Monte Carlo simulations of the backscattering measurements for associated uncertainty

Sayoob Vadakke-Chanat; Palanisamy Shanmugam; Balasubramanian Sundarabalan

doi:10.1364/OE.26.021258

1. Introduction

Scattering is described as the portion of a beam of visible radiation that is scattered out of the ray path in every direction on interaction with particulates in water. The volume scattering function (VSF) is a measure of the angular distribution of scattering at a wavelength which can be integrated over every angle to obtain the total scattering. Backscattering is the part of the scattered light in the backward direction obtained by integrating the volume scattering function in the backward hemisphere (i.e., From 90̊ to 180̊) with respect to the original direction of the incident beam. Scattering, Backscattering and VSF are spectrally variable and are known as inherent optical properties along with absorption and attenuation. Inherent Optical Properties (IOPs) are defined as the optical properties in water that do not depend on the geometrical configuration of the light field and are only dependent on the constituents of the water column.

Unambiguous measurements of IOPs of the water column is critical for most applications that find use with the ocean color remote sensing and for aiding the development of improved biogeochemical algorithms in accuracy and robustness. The scattering and backscattering are primary IOPs of importance required in the field of atmospheric and oceanic optics with application ranging from forward modeling [1] in remote sensing to underwater wireless optical communication [2]. Various devices are available to date for measurement of the scattering and backscattering in the oceanic waters having more or less intrinsic uncertainties associated with them [3–5]. Backscattering is particularly significant because it determines the spectral shape as well as magnitude of the sunlight that is reflected from the water surface and detected by the remote sensing satellite sensors. Determining more accurate values of backscattering becomes more and more important as the remote sensing sensors are being improvised [6] in spectral, radiometric and spatial resolution as well as with better correction procedures in the recent years [7–9]. The quest for optical closure demands that better or more accurate IOP values are measured in the field so that more close to reality values can be expected [10]. Measured backscattering coefficient as well as scattering coefficient find use in various forward modelling approaches in optical oceanography [11,12]. A reduced level of uncertainty in the measurement of backscattering coefficient is also vital in studies related to the spectral variability of the backscattering ratio as the variability is debated and is often found to be within the measurement uncertainty levels [3,13,14] while such a variability has profound implications in the field of ocean color remote sensing [3]. The backscattering is an important property to be determined for various uses in radiative transfer modeling, optical remote sensing of water quality, underwater visibility and quantification of turbidity [8,15–17]. Accurate data collection of the backscattering spectra in water is therefore of critical importance in optical oceanography and ocean color remote sensing.

The measurement of backscattering coefficient ( $b_{b}$ ) is achieved by different commercially available sensors like the ECO-BB9 (WET Labs) and Hydroscat (HOBI Labs) which measure the VSF, β, at a fixed angle. These values are then converted to find the $b_{b}$ . Various other sensors are also available for measuring the VSF, like the LISST-VSF (Sequoia Scientific), for diverse applications. The scope of this study is the ECO-BB9 (WET Labs) which is widely used by the ocean optics research community. ECO-BB9 sensors have LED sources (for various wavelengths of visible and near-infrared radiations) with an angular divergence of 15° and detectors having a field-of-view of 15° half-angle [4]. The source and detector axes are at a distance of 0.008 m and the centroid angle is 120° in pure water. Various studies have established the advantage of angles between 117̊ to 141̊ as an optimum range of angles to measure the VSF for the direct calculation of the backscattering from a single measurement [4,18–20], though the most ideal angle for the measurement of backscattering is 117 ̊ as established by Boss and Pegau [21]. Angles around 120̊ are in general preferred because the variation of VSF normalized to backscattering is remarkably low at these angles [20]. The mean effective in-water centroid angle for measurements using ECO-BB9 is 124 ̊.

The objective of this study is to simulate the working principle of the ECO-BB9 backscattering sensor to improve the measurement accuracy by estimating the uncertainties which can be attributed to the geometric configuration of the instrument and the properties of the water column. Moreover, the instrument design parameters like effective path length and mean centroid angle are validated and the effect of non-water particle absorption on the backscattering measurement is evaluated based on the analytical calculation and Monte Carlo simulations.

2. Data

Measurements of the in situ data were performed in highly contrasting water environments: clear to modestly turbid coastal waters off Chennai (13°7′ 37” N; 80° 22′ 9” E) in January 2015, extremely turbid waters off Point Calimere (10°15′16”N, 80°04′2”E) in January 2014 in the southern part of Tamil Nadu coast, turbid and productive waters of the Ganges River (25°14′25”N; 83°0′39”E) in June 2016 and eutrophic inland lagoon waters of Muttukadu lake (12°48′2”N; 80°14′37”E) in March and November 2014 with dense phytoplankton levels on the southeast coast of India.

The measured parameters used in this study are the absorption coefficient ( $a$ ), the attenuation coefficient ( $c$ ), the backscattering coefficient ( $b_{b}$ ), Salinity ( $S$ ), Temperature ( $T$ ) and Depth. The parameters $a$ and $c$ were measured using an AC-S meter (WET Labs) which was calibrated using the ultra-pure water obtained with a Milli-Q system. The salinity, temperature and depth were measured using the FASTCAT CTD meter (Seabird Electronics). The $b_{b}$ coefficient was measured using an ECO-BB9 sensor (WET Labs). Chlorophyll and turbidity measurements were also made using a FLNTU sensor (WET Labs) to acquire a perception of the range of variability of these biogeochemical parameters. The parameters $a$ and $c$ were corrected for the measurement errors introduced due to temperature and salinity effects while $a$ was additionally corrected for errors due to scattering. These corrections were performed using standard correction procedures followed by the research community [22–24].

The use of in situ data for the Monte Carlo simulation is advantageous over a simulated data as the simulated values of $a$ and $c$ may not represent a wide range of the variation possible in natural waters. Therefore, a realistic variation of the simulation input data is ensured by using in situ data from a wide range of water types. The phase function and the volume scattering function ( $β$ ) for the corresponding data were calculated using the Fournier Forand (FF) phase function model [25] where the input values of the exponent of particle size distribution were calculated from the measured $c$ spectra using the relationship proposed by Boss et al. [26]. Refractive index which is another input for the FF phase function model was also calculated from the above in situ data using the model of Sanjay & Shanmugam [27], which has the advantage of being thoroughly validated for a wide variety of water types and the relatively simpler form of calculation. The total scattering coefficient ( $b$ ) was calculated by subtracting measured $a$ from $c$ .

It should be noted that the natural range of variability of the data used in the present study is more important than the accuracy of individual data in case of the inputs for the Monte Carlo simulation study as the simulation can generate results for any input parameters including simulated data. Therefore, we have not discarded few data for which the magnitude of the measurement data was beyond the 99% confidence level of the instrument.

3. Method

The Monte Carlo simulation works by calculating the trajectory of individual photons in the closest possible geometry defined corresponding to the actual instrument or situation being simulated. A huge number of such photons are generated and tracked in order to statistically calculate the required parameter of interest. A photon is tracked to an event and the weight of the photon is divided with respect to the statistical probability of absorption as an alternative to calculating the absorption probability and terminating the photon. This method of weighting the photon is often used in Monte Carlo simulation of radiative transfer in order to simplify the computation but would not change the final computed results [28]. A flowchart showing the simulation steps for a photon is shown in Fig. 1. In the present simulation, the source of the sensor is a cylinder with diameter of 0.005m which makes an angle of 26° perpendicular to the surface of the sensor face. This would make an elliptical disk at the sensor face. The ellipse described by the source lens is:

\frac{{(x - x_{1})}^{2}}{{(r_{2} / 2)}^{2}} + \frac{y^{2}}{{(r_{1} / 2)}^{2}} = 1,

where r₁ = 0.005 m, r₂ = 0.0056 m and x₁ = 0.0083 m. The source beam of light inside the cylinder is nearly collimated and makes 15° divergences from the source surface in all directions. The detector is also defined as a cylinder with diameter 10mm making an angle of 26° with the perpendicular to the surface of the sensor face in the opposite direction to the source. The schematic of the geometry is provided in Fig. 2. The detector has a Field-of-View 15° diverging from the axis in all direction thus making a solid FoV angle of 30°. The photons are simulated from just below this elliptical disk defined in Eq. (1). Initial direction of photons in terms of directional cosines just above the source elliptical disk is found from:

θ = (x - x') / \sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}

φ = (y - y') / \sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}

ω = (z - z') / \sqrt{{(x - x')}^{2} + {(y - y')}^{2} + {(z - z')}^{2}}

(x’, y’, z’) is the point at which the vectors defining the FoV would have originated if the FoV of the source were to be extrapolated in the backward direction and the coordinates of this point were calculated as (0.0117, 0, −0.0063). Calculating the reduction in weight due to refraction at lens-water interface at the source was based on the θ, φ, ω which were taken as the angular cosines of emergence of the beam and the Fresnel Equations, where ti is the angle of incidence and tt is the angle of transmittance.

t t = \cos^{- 1} (ω)

from Snell ’ s law, t i = \cos^{- 1} (\sqrt{1 - n^{2} \sin^{2} (t t)})

The reflected portion of light is then given by:

r a = \frac{1}{2} {{(\frac{\sin (t i - t t)}{\sin (t i + t t)})}^{2} + {(\frac{\tan (t i - t t)}{\tan (t i + t t)})}^{2}} .

Now the new weight of the photon is 1 – ra times the original weight of the photon which is the transmitted portion of the light. Once a photon interacts with particles, the new weight of photon is calculated based on the absorption and scattering coefficients. Simulations using Monte Carlo methods depend upon the probability density functions which is converted to a cumulative distribution function (CDF) [28]. The new direction of the photon is calculated based on the CDF [28] from the scattering phase function model. In the present study, the FF model is used to calculate the phase function values [25]. The photon is then moved to a limited random distance based on the probability density function defined by attenuation coefficient (_{$c = a + b$}) in the new direction [28]. The position of the photon is constantly traced to find if it has reached within the detector elliptical disk and within the FoV of the detector. The elliptical disk formed by the detector lens can be described by:

\frac{x^{2}}{{(r_{4} / 2)}^{2}} + \frac{y^{2}}{{(r_{3} / 2)}^{2}} = 1,

where r₃ = 0.01 and r₄ = 0.0111. The FoV of the detector can be defined in terms of directional cosines (by the following ways) similar to that of the source FoV.

θ = (x - x ") / \sqrt{{(x - x ")}^{2} + {(y - y ")}^{2} + {(z - z ")}^{2}}

φ = (y - y ") / \sqrt{{(x - x ")}^{2} + {(y - y ")}^{2} + {(z - z ")}^{2}}

ω = (z - z ") / \sqrt{{(x - x ")}^{2} + {(y - y ")}^{2} + {(z - z ")}^{2}}

where (x”, y”, z”) is the point at which the vectors defining the FoV would meet if it was to be extrapolated. This point is found to be (−0.0096, 0, −0.0159). The reduction in weight at the water-lens interface at the detector is also calculated similar to the Eq. (6) & (7). Photons with the incidence angles falling inside this FoV are counted while the rest are rejected. To approximate this, maximum and minimum direction cosines possible were found in each θ, φ, ω using Eq. (9) - (11) and used as limits. The error due to this approximation is insignificant and is ignored. A huge number of photons of the order of 10⁸ to 10⁹ are generated and tracked in this way in the simulation code. The photons reaching the detector elliptical disk in the FoV are counted and then normalized to the number of total photons generated at source. This is the measured count of the photons at the detector which needs to be correlated to the imposed

β

values so as to get the instrument reading in terms of meaningful scientific units. The angles of the incidence of each photon is also recorded and the average centroid angle was found for each of the simulations. The simulation was run in three-dimensional space accounting for the angular diversion of the source beam of photons and the acceptance angle of the detector was also accounted for.

Fig. 1 A flowchart depicting the path of a photon from source to detector in the Monte Carlo simulation.

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Fig. 2 A schematic diagram showing the geometric configuration used in the Monte Carlo simulation of the WETLabs BB9 sensor. Note that the angular divergence of the photon from the source is 30 degrees in both x and y directions.

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Calculating the path length of the instrument should include the geometry of the source sensor configuration. Calculating the path length based on the axes of the source and detector alone may not be representative of the exact path length covered as the configuration of the source detector arrangement makes a varying path length at different points on the source wave front. The optimum path length to the detector increases from the point on the sensor which is closest to the detector towards the farthest point. The angle made by the calculated path while moving from source to detector is in such a way that the central axis makes an angle equal to the centroid angle while the angular divergence of the source rays is applied to the ray path around the axis. However, when the ray path was in such a way that the angular diversion made by it to the detector was beyond the FoV of the detector, we used the maximum angle that could let the ray reach the detector. The instrument geometry used in this simulation is depicted in Fig. 2. The variations of path length calculated assuming the centroid angle to be 110, 115, 120, 124, and 130 degrees are shown in Fig. 3. The x-axis shows the distance of the point on source elliptical disk from the closest point to the detector. It should be noted that in addition to the centroid angle, the FoV of the source as well as the detector are limiting factors in the calculation of the path length. The average path length corresponding to each of these centroid angles were found by integrating these curves and dividing by the distance. These average path length variations with respect to the centroid angles are explained in a later section.

Fig. 3 A diagram showing the variation in path length from x = 0 to 0.56, where x = 0 is at the point where the boundaries of the elliptical disk of the source and the elliptical disk of the detectors coincides. Note the saturation of the maximum possible path length at the right-top corner of this graph. This is due to the limitation induced by the FoV of the sensor. This was calculated for various centroid angles.

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4. Results and discussion

The average path lengths calculated using integration corresponding to each centroid angles are given in Table 1. It was observed (Fig. 4) that a logarithmic relationship exists between the mean path length and the centroid angle:

p a t h l e n g t h = l \times \log (ψ) - m,

where l = 0.0411,

ψ

is the centorid angle in degrees and m = 0.17553 (R² = 0.99, N = 5). This indicates that the path length is actually variable with respect to the centroid angle which in turn is variable with the volume scattering function of the measured volume which may indroduce a non-linearity in the measurement scale factor.

Table 1. The centroid angles and their corresponding average path length found by integration

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Fig. 4 The mean path length versus centroid angles calculated by integrating the corresponding path length variation curves.

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The results of the Monte Carlo simulation based on the configuration and approximations and the details provided here yielded that the mean centroid angle for the entire data simulated was 124° which was calculated from the angles of incidence of photons. The detector counts recorded in the simulation were correlated with the imposed $β$ at corresponding centroid angles obtained from simulation values as shown in Fig. 5, wherein the detector counts are enumerated in the x-axis while the imposed $β$ values are given in the y-axis. It is observed that the imposed $β$ values follow a power function trend in relation to the detector counts. A power relation performs far better in comparison with a linear relationship as evident from these data. Therefore, we have used a power relation to convert the detector counts to scientific units. Our results suggest that for the detector counts being correlated to the actual measurement values or the imposed β values, a power function should be used instead of the commonly used scaling factor. Figure 5 demonstrates that a better correlation between counts received at the detector and the beta values are achieved by using a power function rather than a linear function with zero intercept. It is speculated that a power function would let include more measurement range for the instrument and this would ensure a higher accuracy and wider applicability. A scaling factor / linear function would have significantly overestimated the actual measurement values (Fig. 5). Various factors play in case of a backscattering sensor like BB9; the change of path length with the volume scattering function of the measured volume, the change of path length with respect to each photon according to its position in the source elliptical disk, the change in the actual FoV of the detector with respect to the change in the centroid angle, the change of centroid angles with the volume scattering function. These factors could cause the non-linearity in the sensor measurements. The detector count - power relation is excellent (R² = 0.99; N = 46) and can be expressed as:

β_{m e a s} (124^{\circ}) = f \times g^{h},

where the values found for f = 125.9 , h = 0.58, and g is the detector counts.

Fig. 5 The weighted counts of photons recorded in the detector in the given FoV in relation to the implied β values corresponding to the centroid angle(τ) at each data points.

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The path length calculated from the instrument geomtry using the centroid angle 124 ̊ as described earlier (refer Table 1 & Fig. 3) is found to be equal to 0.023. Using this pathlength (k), the absorption effect on $β$ can be written as [4]:

\frac{β (124^{\circ}, a_{n w} = 0)}{β (124^{\circ}, a_{n w})} = e^{k \times a_{n w}} .

This relationship between between

β (124^{\circ}, a_{n w} = 0) / β (124^{\circ}, a_{n w})

ratio, non-water absorption and path length was validated with the Monte Carlo simulations. The simulation was run for the same input data with and without absorption. The ratio of these readings were plotted against the absorption coefficient to find the effect of absorption on the backscattering measurement. The results from the above simulation and Eq. (14) with the caculated value of k corroborate each other which can be clearly observed in Fig. 6 with a high coefficient of correlation (R² = 0.94, N = 46). However, a small uncertainity of 3.9% was observed in the intercept values. This uncerainity in the intercept which should ideally be 1 can be attributed to the uncorrected scattering effect differences between the simulation runs with and without absorption [4]. Now the corrected VSF,

β_{c o r r}

, is simply another symbol for

β (124 °, a_{n w} = 0)

and

β_{w}

is VSF by pure water alone. To determine the particulate backscattering, the following conversion formula is used:

Fig. 6 The non-water absorption coefficient in relation to the ratio of β when a is set to zero and β when a is finite. The dashed line represents regression line between β_{a = 0}/β and a_nw where it was observed that k = 0.023.

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b_{b p} = 2 π χ (β_{c o r r} - β_{w}) .

Here, $χ$ is a considered as a constant factor with a value 1.1 though it is slightly variable which signifies the correlation of β measurement at an angle with the total backscattering coefficient [20,21]. The factor $χ$ is representative of the $β$ variation in the backward hemisphere and is highly sensitive to the composition, shape and particle size distribution and is projected to vary considerably for various water ecological regimes having varying composition of particulates. However, there exists a consistent angle estimated to be close to 120 ̊ at which the variability of $χ$ factor for different aquatic environments is very little [20,21]. It was observed from the simulation that the mean centroid angle is not exactly a constant for every measurement. It varies with the magnitude of volume scattering and attenuation coefficient which in turn fluctuate the path length. Nevertheless, it was observed that the centroid angles were all below 125 ̊ and averaging to 124 ̊ for our data sets used in the present study. It is observed from the simulation results (Fig. 7) that the centroid angle is proportional to the $β$ values below 0.15 and decreases thereafter with respect to the increasing $β$ . The centroid angle variation with respect to the imposed $c$ is depicted in Fig. 8. A similar trend to the $β$ – centroid angle (Fig. 7) is observed. The centroid angle increases to a certain maximum with respect to the $c$ and is observed to be subsequently decreasing with the increase of $c values$ . The path length variations with respect to the beta and with respect to the attenuation coefficient is depicted respectively in Figs. 9 & 10, wherein the path length was calculated based on the mean centroid angle using Eq. (12). The same trend as observed in Figs. 7 and 8 is seen on these results. This consistent effect in all of these results is due to the limiting condition imposed by the FoV of the detector so that the effective measurement centroid angle is reduced with respect to the expected centroid angle assuming the initial trend. This is for the simulation, while the calculation results for the path-length as demonstrated in Fig. 3 also prove this point as the curves for higher centroid angles can be seen as saturating at the farthest distances because of this limitation put by the FoV of the detector. This effect induces the non-linearity in the measurement of $β$ values from the detector counts and substantiate the use of a power relation for converting detector counts to scientific units in term of $β$ .

Fig. 7 The variation of the centroid angles calculated from the Monte Carlo simulation with respect to the β values calculated from the simulation.

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Fig. 8 The variation of the centroid angles calculated from the Monte Carlo simulation with respect to the implied c coefficients.

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Fig. 9 The variation of the mean path length calculated from the Monte Carlo simulation with respect to the implied β coefficients.

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Fig. 10 The variation of the mean path length calculated from the Monte Carlo simulation with respect to the implied c coefficients.

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5. Conclusion

Through three dimensional simulations using the Monte Carlo method, it was verified that the average centroid angle formed by the ECO-BB9 sensor is 124°. We report that the path length and the effect of absorption on backscattering measurement is in fact less than previously understood. These findings confirm the recent study by Doxaran et al. [4] and the recently updated centroid value in the ECO-BB9 user manual [29]. However, it was found that the effective path-length (0.023) and the coefficient corresponding to the absorption effect in backscattering measurement are different from the one provided in the BB9 user manual as well as the value found by Doxaran et al. [4]. While the variation of the $χ$ value with respect to the perturbations in the centroid angle of measurement [20] and its variation within a given centroid angle may introduce additional uncertainties, though less significant for angles around 120 ̊, they are outside the scope of this study as this work focuses only on the instrument specific configuration and associated parameters. The incapability of the central axes of the source emission and the detector acceptance angular limits to represent the effective path length are the reasons for the requirement of an effective path-length calculation as proposed in this study. The corroboration of the mean path length calculated geometrically and that obtained from the simulation results is a strong validation of the effective path-length estimated in this study. However, a method for estimation of the uncertainty due to the attenuation effect can be formulated for better accuracy in backscattering measurement. This study proposes that a power function be used to relate the detector counts in the instrument to the measurement value in scientific units rather than a scaling factor for better accuracy as observed from this study.

Funding

ISRO-IIT(M) Space Technology Cell (OEC1718175ISROPSHA); Defense Electronics Application Laboratory (DEAL) of the Defense Research and Development Organization (RB1617ELE005DRDOBALA).

Acknowledgments

We would like to thank D. Rajasekhar, The Head, Vessel Management Cell (VMC), and Director of National Institute of Ocean Technology (NIOT) for providing Research Vessels to Indian Institute of Technology (IIT) Madras, Chennai, India for making various physical and bio-optical measurements around Chennai and Point Calimere. We gratefully acknowledge Dr. David Doxaran and Dr. Edouard Leymarie of the Laboratoire d'Océanographie de Villefranche, CNRS-UPMC, France, for their kind assistance. We thank the reviewers for their constructive comments.

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Monte Carlo simulations of the backscattering measurements for associated uncertainty

Abstract

1. Introduction

2. Data

3. Method

4. Results and discussion

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (10)

Tables (1)

Equations (15)

Optics Express

Centroid angle	Mean path length by integration (m)
110	0.018
115	0.020
120	0.022
124	0.023
130	0.024