Based on the shape of the glitches (specifically clusters 1+2 and 4+5), we can reasonably hypothesize they originate from step-like discontinuities in the strain: down-steps for clusters 1+2 and up-steps for clusters 4+5, respectively.
Interestingly, the directions of these jumps appear to be nearly balanced. While the previously reported percentages showed a majority in clusters 1 and 2, Michal has acutely noticed that some of the 25-minute glitches (Cluster 3) may have been misclassified into Cluster 1, which would bring the distribution closer to parity between clusters 1+2 and 4+5.
Regarding the waveform shapes, we can show analytically how a step function leads to the observed patterns in the plots, both in Michal's (with a causal filter) and mine (with a zero-phase filter). Here's a brief derivation:
A unit step function u(t) (equal to 1 for t > 0, 0 otherwise) has a Laplace transform U(s) = L[u(t)]=1/s. The Virgo ASD behaves approximately like a high-pass filter, whose impulse response we denote g(t). Whitening the step function with the ASD, u(t)∗g(t), where L[g(t)] = 1/ASD(s), has a result similar to applying a high-pass filter. We consider, for simplicity, a first-order high-pass filter with transfer function G(s) = s/(s+ω0), where ω0 is the cut-off angular frequency. The transformed output is then: U(s)G(s) =1/(s+ω0), which corresponds to the time-domain function e^{−ω0t} u(t), that is, a decaying exponential starting at the step, the glitch. This matches the right-hand side of the waveforms in the plots for Cluster 4+5 (and is the opposite of Clusters 1+2).
When using a zero-phase (acausal) filter, such as the filtfilt
method (which applies the filter forward and backward), the result becomes symmetric: combining e^{−ω0t} u(t) with its time-reversed counterpart −e^{ω0t} u(−t), forming a double exponential. The same reasoning applies in reverse for down-step inputs.
To explain the oscillatory features seen in Michal's plots, we can model the whitening process using a more realistic second-order filter, with transfer function: s^2 / (s^2 + 2ζω0 + ω0^2), whose response depends also on the damping parameter ζ . There are three main cases when applied to the step function: underdamped (ζ<1, e.g. Butterworth filter with ζ= 1/sqrt(2)), that characterizes an oscillatory response, critically damped (ζ=1) and overdamped (ζ>1), both of which present no oscillations and an approximately exponential decay. For example, for a Butterworth filter, setting ω0 = 1 to simplify the expressions, the filtered step function is: s / (s^2 + sqrt(2) +1) = (s + 1/sqrt(2)) / ((s+1/sqrt(2))^2 +1/2 ) - (1/sqrt(2)) / ((s+1/sqrt(2))^2 +1/2 ), whose inverce Laplace transform can be read from this list: e^{-t/sqrt(2)} [cos(t/sqrt(2)) - sin(t/sqrt(2))] u(t).
Moving to simulated data, I’ve attached:
Figure 1: Plot of a step function,
Figure 2: Its whitened version using the Virgo PSD,
Figures 3 and 4: The filtered function by a high-pass (cut-off = 10 Hz) and band-pass filtering (10 -100 Hz).
For comparison, I also add the same plots for a filtered Dirac Delta function ( ~ 25 minute glitches): Figures 5-8.
In conclusion, the new class of mid-frequency glitches appears to be generated by step discontinuities in the strain. It is therefore quite appropriate to refer to them as "step glitches".
PS: I am no longer able to use codecogs to write equations in the logbook!! :-(