Dragan dam deformations analisys with fourier correlation.

Teusdea, Alin Cristian ; Modog, Traian ; Mancia, Aurora 等

1. INTRODUCTION

Dragan Dam presents a double arch concrete structure featuring 120 m height and 450 m length at the crest. It has 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 inverse pendulum with a very good precision ([10.sup.-2] mm) given by an optical coordiscope. The surveying method readings of dam crust deformations are done with an optoelectronic device called total surveying station. This method involves building a surveying network of reference points, from which sets of readings are measured for the same deformations (Behr, 1998).

For plots 7, 12, 19, 24 and 29, the time series provided from inverse pendulums consist in 2010 readings, from May 2000 until November 2005. This is the reason why the time series provided from the surveying targets (i.e. the control points) consist only in the readings of deformations at control points placed nearest to the measuring points of the inverse pendulum.

This paper presents the time series correlation for five measuring points and their nearest control 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, i.e. 2010 readings for the inverse pendulum and just 12 seasonal readings for the surveying method. The first way is to select only the corresponding 12 dates, from the 2010 dates, which match the dates for the surveying method (figure 1). The second way is to interpolate the 12 dates from the surveying method and to obtain 2010 readings dates, which match the inverse pendulum time series dates.

In this paper, we chose the first way which correlates these two different time series. As mentioned before, the correlation process involves two time series over 12 dates. The inverse pendulum time series (1D+t) are denoted by X (figure 1 - the circle doted line) and the surveying time series are denoted by XT (figure 2--the diamond doted line).

Furthermore, we consider horizontal deformations (X and Y) of plot 19 vertical axis. The vertical axis of plot 19 consists of five measuring/control points (figure 3) spatially distributed along the plot height (2D information)--i.e. one vertical bolded line from figure 3a, 3b.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Time series of the vertical axis horizontal deformations give the (2D+t) surfaces: first upstream-downstream deformations, denoted by HX, for the inverse pendulum readings and second upstream-downstream deformations, denoted by HXT, for the surveying readings.

The correlation process may be a statistical or a Fourier spectral analysis one. The normalized Fourier correlation coefficient, NFCC , can be built from the Fourier analysis, described (Garlasu et al., 1982; Grierson, 1982) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where f (x), g(x) are two functions, F(k),G(k) are their Fourier transforms, x and k are two Fourier-conjugate variables (i.e. t as time and v as frequency) and [F.sup.-1] is the inverse Fourier Transform. From the recommendations provided by literature the best way to correlate pure time series is the Fourier analysis method (Pytharouli, S, 2004; Pytharouliand & Stiros, 2005). When the information is time-spatially distributed, the only way the correlation process can achieve consequent results is by Fourier correlation and not by statistical correlation--as the Pearson coefficient--despite the symmetry of the two definitions of the correlation coefficients.

Statistical significance of the correlation coefficient values are: 0.10 - 0.29 for weak; 0.30 - 0.49 for average, 0.50 - 1.00 for strong.

3. RESULTS AND DISCUSSIONS

Fourier correlations were done in the first step for the (2D) case (i.e. 1D data of five control points of one horizontal dimension distributed as 1D along H) between the HX, HY and HXT, HYT series (table , 1 lines 2-13, column 3 and 4). In the second step the Fourier correlation were done for the (2D+t) case (i.e. 1D data of five control points of one horizontal dimension distributed as 1D along H, over the 12 dates as t) between HX, HY and HXT, HYT (table 1, line 14 and 15, column 3 and 4).

The Fourier correlation of the (2D) series has NFCC minimum values: 0.601 for the HX and HXT pair (upstream--downstream deformations) and 0.414 for HY and HYT pair (left side--right side deformations). This means that the horizontal deformations measured in upstream--downstream direction (HX, HXT time series) are strongly correlated and measured in left side--right side direction (HY, HYT time series) are just averagely correlated.

The (2D+t) correlation was done in two ways. In the first way, the Lp norm (p=2.5) of the (2D) results of correlations is calculated providing the Lp score over all 12 dates. In the second way the pure (2D+t) Fourier correlations of surfaces from figure 3a, 3b are done. The Lp score denotes highly strong correlations between all horizontal deformations measured by the inverse pendulum (HX, HY time series) and by the surveying method (HXT, HYT time series). Instead, the results of Fourier correlation of (2D+t) time series denote that the horizontal deformations measured by the inverse pendulum and by the surveying method are highly strong correlated for (HX vs. HXT) and just strongly correlated for (HY vs. HYT).

4. CONCLUSIONS

Spectral correlation analysis (via Fourier transform) of the structural dam horizontal deformations measured by physical method and by surveying method is presented. As correlation inputs were used: the (2D) time series of the deformations at control points and the (2D+t) time series of the deformations of entire vertical axis of the median plot of the dam. Despite that the (2D) correlation results show an overall averagely correlation, the (2D+t) correlation results show a strongly correlation for both horizontal deformations. This means that the (2D+t) Fourier correlation analysis is more suitable to diagnose the dam's status.

Correlation analysis accomplished in this paper reveals that the surveying method is reliable for periodical monitoring of the dam's deformations and can be used to make accurate diagnosis of the dam's status. In the future research, we plan to make a (3D+t) analysis of all the dam's plots in order to provide a more accurate dam's status diagnosis.

5. REFERENCES

Behr, J. A.; Hudnut, K. & King, N. (1998). Monitoring structural deformation at Pacoima Dam, California, using continuous GPS, Proceedings of the 11th International Technical Meeting of the Satellite Division of the Institute of Navigation ION GPS-98, pp. 59-68, September 1998, Nashville, TN. Available on: http://pasadena.wr.usgs.gov/ scign/group/pacoima_dam/ION/PacoimaGPS.pdf.

Garlasu, St.; Popp, C. & Ionel, S. (1982). Introducerein analiza spectrala si de corelafie / Introduction to spectral and correlation analysis, Ed. Facla, Tirm'soara.

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

Pytharouli, S; Kontogianni, V.; Psimoulis P. & Stiros, S. (2004). Spectral Analysis Techniques in Deformation Analysis Studies, International Conference on Engineering Surveying and FIG Regional Conference for Central and Eastern Europe (INGEO 2004), Bratislava, November 2004, Proceedings-CD, 10 pp.

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, Volume 27, Issue 3, February 2005, Pages 361-370, ISSN 0141-0296.

Tab. 1. The results of the Fourier correlation
of plot 19 time series (* have the significance
of (1D data of one horizontal dimension + 1D
along H) Fourier correlation, where H is the
plot height).

1. Date HX vs HXT HY vs HYT

2. may.2000 0,928 * 0,895 *
3. october.2000 0,601 * 0,950 *
4. july.2001 0,976 * 0,857 *
5. october.2001 0,909 * 0,892 *
6. june.2002 0,988 * 0,891 *
7. november.2002 0,777 * 0,414 *
8. may.2003 0,944 * 0,812 *
9. november.2003 0,642 * 0,753 *
10. june.2004 0,964 * 0,939 *
11. october.2004 0,841 * 0,504 *
12. july.2005 0,988 * 0,865 *
13. november.2005 0,955 * 0,607 *
14. Lp (2D+t) 0,886 0,800
15. NFCC (2D+t) 0,841 0,535