Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 9049260, 10 pages

http://dx.doi.org/10.1155/2016/9049260

## Method of Fusion Diagnosis for Dam Service Status Based on Joint Distribution Function of Multiple Points

^{1}School of Water Resources and Hydropower Engineering, Wuhan University, Wuhan 430072, China^{2}State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China

Received 13 April 2016; Accepted 31 July 2016

Academic Editor: Eric Florentin

Copyright © 2016 Zhenxiang Jiang and Jinping He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2].

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 [3].

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 [4] described the main factors that affected dam deformation as hydraulic press, temperature, and time, in 1956, further studies aimed at the classical monitoring model of single point have been conducted and have been widely used in practical engineering, forming the statistical model [5, 6], the deterministic model [7], and the hybrid model [8]. Meanwhile, with the development of machine learning algorithms [9, 10], the modeling approaches of single point have developed into the nonlinear direction. Stojanovic et al. [11] proposed the artificial neural networks to establish the nonlinear monitoring model between environmental variables and dam displacement, improving the adaptability of model. Ranković et al. [12] used support vector machine to train the original monitoring data, improving the forecast accuracy of model in the nonlinear situation. Some scholars also introduced intelligence algorithms, such as particle swarm optimization [13], ant colony optimization algorithm [14], and genetic algorithm [15], to construct the monitoring model of single point, thereby enriching the modeling approach and achieving accurate result.

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. [16] introduced D-S evidence theory to realize the fusion diagnosis for dam on the basis of expert evaluations. De Sortis and Paoliani [17] used the modulus of elasticity as diagnostic indicator and diagnosed dam behavior on the basis of finite element model and monitoring data. Su et al. [18] used fractal dimension as diagnostic indicator and applied rescaled range analysis to fuse multipoint monitoring information. In addition, some fusion diagnosis methods in other engineering fields are also worth learning. Rafiq et al. [19] proposed Bayesian networks to diagnose the bridge behavior according to expert evaluations. Masoumi et al. [20] constructed finite element model of steel structure and introduced the optimization algorithm to improve diagnostic precision about the modulus of elasticity. Georgoulas et al. [21] diagnosed the rotor bar with Markov model on condition that the monitoring data obey the normal distribution.

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.

#### 2. Research Methods

##### 2.1. Single Point Distribution

In general, the monitoring data of single point are considered normal distribution to facilitate analysis and calculation [22]. However, deviations always exist between the true distribution and normal distribution. The use of approximation may ignore the real characteristic of the monitoring data, thereby leading to calculation error. KDE [23] uses the sample data to estimate the probability density function of the population distribution. The advantage of this method is that the estimation depends on the sample data without a priori assumptions. Therefore, the real trait of monitoring data, which reflect the real feature of single point, can be retained. The expression of KDE can be written as follows: where is the probability density function of a single point (S-PDF), is the measuring value of the sample , is the number of samples, is the kernel function, and is the bandwidth. Equation (1) indicates that the accuracy of KDE is affected by both kernel function and bandwidth. In general, a different kernel function is regarded to have a slight effect on the S-PDF. Gauss kernel function, which is always applied to engineering, is expressed as

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 , and the measuring value is presumably in one observation; thus, the S-CDF value, indicating that the possibility () of is less than , can be expressed as where . is assumed to be the order statistics of the sample to compare the accuracy in a different bandwidth, and the empirical cumulative distribution function of a single point (S-ECDF) can be written as

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 , and low RMSE value means high fitting accuracy. The expression of RMSE is written as

##### 2.2. Multipoint Distribution Based on t-Copula Function

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 [24, 25] provides a theoretical basis to generate the M-CDF based on nonnormal S-CDF. If the number of points is and the expressions of their S-CDF are , then a unique copula cumulative distribution function (C-CDF) connecting the S-CDFs to M-CDF exists. For the measuring values , M-CDF can be expressed aswhere is the parameter of copula function and the domain of is . For any value in the domain, the expression is established.

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 (one matrix) and (one constant), which describe the relationship of various points and degrees of freedom in the function. The parameters can be calculated with maximum likelihood estimation (EML). If the S-PDFs of points are , then M-PDF can be expressed as (7) when the derivation is conducted on (6):where is the probability density function of t-copula. Let be the number of times for observation; thus, the likelihood function can be written as

The corresponding logarithmic likelihood function can be written as

The EML for and can be obtained by solving the maximum value of (9):

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

##### 2.3. Diagnostic Criteria Based on Typical Small Probability Method

As shown in (6), M-CDF, which is the function of measured values collected by different points, has a positive correlation with these measured values. Therefore, M-CDF can be used as an important parameter to diagnose the service status of an entire dam. If the value changes in a fixed interval, then the service status of an entire dam can be considered normal. If the value exceeds the interval, its status can be judged as abnormal. In this study, the monitoring data are divided into two periods: modeling period and diagnosis period. The data of modeling period are applied to establish M-CDF, and the interval can be calculated based on M-CDF values in this period and typical small probability method.

Let the length of the monitoring data in the modeling period be years; thus M-CDF can be established, and the corresponding M-CDF value in each monitoring day can be calculated. Then, the unfavorable M-CDF value in each year indicates that the annual maximum and the annual minimum form the samples of and :

The mean values and and variances and are written as follows:

According to the characteristic values, testing approach is applied to check the distribution pattern of and . Then, the probability density functions and and the cumulative distribution functions and are confirmed. Let the threshold values of M-CDF values be and ; thus, the possibility of or , which indicates the abnormal situation of dam, can be written as

Based on the small probability principle [26], is always set as 0.05. Thus, and can be calculated as (14). If M-CDF exceeds threshold, then the measured values collected by different points are in the small probability event area:

##### 2.4. Diagnosis Process for Service Status of the Entire Dam

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. () An abnormal M-CDF value is caused by multiple points. This phenomenon indicates that the whole dam is in a state of high risk. Attention should be given to the changing trend of the value in the following days, and related measures should be taken to make the value be in the normal range. () An abnormal value is caused by the measuring data of several points. When this phenomenon occurs, the abnormal points should be removed, and a new M-CDF should be reestablished. If the new M-CDF is in a normal state, then the analysis should be focused on these abnormal points. If the new M-CDF value still exceeds the threshold level, then its trend should be observed in the following days. If necessary, the emergency action should also be conducted to recover the value to the reasonable region. The flowchart based on the preceding discussion is illustrated in Figure 1 to describe clearly the diagnosing process for service behavior of dams.