## 1. Introduction

An important aspect of quality control of munitions for military needs is rigorous testing of ammunition to ensure that the inherent scatter of munitions at a specified target range from random variations in weight and explosive charge is within manufacturer specifications. Careful alignment of the rifle or tank mount in a rigid clamping device is required to assure constant firing conditions for testing; however, atmospheric conditions can also have a significant impact on the path of the projectile, making it difficult to isolate manufacturing scatter from scatter due to the inherent atmospheric variability. The most obvious atmospheric effect relates to the wind. In general there will be a translational effect due to mean crosswinds as well as random turbulent fluctuations in wind speed and direction causing random scatter in the test shots. Mean deflections can easily be estimated by inspection of the impact data, but the effect of the turbulent eddies on the impact points of the projectile is more difficult to assess. A technique for estimating the magnitude of the statistical variations of ballistic trajectories due to atmospheric turbulence along the path of the projectile is provided in this paper. These estimates can be used to bound the scatter due to atmospheric processes, and they allow more meaningful interpretations of measured scatter during ballistics testing. These estimates together with suitable measurements can also be used to identify low turbulence conditions that will have a negligible impact on a particular test.

In practice, the ambient atmospheric turbulence conditions can be determined with standard high-quality velocity measurement systems such as 3D sonic anemometers or even 2D wind vane anemometers, provided the sampling rate is sufficiently high. Examination of time series from the anemometer data can provide estimates of the statistical parameters describing the turbulent fluctuations in the frequency domain, such as temporal spectra. If Taylor’s frozen hypothesis is assumed valid, then spatial spectra can be derived from the measured temporal spectra. Simple universal spatial spectral models are available (such as the von Kármán model) that depend only on two parameters: the variance of the velocity fluctuations and a length scale (which is related to a low wavenumber rolloff in the shape of the spectra). These spectral characteristics are discussed for example in Kaimal and Finnigan (1994), and provide a complete statistical description of the turbulent velocity field. These one-dimensional velocity statistics are the required input into standard wind simulation algorithms (Frehlich 1997, 2000, 2001; Frehlich et al. 2001) and can be used in a Monte Carlo approach to drive a ballistics trajectory calculation model to determine quantitative statistics of the shot scatter due to turbulence alone.

In practical applications, it is desirable to use data from sonic anemometers along the path of the projectile to provide real-time estimates of atmospheric mean crosswinds and turbulence parameters. Because of the inherent small-scale atmospheric variability, in general more anemometers will provide more reliable statistics. However, because of cost and logistics considerations, using a large number of sonic anemometers is not always feasible. To quantify the effect of using a finite number of anemometers to estimate turbulence parameters, simulations of the 2D wind fields including the effects of a constant mean crosswind can be used to simulate time-averaged sonic anemometer data from a finite number of locations along the trajectory path. These results can then be used to estimate the contribution of the scatter due to atmospheric turbulence.

The paper is organized as follows: section 2 provides an example of sonic anemometer measurements and turbulence parameter estimates at a ballistics test site at the U.S. Army’s Aberdeen Proving Ground (APG) in Aberdeen, Maryland. Section 3 provides background on the statistical description of turbulence according to the von Kármán spectral model. Section 4 describes the algorithms used for the numerical simulation of turbulent wind fields for 1D geometries (to compute the expected scatter) and 2D geometries (required to evaluate corrections for turbulence using a fixed number of sensors along the target trajectory with various averaging intervals). Section 5 describes the ballistic calculation method and identifies the dependence of ballistic trajectories on the mean plus turbulent wind field. Sections 6 and 7 provide some results of the combined turbulent wind-ballistic trajectory model for various munitions and provides comparison of the combined simulations to actual test data taken at the APG. Section 8 ends the paper with a summary and conclusions.

## 2. Measurements of turbulence

A rigorous evaluation of the effects of atmospheric turbulence on ballistic trajectories requires reliable measurements of the turbulence statistics. 3D sonic anemometers are the preferred instrument for measurements of boundary layer turbulence statistics (e.g., Kaimal and Finnigan 1994). To determine the turbulence statistics that might be typical of actual firing conditions, data were collected at the APG during November 2004 and January and March 2005 using an R. M. Young 3D sonic anemometer (Model 81000 with an accuracy of 0.1% or 0.05 m s^{−1}) with a sampling rate of 10 Hz at a height of 7.3 m above ground level (AGL). An example of the data collected over a 20-min period starting at 1350 LT on 5 November 2004 is shown in the time series for all three velocity components in Fig. 1. Here the individual components are shown in the reference frame of the mean wind vector; that is, in the figure, *u* is the alongstream or longitudinal velocity, *υ* is the cross-stream or transverse velocity, and *w* is the vertical (also transverse) velocity produced with a standard coordinate rotation (Kaimal and Finnigan 1994, p. 236). This 20-min section of data characterizes a fairly stationary interval of turbulence. During this collection period the mean wind speed and direction were about 5 m s^{−1} and 330°, respectively. The mean wind direction defines the alongstream or longitudinal direction, and by definition, the mean transverse velocities, *υ* and *w*, are therefore zero. The standard deviations of the *u*, *υ*, and *w* velocity components *σ _{u}*,

*σ*, and

_{υ}*σ*over the 20-min interval are about 1.65, 1.56, and 0.88 m s

_{w}^{−1}, respectively. Thus the variability of the vertical component is about one-half that of the horizontal components (which are about equal). This is probably due to the relatively close proximity of the measurements to the surface. The locations of this anemometer and the one used for the comparisons with actual shot data to be presented in section 7 are on the same firing range, which has relatively flat terrain located between moderately dense tree lines of height about 20 m situated about 100 m to the northwest and 200 m to the southwest. Thus for this particular case the anemometer was downstream of the tree line.

## 3. The von Kármán turbulence model

*B*

_{LL}) and transverse (

*B*

_{NN}) covariance functions given by (e.g., Monin and Yaglom 1975, p. 19)andwhere

*u*(

*x*) and

*υ*(

*x*) are the velocity components along and transverse to the displacement vector

**r**= (

*x, y, z*), respectively, and 〈〉 denotes an ensemble average. The longitudinal (

*F*) and transverse (

_{L}*F*) one-dimensional spatial spectra of the velocity field are given by the Fourier transform of the covariance function (e.g., Monin and Yaglom 1975, p. 43)andwhere

_{N}*k*is the wavenumber.

*F*and

_{L}*F*are available. A commonly used one is the von Kármán turbulence model (e.g., Hinze 1959), which has the functional formandwhere

_{N}*σ*

^{2}is the variance of the given velocity component,

*L*

_{0}is the outer scale of the turbulence, andis the integral length scale. This model has been shown to be consistent with boundary layer observations under a wide range of unstable and neutral conditions (e.g., Teunissen 1980; Founda et al. 1997; Shiau and Chen 2002), which would be expected at the APG during daytime testing.

*k*[Monin and Yaglom 1975, p. 355;

*E*

_{1}(

*k*) and

*E*

_{2}(

*k*) in their notation] isandwhere ε is the energy dissipation rate (m

^{2}s

^{−3}), and the Kolmogorov constant

*C*is taken as 2.0 (Monin and Yaglom 1975, 483–485). Equating the high wavenumber form (

*L*

_{0}

^{2}

*k*

^{2}≫ 1) of Eq. (3.3a) and Eq. (3.5a) produceswhich connects the turbulence outer scale (a measure of the largest turbulent eddies) to the velocity variance and energy dissipation rate.

*S*(

_{u}*f*),

*S*(

_{υ}*f*), and

*S*(

_{w}*f*) derived from sonic anemometer data by assuming Taylor’s frozen hypothesis is valid (e.g., Hill 1996; Wyngaard and Clifford 1977). Then, with

*k = 2π f*/

*U*, the longitudinal temporal spectrum

*S*(

_{u}*f*) becomesand the two transverse spectra are given bywhere

*U*is the magnitude of the mean velocity.

The velocity spectra *S _{u}*(

*f*),

*S*(

_{υ}*f*), and

*S*(

_{w}*f*) are shown in Fig. 2 for the time series of Fig. 1. To reduce the statistical scatter of the spectral estimates, six spectra from contiguous time series of 200 s each were averaged. Again,

*u, v,*and

*w*are the alongstream, cross-stream, and vertical velocity fluctuations, respectively, not the standard meteorological wind components. These spectra are typical of boundary layer measurements in that they exhibit a well-defined

*f*

^{−5/3}power law over most of the frequency range (indicated by the dashed vertical lines in the figure) resolvable by the sonic anemometers. In this range, the spectral levels of the best-fit

*k*

^{−5/3}power law produce estimates of the energy dissipation rate ε for each component of the velocity field from Eqs. (3.5a), (3.5b), (3.7), and (3.8). The fit cannot be extended to higher frequencies because of aliasing and instrument noise in the measurements. At lower frequencies, the spectra roll off starting at frequencies corresponding roughly to the spatial frequency

*k*defined by the outer scales,

*L*

^{−1}

_{0}

*,*

_{u}*L*

^{−1}

_{0}

*, and*

_{υ}*L*

^{−1}

_{0w}. Equation (3.6) defines the relation of the outer scale to

*σ*

_{u,}

*,*

_{υ}*and ε. For this case,*

_{w}*σ*=1.65,

_{u}*σ*= 1.56,

_{υ}*σ*= 0.88 m s

_{w}^{−1}, ε

_{u}= 0.066, ε

_{υ}= 0.052, ε

_{w}= 0.053 m

^{2}s

^{−3}, and

*L*

_{0}

*, = 64 m,*

_{u}*L*

_{0}

*= 69 m, and*

_{υ}*L*

_{0}

*= 12 m. These von Kármán turbulence parameters completely describe the spatial statistics of the velocity field. Note that the shorter length scale*

_{w}*L*

_{0}

*in the vertical velocity*

_{w}*w*is consistent with the smaller

*σ*in the vertical velocity, apparent in the time series. Applied to the ballistic trajectory problem, the larger velocity variability in the horizontal directions will cause larger scatter in the ballistic trajectories in the cross-track direction than in the vertical direction.

## 4. Simulations of turbulent wind fields

### a. Simulations of atmospheric effects on ballistic trajectories

The effects of atmospheric turbulence on ballistic trajectories are determined statistically by calculating the trajectories for many realizations of the turbulent wind field along the mean trajectory [*x*, *y*, * z*(

*x*)], where

*x*is along range,

*y*is across range, and

*is the average altitude of the trajectory. To simplify the numerical simulations and trajectory calculations of the atmospheric effects, we assume that the velocity components are statistically independent von Kármán fields along the mean trajectory, with parameters determined from the sonic anemometer data. This permits different parameters for each velocity component but does not include correlations between the components. For this application, these correlations have a negligible effect on the ballistics calculations since there is weak coupling between the velocity components in the trajectory equations (see section 5).*z

Standard simulation algorithms are used to generate the statistically independent realizations of *u*, *υ*, and *w* for the calculation of the statistics of the trajectories. Each realization is generated using a spectral technique that produces the exact spatial covariances defined by the parameters of a von Kármán model (Frehlich 1997); that is, the random velocity is produced by generating statistically independent Gaussian random numbers with a mean value of zero for the real and imaginary parts of the Fourier coefficients. The variance of each Fourier coefficient is chosen to produce the desired spatial correlation function. The spatial realizations are calculated from the random Fourier coefficients using the fast Fourier transform (FFT). This simulation algorithm is easily extended to two- and three-dimensional random processes (Frehlich et al. 2001). An example of one realization of the three velocity components along the mean trajectory * z*(

*x*) for the von Kármán parameters derived from Fig. 2 is shown in Fig. 3. Note the overall resemblance to the time series in Fig. 1, from which the von Kármán parameters were originally determined.

### b. Simulations of anemometer data for use in ballistic trajectory calculations

*x*,

*y*,

*(*z

*x*)]. To model the time-averaged sonic anemometer measurements we assume that the mean wind vector

*U*is perpendicular to the initial trajectory angle and assume that Taylor’s frozen hypothesis is valid to produce realizations of the average sonic anemometer data at various locations (

*x, z*) along the trajectory; that is,where

*q*is any velocity component and

*T*is the averaging time for the sonic data. For simplicity, we also assume the anemometer locations are placed along the firing line and that the measurement elevations are very near the mean trajectory elevation.

The 2D turbulence field is approximated by statistically independent realizations of each velocity component in the plane of the mean trajectory, that is, the (*x*, *y*) plane defined by [*x*, *y*, * z*(

*x*)]. The simulation algorithm is an extension of the 1D spectral algorithm and produces the desired spatial statistics in the 2D plane (Frehlich 2000, 2001). An example realization of the simulated

*υ*(crosswind) and

*w*(vertical) velocity fields is shown in Fig. 4 for the von Kármán parameters of Fig. 2. The larger outer scale of the horizontal velocity field is indicated by the larger-scale features in the horizontal direction and the magnitude of the vertical velocity fluctuations is smaller. Also shown in Fig. 4 is the equivalent spatial averaging domain of the sonic anemometer measurements of Eq. (4.1) indicated by the length of the solid bars for an averaging time

*T*= 20 s for a trajectory calculation at the crosswind location

*y*= 100 m at 100-m intervals down range.

## 5. GTRAJ ballistics model

**V**and

**x**are, respectively, the vector velocity and position of the projectile in an earth fixed coordinate system,

*t*is time,

*p*is the axial spin rate,

*is the “yaw of repose,”*

**α**_{R}*ρ*

_{a}is atmospheric density,

**g**is the gravitational acceleration vector,

**Co**is the Coriolis acceleration vector, and

*K*,

_{D}*K*,

_{L}*K*,

_{N}*K*,

_{P}*K*, and

_{R}*K*are munition-dependent coefficients relating to the drag, lift, Magnus forces, and spin damping, rolling, and pitching moments, respectively. The effects of the mean wind and turbulence come in through the third equation in Eq. (5.1), which defines the earth-relative velocity

_{M}**v**(

*x*,

*y*,

*z*,

*t*) in terms of the projectile velocity and the total (mean + turbulent) wind velocity

**W**. In general

**W**is a function of position along the trajectory and time, but for the calculations used here,

**W**is assumed frozen in time during the few seconds of projectile travel. The equation set in Eq. (5.1) is solved using fourth-order Runge–Kutta integration subject to initial conditions at the muzzle exit. All GTRAJ simulations were set up so that the initial elevation angle produced a mean impact point at the target range at about the same elevation as the firing elevation.

Since GTRAJ is set up as an interactive process where one projectile trajectory is simulated for one set of atmospheric conditions, for our purposes the code had to be modified to execute many realizations of turbulence, each realization through a modification to the **W** term in Eq. (5.1), for a given mean wind and munition type. The effects of turbulence on the trajectories of four munition types were simulated with this GTRAJ configuration. In order from the heaviest to the lightest, these were the M865 tank round, M793 tank round, 50-caliber “match” rifle ammunition, and the M852 sniper-rifle ammunition.

## 6. Results of the combined model simulations

The statistics of the ballistic trajectories are determined from many realizations of the combined turbulent wind field–GTRAJ model. An example of the combined model-derived vertical and cross-range scatter at a target 2 km down range from the firing location using 5000 realizations of turbulent wind fields for the turbulence parameters of Fig. 2 for the four different munitions tested is shown in Fig. 5. In all cases the mean crosswind is 5.05 m s^{−1}. As expected, the average cross-range displacement and both the vertical and horizontal scatter increase as the mass of the munition decreases. Note that the scatter varies by two orders of magnitude from the lightest to the heaviest munition. In all cases, the scatter in the vertical direction is smaller than the scatter in the horizontal direction because of the higher turbulence levels in the horizontal velocity component (cf. Fig. 1).

In Fig. 5 the standard deviation of the shots defines the scatter from atmospheric turbulence only for a single crosswind and set of turbulence parameters. During actual firing conditions, anemometer data may be available along the range of fire as indicated schematically in Fig. 4. The simulated anemometer data may be used in the GTRAJ calculations to provide range-dependent winds, including the effects of turbulence, as input to the ballistics calculations by using a velocity profile that is a linear interpolation of the simulated sonic measurements in range. The measurements that would be observed from the sonic anemometers can be simulated using a frozen 2D mean + turbulence velocity field such as shown in Fig. 4. This of course depends on the averaging time of the simulated anemometer data, which is equivalent to spatial averaging of the product of the mean crosswind and the assumed averaging time centered on the reference trajectory (as indicated by the length of the black lines in Fig. 4).

*y*,

*z*) at the target. The first is computed from GTRAJ driven by a realization of the turbulent velocity field to produce an impact point (

*y*,

_{a}*z*). The second is produced by sampling this 2D realization with

_{a}*N*

_{sonics}measurements with a specified averaging time equally spaced along the range and using the sampled data with linear interpolation between the samples to drive GTRAJ to produce an impact point (

*y*

_{sonic},

*z*

_{sonic}). The difference,averaged over many realizations, provides a measure of the feasibility of using anemometer data to estimate the atmospheric turbulence component of impact scatter at the target, and can provide a “turbulence correction” for the total observed scatter. An example of using this procedure is demonstrated by the scatterplot shown in Fig. 6 for the 50-caliber munition and the atmospheric parameters of Figs. 2 –4. Here,

*N*

_{sonics}= 50 with an averaging time of

*T*= 5 s. For the parameters chosen, the inclusion of the simulated sonic anemometer data reduces the scatter significantly in the horizontal direction

*y*(

*σ*

_{Δy}= 0.16 m as compared with

*σ*= 0.58 m). However, in the vertical direction the scatter is only reduced from

_{y}*σ*= 0.16 m to

_{z}*σ*

_{Δz}= 0.15 m because the outer scale of the vertical velocity is much smaller than the outer scale of the crosswind velocity. Consequently, a very large number of sonic anemometer measurements would be required to resolve the vertical velocity eddies.

In the limit of an infinite number of sonic anemometer measurements with very short averaging time intervals *T*, for these ideal simulations the turbulence scatter correction will become perfect and reduce the scatter to zero. In reality, there will be a limit to the accuracy of the corrections that will depend on unknown error sources not included in the simulations. The effect of using simulated data from a finite number of anemometers in reducing the shot impact scatter is shown in Fig. 7 for a 50-caliber munition, Fig. 8 for an M865 munition, and Fig. 9 for an M793 munition. Each figure shows the cross range and vertical scatter (standard deviation of shot location on target estimated with 1000 realizations and the atmospheric parameters of Figs. 2 –5) as a function of the number of sonic anemometers used, *N*_{sonics} for averaging times of *T* = 5, 30, 60, 120, and 300 s. The pure atmospheric scatter obtained directly from the wind realizations is shown as the horizontal line. In all cases, *N*_{sonics} should be larger than about 10 to have a positive effect in reducing the horizontal scatter *σ*_{Δy} below the ambient atmospheric scatter. Larger reductions are associated with shorter averaging times, as expected. With very few sonic measurements, the predicted velocity profile is a poor approximation of the true profile and *σ*_{Δy} is even larger than the ambient atmospheric scatter. For example, a single sonic measurement of the crosswind velocity at 500-m range in Fig. 4 is not representative of the velocity field as a whole, and the use of this single sample would therefore produce a large error in the scatter. Similarly, using a typical averaging time of 300 s (5 min) has very little impact in reducing the scatter from turbulence since the instantaneous spatial variations along the trajectory are not resolved. To reduce the vertical scatter *σ*_{Δz}, the requirements on the sonic measurements are very severe in that both a very large number of anemometers and a short averaging time are required because of the much smaller vertical length scale (*L*_{0}* _{w}* = 12 m in this case).

## 7. Predictions of combined wind–GTRAJ simulations and comparison with actual data

The predictions of the combined wind turbulence–GTRAJ model were evaluated using a special set of M852 sniper munition firing data collected at the APG during January 2006. The wind data from several mechanical wind vane anemometers (R. M. Young model 05103 with 1-Hz sampling rate, wind speed accuracy of 0.3 m s^{−1}, and direction accuracy of 5°) were collected along the test range of length 720 m during each of eight groups of firings. The tests were conducted on five different days, and each test consisted of several rounds (usually 10) fired at different time intervals (cadence, 2 or 3 min). An M24 sniper rifle was clamped to a rigid pedestal to permit repeatable firings. The wind speed and direction from the wind vane anemometers were converted into longitudinal and transverse turbulent fluctuations using standard geometrical conversions (Kaimal and Finnigan 1994). Although the shorter range provides a better chance of having homogeneous turbulence conditions along the path, some cases were nonstationary, and still other cases had incomplete data, so that only a few cases could be compared. An example of the velocity time series from one case (group 4 on 18 January 2006) measured by the wind vane anemometer 100 m down range from the firing position at a height of 2.7 m AGL is shown in Fig. 10. This particular anemometer was chosen because 1) it is close to the height of the projectile, 2) unlike most other anemometers that were covered by a metal shield on the lower portions of the anemometer mast to deflect shock waves from the blast, this one was unshielded, and 3) there were no interruptions in the data flow. The corresponding average spectra from seven contiguous time series of 200 s of the data shown in Fig. 10 are shown in Fig. 11. The longitudinal spectrum has a well-defined *f* ^{−5/3} power-law region and can therefore be used to define the von Kármán turbulence parameters (*σ _{u}* = 2.02 m s

^{−1},

*L*

_{0}

*= 50.5 m). This is consistent with the sonic anemometer data (Figs. 1, 2) taken at nearly the same location but on a different day. In this case the mean wind direction was about 270°, and therefore the wind vane anemometer is again downstream of the tree line. The transverse spectra are dominated by wind direction fluctuations and do not have a well-defined power law because of the inertia of the wind vane. Because of this and also since*

_{u}*w*is not measured by standard wind vane anemometers, isotropic turbulence conditions had to be assumed with the turbulence parameters for all three components defined from the longitudinal spectrum. From the sonic anemometer data shown in Fig. 2, the assumption of horizontal isotropy is probably reasonable, but the vertical velocity statistics are likely very different. Therefore, only the computed horizontal deflections may be representative of actual conditions for this case.

The comparison of simulated (using the wind and turbulence data derived from Figs. 10 and 11) and measured trajectory scatter is shown in Fig. 12. Although the number of rounds fired in this test (10) was too small to allow accurate statistical evaluations of the performance, qualitatively, both the mean deflection and the cross-range scatter produced in the simulations are in good agreement with the scatter of the observations. In addition, the observations contain an extra contribution to the scatter from the random variations in the munition properties such as weight and explosive charge.

The performance of the turbulence correction procedure using simulated sonic anemometer measurements along range for this case is shown in Fig. 13. For the shorter range of 720 m (versus 2000 m used in Fig. 7), fewer sonic anemometers (about seven) would be required to reduce the target scatter below the value with no turbulence corrections. The turbulence correction algorithm could also be evaluated for the vertical displacement if turbulence data were available for the vertical velocity.

## 8. Summary and conclusions

The feasibility of using numerical simulations of realistic atmospheric turbulence conditions combined with 4DOF calculations of ballistic trajectories using a standard ballistics code to estimate the effects of atmospheric turbulence on the horizontal and vertical scatter of the trajectories was demonstrated. A technique was developed and evaluated for estimating the impact of the turbulence-induced scatter on ballistic testing procedures by using a number of sonic anemometer measurements along the trajectory path. The metric used to evaluate the ability of using high-rate anemometer data to characterize the wind field along range to correct for the effects of turbulence-induced scatter was defined as the difference between the target impact scatter produced with and without the use of the down range anemometers in the trajectory calculations. The potential improvement in the testing procedure as measured by this metric was determined as a function of the number of sonic anemometers and the sonic averaging time interval.

Quantitative statistics of the target impact metric were determined with multiple realizations of simulated atmospheric turbulence for various munitions and for actual measured turbulent conditions. These results provide the required measurements (number of anemometers and anemometer averaging time) for sampling turbulent conditions along the range and thereby reducing the scatter estimates due to atmospheric turbulence on threshold levels of acceptance tests. As expected, larger munitions have the least sensitivity to the turbulence for the parameter space of the simulations shown here. The results of an actual ballistics test using a small-caliber munition and a shorter range were in good qualitative agreement with the simulations based on coincident turbulence measurements.

*σ*,

_{y}*σ*) and the improvement metric scatter (

_{z}*σ*

_{Δy},

*σ*

_{Δz}) due to turbulence depends on the number of shots fired during the test. Obviously, firing a larger number of shots will reduce the statistical error bars of the estimates of the scatter. For example, the 90% confidence limits of the estimates

*s*of the true standard deviation

_{y}*σ*for a sample size of say

_{y}*n*= 10 (rounds) are given byas determined from a

*χ*

^{2}distribution with

*n*− 1 = 9 degrees of freedom (e.g., Stuart and Ord 1991, p. 757). Thus in this case, the actual scatter may be as much as 73% larger than the scatter derived from just 10 rounds. For

*n*= 101 rounds this is reduced to 14%. Therefore, a large number of rounds are required for statistically meaningful results. For the less expensive small munitions, a larger number of rounds may be feasible and atmospheric conditions, such as cloudy days with low turbulence, may be selected to permit valid testing to minimize concerns about atmospheric effects. However, a large number of rounds is not feasible for the larger munitions, mainly because of economic reasons. Again, based on guidance from similar simulations, suitable atmospheric conditions can be chosen for these tests. Nevertheless, the procedures outlined here could be used in an operational setting to better isolate the effects of atmospheric turbulence from acceptance statistics. In addition, even more realistic simulations of the atmospheric processes could be performed, such as a full 3D simulation of the wind fields.

## Acknowledgments

This research was funded by the U.S. Army Test and Evaluation Command through an interagency agreement with the National Science Foundation. We thank Dr. Tom Warner for suggestions for improvement on an earlier version of the manuscript. We also thank the three anonymous reviewers, whose comments helped to improve the final version of the paper.

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