1. Introduction

The ocean’s effect on weather and climate are governed largely by processes occurring in the first few tens of meters of water bordering the ocean surface. For example, water warmed at the surface on a sunny afternoon may remain available to warm the atmosphere that evening, or it may be mixed deeper into the ocean not to emerge for many years depending on near-surface mixing processes.

The oceanic mixed layer (ML) is the upper layer of the ocean, in which properties, such as salinity, temperature, and particulate, have been homogenized by turbulence processes from the surface to a given depth [2]. This depth is known as the mixed layer depth (MLD). Since the oceanic mixed layer coincides with the upper boundary layer of the ocean, it is key in driving the interactions between the atmosphere and the ocean interior. It determines the transfer of momentum from wind and convective mixing and also controls the oceanic storage of heat and other seawater components such as carbon dioxide. As it receives sunlight, it is the site of oceanic photosynthesis. The depth and rate of mixing within the oceanic mixed layer are crucial to understanding and quantifying climate dynamics on scales ranging from minutes to interannual.

Given the global importance of the mixed layer, there is a need for a technique that allows for the rapid, widespread determination of the MLD. Such a measurement could, in principle, be obtained from an airborne or spaceborne platform. A review of active and passive optical sensing techniques of MLD measurements was presented in [3]. In general, the active optical approach has an advantage over the passive approach by permitting vertical sampling [3] of the quantity of interest by range gating the returned signal. Lidar is a technique for providing such a range gated signal.

As the marine Lidar technology matures, it ensues an increase in ocean penetration depth - with the Lidar becoming a research tool for investigations of the oceanic mixed layer. Our intent is that the results presented here will provide motivation for future oceanic Lidar systems and technology development for new Lidar ML measurements.

In our approach here we have bypassed the following issues: what optical parameters Lidar can retrieve [4] and how to determine the mixed layer depth from Lidar soundings because as a number of approaches have been postulated in [3, 5]. Rather, we address more fundamental questions associated with oceanic Lidar soundings, namely what percentage of the global mixed layer can be accessed using existing marine Lidar systems?

The ability of marine Lidar to penetrate the ML depends on the local optical conditions and the local depth of the mixed layer. The analysis of seasonal variability of selected Lidar parameters have been carried out in [6, 7]. In this work, we use multiyear MLD data obtained from oceanic in situ observation gathered by the global network of Argo floats [8]. To maximize the MLD coverage, we have focused our investigations on global yearly averaged MLD distribution and their relation to the local long term average Lidar penetration depth.

We carry out our analysis for a period from January 2003 to January 2013. The selected time interval integrates over a number of long term climatic events which modulate the global mixed layer depth - NOAA Oceanic Nin̂o Index (ONI, http://www.cpc.noaa.gov) thus representing truly long term average. We start with a description of the oceanic mixed layer and model of the oceanographic Lidar and then address the Lidar penetration depth. We then address the ability of modeled Lidar to penetrate global mixed layer. We conclude with an analysis of the Lidar penetration depth relative to the local mixed layer depth in the context of a regional measurement using the Gulf of Mexico as a case study.

2. Oceanic mixed layer

The global and climatic importance of the oceanic mixed layer is due to a number of processes that the ML has impact on. The oceanic mixed layer is the uppermost layer of the ocean that mediates all air-sea fluxes [9]. The specific heat of ocean water is much larger than that of air such that the top 2.5m of the ocean holds as much heat as the entire atmosphere above it. The heat required to change a mixed layer of 25m by 1°C is sufficient to raise the temperature of the atmosphere by 10°C. The depth of the mixed layer thus plays a crucial role in determining the temperature range in oceanic and coastal regions and in Earth’s climate.

The oceanic mixed layer also plays an important role in oceanic biological processes. As the mixed layer receives sunlight, it becomes the site of oceanic photosynthesis. If the mixed layer is very deep, the microscopic marine plants known as phytoplankton receive insufficient light to balance respiration with photosynthesis. The shallowing of the mixed layer in the springtime in the North Atlantic is responsible for triggering a spring bloom [10].

Operationally, the depth of the mixed layer is usually determined by hydrography, i.e. by measuring water properties. Two criteria often used to determine the mixed layer depth are changes in temperature or σ_t (density) relative to a reference value (usually the surface measurement). Neither criterion implies that active mixing is occurring at that time. Rather, the operational measurement of mixed layer depth provides a measure of the depth to which mixing has occurred over some period of time. In general the vertical extent of the mixed layer depends on the surface forcing which, in turn, varies with time. It has been observed [11] that at a fixed location, the daytime MLD can change by a factor of 3. In addition the seasonal optical variability [6, 7] in context of Lidar penetration depth can vary by factor of 2. To address a fully resolved combined MLD and optical variability in context of Lidar penetration depth would require local MLD and optical measurements on scales 12 hour which is currently impractical. In order to get handle on ratio of the Lidar penetration depth to the local mixed layer depth - the fractional Lidar penetration depth (FLPD) we have chosen to relate their long term temporal averages recognizing that the daily variability of the FPLD can be same order as FPLD.

In our work here we have employed the MLD measurements obtained by global array of Argo floats. Argo is an array of 3,000 free-drifting profiling floats that measure the temperature and salinity in the upper 2000 m of the ocean. The target of the ARGO program is data coverage of 1 float per 3°×3° grid cell and month over the global ocean. Argo deployments began in 2000 and by November 2007 the array was 100% complete. Currently (11 Aug 2013), 3646 floats are available. The MLD here and following [12] is defined as the depth density increases from 10m to the value equivalent to the temperature drop of 0.2°C [12].

Figure 1 visualizes the color coded number of Argo floats visiting each of 3°×3° grid cell within analyzed time span: January 2003 to January 2013. The density is color coded i.e. red denotes more than 12 floats within 3°×3° and ten year period and the global MLD used in this work is given on 3°×3° grid. For the study here we have only retained the MLD grid points with at least one drifter per month, per cell in ten year period. Due to limited Argo floats, density measurements in some parts of the ocean are not represented well - note the absence MLD data in the Arctic area and few coastal regions. For comparison with satellite data we have re-grided MLD on 9x9 km grid.

 

Fig. 1 The truncated Argo float distribution such that if there were more than 12 floats within 3°×3° and within ten year period then the color remains unchanged.

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The yearly averaged global distribution of the mixed layer depth is presented in Fig. 2 and the corresponding global mean ML depth histogram is in Fig. 7(b).

 

Fig. 2 The global mixed layer distribution based on Argo floats data.

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Globally the mean ML depth distribution is centered on 50 m depth with some notable exceptions such as the Southern Ocean and North Atlantic, which has area with a corresponding ML depth exceeding 200 m. The deepening of the mixed layer in the Southern Ocean is a result of the persistent strong wind [13] and the ML increase in the North Atlantic is the result of the surface cooling associated with wintertime sinking of cold water and rising of warm water.

3. Model of oceanographic Lidar

The oceanic penetration depth by a Lidar system is determined by optical properties of the water column and by the design parameters of the Lidar system.

Our selected Lidar system model is based on the documented specifications for a marine nadir pointing Lidar as introduced in [1]. We have selected that particular Lidar because its experimental performance is well documented [1] for a number of profiles. The Lidar system in that experiment was mounted 10 m above the sea surface, thus minimizing atmospheric issues. This Lidar system is extensively documented throughout literature: [14, 15, 16].

The signal received by the marine Lidar (excluding air side and air/sea interface) is a combination of forward- and back-scattered photons detected by the Lidar receiver. Statistically each Lidar detected photons has undergone at least one backscattering event (backscattered most likely by a particulate) followed/lead by a number of forward scatting events. This situation is schematically presented in Fig. 3. A work by [17]- a Monte Carlo simulation study of a marine Lidar, concluded that the Lidar attenuation coefficient - α (related to fast how Lidar signal attenuates with depth - see Eq. (2), is delimited by the beam attenuation coefficient- c = a + b and the diffuse attenuation coefficient - K_d i.e. K_d ≤ α ≤ c = a + b, here a is the volume absorption coefficient; b is the volume total scattering coefficient- for discussion see [18]. The coefficient K_d is approximately given by K_d ≃ a + b_b [18, 19] where b_b is the volume back-scattering coefficient into the back hemisphere. The last quantity is further partitioned into pure water backscattering - b_bwater and particulate backscattering b_bp i.e.: b_b = b_bwater + b_bp - see [18]. Physically the parameter c represents the extinction coefficient for an infinitely narrow, well collimated light beam. In marine optics, this parameter is also referred to as beamC and throughout this paper we will use the c or beamC interchangeably. The beam attenuation coefficient can be partitioned as the sum of attenuation due to particles c_p, water c_w, and colored dissolved materials (i.e.: detrital particulates and dissolved materials- see [20]) c_cdom: c = c_p + c_w + c_cdom [21]. Since the c_cdom is dominated by its attenuation a_cdom thus c = c_p + c_w + a_cdom.

 

Fig. 3 (a) The twice-scattered signal entering the Lidar detector. (b) The single-scattered signal entering the Lidar detector. R - the receiver footprint.

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Detailed Lidar analysis [17, 4] has established that the Lidar received signal composition is fundamentally related to the size of the receiver footprint at the sea surface - radius R - Fig. 3. The Lidar receiver footprint R, determines the statistics of the Lidar collected photons - i.e. increase of the Lidar footprint permits for the multiple scattered photons to enter the detector. This results in Lidar attenuation coefficient depending on R. The Lidar attenuation coefficient α in the limit of vanishing R → 0, becomes α = c = a + b [17, 4]. As the receiver footprint increases, at R ≥ 1/c the Lidar receiver accept some multiply scattered photons -Fig. 3. The transition between single scattering and multiple scattering is approximately attained [17, 4] when receiver footprint radius R is R = 1/c. In this limit, the Lidar attenuation coefficient approaches K_d. The transition between two scattering regimes has been experimentally validated in the Lidar measurements of [22, 23]. Throughout the paper we will use term single scattering when considering the Lidar system with a narrow receiver footprint radius - when R ≪ 1/c and the quasi-single scattering when referring to a system with a wider receiver footprint of the radius - when R ≃ 1/c.

From operational point of view, the interpretation of the single scattering Lidar signal is relatively straightforward, as the Lidar received signal depends only on the strength of the single backscattering event and the lidar attenuation equals to the beamC. For quasi-single scattering Lidar, the increase of the detector footprint lets more light into the detector, increasing Lidar sensitivity. However, the presence of multiply scattered photons makes interpretation of the Lidar signal more challenging [22, 4] - (in work of [22] the single scattered component of the Lidar signal is called ’coherent component’).

All considered optical quantities are wavelength dependent and throughout this paper we will assume that all optical quantities refer to their values at a wavelength of λ = 530nm unless otherwise stated. This is the most common wavelength for a marine Lidar, due to the available laser options and is the wavelength of the Lidar under consideration [1]. In our analysis, we employ two traditional marine Lidar models [17, 24, 4] for the Lidar with either the narrow or the receiver footprint. For both systems and for the nadir-pointing Lidar, just above the air-sea interface, the depth dependent S(z) received power is:

For the well mixed upper oceanic layer - the Eq. (1) can be further simplified to:

4. Determination of K_d, c and β (π) from satellite observations

In order to determine the global Lidar penetration depths for the analyzed system we need to assemble temporally averaged maps of the relevant parameters entering Lidar Eq. (2) namely: K_d, c_p, a_cdom (since c = c_p + c_w + a_cdom) and β(π), averaged for the analyzed period: January 2003 – January 2013.

For this work, the global gridded parameters: K_d, c and β(π) were derived from the NASA MODIS-Aqua satellite measurements. The MODIS-Aqua satellite, launched in May 2002, carries a medium-resolution, multi-spectral, cross-track scanning radiometer. The MODIS-Aqua radiometer measures Earth reflected Sun light in 21 bands within 0.4–3.0 μm wavelength with the signal-to-noise ratios ranging from 900 to 1300 at 1 km resolution and with absolute irradiance accuracy of 5%. The MODIS-Aqua space based radiometer visits the same spot every 1 to 2 days. There is a number of approaches to deduce approximate values of K_d, c and β(π) from space based optical radiometers measurements. They all have their strengths and weaknesses and all rely on semiempirical relationships supported by a number of experimental insitu validations with varying accuracy - their overview can be found in [25, 19, 18, 26]. To derive spatial distribution of Lidar parameters, we have used three NASA’s Ocean Color Level-3 9x9km gridded products: b_bp(443nm) - particulate backscattering coefficient, K_d(490nm) - diffuse attenuation coefficient and a_cdom(443nm) - total absorption by detrital particulates and dissolved materials [20]. MODIS-Aqua products were provided by the NASA Giovanni web portal (disc.sci.gsfc.nasa.gov/giovanni) and were results of Quasi-Analytical Algorithm (QAA) [25] (b_bp(443nm), K_d(490nm)) or GSM (a_cdom) [20] applied to MODIS-Aqua radiometer dataset. The b_bp(443nm) and K_d(490nm) were averaged from January 2003 to January 2013 while a_cdom from January 2003 to May 2011. The MODIS-Aqua parameters have varying relative accuracy: K_d(490nm) is nominally retrieved with ≃ 0.1 uncertainty (comparison with in situ NOMAD data set) [27], b_bp(443nm) ≃ 0.2 uncertainty (comparison with synthetic data set) and ≃ 0.2 uncertainty (comparison with ’quasi-real’ data [20]).

We have converted K_d(490nm) to K_d(530nm) following relationship [28, 18] and it yields: K_d(530nm) = 0.6924 · K_d(490nm)+ 0.0371, where both quantities have units 1/m with relative conversion accuracy of around 0.01 at 530nm [28].

Oceanic in situ measurements [29] of b_bp and c_p have demonstrated their robust covariance. Authors [29] have collected around 10⁵, b_bp(526nm) and c_p(526nm) samples at a number of sites: Equatorial Pacific, North Atlantic, Subarctic Northeast and Equatorial Pacific, and Mediterranean Sea and found the following relationship in their data: $b_{bp}=0.01\cdot c_p^{0.9768}$ with r² = .92. In addition they showed that the covariance b_bp with c_p is much stronger over observed range than that b_bp with chlorophyll concentration. Based on their observations we have then converted MODIS Aqua b_bp(443nm) product to beamC(532nm) in three steps. First, the MODIS Aqua product b_bp(443nm) was converted to b_bp(532nm) assuming a spectral slope of −1.0337 for particulate backscattering [20] i.e. b_bp(532nm) = (532/443)^1.0337 · b_bp(443nm). The second step is based on observed relationship between b_bp and c_p [29] yielding: $c_p+c_w=111.6b_{bp}^{1.0238}+0.052$. In the final step we have added converted a_cdom(443nm) to a_cdom(532nm) = a_cdom(443nm) · exp(−0.015(532 − 443) = 0.263 · a_cdom(443nm). The combined sum c = c_p + c_w + a_cdom yielded the global mean distribution of beamC(532nm) with an estimated combined relative accuracy of around 0.4 over the dataset range (determined to the first order by uncertainty in the satellite retrievals of b_bp and a_cdom).

The most elusive and the least frequently measured parameter in Eq. (2) is the β(π)- the volume scattering function of the water and its constituents evaluated at the backscattering angle. Following the studies of [30, 31] the backscattering coefficient of particles is commonly estimated from measurement of scattering at a single angle in the backward hemisphere. The data of [31] based on in situ VSM measurements and Petzold [32] averaged VSF suggest relationship between β(π) and b_p as β(π) = A · b_p where coefficient A ≃ 0.18 for Petzold data and A ≃ 0.3 in case [31] shelf water VMS measurements. More recent - analysis by authors [33] of several million particulate volume scattering functions from different field sites around the worlds oceans and coastlines revealed that the shape of the VSF in the backward direction is remarkably consistent, with β(π) ≃ 0.15b_bp (extrapolated to backscattered direction). We acknowledge that this relationship is based on relatively new body of measurements and the variability of A in context of Lidar measurements needs to be further investigated. For this work calculations we will use β(π) ≃ 0.15b_bp relationship with the relative uncertainty of 0.1 - determined by dataset standard deviation [33] and extrapolated to backscattered direction. The above approach led to beamC, K_d and β(π) mean maps from gridded MODIS Aqua radiometer products.

Figure 4 presents the global distribution of the Lidar receiver footprint equal to R = 1/c, assuring that the Lidar operates in quasi single scattering mode. The corresponding histogram is on the Figure 7(a). Notice that to be within the single scattering criterion globally i.e. R·c ≪ 1 the Lidar beam width on the sea surface should be at most 1 m wide.

 

Fig. 4 The Lidar receiver footprint - R in the quasi-single scattering regime.

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To generalize the Lidar system response to different geographical locations we have rearranged Eq. (2) as:

The relative errors associated with estimating z_max from Eq. (4) (see discussion in the Appendix) are around 0.25 for the narrow footprint system and around 0.15 for the wide footprint system.

Notice that the Appendix demonstrates that for the largest values of z_max the uncertainty in β(π) contribution to z_max estimates is at least 3 to 8 times smaller than uncertainty in K_d or beamC. This is consistent with our simplified estimates of β(π) as its variability is relatively less significant when calculating Lidar z_max. Using the above equation we can determine the percentage of global ML available to Lidar soundings.

5. Results

Based on Eq. (3) we have mapped the global Lidar penetration depth z_max for the Lidar operating in the single scattering regime Fig. 5 (top) and for the Lidar in quasi-single scattering regime - Fig. 5 (bottom). Corresponding z_max histograms are presented in the Fig. 7. The modeled Lidar with narrow footprint receiver, on average, penetrates to 20m below the sea surface while the wide receiver counterpart penetrates to over 50m on average. The global pattern of oceanic Lidar penetration depth differs between the Lidar system - this is also mirrored in the different histograms - Fig. 7 (c) and (d). The differences can be attributed to different spatial distribution of beamC and K_d in the ocean.

 

Fig. 5 Top: Lidar penetration depth for the narrow receiver footprint- single scattering regime; Bottom: Lidar penetration depth for the wide receiver footprint- quasi-single scattering regime.

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To compare the Lidar penetration depth with the vertical extent of the oceanic ML depth, we have calculated the ratio of the Lidar penetration depth to the local mixed layer depth -the fractional Lidar penetration depth defined as FLPD = z_max/MLD. The FLPD for narrow and wide footprint Lidars are presented in Fig. 6 and the histograms in Fig. 7. The FLPD is plotted such that the FLPD values larger then 1.2 are set to 1.2 to better illustrate the global ML coverage. The FPLD histogram data in Fig. 7 - were obtained at within Argo measured locations.

 

Fig. 6 Fraction of Lidar penetration depth (FLPD). The FPLD ratio >1 indicates that Lidar penetrates through the entire local mixed layer. Top: single scattering Lidar; Bottom: quasi-single scattering Lidar.

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In general the single scattering Lidar rarely penetrates the full extent of ML with some interesting exceptions. Within the equatorial belt, the single scattering Lidar allows for a nearly full ML characterization. This is very important from an oceanographic and climatic prospective as this is the area directly influencing global and US climate and this is area responsible for hurricane activities. In general, the modeled narrow footprint Lidar system can access 0.4MLD in half of the global ocean - Fig. 7.

For the quasi-single scattering Lidar, the global FPLD is more extensive with the Lidar being capable of sounding the whole MLD in half of the global ocean Fig. 6 and Fig. 7. The Lidar coverage the climatically important Southern Ocean is around 0.2 for the narrow receiver Lidar and with FPLD up to 0.6 for the wide receiver system Fig. 6. In general we conclude based on the histograms in Fig. 7 - and in the absence of clouds, that current Lidar systems can penetrate below the mixed layer for around 50 % of the global ocean area for a wide receiver footprint system. The single scattering Lidar in order to become the global ML sensing tool would require increase in ML penetration depth so as at least to least match quasi-single scattering FPLD. This in turn would require to improve sensitivity of the single scattering Lidar when compared to design of [1] by for example increasing receiving telescope diameter.

6. Northern Gulf of Mexico

The use of marine Lidar to sample the coastal environment can be illustrated using the Northern Gulf of Mexico (GOM) as an example. In general, the GOM is an area of intensive human impact and the ML parameters are important to decision makers and ecosystem managers [34]. The Figure 8(a) - illustrates the estimated Lidar receiver footprint width in the GOM when operating Lidar in quasi-single scattering regime. We observe that for most of the transects perpendicular to the coast, R changes from around 1 m to around 6 m over a distance of 100km - an important observation when carrying out and interpreting results from such transects with a marine Lidar system of fixed detector footprint.

 

Fig. 7 Histograms for the maps in Fig. 2, Fig. 4, Fig. 5 and the Fig. 6. A- Histogram of the Lidar receiver radius R = 1/c. B-Mixed layer depth. C-Single scattering Lidar penetration depth. D-quasi-single scattering Lidar penetration depth. E- FPLD - Single scattering Lidar. F- FPLD - D-quasi-single scattering Lidar. The red line denotes FPLD of 1.

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Fig. 8 Results for the Northern GOM. A- the quasi-single scattering Lidar footprint size R. B- ML depth observed by Argo floats. C- FPLD distribution for the narrow footprint Lidar system. D- FPLD distribution for the wide footprint Lidar system.

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The Argo observed multiyear averaged mixed layer depth in northern GOM is around 25–35 m - Fig. 8(b). From in situ measurements we know that the MLD in the Northern GOM vary [35] varies between 20 m in summer to around 70m in winter. Clearly temporal variation of local MLD and relevant optical parameters have to be analysed on regional scale for better assessing marine Lidar performance there [6, 7].

The MLD and optical variability in that part of the GOM results in an average FPLD of 0.6 –0.8 for the single scattering system (Fig. 8(c)) increasing to over FPLD 1 for the quasi-single scattering system (Fig. 8(d)). This makes even the single scattering Lidar implementation [1] useful for GOM monitoring applications. This is important as the GOM is the site of numerous oil rigs with a potential for an oil spill as illustrated by the recent 2010 DWH spill. The mixed layer dynamics is ultimately responsible for the transport of contaminants resulting from an oil spill. The ability to monitor remotely and rapidly optical parameters of the ML in the GOM region via Lidar sounding is of importance to coastal managers as it allows for estimates of contaminant or its proxy transport within ML. Notice in the Fig. 8(c), the rapidly changing FPLD in vicinity of the Mississippi delta as the result of variability of water column optical properties. This variability results in FLPD change (narrow footprint Lidar) from 0.2 to 0.8 over tens of kilometers.

7. Conclusions

In view of the need for an improved understanding of the mixing intensity and depth of the oceanic mixed layer in context of anthropogenic climate change or the weather prediction, accurate and more extensive measurements are crucial in both a global and local context. In principle, a marine Lidar mounted on a remote platform provides the ability to rapidly cover large swaths of the ocean, providing a tool to research and monitor at least 50% of the cloud free oceanic mixed layer, thus greatly improving our understanding of upper ocean processes. The single scattering marine Lidar, in order to become global ML sounding tool, needs to become more sensitive when compared to its implementation as described in [1].

8. Appendix: uncertainty analysis of the z_max estimates

To estimate the uncertainty of the Lidar penetration depth z_max obtained from Eq. (3), we have calculated the uncertainty contribution to z_max from measured parameters uncertainties. For the analysis we assume that in the first approximation, uncertainty of measured quantities are small (when compared with other terms) and uncorrelated. The relationship defining z_max - Eq. (4) can be rewritten as:

$$x=W(x)\cdot \text{exp}\left[W(x)\right]$$

(5)$$W=\alpha \cdot z_{\mathit{max}}$$

(6)$$x=A\cdot \alpha \cdot b_{bp}^{1/2}$$

where A = 2.6·10⁴m^3/2, W(x) is the principal branch of the Lambert function, b_bp is particulate backscattering and α the Lidar attenuation coefficient, K_d or c, all derived from NASA MODIS Aqua products.

Following the standard linearized-approximation approach, we seek estimate of the relative uncertainty in estimated Lidar penetration depth - $\frac{d{z}_{\mathit{max}}}{z_{\mathit{max}}}$. Based on the Lambert W function property [36]:

(7)$$\frac{dW(x)}{W(x)}=\frac{1}{1+W(x)}\frac{dx}{x}$$

We will evaluate left- and right-hand of the Eq. (7) independently. The dx/x can be found from Eq. (6) as:

(8)$$\frac{dx}{x}=\frac{1}{x}\left[\frac{\partial x}{\partial \alpha }d\alpha +\frac{\partial x}{\partial b_{bp}}d{b}_{bp}\right]=\frac{d\alpha }{\alpha }+\frac{1}{2}\frac{b_{bp}}{b_{bp}}$$

Following the definition of W as W = αz_max we then have:

(9)$$\frac{dW}{W}=\frac{d\alpha }{\alpha }+\frac{d(z_{\mathit{max}})}{z_{\mathit{max}}}.$$

For uncorrelated variables b_bp and α and after substitution of relevant terms in Eq. (9) from Eq. (7) and Eq. (5) we then get:

(10)$$\frac{d(z_{\mathit{max}})}{z_{\mathit{max}}}\le \frac{W}{1+W}\left|\frac{d\alpha }{\alpha }\right|+\frac{1}{2}\frac{1}{1+W}\left|\frac{d{b}_{bp}}{b_{bp}}\right|.$$

where: W is evaluated at z = z_max, thus W = αz_max - Lidar penetration depth in terms of optical depth. The global histogram of W, weighting functions $\frac{W}{1+W}$ and $\frac{1}{2}\frac{1}{1+W}$ used in equation Eq. (10) are presented in Fig. 9.

 

Fig. 9 Top: Histogram of Lidar maximum penetration depth distribution in terms W = αz_max optical depth. Bottom plot of $\frac{W}{1+W}$ and $\frac{1}{2}\frac{1}{1+W}$ of the Eq. (10)

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Note that for shallow Lidar penetration, the uncertainty contribution to Lidar z_max estimate is dominated by $\left|\frac{d{b}_{bp}}{b_{bp}}\right|$ - the backscattering relative uncertainty. The situation becomes different at depth, when the relative uncertainty in Lidar attenuation coefficient - $\left|\frac{d\alpha }{\alpha }\right|$ begins to dominate estimates of the Lidar penetration depth.

Thus the combined relative error in our estimates of z_max when solving Eq. (4) and at around W = 4 is ≃ 0.25 for the narrow footprint system and around 0.15 for the wide footprint system. Note that here in general the largest uncertainty contribution (by a factor 3 to 8) comes from uncertainties associated with the Lidar attenuation coefficient estimates.

Acknowledgments

Analyses and visualizations used in this paper were produced with the Giovanni online data system, developed and maintained by the NASA Goddard Earth Sciences (GES) Data and Information Services Center (DISC) [37]. This research was made possible in part by a grant from BP/The Gulf of Mexico Research Initiative, and in part by JPL/NASA funds.

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