Physical Aspects of Light in the Sea: A Symposium 9780824884918

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TENTH PACIFIC SCIENCE CONGRESS SERIES

TENTH PACIFIC SCIENCE CONGRESS SERIES Tenth Pacific Science Congress, Honolulu, 1961

AGRICULTURE

Soil Conservation in the Pacific—A Symposium and Panel Discussion J. H. Christ, chairman ANTHROPOLOGY

Ryukyuan Culture and Society—A Survey Allan H. Smith, editor BOTANY

Ancient Pacific Floras—The Pollen Story Lucy M. Cranwell, editor ENTOMOLOGY

Pacific Entomology—Report of the Standing Committee Chairman J. J. H. Szent-Ivany GEOLOGY

Geology and Solid Earth Geophysics of the Pacific Basin— Report of the Standing Committee Gordon A. Macdonald, chairman MARINE BIOLOGY

Physical Aspects of Light in the Sea—A Symposium John E. Tyler, editor MEDICINE

Public Health and Medical Sciences in the Pacific—A Forty-year Review J. Ralph Audy, editor

LIGHT IN THE SEA

T E N T H PACIFIC SCIENCE CONGRESS of the Pacific Science Association

HOST INSTITUTIONS

National Academy of Sciences Bernice Pauahi Bishop Museum University of Hawaii

Honolulu, Hawaii, U.S.A. August 21 to September 6, 1961

UNIVERSITY OF HAWAII,

PHYSICAL ASPECTS OF LIGHT IN THE SEA A Symposium J O H N E. TYLER Convener and Editor

UNIVERSITY OF HAWAII PRESS Honolulu, Hawaii, 1964

Copyright 1964 University of Hawaii Press Library of Congress Catalog Card Number: 63-13254

PUBLISHER'S PREFACE

The papers published in this volume were presented at the Tenth Pacific Science Congress o f the Pacific Science Association held August 21 to September 6, 1961, on the campus o f the University o f Hawaii, Honolulu, Hawaii, U.S.A., scene of the first meeting. The Congress was sponsored jointly by the National Academy of Sciences, Bernice Pauahi Bishop Museum, and the University of Hawaii. The publisher is indebted to the chairman for having assembled these papers from the far corners o f the Pacific. In editing the material, American usage has been followed in the main, though the desire to put this material in print as soon as possible after it was assembled has been responsible for some degree o f stylistic inconsistency. Funds toward the issuance o f Tenth Pacific Science Congress papers published by the University o f Hawaii Press have been furnished by the Legislature o f the State o f Hawaii and the National Institutes of Health. It is believed that a useful purpose is served by bringing together in one volume distinguished papers on a common subject as it applies to conditions that prevail in the various countries of this increasingly important segment of the world scene.

vii

CONTENTS PAGE

Introduction JOHN E . TYLER

3

An Undersea Observation Vessel Kuroshio and Its Photographic Apparatus NAOICHI INOUE, SATOSHI NISHIZAWA, MASAAKI FUKUDA, KATSUJI KOBAYASHI, and RYOSAKU OAKI

7

Measurement at Sea of Water Samples ALEXANDRE IVANOFF

11

On the Instruments for Measuring Angular Distributions of Underwater Daylight Intensity TADAYOSHI SASAKI

19

Degeneration of Image Contrast and Resolution in Underwater Photography ALBERT MAY and PAUL H . CORDS, JR

25

Application of Photography to Observations in the Sea HAROLD E . EDGERTON

31

Turbidity of Water Off Mission Beach TRACY F. BALL and E. C. LAFOND

37

Optical Classification of Ocean Water NILS G. JERLOV

45

A Model for Radiance Distributions in Natural Hydrosols RUDOLPH W . PREISENDORFER

51

A Calculation of the Light Scattering Functions for Small Polyhedric Particles MASAAKI FUKUDA

61

ix

LIGHT IN THE SEA

INTRODUCTION SYMPOSIUM ON THE PHYSICAL ASPECTS OF LIGHT IN THE SEA J O H N E. TYLER University of California, Scripps Institution of Oceanography, Visibility Laboratory, San Diego, California

underwater. Such systems include the human eye, cameras, and the television equipment. Effort has been expended to engineer such systems for peak performance in water of known optical properties or to predict the performance of such systems in known waters.

IF ONE EXAMINES the literature relating to light measurements in the sea one finds three major fields using such measurements. 1. BIOLOGY. Biologists were among the first to become interested in light measurements in the sea and in lake waters. The application of light measurements to marine biology starts with the correlation of these light measurements with the primary productivity of the sea. This is an extremely important aspect of light measurements in the sea since it is from the primary productivity that the total population of the sea must grow. In addition there has been a great deal of interest shown in correlations between light measurements and other types of biological activity such as phototaxis, photokinesis, phototropism, photoperiodism, phosphorescence, and bioluminescence. 2.

DESCRIPTIVE OCEANOGRAPHY.

A great deal

3.

IMAGE-PRODUCING

Interest

The Symposium on the Biological and Physical Aspects of Light in the Sea was organized to bring together scientists in these three fields so that they could exchange information and ideas. The symposium had four major objectives: 1. T o bring together biologists, physicists, and geophysicists who, for diverse reasons, had a common interest in the interaction of light with the ocean. 2. T o promote standardization of terminology with respect to radiometric measurements in the sea and of the methods of measurement themselves. 3. To more fully acquaint the nations of the Pacific area with the techniques and instruments now being used in other parts of the world. 4. To further develop the applications of hydrologie optics to engineering sciences, such as the use of television systems for underwater exploration, and to biological productivity. In order to accomplish these objectives two panel discussions were organized; the first was entitled "Biological Aspects" and the second, "Physical Aspects." The intent of these panels was to bring together the outstanding biologists and physicists who had been wofking on light measurements in the sea and to draw them into informal discussion. Dr. George L.

of interest has been shown by oceanographers in describing the ocean by optical methods. Typical descriptions consist of horizontal or vertical sections giving iso-transmission lines or iso-scattering data and thus presenting a type of contour map or contour section. Contour sections of this kind make it possible to delineate such oceanographic features as the major or minor currents, for example, the North Pacific current or the Gulf Stream, the extent of the outfall from rivers, and the areas classified as coastal, offshore, or deep ocean water. SYSTEMS.

has

developed in the observation or recording of the detail of objects underwater and the performance of various image-producing systems 3

4 Clarke of Harvard University, who has worked in this field for many years and has made many contributions, presided at the Biological Aspects panel. The Physical Aspects panel was under the chairmanship of Dr. S. Q. Duntley of the Scripps Institution of Oceanography, who has been active in the study of scatteringabsorbing media for many years and who has contributed to both hydrologie optics and to atmospheric optics. In addition to the panel discussions there were many papers contributed by other scientists interested in phases of hydrologie optics. Participating scientists and their papers were: BIOLOGICAL ASPECTS PANEL

George L. Clarke, Chairman

U.S.A.

Gunnar Thorson Denmark The Control of Breeding, Spawning, Setting and Distribution of Marine Invertebrates by Light J. A. C. Nicol U.K. Luminescence and Vision in Marine Animals Yata Haneda Japan Observation on Luminescence of the Sea G. F. Humphrey Australia Light Measurements Used in a Study of Biological Production Francis T. Haxo U.S.A. Some Implications of Recent Studies on the Role of Accessory Pigments to Photosynthesis in Submarine Daylight John D. H. Strickland Canada Light and Primary Productivity: Some Requirements and the Attempts Being Made to Fulfill Them at Nanaimo Talbot H. Waterman U.S.A. Submarine Polarized Light and the Spatial Orientation of Animals V. J. Chapman New Zealand An Underwater Growth Chamber for Large Algae CONTRIBUTED PAPERS

J. D. H. Strickland, Chairman

Canada

E. Steemann Nielsen Denmark Light and Plankton Photosynthesis Richard A. Vollenweider Italy The Pattern of Underwater Light and its Relation to Primary Productivity

J. N. Sorokin U.S.S.R. The Submarine Illumination and the Primary Production of Photosynthesis in the Sea O. J. Koblenz-Mishke and M. V. Kozlyaninov U.S.S.R. The Interrelation Between Turbidity, Phytoplankton and Primary Production G.GMcLeod U.S.A. The Role of the Accessory Pigments in Photosynthesis V. J. Chapman, Chairman

New Zealand

Shun-ei Ichimura and Yatsuka Saijo Japan Characteristics of Photosynthesis-Light Curve in Marine Phytoplankton and Their Ecological Bearing on the Primary Production in the Ocean Mitsuru Sakamoto Japan Vertical Variation of the Spectral Composition of Light in Water and its Significance to the Photo-Synthesis of Phytoplankton John D. Costlow, Jr., and C. G. Bookhout U.S.A. The Effect of Photoperiod on Larvae Development of Sesarma Dietrich Schneider Germany Phototropic Growth Reaction in the Marine Bryozoan Elizabeth Kampa, B. P. Boden, and B. C. Abbott U.S.A. Some Aspects of Vision in the Galatheid Crustacean, Pleuroncodes pla»ipes Wheeler J. North U.S.A. Modification of Light by Sea Water and its Influence on Certain Shallow Sublittoral Bottom Communities Lois M. Hutchings U.S.A. The Combined Effect of Ultraviolet Irradiation and Heat Upon First Cleavage of Sea Urchin Eggs PHYSICAL ASPECTS PANEL

S. Q. Duntley, Chairman

U.S.A.

Tadayoshi Sasaki Japan On the Instruments for Measuring Angular Distribution of Submarine Daylight Harold E. Edgerton U.S.A. Application of Photography to Observations in the Sea Rudolph W. Preisendorfer U.S.A. Simple Models for Light Fields in Natural Waters

5 Nils G. Jerlov Sweden Optical Classification of Ocean Water Alexandre Ivanoff France Measurement at Sea of Water Samples CONTRIBUTED PAPERS

N. G. Jerlov, Chairman

Sweden

Masaaki Fukuda Japan On the Calculation of the Light Scattering Function for Small Polyhedric Particles Naoichi Inoue, Satoshi Nishizawa, Masaaki Fukuda, Katsuji Kobayashi, Ryosaku Oaki Japan An Undersea Observation Vessel Kuroshio and its Photographic Apparatus Albert May and Paul H. Cords, Jr. U.S.A. Degeneration of Image Contrast and Resolution in Underwater Photography E.C.LaFond U.S.A. Turbidity Structure of the Sea off Mission Beach, California Charles S. Yentsch U.S.A. Attenuation of Visible Lights by Particulate Matter in the Sea J. C.Murchio and M. B.Allen U.S.A. Measurement of Absorption Spectra of Chlorophyll in Algal Cell Suspensions All but two of the panel papers were published in "Abstracts of Symposium Papers,

Tenth Pacific Science Congress, Honolulu, Hawaii, 1961." Because of the symposium policy of informality many of the papers were prepared to provoke discussion and to stimulate thinking rather than for immediate publication. However, some of the Biological Aspects papers have been submitted to appropriate journals, and the Physical Aspects papers which were closely related have been collected and in some cases rewritten for publication in this Pacific Science Congress monograph. Attendance at the Symposium on the Biological and Physical Aspects of Light in the Sea was almost twice as great as had been anticipated and an important exchange of ideas between the disciplines was accomplished. The success of this symposium was in large measure due to the organizational efforts of Harold J. Coolidge, Secretary General of the Congress; to the administrative efforts of the permanent secretariat, Brenda Bishop, and the many persons who assisted her, and of Lenore Smith, who handled applications for funds for travel; and to the generous financial assistance of the National Science Foundation and the Office of Naval Research in supplying such funds through the National Academy of Sciences.

A N UNDERSEA OBSERVATION VESSEL

Kuroshio

AND ITS PHOTOGRAPHIC APPARATUS NAOICHI INOUE, SATOSHI NISHIZAWA, MASAAKI FUKUDA, Faculty of Fisheries, Hokkaido University, Hokkaido, Japan K A T S U J I KOBAYASHI, Tsurumi Shipyard, Nippon Kdkan K. K., Japan and RYOSAKU OAKI Technical Research Institute of N.S.A., Japan

to the sea floor. The windowpanes are of coneshaped optical glass (BK7), of 160 mm inner diameter and 65 mm in thickness. The visual angle of the window is designed to be 23° to its center axis. The window can be positioned as near as 1 m above the ocean floor by shortening the length of the anchor chain. Four projectors of 1 kw are attached to the lower side of the outer hull: two on the port, and two on the starboard, in front of the window at the middle position. Beams of these four projectors can be brought together into the photographing field by a remote control device in the chamber. An electric heater is placed near the inner side of the window to prevent blurs on the glass caused by the high humidity in the chamber.

THE FORMER Kuroshio undersea observation chamber was constructed in 1951. Because it was suspended from the mother boat by means of a steel cable, the operation necessitated the employment of a large mother boat. Operations were not only expensive but often dangerous when a careful shifting of position on the rocky sea bed was desired. The new Kuroshio was remodelled into a submarine vessel at the Tsurumi Shipyard of Nippon Kokan K. K. in June, I960. It is able to maneuver under water by means of an electric engine attached by a long power line to the mother boat. Thus designed, the Kuroshio's mobility as a submarine laboratory is greatly increased because there is no lack of electric power. Such an attachment of the vessel to the mother boat seems to deprive the vessel of perfect maneuverability. On last summer's expedition, however, we found that this defect could be remedied by using a reel system specially designed for paying out the cable and also by maintaining close communication between the vessel and the mother boat. The three principal windows for observation and photographing are situated at the forward, the middle, and the stern position of the inner hull. Each is directed forward with an inclination 45°, 70°, and 90° respectively

The following figures show in detail the Kuroshio and the mother boat. The Kuroshio has a total weight of 12 T, length of 3.2 m, height of 3.2 m, and inner hull diameter of 1.5 m. The mother boat carried a dynamo with a diesel engine of 32 HP. Safe diving depth was 200 m with a complement of 4 to 6 persons and endurance of 24 hours. Last summer we were able to photograph various sections of the sea floor around Hokkaido through these windows. 7

8

FIG. 1. Schematic diagram of the Kuroshio.

LEGEND FOR FIGURES 1 AND 2 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Windows (160 mm diam.). Windows (120 mm diam.). Windows (60 mm diam.). Projectors (1 Kw x 9, 500 W X 1). Taillight. Hatch. Main tank (1.5 m 3 x 4). Ballast tank (210 1 X 2). Horizontal rudder. Vertical rudder (kitchen type). Propeller (800 mm diam. with 3 blades). Electric cable (power, telephone, and television lines 35 mm diam. x 600 m long with 9 mm diam. steel tension member). 13. Connecting gear of the electric cable. 14. Vent bulb. 15. Air tank (150 kg/cm- 2 ).

16. Guide tube for drilling apparatus (5.5 mm diam. X 150 mm long drilling tube). 17. Turntable. 18. Handle for releasing turntable. 19- Rotating mirror. 20. Anchor chain (60 K g hovering chain). 21. Echo-sounders (horizontal and vertical). 22. Magnetic compass. 23. Current meter. 24. Second switchboard. 25. Echo-sounder recorder. 26. Turning handle. 27. Helm indicator. 28. Angle meter. 29. Indicator of magnetic compass. 30. Sounding lead.

FIG. 3. Schematic representation of the mother boat and the Kuroshio. A. Electric cable. B. Glass buoy. C. Cable reel with motor winch (3T power). D. Controlling panel (power, telephone, etc.). E. Loudspeaker.

F. Control room. G. Echo-sounder. H. Frame protecting the propeller from the drifting cable.

FIG. 4. Creeping fish at a depth of 120 m.

10

FIG. 6. Interior of the Kuroshio (the bows).

FIG. 7. Cable reel and controlling panel.

MEASUREMENT AT SEA OF W A T E R SAMPLES ALEXANDRE IVANOFF Museum National d'Histoire Naturelle, Paris, France

In order to determine oxygen, one of the glass windows has been replaced by a plexiglass window provided with rubber discs which make it possible to inject the sample with reagents used in the standard Winkler method for oxygen. This is done with a hypodermic needle (Fig. 4 ) . The iodine thus liberated is measured colorimetrically. For this, the scattering meter has been changed so that it can also be used as a colorimeter (Fig. 5). The first results with this instrumentation were obtained during the cruise aboard the Calypso in March and April of 1961 in the eastern Mediterranean (Fig. 6 ) . These results are scattering data and fluorescent data. Figure 7 gives the relative scattering coefficient at 90° as a function of depth for five deep stations. The scattering coefficient is largest in the photic zone down to 100 m; then it decreases, and is nearly the same in deep water at all the stations, with some slight increase at 300 or 400 m from the bottom. Figure 8 compares for Station 9 a temperature-salinity diagram with the optical diagram suggested some years ago to characterize water masses. In the optical diagram are plotted the scattering coefficient, /?, versus the factor of depolarization, p; the depth in meters is given for various points. Both diagrams show a strong discontinuity between 100 and 150 m. Between these depths the temperature increased by more than 0.6° C. and the scattering coefficient from 1.7 to 2.4. This is due to the intermediate water flowing from the western Mediterranean, which is warmer and apparently less clear. Figure 9 shows relative fluorescence data obtained at Stations 11 to 15 as a function of depth. These data were obtained by illuminating the sample through an ultraviolet filter

DIFFERENT SCATTERING METERS have b e e n de-

signed in the last few years. The best results are obtained with in situ instruments, but many technical problems arise, especially when working in deep ocean water. Consequently, scattering meters for the measurement of collected water samples have been also developed. Figures 1, 2, and 3 illustrate a special water sampler that has already been described at the Oceanographic Congress at New York in 1959. It is a glass tube held in place by a metallic case with glass windows at the ends. The windows can be closed by means of a messenger (Fig. 2). When the sample is recovered (Fig. 3), the tube is taken into the laboratory and, without being opened, is placed in the scattering meter. The sample is illuminated along the axis of the tube and the scattering at right angles can be measured after removing the center section of the metal case. At the same time, the degree of polarization is measured to get information on the size of the suspended particles. This paper describes two improvements to this equipment. First is the use of a mercury lamp, instead of a tungsten lamp, which makes possible either the fluorescence measurement of the sea water sample itself, or the use of this instrument in studies using a fluorescent tracer material. The second improvement makes possible oxygen analysis of the sample after the optical measurements are completed. Because of the cell construction, the sample is never in contact with the atmosphere, but only with the glass and rubber parts of the cell. These are good conditions for oxygen analysis. Moreover, the presence of a bubble in the sample can easily be seen through the glass windows, and affected samples can be detected. 11

12

FIG. 2. Water sampler, showing messenger.

FIG. 1. Water sampler.

13

FIG. 4. Injection of water sample with reagents.

14

FIG. 5. Scattering meter adapted for use as a colorimeter.

FIG. 6. Diffusion-polarization, oxygen, and fluorescence data during the Calypso cruise in March and April, 1961.

15

I

2

I

2 90"

I

2

RELATIVE

I

SCATTERING

2

1

2

,3

FIG. 7. Relative scattering coefficient as a function o f depth at five stations.

2

5

STA. 9

0.20

38

SALINITY

%.

FIG. 8. Comparison o f temperature-salinity diagram with the optical diagram.

16

1I\

r

/

+\

i i-

STA. II

STA. 14

_1_ 2

1 RELATIVE

2

1

_1_ 2

FLUORESCENCE

FIG. 9. Relative fluorescence data obtained at Stations 11 to 15 as a function of depth.

and by putting a green interference filter in front of the multiplier photo tube. The fluorescence increases with depth down to 100 or 200 m, sometimes with secondary peaks, then remains approximately constant in deep water. Table 1 shows the data obtained for relative scattering, {3, and fluorescence, , at Station 15, at the depths shown. Although the instrument sensitivity used was the same, the recording was more stable in fluorescence measurements than in scattering measurements, which suggests that the fluorescence is due to dissolved substances rather than to suspended matter. The lower part of the table illustrates the increase in fluorescence with depth. Figure 10 is a plot of fluorescence, against 90° scattering, /?, for all stations and all depths. The crosses are for data taken at depths equal to or less than 75 m. The circles are for data taken at depths greater than 75 m. Above 75 m there is a correlation coefficient of 0.81,

but below 75 m there is no correlation. A possible explanation is that the fluorescence is due to some decomposition products of suspended matter, and consequently fluorescence increases with the amount of the suspended matter, i.e., with the scattering coefficient, for those lesser depths where decomposition takes place. At greater depths fluorescence remains constant. TABLE 1 Station 15, 8 / 4 / 6 1 DEPTH (m)

SCATTERING ß

1 5 10 25 50 75 100 125 150

1.9 6 1.98 2.04 1.88 2.51 1.60 1.47 1.67 1.48

FLUORESCENCE 1 1 >0.141 < 0


r

Then, in view of (21) and (22), we have for all 0 < 0 < t, 0
)f(ot + k cos 0).

I f ) =

(V4-TT) h ( z ) ,

(25)

where h (z) is the scalar irradiance at depth z: v 2*

h (z) = J J N (z, 0', 4>') sin 0' dQ' d'.

(26)

0 0 By (10), evidentally,

b{z) = h (0) e~kz

(27)

so that (24) may be written

N (z, 0,) = c e~k'/(a + k cos 0),

(28)

where c = sh (0)/4ir. This indicates that for large depths z in a plane-parallel medium which scatters isotropically (and which has an incident radiance distribution at the upper boundary and no additional sources), the radiance distribution N (z, 0 , ) when plotted has the form of an ellipsoid o f revolution with eccentricity k/a, 0 < k/a < 1. It follows that (28) is actually a formal solution of the equation of transfer under the above conditions and for a suitable k. This may be seen by direct substitution of (28) into the equation of transfer (Glasstone and Edlund, 1952; Chandrasekhar, 1950). THE NONLINEAR FEATURES OF N r . The additive combination of two terms involving different exponential functions is the special structure of (19). This structure implies that: for each path (0, 0 , , r) with fixed 0 > r/2 and , a semilogarithmic plotting of Nr (z, 0 , 4>) with respect to, say r, would generally be nonlinear. It follows that, under suitable conditions, Nr increases, reaches a maximum, and then decreases indefinitely. The value rm at which a maximum is attained may be found from (19) by the usual calculus rules. The predicted value of rm is: rm (0, ) =

zm (9, 0 (i.e., that rm (9, ) exist as a positive real number) is that the argument of the logarithm in (29) be greater than unity. This yields, after some manipulation, the equivalent requirement that N* (0, 9, 4>)/a > No (0, 9, 0), 9 > tt/2, i.e., that

(30)

Nq (0, 9, ) > N0 (0, 9, ).

(31)

This is simply the usual requirement that the equilibrium radiance Nq exceed the actual radiance N at a point in order that N should increase locally along the path. That is, from the equation of transfer in the form (1"): dN/dr > 0 if Nq>

N,

dN/dr < 0 i/Ng < N.

(32)

We observe that for paths with 9 < ir/2, it is clear from (21) that Nr plotted against depth z will have a linear semilogarithmic plot with slope —k. Condition (31) occurs frequently. For example, it occurs with a bright, sunny sky and when N0 (0, 9, ) is associated with a point in the sky not on the sun's disk. On the other hand, if the sky is heavily overcast, condition (31) would not be realized and consequently rm would not exist. The experiments referred to in the introduction have verified the predicted values of rm under sunny sky conditions, and have verified the conclusion that rm may not exist for overcast skies; in such cases (31) does not hold. The predicted linearity of Noo (z, 9, 4>), 9 < ir/2, z > 0 with the slope —k is also essentially verified. CONTRAST AND CONTRAST TRANSMITTANCE : Let (z t , 9, , r) be a path in the hydrosol, and let a target of inherent radiance TN0 (zt, 9, ) be placed at the initial point of the path. Further, designate by BN0 (z t , 9, ) the inherent radiance of the target background. Let TNR (z, 9, ) and i,NT (Z, 9, ) be the apparent radiances of target and background at the observation point of the path. Then the inherent and apparent contrasts of the target with respect to its background are defined, respectively, as:

Co (z t , 9, 4>) = [iNo (zt, 9, 4>) and

Ct

(z, 9, *) = [tNr (z, 9, )

(z, 9, )] / bNr (z, 9, ).

(33) (34)

These definitions are general in the sense that their formulation requires no restrictive assumptions on the nature of the radiance function in the optical medium. The contrast transmittance of the path (zt, 9, , r) is defined as: Cr

(z, 9,

)

and by means of (9), (33), and (34) may be expressed as: Cr

(z, 9,

)

/

Co

(zt, 9,

)

=

Tr

(z, 9,

4>)

6No (zt, 9,

4>)

/

bNr

(z, 9,

).

(36)

Using the present model of the radiance distribution, bN0 and bNr are given by (21) and (22) with

58 the assumption that the presence of the target at the initial point does not perturb the radiance of the background. Thus (36) becomes: i

-ia+k cos 0)r -arN-zt

0

Q(zh

N_ z

aec




0

(37)

^

e (z> 0, )

If the asymptotic radiance distribution conditions (23) hold for both the depths z and zt for each © > r / 2 , then (37) reduces to: . j r = (zt — z) sec 0

c, (z, e , * ) / c , (zt, e, ) = e-

(a+h

ej

7o < e < , < < 2tt.

(38)

DERIVATION OF EQUATION 1 7

The expression for N* (z, 9, ) given in (17) is derived from the solution of the two-flow Schuster equation applied to a hydrosol which is optically infinitely deep. First

x / 2 2T

•N* (z, 0, ) = y y 0 0

v/2 at each depth z > 0.

Define

»/2 2» (e, 4>) = y y* ; e ' , 0 sin 0 ' ¿ 0 ' 0 0 V 27T (0, ) = y y 0- (0, ; 0', ') «» 0 ' dQ' dp. x/2 o

Then

^ N , (z, 0, *) =

(0, 4) N (z, + ) +

(0, 4) N (z, —), z —

^

o < ^ < 2r.

Let H (z,+) and H (z,—) be the irradiances induced by N (z,+) and N (z,—) respectively. Hence H (z,±) = ir N (z,±). Solving the Schuster equations for the pair H (z,+), H (z,—), we have: H (z, ± ) = H (0, ± )

=

59 where k = 2 [ < * ( * + a is the volume absorption coefficient and b is the back scattering coefficient which occurs in the Schuster theory. Therefore: (z, e , = cos-1 (cos j cos y — sin j sin y cos 9).

(5)

Therefore equation (4) is reduced to: J J

A

a h3 = f /

H

°

T

(') • T0"> e. y)cos > C0SJ (*• e> y) cos j di d 9

(6)

where J is a function of i, 9, and y. A numerical calculation at every 10° of i, 9, and y was carried out because of the difficulty of theoretical integration. The result is illustrated in Figure 3. The last light component A Hi is distributed over all directions with the following intensity: A Hi = A Ho T(i) R(j') cos i/cos j,

(7)

*/2

J where

R (/') =

"tt"1

2-ir sin j' cos j' dj' l/n

*/2

J o

=

1 - (l/«).

2ie sin /' cos j' dj'

The mean intensity of light which is distributed over all directions after being reflected at A £2 is b- w/2 * ir/2 J J A Ht sin a da d$/ J J sin a da dp. (8) By summing up these three light fluxes, the scattering function is calculated and is shown in Figure 4. The numerical calculation (Fig. 4) of scattering function and Jerlov's experimental function for the particle components (Jerlov, 1961) are compared. The curves calculated for n = 1.1 and the experimental plots are consistent from 10° to 30°. Some deviation of the calculated curve from the observed occurs between 30° to 80° negative, 80° to 140° positive, and 140° to 180° negative. In this calculation, the assumption yields the deviation from observed scattering. This deviation is smaller in forward and backward directions, and is greater in oblique directions. From these discussions, it may be inferred that the numerical calculation of light scattering function is useful for sea water and it is suggested that the refractive index of particles is between 1.1 and 1.15. ACKNOWLEDGMENT

The author wishes to thank Dr. N. G. Jerlov and Professor N. Inoue for valuable advice.

63 and measured by a light meter. Integrating A Ha with respect to i, j', and 0, the intensity of light which is the directed angle y from the axis of incidence is given by the following equation: / / / i f t = / / /

A Ho T(i) • T(j') cos i cos j'/cos j di dj' dQ.

(4)

Here, j' is a function of the angle which is formed between A Hi and A H3. is a function of the incident angle i, the angle y, and the angle 9 which is made by the standard assumed axis AO and the line passing through the element A £2 and point 0 on the plane perpendicular to the incident light axis. Angle is expressed by the following equation and is illustrated in Figure 2. 4> = cos-1 (cos j cos y — sin j sin y cos 9).

(5)

Therefore equation (4) is reduced to: J J

A

a h3 = f /

H

°

T

(') • T0"> e. y)cos > C0SJ (*• e> y) cos j di d 9

(6)

where J is a function of i, 9, and y. A numerical calculation at every 10° of i, 9, and y was carried out because of the difficulty of theoretical integration. The result is illustrated in Figure 3. The last light component A Hi is distributed over all directions with the following intensity: A Hi = A Ho T(i) R(j') cos i/cos j,

(7)

*/2

J where

R (/') =

"tt"1

2-ir sin j' cos j' dj' l/n

*/2

J o

=

1 - (l/«).

2ie sin /' cos j' dj'

The mean intensity of light which is distributed over all directions after being reflected at A £2 is b- w/2 * ir/2 J J A Ht sin a da d$/ J J sin a da dp. (8) By summing up these three light fluxes, the scattering function is calculated and is shown in Figure 4. The numerical calculation (Fig. 4) of scattering function and Jerlov's experimental function for the particle components (Jerlov, 1961) are compared. The curves calculated for n = 1.1 and the experimental plots are consistent from 10° to 30°. Some deviation of the calculated curve from the observed occurs between 30° to 80° negative, 80° to 140° positive, and 140° to 180° negative. In this calculation, the assumption yields the deviation from observed scattering. This deviation is smaller in forward and backward directions, and is greater in oblique directions. From these discussions, it may be inferred that the numerical calculation of light scattering function is useful for sea water and it is suggested that the refractive index of particles is between 1.1 and 1.15. ACKNOWLEDGMENT

The author wishes to thank Dr. N. G. Jerlov and Professor N. Inoue for valuable advice.

64 REFERENCES BURT, W. V. 1955. Interpretation of spectrophotometer readings on Chesapeake Bay waters. J . Mar. Res. 14(1). 1957. On the attenuation of light in the sea. J . Mar. Biol. Ass. U.K. 36. JERLOV, N. G. 1961. Optical measurements in the Eastern North Atlantic. Medd. Oceanogr. Inst. Goteborg 30. TAKENOUCHI, Y. 1949. On the diffusion-ratio of underwater illumination. Oceanogr. Mag. 1.