Spectral estimation, and corresponding time-frequency representation for nonstationary signals, is a cornerstone in geophysical signal processing and interpretation. The last 10-15 years have seen the development of many new high-resolution decompositions that are often fundamentally different from Fourier and wavelet transforms. These conventional techniques, like the short-time Fourier transform and the continuous wavelet transform, show some limitations in terms of resolution (localization) due to the trade-off between time and frequency localizations and smearing due to the finite size of the time series of their template. Well-known techniques, like autoregressive methods and basis pursuit, and recently developed techniques, such as empirical mode decomposition and the synchrosqueezing transform, can achieve higher time-frequency localization due to reduced spectral smearing and leakage. We first review the theory of various established and novel techniques, pointing out their assumptions, adaptability, and expected time-frequency localization. We illustrate their performances on a provided collection of benchmark signals, including a laughing voice, a volcano tremor, a microseismic event, and a global earthquake, with the intention to provide a fair comparison of the pros and cons of each method. Finally, their outcomes are discussed and possible avenues for improvements are proposed.

Time-frequency representation of seismic signals provides a source of information that is usually hidden in the Fourier spectrum. The short-time Fourier transform and the wavelet transform are the principal approaches to simultaneously decompose a signal into time and frequency components. Known limitations, such as trade-offs between time and frequency resolution, may be overcome by alternative techniques that extract instantaneous modal components. Empirical mode decomposition aims to decompose a signal into components that are well separated in the time-frequency plane allowing the reconstruction of these components. On the other hand, a recently proposed method called the “synchrosqueezing transform” (SST) is an extension of the wavelet transform incorporating elements of empirical mode decomposition and frequency reassignment techniques. This new tool produces a well-defined time-frequency representation allowing the identification of instantaneous frequencies in seismic signals to highlight individual components. We introduce the SST with applications for seismic signals and produced promising results on synthetic and field data examples.

The continuous wavelet transform (CWT) has played a key role in the analysis of time-frequency information in many different fields of science and engineering. It builds on the classical short-time Fourier transform but allows for variable time-frequency resolution. Yet, interpretation of the resulting spectral decomposition is often hindered by smearing and leakage of individual frequency components. Computation of instantaneous frequencies, combined by frequency reassignment, may then be applied by highly localized techniques, such as the synchrosqueezing transform and ConceFT, in order to reduce these effects. In this paper, we present the synchrosqueezing transform together with the CWT and illustrate their relative performances using four signals from different fields, namely the LIGO signal showing gravitational waves, a 'FanQuake' signal displaying observed vibrations during an American football game, a seismic recording of the 8.2 Chiapas earthquake, Mexico, of 8 September 2017, followed by the Irma hurricane, and a volcano-seismic signal recorded at the Popocatépetl volcano showing a tremor followed by harmonic resonances. These examples illustrate how high-localization techniques improve analysis of the time-frequency information of time-varying signals.This article is part of the theme issue 'Redundancy rules: the continuous wavelet transform comes of age'.

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We evaluated a semiautomatic method for well-to-seismic tying to improve correlation results and reproducibility of the procedure. In the manual procedure, the interpreter first creates a synthetic trace from edited well logs, determines the most appropriate bulk time shift and polarity, and then applies a minimum amount of stretching and squeezing to best match the observed data. The last step resembles a visual pattern recognition task, which often requires some experience. We replaced the last step with a constrained dynamic time-warping technique, to help guide the interpreter. The method automatically determined the appropriate amount of local stretching and squeezing to produce the highest correlation between the original data and the created synthetic trace. The constraint ensured that stretching and squeezing were kept within reasonable bounds, as determined by the interpreter. Results compared well with the manual method, leading to ties along the entire trace length in contrast to the shorter analysis window in the conventional method. Yet, we advise against unsupervised applications because the method is intended as a guide instead of a fully automated blind approach.

In this paper, the inverse problem of reconstructing reflectivity function of a medium is examined within a blind deconvolution framework. The ultrasound pulse is estimated using higher-order statistics, and Wiener filter is used to obtain the ultrasonic reflectivity function through wavelet-based models. A new approach to the parameter estimation of the inverse filtering step is proposed in the nondestructive evaluation field, which is based on the theory of Fourier-Wavelet regularized deconvolution (ForWaRD). This new approach can be viewed as a solution to the open problem of adaptation of the ForWaRD framework to perform the convolution kernel estimation and deconvolution interdependently. The results indicate stable solutions of the estimated pulse and an improvement in the radio-frequency (RF) signal taking into account its signal-to-noise ratio (SNR) and axial resolution. Simulations and experiments showed that the proposed approach can provide robust and optimal estimates of the reflectivity function.

Tying the synthetic trace to the actual seismic trace at the well location is a labor-intensive task that relies on the interpreter's experience and the similarity metric used. The traditional seismic to well tie suffers from subjectivity by visually matching major events and using global crosscorrelation to measure the quality of that tying. We compared two automatic techniques that will decrease the subjectivity in the entire process. First, we evaluated the dynamic time warping method, and then, we used the local similarity attribute based on regularized shaping filters. These two methods produced a guided stretching and squeezing process to find the best match between the two signals. We explored the proposed methods using real well log examples and compared to the manual method, showing promising results with both semiautomatic approaches. He has received a number of professional awards, including the J. Clarence Karcher Award from SEG in 2001 and the Conrad Schlumberger Award from EAGE in 2011. He devotes part of his time to developing "Madagascar," an open-source software package for geophysical data analysis. Mirko van der Baan received a degree (1996) from the University of Utrecht, the Netherlands, and a Ph.D. (honors) (1999) from the Joseph Fourier University in Grenoble, France, and an HDR (Habilitation) from University Denis Diderot, Paris, France. He then joined the University of Leeds, UK, where he became the reader of exploration seismology. He is a professor at the University of Alberta in the Department of Physics, specializing in exploration seismology. He is the principal SD108 Interpretation /