Array-size dependency of the upper limit wavelength normalized by array radius for the standard spatial autocorrelation method

Abstract

AbstractThis study has shown for the analysis of the standard spatial autocorrelation (SPAC) method that the upper limit wavelength (ULW) normalized by the array radius (normalized ULW, NULW) strongly depends on the array size if we include small (radius r less than a few tens of meters) and very small (r about 1 m or less) microtremor arrays in addition to conventional larger arrays. First, field data of microtremor arrays were analyzed to demonstrate the possible use of small/very small arrays. Specifically, it was shown that, (i) even in the case of a very small array, random errors in the analysis results for very long wavelengths relative to the array radius are kept in an acceptable range for practical use; (ii) the signal-to-noise ratio (SNR) is a crucial factor determining the NULW; and (iii) an equation determining the NULW applies, namely the relation $$(\mathrm{NULW})\propto \sqrt{(\mathrm{SNR})}$$ ( NULW ) ∝ ( SNR ) holds through very small to large arrays. The field data used are those distributed for blind prediction (BP) experiments for an international symposium (BP data), which consist of high-quality microtremor array data with various radii from very small (r = 0.58 m) to large (r = 555 m). It was then shown that SNRs of the BP data, and consequently the NULWs, increase with a decrease of array radius. Statistical data obtained from a few hundred arrays in our previous research also exhibit a similar tendency. The BP data lie around the maximum values of these distributions, showing the high quality of the BP data as well as supporting the array-size dependency of the NULW. Finally, the BP data were processed to identify the characteristics of the soil attenuation. It was found that the array-size dependency of NULW, as well as the large variations in NULW, can generally be explained by soil attenuation. It is plausible that the SNR of small/very small arrays are generally determined by the soil attenuation if the self-noise of the recording system is excluded. A logical conclusion drawn from these results, and also empirically supported, is that the practicality of very small arrays increases as the soil gets softer. Graphical Abstract