Fourier correlations of Dragan Dam horizontal deformation 1D and 2D time series.

Teusdea, Alin Cristian ; Modog, Traian ; Gombos, Dan 等

1. INTRODUCTION

Dragan dam presents a double arch concrete structure featuring 120 m height and 450 m length at the crest, with 33 vertical plots and generates a basin of about 120 million m3 of water. Monitoring the deformations of large concrete dams is important to prevent fatal accidents of dam cracking. The deformations of the dam crust are measured physically with an inverted pendulum with a very good precision (10-2 mm) given by an optical coordiscope. The surveying (topographical) method readings of dam crust deformations are done with a surveying total station. The last method involves building a local surveying network of control points, from which, sets of readings are measured for the same deformations (Hudnut & Behr, 1998) at the target points localized on the dam crust.

For plots 7, 12, 19, 24 and 29, the time series provided by the inverse pendulums consist in 1189 readings, from May 2005 until November 2008. The time series provided by the surveying epochs consist in only 8 readings of deformations at the target points placed near the measuring points of the inverted pendulums.

This paper presents the time series Fourier correlations for five target points and their nearest measuring points, done only for plot 19, which is the middle vertical axis of the dam.

2. METHODS AND SAMPLES

There are two ways to get the correlation information between two time series that have different numbers of readings. The first way is to select only the corresponding 8 readings out of the 1189 readings provided by the inverted pendulum, which match the surveying method dates (figure 1). The second way is to interpolate the 8 readings from the surveying method and to obtain N=1189 readings time series, which match the inverse pendulum time series (figure 2). In this paper, we chose the second way.

There were selected two ways to interpolate the surveying time series (figure 1): Gauss kernel smoothing and Fourier interpolation (W is the low-pass frequency filter window).

The horizontal deformations within the inverse pendulum time series (1D+t) are denoted by x (figure 1, the thin line) and the surveying time series are denoted by xT (figure 1, the bar boxed line). The x direction represents the upstreamdown-stream direction and the y direction represents the left-right side direction, both referenced in a local coordinate system. Furthermore, we consider only the horizontal deformations ( x and y ) measured in five target points of the vertical axis belonging to plot 19.

The vertical axis of plot 19 consists in five target points spatially distributed along the plot height. The (2D+t) time series of the horizontal deformations which were correlated are: Hx and HxT, the upstream-downstream from the inverse pendulum readings and from the surveying readings; consequently, Hy and HyT, the left-right side from the inverse pendulum readings and from the surveying readings.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

The correlation process may be a statistical analysis or a Fourier spectral one. The normalized Fourier correlation coefficient, NFCC can be built from the Fourier analysis, described (Grierson, 2006) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where f(x), g(x) are two functions, F(k), G(k) are their Fourier transforms, t is the time, v is the frequency, [F.sup.-1] is the inverse Fourier transform. When the information is timespatially distributed (2D+t), the only way the correlation process can achieve consequent results is by Fourier correlation (Pytharouli & Stiros, 2005; Pytharouli & Stiros, 2008) and not by statistical correlation.

The statistical significance of the correlation coefficient values is: 0.10-0.29 for weak; 0.30-0.49 for average, 0.50 1.00 for strong (figure 3, 4, 5).

3. RESULTS AND DISCUSSIONS

The (1D+t) case of Fourier correlations was done in two ways: first, between the x, y and Fourier interpolated xT, yT time series, FixT, FiyT--with boxed line in figure 3, 4; second, between the x, y and Gauss kernel smoothed xT, yT time series, GKixT, GKiyT--with circled line in figure 3, 4.

The (2D+t) case of Fourier correlations was also done between Hx and FiHxT, GKiHxT and between Hy and FiHyT, GKiHyT (i.e. vertical axis of plot 19--figure 5).

In both ways of the (1D+t) case the Fourier correlations have NFCC values that qualify them as: highly correlated for upstream-downstream direction (figure 3) and average to strongly correlated for left-right side direction (figure 4).

[FIGURE 5 OMITTED]

The results of Fourier correlation for (2D+t) case time series denote that the horizontal deformations measured by the inverse pendulum and by the surveying method are strongly correlated for (Hx and FiHxT, Hx and GKiHxT ) and just average to strongly correlated for (Hy and FiHyT, Hy and GKiHyT) (figure 5).

4. CONCLUSIONS

Fourier correlation analysis of the structural dam horizontal deformations measured by physical method and by surveying method is presented. As correlation inputs were used: the (1D+t) deformations time series at target points and the (2D+t) deformations time series of the entire vertical axis of the dam median plot. Despite that the (1D+t) correlation results show an overall (2000-2005, 2005-2008) strong correlation, the (2D+t) correlation results show an average to strong correlation of the horizontal deformations (figure 5). This means that the (2D+t) Fourier correlation analysis is more suitable to diagnose the dam's long term behaviour.

From figure 1, one can note that the surveying time series have lost some important dam deformation extreme values presented in inverted pendulum time series. Thus, in order to achieve a better structural dam crust diagnose, the authorized monitoring institutions should double the number of the surveying epochs.

Figure 5 emphasizes the difference between the Fourier correlation results calculated for the two mentioned time intervals. Better correlations are obtained for the period 20052008 in comparison with the period 2000-2005, because of the improved measuring technology.

Future research can involve (3D+t) analysis of all the dam's plots in order to provide a more accurate dam status diagnosis.

5. REFERENCES

Grierson, B. A. (2006). FFT's, Ensembles and Correlations, Available from: http://www.ap.columbia.edu/ctx/ctx.html, Accessed: 2007-08-12

Hudnut, K. W. & Behr, J. A. (1998). Continuous GPS monitoring of structural deformation at Pacoima Dam, California, Seismol. Res. Lett., Vol. 69, No. 4, 1998, pp. 299-308, ISSN 0895-0695

Pytharouli, S. I. & Stiros, S. C. (2005). Ladon dam (Greece) deformation and reservoir level fluctuations: evidence for a causative relationship from the spectral analysis of a geodetic monitoring record, Engineering Structures, Vol. 27, Issue 3, February 2005, pp. 361-370, ISSN 0141-0296

Pytharouli, S. & Stiros, S. C. (2008). Spectral Analysis of Unevenly Spaced or Discontinuous Data Using The Normperiod Code, Computers and Structures, Vol. 86, Issues 1-2, January 2008, pp.190-196, ISSN 0045-7949