# Accuracy Analysis and Appropriate Strategy for Determining Dynamic and Quasi-Static Bridge Structural Response Using Simultaneous Measurements with Two Real Aperture Ground-Based Radars

^{1}

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## Abstract

**:**

_{I}) considerably rises. When using two radars, it is possible to simultaneously determine, for example, vertical and longitudinal displacements and to eliminate the Interpretation Error. The aim of the article is to establish a suitable strategy for determining dynamic and quasi-static response of bridge structures based on the accuracy analysis of measurement by two radars. The necessary theory for displacements determination by means of two radar devices is presented. This is followed by an analysis of errors when measuring with only one radar. For the first time in the literature, mathematical formulas are derived here for determining the accuracy of the resulting displacements by simultaneous measurement with two radars. The practical examples of bridge structures displacements determination by measuring with two radar devices in the field are presented. The key contribution of the paper is the possibility to estimate and plan in advance the achievable accuracy of the resulting displacements for the given radar configurations in relation to the bridge structure.

## 1. Introduction

_{LOS}R/H.

_{I}.

_{I}is shown in Figure 2.

_{I}can be expressed as follows:

_{I}= (d − s

_{y})/d.

_{I}can be calculated based on the geometry shown in Figure 2 according to the relationship.

_{I}and the ratios R/H (radar distance from the measured point/radar distance from the measured point in vertical direction) and s

_{x}/s

_{y}(longitudinal or transversal horizontal displacement/vertical displacement). For clarity, Table 1 shows the values of the Interpretation Error depending on the ratios R/H and s

_{x}/s

_{y}. With the usual size of the ratio of horizontal displacements to vertical s

_{x}/s

_{y}= 0.10 in practice, the value of Interpretation Error E

_{I}= 23% already at the ratio R/H = 2.50. At the ratio R/H = 5.00, E

_{I}= 49%. With a greater ratio of horizontal to vertical displacements, which can occur in some cases, the E

_{I}values are even significantly larger. The size of the Interpretation Error can therefore take on very significant values and in common practice can completely invalidate the measurement results and lead to erroneous conclusions regarding the health of the tested structure. The most important finding regarding the influence of the Interpretation Error E

_{I}is that, with some exceptions, it is not possible to rely on the results of measuring vertical displacements with only one radar.

## 2. Method of GB-RAR with Two Interferometric Radars

#### 2.1. Data Processing

**Proof.**

#### 2.2. Time Synchronization of Two or More Radars

#### 2.3. Accuracy Analysis of Longitudinal and Vertical Component of the Total Displacement

_{bin}) or distance resolution, which can be, for example, ΔR = 0.75m, does not affect the accuracy of determining the displacement vector. This value expresses the uncertainty of the position of the monitored point. It is possible to determine the displacement vector of a point on the bridge very precisely and correctly estimate the accuracy of its determination but, at the same time, it is usually not possible to identify the position of this moving point more precisely than a decimeter resolution allows. This problem is caused not only by the size of the range resolution area ΔR, but also by the different reflectivity of the material and by the complexity (number of details) of the bridge deck construction in the given range resolution area as well.

_{bin}s on the monitored object, it is recommended to create a point cloud of using the laser scanning method. The point cloud must contain both the given object and the positions of both radars during the measurement. Then individual points on the object can be colored according to the distances from the phase centers of the radars at intervals according to the R

_{bin}interface. This identifies the positions of individual R

_{bin}s on the monitored object. It means that the corresponding locations on the monitored object will be assigned to the measured LOS displacements. The creation of a point cloud by laser scanning can of course be replaced by any other geodetic method for creating 3D models of objects, such as, e.g., photogrammetry using UAVs (drones) [33,34].

${\mathit{C}}_{\mathit{s}}$ | … | covariance matrix of displacement vector s; |

${\mathit{C}}_{\mathit{u}}$ | … | covariance matrix of components of vector u; |

${\mathit{J}}_{\mathit{u}}$ | … | Jacobian matrix of mapping $\mathit{u}\mapsto \mathit{s}\left(\mathit{u}\right)$ given by (9); ${\mathit{J}}_{\mathit{u}}:=\frac{\partial \mathit{s}}{\partial \mathit{u}}\left(\widehat{\mathit{u}}\right),\text{}\mathit{s}\left(\mathit{u}\right)\cong \mathit{s}\left(\widehat{\mathit{u}}\right)+{\mathit{J}}_{\mathit{u}}\xb7\left(\mathit{u}-\widehat{\mathit{u}}\right),$ |

$\widehat{\mathit{u}}$ | … | vector of measured or approximately computed values of input quantities ${t}_{1},{t}_{2},{x}_{1},{y}_{1},{x}_{2},{y}_{2},$ i.e., $\widehat{\mathit{u}}=:\text{}\left[{\widehat{t}}_{1},{\widehat{t}}_{2},{\widehat{x}}_{1},{\widehat{y}}_{1},{\widehat{x}}_{2},{\widehat{y}}_{2}\right]=:\left[{\widehat{\mathit{t}}}_{2},{\widehat{\mathit{x}}}_{1},{\widehat{\mathit{x}}}_{2}\right].$ |

#### 2.3.1. The Case When Imprecision of LOS Directions Is Neglected

a | … | main axis size of the ellipse, |

b | … | semi axis size of the ellipse, |

φ | … | orientation of the main axis (in radians), |

D | … | discriminant of the covariance matrix, $D:=\sqrt{{({c}_{1,1}-{c}_{2,2})}^{2}+4{c}_{1,2}^{2}}.$ |

ind | … | dicator function, $\mathrm{ind}\left(\mathrm{true}\right):=1,\text{}\mathrm{ind}\left(\mathrm{false}\right):=0.$ |

#### 2.3.2. The Case When Imprecision of LOS Directions Is Considered

#### 2.3.3. The Case of Placing Radars behind Each Other

#### 2.3.4. Accuracy Analysis in Different Locations of the Monitored Bridge

#### 2.3.5. Accuracy Analysis Separately in Vertical and Longitudinal Directions

#### 2.3.6. Summary of Accuracy Analysis Findings

- Imprecision of LOS directions is neglected—covariance matrix ${\mathit{C}}_{\mathit{t}}$ is given (Section 2.3.1).
- Precision of LOS directions is considered—covariance matrices ${\mathit{C}}_{\mathit{t}}$, ${\mathit{C}}_{xy}$ are given (Section 2.3.2).

#### 2.4. Experimental Measurement in Order to Verify Theory

#### 2.4.1. Experimental Measurement of the Arch Road Bridge “Valy”

#### Description of the Observed Bridge

#### Used Radar Interferometry Equipment

#### Used Standard Measuring Equipment

#### Measurement Configuration

_{bin}s for evaluation. Only those R

_{bin}s that have sufficient signal strength on both radars were selected. Selected R

_{bin}s (crossbeams) are highlighted in Figure 16.

#### 2.4.2. Experimental Measurement of the Railway Bridge “Púchov”

#### Description of the Observed Bridge

#### Used Standard Measuring Equipment

#### Used Radar Interferometry Equipment

#### Used Photogrammetric Digital Image Correlation Equipment

## 3. Results

#### 3.1. Tests on the Bridge “Valy”

#### 3.2. Tests on the Bridge “Púchov”

## 4. Discussion

_{I}(2). This error occurs as a result of assumption about an expected direction of the bridge displacement whenever the expected direction differs from the direction of real total displacement. In common practice, the E

_{I}error can take on values in the tens of % of the magnitude of the determined displacement. In the case of inappropriate geometrical configuration of the radar position and the monitored point, the E

_{I}can acquire unacceptable values, even more than 100%, as shown in Table 1. It is therefore necessary to perform measurements with at least two radar devices that can eliminate the E

_{I}error. This means that when measuring with two radars, a detailed accuracy analysis of determining the total displacements (vertical and longitudinal) according to (4) has to be developed.

_{I}(Figure 23a,c and Figure 28a–c). These E

_{I}errors are generally different for both radars, and therefore, the resulting vertical displacements cannot be simply averaged.

## 5. Conclusions

_{I}that occurs, with exceptions, when measuring with only one radar. The example in Figure 23a can be used to concretely illustrate the significance of the Interpretation Error E

_{I}in practice. At 187.4 s, the single radars determined a vertical deflection of 0.50 mm (Rbin R1 67) and 0.20 mm (Rbin R2 24). By combining both radar measurements, a vertical deflection of 0.27 mm was determined (Figure 23b). The influence of the Interpretation Error when determining vertical displacements separately by single radars is therefore +46% (radar R1) and −35% (radar R2) in this particular case.

- to highlight necessity of simultaneous usage of two interferometric radars to eliminate the Interpretation Error;
- to achieve the highest possible accuracy in determining the resulting total displacements.

- description of the current state and analysis of Interpretation Errors E
_{I}when measuring with single radar (see the Section 1); - presentation of the principles of measurement by two radars with the accuracy analysis of the resulting displacements (see the Section 2.1, Section 2.2 and Section 2.3);
- verification of the results in practice by experimental measurement (see the Section 2.4 and Section 3);

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Line of sight movement (d

_{LOS}) and expected (calculated) vertical movement (d), R—radar distance from the measured point, and H—radar distance from the measured point in vertical direction.

**Figure 2.**Origin of the Interpretation Error E

_{I}when measuring with only one interferometric radar: s—total displacement; s

_{y}—vertical component of total displacement; s

_{x}—horizontal component of total displacement; d

_{LOS}—measured displacement in the range direction; d—calculated vertical displacement; R—radar distance from the measured point; H—radar distance from the measured point in vertical direction.

**Figure 3.**Two basic configurations of the position of the two radars when measuring bridges: (

**a**) the radars measure against each other—at the top, or (

**b**) the radars measure from one side (radars are placed behind each other)—at the bottom.

**Figure 4.**Relationship of displacement vector $\left[{s}_{X},{s}_{Y}\right]$ to measured LOS displacements ${t}_{1},{t}_{2}$ and vertical angles ${\psi}_{1},{\psi}_{2}$ from radars ${R}_{1}$, ${R}_{2}$ to the monitored point.

**Figure 5.**Accuracy of displacement vector $\left[{s}_{X},{s}_{Y}\right]$ when the radars are located at opposite ends of the bridge and vertical angles of radar directions are ${30}^{\circ}$ and ${150}^{\circ}$: Standard deviation of measured LOS displacements is 0.02 mm. Scale of the mean error ellipses is 10:1. (

**a**) Imprecision of the vertical angles of radar directions was neglected; (

**b**) precision of the vertical angles is given by means of standard deviation ${0.5}^{\circ}$.

**Figure 6.**Accuracy of displacement vector $\left[{s}_{X},{s}_{Y}\right]$ when radars are located behind each other, and vertical angles of radar directions are ${10}^{\circ}$ and ${30}^{\circ}$: Standard deviation of measured LOS displacements is 0.02 mm. Scale of the mean error ellipses is 10:1. (

**a**) Imprecision of the vertical angles of radar directions was neglected; (

**b**) precision of the vertical angles is given by means of standard deviation ${0.5}^{\circ}$.

**Figure 7.**Accuracy of displacement vector $\left[{s}_{X},{s}_{Y}\right]=\left[0.0\mathrm{mm},\text{}5.0\text{}\mathrm{mm}\right]$ at various points on the bridge. Positions of the radars are determined with precision $0.2\mathrm{m}$. Standard deviation of measured LOS displacements is 0.02 mm. Scale of the mean error ellipses is 50,000:1. (

**a**) radars are placed at opposite ends of the bridge; (

**b**) radars are placed behind each other.

**Figure 8.**Standard deviations when the radars are located at opposite ends of the bridge (vertical angles of radar directions are ${30}^{\circ}$ and ${150}^{\circ}$ ). The angles are determined with precision ${0.5}^{\circ}$. Standard deviation of measured LOS displacement is 0.02 mm. Numeric values on the axes and on the isolines are in mm: (

**a**) Standard deviation ${\sigma}_{y}$; (

**b**) Standard deviation ${\sigma}_{x}$.

**Figure 9.**Standard deviations when the radars are located behind each other (vertical angles of radar directions are ${10}^{\circ}$ and ${30}^{\circ}$ ). The angles are determined with precision ${0.5}^{\circ}$. Standard deviation of measured LOS displacement is 0.02 mm. Numeric values on the axes and on the isolines are in mm: (

**a**) Standard deviation ${\sigma}_{y}$; (

**b**) Standard deviation ${\sigma}_{x}$.

**Figure 10.**Side view on the third span of the tied-arch bridge (

**a**) photo; (

**b**) schematic with marked positions of accelerometers and Rbins.

**Figure 11.**The view on the load bearing structure in the third span of the bridge (

**a**) photo; (

**b**) schematic with marked positions of accelerometers.

**Figure 12.**A view of the SIRIUS data acquisition station and the left main girder with two 8344 accelerometers attached in the center of the main span of the bridge.

**Figure 13.**A view of the steel crossbeams of the bridge from the position of the R2 radar (near Valy municipality). The crossbeams served as natural signal reflectors.

**Figure 14.**Top view of the location of radars under the third (main) span of the bridge, values are in meters.

**Figure 16.**Three-dimensional model of the bridge with color-highlighted R

_{bin}s for radars R1 and R2, which were evaluated.

**Figure 17.**Side view on the analyzed superstructure BS2 (

**a**) photo; in the background there is the superstructure BS3; (

**b**) schematic with marked positions of DIC target, Rbins, radars, and DIC camera.

**Figure 18.**View on the analyzed horizontal load-bearing structure BS2 that is oriented in the direction of its longitudinal axis (

**a**) photo; in the background there is the superstructure BS3; (

**b**) schematic with marked positions of DIC target, and Rbins.

**Figure 19.**(

**a**) View on the test train moving across the bridge. (

**b**) View on the accelerometer of type 8344 located on the rigid beam.

**Figure 20.**Used interferometric radars: (

**a**) Bottom view on the bridge deck of the structure BS2 with the steel crossbeams from the position of the radar R1; (

**b**) View on both radars: R1 is in the foreground, and R2 is in the background in the photo.

**Figure 21.**Three-dimensional model of the bridge with color-highlighted Rbins for radars R1 and R2, which were evaluated, and with marked positions of both radars below the structure BS2.

**Figure 22.**Photogrammetric digital image correlation equipment: (

**a**) Two active cameras applied during the experiment as part of the 2D-DIC system. (

**b**) The steel boards painted by white color and by a black random pattern fixed at the observed point on the bridge deck.

**Figure 23.**Displacements, including the quasi static component demonstrated on two crossbeams (the first one corresponds to Rbins R1 54 and R2 38, and the second one to Rbins R1 67 and R2 24): at the top during the passage of the vehicle in the direction from Mělice to Valy; at the bottom caused by the regular pace of a group of pedestrians; (

**a**,

**c**) comparison of separately evaluated vertical displacements using Formula (1) measured by radars R1 and R2; (

**b**,

**d**) longitudinal and vertical displacements calculated by combination of both radar measurements (4). The Rbin numbers correspond to the R1 radar.

**Figure 24.**Dynamic displacements evaluated using combination of both radar measurements (by Formula (4)) and corresponding displacements measured by accelerometers: 1st and 2nd row: during the passage of the vehicle in the direction from Mělice to Valy; 3rd and 4th row: caused by the regular pace of a group of pedestrians; 1st and 3rd row: for R1 Rbin 54; 2nd and 4th row: for R1 Rbin 67: 1st column (

**a**,

**d**,

**g**,

**j**): comparison of dynamic displacements with accelerometer measurements; 2nd column (

**b**,

**e**,

**h**,

**k**): comparison of vertical displacements biases between radar and accelerometer measurement with the confidence intervals. Percentage of the biases that belong into the 95% confidence interval is: (

**b**) 99.2%, (

**e**) 94.0%, (

**h**) 100%, and (

**k**) 89.5%; 3rd column (

**c**,

**f**,

**i**,

**l**): comparison of longitudinal displacements biases between radar and accelerometer measurement with the confidence intervals. Percentage of the biases that belong into the 95% confidence interval is (

**c**) 99.4%, (

**f**) 100%, (

**i**) 93.5%, and (

**l**) 97.0%.

**Figure 25.**Comparison of total displacements biases between radar and accelerometer measurement with the confidence ellipses (15). Overlay of point clusters (black points) by 95% confidence ellipses (red curves): (

**a**) percentage of points in the ellipse that represents accuracy of the displacement vector in R1 Rbin 54 is 98.8%; (

**b**) for R1 Rbin 67, the percentage is 95.1%.

**Figure 26.**Effect of the quasi static components on the total displacement. Longitudinal and vertical displacements caused by the regular pace of a group of pedestrians during concurrent passage of a vehicle across the bridge: (

**a**) including the quasi static component calculated by combination of both radar measurements. The irregularity of the oscillation is caused by the vehicle crossing the bridge at the same time; (

**b**) without the quasi static component calculated by combination of both radar measurements. (

**c**) Corresponding longitudinal and vertical displacements converted from accelerometers measurement.

**Figure 27.**Frequency spectra (periodograms)—natural, significant, and distinct frequencies of the observed bridge measured by accelerometer: 1.87, 2.23, 2.95, 3.78, 3.99, 5.52, and 7.22 Hz and by radars (Sy Rbin 54): 1.88, 2.22, 3.32, 3.74, and 3.99 Hz (

**a**) and by accelerometer: 1.61, 1.87, 2.23, 2.46, 2.95, 3.43, 3.70, 3.99, 4.48, and 5.12 Hz and by radars (Sy Rbin67): 1.61, 1.80, 2.22, 3.42, and 3.74 Hz (

**b**).

**Figure 28.**First row: comparison of vertical displacements during the passage of the test train including 2 locomotives: (

**a**) at 40 km/h—direction Bratislava; (

**b**) at 50 km/h—direction Bratislava; (

**c**) at 90 km/h—direction Žilina; second row: comparison of the confidence intervals with biases between radar and DIC measurement of vertical displacements: (

**d**) for the train speed 40 km/h. Percentage of the biases that belong into the 95% confidence interval is 96%; (

**e**) 50 km/h, 94%; (

**f**) 90 km/h, 84%.

**Table 1.**Values of the Interpretation Error E

_{I}for R/H and s

_{x}/s

_{y}ratios commonly used in practice.

R/H\s_{x}/s_{y} | 0.01 | 0.04 | 0.07 | 0.10 | 0.15 | 0.20 | 0.25 | 0.30 | 0.40 | 0.50 |
---|---|---|---|---|---|---|---|---|---|---|

1.00 | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% |

1.20 | 1% | 3% | 5% | 7% | 10% | 13% | 17% | 20% | 27% | 33% |

1.40 | 1% | 4% | 7% | 10% | 15% | 20% | 24% | 29% | 39% | 49% |

1.60 | 1% | 5% | 9% | 12% | 19% | 25% | 31% | 37% | 50% | 62% |

1.80 | 1% | 6% | 10% | 15% | 22% | 30% | 37% | 45% | 60% | 75% |

2.00 | 2% | 7% | 12% | 17% | 26% | 35% | 43% | 52% | 69% | 87% |

2.50 | 2% | 9% | 16% | 23% | 34% | 46% | 57% | 69% | 92% | 115% |

3.00 | 3% | 11% | 20% | 28% | 42% | 57% | 71% | 85% | 113% | 141% |

3.50 | 3% | 13% | 23% | 34% | 50% | 67% | 84% | 101% | 134% | 168% |

4.00 | 4% | 15% | 27% | 39% | 58% | 77% | 97% | 116% | 155% | 194% |

4.50 | 4% | 18% | 31% | 44% | 66% | 88% | 110% | 132% | 175% | 219% |

5.00 | 5% | 20% | 34% | 49% | 73% | 98% | 122% | 147% | 196% | 245% |

5.50 | 5% | 22% | 38% | 54% | 81% | 108% | 135% | 162% | 216% | 270% |

6.00 | 6% | 24% | 41% | 59% | 89% | 118% | 148% | 177% | 237% | 296% |

7.00 | 7% | 28% | 48% | 69% | 104% | 139% | 173% | 208% | 277% | 346% |

8.00 | 8% | 32% | 56% | 79% | 119% | 159% | 198% | 238% | 317% | 397% |

9.00 | 9% | 36% | 63% | 89% | 134% | 179% | 224% | 268% | 358% | 447% |

10.00 | 10% | 40% | 70% | 99% | 149% | 199% | 249% | 298% | 398% | 497% |

Case | ${\mathit{C}}_{\mathit{s}}$ | Substitutions |
---|---|---|

LOS imprecision neglected | ${\mathit{J}}_{\mathit{t}}\cdot {\mathit{C}}_{\mathit{t}}\cdot {\mathit{J}}_{\mathit{t}}^{T}$ | (14), (13) |

LOS precision considered | ${\mathit{J}}_{\mathit{u}}\cdot {\mathit{C}}_{\mathit{u}}\cdot {\mathit{J}}_{\mathit{u}}^{T}$ | (12), (10), (16), (13), (17) |

R1: Radar IBIS—FS Plus | R2: Radar IBIS—RU 172 | |
---|---|---|

Sampling frequency | 200 Hz | 199.2 Hz ^{1} |

Signal range (max. distance) | 75 m | 70 m |

R_{bin} (range resolution area) | 0.75 m | 0.75 m |

^{1}The set value was 200 Hz; the actual value (corrected by the radar control software) was 199.2 Hz.

R1: Radar IBIS—FS Plus | R2: Radar IBIS—RU 172 | |
---|---|---|

Sampling frequency | 200 Hz | 199.2 Hz ^{1} |

Signal range (max. distance) | 100 m | 120 m |

R_{bin} (range resolution area) | 0.75 m | 0.75 m |

Vertical tilt of the radar | 0.0° | 53.9° ^{1} |

^{1}The R2 radar was tilted more upwards to avoid interference between the radars. This resulted in weak reflections from more distant Rbins, which could not be evaluated by the R2 radar.

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## Share and Cite

**MDPI and ACS Style**

Talich, M.; Havrlant, J.; Soukup, L.; Plachý, T.; Polák, M.; Antoš, F.; Ryjáček, P.; Stančík, V.
Accuracy Analysis and Appropriate Strategy for Determining Dynamic and Quasi-Static Bridge Structural Response Using Simultaneous Measurements with Two Real Aperture Ground-Based Radars. *Remote Sens.* **2023**, *15*, 837.
https://doi.org/10.3390/rs15030837

**AMA Style**

Talich M, Havrlant J, Soukup L, Plachý T, Polák M, Antoš F, Ryjáček P, Stančík V.
Accuracy Analysis and Appropriate Strategy for Determining Dynamic and Quasi-Static Bridge Structural Response Using Simultaneous Measurements with Two Real Aperture Ground-Based Radars. *Remote Sensing*. 2023; 15(3):837.
https://doi.org/10.3390/rs15030837

**Chicago/Turabian Style**

Talich, Milan, Jan Havrlant, Lubomír Soukup, Tomáš Plachý, Michal Polák, Filip Antoš, Pavel Ryjáček, and Vojtěch Stančík.
2023. "Accuracy Analysis and Appropriate Strategy for Determining Dynamic and Quasi-Static Bridge Structural Response Using Simultaneous Measurements with Two Real Aperture Ground-Based Radars" *Remote Sensing* 15, no. 3: 837.
https://doi.org/10.3390/rs15030837