AVERAGING BIAS CORRECTION FOR FUTURE IPDA LIDAR MISSION MERLIN

The CNES/DLR MERLIN satellite mission aims at measuring methane dry-air mixing ratio column (XCH4) and thus improving surface flux estimates. In order to get a 1% precision on XCH4 measurements, MERLIN signal processing assumes an averaging of data over 50 km. The induced biases due to the non-linear IPDA lidar equation are not compliant with accuracy requirements. This paper analyzes averaging biases issues and suggests correction algorithms tested on realistic simulated scenes.


INTRODUCTION
Methane is the second most important anthropogenic greenhouse gas after carbon dioxide Error!Reference source not found.. A precise and accurate measurement of atmospheric CH 4 concentration on a global scale is crucial in order to improve the surface flux estimate and thus develop the knowledge of the global methane cycle [2].
The French-German spatial mission MERLIN (Methane Remote Sensing Lidar Mission) aims at measuring the methane dry-air mixing ratio (XCH 4 ) reaching unprecedented accuracy with targeted systematic error that is less than 0.2% relative error for all latitudes and all seasons.Integrated Path Differential Absorption (IPDA) lidar measures the laser light (here short pulses) scattered back from the surface in order to retrieve the column content of a specific trace gas along the line of sight.Differential absorption uses the difference in transmission between the on-line signal with a wavelength set about the center of the CH 4 absorption line and the off-line signal which wavelength is significantly less absorbed.From these two signals the Differential Absorption Optical Depth (DAOD) of CH 4 is computed and the corresponding XCH 4 can be derived.The IPDA lidar equation (eq.( 1) and ( 2)) links the backscattered off-line and on-line measurements to the sounded column of XCH 4 [3].
In order to reach the targeted 1% relative random error on XCH 4 measurements, the signal processing of MERLIN requires an horizontal averaging of data over 50 km along track.However, the non linearity of the IPDA lidar equation in combination with the variability of the observed scene (surface elevation, reflectivity, meteorology) along the averaging window induces a bias on the averaged data.
The present paper analyses and compares the biases of several averaging schemes and suggests correction algorithms.A comparative evaluation of the averaging schemes on realistic scenes is a key element to select the best approach (i.e. the least biased) for MERLIN processing.

Principle of IPDA lidar
For every pair of on-line and off-line shots, the DAOD can be computed as [4]: (1) where the on and off backscattered signal energies are normalized by the ratio of transmitted pulse energies ( is the normalized signals).Then, we can derive the XCH 4 as: (2) where is the pressure where the laser beam is backscattered and is the weighting function computed from external data describing the measurement sensitivity of XCH 4 along the line of sight.
These equations are valid for every shot-pairs but averaging over several successive shots is necessary to reach the targeted precision of the measurement.

Averaging strategies
For the following discussion we will use the triangular bracket notation to denote the arithmetic mean of the quantities associated to each pulse and the will represent the deviation to this arithmetic mean: (3) There are four basic averaging schemes: 1. Averaging of XCH 4 (AVX): (4)

Averaging of signals (AVS): (7)
As these equations are not linear and the physical quantities (surface pressure, surface reflectance...) vary for successive shot-pairs, the four averaging schemes are all biased in their own specific way.Correction algorithms must be applied in order to significantly reduce this averaging bias.AVQ (eq.( 6)) has not been studied in this paper as the noise distribution leads to the highest bias.

Bias on DAOD from noise on :
As the measurements themselves are affected by noise it implies a bias when computing the XCH 4 for every averaging schemes.Indeed, we can write as the sum of their noiseless mean signals and a noise term .From this decomposition we can show that for any shotpair the true DAOD is biased: (8) In the case where the noise follows a normal distribution and has low amplitude (i.e. ), the bias term can be approximated as: Where are the on-line and off-line signal to noise ratios.Providing high enough SNRs the bias produced by the measurement noise can be estimated.

Bias from shot-by-shot geophysical variations:
Variations of surface reflectivity impacts the AVS.Indeed, this scheme is more sensitive to targets with high reflectivity (i.e.high value of ) as shown by the following equation: This sum is the weighted average of the shot-byshot transmissions weighted by the off-signal variations (i.e.proxy of reflectivity variations): (11) So low reflectivity regions contribute less to the average.
Then, when there is a variation of geophysical quantities (e.g.topographic or surface pressure variations) there is a discrepancy between the average DAOD and the single-shot DAOD: (12) We can derive the bias from equation (10) as: (13)

Bias when averaging concentrations:
Directly averaging XCH 4 (eq.( 4)) is not equivalent to averaging DAOD and IWF separately (eq.( 5)).In the first case, we average concentrations and not molecules.So, there is the same contribution from columns with many molecules as from ones with less molecules.
The difference between the two approaches is described by: (14) As presented above, several sources of bias appear when computing the average mixing ratio of CH 4 .
The following section exposes the algorithms that correct for those biases in a practical way.

Correction algorithms 
For AVX, we need to correct the following sources of bias: bias induced by noise on (eq.( 9)) and averaging of concentration instead of molecules (eq.( 14)).
 For AVD and averaging of IWF, we only need to correct the bias induced by the noise on (eq.( 9)).


For AVS and averaging of IWF, several corrections need to be implemented.As AVS is more sensitive to the high reflectance area, the averaging of IWF must be done with consistent weights (eq.( 11)): (15) Furthermore the DAOD from averaging signals is biased according to equation (13).It is possible to evaluate this bias by considering as a first order approximation the DAOD computed from average signals pondered by the relative fluctuations of IWF: (16) This process could be turned into an iterative correction (beyond the scope of this paper).
Then the last correction to take into account is the bias induced by the equivalent noise on the averaged signals.The resulting SNR can be computed by averaging signals and taking the quadratic mean of the standard deviations in order to apply equation (9).

Simulation description
The aim of the simulation is to estimate the biases of estimated XCH 4 .A global description of the simulation is presented on Figure 1.The number of random samples for noise simulation has to be considerable in order to compute the biases with sufficient accuracy.The simulation described above is repeated 300.000 times to reach an accuracy on the bias of approximately 0.1 ppb for the nominal case (reflectivity of 0.1).This simulation has been tested on realistic scenes.

Realistic scenes
The averaging schemes have been tested on three realistic scenes located in France (Toulouse, Millau and Chamonix).The physical data and their sources are: Reflectivity from SPOT5, SRTM topography and Interpolated surface pressure from ECMWF.
The scenes have been chosen to be representative of the main geophysical variations that can produce bias on XCH 4 estimation.Toulouse, Millau and Chamonix respectively present medium, high and extreme altitude variations (cf. Figure 2) and the same trend for reflectivity variations.The mean reflectivity can be tuned.

RESULTS
The results of the simulation are shown in Figure 3.The averaging scheme that is globally the least biased is AVS.For all reflectivity, the bias stays limited to less than 0.7 ppb.The main cause of bias for this scheme is topographic or meteorological variation.AVD and AVX schemes are more affected by the ground reflectivity.The correction given by equation ( 9) is efficient when reflectivity is high.However, for a lower reflectivity the correction is no longer valid because it comes from a Taylor expansion assuming low amplitude of the noise.

CONCLUSIONS
This study shows that it is possible to limit the bias induced by the averaging scheme when an appropriate bias correction is implemented.The AVS scheme is less biased than other schemes in all simulated cases and below the target of 1 ppb allocated for this type of error.An alternative solution would be to take advantage of the two schemes: averaging signals on smaller windows where there is less geophysical variation until we reach a higher SNR that is compliant with the approximation of SNR correction (eq.( 9)).Then averaging the resulting columns together on the 50 km window.Current development of this study tends to show that this combined scheme is not better than AVS results.Further studies will check the impact of an imprecise estimation of SNR on the bias correction (eq.( 9)).

Figure 1
Figure 1 Global description of the simulation Each simulation considers a typical number of 150 pulses per averaging window, approximately corresponding to 50km.It relies on a description of the CH 4 field, surface pressure and reflectivity description.Then, on-line and off-line signals are computed from surface reflectivity and noise simulation.The weighting functions for each pulse are computed using CH 4 absorption crosssections from the 4A radiative transfer model [5] and are integrated vertically.Next, we proceed to the computation of averaged XCH 4 on 50km with the different schemes described in section 2.2 and the correction algorithms described in section 2.4.

Figure 2
Figure 2 Altitude for the 3 studied scenes

Figure 3
Figure 3 Residual Biases on a 50 km window for the three scenes (Toulouse, Millau, Chamonix) -for the three averaging schemes (blue shades for AVS, green shades for AVD and red shades for AVX) -for four different mean reflectivity (from light to dark colors decreasing reflectivity -0.1 vegetation ; 0.05 mixed water/vegetation ; 0.025 sea/ocean ; 0.016 ice/snow) -The biases are displayed over the bars that end outside the frame.