Detecting transient signals in geodetic time series using sparse estimation techniques

Abstract

Citation:Riel, B., M. Simons, P. Agram, and Z. Zhan (2014) Abstract We present a new method for automatically detecting transient deformation signals from geodetic time series. We cast the detection problem as a least squares procedure where the design matrix corresponds to a highly overcomplete, nonorthogonal dictionary of displacement functions in time that resemble transient signals of various timescales. The addition of a sparsity-inducing regularization term to the cost function limits the total number of dictionary elements needed to reconstruct the signal. Sparsity-inducing regularization enhances interpretability of the resultant time-dependent model by localizing the dominant timescales and onset times of the transient signals. Transient detection can then be performed using convex optimization software where detection sensitivity is dependent on the strength of the applied sparsity-inducing regularization. To assess uncertainties associated with estimation of the dictionary coefficients, we compare solutions with those found through a Bayesian inference approach to sample the full model space for each dictionary element. In addition to providing uncertainty bounds on the coefficients and confirming the optimization results, Bayesian sampling reveals trade-offs between dictionary elements that have nearly equal probability in modeling a transient signal. Thus, we can rigorously assess the probabilities of the occurrence of transient signals and their characteristic temporal evolution. The detection algorithm is applied on several synthetic time series and real observed GPS time series for the Cascadia region. For the latter data set, we incorporate a spatial weighting scheme that self-adjusts to the local network density and filters for spatially coherent signals. The weighting allows for the automatic detection of repeating slow slip events.

Single‐station automated detection of transient deformation in GPS time series with the relative strength index: A case study of Cascadian slow slip

“…A series of workshops were recently held on automated detection of transients to aid in regional earthquake hazard studies [Murray-Moraleda and Lohman, 2010]. Proposed methods of automated transient detection have included, but are not limited to, the network inversion filter [Segall and Matthews, 1997] and the subsequent network strain filter [Ohtani et al, 2010], principal component analysis [Dong et al, 2006;Ji and Herring, 2013], covariance descriptor analysis [Kedar et al, 2010], Gaussian wavelet transforms [Melbourne et al, 2005], template matching [Riel et al, 2014], and Multichannel Singular Spectrum Analysis [Walwer et al, 2016]. All of these methods have their individual strengths and weaknesses and aim to perform similar tasks.…”

Greedy Automatic Signal Decomposition and Its Application to Daily GPS Time Series

“…ξ (t) is the noise in the data and is here assumed to be normally distributed. In Riel et al (2014), the unexpected (transient) signal is modeled as a series of B-splines in linear combination with equation (1) and is retrieved by minimizing the number of transient functions that, in addition to the expected functions, solve the optimization problem:…”

“…This workflow is slow and prone to human error when the data sets are very large. Therefore, considerable research has been geared toward automatically separating transient signals of interest: In these studies, the cGPS signal has been decomposed or filtered by a variety of approaches such as singular value decomposition (e.g., principal component analysis/independent component analysis, PCA/ICA; Gualandi et al, ; Kositsky & Avouac, ), multichannel singular spectrum analysis (Walwer et al, ), linear regression (Riel et al, ), and Kalman filtering of state vectors (e.g., the network inversion filter; Segall & Matthews, ). For regression, onset times of sudden displacements (due to earthquakes, antenna changes, or reference frame shifts) are typically fed into the algorithm either by invoking jump times for Heaviside steps or jumps in the time series or by splitting the time series analysis before and after large, nearly instantaneous displacements or jumps.…”

“…Typically, the detection of these emergent transients involves the indentification of a spatially coherent anomaly (e.g., Bartlow et al, ). Of the various approaches to estimating or retrieving transient behavior, linear regression (e.g., Riel et al, ) is the only one that implies an expected signal. The expected background signal, in this case, is the extended trajectory model (ETM) of Bevis and Brown (), which we reformulate here to use exponential rather than logarithmic transients and also to eliminate the explicit use of a reference time: where displacement, x , as a function of time, t , is described using a series of secular, seasonal, step, and decay basis functions.…”

“…ξ ( t ) is the noise in the data and is here assumed to be normally distributed. In Riel et al (), the unexpected (transient) signal is modeled as a series of B‐splines in linear combination with equation and is retrieved by minimizing the number of transient functions that, in addition to the expected functions, solve the optimization problem: where vectors d and p contain the data and model predictions, vectors C a and C b contain the model coefficients for the permanent and sparse functions, and λ a and λ b are the regularization weights. The second and third terms in equation are the regularization of the inversion, and therefore, the B‐spline model parameters are regularized in the L1 sense.…”