Dear Mark,

I am using DINVER code for a joint inversion of dispersion and HVSR data.

The misfit function is defined according to the formula (3.46) given in your PhD thesis, I guess.

I would like to know how the constant alpha is defined.

Thank you very much for your contributions given to the scientific community and for the your attention to my trouble.

Sincerely,

Rosalba

## misfit

### Re: misfit

I guess you referenced equation 3.48 and not 3.46.

The alpha is adjusted on a case by case basis. This highly depends upon the noise of each sub target. On the first run, better to start with the same weight for dispersion and ellipticity. If it appears that one of the two objects is well adjusted at the expense of a bad fit of the other, increase the weight of the worst fit.

As you can see, this is not a fairly objective method. The misfit values will depend upon arbitrary choices made by the user. The approach I prefer is to define a level of acceptance for each individual target. For instance, an acceptable model can be one that has its dispersion curve entirely inside the standard deviations of the experimental curve. It corresponds to a modified misfit value of 1. The modified definition of the misfit includes a minimum value (that you can specify in Dinver). The distance between experimental and theoretical curve is checked for each sample. If it is smaller than the minimum, the misfit value for this sample is set to the minimum. The total misfit for the curve is a RMS of all sample misfit values. Hence, if the theoretical curve is inside the standard deviation, the total misfit for the curve is the minimum value. If at least one sample is outside the standard deviation range, the total misfit of the curve is larger than the minimum. With the classical misfit, a misfit of 1 does not imply automatically that the theoretical curve is fully inside the standard deviation range. One sample can be completely outside but compensated by other well fitted parts of curve, the RMS is averaging the individual sample misfits. This modified misfit definition is activated whenever you specify a minimum misfit in the target panel. The usual minimum is 1. If you choose 0.5, you will get acceptable models that fit within half the standard deviation range. 0 deactivates the option.

Once you have two curves to invert (dispersion and ellipticity), you can set the minimum misfit of each sub-target to 1 and weight to 1 for both. All the models you will obtain with a misfit of 1 will be acceptable with respect to your dispersion and ellipticity curves.

One remark: inverting a Rayleigh ellipticity curve requires that you provide a true Rayleigh ellipticity curve NOT just a HVSR curve. The HVSR curve includes body wave and Love contributions, neglecting them is a big (common) mistake. Using hvtfa module (not yet released) or computing it with the random decrement (Hobiger et al. 2009*) is the recommended method to extract Rayleigh ellipticity from HV records.

* Manuel Hobiger, Pierre-Yves Bard, Cécile Cornou, Nicolas Le Bihan. Single station determination of Rayleigh wave ellipticity by using the random decrement technique (RayDec). Geophysical Research Letters 36 (2009) L14303

The alpha is adjusted on a case by case basis. This highly depends upon the noise of each sub target. On the first run, better to start with the same weight for dispersion and ellipticity. If it appears that one of the two objects is well adjusted at the expense of a bad fit of the other, increase the weight of the worst fit.

As you can see, this is not a fairly objective method. The misfit values will depend upon arbitrary choices made by the user. The approach I prefer is to define a level of acceptance for each individual target. For instance, an acceptable model can be one that has its dispersion curve entirely inside the standard deviations of the experimental curve. It corresponds to a modified misfit value of 1. The modified definition of the misfit includes a minimum value (that you can specify in Dinver). The distance between experimental and theoretical curve is checked for each sample. If it is smaller than the minimum, the misfit value for this sample is set to the minimum. The total misfit for the curve is a RMS of all sample misfit values. Hence, if the theoretical curve is inside the standard deviation, the total misfit for the curve is the minimum value. If at least one sample is outside the standard deviation range, the total misfit of the curve is larger than the minimum. With the classical misfit, a misfit of 1 does not imply automatically that the theoretical curve is fully inside the standard deviation range. One sample can be completely outside but compensated by other well fitted parts of curve, the RMS is averaging the individual sample misfits. This modified misfit definition is activated whenever you specify a minimum misfit in the target panel. The usual minimum is 1. If you choose 0.5, you will get acceptable models that fit within half the standard deviation range. 0 deactivates the option.

Once you have two curves to invert (dispersion and ellipticity), you can set the minimum misfit of each sub-target to 1 and weight to 1 for both. All the models you will obtain with a misfit of 1 will be acceptable with respect to your dispersion and ellipticity curves.

One remark: inverting a Rayleigh ellipticity curve requires that you provide a true Rayleigh ellipticity curve NOT just a HVSR curve. The HVSR curve includes body wave and Love contributions, neglecting them is a big (common) mistake. Using hvtfa module (not yet released) or computing it with the random decrement (Hobiger et al. 2009*) is the recommended method to extract Rayleigh ellipticity from HV records.

* Manuel Hobiger, Pierre-Yves Bard, Cécile Cornou, Nicolas Le Bihan. Single station determination of Rayleigh wave ellipticity by using the random decrement technique (RayDec). Geophysical Research Letters 36 (2009) L14303

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