1. Introduction

Direct-detection lidar is a popular technique for profiling the atmosphere (in this paper it will be simply referred to as “lidar”), exploiting the signal returns from atmospheric layers at varying distance in response to the emission of a light pulse ([1–4] and many others). It offers great advantages such as a high range (distance) resolution, a high temporal resolution, and the ability to monitor the atmosphere continuously for a sustained duration of the order of hours, days and even years. Several types of lidar exist, with various spectral channels designed to target different atmospheric properties, and thus targeting water vapour, trace gases, aerosols, clouds, the temperature profile, etc. All these systems have a number of components in common, and in particular they have an emitter system, which radiates light in a narrow beam into the atmosphere, and a receiver system designed to collect a small amount of that light through a narrow field-of-view, after it has been scattered by the targeted atmospheric layer. For the great majority of lidars (monostatic lidars), the emitter and the receiver are placed in the same location: this set-up allows to simplify considerably the apparatus and its operations.

One of the important requirements for monostatic lidars is that a very fine alignment is required between the emitter and the receiver. An issue, however, is that, even for a perfectly aligned system, there is a near-field region where the emitted beam and the receiver field-of-view don’t overlap or overlap partially. Incomplete overlap arises due to a combination of geometric optics considerations and diffractive effects, and is usually expressed through the overlap function $q(r )$, where r is the range. With a good optical alignment, $q(r )= 0$ in the first layer very close to the lidar, and $q(r )= 1$ for ranges larger than a quantity, ${r_{ovl}}$, called the beginning of the full-overlap range. In-between those two regimes, there exists an intermediate zone where $q(r )$ takes values between 0 and 1 (and occasionally also larger than 1). Whereas no information on the atmosphere can be inferred in the first layer with $q(r )= 0$, it is generally considered good practice to limit use of observations to the “safe” region where $r > {r_{ovl}}$ and $q(r )= 1$.

In order to use observations and determine the optical properties within the intermediate layer where $r < {r_{ovl}}$ and $q(r )> 0$, one must characterise $q(r )$ and apply appropriate data corrections to account for it. This is particularly important for lidar systems that have a large full-overlap range, and for those which are placed very close to the layers that need to be studied (such as for instance systems targeting the boundary-layer which are placed at the Earth’s surface). For example, the Single Calculus Chain within EARLINET community has the option to apply this correction [5]. There are several methods to obtain the overlap function for a ground-based lidar. They can be theoretical computations (e.g., [6–9]) or based on empirical methods. The theoretical models have the advantage of being able to give a constraint on the overlap function based on the instrument design parameters, but on the other hand they require precise measurements of the beam divergence (assumed to have a Gaussian shape), telescope field of view and the tilting angle between the emitter and the receiver. Potential deviations and misalignments in time, if not detected and measured, will not allow computation of the actual overlap function, but rather indicate an ideal one to aspire to with a given lidar system. While the computations may be easier to perform than an experimental determination (although complex), the stability of the system may be questionable while new measurements of the system parameters may not be performed too often. This is why we advocate for a method to determine the overlap function empirically. The empirical methods are based on atmospheric homogeneity [10–12] or polynomial approximation of the logarithm of the range corrected signal [13]. Note that in addition to homogeneity, Sassano et al. [10] considered the transmission term equal with unity (i.e., very clear days), which may be hard to fulfil on regularly basis for ground-based measurements. Such method can be applied after a rainy period. Based on [10], Tomine et al. [11] adapted the method for homogeneous mist. The slope method by [12] is kind of a single angle method where the logarithm of the range corrected signal is analysed versus range, the optical depth being simply the product from the extinction coefficient (considered constant) and the range. In their overlap calculation, the backscatter and extinction are considered constant over the horizontal range and thus, the overlap is retrieved close to the lidar. Some methods make use of one elastic and one Raman channel [14–16]. This method assumes a specific lidar ratio of the atmospheric aerosols, which could induce errors in the retrieved overlap when not accurate. Kovalev [17] discusses the distortions in the lidar signal which influence the overlap function. Regarding the Lufft ceilometers, the manufacturer provides an overlap correction function for each system (determined in the factory, using a reference system) where the complete overlap can be around 1500 m or even 2500 m. For MPL lidars, and similarly for Lufft ceilometers, the overlap function is considered constant in time as the systems do not perform regular alignments. Hervo et al. [18] proposed a correction for the overlap function of these ceilometers taking into account the temperature dependence.

Adam et al. [19] presented a novel method based on multi-angle measurements performed by a ground-based scanning lidar. In this paper we demonstrate a novel technique for the characterisation of the overlap function of an airborne lidar, based on the same ideas. It is important to bear in mind, however, that the technique is general and can be applied to any direct-detection lidar system for atmospheric sensing, provided that a scanning capability is available and that the characterisation can be performed during a “quiet” atmospheric period, when the atmospheric profile can be assumed horizontally homogeneous and constant. The method that we propose can be separately applied to single channels in a multi-channel lidar system, regardless of the specific application, and can therefore be used for ceilometers, elastic lidars, Raman lidars, differential absorption lidars, high spectral resolution lidars, etc. (if they have a scanning capability). The present technique was applied for airborne multi-angle lidar measurements.

The FAAM BAe-146 research aircraft, based in the United Kingdom, serves scientists working in universities funded by the Natural Environment Research Council and the Met Office, the latter having been responsible for operating and maintaining the on-board Leosphere elastic backscatter lidar (http://www.faam.ac.uk/). The FAAM Bae-146 is equipped with a series of instruments performing in-situ measurements of meteorological parameters, trace gases, aerosol properties (absorption, scattering, cloud condensation nuclei counter, concentration), cloud physics (particle state and shape, bulk ice and water content). Moreover, a series of instruments offer a remote sensing capability in the solar and infrared parts of the spectrum [20–22]. Active remote sensing of aerosol properties is achieved using a Leosphere ALS450 UV lidar (with depolarization capability), mounted in a nadir-looking geometry [23–26]. This instrument achieves full overlap at a nominal range of 300 m. However, through the years it has quickly become apparent that the transition to full overlap is more spread out, and empirical corrections based on fits to a molecular atmosphere have been applied on a campaign-by-campaign basis.

The current study presents a methodology to evaluate the overlap function of a lidar on-board aircraft, using multi-angle measurements, which is adapted from the method developed by Adam et al. [19]. Preliminary results of the methodology were shown during the International Laser Radar Conference in 2015 [27].

Section 2 presents the methodology; section 3 presents the application of the methodology for simulated signals while section 4 shows the results based on two different sets of measurements. In section 5 we discuss the results and in section 6 we present our conclusions.

2. Methodology

The multi-angle method is based on measurements taken at several elevation or nadir angles, the main assumption being that the atmosphere can be considered constant and horizontally homogeneous during the whole observation time. The assumption of homogeneity is likely to be fulfilled for an airborne lidar, whose cruise altitude is thousands of meters above ground level (thus, often quite close to a molecular atmosphere). In “clean” conditions, at this altitude we expect more homogeneity as compared with the boundary layer, because the majority of the variations themselves are due to the inhomogeneity of the aerosol field. The schematic of the flying procedure is shown in Appendix A. The representation of the overlap regions is illustrated (Fig. 12) along with the multi-angle measurements (Fig. 13).

For several slant measurements (different elevation or off-nadir angles) of the same altitude layer, the logarithm of the range corrected signal (Z) is plotted versus $x = 1/\textrm{cos}(\varphi )$, where $\varphi $ is the off-nadir angle for downward-looking lidar or the zenith angle ($\varphi = 90^\circ $-elevation angle) for an upward-looking lidar. The slope of the linear regression between these two quantities provides the total optical depth (OD or $\tau $) while the intercept provides the logarithm of the total backscatter coefficient multiplied with the lidar constant.

The lidar equation can be written as a function of range as follows:

For an aircraft flying at altitude ${h_0}$ above sea level and a downward-facing lidar, we define for the layer at altitude h, the vertical distance $\Delta h = {h_0} - h = r\ast cos({{\varphi_j}} )$ from the lidar for off-nadir angle ${\varphi _j}$:

Similar to [19], we define the variables ${x_j}$ and ${y_j}({\Delta h} )$ as:

The lidar Eq. (2) becomes:

In the region with complete overlap (${q_j} = 1$) which will be taken into account in our analysis, the lidar equation becomes:

For each $\Delta h$ (or altitude h), a least squares analysis is applied when we plot ${y_j}({\Delta h} )$ versus ${x_j}$ using lidar data measured along different off-nadir angles and for $r > {r_{ovl}}$. Thus, we obtain the vertical optical depth $\tau ({\Delta h} )$ and the intercept ${A^\ast }({\Delta h} )$. Refer to example (Fig. 4) in Adam et al. [19]. Adam et al. [19] investigated the systematic errors which can arise if the background subtraction is not performed accurately or if the data from incomplete overlap is used (see their Figs. 1 and 2). Thus, when data from incomplete overlap is used in the overlap retrieval, the overlap will be distorted both in the near range (around the beginning of the complete overlap) and few km after (see [19], Fig. 8). When the background subtraction is not accurately estimated, the overlap function is distorted just in the far field (see [19], Figs. 10 and 11). As we focus in the incomplete overlap region, the non-accurate background subtraction basically does not affect overlap function over the incomplete overlap region.

One important criterium in this data analysis refers to the setup of the correct altitude range where we perform the regression between ${y_j}$ and ${x_j}$. The minimum range ${r_{min}}$ from the lidar should be in the region of complete overlap which is not known a priori while the maximum distance ${r_{max}}$ from the lidar should be defined through a threshold yielding a good SNR (signal to noise ratio). The latest research within European Aerosol Research Lidar Network (EARLINET) showed that the application of the telecover test provides us the beginning of the complete overlap [28]. Taking into account that the Z signals for the measurements taken from the ground differ from those taken from an airborne borne lidar, some of the original criteria cannot be applied. Thus, in the current study, ${r_{min}}$ was set up to 600 m and 500 m for the two experiments (where $\Delta {h_{min}} = {r_{min}}\ast \textrm{cos}({\varphi _j}$)) as it can be seen later in Section 4. Given a good SNR from the Leosphere signals, we only applied a restriction on ${r_{max}}$ by restricting the analysis to altitudes higher than 2000 m above sea level. The numerical values for ${r_{min}}$ were determined based on trial and error until no systematic distortion was observed [19,29]. The latter is because we limit anyway to the first 2 - 3 thousand meters where the complete overlap is and the SNR is high. Note that that for ground-based lidar ${r_{min}}$ can be quantified, based on the position of the maximum value in Z [19].

Once the optical depth and the intercept are determined, we can construct the lidar signal without the overlap effect over the full observation range, by applying the above equations:

We will compute the lidar overlap function as the average of the individual overlaps ${q_j}({\Delta h} )$ obtained for different bank angles ${\varphi _j}$ while its dependence is converted back from vertical distance $\Delta h$ to the range r (line of sight). The scheme is shown in Fig. 1.

 

Fig. 1. The workflow of the lidar overlap calculation.

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In order to have more valid measurements of $q(r )$ towards the lidar, we need more measurements at large off-nadir angles for airborne-based lidars. In this case, the latter condition can be fulfilled only up to the maximum off-nadir angle of 60° permitted operationally.

3. Simulations

The experiments were planned using synthetic data, simulating measurements of the molecular atmosphere. Sensitivity studies were performed to find out a suitable set of bank angles capable of yielding to a good estimate of the overlap function. In the following example, we show the simulations for a flying altitude of 4837 m above sea level, which resembles our second experiment. The simulations were performed for the following off-nadir angles: 1°, 10°, 20°, 30°, 40°, 50° and 60° (as for the second experiment).

The synthetic signals were constructed as follows:

The following example shows the results when we choose ${r_{min}} = 400\; m$ (“perfect guess of the beginning of complete overlap”) or other values such as 300 m or 500 m as mentioned above. The following figures show the intermediate and the final results of the methodology. Figure 2 shows the logarithm of the range corrected signal ${y_j}$ versus altitude h. The maximum altitude until we can retrieve data is ${h_{max}} = {h_0} - {r_{min}}\textrm{cos}({30^\circ } )$ due to the requirement of at least 4 points for the linear regression between ${y_j}$ and ${x_j}$. Therefore ${h_{max}} = 4577\; m$, 4490 m, or 4404 m for ${r_{min}} = 300\; m$, 400 m or 500 m, respectively. Thus, based on these choices of ${r_{min}}$ (300 m, 400 m and 500 m) we can see that the overlap function can be retrieved down to a vertical distance ( ≅ range) $\Delta {h_{min}}$ of 260 m, 347 m and 433 m close to the lidar, respectively. As already shown in [19], when ${r_{min}}$ is underestimated (e.g., 300 m) the overlap function is erroneously retrieved in the near range. Our selected altitude range used for retrievals is from ${h_{max}}$ to 2000 m a.s.l. Due to the limitation to $\Delta {h_{min}}$, in order to be able to retrieve the overlap function close to the lidar, we will perform an extrapolation for optical depth and intercept to overcome this limitation.

 

Fig. 2. Logarithm of the range corrected signal (${y_j}$) versus altitude for the simulated experiment. The assumed flying altitude is ${h_0} = 4837m$. The x-axis shows the selected range (from ${h_{max}}$ to the 2000 m a.s.l.). The vertical lines show ${h_{max}}$ (4404 m, 4490 m, 4577 m) corresponding to ${r_{min}}$ (500 m, 400 m, 300 m).

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In the next step, for each altitude h (or vertical distance $\Delta h$), we plot ${y_j}$ versus ${x_j}$ to obtain the slope and the intercept. Figure 3 (left panels) shows the retrievals for ${A^\ast }$ and OD for the case of ${r_{min}} = 400\; m$. The extrapolation was performed from ${h_{max}} = 4490\; m$ to ${h_0} = 4837\; m$ (or $\Delta h$ from 347 m to 0.375 m). The relative error with respect to the original OD is increasing substantially below an altitude of 3000 m. On the contrary, the relative error for the intercept is small and stays below 5% over entire range. Thus, a selected range down to 2000 m is appropriate. As mentioned before, in the cases of choosing an underestimated ${r_{min}}$ (300 m) and overestimated ${r_{min}}$ (500 m), the selected range goes down to $\Delta {h_{min}} = 260\; m$ and $\Delta {h_{min}} = 433\; m$ respectively.

 

Fig. 3. The retrieval of the total optical depth (a, d), intercept ${A^\ast }$ (b, e) and their relative error with respect to the theoretical values (c, f). The vertical line shows where the data begin to be extrapolated towards the lidar (from ${h_{max}}$ to ${h_0} = 4837\; m$). The left panels show the results for ${r_{min}} = 400\; m$ while the right panels show the results for ${r_{min}} = 300\; m$.

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As we select also data from incomplete overlap, ${y_j}(h )$ for off-nadir angles 1° - 40° start to decrease (see ${h_{max}} = 4577\; m$ on Fig. 2). The profiles should ideally be such that the profile at 60° is the smallest while the one at 1° is the highest (Fig. 2). If not fulfilled, it brings an erroneous OD and intercept retrievals as in can be seen in Fig. 3 right panels for h > 4400 m.

Figure 4(a) shows the following step, with the individual overlap functions versus range r from lidar. The lower curves represent the uncertainties, which reach values below 10% after 2000 m. When the noise is small and there are no distortions in the signals, the individual overlap profiles should overlay nicely when plotting versus range. When we plot versus altitude (not shown), the overlap functions get closer to the lidar with increasing off-nadir angle. Figure 4(b) shows the final result, the overlap function as the average of the individual curves (from Fig. 4(a)). As we can see the retrieval of the overlap function is very good, with relative errors with respect to the theoretical function below 2% even at high ranges (see relative error, in red). We can observe that the complete overlap ${r_{ovl}}$ starts at 400 m, as the one assumed in the simulations. ${r_{min,q}}$ represents the minimum range where the overlap can be retrieved if there is no extrapolation (${r_{min,q}} = \Delta {h_{min}}/\textrm{cos}({1^\circ } )\cong \Delta {h_{min}}$).

 

Fig. 4. (a, c) Individual overlap functions for all off-nadir angles range r from lidar. The lower curves represent the uncertainties. (b, d) The overlap function and its relative error with respect to the theoretical value. The vertical line (${r_{min,q}}$) represents the beginning of the extrapolation towards lidar. The left panels show the results for ${r_{min}} = 400\; m$ while the right panels show the results for ${r_{min}} = 300\; m$.

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For the case of ${r_{min}} = 300\; m$, the initial retrieval over the region from 4490 m to 4577 m gives a lower OD and intercept which further brings an erroneously extrapolated OD and intercept (Fig. 3 d-e). Further, the individual overlap functions (Fig. 4(c)) are not correctly retrieved and differ amongst each other for ${r_{min}} = 300\; m$, especially for off-nadir angle of 60°. We omitted the legend as the profiles are almost indistinguishable. The averaged overlap function is shown in Fig. 4(d). The deviation from the theoretical function can be observed for ${r_{min}} = 300\; m$. The complete overlap ${r_{ovl}}$ appears to start around 350 m for ${r_{min}} = 300\; m$. One should note that the case of choosing an overestimated ${r_{min}}$ (500 m) does not bring errors as far as the extrapolation is good and the complete overlap starts at 400 m (as the theoretical one). We chose to not show the figures for ${r_{min}} = 500\; m$ as they are almost identical with the case of ${r_{min}} = 400\; m$.

4. Experiments

For the airborne lidar, scanning was achieved by operating the aircraft in circular orbit patterns with a range of bank or roll angles (nearly equal to the off-nadir angle). As the aircraft spins around 360° with a given bank angle, we have an orbit. The steepest orbit available with the BAe-146 aircraft is one with a 60° bank angle, and it impresses on cabin occupants an acceleration of 2 g. As the lidar is installed with a fixed pointing angle with respect to the aircraft we need to roll the aircraft in a turn in order to change the bank angle. Two flights of the FAAM aircraft (here also called experiments) were dedicated to the present study, on 17 June 2014 and 10 September 2014. The two experiments were planned during campaign periods for FAAM and the two dates were chosen as for cloud free days according to forecasts and confirmed using satellite imagery before takeoff. The flights were performed over the ocean, in the Southwest approaches near the Camborne observation site of the Met Office. Orbits were obtained at mean flying altitudes 3928 m and 4837 m respectively for the two flights, and the bank angles ${\varphi _j}$ were 30°, 40°, 50° and 60° for the first experiment and 10°, 20°, 30°, 40°, 50° and 60° for the second experiment, where j is a subscript denoting each single off-nadir angle. Straight and level runs (horizontal legs) were used for the near-zero off-nadir angle. The scanning pattern is shown in Appendix A (Fig. 13). The median distance of the flight with respect to Camborne was 143 / 84 km, with a minimum and maximum distance of 48 / 99 km and 132 / 173 km for the first / second experiment, respectively. The duration of the experiments from the start to the end of the flight section with orbits (including straight-level runs) was 25 / 75 min, respectively. The flight path for both experiments is shown in Appendix A (Fig. 14). During the observations, the Planetary Boundary Layer (PBL) remained below 1500 m metres, with no elevated aerosol layers, and no clouds were observed in the lidar signal: hence the data used in this paper were obtained within a mainly Rayleigh scattering layer. We will show that in the second experiment an optically thin aerosol was however present in the free troposphere (although not discernible by visual inspection of the lidar signal). Bank angles larger than 60° were not possible operationally with the FAAM aircraft, although they would have been desirable for this study. The aircraft pitch and roll are recorded every second by the on-board navigation system, and can thus be easily linked to the lidar data to compute the off-nadir angle [27].

4.1 17 June 2014

According to the lidar specifications, the nominal full overlap of the airborne Leosphere lidar should be around 300 m. The first flight when orbits were performed for the scientific needs for the present overlap study, took place on 17^th of June 2014. This flight was part of a broader mission, with other scientific objectives. The orbits were planned for the following bank angles: 30°, 40°, 50° and 60°, with a straight and level run at ∼2° as well. The mean off-nadir angles actually obtained in practice were 1.89°, 29.3°, 38.4°, 52.6°, 60.6°. The lidar did not perform optimally during this flight (lidar signal was low and thus SNR was low). The lidar logbook shows that it had been found to have been affected by a coolant leak, which had dirtied the receiver optics, thus lowering the retrieved signal strength. The mean flying (cruise) altitude was 3928 m while the minimum and maximum cruise altitude recorded for the data set were 3782 m and 3976 m. The largest variation of flying altitude was observed for the off-nadir angle of 60° (185 m). The selected measurements were recorded between 11:35 and 12:00 UTC. Despite all this, the method was tested and the result looked promising even if it the data were relatively noisy. We performed a small altitude correction such that the orbits were all brought to the same altitude, by shifting each lidar profile to the mean flying altitude. In other words, the flying altitudes below the mean flying altitude are brought up while the flying altitudes above the mean flying altitudes are brought down.

The number of 2-second profiles averaged for each off-nadir angle are: 142 (1.89°), 82 (29.3°), 56 (38.4°), 37 (52.6°) and 45 (60.6°), reflecting the duration of each orbit. The measurements are taken with a range resolution of 1.5 m and an integration time of 2 s. We also performed spatial smoothing, applying a moving average over 11 range bins.

Figure 5 shows ${y_j}(h )$ versus altitude. The selected range starts at ${h_{max}} = 3425\; m$ ($\Delta h = 503\; m$ from lidar) and ends at h = 2000 m. We observe that the ${y_j}(h )$ profile become noisier towards lower h which further determines a noisier total optical depth $\tau ({\Delta h} )$ and intercept ${A^\ast }({\Delta h} )$ (Fig. 6). The multi-angle linear regression between ${y_j}$ and ${x_j}$ is performed for each single height h separately (not shown here) and the derived regression parameters are shown in Fig. 6. The molecular OD (shown in red in Fig. 6(a)) is computed from the Camborne radiosonde.

 

Fig. 5. Logarithm of the range corrected signal ${y_j}$ versus altitude, for the experiment on 17 June 2014. The vertical line represents ${h_{max}}$ (beginning of the selected range).

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Fig. 6. The retrieval of the total optical depth $\tau ({\Delta h} )$ (a) and intercept ${A^\ast }({\Delta h} )$ (b). The vertical line at ${h_{max}}$ represents the beginning of the extrapolation towards lidar (${h_0} = 3928\; m$).

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We performed a second linear regression on the retrieved parameters, to extrapolate ${A^\ast }(h )$ and $\tau (h )$ in order to retrieve their values for $h > {h_{max}}$. The extrapolated values are shown in blue, starting at the vertical line (${h_{max}}$) and moving towards the lidar. Note that the (weighted) linear fit was performed over the first 150 m down from ${h_{max}}$ for both optical depth and intercept. However, for this experiment, taking into account that the difference between the total optical depth and the molecular optical depth at ${h_{max}}$ is smaller than 5% the extrapolated total optical depth was equal with the molecular optical depth. We observe that the data are noisy while on average, the total OD is similar to the molecular OD (confirming a nearly aerosol-free atmosphere).

Figure 7(a) shows the individual overlap functions with associated uncertainties while Fig. 7(b) shows the (mean) averaged overlap function (dark blue line). From this figure, we note that the complete overlap ${r_{ovl}}$ starts at a range of ∼ 500 m.

 

Fig. 7. (a) Individual overlap functions for all off-nadir angles versus range r from lidar. The vertical line represents the beginning of the extrapolation towards lidar. The lower curves represent the uncertainties. (b) The overlap function determined through the multi-angle method and through the molecular method.

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We tested also the method of getting the overlap function using the molecular approach. Essentially, we construct the Z based on molecular backscatter and extinction coefficients taken from the radiosonde, and accounting for the observation geometry. The ratio of the measured Z to the constructed molecular Z gives us the overlap function. Note that this is valid only if we measure a molecular atmosphere. Figure 7(b) shows both overlap functions (“molecular overlap” in cyan) and we observe that they are very close. This is in line with the observation that the OD was close to the molecular curve (Fig. 6(a)).

4.2 10 September 2014

During a subsequent flight, on 10^th of September 2014, the specific flying strategy for the overlap study was repeated and improved. Following the simulations shown in section 3, in addition to the initial bank angles, the 10° and 20° bank angles were added (to assure more points for the regression between ${y_j}$ and ${x_j}$). However, the cruise altitude was different (∼ 4.8 km). Like the previous one, this flight took also place over the ocean, near the coast of Cornwall. Note that the 30°, 40° and 60° orbits were repeated twice in order to increase the number of available data points (a 60° orbit lasts less than 1 minute). The flying time dedicated to the experiment was three times longer than in the previous experiment, and after the issues encountered the lidar had been cleaned and re-aligned. Thanks to the additional orbits, the better performance of the lidar, and the additional efforts made for this experiment, the data quality was superior than in the first experiment. In addition to the nine orbits, two straight level runs were performed at ∼ 0° and we consider these data as well. The number of profiles averaged for each off-nadir angle are: 210 (1.41°), 196 (11.4°), 119 (19.2°), 153 (29°), 110 (39.5°), 59 (49.3°) and 70 (58.3°). The mean flying altitude was 4837 m and was adhered to as best as possible. The minimum and maximum cruise altitude recorded for the data set were 4784 m and 4892 m. The largest variation of the flying altitude was observed for the off-nadir angle of 60° (108 m). We applied the same altitude correction and smoothing as for the first experiment. The selected measurements were recorded between 13:00 and 14:15 UTC. The PBL height was around 1 km above sea level, and the atmosphere was cloud-free.

Figure 8 shows the logarithm of the Z, ${y_j}(h )$ versus altitude. The selected range starts at 4413 m ($\Delta h = 424\; m$ from lidar) and ends at 2000 m. We observe that ${y_j}(h )$ profiles are less noisy as compared with the first experiment.

 

Fig. 8. Logarithm of the range corrected signal y_j versus altitude for the experiment on 10 September 2014. The vertical line represents h_max (beginning of the selected range).

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The OD and intercept derived with the linear regression between ${y_j}$ and ${x_j}$ are shown in Fig. 9. We can observe values for the total OD that are larger than the molecular, (differently than in the previous experiment), and this is indicative of a non-purely molecular layer. The (weighted) linear fit of the second linear regression was performed first 150 m down from ${h_{max}}$ for both optical depth and intercept. When the extrapolated optical depth became slightly negative, we replaced the values with the molecular values.

 

Fig. 9. The retrieval of the total optical depth $\tau ({\Delta h} )$ (a) and intercept ${A^\ast }({\Delta h} )$ (b). The vertical line (${h_{max}}$) represents the beginning of the extrapolation towards lidar (${h_0} = 4837\; m$).

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Figure 10(a) shows the individual overlap functions with associated uncertainties while Fig. 10(b) shows the average overlap function (dark blue line) which suggests that the complete overlap starts at ∼ 350 m. The overlap function looks very good in the far field (where we expect a unity value). The overlap function determined by the molecular method is shown by the cyan curve, and we observe a close agreement over the first 700 m. Then, the molecular overlap function deviates from the current method. This is explained by the fact that the total OD being larger than the molecular OD the atmosphere cannot be considered purely molecular. This is a good example of where the multi-angle method is superior to the molecular one, in that it permits picking the presence of aerosols, even if optically thin and hard to detect by a qualitative inspection of the Z profile.

 

Fig. 10. (a) Individual overlap functions for all off-nadir angles versus range r from lidar. The lower curves represent the uncertainties. (b) The overlap function determined through the multi-angle method and through the molecular method.

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Figure 11 shows the overlap function determined with the current (multi-angle) method as well as with the molecular approach for both experiments. We observe that the curves are slightly different over 100 m – 400 m with the overlap function from June smaller. Recall that the complete overlap ${r_{ovl}}$ starts at ∼ 500 m in June and at ∼ 350 m in September. The overlap function for June reveals the change in the alignment of the lidar between the experiments, as explained earlier.

 

Fig. 11. The overlap function calculated for both 17 June 2014 and 10 September 2014. For each day, the molecular method was calculated as well. The vertical lines represent the range from where the extrapolation is done (504 m in June and 425 m in September).

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5. Discussion

We have demonstrated an innovative method to determine the overlap function for an airborne lidar. Besides the assumption of a horizontally homogeneous atmosphere, the main constraint in retrieving the overlap function very close to the lidar is the availability of the measurements taken at high bank angles, which is possible up to an aircraft-dependent limit (60° for the FAAM BAe-146). The limitation on the bank angle has been overcome by using a proper extrapolation towards the lidar of the retrieved optical depth and intercept: this allowed the overlap function to be calculated with good precision. In addition, for better results, it is suitable to follow the following requirements: a stable cruise altitude, a large number of profiles to be averaged for each bank angle, a large number of off-nadir angle measurements (including straight-level runs to add measurements at near-zero off-nadir angle) and best efforts to obtain a good SNR. The more measurements can be obtained at large off-nadir angles, and the better we can reach a range closer to the lidar.

For the Leosphere lidar on-board the FAAM aircraft, in the present study we estimated the full overlap to be reached at ∼ 500 m and ∼ 350 m, in June and September 2014, respectively. The September experiment is considered more reliable taken into account the better SNR and more off-nadir angles involved in the calculations. The application of the method for two different time periods allowed to detect that a change of the lidar alignment had occurred. When operating this method, the challenging criterium is to choose the minimum range where to apply the methodology (where the linear regression between ${y_j}$ and ${x_j}$ starts). A rough knowledge of the beginning of the overlap is useful, and for safety, ${r_{min}}$ can be taken a bit further. If the full-range of off-nadir angles is restricted by operational requirements, it becomes advisable to perform a second linear regression, with extrapolation of the optical depth and intercept towards the lidar in order to retrieve the overlap function as close to the lidar as possible. Note that the second linear regression requires that there is no change, or a weak change in the aerosol load to achieve linearity. The requirement is not as strict as with the molecular method (which requires no aerosols present at all) but the overlap calculation flights should not be done in regions with sharp atmospheric inhomogeneities neither in the vertical nor in the horizontal.

To our knowledge, this study presents the first estimation of the overlap function using a multi-angle method for a nadir pointing lidar. The multi-angle measurements (for both ground-based and airborne lidar) provide a good method to determine the lidar overlap function, based on the methodology developed by [19]. As with any other overlap retrieval methods, the overlap function is valid whilst the alignment of the lidar does not change. Taking the ratio of the measured signal to the one expected for a molecular atmosphere is also a good option to determine the overlap function, as far as we are sure that the atmosphere is pure molecular (this is unlikely for a ground-based lidar, for example, but it is reasonable for an airborne lidar flying in the free troposphere). In our first experiment, we demonstrated that the atmosphere was pretty much molecular, and the overlap function retrieved by the multi-angle method was very similar with the one determined by the molecular approach (Fig. 7). On the contrary, in the second experiment, a slightly enhanced signal due to particles was present (Fig. 9) and therefore the overlap function determined with the molecular approach deviated from the one determined by the multi-angle method (Fig. 10). The multi-angle method was needed in this case, to detect the presence of aerosols, otherwise not determined through visual inspection of the range-corrected lidar signal. The main advantage of the multi-angle method is that it can be applied in any kind of atmosphere (Rayleigh or Mie), since as we retrieve the total optical depth. The method can be applied to any scanning system, no matter where the complete overlap starts, as far as we have good SNR for the first few km. For an airborne lidar like the one used here, where operational constraints limit the availability of the full range of bank angles, we suggest to use the extrapolation of the two variables (optical depth and intercept) in order to be able to retrieve the overlap function as close as possible to the lidar. Alternatively, this could be also addressed by building a scanning capability into the airborne lidar system. Last but not least, the multi-angle method is reliable even if the profiles are noisy. For example, for the first experiment (with noisier profiles), the relative error of the range corrected signal at 2000 m altitude was almost 20% for the 60° off-nadir angle. Further, the relative error for the optical depth had occasionally values larger than 100%. Despite the noise, the overlap could be quantified with good accuracy. For the first experiment (noisier), the relative difference with respect to unity is below 5% over 500 m - 2500 m range (the molecular method has a relative difference up to 7%). For the second experiment (better SNR), the relative difference with respect to unity over 500 m – 4000 m range is below 2% (the molecular approach has a relative difference up to 13%).

6. Conclusion

The innovative overlap correction function based on multi-angle measurements for an airborne lidar has been presented. The method presented here has brought statistical significance for the overlap correction function compared with the other experimental methods discussed in the introduction. The retrieval of the intercept and optical depth uses a regression between the logarithm of the range corrected signal and $1/\textrm{cos}(\varphi )$ for a series of measurements taken at several angles ${\varphi _j}$. The drawback of using the multi-angle method is that when angle measurements are not possible close to zenith angle 90° (for ground-based lidar) or off-nadir angle 90° (for airborne lidar), the overlap function cannot be directly retrieved very close to the lidar. Using an extrapolation of the intercept and optical depth, we can overcome this impediment. This extrapolation bears similarity with the slope method used by the other authors. The method was applied for two experiments in 2014 and showed the overlap function of the lidar system with a full overlap starting at ∼ 500 m during June and at ∼ 350 m during September. The method detected the change in alignment between the two periods on one hand, and its superiority with regard to the molecular approach was found when an optically thin aerosol layer was present.

We suggest that operators of airborne lidar systems could consider a protocol for the determination of the instrument’s overlap at specified time intervals, and for detecting changes in alignment. Such a protocol could involve dedicated flights which make use of orbits at several bank angles to enable scanning. The meteorological conditions for such a determination are a key parameter, and ideally one could target cloud-free and (as much as possible) aerosol-free conditions in a sufficiently quiet atmosphere. The multi-angle method does not require a Rayleigh atmosphere (but rather a quiet one), and this represents an advantage over the Rayleigh method. The presence of aerosols can be tolerated as long as one can rely on them being sufficiently homogeneous and constant, which usually would mean that a low aerosol content is desirable. The comparison of the multi-angle method with the one based on a molecular atmosphere gives a useful insight, and the latter could be used for a continuous monitoring of alignment variations when the airplane is flying outside aerosol and cloud layers.

Appendix A: Schematic of the flying procedure

Figure 12 illustrates the overlap regions. Thus, over first few dozen meters, the telescope’s FOV does not see the laser beam (“no overlap”). For the following dozen or hundred meters, the telescope sees partially the laser beam (“incomplete overlap”). After that, we reach the complete (full) overlap region.

 

Fig. 12. Geometric view of the overlap regions. The blue region represents the laser while the yellow region represents the telescope FOV.

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Figure 13 shows the line of sight (LOS) for the off-nadir angle φ. Panel b) shows the LOS for multi-angle measurements, with off-nadir angle ranging from 1° to 60°.

 

Fig. 13. (a) Flying under off-nadir angle φ. PBL stands for Planetary Boundary Layer. (b) multi-angle measurements for off-nadir angles 1°, 10°, 20°, 30°, 40°, 50° and 60°.

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Figure 14 shows the flight path for the two experiments from 17 June 2014 (left) and 10 September 2014 (right).

 

Fig. 14. Flight path for 17 June 2014 (left) and 10 September 2014 (right).

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Funding

Met Office; Ministerul Cercetării, Inovării şi Digitalizării (PN 23 05).

Acknowledgments

When this research was performed, both Mariana Adam and Franco Marenco were employed by the Met Office in Exeter, United Kingdom. The Met Office funded and supported this research. Airborne data were obtained using the United Kingdom BAe-146 Atmospheric Research Aircraft, which at the time of this work was flown by Directflight Ltd., managed by the Facility for Airborne Atmospheric Measurements (FAAM), and was a joint entity of the Natural Environment Research Council (NERC) and the Met Office. The staff of the Met Office Observations Based Research team, FAAM, Directflight Ltd, Avalon Engineering and BAE Systems are thanked for their dedication in making this experiment a success. Part of this work was carried out through the Core Program within the Romanian National Research Development and Innovation Plan 2022-2027, with the support of Ministry of Research, Innovation and Digitalization (MCID), project no. PN 23 05, by the Romanian Ministry of Research, Innovation and Digitalization, through Program 1- Development of the national research-development system, Subprogram 1.2 - Institutional performance - Projects to finance the excellent RDI, Contract no. 18PFE/30.12.2021 and by European Regional Development Fund through the Competitiveness Operational Programme 2014-2020, POC-A.1-A.1.1.1- F- 2015, project Research Centre for Environment and Earth Observation CEO-Terra, SMIS code 108109, contract No. 152/2016.

Disclosures

The authors declare no conflicts of interest.

Data availability

The FAAM aircraft datasets are available from the British Atmospheric Data Centre, Centre for Environmental Data Analysis [34,35]. Processed data presented in this paper may be obtained from the authors upon reasonable request.

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