Toward a numerical deshaker for PFS
F. Schmidt, I. Shatalina, M. Kowalski, N. Gac, B. Saggin, M. Giuranna
TToward a numerical deshaker for PFS
Schmidt , F., Shatalina I., Kowalski M., Gac N., Saggin B., Giuranna ,M. Univ. Paris-Sud, Laboratoire IDES, UMR 8148, Bt 509, Orsay, F-91405, France([email protected]) CNRS, Orsay, F-91405, France Dipartimento di Meccanica, Politecnico di Milano, Campus of Lecco, Via M. d’Oggiono18/a, 23900, Lecco, Italy Laboratoire des signaux et systemes (L2S), UMR8506 Univ Paris-Sud - CNRS –SUPELEC, SUPELEC, 3 rue Joliot Curie, Gif sur Yvette, F-91192, France IFSI, via del Fosso del Cavaliere, 100, 00133 Roma, Italy
Abstract
The Planetary Fourier Spectrometer (PFS) onboard Mars Express (MEx) isthe instrument with the highest spectral resolution observing Mars from orbitsince January 2004. It permits studying the atmospheric structure, major andminor compounds. The present time version of the calibration is limited by theeffects of mechanical vibration, currently not corrected. We proposed here anew approach to correct for the vibrations based on semi-blind deconvolutionof the measurements. This new approach shows that a correction can be doneefficiently with 85% reduction of the artifacts, in a equivalent manner to thestacking of 10 spectra. Our strategy is not fully automatic due to the dependenceon some regularisation parameters. It may be applied on the complete PFSdataset, correcting the large-scale perturbation due to microvibrations for eachspectrum independently. This approach is validated on actual PFS data of ShortWavelength Channel (SWC), perturbed by microvibrations. A coherence checkcan be performed and also validate our approach. Unfortunately, the coherencecheck can be done only on the first 310 orbits of MEx only, until the laser line hasbeen switch off. More generally, this work may apply to numerically “deshake”Fourier Transform Spectrometer (FTS), widely used in space experiments or inthe laboratory.
Keywords:
Keyword: PFS ; Fourier Transform Spectrometer; Micro-vibration; Calibration; Spectroscopy; Blind Inverse problem
1. Introduction
The Planetary Fourier Spectrometer (PFS) is a double pendulum Fouriertransform infrared spectrometer instrument onboard MEx, operating in the 1.2to 5.5 micrometers for the Short Wavelength Channel (SWC), and 5 to 45microns in the Long Wavelength Channel (LWC) (Formisano et al., 2005). It
Preprint submitted to Planetary and Space Science November 4, 2018 a r X i v : . [ a s t r o - ph . I M ] N ov s based on a modified Michelson’s scheme using a double pendulum with cubicreflectors. The optical path difference is defined by the zero crossing of a lasertacking the same optical path asthe signal. The spectra presented in this articleare the numerical Fourier transform of the recorded interferograms.An experimental study of mechanical vibration impact on Fourier-transformspectrometer has been proposed based on PFS example (Comolli and Saggin,2005). Analytical expression of all distortion effects have been formulated sep-arately (Saggin et al., 2007): offset of the reference laser signal, mirrors speedvariation, periodic misalignments, detector non linearity and internal reflections.More recently, a numerical simulation model has been proposed to explore alleffects combined in order to understand the PFS signal (Comolli and Saggin,2010). Perturbations are creating artificial features, called “ghosts”, present insome spectra of the SWC but not in the LWC, thanks to the optimization ofthe pendulum velocity Giuranna et al. (2005b,a). Since the amplitude of ghostsare small (few % of the original signal) and its phase has a stochastic behavior,the worst cases correspond to only few significant ghosts Shatalina et al. (2013).Quantitatively, the ghosts are affecting few % of the total spectrum energy(3% typically; 5% maximum). When single spectra are used, the absolute ra-diometric calibration is degraded, and spurious spectral features may appear inthe spectrum, preventing any surface-related analysis, and introducing possiblelarge uncertainties in the quantitative retrievals of abundances of minor speciesin the atmosphere. When discussing the calibration procedure for the SWC(Giuranna et al., 2005b) and the LWC (Giuranna et al., 2005a), the authorssuggest to stack the data to correct for the effects of the mechanical vibrations.The position of ghosts depends on the frequencies of the external vibrations,which have been found to be quite stable. Since the phase of ghosts is ran-dom and the external frequencies are stable, only the signal should be coherentduring the stack. This idea has been confirmed by numerical modeling of theperturbations (Comolli and Saggin, 2010). Practically, averaging a few spectra(ten or so) is enough to average out the ghosts. However, this will degradethe spatial and temporal resolution of PFS measurements, limiting the inter-pretation of small-scale features and hampering some scientific studies (e.g., thecomposition of ices; detection of minerals at the surface).Typical PFS raw measurements are shown in Fig. 1. One can identify themajor signals from Mars: thermal emission and reflection of solar energy, andthe laser line stray-light. Also the contribution due to mechanical vibrations areshown on the signal, leading to additional energy shifted on left and right almostsymmetrically. The ghost of the laser line is only one sided due to aliasing.Our aim is to provide a new approach to process the PFS instrument withfollowing constraint:1. correct the effect of mechanical vibrations due to both misalignment andoptical path difference errors.2. perform the correction on each spectrum separately.3. validate the approach by using actual PFS observations.In order to avoid unphysical solution, the algorithm is initialized with an a2 igure 1: Typical symmetrized PFS measurement in SWC. Signal and major CO band andlaser lines are noted. The four main ghosts are identified as “Vibration Component” (VC)affecting both signal and laser line.. priori guess of the large scale structure of the spectra, adapted to each measure-ment, reproducing the Martian thermal emission and the reflected solar light(see section 2.2.2).A check of the correction can be done but requires the SWC laser diodeswitched on, to estimate the vibration kernel independently (see section 2.3).
2. Method
This section describes the direct model of the Martian spectra affected byvibrations. Then, an iterative procedure is exposed in order to invert it as wellas some criteria to measure the quality of our estimation. − ) As it can be seen from Fig.1, it is possible to separate the whole spectruminto two wavenumber domains to deal the effects of the mechanical vibrationsapart in each of them. From 0 to 5000 points (5000*1.02 cm − ), we definethe signal domain, where the thermal energy from Mars and the most of thereflected Martian energy are recorded, without significant laser line artefacts.The laser line domain is defined from 5000 cm − to 8330 cm − . It contains alsothe Martian signal but affected by laser line artefacts. Below 1700 cm − thereis no meaningful signal due to the low detector responsivity (see Fig. 15. inGiuranna et al. (2005b)), and this region is characterized only by ghosts of thecontinuum. 3t larger wavenumber than 5000 cm − , the signal is affected by the laserline shape and its ghosts, directly and in aliasing. This domain will be usedto test the coherence of the results. Since the laser has been switched off afterorbit 634, it could not be used for the complete PFS archive (see section 2.3).Using some mathematical reorganization and simplification, the analyticalexpression of mechanical vibration due to periodic misalignment and opticalpath errors can be written as a convolution products in complex form, see Eq.(13) in Shatalina et al. (2013). Assuming that the domain of wavenumber withsignificant signal I Mars around σ ∼ − − is constant ( σ k ∼ I P F S ( σ ) = I Mars ( σ ) + [ σ.I Mars ( σ )] (cid:63) K ( σ ) , (1)simplifies to: I P F S ( σ ) = I Mars ( σ ) (cid:63) [ δ ( σ ) + K ( σ ) .σ k ] . (2)with δ () , the dirac function.By rewriting: I P F S ( σ ) = I Mars ( σ ) (cid:63) K P F S ( σ ) , (3)with I P F S ( σ ) the measured raw spectra, I Mars the contribution of the raw spec-tra from Mars, K P F S the kernel representing the mechanical vibration effects, σ the wavenumber, and K ( σ ) the non-normalized complex kernel (Shatalinaet al., 2013).From Shatalina et al. (2013), the kernel of all frequency of vibrations is: K P F S ( σ ) = δ ( σ ) + A ( σ ) e iϕ A ( σ ) + B ( σ ) e iϕ B ( σ ) . (4)The quantities A, B, ϕ A , ϕ B are unknown and cannot be evaluated quantita-tively due to the lack of knowledge about vibration amplitude and phase. Inpractice, the functions A, B, ϕ A , ϕ B are sparse over σ because the frequencies ofvibrations are sparse. Please note that A, B, ϕ A , ϕ B are not symmetric around σ = 0 due to the relative phase. We propose to estimate those functions usingan inversion procedure described in the next section.The assumption of a reduced wavenumber domain is valid in first approx-imation due to the sensitivity of the detector and the typical Martian signal,leading to a misfit factor of x0.8 to x1.2 that is reasonable for this case. Inaddition, our strategy is to use semi-blind deconvolution algorithm in order toensure the best fit any kind of spectra. This way, the wavenumber domain ofsignificant signal has not to be defined explicitly.Including to our model an additive noise (cid:15) which stands for the others sourcesof acquisition noise besides the mechanical vibrations and the error due to ourPFS modeling by a convolution kernel K P F S , PFS spectra in signal domain asillustrated in Fig. 2, are obtained through: I P F S ( σ ) = I Mars ( σ ) (cid:63) K P F S ( σ ) + (cid:15) . (5)4 PFS (desired spectra) (noise) (acquisition data) I Mars I PFS (instrument)
Figure 3: Modelisation of acquisition by the PFS instrument
The assumption of a reduced wavenumber domain is valid in first approxima-tion due to the sentitivity of the detector and the typical Martian signal, leadingto a misfit factor of x0.8 to x1.2 that is reasonable for this case. In addition,our strategy is to use blind deconvolution algorithm, without a priori shape ofthe spectra, in order to ensure the best fit any kind of spectra. This way, thewavenumber domain of significant signal has not to be defined explicitely.Including to our model an additive noise which stands for the others sourcesof acquisition noise besides the mechanical vibrations and the error due to ourPFS modelisation by a convolution kernel K PFS , PFS spectra in signal domainas illustrated in Fig. 3, are obtained through: I PFS ( σ ) = I Mars ( σ ) K PFS ( σ ) + . (5) − ) Unfortunately, from the laser line domain it is not possible to realize an inver-sion to estimate K PFS because there is twice unknown variables in comparisonto known variable due to aliasing. Under strong hypothesis, it is possible toestimate an approximation of ˆ K approxPFS [11]. Nevertheless, the recent analyticalformulation of the mechanical effects on the laser line, allows us to compute theexact effect of the mechanical vibration on the laser line, knowing the vibrationkernel K PFS (see complex expression in [11]). After the estimation of K PFS ,one simple test of coherence would be to compare the observed laser line ghostto the one predicted. 6
Figure 2: Model of acquisition by the PFS instrument
From the direct model of the PFS instrument described above, see Eq. (5),we propose here a semi-blind deconvolution method to solve the inverse problem:estimation of the desired spectra I Mars from the PFS spectra I P F S although theconvolution kernel K P F S is unknown. We qualified our method as semi-blindbecause the only spectral a priori information is ˆ I Mars , known ab initio. Wealso used two a priori information : ˆ I Mars is smooth and ˆ K P F S is sparse. Thenotation ˆ X, means the estimation of quantity X . A classical approach consistsin introducing a cost function C whose minimum provides an estimation:ˆ I Mars , ˆ K P F S = argmin I Mars ,K PFS C ( I Mars , K
P F S )= argmin I Mars ,K PFS (cid:107) I P F S − K P F S (cid:63) I
Mars (cid:107) + λ K (cid:107) K P F S (cid:107) + λ Mars (cid:107) D (cid:63) I
Mars (cid:107) . (6)Three terms appear in C
1. A data fit term (cid:107) I P F S − K P F S (cid:63) I
Mars (cid:107) that quantifies how well theestimated sources match the measured data. This term takes into accountthe characteristics of the noise supposed to be white and gaussian. Thisdata match term is sensitive to high frequency noise and must be balancedwith regularization term which corresponds to a mathematical prior on theexpected solution (Idier, 2008).2. A sparsity regularization term (cid:107) K P F S (cid:107) is chosen for the kernel, i.e. : the (cid:96) norm (sum absolute value) of the kernel must be low. Indeed, the PFSkernel is supposed to be composed with few diracs at mechanical vibrationfrequencies.3. A smooth regularization term is chosen for the Mars spectra : (cid:107) D(cid:63)I
Mars (cid:107) ,where D is a discrete first-order derivation operator. This prior promotessmooth solution in order to avoid noise improvement.All these terms are balanced with two hyperparameters λ K and λ Mars , bothpositive. The functional 6 is convex for each variables - convex in I Mars when K P F S is fixed and vice versa - but not from the couple ( I Mars , K
P F S ). The5trategy we choose here is a classical alternative procedure: from initial guessesˆ I Mars , an iterative procedure updates successively at each iteration n , the newestimates ˆ K n +1 P F S and ˆ I n +1 Mars . At each iteration n we estimate successively the kernel ˆ K n +1 P F S and the signalˆ I n +1 Mars until iteration N using the following steps:1. First estimation of the kernel ˆ K P F S from filtered ˆ I Mars and I P F S withL1 regularization2. Iterative loop:(a) estimation of the Mars spectra ˆ I n +1 Mars from unfiltered ˆ K nP F S and I P F S with smooth regularization(b) estimation of the kernel ˆ K n +1 P F S from unfiltered ˆ I n +1 Mars and I P F S withL1 regularization3. Last estimation of the Mars spectra ˆ I finalMars from unfiltered ˆ K finalP F S and I P F S
For both estimations, a convex optimization algorithm converges to the solutiondefined by the minimum of a criteria made of a data match and a regularizationsterms. This means that the solution is unique and can be estimated eitheranalytically or iteratively.Since the first step of the iterative procedure is the estimation of the kernelˆ K P F S , the only a priori information of this iterative procedure ˆ I Mars , estimatedab initio. Since ˆ I Mars , can only be estimated at large scale (all absorptionlines may differ from spectra to spectra due to non-homogeneity of chemicalcompounds in the atmosphere/surface of Mars), the first iteration is done in alow-pass filtered space, as described in section 2.2.2.
Estimation of the PFS kernel.
The estimation of the PFS kernel reduce to thefollowing (cid:96) regularized convex (non smooth) problem:ˆ K n +1 P F S = argmin K PFS (cid:107) I P F S − K P F S (cid:63) ˆ I nMars (cid:107) + λ K (cid:107) K P F S (cid:107) , (7)where ˆ I nMars is the estimation of the Mars spectra at the iteration number n .This problem is the well known Lasso (Tibshirani, 1996) or Basis-Pursuit De-noising (Chen et al., 1998) problem, and can be solved efficiently with the FastIterative/Thresholding Algorithm (FISTA) (Beck and Teboulle, 2009). Denot-ing by ˜ I the adjoint of the kernel I and by S λ the so-called soft-thresholdingoperator the algorithm reads:1. Let i = 0 , τ = 1 , k = 1 , Z = K nP F S and L = (cid:107) I Mars (cid:107) .2. K iP F S = S λ K /L (cid:16) Z i + L ( I P F S − Z i (cid:63) ˆ I nMars ) (cid:63) ˜ˆ I nMars (cid:17) S λ ( x ) = x | x | max( | x | − λ, τ i +1 = √ τ i Z i +1 = K iP F S + τ i − τ i +1 ( K iP F S − K i − P F S )5. i = i + 16. Go to 2 until i = i max K n +1 P F S = K i max P F S
From theoretical consideration, the kernel K P F S must be a dirac-shape on zero,so we concentrate the energy around zero into a dirac to create the kernel esti-mation ˆ K nP F S . We would like to emphasize that there is no analytical solutionof eq. 7 so we solve this equation with an iterative procedure, initialized withthe previous step K nP F S . For the first initialization K P F S , we may use ˆ K approxP F S but any other guess (such zero) may apply when the laser line has been switchoff. Nevertheless, closer the initialization, faster the convergence. Estimation of the Mars spectra.
For the Mars spectra, the estimation reducesto a classical Thikonov regularization (Idier, 2008):ˆ I n +1 Mars = argmin I Mars (cid:107) I P F S − ˆ K n +1 P F S (cid:63) I
Mars (cid:107) + λ Mars (cid:107)
D (cid:63) I
Mars (cid:107) (8)Thanks to the fact that a convolution is diagonal in the Fourier domain, andthe Parseval theorem, the solution reads: F ( ˆ I n +1 Mars ) = argmin F ( I Mars ) (cid:107)F ( I P F S ) −F ( ˆ K n +1 P F S ) (cid:12)F ( I Mars ) (cid:107) + λ Mars (cid:107)F ( D ) (cid:12)F ( I Mars ) (cid:107) (9)where (cid:12) is the Hadarmard element-wise product and F the Fourier trans-form. Then, the estimation of the Mars spectra at iteration n + 1 is given inclose form by :ˆ I n +1 Mars = F − (cid:16) F ( I P F S ) (cid:12) ( F ( ˆ K n +1 P F S ) − − λ Mars F ( D ) − ) (cid:17) (10) , where F ( ˆ K n +1 P F S ) − (resp. F ( D ) − ) represents the vector containing the in-verted squared elements of the vector F ( ˆ K n +1 P F S ) (resp. F ( D ) ).We would like to emphasize that equation 10 is the analytical solution of eq8 that did not require initialization. We estimate the Martian spectra large scale feature(noted ˆ I Mars ) by two Planck functions and the major absorption feature, repre-senting (i) the Martian thermal emission and (ii) the solar energy reflected backby Mars and (iii) the 2200-2400 cm − gap, representing the CO absorptionband. The Martian temperature is estimated by fitting the 2500-3000 cm − I n t en s i t y , ( DN ) originalfiltered0 1000 2000 3000 4000 5000 (cid:239) (cid:239) (cid:239) (cid:239) P ha s e , (r ad ) originalfiltered 0 1000 2000 3000 4000 50001234567 SYNTHETIC MARTIAN spectrum I n t en s i t y , ( DN ) originalfiltered0 1000 2000 3000 4000 5000 (cid:239) (cid:239) (cid:239) (cid:239) P ha s e , (r ad ) originalfiltered Figure 3: Raw measurements I PFS (on left) and initial guess of the Martian signal ˆ I Mars (onright) for the PFS measurement ORB0032, No 106. domain, where the ghost seems to be less pronounced. The Planck function ofthe sun is scaled to the 3800-4200 cm − domain. We derive the raw spectrausing the calibrations of detector responsivity and deep space measurements(Giuranna et al., 2005b). This initial guess is only valid at large scale becausethe absorption lines of major and minor gases may change, due to local pressure,atmospheric circulation, surface change and radiative transfer effects.The phase of the initial guess is taken similar to the signal in the domainwhere the ghosts are absent and a constant extrapolation is proposed to theghosted region.Because the iterative procedure is sensitive to initialization, both PFS spec-tra I P F S and mars initial guess ˆ I Mars are filtered with a low-pass filter with acut off frequency of (cid:52) σ , where (cid:52) σ = 1.02 cm − is the spectral resolution, inorder to keep the realistic features.The initial guess ˆ I Mars will force the initial step of the iterative procedure tofind a local minimum around physical solution. Initializing the procedure withrandom or constant signal lead to non physical solutions.
Initial PFS Kernel guess.
Thanks to the approximation from Shatalina et al.,2013, an estimation of the kernel ˆ K approxP F S from the laser line domain can be done(see section 2.3). Neither amplitudes, nor phases are precise but the frequenciesshould be well described by this methodology. This kernel is used as initialguess ˆ K P F S to reach faster the convergence of the first kernel estimation ˆ K P F S .Unfortunately, only Mars Express orbit ¡ 634 are usable for this estimation. Itrepresents 310 orbits out of 6255 orbits currently available i.e., less than 5% ofthe total current orbits. 8 .3. Laser line domain (5000 to 8330 cm − ) Knowing that laser line is almost dirac shaped, we could first hypothesizethat the kernel K P F S can be directly measured in the laser line domain. Unfor-tunately due to aliasing (laser line ghosts from left and right are superposed),from I P F S ( σ ) it is not possible to realize an inversion to estimate K P F S be-cause there are twice unknown variables in comparison to known variable. Un-der strong hypothesis, it is possible to estimate an approximation of ˆ K approxP F S (Shatalina et al., 2013).Nevertheless, the recent analytical formulation of the mechanical effects onthe laser line, allows us to compute the exact effect of the mechanical vibrationon the laser line, knowing the vibration kernel K P F S (see complex expressionin Shatalina et al. (2013)). After the estimation of K P F S , one simple test ofcoherence would be to compare the observed laser line ghost to the one predicted.
We propose several criteria to estimate if the deconvolution is correct.
Because the only ground truth we could have is the real PFS spectra I P F S ,it should be as close as possible to the final simulated PFS spectra ˆ I finalP F S =ˆ I finalMars (cid:63) ˆ K finalP F S . We use the Root Mean Square distance (RMS) of ˆ I finalP F S − I P F S .In this way, we evaluate at the same time the correctness of the estimated Marsspectra ˆ I finalMars and the instrument model ˆ K finalP F S . − ) In the 1 to 1530 cm − wavenumber domain, no signal is expected due tothe very low signal to noise ratio but only the ghosts are present in the rawspectra. Thus, one simple criterion to estimate the efficiency of the correctionis to measure the energy in this domain. − ) The laser line modulated ˆ I finalLM through filter, aliasing and vibrations effectscan be computed from the estimated kernel ˆ K finalP F S using the exact formula-tion of Shatalina et al., 2013. To check the quality of the results, we evaluatethe distance between the actually measured signal and the predicted laser linemodulated with its ghosts. The estimation of the kernel ˆ K approxP F S from the laser line domain can bedone. Neither amplitudes, nor phases are precise but the frequencies shouldwell described by this methodology. The distance between ˆ K approxP F S and ˆ K finalP F S is also a criteria of good results. 9 .4.5. Comparison with vibration frequencies from MEx telemetry and technicalspecification Several sources of vibrations are present in the MEx platform, mainly reac-tion wheels, Inertia Measurement Unit (IMU) dithering. PFS eigenmodes canalso be exited and are considered as “source” of vibrations. Since, these vibra-tions are not unique onboard MEx (cryocooler, other instruments, ...) and theuncertainties on these vibrations frequencies are not known, it is not possible tohave a supervised approach. One also have to note that all vibration frequen-cies may not be present in a PFS spectrum, depending the coupling with PFS.Nevertheless, a comparison between our blind estimation and the actual data isinteresting.From each vibration frequency f d (in Hz), the perturbation is at wavenumber σ = f d /v m (Saggin et al., 2007; Shatalina et al., 2013), with the pendulum speed v m = d zc .f zc where the zero-crossing frequencies f zc is 2500 Hz and zero-crossinglength d zc is 1.2 microns for typical PFS measurements at Mars (Giuranna et al.,2005b). Reaction wheels.
Thanks to telemetry data from ESA, it is possible to estimatethe frequencies of reaction wheels for ORB0032, spectra No 106 at 56.7 Hz, 33.3Hz, 40.6 Hz and 30.3 Hz. Uncertainties are unknown and those frequencies ofmicro-vibrations are expected to change during the mission but can be estimatedfrom telemetry.
IMU.
Astrium technical specification of MEx (MEX.MMT.HO.2379) states thatthe IMU dithering onboard MEx are at 513.9 Hz, 564.3 Hz and 617.4 Hz. Un-certainties are unknown but those frequencies of micro-vibrations are expectedto be constant during the mission.
PFS eigenmodes.
The PFS eigenmodes are around 135 Hz and 160 Hz. Uncer-tainties are unknown but those frequencies of micro-vibrations are expected tobe constant during the mission.
3. Results
Due to the stochastic character of the ghosts and especially their phase, few% of the PFS spectra in the archive, randomly distributed, present significantlevel of perturbations. In some lucky cases, the ghosts are absent but typicalspectra contains few ghosts (Comolli and Saggin, 2010). We propose to illustrateour algorithm on the ORB0032, spectra No 106 of PFS, recorded in particularlyhigh level of disturbances. This spectra contains several obvious ghosts (asshown by the arrows in fig. 5).We find that the optimum inversion is reached with a loop of N=2, withspecial parameter for the first step due using the filtered initialization ( λ K =50)and then usual parameter ( λ Mars =0.001, λ K =1) using the unfiltered spectra .10 .1. Mars spectra and kernel estimations obtained For Mars spectra estimation, the final estimation of the signal ˆ I finalMars is pre-sented in fig. 4 and 5. This figure presents the raw spectra, our corrected spectrain comparison with a synthetic spectra ˆ I Mars (see section 2.2.2) and also thestack of 20 spectra. Our correction clearly removes the ghosts in the region at1-1530 cm − , around 2700 cm − , around 3450 cm − , around 4150 cm − similarto the stacking method. The artifact at 2900 cm − persists, due to pollutionof hydrocarbons in the telescope (Giuranna et al., 2005b). In the 4000-5000cm − domain, our method improves the signal in comparison to the stackingmethod and partly correct the artificial decrease of the signal. The stackingclearly reduces the stochastic noise, that is not removed with our correction.Figure 6 shows the evolution of the average spectra, when stacking 3, 5, 11and 19 spectra. The plots clearly show that our method removes the ghostscontribution, already without stacking. In contrary, the stacking methods re-quire ∼
10 spectra to remove this effect. The signal to noise ratio at small scale,estimated by the standard deviation in the 1-1530 cm − , is not significantlychanged between both methods.The stack of ∼
10 spectra correspond to ∼
10 spots of around 7 km each, sothat the spatial resolution can be improved by one order of magnitude. In termof temporal resolution improvement, it depends mainly on the location due tothe very irregular observation density.
As illustrated in Fig. 4, the lack of fit between the real PFS spectra I P F S and the simulated one from our final guesses ˆ I finalP F S = ˆ I finalMars (cid:63) ˆ K finalP F S is verysmall ( ∼ − ), showing that the solution is compatible with the observation.Our strategy is efficient to remove the norm in the 1 to 1530 cm − domainby a factor of ∼ I finalMars seems to be very smallscale random, as expected (see Fig. 4). The theoretical value of noise standarddeviation is about 0.1 using the estimated Signal to Noise ratio of about 100in the 2000-2400 cm − Giuranna et al. (2005b). The estimated noise standarddeviation of corrected spectra is in agreement with this value (see fig. 6). Inorder to estimate the efficiency of ghost removal, we measure the spectral energyin the 1 to 1530 cm − domain at scale larger than 50 cm − for corrected spectrain comparison to the measured spectra, assuming that the large scale featuresare only due to ghosts. We found that our correction for one single spectraremove 85% of the ghost energy, which is equivalent to the effect of stacking of11 spectra.The laser line modulated ˆ I finalLM through filter, aliasing and vibration kernelˆ K finalP F S are compatible with the observation I LM (see Fig. 7). The four mainspeaks are estimated and also some smaller peaks. The distance is relativelysmall ( ∼ . K finalP F S is close to the initial kernel guess ˆ K P F S =ˆ K approxP F S with a distance ∼ − . As illustrated in Fig. 8, the main vibration fre-11
00 1000 1500 2000 2500 3000 3500 4000 4500 50000510 ORBIT 0032: MARTIAN filtered spectrum − ) M odu l u s , ( DN ) measured: Isimulated: If = If Mars *KfEstimated If
Mars − − − − −
500 0 500 1000 1500 2000 250010 Estimated kernel KfWavenumber (cm − ) l og ( M odu l u s ) , ( DN ) − − − − −
500 0 500 1000 1500 2000 2500 −
202 Wavenumber (cm − ) P ha s e , (r ad )
500 1000 1500 2000 2500 3000 3500 4000 4500 50002468 Estimated spectrum If
Mars
Wavenumber (cm − ) M odu l u s , ( DN )
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 −
202 Wavenumber (cm − ) P ha s e , (r ad ) Figure 4: Final results of the spectra ORB0032 I PFS (blue) simulated ˆ I finalPFS = ˆ I finalMars (cid:63) ˆ K finalPFS PFS spectra (red) and the estimated Martianspectra ˆ I finalMars (black); Lack of fit between I PFS and ˆ I finalPFS is 1.8 10 − ;in the middle: modulus(in log scale) and phase of the final estimated kernel ˆ K finalPFS ; at the bottom: modulus andphase of the final estimated spectrum ˆ I finalMars .
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 − − − − − ) M odu l u s , ( DN ) , s h i ft ed f o r c l a r i t y !! "" " " " " " " " data: I PFS estimation: If
Mars synthetic: If
Mars stack: I
PFS stack: If
Mars
Figure 5: Final results of the spectra ORB0032 − represents the mirror contamination by hydrocarbons, the arrow at 4900 cm − represents an artifact of abnormal small signal, probably due to ghosts.
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 − − − − − − ) M odu l u s , ( DN ) , s h i ft ed f o r c l a r i t y noise : 0.17 ghosts : 100%noise : 0.07 ghosts : 15%noise : 0.05 ghosts : 8%noise : 0.05 ghosts : 6%noise : 0.05 ghosts : 7%noise : 0.04 ghosts : 6% !!! data: I PFS estimation: If
Mars
Mars
Mars
11 stack: If
Mars
19 stack: If
Mars − − − − − − ) M odu l u s , ( DN ) , s h i ft ed f o r c l a r i t y noise : 0.17 ghosts : 100%noise : 0.07 ghosts : 15%noise : 0.1 ghosts : 45%noise : 0.07 ghosts : 24%noise : 0.05 ghosts : 14%noise : 0.03 ghosts : 5% !!! data: I PFS estimation: If
Mars
PFS
PFS
11 stack: I
PFS
19 stack: I
PFS a b
Figure 6: Comparison of our correction versus the stacking method : (a) Stacking of correctedspectra from our method (b) Stacking of PFS spectra. Noise standard deviation from the 1-1530 cm − are expressed for all spectra. Fraction of energy due to ghosts, relative to the rawPFS spectra in 1-1530 cm − is also written for all spectra. Arrows at 2700 cm − representsignificant difference in the signal domain due to ghosts, that persists for stacking of at least5 PFS spectra but well corrected by our method. − M odu l u s , ( DN ) ORBIT 0032 final evaluated from vibrations
Figure 7: Modulus of the simulated laser line modulated ˆ I finalLM through filter, aliasing andvibration kernel ˆ K finalPFS (blue) and the observation I LM (red). The lack of fit is 0.0270 − − − −
500 0 500 1000 1500 2000 250010 − − − − Wavenumber (cm − ) l og ( M odu l u s ) , ( DN ) Estimated and approximated kernel K approx K final from vibrations Figure 8: Modulus in log scale of the vibration kernel ˆ K finalPFS (blue line), the approximatedkernel ˆ K approxPFS (red line)and the reaction wheels vibration (dark grey line), and the combi-nation of reaction wheels (light grey line). The lack of fit is3.2 10 − . quencies estimated in ˆ K approxP F S are present in ˆ K finalP F S . The estimation of ˆ K approxP F S has been done under strong approximation. Especially the unconstrained am-plitude may explain the differences. Also ˆ K finalP F S presents a smooth signal dueto the high frequencies filtering. Other methods without sparsity regularizationdoesn’t succeed to get such a sparse kernel although we believe that the kernel issparse due to limited vibrations in the mechanical environment of PFS onboardMEx (eigenmode of PFS, reaction wheels frequencies, inertia measurement unitdithering frequencies).
4. Discussion and conclusion
We described the approximated direct problem and an algorithm able tocorrect for the mechanical vibration of the PFS instrument. For the first time,we show that it is possible to reduce significantly the ghosts from the observedsignal from 3-5 % of the total energy to 0.4-0.7 %. We show that our estimationis coherent using three quantities: ghosts in the signal domain, laser line ghosts,distance to approximated kernel. Thus the global shape of PFS SWC spectra15an be corrected with our algorithm, allowing to better estimate temperature,and thermal profile on each PFS measurement, improving the few % of spectrawith high χ that could not be processed with current calibration (Grassi et al.,2005). Also, our correction may avoid the continuum removal step in the minorspecies retrieval (Sindoni et al., 2011). When the signal to noise ratio is highenough, our correction will also reduce the stacking procedure.In the future, we would like to propose an algorithm to correct the completearchive that would require: efficient algorithm, timesaving implementation, andfully automatic procedure. Also, new correction procedure must be developedto treat the whole orbits currently available (6405 at the date of writing).In order to correct any shaked FTS, semi-blind deconvolution is possible,knowing ˆ I Mars (but without knowing ˆ K approxP F S from the “laser line domain”)so that the “signal domain” only is required. Thus, any techniques of opti-cal path measurement (laser line, mechanical, etc. . . ) can be corrected withour technique. Nevertheless, the independent estimation of the kernel ˆ K approxP F S significantly improve the convergence of the algorithm. The only limitation toapply this method on other instruments is about the convolution equation. Con-volution is true if the signal from the planet I Mars has a significantly reducedwavenumber domain (as stated in eq. 1-3).
Acknowledgement
We thank Ali Mohammad-Djafari for fruitful discussions. We acknowl-edge support from the “Institut National des Sciences de l’Univers” (INSU),the “Centre National de la Recherche Scientifique” (CNRS) and “Centre Na-tional d’Etude Spatiale” (CNES) and through the “Programme National dePlan´etologie”. We also thank “European Space Agency” (ESA) for providingthe reaction wheels speed of MEx.
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