The relationship between normalised horizontal-to-vertical spectral ratios (HVSRs) of microtremors and the F distribution

Abstract

The Albarello test is a statistical test proposed in 2001 with the aim of quickly measuring microtremors (e.g. 5 min) to estimate horizontal-to-vertical spectral ratios (HVSRs). This test is used to identify the deviation between the empirical distribution of normalised (amplitude-corrected) HVSRs and the F distribution so as to identify a physically meaningful peak from numerous peaks, including local peaks inevitably generated by chance when a short-time record is analysed. A HVSR peak is regarded to be meaningful when a significant deviation is detected between the two distributions. However, a paper published in 2005 empirically checked the reliability of this test itself and concluded that the test was unreliable. In this study, we investigate the cause of the unreliability and evaluate the applicable condition. First, we show theoretically that the empirical distribution of normalised HVSRs is generally expected to resemble the F distribution. Then, we show that these two distributions are very similar to each other on the basis of both simulated and observed microtremors. This means that we tend to fail to identify the deviation (commit a type II error) when the sample size is not adequate. To be specific, a sample size on the order of 102–104 is required for a Kolmogorov–Smirnov test to avoid a type II error. Such a sample size corresponds to records on the order of 100–102 h in length for a representative parameter set for the spectral analyses. The original purpose of this test cannot be achieved because such long records are required to identify the deviation. It may be useful from a practical perspective to consider that normalised HVSRs can approximate the F distribution when developing methods to evaluate the statistical property of HVSRs.