At 94 GHz (3.2 mm, W band), the attenuation by atmospheric gas is mainly due to absorption of water vapor and oxygen (Liebe, 1985; Lenoble, 1993; Liou, 2002). The attenuation A (in dB km-1) by water vapor and oxygen in McRALI codes is computed from Liebe (1985) tabulations. Absorption coefficient σabs (in km-1) is given by σabs=0.2303A. Absorption and scattering are treated separately in McRALI codes, as is done in 3DMCPOL (Fauchez et al., 2014), whereby absorption is considered by a photon weight wabs according to the Lambert–Beer law (Partain et al., 2000; Emde et al., 2011): 15wabs=e-∫0sσabss′ds′, where ds′ is a path element of the photon path.

The Doppler radar community uses the Doppler spectrum S(r,v), a power-weighted distribution of the radial velocities v in the velocity range dv of the scatterers (Doviak and Zrnić, 1984). McRALI-FR codes dedicated to Doppler radar simulations compute S(r,v) by using the first Stokes parameter Ir,f and the Doppler formula v=cf/2f0. We follow the convention that the Doppler velocity is positive for motion away from the radar. The backscattering coefficient profile βr is then given by 16β(r)=∫-∞+∞Ir,vdv.

The reflectivity Zr profile is computed using Eq. (3) and βr. The Doppler velocity profile VDopr is defined as 17VDopr=∫-∞+∞vIr,vdv∫-∞+∞Ir,vdv, and the Doppler velocity spectral width profile σDopr is obtained from 18σDop2r=∫-∞+∞v-VDopr2Ir,vdv∫-∞+∞Ir,vdv.

Figure 4 shows, as an example, a simulation of the Doppler power spectrum, the Doppler velocity, the Doppler velocity spectral width, and the reflectivity profiles for a CPR/EarthCARE-like radar for a homogenous iced cloud layer with fixed 6 m s-1 downdraft at all altitudes (see details of the conditions of the simulation in Table 1) with (σturb=0.5 ms-1) and without (σturb=0 ms-1) sub-grid turbulent wind. In a first step, McRALI-FR codes dedicated to Doppler radar simulations compute the idealized Doppler power spectrum density S(rv). The first Stokes parameter Ir,v of the Doppler spectrums (with and without sub-grid turbulent wind) are shown in Fig. 4a and b, respectively. Then, in a second step, software computes the reflectivity, the Doppler velocity, and the Doppler velocity spectral width profiles with Eqs. (16), (17), and (18), respectively. Multiple-scattering (MS) and single-scattering (SS) Doppler velocity profiles are superimposed on the MS Doppler spectrum. MS and SS Doppler velocity values are constant within the cloud layer (between 9 and 10 km of altitude) and are equal to the “true” 6 ms-1 vertical velocity, whatever the wind turbulence value is. Due to multiple-scattering processes, the apparent Doppler velocity of 6 ms-1 can be observed between the cloud-base altitude and the ground, contrary to the SS apparent Doppler velocity, which appears only in the cloud layer.

In Fig. 4c the MS and SS Doppler velocity spectral width profiles are drawn. Under the SS approximation, the Doppler velocity spectral width σDop is given by (Kobayashi et al., 2003; Battaglia et al., 2013) σDop2=σhydro2+σshear2+σturb2+σmotion2, where σhydro is due to the spread of the terminal fall velocities of hydrometeors of different size, σshear is the broadening due to the vertical shear of vertical wind, σturb is the broadening of the vertical wind due to turbulent motions in the atmosphere, and σmotion is the spread caused by the coupling between the platform motion and the vertical wind shears of the horizontal winds. For a Gaussian circular antenna pattern, assuming zero fall velocities of hydrometeors and no wind shear, σDop is given by (Tanelli et al., 2002) 19σDop2=σturb2+θFOVvsat2ln22, where vsat is the satellite velocity relative to the ground and θFOV is the Gaussian (3 dB) FOV half-angle.

Simulated SS Doppler velocity spectral widths without turbulence and with turbulence are close to 3.58 and 3.62 m s-1, respectively. Both computed values are very close to theory-predicted values. On the other hand, MS processes together with sub-grid turbulent wind are a source of broadening. For example, at 2 km under the cloud base, σDop=3.75 m s-1, which is larger than the SS value.

Vertical profiles of MS and SS reflectivity are shown in Fig. 4d. These profiles are not sensitive to the wind turbulence. MS processes are a source of enhancement of the reflectivity compared to the SS reflectivity and the apparent reflectivity that can be observed under the cloud layer.

The laser transmitter of the ATLID instrument has spectral requirements with a spectral line width below 50 MHz (Hélière et al., 2017). In McRALI-FR codes, the frequency of the emitted radiation is drawn randomly according to a Gaussian law of average f0 with a 1/e half-width σf0=50 MHz.

The current version of McRALI-FR codes assumes that each component of molecular velocity is distributed according to the Maxwell–Boltzmann density function with null mean and standard deviation a given by 20a=kTm, where k is the Boltzmann's constant, T is the temperature, and m is the molecular mass of gas. The multivariate normal distribution is generated using the Box–Muller method. As a next step, we plan to take into account spontaneous Rayleigh–Brillouin scattering.

One of the important features of HSR lidars is the possibility to retrieve profiles of particle extinction and the backscattering coefficient without the need for additional information on the lidar ratio (Shipley et al., 1983; Ansmann et al., 2007, and references therein). HSR technology relies on the principle of measuring the Doppler frequency shift resulting from the scattering of photons by molecules (referred to as molecular scattering or Rayleigh scattering) and by particles (referred to as particulate scattering or Mie scattering). The characteristic shape of the HSR spectrum depends on both these two scattering processes: a broad spectrum of low intensity for molecule scattering and a narrow peak of large intensity for particle scattering.

The spectral width of the particle peak will be determined by the spectral width of the laser pulse itself along with any turbulence present in the sampling volume. The spectral width of the ATLID laser will be on the order of 50 MHz so that the laser line width will be the dominant factor. Thus, the molecular backscatter will be much broader than the particulate-scattering return. This is due to the fact that atmospheric molecules have a large thermal velocity. Assuming a Gaussian molecular thermal velocity distribution with a half-width at 1/e of the maximum, molecular broadening γm can be written (Bruneau and Pelon, 2003) as 21γm=2λ02kTm.

If T=230 K, then γm is of the order of 2 GHz, which is about 40 times larger than the laser line width. Thus, using interferometers (such as the Fabry–Pérot (FP) interferometer equipping the ATLID/EarthCARE lidar) and appropriate signal processing (Hélière et al., 2017), the molecular and particulate contributions of the lidar backscattered signal can be separated. Then particulate and molecular backscattering coefficient profiles (attenuated or apparent attenuated backscattering coefficient or simply attenuated backscatter, also denoted as ATB) can be separately determined. In this study, we suppose that the FP interferometer has the following parameters. The free spectral range is 7.5 GHz, the finesse is 10, and the FP is centered at the wavelength 355 nm. The cross-talk effects were taken into account according to the work by Shipley et al. (1983). The coefficients of the cross-talk correction were computed using an Airy function (see, e.g., Vallée and Soares, 2004), which describes the FP transmission spectrum, assuming a Gaussian molecular thermal velocity distribution with a half-width at 1/e of the maximum γm (Eq. 21). This method determines four calibration coefficients corresponding to the fraction of cloud–aerosol backscatter in the molecular and particulate channels (Cam and Caa, respectively), as well as the fraction of molecular backscatter in the molecular and particulate channels (Cmm and Cma, respectively). The calculation method of these coefficients is described in detail in Shipley et al. (1983). As an indication, for the present study in the ATLID/EarthCARE lidar configuration, these coefficients have the following values: Cmm=0.543, Cma=0.457, Caa=0.998, and Cam=0.002. Note that the cross-talk coefficients used in this paper assume ideal behavior of the ATLID FP interferometer. In practice, the Airy function will be “blurred” due to the effects of nonideal collimation of the beam, frequency jitter, surface roughness, and so on. All these factors combine to decrease the peak transmission and lower the full-width at half-maximum (see the Fig. 9 in Pereira do Carmo et al., 2019). It is important to keep in mind that all the calculations shown in this paper are merely “EarthCARE-like” but with an idealized modeled FP interferometer.

Figure 5 shows particulate and molecular ATB profiles for an ATLID/EarthCARE-like lidar. We consider in this example an ice cloud corresponding to a homogenous layer with an optical depth 3 between 9 and 10 km in altitude (see details of the simulation conditions in Sect. 3.1). In the first step, McRALI-FR codes, dedicated to HSR lidar simulations, compute the HSR spectrum S(rf). The first Stokes parameter Ir,f of the MS HSR spectrum is shown in Fig. 5a. The peak of intensity (in red) centered at 0 GHz between 9 and 10 km of altitude corresponds to the position of the cloud. It is the contribution of the cloud particles (the so-called “Mie contribution”). This spectrum is also characterized by the molecular contribution (Rayleigh contribution). The intensity of the spectrum below the cloud is lower than the intensity of the spectrum above the cloud due to particulate extinction. In Fig. 5b, the MS (computed with McRALI-FR) and SS (computed from SS theory) vertical profiles of spectral width are represented. We note very good agreement between the SS theoretical and MS simulated values at both the cloudy and molecular levels. This suggests that MS effects have very little impact on spectral width. Then, in a second step, a simulated FP interferometer separates the particulate contribution from the molecular contribution and provides the vertical profiles of particulate and molecular ATB as shown in Fig. 5c. The total ATB calculated directly from the spectrum, SS molecular, and SS particulate backscatter profiles is also represented. Above the cloud, particulate ATB is not strictly zero and molecular ATB is not strictly equal to total ATB because of the FP remaining cross-talk effects (see above). In the cloudy part between 9 and 10 km of altitude, the molecular and particulate ATB logically decrease exponentially with depth. The SS backscatter profiles decrease faster with depth than the MS backscatter profiles, revealing that MS effects on ATB are not negligible. Under the cloud, molecular ATB is almost equal to total ATB. It is likely worth pointing out the quasi-exponential decay of the below-cloud molecular return towards single-scattering levels. This result is consistent with the cases shown by Donovan (2016). Particulate ATB is almost zero. Some nonzero values exist due to FP cross-talk effects but also due to Monte Carlo noise and MS processes.

Figure 7a–e show vertical profiles of MS radar Doppler spectra density in the CPR/EarthCARE configuration corresponding to the five positions of the satellite relative to the edge of the box cloud with optical depth set to 3. Regardless of the satellite position, Doppler spectra correspond to negative Doppler velocity values. This is consistent with the convention that the Doppler velocity is positive for motion away from the radar. As the satellite approaches the edge of the cloud and carries on, the NUBF effect decreases. Indeed, the Doppler spectrum becomes more and more symmetrical. The asymmetric shape of the Doppler spectrum is due to zero values beyond a critical value of Doppler velocity vcrit. As an example, vcrit≈-3.4 ms-1 in Fig. 7a. The explanation of the vcrit value is purely geometric. Doppler broadening is dominated by Doppler fading due to satellite motion. Under the SS approximation, neglecting wind velocity, the Doppler shift is given by Δv=1/2vsatk0-k′1, with k0=-k1′. Assuming vsat with an x-horizontal positive component, k0, in the (z–x) vertical plan and with kx as the x-horizontal component, then Δv=-vsatkx. Assuming that satellite is at the x-horizontal d distance to the box-cloud edge and at the z-vertical D distance above the cloud, with a vertical (downdraft) wind velocity vwind fixed at 6 ms-1, then vcrit=-vsatd/d2+D21/2+vwind. For x=-500, x=-250, x=0, x=250, and x=500 m, vcrit=-3.4, vcrit=-1.3, vcrit=6, vcrit=10.7, and vcrit=15.4 m s-1, respectively. These values are very close to those estimated from the five respective power spectra in Fig. 7.

Figure 7f also shows the five reflectivity profiles corresponding to the five positions of the satellite relative to the edge of the box cloud. MS reflectivity profiles are larger than SS reflectivity profiles because MS processes logically increase the reflectivity value. We can also see MS effects on the apparent reflectivity that is non-null under the cloud layer, contrary to the SS apparent reflectivity. At the same time, as the satellite approaches the edge of the cloud and keeps moving forward, SS and MS apparent reflectivity profile values increase due to the fact that the NUBF effect decreases. Many studies have focused on the NUBF effect on rain fields retrieved by radar from space (Amayenc et al., 1993; Testud et al., 1996; Durden et al., 1998; Iguchi et al., 2009). Iguchi et al. (2000) showed that the NUBF effect could be accounted for by a factor determined from horizontal variation of the attenuation coefficient. Our simulations are coherent with the literature. A detailed investigation of the NUBF effect on reflectivity profiles for spaceborne cloud radar will be carried out in a later work.

Figure 7 shows the SS and MS vertical profiles of Doppler velocity superimposed on the five power spectra for the five positions of the satellite relative to the box-cloud edge: x=-500, -250, 0, 250, and 500 m. Since each power spectrum has no value beyond the vcrit velocity, which is due to the NUBF effect, it is obvious that the profile of the apparent Doppler velocity is different from the profile of the vertical wind velocity fixed at 6 ms-1. Figure 8 shows the MS and SS apparent Doppler velocity and Doppler spectrum width computed every 100 m along the horizontal axis. These quantities are estimated at different altitudes (cloud top, middle, and base) and are plotted as a function of the satellite distance to the box-cloud left edge. Differences between apparent Doppler velocities are in general small (around 1 m s-1) whatever the altitude is, and differences between MS and SS Doppler velocities are also small, no larger than 1 m s-1 at the bottom of the cloud. The same conclusions can be drawn for the Doppler spectrum width at which differences are no larger than 0.3 m s-1. This implies that MS processes do not play an important role in the estimation of the apparent Doppler velocity or in the estimation of apparent spectrum width (for the specific conditions of simulation with the box cloud) compared to the NUBF effect. The NUBF Doppler velocity bias between apparent Doppler velocity and “true” vertical wind velocity fixed at 6 m s-1 is around -10, -5, -3, -2, and -1 ms-1 at x=-500, -250, 0, 250, 500 m, respectively.

In general the NUBF bias of Doppler velocity can be expressed as a function of the distribution of the radar reflectivity (Tanelli et al., 2002). An estimate of the NUBF bias of Doppler velocity can be obtained by considering the difference between the Doppler velocity computed with a satellite velocity (i.e., vsat=7.2 kms-1) and the Doppler velocity computed with a satellite velocity set to 0 m s-1 (Battaglia et al., 2018). Sy et al. (2014) showed that the NUBF bias of Doppler velocity is correlated with the horizontal gradient of reflectivity and demonstrated that the theoretical proportional coefficient α value is bounded between 0.165 and 0.219 m s-1 (dBZ km-1)-1. Kollias et al. (2014) estimated this proportional coefficient value close to α=0.23 m s-1 (dBZ km-1)-1 for along-track horizontal integration of 500 m (i.e., Doppler CPR/EarthCARE resolution) and for all their available simulations performed with a cirrus cloud and a precipitation system. Figure 9 shows the Doppler velocity NUBF bias as a function of the horizontal reflectivity gradient for a horizontal integration of 500 m at four positions relative to the box-cloud edge (-200, 100, 0, and 100 m). Computations are carried out for different optical depths of the box cloud of 0.1, 1, and 3. We note that the α value is between 0.14 and 0.16 m s-1 (dBZ km-1)-1 and is almost independent of the position of the satellite relative to the box-cloud edge. If the satellite position is just above the cloud edge, the proportional coefficient value is close to 0.15 m s-1 (dBZ km-1)-1, a value close to that obtained by Sy et al. (2014).

A first study of the effects of multiple scattering on the Doppler velocity vertical profile in the case of vertically heterogeneous wind velocity is carried out for a very specific case in the CPR/EarthCARE configuration. Indeed, for a homogeneous cloud layer with a base altitude of 9 km and with a geometrical thickness of 1 km, the vertical velocity is set to 6 m s-1 (downdraft) and -6 m s-1 (updraft) in the upper and lower part of the cloud layer, respectively. Figure 10 shows vertical profiles of the MS radar Doppler spectrum and the MS and SS Doppler velocity profiles computed with McRALI-FR. The measured Doppler velocity under the SS regime (black dotted line in Fig. 10) is equal to 6 m s-1 (-6 m s-1) in the upper (lower) part of the cloud, and the SS Doppler velocity can be used as the reference of the true velocity. In the upper part of the cloud, the measured MS Doppler velocity is 6 m s-1, which equals the true velocity. In the lower part of the cloud, the measured MS Doppler velocity is biased by multiple-scattering processes with a value not smaller than -3 m s-1, contrary to the true velocity of -6 m s-1 in this cloudy part. For altitudes lower than 7 km, we can also note that multiple-scattering processes can lead to a Doppler velocity lower than -6 m s-1.

In order to investigate the NUBF effects on HSR lidar observables under MS regimes we firstly compare simulation results carried out with the box cloud (full 3D simulation) and an optical depth equal to 3 (hereafter called 3D cloud) to simulations performed with the plane-parallel and homogeneous cloud model (hereafter called PP cloud) as well as with the independent column approximation (or independent pixel approximation) cloud model (hereafter called ICA cloud). PP theory and the ICA assumption are commonly used to assess the radiative effects of inhomogeneous cloud when cloud unresolved variability and net horizontal fluxes, respectively, are ignored (Marshak and Davis, 2005). For the specific case of the satellite position relative to the box-cloud edge x=-4.0 m (see Fig. 11a), and the cloud coverage α inside the lidar receiver FOV is 30 % (see Sect. 3.1). As cloud optical depth (COD) is set to 3, this implies that mean COD weighted by the cloud coverage inside the lidar footprint is 0.9, the assigned value for the optical depth of the PP cloud (Fig. 11b). In other words, the PP profile can be considered a profile computed for a homogeneous cloud with optical depth equal to the mean optical depth of the cloudy part weighted by the cloud cover of the 3D cloud. The ICA simulation is carried out by averaging 30 % of a simulation with a homogeneous cloud with COD of 3 % and 70 % of a simulation in a clear-sky atmosphere (Fig. 11c). In other words, ICA profiles can be considered a profiles averaged over columns (two columns in this case) weighted by the cloud coverage.

Figure 12 shows the HSR spectra simulated by McRALI-FR for the three cloud models at four altitude levels: above the cloud (12 km, Fig. 12a), at cloud top (9.8 km, Fig. 12b) and base (9.2 km, Fig. 12c), and below the cloud (8.5 km, Fig. 12d). In the clear-sky region above the cloud (Fig. 12a), the three frequency spectra line up very well, as expected, because the lidar laser beam has not yet been scattered by the cloud. A similar feature is observed at cloud top (Fig. 12b), whereas a few spikes can be seen in the molecular broad spectrum. Deeper in the cloud (Fig. 12c) and below (Fig. 12d), spikes are more numerous with higher intensity. These spikes are simulation artifacts. They are caused by specific events of multiple scattering, namely by cases when forward scattering is involved during a photon random path. For example, the photons can first be scattered by air molecules, inducing a large frequency shift, then by cloud particles, inducing a large contribution due to the highly forward-peaked phase function. The scattering phase function of ice particles spans about 6 orders of magnitude. Thus, the forward-scattered photons have a weight that is several orders of magnitude larger than those scattered in other directions. At the same time, such cases occur rarely. Consequently, we should carry out simulations with an unrealistic number of photons emitted by the lidar to smooth spikes, which is not possible. Spikes are not observed in simulations under the single-scattering regime (not shown here).

One can clearly see that the intensity of the central Mie (particulate) peak computed by ICA and 3D is lower than the one computed by the PP simulation. The opposite behavior is observed concerning the Rayleigh (molecular) scattering region: the broad and low-intensity spectra for ICA and 3D simulations show larger values (in intensity) compared to the PP simulation and for the full range of frequency shift. The same observation can be made below the cloud.

In order to obtain the molecular and particulate ATB vertical profiles for the three cloud models (Fig. 13b and c), the HSR spectra are filtered by a modeled Fabry–Pérot interferometer (see Sect. 2.5.3 for the filtering parameters) at each altitude level. The ATB vertical profiles (Fig. 13a) exhibit features observed in HSR spectra more clearly: once the cloud is reached (i.e., below 10 km), the three cloud models give very different total ATB. Higher values are observed in clouds for the PP model compared to ICA and 3D cloud models, whereas the opposite is observed below the cloud. This feature shows that PP cloud representation can lead to large discrepancies and is suitable to account for 3D cloud structure. In contrast, ICA cloud models give results rather close to 3D cloud models. In Fig. 13b and c the PP cloud shows the most significant differences, with PP particulate ATB in the cloud larger than ICA and 3D particulate ATB and a PP molecular ATB that is smaller. To a lesser extent, ICA and 3D computations also show differences with 3D total and 3D particulate ATB smaller than the ICA computation. This difference is the opposite for molecular ATB.

In order to quantify the differences coming from the cloud models, PP, ICA, and 3D biases have been computed. These biases, well described in Davis and Polonsky (2005), were first defined in the radiance framework by Cahalan et al. (1994) and were adapted to a lidar signal framework by Alkasem et al. (2017). The 3D bias on ATB (i.e., ΔATB3D) is the sum of the PP bias (i.e., ΔATBPP) and the ICA bias (i.e., ΔATBICA) defined as 22.1ΔATBPP=ATBPP-ATBICA,22.2ΔATBICA=ATBICA-ATB3D, 22.3ΔATB3D=ΔATBPP+ΔATBICA=ATBPP-ATB3D, where ATBICA, ATBPP, and ATB3D are ATB computed by McRALI with the ICA, PP, and 3D cloud models, respectively. The relative biases are also computed and correspond to the biases divided by the reference ATB3D. In Appendix B, it is shown that the PP bias of molecular ATB and particulate ATB is always negative and positive, respectively. It is also shown that the larger multiple scattering, the smaller the PP particulate bias. Note that the PP bias of total ATB is positive but becomes negative with increasing cloud optical depth (Alkasem et al., 2017).

Figure 14 shows that the PP biases are the largest on total ATB as well as the molecular and particulate components. Indeed, they reach 250 %, -60 %, and 1200 % for total, molecular, and particulate ATB, respectively. The ICA biases present lower values (around 25 %, -10 %, and less than 100 % for total, molecular, and particulate ATB, respectively). These results thus show that 3D biases are mainly due to PP biases.

Based on the three cloud models, further simulations have been carried out with varying the cloud coverage inside the lidar FOV from 10 % to 90 % in order to evaluate the impact of cloud coverage on HSR lidar observations. A simulation has been carried out for the study of the NUBF effect on radar observations in a similar way (see Sect. 3.2). The SS bias and the MS relative bias on total, molecular, and particulate ATBs at cloud top (9.8 km), cloud base (9.2 km), and in the middle of the cloud (9.5 km) have been computed for each cloud model in Fig. 15.

Figure 15d and f show that MS total and particulate biases decrease when cloud coverage increases and reach almost zero for 90 % cloud coverage. These relative biases always show the largest values at cloud base, then in the middle of cloud, and then at cloud top, whereas it is not observed for SS bias (Fig. 15a and c). This is due to the division by reference ATB3D values that exponentially increase with cloud altitude. We also find that PP biases are positive, with a maximum bias of 250 % observed for the total relative ATB at cloud base and for cloud coverage of 30 %. These last two observations are consistent with simulation results under the SS regime: PP biases of total ATB and of particulate ATB are generally positive (see Eq. B8) and strictly positive (see Eq. B7), respectively. The maximum bias of ATB also occurs for cloud coverage of 30 %. Otherwise, regardless of cloud coverage, Fig. 15d and f show that MS total and particulate ICA biases are still rather small (<50 % and <200 %, respectively) compared to PP biases (<250 % and <4000 %, respectively). For total and particulate ATB, PP biases are the largest and are mainly responsible for the 3D biases, confirming the findings of the previous section for any cloud coverage in both the SS and MS regimes.

For molecular signals, Fig. 15e shows the negative PP bias that increases in absolute value with increasing cloud coverage, reaching -40 % and -80 % for 90 % cloud coverage in the middle of the cloud and at cloud base, respectively. The negative value of MS molecular PP bias is consistent with the SS molecular PP bias definition (see Eq. B6). Otherwise, MS ICA bias is negative for cloud coverage smaller than 50 % and rather positive for cloud coverage larger than 50 %. This latter observation is consistent with simulation results performed under SS regimes (see Fig. 15b): molecular ICA bias is negative, null, and positive for cloud coverage smaller than, equal to, and larger than 50 %, respectively. Figure 15e shows that MS ICA bias reaches only 15 % and 50 % in the middle of the cloud and at cloud base, respectively. At cloud top, all the molecular biases remain small (between 5 % and -15 %). Because of competition under the MS regime between the rather negative PP bias and the positive and negative ICA bias of the same order of magnitude, no specific trend in the 3D molecular biases according to the cloud coverage can be highlighted, and conclusions are less obvious than for total and particulate ATB.

Finally, our simulation results also show that 3D bias decreases in magnitude when COD decreases due to the PP bias that decreases. For example, if COD is 0.1, 3D total, molecular, and particulate biases are less than 6 %, 15 %, and 100 %, respectively, mainly driven by ICA bias.

We briefly investigate the impact of the size of the field of view (FOV) by carrying out simulations with an FOV 10 times greater than in the simulations performed in the previous sections. Simulations have been carried out for cloud coverage of 50 %, implying COD of 1.5 for the PP cloud model. Figure 16 shows vertical profiles of ATB biases and relative ATB biases computed with this large FOV (i.e., 650 µrad) and those computed with the ATLID FOV (i.e., 65 µrad).

When comparing ATB computed with the three cloud models (i.e., PP, ICA, and 3D cloud model) for the large FOV, the same conclusions as for the ATLID FOV can be made: when reaching the cloud base, total and particulate ATBs show lower values for ICA and 3D models than for the PP model, and the opposite is true for the molecular ATB. Biases also show the same trend as for ATLID FOV, with a maximum bias for the PP model and lower values for ICA models.

When comparing simulations carried out with a large FOV to those performed with a small FOV, we observed that with a larger FOV, ATB values are larger when going into the cloud for the three cloud models. The reason is mainly the multiple scattering, which is obviously more pronounced as the FOV increases. Figure 16 shows that the biases and the relative biases for the large FOV present the same trend as for a small FOV. For total and particulate ATB, we can note that ICA biases are larger whatever the vertical position in the cloud, whereas PP biases are smaller in the upper part of the cloud due to multiple scattering, which becomes more significant as the FOV increases; this latter behavior is coherent with Eq. (B7). The relative biases for the large FOV show slightly smaller values for the three signal components (total, molecular, and particulate). The maximum values for the 3D bias are 150 %, -60 %, and 200 % for the total, molecular, and particulate ATBs, respectively.