William O. Hughes

Glenn Research Center, Cleveland, Ohio

Thomas L. Paez

Sandia National Laboratories, Albuquerque, New Mexico

Application of the Bootstrap Statistical Method in

Deriving Vibroacoustic Specifications

NASA/TM—2006-214446

November 2006

Paper No. 94
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Application of the Bootstrap Statistical Method in

Deriving Vibroacoustic Specifications

NASA/TM—2006-214446

November 2006

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

Ninth International Conference on Recent Advances in Structural Dynamics

sponsored by the Institute of Sound and Vibration Research

Southampton, United Kingdom, July 17–19, 2006

Paper No. 94

William O. Hughes

Glenn Research Center, Cleveland, Ohio

Thomas L. Paez

Sandia National Laboratories, Albuquerque, New Mexico
Acknowledgments

The authors would like to thank the NASA Glenn Research Center and the Sandia National Laboratories for supporting this

work, and Dr. Courtney Burroughs of the Pennsylvania State University. Sandia is a multiprogram laboratory operated by

Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security

Administration under DE-AC04-94AL85000.

Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5285 Port Royal Road

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Available electronically at http://gltrs.grc.nasa.gov

Trade names and trademarks are used in this report for identification

only. Their usage does not constitute an official endorsement,

either expressed or implied, by the National Aeronautics and

Space Administration.

Level of Review: This material has been technically reviewed by technical management.
NASA/TM—2006-214446 1

Application of the Bootstrap Statistical Method in 

Deriving Vibroacoustic Specifications 

William O. Hughes 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Thomas L. Paez 

Sandia National Laboratories 

Albuquerque, New Mexico 87185 

Introduction 

Background 

Numerous types of spaceflight hardware are exposed to random vibroacoustic excitation during the 

three distinct launch events of liftoff, transonic flight and maximum dynamic pressure. Random acoustic 

levels can exceed 160 dB (referenced to 20 μPascals) in overall sound pressure level on large launch 

vehicles. The nonstationary vibroacoustic launch environments are usually analyzed separately, to 

determine the maximum dynamic loading, for each of the three launch events. Acoustic loads occurring at 

liftoff are generated by the turbulent mixing of the rocket engine exhaust gases with the atmosphere. 

Acoustic data measured during several liftoff events for the Titan IV launch vehicle will serve as the data 

source for this paper’s main analytical topic. 

During launch, part of the acoustic energy external to the vehicle is transmitted through the vehicle’s 

outer protection layer. The ensuing internal acoustics excites the payload, resulting in high structural 

vibration responses of the payload. Payloads with large areas and low masses are particularly susceptible 

to this acoustic excitation. Delicate optical or electronic flight hardware may also be damaged by this 

acoustic excitation. 

In order to ensure mission success, it is necessary for the National Aeronautics and Space 

Administration (NASA) to determine this internal acoustic environment for the launch vehicles utilized. 

Knowledge of these environments allows the spaceflight hardware to be properly designed for the 

structural loading that is experienced during launch. Normally, prior to launch the spaceflight hardware 

will undergo dynamic testing on the ground to specified vibroacoustic test levels, thereby providing added 

confidence for mission success. 

Nonstationary Random Vibroacoustic Data Analysis 

Both external (to the launch vehicle) and internal (inside an expendable launch vehicle’s payload 

fairing (PLF) or inside the Shuttle’s cargo bay) microphone measurements of the acoustic pressures may 

be obtained during launch. Measured pressure time histories, x(t), are time-varying data from a 

nonstationary random process. Because each measured signal comes from a random (non-deterministic) 

source it cannot be described by a deterministic expression. If the source was stationary random, it could 

be described by statistics such as its mean (μx), and rms (root-mean-square) value (Ψx), which would be 

temporally invariant. However, these statistical values are time varying for nonstationary random sources. 

It is often convenient to estimate these time-varying averages over short, contiguous segments of the 

measured signal to obtain running averages. 

For example, the one-third octave band (OTOB) sound pressure level (SPL) of a stationary pressure 

time history, x(t) (where x equals pressure), is given in dB by: 
NASA/TM—2006-214446 2

2

10

( ) ( ) 10log ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Ψ

Ψ = ref

x i x i

f L f (1) 

where Ψx(fi) is the rms value of the pressure signal x(t) filtered through a one-third octave bandpass filter 

centered on frequency fi and Ψref is reference rms pressure (20 μPa, here). 

For a nonstationary pressure source a time-varying SPL spectrum may be estimated from a running 

average of time-varying rms value, obtained by replacing Ψx(fi) in equation (1) by 

2

1

2

2

2 ( , ) 1 ( , ) ˆ

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

Ψ = ∫

+

−

x f t dt

T f t i

T

t

T

t x i (2) 

The hat (^) denotes an estimate, x

2

(fi,t) is the instantaneous squared value of the pressure signal passed 

through the ith one-third octave bandpass filter, and T is the averaging time of analysis. For launch 

vehicle vibroacoustic data, T = 1.0 sec with 50 percent overlapping has been found to yield reasonable 

estimates that balance random and bias errors (ref. 1). 

A composite spectrum, referred to as the “maximax” spectrum, is obtained by selecting at each 

OTOB frequency the largest value from all the time-varying SPL spectra, regardless of time slice of 

origin. This maximax SPL spectrum does not represent the instantaneous SPL at any specific time but 

instead has been found to provide a conservative measure of the dynamic environment with respect to the 

damage potential of this signal to spaceflight structures and equipment. A degree of conservatism 

(perhaps substantial) is incorporated into the analysis with the use of maximax spectrum. References 1 

and 2 are excellent sources of information regarding the acquisition and analysis of dynamic data. 

Vibroacoustic Test Specification 

To properly test qualify spaceflight hardware to its launch acoustic environment, test levels are set 

based upon the appropriate maximax SPL spectra available. Zones are defined in a launch vehicle within 

which it is expected that acoustic environments will be reasonably similar.

Different launch vehicles of a particular type display flight-to-flight variations. Some of this 

variability is due to inherent differences between the flights such as different launch pads, payload 

configurations, and weights. But some of this variability is due to the randomness of the launch event 

itself, such as a hot engine burn or a three-sigma maximum dynamic pressure event. Due to limited 

available flight data, it is typical to include in the database as much flight data as is reasonably possible, 

to capture true variability. 

Given a set of data, levels are typically derived that represent the maximum expected environment 

(MEE). This is a level that would typically not be exceeded, and should account for both the expected 

spatial variation within a particular zone as well as the known flight-to-flight variation. A second higher 

level denoted as the Extreme Expected Environment (EEE) is a level that should not be exceeded except 

for the most extreme circumstances. The EEE level is meant to cover known and unknown failure modes 

due to peak loading. 

This paper illustrates two methods to calculate the MEE and EEE test levels. They are: (1) the 

Normal Tolerance Limit method (NTL) and (2) the Bootstrap method. The NTL has traditionally been 

utilized by NASA to calculate its MEE and EEE levels that are used for acceptance and qualification 

testing, respectively. The Bootstrap method is a statistical subsampling method that has wide use in many 

disciplines but has not been used for this application. Both methods will be applied to a set of Titan IV 

liftoff launch vehicle acoustic data and their respective results will be compared in this paper. 
NASA/TM—2006-214446 3

NASA’s Traditional Method of Setting Vibroacoustic Test Levels 

Normal Tolerance Limit Method 

Numerous methods could be applied to a set of measured data to compute vibroacoustic test levels. 

NASA has traditionally used what is called the (NTL) method to compute vibroacoustics test levels 

(ref. 3).

Normal tolerance limits should be applied only to normally distributed random variables. If this 

assumption is appropriate, then the normal tolerance limit (NTLx) for the set of x variables, xi; i = 1, 

2,....n, is given by 

NTL ( , , ) ( * ) x n, , x n X K s β γ = + β γ (3) 

where 

∑

=

=

n

i

xi

n

X

1

1

( )∑

=

− − =

n

i

x xi X

n

s

1

2 ( ) 1

1 (4) 

are the sample mean and standard deviation of x, respectively, Kn,β,γ is the normal tolerance factor, β is the 

minimum portion (probability) of all values that will be less than NTLx (n,β,γ), and γ is the confidence 

coefficient associated with NTLx (n,β,γ). 

The K normal tolerance factors may be easily found in reference 4. A subset of these K factors is 

provided in table 1. The magnitude of the K factor is affected by both the probability desired and the 

confidence desired. This uncertainty in the confidence results from using a sample mean and sample 

standard deviation in lieu of the population’s true mean and standard deviation values. 

TABLE 1.—NORMAL TOLERANCE K FACTORS 

Note that for the case of n = ∞, the confidence is one-hundred percent, since one has the entire data 

population and not just a sample subset. One can therefore calculate with 100 percent confidence the 

population’s true mean and standard deviation. For this special case, the K normal tolerance factors 

become the zα percentage points of the standardized normal distribution. Then 

NTL ( , ) ( * ) x x x ∞ β =μ + zα σ (5) 

where α = 1–β, μx is the true mean, and σx is the true standard deviation of x. 

Lognormal Distributions and the NTL 

As stated, the Normal Tolerance Limit method should be applied to normally distributed random 

variables. There is much evidence that many data sets applicable to spaceflight vibroacoustic data are not 

normal but indeed lognormal (refs. 5 to 11). Therefore, one may still use the NTL method on these data 

by applying a logarithmic transformation, as follows

γ = 0.50 γ = 0.75 γ = 0.90 n 

β = 0.90 β = 0.95 β = 0.99 β = 0.90 β = 0.95 β = 0.99 β = 0.90 β = 0.95 β = 0.99 

10 1.32 1.70 2.41 1.67 2.10 2.93 2.06 2.57 3.53 

17 1.31 1.68 2.37 1.55 1.96 2.74 1.82 2.27 3.14 

50 1.29 1.65 2.34 1.43 1.81 2.54 1.56 1.96 2.74 

∞ 1.28 1.64 2.33 1.28 1.64 2.33 1.28 1.64 2.33 
NASA/TM—2006-214446 4

y = log10 x (6) 

NTL ( , , ) ( * ) y n, , y n y K s β γ = + β γ (7) 

where y and sy are the sample mean and standard deviation of y. The normal tolerance limits in the 

original units of x may then be recovered from 

NTL ( , , ) NTL ( , , ) 10 β γ β γ = n x n y (8) 

The transformation is useful for vibration spectra in g2/Hz and shock response spectra in g when the 

random source of interest is thought to be lognormally distributed. 

Acoustic SPL data in dB can generally be used to obtain NTL directly, without transformation. The 

reason is that pressure sources are assumed lognormal, implying that SPL dB data are normally 

distributed due to their calculation via 

10

2

10 SPL(dB) 10log = 20log ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = pref

p

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

pref

p (9) 

Typical NTL levels used within the aerospace industry (ref. 12) are: 

1. Maximum Expected Environment: NTL (β = 0.95, γ = 0.50) level = P95/C50; 

 MEE levels used as basis for acceptance level testing. 

2. Extreme Expected Environment: NTL (β = 0.99, γ = 0.90) level= P99/C90; 

 EEE levels used for qualification level testing. 

The Bootstrap Method 

Background 

Advances in computational speed and cost in the 1970s permitted numerous advances in statistical 

theories and methods. The Bootstrap, developed by Efron, is one of these methods. In reference 13, Efron 

defines the problem: “given a random sample X = (X1, X2,....Xn) from an unknown probability distribution 

F, estimate the sampling distribution of some prespecified random variable R(X,F), on the basis of the 

observed data X.” The Bootstrap allows one to assess the accuracy and uncertainty of estimated 

parameters from small samples, without any prior assumptions about the underlying distribution. The data 

need not be normally distributed (as in the NTL method). The Bootstrap also has the advantage of 

allowing assessment of parameters that may not be mathematically expressible in simple terms, for 

example, the median of a set of data.

The method consists of repeatedly forming Bootstrap samples (perhaps as few as 100 for parameter 

estimation and up to a few thousand for confidence interval estimation) of the same size as the original 

data sample. The elements of each Bootstrap sample are randomly chosen from the original data, with 

replacements. Thus, a particular sample data point may be chosen several times or perhaps not at all in 

any particular bootstrap sample. The parameter of interest is then evaluated from each of the bootstrap 

samples generated. The numerous bootstrap replicates of the parameter can be used to estimate a 

probability distribution for the parameter. This is an estimate of the parameter sampling distribution, and 

from this distribution, confidence intervals may be approximated. An overview of the Bootstrap method 

is given in reference 14. 
NASA/TM—2006-214446 5

Bootstrap Method 

The following steps define the basic Bootstrap process. 

1. Start with data, X= (x1, x2, ...xn), a sample set taken from the population with unknown 

probability distribution F. 

2. Select a bootstrap sample of the data by randomly sampling, with replacements, the data X. The 

bootstrap replicate, Xb, should be of size n. For example, if X = (x1, x2, x3, x4, x5), n = 5, then one 

bootstrap replicate might be Xb = (x3, x5, x3, x4, x1). 

3. Compute the measure of interest θ; for example if θ = mean (X), then θb = mean (Xb) =

( )

5

1 3 5 3 4 1

1

x x x x x X

n

X

n

j

b bj

+ + + + = ∑ =

=

4. Repeat steps 2 and 3 numerous times to obtain a large number (B) of bootstrap samples and use 

each to compute bootstrap replicates of the measure of interest { } B b b b bB θ = θ ,θ ,θ ,....θ 1 2 3 . 

5. Form an empirical cumulative distribution function (CDF) of the measure of interest. 

6. Confidence limits or intervals for the measure of interest may then be based on the CDF. For 

example, the (1-α) × 100 percent level of confidence is found at the (1-α) × 100 percent 

percentage point of the CDF. 

The Titan IV Liftoff Acoustic Database 

Objective 

For a given set of Titan IV acoustic data, the P95/C50 and P99/C90 specifications are derived using 

the traditional Normal Tolerance Limit method. The goal of this paper is to apply the Bootstrap method to 

the same data to compute the “equivalent” bootstrap derived levels. Finally the results from these two 

methods will be compared. 

Acoustic Database Description 

Seventeen acoustic microphone measurements from liftoff events from six different launches of the 

Titan IV expendable launch vehicle were used for this analysis. This dataset, corresponding to flight 

measurements made inside the Titan IV payload fairing (PLF) at the spacecraft’s location, was studied 

extensively by NASA in the mid-1990s (ref. 15 to 17) to support the launch of the Cassini spacecraft to 

Saturn in 1997. Figure 1 illustrates the Cassini spacecraft within the Titan IV launch vehicle’s PLF. 

Understanding the acoustic environment at PLF zones 7 through 10 was of critical interest. 
NASA/TM—2006-214446 6

Figure 2 shows all 17 SPL measurements plotted together. It is from this collection of data that test 

levels will be derived to set MEE and EEE levels. The mean, using a straight numerical average of the dB 

levels at each frequency, is seen as the lowest line in figure 2. 
NASA/TM—2006-214446 7

Application of the Normal Tolerance Limit Method 

Applying the NTL to this dataset is straightforward. The number of data samples, n, is 17. Referring 

to table 1 the K factors for NTL (n = 17, β = 0.95, γ = 0.50) is 1.68, and for NTL (n = 17,

β = 0.99, γ = 0.90) is 3.14. Knowing the K factor, along with the mean and standard deviation, application 

of the NTL definition (eq. (3)) yields the results shown as the middle (NTL P95/C50) and upper (NTL 

P99/C90) lines in figure 2. 

The assumption in using the NTL method is that the data come from a normal distribution. In this 

case, it is assumed that the SPL data (in dB) are normally distributed. If this is not the case, the resulting 

levels may be in error. 

The P95/C50 level represents the MEE or maximum expected environment. On average (C50) one 

would expect 95 percent (P95) of the data to be at or below this P95/C50 level. There are 23 data points 

that actually exceed the P95/C50 level. Since there are 357 data points in total (17 microphones × 21 

OTOB frequencies) that means that for this particular data set (357–23)/357 or 93.6 percent of the data 

actually lie at or below this P95/C50 level. This is indicative that the NTL P95/C50 level is performing as 

expected. The slight difference from the theoretical 95 percent may be due to the assumption that the SPL 

dB data is normally distributed, or it may simply be due to the randomness of the events themselves. 

For this particular data set, the P99/C90 level should encompass all (or almost all) the flight data if it 

behaves as expected. Per reference 12 for P99/C90 “there is 1 chance in 10 of exceeding the level once in 

100 flights.” Since for this data set we have 357 “flights” the odds are that no flight data should exceed 

the calculated NTL P99/C90 level. Figure 2 indeed bears this out. 

This case points out an advantage of using the NTL P99/C90 level as a qualification ground test level. 

The design of the spaceflight hardware will be tested out (i.e., qualified) to levels that it should rarely see 

in flight. Note however that there is one data point at 40 Hz whose level does approach the NTL P99/C90 

level. The disadvantage of course is the higher test levels. In this particular case, the NTL P99/C90 levels 

range from 7.4 to 16.1 dB above the mean of the data depending on the OTOB frequency, with an average 

increase of 11 dB. 
NASA/TM—2006-214446 8

Application of the Bootstrap Method 

The Bootstrap method was implemented through the use of MATLAB (The MathWorks, Inc.) 

coding. It was written to enable the code to generate the statistics of interest for this paper, namely the 

bootstrap “equivalent” to the NTL P95/C50 and P99/C90 levels. In lieu of a NTL K factor, the bootstrap 

equivalent P95/C50 and P99/C90 levels were computed by using the bootstrap replicates of the statistics 

of interest themselves. 

The statistics of interest for this bootstrap analysis are the P95 and P99 probability levels. This 

requires that the bootstrap mean and bootstrap standard deviation be jointly used as a bootstrap pair to 

compute the desired probability (P) level. In the MATLAB code at each OTOB frequency, a bootstrap 

replicate set is made that consists of 17 data points randomly selected (with replacement) from the 

original 17 Titan IV acoustic SPL data points. Next the mean and standard deviation of this bootstrap 

replicate set is computed. This process of creating bootstrap replicates and generating the bootstrap values 

for the statistics of interest is repeated numerous (nr) times. Figure 3 shows an example of 

nr = 1000 bootstrap replicates (for the 1000 Hz OTOB frequency data). This figure illustrates the 

distribution of the replicate pairs of mean and standard deviation values. 

The distribution of the original Titan IV flight data itself was used to describe the CDF of the data. To 

do this the original set of the 17 flight data measurements were converted to standard normal format for 

each OTOB frequency. This z-value was calculated by taking each data point and first subtracting out the 

sample mean, and secondly by dividing by the sample standard deviation. 

The z-values were calculated for all 17 microphones at all 21 OTOB frequencies. Ideally an 

individual CDF would be formed from the z-values corresponding to each OTOB frequency. However, 

due to the limited number (17) of original measurements, this would not provide the necessary resolution 
NASA/TM—2006-214446 9

required to calculate high probability levels from the CDF. Therefore it was decided to combine the zvalues at all the OTOB frequencies to form the best estimate of the CDF for this entire data set. Doing 

this created 357 z-values (17×21) which does provide enough points in the CDF to resolve the P95 and 

P99 points. Similar approaches of combining data from multiple frequencies have been successfully done 

for previous NASA programs (refs. 17 and 18). 

Combining the z-values at all OTOB frequencies assumes that the data’s distribution is the same for 

all OTOB frequencies. This may be unlikely, but the introduction of this possible error is noted, but 

accepted as necessary in order to proceed. Some recent work indicates that this assumption may be 

reasonable (ref. 19). The cumulative distribution function for the Titan IV acoustic SPL database was 

formed from the z-values. The resulting CDF of the Titan IV data is reasonably similar to that for a 

normal distribution. Table 2 shows a comparison of these CDFs at the probability levels of interest. 

TABLE 2.—PROBABILITY VALUES DERIVED FROM CDF 

Level Titan IV database Theoretical normal 

P50 (mean) –0.9826 0 

P95 1.786 1.645 

P99 2.481 2.33 

The P95 = 1.786 and P99 = 2.481 are the z-values used in the MATLAB code for this Titan IV 

dataset. For each bootstrap replicate the following levels are computed, 

 P95 = bootstrap replicate mean + (1.786 x bootstrap replicate standard deviation) 

 P99 = bootstrap replicate mean + (2.481 x bootstrap replicate standard deviation) 

At this point, the MATLAB code has calculated nr values of the P95 and P99 levels at each OTOB 

frequency. To compute the confidence limit these values are sorted, then the percentage point of the 

distribution corresponding to the desired confidence level is selected. 

Figure 2 shows the Bootstrap P95/C50 result (open triangles) for nr = 1000. The comparison of this 

with the NTL P95/C50 level shows no significant difference. This Bootstrap P95/C50 level would 

represent the MEE or maximum expected environment. As before, one would expect on average (C50) 

that 95 percent (P95) of the data would be at or below this level. There are 20 points that exceed the 

bootstrap P95/C50 level. This leads to (357–20)/357 = 94.4 percent of the data being at or below this 

level. This compares favorably to the 93.6 percent result from the NTL P95/C50 level as described 

previously. A closer inspection of these data shows that the difference between the 

23 exceedences of the NTL level and the 20 exceedences of the bootstrap level is due to insignificant 

changes on the order of 0.2 dB. 

Figure 2 also shows the comparison of the Bootstrap P99/C90 result (open squares) for nr = 1000 

with the NTL P99/C90 level. There is some visible difference, particularly at 160 to 250 Hz, and at 500 to 

630 Hz, but even at these frequencies the levels are within 1.5 dB, and within the test tolerance of most 

reverberant acoustic test chambers. The NTL P99/C90 result tends to be slightly higher, at most but not 

all frequencies. Both P99/C90 levels easily envelope the maximum from the 17 flight data measurements, 

as expected for EEE levels. 

It appears that the Bootstrap method provides results similar to those obtained with the NTL method 

for this particular data set. The advantage of using the Bootstrap method is that it makes no assumptions 

on the distribution of the underlying data. The disadvantage is the additional computational effort 

required to perform the analysis. The MATLAB runtimes are very short, on the order of seconds and 

minutes, however there is setup time involved which is long compared to the “hand-like” calculation time 

of the NTL method. 
NASA/TM—2006-214446 10

Summary and Conclusions 

The focal comparison of this paper examines acoustic test specification via the NTL method versus 

test specification via the Bootstrap. For the data considered here, the two methods produce extremely 

close results as illustrated in figure 2. 

NASA and the aerospace industry have traditionally used the NTL to define the MEE and the EEE 

levels from available flight data. The MEE is used for acceptance testing of flight hardware, and is 

defined to be P95/C50. The EEE is used for qualification testing of hardware designs, and is defined as 

P99/C90. (Note that an alternative method used to set qualification test levels is the statistically derived 

MEE level plus some specified margin, e.g., +3 or +6 dB). By definition the NTL assumes that the data 

are normally distributed. This assumption has been verified with some previous aerospace acoustic SPL 

data and is generally considered safe to assume for acoustic SPL data in dB. If the assumption is not 

correct, as may be the case for random vibration response data, transformation techniques should be 

employed to accurately use the NTL method. 

The Bootstrap Method is a subsampling statistical procedure that uses replicates of the original data to 

allow estimates of parameters and confidence intervals to be made. The Bootstrap makes no prior 

assumption regarding the distribution of the data. 

Both these methods were applied to the Titan IV acoustic SPL liftoff database consisting of 

17 flight acoustic microphone measurements. As shown in this paper, the resulting NTL and Bootstrap 

P95/C50 and P99/C90 levels were remarkably close. This is believed to be due at least in part to the 

apparent normality of the sample flight data utilized. Both methods resulted in (test) levels that would 

perform as MEE and EEE levels should. 

The fact that these results are so close for these two methods provides added confidence in the NTL 

method approach and results that NASA has traditionally used in the past and continues to use today. 

Based on the analysis performed for this paper on this particular flight data set, it appears that both 

the NTL and Bootstrap methods are valid methods to predict the MEE and EEE levels for acoustic SPL 

data in dB. The NTL method is well established within the aerospace industry and is quick to calculate. 

However, unlike the NTL method the Bootstrap method does not assume that the sample data come from 

a normal distribution. 

Future investigations comparing test specification obtained via these two methods on other data sets 

may prove informative. Of particular interest would be applying this to a known non-normal data set to 

see how the bootstrap results would compare with the NTL results. Examples of this data type might be 

random vibration response data, shock response spectrum data, or acoustic pressure (in Pascal units, not 

in dB). If such a comparison were performed, it might, however, be necessary to employ some advanced 

methods of the bootstrap such as Efron’s BCa method (ref. 14), which adjusts the confidence interval 

limits based on two factors called the bias-correction and the acceleration. 

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2. Himelblau, H., Piersol, A., Manning, J., and Rubin, S., NASA-HDBK-7005, Dynamic Environmental 

Criteria NASA Technical Handbook, March 13, 2001. 

3. Kendall, M.G. and Stuart, A., The Advanced Theory of Statistics, Vol. 1, “Inference and 

Relationships,” pp. 128–131, Hafner, NY, 1961. 

4. Owen, D.B., Factors for One-Sided Tolerance Limits and for Variables Sampling Plans, Sandia 

Monograph SC-R-607, Sandia Corporation, Albuquerque, NM, March 1963. 

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Paper No. 94

E–15709

WBS 931164.03.01.02.01

17

Application of the Bootstrap Statistical Method in Deriving Vibroacoustic

Specifications

William O. Hughes and Thomas L. Paez

Statistical analysis; Rocket engine noise; Dynamic structural analysis;

Titan 4 launch vehicle

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Subject Categories: 15, 18, 39, 65, and 71

Prepared for the Ninth International Conference on Recent Advances in Structural Dynamics, sponsored by the Institute

of Sound and Vibration Research, Southampton, United Kingdom, July 17–19, 2006. William O. Hughes, e-mail:

William.O.Hughes@nasa.gov, NASA Glenn Research Center; Thomas L. Paez, e-mail: tlpaez@sandia.gov,

Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico, 87185.

This paper discusses the Bootstrap Method for specification of vibroacoustic test specifications. Vibroacoustic test

specifications are necessary to properly accept or qualify a spacecraft and its components for the expected acoustic,

random vibration and shock environments seen on an expendable launch vehicle. Traditionally, NASA and the U.S. Air

Force have employed methods of Normal Tolerance Limits to derive these test levels based upon the amount of data

available, and the probability and confidence levels desired. The Normal Tolerance Limit method contains inherent

assumptions about the distribution of the data. The Bootstrap is a distribution-free statistical subsampling method which

uses the measured data themselves to establish estimates of statistical measures of random sources. This is achieved

through the computation of large numbers of Bootstrap replicates of a data measure of interest and the use of these

replicates to derive test levels consistent with the probability and confidence desired. The comparison of the results of

these two methods is illustrated via an example utilizing actual spacecraft vibroacoustic data.