C A L C U L A T IO N OF E N E R G Y E Q U IV A L E N T NOISE L E V E L
F R O M T R U N C A T E D S T A T IS T IC A L NOISE L E V E L D IS T R IB U TIO N S
by
R o b e rt R a ck I*
W yle Research
128 M a ry la n d S tre e t
El Segundo, C a lifo r n ia 90245
ABSTRACT
F o rm u la s fo r c a lc u la tin g th e ensem ble energy equ iva ­
le n t noise le vel fo r a p o p u la tio n o f sources are g iven fo r
tru n c a te d u n ifo rm , n o rm a l, and G am m a d is trib u tio n s .
When assessing th e e ffe c tiv e n e s s o f a re g u la tio n lim it in g th e noise em ission
o f a c e rta in p ro d u c t (say, tru c k s , f o r illu s tr a tio n ) , th e fo llo w in g p ro ced ure is o fte n
fo llo w e d .
A s t a t is t ic a l d is trib u tio n o f " c u r r e n t " (i.e ., w ith o u t re g u la tio n ) noise
le vels is o b ta in e d (e.g., F ig u re 1(a)) fo r th e p o p u la tio n o f th a t p ro d u c t. Its ene rg y
e q u iv a le n t ensem ble noise level is c a lc u la te d and used in a noise im p a c t co m p u ­
ta tio n . Then, assum ptions a re m ade on how th e re g u la tio n w ill change th e noise
le v e l d is trib u tio n , and a new e q u iv a le n t noise le v e l is o b ta in e d and fe d in tu rn in to
a noise im p a c t m o d e l. The d iffe re n c e in noise im p a c t m ay be re g a rd e d as th e
e f f e c t o f the re g u la tio n .
N o he L»v*l x , dB
F ig u re I. S ta tis tic a l D is trib u tio n s o f N oise L e v e ls : (a) F u ll, (b) T ru n c a te d
♦ R o b e r t R a c k l, a d o c to ra l g ra du ate o f the U n iv e rs ity o f B r itis h C o lu m b ia , w o rke d
fo r E n v iro n m e n t Canada b e fo re jo in in g W yle L a b o ra to rie s .
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The sim plest assumption about the e ffe c t of a noise level regulation on a
s ta tis tic a l d is trib u tio n o f noise levels fro m a source population is th a t the upper
ta il is tru n ca te d fro m the d is trib u tio n at a known noise level. It is assumed th a t
the param eters o f these d istrib u tio n s have been estim ated and are known q ua nti­
tie s. For the norm al d is trib u tio n , these estim ations are w e ll-know n. For the
Gamma d is trib u tio n , e stim a tio n procedures may be found fo r instance in Bury 1975,
page 311. O nly un ifo rm , norm al and Gamma d istrib u tio n s are tre a te d here since no
o th e r closed fo rm solutions could be found w hich are readily com puted w ith a
program m able pocket c a lc u la to r.
D e noting noise levels by the symbol x w ith units dB, and p ro b a b ility density
by the symbol y, the energy equivalent ensemble noise level L _ (in dB) is defined
'ep
by
I
Le p = 10 log
y(x) z
(I)
dx
where z is a constant equal to 10 ‘ . The area under y(x) must, o f course, equal
u n ity fo r y to be a proper p ro b a b ility density fu n c tio n . The subscript "ep" is chosen
to distinguish this s ta tis tic a l energy equivalent level o f a noise source population,
fro m the energy equivalent level com puted over a tim e in te rva l o f a flu c tu a tin g
noise level signal, usually denoted by l_e .
(a)
U n ifo rm D is trib u tio n
The form ulas fo r u n ifo rm ly d istrib u te d noise levels are as follow s:
x <a
y (x) = fy (x; a , b) =
a < x < b
b< x
Lep
zb - z G
1 0 ,0 9 (b - a) In z
(2 )
When the upper p o rtio n is tru n ca te d above x^, b is sim ply replaced by x^.. Figure 2
shows equation (2) in dimensionless fo rm .
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xt - o
Figure 2=
Dimensionless L
as a Function o f Dimensionless Truncation Value
(U niform D istribution)
(b)
Normal D istribution
The fu ll normal p robability density is
yM
- fN <*; H , a ) - — jL = r exp [ - \ ( ^ )
where H is the mean, and a the standard deviation.
fu nction F ^ is
]
(3)
The cum ulative distribution
x
FN ^x;
=/
mm
-
fN (t; H , o ) d t
60
19
-
W
If the upper ta il o f f ^ is truncated at x^. (Figure 1(b)), y(x) must be corre cted
such th a t the area under y remains equal to u n ity :
y < *>
*
f N
t <x ;
N' *
11 <
^
x . )
’
-
—
7 _______
fo r - 00 < x < Xj. ,
1----------------------------------
FN (xt; » i , a )
L 2V
a
/
]
(5)
J
zero elsewhere.
S u b stitu tin g equation (5) in to equation ( I) yields the fo llo w in g result a fte r some
algebra:
. 2 inz.,.,
V
"
°
"2”
| FN (V “ + ^ inz' ff)
9 { --------FN (xt;
(«
o ) -------
2
The second te rm is the fa m ilia r 0.1 15 a te rm in the expression fo r the
energy equivalent level fo r a norm al d is trib u tio n o f sound levels. The th ird te rm
m ay be called a "tru n c a tio n c o rre c tio n " w hich reduces to zero fo r x — ~°°as should
be expected.
The cu m u la tive norm al d is trib u tio n
is tabulate d in any textbook on
s ta tis tic s . Simple pocket c a lc u la to r programs also exist to evaluate F ^ .
Figure 3 shows equation (6) in dimensionless fo rm . This graph can be used to
evaluate equation (6), in te rp o la tin g between curves when necessary.
(c)
Gamma D is trib u tio n
The fu ll, th re e -p a ra m e te r Gamma p ro b a b ility density fu n c tio n is (Figure 4):
X- 1
(7)
fo r x > (3 ,
zero elsewhere,
where 8 is the location param eter, a the scale param eter, and X the shape
param eter. T (X ) is the com ple te Gamma fu n ctio n defined by
2
(N o rm a l D is tr ib u tio n ) . T ic k M a rks O utsid e R ig h t M a rg in In d ic a te
A s y m p to te s .
N o S w U v * l x , dB
F ig u re 4. G am m a D is trib u tio n
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Note th a t the a rith m e tic mean o f the thre e -p a ra m e te r Gamma d istrib u tio n
is P + o t \ , and the variance is a 2
The cu m u la tive d is trib u tio n fu n ctio n F q is
(8)
P
w hich can be w ritte n
’ •Iit ^? )
FG (x;P, a , X ) = where y
O)
is the "in c o m p le te " Gamma fu n ctio n defined by
A
r x
yxW = y
tX” 1 e "f d t
(10)
0
This leads to the Gamma p ro b a b ility density w ith tru n ca te d upper ta il a t x^:
y<x) -
fG _ t (x;
or, X , x f ) = ------- --
.x p (-
fo r P < x < x^ ,
(ID
zero elsewhere.
Inserting equation ( I I ) in to equation ( I ) yields the fo llo w in g result a fte r some
algebra:
Le p = P " 10X log (1 - a X n z > + 10 log<
7 — —------ ~ >
(12)
The th ird te rm is again the tru n c a tio n c o rre c tio n te rm as s im ila rly observed
w ith the norm al d is trib u tio n (equation (6)) reducing to zero fo r x^——00. W ith one
e xception, L g should always exist fo r fin ite x^.
The exception is th a t the
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c a l c u l a t i o n m e t h o d b r e a k s down when v a lu e s of t h e s c a l e p a r a m e t e r <*> 4.343 (1/ £ n z ), w hich is n o t c o n s id e r e d a s e ri o u s r e s t r i c t i o n . V alues o f a will usu a lly
r a n g e b e t w e e n 0.5 and 2.5 w h e n o p e r a t i n g on e n v i r o n m e n t a l noise d a t a .
A p p e n d ix A gives a n a lg o r i th m fo r c a l c u l a t i n g t h e
f u n c t i o n on a p r o g r a m m a b l e p o c k e t c a l c u l a t o r or a c o m p u t e r .
incom plete
Gam ma
F i g u r e 5 show s c u r v e s r e p r e s e n t i n g e q u a ti o n (12) in d im e n s io n le s s f o r m using
t h e G a m m a s t a n d a r d i z e d v a r i a b le
O ne m a y n o t e t h e b a s ic s im i la r it y of F ig u r e 5 w i t h F i g u r e 3. H o w e v e r , s in c e t h e
G a m m a d i s t r ib u t io n ha s a f i n i t e low er t a i l , t h e c u r v e s s t a y f i n i t e on t h e l e f t side .
F i g u r e 5. D im e n s io n le s s (R e d u c ed ) L ep a s a F u n c t io n o f D im e n s io n le s s ( R e d u c e d )
T r u n c a t i o n V alue (G a m m a D i s tr ib u tio n ). A, B, C a r e E n d p o in ts of
C u r v e s w ith S a m e Shape P a r a m e t e r X (A: X = 2, B: X = 5, C: X =10.)
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The effectiveness of a regulation is best assessed by estimating its e ffe c t on
a population of noise sources as a function of tim e. This has been examined in
Plotkin 1974, Plotkin and Sharp 1974, and Plotkin 1977. These references either
assume normal models, or they use measured noise level distributions without
fitting them with a m athem atical model which is a more accurate and also more
time-consuming technique than the one discussed in this paper. The formulations
developed in this paper are used in Wyle Research 1979 (see references), a
document developed for use by local governments, whereas the previously
mentioned references stress the use by a federal government.
R eferences
Bury, K. V., Statistical Models in Applied Science, John Wiley and Sons, 1975.
Plotkin, K., "A Model for the Prediction of Highway Noise and Assessment of
Strategies for Its A batem ent Through Vehicle Noise Control." Wyle Research
R eport WR 74-5, for U. S. Environmental Protection Agency, September 1974.
Plotkin, K. and Sharp, B., "Assessment of Highway Vehicle Noise Control S trate ­
gies," Inter-Noise 74 Proceedings, p. 497, 1974.
Plotkin, K., "National Exposure to Highway Noise and Effectiveness of Noise
A batem ent Strategies to th e Year 2000," Wyle Research R eport WR 77-13 for U. S.
Environmental Protection Agency, October 1977.
Wyle R esearch, "Community Noise Assessment Manual, Strategy Guidelines,"
Report WR 78-1 for U. S. Environmental Protection Agency, August 1979.
APPENDIX A
ALGORITHM FOR INCOMPLETE GAMMA FUNCTION EVALUATION
Definition of incomplete Gamma function:
x
(Al)
0
where X> 0, x > 0. This is also equal to:
ao
n
x
A(X + 1) . . . (X + n)
n= 0
Algorithm for evaluating equation (A2):
R | , . . . . Ré are storage registers.
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(A2)
NTRY»
Store x în R|
Store X in
Store Allowable Relative Error
€inR6
Calculate x^
Divide This By Rj
Store Result in Rg and R^
Examples:
y }{2)
=0.86466
7 ^ 0 .1 ) = 0.09516
y2(10) =0.99950
y 10(5) = 11549.8
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