The Multi-fractal Scaling Behavior of Seismograms Based on the Detrended Fluctuation Analysis

The multi-fractal scaling properties of seismograms are investigated in order to quantify the complexity associated with high-frequency seismic signals. The third-order MDFA (MDFA3) method is capable of characterising the multi-fractality of earthquake records associated with frequency- and scale-dependent correlations of small and large fluctuations within seismogram. These correlations are related to changes in waveform properties and hence are a measure of the heterogeneities of the medium at different scales, sensed by direct and converted phases in a seismogram with different amplitudes and phases. The non-linear dependence of generalised Hurst and mass exponent with order q confirms the multi-fractal nature of earthquake records. Amongst different types of earthquakes analysed, the multi-fractal properties are more pronounced for signals with distinct P-, S- and coda waves. The multi-fractal singularity spectrum parameters (maximum, asymmetry and width) are used to measure the frequency-dependent complexity of seismograms. The degree of multi-fractality decreases with increasing frequency, and is generally more for the time period windowing dominant seismic phases in the seismogram. Significant difference in spectrum width between the original record and its randomly shuffled surrogates demonstrates that the multi-fractality in earthquake records is predominantly due to long-range correlation of small and large fluctuations within seismogram, although its origin due to broad probability distribution cannot be completely ruled out, based on the values of scaling exponent (H _q  ≈ 0.5) and their weak q-dependency for the surrogates.