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Hi Marc
In the presence of an ellipticity curve with singularities, the HV-DC joint inversion is sufficient to be done considering the right flank of the peak excluding the maximum and minimum values. Is this condition the essential one or is it accepted that the entire ellipticity curve can also be inverted?
In the example that I submit to you, the best misfit is achieved with the inversion of the complete curve and also the ground profile is more in tune with the geological context than the other.
What I observe, and also ask for your evaluation on this, the ellipticity curves calculated with the inversion show in both cases values from infinity to zero for Rayleigh and Love. Is this due to the strong contrast of Vs between bedrock (>2000m/s in the ground profiles) and soft layer or ...?
Finally I noticed that applying the smoothing (konno & ohmachi 40%) to ellipticity curve, obtained with HVTFA m = 8, nppm = 4, the standard deviation bars disappear.
Thanks for your attention
Regards
Luigi

Is this condition the essential one or is it accepted that the entire ellipticity curve can also be inverted?

With HVTFA or Raydec (single station techniques for ellipticity extraction) at high frequency, we usually observe (on synthetics where a ground truth exists) contributions from higher modes, that are not properly separated. Trying to fit them with fundamental mode might be misleading. At low frequency below the peak, the amplitude of the experimental ellipticity might not be correct but it helps in obtaining a correct peak frequency. I did not re-checked but I think that Hobiger et al. (2013 in GJI) discussed this point.

In your "All" case, the dispersion curve looks like better adjusted than in "dx_flank" case, but if you look carefully at the right flank of the ellipticity the inverted results underestimate the measured one. In "dx_flank" case, the experimental ellipticity is perfectly followed whereas the dispersion curve around 4 Hz is not so well reproduced. Because there are more samples to fit in the "All" case and because the complex shape cannot be fully reproduced, the ellipticity misfit component is certainly higher in "All" case than in "dx_flank" case, and it is thus more tolerant which allows a better a fit of the dispersion curve. The better global misfit in "all" case is probably due to the better fit of the dispersion curve. This should be eventually checked on the best model. Extract it for both cases with gpdcreport -best 1. Send it together with the two corresponding .target files. I can re-compute the misfit in the same way as during the inversion and decompose it. In principle, ellipticity angles should not be compared on a relative scale, which is apparently the case when I read the code. Playing with your case might be instructing.

I did not compare the exact details of the shallow part but they look quite similar on both cases corresponding to a nice fit of the dispersion curve above 5 Hz.

This example highlights the difficulties in inverting two kinds of object simultaneously. This is particularly critical when the two objects have distinct types of misfits: one with standard deviations and the other without ("All" case). For usually unpredictable reasons, one of the two objects is better adjusted than the other. Weighting the two misfit components is one solution but it is case dependant. Another solution could be to try the minimum misfit option. For dispersion and ellipticity curves separately, define what you consider as an acceptable misfit. For instance, dispersion curves with standard deviation, a minimum misfit of 1 could be fine: all inverted curves inside the standard deviations are acceptable. Both objects must have a minimum misfit defined in the target panel. When running the inversion with minimum misfit, you have to wait the process to reach the minimum misfit (1 if all minimum misfit are 1 and equal weights). Once this is reached, the inversion starts to randomly sample inside the "acceptable" area of the parameter space. Nr in the status panel should increase. This is the number of equivalently acceptable models.

What I observe, and also ask for your evaluation on this, the ellipticity curves calculated with the inversion show in both cases values from infinity to zero for Rayleigh and Love. Is this due to the strong contrast of Vs between bedrock (>2000m/s in the ground profiles) and soft layer or ...?

In Tokimatsu (1997) the shape of the ratio horizontal/vertical for Rayleigh is shown. You can also check yourself with gpell or gplivemodel. According to the velocity contract, the vertical and horizontal components can vanish at the peak and the trough of the ellipticity curve, respectively (see also Scherbaum et al 2003 in GJI).

I noticed that applying the smoothing (konno & ohmachi 40%) to ellipticity curve, obtained with HVTFA m = 8, nppm = 4, the standard deviation bars disappear.

Where did you smooth? In gphistogram? I'm unsure because the nppm options are still only available in max2curve which does not provide a working smoothing option (if I'm correct).

Thanks Marc for your comprehensive answer.
Unfortunately I don't know how it happened, but the dinver of that data is corrupt and therefore I cannot reopen that dataset and send it to you.
My ideas are a little clearer about it and I have better understood the complexity of the joint inversion problem. My greatest uncertainties are frequently addressed to the dispersion curve because it seems to me to be the "less objective" one compared to the ellipticity curve (or generically H / V) especially when approaching the "low frequencies".
For the considerations relating to ellipticity curves with zero and infinity I have referred to appendix A of deliverable D13.12 SESAME 2004.

One question: how can kmin and kmax be calculated for a linear array? In some work I have seen that they are reported in f-k plots (e.g. Surv Geophys
https://doi.org/10.1007/s10712-018-9473-3)
Greetings
Luigi

For a linear array, K_min and K_max are

• K_min=2*pi/D_max
• K_max=2*pi/D_min

For 2D arrays, it is more complex, based on the theoretical array response. For K_min. we have to estimate the largest size of the central lobe of the array response in all directions. For K_max, it is a little arbitrary. It is the minimum distance on the Kx, Ky map from the origin to the first peak that exceed a given threshold. The value of the threshold is the arbitrary choice. In geopsy, it was set to 0.5. I dropped it to 0.25 to be more selective but it is disputable.