Xiang Li
email: xiangli496@foxmail.com

Rodolphe Vauzelle

Yannis Pousset

Pierre Combeau

Hybrid propagation channel modelling for city area land mobile satellite communications

In this study, a hybrid model is proposed to simulate the realistic channel behavior for urban city land mobile satellite (LMS) communications. It associates the advantages of existing models in that different receiving states are predicted using a deterministic approach, whereas the channel behavior is simulated using adapted statistical laws. Using a geosynchronous satellite transmitter working at 1.5 GHz and a land mobile receiver, the hybrid model is validated for LMS channel. It leads to a high-performance simulation, combining accuracy (global simulation error less than 1 dBW) and small computation time (gain of about 10,000 times compared to a full deterministic reference model).

Introduction

Satellite communication systems are a valuable alternative to terrestrial systems and supports a wide range of applications, such as mobile radio communications, television transmission, radio localization, satellite Internet, military applications and so on. The land mobile satellite (LMS) channel is an important part of the system and is an ever expanding researching field. Many complex factors may influence this channel. In urban areas, for example, radio wave propagation is highly affected by buildings, terrain shape and other obstacles. Shadowing and multipath effects are present and degrade the LMS channel performance.

In order to predict channel behaviors in different propagation conditions, channel models are among the most useful solutions. They are developed to accompany or even to replace measuring campaigns which turn out to be time-and cost-consuming. These models rely on the understanding of channel characteristics and on simplifications, privileging either computation time or accuracy.

Three main families exist in LMS channel modelling: deterministic, statistical and hybrid models. The most used deterministic models are those based on geometric optics [START_REF] Keller | Geometrical theory of diffraction[END_REF][START_REF] Rg Kouyoumjian | A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface[END_REF]. They simulate detailed channel properties and can be directly applied to any 3D maps. However, their main drawback is the computation time due to high complexity in ray tracing algorithms.

Statistical models, on the other hand, are based on measured or simulated data and associated channel behavior analysis. Statistical parameters can be deduced from these data and fed into random generators to simulate the channel. The main advantage of statistical models is the low computation time. Typical LMS channel models can be found in the literature [START_REF] Loo | A statistical model for a land mobile satellite link[END_REF][START_REF] Lutz | The land mobile satellite communication channel-recording, statistics, and channel model[END_REF][START_REF] Ge Corazza | A statistical model for land mobile satellite channels and its application to nongeostationary orbit system[END_REF]. In general, statistical models are often validated for specific area types in a global manner, e.g., a single parameter setup is used in a city area. This may limit the accuracy and the generalization of these models.

Being able to combine the advantages of the two families above, hybrid models have attracted the attention of many researchers. In [START_REF] Oestges | Physical statistical modelling of the land mobile satellite channel based on ray-tracing[END_REF], the channel modelling was based on a conditional Rice distribution. Using a ray tracing technique (deterministic), Rician parameters (statistical) were proved to be related to building height, street width and other terrain parameters. In [START_REF] Perez-Fontan | An enhanced Markov chain based model for the narrowband LMS channel in built-up areas[END_REF][START_REF] Abele | A new physical-statistical model of the land mobile satellite propagation channel[END_REF][START_REF] King | Modelling and Measurement of the Land Mobile Satellite MIMO Radio Propagation Channel[END_REF], virtual propagation environments can be generated according to physical-statistical environment classification. This study, as well as [START_REF] Fontan | Statistical modeling of the LMS channel[END_REF], shows that different receiving states related to Line-Of-Sight (LOS) and Non-Line-Of-Sight (NLOS) may exist and can be modelled by a Markov chain simulator, whereas the global signal variations within each state was predicted by the Loo distribution [START_REF] Loo | A statistical model for a land mobile satellite link[END_REF]. The notion of receiving states was also reported in [START_REF] Berbineau | Satellite channel modelling using a Ray-tracing Tool for train communication[END_REF].

Our work belongs to the hybrid family. It is partially inspired by several existing works cited above and makes new contributions. For example, instead of generating virtual environments, we use some of the environment characteristics in [START_REF] Oestges | Physical statistical modelling of the land mobile satellite channel based on ray-tracing[END_REF][START_REF] Tzaras | A physical-statistical time-series model for the mobile-satellite channel[END_REF] in order to classify our deterministic city environments. Different receiving states, as proposed in [START_REF] Perez-Fontan | An enhanced Markov chain based model for the narrowband LMS channel in built-up areas[END_REF][START_REF] Fontan | Statistical modeling of the LMS channel[END_REF][START_REF] Berbineau | Satellite channel modelling using a Ray-tracing Tool for train communication[END_REF] are also observed in our simulated deterministic signals. Here, the novelty of our study lies in the deterministic state-locating method. Moreover, the NLOS severe state ("blocked"), which was not considered in specific applications like [START_REF] Perez-Fontan | An enhanced Markov chain based model for the narrowband LMS channel in built-up areas[END_REF][START_REF] Berbineau | Satellite channel modelling using a Ray-tracing Tool for train communication[END_REF], is taken into account and thus makes the model general. Statistical parameters are estimated and validated according to different environment classes and satellite positions. The proposed model gives both computation efficiency and satisfying accuracy.

The remainder of this article is organized as follows: Section 2 describes the basis of the proposed model. The design of the hybrid model is presented in Section 3. In Sections 4 and 5, the model is implemented and its performance is evaluated with respect to reference data in terms of accuracy and computation time. Finally, we draw conclusion in Section 6.

Basis of the proposed model

In this section, the LMS channel behavior is analyzed. Based on the multipath nature, a deterministic statelocating method is proposed, followed by a state-specific statistical modelling. In this study, we use the Ergospace software to simulate LMS communications in different configurations. This commercial software is based on ray tracing and is purely deterministic. 3D maps, satellite transmitter, antenna type and receiver course are among its input data. It has to be noted that we only use Ergospace as a tool but the concept is entirely independent of it and can be implemented using any ray tracing software.

LMS channel analysis

The simulations in this study are configured with a geosynchronous satellite working at 1.5 GHz (L-band) at different longitudes and a land mobile receiver travelling through city environments, e.g., dense urban, urban and suburban areas of a city. Figure 1a shows a typical simulated deterministic signal in terms of received power (in dBW) along a receiver course of about 600 m through the district called "Carmes", a dense urban area of Toulouse (France). The Effective Isotropic Radiated Power (EIRP) is 14.77 dBW (30 W). Figure 1b shows the associated cumulative density function (CDF).

Figure 1a and 1b exhibit three receiving states. The LOS state can be found around -168 dBW, with nearly stable signal power. The NLOS weak state lies between -187 and -178 dBW with more signal variations. The NLOS severe state is largely distributed from -240 to 187 dBW, indicating severe fading and rapid signal fluctuations. These observations are conform to previous studies like [START_REF] Perez-Fontan | An enhanced Markov chain based model for the narrowband LMS channel in built-up areas[END_REF][START_REF] Berbineau | Satellite channel modelling using a Ray-tracing Tool for train communication[END_REF] and lead to the studies of the next section.

Deterministic state-locating

In this article, we take advantage of ray tracing-based software to propose a deterministic state-locating technique.

We first assume that the occurrence of these different states is related to the multipath nature. It is known that the origin of the multipath effect is interaction between wave and obstacle. For a given path, radio wave may undergo p reflections (R) and/or q diffractions (D) before arriving at the receiver. These numbers are recorded by ray tracing software as simulation output. If we consider the combination of interactions (CI), this path can be tagged as pRqD. Also, it has to be noted that in satellite communications, the maximum number of interactions to be considered is two for reflection and one for diffraction. In fact, the signal will be too attenuated over this limit and the corresponding path is not calculated in the simulation. This limit is validated by Ergospace in collaboration with CNES (the French Space Agency) based on measured data and used in existent works like [START_REF] Berbineau | Satellite channel modelling using a Ray-tracing Tool for train communication[END_REF]. Hence, six types of CI are valid: 0R0D (direct path), 1R0D (single reflected path), 2R0D, 0R1D, 1R1D and 2R1D.

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Next, for a sample point, we can note a CI group by considering the CI types of the arriving path. For example, a sample point is noted "0R0D + 1R0D + 2R1D" if all received paths are of these three CI types. Note that since the method is based on multipath nature, the number of paths of the same CI is not considered. Now if each CI has a numeric weight, a CI group becomes a sum noted as Σ CI . It reflects the multipath nature on this sample point.

The choice of numeric weights is based on two criteria. Firstly, the CI associated with less attenuation will be assigned with a greater value. The order of magnitude for CIs can be easily determined: paths with more interactions will lose more power, so 0R0D is greater than 1R0D; diffractions result in greater attenuation than reflections [START_REF] Combeau | An optimization in computation time for the prediction of radio coverage zones[END_REF], so 1R0D is greater than 0R1D. Secondly, we have chosen a series of binary weights in order to make unique correspondence between Σ CI values and CI groups. Table 1 shows the chosen weights.

For example, a CI group of "0R0D + 1R0D + 2R1D" corresponds to Σ CI = 32 + 16 + 1 = 49 without ambiguity, according to this table. All possible values of Σ CI range from 1 to 63.

In this manner, each sample point is associated with a Σ CI value, giving birth to the Σ CI evolution along the received signal. The Σ CI evolution can be used to segment and locate different receiving states: we sort the Σ CI evolution from 1 to 63 (Figure 2a) and reorganize the signal sample points accordingly. As a result, sample points with similar multipath nature are now next to each other. Figure 2b shows the reorganized signal samples according to Figure 2a.

In this figure, each point at which Σ CI changes value can be considered as a divide of two different multipath natures. However, it may not be necessary to segment the signal in such detail, as we are mainly interested in locating different receiving states (LOS, NLOS weak and NLOS severe). Indeed, the LOS and NLOS states can be separated by Σ CI = 32 (the blue line in Figure 2) because it indicates the presence of a direct path. In other words, all sample points with Σ CI ≥ 32 are in LOS state.

As for NLOS severe and weak areas, the segmentation can be done for Σ CI = 16. This is justified because the value 16 corresponds to 1R0D, in other words, paths reflected only once and not diffracted. This is the predominant component in the absence of the direct path, as suggested in the studies [START_REF] Berbineau | Satellite channel modelling using a Ray-tracing Tool for train communication[END_REF][START_REF] Combeau | An optimization in computation time for the prediction of radio coverage zones[END_REF]. As a result, all sample points with Σ CI <16 are in NLOS severe areas, while others (16 ≤ Σ CI <32) are in NLOS weak areas. Hence, using the proposed Σ CI method, the signal can be segmented in three states in a deterministic manner.

Another conclusion can be drawn from the above analysis: in order to locate these states, we can simply rely on the presence of two CIs: 1R0D and 0R0D. This can drastically reduce the ray tracing complexity. Unlike the purely deterministic approach which allows up to two reflections and one diffraction, the state-locating only need to allow one reflection. The gain in computation time will be given in Section 5.2.

State-specific statistical modelling 2.3.1 Principles

It can be observed in Figure 2 that the received signal exhibits different degrees of large and small-scale fading variation according to the receiving state. Thus, these different degrees of variation by receiving state induce different degrees of stationarity. So, for each state, we aim at finding a suitable statistical law to model the signal behavior. In other words, we think that the pertinence of the chosen statistical distribution depends on the degree of stationarity of the signal. In order to quantify the stationarity, we propose to study three aspects of the received signal by state:

• the mean signal level (first order statistic);

• the standard deviation (second order statistic); • the covariance statistic within a state.

For the first two quantities, we use the well known sliding-window technique with a window size of 125 samples (about 48 l according to [START_REF] Parsons | The Mobile Radio Propagation Channel[END_REF]) to calculate the local mean and standard deviation in each window. So, more stationary the signal is, more constant these two quantities will be.

Concerning the covariance, we define it by the following expression: where X and X τ are two windows of the received signal, each containing N samples x i and x i+τ such that i = 1, 2, ..., N. These two windows are shifted with an offset noted τ. Moreover, in our study context, we again consider a window size of 48l (corresponding to about 125 samples).

R XXτ (τ ) = Cov (X, X τ ) = N i=1 x i x i+τ -N i=1 x i N i=1 x i+τ N (1)
Thus, more stationary the signal is by state, more stable the covariance will be for a given offset τ in different window pairs (X, X τ ). We conclude on the quasi stationarity of the signal if these covariances are appreciably equal.

Signal stationarity in different states

Figure 3a and 3b indicate the local mean and standard deviation of the sorted signal in Figure 2b, respectively. Compared to the NLOS severe and weak states, the LOS state shows greater mean values with very small dynamic range. If we calculate the standard deviation of these local mean values within a state, we find 0.61, 1.92 and 4.98 for LOS, NLOS weak and NLOS severe, respectively. For local standard deviation values, the dynamic ranges are found to be 0.36, 1.73 and 1.19, respectively.

Figure 4 indicates the covariance analysis with two offsets: 62 samples (≈24l) and 94 samples (≈36l) applied to the same signal. For example, Figure 4a means that the shifted window X τ is defined with an offset τ ≈24l with respect to the fixed window X.

It can be observed that, compared with the NLOS states, the LOS state still has very small covariance values (generally close to 0) with very small variation. Globally, we find an averaged standard deviation value (calculated from all the 125 different offsets) of 0.27 for LOS, compared to 3.04 for NLOS weak and 9.02 for NLOS severe.

Statistical model for the LOS state

From the above study, we can conclude that the signal is quasi stationary in the LOS state. We hence propose to consider a constant global mean Ā received power in LOS areas as the large-scale parameter. Then, we need to find a statistical model for the small-scale fading (which will be added to Ā). This refers to a procedure called law recognition detailed in [START_REF] Pereira | Application of information criteria for the selection of the statistical fast fading model of the radio mobile channel[END_REF]. In the context of satellite communications, the Nakagami-m model [START_REF] Nakagami | The m-distribution. A general formula of intensity distribution of rapid fading[END_REF] turns out to be the most adequate and robust. Equation (2) indicates its probability density function:

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!"#$.%32&2'2()2,*%320,',+,' #$.("#$.%320,',+,'- where =E{ r 2 } = r 2 is the signal variance and m the fading degree parameter. Therefore, three statistical parameters are necessary for LOS areas: constant global mean Ā,mand Ω. For the LOS segment in Figure 2, the signal modelled by Nakagami-m setup results in a RMSE of 0.48 dBW (using CDF comparison) with respect to the original signal.

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Statistical model for the NLOS states

The signal behavior for NLOS states is quite different from LOS state: it has low mean power and the local mean varies quickly. As a result, it can be intricate to separate the large-scale fading within an acceptable confidence interval. According to maximum likelihood estimation, the lognormal model (also used in [START_REF] Perez-Fontan | An enhanced Markov chain based model for the narrowband LMS channel in built-up areas[END_REF]) seems to give the best fit to describe the large-and small-scale fading together. Accordingly, two pairs of parameters should be used, one for NLOS severe (μ s and s s ) and the other for NLOS weak (μ w and s w) . For the NLOS severe and NLOS weak segments in Figure 2, the lognormal setup results in RMSEs of 0.84 and 1.48 dBW, respectively. This is much better than the Nakagami-m setup with RMSEs of 1.95 and 2.70 dBW.

Model design

Based on the principles discussed in the previous section, we present the model's workflow in this section. It works in two phases: the learning phase for parameter estimation and the application phase to actually simulate the "hybrid" signal.

Learning phase: parameter estimation

The learning phase begins by determining adequate statistical parameters of the previous laws (Nakagami and lognormal) in different receiving states. The basis of this step has been covered in Sections 2.2 and 2.3. In practice, we first choose a representative environment as reference scene and launch a purely deterministic simulation with Ergospace allowing a maximum of two reflections and one diffraction (the "classical" configuration). Secondly, the Σ CI method is applied to the simulated deterministic signal in order to find different states. Within each state, statistical parameters are estimated using maximum likelihood estimation for the corresponding law (Nakagami-m or lognormal). The parameter estimation is done only once per environment type.

It is also important to note that these parameters are valid for a given configuration. According to [START_REF] Perez-Fontan | An enhanced Markov chain based model for the narrowband LMS channel in built-up areas[END_REF], the main factors of a configuration are the transmitter position and the environment type. Concerning the former, different geosynchronous satellite positions are taken into account in this study. As for the environment type, a classification approach is necessary to regroup different environments according to their similarity. This is done by finding terrain parameters having a strong impact on radio wave communications. Different classifications can be found in the literature. In our study, we have considered the factors:

• Building density (S): percentage of area covered by buildings

• Meal building height ( h ) • Standard deviation of building height (s h )
In fact, these are the three most important factors in city area communications [START_REF] Mf Ibrahim | Signal strength prediction in built-up areas. Part 1: Median signal strength[END_REF][START_REF] Saunders | A physical-statistical model for land mobile satellite propagation in built-up areas[END_REF] and are also used in generating virtual environments in simulations [START_REF] Tzaras | A physical-statistical time-series model for the mobile-satellite channel[END_REF]. We have studied a total of six districts of Toulouse (Figure 5) and the classification result is given in Table 2. From this table, it can be observed that the building density S (%) decreases from dense urban to suburban. The mean building height h in dense urban is about 20 m. Urban and suburban areas share a smaller but similar mean height of about 16 m. This means the receiving areas become more open from dense urban to suburban areas.

On the other hand, the standard deviation s h is low in dense urban areas, meaning that the buildings are of similar heights. However, from urban to suburban, the building height become more and more variable, indicating a great diversity of building types. s h is also the main difference between urban and suburban areas.

Application phase: hybrid simulation

The hybrid model now proceeds to its application phase which actually simulates the signal, as indicated in Figure 6.

In this step, we choose an application scene in the same environment class as the reference scene. As suggested in Section 2.2, the deterministic module now works in an optimized configuration with a maximum of one reflection. It thus locates only the positions of the three states along a given receiver course. The signal samples are simulated through random generators configured according to statistical parameters estimated in the learning phase for a given environment class and satellite position. For example, the samples of the LOS state will be generated using the Nakagami-m random generator, overlapped onto a constant global mean. The NLOS states are generated agreeably to two different lognormal distributions (NLOS weak and NLOS severe). Finally, all generated samples are associated with sample locations indicated by the deterministic module.

Learning phase implementation

In order to apply the hybrid model, we choose two representative environments of each class in Figure 5, one serves as reference scene and the other as application scene. The model's learning phase is presented in the section.

Global state occurrence

Figure 7 shows the state occurrence rate for 13 different satellite positions in each of the three environment types of Figure 5a, c and 5e. The position of a geosynchronous satellite is indicated by its longitude. Note that we use the notion of longitude in our study because it is a directly configurable satellite-related parameter. Seen from Toulouse (latitude: 43°37'N, longitude: 1°26'E), from 300°to 360°, it is like that the satellite emerges and "goes up" until the highest position. Then, from 0°( or 360°) to 60°, it gradually "goes down" and disappears.

Dense urban areas

We observe that in dense urban areas with a low angle, e.g., 300°, a large part (79.6%) of the received signal is found in NLOS severe state, whereas LOS and NLOS weak states are less often present. These two states gradually increase with the longitude. At a high angle like 360°, the occurrence of the three states seems to be well balanced. Also note the increasing trend of NLOS weak state for dense urban areas. In this case, radio waves have more chance to arrive (through simple reflection) at the receiver when the satellite is high. Otherwise, multiple reflection is needed and the sample point is in NLOS severe state.

Urban and suburban areas

The state occurrence rates of these two areas are in general very different from dense urban areas. Even in the worst case (300°), we find a minimum of 42.2% of LOS state. This is mainly due to the difference in building density (S) and mean building height ( h ). These two areas share similar state occurrence statistics. This can be explained in referring to Table 2: urban and suburban areas are globally similar in terms of S and h while the factor s h makes them different. But as satellite transmitters are very far from the Earth, the building height variation becomes a minor factor on the state occurrence.

Statistical behavior in each state

LOS state

The Figure 8 shows that the global mean power Ā increases with the longitude and seems to be seldom affected by the environment class.

It has been pointed out in Section 2.3 that the signal mean power is considered to be constant due to the quasi-invariant transmitter-receiver separation. This is only true for a given satellite position. In fact, a change in transmitter position also modifies the angle of departure (off-nadir angle) and the angle of arrival (site angle or elevation) and thus results in different antenna gains. It is therefore necessary to study how the off-nadir angle and the elevation angle change with longitude. We find that due to the great distance of a geosynchronous satellite, the off-nadir angle does not change significantly. The transmitter gain is 12.9 dB at 300°and 12.4 dB at 360°. The receiver gain, on the other hand, can be determined using the antenna radiation pattern and the elevation angle. Figure 9 shows the radiation pattern of the studied antenna, with the elevation range (between 11.7°and 39.7°as calculated according to the longitude range) marked in gray. Generally speaking, higher satellite elevations (closer to zenith) result in higher receiver antenna gains. If we calculate and remove the influence of antenna gains (both transmitter and receiver), the curves in Figure 8 become flat with the signal power around -176 dBW, corresponding to the signal power received through freespace. !"#"$ %&'#"()* !(+,%(-($. .$"$/+()(0"(' 0(+1%#"(&)

!"#"(0"('#% ,#/#+$"$/0 2#).&+ *$)$/#"&/0 345/(. 0(*)#% 6$"$/+()(0"('7+&.1%$ !"#"(0"('#%7+&.1%$ Concerning the small-scale fading, Figure 10 indicates the evolutions of Nakagami-m parameters. When the satellite goes higher, we find m increase (less severe fading) while Ω decreases (less signal variations). Both observations confirm that when the transmitter is high, the influence of multipath decreases.

The influence of environment class is mostly reflected by signal variance, where dense urban areas have in general greater Ω values. In fact, more paths are created in dense urban areas due to tightly surrounding buildings.

NLOS severe state

The parameter evolutions for NLOS severe state are indicated in Figure 11. They seem to be similar in urban and suburban areas, but different in dense urban areas.

This difference is explained by the Figure 7. It shows that the state occurrence rates in dense urban environment is clearly different from those in urban and suburban ones. In particular, the occurrence of the NLOS severe state is significantly more important. This induce different dominant propagation phenomena: in the dense urban case, the probability of building blockage is very important as the environment is highly cluttered. Note that the occurrence of the NLOS severe state decreases for high satellite longitudes. Similar signal behaviors were in studies like [19]. In this article, the authors considered the Loo distribution to model the experimental data and defined three states named "LOS", "heavy shadow" and "light shadow" in different environments. They showed that the occurrence of the heavy shadow (corresponding to NLOS severe in our work) state and the channel model parameters are very different in dense urban areas.

NLOS weak state

We find greater values for both μ w and s w in dense urban areas (with respect to urban and suburban). This again can be explained by the abundance of multipath. More received paths contribute to higher signal power but also more rapid fluctuations.

However, for a certain environment class, the evolution of μ w and s w with respect to the longitude does not seem to be consistent, as shown in Figure 12. As no clear conclusions may be drawn from the data, detailed study is needed in the future to understand the statistical behavior of this transitional state.

Remark. From Figures 8, 10, 11 and 12, we see that the statistical parameters globally evolve in a parabolic manner. One possibility to process data is to apply polynomial fitting to these curves in a least-square sense. As a result, statistical parameters are modelled by polynomial laws and can be calculated for any longitude. In the upcoming application phase, however, as the state occurrence rate and the statistical parameter evolutions all have a certain symmetric behavior, we propose to consider only three representative satellite positions: 300°(low angle), 330°(mid angle) and 360°(high angle). Statistical parameters are calculated using the fitted coefficients and listed in Table 3.

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As Table 3 indicates, urban and suburban areas share similar statistical parameters. This follows well the discussions based on Figures 8, 10, 12 and 11 or Table 2. It is possible to simplify further the hybrid model if these two environments are regrouped by taking the mean value of related parameters in Table 3, provided that the accuracy is not greatly reduced. Further study is made in Section 5.1.

Application phase and performance evaluation

In the application phase, the hybrid model actually simulates the received signal, as described in Section 3.2. The application scenes of Figure 5b, d and 5f are used. This section presents the accuracy and computation time evaluations for the model.

Accuracy evaluation

In order to estimate the simulation accuracy, we plot the CDF of the simulated hybrid signal. The reference is the CDF of the purely deterministic signal simulated by Ergospace with classical 2R1D configuration along the same receiver course in the application scenes. Moreover, CDFs of the regrouped hybrid signals are plotted for urban and suburban environments using averaged statistical parameters of there two environment types in Table 3. All the simulations results are indicated in Figures 13,14 and 15. In order to quantify the simulation quality, we calculate the root mean square error (RMSE) in terms of dBW with respect to the deterministic CDF. The results are listed in Table 4.

For dense urban areas, it can be observed that the hybrid signal closely follows the deterministic signal. The RMSEs for 330°and 360°are 1.87 and 1.58 dBW, respectively. As for 300°, a RMSE of 3.55 dBW is relatively high, it is mainly due to the difference of the NLOS tail in Figure 13a.

The hybrid simulations are highly accurate as well in urban areas. The RMSEs for modelled hybrid signals are found to be 1.83, 0.92 and 0.90 for 300°, 330°and 360°, respectively. Regrouped hybrid signals, on the other

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To conclude, the modelled hybrid gives very satisfying simulation results in terms of accuracy. The regrouped hybrid may increase or decrease the simulation error. Therefore, the grouping of urban and suburban areas is feasible and gives acceptable approximations if we seek to further simplify our model.

Computation time evaluation

The simulation complexity of the hybrid model can be compared to a reference model. As discussed in Section 2.2, the reference is the ray tracing software, Ergospace, in its classical configuration. Basically, in ray tracing methods, allowing more interactions (especially diffractions) greatly increases the computation time. Ergospace, for example, requires 280 seconds to simulate a course of 7,000 samples on a test machine with a Pentium IV 3.0 GHz processor and 2 GB RAM.

The hybrid model consists of two parts: the statistical model is very fast thanks to random generators which take only 2 ms to simulate 7,000 samples on the same machine. On the other hand, the deterministic model is optimized as well with a maximum of one reflection and only locates different states. This simplification efficiently reduces the computation complexity. Neither the calculation of pathloss due to interactions nor the sum of all vector multipaths at the receiver is needed. As a result, the optimized deterministic state-locating for 7,000 samples only takes 25 ms. Combining two modules together, a gain of about 280 (25+2)×10 -3 ≈ 10370 times can be considered in comparison to a purely deterministic model.

Conclusion

We have presented in this paper a novel hybrid LMS channel model. Working on a two-phase basis, the model combines the accuracy of a deterministic approach and the speed of a statistical one. In the learning phase, the channel behavior is modelled using adequate statistical laws with adapted parameters. These parameters were modelled by polynomial fitting and reused in similar environments in the application phase. The model is highly accurate and gives insignificant simulation errors. It is also very optimized to greatly reduce the computation time.

Our studies can be further compared to measured data for its validity, this is part of the future work of this article.

Figure 1

 1 Figure 1 Example of a simulated signal in a dense urban environment of Toulouse: (a) signal envelop; (b) CDF of the signal.

Figure 2

 2 Figure 2 Application of the Σ CI segmentation method: (a) sorted Σ CI evolution; (b) associated signal.

Figure 3 Figure 4

 34 Figure 3 Statistical analysis of the simulated signal in Figure 2b: (a) First order; (b) second order. Red line: NLOS severe/weak separator; Blue line: NLOS/LOS separator.

Figure 5

 5 Figure 5 City environments of Toulouse: (a) Carmes; (b) Capitole; (c) St-Agne; (d) St-Michel; (e) Arènes; (f) Empalot. Receiver courses are marked in yellow.

Figure 6

 6 Figure 6 Model flowchart.

Figure 7

 7 Figure 7 State occurrence rate: (a) dense urban; (b) urban; (c) suburban.

) 7 Figure 8

 78 Figure 8 Global mean signal power evolution as a function of longitude.

Figure 10

 10 Figure 10 Nakagami-m parameter evolutions: (a) m; (b) Ω.

Figure 11 Figure 12

 1112 Figure 11 Lognormal parameter evolutions in NLOS severe state: (a) μ s ; (b) s s .

Figure 13 Figure 14 9 !Figure 15

 1314915 Figure 13 CDF comparison of dense urban ("Capitole") simulation results: (a) 300°; (b) 330°; (c) 360°.

Table 1

 1 Types of CI and associated weights

	CI	0R0D	1R0D	2R0D	0R1D	1R1D	2R1D
	Weight	32	16	8421

Table 2

 2 Environnement classification of six city areas

	Environnement	S (%)	h (m)	s h (m)
	Dense urban
	Carmes	50	19	9
	Capitole	55	20	10
	Urban
	St-Agne	27	14	13
	St-Michel	35	18	16
	Suburban
	Arènes	22	14	26
	Empalot	17	16	24

Table 3

 3 Statistical parameters

		LOS		NLOS weak		NLOS severe
	m	Ω (dBW)	Ā(dBW)	μ w (dBW)	s w (dBW)	μ s (dBW)	s s (dBW)
	Dense urban
	300°0.8	4.8	-171.8	-183.3	8.2	-196.2	9.1
	330°1.0	2.1	-167.5	-182.9	7.6	-193.0	10.7
	360°1.0	1.1	-165.9	-178.4	7.3	-192.2	11.3
	Urban
	300°0.8	3.5	-172.1	-183.5	8.7	-194.6	9.4
	330°1.0	1.0	-167.6	-183.5	6.0	-201.1	8.4
	360°1.1	0.0	-165.9	-186.1	4.9	-203.5	8.0
	Suburban
	300°0.9	2.1	-172.0	-187.5	6.7	-199.1	9.3
	330°1.1	0.3	-167.4	-187.4	5.9	-203.4	8.5
	360°1.1	0.0	-165.8	-183.4	5.8	-204.4	8.3

Table 4

 4 RMS simulation errors (in dbw)

	Dense		Urban		Suburban
	Hybrid	Hybrid	Regrouped	Hybrid	Regrouped
	300°3.55	1.83	1.81	1.54	1.69
	330°1.87	0.92	1.40	1.01	1.13
	360°1.58	0.90	1.47	0.72	0.80

Competing interests

The authors declare that they have no competing interests.


D Jacquemart

E Lellouch

B Bézard

C De Bergh

A Coustenis

N Lacome

B Schmitt

M Tomasko

New laboratory measurements of CH 4 in Titan's conditions and a reanalysis of the DISR near-surface spectra at the Huygens landing site

Laboratory spectra of methane-nitrogen mixtures have been recorded in the near-infrared range (1.0 -1.65 µm) in conditions similar to Titan's near surface, to facilitate the interpretation of the DISR/DLIS spectra taken during the last phase of the descent of the Huygens Probe, when the surface was illuminated by a surface science lamp. We used a 0.03 cm -1 spectral resolution, adequate to resolve the lines at high pressure (p N2 ~ 1.5 bar). By comparing the laboratory spectra with synthetic calculations in the well-studied ν 2 + 2ν 3 band (7515-7620 cm -1 ), we determine a methane absorption column density of 178±20 cm-am and a temperature of 118±10 K in our experiment. From this, we derive the methane absorption coefficients over 1.0-1.65 µm with a 0.03 cm -1 sampling, allowing for the extrapolation of the results to any other methane column density under the relevant pressure and temperature conditions. We then revisit the calibration and analysis of the Titan "lamp-on" DLIS spectra. We infer a 5.1±0.8 % methane mixing ratio in the first 25 m of Titan's atmosphere. The CH 4 mixing ratio measured 90 sec after landing from a distance of 45 cm is found to be 0.92±0.25 times this value, thus showing no post-landing outgassing of methane in excess of ∼ 20 %. Finally, we determine the surface reflectivity as seen from 25 m and 45 cm and find that the 1500 nm absorption band is deeper in the post-landing spectrum as compared to pre-landing.

Introduction

The Descent Imager /Spectral Radiometer (DISR) investigation onboard the Huygens Probe [START_REF] Tomasko | The Descent Imager/Spectral Radiometer (DISR) experiment on the Huygens entry probe of Titan[END_REF] has revolutionized our understanding of Titan's surface/atmosphere system, revealing traces of past flows on Titan's surface, probably driven by intense and episodic methane rain [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF]. Beyond this spectacular discovery, DISR also contributed in characterizing essential aspects of Titan's atmosphere, such as the distribution and properties of the haze (particle size, optical depth) and the wind profile as a function of altitude below 150 km. In addition to direct imaging, surface science with DISR included measurements of the surface spectrum. For this purpose, the DISR downward-looking spectrometers (DLVS and DLIS), operating respectively at 500-980 nm and 840-1700 nm, were associated with a 20 W surface science lamp (SSL) which was turned on in the last phase of the descent. The SSL illuminated the surface, providing light even in strong methane bands where solar radiation had been completely absorbed in Titan's atmosphere. Although the lamp was turned on near 620 m altitude, about 136 seconds before landing, its signal became visible only below about 60 m from the surface. Between this altitude and the landing, two DLVS and two DLIS spectra were recorded. After landing, DLIS and DLVS continued to acquire surface spectra from a distance of about 50 cm.

Even from an altitude of only a few tens of meters, and at the low resolution of the DISR spectrometers, methane absorption is clearly visible in the lamp spectra through signatures centred at 890, 1000, 1120-1160, and 1340-1420 nm. Retrieving a methane abundance from these features requires an appropriate knowledge of the methane absorption properties. Methane spectroscopy in the near-infrared (shortward of 1600 nm) is insufficiently known, as line-by-line information necessary for synthetic spectra calculations in any temperature, path, and pressure conditions (i.e. including line position, intensity, and lower energy level) is restricted to a few narrow intervals. This problem, which has in fact hindered analyses of near-infrared spectra of the giant planets and Titan for several decades, is usually being circumvented by the use of low-resolution "absorption coefficients". These are obtained from band models fit to laboratory data and recast in the form of exponential-sum coefficients. In effect, this approach accommodates the fact that at low resolution, the absorption does not follow Beer's law. Still, as demonstrated in Fig. 3 of [START_REF] Sromovsky | Near-infrared methane absorption in outer planet atmospheres: improved models of temperature dependence and implications for Uranus cloud structure[END_REF], no laboratory measurements, even at low spectral resolution, yet sample the full range of pressurestemperatures-CH 4 paths relevant to outer planet conditions. For the analysis of the entire DISR dataset, [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] used absorption coefficients based on the results of [START_REF] Karkoschka | Methane, ammonia, and temperature measurements of the jovian planets and Titan from CCD-spectrophotometry[END_REF] at wavelengths shorter than 1050 nm and on a modified version of the [START_REF] Strong | Spectral parameters of self-and hydrogen-broadened methane from 2000 to 9500 cm -1 for remote sounding of the atmosphere of Jupiter[END_REF] Goody-Voigt model at longer wavelengths. These exponential-sum coefficients were matched (i.e. spectrally convolved) to the spectral pixels of the DISR spectrometers, and calculated on a temperature/pressure grid appropriate to Titan. With this set of absorption coefficients (referred to as "the Karkoschka model" hereafter), [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] analyzed the highest S/N lamp-on DLIS spectrum --the last one taken before landing --and inferred a 5+/-1 % methane abundance in the first 21 m of Titan's atmosphere. They also found that the gas transmission model did not provide a perfect match of the absorption bands in the observed lamp spectrum, leaving in particular, residuals in the 1000 nm band and in the absorption shoulder near 1320 nm.

Recently, new absorption coefficients suitable to low temperature and long path conditions have become available longwards of 1050 nm [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF]. They were calculated from a combined reanalysis of the [START_REF] Strong | Spectral parameters of self-and hydrogen-broadened methane from 2000 to 9500 cm -1 for remote sounding of the atmosphere of Jupiter[END_REF] data and of recent, lower temperature (100-140 K) short-path measurements obtained by [START_REF] Sihra | Laboratory measurements of near-infrared methane bands for remote sensing of the jovian atmosphere[END_REF]. [START_REF] Sromovsky | Near-infrared methane absorption in outer planet atmospheres: improved models of temperature dependence and implications for Uranus cloud structure[END_REF] demonstrated the better agreement of the new coefficients with both the laboratory measurements of [START_REF] Mckellar | The spectrum of gaseous methane at 77 K in the 1.1-2.6 μm region: A benchmark for planetary astronomy[END_REF] at 77 K and the near-IR spectrum of Uranus. As discussed later, these coefficients (see http://www.atm.ox.ac.uk/user/irwin/methane.html), when used to calculate the atmospheric transmission in conditions typical of Titan's near surface (T = 94 K, p = 1.5 bar) predict less absorption in the core of the strong 1150 and 1360 nm than the Karkoschka model, as well as small differences throughout the spectrum. This of course has implications for the retrieved methane abundance.

The overall validity and merits of the existing sets of absorption coefficients has been explored by [START_REF] Negrão | Titan's surface albedo variations over a Titan season from near-infrared CFHT/FTS spectra[END_REF], based on a comparison with ground-based observations. Furthermore, they can be tested against the entire DISR spectral dataset, which offers the unique opportunity to probe the atmosphere in successive "slices". Such an analysis, performed by [START_REF] Tomasko | Measurements of methane absorption by the Descent Imager/Spectral Radiometer (DISR) during its descent through Titan's atmosphere[END_REF], has the ultimate goal of providing the community with reliable methane absorption coefficients in Titan's conditions (albeit at low spectral resolution), a task of utmost importance especially in view of the huge Cassini/VIMS dataset. With the more restricted objective of analyzing the DISR lamp-on surface spectra, another, simpler, approach is to acquire laboratory measurements of the methane absorption properties in conditions as similar as possible to the near surface of Titan, enabling a "direct" comparison with the DISR observations. This is the goal of the present paper. Laboratory measurements of CH 4 -N 2 mixtures are presented in the next section, and the absorption coefficients are extracted. Results are then applied to the DISR/DLIS spectra acquired just before and after landing, and results on the near-surface methane mixing ratio and surface spectrum are presented.

Laboratory spectra of methane-nitrogen mixtures

The rapid scan Bruker IFS 120 HR interferometer of the LADIR (Laboratoire de Dynamique, Interactions et Réactivité) was used to record laboratory spectra of methane-nitrogen mixtures, using a maximal optical path difference of 18 cm. The interferometer was equipped with a CaF 2 /Si beamsplitter, an InGaAs detector, and a Tungsten source. No optical filter was used, so that the covered spectral region was between 1000 and 1650 nm. For all spectra the whole optical path was under vacuum. A multipass cell of one meter base length was used for a total absorption path of 2015 ± 1 cm. The cell was equipped with KCl windows. The commercial methane gas sample, furnished by Air Liquid, with a stated purity of 99.99 % in natural abundances, was used. The temperature of the gas was recorded with four platinum probes located at different places inside the cell. In order to cool the gas, liquid nitrogen was directly injected in the second envelope of the cell. The pressure of the gas was measured with capacitance MKS Baratron manometers. The spectra were not numerically apodized, but they were slightly over sampled (over sampling ratio equal to 2) by post-zero filling the interferograms. Averaging 100 scans, the signal-to-noise ratio (peak to peak) is nearly equal to 200 around 1650 nm, but is decreasing to 20 around 1000 nm. This is mainly due to the response of the InGaAs detector. Note that in the pressure conditions of our spectra (1.5 atm of N 2 ), the apparatus-function width (FWHM = 0.033 cm -1 ) is negligible compared to the Lorentz line widths of methane, especially at low temperature (FWHM ~ 0.4 cm -1 around 100 K and 0.2 cm -1 around 300 K). The 100 % transmission level of the laboratory spectra was derived using spectra recorded with no gas, and in the same temperature and optical conditions.

We tried to reproduce as closely as possible the Titan surface conditions (p N2 ~1.4 bar, p CH4 ~ 70 mbar, T ~ 94 K). To record a low temperature spectrum, the cell was first filled at room temperature with methane and N 2 . Liquid nitrogen was then introduced in the second envelope of the cell. As the gas cools, the pressure in the cell decreases, so that N 2 was reinjected by steps until the appropriate experimental conditions were reached. However, several difficulties were encountered in this procedure. Due to direct injection of liquid nitrogen in the secondary envelope of the cell, complete thermalization of the cell was not achieved, leading to a temperature gradient. Indeed, the temperature sensors indicated a gradient of 20 K ranging from 83 K (for the sensor closest to the walls of the cell) to 103 K (for the sensor farthest from the walls). Because there was no temperature sensor in the middle of the cell, the average temperature of 93 K is certainly not representative of the actual mean temperature of the absorbing methane. Moreover, liquid nitrogen in contact with the external envelope of the cell led to non-negligible condensation of methane. Therefore the effective temperature and partial pressure of methane in the cell had to be determined a posteriori, as discussed below.

In view of a longer-term study of the line parameters of methane, we also acquired a number of spectra at room and intermediate temperatures. As explained below, the room temperature spectrum (500 scans) is of use for the present study. Its conditions are the following: T = 297 K, P CH4 = 0.1053 atm, P N2 = 0.894 atm. Unlike the low temperature spectrum, these conditions are well determined.

Analysis and determination of methane absorption coefficients

The first step of the analysis consists in determining the actual effective conditions (column density of methane, temperature) in the low temperature spectrum. For this, we compared the measurements with synthetic spectra in selected spectral ranges, based on the HITRAN database [START_REF] Rothman | The HITRAN 2004 Molecular Spectroscopic Database[END_REF]). As noted above, this and all other databases are extremely incomplete. For example the energy of the lower state (noted E") of the 28226 transitions present between 900 and 1600 nm is only known for 205 lines. This results from the high number of levels involved, leading to serious difficulties in transition assignments, even for cold bands. This limitation makes it impossible to calculate synthetic spectra at low temperature, except in restricted spectral intervals where assigned transitions are well isolated from other, non-assigned, transitions. We found that the 7515-7620 cm -1 portion of the ν 2 + 2ν 3 band matches this requirement. In this spectral range, 44 assigned transitions are present in the current version of HITRAN and come from the work of Hippler and Quack [START_REF] Hippler | High-resolution Fourier transform infrared and cw-diode laser cavity ringdown spectroscopy of the ν 2 +2 ν 3 band of methane near 7510 cm -1 in slit jet expansions and at room temperature[END_REF]; their quantum number J" varies from 0 to 9. Since the temperature dependence of line intensities varies significantly with J", this allows one to simultaneously determine the temperature and the column density of methane in the conditions of the spectrum. Synthetic spectra were calculated using a Lorentz profile. We first validated the calculation by fitting the laboratory spectrum recorded at room temperature. For this, knowing the lower energy levels is not necessary, so the HITRAN database can be used without modification. In the studied spectral range, all transitions in HITRAN have default constant values for broadening coefficients and their temperature dependence. Pressure shifts from N 2 cannot be neglected at the resolution of our measurements, and the pressure-shift coefficient of HITRAN (Rothman et al. 2004), about -0.014 cm -1 atm -1 between 7520 and 7620 cm -1 , was used. As can be seen in Fig. 1 (top panel), the synthetic spectrum (red curve) matches almost perfectly the experimental spectrum (black curve) recorded at 297 K with 0.1053 atm of methane and 0.894 atm of N 2 . For comparison, the calculation with only the assigned transitions from HITRAN is plotted as the green line in Fig. 1

(top).

One can observe that some of the transitions are not assigned. It is the case in particular of some lines around 7586, 7595, 7609, and 7618 cm -1 , that are observed in spectra at lower temperatures. We tried to estimate their lower energy levels E" from low temperature spectra. In the studied spectral region, the ν 2 + 2ν 3 band is well isolated, so we can assume that transitions with the same value of J" (same value of E") lie close to each other. This assumption seems to be confirmed by a new spectroscopic analysis of this region (Rey, Boudon, and Quack, private communication). We estimated the lower states of 45 transitions as having J" values between 1 to 9; some of them are certainly from the ν 2 + 2ν 3 band, while others may belong to other weaker cold bands, such as the 3ν 2 + ν 3 band (same priv. comm.). No true assignment has been performed in this work; we focused only on estimating the lower energy level of the transitions. Note that most of these new "assigned" transitions are quite weak, but will improve the global modelling of the low temperature spectrum. The line positions, intensities, and lower energy levels of the assigned transitions present in HITRAN, plus the line positions and intensities from HITRAN of the 45 transitions for which the lower energy level has been estimated, constitute what we will call HITRAN*. For this reduced line list, the default values for N 2 -broadening coefficients and their temperature dependence were replaced by values depending on J". The N 2 -broadening coefficients were taken from [START_REF] Perrin | Approximate radiative properties of methane at high temperature[END_REF], except for J" = 0, 1 and 2, for which average data of [START_REF] Pine | Multispectrum fits for line mixing in the ν 3 band Q branch of methane[END_REF] were used. Concerning the temperature exponent, values between 0.9 and 0.7 were used for values of J" between 0 to 9. Calculation at room temperature using HITRAN* transitions is plotted in green in Fig. 1(b); for comparison, a calculation including in addition the non-assigned transitions from HITRAN is shown in red.

We then turned to the low temperature spectrum. A grid of calculations using HITRAN* and various column densities and temperatures of methane is plotted on Fig. 2, demonstrating how the column density of methane and the temperature can be retrieved simultaneously. Using HITRAN* instead of HITRAN allowed to better model the groups of lines with J" = 0 to 9. As can be seen from the residuals in Fig. 2, the line list is still not perfect. The best fit solution is obtained for a column density of 178 cm am, and a temperature of 118 K (corresponding to a methane partial pressure of 0.038 atm). Note that the pressure of 0.038 atm corresponds to the vapour pressure at a temperature equal to 84 K [START_REF] Prialnik | Modeling the Structure and Activity of Comet Nuclei[END_REF], close to the coldest temperature in the cell. However, due to the absolute uncertainty of our temperatures measurements (± 3K), no accurate estimation of methane pressure can be performed using the lowest recorded temperature. We estimate an uncertainty of ± 11 % on the column density and ± 10 K on the temperature. These uncertainties may seem a little conservative given the grid of models of Fig. 2, but we here also account for the effect of the uncertainty on the 100 % transmission level. Note that between 7515 and 7520 cm -1 , no lines are assigned in HITRAN or HITRAN*, which explains why the calculation in Fig. 2 does not reproduce the experimental spectrum in this spectral range.

The low-temperature spectrum shows 4 absorption regions due to methane. However, only the 1100-1200 and 1300-1500 nm regions (respectively the triacontad and the icosad) will be used. The tetradecad region at 1650-1900 nm (on the edge of the detection possibilities for the InGaAs detector) falls outside of the DISR range. Fig. 3 shows the 900-1600 nm portion of the spectrum. The 1000 nm absorption region is also not used because of insufficient signal-to-noise in the laboratory spectrum. At wavelengths where the methane absorption is not unambiguously detected, the transmission is fixed to 1 (see Fig. 3). As mentioned previously, the instrumental convolution is negligible, so the measured transmission can be at each frequency converted to an absorption coefficient using Beer's law.

Application to DISR spectra a. Data selection and calibration

As discussed in [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], nine DLIS spectra with the Surface Science Lamp (SSL) on were recorded below 700 m altitude. Each of these spectra has a sampling time of 1.00 sec. Following [START_REF] Schröder | The reflectance spectrum of Titan's surface at the Huygens landing side determined by the Descent Imager / Spectral Radiometer[END_REF], we refer to these spectra by using their DISR internal number (#), mission time, and/or altitude, according to the trajectory reconstruction of [START_REF] Karkoschka | DISR imaging and the geometry of the descent of the Huygens probe within Titan's atmosphere[END_REF] (see Table 1). As outlined in [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], as soon as the lamp was turned on, the spectra showed an increase of signal at all wavelengths, due to scattered light from the lamp within the instrument. In addition, all spectra show a large contribution due to the diffuse solar field. This component actually dominates over the signal due to the lamp reflection from the surface at all wavelengths except in the strong methane bands. Only two (# 206 and # 210) out of the nine spectra show significant lamp signal reflected off the surface. They were acquired respectively at 55 and 24.8 m above the surface, where the altitude corresponds to the middle of the exposure. We focus on spectrum # 210, in which the surface signal is ~ 5 times stronger. To extract a "lamp-only" spectrum, we essentially followed the procedure outlined in [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], on which we elaborate here, with some minor differences.

In order to eliminate the instrument-scattered light and the diffuse solar light, the simplest approach is to subtract a "background" spectrum from the spectrum of interest. Ideally, the background spectrum should be a "lamp-off" spectrum of the same area taken at the same altitude and with the same solar azimuth with respect to the center of the instrument field-ofview. As this is not possible in practice, we proceeded empirically by exploring combinations of the other lamp-on spectra that would provide an "optimum" background. Because the solar light dominates so much over the lamp signal in regions of weak methane absorptions, in particular near 1130 and 1280 nm, a criterion for selecting the background was the requirement that the background-subtracted spectrum should be reasonably flat in these spectral regions, as it should be if all solar light has been properly eliminated. We found that spectra # 199 and # 202, taken respectively at 109 and 86 m altitude, provided appropriate background, and consistent with [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] that the mean of these two spectra allowed an optimum correction of the solar field. Fig. 4 illustrates the sensitivity of the resulting lamp-only spectrum to slightly different choices. Note that [START_REF] Schröder | The reflectance spectrum of Titan's surface at the Huygens landing side determined by the Descent Imager / Spectral Radiometer[END_REF] used instead spectrum # 202 multiplied by 1.01 for background reference.

Once recovered in this manner, the lamp-only spectrum must be divided by the spectral response of the SSL, which has been measured on the ground during instrumental calibration. We use here the relative spectral response derived by [START_REF] Schröder | The reflectance spectrum of Titan's surface at the Huygens landing side determined by the Descent Imager / Spectral Radiometer[END_REF] from a spline fit of the original measurements and correction for an absorption band at 1380 nm, caused by atmospheric water. This SSL spectrum also does not show a 1280 nm absorption that was present in the SSL response used by [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], and shows a little more decrease of power from 800 to 1600 nm.

For calibration of the lamp-only Titan spectrum in terms of an I/F reflectivity, we did not attempt to use the absolute SSL fluxes measured in the laboratory, as [START_REF] Schröder | The reflectance spectrum of Titan's surface at the Huygens landing side determined by the Descent Imager / Spectral Radiometer[END_REF] did. Instead, we followed and improved on the method used by [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF]. It consists of calculating the I/F reflectivity in regions of negligible methane absorption as the ratio of the upward intensity to the downward flux (πF) divided by π. The upward intensity is determined from the DLIS measurements, averaged over all lamp-on spectra except # 210 (i.e. spectra # 194, 196, 199, 202, 204 and 206). To estimate F, [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] averaged the two ULIS-measured spectra (among the nine taken in the last phase of the descent) having the strongest and the weakest intensity, respectively. As azimuth information is now available (Table 1), this approach can be refined. We averaged the three ULIS spectra with the Sun in the field-of-view (# 202, 206, and 210, taken at 86, 55 and 24.8 m, and with azimuths of 322, 5 and 45° respectively). This yielded a mean spectrum S 1 . We also averaged the two ULIS spectra with the Sun most distant from the FOV (# 190 and 194, taken at 708 and 200 m, and with azimuths of 146 and 166°), yielding a mean spectrum S 2 . Based on a set of synthetic calculations, including the spatial response of the ULIS diffuser, we found that the combination 0.4 x S 1 + 0.6 x S 2 provided a very good approximation to F. Compared to the initial analysis of [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], the slight change in the relative weights given to the "Sun" and "no-Sun" spectra can be understood from the ULIS diffuser response vs. zenith angle. With the DLIS and ULIS data used in this way, we finally determined I/F reflectivities of 0.166, 0.165, 0.155, 0.140 and 0.727 at wavelengths of 837, 938, 1079, 1282, and 1553 nm, with an estimated precision of ±10 %. As illustrated in Fig. 4, scaling the lamp-only spectrum by a constant factor provided a satisfactory match of these continuum reflectivities, and we will onwards refer to this calibrated spectrum as "the spectrum at 25 m". Beyond the methane absorptions, the spectrum shows a significant blue slope, although somewhat less than in its version originally shown by [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], a result of the newly adopted relative spectral response of the SSL. As compared to the spectrum presented by [START_REF] Schröder | The reflectance spectrum of Titan's surface at the Huygens landing side determined by the Descent Imager / Spectral Radiometer[END_REF], the main difference is in the absolute calibration, as our I/F reflectivities are about 40 % lower than theirs. The origin of this discrepancy is uncertain, although Schröder and Keller speculate on a possible phase angle dependence of the surface reflectivity.

We also considered DLIS spectra taken after landing. Specifically, we reanalyzed the first non-overexposed spectrum (# 249), recorded about 90 sec after landing and with a sampling time of 16 msec. About 20 other spectra were taken subsequently, but they all appear to be identical. Spectrum # 249 was corrected with the same "background" spectrum as spectrum # 210, and divided by the SSL relative spectral response. Because the lamp signal from a distance of ~50 cm is so overwhelmingly dominant over the solar light, the background correction is in fact insignificant. For absolute calibration, we simply rescaled the spectrum to the I/F continuum values of the spectrum at 25 m, referring to the result as "the post-landing spectrum". The 1050-1250 nm range of this spectrum is shown in Fig. 5, and Fig. 6 shows the entire post-landing spectrum after correction of its methane absorption. b) Determination of CH 4 near-surface mixing ratio and surface reflectance spectrum Method Determining the CH 4 mixing ratio in the Titan spectra from the laboratory spectra is straightforward in principle. The laboratory spectra allow one to determine "monochromatically" (i.e. at the resolution of the laboratory spectra) the absorption coefficient (in cm -1 -am -1 ) as a function of wavelength, k λ ,, from k λ = -1/a ln(T λ ), where a is the methane column density in the laboratory spectrum (178 cm-am). The "monochromatic" transmittance for any column b is then calculated as T ' λ = exp (-k λ b), and convolved by the DLIS instrumental function, which can be approximated with a Gaussian function with 15 nm FWHM. The methane column b is adjusted until agreement with the CH 4 bands in the Titan spectra is reached, and converted into a methane near-surface mixing ratio q (i.e. the ratio of the number of methane molecules per unit volume to the total number of molecules per unit volume at the surface level, also known as the mole fraction). Due to the short-wavelength cutoff of the laboratory spectra, we restrict ourselves to the 1050-1600 nm range. As the absorption in the lamp spectra is two-way and the DLIS spectrometer looks at a nadir angle of 21.4°, the absorption length is L = 2 z / cos(21.4°), where z is the altitude of the spectrum of interest. q is related to b through to b = q x p x 273.15/T x L, where p = 1.47 bar and T=93.7 K at Titan's surface [START_REF] Fulchignoni | In Situ measurements of the physical characteristics of Titan's environment[END_REF].

Post-landing spectrum [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] noted that the post-landing spectrum shows hints for methane absorption features but did not attempt to model them in the absence of information on the absorption path length. Based on the [START_REF] Karkoschka | DISR imaging and the geometry of the descent of the Huygens probe within Titan's atmosphere[END_REF] analysis, the DLIS is at 45 cm above the surface after the landing. With this distance, the clearly detected 1150 nm CH 4 absorption band indicates a methane mixing ratio q = 4.7±0.8 % (b = 19.5±3.2 cm-am, i.e. 9.1 times less than in the laboratory spectrum) where the uncertainty reflects only fitting errors (Fig. 5). To fit the band, which superimposes on a blue slope, a simple linear continuum was assumed. In contrast, the expected 1300-1450 nm absorption is not obviously present in the spectrum, as noted by Schroeder and Keller (2007), and is presumably masked by weak surface reflectance features.

Dividing at all wavelengths the post-landing spectrum by the methane transmission provides the surface reflectivity at the landing point (Fig. 6) from 840 to 1600 nm. Shortward of 1050 nm (where we do not have reliable laboratory measurements), it can be safely assumed that the methane transmission for such a small path is equal to 100 %. The surface reflectivity is similar to the one originally presented by [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], with minor differences due to the different choice of the SSL spectrum (in particular, the new surface spectrum shows a somewhat less blue slope and no emission "bump" near 1280 nm) and to this atmospheric correction. Except for a deep absorption longward of 1450 nm, the spectrum is remarkably featureless. The only other (possibly) significant feature is a reflectance maximum near 1360 nm. This feature is present in the post-landing spectrum uncorrected for methane, and is somewhat amplified by the atmospheric correction, as methane has absorption peaks at 1360 and 1400 nm and a local absorption minimum near 1375 nm.

Spectrum at 25 m

Figure 7 shows the fit of the spectrum at 25 m with various CH 4 abundances. Such a fit requires an assumption on the surface reflectance, and we first used the reflectance inferred from the post-landing spectrum. However, this approach led to an underestimate of the observed spectrum in its long-wavelength edge. To achieve a good fit, it was necessary to increase the surface reflectance at wavelengths longer than 1420 nm (Fig. 6), resulting in a slightly shallower 1500 nm absorption. As the DLIS was looking at different terrains from distances of 25 m and 45 cm, the difference -which was visible in Fig. 15a of [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] but not discussed -may conceivably be real. The detailed interpretation of the surface spectra, and particularly the origin of the 1500 nm band, will be presented in Schmitt et al. (2007, in preparation). As noted by [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], this band is reminiscent of water ice absorption, but the identification is not unique. In particular, bright yellow-orange tholins produced in laboratory experiments (and which are therefore plausible candidates on Titan's surface) also show absorption in the same range. Whatever the material responsible for this band, a different band depth may indicate a variation of abundance, grain size, and/or state of ice. Beyond this absorption, no significant difference can be found between the surface reflectance as inferred from the spectrum at 25 m and the post-landing spectrum. In particular, a 1320 nm mismatch, suggested by [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] on the basis of the "Karkoschka" coefficients does not seem to be significant any more.

With the adopted surface reflectance, the spectrum at 25 m indicates a best fit CH 4 column abundance b = 1165±90 cm-am, i.e. a mixing ratio of 5.1±0.4 % (Fig. 7). This column corresponds to an extrapolation by a factor of 6.5 from the laboratory conditions, but this is not a problem because the lines are fully resolved. The error bar is estimated from the fact that optimum fit of the 1150 nm band is achieved for 4.7 %, while the 1400 nm band would favour a 5.5 % mixing ratio. Additional errors, due to the uncertainty of the methane column density and temperature in the laboratory data, are discussed hereafter. Being systematic, these errors affect the methane determination in the post-landing and the 25 m spectrum in the same way. Thus the ratio of the post-landing to the pre-landing methane mixing ratio is 0.92 +0.25/-0.22, i.e., its upper limit is 1.17. The result is thus not consistent with the GCMS observation of a 40 % increase in the methane mixing ratio over the 2 minutes following the Huygens landing that was presumed to be due to the surface heating by the instrument and attendant evaporation from a liquid methane containing surface. Schroeder and Keller ( 2007) also find that the pre-and post-landing spectra give consistent amounts of methane.

Uncertainties

In addition to the ~ 8 % uncertainty associated with data fitting, the methane surface mixing ratio is affected by the adopted 11 % uncertainty on the methane amount in laboratory data. A third source of uncertainty, more difficult to estimate, comes from the fact that we are using absorption coefficients (in cm-am -1 ) determined at 118±10 K to model a spectrum at 94 K. To get a handle of the possible resulting error, we performed synthetic calculations on other methane bands for which line-by-line information is complete and reliable. Fig. 8 shows models in the region of the pentad (2400-3300 cm -1 ) and of the octad (3700 -4800 cm -1 ), at 118 K and 94 K. The models were convolved to a spectral resolution comparable to that of the DISR/DLIS resolution, and the amount of methane was adjusted in order to get bands of comparable depths than those observed on Titan. Namely, we used 15 cm-am for the pentad, and 80 cm-am for the octad, ensuring a minimum transmission of about 0.1, as in the Titan spectrum at 25 m. Fig. 8 shows the effect of the gas temperature on the band shape. Overall, the bands are slightly more saturated at 94 K than at 118 K (since fewer levels are being populated), which leads to a slightly smaller absorption equivalent width for the same column. Recovering the equivalent width calculated at 118 K with the synthetic spectrum at 94 K would require an increase of the CH 4 column by 10 % for the pentad and 12.5 % of the octad. This gives an estimate of the maximum error that would result from using absorption coefficients at 118 K to model an observed spectrum at 94 K in the region of the pentad and the octad. Note however the strong difference in shape between the 94 K and 118 K cases. In the case of Titan, the 1150 and 1400 nm band shapes are very well reproduced by our 118 K model. This argues for a much smaller effect than in the region of the icosad and triacontad. Perhaps still the effect is different in the 1150 and 1400 nm bands, possibly explaining why the two bands are best fit with slightly different CH 4 mixing ratios. In an attempt to account for this effect, we include an additional 5 % uncertainty, which is smaller than the other two sources of errors. Altogether, combining the error bars quadratically, we estimate a total 15 % uncertainty on the surface mixing ratio, which is finally estimated to be equal to 5.1±0.8 %.

For the post-landing spectrum, the individual error bars (17 % , 11 % and 5 %) combine to produce a 21 % uncertainty, and the final estimate of the methane mixing ratio is 4.7±1 %.

c) Comparison of absorption coefficients

To try and assess the quality of the absorption coefficients proposed by various authors, we compare in Fig. 9 fits of the 25 m spectrum using the "Karkoschka" model, the [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF] coefficients, and the coefficients derived in the present work. For all these models, we use a methane column density of 1165 cm-am. Clearly apparent is the fact that the Karkoschka model, which was used by [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF], produces slightly more absorption than the other models, and requires accordingly a ~20 % smaller methane amount to fit the spectrum. (Indeed, [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] determined a 960 cm-am best fit column. This still corresponded to a 5±1 % mixing ratio, consistent with our present value, because they assumed an altitude of 21 m for this spectrum). Fig. 9 also shows that the Karkoschka model leads to a mismatch in shape near 1320 nm, already noted by Tomasko et al. The use of the [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF] or of our own coefficients solves this problem for the most part. Longwards of 1050 nm, the Karkoschka model is based on the [START_REF] Strong | Spectral parameters of self-and hydrogen-broadened methane from 2000 to 9500 cm -1 for remote sounding of the atmosphere of Jupiter[END_REF] laboratory measurements, which were restricted to temperatures higher than 190 K. On the other hand, [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF] derived their coefficients from a joint analysis of the Strong et al. data with more recent measurements of pure methane down to temperatures of 100 K, obtained by [START_REF] Sihra | Laboratory measurements of near-infrared methane bands for remote sensing of the jovian atmosphere[END_REF]. The much better agreement of the [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF] model with the Titan spectrum is therefore not surprising, given also the fact that the Sihra measurements included amounts of methane (up to 273 mbar at 100 K for a cell length of 1326 cm, i.e. a column of 990 cm-am) appropriately similar to that present in the 25 m spectrum, and a pressure not too different. As discussed above, our coefficients were derived for methane-nitrogen mixtures, and at an effective temperature of 118 K. We therefore show a comparison of our model with two versions of the [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF] models, respectively at 118 K and 94 K. The agreement between the three models is generally satisfactory (except for a small deviation at 1100 nm, where our model gives a better match), although curiously, we find that the Irwin et al. model at 94 K starts to show some deviation with the data near 1340 nm. Note that using a slightly different version of the 25 m spectrum and the Irwin et al. coefficients, Schröder and Keller (2007) derive a mixing ratio 4.5±0.5 % from an overall fit of the four methane complexes, and achieve a very good match of the 1150 and 1350 nm absorptions.

In summary, we find that, in the conditions of Titan's 25 m spectrum, the [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF] coefficients provide predictions comparing generally well with our laboratory measurements, and do represent a significant improvement over the Karkoschka model. This however, does not necessarily mean that these coefficients are valid throughout Titan's atmosphere. In fact, while [START_REF] Negrão | Titan's surface albedo variations over a Titan season from near-infrared CFHT/FTS spectra[END_REF] found that the [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF] coefficients also permit a better fit of Titan ground-based spectra compared to the Karkoschka model, [START_REF] Tomasko | Measurements of methane absorption by the Descent Imager/Spectral Radiometer (DISR) during its descent through Titan's atmosphere[END_REF] point out serious deficiencies of the Irwin et al. coefficients, which must be scaled by factors of ~0.5 (in band centers) to several (in regions of weak absorption and windows) to fit the ensemble of DISR data. This does not contradict our finding that the Irwin et al. coefficients are adequate to match the "lamp-only" spectra, given that the Tomasko et al. study pertains to either short paths at low pressures (in band centers) or to very long paths (in the methane windows).

Summary

In order to fully analyze the DISR/DLIS spectra taken during the last phase of the descent of Huygens, we have recorded laboratory spectra in conditions similar to those of Titan's atmosphere. Extrapolating the spectra to the appropriate path lengths, we determine the CH 4 mixing ratio to be to 5.1±0.8 % in the first 25 m of the atmosphere, and 4.7±1.0 % for a spectrum taken 90 sec after landing. These figures agree with the initial estimate by [START_REF] Tomasko | Rain, winds, and haze during the Huygens probe's descent to Titan's surface[END_REF] (5±1 % for the pre-landing spectrum, then thought to be at 21 m altitude), based on the "Karkoschka" model, as well as from a more recent determination by Schröder and Keller (4.5±0.5 %) from the same dataset, using the [START_REF] Irwin | Improved near-infrared methane band models and k-distribution parameters from 2000 to 9500 cm -1 and implications for interpretation of outer planet spectra[END_REF] absorption coefficients. All these numbers agree with the Huygens/GCMS determination of 4.92±0.25 % near the surface [START_REF] Niemann | The abundances of constituents of Titan's atmosphere from the GCMS instrument on the Huygens probe[END_REF]. We also infer the surface reflectivity as seen from 25 m altitude and just after landing, and find a variation in the 1500 nm band depth.

Although the approach we have used is in principle the most straightforward, the precision of our determination has been in practice limited by the uncertainty in the actual conditions of our low-temperature laboratory spectrum. Ultimately, an accurate modelling of the methane absorptions in Titan spectra will demand a line-by-line knowledge of the methane parameters in the entire near-infrared and visible range. From the experimental as well as theoretical point of view, this is a formidable task. The spectra we have acquired for the present study will contribute to this effort. We plan to complete them by other laboratory measurements, especially in pure methane at low pressures and several low temperatures, and at a even higher spectral resolution. Fig. 1. Modelling of the room temperature spectrum in the region of the ν 2 + 2ν 3 band. The experimental spectrum recorded at 297 K with 0.1053 atm of methane and 0.894 atm of N 2 , is plotted in black. In the upper panel, the calculation using all HITRAN transitions is plotted in red, and the one using only assigned transitions of HITRAN is plotted in green. In the lower panel, the calculation using HITRAN* (see text) plus the non assigned transitions from HITRAN is plotted in red, and the one using only HITRAN* is plotted in green. Differences between measurements and models are shown. The lower quantum number J'' for series of transitions of the ν 2 + 2ν 3 band has been indicated on the lower plot. For more visibility, only the even number of J'' have been reported.

Fig. 2 .

 2 Fig.2. Modelling of the low-temperature laboratory spectrum (shown in black) at 7515-7620 cm -1 . Synthetic spectra of methane broadened by N 2 are calculated using HITRAN* (see text) on a grid of temperature and methane column density. The absorption path is equal to 2015 cm, and the pressure of N 2 is equal to 1.47 atm. Horizontal lines (from top to bottom) correspond to methane column densities of 158, 178 and 197 cm-am, respectively. Vertical columns (from left to right) correspond to temperatures of 108, 118, and 128 K, respectively. In each panel, the curve at the bottom shows the (obs.-calc.) residuals multiplied by 2. Best fit is achieved for 178 cm-am and 118 K (central panel).

Fig. 3 .Fig 4 .

 34 Fig. 3. Low temperature transmission spectrum used to reproduce the DISR/DLIS spectra.The experimental conditions corresponding to this spectrum are P N2 = 1.47 atm, L = 2015 cm, T = (118±10) K, and a methane column density of 178+/-20 cm-am, i.e. P CH4 = (0.038±0.004) atm.

Ackowledgments:

We acknowledge financial support from the French "Programme National de Planétologie". We also acknowledge Stefan Schröder for "illuminating" discussions, and for a digital version of the Surface Science Lamp spectrum. We would like to thank gratefully Michaël Rey, Vincent Boudon and Martin Quack for helpful discussions and giving us recent results that have not yet been published.