Retrieving the seawater volume scattering function at the 180&#x00B0; scattering angle with a high-spectral-resolution lidar

Yudi Zhou; Dong Liu; Peituo Xu; Chong Liu; Jian Bai; Liming Yang; Zhongtao Cheng; Peijun Tang; Yupeng Zhang; Lin Su

doi:10.1364/OE.25.011813

1. Introduction

The ocean covers more than 71 percent of the earth. Studies on the vast ocean, e.g. primary productivity, and biogeochemical cycles, etc., are of great significance to resources utilization and climate change [1, 2]. Several methods are employed for detecting the interior of the ocean. In situ methods, such as the transmissometer and scattering meter, can obtain the marine information accurately, yet are limited to the efficiency [3]. The ocean color remote sensing based on the plane or satellite, like Sea-viewing Wide Field-of-view Sensor [4], is able to collect the global data over a long term efficiently. Nevertheless, the limited information about the depth and the dependence on the natural light restrict its applications [5]. Acoustics are widely used for the profiling of the seawater, but they can only work under water due to a high loss in the air-water interface. Lidar is an effective method to deal with the profiling of the seawater with fewer limitations of platforms and natural light. High temporal-spatial resolution in the measurement, applicable coverage from the water surface to several tens meters depth, and possibility of the marine observation at ambient conditions make up the attractiveness of lidar instruments [6]. So far, oceanographic lidars have been employed in detecting fisheries, phytoplankton layers and internal waves, etc., in the upper ocean layer [5].

The lidar return contains two properties of the seawater—the volume scattering function at 180° scattering angle $β^{π}$ and lidar attenuation coefficient $α$ . Standard backscatter lidars are limited to the cases where properties of the water column vary slowly with the depth [5]. In complicated water columns, a large retrieval error is inevitable due to the assumption of lidar ratio, which is the ratio of $α$ and $β^{π}$ [5]. Raman lidars use molecular Raman scattering as the reference signal to retrieve $β^{π}$ without the assumption of lidar ratio [7]. However, the low signal-to-noise ratio (SNR) has always been a problem as a result of the low strength of Raman scattering per unit wavelength. The total signal strength of molecular Mandelshtam-Brillouin (MB) scattering is nearly the same as Raman scattering, but focuses in an ultra-narrow spectral band. Its spectral distribution helps to decrease the noises of external sources, such as sunlight, fluorescence, etc [8]. Hence, it is of great potential to obtain seawater optical properties using a lidar employing MB scattering.

High-spectral-resolution lidar (HSRL) can separate the MB component from the total signals through an ultra-narrow spectral discriminator [9]. So far, the iodine absorption cell, which is the most common discriminator that has been used in the atmosphere and ocean, can only work at 532 nm [10]. However, optical properties of different marine constituents vary with the light wavelength, such as the absorption of chromophoric dissolved organic matter (CDOM) [11]. Therefore, a fixed-wavelength discriminator can hardly satisfy the demands of the HSRL for deeper information and higher accuracy, while wavelength-tunable discriminators, e.g. interferometers, provide stronger possibility. Nevertheless, the oceanographic HSRL generally demands a large field of view (FOV) [12], which is a challenge to most interferometers, for example, the Fabry-Perot interferometer [13]. Our early works propose a field-widened Michelson interferometer (FWMI) discriminator for the atmospheric HSRL [14, 15]. It has a large FOV due to the angle-insensitive optical path difference and good tolerances with the cumulative wavefront error and frequency locking error. In this paper, an FWMI-based HSRL is proposed to retrieve $β^{π}$ without the assumption of the lidar ratio in the ocean. The theoretical framework of the retrieval and error estimation is established considering a dual-channel HSRL configuration. Simulation on the retrieval and error estimation presents a high performance of the proposed HSRL.

The rest of the paper is constructed as follows. In Section 2, the theoretical framework of the retrieval and error estimation is presented. The spectral discrimination of the FWMI is discussed in Section 3. In Section 4, the retrieval and error estimation are implemented using Monte Carlo (MC) analysis. Concluding remarks of the oceanographic HSRL are summarized in the last Section.

2. Principle

2.1 HSRL signals

The proposed oceanographic HSRL transmits a laser pulse into the seawater and detect the lidar returns for the analysis of seawater optical properties. The lidar returns contain particulate scattering, Rayleigh scattering (water), MB scattering (water), Raman scattering (water), and fluorescence (CDOM), etc. Raman scattering and fluorescence can be easily filtered out by interference filters, due to their shifted wavelengths [7, 16]. HSRL technique relies on the spectral differences of the other components, as shown in Fig. 1. Particulate scattering (blue dotted line) and Rayleigh scattering (blue solid line) are both centered at the laser spectrum. MB scattering (blue broken line) is slightly shifted by ~7-8 GHz in the backward direction. The ratio of the total strengths between Rayleigh scattering and MB scattering, known as “Landau-Placzek ratio”, is generally less than 2% [17]. The FWMI discriminator of the HSRL (black solid line) can be employed to suppress particulate scattering and Rayleigh scattering, but to transmit MB scattering.

Fig. 1 Schematic illustrations of the spectral discrimination: the FWMI discriminator (black solid line) suppresses particulate scattering (blue dotted line) and Rayleigh scattering (blue solid line) but transmits MB scattering (blue broken line); Shaded areas represent signals passing the FWMI.

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Structure diagram of the proposed HSRL is shown in Fig. 2. The lidar returns from the ocean are collected by the telescope. After collimation, filtering and beam splitting, they are detected by the combined channel and molecular channel. Specifically, the combined channel passes all frequencies of the lidar returns, and the molecular channel, which is equipped with an ultra-narrowband FWMI discriminator, rejects particulate and Rayleigh components yet pass the MB component. Signals of the combined channel and the molecular channel are written as, respectively,

{\begin{cases} B_{c} = S_{p} + S_{M B} + S_{R} \\ B_{m} = T_{p} S_{p} + T_{M B} S_{M B} + T_{R} S_{R} \end{cases},

where

S_{i}

is the lidar return,

T_{i}

is the transmittance through the FWMI, the subscript

i = M B, R, p

refers to MB, Rayleigh or particulate components, which will be also used in the following text.

Fig. 2 Structure diagram of the oceanographic HSRL: the combined channel and the molecular channel are main channels to obtain signals, and other channels are reserved for Raman scattering, and fluorescence, etc. detections in the future.

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2.2 Multiple scattering

The lidar returns of MB, Rayleigh and particulate components in Eq. (1) under the single-scattering approximation can be described as [5,7]

S_{i} (z) \propto β_{i}^{π} (z) \exp [- 2 τ_{i} (z)],

where

z

is the water depth,

β_{i}^{π} (z)

is the volume scattering function at the 180° scattering angle. Because of the small frequency shift, it is assumed that the optical depths of MB, Rayleigh and particulate components are equivalent, namely

τ_{M B} (z) = τ_{R} (z) = τ_{p} (z)

. Then the retrieval of

β^{π}

will be simplified as the form in [18]. However, multiple scattering, which is not included in the single-scattering approximation, becomes an essential part in the lidar returns of the seawater. Therefore, the method in [7,19] considering multiple scattering is employed for modeling the lidar returns, which takes into account the small-angle quasi-single scattering approximation. The lidar returns can be described as

S_{i} (z) = C_{i} β_{i}^{π} (z) \int d n P_{i} (z, π - | n |) W_{i} (z, r = 0, n) / P_{i} (z, π),

where

C_{i}

is the system constant,

P_{i} (z, θ)

is the phase function,

W_{i} (z, r, n)

is the spatial-angular distribution of the radiance produced by the transmitter and receiver,

(r, n)

gives the position and direction in the scattering plane. Exact calculations of

W_{i} (z, r, n)

rely on optical properties of the water column, e.g. the phase function, beam attenuation coefficient and scattering coefficient, which will be used in Section 4 [19]. This model ignores the effect of forward pulse stretching, since it only makes sense in a large optical thickness [7].

Tracing the histories of the lidar returns from the depth $z$ is necessary for understanding Eq. (3). The MB component at the depth $z$ is used as an example:

A. The frequency shift does not happen in the forward multiple scattering, when the laser pulse arrives at the scatterer from the transmitter [20].
B. The single backscattering process produces a MB frequency shift.
C. Photons with an MB frequency shift do not change their frequency in the forward direction, when they come back to the receiver from the depth $z$ .

The scheme is similar to that of Rayleigh and particulate components in processes A and C, which are described by $W_{i} (z, r, n)$ . Therefore, it is credible to give the equation

W_{M B} (z, r = 0, n) = W_{R} (z, r = 0, n) = W_{p} (z, r = 0, n) .

Furthermore, dependences of phase functions of MB, Rayleigh and particulate components on the scattering angle are weak in the nearly backward direction [21]. Therefore, it is possible to employ the approximation when

θ

is close to

π

, that is,

\frac{P_{M B} (z, θ)}{P_{M B} (z, π)} \approx \frac{P_{R} (z, θ)}{P_{R} (z, π)} \approx \frac{P_{p} (z, θ)}{P_{p} (z, π)} \approx 1.

Consequently, an approximation combining Eqs. (3)-(5) is given by

\frac{S_{M B} (z)}{β_{M B}^{π} (z)} \approx \frac{S_{R} (z)}{β_{R}^{π} (z)} \approx \frac{S_{p} (z)}{β_{p}^{π} (z)} .

2.3 Retrieval

Taking into account Eqs. (1) and (6), the volume scattering function at the scattering angle $180^{\circ}$ of the seawater $β^{π}$ , which is an inherent optical property, is expressed as

β^{π} = β_{M B}^{π} + β_{R}^{π} + β_{p}^{π} = \frac{β_{R}^{π} (T_{p} - T_{R}) + β_{M B}^{π} (T_{p} - T_{M B})}{T_{p} - 1 / K},

where

K = B_{c} / B_{m}

. Since

β_{M B}^{π} (T_{p} - T_{M B})

is more than 10³ times of

β_{R}^{π} (T_{p} - T_{R})

, Eq. (7) can be simplified as

β^{π} \approx β_{M B}^{π} \frac{T_{p} - T_{M B}}{T_{p} - 1 / K} .

Actually, the additional information about the lidar attenuation coefficient can be derived simultaneously. However, it does not belong to the inherent optical properties and is perhaps the research focus of the multiple field-of-view lidar [22].

The classical error-propagation laws are used to estimate the relative error of $β^{π}$ . Errors from calibrations of system constants and geometrical form factor etc. will not be included here as they are common in all lidars [9]. The measurement error of $K$ and calibration errors of $T_{M B}$ and $T_{p}$ are considered, and hence the relative error of $β^{π}$ is expressed as [18],

ε_{β^{π}}^{2} = ε_{K}^{2} + ε_{T_{p}}^{2} + ε_{T_{M B}}^{2},

where

{\begin{cases} ε_{K}^{2} = {(\frac{\partial β^{π}}{β^{π} \partial K})}^{2} (^{σ_{K}) 2} = {(1 + \frac{R}{SDR - 1})}^{2} (\frac{1}{{SNR}_{c}^{2}} + \frac{1}{{SNR}_{m}^{2}}) \\ ε_{T_{p}}^{2} = {(\frac{\partial β^{π}}{β^{π} \partial T_{p}})}^{2} (^{σ_{T_{p}}) 2} = {(\frac{R - 1}{T_{M B} - T_{p}})}^{2} σ_{T_{p}}^{2} \\ ε_{T_{M B}}^{2} = {(\frac{\partial β^{π}}{β^{π} \partial T_{M B}})}^{2} (^{σ_{T_{M B}}) 2} = {(\frac{1}{T_{M B} - T_{p}})}^{2} σ_{T_{M B}}^{2} \end{cases},

where

σ

is the statistic standard deviation for each parameter,

R = β^{π} / β_{M B}^{π}

is introduced to quantify the particulate load in the ocean, the ratio of the MB and particulate transmittances is defined as spectral discrimination ratio (SDR),

{SNR}_{c}

and

{SNR}_{m}

are signal-to-noise ratios of the combined channel and the molecular channel, respectively. In practical conditions,

ε_{K}

is caused by random noises in the measurement, mainly including the shot noise, dark current, background fluctuation and excess noise of photodetectors, etc.

ε_{T_{p}}

and

ε_{T_{M B}}

are due to the calibration of the spectral discrimination.

The retrieval accuracy is influenced by the spectral discrimination parameters, e.g. SDR and the MB transmittance, according to Eq. (10) [18]. A large SDR can decrease the effect of $R$ on the term $ε_{K}$ , especially in a turbid water column. Generally, we suggest an SDR that is five times of $R$ to achieve the minimum of $ε_{K}$ . $R$ in the turbid harbor is about 30 from the typical experiment data of Petzold [23] and then an SDR larger than 150 is enough. Furthermore, a large value of ${SNR}_{m}$ is beneficial to the suppression of $ε_{K}$ as well. The SNR is not only influenced by the laser energy, water attenuation, optical efficiency of the instrument and so on, but also dependent on $T_{M B}$ . A large $T_{M B}$ helps to increase the signal intensity and then promote ${SNR}_{m}$ , for the fact that lidar operates in the shot-noise-limit mode and thus SNR is equal to the square root of the signal. Notably, although these discussions and conclusions are about the term $ε_{K}$ , large $T_{M B}$ and SDR are desirable as well to reduce values of $ε_{T_{_{p}}}$ and $ε_{T_{_{M B}}}$ .

2.4 Spectral discrimination parameters

Large spectral discrimination parameters are needed to reduce the retrieval errors, from the results in Section 2.3. The MB and particulate transmittances can be expressed as

T_{i} = \frac{\int_{- \infty}^{+ \infty} L (Δ W) d (Δ W) \int_{0}^{θ_{d} / 2} M (θ) θ d θ \int_{- \infty}^{+ \infty} I_{i} (υ) F (υ, Δ ϕ) d υ}{\int_{- \infty}^{+ \infty} L (Δ W) d (Δ W) \int_{0}^{θ_{d} / 2} M (θ) θ dθ \int_{- \infty}^{+ \infty} I_{i} (υ) d υ},

where

θ_{d}

is the full divergence angle of incident light, spectral distributions of MB and particulate components in Fig. 1 can be expressed as, respectively,

{\begin{cases} I_{M B} (υ) = {I^{'}}_{M B} (υ) * I_{L} (υ) \\ I_{p} (υ) \approx I_{L} (υ) \end{cases},

where

υ

is the light frequency relative to laser, the symbol

(*)

represents the convolution,

{I^{'}}_{M B} (υ)

is the ideal fine distribution of MB scattering without regard to the laser linewidth [20] and

I_{L} (υ)

is the laser spectral distribution. The distributions can be described as respectively

{\begin{cases} {I^{'}}_{M B} (υ) = \frac{1}{π Δ υ_{M B}} \cdot \frac{1}{1 + {[2 (υ \pm υ_{M B}) / Δ υ_{M B}]}^{2}} \\ I_{L} (υ) = \frac{1}{Δ υ_{L} \sqrt{π}} \exp (- \frac{υ^{2}}{Δ υ_{L}^{2}}) \end{cases},

where

Δ υ_{L}

is the

1 / e

half-width of the laser spectrum,

υ_{M B}

and

Δ υ_{M B}

are the frequency shift [8] and full width at half maximum (FWHM) of MB scattering [24], respectively. The spectral function of the FWMI is expressed as a cosine function, that is [14, 15],

F (υ, Δ ϕ) = I_{1} + I_{2} - 2 \sqrt{I_{1} I_{2}} \cos (2 π υ / FSR + Δ ϕ),

where

I_{1}

and

I_{2}

are the light intensities of two beams,

Δ ϕ

is the phase shift caused by the incident angle

θ

, the cumulative wavefront error

Δ W

and the frequency locking error

Δ γ

[13],

L (Δ W)

and

M (θ)

is the probability distribution of

Δ W

and

θ

.

Several approximations can be employed to decrease the complexity of the Eq. (11). Signals are supposed to be homogeneous along the whole light angles, which gives $M (θ) = 1$ . Besides, the absorption and reflection loss of the FWMI are not considered, so that $I_{1}$ and $I_{2}$ are equal to 0.25. Then, the Pseudo-Voigt function is employed to simplify the convolution in Eq. (12) and the parameter $η$ is used to describe the linear combination of the Gaussian and Lorentzian curves [25]. Therefore, the MB and particulate transmittances can be simplified as respectively

T_{M B} = \frac{1}{2} - \frac{Ψ}{2} \cos (\frac{2 π υ_{M B}}{FSR}) [η \exp (- \frac{π Δ υ_{M B}}{FSR}) + (1 - η) \exp (- \frac{π^{2} Δ υ_{L}^{2}}{{FSR}^{2}})],

and

T_{p} = \frac{1}{2} - \frac{Ψ}{2} \exp (- \frac{π^{2} Δ υ_{L}^{2}}{{FSR}^{2}}),

where

Ψ

is the overall effect of the phase shift

Δ ϕ

, that is,

Ψ = 8 θ_{d}^{- 2} \int_{- \infty}^{+ \infty} L_{i} (Δ W) d (Δ W) \int_{0}^{θ_{d} / 2} θ \cos (Δ ϕ) dθ

3. Spectral discrimination performance

The general concept of the oceanographic HSRL has been established and the importance of the spectral discrimination performance has been illustrated in Section 2. Large values of the MB transmittance and SDR are required for the reduction of the retrieval errors. In the following studies, the spectral discrimination performance of the FWMI at the wavelength of 532 nm will be used as an example to verify the practical feasibility of the proposed HSRL.

Spectral functions of the lidar returns and the FWMI are of great importance to the spectral discrimination performance, as can be seen in Eqs. (15) and (16). The lidar returns are concerned with $υ_{M B}$ , $Δ υ_{M B}$ and $Δ υ_{L}$ . Injection-seeded technique can promise an ultra-narrow bandwidth of the laser, so that $Δ υ_{L}$ hardly influences the spectral discrimination performance. $υ_{M B}$ and $Δ υ_{M B}$ are functions of the seawater temperature, which should be discussed in detail. The spectral function of the FWMI is sensitive to the FSR mismatch, frequency locking error, full divergence angle of the incident light, and cumulative wavefront error, etc. In the following part, we are going to give an initial value for each factor according to Table 1 and then change only one factor each time to evaluate spectral discrimination sensitivity. Initial values are all produced from standard or well-performed conditions, so that the following analysis is thorough under the practical conditions.

Table 1. Factors in the spectral discrimination performance

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3.1 Sensitivity to the spectral functions of the lidar returns

MB scattering properties, e.g. $υ_{M B}$ and $Δ υ_{M B}$ , are important parameters in the spectral functions of the lidar returns. They have been studied in [8] and [24], which are closely related to the seawater temperature. Discussions of their dependences on the salinity and pressure are neglected for reasons that: 1) the salinity is fairly stable and the historical data can provide useful information for the correct of the salinity; 2) the pressure is a well-known function of depth and can also be included later [27]. Therefore, studies on the sensitivity to $υ_{M B}$ and $Δ υ_{M B}$ will only consider the seawater temperature. Through analyzing the temperature from Argo data within the 50 m depth covering marine areas from 120E to 70W in longitude and from 60S to 60N in latitude between Jan. 2004 and Dec. 2011 [28], the probability density of the temperature is shown in Fig. 3. Generally, the temperature in 99 percent of the lidar detecting range is within the range 0~30 $^{o} C$ .

Fig. 3 Probability density function of the temperature from the Argo data within the 50 m depth covering the area from 120E to 70W in longitude and from 60S to 60N in latitude between Jan. 2004 and Dec. 2011 [28].

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The temperature effects on $T_{M B}$ and SDR are shown in Fig. 4. Notably, $T_{M B}$ larger than 80% and SDR larger than 200 hold all the time. For cases that the temperature is larger than 10 ${}^{\circ}C$ , whose probability of occurrence is about 80 percent, $T_{M B}$ is not less than 90%. However, the large variation of the MB transmittance may cause a large $ε_{T_{M B}}$ without the real-time calibration of the seawater temperature. Fortunately, the surface temperature can be measured remotely employing microwave radiometer and thermal-Infrared Radiometer [29]. Previous studies show that, temperature variations from the surface to 50 m depth are generally small in winter, such as 7~12 $^{\circ} C$ , but large in summer, such as 12~28 $^{\circ} C$ , due to the thermocline [30]. If the FSR is well-selected, e.g. 14.6 GHz, MB transmittance variations with the temperature are large in the low temperature, yet small in the high temperature, as can be seen in Fig. 4. Therefore, when the surface temperature is well-obtained, the uncertainty of the MB transmittance is less than 3% within the lidar detecting range whatever the season is. Certainly, more accurate calibration of $T_{M B}$ can be obtained, if Brillouin temperature lidar or in situ methods, e.g. CTD, are used for the temperature profiling synchronously [31].

Fig. 4 Temperature effects on T_MB and SDR

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3.2 Sensitivity to the spectral function of the FWMI

Figure 5 shows the sensitivity of the spectral discrimination performance to the spectral function of the FWMI. The analysis is split into four parts including the FSR mismatch, frequency locking error, full divergence angle of the incident light, and cumulative wavefront error. Figure 5(a) describes the relationship between the spectral discrimination performance and the FSR error. The FSR error is relative to the baseline value of the FSR 14.6 GHz, which contributes to the spectral mismatch between the peaks of the MB return and spectral function of the FWMI. The results show that, the spectral discrimination parameters are nearly constant within the FSR error $\pm$ 100 MHz. Fortunately, limiting the FSR error to the value $\pm$ 100 MHz is not a big problem for the practical fabrication and adjustment of the FWMI.

Fig. 5 T_MB and SDR are influenced by (a) FSR error; (b) frequency locking error; (c) full divergence angle of incident light; (d) cumulative wavefront error RMS including defocus, tilt and random types.

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Another aspect of the spectral mismatch is concerned with the frequency locking error, which refers to the deviation between the central frequency of the laser and the valley of the spectral function of the FWMI. A high frequency locking accuracy is beneficial to the suppression of the particulate signals and promotion of SDR. From Fig. 5(b), the MB transmittance decreases smoothly with the increasing of the frequency locking error, while SDR decreases rapidly. Therefore, we only need to concentrate on the SDR variation with the frequency locking error, without the consideration about the MB transmittance. The SDR is over 200 when the frequency locking error is less than $\pm$ 120 MHz, as can be seen in Fig. 5(b). Our previous works have employed a heterodyning-based frequency locking method for the FWMI application [26]. The method limits the frequency locking error below 15 MHz, which satisfies the locking accuracy here sufficiently.

The FOV of the telescope is generally required to be large for oceanographic lidars except the lidar in an extremely high operating altitude, because more lidar returns from multiple scattering can be obtained. Therefore, the divergent angle of the incident light obtained by the FWMI is also large through the optical system of the telescope and collimation lens. However, it is a difficult problem for interferometric discriminators that the sensitivity of the spectral discrimination performance to the incident angle stays constant. Our early work proposed a special compensation technique to solve the angle problem, which is the highlight of the FWMI discriminator. Figure 5(c) shows the angle sensitivity of the FWMI designed for the oceanographic HSRL that, the divergent angle less than 100 mrad hardly changes the values of $T_{M B}$ and SDR. The results can meet most demands for FOVs of oceanographic lidars through configuring a proper focal length ratio between the telescope and collimation lens [12]. The calculation in Fig. 5(c) is under the assumption that the incident light is homogeneous along the whole divergent angle. In practical conditions, the intensities of the lidar returns decrease with the increasing of the incident angle, which gives a low weight for the large incident angle and therefore makes the above assumption more credible.

The cumulative wavefront error, which is the distortion of the wavefront through the FWMI, is one of the most important technical factors for the spectral function of the FWMI. As in [13], three types of cumulative wavefront error are introduced: 1) defocus wavefront error, which is caused by the mismatch of the field-widen position of the FWMI; 2) tilted wavefront error, and 3) random wavefront error. The root mean square (RMS) is used to describe the uncertainty of cumulative wavefront error. Figure 5(d) shows that, the spectral discrimination performance is independent of types of the cumulative wavefront error, but only concerned with the RMS value. Therefore, we can only focus on the RMS value of the cumulative wavefront error, without the discussion about the specific cumulative distributions in practical machining conditions. From Fig. 5(d), the MB transmittance decreases smoothly with the RMS value, while SDR decreases rapidly due to the increasing of the Particulate transmittance. Obviously, better practical machining conditions are beneficial to the spectral discrimination performance, which makes an SDR greater than 1500 if the RMS value is limited to $0.01 λ$ . Even though the RMS value rises to 0.02 $λ$ , SDR is about 255 and also exceeds the demand for the turbid water, as shown in Section 2.3. Moreover, the RMS value 0.02 $λ$ can be obtained easily in practical machining conditions.

The above results show that, an SDR larger than 200 and a $T_{M B}$ larger than 90% in most cases are not a difficult work. We have concluded in Section 2.3 that, an SDR larger than 150 is enough for the turbid water in Perzold’s experiments. In addition, the extremely-large $T_{M B}$ can reserve most of the information signal of the MB component and is beneficial to the SNR of the molecular channel. Notably, the spectral discrimination can be promoted further by the suppression of the cumulative wavefront error RMS, if the detection of the extremely-turbid water is needed. In conclusion, the FWMI has a good spectral discrimination performance and can be equipped in the oceanographic HSRL.

4. Simulation

This section uses simulations to obtain HSRL signals and verify the technical practicability of the retrieval and error estimation methods proposed in Section 2. Water properties under the standard condition in [8] and [17] are cited, e.g. $β_{M B}^{π}$ is 2.4 × 10⁻⁴ m⁻¹sr⁻¹ and “Landau-Placzek ratio” is 2%. The Dolin’s model for approximating small-angle component of the phase function is employed and its parameter m is 6 for the open ocean [32]. Then, seawater Jerlov Type IB is used, and a phytoplankton layer is simulated between 5 and 9 m [21], the CDOM is considered between 10 and 14 m [21], the detritus is simulated between 16 and 20 m [23]. The inherent optical properties of the water column, such as volume scattering function at the angle $180^{\circ}$ $β^{π}$ , beam attenuation coefficient $c$ and scattering coefficient $b$ , are shown in Fig. 6(a). The values are essential for signal simulations under Eqs. (1) and (3).

Fig. 6 Simulation of the oceanographic HSRL. (a) Water inherent optical properties: volume scattering function at the scattering angle 180° β^π (m⁻¹sr⁻¹), beam attenuation coefficient c (m⁻¹) and scattering coefficient b (m⁻¹); (b) a set of typical HSRL signals based on the ship from the combined and molecular channels.

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A set of typical shipborne HSRL signals of combined and molecular channels is calculated by the MC method and normalized by the maximum value of the combined channel under Eqs. (1) and (3), as shown in Fig. 6(b). The specifications of the oceanographic HSRL in the simulation are shown in Table 2. Noises are introduced to simulate real signals that have been contaminated. Relative uncertainties of $T_{p}$ and $T_{M B}$ are assumed as 5% and 2%, respectively. Shot noises from the background and information signals are considered, which limit the SNRs of the two channels. The background radiance during the daytime is assumed as 0.14 Wm⁻²sr⁻¹nm⁻¹, and then the background signal is calculated from the background radiance, bandwidth of the interference filter, receiving area and FOV of the telescope, etc. The signals are integrated over 10 times to decrease the random noises. Signals within 2 m depth are eliminated using the gating technique to avoid the disturbance of the water surface [12]. $T_{M B}$ and SDR are set as 90% and 200, which are achievable according to the discussion in Section 3. Therefore from Fig. 6(b), most of the particulate signal in the molecular channel is suppressed, as a result of the good spectral discrimination performance.

Table 2. Shipborne oceanographic HSRL specifications in the simulation

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200 retrieval profiles of $β^{π}$ from the MC method are plotted in red dots in Fig. 7(a). True values (green solid line) are also shown in Fig. 7(a) along with the 3-σ retrieval error limits (blue broken line) calculated by the theoretical model of Eq. (9). It can be seen that, most of the retrieval values are limited within the predicted range, which shows excellent agreement between the theoretical model and MC method. The condition in the above simulation is considered as Condition A. In order to compare the effects of calibration and measurement errors on the overall retrieval error of $β^{π}$ , Condition B and C are introduced in Fig. 7(b). Relative uncertainties of $T_{p}$ and $T_{M B}$ are changed to 10% and 4%, respectively, for Condition B and the single pulse energy in Condition C is 40% of that in Condition A, while other parameters do not change. Figure 7(b) shows the statistical error RMS from the MC simulation along with the theoretical error calculated straightforwardly by Eq. (9). Obviously, the theoretical model is in good agreement with the MC method, which verifies the theoretical model again. Furthermore, the retrieval error within the 20 m depth is dependent on the calibration error of $T_{M B}$ and $T_{M}$ from the comparison between Condition A and B. The retrieval error becomes large inevitably with the increasing of the depth, since shot noises cannot be neglected when the SNRs of the lidar channels are low in the deep water. Therefore, the inflection point of the retrieval error in Condition C is shallower than that in Condition A, due to the lower single pulse energy. In conclusion, the retrieval values are reliable when the SNRs are not too poor and the transmittances can be well-calibrated, e.g. Condition A.

Fig. 7 Shipborne oceanographic HSRL retrieval: (a) β^π under Condition A; (b) relative error of β^π using the theoretical model and MC method under Condition A, B and C

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Another case of the retrieval results based on an airborne HSRL is presented in Fig. 8. Since an altitude of 1000 m is considered, some specifications in Condition A are changed to obtain stronger signals or decrease the influence of the background light, e.g. a single pulse energy of 50 mJ and a telescope FOV of 10 mrad. Then, Condition B changes relative errors and Condition C changes the single pulse energy like in Fig. 7. In the airborne case, the return from near the surface will be lower, but the attenuation will also be lower as a result of multiple scattering. Therefore, although the SNRs play important roles below the depth 5 m, the relative error of $β^{π}$ increases slowly due to the low attenuation, from Fig. 8(b). Notably, the sudden increase of the error between 16~20 m is caused by the attenuation of the detritus layer rather than the low SDR. We employ Condition D, whose SDR is 900 compared with an SDR 200 of Condition C, to illustrate the fact. The line of Condition D is nearly the same as that of Condition C. The fact reemphasize that, the SDR of 200 is enough for the particulate load below 50, even though the SNRs are low in the airborne HSRL. Notably, the sunlight influence always exists in the simulations and a high performance of the HSRL can be expected at the weak background light.

Fig. 8 Airborne oceanographic HSRL retrieval: (a) β^π under Condition A; (b) relative error of β^π using the theoretical model and MC method under Condition A, B, C and D

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5. Conclusion

Standard backscatter lidar is an effective method to deal with the profiling of the seawater. However, a standard backscatter lidar actually measures the product of the volume scattering function at 180° scattering angle $β^{π}$ and the two-way transmission between the lidar and the backscatter volume. The assumption of the lidar ratio is employed during the retrieval of $β^{π}$ , and retrieval error will be yielded then. The FWMI-based HSRL, whose working wavelength is tunable, is proposed to estimate $β^{π}$ independently without the assumption. The HSRL technique employs the spectral discrimination of the different spectral components in the lidar returns, e.g. particulate scattering and water molecular MB scattering .

The theoretical framework to retrieve $β^{π}$ and analyze its errors is presented, which stays valid even under multiple scattering conditions. The analysis presents that, the spectral discrimination performance of the FWMI is important for the reduction of the retrieval error. Then, the examination on the spectral discrimination performance shows good results that, the MB transmittance is larger than 90% and SDR is larger than 200 in most cases. From the simulation on the retrieval and error estimation, the proposed oceanographic HSRL based on the ship or aircraft can perform well to extract the profile of $β^{π}$ and has a real potential in the oceanographic remote sensing.

Funding

National Key Research and Development Program of China (2016YFC0200700, 2016YFC1400902), National Natural Science Foundation of China (41305014, 61475141), Public Welfare Project of Zhejiang Province (2016C33004), the Fundamental Research Funds for the Central Universities(2017QNA5001), the State Key Lab. of Modern Optical Instrumentation Innovation Program (MOI2017QN01) and Open funds of State Key Lab. of Remote Sensing (OFSLRSS201614).

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Factors	Baseline values	Variations
Seawater temperature (°C)	20 ^a	0~30
Free spectral range (GHz)	14.6 ^b	14.5~14.7
Locking error (MHz)	20 ^c	0~200
Full divergence angle of incident light (mrad)	100 ^d	0~200
Cumulative wavefront error RMS (λ)	0.02 (for random type) ^e	0.01~0.05

Specification	Value	Specification	Value
Laser wavelength (nm)	532	Spectral bandwidth (nm)	1
Single pulse energy (mJ)	5	Total optical efficiency of combined and molecular channel	15%/35%
FWHM of laser (GHz)	0.1	MB transmittance	90%
Telescope diameter (mm)	80	Spectral discrimination ratio	200
Telescope FOV (mrad)	20	Quantum efficiency of PMTs	10%
Telescope magnification	4	Sampling interval (m)	0.1
Altitude (m)	10	Electrical bandwidth (MHz)	100

Factors	Baseline values	Variations
Seawater temperature (°C)	20 ^a	0~30
Free spectral range (GHz)	14.6 ^b	14.5~14.7
Locking error (MHz)	20 ^c	0~200
Full divergence angle of incident light (mrad)	100 ^d	0~200
Cumulative wavefront error RMS (λ)	0.02 (for random type) ^e	0.01~0.05

Specification	Value	Specification	Value
Laser wavelength (nm)	532	Spectral bandwidth (nm)	1
Single pulse energy (mJ)	5	Total optical efficiency of combined and molecular channel	15%/35%
FWHM of laser (GHz)	0.1	MB transmittance	90%
Telescope diameter (mm)	80	Spectral discrimination ratio	200
Telescope FOV (mrad)	20	Quantum efficiency of PMTs	10%
Telescope magnification	4	Sampling interval (m)	0.1
Altitude (m)	10	Electrical bandwidth (MHz)	100

Retrieving the seawater volume scattering function at the 180° scattering angle with a high-spectral-resolution lidar

Abstract

1. Introduction

2. Principle

2.1 HSRL signals

2.2 Multiple scattering

2.3 Retrieval

2.4 Spectral discrimination parameters

3. Spectral discrimination performance

3.1 Sensitivity to the spectral functions of the lidar returns

3.2 Sensitivity to the spectral function of the FWMI

4. Simulation

5. Conclusion

Funding

References and links

Cited By

Figures (8)

Tables (2)

Equations (17)

Optics Express