# Analysis on Effective Intensity of Flashing Lights 

# And Modification of Allard Method

Yoshi Ohno

National Institute of Standards and Technology, USA

100 Bureau Drive, Stop 8442, Gaithersburg, MD 20899-8442

Email: ohno@nist.gov, Tel: 301-975-2321

Dennis Couzin

Avery Dennison

6565 Howard, Niles, IL 60714

Email: dcouzin@stimsonite.com, Tel: 847-588-7261

Type of proposal; Research paper

(Intend to expand my proposal for a full paper publication)

**ABSTRACT**

For the measurement of effective intensity of flashing lights, the
Blondel-Rey, Form Factor, and Allard methods are adopted in various
applications. These methods produce significantly different results
depending on the waveform of pulses, and a need for international
standardization has been addressed. These three methods have been
studied in detail by computation using ten different waveforms of pulses
including train of pulses and a modulated pulse at varied duration from
0.001 s to 100 s. The results showed some failures of the Blondel-Rey
and Form Factor methods for a train of pulses. Blondel-Rey method also
showed a failure for modulated pulse, and Form Factor method showed a
failure for a slow pulse with a sharp noise added. Allard method showed
reasonable results for all forms of pulses except that it showed results
20 % to 30 % higher than the Blondel-Rey results for most of the pulses
at 0.1 to 1 s duration, while the Blondel-Rey equation is believed to be
accurate for rectangular pulses. This problem has been solved by
modifying the Allard method. Allard method is based on convolution of
the flash pulse with a visual impulse response function *q*(*t*), which
is an exponential decay function. This function has been modified so
that the calculation results for rectangular pulses would match the
results by the Blondel-Rey equation. A function for *q*(*t*) has been
found that perfectly matches the Blondel-Rey results, and an
approximation of the function has been achieved by two exponential
functions that can be realized by simple resistor-capacitor filter
circuits. This modified method is practically equivalent to the
Blondel-Rey equation for rectangular pulses, yet, solving all the
problems identified with the other two methods for non-rectangular
pulses. It is considered that this modified Allard method can further
improve the accuracy in measuring effective intensity of various
waveforms of flashing lights from what has been achieved by Blondel-Rey
and Form Factor methods. In addition, this method has a practical
advantage in producing photometers with simple analog circuitry.

**1. INTRODUCTION**

Flashing lights are widely used in many signaling applications in
aviation, marine navigation, and land transportation. The visibility or
conspicuity of flashing lights varies depending on the duration and
waveform of flashes for the same physical energy and spectrum of the
flashes. To take into account such visual effects, the term, *effective
intensity,* is used to specify the intensity of flashing lights for
signaling applications. Effective intensity is the luminous intensity
\[cd\] of a steady light source that has the same visual range as the
flashing lights in question.

The three well-known methods to define effective intensity are Allard
method (1876) \[1\], Blondel-Rey equation (1911) \[2\], and Form Factor
method (1971) by Schmidt-Clausen \[3\]. Blondel-Rey equation or its
extended form, Blondel-Rey-Douglas equation \[4\], has been most widely
accepted in many application areas \[5,6,7\]. The Form Factor method is
recently well recognized and adopted in some application areas \[8\].
Allard method did not prevail probably due to difficulty of calculation
in the old days, but is recommended for a train of pulses \[9\]. These
three methods give significantly different results depending on the
waveform of pulses. International Association of Lighthouse Authorities
(IALA) compared these three methods by calculation and published a
recommendation in 1977 \[9\], which put all the three methods on the
equal footing except for trains of pulses, and left users to select a
method. None of these methods has been adopted as standards by
Commission Internationale de l\'Éclairage (CIE).

To address such needs for standardization of effective intensity of
flashing lights, a task is undertaken in Commission Internationale de
l\'Éclairage (CIE) \[10\] to develop an international recommendation on
photometry of flashing lights, and in American Society for Testing and
Materials (ASTM) \[11\] to recommend a definition of the effective
intensity for traffic applications. In these committees, Couzin reported
a failure of the Form Factor method for a certain form of pulses, and,
to solve the problem, proposed a method based on convolution of the
flash pulse with a certain visual impulse response function \[12\],
which turned out to be a similar method as Allard method. Couzin also
showed that the convolution in Allard method could be achieved by a
simple resistor-capacitor filter circuit.

To examine the differences between the three methods mentioned above,
identify problems, and to provide directions to the committees, analyses
were made by computation on ten different theoretical waveforms of
pulses including a modulated pulse and trains of pulses. The results
have identified some failures of the Blondel-Rey and Form Factor methods
for some forms of pulses including a train of narrow pulses. Allard
method showed reasonable results for all forms of pulses except that it
showed significant discrepancy from the Blondel-Rey results for all
pulses at 0.1 s to 1 s duration. The authors developed a solution for
this problem by modifying the Allard method. This modified method, which
can still be realized by resistor-capacitor circuits, is practically
equivalent to the Blondel-Rey equation for rectangular pulses, yet,
solving those problems identified with the other two methods for a
variety pulses. The results of the analysis and the modified Allard
method are presented.

**2. DEFINITIONS IN THE CONVENTIONAL THREE METHODS**

In the Blondel-Rey equation, the effective intensity *I*~e~ of flashing
lights having instantaneous luminous intensity *I*(*t*) is obtained by

$I_{e} = \frac{\int_{}^{}{I(t)dt}}{a + (t_{2} - t_{1})}$ (1)

where (*t*~2~-*t*~1~) is the duration of the flash, and *a* is a time
constant of 0.2 s, known as the Blondel-Rey constant. For
non-rectangular pulses, *t*~1~ and *t*~2~ are determined in such a way
that *I*~e~=*I*(*t*~1~)=*I*(*t*~2~). This can be solved
as![](media/image1.pct){width="1.6104166666666666in"
height="1.2416666666666667in"}

$\int_{}^{}{(I(t) - I_{e})dt = a \cdot I_{e}}$. (2)

For a train of pulses, the following Blondel-Rey-Douglas equation \[4\]
is used. The effective intensity of a train of pulses as shown in Fig. 1
is given by

$I_{e} = \frac{\int_{}^{}{I(t)dt + \int_{}^{}{I(t)dt}}}{a + (t_{2} - t_{1})}\text{ ,    where   }I_{e} = I(t_{1}) = I(t_{2})$
(3)

In the Form Factor method, the effective intensity *I*~e~ for a flash
pulse *I*(*t*) is given by

$I_{e} = \frac{I_{\text{max}}}{1 + \frac{a}{F \cdot T}}\text{; }\text{    }F = \frac{\int_{}^{}{I(t)dt}}{I_{\text{max}} \cdot T}$
(4)

where *I*~max~ is the maximum of the instantaneous effective intensity.

In Allard method, the instantaneous effective intensity *i*(*t*) for a
flash pulse *I*(*t*) is given by
$\frac{di(t)}{dt} = \frac{I(t) - i(t)}{a}\text{;   }a = 0\text{.}2s$ (5)

This differential equation is solved as a mathematical convolution of
*I*(*t*) with a visual impulse function *q*(*t*) as given by

$i(t) = I(t) \ast q(t)\text{;  }\text{   }q(t) = \frac{1}{a}\text{ e}^{- \frac{t}{a}}$
(6)

The effective intensity *I*~e~ is given as the maximum value of
*i*(*t*). This convolution is achieved electronically by an R-C filter
circuit.

**3. ANALYSIS ON THE CONVENTIONAL THREE METHODS**

To evaluate the differences between the three methods, analyses were
made by calculating the effective intensity of 10 different waveforms of
pulses shown in Fig. 5 with varied duration. Pulse (2) is a 1:2
trapezoid, (6) is a sine-squared function, (7) is 20 cycles of sine-wave
oscillation with amplitude from 1/7 to 1 (peak to peak) modulated by the
sine-squared function, simulating a discharge lamp emission, (8) is a
sine-square function with a peak height of 0.3, on top of which a narrow
peak (1/500 width) with a height of 0.7 is added. (9) is a train of four
short pulses of real xenon flash with a half-maximum width of \~1 ms at
interval of 0.1 s. (10) is a train of four sine-squared pulses. The
duration of these pulses (zero second to the time of the last non-zero
value) are changed at 13 points from 0.001 s to 100 s.
![](media/image2.pct){width="3.1791666666666667in"
height="1.8590277777777777in"}

The results for the rectangular pulse are presented in Fig. 3 (a) and
(b). The two graphs are different presentations of the same results. The
upper figure (a) is the plots of the effective intensity *I*~e~ for the
pulses with constant height (*I*~max~=1) with varied duration. As the
duration increases and exceeds 1 s, *I*~e~ saturates to 1, where *I*~e~
equals the luminous intensity of steady light. The lower figure (b) is
the plots of the effective intensity normalized for constant energy or
integral *J* of the pulse, where
![](media/image3.pct){width="3.165277777777778in"
height="4.0159722222222225in"}

$J = \int_{}^{}{I(t)dt} = 1$. (7)

When the pulse duration is much shorter than the time constant (0.2 s),
the effective intensity *I*~e~ is 5 times (reciprocal of the time
constant) of *J*. As the duration becomes longer, *I*~e~ decreases and
diminishes at longer duration. The plots of *I*~e~(*I*~max~=1) present
characteristics at longer duration well and *I*~e~(*J*=1) does well for
shorter duration. Both figures show four curves for the results by the
different methods. Modified Allard results are discussed in next
section.![](media/image4.pct){width="3.265277777777778in"
height="2.115972222222222in"}

Fig. 3 (a) and (b) show that Allard results are 20 % to 30 % higher than
the Blondel-Rey-Douglas results at duration of 0.1 s to 1 s. This
tendency is observed for all other pulses, and is confirmed as the
problem of the Allard method, since the Blondel-Rey equation was proved
to be accurate for rectangular pulses \[13\].

Fig. 4 shows the results (*I*~e~ for *I*~max~=1) for the modulated
pulse. It shows significantly large differences between Form Factor and
Blondel-Ray-Douglas (by a factor of 2 at 0.5 s duration). Considering
the oscillation period of the pulse to be 5 s at duration 100s, *I*~e~
should converge to nearly 1. Blondel-Rey-Douglas is failing showing
*I*~e~ values too low. The results for No. 10 pulse (train of four
sine-squared pulses) showed the same failure of Blondel-Rey-Douglas.

Fig. 5 shows the results (*I*~e~ for *J*=1) for a train of short pulses.
Form Factor shows dramatically higher values than the others at longer
durations, starting to fall off only at \~1 s, while the other methods
show a falloff starting at 0.01 s. At duration of 0.3 s, where the
interval of pulses is 0.1 s (found in some aircraft anticollision
lights), Form Factor shows a value twice as high as Blondel-Rey-Douglas.
At 3 seconds of the total pulse duration, for example, the interval
between narrow pulses is 1 second, when the visual response is
considered to be determined by each single pulse rather than four
pulses. The *I*~e~ (for *I*~max~=1) for the same single pulse is
calculated and compared with these results. The ratio (train of 4 pulses
/ single pulse) was 1.01 for Allard, 3.3 for Form Factor, and 0.30 for
Blondel-Rey-Douglas. Since the ratio should be nearly 1 based on
knowledge of visual response, both Form Factor and Blondel-Rey-Douglas
are shown to fail, in spite of the report in 1982 that examined the Form
Factor method for a train of pulses and concluded its validity \[14\].
The failure of the From Factor method for a train of narrow pulses is
explained from its equation, which ignores the interval between the
pulses, and its results only depend on the duration of each short pulse.
![](media/image5.pct){width="3.217361111111111in"
height="2.109722222222222in"}

The results for No. 8 pulse (a narrow pulse added on a sine-squared
pulse) also clearly showed the failure of Form Factor method, which
produces *I*~e~ that is higher by factor of two than Blondel-Rey-Douglas
or Allard at duration of 0.5 to 100 s.

For all triangular pulses (No. 3, 4, 5 pulses), the differences between
Form Factor and Blondel-Ray-Douglas are fairly large, with Form Factor
showing about 30 % higher *I*~e~ in the 0.2 to 2 s duration. Allard
results vary depending on the form of triangle. At the moment, no past
data is available to indicate which result is the most accurate.
However, the three different triangular pulses are measured with no
difference by Form Factor or Blondel-Ray-Douglas. Allard gives different
results for different triangles. Intuitively, the visual response for
the rising triangle and falling triangle is considered different, which
leaves a hope that Allard results may match visual response.

**4. MODIFICATION OF ALLARD METHOD**

The results indicated that Allard method has no problems with any forms
of pulses at any duration, except that it gives too high values in the
0.1s to 1 s duration. To solve this problem, attempts were made to
modify the visual impulse response function to match the Allard's
results to Blondel-Rey for rectangular pulses. Fig. 3 indicated that
there is a much longer time constant component in the Blondel-Rey
results. The Allard's visual impulse response function *q*(*t*) has been
modified to be composed of two exponential functions with different time
constants. The two time constants have been optimized in such a way that
the difference of the response function to rectangular pulses from that
of Blondel-Rey equation was minimized, while keeping the total time
constant to be 0.2 s and the *I*~e~ for steady light to be equal to
*I*~max~. These requirements are met when the maximum value of *q*(*t*)
is 1/*a* (*a*=0.2 s) and the integral of *q*(*t*) is 1. The modified
impulse response function is given by

$q(t) = \frac{w_{1}}{a_{1}}e^{- \frac{t}{a_{1}}} + \frac{w_{2}}{a_{2}}e^{- \frac{t}{a_{2}}}$,
where
$\frac{w_{1}}{a_{1}} + \frac{w_{2}}{a_{2}} = \frac{1}{a}\text{and}w_{1} + w_{2} = 1\text{;  }a = 0\text{.}2s$.
(8)

and $q(t) = 0\text{ when }t\text{<0}$

One of the solutions for the optimization was *a*~1~=0.113,
*a*~2~=0.869, *w*~1~=0.5, *w*~2~=0.5. With these parameters, the results
agreed to Blondel-Rey for rectangular pulses to within 5 %. These
results are shown as the curves labeled "Modified Allard" in Figs. 3 --
5. Fig. 3 shows that Modified Allard's results are practically
equivalent to Blondel-Rey for rectangular pulses, yet showing no
problems in Figs. 4 and 5. This modified Allard method can be easily
realized as a photometer by two R-C filter circuits and a peak-hold
circuit.

Further theoretical analysis discovered a *q*(*t*) function that
perfectly matches the results of Blondel-Rey for rectangular pulses, as

$q(t) = \frac{a}{(}a + t)^{2}$ ; $q(t) = 0\text{ when }t\text{<0}$ (9)

No analog circuits have been found to achieve the convolution with this
function perfectly, however, this function may be useful to define the
method, or to fit the method to experimental data in different
conditions by changing only one time constant.

## 5. CONCLUSIONS

The analysis revealed some failures of Blondel-Rey-Douglas and Form
Factor methods for certain forms of pulses including a train of short
pulses. The modified Allard method proposed by the authors gives nearly
equivalent results to Blondel-Rey for single rectangular pulses, yet
solving all the problems identified with the other two methods. In
addition, the modified Allard method has a practical advantage in
instrument realization using simple analog circuitry. Although the
modified Allard method shows reasonable results for various forms of
pulses, the accuracy of the method to visual response should be
experimentally proved.

**REFERENCES**

1\. Allard E., "Mémoire sur l'intensité et la portée des phares", 62-73,
Imprimerie Nationale, Paris (1876)

2\. Blondel, A. and Rey, J., "Sur la perception des lumiéres bréves à la
limite de leur portée", *Journal de Physique*, juillet et aout, 643
(1911)

3\. Schmidt-Clausen, H. J., "The influence of the angular size,
adaptation luminance, pulse shape and light colur on the Blondel-Rey
constant *a*", *The perception and application of flashing lights*,
95-111, Adam Hilger, London (1971)

4\. Douglas, C. A., "Computation of the effective intensity of flashing
lights", *Illuminating Engineering*, Vol. 52, No. 12, 641-646 (1957)

5\. IES Guide for Calculating the Effective Intensity of Flashing Signal
Lights, *Illuminating Engineering*, November 1964, pp. 747-753 (1964)

6\. ITE, Purcase Specification for the Flashing and Steady Burn Warning
Lights, ITE Publication, Equipment and Material Standards of ITE,
ST-017, revision 2000

7\. SAE, Aerospace Recommended Practice 5029, Measurement Procedures for
Strobe Anticollision Lights (1998)

8\. ECE Regulation No. 65, Addendum 64, Uniform Provisions Concerning
the Approval of Special Worning Lights for Motor Vehicles (1986)

9\. IALA, Recommendations on the determination of the luminous inensity
of a marine aid-to-navigation light, December 1977

10\. CIE Technical Committee 2-49 Photometry of Flashing Lights, Y.
Ohno, chair.

11\. ASTM E12 Color and Appearance, Subcommittee 11 Visual Methods, WG
05 Flashing Lights, Y. Ohno, chair.

12\. Document "Re: Allard's method", presented at the ASTM E12.11.05
meeting in Rochester, June 19, 2001, published at
<http://cie2.nist.gov/ASTM_E12_WG05/home.html>

13\. Schmidt-Clausen, H. J., Concerning the perception of various light
flashes with varying surrounding luminances, Darmstädter Dissertation
D17, Darmstadt University of Technology, 1968.

14\. Schmidt-Clausen, H. J., A comparison of different methods for the
determination of the effective luminous intensity of singal lights in
the form of Multiple pulses, *CIE-Journal*, 1-1 (1982)