We use Monte Carlo time-dependent simulations of light pulse propagation through turbulent water laden with particles to investigate the application of Multiple Field Of View (MFOV) lidar to detect and characterize oceanic turbulence. Inhomogeneities in the refractive index induced by temperature fluctuations in turbulent ocean flows scatter light in near-forward angles, thus affecting the near-forward part of oceanic water scattering phase function. Our results show that the oceanic turbulent signal can be detected by analyzing the returns from a MFOV lidar, after re-scaling the particulate back scattering phase function.
© 2007 Optical Society of America
Most oceanic flows are turbulent and therefore characterized by a wide range of coexisting scales of motion, from millimeters to meters. Turbulent fluctuations of passive scalars (such as temperature or salinity) in the water cause fluctuations in fluid density, and consequently in its refractive index. In the ocean, the nearforward part of the oceanic phase function is determined by light interactions with turbulent inhomogeneities of the refractive index field , mostly due to temperature fluctuations . These turbulent-induced inhomogeneities scatter light in near-forward angles and provide an opportunity to use optics to quantify the turbulent oceanic flow. Both experimental studies of small angle scattering and measurements of the oceanic Volume Scattering Function (VSF)in the range of 10-7 to 10-3 rad demonstrate that the total scattering coefficient, b, due solely to oceanic turbulence, can be on order of 10 m-1. Thus the small-angle scattering function due to turbulence is, in most cases, significantly larger than typical values due to the combined contribution of oceanic particles and the ’pure water’ scattering function, which is typically less than 0.25 m-1 and 0.05 m-1 in coastal and deep ocean waters respectively [1, 3]. Homogeneous and isotropic turbulent flows are characterized by two parameters: χ, the temperature variance dissipation rate, which expresses the strength of the temperature gradient, and ε, the rate of dissipation of turbulent kinetic energy, which is inversely proportional to the size of the smallest flow structure. Typically χ ranges from 10-2 °C 2 s -1 a few meters below the surface  to 10-10 °C 2 s -1 in the deep ocean . The ε attains values of 10-4 m 2 s -3 in the fairly energetic upper layer to 10-11 m 2 s -3 in the mid-water column . The corresponding value of the light scattering coefficient due to turbulence bturb ranges between 10 and 30 m-1 a few meters below the surface to bturb ≃ 0.1m-1 in the deep and quiescent part water column . The depth-dependent light extinction α(in m -1) in the water column is given by the sum of absorption, a, and scattering, b, i.e. α(z) = a(z)+ b(z).
The large value of the turbulent scattering coefficient, (bturb > 1 m -1), implies that the photon mean path length given by lphot ≃ 1/b < 1m is short. Consequently, most remotely sensed photons emerging from any depth > 1m will undergo multiple forward scattering events on turbulent flow before reaching a remote detector. Here we explore the potential optical consequences of a large scattering coefficient due to turbulence in the energetic upper ocean. Namely, can surface layer turbulence be detected and quantified by remote measurements? The enhanced forward scattering on turbulence can be likened to scattering on atmospheric clouds with large water droplets. For clouds, the single-scatter lidar equation predicts a small signal variation as a function of the FOV (Field Of View) of the receiver. However, multiple scattering leads to a strong FOV dependence on signal return from inside the cloud as suggested in the pioneering work of  and subsequently explored by others [8, 9]. This suggests the possibility of using the MFOV (Multiple Field Of View) lidar technique  to remotely characterize the near surface turbulent layer. The MFOV lidar uses multiple fields of view (apertures) combined with time-gated lidar pulses to remotely measure the scattered energy and ultimately characterize the scattering media. Using multiple FOV to measure the returned signal allows detection of either a single scattering signal (small FOV - direct backscatter - a conventional lidar signal) or multiple scattering (large FOV). The multiply scattered signals combined with the direct back-scatter (single scattering) signal, contain more information about the media than is available from a conventional single field-of-view lidar.
In this paper we present results from Monte Carlo numerical simulations of a collimated lidar pulse propagating through a turbulent layer embedded in turn in a particle laden layer across a flat ocean surface. We postulate that the signal, which has been scattered by combined turbulence and by particle, measured by the MFOV lidar can be used to characterize the strength of turbulence in the upper layer.
2. The setup of the Monte Carlo numerical runs
We use the Monte Carlo numerical method to solve the time dependent Radiative Transfer (RT) equation  for an initially narrow,infinitely-short pulsed light beam impinging perpendicular to the flat ocean surface and then propagating through a layered water column. The Monte Carlo approach models the continuum propagation of radiation along a very large number of possible distinct trajectories (photons or histories in the Monte Carlo jargon). Return power is then calculated as the number of trajectories within the receiver acceptance angle, normalized by the total number of trajectories considered. We use a time dependent 3-D Monte Carlo radiative transfer code, designed for a stratified ocean. The code is a modified version of that presented in Piskozub et al. . In the current configuration, the code permits the definition of a number of layers, each possessing a different value for the coefficients of absorption or total scattering and a prescribed phase function. The photons in our algorithm are traced in the forward direction. Absorption ends a given photon’s history, i.e. no partial photons are traced. We also assume incoherent addition of multiply scattered radiation - such that the particle and turbulent phase functions can be added to describe the medium, and we do not address any possible coherent effects. The assumption of no coherent effects, for the forward light scattering is consistent with oceanic observations  while for atmosphere, the backscattered light, due to coherent effects has been postulated to be enhanced up to a factor of 3  No such observations (that we know of) are available in case of oceanic measurements. For each case, calculations we traced over 1012 histories. Runs were executed on the USC Linux cluster, distributed among 1024 nodes. In the following figures we convert the flight-time of the returned lidar signal to an equivalent depth in 1m increments. For this work we analyzed three cases. In each of the three cases (Fig. 1) the narrow Gaussian beam, with a 10-5 rad divergence and wavelength λ = 550nm, impinges normally on a flat ocean surface. The relevant optical parameters corresponding to the medium and turbulence for each case are presented in Table 1. We additionally calculated the equivalent photon diffusive distance, defined as ldiff = 1/α · 1/(1 - g) where g > 0 is the average cosine of the forward scattering angle . The equivalent diffusive distance represents the propagation distance after which the initially collimated photon has ’forgotten’ its original direction. The particles used in our calculations have a scattering coefficient b = 0. 1m -1 and an absorption coefficient a = 0.1/m -1 The underlying pure water follows the Rayleigh scattering phase function with bRayleigh, = 1.9 · 10-3 m -1  and absorption aRayleigh, = 5.7 · 10-2/m -1. In calculations we have used a turbulent light scattering phase function corresponding to relatively large but realistic oceanic turbulence levels, with the rate of temperature variance dissipation -χ = 10-3 deg C 2 s -1, and a rate of dissipation of turbulent kinetic energy -ε = 10-6/m -6 s -3. The turbulent phase function was obtained experimentally [2, 1]. Below the turbulent layer the water was assumed quiescent, i.e. with vanishing ε and χ. For computational efficiency,i.e. to shorten the computation time, we enhanced the backscattering of the particulate VSF relative to oceanic observations  near 180° by a water-type varying factor of 100 to 1000 - see the bottom part of Fig, 2. The larger value used for backscatter at or near 180° allows more photons to return to the source thus decreasing the necessary simulation time to build the statistics of the returning photons. As it will be argued below (see Eq 1 and Eq 2) the backscattering at or near 180° is a multiplicative constant of the problem considered here and the obtained results are ∝ VSF(180°).
The particle phase function (blue curve in the Fig. 2) used in our calculations was assumed to follow the formulation derived by Haltrin  which extends the Henyey-Greenstein phase function and allows adjustment of the backscattering coefficient values. The VSF used in calculations (for the layer where turbulence was present) was combined particle and turbulent VSF - Fig. 2. In addition Fig.2 compares the VSF (blue color) used in our calculations to in situ ocean data (plotted with red indicating the particles with low refractive index and green indicating those with higher refractive index) obtained from Mie scattering calculations for realistic oceanic size distributions and different indices of refraction for particles  (particle data provided by Dr. Stramski). This set of refractive indices represents the two end members of oceanic water; the smaller value of the refractive index n = 1 .04 corresponds to waters with optical properties dominated by chlorophyll in the water column while the larger value of n = 1.18 corresponds to waters dominated by inorganic sediment . Comparison of the equivalent diffusive distance (Table 1) is in agreement with observations that the presence of turbulence does not modify a wide beam light field (eg. solar irradiance). This has been pointed out in the simulations of Gordon , where the modifications of oceanic phase function between 0° and 15°, i.e. near-forward angles, does not significantly modify the solar irradiance observed underwater.
3. Theory of lidar scattering - quasi-small angle (QSA) approximation theory
Throughout the paper, we interpret the returned lidar signal in the framework of the quasi-small angle (QSA) approximation , in which all multiple-scattering processes that contribute to the lidar return are small-angle forward scattering events, except for a single particle backscattering event close to 180°. This condition is valid for particles that are large compared with the lidar wavelength and requires that the lidar FOV footprint be smaller than the mean free path between scattering events. The latter condition ensures that only small angle scattering contributes to the received signal and that the range resolution based on the measured time of flight is maintained to a good approximation. In our simulation, we vary the FOV between 10-5 rad to 3·10-2 rad and the analyzed range is 20m. Since 1/bparticle = 10m ≫ 0.6m=3·10-2·20m, then the single backscattering condition is met. In addition turbulent scattering does not contribute to large angle scattering and characteristic turbulence lengthscale is much smaller than lidar FOV footprint. The QSA approximation offers a useful theorem to interpret the received lidar signal. We will apply this theorem to analyze our results obtained from the Monte Carlo simulations. Katsev et al.  demonstrated that, within the QSA framework and for a medium with uniform scattering coefficient over angles close to 180° (backscattering), for the radiative transfer calculations, the water column can be replaced by a fictitious water column possessing twice the extinction coefficient for the purposes of lidar calculations. Then the angular properties of the scattered beam are modeled only on the downward propagation leg. This result has been extensively verified in atmospheric lidar experiments [9, 8]. Following this theorem, the lidar equation can be written as :
where z is the range, θ is the FOV of the receiver, Pss is the received power due to a single-scattering contribution, and M(z,θ) represents the multiple scattering contribution at variable FOV. The single scattering contribution Pss(z) (by definition FOV independent) is given by:
where K is an instrument constant, P 0 is the laser pulse power, β(z, π) is the particle backscatter coefficient (at 180°), and τ(z) = ∫z 0 α(z′)dz′ is the integrated depth dependent optical depth or thickness. We use an approximate expression for the multiple scattering contribution M(z, θ) developed by Eloranta :
where x is defined as: x = zθ/[(z - z 0)θ 0] , the constant z 0 is the distance between the scattering layer and the receiver, z the total range, θ is the FOV of the receiver, and θ 0 is the mean-square width of the particle scattering peak, the red or the green curve, Fig. 2. The above equation, Eq. 3, is an approximate expression for double (k = 1) and higher order (k > 1) scattered light . The Erf is the error function. In our calculations the turbulent phase function has θ 0 ≃ 10-4 rad (Fig. 2). We can estimate Monte Carlo RT accuracy comparing the results of Case C (in the absence of turbulence) to predictions of Eq. 3. The ratio of the received lidar signal at different FOV normalized to its single scattering value is compared in Fig. 3. The blue and red lines denote the RT Monte Carlo data and the black lines the results of Eq. 3. The results of RT Monte Carlo are consistent with theoretical predictions of Eq 3. In addition, we note the effects of double and multiple scattering which are evident at FOV of 8 · 10-4 rad as the returned signal approximately follows a straight line obtained from Eq 3. The effect of higher order scattering becomes pronounced for larger FOV of 3 · 10-2, where the returned signal follows a k > 1 power law; the doubly scattered return obtained from Eq 3 is plotted as reference - the black line.
4. Results of the RT calculation for an embedded turbulent layer
The turbulent medium is characterized by a highly peaked forward scattering phase function, Fig. 2. In such a medium (with highly peaked forward scattering phase function and relatively flat in the backscatter direction), the FOV lidar detected irradiance at a given FOV can be effectively inferred from formulas for the propagation of collimated light  via the QSA effective medium theorem . Following the formula developed by (Walker , p.66) for the irradiance of the collimated light beam in the asymptotic limit of the long propagation distance, the multiple scattered component M(z,θ) of the FOV lidar can be expressed as:
where θ 0 is the mean-square width of the highly peaked forward scattering phase function, i.e. the turbulent scattering VSF with θ 0 ≃ 10-4 rad (Fig. 2), θ is the receiver FOV, α 0 the constant turbulent layer extinction coefficient, g- is the mean cosine, and z 0 is the distance to the scattering turbulent layer from the receiver. In Eq 4, the initially collimated beam spread angle is given by θbeam(z)2 ∝ (z - z 0) · θ 2 0. This dependence of the spreading angle on the propagation distance is a hallmark of a random walk or the diffusion with distance as independent variable and the angle as a dependent variable (i.e. the diffusion in the angle space). In Eq 4, the asymptotic limit of large propagating distance is usually reached quickly. It has been shown  that when analyzing solutions of Monte Carlo RTE and an approximate RTE solution based on the Fokker-Planck equation,the asymptotic behavior of Eq 4 is attained within 5 times the total photon mean free path. This means that for conditions corresponding to our simulations, Eq 4 becomes valid over the propagation distance of a 0.2m. To compare the formula of Eq 4 to our RT calculations we expand it for small FOV (i.e. θ) as:
From Eq 5, the leading term of the multiple scattering signal at a fixed small FOV depends on the beam travel distance within the turbulent layer ∝ 1/(z - z 0) and does not depend on any turbulent parameters. This is what we observe at the smallest FOV of 8·10-4 rad in Fig. 4. (Note the stretch of vertical coordinates on the Fig. 4, depends on the analyzed FOV. The lidar return slopes, Fig. 4, at the onset of turbulent layer in absence of coordinate stretching are consistent with the first two terms of Eq 5). At the larger FOV (3 · 10-2 rad) the multiple scattering contribution, M(z,θ), corresponding to the term containing ∝ θ -2 0, of Eq 5 becomes dominant. In general, at larger and fixed FOV, the slope of the multiple scattering contribution M(z,θ = FOV) depends on the width (θ 2 0) of the turbulent scattering phase function (Fig. 2). In turn, the width of the turbulent VSF was shown  in to depend directly on the turbulent temperature spectra as:
where ET(k) is the depth dependent temperature variance spectrum. The quantity 〈θ 2 0〉 is affected by the smallest scale region of the temperature spectrum ET(k) as discussed previously . Away from the turbulent layer the lidar returns follow the double/multiple particle scattering formula as predicted by Eq 3 in both Cases A and B (Fig. 4). The comparison of calculated and observed double/multiple particle scattering (away from turbulent layer) is presented in Fig. 4. The turbulent scattering observed in the MFOV lidar is due to the fact that the turbulent layer acts as a very effective lidar beam scatterer by widening the collimated light beam and very efficiently removing photons from the ballistic beam. The effect of varying strength in turbulence (temperature dissipation rates) is reflected in the slope of the lidar detected signal with depth within the turbulent layer following the second term of the expansion Eq 5. The realistic observed signal to noise (S/N) of the lidar system signal originating within the turbulent layer will depend on the optical properties and concentration of particles in that layer. We speculate, on the basis of Eq 2, that re-scaling of the data for Fig. 4 by a factor of 100 (inorganic particle dominated water column) or a factor of 1000 (chlorophyll dominated water column) should corresponds to the MFOV experimentally observed signal. In some cases - Northwestern Atlantic off New Jersey this factor can be as low as 10. Hence in the worst case scenario - for the chlorophyll dominated water column, we should observe the MFOV normalized lidar signal change by a factor of order 2 or more in the presence of the turbulent layer. In the ongoing investigations we will experimentally verify these predictions. We plan to explore the differences of a turbulent and a particle lidar return to obtain the information about the turbulent layer
Here we show that lidar returns measured at MFOV depend on the presence and depth of the turbulent layer in the upper ocean. Consequently, they can be used to estimate turbulence characteristics of the surface mixed layer. Realistic lidar returns from the turbulent layer will be a function of the local temperature variance spectra at the smallest spatial scales. The separation of the effect of turbulence and particulates in the angular domain (in the scattering context) is the main physical reason allowing the turbulence scattering to be distinguished from the particle scattering. Having demonstrated the ability of MFOV to distinguish between turbulent and quiescent surface and subsurface layers, future work will aim to define the requirements of a detector capable of sensing the turbulence levels for representative oceanic mixed layers (turbulence levels, and profiles of temperature, salinity and particulates).
This work was supported by ONR (Contract Monitor Dr. S. G. Ackleson) and the NASA Oceanography Program. Computations reported in this paper were performed at the High Performance Computing and Communications facility at the University of Southern California. Portions of this work were carried at the Jet Propulsion Laboratory, California Institute of Technology under contract from NASA. We thank Dr. Zaneveld and Dr. Jonasz for suggestions and comments regarding the manuscript. We thank anonymous reviewers for thorough comments on a previous version of this manuscript.
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