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<p>Irregularly sampled geophysical time series are common in practice due to data gaps caused by sensor outages, environmental disturbances, or quality control procedures. Conventional digital filters, such as Fourier filters, require complete time series data and therefore cannot be directly applied to unevenly spaced noisy data without prior interpolation. In this study, we demonstrate that filtering unevenly spaced time series using Fourier filtering is inherently an ill-posed problem, manifested as rank deficiency in the associated parametric model. Building on this insight, we propose a minimum norm least squares Fourier filtering (MFF) that processes unevenly spaced time series without the need for preliminary data interpolation. Additionally, the prior covariance matrix of the time series is incorporated to further improve the filtering performance. We first apply the proposed method to extract deformation signals from daily position time series of 27 global navigation satellite system (GNSS) monitoring stations across the mainland China spanning from 1999 to 2024. The performance of MFF is compared with conventional Fourier filtering (CFF) with interpolation. The results demonstrate that MFF outperforms CFF, especially when prior precision is considered, as evidenced by a smaller fitting error of the extracted signals. Simulations confirm that signals recovered by MFF are closer to the true signals, with root mean square error (RMSE) reductions of 12.3 to 19.4% across the 27 stations, depending on the percentage of missing data. Incorporating formal errors provides an additional average RMSE reduction of 2.9%. Finally, we apply MFF to retrieve mass change signals from monthly gravity recovery and climate experiment (GRACE) and GRACE-FO gravity field solutions. The results agree with those from GNSS time series and show that MFF outperforms CFF in extracting components within desired frequency bands.</p>

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Filtering Unevenly Spaced Geophysical Time Series as an Ill-Posed Problem

  • 期刊论文

  • 发表日期：

    2026年1月13日

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  • Kunpu Ji,
  • Lin Zhang,
  • Fengwei Wang

摘要

Irregularly sampled geophysical time series are common in practice due to data gaps caused by sensor outages, environmental disturbances, or quality control procedures. Conventional digital filters, such as Fourier filters, require complete time series data and therefore cannot be directly applied to unevenly spaced noisy data without prior interpolation. In this study, we demonstrate that filtering unevenly spaced time series using Fourier filtering is inherently an ill-posed problem, manifested as rank deficiency in the associated parametric model. Building on this insight, we propose a minimum norm least squares Fourier filtering (MFF) that processes unevenly spaced time series without the need for preliminary data interpolation. Additionally, the prior covariance matrix of the time series is incorporated to further improve the filtering performance. We first apply the proposed method to extract deformation signals from daily position time series of 27 global navigation satellite system (GNSS) monitoring stations across the mainland China spanning from 1999 to 2024. The performance of MFF is compared with conventional Fourier filtering (CFF) with interpolation. The results demonstrate that MFF outperforms CFF, especially when prior precision is considered, as evidenced by a smaller fitting error of the extracted signals. Simulations confirm that signals recovered by MFF are closer to the true signals, with root mean square error (RMSE) reductions of 12.3 to 19.4% across the 27 stations, depending on the percentage of missing data. Incorporating formal errors provides an additional average RMSE reduction of 2.9%. Finally, we apply MFF to retrieve mass change signals from monthly gravity recovery and climate experiment (GRACE) and GRACE-FO gravity field solutions. The results agree with those from GNSS time series and show that MFF outperforms CFF in extracting components within desired frequency bands.