# Numerical Analysis of Ultrasonic Multiple Scattering for Fine Dust Number Density Estimation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background on Multiple Scattering of Acoustic Waves

_{S}), as illustrated in Figure 1. Here, l

_{S}is the characteristic length of the exponential decay of a coherent wave that propagates in its initial direction after taking ensemble averaging over multiple realizations of disorder [16].

_{S}, unscattered and singly scattered wave components (presumably coherent waves) are dominant in the measured response. In this regime, the medium can be considered quasi-homogeneous, but with a renormalized wave speed. When the value of L is equivalent to a few mean free paths, single scattering becomes unimportant, and the wave propagation lies in the intermediate multiple scattering regime (Regime 2). In this regime, multiply scattered wave components are developed, but coherent waves are still observable with sufficient ensemble averaging. As L further increases to become much greater than l

_{S}, most wave energy is diffusively transferred by incoherent waves (Regime 3). In this regime, the transfer of the average wave intensity can be approximated by the diffusion equation. In this study, we focus on using coherent waves under Regimes 1 and 2 to estimate the number density of fine dust particles in the air, considering that acoustic wave propagation in the air–fine dust system typically lies in Regimes 1 and 2. Moreover, the independent scattering approximation (ISA) method, the basis of the proposed approach, works for Regimes 1 and 2 but not for Regime 3.

**r**given a point source location

**r**’ can be represented as the summation of the incident wave and all possible waves scattered by the scatterers (e.g., random particles) [16]. It is given by:

**k**) domain as:

_{S}, $\mathrm{G}\left(\omega ,\text{}{\mathit{r}}_{1},{\mathit{r}}^{\prime}\right)$ in Equation (1) can be approximated as ${G}_{0}\left(\omega ,{\mathit{r}}_{1}-{\mathit{r}}^{\prime}\right)$, which is called the Born approximation. However, as the distance ($\left|\mathit{r}-{\mathit{r}}^{\prime}\right|$) increases, the Born approximate solution diverges from the actual solution. To obtain an accurate solution that considers the multiple scattering effects, the integral equation (Equation (1)) needs to be expanded to nearly infinite orders with respect to ${G}_{0}$, which is computationally intractable.

_{0}, unlike the Green’s function G shown in Equation (1). Then, the mean Green’s function in the

**k**domain is given by

_{e}) [18,19,20,21,22]. In this study, we incorporate the independent scattering approximation (ISA) method [22] to estimate the number density of fine dust in the air from the measured acoustic wave responses.

**k**. In this regime, the random medium can be considered a quasi-homogeneous medium (effective medium) with an effective wavenumber ${k}_{e}$ given by

_{S}given by

_{S}is further related to the number density n, given by

^{2}), and ${\sigma}_{t}$ (m). In 3-D, n and ${\sigma}_{t}$ have a unit of 1/m

^{3}and m

^{2}, respectively. Equation (8) depicts the fundamentals of the proposed fine density estimation technique: the number density of scatterers can be estimated using α obtained from the ensemble average wave responses and the analytically computed ${\sigma}_{t}$. More detail about the ISA theory can be found in [22]. The proposed fine dust density estimation technique is detailed further in the following section.

## 3. Number Density Estimation Approach Based on Independent Scattering Approximation

## 4. Numerical Simulations

## 5. Results and Discussion

#### 5.1. Multiply Scattered Acoustic Waves

#### 5.2. Number Density Estimation Results

#### 5.3. Effects of Particle Volume Fractions

^{3}per day in the unit of mass concentration [30]. In other words, the volume fraction of fine dust, in reality, may be lower than 1%.

#### 5.4. Effects of Particle Shapes

## 6. Conclusions

- Independent scattering approximation can be used to estimate the number density, especially when the volume fraction of fine dust particles is low.
- The proposed ultrasonic wave data processing approach enables the estimation of the number density of fine dust particles with an average error of 43.4% in the frequency band 1–10 MHz (ka ≤ 1) at a particle volume fraction of 1%.
- Variation of fine particle shapes is a cause of uncertainty in number density estimation.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Shridhar, V.; Khillare, P.S.; Agarwal, T.; Ray, S. Metallic species in ambient particulate matter at rural and urban location of Delhi. J. Hazard. Mater.
**2010**, 175, 600–607. [Google Scholar] [CrossRef] - Jim, C.Y.; Chen, W.Y. Assessing the ecosystem service of air pollutant removal by urban trees in Guangzhou (China). J. Environ. Manag.
**2008**, 88, 665–676. [Google Scholar] [CrossRef] - Nowak, D.J.; Crane, D.E.; Stevens, J.C. Air pollution removal by urban trees and shrubs in the United States. Urban For. Urban Green.
**2006**, 4, 115–123. [Google Scholar] [CrossRef] - Pope, C.A.; Dockery, D.W. Health effects of fine particulate air pollution: Lines that connect. J. Air Waste Manag. Assoc.
**2006**, 56, 709–742. [Google Scholar] [CrossRef] [PubMed] - Center for Disease Control and Prevention Particle Pollution. Available online: https://www.cdc.gov/air/particulate_matter.html (accessed on 10 December 2020).
- Amaral, S.S.; de Carvalho, J.A.; Costa, M.A.M.; Pinheiro, C. An overview of particulate matter measurement instruments. Atmosphere
**2015**, 6, 1327–1345. [Google Scholar] [CrossRef] [Green Version] - Hauck, H.; Berner, A.; Gomiscek, B.; Stopper, S.; Puxbaum, H.; Kundi, M.; Preining, O. On the equivalence of gravimetric PM data with TEOM and beta-attenuation measurements. J. Aerosol Sci.
**2004**, 35, 1135–1149. [Google Scholar] [CrossRef] - Giechaskiel, B.; Maricq, M.; Ntziachristos, L.; Dardiotis, C.; Wang, X.; Axmann, H.; Bergmann, A.; Schindler, W. Review of motor vehicle particulate emissions sampling and measurement: From smoke and filter mass to particle number. J. Aerosol Sci.
**2014**, 67, 48–86. [Google Scholar] [CrossRef] - Chow, J.C.; Watson, J.G.; Park, K.; Lowenthal, D.H.; Robinson, N.F.; Park, K.; Maglian, K.A. Comparison of particle light scattering and fine particulate matter mass in central california). J. Air Waste Manag. Assoc.
**2006**, 56, 398–410. [Google Scholar] [CrossRef] - Chimenti, D.E. Review of air-coupled ultrasonic materials characterization. Ultrasonics
**2014**, 54, 1804–1816. [Google Scholar] [CrossRef] [PubMed] - Safaeinili, A. Air-coupled ultrasonic estimation of viscoelastic stiffnesses in plates. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**1996**, 43, 1171–1180. [Google Scholar] [CrossRef] - Choi, H.; Song, H.; Tran, Q.N.V.; Roesler, J.R.; Popovics, J.S. Concrete International. 2016. Available online: https://www.concrete-international.nl/en/ (accessed on 20 December 2020).
- Zhang, J.; Drinkwater, B.W.; Wilcox, P.D. Defect characterization using an ultrasonic array to measure the scattering coefficient matrix. IEEE Trans. Ultrason. Ferroelectr. Freq. Control
**2008**, 55, 2254–2265. [Google Scholar] [CrossRef] - Fu, S.; Lou, W.; Wang, H.; Li, C.; Chen, Z.; Zhang, Y. Evaluating the effects of aluminum dust concentration on explosions in a 20L spherical vessel using ultrasonic sensors. Powder Technol.
**2020**, 367, 809–819. [Google Scholar] [CrossRef] - Kazys, R.; Sliteris, R.; Mazeika, L.; Van den Abeele, L.; Nielsen, P.; Snellings, R. Ultrasonic monitoring of variations in dust concentration in a powder classifier. Powder Technol.
**2020**, in press. [Google Scholar] [CrossRef] - Tourin, A.; Derode, A.; Peyre, A.; Fink, M. Transport parameters for an ultrasonic pulsed wave propagating in a multiple scattering medium. J. Acoust. Soc. Am.
**2000**, 108, 503–512. [Google Scholar] [CrossRef] - Sheng, P.; van Tiggelen, B. Introduction to wave scattering, localization and mesoscopic phenomena. Second edition. Waves Random Complex Media
**2007**, 17, 235–237. [Google Scholar] [CrossRef] - Foldy, L.L. The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers. Phys. Rev.
**1945**, 67, 107–119. [Google Scholar] [CrossRef] - Lax, M. Multiple scattering of waves. II. the effective field in dense systems. Phys. Rev.
**1952**, 85, 621–629. [Google Scholar] [CrossRef] - Waterman, P.C.; Truell, R. Multiple scattering of waves. J. Math. Phys.
**1961**, 2, 512–537. [Google Scholar] [CrossRef] - Lloyd, P.; Berry, M.V. Wave propagation through an assembly of spheres: IV. Relations between different multiple scattering theories. Proc. Phys. Soc.
**1967**, 91, 678–688. [Google Scholar] [CrossRef] - Lagendijk, A.; Van Tiggelen, B.A. Resonant multiple scattering of light. Phys. Rep.
**1996**, 29, 143–215. [Google Scholar] [CrossRef] - Tourin, A.; Fink, M.; Derode, A. Multiple scattering of sound. Waves Random Media
**2000**, 10, R31–R60. [Google Scholar] [CrossRef] - Graff, K.F. Wave Motion in Elastic Solids; Dover Publications Inc.: New York, NY, USA, 1975; ISBN 0486667456. [Google Scholar]
- Derode, A.; Tourin, A.; Fink, M. Random multiple scattering of ultrasound. I. Coherent and ballistic waves. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top.
**2001**, 64, 7. [Google Scholar] [CrossRef] - Treeby, B.E.; Cox, B.T. k-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields. J. Biomed. Opt.
**2010**, 15, 021314. [Google Scholar] [CrossRef] [PubMed] - Berenger, J.P. A Perfectly Matched Layer for the Absorption of Electromagnetic Waves. J. Comput. Phys.
**1994**, 114, 185–200. [Google Scholar] [CrossRef] - Chekroun, M.; Le Marrec, L.; Lombard, B.; Piraux, J. Time-domain numerical simulations of multiple scattering to extract elastic effective wavenumbers. Waves Random Complex Media
**2012**, 22, 398–422. [Google Scholar] [CrossRef] [Green Version] - Kim, J.-Y. Models for wave propagation in two-dimensional random composites: A comparative study. J. Acoust. Soc. Am.
**2010**, 127, 2201–2209. [Google Scholar] [CrossRef] - World Health Organization. WHO Air Quality Guidelines for Particulate Matter, Ozone, Nitrogen Dioxide and Sulfur Dioxide: Global Update 2005: Summary of Risk Assessment; World Health Organization: Geneva, Switzerland, 2006. [Google Scholar]

**Figure 1.**Conceptual illustration of multiple scattering of acoustic waves. An example time-domain signal is displayed below the conceptual illustration of each regime. The scattering mean free path (l

_{S}) is the characteristic length of the exponential decay of a coherent wave that propagates in its initial direction after taking ensemble averaging over multiple realizations of disorder.

**Figure 3.**The overall structure of the numerical simulation model. At each volume fraction case, two models that had different scatterer distributions were simulated to obtain acoustic wave responses across multiple realizations.

**Figure 4.**Acoustic wavefields for the 1% volume fraction case: (

**a**) wavefield for a single realization and (

**b**) ensemble-average wavefield. Acoustic wave signals at the first sensor position for the two cases are shown in (

**c**) and (

**d**), respectively.

**Figure 5.**Coherent wave attenuation for the 1% volume fraction case. An analytical attenuation curve computed using independent scattering approximation (ISA) is displayed as a reference. A smoothed coherent wave attenuation curve obtained by applying moving averaging is also displayed.

**Figure 7.**Scatterer number density estimation results for the 1% volume fraction case: (

**a**) computed number density and (

**b**) estimation error. A zoom-in plot of the computed number density in the frequency region (1–10 MHz) is displayed in (

**a**).

**Figure 8.**Number density estimation results (

**left**) and estimation error (

**right**) for different scatterer volume fractions: (

**a**) 2%, (

**b**) 5%, (

**c**) 10%, and (

**d**) 15%.

**Figure 9.**Variations of scattering cross-section under varying scatterer shapes: (

**a**) six different particle shapes and (

**b**) their corresponding total scattering cross-sections.

Material Properties | ||

Wave Speed (m/s) | Mass Density (kg/m^{3}) | |

Air | 343 | 1.2754 |

Fine dust particle | 343 | 1050 |

Simulation Parameters | ||

Number of grid points (N_{x} × N_{y}) | 25,000 × 1000 | |

Grid spacing (dx and dy) | 0.057 μm | |

Time step (dt) | 0.0166 ns | |

Time duration (T) | 8 μs | |

Number of sensing points at each medium | 1050 (35 × 30) | |

Sensor spacing along x axis | 36 μm | |

Sensor spacing along y axis | 1.65 μm | |

Fine dust particle volume fractions | 1, 2, 5, 10, and 15% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Song, H.; Woo, U.; Choi, H.
Numerical Analysis of Ultrasonic Multiple Scattering for Fine Dust Number Density Estimation. *Appl. Sci.* **2021**, *11*, 555.
https://doi.org/10.3390/app11020555

**AMA Style**

Song H, Woo U, Choi H.
Numerical Analysis of Ultrasonic Multiple Scattering for Fine Dust Number Density Estimation. *Applied Sciences*. 2021; 11(2):555.
https://doi.org/10.3390/app11020555

**Chicago/Turabian Style**

Song, Homin, Ukyong Woo, and Hajin Choi.
2021. "Numerical Analysis of Ultrasonic Multiple Scattering for Fine Dust Number Density Estimation" *Applied Sciences* 11, no. 2: 555.
https://doi.org/10.3390/app11020555