost all locations within the city during the 5 years. Significant features observed were that the green/background location showed a declining trend in PM2.5 concentrations. However, the city's industrial area indicated an increase in PM2.5 concentrations over the years. The seasonal bivariate plot of PM2.5 showed that the winter season has the highest concentration, and also at low wind speeds a high concentration was observed, indicative of the local sources. Concentrated weighted trajectory analysis indicated that regional sources due to long-range transport also played a role in the PM2.5 mass concentration. The wavelet power spectrum of PM2.5 showed 2–4 day oscillations and 30–50 day oscillations associated with Madden-Julian oscillations.
... The statistical significance of correlations is computed accounting for the effective degrees of freedom of the series following Dawdy and Matalas (1964). Wavelet analysis is performed using the toolbox by Torrence and Compo (1998) and Grinsted et al. (2004). The Morlet function (w 0 = 6) is used as the mother wavelet. ...
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A vivid scientific debate exists on the nature of the Atlantic Multidecadal Variability (AMV) as an intrinsic rather than predominantly forced climatic phenomenon, and on the role of ocean circulation. Here, we use a multi-millennial unperturbed control simulation and a Holocene simulation with slow-varying greenhouse gas and orbital forcing performed with the low-resolution version of the Max Planck Institute Earth System Model to illustrate thermohaline conditions associated with twelve events of strong AMV that are comparable, in the surface anomalies, to observations in their amplitudes (~ 0.3 °C) and periods (~ 80 years). The events are associated with recurrent yet spatially diverse same-sign anomalous sea-surface temperature and salinity fields that are substantially symmetric in the warm-to-cold and following cold-to-warm transitions and only partly superpose with the long-term spatial AMV pattern. Subpolar cold-fresh anomalies develop in the deep layers during the peak cold phase of strong AMV events, often in association with subtropical warm-salty anomalies yielding a meridional dipole pattern. The Atlantic meridional overturning circulation (AMOC) robustly weakens during the warm-to-cold transition of a strong AMV event and recovers thereafter, with surface salinity anomalies being potential precursors of such overturning changes. A Holocene simulation with the same model including volcanic forcing can disrupt the intrinsic AMV–AMOC connection as post-eruption periods often feature an AMOC strengthening forced by the volcanically induced surface cooling. Overall, our results support the AMV as a potential intrinsic feature of climate, whose episodic strong anomalous events can display different shades of spatial patterns and timings for the warm-to-cold and subsequent cold-to-warm transitions. Attribution of historical AMV fluctuations thus requires full consideration of the associated surface and subsurface thermohaline conditions and assessing the AMOC–AMV relation.
... The wavelet transform is used to analyze time series containing nonstationary power with 174 different frequencies (Torrence and Compo, 1998). The wavelet transform of a variable xn is defined 175 as the convolution of xn with the scaled and normalized wavelet 176 ...
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This study aims to investigate characteristics of continental shelf wave (CSW) on the 15 northwestern continental shelf of the South China Sea (SCS) induced by winter storms in 2021. 16 Mooring and cruise observations, tidal gauge data at stations Hong Kong (HK), Zhapo (ZP) and 17 Qinglan (QL) and sea surface wind data from January 1 to February 28, 2021 are used to examine 18 the relationship between along-shelf wind and sea level fluctuation. Two events of CSWs driven by 19 the along-shelf sea surface wind are detected from wavelet spectra of tidal gauge data. The signals 20 are triply peaked at periods of 56, 94 and 180 h, propagating along the coast with phase speed 21 ranging from 6.9 to18.9 m s-1. The dispersion relation shows their property of the Kelvin mode of 22 CSW. We develop a simple method to estimate amplitude of sea surface fluctuation by along-shelf 23 wind. The results are comparable with the observation data, suggesting it is effective. The mode 2 24 CSWs fits very well with the mooring current velocity data. The results from rare current help to 25 understand wave-current interaction in the northwestern SCS. 26 27
... In fact wavelet transform displays a variable timefrequency resolution and is able to highlight the changes over time (i.e. the non-stationarity) of the frequencies contributions. [27][28][29][30][31][32][33][34][35] Wavelets can detect in the time-frequency (scale) plane both long-period backgrounds (trends) and short-period discontinuities (anomalies). So, anomalies represent high-frequency hidden signals and -despite their limited temporal location -possess a huge amount of information content. ...
Chapter
The Covid-19 pandemic has spread across the world at a rate never seen before, affecting different countries and having a huge impact not only on health care systems but also on economic systems. Never as in this situation the continuous exchange of views between scientists of different disciplines must be considered the keystone to overcome this emergency. The dramatic global situation has prompted many researchers from different fields to focus on studying the Covid-19 pandemic and its economic and social implications in a multi-facet fashion. This volume collects the contributions to the COVid-19 Empirical Research (COVER) Conference, organized by the Centre of Excellence in Economics and Data Science of the Department of Economics, Management and Quantitative Methods, University of Milan, Italy, October 30th, 2020. This conference aimed to collect different points of view by opening an interdisciplinary discussion on the possible developments of the pandemic. The conference contributions ranged in the social, economic and mathematical-statistical areas
... (Zakaria et al., 2016) stated that the wavelet transform helps investigate transient phenomena because it can extract temporal and frequency data from the transient signal. The concept of wavelet analysis and contrast made using Fourier analysis has been recorded in research conducted by (Torrence & Compo, 1998). The wavelet transform was established, and it has been employed in several lightning-related research. ...
Conference Paper
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In this study, only the first return stroke of CGs and NBPs lightning waveform are thoroughly analyzed using two different methods in wavelet and temporal. 4 parameters which are the pulse duration (PD), rise time (RT), zero-crossing time (ZCT) and full width half maximum (FWHM) for temporal analysis meanwhile the frequency and power spectrum are the parameters analyzed for wavelet analysis respectively. Comparison according to criteria were included as well as to differentiate lightning properties. 50 data from negative cloud to ground (NCG), narrow negative bipolar pulse (NNBP) and narrow positive bipolar pulse (NPBP) were used in wavelet transform along with 40 data from positive cloud to ground (PCG). The result of the wavelet has shown that narrow bipolar pulses (NBPs) radiate energy at a higher frequency than any other cloud to ground (CGs) lightning. However, CGs has 4 times higher peak power spectrum than NBPs. In temporal analysis, 100 data from NCG, NNBP and NPBP have been analyzed accordingly, while PCG has only 40 data available to be analyzed. From the time domain characteristics, in most of the above-mentioned parameters the NBPs have much smaller value compared to the CGs.
... Wavelet transform is a widely applied method for the periodic phenomenon in nonstationary time series. The wavelet transform is a local transformation of frequency and time through multi-scale analysis using scaling and translation [12,46]. The method divides the time series into a subset of continuous or discrete wavelets, and each sub-signal plays a different role and has a unique behavior. ...
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Reliable long-term (decadal scale) streamflow prediction would provide significant planning information for water resources management, particularly in areas marked by significant variability at those time scales. In this study, a multi-model for prediction using four models that incorporate preprocessing methods along with data-driven forecast models coupled using the least absolute shrinkage and selection operator (LASSO) regression method is proposed. Models utilized complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) and wavelet transform (WT) as the decomposition methods and autoregressive (AR) and hidden Markov models (HMM) as the predictive method. The model is evaluated in a comparative analysis with a variety of models previously proposed for hydrological time series prediction. We compare the predictive skill of alternative data-driven models for average annual streamflow (3 ~ 15 years) prediction. Results indicate that the multi-model performed better than the other models, presenting lower values of MAE and RSME. This multi-model can be a reliable tool for forecasting, which can be explored for hydrological data that have remarkably nonlinear and nonstationary features.
... Wavelet coherence analysis was used to analyse the causality of two variables along a time series. It was popularised by Christopher Torrence, who initially applied it for meteorology in 1998 [31]. Over time, it has been applied to diverse fields, from Economics [32] to Medicine [33]. ...
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The aim of this study is to explore the causal relationship between the economy and the elderly population globally as well as continent-wise. This research was designed as a continent-wide study to investigate the differences between several regions simultaneously. The economy was measured by the Gross Domestic Product (GDP) per capita growth rate while the population aged above 65 as a percentage of the total was considered the elderly population. A panel dataset published by the World Bank for a period of six decades from 1961 to 2020 covering 84 countries was used as data for the analysis. Wavelet coherence was the methodology used for the study since it was considered suitable to present causality as well as the causal direction between the two variables for different sections during the six decades. Thereafter, Granger causality was applied for a cross-country analysis to gain further insights on the causality of individual countries over the years. Findings of the study reveal that the causality and its direction have been changing over time for most continents. Negative correlations with the leading variable interchanging with time are evident for the majority of the regions. Nevertheless, results indicate that in a global perspective, elderly population predominantly leads the economic growth with a positive correlation. Research approach allows ascertaining the short-term and medium-term changes that occurred concerning the direction of the relationship throughout the stipulated period of the study, which could not be drawn by any previous study. Even though region-wise literature is available on this topic, global studies for decades have not been conducted yet.
... Wavelet analysis has been applied to lidar data before to separate GWs of different orientations . Assuming nonstationarity on the straight flight legs a 2-D Morlet continuous wavelet transform (e.g., Torrence & Compo, 1998) was computed according to Chen and Chu (2017), using a Morlet oscillation parameter (k factor) of 2/π. We found a discretization starting at a spatial scale of 40 km while using 20 spatial and 30 angular scales sufficient in order to separate different slants of GW phase fronts. ...
Article
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Horizontal gravity wave (GW) refraction was observed around the Andes and Drake Passage during the SouthTRAC campaign. GWs interact with the background wind through refraction and dissipation. This interaction helps to drive midatmospheric circulations and slows down the polar vortex by taking GW momentum flux (GWMF) from one location to another. The SouthTRAC campaign was composed to gain improved understanding of the propagation and dissipation of GWs. This study uses observational data from this campaign collected by the German High Altitude Long Range research aircraft on 12 September 2019. During the campaign a minor sudden stratospheric warming in the southern hemisphere occurred, which heavily influenced GW propagation and refraction and thus also the location and amount of GWMF deposition. Observations include measurements from below the aircraft by Gimballed Limb Observer for Radiance Imaging of the Atmosphere and above the aircraft by Airborne Lidar for the Middle Atmosphere. Refraction is identified in two different GW packets as low as ≈4 km and as high as 58 km. One GW packet of orographic origin and one of nonorographic origin is used to investigate refraction. Observations are supplemented by the Gravity‐wave Regional Or Global Ray Tracer, a simplified mountain wave model, ERA5 data and high‐resolution (3 km) WRF data. Contrary to some previous studies we find that refraction makes a noteworthy contribution in the amount and the location of GWMF deposition. This case study highlights the importance of refraction and provides compelling arguments that models should account for this.
... The cross wavelet transform can detect the time-domain and frequency-domain variations and coupling oscillations of two given time series. The coherent wavelet transform focuses on the correlation of low-energy regions, which is used to measure the local correlation of two time series in time-frequency space [25]. A wavelet packet is provided in MATLAB. ...
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Vegetation information is a critical factor in regional environment management under climate change. In this study, a typical arid and semi-arid watershed on the Loess Plateau, the Zu Li River Basin (ZRB), was selected to study the long-term changes in vegetation cover and its drivers and impacts. Unlike existing normalized vegetation index (NDVI) products, which have coarse spatial resolution and short time horizons, this study used the 30 m Landsat dataset analyzed in the Google Earth Engine (GEE) to generate high-resolution and long-term NDVI data, which are the most ideal for monitoring vegetation dynamics using long-time-series data products. The results showed that the annual mean maximum NDVI (normalized vegetation index) in the ZRB increased during 1987–2021, with a significant (p < 0.05) increasing trend in most areas. Upstream vegetation cover increased more than midstream and downstream, but the increase was smaller. Precipitation in the ZRB area was significantly (p < 0.05) correlated with the NDVI series, except for the upstream pass area, where human activities played an important role. NDVI was significantly (p < 0.05) negatively correlated with runoff coefficient and sand content, indicating that vegetation cover was an important reason for the decrease in runoff coefficient and sand content.
... Most of the movement or the co-movement of the factor remained to be in low intensity. The black contour, based on the Monte Carlo simulations is represented on the far right corner of the figure and depicts the scarce significance level (Torrence and Compo 1998). These results of all three markets suggest that the fluctuations and the co-movement between the markets have been volatile at large and has had significant. ...
Conference Paper
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This research study compares and explores students’ perceptions (P), expectations (E), and quality gaps (QG) regarding service quality (SQ) of a public and private business school of Pakistan using SERVQUAL. Other objectives also included investigating the impact of demographic factors upon the students’ perceptions, institutional difference, and test the validity of SERVQUAL in Pakistan context. Using Random sampling technique, data was obtained from 190 respondents (students) from both B-schools through a structured instrument. A sample of 190 respondents including 100 respondents from Public B-school and 90 from private B-school was selected for this study. Cross Sectional Study design and structured SERVQUAL questionnaire was used to get feedback from respondents across five dimensions (regarding E and P) of SQ. Using Quantitative approach, data was analyzed through SPSS 23.0 by applying descriptive and inferential statistics. Results showed overall negative QG in public and private B-school with mean scores of -0.48 and -0.42 respectively and gap was also negative in 5 SERVQUAL dimensions. In public B-school, highest negative gap was reported in responsiveness (-0.63); followed by empathy (-0.62); reliability (-0.41); tangibles (-0.38); and assurance (-0.34). Similarly, in private B-school, highest gap was reported in empathy (-0.63); followed by responsiveness (-0.51); reliability (-0.50); assurance (-0.25); and tangibles (-0.23). Paired t-test results reported statistically significant difference between students’ P and E of SQ dimensions in both B-schools. The study findings revealed that students’ P are lower than their E resulting in QG and dissatisfaction. Hypothesis testing reported statistically no significant difference in terms of gender and qualification groups. Results also reported statistically significant institutional difference. As students’ P and E scores in private B-school were higher than public B-school. It was also concluded that SERVQUAL can be applied for quality assessment, gap analysis, and identification of problem areas (critical quality dimensions) in B-schools of Pakistan as findings supported previous study results. Findings of this study can guide management/policymakers of Business schools to improve quality and allocate resources based upon identified problem areas/gaps in SQ dimensions. Key Words: Service Quality, Expectations, Perceptions, Quality Gap, SERVQUAL, Business School.
... with a Morlet wavelet (e.g., Torrence and Compo, 1998) to compute spectral power and estimate the dominant vertical wavenumber at each altitude. The spectral power of [exp(−z/2H )(θ /θ )] is proportional to potential energy per unit volume. ...
Article
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Sudden stratospheric warmings (SSWs) have a long-lasting effect within the stratosphere as well as impacts on the underlying troposphere. However, sub-seasonal forecasts of the winter polar stratosphere fail to use their full potential for predictability as they tend to underestimate the magnitude and persistence of these events already within the stratosphere. The origin of this underestimation is unknown. Here, we demonstrate that the associated polar stratospheric cold bias following SSW events in sub-seasonal hindcasts can be halved by increasing vertical model resolution, suggesting a potential sensitivity to gravity wave forcing. While the predictability of the planetary Rossby wave flux into the stratosphere at lead times longer than a week is limited, the existence of a critical layer for gravity waves with a low zonal phase speed caused by the disturbed polar vortex provides predictability to the upper stratosphere. Gravity wave breaking near that critical layer can, therefore, decelerate the zonal flow consistently with anomalous subsidence over the polar cap leading to warmer temperatures in the middle polar stratosphere. Since the spectrum of gravity waves involves vertical wavelengths of less than 4000 m, as estimated by wavelet analysis, a high vertical model resolution is needed to resolve the positive feedback between gravity wave forcing and the state of the polar vortex. Specifically, we find that at a spectral resolution of TCo639 (approximate horizontal grid spacing of 18 km) at least 198 levels are needed to correctly resolve the spectrum of gravity waves in the ECMWF Integrated Forecasting System. Increasing vertical resolution in operational forecasts will help to mitigate stratospheric temperature biases and improve sub-seasonal predictions of the stratospheric polar vortex.
... The wavelet power spectrum on each ripple wavelength domain can give us the location and amplitude of each power spectrum, which illustrates the dominant wavelength. Detailed methods of the wavelet transform are introduced by Torrence and Compo (1998). Wavelet analysis was applied to each spatial time series of crest position of the ripple. ...
Article
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We investigate pathways of sediment diffusion for a Gaussian‐shaped sand mound subjected to monochromatic waves. Our unique results nearly close the sediment budget by quantifying each of the sediment transport processes responsible for mound diffusion associated with sediment flux due to slope driven transport and ripple migration. Downslope ripple progression was observed as ripples formed at the mound top advanced down the side slopes in a direction perpendicular to wave propagation. Once ripples formed on the sides of the mounds the ripples became pathways for sediment flux from the top to the bottom of the mound, persisting even after ripples reached the base of the mound as sediment avalanching due to gravity and mound slope. Lateral ripple migration caused ripples to migrate along the sides of the sand mound in a direction parallel to wave propagation. Once ripples reached the base of the mound, lateral migration of ripples caused spreading of sand around the sides of the mound. Lateral ripple migration was largely driven by ripple splitting caused by a large downslope sediment flux from the center of the mound that generated ripples with longer wavelengths than wave orbital hydrodynamics could support. To restore equilibrium between sediment and flow conditions, ripples with longer wavelengths continuously split and migrated laterally around the mound. Our results reflect the importance of slope driven transport, bed fluidization, and ripple dynamics on the larger scale diffusivity and suggest that slope driven and ripple driven sediment fluxes should be more explicitly included in sediment transport formulations.
... Wavelet analysis (Torrence and Compo 1998) of the reconstructed heavy precipitation totals indicates concentrations of time series variance in the El Nino/Southern Oscillation (ENSO) band from 3-to 8-years, but statistical significance is only episodic over the past 440years, often during the regimes of elevated frequency and intensity of precipitation extremes (e.g., 1650s, 1730s, 1780s, and 1860s; Figure 5d). No significant multi-decadal wavelet power is detected in the reconstruction. ...
Article
The variability of water year precipitation and selected blue oak tree-ring chronologies in California are both dominated by heavy precipitation delivered during just a few days each year. These heavy precipitation events can spell the difference between surplus or deficit water supply, and elevated flood risk. Some blue oak chronologies are highly correlated with water year precipitation (r = 0.84) but are equally well correlated (r = 0.82) with heavy precipitation totals ≥25.4 mm (one inch, ≈95 th percentile of daily totals, 1949-2004). The blue oak correlation with non-heavy daily totals is much weaker (<25.4 mm; r = 0.55). Consequently, some blue oak chronologies represent selective proxies for the temporal and spatial variability of heavy precipitation totals and are used to reconstruct the amount and number of days with heavy precipitation in northern California from 1582-2021. Instrumental and reconstructed heavy precipitation totals are strongly correlated with gridded atmospheric river related precipitation over the western United States, especially in central California. Spectral analysis indicates that instrumental heavy precipitation totals may be dominated by high frequency variability and the non-heavy totals by low frequency variance. The reconstruction of heavy precipitation is coherent with instrumental heavy totals across the frequency domain and include concentrations of variance at ENSO and biennial frequencies. Return period analyses calculated using instrumental heavy precipitation totals are representative of the return periods in the blue oak reconstruction despite the large differences in series length. Decadal surges in the amount, frequency, and inter-annual volatility of heavy precipitation totals are reconstructed, likely reflecting episodes of elevated atmospheric river activity in the past.
... In practice, the cross-wavelet power and the cross-wavelet transform must first be defined. To better understand the cross-wavelet transform, Torrence and Compo (1998) suggest that two-time series and as follows: ...
Article
We investigate the impact of macroeconomic surprise and uncertainty on G7 financial markets around COVID-19 pandemic using two real-time, real-activity indexes recently constructed by Scotti (2016). We applies the wavelet analysis to detect the response of the stock markets to the macroeconomic surprise and an uncertainty indexes and then we use NARDL model to examine the asymmetric effect of the news surprise and uncertainty on the equity markets. We conduct our empirical analysis with the daily data from January, 2014 to September, 2020. Our findings indicate that G7 stock markets are sensitive to the macroeconomic surprise and uncertainty and the effect is more pronounced at the long term than the short term. Moreover, we show that the COVID-19 crisis supports the relationship between the macroeconomic indexes and the stock prices. The results are useful for investment decision-making for the investors on the G7 stock indices at different investment horizons.
... Finally, following the method of Meyers, Kelly & O'Brien (1993), the relationship between the equivalent Fourier period and the wavelet scale can be derived analytically by substituting a cosine wave of known frequency into (4.3) and computing the scale k at which the wavelet power spectrum reaches its maximum. More details about the operative algorithms adopted to compute the wavelet transform in this work can be found in Torrence & Compo (1998). ...
Article
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A direct numerical simulation of an oblique shock wave impinging on a turbulent boundary layer at Mach number 2.28 is carried out at moderate Reynolds number, simulating flow conditions similar to those of the experiment by Dupont et al. (J. Fluid Mech., vol. 559, 2006, pp. 255–277). The low-frequency shock unsteadiness, whose characteristics have been the focus of considerable research efforts, is here investigated via the Morlet wavelet transform. Owing to its compact support in both physical and Fourier spaces, the wavelet transformation makes it possible to track the time evolution of the various scales of the wall-pressure fluctuations. This property also makes it possible to define a local intermittency measure, representing a frequency-dependent flatness factor, to pinpoint the bursts of energy that characterise the shock intermittency scale by scale. As a major result, wavelet decomposition shows that the broadband shock movement is actually the result of a collection of sparse events in time, each characterised by its own temporal scale. This feature is hidden by the classical Fourier analysis, which can only show the time-averaged behaviour. Then, we propose a procedure to process any relevant time series, such as the time history of the wall pressure or that of the separation bubble extent, in which we use a condition based on the local intermittency measure to filter out the turbulent content in the proximity of the shock foot and to isolate only the intermittent component of the signal. In addition, wavelet analysis reveals the intermittent behaviour also of the breathing motion of the recirculation bubble behind the reflected shock, and allows us to detect a direct, partial correspondence between the most significant intermittent events of the separation region and those of the wall pressure at the foot of the shock.
... For example Holm (2014) describes in great detail how delicate it can be to interpret spectral anal ysis results in terms of non-stationarity problems. Therefore, some analysis tools, as for example the wavelet analysis (Daubechies, 1992, Chui, 1992, Torrence & Compo 1998, were established. They assume stationarity only for short parts of the time series. ...
Chapter
The effects of in-cloud turbulence on the growth of cloud droplets by condensation and coalescence in low level water clouds are reviewed. It is shown that while the entrainment of dry air into clouds may explain the observed variability in the droplet spectra, and particularly in droplet concentration, doubt still exists as to whether the entrainment of subsaturated air can give rise to enhanced droplet growth by condensation, that is growth in excess of that expected in an adiabatic parcel. It is also shown that the increase in the length of the path of some of the droplets in stratocumulus due to turbulent updraughts can significantly increase the rate of production of drizzle sized droplets by coalescence. The effects of turbulence are generally believed to increase the collection kernels of cloud droplets, although the magnitude of the increase is in doubt. The impact of such changes on the growth of a population of cloud droplets by coalescence is sensitive to their magnitude. In particular, such changes may be capable of partially overcoming the barrier to growth which occurs between the condensation and coalescence processes in still air at a droplet radius of around 20 μm, although further detailed studies are required.
... To observe the waves, we use the dVTEC to do spectral analysis. To get information about the spectral content of the signal, we use a continuous wavelet transform based on the complex Morlet wavelet function to examine how it changes with time (Torrence & Compo, 1998). The wave-like fluctuation in VTEC which are generated within the Cone of Influence (COI) are TIDs. ...
Article
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We present a statistical study of global lightning activity and correlate this activity with ionospheric fluctuations in the Total Electron Content (TEC). Thunderstorms play a significant role in modulating the ionosphere's plasma distribution. Using the well‐known World Wide Lightning Location Network (WWLLN) system, we investigate the diurnal, monthly, and seasonal variations in lightning stroke numbers across six continents. Typically, the monsoon causes a significant increase in lightning occurrences, as reflected in the WWLLN observations. We chose a time frame of July–December 2019, which usually covers the peak and post‐monsoon seasons on all the continents. The diurnal lightning activity is observed to be highest in the early afternoon for all the locations. In addition, we observe hemispherical changes in lightning activity that appear opposite in nature. Using the Global Navigation Satellite System (GNSS)‐International GNSS Service data, we use the Vertical TEC (VTEC) and Rate of change Of TEC Index as proxies for the amplitude scintillation S4 index. Using the wavelet analysis of the small‐scale fluctuations in the VTEC profile, we provide a time series analysis of wave‐like TEC fluctuations alongside lightning stroke count as a proxy for thunderstorm activity. It is found that, in most cases, Traveling Ionospheric Disturbances (TIDs) or a local maximum of small‐scale fluctuation in VTEC (dVTEC) that could be coupled to atmospheric gravity waves are observed during periods of peak thunderstorm activity. Most of the locations show that the spectral power of TIDs and the stroke energy density are highly correlated.
... z v (p) is the confidence level related to the probability p, and v is the degree of freedom. The calculation program of cross wavelet analysis can be found in the study of Torrence and Compo [38], and the code can be downloaded from http:// noc.ac.uk/using-science/crosswavelet-wavelet-coherence. ...
Chapter
Analyzing the hydrological sequence from the non-stationary characteristics can better understand the responses and significances of changes in extreme rainfall to global climate change. Taking the plain area in the middle and lower reaches of the Yangtze River basin (MLRYRB) as the study area, this study adopted a set of extreme rainfall indices and used the Bernaola-Galvan Segmentation Algorithm (BGSA) method to test the non-stationarity of extreme rainfall events. The General Pareto Distribution (GPD) was used to fit extreme rainfall under different thresholds and the range of return period was calculated to select the optimal threshold of extreme rainfall. In addition, the cross-wavelet technique was used to explore the correlations of extreme rainfall with El Niño-Southern Oscillation (ENSO) and Western Pacific Subtropical High (WPSH) events. The results show that: (1) extreme rainfall under different thresholds had different non-stationary characteristics, and 40-60 mm was more suitable as the optimal threshold for extreme rainfall. (2) The GPD distribution could well fit the extreme rainfall in the MLRYRB. By comparing the uncertainty of the return period, 40-60 mm was also considered as the suitable optimal threshold for extreme rainfall; however, different sub-regions had different optimal thresholds. (3) ENSO and WPSH had significant periodic effects on the extreme rainfall in the MLRYRB. These findings highlighted the significance of non-stationary assumptions in hydrological frequency analysis, which were of great importance for hydrological forecasting and water conservancy project management.
... Periodic components in the GR log were detected using wavelet analysis (Torrence and Compo, 1998). Then, 80 m Locally Weighted Scatterplot Smoothing (LOESS) was conducted to remove the longterm trend (Fig. 2a). ...
... This Fourier transform can only decompose the series into their frequencies with no chance to distinguish between superimposed signals over its entire domain. Put differently, this transform is inapplicable to characterise the time-varying signal (Torrence & Compo, 1998). In this paper, we propose a new Wavelet-Time Varying Parameter-VAR (W-TVP-VAR) approach with the ability to simultaneously measure connectedness at multi-scale horizons. ...
Article
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This paper investigates the asymmetric, time, and frequency-based volatility spillovers in global IT industries. To this end, we introduce a new Wavelet-Time Varying Parameter-VAR (W-TVPVAR) approach to compute connectedness combined with the asymmetrical connectedness of (Barndorff-Nielsen et al., 2010) and (Baruník et al., 2016, 2017) at different frequencies. Daily stock prices of the IT sector in thirteen countries representing the top technologically advanced countries ranging from January 15, 2016, until June 24, 2022, are used. The empirical results show that the aggregate volatility is slowly transmitted across markets with an effect lasting more than twenty days. The result also supports the presence of asymmetrical transmission as downside spillovers dominate upside spillovers, regardless of the frequency. Furthermore, the time-varying spillover shows the dominance of downside spillovers in various crisis periods, especially during the pandemic. The time and frequency-based spillover indicate that the overall spillover increased during the recent COVID-19 pandemic crisis period, which is mostly driven by the short-term, suggesting that panic decisions and herd behavior result in extreme connectedness. These findings are helpful to participants and policymakers.
... Furthermore, there was also a significant coherence on the 2-3 day frequency band. A wavelet analysis was carried out with the available tool from Torrence and Compo (1998) for the MATLAB software. This tool can have a bias on short-period peaks and can produce a distorted power spectrum, underestimating these peaks. ...
... Finally, we employed the WTC analysis by the framework of Torrence and Compo (1998), given as follows: ...
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The wavelet transform is of interest for analysing non-stationary signals. The squared modulus of the wavelet transform leads to the wavelet spectrogram or scalogram. When signals are embedded in additive noise, it is important to study the estimation accuracy in terms of bias and variance. The mean and variance statistical properties of the wavelet spectrogram of a signal embedded in additive gaussian white noise are derived in this paper. Examples and simulation results are also presented.