# Shooter Localization in Wireless Microphone Networks

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## Abstract

Shooter localization in a wireless network of microphones is studied. Both the acoustic muzzle blast (MB) from the gunfire and the ballistic shock wave (SW) from the bullet can be detected by the microphones and considered as measurements. The MB measurements give rise to a standard sensor network problem, similar to time difference of arrivals in cellular phone networks, and the localization accuracy is good, provided that the sensors are well synchronized compared to the MB detection accuracy. The detection times of the SW depend on both shooter position and aiming angle and may provide additional information beside the shooter location, but again this requires good synchronization. We analyze the approach to base the estimation on the time difference of MB and SW at each sensor, which becomes insensitive to synchronization inaccuracies. Cramér-Rao lower bound analysis indicates how a lower bound of the root mean square error depends on the synchronization error for the MB and the MB-SW difference, respectively. The estimation problem is formulated in a separable nonlinear least squares framework. Results from field trials with different types of ammunition show excellent accuracy using the MB-SW difference for both the position and the aiming angle of the shooter.

### Keywords

Shock Wave Root Mean Square Error Wireless Sensor Network Detection Error Synchronization Error## 1. Introduction

Several acoustic shooter localization systems are today commercially available; see, for instance [1, 2, 3, 4]. Typically, one or more microphone arrays are used, each synchronously sampling acoustic phenomena associated with gunfire. An overview is found in [5]. Some of these systems are mobile, and in [6] it is even described how soldiers can carry the microphone arrays on their helmets. One interesting attempt to find direction of sound from one microphone only is described in [7]. It is based on direction dependent spatial filters (mimicking the human outer ear) and prior knowledge of the sound waveform, but this approach has not yet been applied to gun shots.

Indeed, less common are shooter localization systems based on singleton microphones geographically distributed in a wireless sensor network. An obvious issue in wireless networks is the sensor synchronization. For localization algorithms that rely on accurate timing like the ones based on *time difference of arrival* (TDOA), it is of major importance that synchronization errors are carefully controlled. Regardless if the synchronization is solved by using GPS or other techniques, see, for instance [8, 9, 10], the synchronization procedures are associated with costs in battery life or communication resources that usually must be kept at a minimum.

In [11] the synchronization error impact on the sniper localization ability of an urban network is studied by using Monte Carlo simulations. One of the results is that the inaccuracy increased significantly ( Open image in new window 2 m) for synchronization errors exceeding approximately 4 ms. 56 small wireless sensor nodes were modeled. Another closely related work that deals with mobile asynchronous sensors is [12], where the estimation bounds with respect to both sensor synchronization and position errors are developed and validated by Monte Carlo simulations. Also [13] should be mentioned, where combinations of directional and omnidirectional acoustic sensors for sniper localization are evaluated by perturbation analysis. In [14], estimation bounds for multiple acoustic arrays are developed and validated by Monte Carlo simulations.

In this paper we derive fundamental estimation bounds for shooter localization systems based on wireless sensor networks, with the synchronization errors in focus. An accurate method independent of the synchronization errors will be analyzed (the *MB-SW model*) as well as a useful bullet deceleration model. The algorithms are tested on data from a field trial with 10 microphones spread over an area of 100 m and with gunfire at distances up to 400 m. Partial results of this investigation appeared in [15] and almost simultaneously in [12].

The outline is as follows. Section 2 sketches the localization principle and describes the acoustical phenomena that are used. Section 3 gives the estimation framework. Section 4 derives the signal models for the muzzle blast (MB), shock wave (SW), combined MB;SW, and difference MB-SW, respectively. Section 5 derives expressions for the root mean square error (RMSE) Cramér-Rao lower bound (CRLB) for the described models and provides numerical results from a realistic scenario. Section 6 presents the results from field trials, and Section 7 gives the conclusions.

## 2. Localization Principle

Two acoustical phenomena associated with gunfire will be exploited to determine the shooter's position: the *muzzle blast* and the *shock wave*. The principle is to detect and time stamp the phenomena as they reach microphones distributed over an area, and let the shooter's position be estimated by, in a sense, the most likely point, considering the microphone locations and detection times.

The muzzle blast (MB) is the sound that probably most of us associate with a gun shot, the "bang." The MB is generated by the pressure depletion in effect of the bullet leaving the gun barrel. The sound of the MB travels at the speed of sound in all directions from the shooter. Provided that a sufficient number of microphones detect the MB, the shooters position can be more or less accurately determined.

The shock wave (SW) is formed by supersonic bullets. The SW has (approximately) the shape of an expanding cone, with the bullet trajectory as axis, and reaches only microphones that happens to be located inside the cone. The SW propagates at the speed of sound in direction away from the bullet trajectory, but since it is generated by a supersonic bullet, it always reaches the microphone before the MB, if it reaches the microphone at all. A number of SW detections may primarily reveal the direction to the shooter. Extra observations or assumptions on the ammunition are generally needed to deduce the distance to the shooter. The SW detection is also more difficult to utilize than the MB detection, since it depends on the bullet's speed and ballistic behavior.

Of course, the MB and SW (when present) can be used in conjunction with each other. One of the ideas exploited later is to utilize the *time difference* between the MB and SW detections. This way, the localization is independent of the clock synchronization errors that are always present in wireless sensor networks.

## 3. Estimation Framework

It is assumed throughout this work that

- (1)
the coordinates of the microphones are known with negligible error,

- (2)
the arrival times of the MB and SW at each microphone are measured with significant synchronization error,

- (3)
the shooter position and aim direction are the sought parameters.

Notation. MB, SW, and MB-SW are different models, and L/N indicates if model parameters or signals enter the model linearly (L) or nonlinearly (N).

Variable | MB | SW | MB-SW | Description |
---|---|---|---|---|

Number of microphones | ||||

Number of microphones receiving shock wave, Open image in new window | ||||

N | N | N | Position of shooter, Open image in new window ( Open image in new window ) | |

N | N | N | Position of microphone Open image in new window , Open image in new window ( Open image in new window ) | |

L | L | L | Measured detection time for microphone at position Open image in new window | |

L | L | Rifle or gun firing time | ||

L | N | N | Speed of sound | |

N | N | Speed of bullet | ||

N | N | Shooting direction, Open image in new window ( Open image in new window ) | ||

L | L | Synchronization error for microphone Open image in new window | ||

L | L | L | Detection error at microphone Open image in new window | |

N | N | Bullet speed decay rate | ||

Point of origin for shock wave received by microphone Open image in new window | ||||

Mach angle, Open image in new window | ||||

Angle between line of sight to shooter and shooting angle |

Here, Open image in new window denotes the minimizing argument, Open image in new window the minimum of the function, and Open image in new window denotes the Open image in new window -norm, that is, Open image in new window . Whenever Open image in new window is omitted, Open image in new window is assumed. The loss function norm Open image in new window is chosen by consideration of the expected error characteristics. Numerical optimization, for instance, the Gauss-Newton method, can here be applied to get the NLS estimate.

*the separable least squares method*; see, for instance [17]. By doing so, the NLS search space is reduced which in turn significantly reduces the computational burden. For that reason, the signal model (1) is rewritten as

This general separable least squares (SLSs) approach will now be applied to four different combinations of signal models for the MB and SW detection times.

## 4. Signal Models

### 4.1. Muzzle Blast Model (MB)

*muzzle blast (MB) sound*is assumed to reach Open image in new window at the time

*inverse*of the speed of sound enters linearly. The Open image in new window notation indicates that Open image in new window is part of a linear relation, as described in the previous section. With Open image in new window and Open image in new window , (6) gives

Here, Open image in new window depends on Open image in new window as given in (9b).

### 4.2. Shock Wave Model (SW)

*Mach angle*defined as

according to the clock at microphone Open image in new window . Note that the variable names Open image in new window and Open image in new window for notational simplicity have been reused from the MB model. Below, also Open image in new window , and Open image in new window will be reused. When there is ambiguity, a superscript will indicate exactly which entity that is referred to, for instance, Open image in new window .

which together with (12) implicitly defines Open image in new window . We have not found any simple closed form for Open image in new window ; so we solve for Open image in new window numerically, and in case of multiple solutions we keep the admissible one (which turns out to be unique). Open image in new window is trivially induced by the shooting direction Open image in new window (and Open image in new window , Open image in new window ). Both these angles thus depend on Open image in new window implicitly.

and Open image in new window is the admissible solution to (12) and (15).

### 4.3. Combined Model (MB;SW)

### 4.4. Difference Model (MB-SW)

*MB-SW model*:

for Open image in new window . This rather special model has also been analyzed in [12, 15]. The key idea is that Open image in new window is by cancellation independent of both the firing time Open image in new window and the synchronization error Open image in new window . The drawback, of course, is that there are only Open image in new window equations (instead of a total of Open image in new window ) and the detection error increases, Open image in new window . However, when the synchronization errors are expected to be significantly larger than the detection errors, and when also Open image in new window is sufficiently large (at least as large as the number of parameters), this model is believed to give better localization accuracy. This will be investigated later.

which requires numerical optimization. Numerical experiments indicate that this optimization problem is more prone to local minima, compared to (10a) for the MB model; therefore good starting points for the numerical search are essential. One such starting point could, for instance, be the MB estimate Open image in new window . Initial shooting direction could be given by assuming, in a sense, the worst possible case, that the shooter aims at some point close to the center of the microphone network.

### 4.5. Error Model

Note that the correlation structure of the clock synchronization error Open image in new window enables estimation of these. Note also that the (assumed known) total error covariance, generally denoted by Open image in new window , dictates the norm used in the weighted least squares criterion. Open image in new window also impacts the estimation bounds. This will be discussed in the next section.

### 4.6. Summary of Models

Four models with different purposes have been described in this section.

- (i)
*MB*. Given that the acoustic environment enables reliable detection of the muzzle blast, the MB model promises the most robust estimation algorithms. It also allows global minimization with low-dimensional exhaustive search algorithms. This model is thus suitable for initialization of algorithms based on the subsequent models. - (ii)
*SW*. The SW model extends the MB model with shooting angle, bullet speed, and deceleration parameters, which provide useful information for sniper detection applications. The SW is easier to detect in disturbed environments, particularly when the shooter is far away and the bullet passes closely. However, a sufficient number of microphones are required to be located within the SW cone, and the SW measurements alone cannot be used to determine the distance to the shooter. - (iii)
*MB;SW*. The total MB;SW model keeps all information from the observations and should thus provide the most accurate and general estimation performance. However, the complexity of the estimation problem is large. - (iv)
*MB-SW*. All algorithms based on the models above require that the synchronization error in each microphone either is negligible or can be described with a statistical distribution. The MB-SW model relaxes such assumptions by eliminating the synchronization error by taking differences of the two pulses at each microphone. This also eliminates the shooting time. The final model contains all interesting parameters for the problem, but only one nuisance parameter (actual speed of sound, which further may be eliminated if known sufficiently well).

Summary of parameter vectors for the different models Open image in new window , where the noise models are summarized in (29a), (29b), (29c), (29d), and (29e). The values of the dimensions assume that the set of microphones giving SW observations is a subset of the MB observations.

Model | Linear Parameters | Nonlinear Parameters | ||
---|---|---|---|---|

MB | ||||

SW | ||||

MB;SW | ||||

MB-SW |

## 5. Cramér-Rao Lower Bound

where Open image in new window is Fisher's information matrix evaluated at the correct parameter values Open image in new window . Here, the location Open image in new window is for notational purposes part of the parameter vector Open image in new window . Also the sensor positions Open image in new window can be part of Open image in new window , if these are known only with a certain uncertainty. The Cramér-Rao lower bound provides a fundamental estimation limit for unbiased estimators; see [20]. This bound has been analyzed thoroughly in the literature, primarily for AOA, TOA, and TDOA [21, 22, 23].

The bound is evaluated for a specific location, parameter setting, and microphone positioning, collectively Open image in new window .

The RMSE bound can be used to compare the information in different models in a simple and unambiguous way, which does not depend on which optimization criterion is used or which numerical algorithm that is applied to minimize the criterion.

### 5.1. MB Case

### 5.2. SW, MB;SW, and MB-SW Cases

The estimation bounds for the SW, MB;SW, and MB-SW cases are analogously to (33), but there are hardly any analytical expressions available. The Jacobian is probably best evaluated by finite difference methods.

### 5.3. Numerical Example

and that Open image in new window m/s, Open image in new window m/s, and Open image in new window . The scenario setup implies that all microphones detect the shock wave, so Open image in new window . All bounds presented below are calculated by numerical finite difference methods.

MB Model

SW Model

The SW model is disregarded here, since the SW detections alone contain no shooter distance information.

MB-SW Model

The dashed lines in Figure 5 correspond to the RMSE bound for four different values of Open image in new window . Here, the MB-SW model gives at least twice the error of the MB model, provided that there are no synchronization errors. However, in a wireless network we expect the synchronization error to be 10–100 times larger than the detection error, and then the MB-SW error will be substantially smaller than the MB error.

MB;SW Model

The expression for the MB;SW bound is somewhat involved; so the dependence on Open image in new window is only presented graphically, see Figure 5. The solid curves correspond to the MB;SW RMSE bound for the same four values of Open image in new window as for the MB-SW bound. Apparently, when the synchronization error Open image in new window is large compared to the detection error Open image in new window , the MB-SW and MB;SW models contain roughly the same amount of information, and the model having the simplest estimator, that is, the MB-SW model, should be preferred. However, when the synchronization error is smaller than 100 times the detection error, the complete MB;SW model becomes more informative.

These results are comparable with the analysis in [12, Figure 4a], where an example scenario with 6 microphones is considered.

### 5.4. Summary of the CRLB Analysis

The synchronization error level in a wireless sensor network is usually a matter of design tradeoff between performance and battery costs required by synchronization mechanisms. Based on the scenario example, the CRLB analysis is summarized with the following recommendations.

- (i)
If Open image in new window , then the MB-SW model should be used.

- (ii)
If Open image in new window is moderate, then the MB;SW model should be used.

- (iii)
Only if Open image in new window is very small ( Open image in new window ), the shooting direction is of minor interest, and performance may be traded for simplicity, then the MB model should be used.

## 6. Experimental Data

Armament and ammunition used at the trial, and number of rounds fired at each shooter position. Also, the resulting localization RMSE for the MB-SW model for each shooter position. For the Luger Pistol the MB model RMSE is given, since only one microphone is located in the Luger Pistol SW cone.

Type | Caliber | Weight | Velocity | Sh. pos. | # Rounds | RMSE |
---|---|---|---|---|---|---|

308 Winchester | 7.62 mm | 9.55 g | 847 m/s | 1, 2, 3 | 4, 4, 6 | 19, 6, 6 m |

Hunting Rifle | 9.3 mm | 15 g | 767 m/s | 1, 2, 3 | 4, 4, 4 | 6, 5, 6 m |

Swedish Mauser | 6.5 mm | 8.42 g | 852 m/s | 1, 2, 3 | 4, 4, 4 | 40, 6, 6 m |

Luger Pistol | 9 mm | 6.8 g | 400 m/s | 3 | —, —, 4 | —, —, 2 m |

Light wind, no clouds, and around Open image in new window C are the weather conditions. Little or no acoustic disturbances are present. The terrain is rough. Dense woods surround the test site. There is light bush vegetation within the site. Shooter position 1 is elevated some 20 m; otherwise spots are within Open image in new window 5 m of a horizontal plane. Ground truth values of the positions are determined with less relative error than 1 m, except for shooter position 1, which is determined with 10 m accuracy.

### 6.1. Detection

The MB and SW are detected by visual inspection of the microphone signals in conjunction with filtering techniques. For shooter positions 1 and 2, the shock wave detection accuracy is approximately Open image in new window s, and the muzzle blast error Open image in new window is slightly worse. For shooting position 3 the accuracies are generally much worse, since the muzzle blast and shock wave components become intermixed in time.

### 6.2. Numerical Setup

For simplicity, a plane model is assumed. All elevation measurements are ignored and Open image in new window and Open image in new window . Localization using the MB model (7) is done by minimizing (10a) over a 10 m grid well covering the area of interest, followed by numerical minimization.

Localization using the MB-SW model (25) is done by numerically minimizing (28). The objective function is subject to local optima; therefore the more robust muzzle blast localization Open image in new window is used as an initial guess. Furthermore, the direction from Open image in new window toward the mean point of the microphones (the camp) is used as initial shooting direction Open image in new window . Initial bullet speed is Open image in new window m/s and initial speed of sound is Open image in new window m/s. Open image in new window is used, which is a value derived from the 308 Winchester ammunition ballistics.

### 6.3. Results

Localization RMSE and theoretical bound (34) for the three different shooter positions using the MB and the MB-SW models, respectively, beside the aim RMSE for the MB-SW model. The aim RMSE is with respect to the aim at Open image in new window against the target, Open image in new window , not with respect to the true direction Open image in new window . This way the ability to identify the target is assessed.

Shooter position | 1 | 2 | 3 |
---|---|---|---|

RMSE( Open image in new window ) | 105 m | 28 m | 2.4 m |

MB Bound | 1 m | 0.4 m | 0.02 m |

RMSE ( Open image in new window ) | 26 m | 5.7 m | 5.2 m |

MB-SW Bound | 9 m | 0.1 m | 0.08 m |

RMSE( Open image in new window ) |

#### 6.3.1. Synchronization and Detection Errors

Since all microphones are recorded by a common recorder, there are actually *no* timing errors due to inaccurate clocks. This is of course the best way to conduct a controlled experiment, where any uncertainty renders the dataset less useful. From experimental point of view, it is then simple to add synchronization errors of any desired magnitude off-line. On the dataset at hand, this is however work under progress. At the moment, there are apparently other sources of error, worth identifying. It should however be clarified that in the final wireless sensor product, there will always be an unpredictable clock error. As mentioned, detection errors are present, and the expected level of these (80 Open image in new window s) is used for bound calculations in Table 4. It is noted that the bounds are in level with, or below, the positioning errors.

There are at least two explanations for the bad performance using the MB-SW model at shooter position 3. One is that the number of microphones reached by the shock wave is insufficient to make accurate estimates. There are four unknown model parameters, but for the relatively low speed of pistol ammunition, for instance, only one microphone has a valid shock wave detection. Another explanation is that the increased detection uncertainty (due to SW/MB intermix) impacts the MB-SW model harder, since it relies on accurate detection of both the MB and SW.

#### 6.3.2. Model Errors

No doubt, there are model inaccuracies both in the ballistic and in the acoustic domain. To that end, there are meteorological uncertainties out of our control. For instance, looking at the MB-SW localizations around shooter position 1 in Figure 7 (squares), three clusters are identified that correspond to three ammunition types with different ballistic properties; see the RMSE for each ammunition and position in Table 3. This clustering or bias more likely stems from model errors than from detection errors and could at least partially explain the large gap between theoretical bound and RMSE in Table 4. Working with three-dimensional data in the plane is of course another model discrepancy that could have greater impact than we first anticipated. This will be investigated in experiments to come.

#### 6.3.3. Numerical Uncertainties

Finally, we face numerical uncertainties. There is no guarantee that the numerical minimization programs we have used here for the MB-SW model really deliver the global minimum. In a realistic implementation, every possible a priori knowledge and also qualitative analysis of the SW and MB signals (amplitude, duration, caliber classification, etc.) together with basic consistency checks are used to reduce the search space. The reduced search space may then be exhaustively sampled over a grid prior to the final numerical minimization. Simple experiments on an ordinary desktop PC indicate that with an efficient implementation, it is feasible to, within the time frame of one second, minimize any of the described model objective functions over a discrete grid with Open image in new window points. Thus, by allowing—say—one second extra of computation time, the risk for hitting a local optima could be significantly reduced.

## 7. Conclusions

We have presented a framework for estimation of shooter location and aiming angle from wireless networks where each node has a single microphone. Both the acoustic muzzle blast (MB) and the ballistic shock wave (SW) contain useful information about the position, but only the SW contains information about the aiming angle. A separable nonlinear least squares (SNLSs) framework was proposed to limit the parametric search space and to enable the use of global grid-based optimization algorithms (for the MB model), eliminating potential problems with local minima.

For a perfectly synchronized network, both MB and SW measurements should be stacked into one large signal model for which SNLS is applied. However, when the synchronization error in the network becomes comparable to the detection error for MB and SW, the performance quickly deteriorates. For that reason, the time difference of MB and SW at each microphone is used, which automatically eliminates any clock offset. The effective number of measurements decreases in this approach, but as the CRLB analysis showed, the root mean square position error is comparable to that of the ideal stacked model, at the same time as the synchronization error distribution may be completely disregarded.

The bullet speed occurs as nuisance parameters in the proposed signal model. Further, the bullet retardation constant was optimized manually. Future work will investigate if the retardation constant should also be estimated, and if these two parameters can be used, together with the MB and SW signal forms, to identify the weapon and ammunition.

## Notes

### Acknowledgment

This work is funded by the VINNOVA supported Centre for Advanced Sensors, Multisensors and Sensor Networks, FOCUS, at the Swedish Defence Research Agency, FOI.

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