Utilizing the components of vector irradiance to estimatethe scalar irradiance in natural waters

J. H. Jerome, R. P. Bukata, and J. E. Bruton

A Monte Carlo computer simulation has been used to determine the ratio of the scalar irradiance E0 to thedownwelling irradiance Ed. These EO/Ed ratios were calculated at depths corresponding to the 100, 10, and1% downwelling irradiance levels. A range of volume reflectance 0 < R < 0.14 was considered, as were sixconditions of incident radiation (collimated beams with incident angles 0 S 0' < 890 plus a diffuse cardioidaldistribution). Mathematical expressions were curve fitted to the Monte Carlo outputs to yield relationshipsbetween EO/Ed and R for the depths and incident conditions considered. It was found that in many cases asingle relationship would not accommodate the entire range of volume reflectances and that R = 0.055provided an appropriate demarcation for mathematical curve fitting. Curves, tables, and equations arepresented which indicate (a) for all R > -0.02, the EO/Ed ratio at the 1% downwelling irradiance depth is thesame for 0' = 0 as for diffuse cardioidal incidence, and (b) for R > -0.08, the EO/Ed ratio at the 10%downwelling irradiance depth for 0' = 00 is nearly the same as the EO/Ed ratio at the 1% downwelling irradiancedepth for diffuse cardioidal incidence.

1. Introduction

The subsurface downwelling and upwelling irra-diances (the components of the vector irradiance) havelong been the commonly measured in situ quantities inoptical studies of natural water masses. The populari-ty of such optical properties is due, in no small part, tothe relative ease encountered in their measurementand the availability of reliable instrumentation withwhich to perform such measurements. However, sincethe subsurface downwelling and upwelling irradiancesare vector quantities, their measured magnitudes arestrongly influenced by the directions of the incidentphotons, and consequently they are not the most ap-propriate parameters for monitoring the energy towhich an algal cell or photodegradable contaminant isexposed.

The scalar irradiance is the total energy per unit areaarriving at a point from all directions when all direc-tions are equally weighted. Thus, when divided by thespeed of light in water, the scalar irradiance yields theradiant energy density, i.e., the radiant energy per unit

The authors are with Canada Centre for Inland Waters, NationalWater Research Institute, Rivers Research Branch, P.O. Box 5050,Burlington, Ontario L7R 4A6, Canada.

Received 11 January 1988.0003-6935/88/194012-07$02.00/0.© 1988 Optical Society of America.

volume at a point within the water column. The ap-propriateness of utilizing scalar irradiance measure-ments to monitor the available energy for photosyn-thesis has long been advocated.1' 2

This paper presents the results of employing MonteCarlo simulations of photon propagation through nat-ural water columns to determine the functional rela-tionships among upwelling and downwelling irra-diances (i.e., components of the vector irradiance) andthe scalar irradiance.

II. Methodology

A fixed scattering to absorption ratio (b/a) and afixed volume scattering function [(0)] were suppliedas inputs to the Monte Carlo simulation of photonpropagation. The water column was assumed to behomogeneous. For a given incident light distributionthe photons were tracked and recorded to give thescalar irradiance levels E0 at the depths Zloo, Z1 o, andZ1 corresponding to the depths of the 100, 10, and 1%downwelling irradiance levels Ed. In addition, theMonte Carlo simulation was used to determine theupwelling irradiance E at Z100 (i.e., just beneath theair-water interface).

From these values, the ratios E/Ed were readilydetermined at Z1oo, Zo, and Z, as was the volumereflectance R = E/Ed at Z100 (i.e., R just below thesurface). (This value of R at Z100 is the variable con-sidered in all subsequent equations here. The rela-tionship of this variable to the inherent optical proper-ties of the water is discussed in the Appendix.) R, E0,

4012 APPLIED OPTICS / Vol. 27, No. 19 / 1 October 1988

Eu, and Ed are functions of wavelength. To simplifythe ensuing nomenclature, such wavelength depen-dency, while understood, will not be explicitly ex-pressed.

To illustrate that the appropriate choice in indepen-dent variable for defining the quantity EO/Ed is thevolume reflectance R (proportional to Bb/a, where B isthe backscattering probability) rather than the scat-tering to absorption ratio b/a, the Monte Carlo simula-tion was utilized for a single fixed value of b/a andthree volume scattering functions3 with backscatter-ing probabilities B = 0.013, 0.025, and 0.044. Thisresulted in three distinct values of EO/Ed for the singlevalue of b/a. These three values of EO/Ed, however,were linearly related to Bb/a, indicating in agreementwith the work of Prieur and Sathyendranath, 4 that theratio EO/Ed is a function of the volume reflectance R.Thus, for the purpose of this work, the volume reflec-tance was considered as the controlling parameter, andits value was varied by ascribing a fixed value of B (inthis analysis B = 0.025) to a set of b/a values whichincluded b/a = 1, 3, 5,8, 10, 12, and 18. Six conditionsof incident radiation were considered, correspondingto collimated beams with incident zenith angles 6' of 0,30, 45, 60, and 890 as well as a diffuse cardioidal inci-dent distribution.

III. Analyses

A. For Z100 (i.e., the 100 % Downwelling IrradianceDepth)

Figure 1 illustrates the value of EO/Ed resulting fromthe Monte Carlo simulation for collimated verticalincidence (' = 00) and Z1oo as a function of volumereflectance R. (R is the value appropriate to Z1oo and0' = 0°. A method for obtaining this R from a volume

reflectance associated with a nonvertical incident radi-ation field is given in the Appendix.) The least-squares fit to the values of Fig. 1 yields

Eo (R,00) = 1 + 3.13R,Ed (1)

with a coefficient of variation r2= 0.99.

Prieur and Sathyendranath 4 express the Ea/Ed ratioas

-(R) = 1 +Ed Ad Au

(2)

where i-d and iiu are the mean cosines for the downwell-ing and upwelling irradiance fields, respectively.

Comparing Eqs. (1) and (2) would suggest that 1/1ud= 1.0 and 1/~,u = 3.13. For the diverse radiant fieldsand depths considered by Prieur and Sathyendran-ath,4 the values of 1/uu varied from 2.3 to 3.0. Thisplaces the 1/j,, value suggested by Eq. (1) just outsidethis range. This is a consequence of 1/Md not beingexactly equal to 1.0 for vertical incidence due to inter-nal reflections of the upwelling irradiance. Since theamount of total internally reflected energy will in-crease with increasing R, this departure of Md from avalue of 1.0 (corresponding to the condition of vertical

E (FEfd

1.

1A

1.3-

1.2

E (R) 1.00 + 3.13 R

1.1-

1.00 0.02 0.04 0.06 0.08 O.iO 0.12 0.1'

Volume reflectance., R

Fig. 1. Relationship between the ratio EO/Ed and volume reflec-tance for ZQ00 and vertical incidence.

° (R,9Ed

Ed (R,'EdT

1.0 1.1 1.2 1.3 1.4 1.5

I/Po

Fig. 2. Ratio Eo/Ed for Z1QQ and incident angle 0' to Eo/Ed for Zlooand vertical incidence plotted as a function of the inverse of the

cosine of 0o.

incidence with no volume reflectance) is incorporatedinto the constant of 3.13. Consequently, it must beemphasized that the two terms on the right-hand sideof Eq. (1) are not directly comparable with the twoterms on the right-hand side of Eq. (2).

Figure 2 displays the dependence of the Eo/Ed ratioon the directionality of the incident radiation field.Herein is plotted the ratio of the value of EO/Ed for agiven angle 0' to its value for vertical incidence (i.e., 6' =0°) as a function of 1/Mo (the inverse of the cosine of thein-water angle of refraction 00). The plotted ordinatevalue for each of the five incident angles of Fig. 2 is theaverage of the ordinate values for each of the sevenvolume reflectance values considered in this work.

1 October 1988 / Vol. 27, No. 19 / APPLIED OPTICS 4013

1.5-

1.4-

1.3

1.2-E0 (R )

1.068 _ 0.068

E (0) 1Ia

1.1d (R'°E

A.-

0.

,/oFig. 3. Total internal reflection PINT plotted as a function of the

inverse of the cosine of 00.

0 .02 .04 .06 .08 .10 .12 .14

Volume reflectance, R

Fig. 4. E/Ed vs R curves fitted to Monte Carlo simulation outputsfor Z00 and 0' = 0, 30, 45, 60, and 890.

The maximum standard deviation observed for theseaverages is 0.012.

The linear relationship of Fig. 2 is given byEQ (1068 EQ-(R,0') = - 0.068 (R,0 0), (3)

Ed A0 Ed

with r2 = 0.99.The slope of Fig. 2, being greater than 1, indicates

that the value of EO/Ed as a function of R and ' in-creases with increasing 0 at a rate greater than thatwhich would result from the cosine effect alone. Reit-erating, this is likely attributable to total internal re-flection. Figure 3 illustrates the total internal reflec-tion PINT experienced by the upwelling subsurfaceirradiance plotted as a function of /Mo. The relation-ship of Fig. 3, which emerged from the Monte Carlosimulation, may be expressed as

0.249PINT = -- + 0.271 (4)

with r2 = 0.98. For the condition of vertical incidence(Mi = 1) Eq. (4) yields a value of PINT = 0.520, whichmay be compared to the value of 0.485 obtained byGordon5 assuming a diffuse upwelling field. Thus, fora given volume reflectance R, the total internal reflec-

tion PINT increases with increasing incidence angle 6',contributing to the slope >1 of Fig. 2.

Combining Eqs. (1) and (3) the ratio EO/Ed at Z100 isgiven by

Eo (R,0') =(1068 _ 0.068) [1 + 3.13R].Ed (AO

(5)

It is understood in Eq. (5) that while the ' valuesmay vary between 0 and 890, the R values are con-strained to the value for 0' = 0°. Figure 4 illustratesthe Eo/Ed vs R curves resulting from Eq. (5) for Zi 0oand angles ' = 0, 30, 45, 60, and 890 (reading frombottom to top, respectively). For comparison, the val-ues resulting from the Monte Carlo simulation pro-gram are shown as points. The average magnitude ofthe differences between the Monte Carlo and Eq. (5)outputs is -0.5% with a maximum difference of ±1.3%.

B. For Z10 (i.e., the 10% Downwelling Irradiance Depth)

It was assumed that an appropriate equation relat-ing Eo/Ed to R was of the form

Ed (R,O') = [1 + (C1 + C2R + C3R2)R]C4, (6)

where Cl, C2, C3, and C4 are constants, and R is thevolume reflectance value for Z10 0 and 6' = 0°.

Equation (6) appropriately reduces to EO/Ed = 1/Moin the absence of volume reflectance (i.e., for the condi-tion of b = 0).

However, great difficulty was encountered when at-tempting to fit Eq. (6) to the outputs of the MonteCarlo simulation as a single continuous curve over theentire range of R considered herein, strongly sugges-tive of the need to consider curve fitting to limitedranges of volume reflectance.

Furthermore, it was noted that for a given R changesin Eo/Ed varied linearly with 1/,Mo. It was deemedappropriate, therefore, to curve fit Eq. (6) (segmental-ly in R) for the enveloping ' conditions of 0' = 0 and 0'= 89° and interpolate for the intermediate values of 6'.The results of such curve fitting were as follows:

For the case of ' = 00,

EQ QEd (R,00) = [1 + (28.0 -50.5R)R]°Ed

(7)

for 0 S R S 0.14, i.e., for the entire range of volumereflectance values considered by this work to representlogically that which would normally be observed inboth inland lake and ocean waters for 0' = 00. Extrap-olation of Eq. (7) beyond R = 0.14 would be consideredtenous.

For the case of ' = 890 the volume reflectance rangewas considered to be appropriately divided into a non-linear range (0• R S 0.055) and a linear range (0.055 SR 0.14). Utilizing this demarcation of volume re-flectance into a lower and a higher range, the resultingpair of equations became

- (R,890) = 1.512[1 + (7.39 - 149R + 1376R2 )R]Ed

for 0 S R S 0.055 and

(8)

4014 APPLIED OPTICS / Vol. 27, No. 19 / 1 October 1988

E (R,89 0) = 1.60 + 3.43REd (9)

for 0.055 S R 0.14.As mentioned earlier, the estimation of Eo/Ed for

values of ' intermediate to 0 and 89° was accom-plished by interpolation using a linear function of 1/Mto.Thus

E (R,0') =E (R,00) + cos486° ) ( toEd Ed 1 - cos48.6' / \ 1AQ /

X EQ EQ 1XI - (R,890) - -(R,00)1,[Ed Ed J (10)

where Eqs. (7) and (8) are inserted for the volumereflectance range 0 R 2 0.055, and Eq. (7) and (9) areinserted for the volume reflectance range 0.055 S R '0.14.

Figure 5 shows the curves resulting from the use ofEqs. (7), (8), and (9) and their interpolation Eq. (10).Also shown are the (Eo/Ed,R) pairs resulting from theMonte Carlo simulation. The average magnitude ofthe differences between the curves and Monte Carlopoints of Fig. 5 is 0.7% with a maximum difference of±2.8%, indicating an excellent agreement among thecurve-fitting techniques and Monte Carlo analyses forZ1 o.

C. For Z1 (i.e., the 1 % Downwelling Irradiance Depth)

At the depth of the 1% downwelling irradiance levelit was again deemed appropriate to segment the vol-ume reflectance curve fitting at R = 0.055. The equa-tions resulting from the regression analyses for thelower range of volume reflectance values (0 S R <0.055) were

° (R,00) = [1 + (39.1 - 176R)R]05 , (11)Ed

E(R,89°) = 1.512[ + (-2.10 + 117R-582R )R] (12)

Ed (R, 0) =Ed (R,00) + cos48.60 )l 1 - AAEd Ed ki - cos48.60 ) Ao "

rEQ EQ X (R.890 )- - (R,00 ) * (13)

[Ed Ed J

For the higher range of volume reflectance values(0.055 R 0.14) it was found that for a given R theMonte Carlo simulation yielded Eo/Ed values (for theentire 0' interval) within such a narrow range that thestatistical scatter resulted in the EO/Ed values beinginterspersed independently of Mo. Consequently, for0.055 S R S 0.14 a single linear relationship betweenEo/Ed and R was considered in the curve-fitting activi-ty, utilizing the average of the EO/Ed values obtainedby varying Mo at a fixed value of R. For each of the fourvalues of R > 0.055, the standard deviation of theaverage was S0.04, while the maximum deviation was±0.045 from the average Eo/Ed value. Therefore, the

0 .02 .04 .06 .08 .10 .12 .14

Volume reflectance, R

Fig. 5. E/Ed vs R curves fitted to Monte Carlo simulation outputsfor Z10 and 0' = 0, 30, 45, 60, and 890.

Ed (R, 05 IEd

.04 .06 .08Volume reflectance, R

Fig. 6. E/Ed vs R curves fitted to Monte Carlo simulation outputsfor Zj and 0' = 0,30,45,60, and 890. ForR >0.055, the Eo/Ed valueshave been shown as the average values of the Monte Carlo outputs

for all incident angles along with its standard deviation.

incident angle has very little effect on the Eo/Ed ratioat the 1% downwelling irradiance depth Zi for watermasses characterized by volume reflectances of20.055.

As discussed more fully in the next section, the sin-gle linear equation for EO/Ed at Z for 00 S 0' 890 and0.055 S R S 0.14, which was curve fitted as

E (Rf') = 1.37 + 4.93REd

(14)

is in actuality the equivalent relationship appropriateto Z1 and Z1o for the condition of diffuse cardioidalincident radiation.

Figure 6 illustrates the curves resulting from the useof Eqs. (11), (12), and (14) and their interpolation Eq.(13). Also shown are the (EO/Ed,R) values resultingfrom the Monte Carlo simulation. The average mag-nitude of the difference between the equations and theMonte Carlo simulation is 1.4% with a maximum dif-ference of +3.7%.

1 October 1988 / Vol. 27, No. 19 / APPLIED OPTICS 4015

D. For Diffuse Cardioidal Incidence

Employing a cardioidal distribution for the incidentradiation field, the Monte Carlo simulation yielded arelationship between EO/Ed and R at Z100 of the form

° (R) = 1.177(1 + 3.13R), (15)Ed

which, by comparison with Eq. (1), can be seen to be1.177 times the relationship appropriate for verticalincidence (' = 0°).

Setting Eq. (15) equal to Eq. (5) (Zl00 for verticalincidence) and solving for o, a value o = 0.858 isobtained which is in excellent agreement with the val-ue of 0.859 given by Prieur and Sathyendranath 4 as theaverage in-water cosine value for a cardioidal incidentdistribution.

Furthermore, the use of a cardioidal distribution forthe incident radiation field resulted in the Monte Car-lo simulation yielding E/Ed values for Z10 and Zwhich were nearly equivalent for a given value of R.

Once again the segmentation of the volume reflec-tance into two ranges was strongly suggested by theMonte Carlo output, and for 0 S R S 0.055 and diffusecardioidal incidence, curve fitting the Monte Carloresults at both Z10 and Z1 yielded the relationship

E0° (R) = 1.177[1 + (11.8 + 96.8R)R]05 . (16)

Ed

Table I lists the Eo/Ed ratio values resulting from theMonte Carlo simulation for the four R values of >0.055at the Z10 and Z1 depths assuming a diffuse cardioidalincident radiation field. Also, for comparison, Table Ilists the average of the Eo/Ed ratios resulting from theMonte Carlo simulation for these R values at the Z1level assuming a series of collimated incident radiation

Table 1. Values of the Ratio EO/Ed for Volume Reflectances >0.055

Eo/Ed ratioDiffuse Diffuse Average of

Volume incidence incidence angularreflectance and and incidence

RH Z1 0 Z and Z

0.0640 1.670 1.701 1.7080.0806 1.755 1.769 1.7710.0937 1.858 1.870 1.8590.1331 2.009 2.050 2.012

fields with incident zenith angles of 0, 30, 45, 60, and890. Clearly, the three sets of numerical entries inTable I are very similar, further accentuating the mini-mal role played by the incident angle 0' in the numeri-cal value of the Eo/Ed ratio at Z1 for volume reflec-tances of >0.055. It was, therefore, considered notinappropriate to represent all three sets of entries inTable I by the same linear relationship.

Thus, for 0.055 S R S 0.14, for both depths Z10 andZ1 and a diffuse cardioidal incident radiation, as well asfor depth Z1 and a collimated incident radiation, Eq.(14), as alluded to earlier, is considered a multidutyrelationship. The average magnitude of the differ-ences between the Eo/Ed values emerging from Eqs.(14), (15), and (16) and the Monte Carlo simulationsfor Z1oo, Zio, and Z1 and conditions of cardioidal inci-dent distribution is -0.8% with a maximum differenceof ±2.0%.

IV. Discussion

Table II lists the values of the Eo/Ed ratio for verticalincidence at Z100, Z10, and Z1 and for diffuse cardioidalincidence at Z1. These parameters are listed as thenumerical solutions from the Eo/Ed equations for sev-eral values of R. Also tabulated are the ratios of eachof Z1 and Z10 with vertical incidence to Zi with diffusecardioidal incidence [i.e., the ratios of the Eo/Ed values,designated as Zi(00 )/Zl(diff) and Zio(00)/Zl(diff)]. Itis seen from Table II that

(a) the E/Ed ratio at Z for vertical incidence isvirtually equivalent to the Eo/Ed ratio at Z1 for diffuseincidence when R > -2%;

(b) the Eo/Ed ratio at Z10 for vertical incidence isalmost equal to the Eo/Ed ratio at Z1 for diffuse inci-dence when R > -2%. This is particularly true for R >-8% where the Z1 o(0")/Zj(diff) ratios are 0.97.

Table II, therefore, suggests that when R > -8% theradiance distribution at Z10 is not substantially differ-ent from the asymptotic radiance distribution. It maynot be unreasonable, therefore, to suggest further thatmany of the relationships derived for conditions ofasymptotic radiance distribution, while appropriate todepths as shallow as Z1 for all values of R > -2%, mayalso be appropriate to more shallow depths Z10 for allvalues of R > -8%.

Prieur and Sathyendranath 4 have derived an ex-pression for determining the absorption coefficienta(Z) in terms of a number of parameters, one of theparameters being the ratio Eo/Ed. Each of these pa-

Table 11. Values of the Ratio E0/Ed at Various Depths for Conditions of Vertical and Diffuse Incidence

EQ/Ed ratioVolume Z100(O) Z10(O0 ) Z1 (00 ) Zj(diff) Z 1(00) Z10(00)

reflectance (vertical) (vertical) (vertical) (diffuse) Zj(diff) Zi(diff)

0.0078 1.024 1.102 1.138 1.233 0.92 0.890.0237 1.074 1.279 1.352 1.359 0.99 0.940.0406 1.127 1.433 1.516 1.507 1.01 0.950.0640 1.200 1.608 1.686 1.686 1.00 0.950.0806 1.252 1.711 1.767 1.767 1.00 0.970.0937 1.293 1.783 1.832 1.832 1.00 0.970.1331 1.417 1.958 2.026 2.026 1.00 0.97

4016 APPLIED OPTICS / Vol. 27, No. 19 / 1 October 1988

Bb

Fig.7. Relationship between R(0') and Bb/a for the case of verticalincidence (i.e., 0' = 00).

rameters, with the exception of Eo/Ed, is determinablefrom profiles of upwelling and downwelling irra-diances. The EO/Ed equations and graphic represen-tations included within this communication afford theopportunity of also being able to determine Eo/Ed fromprofiles of upwelling and downwelling irradiances.

Appendix: Subsurface Volume Reflectance R at Z1 0 0

A Monte Carlo simulation was used to determine therelationships among subsurface volume reflectance atZ1oo, the inherent optical properties of the water, andthe incident radiation distribution. Values of b/a (ra-tio of the scattering coefficient to the absorption coef-ficient of the water) were taken to be 1, 3, 5, 8, 10, 12,and 18. A volume scattering function with a backscat-tering probability B = 0.0253 was taken from Petzold.3For such a volume scattering function, values of thevolume scattering distribution function were deter-mined at 1° intervals up to 1800. The incident radia-tion distributions included collimated fields definedby 0' = 0,30,45,60, and 890 (' is the angle of incidenceof the impinging radiation measured from the vertical)as well as a diffuse cardioidal distribution. Photonswere counted until the standard deviation of the errorwas <1.0%.

Figure 7 illustrates our Monte Carlo relationshipbetween R(0') and Bb/a for the case of vertical inci-dence (i.e.,0' = 00). To accommodate the Monte Carlooutputs (shown as points in Fig. 7) with a linear rela-tionship, it is necessary to segment the range of Bb/avalues. A demarcation value of Bb/a = 0.25 is readilysuggested. The two linear relationships depicted inFig. 7 are

R(O) = 0.319 Bb (Al)a

for 0 Bb/a < 0.25 and

R(O) = 0.267 Bb + 0.013 (A2)a

for 0.25 < Bb/a S 0.50.The range of volume reflectances (0 S R < 0.15)

considered in Fig. 7 encompasses that which would be

Fig. 8. Relationship between the ratio R(0')/R(00 ) and the inverseof the cosine of the in-water angle of refraction (1/uo).

observed for most naturally occurring ocean and/orinland waters.

Figure 8 illustrates the response of volume reflec-tance R(6') to varying the angle of incidence 0'. Hereinare plotted the values of the ratio R(0')/R(0 0) as afunction of 1/,Mo, where ,0 is the cosine of the in-waterrefracted angle 00 for the incident radiation distribu-tion characterized by 0'. These values of R(0')/R(0 0 )are taken as the arithmetic mean of this ratio for theseven values of Bb/a considered in this work. Thestandard deviations of these R(0')/R(0 0) ratios are in-dicated by the vertical bars in Fig. 8. The largerstandard deviation for 1/Muo = 1.315 and 1.512 are aconsequence of an apparent second-order relationshipbetween R(0') and Bb/a, which increases with increas-ing 0'. The linear relationship of Fig. 8 is defined by

R(0') = R(° ) (A3)

where Mto = cosOo. The second-order effect was notdetermined as it resulted in deviations from this rela-tionship-of <5%.

In a similar manner, our Monte Carlo simulation,considering a diffuse cardioidal distribution as theincident radiation field, yields the relationship

RD = 1.165R(OO), (A4)

where RD indicates the subsurface volume reflectanceassociated with a cardioidal incident field. Compar-ing Eqs. (A3) and (A4) suggests the value of Mo to be0.858, in excellent agreement with the value of 0.859calculated by Prieur and Sathyendranath 4 as beingappropriate for cardioidal incidence.

Equations (A1)-(A4) predict values of volume re-flectance that fall between those that would be pre-dicted by Kirk6 and Gordon et al.7 for waters with thesame inherent optical properties.

1 October 1988 / Vol. 27, No. 19 / APPLIED OPTICS 4017

References

1. W. R. G. Atkins and H. H. Poole, "A Cubical Photometer forStudying the Angular Distribution of Submarine Daylight," J.Mar. Biol. Assoc. U.K. 24, 271 (1940).

2. W. R. G. Atkins and H. H. Poole, "Cube Photometer Measure-ments of the Angular Distribution of Submarine Daylight and theTotal Submarine Illumination," J. Int. Council Exploration Sea23, 327 (1958).

3. T. J. Petzold, Volume Scattering Functions for Selected Waters(Scripps Institution of Oceanography, U. California at San Diego,1972), SIO Ref. 72-78.

4. L. Prieur and S. Sathyendranath, "An Optical Classification of

Coastal and Oceanic Waters Based on the Specific Spectral Ab-sorption Curves of Phytoplankton Pigments, Dissolved OrganicMatter and Other Particulate Materials," Limnol. Oceanogr. 26,671 (1981).

5. J. I. Gordon, Directional Radiance (Luminance) of the Sea Sur-face (Scripps Institution of Oceanography, U. California at SanDiego, 1969), SIO Ref. 69-20.

6. J. T. 0. Kirk, "Dependence of Relationship Between Inherentand Apparent Optical Properties of Water on Solar Altitude,"Limnol. Oceanogr. 29, 350 (1984).

7. H. R. Gordon, 0. T. Brown, and M. M. Jacobs, "ComputedRelationships Between the Inherent and Apparent Optical Prop-erties of a Flat Homogeneous Ocean," Appl. Opt. 14, 417 (1975).

Meetings continued from page 4011

1988October

24-27 II Workshop on Dynamics of Non-Linear Optical Sys-tems, Cantabria U. of Cantabria, Cantabria, Spain

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6-11 Cambridge Symp. on Advances in Intelligent RoboticsSystems, Cambridge SPIE, P.O. Box 10, Belling-ham, WA 98227

7-10 Int. Cong. on Applications of Lasers & Electro-Optics,Orlando Laser Inst. Am., 5151 Monroe St., Toledo,OH 43623

7-11 Visual Communications & Image Processing III, Cam-bridge SPIE, P.O. Box 10, Bellingham, WA 98227

9-11 Optical Storage for Small Systems Mtg., Los AngelesTOC, P.O. Box 14817, San Francisco, CA 94114

13-16 Symp. on Business Issues in Photographic & ElectronicImaging, NY SPSE, 7003 Kilworth La., Springfield,VA 22151

13-17 Biostereometrics Mtg., Basel SPIE, P.O. Box 10, Bel-lingham, WA 98227

14-18 Laser Safety course, Los Angeles Eng. Tech. Inst., P.O.Box 8859, Waco, TX 76714

continued on page 4026

4018 APPLIED OPTICS / Vol. 27, No. 19 / 1 October 1988