1. Introduction

Light detection and ranging (LiDAR) is a mature technology for remote sensing of ocean and coastal waters. While bathymetry has been the dominant focus of oceanographic LiDAR since its early development [1], particularly over the last decade or so there has been an accelerating focus on its use for remote sensing of the water column [2], with the aim of making depth-resolved measurements of ocean “layers”, fish, bubbles, chlorophyll and more. To this end, more complex LiDAR technologies have recently been proposed involving the use of dual laser wavelengths, polarisation channels and, of particular relevance to this paper, a channel for Raman emission [3,4].

While conventional LiDAR detects elastic scattering of the laser beam as it propagates through the water column, a small amount of the beam is inelastically scattered from the water column, such that its frequency is shifted via the processes of Raman scattering or Brillouin scattering. The shift associated with Brillouin scattering is small (0.2 cm^-1, or 0.006 nm shift at 532 nm), though its use for oceanographic remote sensing has been considered [5]. The Raman shift is far larger: dominated by the OH stretching band, it is centred at 3400 cm^-1 (117 nm shift at 532 nm), and has a spectral width of about 500 cm^-1 at full-width half maximum [6]. It is well-known that the Raman spectrum of water is sensitive to temperature and salinity [7] and these systematic dependences can be seen in [8]. The potential to use this effect to measure depth-resolved subsurface water temperature was investigated originally in [9,10], and remains an area of active research [10–12].

There are a few instances where Raman signals have already been used to complement elastic LiDAR returns. In particular, the SHOALS airborne laser bathymetry LiDAR system included a Raman channel in addition to elastic channels at 532 nm and near-infrared [13]. The Raman channel is effective in validating the air-sea and land-sea interfaces [14], since it can only originate from the water volume. The Raman return has also been used to detect surface oil [15].

Numerical modelling and simulations play an important role in the development of LiDAR systems, given that analytic models cannot handle multiple large-angle scattering events. For general applicability, numerical Monte Carlo methods are the tool of choice, able to simulate a wide range of parameters with few assumptions. Monte Carlo methods for simulating elastic scattering oceanic LiDAR are well developed, tracking the random paths of a large number of photons packets as they propagate, scatter, and attenuate in the water column [16–18]. The key inputs to such a model are the inherent optical properties of the water column: absorption and scattering strength, and the elastic scattering phase function. Models can be used to guide the choice of experimental parameters such as field-of-view, can enable the extraction of ocean properties from measured data [19–21], and inform LiDAR use for bathymetry [22]. There are a range of tutorial-style resources available (Ocean Optics Webbook, and [16]).

On the other hand, to our knowledge no work has been published on predicting Raman returns from a LiDAR experiment. Raman scattering in the water column has been numerically investigated for solar illumination, looking at the effect of Raman scattering on the intensity and spectrum of the in-water and upwelling spectral intensity [23–25]. Poole and co-workers have simulated the relative strength of integrated Raman returns and fluorescence returns under laser illumination [26], with a view to correcting field measurements of fluorescence for varying intrinsic absorption [27]. That work used a Monte Carlo model that included Raman scattering, but the authors did not simulate the time-resolved properties of the Raman signal nor consider its differences to the elastic signal.

In this paper we present Monte Carlo modelling of elastic and Raman LiDAR returns. We build on well-known methods for predicting elastic LiDAR returns, and detail our methods for predicting Raman returns. We compare and contrast the characteristics of the two types of returns, and demonstrate the usefulness of the model by comparing elastic and Raman returns for two different simulations with different laser wavelengths and detector field-of-view.

2. Monte Carlo model including Raman scattering

We use a semi-analytic Monte Carlo method to simulate the elastic and Raman interactions with the water column. In the standard approach for elastic scattering [16], ‘packets’ of photons are launched and tracked through a sequence of elastic scattering events. Absorption events are not simulated but rather handled by reducing the packet weight at each elastic scattering event by an albedo $b/({a + b} )$, where b is the elastic scattering coefficient, and a is the absorption coefficient. In elastic-only simulations, Raman scattering is implicitly included in the absorption term; we explicitly simulate Raman scattering, changing the albedo to $({b + {b_{Rq}}} )/({a + b + {b_{Rq}}} )$ and stochastically choosing between Raman and elastic scattering at each event using a Raman probability ${P_R} = {b_{Rq}}/({b + {b_{Rq}}} )$. The coefficient ${b_{Rq}}$ is the quantum Raman scattering coefficient appropriate for conversion of photons, described below. At Raman events, the entire packet is considered to scatter from the laser wavelength ${\lambda _l}$ to the central Stokes wavelength given by $1/{\lambda _s} = 1/{\lambda _l} - {\bar{\nu }_R}$, with ${\bar{\nu }_R} = 3400$ cm^-1 [7]. Subsequent propagation and elastic scattering are determined using the ${a_s}$ and ${b_s}$ coefficients at the Stokes wavelength. Further Raman scattering events to the second-Stokes wavelength are not simulated but rather included in the ${a_s}$ absorption coefficient. Detected signals at laser and Stokes wavelengths are determined semi-analytically utilising the so-called SALMON method [18]. In this method a fractional return to the detector is calculated at each scattering event: we record a signal at the laser wavelength if a packet has not Raman scattered, and at the Stokes wavelength otherwise. Detected signals are recorded with their arrival time, their angle of incidence at the detector, and for Stokes signals the depth at which the single Raman scattering event for that packet occurred. Packets are simulated until they either exit the surface of the water or exceed a detection time cut-off.

We take our parameters from Solonenko et al. [28], who found sets of parameters that fit Jerlov’s measurements of inherent optical properties of a range of water types [29,30]. For each simulation involving a chosen water type, we take $a\; $and $b\; $ coefficients at both the laser and Stokes wavelengths. To model the distribution ${\tilde{\beta }_E}(\theta )$ of the elastic scattering deviation angle $\theta $, we combine a relatively isotropic molecular component (${\tilde{\beta }_M}(\theta )\propto 1 + 0.84{\cos ^2}\theta $ [17]) with a forward-peaked particulate component (${\tilde{\beta }_P}(\theta )$, modelled using the Fournier-Forand function [31] with $n = $1.10, $\mu = 3.5835$ to fit the Petzold measurements [32]) to get an overall phase function ${\tilde{\beta }_E} = \eta {\tilde{\beta }_M} + ({1 - \eta } ){\tilde{\beta }_P}$. The relative weight $\eta $ of these components depends on our chosen water type [28], with small $\eta $ corresponding to strongly-forward scattering. We calculate the quantum Raman scattering coefficient ${b_{Rq}}({{\lambda_l}} )\; $ for the simulated laser wavelength from a reference power scattering coefficient ${b_R}({{\lambda_l} = 488\; \textrm{nm}} )= 2.6\; \times {10^{ - 4}}\; $m^-1 [33,34], assuming a $\lambda _s^4$ dependence of this coefficient on Stokes wavelength, and the relation ${b_R} = {b_{Rq}}\; {\lambda _l}/{\lambda _s}$. For the Raman scattering phase function we use ${\tilde{\beta }_R}(\theta )\propto 1 + 0.55{\cos ^2}\theta $ [17]. We do not currently model the spectral spread of the Raman returns, and do not model depolarization [35].

For all water types, and particularly for more turbid water, the number of packets returning to the detector at the Stokes wavelength is less than at the laser wavelength, leading to a larger simulation noise on the Stokes signals. To equalise the simulation noise, thus making most efficient use of the computing time, we oversample the Raman events [16,36], accounted for by adjusting the albedos.

3. Monte Carlo simulations

For our simulations, the photon packets are launched from height $H = 400\; \textrm{m}$, and enter the water at a point travelling vertically downwards. We record returned signal power per Joule of laser energy (W/J), at a detector of radius $r = 0.15\; \textrm{m}$ colocated with the source at $H = 400\; \textrm{m}$. This signal strength can in principle be post-processed to account for real detector properties (e.g. V/W sensitivity, electronic and shot noise) and background light, and the time dependence can be convolved with a response function to account for laser pulse duration and detection bandwidth. For this first investigation of the properties of Raman returns, we do not post-process the data, in order to better focus on the underlying physics. We define $t = 0$ as the time at which the first returns arrive at the detector. We do not simulate the return from the air/water interface, which would appear at $t = 0$ and only in the elastic signal.

We present two simulations in the present work. Table 1 lists the model inputs for both simulations. Both simulations are for Jerlov coastal water Type 1, referred to hereafter as J1C [28]; this is the clearest coastal water. For J1C water, scattering is strongly peaked in the forward direction ($\eta = 0.005$ corresponds to 71% of scattering events having a deviation of under 10°, and 90% of events deviating under 30°). We assume that ocean properties are uniform with depth. We present simulation results for two scenarios: using a 532 nm laser wavelength and a detector full-angle field of view (FOV) of 30 mrad, and using a 480 nm laser wavelength with an FOV of 8 mrad. The first scenario is not ideal for measuring Raman returns, but illustrates well the physics that affects the Raman returns; the second scenario is chosen to give a sense of what can be achieved with the parameters closer to optimum. Both simulations modelled 3 × 10¹¹ photon packets.

Table 1. Parameters for the detector, laser, and water type for the two simulations presented [28].^a

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Simulation 1: J1C water, 532 nm laser wavelength and 30 mrad FOV

Figure 1 shows output of the Monte Carlo simulations for J1C water, with a laser wavelength of 532 nm, and detector FOV of 30 mrad. Table 1 indicates that J1C water at 532 nm has a diffuse attenuation coefficient of 0.126 m^-1. Note that elastic scattering $b\; $ is about six times stronger than absorption a. Figure 1(a) shows the elastic scattering return to the detector (at the laser wavelength) and the Raman scattering return (at the Stokes wavelength). We see that Raman returns at early times, i.e. from near the surface, are two orders of magnitude weaker than the elastic returns at the same time, and that the Raman returns decrease more rapidly with time. To give a sense of magnitudes, 1.8% of photons exit the water at the laser wavelength, and 2 × 10⁻⁷% are collected by the receiver; 9.3 × 10⁻³% of the photons exit at the Stokes wavelength, and 1.1 × 10⁻⁹% are collected.

 

Fig. 1. Results for Simulation 1, with parameters from Table 1. There are four panels: (a) shows elastic and Stokes return signal as a function of time, compared with analytic models; (b) shows Stokes signal strength as a function of time and depths of the Raman scattering event; (c) shows, for a series of different times, from what depths the Raman signal emanates; (d) shows the median depth of the Raman event as a function of time, and the depth resolution defined in the text. The dotted line shows $z = ct/2n.$ Note that in (c) and (d), particularly for later times, there is significant variance; there is no genuine fine structure expected in the data and so in principle this could be smoothed.

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We compare our simulation results to simple theory, to see where agreement and differences lie. We can make a simple theoretical prediction for elastic signal return as a function of depth z assuming a single elastic backscatter event and otherwise only strongly-forward-peaked elastic scattering. This is the so-called quasi-single scattering model developed by Gordon [37]

Figure 1(a) shows that the Monte Carlo simulation of elastic return signal initially follows the quasi-single scattering model, but then falls below it for later times. This is attributed to some fraction of the photons scattering out of the field of view: the simulation shows that at 100 ns, detected elastic photons have elastically scattered an average of 10 times. There is a substantial literature on how the slope of the elastic return depends on $a,\; b,\; {K_d}$ and FOV following the original work of Gordon [2,20–40], with predictions of slope vs depth having complex shapes [41].

The simplest analytic model for Raman returns assumes strongly-forward elastic scattering and a single backscatter event – a Raman event – and so the beam attenuates according to the laser-wavelength a coefficient on the transit down to depth z and attenuates at the Stokes-wavelength ${a_s}$ coefficient on the way up [11]:

Looking again at Fig. 1(a), this is again expected to be an upper limit that assumes that forward scattering does not present as a loss. Raman scattering returns are far weaker than elastic returns at $t = 0$, reflecting the fact that ${b_{rq}}{\beta _R}({180^\circ } )\ll b{\beta _E}({180^\circ } )$. Indeed ${b_{rq}}$ is approximately an order of magnitude weaker than even the molecular contribution to b, and so Raman returns will be weaker for all water types. The slope of ${R_R}$ is also much steeper than ${R_E}$, reflecting the 5-times-higher absorption at the Stokes wavelength. The model agrees reasonably with the Monte Carlo simulation for early times, but for later times the simulated Raman return signal is far stronger than the supposed ‘upper limit’ theory, indicating a breakdown of the simple analytic model assumption. A Monte Carlo approach is particularly valuable here as it provides insights that cannot be matched by other means. We can investigate the more complex scattering pathways that contribute to the return signal, and explore how that behavior can be used to infer water properties.

Each photon returning to the detector at the Stokes wavelength must have Raman scattered once and only once, and may have also elastically scattered any number of times. We can label each simulated photon packet with both its detection time t and the depth at which it was Raman-scattered, ${z_R}$. This allows us to consider more carefully the relationship between time and depth for the Raman signal. Figure 1(b) shows a colourmap plot vs t and ${z_R}$ with the colour representing the detected power (W J^-1m^-1) on a logarithmic scale. Signals are only seen in the upper-right half of the plot: for a given time t photons can at most have travelled down to ${z_R} = ct/2n$. For early times, the strongest returns indeed lie close to this diagonal, indicating that the signal is dominated by photons that have taken the simplest path: travelled down, Raman scattered, and returned to the detector, all without substantial deviation by elastic interactions. However, at later times, the photons arriving at a particular time have their Raman events spread in depth, and for $t > 300\; \textrm{ns}$, the signal is dominated by photons that Raman scatter near the surface. Figure 1(c) shows a different view of the data in 1b by highlighting particular detection times: each curve shows the range of depths from which photons arrive at that time. We see that for $t = 250$ ns, the return from near the surface is actually stronger than that from ${z_R} = ct/2n$.

The reason that near-surface Raman scattering dominates for later times is the relative strength of the absorption coefficients for the laser and Stokes wavelengths. At later times, there is a significant probability that, in addition to the single Raman-scattering event, one or more large-angle elastic scattering events takes place. This leads to detected photons taking more convoluted photon paths through the water. At 532 nm, ${a_s} = 4.9\; a$, so photon packets that Raman scatter towards the beginning of their path will be more strongly attenuated than those that Raman scatter towards the end of the path. Thus, the dominant photon paths at late times are ones for which the photon undergoes large-angle elastic scattering (at least once, and no deeper than ${z_E} = ct/2n$) and only then Raman scatters just before leaving the water. Simulation data shows that Stokes photons detected at 200 ns have on average elastically scattered 16 times before their Raman scattering event and only 2.5 times afterwards.

The degradation and eventual loss of the correspondence between detection time $t\; $ and depth of the Raman event ${z_R}$ is a potential problem for some applications of Raman-scattering LiDAR, such as the proposed measurement of temperature and salinity [8]. Properties of the local environment are encoded at each Raman event in the spectrum and polarisation of the photons, which can be measured as a function of arrival time at the detector; at each time though, arriving photons have Raman scattered at a range of depths, limiting the depth resolution of our measurement of properties. To quantify this effect, for each detection time bin we calculate the median photon depth (the depth for which half of the detected energy comes from above and half from below) and the depth resolution: this is defined as the smallest range of depths that encompasses 80% of the detected energy. These parameters are shown in Figure 1(d).

For $t < 80$ ns, the median depth is close to, but less than, $ct/2n$, shown as a dotted line, and the depth resolution is better than 3 m (i.e 80% of the photons originate from a 3 m layer of water, encompassing the median). In this time range the concept of depth-resolved temperate and salinity measurements is valid. Between times 80 and 140 ns, the median depth continues to increase, but the depth resolution degrades, with significant returns both from deep waters and from near the surface. For $t > 140\; \textrm{ns}$, depth resolution ‘improves’ as the photon returns become dominated by those that Raman scatter towards the end of their paths close to the surface, but the correlation between t and ${z_R}$ is entirely lost. So, while Raman signals might be detected at such late times, those signals do not encode information about the depth $z = ct/2n$.

Simulation 2: J1C water, laser wavelength at 480 nm and 8 mrad FOV

Fortunately, there are a number of parameters to optimize if we wish to improve signal strength, depth penetration and depth resolution of the returned Raman signal, the most obvious being the laser wavelength and the detector FOV. Laser wavelengths shorter than 532 nm are sometimes employed for elastic LiDAR, since for less turbid water types the a and b coefficients can be lower. Solonenko’s analysis of Jerlov types suggest that the minimum diffuse attenuation coefficient lies at 486 nm or shorter for oceanic types (Fig. 2 in [28]), and at longer than 486 nm for coastal types (Fig. 1 in [28]). A recent study suggested that 490 nm might be near-optimal for a large fraction of water bodies [42], and centering the laser wavelength specifically at 486.15 nm has the additional advantage of somewhat weaker daylight background due to a Fraunhofer line at that wavelength. Here we consider a laser wavelength of 480 nm; the corresponding set of inputs is given in Table 1. The Raman signal is now centred at 574 nm, for which we have ${a_S} = 0.090$ m^-1, almost four times lower than for the Raman returns at 650 nm from a 532 nm laser. Furthermore, by now having similar absorption coefficients for the laser and Stokes photons (0.090 and 0.073 m^-1 respectively), we can hope to avoid the catastrophic loss of correlation between t and ${z_R}$.

 

Fig. 2. Results for Simulation 2, with parameters from Table 1. The four panels are as described in Fig. 1 caption.

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The factors affecting choice of field of view are well studied for elastic LiDAR [2,20–41]. A decrease in FOV discards photons returning from large angles and so, on average, favours photons with lower deviation, tending to tighten the correlation between time and depth [1,41] at the expense of lowering signal (though it may increase the signal to background ratio). We can expect the same benefits for Raman returns.

Figure 2(a) shows simulations of the returned signals using a shorter laser wavelength of 480 nm and a smaller field of view of 8 mrad, for the same J1C coastal water type. Compared to 532 nm, the near-surface elastic signal is slightly stronger (since b is 20% larger), but the signal attenuates somewhat more strongly with time ($a$ is just larger by 4%, but the decrease of FOV is the dominant cause evidenced by the larger deviation from the quasi-single scattering model). The initial Raman return is also larger for a 480 nm laser wavelength than for 532 nm, due to the 62% higher ${b_{Rq}}$ at the shorter wavelength; the attenuation is much less with time now, despite the smaller FOV, owing to the strongly reduced absorption ${a_s}$ at the Stokes wavelength.

Importantly, the similar absorption values at the laser and Stokes wavelengths no longer favours Raman scattering late in the trajectory. We see in Figure 2(b) and 2(c) that the Raman returns at a given time remain dominated by trajectories with Raman scattering at near-maximum depth. Figure 2(d) shows that the depth resolution remains better than 4 m until after 200 ns; at this time the median return depth is 21 m.

4. Discussion

We have, for the first time to our knowledge, presented a Monte Carlo model that predicts Raman returns as a function of time. Interpreting Raman returns from LiDAR systems is more complex than elastic returns, with a larger set of input parameters and different scattering phase functions for the elastic and Raman scattering processes. We see in the two scenarios presented here that the Raman return has different behavior to the elastic return as a function of time, and there is clearly scope for retrieving more information about the water column by collecting both returns.

For returns at the Stokes wavelength, the single Raman scattering event is a special location in the track of each photon. In the first scenario, we show that a difference in water parameters at the laser and Stokes wavelength can cause the location of this event to be skewed, such that the median depth depends strongly on water type and on time spent in the water column. This understanding is vital to interpret the form of the Raman LiDAR signal. It also has implications for uses of Raman LiDAR that seek to measure water parameters encoded in the properties of the Raman event. The first scenario showed a complete breakdown of the naïve assumption that the average depth of the Raman event would correlate with time. Fortunately, the second scenario showed simpler behavior with a good correlation between depth of the Raman event and time. This gives confidence that with careful system design, Raman LiDAR returns can indeed hope to provide a new channel for retrieving information from the water column as a function of depth.

For elastic LiDAR there is no immediate analogue to the Raman depth and depth resolution, since there may be many elastic scattering events for each photon. One could choose to track the deepest scattering event for each photon track, or the tracks’ median depth; either will deviate from that expected from simple correlation with time, with the degree of deviation dependent on the water type. This does not appear particularly well reported for non-bathymetric applications of LiDAR, though the analogous metric of tracking the range of arrival times of photons reaching a particular depth is well studied for bathymetric applications [1,41,43]. The standard practice of plotting LiDAR signals vs depth using the limiting value of $z = ct/2n$, while convenient for visualizing the scale of the penetration, perhaps encourages the reader to assume too much.

Raman LiDAR is an emerging technology, and Monte Carlo modelling of Raman returns will be useful in optimizing its performance and establishing whether it will provide useful intelligence about the water column. There is a large range of investigations possible with a Raman Monte Carlo model, including modelling of temperature and salinity measurement using shifts in wavelength and/or depolarization, and devising methods to use the Raman returns to add to or enhance the information that can be retrieved from the elastic returns. Our code can be extended to simulate actual returns for a specific LiDAR system by convolution with a real laser pulse shape and detector response. In the future, we will model signal-to-background and signal-to-noise ratios to determine the depth from which Raman signals can be usefully measured. Raman signals will in general be weaker and so will be more susceptible to noise. Filter bandwidth will in general need to be larger for Raman returns given the Stokes photons are spread over a significant wavelength range, and so night operation will be particularly advantageous.