The relationship between the range-resolved backscattered power, P(r), and the atmospheric scattering and attenuation properties is described by the lidar equation and may be expressed as follows: C⋅P(r)r2=(βm(r)+βp(r))⋅exp[-2∫0r(σm(r′)+ησp(r′))dr′], where C is a calibration constant depending on the lidar system and the atmospheric profile. β(r) and σ(r) represent the backscattering and extinction coefficients, respectively, and both contain a contribution arising from purely molecular backscattering (βm(r)) or extinction (σm(r)) and a contribution arising from cloud or aerosol particles that may be present in the atmosphere (βp(r) and σp(r), respectively). The factor η accounts for multiple-scattering processes. For the remainder of this paper we assume that η=1 for aerosols and η=0.75 for cirrus clouds. The value of 0.75 for cirrus clouds has been chosen based on the PhD thesis of in which the multiple-scattering factor has been evaluated by comparing the optical depth retrieved from measurements of the micro-pulse lidar in Lille to the optical depth retrieved from CALIOP and adjusting η to find a coherent retrieval. The multiple-scattering factor used for the CALIOP version 3 retrievals is η=0.6 . For ground-based lidars the multiple-scattering effect is less important since they have a much smaller FOV in combination with a shorter distance to the cloud, although it should not be neglected because large ice crystals may considerably increase the forward scattering of the laser beam . However, since our knowledge of this parameter is rather poor we assign a large error to it in our optimal estimation algorithm (see Sect. ).

The retrieval of particle optical properties from elastic lidar measurements alone is challenging since there is an intrinsic ambiguity between the effects of backscattering and extinction arising from the combination of scattering and absorption processes in the atmosphere. In Eq. (), the backscattering coefficient of aerosol or cloud particles, βp(r), can be replaced by βp(r)=k(r)⋅σp(r), where k(r) represents the range-dependent backscatter-to-extinction ratio. Since we use a simple micro-pulse lidar, we need to introduce some assumptions for k(r) to retrieve profiles of extinction by aerosol and cloud particles. Unfortunately, this parameter is highly variable and depends strongly on the type, size and shape of the atmospheric particles.

In this study we are focusing on the retrieval of cirrus cloud properties. Thus, the backscatter-to-extinction coefficient for aerosols is assumed to be constant and is fixed to 64 sr. This value originates from the Optical Properties of Aerosols and Clouds (OPAC) database and corresponds to a water-soluble urban aerosol. Since our measurement site is located in an urban/industrial area, we used the optical properties for this aerosol type (for the visible lidar wavelength as well as for the TIR as will be discussed in Sect. ) from the OPAC database to test our new algorithm. This parameter needs to be refined in future studies depending on the aerosol type that is actually present during the measurement obtained from additional information. However, as discussed above for the multiple-scattering factor, we also assign a large uncertainty to the lidar ratio of aerosols to account for our rather poor knowledge of it. Unfortunately, this reduces the quality of the information returned by our algorithm.

The backscatter-to-extinction ratio for cirrus clouds is calculated using the definition of : k=ϖ0⋅P11(π), where ϖ0 is the particle single-scattering albedo and P11(π) the phase function in the exact backscattering direction.

We obtain the single-scattering properties (scattering coefficient, absorption coefficient and asymmetry parameter) for each cloud layer from the parametrization of , which is based on the ensemble model for cirrus introduced by . The idea of this model is to represent the variability of ice crystal sizes and shapes inside a cirrus cloud by assuming a distribution of some idealized shapes rather than assuming just a single geometrical form throughout the whole size spectrum. Observed ice crystal shapes in cirrus clouds range from simple pristine particles such as hexagonal ice columns and bullet rosettes (associated with small particles) over aggregates of these particles to aggregate chains, while the complexity of the crystal tends to increase with increasing size. The ensemble model of attempts to reproduce these observations by using the six members shown in Fig. . The smallest ice crystals are represented by the first two members, which are simple hexagonal ice columns and bullet rosettes. The following members represent larger and more complex ice crystals by arbitrarily attaching up to 10 hexagonal elements to create chain-like structures. For the calculation of the bulk optical properties, the particle size distribution (PSD) of is assumed, which is independent of assumptions about the ice crystal shape and depends only on the in-cloud temperature and the IWC. This parametrization has been constructed based on a large number of in situ measured PSDs and does not include measurements of ice crystal sizes less than 100 µm due to the shattering problem . For particles smaller than 100 µm an exponential PSD is assumed. used the parametrization to calculate the single-scattering properties for the ensemble model as functions of IWC and in-cloud temperature for a total of 20 662 in situ measurements from different aircraft-based field campaigns located in the tropics and in the midlatitudes. They created a database of optical ice cloud properties comprising 145 wavelengths between 0.2 and 120 µm. This database was used by to develop a new ice cloud parametrization that predicts the single-scattering properties named above as functions of the in-cloud temperature and IWC without the need of a priori information on the shape and the effective diameter of the ice crystals. For the remainder of this paper, we will call this microphysical model BV2015.

Since our algorithm seeks to retrieve the IWC for cirrus cloud layers, the extinction required in the lidar equation (Eq. ) is calculated from the scattering and absorption coefficients obtained from the BV2015 parametrization as a function of the IWC of each cloud layer. The necessary temperature information is obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis by matching atmospheric temperature profiles to the corresponding cirrus cloud altitude. The single-scattering albedo required in Eq. () is also calculated from the scattering and absorption coefficients, and the scattering phase function is generated from the asymmetry parameter using the analytic phase function of which is a linear piecewise parametrization of the Henyey–Greenstein phase function depending only on the asymmetry parameter. It is kept smooth and featureless since atmospheric ice crystals may be distorted, be roughened or contain inclusions of air bubbles or aerosols. All these processes would remove or reduce the optical features of the phase function like the halos at 22 or 46∘ and the backscattering peak e.g.,. demonstrated that their parametrization reproduces short-wave multi-angle satellite and aircraft observations, and showed a good agreement with high-resolution infrared observations between 3 and 18 µm. It has also been shown to be in good agreement with the backscattering features observed from POLarization and Directionality of the Earth's Reflectances (POLDER) measurements . The scattering phase function, especially in the exact backscattering direction, is a crucial parameter in our algorithm since it defines the lidar backscatter-to-extinction ratio. It will be discussed in more detail in Sect. .

As mentioned above, we apply an optimal estimation method to invert the lidar equation following . Optimal estimation is based on a Bayesian approach which uses probability density functions to link the measurement space to the state space accounting for their uncertainties . This approach allows us to find the most likely solution that is consistent with both the measurement and any given prior knowledge of the state within the range of their uncertainties. In general, the measurement vector y can be related to the state vector x via the forward model F by y=F(x)+ϵ, where ϵ represents the uncertainties arising from the measurements and the forward model. The aim of every inversion method is to invert the connection between the state vector and the measurement vector, which is given by the forward model, in order to retrieve the elements of the state vector using the information provided by the measurement vector.

Following , the best estimation of the state vector can be obtained by minimizing the following cost function: Φ=[y-F(x)]TSϵ-1[y-F(x)]+[x-xa]TSa-1[x-xa]. There are two contributions in this cost function: the first term on the right-hand side of Eq. () represents the contribution arising from the forward model and the measurement, where Sϵ is the sum of the variance–covariance matrices of the forward model and the measurement, and the second term represents the contribution from the so-called a priori state vector which contains the prior knowledge of the state vector before the measurement has been performed. Sa is the variance–covariance matrix of the a priori state vector which, in our case, was chosen to be sufficiently large to reduce the influence of the a priori assumptions on the final retrieval.

To find the best estimate of the state vector x^ that minimizes the cost function Φ, an iterative method was applied following the approach of Levenberg–Marquardt , which is described in detail by . This approach is based on the Newton–Gauss method to which the parameter γ is added that regulates the size of each iteration step in order to diminish the cost function compared to the previous step. The equation for this iteration may be expressed by xi+1=xi+[(1+γ)Sa-1+KiTSϵ-1Ki]-1{KiTSϵ-1[y-F(xi)]-Sa-1[xi-xa]}, where K is the Jacobian containing the sensitivities of each of the parameters of the state vector to each individual measurement. K acts as a matrix of weights in Eq. () and may also be referred to as the weighting matrix or kernel. When convergence is reached, the variance–covariance matrix of the retrieved state vector x^ is given by Sx^=(Sa-1+KTSϵ-1K)-1, where K and Sϵ correspond to the last iteration. The matrix Sx^ allows us to identify the error on each retrieved parameter. Convergence is obtained when the following convergence test is true [y-F(x^)]TSϵ-1[y-F(x^)]<N, where N is the number of elements in the measurement vector.

The application of this theoretical framework to the lidar retrieval problem requires the definition of all necessary elements described above. The state vector x contains the desired quantities to be retrieved. These are in our case a profile of extinction (denoted by σp) outside the cirrus cloud and a profile of IWC inside the cloud layer, x=[σp(r1),σp(r2),⋯,σp(rj_bot-1),IWC(rj_bot),⋯,IWC(rj_top),σp(rj_top+1),⋯,σp(rN)]T, where the subscripts j_bot and j_top denote the range index of the bottom cloud layer and the top cloud layer, respectively. The measurement vector y consists of the logarithm of the calibrated range-corrected lidar signal, y=[ln(C⋅P(r1)r12),ln(C⋅P(r2)r22),⋯,ln(C⋅P(rN)rN2)]T.

As mentioned in Sect. , the measurements provide information about the backscattering particles every 15 m and the same constant vertical resolution is used in the state vector.

Following , the forward model F is given by the lidar equation in its logarithmic and discretized form: F(xj,bj)=ln(βm(rj)+k(rj)σp(rj))-2∑l=1j[σ¯m,l+ησ¯p,l]ΔR, defined at each range rj, where j=2,…,N. The overline indicates layer mean values. As discussed in Sect. , the multiple-scattering factor η is set to unity for aerosols and 0.75 for cirrus clouds, and ΔR is the range resolution of 15 m of the lidar system. The state vector x defined in Eq. () contains the IWC inside the cloud; hence the extinction σp(rj), which is required in Eq. () is calculated as a function of IWC, σp(IWC(rj)), for all rj values inside the cloud layer with the BV2015 microphysical model described in Sect. . Vector b in Eq. () represents the non-retrieved parameters and is defined below (see Eqs.  to 18).

The Jacobian K given in Eq. (12) contains the sensitivities of the forward model to each element of the state vector, for which the terms F(xj,bj), σp(rj) and IWC(rj) have been shortened to Fj, σp,j and IWCj, respectively, to increase the readability. This short notation will be used for all variables that are a function of range for the remainder of this article. Inside cirrus cloud layers, the partial derivatives are expressed by Kij=∂Fi∂IWCj=∂Fi∂σp,j∂σp,j∂IWCj, and can be obtained from the BV2015 parametrization. The partial derivatives with respect to extinction are calculated by differentiating Eq. () with respect to σp: Kij=0fori<j-2ηΔRfori>jkiβm,i+kiσp,i-2ηΔRfori=j.

It should be noted that in the case of opaque cirrus clouds that completely attenuate the lidar signal, the size of the measurement vector and consequently the size of the state vector are reduced. In this case, only the altitudes until full attenuation of the lidar signal are considered. Thus, the size of the Jacobian is reduced as well and it contains only Natt lines (and columns), where Natt is the number of levels until the altitude of full attenuation.

Following , all variance–covariance matrices are assumed to be diagonal. Hence, the variance–covariance matrix of the a priori state vector can be defined by Sa,ii=σa,i2, where σa,i represents the variances of each of the elements of the a priori state vector. In this study, we have assigned sufficiently large variances to the a priori state vector in order to mainly rely on the information contained in the measurement vector.

As mentioned above, vector b in Eq. () represents the non-retrieved parameters, which are each a function of altitude, b1={βm,j;j=1,⋯N},b2={kj;j=1,⋯N},b3={ηj;j=1,⋯N}. The molecular extinction σm does not need to be considered a non-retrieved parameter because it can be obtained from βm by multiplication with the constant molecular backscatter-to-extinction ratio of 3/8π . The variance–covariance matrix of the forward model and the measurement is then defined by Sϵ,ii=σy,i2+σb1,i2+σb2,i2+σb3,i2, where σy,i represents the measurement error and σb1,i, σb2,i and σb3,i represent the errors on the non-retrieved parameters in the forward model calculated via σb1,i=pβ(%)⋅βm,iβm,i+kiσp,i,σb2,i=pk(%)⋅kiσp,iβm,i+kiσp,i,σb3,i=pη(%)⋅(-2ηΔR), where pβ(%), pk(%) and pη(%) represent the percentage errors assumed for the molecular backscattering profile, the backscatter-to-extinction ratio and the multiple-scattering factor, respectively. As discussed in Sect. , we chose large errors on the multiple-scattering factor for ice clouds and the backscatter-to-extinction ratio for aerosols and quantified them to pη(%)=25% and pk,aer(%)=25%, respectively. The same error has been attributed to the backscatter-to-extinction ratio for ice clouds since the knowledge of the phase function in the exact backscattering direction which is used in Eq. () is rather poor as well (pk,ice(%)=25%). The error on the molecular backscattering profile, which is obtained from the empirical equation of using the atmospheric temperature and pressure profiles from ECMWF reanalysis, is set to pβ(%)=2%. The error on the lidar measurement depends on the altitude since the measurement noise increases with increasing altitude. It is calculated as the standard deviation around the mean over a vertically sliding window of 20 gates.

It should be noted that the characterization of the errors related to the BV2015 parametrization is very challenging. Thus, no error for the microphysical model is currently taken into account. However, this issue needs to be addressed in future studies, and an evaluation of the uncertainty arising from the parametrization, which has to be integrated in future versions of our algorithm, is planned. This evaluation could be performed by comparing the single-scattering properties calculated directly from the ensemble model of with the results from the parametrization. Unfortunately, this is very costly to realize for all couples of IWC and temperature, particularly since the parametrization has not been developed by us. Furthermore, the uncertainty arising from this parametrization is assumed to be smaller than 5 % (Anthony J. Baran, personal communication, 2018). Consequently, the present study neglects an error related to the microphysical model.

The synergy algorithm is an expansion of the lidar-only algorithm, which integrates the TIR radiometer measurements in the optimal estimation method. The new state vector contains, in addition to the elements of the previous state vector given by Eq. (), the correction factor κ for the phase function in the exact backscattering direction: x=[σp(r1),σp(r2),⋯,σp(rj_bot-1),IWC(rj_bot),⋯,IWC(rj_top),σp(rj_top+1),⋯,σp(rN),κ]T, where the new phase function P11′(π) in the backscattering direction is related to the previous one via P11′(π)=κ⋅P11(π).

The measurement vector, initially containing the logarithm of the calibrated range-corrected lidar signal, is expanded by the measured radiances from the two channels of the CLIMAT instrument discussed above: y=[ln(C⋅P(r1)r12),ln(C⋅P(r2)r22),⋯,ln(C⋅P(rN)rN2),LC11,LC12]T.

The forward model for the lidar is the same as in the case of the lidar-only algorithm, given by the lidar equation in the form of Eq. (), with the only modification that the backscatter-to-extinction ratio for ice cloud layers is now calculated by k′=ϖ0⋅P11′(π)=ϖ0⋅κ⋅P11(π).

The forward model for the TIR radiances is the abovementioned radiative transfer model LIDORT . The advantage of this model is that it provides not only radiances but also weighting functions for atmospheric and surface parameters. That means the Jacobians for surface parameters such as emissivity or temperature; profiles of Jacobians for the temperature, atmospheric gases, or IWC profiles; and column Jacobians for the integrated quantities, for example the (Jacobian about the) integrated water vapor in the whole atmospheric column, can be obtained together with the radiances from one single simulation. Therefore the use of this model considerably reduces the computation time of the algorithm in comparison to finite-difference calculations to obtain Jacobians. This numerical efficiency allows the use of a fine vertical resolution in the radiative transfer calculations without exceeding reasonable computation times. Hence, the radiative transfer calculations can be realized on a vertical grid corresponding to the lidar resolution inside the cirrus cloud (outside the cloud the vertical resolution is defined by the ECMWF reanalysis profiles on 137 levels). That means for the radiative transfer calculations in our algorithm, the extinction profile inside the cirrus cloud is calculated with the BV2015 microphysical model from the IWC given on the lidar resolution. As a consequence, thanks to the synergy with the lidar measurements and to the numerical efficiency of the radiative transfer model, our algorithm does not have to assume a homogeneous cloud like most inversion algorithms for cloud properties do.

The Jacobian of the synergistic algorithm contains, in addition to the Jacobian of the lidar-only algorithm, two new rows for the sensitivity of the TIR forward model to each state vector parameter and one new column for the sensitivity of the forward model to the new state vector element κ (Eq. 28).

The sensitivities of the TIR radiances to the extinction profile outside the cloud are set to zero since they are assumed to be small. The correction factor κ does not have a direct influence on the TIR radiances; thus the last two elements in the last column of the Jacobian matrix (K) are also set to zero (see Eq. 28). The sensitivities of the TIR radiances to the IWC profile inside the cloud are calculated directly in LIDORT. Finally, the partial derivatives of the lidar forward model with respect to κ are set to zero outside the cloud and calculated analytically as a derivation of the forward model (Eq. ) for the ice cloud layers, ∂Fj∂κ=ϖ0⋅P11(π)⋅σp,jβm,j+ϖ0⋅P11(π)⋅κ⋅σp,j, where the layer extinction σp,j is calculated from the IWC with the BV2015 parametrization.

As in the lidar-only algorithm, the variance–covariance matrices in the synergy algorithm are also considered to be diagonal. Concerning the lidar, they are defined in the same way as in the lidar-only algorithm (see Sect. ) with the only difference that the error for the backscatter-to-extinction ratio in ice cloud layers is no longer considered since with the new algorithm we retrieve a correction factor for the phase function in the backscattering direction that is directly related to the backscatter-to-extinction ratio. Instead, an error of 1 % on the single-scattering albedo is integrated (compare with Eq. ). For the variance–covariance matrix of the TIR forward model the considered non-retrieved parameters (and the errors attributed to them) are the following: surface emissivity (2 %), surface temperature (1 K), and the profiles of atmospheric temperature (1 K for each layer), water vapor (10 % for each layer) and ozone (2 % for each layer). The standard deviations are calculated via σbj=∂F∂bj⋅bj⋅pbj(%)100, where bj represents the considered non-retrieved parameter, pbj(%) its error in percent and ∂F∂bj the sensitivity of the forward model to this parameter. As mentioned above, the latter can be calculated directly in LIDORT for all desired parameters (for a detailed description of the calculation of Jacobians in LIDORT the reader is referred to the LIDORT User's Guide; ). The elements of the diagonal variance–covariance matrix are then given by Sϵ,iiTIR=σy,iTIR2+∑jσbj,iTIR2, where σy,iTIR2 represents the measurement errors for each of the two channels of the TIR radiometer. This error depends on the calibration procedure and on the temperature of the instrument during the measurement because its sensitivity is a function of temperature. The calibration of the instrument is performed in the laboratory at room temperature. Unfortunately, our instrument does not have a thermal enclosure system and during field measurements it is exposed to atmospheric temperature influences. On 30 November 2016, the atmospheric temperature was low. Due to the poor knowledge of a coefficient to correct for the instrument's temperature, the assumed errors on the measured radiances are rather large. In particular, for channel C11 the error arising from this temperature correction ranges between 15 % and 30 % depending on the value of the radiance. The largest error percentages occur for small normalized radiances in combination with a cold instrument temperature. The radiances measured by channel C12 are larger and hence the error on the measurements of this channel is smaller and ranges between 5 % and 8 %.

This section presents some preliminary results of our new algorithm. Figure  shows the same example as given in Fig.  but obtained from the synergy algorithm. The a priori assumptions for the extinction and IWC profiles in the synergy algorithm are the same as in the lidar-only algorithm (see Sect. ). Figure b shows that once the algorithm converged, the lidar forward model and the measured lidar signal overlay each other almost perfectly. As in the lidar-only algorithm, the relative difference between the forward model and the measurement is smaller than 1 % for all layers. Thus, the good convergence found in the lidar-only algorithm is confirmed in the synergy algorithm. Table  summarizes the radiometer measurements and the TIR forward model (LIDORT) corresponding to this profile before and after the algorithm's convergence (expressed in normalized radiances in Wm-2sr-1µm-1). Since the values of the TIR forward model after the iteration process are within the error range of the measurements, it can be concluded that the algorithm converged in the TIR as well.

The retrieved value for the correction factor κ for the phase function in the backscattering direction is 1.48 ± 0.33, which is close to the range of 1.5 to 2.0 found in the literature and confirms the result shown in Fig. . The corresponding retrieved IWC and extinction profiles are shown in Fig. . By comparing them to the result of the lidar-only algorithm (Fig. ), it is obvious that the IWC and the extinction are smaller for the synergy algorithm because the backscatter-to-extinction ratio in the ice cloud is larger. The resulting IWP is 6.13±2.19 gm-2 compared to the initial IWP of 10.32±2.47 gm-2 from the lidar-only algorithm. As a consequence, the COT is also considerably reduced to a value of 0.239±0.085 compared to 0.402±0.096 from the lidar-only algorithm and is in good agreement with the COT of 0.267±0.126 derived from the transmission method considering the error ranges. This indicates that the result of the synergy algorithm is more coherent than the result of the lidar-only algorithm because the backscatter-to-extinction ratio has been characterized more realistically.

The application of the synergy algorithm to the profile measured at 16:20 UTC on 30 November 2016 results in a factor κ of 2.15±0.33. This value for the correction factor corresponds to the region where the simulated TIR radiances converge to the measurements in Fig. . Table  shows the radiometer measurements and the TIR forward model after the iteration. As for the profile at 18:11 UTC, the values of the TIR forward model after the iteration process are within the error range of the measurements. Thus, it can be concluded that the algorithm found a solution allowing both the lidar and the TIR forward model to converge towards the corresponding measurements. For the retrieved correction factor, the large IWC peak at the cloud top obtained from the lidar-only algorithm for a correction factor of κ=1.0 is considerably reduced, which results in a more realistic shape of the IWC profile. The IWP obtained from the synergy algorithm is 7.79±2.54 gm-2 compared to the abovementioned value of 32.21±8.93 gm-2 from the lidar-only algorithm. Hence, the retrieval with the synergy algorithm results in an important decrease in the IWP. However, the COT of 0.304±0.099 obtained from the synergy algorithm underestimates the COT of 0.608±0.186 obtained from the transmission method.

Finally, Fig.  presents the temporal evolution of the retrieval results from the synergy algorithm for the time period from 15:00 to 20:00 UTC on 30 November 2016. Figure a reiterates the measured lidar signal already shown in Fig. a, Fig. b shows the retrieved IWC profiles, and Fig. c shows the retrieved correction factor κ for the phase function in the backscattering direction (blue) and the corresponding lidar ratio in steradians (red), which might be easier to interpret. Figure d shows the comparison of the COT obtained from the synergy algorithm (blue) and the transmission method (red), and Fig. e presents the TIR radiometer measurements for channels C12 and C11 in green and blue, respectively, and the forward model after convergence for channel C12 in red and for channel C11 in violet, including their uncertainties. This plot indicates that the majority of retrievals converge well in the TIR. Furthermore, retrieval results are only shown in the other plots of this panel if the normalized cost function is much smaller than unity. Hence, the large number of results shown in this figure also indicates the overall good convergence of our algorithm. The only retrievals that did not converge correspond to either optically very thin clouds or to clouds that are thick enough to attenuate the lidar signal completely (e.g., around 19:48 UTC). In the second case, this could be related to the reduced size of the Jacobian mentioned in Sect. . However, we believe that this non-convergence is more likely due to physical reasons since the cloud base at this time was located in low altitudes (around 6 km) and the temperature at this altitude was rather warm for a cirrus cloud (between 245 and 250 K). In this temperature range, the presence of supercooled liquid droplets is possible, which is not included in the BV2015 microphysical model. Hence, this model probably does not represent the optical properties of this cloud accurately enough. Nevertheless, compared to the lidar-only algorithm the synergy algorithm converged for more profiles between 19:24 and 20:00 UTC and the retrieved COT compares well to the COT from the transmission method during this period. Hence, the TIR helped to constrain the backscatter-to-extinction ratio through the IWP, allowing a better coherence between the visible and the TIR forward model.

However, between 16:00 and 18:18 UTC the COT obtained from the synergy algorithm underestimates the COT derived from the transmission method for most of the retrievals (except around 18:12 UTC). It should be noted that the COT obtained from the transmission method is an effective COT * and that the real COT depends on the multiple-scattering factor for which COT*=η⋅COT. Hence, for Fig. c, the effective optical thickness obtained from the transmission method has been divided by the assumed multiple-scattering factor for ice clouds (η=0.75) in order to be consistent with the retrievals from the synergy algorithm. Thus, this corrected COT depends strongly on the assumed multiple-scattering factor. Applying a larger value of η (and hence reducing the effect of multiple scattering) would reduce the optical thickness obtained from this method and would result in a value that is closer to the result from the synergy algorithm. Conversely, the COT retrieved with the synergy algorithm is constrained by the TIR radiometer measurements and remains constant when applying another multiple-scattering factor. In the synergy algorithm, the retrieval of the correction factor κ for the phase function in the backscattering direction would change and hence the microphysics of the cirrus cloud would change. The influence of the multiple-scattering factor on the retrievals of our synergy algorithm has to be further investigated in future studies in order to draw more sophisticated conclusions and the retrievals shown here should be understood as a first test to show the potential of the algorithm.

However, the multiple-scattering factor alone cannot explain the inconsistency between the COT retrieved with the synergy algorithm and the COT derived from the transmission method. Another possible reason for this discrepancy may arise from the uncertainty in the transmission method itself because it depends on a good characterization of the molecular signal above the cloud and a good estimation of the cloud-top altitude. These parameters are related to rather large uncertainties due to the quite noisy micro-pulse lidar signal in the high altitudes of cirrus clouds. Furthermore, the discrepancy between the two COTs could also originate from a potential bias in the TIR radiometer measurements due to an inaccurate temperature correction as mentioned in Sect.  or from a potential bias in the TIR forward model due to an inaccurate description of the atmospheric water vapor profile since the TIR radiometer measurements are very sensitive to water vapor . Finally, the difference in the COTs from the synergy algorithm and the transmission method could also originate from the microphysical model, which might not be perfect. The extinction at the lidar wavelength, which is calculated based on the IWP constrained by the TIR radiometer measurements, could be slightly underestimated. Figure c shows that if the IWP is larger, the difference between the COTs from the two methods becomes smaller. This can be explained by the fact that the contribution of the water vapor in the TIR radiometer measurements is more important for thin clouds than for thick clouds, leading to an underestimation of the IWP and consequently an underestimation of the extinction, especially in the case of thin cirrus clouds.

Despite these limitations, the retrievals of the lidar ratio shown in Fig. d are promising. For the geometrically and optically thicker cloud between 16:00 and 17:12 UTC when a considerable increase in the measured TIR radiances was observed (see Fig. ), the average value of the retrieved lidar ratio is 35.6 ± 4.5 sr, which is in agreement with the lidar ratios for cirrus clouds reported in the literature ranging between 20 and 40 sr e.g.,. Between 17:12 and 18:12 UTC the retrieved lidar ratios are much higher (on average 52.2 ± 17.0 sr) but the cloud observed during this period is optically very thin and the signal in the TIR radiances very small, so it is not surprising that our algorithm reaches its limit here. For the optically thicker cloud between 19:12 and 20:00 UTC the average of the retrieved lidar ratio is 35.8±9.1 sr, corresponding to the literature again. However, as discussed above, the retrievals of the correction factor κ and thus the lidar ratio depend on the assumed multiple-scattering factor.

Nevertheless, this new synergistic algorithm suggests that using information from both, active in the visible part of the electromagnetic spectrum and passive in the TIR part, allows us to obtain new information on bulk ice optical properties, especially on the amount of ice and its capability to backscatter the visible light. Moreover, it allows us to test existing microphysical models, particularly the BV2015 model and its original representation of bulk optical properties as a function of the in-cloud temperature and IWC. The results of this study point out the overall good coherence of the BV2015 model but also its limitations in representing all the different measured profiles, especially due to the poor representation of the exact backscattering characteristics of the bulk ice.