Restoration of clipped seismic waveforms using projection onto convex sets method

Abstract

The seismic waveforms would be clipped when the amplitude exceeds the upper-limit dynamic range of seismometer. Clipped waveforms are typically assumed not useful and seldom used in waveform-based research. Here, we assume the clipped components of the waveform share the same frequency content with the un-clipped components. We leverage this similarity to convert clipped waveforms to true waveforms by iteratively reconstructing the frequency spectrum using the projection onto convex sets method. Using artificially clipped data we find that statistically the restoration error is ~1% and ~5% when clipped at 70% and 40% peak amplitude, respectively. We verify our method using real data recorded at co-located seismometers that have different gain controls, one set to record large amplitudes on scale and the other set to record low amplitudes on scale. Using our restoration method we recover 87 out of 93 clipped broadband records from the 2013 Mw6.6 Lushan earthquake. Estimating that we recover 20 clipped waveforms for each M5.0+ earthquake, so for the ~1,500 M5.0+ events that occur each year we could restore ~30,000 clipped waveforms each year, which would greatly enhance useable waveform data archives. These restored waveform data would also improve the azimuthal station coverage and spatial footprint.

On Unlimited Sampling and Reconstruction

“…Scenarios such as moving out of a tunnel in the day [11] or a headlight in the line-of-sight can cause the imaging sensors to saturate. Similarly, in scientific imaging systems such as ultrasound [12], radar [13] and seismic imaging [14], strong reflections or pulse echoes blind the sensor. In communication systems, clipping results in performance degradation [15].…”

“…and by plugging the same in (14), we obtain (15). 1) Recovering Higher Order Differences from Modulo Samples: The result of Lemma 2 and in particular, the inequality in (15) will be the key to our proposed recovery method.…”

Aerial Seismology Using Balloon-Based Barometers

“…The ground motion signal was found to be saturated in several of the geophones close to the hammer due to insufficient dynamic range. Although most of the geophones within a 12-m radius of the hammer were rendered unusable, we were able to recover usable data from many of the saturated geophones using the Projection Onto Convex Sets (POCSs) technique [23], [24]. Ground motion traces from multiple shots are shown in Fig.…”

Unlimited Sampling of Sparse Signals

“…Clipping of a bandlimited signal results in discontinuities which manifest as aliasing due to high frequency distortion in the Fourier domain [21]. In view of this, a number of numerical methods have been proposed in the literature [22][23][24][25], however, the exact link to sampling theory of bandlimited or sparse signals remains largely unclear. This problem is of specific practical relevance in the context of calibration, namely, the knowledge of the unknown kernel ψ is critical for accurate recovery of s K in sparse sampling models such as (1).…”