Introduction

Spectral estimation and corresponding time-frequency analysis for nonstationary signals is a cornerstone in geophysical data analysis (Tary et al., 2014), and has been widely applied in various geophysical data processing tasks (Yao and Liu, 2022; Chen, 2021; Bai et al., 2022; Mohammadigheymasi et al., 2022; Tan et al., 2022). The short-time Fourier transform (STFT) is a fundamental technique for time-frequency analysis (Liu et al., 2016; Allen, 1977; Lu and Li, 2013). This method involves applying a series of short tapers to the input signal and subsequently performing a Fourier transform on each tapered segment. The spectrum provides localized time-frequency distributions for each data point and displays a detailed view of the frequency characteristics over time. The continuous wavelet transform (CWT) (Chakraborty and Okaya, 1995) is another widely used alternative for time-frequency analysis. CWT replaces the length-fixed taper in STFT with mother wavelets that can be shifted in time and stretched by a scale, thus yielding a variable and concentrated time-frequency map of the input time series. The S-transform (Stockwell et al., 1996; Liu et al., 2019; Wu et al., 2023) combines the properties of both STFT and CWT, and can be viewed as a middle ground between these two methods. It employs a time-shifted and squeezed Gaussian taper, which is analogous to the mother wavelet used by CWT, while keeps a uniform frequency sampling and retains the original signal phase Stockwell (2007). Recently developed methods, such as matching pursuit (MP) Mallat and Zhang (1993), basis pursuit (BP) (Chen et al., 2001), Wigner-Vile distribution (Boashash and Black, 1987), empirical mode decomposition (EMD) (Huang et al., 1998; Zhang et al., 2020; Flandrin et al., 2004), the synchrosqueezing transform (Daubechies et al., 2011), sparse time-frequency transform (Yang et al., 2021; Gholami, 2012) and the deep learning (DL)-based method (Yang et al., 2022; Liu et al., 2023; Qian et al., 2022) have been designed to mitigate issues such as spectral smearing and leakage. These methods aim to achieve superior time-frequency localization, which enhances the precision and clarity of signal analysis. Although these methods can be powerful tools for spectra estimation, their fixed frequency sampling and stationary time-frequency localization limits their applications in nonstationary seismic data processing. Additionally, they often suffers in high computational costs, which make them less suitable for complex tasks such as massive seismic data processing.

Local attributes method employs regularized nonstationary regression to decompose input data into a number of nonstationary components (Fomel, 2007a), and is proved an effective tool to measure nonstationary seismic data characteristics and has been successfully applied to seismic data stacking (Liu et al., 2011a), noise attenuation (Liu et al., 2011a; Zheng et al., 2022), image registration (Fomel and Jin, 2009) and phase detection (Fomel and van der Baan, 2010). Liu et al. (2011b) applied shaping regularized nonstationary regression for computing nonstationary local frequency attributes. They used the Gaussian smoothing operator to control the time-frequency localization and avoids the window strategy used in those conventional methods. Liu and Fomel (2013) expanded the local frequency attributes (Liu et al., 2011b) by designing an invertible nonstationary time-frequency decomposition, which is effective in practical processing tasks. Chen (2021) further developed the nonstationary local time-frequency transform method by applying smoothness constraints in all physical dimensions, which ensures a higher resolution and antinoise ability of the LTF decomposition method. Although the local attributes and LTF analysis methods prove to be effective in seismic data analysis and processing (Liu et al., 2011b; Chen, 2021; Liu and Fomel, 2013), their high computational cost brought by iterative inversion framework limits their practical applications. To improve calculation efficiency, Geng et al. (2024) employed the streaming algorithm (Fomel and Claerbout, 2016) to estimate local seismic attributes. Instead of using the shaping regularization, Geng et al. (2024) incorporated streaming computation with local attributes to obtain time-frequency map. However, the direct application of the streaming algorithm leads to a global frequency map rather than a local one, so Geng et al. (2024) implemented a window strategy to ensure localization.

The streaming method, first proposed by Fomel and Claerbout (Fomel and Claerbout, 2016,2024), is an efficient non-iterative approach to compute nonstationary prediction-error filters (PEFs). Geng et al. (2024) proposed an efficient streaming seismic attributes method by employing the streaming method to solve the nonstationary regression problem in local frequency attributes (Liu et al., 2011b). In this paper, we expand the streaming seismic attributes method (Geng et al., 2024) by improving the streaming calculation and designing an invertible streaming local time-frequency transform (SLTFT). We tailor the streaming computation by adding a localization scalar to control the localization of the time-frequency spectrum, which avoids the window strategy and provides an adaptively localized time-frequency map. The proposed SLTFT is computational efficient with its forward and inverse transform requiring only elementary algebraic operations and no iterations. The benchmark signal example indicate that the proposed SLTFT can provide more flexible time-frequency resolution than the classical S-transform and STFT. Field data tests including ground-roll attenuation, inverse-Q filtering and multicomponent data registration indicate that the proposed SLTFT provides an effective tool for analyzing and processing nonstationary seismic data.