Under the assumption that the appropriate distribution is lognormal, the resulting appropriate mean of two horizontal records is the geometric mean. Additionally, most historical ground motion prediction equations have adopted the geometric mean as the dependent variable.

Nakamura's Technique determines average soil properties. These average soil properties are resemblance of soil property variation in both horizontal directions. Thus, it is better to take mean of both the horizontal records.

In Sesame project was used quadratic mean for horizontal components (NS and EW). It is difficult to explain an average geometrical. Perhaps in choosing of method of averaging horizontal component still there is potential for the development of method.

Hi Lorenzo,

the rationale for using the geometric mean goes back to an argument made by Bard, 1999 (you find the paper in the attached link below, see page 1265 of that paper). Basically same reason given by Timothy above.

Most practitioners have however adopted the arithmetic mean. A good overview can also be found in the appendix of Saenger et al., 2009 (GEOPHYSICS,VOL. 74, NO. 2; P. O29–O40).

Hope that helps?

Conference Paper Microtremor measurements: A tool for site effect estimation?

Dear Alex,

thank you for the paper by Bard: I was really looking for it. I go through it and let you know what I think. I already know the paper by Saenger.

Regards, Lorenzo

I would, respectfully, disagree with the statement that "most practitioners have adopted the arithmetic mean." Design spectra are typically geometric mean based. Current trends in the building industry to develop maximum component-based design spectra are an exception. But even in that case, the arithmetic mean is not used. When computing the mean of a suite of record pairs, then yes, the arithmetic mean is sometimes used. But when computing the effect of horizontal components of the same record, geometric mean is used.

you might be right, Timothy. The SESAME report from 2006 actually recommends using the geometric mean (p. 25; link below).

Despite, it does not seem to be used consistently in the literature I have access to (like, e.g., Saenger et al.)

Should be mentioned perhaps that Bard (1999, see above) states that it does not make a difference in practice.

ftp://ftp.geo.uib.no/pub/seismo/SOFTWARE/SESAME/USER-GUIDELINES/SESAME-HV-User-Guidelines.pdf

Your point is well taken, Alex. It seems to me that the most important thing is to be very specific in any reported results. However one computes the mean, it should be explicitly stated. There does seem to me to be some benefit in using the geometric mean so that results might be more consistent across all disciplines - structural engineering, geotechnical engineering, signal processing, seismology, etc.

Hi everybody

be careful, my 1999 paper should not be taken as the ultimate reference and I think both are acceptable ! Actually, as mentioned by Alex, it does not make any difference in practice if what you are interested in is simply picking the H/V fundamental frequency. However, if you also want to use H/V amplitude (which raises a lot of open issues), it is different: for instance,; for an isotropic noise, quadratic H/V will be 1.414 times larger than geometric mean. 

Best regards

py

Dear all,

I want to explain my original curiosity about the combination of spectral amplitudes of horizontal components by geometric mean: it arises just from the SESAME guidelines report in which the geometric mean is recommended, but no motivation is given for it. My concern about the geometric mean is the following: suppose that, for whatever reason, at a frequency (or band) the noise is almost all on one horizontal component. Then, the H from geometric mean is almost zero at that frequency (or band) and so the H/V and you lose all the information about noise that in reality is non-zero. This does not happen with other averaging methods that anyway preserve the noise information. Then, I do prefer quadratic mean, i.e. sqrt((H1^2+H2^2)/2) because it seems more related to noise energy. So, I started to look around and found that different authors use different combination methods and nobody gives a real motivation of his choice. Only, they give the processing steps they used. Then, my conclusion is that one can choose the method he prefers, the important thing is to be clear about the followed procedure, as Timothy said.

Anyway, note that today on the Geopsy website (which is the heir of the SESAME project), the geometric mean is no more supported (see the following link http://www.geopsy.org/wiki/index.php/Geopsy:_H/V_and_Spectrum_Toolboxes:_Processing_Tab#Horizontal_components_section_.28only_available_for_HV_toolbox.29)

I want to say that in the paper by Bard, the geometric mean of the horizontal components is presented as a possible alternative, but no reason is given. The link to the lognormal distribution is made for a different item, i.e. the averages of the H/V ratios obtained in the single windows, on the hypothesis, to be demonstrated, that H/Vs have the same distribution of the earthquake spectral ratios, found to be lognormal. So, not related to the distribution of the horizontal components themselves.

About the latter, I didn’t know about the lognormal distribution of the spectral amplitudes because I found the very interesting paper from Albarello & Lunedei  Geophys. J. Int. (2013) 194, 936–951, in which they claim for a normal distribution of noise spectral amplitudes in the three directions. Moreover, they demonstrate that all the different methods used to average the horizontals converge to the same H/V value (except for the root of squares, which is 1.41 times the others), if the number of windows on which to average the results is more than 10. It confirms the point of Bard.

Thank you everybody for the very helpful contributes to the discussion. Please, let me know other hints to think about.

Lorenzo