Risk of Hearing Loss Injury Caused by Multiple Flash Bangs on a Crowd

A flash bang is a non-lethal explosive device that delivers intensely loud bangs 
and bright lights to suppress potentially dangerous targets. It is usually used in 
crowd control, hostage rescue and numerous other missions. We construct a 
model for assessing quantitatively the risk of hearing loss injury caused by 
multiple flash bangs. The model provides a computational framework for incorporating 
the effects of the key factors defining the situation and for testing 
various sub-models for these factors. The proposed model includes 1) uncertainty 
in the burst point of flash bang mortar, 2) randomness in the dispersion 
of multiple submunitions after the flash bang mortar burst, 3) decay of acoustic 
impulse from a single submunition to an individual subject along the 
ground surface, 4) the effective combined sound exposure level on an individual 
subject caused by multiple submunitions at various distances from the 
subject, and 5) randomness in the spatial distribution of subjects in the crowd. 
With the mathematical model formulated, we seek to characterize the overall 
effect of flash bang mortar in the form of an effective injury area. We carry 
out simulations to study the effects of uncertainty and randomness on the risk 
of hearing loss injury of the crowd. The proposed framework serves as a 
starting point for a comprehensive assessment of hearing loss injury risk, taking 
into consideration all realistic and relevant features of flash bang mortar. 
It also provides a platform for testing and updating component models.


Introduction
In recent years, armed gangs, militias, and terrorist cells have become more prevalent in modern asymmetric conflict and irregular warfare. In the 1950s noncombatants accounted for about one-half of all US military operation casualties and the rate rose to about 80% in the 1980s [1]. Military forces today must be able to execute missions across a large range of military operations. This spans from stability operations, disaster response and humanitarian assistance to full-scale armed combat. Non-lethal weapons can allow for tailored responses to targets and situations across this continuum and can provide commanders the flexibility with escalation-of-force options to minimize civilian casualties and collateral property damage [2]. Further, according to the US Department of Defense Non-Lethal Weapons program, while non-lethal weapons traditionally have supported operations such as peacekeeping and humanitarian assistance, there is a growing appreciation for these weapons, devices and munitions in irregular warfare operations such as counterinsurgency, counterterrorism, stability operations, and counter-piracy [3].
Non-lethal weapons have been successfully employed in the engagements with potential threats, including counterinsurgency operations, peacekeeping operations, humanitarian efforts, crowd and riot control, and crisis management [6] [7]. For example, in 1995 US forces in Somalia successfully utilized non-lethal weapons to preclude injury to civilians in support of humanitarian operations.
In 2000, a US military police unit used non-lethal weapons to disperse a violent rocking-throwing and stick-wielding crowd and to provide protection for the peacekeeping personnels during international peacekeeping operations in Kosovo.
Non-lethal weapons are explicitly designed and primarily employed to incapacitate targeted personnel or material immediately, while minimizing fatalities, permanent injury to personnel, and undesired damage to property in the target environment. A key characteristic of non-lethal weapons is that they are intended to have reversible effects on personnel and material [4]. Generally speaking, conventional lethal weapons, such as explosive-filled warheads, damage or kill their targets through blast, penetration and fragmentation [5]. In contrast, non-lethal weapons employ means other than catastrophic physical destruction to interrupt the opponent's normal functions. In order to use non-lethal weapons judiciously and effectively, it is important to be able to assess the risks and damages associated with applying various non-lethal weapons.
From a mathematical point of view, predicting the outcome of an area non-lethal weapon used against a crowd can be challenging since in both achieving the desired effect and causing undesired injury, human effects play an important role and need to be taken into consideration [8].
One widely used type of non-lethal weapon is the flash bang munition. Flash bang devices are designed to deny access into and out of an area, move individuals through an area, or cause suppressive effects. These devices deliver a bright Recently hearing loss injury associated with multiple acoustic impulses was examined in a study where an empirical logistic dose-response model was developed [9]. This empirical injury model was theoretically interpreted in our previous studies, from the point of view of immunity [10] and biovariability [11] [13], respectively.
In this study we aim at building a mathematical framework that takes into

Background and Formulation
A flash bang munition is a non-lethal explosive device that emits a dazzling flash of light and a thunderous noise impulse to temporarily disorient the senses of affected individual subjects. In particular, the bright flash can induce temporary flash blindness that lasts seconds whereas the loud blast is at 170 decibels or more and can cause temporary large threshold shifts in hearing [14]. The US military is developing a long range non-lethal mortar round that delivers a flash bang payload. When the mortar round gets close to the targeted area, it bursts to release multiple submunitions that are dispersed over an elliptical area. After falling to the ground, the submunitions are ignited to generate optical and acoustic impulses. In this study, we focus on the hearing loss due to the acoustic impulses from the submunitions. Below we first describe the model components for constructing a comprehensive modeling framework.
1) The ground surface and the spatial distribution of subjects in the crowd.
2) The burst of flash bang mortar and the spatial dispersion of submunitions.
3) The decay of acoustic impulse vs distance along ground surface. 4) Effective combined SELA caused by multiple submunitions.

5)
Logistic dose-response relation predicting injury probability from SELA.
We then assemble these components into a computational framework for assessing the risk and fluctuations in the occurrences of hearing loss injury of a crowd, caused by multiple submunitions dispersed in the air over the crowd.

The Ground Surface and the Crowd
We establish the coordinate system as follows: the x-axis is the range direction from the mortar launch position to the target (center of the crowd), the y-axis the deflection direction, and the z-axis the vertical direction. We consider a ground surface, with possible variation in height given by function A crowd on the ground surface is distributed in the ( ) , x y dimensions according to a given probability density, such as, a normal distribution, or a uniform distribution inside a bounded region. As an example, we consider a crowd uniformly distributed in a circle. We put the origin of the coordinate system at the center of crowd. Let 1) c n : number of subjects in the crowd.
2) c d : diameter of the circle formed by the crowd.
3) ear h : height of ears from the ground surface, for a random subject in the crowd.
The uniform distribution of c n subjects inside the circle of diameter c d is sampled as For each subject, the point of focus for assessing hearing loss injury is the ears.
The z-coordinate of ears of each subject is specified by

Burst of Mortar Round and Dispersion of Submunitions
Next we describe the spatial distribution of the multiple submunitions released when a mortar round bursts. After being released and dispersed, the submunitions controlled by time delay fuses, by design, will fall to the ground surface before being ignited to discharge optical and acoustic energy. In the current study, we focus on this idealized situation. In subsequent studies, we will include the possible failure of time delay fuses, which leads to uncertainty in the altitude of submunition ignition. Here, we only need to consider the ( ) , x y coordinates of a submunition at its ignition, which is the outcome of the random dispersion; its z coordinate at ignition is calculated from ( ) 2) σ : standard deviation of Gaussian aiming error in each dimension of d X .
3) s n : number of submunitions released from a mortar round.  x y x y d d where δ is the relative breath of the annulus, which is the ratio of breadth to the outer radius of the annulus. The relative breadth δ measures how fat the annulus is. Figure 1 illustrates the dispersion of submunition as given in (3) with parameters rng 25 m d = , defl

Decay of Acoustic Impulse vs Distance along Ground Surface
The A-weighted sound exposure level (SELA) is an effective single metric for predicting the injury risk [15]. For that reason, SELA is selected as the dose in the dose-response relation for predicting injury probability. At a subject's ears, the SELA value caused by a single submunition varies with the distance between the two. We consider two models. Model A is an empirical model from [16], which does not explicitly count for the energy dissipation in acoustic wave propagation. We propose Model B, a revised model that includes both the power law decay of energy per area due to the expansion of the spherical wave and the exponential decay due to energy dissipation in wave propagation. We fit Model B to the experimental data from [16] to determine the model parameters.
Let r denote the distance between the sound source and the target. To distinguish models A and B, the SELA at the target is denoted, respectively, as propagates out as a spherical wave. At distance r from the point source, the area of spherical wave front is proportional to r 2 . In the absence of energy dissipation, the energy per area at distance r is inversely proportional to r 2 . The sound energy dissipation does occur during wave propagation, partly due to viscous dissipation in the fluid medium (the air). When both the sound source and the receiving target are on the ground, however, the interaction between the sound wave and the ground surface may cause a sound energy loss significantly larger than that attributed to the viscous dissipation in open air. We model the sound energy loss phenomenologically as an exponential decay with the distance traveled for all relevant frequencies in human hearing loss injury. A more detailed model for acoustic wave propagation would have a frequency dependent energy dissipation [12]. Combining the spherical wave expansion and the energy dissipation, we model the sound energy per area at distance r as To determine the coefficients 0 c and 1 c , we fit the proposed model (6) to the data measured in [16]. The result is Model B displayed below.
( ) ( ) 10 Model B : 20 log 0.5 148.5 Figure 2 plots SELA vs distance, respectively, for the experimental measurements from [16], for Model A from [16], and for Model B proposed in this study. The comparison in Figure 2 is for distance between 0.5m and 10m, the distance range of the experimental measurements in [16]. Over this short distance, the difference between Model A and Model B is insignificant. Both models match the data points fairly well. Over a longer distance, however, the linear decrease component of Model B in SELA (corresponding to exponential decay in energy) will set the two models apart. Figure  Comparison of experimental data from [16], Model A from [16], and Model B proposed in this study. Over the distance range of experiments (0 -10 m), the difference between the two models is small. The exponential decay of acoustic energy in Model B will have a more prominent effect over longer range of distance (see Figure 3).

Effective Combined SELA Caused by Multiple Submunitions
After being dispersed in the air over the crowd and falling to the ground, the s n submunitions are ignited individually to generate acoustic and optical impulses.
The ignitions of s n submunitions are not completely synchronized but they occur within a short period of time (typically within 3 seconds [16]). Let : effective combined SELA on subject k from all s n submunitions.
is determined by the distance between the submunition and the subject.
, is calculated using the dose combination rule [10].
where, for combining 25 or fewer impulses, coefficient λ takes the value

Logistic Dose-Response Relation
We review the injury model from [9]. Here we make it into a short sub-section for readers' convenience and for facilitating the flow of presentation.
In [9], an empirical dose-response relation was developed based on extensive data from hearing loss experiments using a chinchilla model [17] [18] [19]. The model on chinchillas was then calibrated with data of rifle noise on human and scaled up for describing human hearing loss. In the injury model, the dose is the effective combined SELA comb S , and the response is, probability of a subject exposed to dose comb S .
The logistic dose-response relation established in [9] has the form : where ID 50 is the median injury dose and coefficient α controls the steepness of function. A hearing loss injury is characterized as a permanent threshold shift (PTS) above a given cut-off, for example, PTS ≥ 30 dB. Table 1 lists the median , the effective combined SELA of all submunitions, using the logistic dose-response relation:

A Computational Framework
We assemble the modeling components set up in previous sub-sections into a computational framework for assessing the risk of hearing loss injury of a crowd, caused by multiple submunitions dispersed in the air over the crowd.
, the positions of c n subjects in the crowd be the indicator function that subject k in the crowd is injured.
For each 1, 2, , c k n = , generate a sample of ( )

Setup and Parameters
Throughout this study, we use the setup and parameters below. 1) Ground surface: dissipation, fitted to data in [16]).

Injury Area Characterizing the Flash Bang Mortar's Potential in Causing Injury
With the framework established, we carry out simulations to study the injury causing effect of multiple submunitions dispersed from a flash bang mortar. The objective of the simulations in this section is to study the risk of significant injury (RSI) as a function of location ( ) , x y . In terms of risk function ( )

RSI ,
x y , we compare the two models of SELA vs distance, introduced in previous section. Based on ( )

RSI ,
x y , we seek to find as a single metric to quantify the flash bang mortar's potential in causing injury. We study the effective injury area as a candidate for the metric.

Risk of Significant Injury (RSI) as a Function of Location
We consider the injury caused by a flash bang mortar with multiple submunitions.
We first introduce the risk of significant injury (RSI) at ( ) , x y , which is defined as the conditional probability of injury for a subject located at ( ) , x y given a realized dispersion of submunitions, averaged over all realizations of dispersion.
We study the effect of submunition dispersion with respect to the mortar burst position and the effect of SELA vs distance decay. For that purpose, we shall set the aiming error to zero ( 0 σ = ). Function

( )
RSI , x y is the injury risk for a subject fixed at ( ) , x y , which is unaffected by the number of subjects in the crowd ( c n ) or the radius of crowd distribution ( c d ). As a result, parameters c n and c d are irrelevant in the calculation of ( ) RSI , x y . All other parameters are as described in the previous section.  on an asymptotic analysis in the simplified case of a single submunition. The asymptotic analysis will be given in next subsection when we discuss effective injury areas. In contrast to the exponential decay of RSI for Model B, the RSI of Model A decreases much slower. The discrepancy in RSI between the two models is more pronounced at larger distance. This discrepancy is the mechanism behind an observation in next subsection that for a large crowd distributed with a constant density over a large area, Model A yields a diverging total number of injuries while in Model B, the total number of injuries remains finite, regardless of how large the area is.

Effective Injury Area of a Flash Bang Mortar
We use an effective injury area relative to the burst point as a single metric to characterize the injury causing effect of the flash bang mortar. The injury area is defined based on the risk function ( )

RSI ,
x y . In the simulations for studying injury areas, we shall set the aiming error to zero ( 0 σ = ) since the injury area is relative to the burst point. Also, as pointed out in the previous subsection, parameters c n (number of subjects) and c d (radius of crowd distribution) are Therefore, for a large crowd distributed over a large area with a fixed constant density, when SELA vs distance is described by Model A, the total number of injuries diverges to infinity. In contrast, Model B of SELA vs distance yields a well defined total number of injuries, depending only on the crowd distribution density, regardless of how large the crowd size is. This nice mathematical property of Model B is again demonstrated with an asymptotic analysis on the injury probability at large distance from the submunition.
With Model B of SELA vs distance, the SELA at ( ) In conclusion, cookie-cutter injury area is finite only for Model B. On the other hand, injury risk contour area is always defined and finite for both models but it may be empty if the injury risk threshold is set too high. Figure 6 shows injury risk contour area and cookie-cutter injury area, respectively, of 2%, 5% and 10% risk. All five injury areas displayed in Figure 6 are based on Model B Figure 6. Injury risk contour area and cookie-cutter injury area of various injury risk thresholds based on Model B for SELA vs distance. In general, cookie-cutter injury area is larger than injury risk contour area of the same risk threshold.
for SELA vs distance. Injury risk contour area of 10% risk is empty since the maximum of risk function ( )

RSI ,
x y is only 8.33%, below the 10% threshold specified.

Results and Discussion
We study the hearing loss injury risk of a crowd caused by multiple submunitions dispersed from a flash bang mortar round. We carry out simulations to calculate the fraction of injured caused by a flash-bang mortar of 20 s n = submunitions on a crowd of c n subjects uniformly distributed in a circle of diameter c d . We will examine both the average fraction of injured (RSI) and Monte Carlo samples of the actual injury fraction based on individual realizations of flash bang mortar burst location, submunitions dispersion and crowd subjects distribution. The problem setup and the parameters used in simulations are described in Section 2. Table 2 displays the average injury fraction as a function of two variables: 1) PTS cut-off, and 2) diameter of the crowd distribution (d c ). Each value of average injury fraction is calculated based on 100000 Monte Carlo iterations. Table 2 Table 2 shows that when the crowd diameter is in the range of 25 m  Table 3 lists the average injury fraction as a function of 1) standard deviation of aiming error (σ), and 2) diameter of the crowd distribution (d c ) for hearing loss injury of PTS 30 dB ≥ . As in the case of Table 2, each value in Table 3 is based on 100,000 Monte Carlo iterations. From the results shown in Table 3, we see that an aiming error of standard deviation 5m does not seem to change the average injury fraction appreciably. An aiming error of larger magnitude reduces the average injury fraction monotonically. At a fixed crowd distribution diameter, the larger the

Fluctuations in the Actual Number of Injured
We study the actual numbers of injured among individual Monte Carlo  The main effect of aiming error is to shift the distribution of actual injury fraction toward the lower end, and to reduce the overall average injury fraction.

Predictions Based on the Average Injury Fraction and the Binomial Distribution Model
For a crowd of 100 c n = subjects, Figure 8 compares the Monte Carlo simulation results and theoretical predictions using binomial distribution in four cases of different aiming errors. Based on the results shown in Figure 8, the binomial distribution approximation is accurate when the aiming error is small. As the aiming error increases, the binomial distribution approximation deviates from the true distribution of actual injury number. Figure 9 does the same comparison for a crowd of only 10 c n = subjects, instead of 100 subjects. The results of Figure 9 suggest that for smaller number of subjects, the binomial distribution approximation is relatively more accurate. The mathematical theory behind this seemingly unintuitive valid approximation for small number of subjects will be investigated in a subsequent study, in which we will also explore alternative approximations when the binomial distribution approximation breaks down.   Table 4).
We study the probability of the actual injury number exceeding a prescribed threshold TLV k among individual realizations. Table 5 compares the values of this probability calculated in Monte Carlo simulations and those predicted based on binomial distribution with parameters c n and RSI. The predicted probability has the expression The results displayed in Table 5 indicate that for small or no aiming error, the

Concluding Remarks
Flash bangs are one of the commonly used anti-personnel non-lethal weapons with dual civil-military applications. In this paper, we have developed a mathematical model for computing the risk of hearing loss injury caused by multiple flash bang submunitions on a crowd. Our model includes the effects of 1) aiming error in the burst point of flash bang mortar, 2) uncertainty in the dispersion of multiple submunitions after the burst of flash bang mortar, 3) propagation of acoustic impulse from a single submunition to an individual subject along the ground surface, 4) effective combined sound exposure level on an individual subject caused by multiple submunitions ignited at various distances from the subject, and 5) randomness in the spatial distribution of subjects of the crowd.
Based on the mathematical model, we explored two effective injury areas as two candidates for a single metric characterizing the overall injury causing potential of flash bang mortar. We conducted numerical simulations to study the dependence of the average injury fraction on i) magnitude of aiming error, ii) diameter of crowd distribution, and iii) PTS cut-off in defining injury. We examined the random actual injury fraction among individual realizations of mortar burst position, submunitions dispersion, and crowd subjects distribution. In the case of small or no aiming error, we found that the behavior of actual injury fraction is well predicted using a binomial distribution. This observation gives us a simple way of characterizing the actual injury fraction among individual realizations: when the binomial distribution approximation is valid, the behavior of random actual injury fraction is completely described by the number of subjects in the crowd and the overall average injury fraction. The proposed mathematical framework serves as a starting point for a comprehensive assessment of hearing loss injury risk, taking into consideration all realistic and relevant features of flash bang mortar. It provides a platform for testing and updating component models for various aspects in flash bang's injury causing process.