Visibility

Visibility C. BECK

A clear, precise definition of visibility is given. It depends on the modulation transfer function of the light transmitting medium (or contrast loss ratio), the object-to-image distance, the spatial frequency, the medium dispersion (not scattering) coefficient and the medium absorption coefficient. The absorption coefficient, a, and the dispersion coefficient, 0, are computed from the experimental modulation transfer function for a region of the ocean. The results obtained confirm the validity of the visibility defining equation.

a, c or E

attenuation

L

(extinction)

coefficient

coefficient

B

distance dispersion coefficient,

Z

object-image

rad’m-’

K,Q

absorption

0,s orb

scattering coefficient

w

spatial frequency,

rad m-l

MTF

Modulation

WZ

spatial frequency,

at distance z rad rad-’

Transfer Function

distance, m

CO

object contrast at distance zero

ZV

visibility or visible range, m

CZ

image contrast at distance z

f

cycles rad-’

C&O

contrast loss ratio at distance z

%

2nf

-

The definitions of visibility (or visual range) are too numerous and varied to discuss and in many cases do not lead to quantitative or meaningful results. The best single source for the presentation of discussions on visibility and the many frustratingly ingenious devices for measuring visibility to accord with a specific definition is ‘Vision Through the Atmosphere’ by W.E. K. Middleton’. Near the end of his volume, the author writes, ‘It is now his considered belief that there is only one way in which meterological observations of this element (visibility) can be rescued from complete futility: abandon the entire scheme of marks and estimates, make good instrumental measurements of the extinction coefficient, and then calculate something which will be of interest to the user of the datum’. Middleton continues in a similarly defeatist and dejected vein, not altogether unjustified by the groping efforts that encumber the major part of the text. Another valuable book on visibility is ‘Marine Optics’ by N.G. Jerlov.’ Most common error in visibility

definition

Nearly all investigators of the subject have attempted and sometimes succeeded, although in an arbitrary manner to The author is an independent Brentwood, Road, Abington, 18 December 1979.

Environmental Consultant at 1855 Pennsylvania 19001, USA. Received

0030-3992/79/040215-03 OPTICS AND LASER TECHNOLOGY.

AUGUST

1979

connect their definition of visibility - or visual range in a medium to the total beam volume attenuation (extinction) coefficient, variously designated 01,c or E. This is made up of the sum of the medium absorption coefficient and the medium scattering coefficient, the latter having been shown to be inadequately defined.’ In view of the last fact, the attenuation (extinction) coefficient also has a chimerical meaning, in spite of its almost slavishly universal empirical application. In the words of M.V. Kozlyanininov, ‘The processes of absorption and scattering of radiation propagating in the sea cannot be quantitatively estimated by means of classical primary optical characteristics ie the indices of absorption x (absorption coefficient, a) scattering u (scattering coefficient, s or b) and attenuation (extinction coefficient, 01or c) and the scattering indices (volume scattering function).4

When is an object visible? An object is visible when light from it reaches the eye unscattered (sharp image) or when the light is scattered through a narrow angle but reaches the eye (blurred image). The two phenomena may, of course, occur simultaneously to different photons. If the radiation from the object is either absorbed before it reaches the eye or if the light is scattered through a wide enough angle, the scattered light never does reach the eye and the object is not visible. $02.00

0

1979 IPC BusinessPress 215

Equation (1) becomes ~0

MTF = e x p ( - a z ) e x p ( - / 3 w 2 z

05

3)

(2)

If the distance z is in metres, then for small angles, multiplying the object spatial frequency co (rad m -1) by the object-to-image distance z metres converts the spatial frequency co to ~z (rad rad -1) at the image.

02 [12

Equation (2) becomes:

0.05

MTF

= e x p ( - az)exp(-/3Wz2Z)

(3)

MTF

= e x p [ - ( a +/3~2z)Z ]

(4)

0.02 005

I

L

0.1

0.2

I

0.15

1.0

I

210

51.0

I0

210

50

Ln(MTF)

Spatial frequency(k cyclesrod-I) Fig. 1

= - ( a +/kO2z)Z

(4a)

Visibility is defined as the distance

Curve showing relative response vs spatial frequency

Zv_ The two important clues here as to whether or not the object light reaches the eye are the extent of light absorption by the medium and the degree of scattering of the unabsorbed light. Too great a change in unabsorbed light direction as a result of scattering will make the object light miss the eye and hence not be visible. The object will also not be visible if the object contrast difference is too small.

Properties that determine visibility The properties that determine visibility may now be listed:

Ln(MTF)

(5)

-(a + t3~4) where Zv = visibility or visible range, m. Equation (5) states that, given an MTF of the intervening medium with absorption coefficient a ( m -]) and dispersion coefficient 13(rad 2 m -1) an object of known angular size (given by spatial frequency determined by object size relative to object-image distance, z) the prevailing visibility will be Z vFrom equation (5), it follows that the absorption coefficient, a, and the dispersion coefficient/3 may be determined from the coordinates of any two points on the MTF versus COz curve when it is known for any medium.

1. absorption of the intervening medium; 2. degree of change in direction of light per scattering event;

Visibility equation confirmation The coefficients a and j3 will be calculated from ocean experimental data shown in the curve .6 The relative response is the MTF.

3. size of the object; 4. object-to-image distance; 5. modulation transfer function (contrast loss ratio) at imaging distance.

Ln(MTF)

These factors will determine whether an object in the atmosphere of ocean is visible. It should be noted that recognition (identification) is not referred to since it is a more complex phenomenon influenced by other factors.

Cz

-

exp(-/3co2z 3)

(6)

The distance Zv is 10.2 metres. Three points were selected from the MTF curve corresponding to spatial frequencies 5 000, 10 000 and 20 000 cycles per radian. Readings from the experimental curve and the corresponding results calculated by substitution in equation (6) are shown in the table below. Substitution of tabulated vaiues result in independent equations in the unknowns a and/3,

(1)

Co where Co = object contrast; Cz = image contrast at distance z; z = object-image distance, m; co = spatial frequency at the object, rad m-l; MTF = modulation transfer function; Cz/Co = contrast loss ratio at distance z. It will be noticed that absorption is not indicated in (1). Since narrow angle scattering was assumed, the scattered photons from the object travel about the same distance z as the unscattered or hardly scattered ones. Thus, the MTF is additionally reduced by absorption exponentially (Bouguer's law).

216

a - 47r2f 2t3

where Wz = 27rf; f = cycles per radian.

The new feature in the definition of visibility is/3, the dispersion (not scattering) coefficient derived and explained elsewhere, s It is shown that the contrast loss ratio and the modulation transfer equation are the same and given by -

_

Zv

Derivation of visibility definition equation

MTF

Equation (5) may be rewritten

A

0.0586

=

B

0.1029

= a + 4 x 109/3

C

0.3040

= a + 16 x 109/3

Zv (m)

a +

f (cyclesrad_l) f2

109/3

MTF

Ln(MTF)

1_ Ln(MTF) Zv

A

10.2

5 x 103

25 x 106 0.550

-0,5978

-0.0586

B

10.2 10.x 103

100 x 10e 0.350

-1,050

-0.1029

C

10.2 20.x103

400 x 10e 0.045

-3,101

-0.3040

OPTICS AND LASER TECHNOLOGY. AUGUST 1979

Discussion o f the misunderstanding of the inadequacy of the scattering coefficient is beyond the scope of this paper, which has the sole purpose of attempting to introduce the proper and most useful definition of visibility in a dispassionate way.

Any pair of equations can be solved for a and/3. From the simultaneous equation solutions of the three paired equations, the resulting a and/3 are tabulated

Equations

a (m-1 )

/3(rad2 m-1 )

A and B

0.042

0.015 x 10-9

B and C

0.037

0.017 x 10-9

A and C

0.042

0.016 x 10-9

1

AVERAGE

0.040

0.016 x 10-9

2

References

3 The three values for b o t h the absorption coefficient a m -1 and the dispersion coefficient/3 r a d : m -1 are close enough to consider the theory propounded confirmed, particularly since the author did not take the original data himself but had to rely on others 6 whose measurements were made in sea water that was fairly constant.

4

5

6

It should be noticed that nowhere has the scattering coefficient nor the futile efforts to measure this coefficient by transmissometer 7 been referred to.

7

Middleton, W.E.K. 'Vision through the Atmosphere' (University o1 Toronto Press, 1968) Jerlov, N.G. 'Marine Optics' (Elsevier Scientific Publishing Co, New York, NY, 1976) Beck, C. 'Inadequacy of the Scattering Coefficient' Photogram Engin and Rein Sens 42 No 10 (Oct 1976) 1261-1264 Kozlyaninov, M.V. 'Osnovnye zavisimost; mezhdu gidrooticheskimi: Karackteristikami. Parametry svetovogo polya v more i gidrooptichesKiye Karaktiki;' Optics of the Ocean and Atmosphere (Optika okeana i atmosfery) 'Nauka' Publishing House Leningrad, 1972) 17 Beck, C. Opt and Laser Tech 9 No 2 (1977) 81-85 Replogle, F.S. Tm No 343-70-76, Naval Underwater Systems Center, New London, CT, USA Williams, J. 'Alpha Meter Design Considerations' ISA Transactions 11 No 2 155

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