Abstract: L. Bodri, V. Čermák (2005)

In the present work, we focus on the multifractal structure of the microtemperature time series monitored at depth in boreholes, namely in two holes drilled in Kamchatka (Russia). Two monitoring series were performed for approximately two weeks with a 5-second reading interval; thus, each series contains about 230,000 data points. The observed temperatures displayed sharp gradients and large fluctuations over all observed time ranges, reflecting fine structure of the heat transfer process in the shallow subsurface. Recurrence plot (RP) technique was applied for detecting hidden rhythms that generated the time series. The most characteristic feature of the RPs is their web-like structure, indicating quasi-periodic occurrence of the sharp changes. The spectral and the local growth of the second moment techniques were used for distinguishing between the potentially different heat transfer processes. Both spectra show "red noise" behavior, however, exhibit distinct scaling exponents for different frequency domains from near 1.33 for low frequencies to near 3 at the high frequency end of the spectra. Local growth of the second moment technique has revealed the presence of temperature forming process with the characteristic time of approximately 1 minute, that smoothes out generally anti-persistent behavior on shorter time scales, but has a little effect on longer time scales.

The investigation was accomplished by the calculation of universal multifractal indices, which characterize temperature fluctuation upon scales in the range from minutes to weeks. Both time series show very similar multifractal behavior. The Hurst exponent, characterizing the degree of non-stationarity, equals to 0.18-0.20, the measure of intermittence C₁ amounts to 0.10, and the α-index, characterizing the degree of multifractality, amounts to 1.32 and 1.24 for both boreholes. We speculate on the origin of these common features intervening all over the observed range of time scales in a borehole.

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Multifractal analysis of temperature time series: Data from boreholes in Kamchatka