Monte Carlo study of the scattering error of a quartz reflective absorption tube

arXiv:physics/0006061v1 [physics.ao-ph] 23 Jun 2000 Monte Carlo study of the scattering error of a quartz reflective absorption tube Jacek Piskozub Institute of Oceanology PAS Powsta´nc´ow Warszawy 55 81-712 Sopot, Poland piskozub@iopan.gda.pl Piotr J. Flatau Scripps Institution of Oceanography, University of California, San Diego La Jolla, CA 92130-0221 pflatau@ucsd.edu J. V . Ronald Zaneveld College of Oceanic and Atmospheric Sciences, Oregon State University 104 Ocean Admin Bldg Corvallis, OR 97331-5503 October 30, 2018 In Press: Journal of Atmospheric and Oceanic Technology, 2000 Abstract A Monte Carlo model was used to study the scattering error of an absorption meter with a divergent light beam and a limited acceptance angle of the re ceiver. Reflections at both ends of the tube were taken into account. Calculations of the effe ct of varying optical properties of water, as well as the receiver geometry, were performed. A weighting function showing the scattering error quantitatively as a function of angle w as introduced. Some cases of the practical interests are discussed 1 INTRODUCTION Light absorption is the essential element required for mari ne phytoplankton growth. The re- gion of the upper ocean illuminated by sunlight is responsib le for most of marine primary production. Absorption and scattering of phytoplankton co ntrol the light field (at least for 1

Case 1 waters which comprise a large majority of ocean waters ), and thus, affect the spectral reflectance of the ocean surface. Such spectral changes provide information about ocean color (Gordon and Morel, 1983) In all radiative transfer modeling of the marine environment the inherent optical properties (IOPs) are either the necessary input parameters, or in the case of inverse problems, the output of the calculations. This makes them important in marine opt ics studies. The measurement of the attenuation coefficient is relatively easy; an attenuation meter consists of a collimated light source and a collimated light detector at a known distance. T he only major problem in the attenuation measurement are photons from outside sources or the ones reflected on the instru- ment scattered into the light beam, causing an underestimat ion of the attenuation coefficient value. The detector has a finite aperture so that some of the fo rward scattered photons will be counted as part of the direct beam, also causing an underes timation. However, the mea- surement of the components of the attenuation, the absorpti on and scattering coefficients, is inherently much more difficult, and researchers have been st riving to minimize the measure- ment errors of these parameters for most of this century. The central idea behind any absorption measurement is to pro ject a beam of light through an absorbing medium. If one could measure all of the unabsorb ed light of the direct beam, as well as all of the scattered light, the only light lost woul d be the absorbed light. Thus, absorption meters tend to be arranged to collect as much of the scattered light as possible. The absorption coefficient measured in any optical device has tw o major sources of error, both of which are due to the fact that natural suspensions tend to sca tter light as well as absorb it. In practice, the scattered light traverses a longer path thr ough the absorbing medium and so is more likely to be absorbed (the path length amplification f actor). Secondly, not all of the scattered light is collected due to the geometry of the absorption meter (the scattering error). One of the choices to measure in situ absorption is the cylind rical reflection tube. Such a tube needs to be be long enough to provide a sufficient optica l path to measure the low absorption values typical for Case1 waters below 580nm. Any photon scattered forward off the instrument axis is, at least in theory, reflected by the tube walls until it reaches the detector. An ideal instrument of this kind should also have a reflector inside of the source end to collect the backscattered photons. The first working prototypes of such a device were developed by Zaneveld and co- workers (Zaneveld and Bartz, 1984; Zaneveld et al., 1990). One of the greatest problems of making a practical reflective tube is the reflectivity coefficient of the walls. A Monte Carlo study of the performance of a reflective tube absorption meter (Kir k, 1992) shows that the results quickly deteriorate with reflectivity decreasing from 100% . As it is virtually impossible to produce perfect reflecting walls, especially ones that would not deteriorate with prolonged use of the instrument, the concept of a quartz glass tube surroun ded by air was proposed instead (Zaneveld et al., 1990). Assuming smooth surfaces of the tub e, all photons encountering the wall at an angle to the wall surface smaller than the critical angle, 41◦, must be internally reflected. Therefore, a clean quartz reflection tube should c ollect all photons scattered in the angular range between 0◦ and 41◦ (if multiple scattering is neglected). However, the large losses of photons scattered at angles above 41◦ are the main theoretical source of error for quartz tube absorption meters. The Monte Carlo calculations by Kirk (1992) were conducted f or a prototype absorption tube with an almost parallel light beam resembling the Wetla bs ac-9 absorption meter. The 2

results showed that the relative error of absorption is alwa ys positive and increases linearly with the ratio of scattering to absorption. The error increa ses with decreasing wall reflectance and acceptance angle of the receiver. Another study by Hakvo ort and Wouts (1994) used a Lambertian light source. In this case the absorption error d ecreases with the decreasing angle of photon acceptance. However, early prototypes of HiStar, the new Wetlabs spectrophotome- ter, has a diverging beam limited to 20◦. The receiving end of the tube is illustrated in Fig. 1. It consists of a Light Shaping Diffuser (LSD) in front of a len s. The fiber transmits light to a spectrometer. Such an arrangement results in a large receiv ing area which through the use of the LSD and the lens translates into the small acceptance angle needed by the spectrometer. In this paper we take account of the reflections at both ends ofthe tube, that were neglected by previous studies of this kind. We also extend previous results by considering a more realistic arrangement, by introducing weighting functions that quan titatively show the scattering error as a function of angle, and by providing calculations for some cases of practical interest. 2 CALCULATION SETUP The Monte-Carlo code used was adopted from the code written to determine the effects of self- shading on an in-water upwelling irradiance meter (Piskozub 1994). This is a forward Monte Carlo algorithm, meaning that the photons are traced in the forward direction starting from the light source. An absorption event ends a photon’s history. The low values of the optical depth inside the absorption tube make this approach comparatively efficient. The absorption tube studied (called henceforth α-TUBE) is a cylindrical shell of inner radius r=0.006 m and length d=0.23 m which corresponds to the dimensions of the HiStar quartz reflective tube (see Fig. 1). Thickness of the quartz wall is ∆r=0.002 m. The indices of refraction used are 1.33 for water and 1.41 for quartz. The photons inside the α-TUBE are traced along their three dimensional trajectories. The scattering or absorption events inside the volume of the wat er sample are defined by inherent optical properties of the medium: the absorption coefficien t of the medium a, scattering coef- ficient b, and the scattering phase function β(θ). Random numbers are used to choose whether a given photon traveling from point (x,y,z) towards direction (θ, ϕ) will reach the wall or the end of the cylinder. Otherwise, it is assumed that the photon ends its trajectory inside the liq- uid medium, and the type of the event (scattering or absorption) is determined by comparing a random number to the single scattering albedo ω0 value. All random number used in the code are uniformly distributed in the open interval (0, 1). We define the Cartesian coordinate system such that the axis of the tube is co-located with the z-axis, and the center of the source end of the tube defines the origin of the coordinate system. Therefore, θ is the angle between the photon direction and the tube axis an d ϕ is the angle of projection of the photon direction onto the x-y plan e. In every scattering event the new direction of the photon is chosen using the relevant phase function. The source of photons is assumed to be within a circle of the ra dius r = 1.5 mm emitting photons into a cone of 25◦ half-width. This represents the fiber head. The angular distribution of photons is assumed to be Lambertian up to the 25◦. The position of the photon entering the tube itself is calculated, taking into account the dista nce from the fiber head to the tube entrance (0.019 m) and refraction from air to water at the mouth of the reflective tube. 3

Figure 1: Schematic representation of the reflective tube ab sorption meter setup discussed in the paper Such an α-TUBE, defined above, is similar to the prototype WET Labs HiStar, but it does not precisely model the detailed radiance structure surrounding both the source and the receiver ends of the reflective tube. We believe that those details areof minor importance in comparison to neglecting the albedo of the tube ends. Therefore we intro duced the albedo of the source end for photons returning to it from inside the tube, for whic h we estimated the albedo as 0.3 (0.2 being the diffuse albedo and 0.1 the specular albedo). T his is a rough assumption of the combined effects of the input window and the silver colored m etal surface around the fiber head. The quartz walls are treated as ideally smooth, reflecting and refracting photons according to geometrical optics. The direction of a refracted photon is determined in the code by Snell’s law and probability of reflection and refraction by Fresnel’ s law. However, no polarization effects are included. The photons are also followed inside t he quartz, which is assumed to be non-absorbing. All photons leaving the outer surface of the quartz tube are treated as lost. The tube is assumed to be surrounded by an ideally black medium. The receiver end of the tube is treated as a smooth surface refl ecting 8% of the incident photons. This corresponds to the angular average of reflecti on for the glass window and the Light Shaping Diffuser (LSD) used. It is assumed that a photon hitting the diffuser plate at an angle θ reaches the receiver with a probability proportional to theangular diffusion distribution (or scattering angle profile) of the LSD. The Light Shaping Diffuser used in the instrument is a Physical Optics Corporation 20◦ LSD. Also, we use in the calculations two other diffusers:60◦ LSD and 95◦ LSDs. The angles are half-widths of the angular diffusion fu nction. This means that, for example, a 60◦ LSD has a 50% transmission efficiency at angle 30◦. The angular diffusion distribution used was determined by the least-sq uare fits of the experimental curves provided by the producer. The 20◦ LSD was approximated by two Gaussian functions (with all angles in degrees): L20(θ) = A exp [ −0.5 ( θ B ) 2] + (1 − A) [ −0.5 ( θ C ) 2] (1) where A = 0.85, B = 8.0◦, C = 15.3◦. The 60◦ LSD was approximated by a single Gaussian 4

Figure 2: Light diffusing characterisitics of the Light Shaping Diffusers for three different types of diffusers. function: L60(θ) = A exp [ −0.5 ( θ B ) 2] (2) where A = 1 .0 and B = 24 .5◦. Finally, the 95◦ LSD was approximated by a Gompertz function (Gompertz, 1825): L95(θ) = A exp [ − exp ( − ( θ − θ0 B ))] (3) where A = 1.0, B = 7.33◦ and θ0 = 50.75. The shapes of the three LSD diffusion angle profiles are presented in Figure 2. The standard Petzold scattering phase function for open turbid waters wa s used (Petzold 1972) except for the calculations of the effect of the phase function shape on the measurement error where Henyey-Greenstein phase functions (Henyey and Greenstein, 1941) were used. Each Monte Carlo calculation was performed twice: one to det ermine how many photons are recorded if there is no absorption and scattering ( P0) and the other for the IOPs (absorp- tion, scattering and phase function) of the aquatic medium b eing studied (P). We define the 5

measured absorption am as Kirk (1992): am = (1/d) ln(P0/P ) (4) where d is the length of the cylinder. It must be noted that the first program run corresponds to cali bration of the instrument in “ideal water,” not in air. We use the water index of refractio n for this medium so as not to influence the new photon direction in refraction events. The refore, the results are different from what one would obtain if the tube was filled with air. Such a non-scattering and non- absorbing liquid medium does not exist. However, it is the mo st obvious reference value to use in the calculations aimed at studying the calibration of the instrument, because it does not introduce any photon losses due to scattering (b=0) and d oes not change the behavior of photons on the quartz-liquid border. 3 RESULTS 3.1 SEMI-ANALYTICAL CONSIDERATIONS In this section we present qualitative arguments related to the optical phenomena influencing the fate of photons in the absorption tube. The purpose of this is to provide approximate results that can be compared with exact Monte Carlo results later. Th e semi- analytic results give a better understanding of the underlying physics. If, for exa mple, the IOPs of the studied water sample are assumed to be: b=0.8 m − 1, a=0.2 m− 1, and the Petzold San Diego Harbor (turbid water) phase function, one can expect that 4.5% ( exp(−ad)) of all photons will be absorbed and 16.8% (exp(−bd)) will be scattered in the water volume inside the quartz tube. The phase function used, determines that 92.9% of all scatte ring events take place in the 0-41◦ range. Assuming for simplicity, that before scattering, al l photons traveled parallel to the tube axis, all those scattered photons will be reflected b ack into the tube on all encounters with the tube wall. Similarly, all photons backscattered in to the range of 139-180◦ stay in the tube on their way back to the source end. These represent 0.3% of all photons. The remaining 6.8% of all scattered photons are scattered into the range 41-139◦. These photons have a large probability of leaving the tube on the first encounter with th e wall. The angles to the z-axis at which those photons travel make it virtually certain that they will leave the tube before reaching either end, unless they are scattered close to 41◦. For example at θ = 45 ◦, a photon scattered in the very center of the tube will hit the wall 9 tim es before reaching the end of the tube making the probability of not leaving it < 10− 8. The 6.8% of scattered photons that are lost translates into 1.1% of all traced photons being lost through the walls. After the absorption and scattering losses are taken into ac count, 94.3% of all traced pho- tons reach the receiver end of the tube. Thus, the assumption of 8% specular albedo of the receiver end of the tube means that 7.5% of all traced photons are reflected from the receiver end of the tube. Absorption and scattering losses result in o nly 7.1% of all initial photons coming back to the source end after being reflected. The backs cattered photons, which are attenuated in a pathlength approximately equal to the tube l ength, increase this number to 7.2%. 6

Another important aspect of a quartz absorption tube is the length of the path that the pho- tons travel in quartz instead of water. Any distance traveled in quartz decreases the attenuation due to the negligible absorption coefficient in the visible r ange of the spectrum (Kirk 1992). This decreases the measured absorption values because the r eceiver is reached by more pho- tons due to the shorter path through the absorbing medium. In the studied tube geometry, the ratio of tube diameter to wall thickness is only 6:1, and considering that every time the photon goes across the water volume it needs to cross the wall twice ( out and back into the tube) it would seem that this effect must be overwhelming. However, d ue to Snell’s law and the nar- row angles between the direction of most photons and the wall, the average path of photons in quartz is much smaller. 3.2 SANITY CHECKS OF THE MONTE CARLO RESULTS Even if rough approximations of the results are possible wit h simple arithmetic calculations, Monte Carlo modeling includes all optical phenomena taking place inside the tube. This can be illustrated by comparing the approximate values from the previous section to the results of a Monte Carlo run using 40 million photons with the same IOPs. The number of photons lost through the walls is 1.74% (estimated above as 1.1%). The dis crepancy can be explained by the divergence of the input beam: photons entering the tube a t an angle may leave it through the walls, even if scattered from its original direction at l ess than the angle of total internal reflection, resulting in higher scattering losses. The Monte Carlo derived absorption losses percentage is 4.0 9% (semi-analytically esti- mated as 4.5%). The reason for this is that the path length of t he photons is in quartz instead of in water. The Monte Carlo code makes it possible, by tracin g each photon individually, to calculate the average path traveled by the photons in wate r to be 21.15 cm and in quartz: 2.83cm. This means that the path in water is 8% shorter than th e length of the reflective tube, a value much higher than that calculated by Kirk (1992) for a parallel beam in which only the scattered photons encounter the walls. The losses by absorp tion on the front end of the tube are 4.84% of all photons (estimated as 5.0%). The small difference is the net result of smaller absorption losses and greater scattering losses. The effect of the shortened optical path, described above, i s not very grave if the way the instrument is calibrated is taken into account. All calibration techniques involve a comparison of the received signal (that is the number of photons reachin g the receiver) for the measured sample with pure water or air values. Using pure water almost completely removes the error due to non-scattered photons traveling partly inside the wall, as the effect will be identical for water of any IOPs. Calibrating the instrument in air leaves s ome error due to path length in quartz because air has a different index of refraction making the path length in quartz shorter. There is, however, a much bigger source of error in calibrati ng the instrument without water, which is the removal of total internal reflection on the quartz wall by surrounding both its sides by the same medium. 3.3 V ARIATION IN THE SCATTERING-ABSORPTION RATIO It was shown by Kirk (1992) that for a reflective tube absorption meter propagating an almost parallel beam, that the ratio of the measured to the true valu e of the absorption coefficient 7

am/a increases linearly with b/a at a rate depending on the phase function used. We decided to test whether such a relationship will hold for the relatively divergent light beam and limited acceptance angle of the α-TUBE. Figure 3 shows that this is indeed the case for all stud ied angles of acceptance (depending on the LSD type used). The slope coefficients w of the linear fit am = a + wb (5) are 0.194 for 20◦ LSD, 0.139 for 60◦ LSD, and 0.0957 for 95◦ LSD. These results suggest that the scattering losses of the instrument increase with the de creasing angle of acceptance of the receiver. It must be noted that the values of the coefficient w depend on the scattering phase function used. Figure 3 shows that there is a small offset in Equation 5. Its general version am = a+wb+o (where o is the offset) can be transformed to a = am − wc − o 1 − w (6) which, for a given phase function and absorbing tube geometr y, allows for determining the true a value if attenuation c is measured independently. The parameters w and o may be calculated for the given absorption meter geometry if the ph ase function is known at least approximately. The values of the coefficient w, for the physically important range of Henyey- Greenstein asymmetry parameter g and the 20◦ LSD receiver are shown in Figure 4. 3.4 V ARIATION IN THE SCATTERING PHASE FUNCTION The measurement error in absorption depends on the scatteri ng phase function of the aquatic medium (Kirk, 1992). To study the effect of the scattering ph ase function on absorption mea- sured by the given instrument setup with 20◦ LSD, we performed a sequence of Monte Carlo calculations for the Henyey-Greenstein phase function (Henyey and Greenstein 1941) by vary- ing the asymmetry parameter g β(θ) = 1 − g2 (1 + g2 − 2g cos θ)3/2 (7) where g is a parameter determining the shape of the phase func tion. In this paper we assume range of g from g = 0 (isotropic scattering) to g = 1 (forward scattering). The results in Fig. 5 show am/a as a function of the asymmetry parameter. They were obtained by running the code for the studied tube equipped wi th three different LSD plates (20◦, 60◦, and 95◦). As expected, for g approaching 1 the am/a ratio is close to 1 as all photons are scattered almost in the direction that they were headed before the scattering. For a more realistic range of g values between 0.8 and 0.9 (Mobley, 1994) the measured absorption increases when g decreases due to the greater scattering los ses of the photons. The value of am/a < 1 for g = 0 .95 and 95◦ LSD is not a statistical error, but the effect of the slightly longer path of scattered photons in the quartz, in comparison to the non-scattering ”ideal water” which results in decrease of absorption. For small g values the am/a values (Fig. 5) approach the maximum (a+b)/a at which all the scattered photons are lost and the measured value of absorption equals the total attenuation. 8

Figure 3: Ratio of the measured to the true absorption coeffic ient am/a as a function of the scattering-absorption ratio b/a and a = 0 .2m− 1, Petzold ”turbid” phase function for three dif- ferent Light Shaping Diffusers in front of the receiver lens. 9

Figure 4: The slope coefficient w as a function of Henyey-Gree nstein asymmetry factor g for the 20◦ LSD receiver geometry. 10

Figure 5: Ratio of the measured to the true absorption coefficient am/a as a function of the Henyey- Greenstein scattering phase function asymmetry parameter g; a = 0 .2m− 1, b = 0 .8m− 1, Petzold turbid phase function, for three different Light Shaping Diffusers in front of the receiver lens. Solid lines are third order polynomial fit. 11

Figure 6: Ratio of the measured to the true absorption coefficient am/a as a function of the accep- tance angle of the receiver, a = 0.2m− 1, b = 0.8m− 1, Petzold turbid phase function. In the case studied (a+b)/a = 5. However, even for an isotropic scattering phase function (g=0) the value of the scattering error does not reach this maximum value because some photons are scattered at angles small enough to be recorded by the receiv er. Again, the figure shows that increasing the angle of acceptance of the receiver decreases the absorption error for all realistic phase functions. 3.5 V ARIATION IN THE ACCEPTANCE ANGLE Results presented by Kirk (1992) for a reflective tube absorption meter with an almost parallel beam show a rapidly increasing value of am/a with the acceptance angle decreasing below 90◦. On the other hand, Hakvoort and Wouts (1994) suggest that fo r the reflective tube with Lambertian light source and metallic reflective walls, the m easurement error increases with the acceptance angle of the receiver. The Monte Carlo code wa s run for several values of the receiver angle of acceptance to test if the similiar situati on arises for the α-TUBE. No Light Shaping Diffuser was used in this case. The results are presented in Figure 6. It can be seen that for large acceptance angles (> 40◦) the am/a values have a similar shape 12

to that of a parallel beam case. However, for acceptance angles in the 0−40◦ range a minimum can be observed, instead of a high peak as in the case of a paral lel beam. The reason for that is that the source beam is more divergent than the angle of acc eptance. For small acceptance angles the receiver accepts some scattered photons that would not be accepted if they reached the receiver without a scattering event. This is because the ir angle of incidence is reduced by the scattering event. This reduces the scattering error of t he absorption measurement. Unlike the Lambertian source case, there is no major increase of am/a values for large acceptance angles (although a small increase above 120◦ may be discerned). It should be noted that the LSD plate used in the instrument corresponds roughly to an acceptance angle of 20◦. Although this angle is close to a local minimum of am/a, it is still within the range of the highest levels of absorption error. 3.6 ANGULAR FUNCTION OF PHOTON LOSS PROBABIL- ITY W (θ) Full understanding of the mechanism of scattering loss is not possible without determining the angular relationship of the scattering error of the reflective tube absorption meter. The fraction of scattered light lost because of the absorption after bein g scattered at a given angle can be defined as a loss function W (θ). The angular integral of this function multiplied by the pha se function is the error of the measured absorption value ε = am − a ε = ∫ π 0 W (θ)β(θ) sin(θ)dθ (8) The advantage of the W function is that it is not dependent on t he shape of the phase function but only on the tube geometry. This is true only if mu ltiple scattering is neglected because a second scattering event influences the fate of a pho ton already scattered. In order to differentiate among the possible sources of the scatteri ng loss error of the absorption value we decided to calculate three variants of the W function (a) W0(θ) for scattering losses due to photons lost through the tube walls, (b) W1(θ) same as W0(θ) and for scattering losses due to additional photons absorbed by the source (”front”) end of the tube, (c) W2(θ) same as W1(θ) and for scattering losses due to additional photons lost on the LSD. By ”additional photons” we mean that only the difference between the number of photons lost by the absorption and scattering and the ”ideal” water i s taken into account. In each of the three cases the actual value of W for angle range is deri ved by dividing the number of photons scattered into the angular sector and subsequentlylost by the sum of these photons plus the number of photons scattered into the same sector which ar e subsequently recorded by the receiver. Fig. 7 presents the W functions calculated by running the code for 40 million photons with a = 0 .2m− 1, b = 0 .8m− 1, Petzold turbid phase function, and 20◦ LSD. In this case the number of multiple scattered photons is 7% of all which are sc attered. That means that the influence of the phase function shape on the W function is of secondary importance. All three functions have the sigmoid shape for the forward scattered p hotons between 0 − 90◦. There is no sharp step at 41◦ and 139◦ because the divergence of the beam causes some photons to travel at an angle to the tube axis before scattering which allows photons to escape through the wall even if scattered at an angle smaller than the total inte rnal reflection angle. On the other hand, a photon scattered at more than 41◦ back towards the axis may survive the encounter 13

with the wall without being refracted out of the tube. The sym metrical shape of W0 is caused by the fact that losses on the ”front” end are not taken into ac count. A photon backscattered at angle 180 − φ has the same chance of being lost through the walls as the one s cattered at φ. The symmetry is broken only by the assumed diffusive albedoof the light source end of the tube because some photons leave the tube by the wall after being diffused on its source end. Adding the photon losses on the light source end of the tube (f unction W1) changes the symmetry. Most of the backscattered photons are lost in this way as well as some of the forward scattered. The latter is caused by two reasons. Some forward-scattered photons are reflected by the receiver end of the tube back towards the sour ce end. However, it is even more important that some of the forward scattered photons ar e absorbed on the ”front” end because they were scattered after they were reflected on the receiver end. The reverse direction of those photons before scattering complicates the W (θ) function, but we decided to include these scattering events in the statistics used to calculate the functions because they contribute to the total scattering losses. Receiver end losses are caused by more photons missing the re ceiver due to reaching it at a wider angle after a scattering event. This changes the pict ure by, paradoxically, decreasing the photon losses for photons that were scattered in the forward direction (function W2). This means that scattering a photon by a small angle increases its probability of being accepted by the receiver compared to the average of non-scattered photo ns. This is caused again by the source beam being wider than the acceptance angle of the rece iver. The discrepancy between W1 and W2, especially at scattering angles close to 0, is also partly caused by the inclusion of photons scattered on their way back to the source end of the tu be. One of its consequences is that the population of photons scattered at close to 0 degree is different from the population of non-scattered photons. The shape of the W functions for angles greater than90◦ is not important for the scattering error estimates because only a fraction of all the photons isbackscattered. For the W2 function which includes all the photon losses due to scattering, the l oss of backscattered photons is close to 100%. Therefore, the best fit approximations to W0, W1 and W2 were calculated for the range 0 − 90◦ only. The three functions were fitted with a 4-parameter sigmoid W (x) = y0 + A (1 + exp(−(x − x0)/B)) (9) where for W0 we have A = 0.935, B = 2 .53◦, x0 = 26 .72◦, y0 = 0 .061, for W1 we have A = 0.841, B = 3.35◦, x0 = 25.75◦, y0 = 0.157, and for W2 we have A = 0.901, B = 2.58◦, x0 = 26.79◦, y0 = 0.095. 4 CONCLUSIONS Monte Carlo calculations of the scattering error for the HiStar prototype absorption meter show that the different blueprint of the instrument (more divergent light beam and a limited angular view of the receiver) in comparison to other reflective tube a bsorption meters does influence the error. Due to additional scattering losses by the view li mited receiver the absorption error is greater for all combinations of moderate optical parameters. It is possible to correct the error by careful calibration, and the results presented in this paper for the turbid water Petzold phase 14

Figure 7: The photon loss probability W (θ) as a function of the scattering angle, a = 0 .2m− 1, b = 0 .8m− 1, Petzold turbid phase function. These three functions repr esent (a) W0 which defines losses of photons leaving the tube through the quartz walls, (b) W1 same as W0 and losses on the light source end of the cylinder, (c)W2 which is the same asW1 but with losses on the receiver-end. 15

function are a step in this direction. Independent measurements of attenuation may improve the error correction for water samples of known (at least approximately) phase functions as shown in the paper. However, the source of inter-instrumental discrepancy in absorption measurement that is more difficult to correct is their different response s to phase function variability. One possible solution is to use the photon loss function W. If the phase function of the studied sea-water is known (or is possible to be estimated by compari ng it to known phase functions of similar water samples), the W function makes it possible t o calculate directly the value of the scattering error for the sample. Our calculations show that the main source of the absorption error is the view limited receiver. Therefore, we suggest that if technically possib le (it would diminish the amount of light collected by the receiver optical fiber) a diffuser of w ider angular characteristics should be used in the instrument receiver setup. 5 REFERENCES Bricaud A., Babin M., Morel. A., and Claustre H., 1995: ”Vari ability in the chlorophyll- specific absorption coefficients of natural phytoplankton: analysis and parameterization,” J. Geophys. Res. , 100, 13321–13332. Gompertz B., 1925: ”On the Nature of the Function Expressive of the Law of Human Mortal- ity,” Phil. Trans. Roy. Soc. London., 115, 513. Gordon H. R. and Morel, A. 1983: ”Remote assessment of ocean c olor for interpretation of satellite visible imagery, a review” in Lecture notes on coa stal and estuarine studies, vol. 4, Springler Verlag, New York, 114pp. Hakvoort J.H.M, Wouts R. 1994: ”Monte Carlo modelling of the light field in reflective tube type absorption meter,” Proc. SPIE , 2258, Ocean Optics XII, 529–538. Henyey, L. G., and J. L. Greenstein, 1941: ”Diffuse radiation in the galaxy,” Astrophys. J., 93, 70–83. Kirk, J.T.O, 1992: ”Monte Carlo modeling of the performance of a reflective tube absorption meter,” Appl. Opt., 31, 6463–6468 Mobley, C. D., 1994, Light and water: radiative transfer in n atural waters, Academic Press, San Diego. Petzold, T. J., 1972: ”V olume scattering functions for selected ocean waters,” SIO Ref. 72-78, Scripps Institution of Oceanography, Univ. of California, San Diego. Piskozub J., 1994: ”Effects of surface waves and sea-bottom on self-shading of in-water opti- cal instruments” in Ocean Optics XII, Proc. SPIE, 2258, 300–308. 16

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