# Numerical Analysis of Ultrasonic Multiple Scattering for Fine Dust Number Density Estimation
## 1 Introduction  
Airborne particulate matter, such as fine dust, is a global concern owing to its detrimental impact on human health [1; 2; 3; 4]. Particles with a diameter of 10 \(\upmu\)m or less (PM10) include dust from construction sites, industrial sources, and wind-blown dust from open lands. PM10 can cause eye, nose, and throat irritation and causes adverse health effects such it is inhaled into the lungs [5]. Particles with a diameter of 2.5 \(\upmu\)m or less (PM2.5) can be emitted from the combustion of gasoline, diesel, and wood produce. The smaller particles (PM2.5) are more dangerous as they can enter the deep parts of the lungs and blood [5]. In this regard, the measurement of fine dust concentration has been recognized as a critical task for maintaining human health and there is an increasing need for accurate measurements of fine dust concentration.  
Studies have been conducted to measure fine dust concentrations [6]. The available measurement techniques include the beta-attenuation method, gravimetric method, and light scattering method [7; 8; 9]. Among the various measurement methods, the light scattering method, which measures fine dust concentration by measuring scattered light intensity by airborne particles, is arguably the most widely accepted. This method allows for online measurements as well as low-cost, compact instrumentation. However, the light scattering method is an indirect method; the measured quantity is not an absolute physical quantity and requires additional calibration work to estimate the fine dust concentration. Moreover, it is susceptible to environmental factors, such as relative humidity, since light is an electromagnetic wave.  
Unlike light, mechanical waves are less susceptible to changes in the electromagnetic properties of a medium (e.g., relative humidity), but they are sensitive to mechanical changes in a medium. Numerous studies have demonstrated the successful use of mechanical waves to characterize the mechanical properties of various media [10; 11; 12; 13]. Mechanical waves have been employed to estimate particle concentration in various industrial applications [14; 15]. Despite its potential, the authors were unable to find a work that reports the use of mechanical waves in measuring fine dust concentration.  
In this study, we propose a mechanical wave-based method to estimate the number density of fine dust particles (the number of particles per unit area) in air. Here, the number density is defined as the number of particles per unit area rather than the number of particles per unit volume (in 3-D), because we performed 2-D simulation studies. The proposed method uses ultrasonic acoustic waves multiply scattered by fine dust particles in the air and the number density of fine dust particles is computed by processing the measured multiple scattering data based on the independent scattering approximation theory. The proposed number density estimation method was validated using ultrasonic wavefield data generated by a series of numerical simulations. The unique contributions of this study are: (1) the effective use of mechanical waves to estimate the number density of fine dust particles, (2) development of an ultrasonic wavefield data processing technique that directly computes the number density from measured wavefield data, and (3) understanding of the physical interaction between ultrasonic acoustic waves and fine dust particles randomly distributed in the air.  
The remainder of this article is organized as follows: in Section 2 the theoretical background on the multiple scattering of acoustic waves in a discrete random medium is introduced; Section 3 describes the proposed ultrasound data processing approach to estimate the number density of fine dust particles; Section 4 provides details on the numerical simulations of multiply scattered ultrasonic wavefields; in Section 5 numerical simulation results of the number density estimation are demonstrated; in Section 6 we discuss several factors that affect the accuracy of the number density estimation.