Roberto Henry Herrera

Spectral estimation—What is new? What is next?

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.

Body Wave Separation in the Time-Frequency Domain

Separation of a seismogram into its individual constitutive phases (Pand S-wave arrivals, surface waves, etc.) is a long-standing problem. In this letter, we use a high-resolution time-frequency transform to achieve this and reconstruct their individual waveforms in the time domain. The procedure is illustrated using microseismic events recorded during a hydraulic fracturing treatment. The synchrosqueezing transform is an extension of the continuous wavelet transform combined with frequency reassignment. Its high-resolution time-frequency decompositions allow for separation and identification of Pand S-waves with subtly different frequency contents that would not be recoverable using short-term Fourier transforms due to its smearing in the frequency domain. It is an invertible transform, thus allowing for signal reconstruction in the time domain after signal separation. The same approach is applicable to other seismic signals such as resonance frequencies and long-period events and offers promising new possibilities for enhanced signal interpretation in terms of underlying physical processes.

Automatic approaches for seismic to well tying

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.

Short-time homomorphic wavelet estimation

Successful wavelet estimation is an essential step for seismic methods like impedance inversion, analysis of amplitude variations with offset and full waveform inversion. Homomorphic deconvolution has long intrigued as a potentially elegant solution to the wavelet estimation problem. Yet a successful implementation has proven difficult. Associated disadvantages like phase unwrapping and restrictions of sparsity in the reflectivity function limit its application. We explore short-time homomorphic wavelet estimation as a combination of the classical homomorphic analysis and log-spectral averaging. The introduced method of log-spectral averaging using a short-term Fourier transform increases the number of sample points, thus reducing estimation variances. We apply the developed method on synthetic and real data examples and demonstrate good performance.

Attenuation estimation using high resolution time–frequency transforms

Abstract Wave attenuation is often measured using spectral techniques such as the spectral ratio method and the frequency shift method, comparing the spectral content of pairs of waveforms along the ray path. The recent introduction of novel highly-localized time–frequency transforms leads to high-resolution but discontinuous spectra. It prevents the use of these time–frequency transforms with conventional attenuation measurement methods. We show how three highly-localized time–frequency transforms, namely basis pursuit, the synchrosqueezing wavelet transform, and complete ensemble empirical mode decomposition, can still be used to estimate attenuation using the peak frequency method. Assuming a Ricker source wavelet, the decrease in peak frequency of a wave spectrum as it propagates in a given medium is used to estimate attenuation. When applied to a synthetic benchmarking signal corrupted by Gaussian white noise, the three transforms show different degrees of performance and robustness for different signal-to-noise ratios. The developed methodology is suitable for geophysical investigations, but may also find application in other fields such as biomedicine, acoustics and engineering.

Automated Seismic-to-well Ties?

The quality of seismic-to-well tie is commonly quantified using the classical Pearsons correlation coefficient. However the seismic wavelet is time-variant, well logging and upscaling is only approximate, and the correlation coefficient does not follow this nonlinear behavior. We introduce the Dynamic Time Warping (DTW) to automate the tying process, accounting for frequency and time variance. The Dynamic Time Warping method can follow the nonlinear behavior better than the commonly used correlation coefficient. Furthermore, the quality of the similarity value does not depend on the selected correlating window. We compare the developed method with the manual seismic-to-well tie in a benchmark case study.

Spectral Decomposition by Synchrosqueezing Transform

The synchrosqueezing transform (SST) is a wavelet-based time-frequency reassignment method, which has a grounded mathematical foundation. It produces a well defined time-frequency representation allowing the identification of instantaneous frequencies to highlight individual components. The field data examples demonstrate the high time-frequency resolution feature of SST, therefore render this technique is promising for seismic processing and interpretation.

Analysis of time-varying signals using continuous wavelet and synchrosqueezed transforms

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 Mw 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’.

Comparison of Two Semi-automatic Techniques for Seismic-to-well Tying

Tying well logs to seismic data is a highly subjective task that relies on the interpreters experience and the similarity metric used. Automated alternatives could help reduce this degree of subjectivity by making the tie reproducible. In this paper we compare two automated techniques: the dynamic time warping method and the local similarity attribute based on regularized shaping filters. These two methods produce superior tying in a guided stretching and squeezing framework. Results using a real well log example validate both approaches. Automated seismic-to-well tie algorithms can greatly aid in seismic interpretation. It is important to emphasize however that they are based on goodness-of-fit criteria and do not measure correctness of a fit. Best practices in well-tying have to be followed for their results to be meaningful.

Spatial Resolution of the Nonstationary Phase Estimation Method - Ketzin Case Study

Local phase analysis can serve as a complementary tool in seismic interpretation because amplitude, peak frequency and phase of the locally observed wavelet are determined by the local reflectivity that is by layer thickness, type of impedance contrast, and boundary shape. To estimate the local phase, statistical methods like kurtosis-based phase estimation, can be applied. Their advantage is that they do not require well logs and analyze the seismic data directly, which allows for instance to analyze whether spatial and/or temporal variations occur in the amplitude and phase spectrum of the seismic wavelet. Here, we investigate the spatial resolution of the kurtosis-based phase estimation for the Ketzin 3D seismic data set from the Northeast German Basin and show that the decrease of phase estimation block size allows to estimate the spatial variations in the local phase and following also the local geology with high resolution. The phase distribution follows the geological structure of the anticline visible already in the amplitudes and may reveal potentially additional details to conventional amplitude imaging.