Multiple scattering is an inevitable effect in spaceborne oceanic lidar because of the large footprint size and the high optical density of seawater. The effective attenuation coefficient klidar in the oceanic lidar equation, which indicates the influence of the multiple scattering effect on the formation of lidar returns, is an important parameter in the retrieval of inherent optical properties (IOPs) of seawater. In this paper, the relationships between klidar of the spaceborne lidar signal and the IOPs of seawater are investigated by solving the radiative transfer equation with an improved semianalytic Monte Carlo model. Apart from the geometric loss factors, klidar is found to decrease exponentially with the increase of depth in homogeneous waters. klidar is given as an exponential function of depth and IOPs of seawater. The mean percentage errors between klidar calculated by the exponential function and the simulated ones in three typical stratified waters are within 0.5%, proving the effectiveness and applicability of this klidar-IOPs function. The results in this paper can help researchers have a better understanding of the multiple scattering effect of spaceborne lidar and improve the retrieval accuracy of the IOPs and the chlorophyll concentration of case 1 water from spaceborne lidar measurements.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The ocean is a complex biological system that contains a vast number of biological species. Phytoplankton play a very important role in the material recycle and radiation budget of the marine ecosystem . Satellite-based ocean color observations have been widely used to understand the distribution and variable characteristics of global phytoplankton . The current ocean color measurements were made by passive sensors, which have several fundamental limitations: retrievals are highly sensitive to the atmospheric correction errors; information about the vertical distribution of ocean layers is missing; observations during nighttime, at high solar zenith angles and through thick cloud or absorbing aerosols are unavailable . The Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) has demonstrated advantages in day-night continuous sampling, observations through absorbing aerosols and thin cloud layers , capabilities of vertical profiling through the subsurface [5–7], and measurements at high-latitude regions . However, CALIOP has limited spatial coverage compared to most passive sensors, and can only provide an integrated view of optical properties in the upper ocean. Specifically, CALIOP observation is limited by its coarse vertical resolution (i.e., 22.4m in the water) [9,10]. To improve the retrieval accuracy of spaceborne lidars, more information is needed to better understand factors involved in lidar functions.
Lidars obtain depth-resolved return profiles by collecting backscattered signals from molecules and particles. It is essential to establish an accurate model of the radiative transfer process. The return signal of spaceborne oceanic lidar is strongly affected by multiple scattering due to the large footprint size and the high optical density of seawater .To characterize the multiple-scattering radiative transfer process, an effective attenuation coefficient klidar is introduced into the oceanic lidar equation . The relationships between klidar and the optical properties of water and the lidar system parameters were extensively studied. Gordon  found that the ratio of the radius of the spot on the sea surface viewed by the lidar receiver optics to the mean free path of photons in water is a significant parameter which is related to klidar. When this ratio is near zero, klidar is given by the beam attenuation coefficient. If the ratio is greater than 5, klidar is given by the diffuse attenuation coefficient Kd. Phillips et al.  proposed a slightly different assumption that the upper and lower limits of klidar is given by the beam attenuation coefficient and the absorption coefficient, respectively, depending on field of view (FOV). Lee et al.  and Schulien et al.  observed strong correlations between the in situ Kd and klidar derived from airborne lidar experiments. Collister et al.  exhibited the similarity between klidar and the absorption coefficient under wide FOV using shipborne lidar measurements. The aforementioned studies are based on airborne or shipborne platform, and the depth characteristics of klidar are ignored. The values of klidar at different depths in homogenous water are not constant, due to the multiple scattering effect [17,18], especially near the ocean surface. This phenomenon leads to errors in the retrieval of optical properties, which further affects the assessment accuracy of the phytoplankton biomass and primary productivity. Thus, the depth characteristics of klidar and the relationship between klidar and the optical properties of seawater need to be studied in detail.
In this paper, the potential of using an improved semianalytic Monte Carlo radiative transfer simulation model (SALMON) to accurately identify the characteristics of both homogeneous and inhomogeneous waters is investigated. The use of SALMON is proposed by Poole et al.  and widely applied in oceanic lidar, therefore, it would be beneficial to develop an improved SALMON that can better simulate the inhomogeneous waters in the real oceans. In Section.2, we present the principle and method of proposed SALMON. In Section 3, we use this improved model to investigate the characteristics of lidar return profiles and the corresponding klidar with analysis of the influence of the lidar system parameters on klidar and the depth characteristic of klidar for homogenous water. In addition, an exact relationship between the depth-resolved klidar and IOPs of seawater is derived based on the statistical properties of case 1 waters. In Section 4, we compare klidar calculated with our function and Kd from bio-optical model with the simulation results in three vertical stratified waters to verify the applicability of this function in real ocean environment.
2. Principle and method
2.1Return signals of oceanic lidar
The schematic diagram of the theoretical model of laser backscatter from water is illustrated in Fig. 1. The laser pulse transmits downward towards the flat horizontal water surface at the height of . The photons are scattered and absorbed by the molecules and particles in seawater, and the backscattered photons are received by the detector forming the lidar return signal, which can be described by the lidar equation [3,20]:
Rewriting Eq. (1), we have the expressed by
Thus, the effective attenuation coefficient can be written as
klidar is an important parameter for retrieving optical properties of seawater. The IOPs considered here include the scattering coefficient(m−1), the absorption coefficient (m−1), and the beam attenuation coefficient(m−1). The relation among those three parameters are expressed as
The IOPs of case 1 water can be parameterized in terms of the chlorophyll concentrationand wavelength . The case 1 water is chlorophyll dominated water which includes almost 98% of the world's open ocean and coastal waters. Therefore, lidar remote sensing of the case 1 water plays an important role in estimating global ocean primary productivity.
According to the bio-optical model of case 1 water , the absorption coefficient could be expressed as21]. The scattering coefficient can be written as 22]
When [Chl] is expressed in the unit of mg/m3 and is in the unit of nm, the resulting and are in the unit of m−1. In this paper, the working wavelength is at 532 nm, which is commonly used in oceanic lidars.
The diffuse attenuation coefficient Kd, one of the apparent optical properties (AOPs) of case 1 water, is characterized by [Chl] . For the wavelength of 530 nm, Kd is given by
In the case of single scattering, klidar is equal to c [12,13], whereas for the multiple scattering, the express of klidar is more complicated. In this paper, an improved SALMON is used to trace the multiple scattering behavior of photons in water to get the lidar return profiles and the slope method was introduced to calculate the depth-dependent klidar.
2.2 An improved semianalytic Monte Carlo radiative transfer simulation model
Monte Carlo method is an a numerical method of solving the radiative transfer which is used in optical communication , biological optics [25,26] and optical remote sensing . In 1981, Poole et al.  proposed a semianalytic Monte Carlo radiative transfer simulation model (SALMON) for atmosphere lidar, which is widely used in the current oceanic lidars. The improved SALMON proposed here includes a Gaussian beam emission model and it is applicable for both homogeneous and inhomogeneous waters.
A: Gaussian beam emission model
The beam divergence angle of lidar is always neglected in airborne lidar simulations because it is very small compared with the FOV of airborne lidar receiving system [12,13]. However, it is not negligible in the space lidar simulation, because the platform of spaceborne lidars usually has an altitude of several hundred kilometers; as a result, the spot diameter of the laser beam incident on the sea surface could be several tens of meters. A Gaussian distribution sampling model is introduced in this paper to determine the coordinates and directions of photons incident on the water surface .
The intensity perpendicular to the transmission direction of the Gaussian beam is expressed asEq. (9) in our Monte Carlo simulations, a sampling model given in Eqs. (10) and (11) is used to describe the distance from the photon to the center of light spot r and the azimuth angle of photons, respectively.
Therefore, the coordinates of photons incident on the water surface can be expressed by
Because the altitude of the spaceborne lidar, H, is much greater than the Rayleigh length of the Gaussian beam, the wave front is approximately spherical near the sea surface and the normal direction of each point on the sphere is the direction of photons, which is given by
B: Photon behavior in seawater
The SALMON model simulates the random collision and propagation in water of individual photons from the transmitted beam through a series of absorption and scattering until the photons move outside of the field of view of the lidar receiver or reach the boundaries of medium. For each segment of a photon trajectory, the distance travelled before collision and the propagation direction are determined by making random selections from cumulative probability distribution function of these parameters. These two parameters play a critical role in describing the photon behavior in seawater. The details about the definitions of these two parameters and the formation of the lidar return signals are given in the following paragraphs.
The distance travelled before collision is the step size (s) in dimensionless unit and defined as the integration of the attenuation coefficient over the photon pathway . In the SALMON model, is calculated by , where R is a random value between 0 and 1. The larger the attenuation coefficient c, the shorter distance the photon traveled before it is scattered or absorbed. For the stratified inhomogeneous waters, the IOPs are assigned separately for each layer. If the photon passes through more than one layer, c will change along the pathway. Therefore, the actual step size in layer could be calculated as ,where d is the geometrical distance along the ray in layer i where the scattering took place. The sign indicates the photon propagates up (-) or down ( + ). The photon travels until the initial step size has been consumed ().
The propagation direction of the photon after each collision can be expressed by the scattering angle and the azimuth angle. The value of is distributed uniformly between 0 and 2, and the distribution of angle is modeled by the scattering phase function . Both the pure water molecules and particles contribute to the total scattering phase function , i.e.,22], i.e.,29], i.e.,24]. Therefore, a weighted sum of HG phase function, called the two-term Henyey-Greenstein (TTHG), is used here to solve the issue, which is given by Kattawar :31]. Hence, we set ,and , which gives according to the exact relationships between ,,andgiven by Haltrin .
In the process of tracking the photons, each photon is conceived as a large packet of identical photons traveling in the same direction. The photon packet has an initial weight . After each collision, the current weight is multiplied by the single scattering albedo , which is given by . A roulette procedure is employed when the weight w is below the threshold of . The expected value of the fraction of photons collected by the lidar receiver without further interactions at each collision point in layer i is given by
3.1 Influence of the lidar system parameters on klidar
The FOV of the lidar receiver is an important parameter that influences the order of scattering in the lidar return signals. For the airborne lidars, the FOV ranges from tens of milliradians to several hundred milliradians depending on different detection purpose. For a spaceborne lidar, smaller FOV is preferred in order to reduce the impact of the background noise from sun light. The FOV of the lidar receiving system should be larger than the divergence angle of the transmitting laser beam due to the divergence of the Gaussian beam. Besides, the altitude of satellite platform also affects the multiple scattering signal. In this paper, we analyze four satellite orbits of the actual spaceborne atmospheric lidar systems: 705 km for CALIOP on the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) satellite , 550 km for the Japanese Experimental Lidar in Space Equipment (ELISE) , 415 km for the Cloud-Aerosol Transport System (CATS), which is a lidar remote sensing instrument on the International Space Station (ISS) , as well as 400 km for the Earth Clouds, Aerosol and Radiation Explorer (EarthCARE) satellite to be launched in 2021 . In these simulations, the divergence angle of the laser beam is assumed to be 0.1 mrad and the diameter of the telescope is 1.5 m . The depth resolution of the signals is set to be 0.3 m. The ocean is assumed to have a depth of 50 m with a black bottom, considering the effective attenuation optical depth of three, and a small chlorophyll concentration of 0.35 mg/m3 that is uniformly distributed from the surface to the bottom. See Table 1 for the values of input parameters of the calculations.
The klidar for different system parameters, such as FOV of the receiving telescope and orbit altitude, are compared in Fig. 2. The orbit altitude in Fig. 2(a) is 400 km, and the FOV is in the range of 0.1 to 0.5 mrad, then the corresponding footprint of the receiver is in the range of 20 to 100 m and the diameter of laser spot on the ocean surface is 40 m. Figure 2(a) shows that klidar at different optical depths decreases with the increase of FOV, but remains almost constant when the FOV is larger than a critical value, indicating that the lidar receives the full-multiple scattering signals. For the orbit altitude of 400 km, FOV of 0.4 mrad is wide enough to satisfy the full-multiple scattering statement at optical depth (,) from 1 to 5. Figure 2(b) presents the klidar for four orbit altitudes, namely 400 km, 415 km, 550 km and 705 km, with FOV equals to 0.4 mrad. The result indicates that klidar of the full-multiple scattering return signals is independent of H. According to our simulation, this result is still valid even when the divergence angle of the incident beam increases to 0.2 mrad because the beam divergence in water is mainly due to the multiple scattering. Therefore, klidar of spaceborne lidar is only related to the IOPs of water, and system parameters for the full-multiple scattering statement are adopted in the following sections.
3.2 Relationships between IOPs and klidar
A. Characteristics of lidar return signals for homogeneous case 1 waters
The effective attenuation coefficient klidar in lidar equation is the reflection of the IOPs of seawater. Determining the value of klidar is beneficial to improve the retrieve accuracy of IOPs from lidar equation. In terms of the orbit altitude of 400 km, the characteristics of lidar return signals from homogeneous case 1 water with various chlorophyll concentrations are studied. The normalized lidar return signal power and the associated effective attenuation coefficients of eight groups of seawaters with [Chl] between 0 to 2 mg/m3, which covers most open ocean and productive coastal environment, are provided in Fig. 3. The normalized lidar return profiles in Fig. 3(a) shows that the linearity of the lidar return signal decreases with the increase of [Chl]. For seawaters with high [Chl], klidar is not constant even if the IOPs are uniform in the vertical direction [Fig. 3(b)]. klidar decreases with the increase of depth at first, then it becomes more stable. The full range of the nonlinearity in klidar depends on the sensitivity of the receiving system. If this nonlinearity in klidar is ignored in homogeneous waters, it may cause error in the retrieval of lidar signal.
Figure 4 illustrates the statistical estimates of the normalized lidar return signal power with different order of scattering and the percentage for each order of scattering in the total scattering signal for [Chl] = 0.35 mg/m3. The signal of the first order of scattering decreases exponentially with the increase of depth, and the slope of the signal is c. The presence of multiple scattering deforms the waveform of the total lidar return signal, and the rapid increase of high-order scattering signal causes the decrease of klidar. It is shown in Fig. 4(b), that the percentage of high-order scattering in the total signal increases with depth. When the depth is larger than about 10 meters, signal with scattering order more than 1 becomes dominant in the total signal power. Due to the absorption of water, the intensity of the high-order scattering signal declines and klidar shows a characteristic of stability and invariability.
The nonuniformity of klidar caused by multiple scattering is inevitable and it increases the uncertainty of the retrieval. However, in the real ocean environment, the IOPs is not homogenously distributed. Therefore, it is necessary to make a further research of the relationship between the klidar(z) and IOPs of water.
B. The relationships between the klidar(z) and IOPs of water
Based on the results shown in Fig. 3(b), an exponential formula is proposed to describe the depth characteristic of klidar, which is given by
The fitting results of the effective attenuation coefficients for 16 types of case 1 waters with [Chl] ranging from 0.005 to 2 mg/m3 are listed in Table 2. R2 is the criterion for the goodness of fit. All the R2 values for the fitting results of each group of seawater are greater than 0.99, which confirms the validity of Eq. (19). As shown in Eq. (19), klidar will be given by is an asymptotic parameter p, when z is large enough to make the exponential term close to zero. The fitting results show that the asymptotic value for each klidar is a little bit smaller than Kd.
The slope and the asymptotic value of klidar in Eq. (19) are determined by m, n and p, respectively. Moreover, klidar involves the IOPs of seawater, indicating these three parameters are related to the IOPs of seawater.
The influence of the IOPs on the qualitative behavior of klidar is analyzed using a series of calculations with one variable at a time (Fig. 5). Figure 5(a) shows that the curves of klidar for different (with b = 0.2 m−1) are parallel to each other, which means that the absorption coefficient a only affects the parameter p. Figure 5(b) shows that the scattering coefficient b influences both the slope and the asymptotic value of klidar. Figure 5(c) shows that the phase function mainly affects the slope of the klidar but has little influence on the asymptotic value of klidar. The dependence of klidar on the phase function and b can be summarized by a backscattering coefficient bb, which is obtained by integrating the phase function in the range from to .
In summary, the relationships among the parameters m, n, p and IOPs can be derived based on the fitting results in Table 1 (Fig. 6). Both m and n show a linear relation with bb, and p is a linear function of a and bb. The linear functions can be expressed by
klidar for the phytoplankton bloom case 1 water with [Chl] of 10 mg/m3 and 20 mg/m3 are also calculated, and the fitting results are coincident with the fitted curves in Fig. 6, which broadens the application range of Eq. (20). It is easy to obtain the relationship of IOPs and depth resolved klidar by substituting Eq. (20) into Eq. (19). For convenience, this function is termed and referred to as klidar-IOPs model in the following sections. This model depends on the IOPs’ bio-optical model and the scattering phase function of particle, especially the backscattering properties of particle. Since the scattering properties are variable in real waters, the goodness of fit of this klidar-IOPs model will be sensitive to particles with very strong or weak backward scattering.
4. Applicability of the relationship between IOPs and depth resolved klidar in real seawaters
In the real ocean environment, the distribution of optical properties is normally inhomogeneous, so the validity of the klidar-IOPs model needs to be tested in the practical seawaters. We build a model of inhomogeneous water by defining multiple parallel thin layers, which have the same optical properties in the horizontal direction but different in the vertical direction. The thickness of each layer is assumed to be the same as the resolution of the lidar return profiles. According to Uitz et al. , the vertical distribution of chlorophyll a concentration in the global open ocean conforms to the Gaussian distribution given by
Uitz et al.  classified the stratified case 1 waters into nine trophic categories, named from S1 to S9, with respect to the Chlorophyll a concentration within the surface layer . In order to examine the relationship between IOPs and depth resolved klidar derived in Section 3 in actual seawaters, three trophic categories, with low (<0.04 mg/m3), medium (0.2-0.3 mg/m3), and high (2.2-4 mg/m3), namely S1, S5 and S9, are selected based on the experimental and statistical results of stratified waters. These three waters can represent the trophic states of most open ocean waters in the world. The parameters in Eq. (21) of these three typical trophic class of stratified waters are outlined in Table 3. The distributions of IOPs and Chla of these three waters are shown in Fig. 7.
Comparisons of the Kd computed by the bio-optical model given as Eq. (8) and klidar obtained from the calculation model proposed in this paper with simulated effective attenuation coefficients for S1, S5 and S9 waters are shown in Fig. 8. The lidar is expected to penetrate about 50 m in S1 water, 40 m in S5 water and 20 m in S9 water, with the effective detection depth of three optical thicknesses. For all these three waters, the effective attenuation coefficient klidar of simulating return signals is found to be different from Kd (Fig. 8). The values ofwhich is the attenuation coefficient calculated using the klidar-IOPs model are very close to the simulated ones.
To quantify the accuracy of klidar-IOPs model, a mean percentage error within the effective detection depth range is defined, which is expressed by
In this paper, the multiple scattering effect on the spaceborne oceanic lidar returns is investigated with an improved SALMON, which includes a Gaussian beam emission model and is applicable for both homogeneous and inhomogeneous waters. The effective attenuation coefficient decays exponentially with the increase of depth in water according to the simulations of lidar return signal from case 1 waters, because high order scattering signals increase with depth and predominate the total signal at shallow depth. An exponential function is proposed to describe the depth characteristic of klidar and the parameters in this exponential function is related to IOPs. The exact function of klidar, depth and the IOPs of seawater is given and examined using real ocean waters simulated by a model of inhomogeneous water.
Comparison of the simulated and calculated results of three typical stratified case 1 waters shows that the errors between klidar-c and klidar are less than 0.5%. The better agreement shows that the exponential klidar-IOPs function is feasible for oceanic lidar measurement in theory. Using this function, one can estimate the inherent optical properties and retrieve the chlorophyll concentration of case 1 water, which improves the accuracy of spaceborne lidar data retrieval and assessing global phytoplankton biomass and primary productivity.
National Key Research and Development Program of China (2016YFC1400902, 2016YFC1400905, 2016YFC0200700); National Natural Science Foundation of China (NSFC) (41775023, 61475141).
The authors thank Prof. Heather Bouman in the Department of Earth Sciences, University of Oxford for providing the knowledge of chlorophyll concentration.
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