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The traditional methods of diagnosing dam service status are always suitable for single measuring point. These methods also reflect the local status of dams without merging multisource data effectively, which is not suitable for diagnosing overall service. This study proposes a new method involving multiple points to diagnose dam service status based on joint distribution function. The function, including monitoring data of multiple points, can be established with t-copula function. Therefore, the possibility, which is an important fusing value in different measuring combinations, can be calculated, and the corresponding diagnosing criterion is established with typical small probability theory. Engineering case study indicates that the fusion diagnosis method can be conducted in real time and the abnormal point can be detected, thereby providing a new early warning method for engineering safety.

A dam is a kind of important infrastructures that form a reservoir, and its safety, which is a major public safety issue, is related not only to reservoir benefits but also to human lives, national economic development, social stability, and many other aspects. However, some dams confront some security risks and crash risks because of the complexity of their surroundings and because of other deficiencies in management and engineering; in addition, dam burst events have also occurred occasionally in the world [

Dam safety monitoring is an important means to ensure the normal operation of dams. The service status of dams can be reflected and the abnormal situation can be detected by analyzing the effect variable of dams, such as deformation, seepage, and stress, thereby providing a useful way to monitor these structures and to provide an early warning system [

At present, the work of analyzing monitoring data and diagnosing dam status mainly relies on the monitoring model of single point. Since the Italian scholar Tonini [

However, the monitoring model of single point is only a reflection of the local structure of a dam, and the overall status of dam cannot be described easily by this method. In general, the monitoring information of each measuring point has a strong correlation. Thus, the overall service state of a dam should also be diagnosed through the fusion of multipoint monitoring information. The current research in this area is relatively scarce. He et al. [

Although a useful attempt to diagnose dam services was derived from the results of these studies, some shortcomings remain; for instance, expert evaluations are always subjective, and the diversity of different experts may affect the diagnosis. Then the diagnostic model based on these methods regards the characteristic parameter (such as fractal dimension and modulus of elasticity) as diagnosis basis. Therefore, making a real time diagnosis for the whole dam based on the measuring values of each point in each monitoring day is impossible. In addition, the monitoring values should presumably obey the normal distribution when the aforementioned methods are used to conduct diagnosis of whole dam. However, these values do not strictly obey the normal distribution. In particular, the distribution is not normal when the abnormal data exist in the monitoring series. Therefore, the fusion diagnosis methods for dam behavior under the multiple points need to be further studied.

This study aims at the aforementioned shortcomings and proposes a new fusion diagnostic method related to joint distribution function based on the in situ monitoring data of dams. The distribution of a single point can be calculated with kernel density estimation (KDE), thereby obtaining the relatively real distribution of each point. Then, the distribution of different single points can be connected to the multidistribution function with t-copula function, and the possibility of measuring values in different combinations can be calculated, thereby providing benefits in analyzing the synergetic changing feature of multiple points. Then, the diagnostic criterion is established with the typical small probability method, and the diagnostic process is described, thereby providing a real time and efficient way to diagnose the overall service status of dams. In addition, the proposed method can also be applied to the fusion diagnosis of other engineering fields.

In general, the monitoring data of single point are considered normal distribution to facilitate analysis and calculation [

Therefore, the precision of S-PDF mainly relies on the bandwidth, which is a key factor determining the shape and smoothness of the S-PDF curve. The cumulative distribution function of single point (S-CDF) can be obtained with the integration of S-PDF. The expression of S-CDF is presumably

The fitting goodness can be estimated with root mean square error (RMSE) by comparing the S-CDF value with the S-ECDF value in the sample values

The S-PDF and S-CDF only reflect the operational state of one point. However, the probability density function of multiple points (M-PDF) and the cumulative distribution function of multiple points (M-CDF) should be constructed based on S-PDF and S-CDF to diagnose the overall service status of dams. Sklar theorem [

Sklar theorem has shown M-CDF can be decomposed into several S-CDFs and one C-CDF that describes the relevant information among these variables. C-CDF has many forms. In this study, t-copula function is chosen as the C-CDF because of its symmetrical tail and sensitive feature in capturing tail correlation of variables. For t-copula function, the parameters are

The corresponding logarithmic likelihood function can be written as

The EML for

The fitting accuracy can also be evaluated by comparing the M-CDF values and the empirical cumulative distribution function of multiple points (M-ECDF).

As shown in (

Let the length of the monitoring data in the modeling period be

The mean values

According to the characteristic values,

Based on the small probability principle [

M-CDF reflects the joint distribution of various points based on historical monitoring data. Therefore, if the value of the diagnostic period exceeds the threshold level, then two cases should be considered. (

Flowchart of diagnosing process.

Wanan Water Conservancy Project is located in the middle reaches of Gan River in Jiangxi province, China. This project is constituted by a concrete gravity dam, earth- and rock-filled dams, and a ship lock. Figure

Wanan Water Conservancy Project.

Diagram of wire alignment in dam crest.

The process lines in the modeling period and diagnosing period of the four points are shown in Figure

Recorded static horizontal deformation of EX422, EX423, EX424, and EX425.

According to (

RMSE in different bandwidths of different points.

Bandwidth | EX422 | EX423 | EX424 | EX425 |
---|---|---|---|---|

1.0 | | | | |

0.5 | | | | |

0.1 | | | | |

S-PDF of EX422, EX423, EX424, and EX425 in different bandwidths.

S-PDF of EX422

S-PDF of EX423

S-PDF of EX424

S-PDF of EX425

The corresponding S-CDF can be written as

Then, the M-CDF values of the four points are established with (

Figure

M-CDF value and M-ECDF value.

According to (

Maximum and minimum values of the four-point M-CDF.

Year | | |
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1999 | | |

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2003 | | |

2004 | | |

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2006 | | |

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2008 | | |

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2011 | | |

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2013 | | |

Process line of the four-point M-CDF in diagnosis period.

As shown in Figure

The results of

Characteristic parameter of three-point M-CDF.

M-CDF | | | Logarithmic | | ||
---|---|---|---|---|---|---|

| | | | |||

EX422, EX423, and EX424 | | | | | | |

EX422, EX423, and EX425 | | | | | | |

EX422, EX424, and EX425 | | | | | | |

EX423, EX424, and EX425 | | | | | | |

Process line of the three-point M-CDF values in the diagnosis period.

M-CDF of EX422, EX423, and EX424 M-CDF

M-CDF of EX422, EX423, and EX425

M-CDF of EX422, EX424, and EX425

M-CDF of EX423, EX424, and EX425

As shown in Figures

Measured information of various single points on March 13.

Point | ||||
---|---|---|---|---|

EX422 | EX423 | EX424 | EX425 | |

Measured value (mm) | 13.35 | 3.84 | 4.92 | 4.42 |

S-CDF value | | | | |

M-CDF values on March 13.

EX422, EX423, | EX422, EX423, and EX424 | EX422, EX423, and EX425 | EX422, EX424, and EX425 | EX423, EX424, and EX425 | |
---|---|---|---|---|---|

M-CDF | | | | | |

Based on (

The M-CDF involving EX423, EX424, and EX425 is in normal region on March 13. Therefore, under this condition, when the M-CDF values of the four points reach

Then, the conditional probability for EX422, which means the S-PDF value, is in the region of

According to the analysis, the measured value of EX422 is in the region of

In this study, a diagnosis method for the service status of the whole dam based on joint distribution of multiple points is proposed. The distributions of single point and multiple points are researched. The method of parameter estimation and the accuracy of S-CDF and M-CDF are discussed. Then, the diagnostic criteria and processes are established with typical small probability method. The following conclusions are obtained.

KDE can calculate the S-PDF on the basis of protecting in situ monitoring information effectively. When the bandwidth is set as 0.1, S-PDF has a high fitting precision, which can overcome the shortage of a priori assumption and can provide a new method to estimate the distribution of single point.

M-CDF based on t-copula can reflect the relationship of multiple points. It also has high fitting precision for joint distribution. The M-CDF effectively fuses the measuring values of multiple points. Therefore, it can be considered an important index to diagnose the service status of an entire dam.

The diagnosis criteria based on the typical small probability method reflect the distribution of the extremum of M-CDF. This extremum is beneficial to diagnose the service status of an entire dam and to identify abnormal points.

The authors declare that they have no competing interests.

This research is supported by the National Natural Science Foundation of China (51379162 and 51079114); Water Conservancy Science and Technology Innovation Project of Guangdong Province (2016-06); and the Fundamental Research Funds for the Central Universities (2042016kf1081).