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Abstract
The forward volume scattering function (VSF) is an inherent optical property important in ocean lidar and underwater imaging and communication. The scattered power within 60° contains >90% of total scattered power, making it essential for determining the asymmetry parameter g. Thus, the new oblique-incidence-design Bettersize BT-3000 particle sizer was utilized to measure forward VSF (0.03°–60°) synchronously. A double-exponential model was then used to construct the full-angle-range VSF (0°–180°). The g value calculated therefrom had an uncertainty of <1%. Calibration was implemented using 11-µm beads alone, and the BT-3000’s performance was validated.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The volume scattering function (VSF) (β(θ), m−1sr−1) is a fundamental inherent optical property of the ocean and the most important parameter in solving the integral–differential radiative transfer equations for predicting the light field in the ocean [1–3]. It describes the angular spread of the scattered light resulting from an incident beam interacting with an infinitesimally small volume of water and is mathematically formulated as
where E is the irradiance of an incident beam, and dI is the radiant intensity scattered from a volume element dV at a scattering angle θ relative to the incident beam. The more routinely measured scattering coefficient, b, can be directly related to the VSF through the following equation, assuming azimuthal symmetry: Moreover, the VSF is often normalized by b to yield the phase function ($\bar{\beta }(\theta )$, sr−1) as The high variability of the propagation of collimated light (such as a laser pulse) in the ocean environment is complicated by the oceanic particulate VSF, which is the major challenge in the implementation of lasers for underwater ranging, communication, and imaging [4,5]. The particulate volume scattering causes dispersion of the laser beam or pulse both spatially and temporally [4,6] and thus decreases sensor performance in applications demanding high spatial and temporal resolution, such as laser bathymetry, profiling of scattering layers, and detection of submerged targets [6,7]. Because the phase functions of most oceanic constituents are characterized by well pronounced forward scattering, the significant variability in the forward VSF is a major influence during the determination of beam spreading or pulse stretching in the ocean.The beam spread function (BSF) in the anisotropic scattering medium of the ocean is usually solved analytically from radiative transfer equations by techniques such as small-angle approximation [8–10] or the method of moments [11]. In all of these techniques, the BSF is expressed by the quantities directly related to VSF, such as the mean value of the cosine function averaged over the single-scatter VSF, commonly referred to as the asymmetry parameter g:
The angular distribution of VSF in the forward direction is governed by the physical properties of the oceanic particles, such as size, shape, and composition, in a complex way [12–14] and naturally shows significant variability for different ocean environments. In the past, measurements of VSF have been performed with custom-built instruments such as the Multispectral Volume Scattering Meter (MVSM) [15], Multi-Angle Scattering Optical Tool (MASCOT) [16], and Polarized Volume Scattering Meter (POLVSM) [17]. These instruments only yield coarse angular resolution for near-forward angles, and most of them provide low rates of measurement, owing to usage of a single rotating detector for providing a wide range of measurement angles. To our knowledge, the most efficient and successful VSF measurements in the forward direction to date have been performed with the Laser In Situ Scattering and Transmissometry (LISST) series of instruments. The physical motivation behind the LISST instruments is the effectiveness of using a laser diffraction technique to measure the VSF in the forward direction; the particle size distribution is then derived from the small-angle scattering measurements using inversion methods [13]. The general working principle is that a laser beam of known power is transmitted through a sample chamber, and the forward-scattered light passes through a lens and onto 32 logarithmically spaced detector rings placed at the lens’s focal length [18]. Slade and Boss [18] used polystyrene beads to calibrate the LISST-100X to yield the VSF, correct in both shape and magnitude, for angles 0.08°–15°, and recently Sandven et al. [19] used the LISST-200X (the most recent successor of the LISST-100X) to measure VSF for angles 0.04°–13° at 670 nm. However, the number of published studies reporting results of VSF in the forward direction measured using LISST is still limited [19–22].
More importantly, according to VSF observations reported by Petzold [23], only 69%–83% of scattered power is contained in θ < 15°, and the scattered power within 60° can take up more than 90% of the total scattered power. Therefore, forward large-angle VSF measurements are crucial for the accurate calculation of g and determination of the BSF. The new commercial LISST-VSF instrument added an additional eyeball component to measure the VSF at θ values ranging from 15° to 60° and even up to 150°. Although a wide angular range (from 0.1° to 150°) was reached, the LISST-VSF has a more complicated structure and requires additional calibration for the eyeball component [24]. In view of these limitations, we investigated the oblique-incidence method (OIM) as a more convenient alternative for achieving good performance with VSF measurements from near 0° to 60°. Although some commercial particle sizers, such as the Malvern Mastersizer 3000 (Malvern Instruments Ltd.) and the Bettersizer 3000 (Bettersize Instruments Ltd.), have adopted the OIM to measure the distribution of smaller submicron particle sizes, they have not been used for VSF measurements.
In this study, the Bettersizer 3000 (BT-3000) particle sizer, which exploits oblique incidence, was used to record the forward VSF between 0.03° and 60°. The feasibility of calibration using standard polystyrene beads of various sizes to obtain both the shape and magnitude of the forward VSF was evaluated. Measurements on natural waters were conducted in three locations: the East China Sea (ECS), Xiangshan Harbor, and the Wenzhou coastal areas in China. Comparison measurements using LISST-VSF were conducted in Xiangshan Harbor. Finally, we investigated construction of the full-angle-range VSF based on the forward scattering to estimate the value of the asymmetry parameter.
2. Methods and materials
2.1 Oblique-incidence design of the BT-3000 particle sizer
The oblique-incidence method has more advantages than the normal-incidence method and can measure the forward scattering in a wider angular range. With the normal-incidence design, shown in Fig. 1(a), the incident light is perpendicular to the receiving window (RW). The maximal scattering angle θs is determined by the numerical aperture (NA) of the lens (L), denoted as θNA. However, when the incident light has an oblique angle α with respect to the optical axis (OA), the maximal scattering angle is equal to θs plus α (Fig. 1(b)). This is the feature whereby the oblique-incidence design may broaden the detection angular range for detection of scattering. In this study, the BT-3000 particle sizer with this oblique-incidence design was used to synchronously measure forward VSF from approximately 0.03° to 60°.
Fig. 1. Comparison between normal incidence and oblique incidence. Gray and green solid lines represent light paths. The thicker lines represent the incident light; the thinner lines represent the light scattered at the maximal angle. Both designs include a receiving window (RW), lens (L), and ring detector (RD). The lens focuses the light on the ring detector located in the focal plane of the lens. (a) Normal-incidence design: the incident light is perpendicular to the window; the maximal scattering angle is θs. (b) Oblique-incidence design: α is the oblique angle with respect to the optical axis (OA); the maximal scattering angle is θs + α.
A schematic of the BT-3000 particle sizer is shown in Fig. 2(a). The BT-3000 consists mainly of an obliquely placed laser source (LS) centered on 515 nm (or 532 nm), a pair of lenses (L1 and L2), a cuvette (CV), and a ring detector (RD). The laser source is positioned at the focal length of Lens 1. Lens 1 turns the divergent laser beam into a parallel beam that illuminates the water sample. Lens 2 converges the direct beam into one relatively tiny spot for optical power measurement of this beam. At the same time, Lens 2 converges the scattered light, especially the light scattered at large angles. The detection area of the detector ring is smallest in the forward direction (near 0°) and largest at angles near 60°; thus, the strong forward-scattering signal is captured more accurately, while as much of the weak signal of light scattered near 60° is collected as possible. There are 76 rings between 0.03° and 60°. The signal detected by the first ring represents the power of the direct light and is used to calculate the attenuation coefficient c for the sample. The path length of the light passing through the basin is 12 mm, and the scattering volume is approximately 0.09 cm− 3. All of these components are contained in a black-anodized housing to avoid interference from ambient light. A cuvette (CV), centrifugal pump (CP), circulation chamber (CC), and ultrasonic degassing module (UD) constitute the water circulation system. The internal optical structure of BT-3000 is shown in Fig. 2(b). When BT-3000 works, the light is emitted by the laser source (LS) and converges into parallel light through the lens 1 (L1). When the parallel light passes through the cuvette (CV), it interacts with the particles in the cuvette (CV) producing the scattered light. The scattered light is accepted by the ring detector (RD) after passing through the lens 2 (L2). The direct light continues to transfer forward and is also accepted by the ring detector (RD) after passing through the lens 2 (L2). The overall of BT-3000 is shown in Fig. 2(c).
Fig. 2. (a) Schematic of BT-3000 particle sizer. The laser source (LS)’s central wavelength is 515 nm (or 532 nm). It is positioned at the focal length of Lens 1 (L1). Lens 1 turns the divergent beam into a parallel beam. The water sample is stored in the cuvette (CV), and the blue part is the scattering volume (SV). Lens 2 (L2) converges the scattered light. The ring detector (RD) is placed at the focal length of Lens 2 (L2). The part marked in red is used to detect the power of the direct light and also to determine the attenuation coefficient of the sample. Other irregular rings are used to detect the scattering signal. There are 76 detectors between 0.03° and 60°, spaced logarithmically. A cuvette (CV), centrifugal pump (CP), circulation chamber (CC), and ultrasonic degassing module (UD) constitute the water circulation system. (b) The internal optical structure of BT-3000. (c) Photograph of BT-3000.
2.2 Determination of particulate VSF
To accurately obtain the particulate VSF, a series of work were carried out using standard polystyrene beads of known particle sizes and refractive indexes. First, the contribution of the VSF of pure water was subtracted from the total VSF for each group of samples to allow determination of the particulate contribution alone. Pure water was prepared using the Milli-Q Advantage A10 water purification system (EMD Millipore Inc.) and then filtered through a polycarbonate cartridge filter of pore size 0.2 µm (PN 12991, Pall Corp.) to further remove residual particle contamination. In addition, before the measurement of pure water was performed, the ultrasonic degassing module inside the BT-3000 was used to eliminate bubbles to avoid measurement error caused by bubbles.
Each detector ring corresponds to a specific scattering angle. The accuracy of this angle correspondence needed to be verified to ensure that the directional properties of the particles would be properly characterized. The verification results are shown in Section 3.2. The area of each detector ring is different, and the distance between each ring and the scattering volume is different, resulting in a different solid angle for each ring. Using the definition of VSF, the signals measured by each ring were normalized according to the following equation:
where VN(θ) is the normalized voltage, V(θ) is the measured raw voltage, S(θ) is the area of the detector ring (which was provided by the manufacturer, Bettersize Instruments Ltd.), and r is the distance between the detector ring and the scattering volume. The path length of light propagation in the sample is very small (12 mm), and the scattering volume is in the receiving field of view of each ring. Therefore, it was considered that the scattering volume is roughly the same for each scattering angle.The most important step was that of converting the units of normalized voltage (V sr−1) into geophysical units (m−1 sr−1). Polystyrene microsphere beads with precisely known diameters, particle size distributions, and refractive indexes are commonly used for calibration because their absolute VSF can be calculated from Mie theory. Mie theory considers 1) the wavelength of light; 2) particle size (mean size and distribution of sizes); 3) size parameter, defined as x = 2πr/λ, where r and λ are the sphere radius and wavelength of light (in the medium surrounding the particle), respectively; and 4) the complex index of refraction of the spheres (np = nr + ni) relative to the surrounding medium (nw, assumed non-absorbing), n = np/nw. The real index of refraction for polystyrene was calculated based on the results of Ma et al., and we used a value of 0.00035 ± 0.00015 for the imaginary part of the refractive index [25]. The index of refraction of pure water at room temperature is 1.337 at 515 nm (1.336 at 532 nm). The polystyrene beads (Duke StandardsTM) used for calibration (11-µm) and validation (2-, 5-, 11- and 20-µm) of BT-3000 were ordered from Thermo Fisher Scientific Inc.; their specifications are listed in Table 1.
Table 1. Specifications of polystyrene beads used in this study. Beads of a nominal diameter µND are assumed to be normally distributed with an actual mean diameter of µD and a standard deviation of σD. δD represents the uncertainty in determining µD at the 95% confidence level. The complex refractive index (np) values at 515 nm and 532 nm are also shown.
The specific method of conversion is that using the following equation to calculate the calibration coefficient [18,24]:
where ${\bar{\beta }_{Mie}}(\theta )$ is the phase function calculated from Mie theory, cBT is the attenuation coefficient as measured by the BT-3000, k(θ) is the calibration coefficient, and VN(θ) is the normalized voltage.Slade and Boss [18] adopted three kinds of beads with different sizes to determine the calibration coefficient k for different angle ranges based on the following reasoning. In a certain angle range, beads that provide a much flatter scattering response are more suitable for determining the value of k in this angle range because the ripples in the angular scattering that are very sensitive to the precise size and refractive index of the beads cause k to have greater uncertainty; large beads show strong forward scattering and thus were used to determine the value of k in the near-forward-angle range. This method was successfully used to obtain accurate values of k according to the scattering patterns of different beads in different angle ranges.
In this study, exploiting the design and capabilities of the BT-3000, we used a simpler method to determine k using only 11-µm beads. As shown in Fig. 3(a), 11-µm beads provide relatively featureless patterns of angular scattering in the angles within 0.8° and from 6.6° to 60°, whereas 20-µm beads provide flat and stronger scattering in the most near-forward angles within 0.08°, and only 2-µm beads produce a much flatter angular scattering from 0.8° to 6.6°. According to the method proposed by Slade and Boss [18], data from beads of the three sizes should be used to calibrate the different detector rings. In other words, letting k2, k11, and k20 represent the calibration coefficients determined using 2-, 11-, and 20-µm beads, respectively, k(0.8°–6.6°) = k2(0.8°–6.6°), k(6.6°–60°) = k11(6.6°–60°), and k(<0.08°) = k20(<0.08°). This method is certainly reasonable but is time consuming. Based on an analysis of the calibration results using these different bead sizes, that k20(0.03°-0.08°) = 207.8 ± 16.3 sr−1 m−1 V−1, k11(0.08°–0.8°) = 207.4 ± 6.8 sr−1 m−1 V−1, k2(0.8°–6.6°) = 210.6 ± 6.3 sr−1 m−1 V−1 (see Fig. 6(d) in section 3.1), these results are close to each other. Then we found that the value of k varied only slightly across the entire angle range, which could also be confirmed by the fact that the photoelectric efficiency of each detector ring is close to each other. Therefore, there should be little difference between k2, k11, and k20. k2(0.8°–6.6°) and k20(<0.08°) were nearly equal to the mean value of k11(0.08°–0.8°). Therefore, the final value for k was determined as follows
Fig. 3. (a) Theoretical phase function for 2-, 11-, and 20-µm beads at 515 nm calculated using Mie theory. All three kinds of beads provide relatively flat scattering response within 0.8°. 2-µm beads provide relatively flat scattering response from 0.8° to 6.6°. 11-µm beads provide relatively flat scattering response from 6.6° to 60°. (b) Polarization factor p(θ) = |m12(θ)/m11(θ)| calculated using Mie theory for the standard beads.
The impacts of the uncertainties (measured as the coefficient of variation) at 515 nm within 60° in the mean diameter varying within µD ± δD on the ${\bar{\beta }_{Mie}}$ calculation were 1.4%, 3.6%, and 5.7% for the 2-, 11-, and 20-µm beads, respectively. The imaginary part of the refractive index is small in the visible wavelengths, and its effects on scattering within 60° can be neglected [15,17,25,26]. In addition, the VSF, which corresponds exactly to the m11 element of the Mueller scattering matrix [17], describes the scattering distribution of unpolarized incident light. However, the laser installed in the BT-3000 is entirely linearly polarized, and the ring detectors are non-polarized receivers. Therefore, the polarization angle of the laser sources is nominally set to 45° with respect to the vertical axis. However, as it is difficult to achieve a perfect 45°, meaning that the measured signals will incorporate the m12 and m13 elements of the Mueller matrix [18], it is necessary to consider the influence of polarization on the calibration. Because m13 approaches zero in the case of spherical scattering [17], even in natural samples, the polarization factor was calculated as p(θ) = |m12(θ)/m11(θ)| to indicate the degree of linear polarization of the scattered light. As shown in Fig. 3(b), the p values for the three beads as calculated from Mie theory are all very small (less than 0.1) within 6.6°. For the 11-µm beads in particular, p is less than 0.02 within 6.6° and increases after 6.6° but remains less than 0.09. The error introduced by use of the linearly polarized laser is thus relatively small when 11-µm beads are used for calibration.
During the calibration procedure, a master solution of each kind of bead was prepared and was agitated on a vortex mixer to homogenize the suspension and break apart possible aggregations of particles. Subsequently, controlled amounts of bead master solution were added to the chamber sequentially. For each kind of bead, five groups of solutions with different concentrations were measured. The optical thickness of each sample was controlled to be less than 0.1 to meet the requirement for a single-scattering regime [27]. After each addition, the circulation system was kept operating for 1 min to homogenize the beads in the chamber. Each sample was measured by 30 times.
Calibration for VSF at 532 nm followed the same method. The scattering pattern of standard beads within 60°, the impacts of the uncertainties within 60° in the mean diameter varying within µD ± δD on the ${\bar{\beta }_{Mie}}$ calculation, and the error introduced by the linearly polarized laser at 532 nm were similarly considered.
2.3 Data collection
Forward-scattering VSF measurements using the BT-3000 were conducted on three cruises: in Xiangshan Harbor in January 2021, in the Wenzhou coastal areas in May 2021, and in the East China Sea in August 2021. The sampling sites on the three cruises are shown in Fig. 4(a)–(c). The measured samples covered various types of water bodies, which had c values ranging from 0.62 m−1 to 10.87 m−1.
Fig. 4. (a) Map of sampling locations in Xiangshan Harbor (January 2021). (b) Map of sampling locations in the Wenzhou coastal areas (May 2021). (c) Map of sampling locations in the East China Sea (August 2021). The gray part represents the land, and the white part represents the sea.
Two instruments in addition to the BT-3000 were used to measure VSFs: a LISST-VSF and the VSFlab. LISST-VSF (Sequoia Scientific, Inc.) is a commercially mature, widely used VSF meter [28]. It consists of two components: one called “ring detectors,” with a normal-incidence design, which covers scattering angles from 0.1° to 15°, and the other called “eyeball,” a rotating detector, which covers a wide range of scattering angles, from 15° to 150°. This instrument was used for validation of the BT-3000 at Xiangshan Harbor.
The VSFlab (shown in Fig. 5(a)) is a custom device established based on the principle of the double periscopic optical system and rotating detector to measure the VSF for 1°–178.5° at 532 nm (or 520 nm) [29]. The specific calibration and validation results of VSFlab could be seen in S1 of supplemental document. Both the BT-3000 and the VSFlab have a 532-nm wavelength. VSF measurements using both the BT-3000 and the VSFlab were implemented in the Wenzhou coastal areas and the East China Sea; these VSFs were then combined to produce a single VSF from near 0.03° to 178.5° at 532 nm. The number of samples measured by each VSF instrument on each cruise is shown in Table 2 along with the detection wavelength used with the BT-3000.
Fig. 5. (a) Photograph of VSFlab. The two valves on the outer wall of the basin are the interface to the circulation system. (b) Laboratory installation of LISST-VSF: A, opaque cloth; B, polycarbonate cartridge filter, pore size 0.2 µm; C, Liqui-Cel Membrane Contactor; D, diaphragm pump; E, LISST-VSF. The polycarbonate cartridge filter, diaphragm pump, and Liqui-Cel Membrane Contactor were connected via a silica gel tube to form a filtration/degassing/circulation system.
Table 2. Number of samples measured by three instruments during each observation cruise and detection wavelength used with BT-3000.
The calibration and measurement procedures for the BT-3000 are described fully in Section 2.2. Similar procedures including the baseline correction using pure seawater, bubble removal, and calibration with standard beads were implemented for the other two instruments as well. For calibration of the LISST-VSF, the instrument was installed on a laboratory table according to the instructions, as shown in Fig. 5(b). Before each measurement, we cleaned the inner endcaps and the windows, drained the test chamber, and wiped and dried the windows. The pre-filtered seawater was poured into the sample volume. The polycarbonate cartridge filter of pore size 0.2 µm (PN 12991, Pall Corp.), the diaphragm pump (EMD Millipore Inc.), and the Liqui-Cel Membrane Contactor (3M Co.) were connected via a silica gel tube to form a circulation system. Filtration and bubble removal lasted 1 h. Finally, degassed and particle-free seawater was prepared. Because the LISST-VSF ring detector is sensitive to ambient light, the sample volume was covered with an opaque cloth during measurements. Then, standard measurements were implemented to determine the calibration coefficients according to the method proposed by Slade and Boss [18] and Hu et al. [24]. For the VSFlab, two valves on the outer wall of the basin were connected with the silica gel tube of the circulation system (shown in Fig. 5(b)) to complete the filtration and bubble removal. Calibration followed the same procedures as for the LISST-VSF but using 0.203-µm beads to determine the calibration coefficients. Each sample was fully mixed in a magnetic stirrer before being measured. There is no magnetic stirrer installed in the VSFlab; however, the particles were thought to be evenly distributed in the solution as a complete measurement only requires 8 s. Calibration was performed each day for each of the three instruments to ensure their measurement accuracy.
In addition, a detector for monitoring the electrical signals is installed inside the BT-3000. For performing the sample measurements, when the relative standard deviation of this signal was within ±5‰, the instrument was considered stable, and measurement was begun. Each sample was mixed well via the circulation system inside the BT-3000. The optical thickness τ of seawater samples should be less than 0.1 before measurement. For the BT-3000, τ is calculated according to
where s is the shading rate, representing the percentage of the energy loss of the incident light. s was measured by the BT-3000. If τ exceeded 0.1, the sample was diluted with pure water to meet the requirement for the single-scattering regime. Similarly, measurements using the other two instruments were also carried out in the single-scattering regime.3. Results and discussion
3.1 Calibration coefficient k determined by 11-µm beads
Measurements using different concentrations of solutions of 20-, 11-, and 2-µm beads were performed to validate the calibration coefficient k as determined using 11-µm beads. By applying a robust linear regression model [30], the calibration coefficient k at each angle was derived as the slope between the gained signal (voltage, V) and VSF (β, m−1 sr−1) calculated from Mie theory for each angle. Figure 6(a)–(c) shows the correlation for 20-, 11-, and 2-µm beads between gained voltage and β at 0.04°, 0.5°/60°, and 2°, respectively. β exhibited a tightly linear relationship with the gained voltage: all coefficients of determination (R2) reached 0.99.
Fig. 6. Mie-derived VSF plotted against VN(θ) for (a) 20-µm beads at 0.04°, (b) 11-µm beads at 0.5°/60°, and (c) 2-µm beads at 2°. The values (voltage and β) at 60° for 11-µm beads are multiplied by 30,000 for an unambiguous presentation. Horizontal and vertical error bars represent standard deviations estimated, respectively, from the 30 measurements of VN(θ) at each concentration and ${\beta _{\textrm{Mie}}}(\theta )$ calculated by accounting for uncertainties in the mean diameter of the beads and measured cBT. The robust fit regression line (black line) is also plotted. (d) Calibration coefficients k. The blue line (k20), red line (k11), and green line (k2) represent the k values estimated using 20-, 11-, and 2-µm beads, respectively. Vertical error bars represent standard deviations of the estimated coefficients. The black line (k) represents the corrected k value.
Figure 6(d) shows the estimated values of k2, k11, and k20 and the final k value. In the range 0.03°–0.08°, only k20 was available, having the value 207.8 ± 16.3 sr−1 m−1 V−1, because 20-µm beads scatter strongly in the most near-forward rings. The 11- and 20-µm beads yielded consistent calibration results from 0.08° to 0.8°; k11 and k20 were 207.4 ± 6.8 and 214.8 ± 8.5 sr−1 m−1 V−1, respectively. Although the near-forward scattering of 2-µm beads was weak and k2 was unreliable within 0.8°, k2 exhibited a flatter curve in the range 0.8°–6.6°, having the value 210.6 ± 6.3 sr−1 m−1 V−1. In contrast, in the range 0.8°–6.6°, the curves of k20 and k11 exhibit ripples, indicating that 20- and 11-µm beads are not suitable for calibration in this angle range. This is because of the larger deviations due to increasing ${\bar{\beta }_{Mie}}(\theta )/d\theta $ (Fig. 3). More importantly, it was found that k2 from 0.8° to 6.6° is nearly the same as k11 from 0.08° to 0.8°. These results suggest that k changed only slightly in the angle range where the theoretical ${\bar{\beta }_{Mie}}(\theta )$ curves change gently. There is little difference between the four k values (207.8, 207.4, 214.8, and 210.6 sr−1 m−1 V−1); the relative standard deviation is approximately 1.6%. This indicates that the photoelectric efficiency of each detection ring was equivalent, and the k curve could be considered as a straight line within 6.6°. Therefore, the k value within 6.6° was determined as the mean of k11 in the range 0.08°–0.8°, or 207.4 sr−1 m−1 V−1. On the other hand, the k coefficients in the range 6.6°–60° were also the same as k11, which decreased gently with increasing scattering angle. The same reduction of the calibration coefficient can be seen in Slade and Boss’s work [18]. Based on k11 alone, the calibration coefficient k was finally determined for the calibration of other measured VSF curves.
3.2 Performance validation
3.2.1 Validation with 20-, 11-, 5-, and 2-µm beads
In order to evaluate the accuracy of the BT-3000, a comparison between VSF (βBT) as measured by the BT-3000 and the theoretical VSF (βMie) calculated from Mie theory using 20-, 11-, 5-, and 2-µm beads of different concentrations was carried out using the k value derived from 11-µm beads. The contribution of pure water has been subtracted from all results. Figure 7(a)–(d) shows the results for 20-, 11-, 5-, and 2-µm-bead samples with scattering coefficient (b) values of 7.52, 4.63, 3.55, and 4.71 m−1, respectively. As the optical thickness of these samples was less than 0.06, the single-scattering assumption was valid for these measurements. All βBT values agree well with the βMie values within 60°, and the mean absolute percentage difference (APD) values for the 20-, 11-, 5-, and 2-µm beads were 14.15%, 7.44%, 13.01%, and 17.54%, respectively (Fig. 7(e)). These results indicate that the BT-3000 is capable of delivering accurate VSF measurements with polystyrene beads within 60°. More importantly, this measurement is for the forward direction, in contrast with measurements by the LISST-VSF’s ring detector (0.1°–15°) and eyeball (15°–60°) components and was realized with a single measurement strategy.
Fig. 7. (a)–(d) Validation results comparing the VSFs measured by BT-3000 (βBT, red line) and Mie-simulated VSFs (βMie, black line) for 20-, 11-, 5-, and 2-µm beads. (e) Angular evaluation of the validation results in terms of absolute percentage difference (APD) between βBT and βMie. The mean APDs for 20-, 11-, 5-, and 2-µm beads were 14.15%, 7.44%, 13.01%, and 17.54%, respectively. (f) Scatterplot of βBT and βMie for 20-, 11-, 5-, and 2-µm beads, each at five concentrations. The total number of points (N) is 1245. The overall APD is 12.37%, and the Pearson correlation coefficient r is 0.99.
At the forward angles near 0°, the scattering signal for the smaller beads (i.e., 2 and 5 µm) was smaller than the theoretical values and sometimes could not even be detected. This is because the smaller beads provide weaker forward-scattering intensity than larger beads (i.e., 20 and 11 µm). At the forward angles near 0°, where the receiving area of the rings is small, the signal-to-noise ratio (SNR) of the detection signal is unavoidably smaller than that at other angles. In the range of angles from 0.8° to 6.6°, there was deviation between βBT and βMie for the 20- and 11-µm beads, especially at the positions of peaks and valleys. The absolute percentage difference (APD) of both 20- and 11-µm beads in this angle range was extremely high (Fig. 7(e)). This is because the k20 and k11 values were replaced by a mean value for k11 in this angle range, where βMie curves fluctuate. The average APDs for 20- and 11-µm beads excluding the range of angles from 0.8° to 6.6° were 7.06% and 7.13%, respectively. The ring detectors are arrayed in steps whose detection areas are related logarithmically, and thus the angular resolution is high for the forward direction but low between 15° and 60°. Therefore, the measured VSF curves were unable to reflect the fluctuation of the theoretical VSF for 2- and 5-µm beads from 15° to 60°, with the result that the APD for the 2- and 5-µm beads was a bit higher (Fig. 7(e)). However, this does not matter because such fluctuation would likely not be a pattern for the VSF of natural samples. The natural samples provide a much flatter scattering response compared with 2- and 5-µm beads. The mean APDs for 5- and 2-µm beads excluding the range of angles from 15° to 60° were 9.76% and 12.02%, respectively. By applying this corrected k value, the βBT values for 20-, 11-, 5-, and 2-µm beads of five concentrations exhibit a tightly linear relationship with βMie. The linear fit, which includes a total of 1245 data points, displays a Pearson correlation coefficient r of 0.99 and an overall APD of 12.37% (Fig. 7(f)).
In general, good validation results were achieved by applying the proposed k. These results demonstrate that one size of bead (11 µm) is sufficient for completing the calibration procedure for the BT-3000, which is simpler than other methods using standard beads of various sizes. This calibration method was applied for the natural sample measurements.
3.2.2 Comparison with LISST-VSF
First, 2-µm beads were used to validate the LISST-VSF calibration results by comparing them with theoretical results. The APD for the LISST-VSF measurements was near 7.17%, showing good consistency between theoretical and measured VSFs (Fig. 8(a)). Then, experiments to compare BT-3000 and LISST-VSF were carried out at ten stations in Xiangshan Harbor. The attenuation coefficient c for these stations varied from 2.13 to 10.87 m−1. All VSF results were normalized by c for the comparison. The actual wavelength of the laser in the LISST-VSF used here is 519 nm, which differs slightly from the nominal wavelength of 515 nm. An assessment found the mean APD for 20-, 11-, 5-, and 2-µm beads between the theoretical results at 515 nm and 519 nm to be approximately 8.13%.
Fig. 8. (a) Comparison between the VSFs measured by LISST-VSF and Mie-simulated VSFs for 2-µm beads. (b)–(d) Results of VSF measurements at Xiangshan Harbor: (b), (c) Examples of comparison of VSFs (βBT/c) measured with BT-3000 and VSFs (βLISST/c) measured with LISST-VSF. Blue dots are the results from 0.1° to 15° measured with LISST-VSF’s ring component. Black dots are the results from 15° to 60° measured with LISST-VSF’s eyeball component. The red line is the result from 0.04° to 60° measured with BT-3000. (d) Scatterplot of βBT/c and βLISST/c for all ten stations. The overall APD is 24.04%, and the Pearson correlation coefficient r is 0.98.
Figure 8(b), (c) shows comparisons of two stations’ results as examples. Generally, the results as measured with the BT-3000 are consistent with those measured with LISST-VSF’s ring detector and eyeball components, having APD values of 13.9% and 18.1%, respectively. The VSFs measured with LISST-VSF’s ring detector near 0.1° were higher than those measured with the BT-3000 (Fig. 8(c)). The VSFs of 11-µm beads measured by Hu et al. with LISST-VSF were also higher than the theoretical VSFs near 0.1° [24]. Koestner et al. [31] found that the VSFs measured using standard beads near 0.1° were less than the theoretical VSFs. This is because 0.1° is the minimum detection angle of LISST-VSF, and the measurement uncertainty near 0.1° is relatively high. Similarly, for BT-3000, the measurement uncertainty is likewise high at angles close to 0.03°, and thus the measurement curves sometimes exhibit ripples near 0.03°. A comparison of results for all ten stations shows that there is a tightly linear relationship between BT-3000 measurement results and LISST-VSF measurement results (Fig. 8(d)). The Pearson correlation coefficient is 0.98, and the mean APD is 24.04%. In summary, the BT-3000 has good performance in measuring VSF and was demonstrated to be a reliable VSF meter.
3.3 Contribution of forward VSF to g
The asymmetry parameter g is a convenient measure of the shape of the VSF. For example, if VSF is very large for small angles, g is near 1; in this case, g would be determined mainly by forward VSF for ocean water with strong forward scattering. In this section, the contribution of forward VSF to g is analyzed.
The calculation of g requires the full-angle-range (0°–180°) VSF. At present, there is no single strategy for obtaining the full-angle-range VSF. Typically, two instruments are proposed for obtaining VSFs in two angle ranges, and then these results are combined to obtain a single VSF curve [21,23]. Thus, the general strategy is to use two instruments to measure VSF, one for near-forward angles and one for a wide angular range, to obtain the full-angle-range VSF. The forward VSF (0.03°–60°) and wide-angle-range VSF (1°–178.5°) were measured at the same time with the BT-3000 and VSFlab at 532 nm, respectively, at 20 stations in the Wenzhou coastal areas and the East China Sea. After calibration, the two VSFs were spliced to form an approximately full-angle-range VSF (0.03°–178.5°). Then, we extrapolated the VSF in the near-forward direction using an exponential model and in the backward direction using a linear model to obtain the full-angle-range VSF (0°–180°) [17], thereby allowing g to be calculated as mathematically defined. The validation of full-angle-range VSF obtained could be seen in S2 of supplemental document. Typical ocean waters (clear ocean, coastal ocean, turbid harbor and particulate) have g values of 0.869, 0.94, 0.916 and 0.921 (e.g., samples 21–24 in Fig. 9(b)), and g measured in the Wenzhou coastal areas and in the ECS ranged from 0.91 to 0.96. These values of g, near 1, indicate that the ocean water in these regions exhibits strong forward scattering; this can also be seen in Fig. 9(a).
Fig. 9. (a) VSFs measured between 0° and 180° at 20 stations in the Wenzhou coastal areas and the East China Sea (blue lines), and other types of VSF (turbid harbor, particulate, coastal ocean, and clear ocean) as measured by Petzold. (b) g values calculated from the VSFs in (a). (c) Probability (P) for the VSFs in (a). (d) Asymmetry parameter g(θ) calculated based on the VSFs in (a) within θ.
Then, the contribution of the forward scattering to the total scattering and to g were assessed by calculating P(θ) and the asymmetry parameter g(θ), respectively, as functions of the angle, based on the measured VSFs (βm), where P(θ) is a probability representing the proportion of the power scattered into angles less than θ relative to the total scattered power:
3.4 Modeling VSF from forward scattering
Considering the high contribution of forward VSF to the total scattering and to g, we explored the possibility of using the forward VSF data alone to construct the full-angle-range VSF through a double-exponential model:
The measured full-angle-range VSF was taken as the true value for evaluation of the four models. The modeling results are shown in Fig. 10(a)–(d). The four constructed VSFs are in good basic agreement with the measured VSFs. The mean APD between the modeled VSFs and the measured VSF for F-60, F-45, F-30, and F-15 were 24.79%, 33.18%, 43.95%, and 82.62%, respectively (Fig. 10(e)). F-60 yielded the highest accuracy. It can be seen that the wider the angle range was, the more data were available for modeling and the smaller was the APD. With regard to the accuracy of g, the mean APD decreased as the angular range of the model increased (Fig. 10(f)). The mean APD values for g calculated from F-60, F-45, F-30, and F-15 were 0.38%, 1.03%, 2.11%, and 3.27%, respectively. F-60 yielded the best performance in estimating g, and the APD of less than 1% satisfies the error limit for the determination of zD. The methods (F-60, F-45, F-30, and F-15) were also applied to VSFs (clear ocean, coastal ocean, turbid harbor and particle) measured by Petzold. Taking coastal ocean as an example, gc of the coastal ocean water calculated from four methods (F-60, F-45, F-30, and F-15) was 0.944, 0.945, 0.95 and 0.96. The mean APD values were 0.45%, 0.62%, 1.26%, and 2.5% for four methods, respectively. Therefore, constructing a full-angle-range VSF by using forward VSF (0.03°–60°) obtained by the BT-3000 is a simple and reliable strategy for the accurate estimation of g.
Fig. 10. (a)–(d) VSFs modeled using a double-exponential model based on the VSF data within 60°, 45°, 30°, and 15°, respectively. In (a), for example, the new constructed VSFs consist of the measured VSFs from 0° to 60° and the modeled VSFs from 60° to 180°. The four model methods are designated F-60, F-45, F-30, and F-15, respectively. (e) Absolute percentage difference (APD) between the modeled VSFs and the measured VSFs at each angle. (f) APD between the g values calculated based on the constructed VSFs and measured VSFs for each sample.
4. Conclusion
Our findings demonstrate that the oblique-incidence particle sizer BT-3000 can be utilized to measure the forward VSF. Because the instrument’s oblique-incidence design expands the angular range for single measurements, it enables the measurement of VSF between 0.03° and 60°. Through validation using different standard beads and comparison with the LISST-VSF, it was demonstrated that the task of calibration can be completed using 11-µm beads alone. The probability was calculated based on the measured VSF; 95% of the scattering energy is distributed in the angle within 60°. The g value calculated by using the VSF within 60° is close to the actual g value, demonstrating the importance of the forward scattering, especially the VSF within 60°. Finally, VSFs for four angular ranges (within 15°, within 30°, within 45°, and within 60°) were used to construct the full-angle-range VSF using a double-exponential model; the four methods are designated F-15, F-30, F-45, and F-60, respectively. g was then calculated according to the VSF thereby constructed. A comparison with g values calculated from the measured VSFs found that F-60 is the most accurate of the four methods, with an APD of 0.38%. In summary, the oblique-incidence VSF measurement strategy provides a simple and effective solution for the accurate estimation of the asymmetry parameter g for seawater. This strategy uses detector rings to measure the VSF in the range 0.03°–60° at one time; hence, it offers the potential of being applied to analyze the optical field distribution within 60° and to realize rapid in situ measurement of g. The small path length of light propagation in the sample is unable to meet the measurement of waters with low scattering. In future work, given the low scattering intensity of clear ocean water, it would be useful to allow adjustment of the path length to increase the scattering volume. It should be noted that the small-angle forward VSF is highly variable as it is determined by the particle size distribution (PSD). The PSD is not constant in water, and variable in both time and space; thus, the double exponential function used here may be not a canonical form of VSF in the other areas, and other forms of approximate models should be considered
Funding
The Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (GML2019ZD0602); Global Change and Air-Sea Interaction II Program (GASI-01-DLYG-EPAC02, GASI-01-DLYG-WIND02); National Natural Science Foundation of China (41876033, 41976179, 61991454).
Acknowledgments
We thank Yaorui Pan, Changpeng Li and Youzhi Li from the second institution of oceanography for help carrying out experiments for this work
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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