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Bio-optical models are used to develop a model of the lidar extinction-to-backscatter ratio applicable to oceanographic lidar. The model is based on chlorophyll concentration, and is expected to be valid for Case 1 waters. The limiting cases of narrow- and wide-beam lidars are presented and compared with estimates based on in situ optical measurements. Lidar measurements are also compared with the model using in situ or satellite estimates of chlorophyll concentration. A modified lidar ratio is defined, in which the properties of pure sea water are removed. This modified ratio is shown to be nearly constant for wide-beam lidar operating in low-chlorophyll waters, so accurate inversion to derive extinction and backscattering is possible under these conditions. This ratio can also be used for lidar calibration.
© 2014 Optical Society of America
1. Introduction
Increasingly, backscatter lidar is being used for a variety of oceanographic measurements [1]. Examples include fisheries [2, 3], plankton layers [4, 5], and internal waves [6, 7]. While these examples used airborne lidar, measurements have also been made from ships [8, 9]. More recently, the sub-surface ocean signals from a lidar in space have been investigated [10, 11]. The breadth of these studies suggests that improved lidar inversion techniques could be generally useful.
The lidar signal from a given range in the ocean depends on two optical properties of the water. One is the volume scattering coefficient at the direct backscatter direction at that range, and the other is the total integrated attenuation of the propagation to that range and back. If the optical properties of the water do not vary with depth, the lidar signal will decay exponentially with depth, the rate of decay provides the attenuation, and the scattering parameter can then be determined. This technique can also be applied to profiles where segments of exponentially-decreasing signals can be identified, but, in general, the lidar signal alone is not enough information to unambiguously determine either the attenuation coefficient or the volume backscattering coefficient.
The lidar ratio, defined as the ratio of the lidar attenuation coefficient to the volume scattering coefficient at the lidar scattering angle of π sr, has proven to be a useful quantity for interpreting atmospheric lidar signals. For water clouds, the ratio is near 19 sr [12]. For ice clouds, measurements are in the range of 29 ± 12 sr [13]. Measured values for aerosols in the atmosphere are generally higher [14–16]. The models used to interpret data from a lidar on a satellite use six aerosol types with values as follows: clean continental (35 sr), clean marine (20 sr), dust (40 sr), polluted continental (70 sr), polluted dust (65 sr), and smoke (70 sr) [17]. All of these values are for a lidar wavelength of 532 nm.
For Case 1 waters, where the optical properties are largely determined by constituents related to biological processes, average values of the optical properties have been modeled as functions of chlorophyll-a. These models were used to construct a model for the lidar ratio for a lidar at the most common lidar wavelength of 532 nm.
Depending on beam divergence and initial beam width, the attenuation coefficient of an oceanographic lidar will generally be between the beam-attenuation coefficient, c, and the diffuse-attenuation coefficient, Kd. For an airborne lidar, this depends on the diameter of the lidar footprint on the surface and the single scattering albedo [18], but we will consider the two limiting cases in our analysis.
2. Bio-optical model
The lidar ratio is defined as [19]
where α is the lidar attenuation coefficient, β(π) is the volume scattering coefficient at a scattering angle of π radians, b is the scattering coefficient, and(π) is the scattering phase function at that angle. This is the only angle of interest for lidar applications, and the functional dependence will be dropped in the following development.The model for diffuse-attenuation coefficient is (note that all optical coefficients in the following are for 532 nm unless noted)
where Kdw is the diffuse-attenuation coefficient of pure sea water (0.0452 m−1) and C is the concentration of chlorophyll-a in mg m−3 [20].The beam attenuation is given by
where a is the absorption coefficient and b is the scattering coefficient.The absorption can be modeled as [21, 22]
which includes the contribution of 0.052 m−1 from pure sea water. We will use this model as published, but a value for the absorption by pure sea water of aw = 0.0448 m−1 [23], elsewhere.A model for the scattering coefficient is [24]
This equation was actually developed for a wavelength of 550 nm, but the difference is small (< 4% for C = 0.01 mg m−3 and decreasing with increasing C until it is 0 for C > 2 mg m−3) and will be neglected. The water contribution, bw = 1.7 × 10−3 m−1, is less than 10% for C > 0.01 mg m−3. A similar model, also for 550 nm, is given by [20, 25]This more recent model will be used.There are two main models for the scattering phase function. Petzold measured volume-scattering coefficients using a 75 nm spectral band centered at 514 nm [26, 27], which covers the lidar wavelength of interest. From these measurements, an average particulate phase function was estimated [28], which has a value of 3.15 × 10−3 sr−1 at a scattering angle of π sr. A more recent model considers the particulate phase function to be a weighted sum of two separate functions that represent small and large particles and have values of 2.2 × 10−3 sr−1 and 2.2 × 10−4 sr−1, respectively [29]. These phase functions were calculated for a wavelength of 550 nm, but are not expected to be much different at 532 nm. The result is
where the fraction of small particles is given byThe reported range of validity for this model is 0.1 mg m−3 < C < 10 mg m−3. An extrapolation of the results of [30] to the scattering angle of interest suggest a phase function given bywhere bbp is the particulate contribution to the backscattering coefficient. The ratio on the right side of this equation can be approximated by [20]These two models provide very similar results, and the second will be used here.Taking all of this together, we see that the lidar ratio will be between the limiting cases of SKd, for which the attenuation is equal to Kd, and Sc, for which the attenuation is given by c. In addition to the conventional definition, we will also consider a modified definition in which the known pure-sea-water values are subtracted from both attenuation and volume backscatter coefficients. The four resulting models (Fig. 1) are given by:
This Fig. shows the reason for modified definition. In both limiting cases, there is much less variability of the lidar ratio with chlorophyll using the modified definition. This implies that estimated values can be used in lidar inversions to derive extinction and β(π) with much less error, especially for a broad lidar beam.
Fig. 1 Plot of lidar ratio, SKd (black, left vertical axis), for the limiting case of α = Kd, and Sc (red, right axis) for the limiting case of α = c. Solid lines represent the conventional definition and dashed lines the modified definition.
In general, the results of [18] can be used to obtain values for α between Kd and c. This would only be necessary for a rather narrow range of parameters where the spot diameter on the surface and c−1 are of the same order and the single scattering albedo is high. Note, for example, that the values for pure sea water are not too different, SKd = 233 sr and Sc = 292 sr, because the attenuation is largely a result of absorption.
3. In situ measurements
In situ measurements were collected from several stations at five different coastal and open-ocean environments selected to represent a diverse range of environments and optical values. The locations chosen were the New Jersey bight off the Northeast USA coast (1.0 < C < 8.1); the San Diego, California surf-zone (0.014 < C < 1.3); the Southern Ocean near South Georgia Island (0.046 < C < 0.71); the Ligurian Sea near Italy (0.035 < C < 1.4); and Monterey Bay, California (1.7 < C < 110). These data sets span both Case I and Case II waters. Vertical profiles were collected using a profiling package equipped with, among other instruments, a WET Labs AC-9, Seabird SBE49 CTD and a WET Labs ECO-BB3 backscattering sensor. Data from the instruments were merged together using common instrument time. After merging, all data were averaged to 1 meter depth bins. The AC-9 measures the absorption and beam attenuation of all in-water constituents except water, denoted apg and cpg, respectively. The instrument measures these parameters at nine wavelengths, although only the 532 nm channel was used here. The AC-9 was pure water calibrated before and after each cruise and corrections applied for temperature and salinity dependence as described by Twardowski et al. [31]. Scattering errors in the AC-9 absorption channel was corrected using the proportional correction algorithm of Zaneveld et al. [32]. To obtain the total absorption and attenuation coefficients, a and c, the pure water absorption coefficients of Pope and Fry [23] were added to the corrected apg and cpg measurements. The ECO-BB3 is a three wavelength (including 532 nm) backscattering sensor with a nominal centroid angle (θ) of 124°. The volume scattering function (β) and backscattering coefficient (bb) at 532 nm were calculated from the ECO-BB3 data according to Sullivan et al. [33]. For modeling inputs, β(π) was estimated by transforming the ECO-BB3 β(θ) values to β(π) using a transform factor based on the global particulate phase function derived by Sullivan and Twardowski [30]. Lidar attenuation, α, was estimated by c or by the Lee et al. [34] Kd model with a solar zenith angle of 0.
Chlorophyll concentration was estimated from the AC-9 data as in [34]. The chlorophyll absorption was estimated as the difference between the absorption at the chlorophyll peak wavelength (676 nm) and a nearby value (650 nm). This difference was converted to chlorophyll concentration using an in vivo specific absorption for chlorophyll, aph*(676), of 0.014 m2 mg−1. Errors in these estimates are caused by limits to the precision of the AC-9 measurements and by variability in the specific absorption. Chlorophyll specific absorption coefficients among diverse phytoplankton populations have been shown to vary between ~0.01 to 0.02 m2 mg−1 [35]. Overall, relative errors are expected to be less than ± 50% [36].
In all, 904 measurements from all depths sampled were binned by chlorophyll concentration, with five bins per decade. The mean values of S and error bars representing ± one standard deviation presented in Fig. 2 were computed exclusively from the in situ measured optical properties. Estimated errors in C, not shown in the Fig., are roughly ± one bin. For each 1-m binned set of measurements, the assumption is made of a homogeneous water column with those optical properties to test the bio-optical modeling of S. Below 2 mg m−3, measured S 'Kd values are very close to the model, while SKd values are generally below the model values by as much as 30%. Above 2 mg m−3, the measured values of all four ratios tend to be higher than the model and exhibit higher variation. Most of these measurements were from Monterey Bay, and the assumption of Case I waters may not be valid for all values.
Fig. 2 Plot of lidar ratios, SKd (left panel) and Sc (right panel), calculated from in situ measurements. Black lines and symbols represent the theory and measurements using the conventional definition, and red lines and symbols using the modified definition.
4. Lidar measurements
Data from the NOAA dual-polarization lidar from two different locations were used. The first case uses lidar data collected off the west coast of the United States, described in detail in [37], and chlorophyll concentration inferred from satellite measurements. The second uses lidar data collected in an inland fjord, described in detail in [38], and chlorophyll concentration inferred from in situ measurements. In both cases, the co- and cross-polarized lidar returns were added together, and the attenuation and volume backscatter of the combined signal were used to estimate the lidar ratio. The lidar footprint is relatively large for this system (D = 5 m at an altitude of 300 m), and measured attenuation coefficients were generally within about 10% of Kd estimated from satellite ocean color in the first case [39] and from in situ measurements in the second [38]. Thus, SKd is most representative for our lidar. The modified lidar ratio was estimated by subtracting 0.0452 m−1 from the measured attenuation and 1.94 × 10−4 m−1 sr −1 from the volume backscatter.
For the first case, lidar data were collected from the coast out to a distance of 350 km, and include data in and around the Columbia River plume, comprising both Case I and Case II water types. These data were chosen for the relatively low chlorophyll values in the offshore flight segments. For this study, a linear fit to the log of the signal between 5 and 10 m depths was used to estimate the lidar parameters. Chlorophyll concentrations were obtained from the daily MODIS-Aqua, Level 3 binned data at 4 km resolution, noting that uncertainties in satellite chlorophyll retrievals can be as high as 40% in Case I waters. The closest chlorophyll value was matched within 10 days of each lidar measurement. This was necessary to provide enough matches for comparison, but added scatter to the data.
For the second case, lidar data were collected in East Sound, Washington during a strong diatom bloom. All of the data from this study are considered Case II. Optical properties were measured in situ with a high-depth-resolution profiling package, including a Seabird SBE-25 CTD with a dissolved oxygen sensor, a WET Labs AC-9 and AC-S, a WET Labs ECO-VSF, a WET Labs ECO-BB3, and WET Labs chlorophyll and CDOM fluorometers. The parameter Kd was again estimated from Lee et al. [34], but with depth-averaged inherent optical properties over the 5 – 10 m comparable lidar range. The nearest five lidar profiles to each cast were averaged together for comparison with the chlorophyll concentration inferred from the AC-9 absorption spectrum. Most of the lidar profiles were within a few tens of m of the in situ data with a time difference of less than two hours.
Even with the observed scatter, data show good agreement with the model (Fig. 3), despite the relative uncertainties in the input parameters and the application of substantial data from Case II waters. The relative difference (defined as the model value minus the measured value, normalized by the model value) had an average value of 9.0% for the conventional definition and −8.9% for the modified definition. Although close, these values are not significantly different from zero, with p-values of 0.06 and 0.09, respectively. The standard deviation of the error values was 22% for the conventional values and 34% for the modified values. The difference in the standard deviations of the relative values is a result of the lower mean values when the modified definition is used. The standard deviation of the unnormalized differences is the same for both definitions.
Fig. 3 Plot of lidar ratio, SKd, estimated from lidar measurements. Black lines and symbols represent the theory and measurements using the conventional definition and red lines and symbols using the modified definition.
5. Discussion
Based on the in situ measurements, the model seems to work best in water with relatively low chlorophyll concentration. The likely reason is that the assumptions of the bio-optical models are best satisfied in oceanic waters with low chlorophyll concentration and even lower concentrations of suspended sediments, where all optically significant constituents approximately co-vary and are, to first order, driven by biological processes. Improved bio-optical models, of course, will lead to better estimates of lidar ratio.
In atmospheric lidar, the lidar ratio has been useful for inverting the signal to simultaneously obtain extinction and backscattering. The general approach is to find a range at which one of these parameters is known and proceed iteratively to other ranges. For example, one might use a point at high altitude where the scattering is dominated by molecules and can be accurately calculated from the density. For airborne oceanographic lidar, it will probably be more convenient to start from the sea surface, where the total integrated attenuation is negligible. For a series of measurements of the attenuated backscatter, γn, the values are found iteratively starting with sample number 0 at the surface as follows:
where Δz is the depth between samples and S is the appropriate lidar ratio.The measured lidar ratio has been used in the atmosphere to characterize cloud and aerosol types. This may prove to be more difficult in the ocean, where there are many more different types of scattering particles mixed together. These differences are contained within the variations of the measured values in this work, and more detailed models of lidar ratio will be required for this application.
It has also been suggested that the lidar ratio can be used in the radiometric calibration of atmospheric lidar, since the attenuation can be measured without calibration [40–42]. A similar technique should be possible for oceanographic lidar. The objective of calibration is to find the parameter A such that the product of A and the measured photocathode current I is an estimate of the attenuated backscatter. From the definitions of lidar ratio, we can find A from a measurement of I at depth z from
as long as the water properties are uniform from the surface to depth z. The most favorable case is when relatively clear water is used to calibrate a wide-beam lidar. For this case, we can use S ' = 105 sr and the error introduced by not knowing the exact value of C will be less than 2% as long as C is less than 1 mg m−3.6. Conclusions
A model of the lidar extinction-to-backscatter ratio was developed for Case 1 waters and limiting cases of broad- and narrow-beam oceanographic lidars. This model depends only on chlorophyll concentration and the well-known properties of pure sea water. A modified definition, in which the contributions from pure water are removed, was suggested to reduce the sensitivity to uncertainty in chlorophyll concentration. This reduction was greatest for the case of a wide lidar beam in waters with chlorophyll concentration below 1 mg m−3. Because chlorophyll concentration does not need to be known precisely, the lidar ratio will be useful for lidar inversions and for radiometric calibration.
The resulting model provides good agreement with both in situ and airborne lidar measurements. The differences are smaller than might be expected from the scatter in the data that were used to develop the bio-optical models. This scatter is a result of the wide variation in the size, shape, and internal structure of phytoplankton cells that can produce the same overall chlorophyll concentration. The scattering properties are particularly sensitive to these factors. Some of the differences are reduced in the lidar ratio, because scattering affects both the numerator and denominator of the ratio.
Acknowledgments
Open-ocean lidar data collection was partially supported by the National Ocean Partnership Program and the NOAA Office of Ocean Exploration. East Sound lidar data collection was partially supported by the Office of Naval Research Optics and Biology Program under Award Nos. N0001410IP20035 and N0001409IP20039. The authors acknowledge the MODIS Science team for the Science Algorithms, the Processing Team for producing MODIS data, and the GES DAAC MODIS Data Support Team for making MODIS data available to the user community. In situ optical data collection was supported by the NASA Ocean Biology and Biogeochemistry program under award nos. NNX06AH32G and NNX06AH86G.
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