Enhancing Dam Safety: Statistical Assessment and Kalman Filter for the Geodetic Network of Mosul Dam

Abstract

Dams play a pivotal role in providing essential services such as energy generation, water supply, and flood control. However, their stability is crucial, and continuous monitoring is vital to mitigate potential risks. The Mosul Dam is one of the most interesting infrastructures in Iraq because it was constructed on alternating beds of karstified and gypsum which required continuous grouting due to water seepage. Therefore, the ongoing maintenance issues raised international concerns about its stability. For several years the dam indicated a potential for disastrous failure that could cause massive flooding downstream and pose a serious threat to millions of people. This research focuses on comprehensive statistical assessments of the dam geodetic network points across multiple epochs of long duration. Through the systematic application of three statistical tests and the predictive capabilities of the Kalman filter, safety and long-term stability are aimed to be enhanced. The analysis of the dam’s geodetic network points shows a consistent trend of upstream-to-downstream movement. The Kalman filter demonstrates promising outcomes for displacement prediction compared to least squares adjustment. This research provides valuable insights into dam stability assessment, aligns with established procedures, and contributes to the resilience and safety of critical infrastructure. The outcome of this paper can encourage future studies to build upon the foundation presented.

1. Introduction

Dams are one of the most important strategic national infrastructures that provide essential services such as energy generation, water supply, and flood control. However, if the dams are not effectively managed and monitored will present essential risks that can lead to catastrophic outcomes like flooding, habitat destruction, and disruptions to water supply and energy generation. Consequently, there is a need for efficient techniques for assessing and mitigating these risks influencing dam safety.

Many existing dams are aging and may not meet present safety standards. Managing the unique challenges of aging infrastructure, coupled with external factors such as geological conditions, tectonic movement, seismic activity, and renovation quality, is essential to ensure dam stability. This research aims to offer a range of statistical techniques optimized for assessing the compatibility of geodetic network points across multiple epochs in dam monitoring. These methods offer early warning systems and improve decision-making for dam operators and authorities.

Why are statistical analysis methods needed? Because a more objective and rigorous assessment of whether the observed differences in coordinates are significant or could have been due to measurement errors can then be provided. Accordingly, scientific validity is added for checking compatibility or deformation in geodetic networks through statistical tests.

Several research papers published in recent decades have explored the utilization of statistical hypothesis testing for strategic infrastructure, including dams. These tests have been applied to both 2D and 3D geodetic networks, encompassing traditional surveying methods involving measurements of distances and angles, as well as modern 3D networks facilitated by Global Navigation Satellite Systems (GNSS). The research papers examined how these methods perform in assessing the compatibility and deformations of geodetic networks used in critical infrastructure [1,2,3,4,5,6,7,8,9,10]. On one hand, the statistical methods (F-test, t-test, Z-test) enable objective and rigorous assessments of whether observed differences in coordinates are significant and then aid in the detection of movements and incompatibilities. On the other hand, each statistical test has specific sensitivity levels and might not detect changes in the same set of points that can result in variations.

Building upon the body of research that has explored the application of statistical hypothesis testing in the context of critical infrastructure, including dams, the focus of this study is on the Mosul Dam (Figure 1).

Mosul Dam, which was built in 1980, is a large earthen embankment on the Tigris River in northern Iraq near Mosul City. The dam has a storage capacity of 11.11 billion cubic meters at 330 m above sea level at normal operational levels [11]. It is one of the largest dams in the Middle East and has strategic importance for Iraqis. The dam aims to provide water for irrigation, generate hydroelectricity, and regulate the flow of the Tigris River. The dam has a hydroelectric power plant that has a capacity of about 1000 MW from four turbines, contributing to the electricity national grid in Iraq [12]. Since the Mosul Dam is located in a gypsum area and has ongoing maintenance issues, it raised international concerns about its stability [13]. For several years, the dam indicated a potential for disastrous failure, which could cause massive flooding downstream and pose a serious threat to millions of people.

The security challenges because of the presence of extremist organizations added to the political tensions have caused in the last decade serious problems in the area around the Mosul Dam. Hence, the Mosul Dam case reflects the complicated interactions among engineering, geopolitics, and local security while handling significant infrastructure projects.

As a result, the Mosul Dam requires constant maintenance and grouting to address seepage and stability issues. Maintaining the integrity of the Mosul Dam has been always a significant engineering challenge. In 2016, following the Iraqi government’s reacquisition of control over the dam, the Ministry of Water Resources collaborated with the Italian firm Trevi and the U.S. Army Corps of Engineers in a three-year comprehensive initiative. This program aimed to acquire new equipment and implement aggressive cement treatments on the rock foundation, with the primary goal of ensuring the stability of the dam [14,15,16,17]. According to [18], the state of Mosul Dam was declared safe and stable because of successive grouting and other curative measures.

The Mosul Dam’s monitoring geodetic network has been thoroughly and periodically observed by the Iraqi survey authority since the dam was established. Unfortunately, there was no appropriate statistical testing applied to check the stability of the dam network. This research is driven by the requirement to conduct comprehensive statistical assessments on the archived observations of the dam that align with the established procedures of the Iraqi survey authority. The objective is to properly assess the compatibility of dam points by systematically using statistical tests which are augmented by the predictive capabilities of the Kalman filter [19,20]. Several papers were published demonstrating the application of the Kalman filter for structural deformations [21,22,23,24]. The main objective of this research is to offer data-driven, educated recommendations for analyzing the Mosul Dam movement. This study guarantees that the presented techniques continue to follow established standards while continuing initiatives to increase the Mosul Dam’s long-term stability. Moreover, this research will apply these statistical tests to archived observations of the geodetic network of the Mosul Dam over eight years, revealing lessons learned in assessing dam stability and enabling more precise, data-informed decisions. By developing and implementing innovative monitoring techniques, this study contributes to the ongoing efforts to ensure its resilience and reinforce the safety of the communities and infrastructure it serves.

Figure 1. Aerial view of the Mosul Dam site taken from Google Earth [25].

Figure 1. Aerial view of the Mosul Dam site taken from Google Earth [25].

Accordingly, this paper aims to answer the following questions: How effective is the testing hypothesis in detecting compatibility or deformations in the monitored points at the Mosul Dam site over time? What is the feasibility and efficiency of the Kalman filter in predicting displacement at the Mosul Dam points, and how does it compare to traditional least squares adjustments? We have applied a comprehensive analysis of Mosul Dam’s geodetic network to answer these questions. The paper is structured as follows: first, an overview of the study area and geodetic network setup is given. Then, it presents statistical methods such as hypothesis tests and the Kalman filter approach that were applied. Lastly, results are presented showing how effective the test was in detecting deformations and displacement prediction by the Kalman filter. Ultimately, findings are discussed and compared with traditional ones alongside future research recommendations with practical implications for dam monitoring and safety.

The paper is organized as follows: Section 2 outlines the methodology, Section 3 presents the results, Section 4 provides a detailed discussion, and Section 5 presents the conclusions.

2. Materials and Methods

Statistically, monitoring the compatibility of points (or congruence) is used to measure whether there are significant differences between two sets of coordinates for the same points that are typically collected at different epochs. In the case of monitoring dams, it needs to be determined whether there are statistically significant differences ($dX,dY$) in the coordinates between two epochs, indicating potential movement or incompatibility of points.

Initially, if the difference ‘$d$’ between ‘$n$’ coordinates (Equation (1)) resulting from the two epochs is small compared to the accuracy of the measurements and correction results, the network is considered compatible and vice versa.

$$d=\left[\begin{array}{c}{X}_1^{\prime }\\ Y_1^{\prime }\\ ⋮\\ X_i^{\prime }\\ Y_i^{\prime }\end{array}\right]-\left[\begin{array}{c}{X}_1\\ Y_1\\ ⋮\\ X_i\\ Y_i\end{array}\right]=\left[\begin{array}{c}{dX}_1\\ {dY}_1\\ ⋮\\ {dX}_i\\ {dY}_i\end{array}\right] i=1:n$$

(1)

where

$(X_i^{\prime },Y_i^{\prime }):$ 2D coordinates of the monitoring point $i$ at the current epoch.

$(X_i,Y_i):$ 2D coordinates of the monitoring point $i$ at the previous epoch.

$({dX}_i,{dY}_i):$ The difference in $XY$ coordinates between two epochs at point $i$.

Certainly, the deformation distance between the coordinates in the two epochs for each point, $X$ and $Y$, can be calculated as follows:

$$∆=\left[\begin{array}{c}{d}_1=\sqrt{{dX}_1^2+{dY}_1^2} \\ ⋮\\ ⋮\\ d_i=\sqrt{{dX}_i^2+{dY}_i^2}\end{array}\right]$$

(2)

It should be remembered that adjustment with least squares for point P, for example, at the two successive epochs will be as follows:

$$\left.\begin{array}{c}{\widehat{P}}_1=P_1^o+{{\left(B^{t}WB\right)}^{-1}\left(B^{t}WF\right)}_{epoch\mathrm{}1}\\ {\widehat{P}}_2=P_2^o+{{\left(B^{t}WB\right)}^{-1}\left(B^{t}WF\right)}_{epoch\mathrm{}2}\end{array}\right\}$$

(3)

where $P_1^o,P_2^o$ represent the point coordinates of two epochs before adjustment, matrices $B,W,\mathrm{a}\mathrm{n}\mathrm{d} F$ represent the least squares adjustment matrices of the partial derivates and weights. There are multiple statistical methods for testing the compatibility of networks, all of which share the common principle of testing statistical hypotheses [26]:

-

Null Hypothesis ($H_0$): There is no significant difference between the two sets of observations (no movement or incompatibility).

$$H_0:d=0 \mathrm{null} \mathrm{hypothesis}$$

(4)

-

Alternative Hypothesis ($H_a$): There is a significant difference between the two sets of observations (movement or incompatibility is present).

$$H_a:d\ne 0 \mathrm{alternative} \mathrm{hypothesis}$$

(5)

In addition to these statistical methods, the advanced technique of the Kalman filter will be used to predict and evaluate the potential movement of dam points over different epochs. The application of the Kalman filter enables the estimation of the dynamic model for dam points based on historical epochs.

The statistical testing for compatibility, in conjunction with the Kalman filter predictions, forms a comprehensive approach to monitor dam stability and detect movements. The presented approach in this paper not only helps to evaluate whether there are significant differences between two sets of coordinates but also provides a deeper understanding of the dynamics of dam points over time.

2.1. Kalman Filter

The Kalman filter is an important sensor and data fusion algorithm which is introduced by Rudolf E. Kalman in his 1960 research [19]. The Kalman filter is instrumental in estimating system parameters over time epochs and is recognized for its computational efficiency, recursive properties, and suitability for real-time implementation. Hence, the Kalman filter stands as the ideal algorithm for conducting estimation by connecting real-time measurements and accurately estimating the system’s state parameters over successive time epochs. Why recommend the Kalman filter for the displacement prediction? Because it is a recursive algorithm that is continually updating its estimates based on new data for monitoring dam movements. Additionally, it provides optimal estimates when dealing with linear dynamic systems and Gaussian error statistics and can effectively handle measurement noise and uncertainty. Finally, the Kalman filter can efficiently integrate historical data with recent measurements while providing a comprehensive view of the dam’s behavior over time.

Kalman filter is composed of two phases as shown in Figure 2 [20]:

• The predicted measurement $=\left(B{\widehat{x}}_k,{BP}_k{B}^t\right)$;
• The observed measurement $=\left({\overrightarrow{z}}_k,R_k\right)$.

Computationally, the following steps should be applied.

$$B{\widehat{x}}_k^{+}=\mathrm{}B{x}_k+K({\overrightarrow{z}}_k-B{\widehat{x}}_k)$$

(6)

$$B{P}_k^{+}{B}^t=\mathrm{}B{P}_k{B}^t-KB{P}_k{B}^t$$

(7)

$$K=\mathrm{}B{P}_k{B}^t\mathrm{}{\left(B{P}_k{B}^t+R_k\right)}^{-1}$$

(8)

where

$B$ is the Jacobian matrix of partial derivatives to unknowns.

${\overrightarrow{z}}_k$ is the probability of sensor reading for the current state $x_k$ with its uncertainty $R_k$.

$B{\widehat{x}}_k$ is the probability of prediction from the previous state with its uncertainty of $P_k$.

$K$ is the Kalman gain matrix.

From Equations (7) and (8), a common factor of the matrix $B$ is derived and substituted as follows:

$$K^{\prime }=\mathrm{}{P}_k{B}^t\mathrm{}{\left(B{P}_k{B}^t+R_k\right)}^{-1}$$

(9)

$${\widehat{x}}_k^{+}=\mathrm{}{x}_k+K\prime ({\overrightarrow{z}}_k-B{\widehat{x}}_k)$$

(10)

$$P_k^{+}=\mathrm{}{P}_k-K\prime B{P}_k$$

(11)

Equations (9)–(11) are used for the update of the system in a repetitive way where ${\widehat{x}}_k^{+}$ is the new best estimate of the state and together with $P_k^{+}$ are substituted back into a new iteration of prediction and updated until stopping criteria are reached [20].

In this study, the Extended Kalman filter will be employed to process the measurements of the dam pillars at each epoch, enabling the prediction of their displacements over future time intervals. This approach is expected to offer a robust and systematic means of estimating and assessing the dam’s behavior, particularly regarding potential displacements. It must be mentioned that the dynamic model for the state case ${\widehat{x}}_k^{+}$ at $∆t$ time interval will be as follows:

$$X_k=X_{k-1}+{\dot{X}}_{k-1}∆t+{\displaystyle \frac{1}{2}}{\ddot{X}}_{k-1}{(∆t)}^2$$

$$Y_k=Y_{k-1}+{\dot{Y}}_{k-1}∆t+{\displaystyle \frac{1}{2}}{\ddot{Y}}_{k-1}{(∆t)}^2$$

(12)

where

$X_k,Y_k$: 2D coordinates of dam points at epoch $k$.

$X_{k-1},Y_{k-1}$: 2D coordinates of dam points at epoch $k-1$.

$\dot{X}$,$\dot{Y}:$ velocity of point movement over epochs.

$\ddot{X}$,$\ddot{Y}:$ acceleration of point movement over epochs.

Equation (12) illustrates that the prediction of future displacement can be calculated by estimating the dynamic model from the previous epoch using the Kalman filter.

Hence, the prediction for each dam point in the Kalman filter, at a time interval $∆t$, employing the dynamic model of Equation (12) can be expressed in matrices as

$${\widehat{x}}_k=T_k{\widehat{x}}_{k-1}=\left[\begin{array}{c}1\\ 0\end{array}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\begin{array}{c}0\\ 1\end{array}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\begin{array}{c}∆t\\ 0\end{array}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\begin{array}{c}0\\ ∆t\end{array}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\begin{array}{c}{∆t}^2/2\\ 0\end{array}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\begin{array}{c}0\\ {∆t}^2/2\end{array}\right]\left[\begin{array}{c}\begin{array}{c}{X}_{k-1}\\ Y_{k-1}\end{array}\\ {\dot{X}}_{k-1}\\ {\dot{Y}}_{k-1}\\ {\ddot{X}}_{k-1}\\ {\ddot{Y}}_{k-1}\end{array}\right]$$

(13)

2.2. The t-Statistic Test

The t-statistic test is one of the important statistical hypothesis testing methods to compare two samples. The test is used to determine if there is a statistical difference between the two samples by testing the hypotheses of $H_0$ and $H_a$ of the aforementioned Equations (4) and (5).

The t-statistic test can be a valuable technique for determining any significant changes in the coordinates of dam points during many epochs, which can help determine structural stability. Even if the displacements are small, they may indicate incompatibilities that affect the dam’s safety. However, the displacements can be affected by random errors. Accordingly, the t-statistic test can be applied to examine if the observed coordinate differences are significant or result from random errors. It is worth noting that a one-tail t-test is adopted in this study because $\left(d\right)$ is always positive in the adopted method.

The adopted procedure of using the t-statistic test for incompatibility check in this paper is as follows

• Hypothesis Testing: The t-statistic test is a form of hypothesis testing used to evaluate two competing hypotheses of $H_0$ and $H_a$.
• Observational Data: The t-test is based on the coordinate’s differences from two different epochs. This means the difference between the adjusted coordinates of the 1st epoch ${(X}_i,Y_i)$ and the 2nd epoch $\left({X^{\prime }}_i{,Y^{\prime }}_i\right)$ as shown in Equation (1). Accordingly, the t-test assesses whether the differences in coordinates $d$ are statistically significant.
• Combined Error: In the context of the dam monitoring, standard deviation values denoted as ${\sigma }_{o1}$ and ${\sigma }_{o2}$ for both epochs should be known after the least square adjustment. For assessing compatibility, it is required to compute the combined uncertainty when comparing the coordinate differences between the two epochs. Equation (14) is used to calculate the combined standard error ${\sigma }_o$ which represents the expected uncertainty associated with each of the coordinate differences.

  $${\sigma }_o=\sqrt{{{\sigma }_{o1}}^2+{{\sigma }_{o2}}^2}$$

  (14)
• Degrees of Freedom DOF: For compatibility testing using the t-statistic, DOF represents the number of coordinates $X$, and $Y$ of each point.
• Critical t-Statistic Value: As in other hypothesis testing methods, the test involves comparing the calculated t-statistic with a critical (or tabular) value $T$. The critical value depends on the desired significance level ($\alpha$) and the DOF as shown in Equation (15). Figure 3 shows an illustrative plot of the t distribution at (0.10, 12) where the critical value is shaded.

  $$T(\alpha ,DOF)$$

  (15)
• Hypothesis testing: $t\_computed$ (Equation (16)) compares the observed differences $d$ to the expected variability (${\sigma }_o$) to determine if the changes are statistically significant or within the range of expected variability. Accordingly, the null hypothesis ($H_0$) will be rejected if $t\_computed$ is greater than the critical value $T$. This means statistical evidence of a significant difference between the two epochs has been found.

  $$t\_computed=d/{\sigma }_o$$

  (16)

  $$\mathrm{If} {\widehat{t}}_{i }>T\to H_0 reject$$

  (17)

  $$\mathrm{If} {\widehat{t}}_{i }\le T\to H_0 accept$$

  (18)

Table 1. Pseudocode—t test.

Input:

- Epoch1: An array of adjusted coordinates for the first epoch (x1,y1). - Epoch2: An array of adjusted coordinates for the second epoch (x2,y2). - Variance-covariance matrix of coordinates in epoch 1 - Variance-covariance matrix of coordinates in epoch 2 - Significance Level: The desired significance level $\mathit{\alpha }$.

Output:

- t Statistic: The calculated t-statistic for the coordinate differences. - Critical Value: The critical value from the t-distribution table for the given α. - Significant Change: A Boolean indicating whether significant changes were detected.

Begin:

Degrees Of Freedom DOF = 2 # derive standard deviations from the covariance matrices Sd_Epoch1 = Standard Error Of (Epoch1) Sd_Epoch2 = Standard Error Of (Epoch2) # Calculate the t-statistic for coordinate differences Dx = sqrt(x1^2 − x2^2); Dy = sqrt(y1^2 − y2^2); Qd = sqrt((SE_Epoch1^2 + SE_Epoch2^2)); t_computed_x = (Dx)/Qd_x; t_computed_y = (Dy)/Qd_y; # Find the critical value from the t-distribution table Tabular_t = Lookup Critical Value($\alpha$, DOF) # Compare the t-statistic with the critical value If t_computed > Tabular_t Then     Significant Change = True Else     Significant Change = False Return T_Statistic, Tabular_t, Significant Change     End Procedure

illustrates the pseudocode of applying the t-test for checking the dam points compatibility.

2.3. The F-Statistical Test

After performing the least squares adjustment between two epochs, there are two stages of network compatibility testing: the Global test and the Local test.

The Global test for compatibility involves assessing the movement of all network points as a whole to determine whether there is overall network compatibility. If there is an indication of incompatibility in the Global test, it paves the way for a more detailed Local test. The Local test entails analyzing individual network points one by one to detect points that exhibit deformation. Subsequently, appropriate corrective measures can be taken later, such as re-surveying, or excluding the problematic point from further analysis.

• Hypothesis Testing: The F-statistic test is a form of hypothesis testing used to evaluate two competing hypotheses of $H_0$ and $H_a$.
• Observational Data: The F-test is based on the observations from two different epochs. This means the adjusted coordinates of the 1st epoch ${(X}_i,Y_i)$ and the 2nd epoch $\left({X^{\prime }}_i{,Y^{\prime }}_i\right)$.
• Variance Comparison: The F-statistic test focuses on the variances of the two datasets. It calculates the ratio of the variances of the two datasets, which represents the spread or variability of the data.
• Degrees of Freedom: The degrees of freedom for the F-statistic depend on the sample sizes of both datasets. It is essential for calculating the F-statistic tabular value.
  For the Global test:

  $$DOF1=2u$$

  (19)

  $$DOF2=n\mathrm{}+\mathrm{}n\prime -\mathrm{}2u$$

  (20)

  where $n$ and $n\prime$ represent the number of observations in the two epochs. $u$ and $u\prime$ represent the number of non-fixed point coordinates. It must be noted that when dealing with free network adjustment, the $DOF$₂ will be $\infty$ while $DOF$₁ will be $(2u-4)$ [27].
• Pooled Variance: This is the weighted average of the variances of the two datasets. It takes into account the sample sizes and variances of both datasets.

The difference in coordinates for the same network points between two epochs will be calculated as shown in Equation (1). The matrix of weight coefficients, $Q_d$, obtained after the adjustment between the two epochs, will be computed as follows:

$$Q_d=Q_{{\widehat{P}}_1}+Q_{{\widehat{P}}_2}$$

(21)

It is also needed to calculate what is referred to as the residual covariance matrix ($\Omega$, $\Omega \prime$) for the observations in the two consecutive epochs, utilizing the results of the adjustment as follows:

$$\Omega =v^{t}Wv$$

(22)

$$\Omega \prime ={v\prime }^{t}W\prime v\prime$$

(23)

where $v$ refers to residuals after least square adjustment and $W$ refers to weights.

Additionally, the pooled variance, often referred to as the posterior unit variance, can be calculated as follows:

$${\overline{\sigma }}_o^2={\displaystyle \frac{\Omega +{\Omega }^{\prime }}{\left(n-u\right)+(n^{\prime }-u^{\prime })}}$$

(24)

6.

Global Test: The F-statistic is used as a Global test to assess overall differences between the coordinates of dam points in the two epochs. The F-statistic is calculated based on the variance ratio.

Of course, for the Global testing, the statistical compatibility test is calculated as follows and compared to the tabular $F$ value:

$$\widehat{F}={\displaystyle \frac{d^t{Q}_d^{-1}d}{u {\overline{\sigma }}_o^2}}$$

(25)

-

Find the critical value from the F-distribution table for a given significance level ($\alpha$) as shown in Equation (26). Figure 4 shows an illustrative plot of the F-distribution at (0.1, 12, 258) where the critical value is shaded.

$$F(\alpha ,{DOF}_1,{DOF}_2)$$

(26)

-

Compare the computed F-statistic ($\widehat{F}$) with the tabular $F$. If $\widehat{F}$ is greater, it indicates that there is a significant change or incompatibility in the network.

$$\mathrm{If} \widehat{F}<F\to H_0 accept\mathrm{}$$

(27)

$$\mathrm{If} \widehat{F}\ge F\to H_0 reject$$

(28)

7.

Local Test: When the Global test indicates incompatibility, Local tests are performed for individual points. The variance of each point is calculated and compared to a critical value $F(\alpha ,2,{DOF}_2)$ to classify points as moved or stable.

$${\widehat{F}}_i={\displaystyle \frac{{di}^t{Q}_{di}^{-1}di}{2\mathrm{}{\overline{\sigma }}_o^2}}$$

(29)

$$\mathrm{If} {\widehat{F}}_i<F\to H_0 accept$$

(30)

$$\mathrm{If} {\widehat{F}}_i\ge F\to H_0 reject$$

(31)

Table 2. Pseudocode—F test.

Input:

- Epoch1: An array of adjusted coordinates for the first epoch (x1,y1). - Epoch2: An array of adjusted coordinates for the second epoch (x2,y2). - Variance-covariance matrix of coordinates in epoch 1 or Q1 - Variance-covariance matrix of coordinates in epoch 2 or Q2 - Significance Level: The desired significance level α.

Output:

- F-Statistic: The calculated F-statistic for the variances of the two datasets. - Critical Value: The critical F-value from the F-distribution table for the given α. - Significant Change: A Boolean indicating whether significant changes were detected.

Begin:

Degrees Of Freedom DOF1 = 2 × unknowns Degrees Of Freedom DOF2 = n1 + n2-DOF1 # Compute the displacement for each point   dx = sqrt(x1^2 − x2^2)   dy = sqrt(y1^2 − y2^2)   d = [dx, dy] # Derive variances from the covariance matrices Variance_Epoch1 = Variance Of (Epoch1) Variance_Epoch2 = Variance Of (Epoch2) # Calculate the pooled variance Pooled Variance = (Variance Of (Epoch1) + Variance Of (Epoch2)) # Compute the F-statistic Wd = inv(Q1 + Q2) Sd = (d’×Wd×d)/DOF1

# Run the Global test

F_computed = Sd/(DOF1×Pooled Variance);     # Find the critical value from the F-distribution table     Tabular_F = Lookup Critical Value (α, DOF1, DOF2)     # Compare the F-statistic with the critical value If F_computed > Tabular_F Then Global Test Result = True Else Global Test Result = False End If

# Run the Local test if Global Test Result = True

For i = 1 to Length Of d       # Calculate the variance of displacement for point i       sd(i) = (d(i) × Wd(i) × d(i))       # Calculate the F-statistic for the Local test       F_comp(i) = sd(i)/(2 × Pooled Variance)       # Set the confidence level = 0.99       # Find the critical value from the F-distribution table       F_tabular = Lookup Critical Value (confidence, 2, DOF2)       # Compare the F-statistic with the critical value for the Local test       If F_comp(i) < F_tabular Then             Move(i) = 1 # Indicates compatibility       Else             Move(i) = 0 # Indicates incompatibility       End If, End For End Local F-Test

illustrates the pseudocode of applying the F-test for checking the dam points’ compatibility.

2.4. Z-Score Test

The third statistical test adopted in this paper is the confidence interval approach using the Z-score. This test is applied by explicitly calculating confidence intervals for both coordinate differences in Easting (ΔE) and Northing (ΔN). If these confidence intervals do not include displacement, then it indicates a significant change in coordinates. In other words, it is concluded (at the specified confidence level) that the true displacement in coordinates holds statistical significance. The adopted procedure of using the Z-score test for incompatibility check in this paper is as follows.

• Hypothesis Testing: The Z-statistic test is a form of hypothesis testing used to evaluate two competing hypotheses of $H_0$ and $H_a$.
• Observational Data: The Z-score test is based on the coordinate’s differences from two different epochs (Equation (1)). Accordingly, the Z-test assesses whether the differences in coordinates $d$ are statistically significant.
• Critical Z-score: Z-scores are used to calculate desired confidence intervals and it depends on the desired significance level ($\alpha$) which is adopted in this paper at a probability of 99.9% (Figure 5).

  $$Z_{\alpha /2}=\mathrm{}Z\_score\mathrm{}=\mathrm{}3.29\mathrm{}$$

  (32)

The selection of a very high probability Z-score of 3.29 is connected to the concept of committing Type-I error in hypothesis testing [28]. Type-I error represents the probability of making a false positive error which means rejecting $H_0$ when it is actually true. Setting a 99.9% confidence level means a very high threshold for statistical significance. Accordingly, observing displacements $d$ that exceed this threshold indicates a high confidence that the change is significant and then makes it less likely to commit a Type-I error.

4.

Confidence interval test: The test involves comparing the calculated confidence with the displacement calculated ($d$). The confidence intervals can be written in Equation (33):

$$\overline{x}-z_{\alpha /2}{\displaystyle \frac{\sigma }{\sqrt{n}}}\mathrm{}\le \mathrm{}d\le \overline{x}+z_{\alpha /2}{\displaystyle \frac{\sigma }{\sqrt{n}}}\mathrm{}$$

(33)

5.

Statistical decision: The statistical test is applied as follows.

$$\mathrm{If} d_i<Z\_score\mathrm{}\to H_0 accept$$

(34)

$$\mathrm{If} d_{i\mathrm{}}\ge Z\_score\to H_0 reject$$

(35)

Table 3. Pseudocode—Z-test.

Input:

- Epoch1: An array of adjusted coordinates for the first epoch (E1, N1). - Epoch2: An array of adjusted coordinates for the second epoch (E2, N2). - Variance-covariance matrix of coordinates in epoch 1 - Variance-covariance matrix of coordinates in epoch 2

Output:

- Z Statistic: The calculated Z-score for the variances of the two datasets. - Significant Change: A Boolean indicating whether significant changes were detected.

Begin:

# Compute the displacement for each point      DeltaE = sqrt(E1^2 − E2^2)      DeltaN = sqrt(N1^2 − N2^2)      d = [ DeltaE, DeltaN]      Z_tabular = 3.29     For i = 1 to Length Of d     # Derive std deviation from the covariance matrices     std_DeltaE = sqrt(std_E1(i)^2 + std_E2(i)^2)     std_DeltaN = sqrt(std_N1(i)^2 + std_N2(i)^2)     s_E = ([Z_tabular * std_DeltaE])     s_N = ([Z_tabular * std_DeltaN])     d = combine (sE, sN)

For each entry in s_E and s_N:

if abs (DeltaE[i]) < s_E[i]:           move [i] = 1      else:           move [i] = 0      if abs (DeltaN[i]) < s_N[i]:           move [i + size(s_N)] = 1      else:           move [i + size(s_N)] = 0

illustrates the pseudocode of applying the Z-test for checking the dam points compatibility.

3. Results

In this section, the results of the comprehensive analysis of the horizontal geodetic network at Mosul Dam over eight years (2005–2013) are discussed. The geodetic network is strategically designed to monitor potential movements and incompatibilities within the dam structure. Additionally, the significance of adjusting the weight of measurements in the network is investigated, highlighting potential variations in adjustment outcomes. The influence of control points on the network’s reliability and stability is emphasized, promoting the selection of control points with smaller uncertainties. Subsequently, the compatibility of the dam points is assessed using statistical tests in both constrained and free network adjustments. Finally, the application of the Kalman filter for displacement prediction is explored, providing insights into its precision and potential impact.

3.1. Geodetic Network of Mosul Dam

The geodetic network of Mosul Dam consists of a horizontal 2D network and a vertical 1D network and is composed of strategically positioned pillars and benchmarks over the dam site. The primary objective of this network is to monitor any potential movements or incompatibilities within the dam structure over time. The main network, encompassing pillars, forms the backbone of this system, while the extensive network incorporates both pillars and benchmarks to ensure comprehensive monitoring.

The Iraqi General Authority of Survey assumes the fixed positions of two key pillars, P42 and P44 [29,30], which serve as reference points within the horizontal geodetic network while assuming all the other pillars in the network as non-fixed. However, in this paper, all the pillars are distributed across the surrounding terrain, and the dam’s structure is assumed as fixed points (Figure 6). The logic behind this classification is based on the principle that significant movements might primarily affect the dam body while having negligible impacts on the surrounding ground which includes all the pillars beside P42 and P44. This framework laid the foundation for the adopted assessment of the compatibility of dam pillars (P61 up to P66) within the network.

In this paper, observational data spanning eight years (2005–2013) was accessed which is generously provided by the General Authority of Survey. This dataset is valuable for assessing the stability and compatibility of the pillar points of the Mosul Dam over successive epochs. The observational data are particularly significant for tracking any displacements, especially considering the inherent complexities of the dam’s geological foundations and the potential implications of instability.

It is worth noting that due to different reasons, the network adjustment results did not provide constant accuracy measures over the network. Environmental conditions, the quality and calibration of measurement instruments, and the proficiency of the surveyors involved all have a substantial influence on the results. These diverse factors introduce variability and can result in a noticeable difference in the accuracy of geodetic network adjustments. Figure 7 shows the ellipse of error plots at the same exaggeration scale for two epochs where the difference in the uncertainty of the adjusted points can be noticed.

Considerably, the variations in the quality of the adjusted dam pillars across epochs will be considered in the statistical analysis by incorporating standard deviations, as detailed in Section 2. This allows an efficient detection of significant displacements in the dam points. Figure 7 illustrates how the discrepancies and limitations in network measurements throughout epochs impact the precision of estimated accuracy at the adjusted dam pillars.

Another noteworthy consideration concerns the weighting of measured angles and distances within the network. As per the General Authority of Survey’s guidelines, standard deviations of 10” for angles and 1 cm for distances have been adopted, aligning with the specifications of the total station instruments in use. However, it is crucial to recognize that altering these standard deviations can yield distinct adjustment outcomes. Figure 8 presents the adjusted network at epoch 37 under the conventional weight specifications and contrasts it with the results obtained when a tighter confidence level of 5” for angles is employed.

The same important observation related to the geodetic network of Mosul Dam is the selection adopted by the General Authority of Survey of two control points P42 and P44 among the other network pillars. Those two pillars are assumed fixed when running the geodetic network constrained-adjustment. In this paper, a free adjustment of the network was applied to assess the internal reliability of the network and make a robust decision in terms of quality and reliability about which pillars should be adopted as control points. Generally, control points that have smaller uncertainties (ellipses of errors) should indicate higher precision and reliability. Using these precise points as constraints in the adjustment will ensure that the other adjusted points in the network are as accurate as possible and is expected to reduce systematic errors in the adjustment. In other words, free-adjusted points with smaller ellipses of errors have higher internal reliability and are not expected to introduce inconsistencies or incompatibilities in the network during the constrained adjustment. Therefore, selecting the minimal error points as control leads to a more stable and reliable adjustment.

However, the selection is also governed by the network points’ spatial distribution, number and type of measurements besides other site conditions and requirements.

Moreover, the free network adjustment was applied at different epochs and estimated the quality of the adjusted points of the network. Figure 9a–c illustrate the results of the free net adjustment in trilateration, triangulation, and combined observations, respectively, where the ellipse of errors is plotted at an exaggeration scale. Evidently, P42 and P44 are not the best choice among the network points to be adopted as control points in the constrained adjustment of the network while points P51, P52, and P55 showed a more reliable choice to be adopted as control points. To evaluate the validity of replacing P42, and P44 with points P51 and P55 for the constrained adjustment, the adjustment process is conducted twice: once using the conventional points and then using P55 and P51. The outcomes are presented in Figure 9d,e where the ellipses of errors for the network points in both scenarios are shown. The results provide a convincing indication that employing P51 and P55 as control points leads to a more robust adjustment network.

3.2. Testing Dam Points Compatibility

In this section, the rigorous process of assessing the compatibility of the dam’s geodetic network pillars was explored using the statistical test described in Section 2. This compatibility evaluation involves two key approaches: the constrained network adjustment, which is explored in Section 3.2.1, and the free geodetic network adjustment, elaborated on in Section 3.2.2.

The visual representation of dam pillar displacement directions from epoch 36 (in 2005) to epoch 50 (in 2013) is presented in Figure 10. In Figure 10a, the exaggerated displacement vectors at each pillar are depicted, utilizing constrained network adjustment. On the other hand, Figure 10b illustrates the displacement vectors resulting from the free network adjustment. These visualizations provide valuable insights into the changes and movements observed in the dam pillars over the specified period. However, following compatibility testing is important to evaluate the significance of the shown displacements between measured epochs.

3.2.1. Constrained Geodetic Network

To assess the compatibility of the geodetic network pillars located on the dam, these pillars were treated as ‘unknown’ or ‘unfixed’ during the constrained least square adjustment and subsequent statistical post-analysis. In contrast, all other pillars beyond this subset were assumed to be “fixed” due to the presumption of minimal movement in the surrounding terrain. Three statistical tests were applied including the F-test, t-test, and Z-score to assess the compatibility of pillars P61 up to P66 across successive epochs (2005 to 2013). Generally, these statistical tests allow data-driven determinations of significant movements or incompatibilities within the geodetic network.

The compatibility checks using F-statistic (Section 2.3) starting at epoch 36 measured in 2005 and ending at epoch 50 measured in 2013 are shown in Figure 11a. Two alarms are indicated after running the F-test between epochs 38–42 and epochs 49–50.

Then, compatibility checks using t-statistic (Section 2.2) for the same epochs are applied and the same alarms are indicated as before between epochs 38–42 and epochs 49–50, as shown in Figure 11b.

A final compatibility check using Z-score (Section 2.4) for the same tested epochs is applied at probability level 99.9% (Z-score = 3.29) and probability 99% (Z-score = 2.33). Figure 12a illustrates the result of the Z-score test at a threshold of 3.29 and Figure 12b shows the results of the test at Z-score 2.33.

3.2.2. Free Geodetic Network

Preferably, when assessing the compatibility of a monitoring geodetic network across multiple epochs, the initial approach often involves a free network adjustment. This choice stems from the assumption that the network’s control points may exhibit variations in their datums over time, necessitating an initial independent adjustment for each epoch. Various methods can be adopted to tackle the difficulties posed by these varying data and ensure a rigorous compatibility assessment.

One such method is called the Boljen method using F-statistic as presented in [31], which offers a structured approach to address the challenge of differing datums in a multi-epoch geodetic network. Coordinate transformations (e.g., Helmert transformation) are required to align the coordinates from different epochs and then assess the quality of these transformations.

To perform the compatibility check using the freely adjusted network of Mosul Dam across multiple epochs, similar to the approach discussed in Section 3.2.1, it is important to emphasize that this assessment will encompass not only the dam pillars (P61 to P66) but all the network points. However, the primary focus remains on the dam pillars. This emphasis on dam pillars is vital for a meaningful comparison with the results obtained from the constrained adjustment, as presented in the preceding section. The F-statistic and t-statistic tests are both applied to the same epochs. Figure 13a shows the compatibility checks using the F-statistic and Figure 13b shows the compatibility checks using the t-statistic.

3.3. Displacement Prediction Using Kalman Filter

A displacement prediction for the dam pillars is applied in this research using a Kalman filter with a total of ten epochs. In practice, more epochs should certainly be used in building the Kalman prediction model considering only significant movements indicated in the compatibility tests. The result of the Kalman filter comprises predictions for dam pillar displacements, which can be predicted for any future time interval using the dynamic model specified in Equation (12). These predictions are visually depicted on a graph for epoch 50 (recorded in 2013). The predicted state is put together with the adjusted coordinates computed from the least square adjustment to illustrate the Kalman filter’s precision in predicting the pillar’s displacement, as illustrated in Figure 14a. Furthermore, the velocity and heading (displacement) of the dam points can be predicted using the estimated velocity components $\dot{X},\dot{Y}$, as shown in Figure 14b.

4. Discussion

In Section 3, different statistical analyses were applied to the configuration of the geodetic network including the control points and the compatibility checks using constrained and free adjustment. Furthermore, the Kalman filter is applied to investigate its feasibility for future displacement prediction. The comprehensive approach of the statistical tests allowed for a thorough investigation into the stability of the critical dam components and can provide insights into long-term trends and movements. The following list of discussion points is found:

• As shown in Figure 10, the 8-year displacement analysis of the dam’s pillars through both constrained and free adjustments reveals a consistent trend of movement from upstream to downstream, as anticipated.
• Implementing the Kalman filter for displacement prediction at the dam points has demonstrated promising outcomes (Figure 14). Nevertheless, practical applications require more epochs to ensure a robust and reliable prediction. The Kalman filter offers continuous and dynamic estimation by updating predictions based on measurement epochs which is more suitable for monitoring dynamic systems. In contrast, the least squares adjustment provides static estimations independently for each epoch. Accordingly, it is preferred to use the Kalman filter for dam monitoring and dynamic prediction. Figure 14a revealed a less optimistic prediction for the movement of dam pillars when compared to the least square-adjusted points based on actual measurements. In contrast, Figure 14b depicted a uniform velocity and angular heading among all the dam points, persisting from 2005 to 2013, moving consistently from upstream to downstream at an average velocity of 1.5 cm/year.
• Reducing the weight of measured angles or distances in the least squares adjustment will result in a higher uncertainty (unit variance) and more indication of incompatibility in pillars. Accordingly, if the measurements are assigned higher weights (low uncertainty), the errors can then propagate into the final adjusted coordinates, leading to increased variance and larger error ellipses. In simpler terms, more variability is introduced into the data by decreasing the weight of the measurements or increasing the time between measurements (for example, between epoch 38 and epoch 42). Therefore, the adjustment results should be interpreted in the context of the data collection process. The choice of measurement intervals and their associated uncertainties can significantly impact the detection of movement or incompatibility in the dam pillars.
• In the statistical compatibility testing, the Z-statistic showed less reliability compared to the t-statistic and F-statistic. Lowering the Z-score probability threshold enhanced the sensitivity to errors, for example, setting it at 99% (Z-score = 2.33) as shown in Figure 12. In the constrained adjustment of epochs, both the F-test and t-test yielded similar indications of incompatibility for dam pillars when comparing epoch 49 to epoch 50. However, the Z-score test did not detect any significant change at 99.9% while adjusting the Z-score threshold to 99% (Z-score = 2.33), it did identify incompatibility. In principle, all three statistical tests agree on the presence of incompatibility between epochs, yet they may not identify changes in the same set of dam points.
• The observed incompatibility between epochs 49 and 50 in the constrained adjustment could be attributed to the temporal differences between the two epochs. Epoch 49 was recorded in June during the summer season when the reservoir water level was relatively low. In contrast, epoch 50 was recorded in the winter of 2013, with higher water levels. This seasonal variation in water levels may well account for the observed differences. However, it is noteworthy that the free-adjusted networks indicated a compatibility of the dam points between the mentioned epochs. The constrained adjustment relies heavily on control points, and significant discrepancies might be flagged as incompatibilities. In contrast, the free adjustment optimizes all points, and better determination of their coordinates can lead to a perception of compatibility between epochs.
• An additional finding from our analysis revealed that pillars P51, P52, and P55 were the most reliable choices as control points. These pillars presented the smallest uncertainties, graphically represented by the ellipses of errors (Figure 9c). This indicates higher precision and stability that aligns with the principles of zero-order network design optimization.

5. Conclusions

Dams are essential infrastructures, and their long-term stability is of high importance to prevent disastrous consequences. The Mosul Dam in Iraq is situated in a gypsum area which poses stability challenges due to the breakdown of gypsum in its foundation. Accordingly, international concerns have always been raised about its safety. This research addresses the critical need for comprehensive statistical assessments of the geodetic network points at the Mosul Dam. Valuable insights have been found through this research about the stability of the dam’s geodetic network through a rigorous analysis by the examination of the data spanning multiple epochs (2005–2013) revealing a consistent trend of upstream-to-downstream movement that aligns with the expectations.

Three different statistical tests have been investigated including the F-statistic, t-statistic, and Z-score to rigorously assess the compatibility and deformations within the geodetic network of the Mosul Dam over multiple epochs. The F-statistic and t-statistic revealed their effectiveness by detecting critical changes in the network signaling alarms between epochs 38–42 and epochs 49–50 and offering an insightful perspective on the dynamic behavior of the dam’s pillars. On the other hand, the Z-score showed less reliability in the compatibility checks but still demonstrated its utility by identifying significant changes in the network when selecting appropriate probability levels. These statistical tests enable for comprehensive analysis of the stability of the critical dam components and provide valuable insights into the long-term trends and movements that impact the Mosul Dam’s safety and stability.

Ultimately, the Kalman filter was applied for displacement prediction at the dam points over 10 epochs which has shown efficient computations with an average velocity of 1.5 cm/year from upstream to downstream. Practical applications may necessitate more epochs to ensure robust and reliable predictions. The Kalman filter dynamic and continuous estimation based on measurement epochs makes it more suitable for monitoring dynamic systems compared to the static estimations provided by the least squares adjustment. Therefore, we recommend the utilization of the Kalman filter for dam monitoring and prediction to enhance the safety of this critical infrastructure.

This research outcome ensures the resilience and safety of this vital infrastructure by contributing to the body of knowledge on dam stability assessment and following established procedures. It is anticipated that the findings presented in this paper will lay the foundation for future studies with more recent epochs. The described methodology can serve as a guide for other researchers to follow in assessing the stability of other infrastructures. Future studies for dam movement prediction will investigate the use of machine learning techniques like neural networks to be trained and predict dam point movement based on historical data.