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Estimation of suspended sediment concentrations in natural water bodies from Secchi disk measurements. Lewis, Charles Peter 1970

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ESTIMATION OF SUSPENDED SEDIMENT CONCENTRATIONS IN NATURAL WATER BODIES FROM SECCHI DISK MEASUREMENTS by C. PETER LEWIS B.A., U n i v e r s i t y of Waterloo, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS In the Department of Geography We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1970 In presenting t h i s thesis in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it f r e e l y available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for f i n a n c i a l gain shall not be allowed without my written permission. Depa rtment The University of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The Secchi disk i s one of the simplest and l e a s t expensive instruments f o r measuring the rate of attenuation of l i g h t i n natural water bodies. Because t h i s rate of attenuation i s often c l o s e l y r e l a t e d to the concentration of suspended sediment i n the water, disk measurements may, i n some s i t u a t i o n s , provide a u s e f u l and inexpensive a l t e r n a t i v e to the standard gravimetric methods f o r determining sediment concentration. Published empirical .studies i n d i c a t e that s a t i s f a c t o r y r e l a t i o n -ships between Secchi disk transparency and concentration can be estab-l i s h e d but that these r e l a t i o n s h i p s d i f f e r from area to area and become more complex as t u r b i d i t y increases. Results presented i n t h i s paper based on data from the Mackenzie d e l t a , N.W.T. support these conclusions. These r e s u l t s cover a range of t u r b i d i t i e s which extends w e l l above those previously examined. Phy s i c a l i n t e r p r e t a t i o n of these empirical r e s u l t s i s made d i f f i c u l t by the large number of uncontrolled variables associated with Secchi disk readings. This i n t e r p r e t a t i o n i s f a c i l i t a t e d by the i n t r o -duction of objective attenuation c o e f f i c i e n t s (more variables controlled) as v a r i a b l e s intermediate between transparency and concentration. The two c o e f f i c i e n t s most commonly used are the volume attenuation c o e f f i c i e n t oc and the d i f f u s e attenuation c o e f f i c i e n t f o r downwelling i r r a d i a n c e K^. Empirical r e l a t i o n s h i p s between e i t h e r of these objective i i . i i i attenuation c o e f f i c i e n t s and disk transparency are r e c i p r o c a l i n form but depart from the simple r e c i p r o c a l model at high sediment concen-t r a t i o n s . T y l e r (1968) has shox«i t h e o r e t i c a l l y that transparency i s dependent upon both <* and K^. Since the r a t i o <*/K<j increases with t u r b i d i t y , the observed departures from the simple r e c i p r o c a l model can be explained. Burt (195^» 1955) has developed a t h e o r e t i c a l expression r e l a t i n g c< and sediment concentration i n which he shows that oc and thus Secchi disk transparency are dependent upon t o t a l p a r t i c l e surface - i e . , on p a r t i c l e s i z e as well as concentration. This dependence on p a r t i c l e s i z e provides an explanation f o r the in c r e a s i n g complexity of empirical transparency-concentration r e l a t i o n s h i p s as the t u r b i d i t y range considered widens and f o r the change i n r e l a t i o n s h i p s with l o c a t i o n . T y l e r ' s and Burt's expressions have been combined i n t h i s paper to enable estimations of p a r t i c l e s i z e to be made given only an empirical transparency-concentration r e l a t i o n s h i p and knowledge of the r a t i o c</Kd. The r e s u l t i n g equation has been applied to the Mackenzie de l t a data and, considering the large number of untested assumptions made, the p a r t i c l e size estimations are remarkable close to those obtained using Postma's (1961) purely empirical r e l a t i o n s h i p . Secchi disk measurements, then, can provide a s u r p r i s i n g amount of information about sediment concentration and p a r t i c l e s i z e . There are, of course, many l i m i t a t i o n s to the method and the theory associated with i t i s l a r g e l y untested. Within the context of these l i m i t a t i o n s , however, the disk can be very u s e f u l . I t i s simple to use i n the f i e l d , i s remarkable i n s e n s i t i v e to changes i n the underwater l i g h t f i e l d and . i v . not greatly influenced by observer bias \ TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES v i i LIST OF FIGURES v i i i ACKNOWLEDGMENTS i x SYMBOLS AND ABBREVIATIONS x Chapter I . INTRODUCTION 1 I I . SECCHI DISK TRANSPARENCY AND THE ATTENUATION OF LIGHT IN WATER 3 A. COMPONENTS OF THE PROBLEM 3 B. ATTENUATION COEFFICIENTS if 1. The Volume Attenuation C o e f f i c i e n t 2. The Diffuse Attenuation C o e f f i c i e n t I I I . EMPIRICAL RELATIONSHIPS 10 A. PUBLISHED INVESTIGATIONS 10 1. Relationships between Secchi Disk Trans-parency and Attenuation C o e f f i c i e n t s 2. Relationships between Attenuation C o e f f i -c i e n t s and Concentration of Suspended Sediment 3. D i r e c t Relationships between Secchi Disk Transparency and Concentration of Sus-pended Sediment if. Related Studies B. THE MACKENZIE DELTA, N.W.T 29 1. The F i e l d Area 2. Measurement Techniques 3. The Functional Relationship if. Observer Bias 5. C o l l o i d a l and Dissolved Load 6. Conclusions IV. IN THE CONTEXT OF UNDERWATER OPTICAL THEORY if? . v . v i . Chapter Page A. SECCHI DISK TRANSPARENCY AND ATTENUATION hi B. ATTENUATION AND CONCENTRATION OF SUSPENDED LOAD . . 51 C. ESTIMATION OF PARTICLE SIZE 53 V. SUMMARY AND CONCLUSIONS 57 LITERATURE CITED 60 LIST OF TABLES Table Page I . E m pirical Relationships between Secchi Disk Transparency and the Diffuse Attenuation C o e f f i c i e n t K<j 11 I I . Empirical Relationships between Secchi Disk Transparency and the Volume Attenuation C o e f f i c i e n t o< . . . . . . . . 17 I I I . Relationships between Secchi Disk Transparency and Concen-t r a t i o n of Suspended Sediment i n the Lake-Channel System . j6 IV. Test f o r Coincidence of Functional Relationships f o r D i f f e r e n t Observers V. P a r t i c l e Size Estimation 55 . v i i LIST OF FIGURES Figure Page 1. E m p i r i c a l Relationships between Secchi Disk Transparency and the Diffuse Attenuation C o e f f i c i e n t 1 2 2 . Empirical Relationships between Secchi Disk Transparency and the Volume Attenuation C o e f f i c i e n t c< 18 3 . Relationships between the Volume Attenuation C o e f f i c i e n t and Concentration of Suspended Sediment 21 k. The Influence of P a r t i c l e Size on the Secchi Disk -Sediment Concentration Relationship . . . 2 6 5 . The Lake-Channel System 31 6 . Models of the Relationship between Secchi Disk Trans-parency and Concentration of Suspended Sediment i n the Lake-Channel System 3 7 7 . Residuals of Expected Values - The Reciprocal Model . . . J& 8. Residuals of Expected Values - The Power Function Model . 3 9 9 . Residuals of Expected Values - The Complex Model 40 . v i i i . ACKNOWLEDGMENTS Data c o l l e c t i o n f o r t h i s study was c a r r i e d out during the summers of 1968 and 1969. The work formed part of a l a r g e r project which was supported by the Quaternary Research and Geomorphology D i v i s i o n of the Geological Survey of Canada, the Inuvik Research Laboratory of the Department of Indian A f f a i r s and Northern Development, the National Research Council of Canada, the Department of Indian A f f a i r s and Northern Development (through the A r c t i c and Alpine Committee, U n i v e r s i t y of B r i t i s h Columbia), Research Funds (U.B.C.), and the Department of Geography of U.B.C. For t h e i r assistance i n the f i e l d , the author would l i k e to thank Mr. Alan Graves, Mr. Jerr y A l l e n and Mr. Peter Hope. The sediment concentration analyses were c a r r i e d out by Mr. John O s t r i c k of the Inuvik Research Laboratory and the author extends p a r t i c u l a r thanks to him. Chemical analyses of selected water samples were c a r r i e d out by the Water Quality D i v i s i o n of the Inland Waters Branch, Department of Energy, Mines and Resources. F i n a l l y , the author wishes to thank Dr. J, Ross Mackay, both f o r h i s support as research supervisor and f o r i n i t i a l l y suggesting that the Secchi disk be used to estimate water t u r b i d i t y . i x . SYMBOLS AND ABBREVIATIONS Symbol a Total absorption c o e f f i c i e n t b Back-scattering c o e f f i c i e n t c Suspended sediment concentration C Contrast d P a r t i c l e diameter H Irradiance K Diffuse attenuation c o e f f i c i e n t K s E f f e c t i v e area c o e f f i c i e n t m Relative r e f r a c t i v e index n T o t a l number of p a r t i c l e s N Radiance N* Path function representing gain of radiance by s c a t t e r i n g r Distance R Reflectance s T o t a l s c a t t e r i n g c o e f f i c i e n t S T o t a l projected p a r t i c l e area on a plane surface SD Secchi disk transparency (Secchi depth) V P a r t i c l e volume per u n i t volume of water-sediment mixture z Depth normal to a h o r i z o n t a l water surface oc Volume attenuation c o e f f i c i e n t • Angle of path from zenith -A Wavelength . x i . Symbol X Mean wavelength e P a r t i c l e mass density <j> Azimuth of path Au> Increment of s o l i d angle 0o, p i Linear model population parameters b Q , b^ Least squares estimates of 0 and jff ^  e Residual error Subscripts b Background d Downwelling do Dissolved organic compounds ds Dissolved s a l t s o Source ob Observer p Suspended p a r t i c l e s r Distance r from source R Red l i g h t SD Secchi disk t Target T Tungsten l i g h t u Upwelling w Pure (or d i s t i l l e d ) water ws Relat i v e to d i s t i l l e d water standard Abbreviation o cm. Centimeter ( 10" m.) . x i i Abbreviation const. Constant cos Cosine d Derivative df Degrees of freedom e Exponential (2.7183) est. Estimated f Function f t . Foot (approximately 0.30 m.) F Variance r a t i o (MS/MS R e s) gm. Gram km. Kilometer (10^ m.) 1. L i t e r I n Natural logarithm (base e) m. Meter mg. Mill i g r a m (10 gm.) min. Minute (60 sec.) ml. M i l l i l i t e r (10~3 1.) -9 mju M i l l i m i c r o n (10 m.) MS Mean square of deviations (SS/df) No. Number of observations Res Residual RSQ Proportion of t o t a l variance explained by a l i n e a r model (assuming a b i v a r i a t e normal d i s t r i b u t i o n ) se Standard e r r o r sec. Second SS Sum of squares of deviations Abbreviation Micron (10~^ m.) TT P i (3.1416) I Sum X Chi-square value CHAPTER I INTRODUCTION Standard methods of determining suspended sediment concentrations i n n atural water bodies require c o l l e c t i o n of water samples and subsequent laboratory analysis of the samples. This can be both time consuming and expensive i f a la r g e number of determinations are required. Water transparency measurements, on the other hand, require no samples or laboratory analyses and, i n the case of Secchi disk transparencies, no elaborate instrumentation. The existence of a quantitative r e l a t i o n s h i p between Secchi disk transparency and concen-t r a t i o n of suspended sediment, therefore, would be of s i g n i f i c a n t use i n many studies. The Secchi disk transparency i s simply "the depth at which a white disk l e t down from the surface j u s t disappears from view" (Hutchinson, 1957» P«399). Recognition of the point of disappearance involves perception of the difference between the radiance o f the disk and that of i t s surroundings and, as such, i s a problem of contrast transmittance. This, i n turn, depends p r i m a r i l y on c h a r a c t e r i s t i c s of the disk, on possible observer bi a s , and on the i n t e r a c t i o n of radiant energy i n the v i s i b l e wavelengths with the water and with the materials, both p a r t i c u l a t e and dissolved, i n the water. The problem with which t h i s paper i s most concerned, then, i s that the concentration of suspended sediment ( p a r t i c u l a t e matter i n the water) i s but one of the many fa c t o r s which determine a Secchi disk . 1 . • 2# • reading. I f a reasonable d i r e c t r e l a t i o n s h i p between the two i s to e x i s t , a l l other influences on the reading must be e i t h e r constant or i n s i g n i f i c a n t over a known range of p h y s i c a l conditions. The approach taken here i s as follows: a) To review and expand upon empirical r e l a t i o n s h i p s which have been established between Secchi disk transparency and concentration of suspended load. b) To discuss and i n t e r p r e t these r e l a t i o n s h i p s i n the context of the p r i n c i p l e s of underwater o p t i c s . c) In l i g h t of the above, to consider the v a l i d i t y of the proposed technique, with emphasis on the r e s t r i c t i o n s and precautions which must be taken i n t o account i f i t i s to be c o r r e c t l y used. CHAPTER I I SECCHI DISK TRANSPARENCY AND THE ATTENUATION  OF LIGHT IN WATER A. COMPONENTS OF THE PROBLEM The propagation i n water of radiant energy i n the v i s i b l e wave-lengths i s c o n t r o l l e d by the magnitude and geometry of the underwater l i g h t f i e l d and by the o p t i c a l properties of the water medium. In general, the energy f l u x decreases or attenuates with i n c r e a s i n g distance of t r a v e l through a body of water. Energy i s converted i n t o other forms and other wavelengths by absorption or i s scattered out of the f l u x path. The rate of f l u x attenuation can, i n most s i t u a t i o n s , be expressed as a simple attenuation c o e f f i c i e n t . I n f a c t , several types of attenuation c o e f f i c i e n t can be determined f o r any natural water body. The p a r t i c u l a r properties of each depend upon which parameters of the l i g h t f i e l d are held constant by the measurement technique. A Secchi disk reading may be regarded as a rough index of the rate of l i g h t attenuation i n water and, therefore, as an attenuation c o e f f i c i e n t . This c o e f f i c i e n t i s dependent, however, not only on the nature of the underwater l i g h t f i e l d and the properties of the water medium but upon conditions i n the a i r above the water body and at the air-water i n t e r f a c e , on the c h a r a c t e r i s t i c s of the disk i t s e l f and on possible observer b i a s . Suspended sediment concentration, on the other hand, i s a property of the water medium. I t i s , as has been stated . 4 . previously, but one of the many f a c t o r s which may influence Secchi disk transparency. Other measures of the rate of l i g h t attenuation e x i s t , though, which, i n terms of measurement technique, are more objective than Secchi depths and which depend to a much greater extent only on the properties of the water medium. They are, therefore, more d i r e c t l y r e l a t e d to the concentration of suspended sediment. In t h i s context, a common approach taken i n both t h e o r e t i c a l and empirical studies of the r e l a t i o n s h i p between Secchi disk transparency and concentration of suspended sediment has involved d i v i s i o n of the problem i n t o two components. These are: a) The r e l a t i o n between the Secchi disk reading and an objective attenuation c o e f f i c i e n t f o r l i g h t i n water. b) The r e l a t i o n between the attenuation c o e f f i c i e n t and the concentration o f suspended sediment. This d i v i s i o n permits comparison of Secchi r e s u l t s with those obtained by the more objective but instrumentally more complex methods. I t also enables separate evaluation of the influence of the water medium on Secchi disk readings. B. ATTENUATION COEFFICIENTS Of the several i n d i c e s which can be used to describe the rate of attenuation of l i g h t i n na t u r a l water bodies, two have been chosen by researchers concerned with Secchi disk v i s i b i l i t y . These are: a) The volume attenuation c o e f f i c i e n t b) The d i f f u s e attenuation c o e f f i c i e n t f o r downwelling i r r a d i a n c e Both are r e l a t i v e l y easy to measure and have been shown to be t h e o r e t i c a l l y r e l a t e d to the depth of disappearance of a Secchi disk. They d i f f e r s i g n i f i c a n t l y i n t h e i r properties, however, and no simple r e l a t i o n s h i p need e x i s t between them. Their d e f i n i t i o n s , properties and d i f f e r e n c e s are discussed below. 1. The Volume Attenuation C o e f f i c i e n t Suppose a h i g h l y collimated underwater beam of a r t i f i c i a l mono-chromatic l i g h t has r a d i a n c e 1 N 0 at i t s source. L e t N r be the radiance of the beam at distance r from the source. N r represents only that radiance which has not been a l t e r e d or scattered during i t s passage through the water (monopath f l u x ) . As r increases, N r has been found experimentally to decrease according to the r e l a t i o n N r = N 0 e " * r (1) assuming tha t the water medium i s o p t i c a l l y homogeneous along the length r . The constant c< i s the volume attenuation c o e f f i c i e n t ( T y l e r and Preisendorfer, 1962, p.401) and gives the attenuation per u n i t length of the beam. Solving Equation ( l ) f o r c< , we get cx = - ± ln( -j£ ) ( 2 ) or, i n d i f f e r e n t i a l form, 1 Radiance i s the energy f l u x per u n i t projected area per u n i t s o l i d angle i n a s p e c i f i c d i r e c t i o n ( T y l e r and Preisendorfer, 1 9 & 2 , P. 3 9 7 ) . The volume attenuation c o e f f i c i e n t ex i s an inherent o p t i c a l  property of the water medium - i e . , i t s operational value "at a given point i n a given medium i s i n v a r i a n t under a l l changes of the radiance d i s t r i b u t i o n at that point" (Preisendorfer, 196l, p.12). ex, there-f o r e , i s dependent only on the absorbing and s c a t t e r i n g properties of the medium and i s independent of N 0, of the o r i e n t a t i o n of the beam and of e x i s t i n g l i g h t i n g conditions. More s p e c i f i c a l l y , c< i s the sum of two independent terms: s, the t o t a l s c a t t e r i n g c o e f f i c i e n t and a, the t o t a l absorption c o e f f i c i e n t - i e . , cx = a + s . Because both a and, to a l e s s e r degree, s are wave-length dependent i n n a t u r a l water, cx v a r i e s with the wavelength of the beam of l i g h t used i n i t s measurement. To summarize, then, o< represents the rate of attenuation of monopath radiance of a given wavelength. I t i s dependent only on the properties of the water medium and might be expected, therefore, to be more c l o s e l y r e l a t e d to the concentration of suspended sediment i n the water than would a Secchi disk transparency, e x may be determined i n s i t u at any depth by means of a beam transmittance meter. 2. The Diffuse Attenuation C o e f f i c i e n t The second index of l i g h t attenuation which has been used i n empirical studies of Secchi disk transparency i s the d i f f u s e attenuation c o e f f i c i e n t f o r downwelling i r r a d i a n c e . Determination of t h i s i r r a d i a n c e i s the radiant f l u x per u n i t area. I t i s r e l a t e d to the radiance d i s t r i b u t i o n by the summation H = I I Ncos(e)Aw where H - i r r a d i a n c e e = angle from zenith e w N = radiance u> = increment of s o l i d angle ( T y l e r and Preisendorfer, 1962, p.399) . 7 . c o e f f i c i e n t involves measurement of the n a t u r a l daylight i r r a d i a n c e at selected l e v e l s i n a water body rather than of the a r t i f i c i a l mono-chromatic l i g h t used to obtain values of o<, the volume attenuation c o e f f i c i e n t . The basic assumption that n a t u r a l water bodies are o p t i c a l l y homogeneous i n the h o r i z o n t a l plane enables conceptual decomposition of the underwater l i g h t f i e l d i n t o two counter-moving streams: an upwelling and a downwelling stream (Preisendorfer, 196l, p.19). Thus the i r r a d i a n c e produced i n such bodies by na t u r a l daylight may be subdivided i n t o H u(z) upwelling i r r a d i a n c e and H^(z) downwelling i r r a d i a n c e at any depth z. Consider the i r r a d i a n c e H d(z) r e s u l t i n g from the downward f l u x of n a t u r a l daylight i n a water body. Suppose a photometer i s used to measure t h i s i r r a d i a n c e at two depths, z^ and Z£» where Z2 > z^. Then, assuming that the water i s o p t i c a l l y homogeneous between z^ and Z£, the rate of attenuation of the f l u x can be determined from the f o l l o w i n g expression, H d ( z 2 ) = H d ( Z l ) e - Rd< z2 - z l > (4) where K d i s a d i f f u s e attenuation c o e f f i c i e n t f o r downwelling i r r a d i a n c e . Solving f o r K d, Kd = 7 - — — l n ( |4^4 ) (5) (Z2 - z i ) H d U i ) or, i n d i f f e r e n t i a l form, 8 The s i m i l a r i t y i n form o f Equations (1) and (4) i s apparent. In f a c t , Equation (1) could equally well have been written i n terms of i r r a d i a n c e . This does not mean, however, tha t cx and K d are equivalent. R e c a l l that cx i s an index only of the attenuation of monopath f l u x of a given wavelength. K d, on the other hand, i s r e l a t e d to the attenuation of multipath or d i f f u s e f l u x as w e l l . To i l l u s t r a t e , imagine a h o r i z o n t a l u n i t area at some given depth i n a body of water. Natural l i g h t of a l l wavelengths penetrating the' surface of the body w i l l be both scattered and absorbed by the medium. A portion of the i n c i d e n t l i g h t may reach the given area unaltered by i t s passage through the water. However, another and often l a r g e r portion reaching the area w i l l have been scattered or absorbed at l e a s t once during i t s passage. This i s the multipath or d i f f u s e f l u x . Because of t h i s a d d i t i o n of multipath f l u x , K d w i l l always be l e s s than cx. The volume attenuation c o e f f i c i e n t cx has been shown to be the sum of the t o t a l s c a t t e r i n g c o e f f i c i e n t s and the t o t a l absorption c o e f f i c i e n t a. In a stable l i g h t f i e l d K d, i n contrast, i s a function of the absorption c o e f f i c i e n t f o r downwelling i r r a d i a n c e a d ( z ) and the back-scattering c o e f f i c i e n t s b d ( z ) and t^Cz). A f t e r Preisendorfer (1961, p.24), K d = a d ( z ) + b d ( z ) - b ^ z j R d U ) (7) where R d ( z ) = = reflectance K d i s an apparent rather than an inherent o p t i c a l property. I t i s not i n v a r i a n t under changes of the radiance d i s t r i b u t i o n i n the water body. For example, xd.ll vary with changes i n the p o s i t i o n of the sun i n the sky. I t w i l l also be influenced by waves on the water surface. The d i f f u s e attenuation c o e f f i c i e n t f o r downwelling i r r a d i a n c e , then, i s dependent upon f a r more f a c t o r s than i s the volume attenuation c o e f f i c i e n t . These a d d i t i o n a l influences also a f f e c t a Secchi disk reading. We might expect, therefore, t h a t w i l l be more c l o s e l y r e l a t e d to Secchi depth but l e s s c l o s e l y r e l a t e d to suspended sediment concentration than w i l l o<. CHAPTER I I I EMPIRICAL RELATIONSHIPS Applied studies i n underwater o p t i c s have been conducted p r i m a r i l y by oceanographers. Published r e s u l t s on Secchi disk trans-parency and suspended sediment concentration are based, therefore, l a r g e l y on data c o l l e c t e d i n oceanic and offshore c o a s t a l environments where concentrations of sediment are low. Knowledge of sediment concentrations i n the often more t u r b i d f l u v i a l environments i s also of great importance, however, and a simple method such as Secchi disk transparency f o r estimating such concentrations would be of equal value. For these reasons, r e l a t i o n s h i p s based on data c o l l e c t e d i n channels and lakes of the Mackenzie d e l t a are discussed i n d e t a i l i n t h i s paper. These r e s u l t s cover a range of concentrations extending w e l l above those which are found i n oceans and seas. A. PUBLISHED INVESTIGATIONS 1. Relationships between Secchi Disk Transparency and Attenuation  C o e f f i c i e n t s Perhaps because of i t s r e l a t i v e ease of measurement, the d i f f u s e attenuation c o e f f i c i e n t f o r downwelling i r r a d i a n c e , K d, was the f i r s t o bjective measure of l i g h t attenuation to be e m p i r i c a l l y r e l a t e d to Secchi disk transparency. Poole and Atkins (1929) appear to have established the e a r l i e s t r e l a t i o n s h i p of t h i s type. A modified form of t h e i r equation together with others established subsequently are l i s t e d i n Table I and p l o t t e d i n Figure 1. . 10 . TABLE I EMPIRICAL RELATIONSHIPS BETWEEN SECCHI DISK TRANSPARENCY  AND THE DIFFUSE ATTENUATION COEFFICIENT Kd Author Location Equation No. RSQ Rangea Poole and Atkins, 1929 English Channel K d = 0.02! + Ig. 14 0.80 6.5-22 m. Clarke, 1941 South and Southeast of Bermuda K d = .0.006 + ^ 25 0.84 13-47 m. Tibby and Barnard, 1963 Coast of Southern California K d = 1.40 SD 80-100 est. - 1.5-26 m. Graham, 1966 Cruises: a. Smith 52 Central and Eastern North Pacific Kd = -0.08a + * § > 26 0.53 14-40 m. b. Manning 32 K d = -0.005 + 42 0.72 6.5-33 m. c. Manning 36 K d = -0.075 • 39 0.74 6.5-22 m. Hanaoka et al, I960 Bay Waters of Japan K d = bo SD - ° ' 3 - - -aRange of Secchi disk transparencies 11 0-72 0-64 0-56 i . E * 0-48 S 0-40 o o i 0-32 <D £ 0-24 a 016 0 0 8 0 0 0 y / ! L E G E N D Poole and Atkins, 1929 \ Clarke, 1941 Tlbby and Barnard, 1963 \Graham, 1966 (Smith 52) — Graham, 1966 (Manning 32) Graham, 1966 (Manning 36) Hanaoka et al, I960 (assuming b o=l-0) Length of linear plot gives approximate data range 1 I 1 0-10 0-20 0-30 0-40 0-50 0-60 Reciprocal of Secchi Disk Transparency l/SD (m."') FIGURE 1: EMPIRICAL RELATIONSHIPS BETWEEN SECCHI DISK TRANSPARENCY AND THE DIFFUSE ATTENUATION COEFFICIENT K d . 12 . . 13 . Poole and Atkins compared Secchi disk readings i n the English Channel with the mean value of K d f o r the f i r s t 20 m. of water depth. The data was f i t t e d with a r e c i p r o c a l r e l a t i o n s h i p of the form K d = y (8) where SD = Secchi disk transparency (m.) Clarke (1941) found a s i m i l a r r e l a t i o n s h i p i n the west A t l a n t i c south of Bermuda, using mean values of K d f o r 2 to 50 ™« Tibby and Barnard (1963) also used a simple r e c i p r o c a l transformation but found that the constant was 1.4 rather than 1.7 f o r t h e i r data c o l l e c t e d o f f the coast of southern C a l i f o r n i a . The r e c i p r o c a l model i s based on the assumption that the percentage of downwelling i r r a d i a n c e at each Secchi depth r e l a t i v e to some standard ( u s u a l l y a simultaneous reading above the water surface) i s constant. I f t h i s i s so, the term l n ( H d(Z2)/H d(z-^) ) i n Equation (5) i s also constant and K d = ^ T " ( 9 ) a z 2 - z t I f z-^  i s taken as the water surface, i e . = 0, then * d d o ) Poole and Atkins found that an average of 15.8$ of the surface i r r a d i a n c e remained at t h e i r 1929 Secchi depths (range: 9.6 to 22.6$). Clarke made s i m i l a r computations and obtained a mean value of 15.2$ with a range of 8.0 to 26.0$. The simple r e c i p r o c a l model w i l l hold i f the variances from these means are randomly d i s t r i b u t e d with respect to . 14 . N K d and SD. The near equality of the two mean i r r a d i a n c e percentages suggests that the slopes of the two equations r e l a t i n g K d to SD should be s i m i l a r and t h i s , i n f a c t , i s the case. I m p l i c i t i n r e l a t i o n s h i p s of the form used by these writers i s the assumption that, as K d approaches zero, l/SD approaches zero. The data of Poole and Atkins and of Clarke has been reanalyzed using l e a s t squares techniques. The r e s u l t i n g equations, given i n Table I , are estimated f u n c t i o n a l r e l a t i o n s h i p s assuming K d to have been measured without e r r o r . In these two examples the i n t e r c e p t s do not appear to d i f f e r s i g n i f i c a n t l y from zero. Graham (1966), however, di d f i n d i n t e r c e p t s which d i f f e r e d s i g n i f i c a n t l y from zero at the 99$ l e v e l of confidence f o r both Smith Cruise 52 and Manning 36. Unfortunately, stormy weather conditions during data c o l l e c t i o n severely r e s t r i c t the usefulness of the r e l a t i o n s h i p s established by these two c r u i s e s . T y l e r (1968J P»4) comments that a constant Secchi reading of 30 m. was obtained f o r a range of K d values from 0.074 m."* to 0.044 m.~* during Manning Cruise 36. There i s no reason, therefore, to r e j e c t the assumption that p l o t s of K d versus l/SD must pass through the o r i g i n , p a r t i c u l a r l y since i t appears, as w e l l , to be j u s t i f i e d t h e o r e t i c a l l y (Equation (10)). As we have seen, the a p p l i c a b i l i t y of the simple r e c i p r o c a l model hinges on the variance of percentage i r r a d i a n c e at a l l Secchi depths i n a given area being randomly d i s t r i b u t e d about the mean. This does not appear to be true, e s p e c i a l l y i n very t u r b i d water. Tibby and Barnard (1963> PP.873-874) found the percentage of l i g h t remaining at Secchi depth to be a d i r e c t f u nction of water t u r b i d i t y and thereby, . 15 . apparently unknowingly, destroyed the r a t i o n a l e behind t h e i r own model. Closer examination of the data of Clarke and of Poole and Atkins reveals a s i m i l a r trend. Tibby and Barnard (p.874) suggest that the discrepancy i s due to more pronounced s c a t t e r i n g (as opposed to absorption) i n t u r b i d water. Regardless of cause, however, the existence of such a trend casts doubt on the general a p p l i c a b i l i t y of the simple r e c i p r o c a l model. I t may explain, though, the inverse power function K d = b Q SD " 0 , 3 (11) required by Hanaoka et a l (i960) to f i t t h e i r data from the r e l a t i v e l y t u r b i d c o a s t a l waters of Japan. I f we assume that the percentage of i r r a d i a n c e remaining at Secchi depths increases with t u r b i d i t y , the slopes of l i m i t e d range r e c i p r o c a l p l o t s should decrease correspond-i n g l y (by Equation ( 5 ) ) . This condition i s s a t i s f i e d by the inverse power function (Figure l ) but not by the r e c i p r o c a l model. Although Tibby and Barnard have used the simple r e c i p r o c a l f o r data which extends to r e l a t i v e l y high t u r b i d i t i e s , t h e i r extreme value of SD = 1.5 m. and K d = 0.60 m.-* l i e s f a r to the r i g h t of the p l o t of t h e i r equation i n Figure 1. This may i n d i c a t e that a power function would have provided a better f i t to t h e i r data. On the basis of the foregoing discussion, we might expect the slopes of r e c i p r o c a l p l o t s i n Figure 1 to be l e a s t f o r equations based on data from t u r b i d water. No general trend i s evident, however, although Clarke's equation f o r very c l e a r ocean water (Table I ) i s of s l i g h t l y greater slope than that of Poole and Atkins f o r the somewhat . 16 . more t u r b i d water of the English Channel. Because of t h e i r doubtful v a l i d i t y , the r e l a t i o n s h i p s of Tibby and Barnard and of Graham must be disregarded. The r e c i p r o c a l model has also been used to r e l a t e Secchi disk transparency and the volume attenuation c o e f f i c i e n t ex. E mpirical equations established by Jones and W i l l s (195&) and Otto (1966) are given i n Table I I and p l o t t e d i n Figure 2. I n these cases the p l o t i n t e r c e p t s are meaningless. Otto's data i s standardized only against the c l e a r e s t water found i n the p a r t i c u l a r area to which each of h i s equations apply. As w e l l , the slopes of Otto's p l o t s are not d i r e c t l y comparable with that of Jones and W i l l s . The volume attenuation c o e f f i c i e n t s measured by Otto are f o r red l i g h t while those of Jones and W i l l s are f o r tungsten l i g h t ( X = 550 m> i ) . T h e o r e t i c a l l y both Secchi disk transparency and the d i f f u s e attenuation c o e f f i c i e n t K d are dependent upon the geometry of the underwater l i g h t f i e l d . Since cx i s not, we might expect i t s r e l a t i o n -ship to disk v i s i b i l i t y to be l e s s precise than that of K d. However, a comparison of Tables I and I I shows that t h i s i s not the case. RSQ values i n v o l v i n g cx are commonly as good or better than those i n v o l v i n g K d. A p h y s i c a l explanation of t h i s i s d i f f i c u l t . Perhaps Secchi disk v i s i b i l i t y i s a c t u a l l y i n s e n s i t i v e to the c h a r a c t e r i s t i c s of the l i g h t f i e l d . Poole and Atkins (1929> p.310) do state that disk v i s i b i l i t y "was not a f f e c t e d by the approach of n i g h t f a l l u n t i l the l i g h t became so poor that the adaptive power of the eye could no longer compensate f o r the reduction." The slopes of p l o t s i n Figure 2 do not appear to be r e l a t e d to TABLE II EMPIRICAL RELATIONSHIPS BETWEEN SECCHI DISK TRANSPARENCY AND THE VOLUME ATTENUATION COEFFICIENT (X Author Location Equation No. RSQ Range3, Otto, 1966 D 4 61 a. Central North Sea cxR = -0.127 + 3.96 b. Malzwin, Wadden Sea cx^ = -2.21 + 4 6l c. Lightvessel Texel ocR = -0.3&7 + gp 3 29 d. North Sea West of cx0 = -0.097 + Texel R " SD 6.31 e. North Sea West of cx = -0.919 + on Ijmuiden 2 5? f. North Sea, Southern cxR = 0.051 + Bight S D 8 0.31 12-20 m. 24 0.81 0.5-1 m. 6 0.78 5-8 m. 32 0.64 3-15 m. 5 0.94 1.8-5 m. 10 0.74 2-6 m. c 26 Jones and Wills, Thames Estuary and cx^T = -0.189 + 29 0.97 0.75-21 m. 1956 Estuaries near Plymouth ^ange of Secchi disk transparencies b cX^ = volume attenuation coefficient for red light c cx,p = volume attenuation coefficient for tungsten light • 17 . —r* / / / / / / / / / / / / / / / / / / / / LEGEND Otto, 1966 (Central North / Sea) / Otto, 1966 (Malzwin, f Wadden Sea) Otto, 1966 (Lightves8el Texel) - — Otto, 1966 (North Sea West of Texel) — Otto, 1966 (North Sea West of Ijmuiden) —•- Otto, 1966 (North Sea, . Southern Bight) Jones and Wills, 1956 Length of plot gives approximate data range I I 1 L 0-4 0-8 1-2 1-6 2 0 2-4 Reciprocal of Secchi Disk Transparency l/SD (m."1) FIGURE 2: EMPIRICAL RELATIONSHIPS BETWEEN SECCHI DISK TRANSPARENCY AND THE VOLUME ATTENUATION COEFFICIENT ex . 1.8 . . 19 . e i t h e r the extremes or lengths of the data ranges. Otto (p.32) suggests "that d i f f e r e n c e s i n the o p t i c a l properties of the water masses (wavelength s e l e c t i v i t y ? ) may be important but goes i n t o no f u r t h e r d e t a i l . A v a ilable data also gives no i n d i c a t i o n of a need f o r an inverse power function model f o r large t u r b i d i t y ranges. Jones and W i l l s data can be f i t t e d with a power function of the form cX T = 4.90 SD * 1 , 0 6 RSQ = 0.98 (12) but the exponent i s so near - 1.0 that the simple r e c i p r o c a l model seems preferable. They do note, however, that the values which d i f f e r most from the simple model are those from the most t u r b i d water (p.W-0). In waters of low t u r b i d i t y , then, both oc and K d can be r e l a t e d to Secchi disk transparency by a r e c i p r o c a l model. The r e l a t i o n s h i p becomes more complex i n areas of higher sediment concentration and, f o r the d i f f u s e attenuation c o e f f i c i e n t at l e a s t , an inverse power function model may become necessary. Empirical r e s u l t s provide no simple explanation f o r d i f f e r e n c e s i n the slopes of p l o t s i n v o l v i n g e i t h e r attenuation c o e f f i c i e n t and the excellent c o r r e l a t i o n s between disk transparency and the volume attenuation c o e f f i c i e n t ex require f u r t h e r inve s t i gation. 2. Relationships between Attenuation C o e f f i c i e n t s and Concentration of  Suspended Sediment Most published empirical research on r e l a t i o n s h i p s between rate of attenuation and suspended sediment concentration i n natural water bodies has involved the volume attenuation c o e f f i c i e n t cx rather than the d i f f u s e c o e f f i c i e n t K d. Since cx , u n l i k e K^, i s t h e o r e t i c a l l y dependent only on the properties of the water medium, t h i s i s not 20 s u r p r i s i n g . By c o n f i n i n g a study to r e l a t i o n s h i p s between cx and sediment concentration, only the attenuation of monopath radiance need be considered. The complexities added by multipath radiance can be ignored. Jones and W i l l s (1956) measured the attenuation of tungsten l i g h t i n sea and estuary waters and i n prepared mixtures of k a o l i n and Thames mud i n tap water (Figure 3)« Except at very low concentrations, the r e l a t i o n between cx and suspended sediment concentration was found to be l i n e a r f o r the laboratory mixtures. The same was not true f o r the sea and estuary waters, though. Here the rate of increase of « per u n i t increase i n concentration decreased with i n c r e a s i n g t u r b i d i t y to give a p l o t of the form shown i n Figure 3* This was a t t r i b u t e d by the authors to " v a r i a t i o n s i n the nature or p a r t i c l e s i z e d i s t r i b u t i o n of the suspended m a t e r i a l " (p.439). In suspensions dominated by inorganic sediments of nearly constant density and r e f r a c t i v e index, the e f f e c t of p a r t i c l e s i z e on attenuation i s of p a r t i c u l a r i n t e r e s t . Consider a volume of sediment-water mixture of u n i t area and incremental thickness. L i g h t energy passing through t h i s volume normal to i t s surface w i l l be attenuated by both the sediment p a r t i c l e s and by the water i t s e l f . Assuming that the e f f e c t of the water on attenuation i s constant f o r a l l concentrations with which we are concerned, l e t us hypothesize that the rate of attenuation i n the volume i s proportional to the t o t a l p a r t i c l e surface capable of i n t e r c e p t i n g l i g h t energy - i e . , cx = f(S) (13) where S = t o t a l projected p a r t i c l e area on the plane surface Concentration of Suspended Sediment c (mg./l.) FIGURE 3: RELATIONSHIPS BETWEEN THE VOLUME ATTENUATION COEFFICIENT AND CONCENTRATION OF SUSPENDED SEDIMENT 22 . f a c i n g the l i g h t source Assume f u r t h e r that the p a r t i c l e s are s p h e r i c a l and of uniform diameter d. Then, f o r no p a r t i c l e overlap, S = ^ n d 2 (14) where n = t o t a l number of p a r t i c l e s i n the volume Now, sediment concentration, c, i s simply mass of sediment per u n i t volume of mixture. For s p h e r i c a l p a r t i c l e s and u n i t volume, c = f e n d 3 (15) where f = p a r t i c l e mass density S u b s t i t u t i n g back i n t o Equation (14), we get " J h  (l6) Therefore, since ? has been assumed constant, we may say (from Equation (13)) that °< = f ( f ) (17) Thus, f o r a given concentration, ex. decreases as d increases - i e . , the l a r g e r the p a r t i c l e s the l e s s the rate of attenuation. Equation (1?) i s obviously only a gross representation of the r e l a t i o n s h i p between attenuation, concentration and p a r t i c l e s i z e . I t does, however, introduce p a r t i c l e s i z e as a major parameter i n the attenuation - concentration r e l a t i o n s h i p and r e i n f o r c e s the conclusions drawn by Jones and. W i l l s about t h e i r p l o t s i n Figure 3. The s t r a i g h t -l i n e r e l a t i o n s f o r the laboratory mixtures suggest constant mean . 23 . particle size while the relation for natural water indicates an increase i n particle size with concentration. Numerous other empirical relationships have been established between c* and the concentration of suspended sediment. Ochakovskiy (1966) found a moderate linear correlation for samples from very clear water (c = 0.5 to 2.0 mg./l.) i n the Mediterranean Sea. Otto (1966, 1967) also used simple linear models to f i t his data from the Wadden Sea, North Sea and English Channel. Concentrations i n these areas ranged from 1 to 110 mg./l. In the Wadden Sea the slopes of the equations f e l l into two groups: one with mean 0.060 and range 0.048 to 0.074 and the other with mean 0.129 and range 0.108 to 0.152. Otto concluded that two kinds of suspension had been encountered, each of f a i r l y constant but different particle size distribution. By Equation (17), mean particle size should be greatest i n the f i r s t group. The relationship between the diffuse attenuation coefficient and sediment concentration, as was noted, has received l i t t l e attention i n the literature. Hanaoka et a l (i960) have f i t t e d a power function of the form K d = b0» c °' 2 (18) where b c' i s a parameter which varies with b 0 (Equation (11)) according to V = -0-55 + 0.64 b d (19) Since b Q was said to vary with particle size, this equation does not . 24 . contradict any of the conclusions drawn with regard to cx . The u n i v e r s a l a p p l i c a b i l i t y of the power function model and, more s p e c i f i c -a l l y , of the exponent 0.2 i s doubtful, however. This value w i l l depend on many f a c t o r s , i n c l u d i n g how the size d i s t r i b u t i o n changes with concentration. Most probably there i s l i t t l e difference between the form of r e l a t i o n s h i p s i n v o l v i n g and those i n v o l v i n g cx . Empirical studies, then, do not c o n t r a d i c t the statement of J e r l o v (I963, p.98) that, f o r a given type and si z e of p a r t i c l e , ex w i l l be a l i n e a r function of p a r t i c l e concentration. The importance of p a r t i c l e s i z e , however, does introduce an added complication to the problem of e s t a b l i s h i n g a d i r e c t r e l a t i o n s h i p between Secchi disk transparency and concentration of suspended sediment. P a r t i c l e s i z e w i l l vary not only between water bodies but w i l l probably change with concentration at i n d i v i d u a l sampling l o c a t i o n s . 3. D i r e c t Relationships between Secchi Disk Transparency and Concentra- t i o n of Suspended Sediment D i r e c t empirical r e l a t i o n s h i p s between Secchi disk transparency and suspended sediment concentration can u s u a l l y be e a s i l y established but may be d i f f i c u l t to analyze. The greatest problem involved occurs when the data cannot be f i t t e d by a r e c i p r o c a l model. At high t u r b i d i t i e s r e c i p r o c a l n o n l i n e a r i t y may be due to any combination of changes i n p a r t i c l e nature or si z e d i s t r i b u t i o n or to l i g h t s c a t t e r i n g e f f e c t s . Separation of the r e l a t i v e i n fluences of each requires a d d i t i o n a l data. Postma (196l) has provided some of t h i s data. Most importantly, he has shown e m p i r i c a l l y that the e f f e c t of changing p a r t i c l e s i z e on Secchi disk transparency (attenuation) i s s i m i l a r to that postulated i n . 25 . Equation (17). His r e s u l t s are summarized i n Figure k. For every s i z e f r a c t i o n the r e l a t i o n s h i p between the r e c i p r o c a l of Secchi depth and sediment concentration i s approximately l i n e a r except at very high concentrations. The curvature i n the p l o t s where p a r t i c l e concentra-t i o n i s great may be due to l i g h t s c a t t e r i n g e f f e c t s since, as we have seen, the simple r e c i p r o c a l r e l a t i o n s h i p between Secchi depth and the objective attenuation c o e f f i c i e n t s may not apply i n t h i s s i t u a t i o n . Of greatest s i g n i f i c a n c e i n t h i s context i s that t h i s n o n l i n e a r i t y i s almost inconsequential i n comparison to the e f f e c t of changes i n p a r t i c l e s i z e . For any given Secchi depth, the r e l a t i o n s h i p between concentration and grain s i z e i s nearly l i n e a r . Postma (p.363) found that h i s data was best f i t t e d by the equation = const, (20) Assuming constant p a r t i c l e shape and density, t h i s equation leads to the r e s u l t = f(J) = f(S) (21) where V = p a r t i c l e volume per u n i t volume of mixture Thus l/SD i s proportional to the t o t a l p a r t i c l e surface S. Equation (21) provides empirical v e r i f i c a t i o n of the basic assumption used i n the d e r i v a t i o n of Equation (17). The experiments i l l u s t r a t e d i n Figure 4 were concerned only with suspended inorganic matter of uniform density. Postma (p.365-367) also considers p o s s i b l e e f f e c t s of the presence i n most n a t u r a l waters of varying amounts of organic m a t e r i a l . The density of dry organic matter 0-24 0-20 0-16 012 0 0 8 0-04 ?/ / / / / / • r \ i , ' / I 11* s s lb •So Particle diameters in microns — (After Postma, 1961, p.36l) 1 1 1 1 1000 2000 3000 4000 Concentration of Suspended Sediment c (mg./l.) 5000 6000 FIGURE 4: THE INFLUENCE OF PARTICLE SIZE ON THE SECCHI DISK - SEDIMENT 'CONCENTRATION RELATIONSHIP . 27 . i s about 1.5 gm./ml., i n contrast to the 2.65 gm./ml. of most inorganic sediment. I n nat u r a l suspensions the density d i f f e r e n c e i s even more extreme. Suspended organic matter i s not dry. I t s water content may exceed 80$ by weight, g i v i n g an e f f e c t i v e density of l e s s than 1.07 gm./ml. The volume of a given dry weight of organic material i n water, therefore, w i l l be much greater than the same amount of inorganic matter. This, i n turn, w i l l l e a d to a greater t o t a l surface and thus to an increased rate of attenuation per u n i t dry weight. This r e s u l t follows d i r e c t l y from Equation (16). The e f f e c t may be lessened by the greater transparency of organic material but Postma's (p.374) data from natural waters i n d i c a t e that the compensation i s f a r from complete. For equal dry weights, then, organic matter w i l l u s u a l l y have a greater influence on Secchi disk readings than w i l l inorganic. 4. Related Studies The empirical research discussed to t h i s point i n d i c a t e s that Secchi disk transparencies may, i n c e r t a i n s i t u a t i o n s , be e f f e c t i v e l y used to estimate the t o t a l surface of p a r t i c u l a t e matter i n nat u r a l suspensions. There are, however, a number of assumptions, i n ad d i t i o n to those already mentioned, which place f u r t h e r l i m i t a t i o n s on the usefulness of such r e l a t i o n s h i p s . The most s i g n i f i c a n t of these hidden assumptions are r e l a t e d to the properties of the various materials found i n nat u r a l water bodies. As such, they can most e a s i l y be discussed i n terms of components of the volume attenuation c o e f f i c i e n t . The volume attenuation c o e f f i c i e n t i s , as we have seen, the sum of the t o t a l absorption c o e f f i c i e n t a and the t o t a l s c a t t e r i n g c o e f f i c i e n t s. Each of these c o e f f i c i e n t s , however, may be subdivided . 28 . on the basis of the various types of material which may be present i n any n a t u r a l water body. These materials include the water i t s e l f , d i ssolved s a l t s , d i s s o l v e d organic compounds and suspended organic and mineral p a r t i c l e s , cx , therefore, may be expressed as cx = ^ + s w + a d s + s d s + a d o + s d 0 + a p + s p (22) where w = pure water ds = di s s o l v e d s a l t s do = d i s s o l v e d organic compounds p = suspended p a r t i c l e s I n our discussion of r e l a t i o n s h i p s between measures of attenuation and sediment concentration, we have i m p l i c i t l y assumed that the f i r s t s i x terms on the r i g h t hand side of Equation (22) are e i t h e r i n v a r i a n t or i n s i g n i f i c a n t i n a l l n a t u r a l waters. To the order of accuracy required f o r the problem with which we are concerned, the attenuation caused by pure water may, i n f a c t , be considered i n v a r i a n t f o r any given wavelength of i n c i d e n t l i g h t ( J e r l o v , 1964, p.45). There i s also no evidence that the presence of dissolved s a l t s has any e f f e c t on attenuation throughout the photopic ( v i s i b l e ) band ( J e r l o v , 1968, p.53)« F i n a l l y , s c a t t e r i n g by dissolved organic matter i s also i n s i g n i f i c a n t r e l a t i v e to i t s absorbing properties and to p a r t i c l e s c a t t e r i n g . I f , then, cx i s determined r e l a t i v e to a d i s t i l l e d or f i l t e r e d water standard, Equation (22) may be compressed to ( J e r l o v , 1953> P«6l) °<s = °< - ~w = ado + < ap + sp> (23) . 29 . The presence i n large q u a n t i t i e s of dissolved organic compounds, however, may greatly influence any r e l a t i o n s h i p between rate of attenuation and t o t a l p a r t i c l e surface. The chief agent i n such absorption i s "yellow substance" ( J e r l o v , 1964, p.45) or, equivalently, "Gelbstoff" (Otto, 196?, p.535). To quote Je r l o v (1964, p .45), "Yellow substance i s formed from free carbohydrates and free amino acids, i e . , where organic matter i s d i s i n t e -grating. The end product i s composed of carbohydrate-humic acids or melanoidines which are yellow and f a i r l y stable i n sea water." In fresh water the brown humic acids predominate ( J e r l o v , 1968, p.55)» The absorption curve f o r yellow substance increases almost exponenti-a l l y toward the shorter v i s i b l e wavelengths. I t s e f f e c t may be l a r g e l y overcome i n d i r e c t measurements of the volume attenuation c o e f f i c i e n t by using a red l i g h t source. This, of course, i s not possible f o r Secchi disk transparencies. In some environments, therefore, s a t i s f a c t o r y r e l a t i o n s h i p s between Secchi depth and concentration of suspended sediment may not e x i s t because of the influence of large and hi g h l y v a r i a b l e q u a n t i t i e s of yellow substance. B. THE MACKENZIE DELTA, N.W.T. During the summers of 1968 and 1969 a set of 296 comparative observations of Secchi disk transparency and suspended sediment concentration were made i n a selected lake-channel system of the Mackenzie d e l t a , N.W.T. Although the purpose of the study was simply to develop a p r e d i c t i v e equation f o r estimating sediment concentration i n t h i s p a r t i c u l a r area, the r e s u l t s are of more general i n t e r e s t because of the high t u r b i d i t i e s encountered. Measured sediment concen-t r a t i o n s ranged from 1 to II56 mg./l., t h i s l a t t e r f i g u r e being w e l l . 30 . above any discussed i n the published l i t e r a t u r e . As we have seen, the r e l a t i o n s h i p between transparency and concentration can become complex at high t u r b i d i t i e s . 1. The F i e l d Area The modern Mackenzie d e l t a ranges from 6?° to 70° N l a t i t u d e and 134° to 136 0 W longitude i n the northwest corner of the Northwest T e r r i t o r i e s of Canada. I t has a north-south length of approximately 210 km., a mean width of about 65 km. and a t o t a l surface area of over 12,000 km. (Mackay, 1963» P«98). The d e l t a i s composite i n o r i g i n but most of i t s area has been b u i l t up by the Mackenzie River which has an estimated mean annual discharge of 11,230 m.^/sec. and a mean annual f l o o d peak of 26,800 m.^/sec. (MacKay, 1967» p.19). The dominant s u r f i c i a l features of the d e l t a p l a i n are the myriad of d i s t r i b u t a r y channel and f l o o d p l a i n lakes. The lake-channel system i n which data was c o l l e c t e d f o r t h i s study i s located near the east side of the d e l t a approximately 50 km. northwest of the v i l l a g e of Inuvik. The system may be divided i n t o three components (Figure 5)s a) A d e l t a d i s t r i b u t a r y channel of width 200 m. and maximum depth 4.5 m. at the sample cross-section. The mean flow v e l o c i t y i n t h i s channel i s u s u a l l y l e s s than 1 m./sec. b) A d e l t a lake of north-south length 780 m., width 800 m. and maximum depth 2.5 m. c) A small reversing-flow channel of mean width 10 m. and depth 2 m. which connects the lake and d i s t r i b u t a r y . During periods of r i s i n g stage, t u r b i d d i s t r i b u t a r y SCALE 1000 500 0 1000 © SAMPLING LOCATION t M M&S&ca gflgaw* Feet FIGURE 5: THE LAKE - CHANNEL SYSTEM . 32 . water flows i n t o the lake through t h i s channel. When the stage f a l l s , the flow d i r e c t i o n reverses and c l e a r water i s c a r r i e d out of the lake. Mean flow v e l o c i t i e s can approach 1 m./sec. i n e i t h e r d i r e c t i o n but are u s u a l l y considerably l e s s . The water depths given above are co r r e c t r e l a t i v e l y and represent average mid-summer values. During spring f l o o d depths may be several meters greater. Transparency and sediment concentration measurements were made i n each of the three components during both inflow and outflow conditions. S p e c i f i c sampling l o c a t i o n s are shown i n Figure 5» 2. Measurement Techniques The Secchi disk used was painted white on i t s upper surface and black on the lower, was 3° cm« i n diameter and was attached to a r i g i d graduated rod. The procedure followed i n taking readings was that recommended by Hutchinson (1957, p.399). Each recorded value was the mean, to the nearest centimeter, of the point of disappearance of the disk and the point of reappearance as i t was r a i s e d to the water surface. A l l readings were taken i n shadow to minimize variance i n surface r e f l e c t i o n . The p r e c i s i o n of each observation was estimated to be t 5$ under smooth water surface conditions. This e r r o r was magnified g r e a t l y when the surface was rough. Water samples f o r sediment load analysis were c o l l e c t e d by two d i f f e r e n t methods. In the channels a US DH48 depth-integrating sampler was used; i n the lake each sample was taken at 0.6 t o t a l water depth with a Kemmerer tube sampler of length 30 cm. Neither method gives samples which are s t r i c t l y comparable with Secchi depths - i e . , . 33 . integrated samples from the surface to the Secchi depth. However, because of shallow water depths and almost uniform v e r t i c a l d i s t r i b u -t i o n s of sediment concentration, no serious error i s believed to have been introduced. E r r o r s due to unavoidable v a r i a t i o n s i n measurement technique and to sample contamination during storage may also be present but t h e i r combined e f f e c t i s d i f f i c u l t to evaluate on the basis of the data at hand. The samples were analyzed i n a laboratory f o r both t o t a l and suspended load. Suspended load was determined by f i l t r a t i o n of a 50 ml. p o r t i o n of each sample through paper of about 1 micron pore diameter. The p r e c i s i o n of the method, on the basis of analyses on second 50 ml« portions of 60 samples, was found to be approximately 1 4$ f o r samples of greater than 20 mg./l. concentration. Some improvement was noted as concentration increased. In 13 cases separate samples c o l l e c t e d concurrently were analyzed i n a d i f e r e n t laboratory using 500 ml. a l i q u o t s . The mean c o e f f i c i e n t of v a r i a t i o n f o r suspended sediment concentration was only 6.5$ (comparing with concentrations determined f o r the primary samples). This i s a gross i n d i c a t i o n of the p r e c i s i o n of the sampling technique and of the accuracy of the method of analysis. T o t a l load was determined by evaporation of 100 ml. portions of each sample. The p r e c i s i o n of the method was of the same order as f o r suspended load. The accuracy of the r e s u l t s , however, i s much more dubious. Analyses of 23 separate samples i n a d i f f e r e n t laboratory i n d i c a t e d a widely varying r e l a t i o n s h i p between corresponding samples. C o e f f i c i e n t s of v a r i a t i o n ranged from + 23$ to - 26$ r e l a t i v e to the . 3k . primary samples. No simple explanation f o r t h i s i s apparent since the evaporation method was used i n both l a b o r a t o r i e s . 3« The Functional Relationship The r e l a t i o n s h i p between Secchi depth and sediment concentration was determined by simple l e a s t squares a n a l y s i s . Estimation of the parameters of l i n e a r models using t h i s technique demands no assumptions about the underlying p r o b a b i l i t y d i s t r i b u t i o n s associated with each v a r i a b l e (Lindley, 19^7* p.24l). However, the many s t a t i s t i c a l measures (standard er r o r s , s i g n i f i c a n c e p r o b a b i l i t i e s , etc.) commonly associated with l e a s t squares equations do (Draper and Smith, 1966, p.17). A s a t i s f a c t o r y model, then, must both f i t the data and s a t i s f y these assumptions. P a i r s of Secchi depth and sediment concentration observations were selected randomly i n the f i e l d . No attempt was made to c o n t r o l e i t h e r v a r i a b l e . An important property of t h i s s i t u a t i o n i s the asymmetry of the two possible l e a s t squares equations, depending upon which v a r i a b l e i s considered independent (Madansky, 1959» P.175). The choice of independent and dependent v a r i a b l e s i s , therefore, meaningful s t a t i s t i c a l l y and must be based upon the use which i s to be made of the r e s u l t i n g equation. Since our purpose i n developing t h i s r e l a t i o n s h i p f o r the Mackenzie d e l t a data i s comparison with t h e o r e t i c a l and other empirical studies, we require a f u n c t i o n a l r e l a t i o n s h i p between "true" values of both v a r i a b l e s rather than a regression r e l a t i o n s h i p between "observed" values (Williams, 1959 > p.19^). The f u n c t i o n a l parameters of a l i n e a r model are d i f f i c u l t to estimate unless the independent v a r i a b l e i s . 35 . measured without erro r . The discussion of the previous section ( I I I , B, 2) i n d i c a t e d that, although both transparency and sediment concen-t r a t i o n observations were affe c t e d by measurement er r o r , the l a t t e r are more precise, p a r t i c u l a r l y since transparency i s to be considered as an index of concentration. Therefore, concentration of suspended load (c) has been tr e a t e d as the independent v a r i a b l e and Secchi disk transparency (SD) as the dependent. The measurement errors associated with values of c were assumed to be i n s i g n i f i c a n t . The parameters of a f u n c t i o n a l r e l a t i o n s h i p are r e l a t i v e l y i n s e n s i t i v e to such errors unless they are very large (Lindley, 1947, p.223). Three l i n e a r models, each i n v o l v i n g d i f f e r e n t v a r i a b l e transform-ati o n s , were tested against the observed data. These were: a) Reciprocal l/SD =/3o+0±c + €. b) Power Function ln(SD) = l n ( / ? 0 ) + ±ln(c) + l n ( e ) c) Complex ln(SD) = l n ( £ 0 ) + ^ 1 ( l n ( c ) ) 2 + ln(e) where /3Q, /3 ^ = f u n c t i o n a l population parameters 6 = r e s i d u a l error SD = Secchi disk transparency (cm.) c = concentration of suspended sediment (mg./l.) The estimated f u n c t i o n a l r e l a t i o n s h i p s f o r a l l data are given i n Table I I I and p l o t t e d i n Figure 6. The r e c i p r o c a l model was suggested by both published i n v e s t i g a -t i o n s and by simple theory. Table I I I i n d i c a t e s that i t explains a s i g n i f i c a n t amount of the t o t a l variance i n Secchi disk transparency. Unfortunately, however, t h i s model v i o l a t e s the i n i t i a l requirement of curve f i t t i n g . As the r e s i d u a l p l o t i n Figure 7 shows, i t does not f i t TABLE I I I RELATIONSHIPS BETWEEN SECCHI DISK TRANSPARENCY AND CONCENTRATION OF SUSPENDED SEDIMENT IN THE LAKE-CHANNEL SYSTEM No. E q u a t i o n s e ( y ) se(b0) seCb^ ) RSQ A l l Data a . R e c i p r o c a l 296 l/SD = 0.0232 + 0.000172 c 0.0168 0 . 0 0 1 1 4 5 . 4 7 x 10" 0 . 7 7 b. Power F u n c t i o n 296 l n ( S D ) = 5.16 - 0 .457 l n ( c ) O.3O8 O.O505 0 . 0 1 2 ? 0 . 8 2 c . Complex 296 l n ( S D ) = 4 . 4 2 - 0 . 0 6 0 8 ( l n ( c ) ) 2 0 . 2 7 5 0 . 0 2 8 4 0.00147 O.85 Complex (One L o c a t i o n ) a . O v e r - a l l 186 l n ( S D ) = 4 . 4 7 - 0 .0622 ( l n ( c ) ) 0 . 2 5 8 0.0379 0.00234 0.79 b . O b s e r v e r N o . 1 94 l n ( S D ) = 4.49 - 0 . 0 6 2 5 ( l n ( c ) ) 2 0.234 0 . 0 4 5 7 0.00300 0 . 8 2 c . O b s e r v e r N o . 2 65 l n ( S D ) = 4 .45 - 0 . 0 6 4 7 ( l n ( c ) ) 2 0.293 0 . 0 7 6 1 Q.00493 0 . 7 3 d . O b s e r v e r N o . 3 27 l n ( S D ) = 4.50 - 0 . 0 6 0 4 ( l n ( c ) ) 2 0 . 2 4 1 0 . 1 2 1 0.00584 0 . 8 1 se('y) = s e ( l / S D ) o r s e ( l n ( S D ) ) as a p p l i c a b l e . A l l v a l u e s o f b^ are s i g n i f i c a n t l y d i f f e r e n t from zero a t t h e 95$ l e v e l o f c o n f i d e n c e . . 36 . 0-24 100 , 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 .Vv Concentration of Suspended Sediment c (mg./l.) 9 0 0 1000 FIGURE 6: MODELS OF THE RELATIONSHIP BETWEEN SECCHI DISK TRANSPARENCY AND CONCENTRATION OF SUSPENDED SEDIMENT IN THE LAKE - CHANNEL SYSTEM , :. + 0-60 V>3 CO . +0-40 - ., +0-20 • o •' ~ 000 "5 CO K -0-20 -0-40 -0-60 +se(l/SD)" 10-04 0 008 "012 i '016 I . I l/SD 0-20 .SQ(I/SD) FIGURE 7: RESIDUALS OF EXPECTED VALUES - THE RECIPROCAL MODEL. + 1-2 +0-8 + 0-4 c -=: 00 •0 V • o »Jt A t • —«, p -^s* p " fl I * " .3 1° :—iH • \ • . • . • • . • • . • . . . . -. -~ - • • - • • • + se(ln(SD))' In(SD) o TJ "co CD .or -0-4 -0-8 - | . 2 i _ 4-8 ;_ -o.- ________ A.--'r__!_!___ _JL__, _ _ J . V * * ' ' -se ( l n ( S D ) ) . e • -•/•FIGURE 8: RESIDUALS OF EXPECTED VALUES - THE POWER FUNCTION MODEL . 3 9 . + 1-2 + 0-8 + 0-4 c o 'Lo © or 0-0 • •• • 9 • • a « • V-'-, . i - - 4 ^ • \Z-4\'~' •••• '3-2 • e» * g * • * • • • ._ _« « • ' | • , + se_(]n_(SD))_ I In(SD) 1-6 -0-4 -0-8 ... ... • '4-8 -38(ln(SD)) - I - 2 1 -FIGORE 9: RESIDUALS OF EXPECTED VALUES - THE "COMPLEX" MODEL . 40 . . 41 . the data. Because of t h i s , the c a l c u l a t e d standard e r r o r s , RSQ value ' and s i g n i f i c a n c e t e s t are meaningless. The power function model, tested because of the high sediment concentrations encountered, i s i n v a l i d f o r the same reason. Residuals f o r t h i s model are p l o t t e d against expected values of ln(SD) i n Figure 8. Figure 8 i n d i c a t e s a tendency f o r the power function model to overestimate both high and low values of transparency and to under-estimate intermediate ones. In a d d i t i o n , along the curve which the r e s i d u a l s form, variance appears to increase s l i g h t l y with i n c r e a s i n g transparency (and thus to decrease with i n c r e a s i n g concentration). A corr e c t model should show constant r e s i d u a l variance f o r a l l values of the independent v a r i a b l e . Both of these problems can l a r g e l y be eliminated by r a i s i n g the independent v a r i a b l e , l n ( c ) , to some power greater than 1.0. In t h i s case the best f i t equation (highest F - r a t i o ) was achieved by using ( l n ( c ) ) ^ * ' ' - V However, because the simpler ( l n ( c ) only marginally decreased the f i t , the equation given i n Table I I I as the complex r e l a t i o n s h i p was used instead. This equation may also be expressed as SD = 83.1 e " °* 6 1 ( l n(°)) 2 (24) The r e s i d u a l p l o t (Figure 9) f o r the complex r e l a t i o n s h i p shows both the improvement i n f i t and s t a b i l i z a t i o n of r e s i d u a l variance. As w e l l , a X 2 t e s t (95$ l e v e l ) i n d i c a t e d no reason to r e j e c t the hypothesis that the r e s i d u a l s are normally d i s t r i b u t e d about the expected values of Secchi disk transparency. Therefore, the c a l c u l a t e d standard errors and s i g n i f i c a n c e t e s t s are meaningful f o r t h i s r e l a t i o n s h i p . The RSQ . 42 . value of 85$ variance explained probably is as well but no test for bivariate normality was carried out. The model is far from perfect, though. Data points for extremely low Secchi depths (very high concen-trations) show considerable deviation from predicted values. Extrapo-lation in this direction, therefore, would be completely unjustified. 4. Observer Bias The influence of possible observer bias on Secchi disk trans-parencies has not been investigated in the published literature. Since transparencies in the lake-channel system were determined by three different observers, the present data provides an opportunity to examine this factor. In order to minimize the effect of other sources of variation, the analysis in this section has been restricted to observations made at a single sampling location in the reversing-flow channel. A l l water samples were collected by the DH48 depth-integrated method and a l l determinations of suspended sediment load were carried out by the same person. Again, the functional relationship is required (Williams, 1959» pp.196-197). The data has been divided by observer into three independent groups and separate equations, using the complex model, calculated for each. These equations, together with the over-all relationship for the sampling location, are given in Table III. Coincidence of the slopes ( 0 ^ ) and the intercepts ( f t Q ) of the relationships was tested by analysis of variance (Table IV), using the method given by Williams (pp.129-133). No reason to reject the hypothesis (Fy Table IV) that the slopes of individual equations were equal to the over-all slope was . 43 . found. As w e l l , there was no reason to doubt the equality of i n t e r -cepts (Fg, Table I V ) . F^ in d i c a t e s that the o v e r - a l l slope i s s i g n i f i c a n t l y d i f f e r e n t from zero. A l l hypotheses were tested at the 95$ l e v e l of confidence. Therefore, we may conclude that the fu n c t i o n a l r e l a t i o n s h i p s f o r each observer are coincident with the o v e r - a l l r e l a t i o n s h i p and that observer bias i s not s i g n i f i c a n t f o r t h i s data. TABLE IV TEST FOR COINCIDENCE OF FUNCTIONAL RELATIONSHIPS FOR DIFFERENT OBSERVERS Source of Variance DF SS. MS F O v e r - a l l Equation 1 46.8236 46.8236 F l = 710.52 Difference of 2 0.3231 0.l6l6 F2 = 2.45 Intercepts (/9Q) Difference of 2 0.0018 0.0009 F3 = 0.01 Slopes ( P±) 11.8631 0.0659 j Combined Residual 180 T o t a l 185 59.0116 This conclusion must, of course, be r e s t r i c t e d to the range of transparencies encountered and to the p a r t i c u l a r observers involved. At very low sediment concentrations transparency becomes a much more se n s i t i v e i n d i c a t o r of attenuation. As a r e s u l t , i f the r e l a t i o n s h i p i s used to estimate sediment concentration, observer bias may cause a large r e l a t i v e e r r o r when t u r b i d i t y i s low. In absolute terms, however, t h i s e r r o r w i l l probably be quite small. . 44 . 5. C o l l o i d a l and Dissolved Load The di f f e r e n c e between t o t a l and suspended loads f o r each sample was assumed to be the c o l l o i d a l and d i s s o l v e d load. This would include organic and mineral p a r t i c l e s of l e s s than 1 micron diameter plus diss o l v e d s a l t s and yellow substance. Dissolved s a l t s have no e f f e c t on l i g h t attenuation and thus none on Secchi disk readings. However, the presence of f i n e p a r t i c u l a t e matter and of yellow substance might account f o r a portion of the variance i n transparency not explained by sediment concentration. Available data provides l i t t l e evidence to support t h i s hypothesis. No trend was apparent i n a scatter diagram of c o l l o i d a l and dissolved l o a d versus transparency and product-moment c o r r e l a t i o n c o e f f i c i e n t s f o r the r e c i p r o c a l , power function and complex models were very close to zero and i n s i g n i f i c a n t at the 95$ l e v e l of confidence. The absence of any r e l a t i o n s h i p may be misleading, however. The previously discussed inaccuracies i n t o t a l load determinations and the presence, i n l a r g e r e l a t i v e q u a n t i t i e s , of dissolved s a l t s may have masked any trend. For 23 samples on which complete chemical analyses were run, d i s s o l v e d s a l t s formed 75 to 100$ by weight of the c o l l o i d a l and d i s s o l v e d load. Studies by many i n v e s t i g a t o r s suggest that f i n e mineral and organic p a r t i c l e s are u n l i k e l y to have a f f e c t e d transparency ( J e r l o v , 1968, p.41). The same i s not true f o r yellox* substance, as we have already seen. Residues on i g n i t i o n of the c o l l o i d a l and d i s s o l v e d load f o r the same 23 samples (above) i n d i c a t e d that up to 20$ may have been organic i n nature. Of t h i s , a portion may have been dissolved (yellow . 45 . substance) rather than p a r t i c u l a t e . Although the amount present was too small to have influenced the scat t e r diagram, i t may have aff e c t e d transparency measurements, p a r t i c u l a r l y when suspended sediment concentrations were low. Again, the absolute e r r o r i n estimated sediment concentration w i l l probably be small. 6. Conclusions The r e s u l t s of the Mackenzie d e l t a study are of p a r t i c u l a r i n t e r e s t f o r several reasons. F i r s t , the existence of a r e l a t i o n s h i p between Secchi disk transparency and concentration of suspended load at t u r b i d i t i e s w e l l above those encountered i n other f i e l d i n v e s t i g a t i o n s has been demonstrated and i t s complexity confirmed. Second, observer bi a s has been examined e m p i r i c a l l y f o r the f i r s t time and, f o r t h i s study at l e a s t , has been shown to be i n s i g n i f i c a n t . T h i r d , the lack of influence of v a r i a t i o n s i n the concentration of dissolved s a l t s on r e l a t i o n s h i p s of t h i s type has also been confirmed. F i n a l l y , and most importantly, the usefulness of Secchi disk measurements i n low energy f l u v i a l environments i s w e l l i l l u s t r a t e d by the strength of the established r e l a t i o n s h i p (85$ variance explained). The f u n c t i o n a l equation developed f o r the sample lake-channel system (Table I I I and Equation (24)) should not be used to p r e d i c t sediment concentrations. I t i s u s e f u l only f o r comparison with theory and with other empirical r e l a t i o n s h i p s . For p r e d i c t i o n , we require a regression r e l a t i o n s h i p between "observed" values of concentration and transparency (Williams, 1959* P.195)» Measurement errors are no longer a consideration so transparency should be used as the independent v a r i a b l e . The r e s u l t i n g p r e d i c t i v e equation . 46 . (ln(c)) 2 = 64.36 - 14.02 ln(SD) (25) is plotted in Figure 6. CHAPTER IV IN THE CONTEXT OF UNDERWATER OPTICAL THEORY Although some simple theory has been introduced, the preceding discussion of r e l a t i o n s h i p s between Secchi disk transparency and concentration of suspended load has been based l a r g e l y upon empirical studies. The problem can, however, be approached from a more t h e o r e t i -c a l point of view. Duntley (1962, 1963) has formed a general theory of underwater v i s i b i l i t y based on the r a d i a t i v e t r a n s f e r equation. This theory has been applied d i r e c t l y to Secchi disk v i s i b i l i t y by T y l e r (1968). In addition, Burt (195*+, 1955» 1956, 1957) has used the Mie theory f o r the perturbation of an electromagnetic wave by s p h e r i c a l p a r t i c l e s to form an expression r e l a t i n g the volume attenuation c o e f f i c i e n t cx to the t o t a l surface of p a r t i c l e s i n a suspension. These theories are b r i e f l y reviewed below and, i n the l a s t section of t h i s chapter, are applied to the Mackenzie d e l t a data to get estimates of p a r t i c l e s i z e . A. SECCHI DISK TRANSPARENCY AND ATTENUATION As was noted i n the i n t r o d u c t i o n to t h i s paper, the v i s i b i l i t y of a Secchi disk i n water can be treated as a problem of contrast transmittance. The contrast of an object which emits radiance N and i s seen against a uniform background of radiance N^ i s defined as ( J e r l o v , 1968, p.137) c = * L ^ I k (26) Nb . 47 . . 48.. In natural water bodies in daylight a background always exists because of scattering. Duntley (1962, 1963) has developed a general expression for the visibility of any underwater object by introducing two types of contrast: the inherent contrast C 0 when the object is viewed from zero distance and the apparent contrast C r when the object is viewed from distance r. Visibility depends upon C r which, in turn, is dependent upon C 0. Duntley uses the definition of the diffuse attenuation coefficient N(zob,e,tf) =N(zt,e,c/) e (27) where z o b = depth of observer z-t = depth of object e = zenith angle of the path of sight <p = azimuth of the path of sight (a more general form of Equation (4) expressed in terms of radiance) and the equation of transfer for spectral field radiance = N*(z,e,^) - «xU)N(afe,rf) (28) where N* = path function representing the gain of radiance by scattering to derive an expression relating C r to C0. For the special case of an object suspended in deep water - ie., assuming the effect of bottom reflection is negligible, r (n * A\ - n ( « * A\ ~ " <*( z) r + K(z,e,fOrcos(e) ,OQs cr\ zob ' e * P J C0(zt,e,c4; e (29; . 49 . A Secchi disk is observed downward along a vertical path. In this situation e = 180° and C0, C r and K = K d are independent of <jf If we assume z o b = 0 (the water surface) and z t = SD (the Secchi depth), Equation (29) reduces to c r - c 0 . - < 0 < + I a ) r (30) A disk reading is simply the value of r (ie., SD) when C r reaches the threshold of visual contrast. The applicable values of cx and K d are means for photopic band width between the water surface and the Secchi depth. A method for estimating C 0 in Equation (30) is presented by Tyler (1968, p.2). Assuming no surface reflection, the irradiance on a Secchi disk at depth r = SD will be (from Equation (4)) Hd(SD) = H d(0) e " K d S D ( 3 D where Hd(SD) = downwelling irradiance at depth SD H^(0) = downwelling irradiance at the water surface The radiant emittance of the disk will be Hd(SD)Rsrj where RgD is its measured reflectance. The radiant emittance of the water background at depth SD will equal the upwelling irradiance HU(SD) at that depth. This can be obtained from the reflectance factor Rd(SD) = Hu(SD)/Hd(SD) for water which, for photopic band width, usually has the value 0.02. If the same directional reflectance characteristic is assumed for the disk and for the water background, the inherent spectral contrast of the disk can be expressed as (after Equation (26)) . 5 0 . = H d(SD)R S D - 0 . 0 2 H d(SD) 0 . 0 2 H d(SD) _ R Sp - 0 » 0 2 0 . 0 2 ( 3 2 ) T y l e r attempted to t e s t h i s approach by using the data from Manning Cruise 3 6 (Graham, 1 9 6 6 ) . He assumed a value of 0 . 8 2 f o r R S D and determined a value of 0 . 0 0 6 6 f o r C r. This l a t t e r f i g u r e i s based on tabulated data f o r contrast thresholds of the human eye given the angular subtense of the submerged Secchi disk and the reported l i g h t l e v e l . With these numbers Equation ( 3 0 ) becomes ( T y l e r , 1 9 6 8 , p . 3 ) The form of Equation ( 3 3 ) i s of major importance because i t i n d i c a t e s that Secchi disk transparency depends upon both the d i f f u s e and volume attenuation c o e f f i c i e n t s . The e m p i r i c a l l y derived r e c i p r o c a l models discussed e a r l i e r i n t h i s paper include one or the other but not both. These models require the i m p l i c i t assumption, therefore, that <x/Kd = constant. This i s not u s u a l l y true and, because i t i s n ' t , the empirical power function r e l a t i o n s h i p s can be r a t i o n a l i z e d . As water t u r b i d i t y increases, the proportion of m u l t i -path (scattered) to monopath ir r a d i a n c e also increases. Since, as we have seen, o< w i l l equal K d only when there i s no multipath i r r a d i a n c e , the r a t i o oc/Kd w i l l also increase with t u r b i d i t y . This causes a decrease i n the depth of disappearance of a Secchi disk greater than can be accounted f o r s o l e l y by an increase i n K d. The discrepancy occurs because the rate of increase of cx must exceed that of K d i f the r a t i o °</K d i s to increase simultaneously (by Equation ( 3 3 ) ) . This ( O C + K d) = 8 . 6 9 SD ( 3 3 ) . 51 • condition i s met by the power function model (See pp.14-15 and Figure 1, above.). I t i s not met by the simple r e c i p r o c a l model i n v o l v i n g only one attenuation c o e f f i c i e n t . T y l e r ' s study r a i s e s a point which has not yet been considered i n t h i s paper. His value of C r i s dependent upon the angle subtended by the Secchi disk - i e . , on disk diameter. Empirical research has l e d to divergent opinions on t h i s e f f e c t . Juday and Birge (1933» P«205) found only a 5$ difference between readings taken with disks of 10, 25, 50 and 75 cm. diameter and concluded that disk diameter has no s i g n i f i -cant influence on Secchi depth. V i s s e r (1967), however, noted a l a r g e r d i f f e r e n c e between 1 f t . and 1 m. disks. The extent of influence of disk diameter appears to hinge on two points. Disk diameter w i l l be important i f the angle i t subtends i s greater than the minimum angle which the eye can resolve (approximately 1 min. of arc) and/or i f the water surface i s rough. For V i s s e r ' s data t h i s minimum c r i t i c a l angle was not exceeded by e i t h e r disk but surface roughness was found to be of major importance. T y l e r ignores surface roughness but considers a disk which subtends 22 min. of angle. Juday and Birge's conclusion i s probably i n c o r r e c t given the r e l a t i v e l y small Secchi depths i n t h e i r lakes. B. ATTENUATION AND SUSPENDED LOAD Mie theory has been used by Burt (1954, 1955) to derive an expression r e l a t i n g the p a r t i c l e volume attenuation c o e f f i c i e n t \<Xp to the number of p a r t i c l e s i n a monodisperse suspension. The treatment assumes that the p a r t i c l e s are s p h e r i c a l , of uniform diameter and of known r e f r a c t i v e index, that the scattered l i g h t has the same wavelength . 52 . as the i n c i d e n t l i g h t , and that the p a r t i c l e s are independent. Independence requires the spheres to be separated by a distance greater than three times t h e i r radius. M u l t i p l e s c a t t e r i n g i s assumed to be n e g l i g i b l e so t o t a l s c a t t e r i n g i s proportional to the number of p a r t i c l e s . ( J e r l o v , 1968, pp.28-29) The r e s u l t i n g r e l a t i o n s h i p c X p = n K s n d 2 (34) i s equivalent to that proposed i n Equations (13) and (14) of t h i s paper except f o r the int r o d u c t i o n of the e f f e c t i v e area c o e f f i c i e n t K s. Burt (1955* P'35) defines t h i s c o e f f i c i e n t as "the dimensionless number by which the geometric cross-s e c t i o n a l area of a p a r t i c l e ( i n the plane perpendicular to the d i r e c t i o n i n which the o r i g i n a l d i r e c t l i g h t beam i s t r a v e l l i n g ) must be m u l t i p l i e d i n order to determine i t s effectiveness i n removing or d e f l e c t i n g l i g h t from a d i r e c t beam by processes of s c a t t e r i n g and absorption." Values of K s can be determined from the Mie equations, where K s = f ( d , \, m(l - a p i ) ) (35) and d = p a r t i c l e diameter X= wavelength of the i n c i d e n t l i g h t m = r e f r a c t i v e index of the ma t e r i a l i n the p a r t i c l e r e l a t i v e to that of the suspending medium a p = p a r t i c l e absorption c o e f f i c i e n t i = V^ T For n a t u r a l waters, Burt (1955* P»3^) suggests that no appreciable error i s introduced by the presence of non-spherical p a r t i c l e s . The Mie theory may be used under these conditions as long . 53 . as the r e l a t i v e r e f r a c t i v e index m i s l e s s than 2.0. Both Burt (1955» p.36) and Ochakovskiy (19&6, p.12) found the mean value of m f o r mineral p a r t i c l e s to be about 1.15. Burt (1955> P«35) also makes the more unreasonable assumption that no p a r t i c l e absorption occurs. Given t h i s , the imaginary term api i n Equation (35) drops out and Ks becomes a function of the r e a l index m. This assumption could introduce appreciable e r r o r i n r e s u l t s based upon Equation (3*0. The change i n water colour from blue to brown with i n c r e a s i n g sediment concentration i s due p r i m a r i l y to s e l e c t i v e absorption by p a r t i c l e s and yellow substance. Equations (14), (16) and (34) can be combined to give an expression f o r cXp i n terms of sediment concentration and p a r t i c l e diameter: This equation assumes non-absorbing p a r t i c l e s of known density ( ? ) and r e f r a c t i v e index (m). Values of Ks f o r given m, d and X can be obtained from a diagram presented by Burt (1956, p.78). For i n c r e a s i n g p a r t i c l e radius (d/2), the c o e f f i c i e n t increases r a p i d l y f o r small r a d i i , approaches a maximum f o r sizes of the same order as X, and tends toward a constant value of 2.0 f o r large sizes (d greater than 2>t) i r r e s p e c t i v e of the wavelength ( J e r l o v , 1968, p.32). C. ESTIMATION OF PARTICLE SIZE Ty l e r ' s and Burt's t h e o r e t i c a l treatments can be employed to get estimates of p a r t i c l e s i z e given only an empirical r e l a t i o n s h i p between Secchi disk transparency and sediment load and some knowledge of the . 5k . r a t i o cx/K d. The Mackenzie d e l t a r e s u l t s are used below to i l l u s t r a t e the procedure. The r a t i o of cx to K<j i s not known f o r the Mackenzie data and i s probably not constant i n any case. We do know, however, that the water involved was quite t u r b i d and that cx/K^ w i l l be r e l a t i v e l y large under these conditions. As w e l l , changes i n p a r t i c l e s i z e have been shown to be much more important i n determining the form of a Secchi depth-concentration r e l a t i o n s h i p (Jones and W i l l s , 195&; Postma, 196l) than are v a r i a t i o n s i n the cx/K d r a t i o . L e t us assume, then, f o r the sake of example, that cx/K^ i s constant and equal to 3*0» Equation (33) reduces to i - 0 . 1 5 3 < * (37) where cx i s a mean value between the surface and depth SD f o r photopic band width. Equation (36) i s expressed i n terms of the p a r t i c l e volume attenuation c o e f f i c i e n t c<p rather than the t o t a l c o e f f i c i e n t ex. used i n Equation (37). The e f f e c t of the water i t s e l f i s not included i n cXp. Previous studies have shown, however, that i n water of moderate to high t u r b i d i t y , attenuation i s due almost e n t i r e l y to sc a t t e r i n g by suspended p a r t i c l e s (Otto, 1966, p.39). Therefore, we may combine Equations (36) and (37) to get JL = 0.230 K s C (38) SD J ? d K J J Scattering i s also thought to be produced c h i e f l y by p a r t i c l e s of greater than Zjx diameter ( J e r l o v , 1968, p.41). Thus K s may be 55 assigned the value 2.0. I n addition, since organic content was low i n the Mackenzie water, appreciable error w i l l probably not be introduced by assuming a l l the suspended matter to have been mineral i n composition and of density 2.65 gm./ml. Equation (38) then becomes X = 0.00174 |- (39) with u n i t s SD (cm.) c (mg./l.) d 00 TABLE V PARTICLE SIZE ESTIMATION Sediment , H X Secchi , s Predicted P a r t i c l e Size 00 Concentration g* * Depth C m ' T h e o r e t i c a l Postma 50 33 2.9 2.5 100 23 4.0 3.5 300 r 12 6.3 5.2 500 8.0 7.0 6.0 1000 4.6 8.0 6.9 Given values of SD and c, Equation (39) can be used to compute p a r t i c l e diameters d. Table V l i s t s some p a r t i c l e sizes computed f o r selected concentrations and e m p i r i c a l l y derived (Equation (24)) Secchi depths. The p h y s i c a l meaning of these p a r t i c l e diameters i s d i f f i c u l t to evaluate since a c t u a l p a r t i c l e s i z e d i s t r i b u t i o n s are not known f o r t h i s data. They might best be described as e f f e c t i v e o p t i c a l diameters a f t e r Postma (196l, p.367). . 56 . The t h e o r e t i c a l l y predicted p a r t i c l e sizes are compared i n Table V with those computed from Postma's (196l) completely empirical equation. The form taken by h i s r e l a t i o n s h i p has already been discussed (Equation (20)). The actual equation, given here f o r the same u n i t s as Equation (39)» i s = 0.0015 I (40) Given the many assumptions required f o r the t h e o r e t i c a l approach, the close s i m i l a r i t y of constants i n Equations (39) and (40) i s undoubtedly somewhat c o i n c i d e n t a l . The value f o r cx/K d, f o r example, was chosen almost at random. There i s , at present, i n s u f f i c i e n t data a v a i l a b l e f o r a quantitative study of the e f f e c t s of each of these assumptions under changing natural conditions. CHAPTER V SUMMARY AND CONCLUSIONS The discussion i n t h i s paper leaves no doubt that Secchi disk transparencies can be used to estimate concentrations of suspended load and that s t a t i s t i c a l l y s i g n i f i c a n t r e l a t i o n s h i p s can be established i n environments as d i f f e r e n t as c l e a r oceanic zones and quite t u r b i d , low energy f l u v i a l environments. As t u r b i d i t y increases, however, empirical r e l a t i o n s h i p s become more complex i n form and, because of t h e i r general r e c i p r o c a l nature, the s e n s i t i v i t y of transparency as an i n d i c a t o r of sediment concentration decreases r a p i d l y . At concentrations of over 1000 mg./l., an almost imperceptible change i n Secchi depth may mean a change i n sediment load of several hundred mg./l. There i s , therefore, an upper bound i n t u r b i d i t y beyond which the Secchi disk becomes useless as an i n d i c a t o r . One of the major advantages of the Secchi disk i s i t s s i m p l i c i t y . This very s i m p l i c i t y , however, and the r e s u l t a n t large number of uncontrolled v a r i a b l e s f o r any set of readings, makes ph y s i c a l i n t e r p r e t a t i o n of empirical r e s u l t s d i f f i c u l t . Such i n t e r p r e t a t i o n i s f a c i l i t a t e d , though, by the int r o d u c t i o n of the objective attenuation c o e f f i c i e n t s as v a r i a b l e s intermediate between transparency and sediment concentration. In t h i s way the influence of changes i n the underwater l i g h t f i e l d can be separated from that of changes i n the water-sediment mixture. The most s i g n i f i c a n t r e s u l t to be brought out by t h i s approach i s the dependence of Secchi disk transparency on t o t a l . 57 . . 58 . p a r t i c l e surface and thus on p a r t i c l e size as well as concentration. This dependence on p a r t i c l e s i z e means that empirical transparency-concentration r e l a t i o n s h i p s apply only to the p a r t i c u l a r areas f o r which they have been determined. The t h e o r e t i c a l treatments of T y l e r (1968) and Burt (195^, 1955) provide valuable i n s i g h t s i n t o the nature of transparency-concentration r e l a t i o n s h i p s and, as w e l l , have p o t e n t i a l l y important a p p l i c a t i o n s . Estimation of p a r t i c l e size i s an excellent example of t h i s l a t t e r p oint. Both theories, however, require that major and, as yet, l a r g e l y untested assumptions be made before they can be applied to nat u r a l s i t u a t i o n s . Burt, f o r example, does not consider absorption by water, p a r t i c l e s or dissolv e d organic compounds (yellow substance). Tyler's equation (Equation (33)) demands knowledge of the cx/K d r a t i o . This r a t i o can vary with t u r b i d i t y i n a given area as we l l as between areas. At the present time, then, e m p i r i c a l l y derived equations are of more p r a c t i c a l value. Further study i s necessary before the t h e o r e t i c a l equations can be applied with confidence. This leads to c e r t a i n r e s t r i c t i o n s , however. The a r e a l l i m i t a t i o n of i n d i v i d u a l empirical equations has already been discussed. As w e l l , the establishment of such equations requires that a greater number of actual measurements of sediment concentration be made than would be necessary i f a proven t h e o r e t i c a l r e l a t i o n s h i p could be applied. The lack of a t h e o r e t i c a l base f o r these r e l a t i o n s h i p s also means that extrapolation beyond measured concentration extremes would be very dangerous. There are, i n addition, a number of l i m i t a t i o n s on where Secchi depths can or should be used to estimate sediment concentrations. The . 59 . existence of an upper concentration bound has been mentioned. The method i s also i n a p p l i c a b l e i n h i g h l y turbulent flow and i n water bodies whose depths are so shallow that the disk c l o s e l y approaches the bottom before the point of disappearance (because of the e r r o r introduced by bottom r e f l e c t i o n ) . The presence i n a water body of large and v a r i a b l e quantities of yellow substance w i l l diminish the strength of or even destroy any r e l a t i o n s h i p between Secchi depth and sediment concentration. F i n a l l y , since only mean concentration values between the surface and the Secchi depth can be obtained, disk measurements should not be used where v e r t i c a l v a r i a t i o n s i n concentration are i r r e g u l a r or where point values are required. A beam transmittance meter would be more adequate. In s p i t e of these r e s t r i c t i o n s and l i m i t a t i o n s , however, the Secchi disk can be a most valuable instrument. I t i s dependable and simple to use i n the f i e l d , i s remarkably i n s e n s i t i v e to changes i n the c h a r a c t e r i s t i c s of the underwater l i g h t f i e l d , and i t does not seem to be greatly influenced by observer b i a s . Most importantly, of course, as i s i n d i c a t e d by the Mackenzie data, transparency can be strongly r e l a t e d to sediment concentration over a wide range of t u r b i d i t i e s . Where a l a r g e number of average concentration estimates are necessary and where time and/or expense are important, the Secchi disk i s unexcelled. LITERATURE CITED Burt, W.V. 1954. "Specific Scattering by Uniform Minerogenic Suspen-sions," Tellus, 6, 229-231. . 1955« "Interpretation of Spectrophotometer Readings on Chesapeake Bay Waters," J. Marine Res., 14(1), 33-46. . 1956. "A Light-Scattering Diagram," J. Marine Res., 15(1)» 76-80. . 1957. "On the Attenuation of Light in the Sea," J. Marine Biol. Assoc. U.K., 36(2), 223-226. Clarke, G.L. 1941. 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