User:Aerowater

HiCum (Histogram Cumulation)

By ZHU Ping & Pr. Michel van Ruymbeke

Introduction
Detect different periodical signals is an important issue in nature science, especially in the geophysical domain. Periodicities of signals have been widely studied since Fourier transform algorithm was published.Earth is continuously moving inside the gravitational gradient induced by the Moon and Sun. The orbital parameters of the motion are strictly defined by the celestial mechanics so that the reaction of the Earth to each outside body produces many periodical signals and most of them could be separated using stacking method if long enough time series are available.

Time based observation is a combination of periodical signals with nonlinear factors, so that main frequencies of the contained signal could be localized by the Fourier Transform, an important hypothesis for the HiCum method being that the period is precisely priori defined. A periodical signal could be expressed as:

$$f(t)=\frac{1}{N_S}\sum_{i=1}^{N_S} \sum_{j=1}^T y(t_j)+\varepsilon$$   i=1,T,2T,3T,...Ns    j=1,2,3,...,T    ...(1)

$$\frac{2\pi}{\omega} $$ ....(2)

f(t):Stacing results, t:time, y(t): observed data, T:selected period, Ns: stacking times, $$ \omega $$:angular speed, $$\varepsilon$$:the uncertainties and errors

$$ s(t)=A \sin(\omega t+\alpha)$$...(3)

where A is the amplitude and $$ \alpha $$: the phase

$$A=\sqrt{ \sum_{j=1}^T \left( \left(f(t_j) \sin(\omega t_j) \right)^2+ \left( f(t_j) \cos(\omega t_j) \right)^2 \right)} $$ …(4)

$$\alpha=\tan^{-1} \left( \frac{\sum_{j=1}^T f(t_j)\sin(\omega t_j)}{\sum_{j=1}^T f(t_j)\cos(\omega t_j)}\right)$$…(5)

$$ RMS=\sqrt {\frac{1}{T} \sum_{j=1}^T \left( f(t_j)-s(t_j) \right) ^2 } $$ …(6)

For instance, if one year gravity records with minute sampling is stacked in the lunar period M2 (central frequency 1.9504 cycle per day, corresponding to 745 minutes the period), the M2 wave could be separated by 705 (365.15day*1440minutes/745minutes) times stacking and the mean square error will be reduced about  $$ \sqrt{N_S} $$times (this example is $$ \sqrt{705} $$  times).

Sometimes, for a high quality data bank with high sampling rate compare to the signal’s period, the signal could be easily separated by several stacks. It provides a possibility to check the amplitude and phase variation for the signal in an approximated time interval.

So that, the static stacking function is developed into a dynamic one by a sliding windowed stacking function:

$$ f_W(t)=\sum_{k=1}^N f(t) $$ … (7)

$$ f_W(t)=\sum_{k=1}^N \left( \frac{1}{N_S}\sum_{i=1}^{N_S}\sum_{j=1}^T y(t_j)+ \varepsilon \right) $$ ...(8)

where $$ f_W(t)$$ is a windowed stacking result, N the number of the windows, $$ N_S $$ the sliding window length and $$ W \geq T $$. Through equations (1), (3), (6), we can build up a matrix E with the stacking times Ns and its corresponding stacking results errors RMS:

$$ E=\left(N_SRMS \right) $$ ...(9)

Super Conductor Gravimeters Record
Fourier spectrum and HiCum stacking on 365 days data of theoretical tides, gravity residuals and barometric pressure was compared. The synthetic tides was calculated by Tsoft with a local mode adjusted by long term SGs observations [Van Camp & Vauterin, 2005]. The gravity residuals are directly computed by subtraction of the theoretical value from the observed tide. Diurnal, semi-diurnal and ter-diurnal energies appear obviously in the spectrum of observed and synthetic tides. There are very small peaks in the semi-diurnal and ter-diurnal frequency band of the residuals and barometric Fourier spectrum (Figure1) by which it’s difficult to figure out the barometric pressure effect on the SGs record. However, the pressure effect becomes very clear in the S2 period stacking results when the same data set is stacked in by HiCum. We selected the four waves which share 70% total energy of gravity tides coming from lunar and solar attraction (Figure2,3).

Sea Gauge
A sea gauge sensor records the sea level in the first open lake of the lava tunnel. Very smooth motion appears due to the filtering of short period sea waves through the few openings existing between the open-sea and the tunnel. Fourier spectrum confirms the nonlinearity of the water motion inside the lava tunnel (Figure 4). The HiCum is applied to five years records. M2 amplitude is 40.528 cm of sinusoidal fit with RMS error 0.8261 and S2 amplitude 4.972 cm of sinusoidal fit with RMS error 0.2132. It shows the efficiency of stacking to separate harmonic components on long registrations (Figure 5).

Strain Meter
the HiCum analysis on a file covering 800 day’s records with four channels. The first channel is a glass stain meter RA1 located on a vertical crack. The second and the third channels are two thermometers monitoring temperature of the surface rock and environment in the gallery nearby the strain-meter. The fourth channel is a barometer. The spectral analysis of the four channels shows different peaks in diurnal, semi-diurnal and ter-diurnal frequency bands (Figure 6). For the strainmeter, diurnal and semi-diurnal activities appear clearly without harmonics. Contrarily for rock temperature and barometric pressure, semi-diurnal and ter-diurnal peaks exist and seem to be correlated. We subtract the raw dada with a third degree polynomial function to eliminate long term instrument drift before stacking the data (Figure 7).

Borehole Ground Water Record
records the change of the water levels in three tubes plunging separately in three independent aquifers located at different depths (B1:-35m, B2:-64m, B3:-95m). The upper aquifer is an opened one, the second a semi-confined, and the third a confined one. The original records were pre-treated with the same procedure as the previously studied Rochefort one. Main conclusions from the spectral analysis concern interaction of water levels with simultaneous atmospheric pressure and tidal gravity changes. However the signal to noise ratio for spectrums is too low for understanding the relations between these two effects (Figure 8). We apply the HiCum method on same series of data. For the first borehole and the barometric channel, HiCum shows a common large amplitude S2 and negligible modulations on M2. The gravitational effect appears on the B3-aquifer and smoothly on B1. Induction of atmospheric pressure looks negligible for the B2 and B3 boreholes. It is another example of this method to compare by stacking, two kinds of actions by the influences detected in reactions (Figure9).

Conclusion
A stacking method (HiCum) is introduced which could transform time based observation into different periodical based signals by stacking. One important issue in Earth science is to find periodical phenomena and to explain the physical meanings behind it. HiCum stacking method is another way to precisely study the performance of different signals in the frame of a known period T.