extracting raw dt (time shift) or Vref

[wgajek]

Hello,
I am working with a dataset where interstation distance varies over time. Hence, to obtain a correct dv/v value, it is necessary to update the distance for each day. For this, I need the measured Δt, however, I was unable to find where Δt is stored. So my question is:
If absolute Δt values are stored somewhere, I only need the information on how to access them. Are they stored?

If not stored, it is still possible to recompute the Δt values using the formula if knowing reference velocity:
dv_v =− Δt/t = − Δt⋅v_ref/Dist,
hence:
Δt=− V_ref ⋅ dv_v ⋅ Dist,
where:
Δt is the time shift, which I am looking for,
t is the traveltime
dv_v is dv.value, a resulting dv/v
Dist is an interstation distance in km

I could get the Δt value if I knew Vref. So here comes the second question: How is the Vref computed, and if it is stored anywhere? Or the code is taking V_ref from the params.yaml file, even if compute_tt is set to False?

In params.yaml, handling the velocity is defined as follows, suggesting with compute_tt = False, rayleigh_wave_velocity is ignored:

compute_tt : False  # Computes the travel time and adds it to tw_start (tt is 0 if not xstations). If true a rayleigh wave velocity has to be provided
rayleigh_wave_velocity : 1.65  # rayleigh wave velocity in km/s, will be ignored if compute_tt=False

I understand this is probably not a typical application of the code. Many thanks for any tips on tackling this problem.

[PeterMakus]

Hi,
Oh that is an interesting question!

The methodology does not compute v_ref, nor is it required.

I think you will unfortunately not get around making some changes to the source code.

So here is what you want to do:
As the distance is changing, the theoretical time of arrival will change. For the stretching technqiue, this means you will want to have a dynamic tw_start parameter, where tw_start=time_of_arrival(t)+tw_start_static. (tw is the time-window, in which the stretching is applied).

The problem is that your time of arrival needs to be determined precisely. I would suggest doing this by aligning (i..e, shifting) correlation functions over delay times. You can have a look at SeisMIC's implementation to determine clock-drifts, which solve a similar problem (although not identical).
Also, I would have a look at the first few lines of Monitor.compute_velocity_change() in seismic.monitor.monitor, where the arrival time correction is executed (albeit for a constant time of arrival)

I am afraid there is no out-of-the-box way to solve this and it is really not a super trivial issue. I hope this small explanation helped.

[wgajek]

I appreciate your response. Took me some time to cross-check some assumptions.

You suggested manipulating the tw_start. Actually, the change of interstation distance in my case is small - up to 2%, hence, comparable to typically expected dv/v (I forgot to mention it before). That brings me then to another question:
As the stretching technique looks for most suitable stretch factor to match the waveforms, is it possible that a 2% expansion of interstation distance (assuming the medium parameters constant and tw_start and tw_length long enough to accommodate delayed waveforms) will not result in any observable dv/v variation? (waveform will be still matched with stretching factor of 0, however, with a certain delay = dDist/v)
If this reasoning is correct (please correct me if I'm mistaken) then the resulting dv/v should be accounted as changes in the medium (resulting from medium expansion, like e.g., cracking) without the influence of distance expansion (which is exactly what I am looking for).

[PeterMakus]

I realised that my initial answer was perhaps misleading. Actually, dv/v and ds/s (the distance change) will be practically indistinguishable. Ultimately, we stretch the time domain of the waveform (i.e., x(psi t)), where psi=1+kappa (kappa is the stretching factor). As t=v*s (assuming homogeneous medium), psi is constituted by both the change in distance and the change in velocity. The common assumption would be that s is constant -> kappa=-dv/v (approximately, see the supplement of our SeisMIC paper in Seismica). If that is not the case, then t' (the stretched time domain) can be approximated as t'=(1-dv/v)*v*(1+ds/s)*s, so psi=(1-dv/v)*(1+ds/s)=1-dv/v+ds/s-dv/v*ds/s -> kappa=-dv/v+ds/s-dv/v*ds/s. The code extracts kappa (referred to as value). But in the end, you are left with two unknowns.

My suggestion would be to use three stations to distinguish between the two. OR: combine autocorrelations/self-correlations and cross-station-correlations (distance changes don't show up on auto and self-correlations :) ).

I hope that this provided some enlightenment.

[wgajek]

Thanks a lot for the detailed explanation.
I got all I needed.