# Algorithm to Correct Measurement Offsets Introduced by Inactive Elements of Transducer Arrays in Ultrasonic Flow Metering

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## Abstract

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## 1. Introduction

## 2. Measurement Offsets Introduced by Inactive Transducer Array Elements

#### 2.1. On the Shape of the Beam

#### 2.2. On the Travel Path

## 3. Algorithm

## 4. Validation of the Algorithm

#### 4.1. Simulation

#### 4.1.1. Settings

#### 4.1.2. Results

#### 4.2. Experiment

#### 4.2.1. Setup

#### 4.2.2. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

UFMs | Ultrasonic flow meters |

SNR | Signal-to-noise ratio |

t-x | time-space |

f-k_{x} | frequency-wavenumber |

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**Figure 1.**(

**a**) Spatial aperture of a 96−element transducer array with fully working elements (in blue) and 10 inactive elements (in red). (

**b**) Fourier transform of the spatial apertures shown in (

**a**).

**Figure 2.**Simulation results obtained using FIELD II [28,29] to generate acoustic beams in water steered at an angle of $\theta =8.6{}^{\circ}$ relative to the normal of the surface of a 37-element transducer array. The flow effect was included by defining the sound speed of the medium as $c=1480\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}\mp {v}_{f}cos\left(8.6{}^{\circ}\right)$, with ${v}_{f}=0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}$ being the flow speed, and where ∓ defined the acoustic wave propagating upstream and downstream, respectively. The beams were measured in the far field. (

**a**) Location, relative to the surface of the transducer array, of the peak amplitude of each illgenerated beam as a function of the position at which the inactive element is located on the array. The dashed horizontal lines show the ground truth locations, i.e., those obtained with fully working arrays. (

**b**) Transit time difference offset introduced by the distorted travel paths described by the ill-defined beams. The dashed horizontal line indicates the ground truth transit time difference, i.e., the one obtained with fully working arrays.

**Figure 3.**Flowchart of the designed algorithm to correct for the measurement offsets introduced by inactive (non-functioning/broken/damaged) elements during transducer array-based ultrasonic flow metering.

**Figure 5.**Simulation results of (

**a**) mean transit time difference for 100 random configurations for the cases of a total of 10, 20, 40 and 50 inactive array elements situated at random locations along the aperture of 96−elements arrays. Per configuration, a data point represents the mean of 50 “virtual” transit time difference values obtained from 50 simulated noisy wavefields. The blue dots represent the values obtained without implementing the proposed algorithm (i.e., after just beamforming, obtaining and averaged signal per array and cross-correlating them to obtain the transit time difference), and the red circles represent the values obtained after implementing the proposed algorithm. The horizontal black line represents the value obtained without inactive elements, i.e., $\Delta {t}_{mean}=-0.93\phantom{\rule{0.166667em}{0ex}}\mathrm{ps}$. (

**b**) Standard deviation of the $\Delta {t}_{mean}$ value of each configuration reported in (

**a**). Each data point represents the standard deviation of 50 “virtual” transit time difference values obtained per configuration.

**Figure 6.**Mean value of the distributions of standard deviation shown in Figure 5b, which were obtained from simulation results.

**Figure 7.**Mean and standard deviation of the transit time difference obtained when considering groups of inactive elements located (

**a**) at the center of the aperture of the arrays, and (

**b**) at the center and both edges of the aperture of the arrays (i.e., 1/3 of the total number of inactive elements at each location). Per number of inactive elements, a data point of $\Delta {t}_{mean}$ represents the mean of 50 “virtual” transit time differences values obtained from 50 simulated noisy wavefields, and a data point of $\sigma $ represents the standard deviation of those 50 “virtual” transit time differences values. The dashed black line on the graphs for $\Delta {t}_{mean}$ represents the value obtained without inactive elements, i.e., $\Delta {t}_{mean}=-0.93\phantom{\rule{0.166667em}{0ex}}\mathrm{ps}$.

**Figure 8.**(

**a**) Setup to carry out measurements on which to test the proposed algorithm. The probes are two P4−1 transducer arrays clamp-on on the outside of a square stainless steel pipe filled with water. (

**b**) Typical measured wavefield. Measurements were carried out at zero-flow conditions.

**Figure 9.**2D Fourier transform of the spatio-temporal data shown in Figure 8b. The red circles frame the bandwidth of energy associated to the wave mode of interest. All other bandwidths of energy are associated to spurious wave modes.

**Figure 10.**Measurement results of (

**a**) mean transit time difference for 100 random configurations for the cases of a total of 10, 20, 40 and 50 inactive array elements situated at random locations along the aperture of 96−elements arrays. Per configuration, a data point represents the mean of 50 transit time difference measurements carried out. The blue dots represent the values obtained without implementing the proposed algorithm (i.e., after just beam-forming, obtaining and averaged signal per array and cross-correlating them to obtain the transit time difference), and the red circles represent the values obtained after implementing the proposed algorithm. The horizontal black line represents the value obtained without inactive elements, i.e., $\Delta {t}_{mean}=-0.76\phantom{\rule{0.166667em}{0ex}}\mathrm{ns}$. (

**b**) Standard deviation of the $\Delta {t}_{mean}$ value of each configuration reported in (

**a**). Each data point represents the standard deviation of 50 transit time difference measurements carried out per configuration.

**Figure 11.**Mean value of the distributions of standard deviation shown in Figure 10b, which were obtained from measurement results.

**Figure 12.**Simulation results of mean (

**left**) and standard deviation (

**right**) of the transit time difference for 100 random configuration for the case of 10 inactive array elements situated at random locations along the aperture of 96− elements arrays and considering a flow speed of ${v}_{f}=0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}$. Blue dots represent the values obtained without implementing the proposed algorithm (i.e., after just beam-forming, obtaining and averaged signal per array and cross-correlating them to obtain the transit time difference), and the red circles represent the values obtained after implementing the proposed algorithm. The horizontal black line represents the value obtained without inactive elements, i.e., $\Delta {t}_{mean}=72.78\phantom{\rule{0.166667em}{0ex}}\mathrm{ns}$.

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**MDPI and ACS Style**

Massaad, J.; van Neer, P.L.M.J.; van Willigen, D.M.; Pertijs, M.A.P.; de Jong, N.; Verweij, M.D.
Algorithm to Correct Measurement Offsets Introduced by Inactive Elements of Transducer Arrays in Ultrasonic Flow Metering. *Sensors* **2022**, *22*, 9317.
https://doi.org/10.3390/s22239317

**AMA Style**

Massaad J, van Neer PLMJ, van Willigen DM, Pertijs MAP, de Jong N, Verweij MD.
Algorithm to Correct Measurement Offsets Introduced by Inactive Elements of Transducer Arrays in Ultrasonic Flow Metering. *Sensors*. 2022; 22(23):9317.
https://doi.org/10.3390/s22239317

**Chicago/Turabian Style**

Massaad, Jack, Paul L. M. J. van Neer, Douwe M. van Willigen, Michiel A. P. Pertijs, Nicolaas de Jong, and Martin D. Verweij.
2022. "Algorithm to Correct Measurement Offsets Introduced by Inactive Elements of Transducer Arrays in Ultrasonic Flow Metering" *Sensors* 22, no. 23: 9317.
https://doi.org/10.3390/s22239317