GOMOS High Level Algorithms Definition Document

Transcription

1 GOMOS High Level Algorithms Definition Document Copyright: VTT/Finland Issue: /04/1998 ACRI S.A. Finnish Meteorological Institute Service d Aeronomie Institut d Aéronomie Spatiale de Bruxelles

2 1. INTRODUCTION AIM OF THE DOCUMENT STRUCTURE OF THE DOCUMENT PHYSICS OF THE GOMOS MEASUREMENTS SOURCES OF LIGHT Stellar sources Sun Planets, Moon, and comets Auroral lights and other natural emissions PROPAGATION OF STELLAR LIGHT THROUGH THE ATMOSPHERE PROPAGATION OF SOLAR LIGHT THROUGH ATMOSPHERE DETECTION OF LIGHT BY THE GOMOS INSTRUMENT GOMOS instrument Imaging properties of GOMOS Detection properties of GOMOS GOMOS LEVEL 0 AND LEVEL 1B DATA PROCESSING DATA PROCESSSING OF LEVEL 0 DATA LEVEL 1B PROCESSING: WAVELENGTH ASSIGNMENT, GEOLOCATION AND DATATION LEVEL 1B PROCESSING ALGORITHMS FOR SPECTROMETERS Detection and correction of anomalies and outliers Corrections to measured counts Identification of stellar and background signals Full transmission calculation FROM LEVEL 0 TO LEVEL 1B FOR FAST PHOTOMETERS LEVEL 1B DATA PRODUCTS GOMOS LEVEL 2 DATA PROCESSING LEVEL 2 DATA PROCESSING ALGORITHMS FOR SPECTROMETERS Introduction Scintillation and dilution correction Chromatic refraction correction Spectrometer A spectral inversion Spectrometer B spectral inversion Line densities smoothing Vertical inversion Level 2 loops FROM LEVEL 1B TO LEVEL 2 FOR PHOTOMETERS High resolution temperature profile LEVEL 2 PRODUCTS

3 ACRONYMS DPM Detailed Processing Model ESL (GOMOS) Expert Support Laboratory FMI Finnish Meteorological Institute GOMOS Global Ozone Monitoring by Occultation of Stars GOSS GOMOS System Simulator GOPR GOMOS Prototype (GOMOS gaound segment simulator) IASB Institut d Aéronomie Spatiale de Bruxelles IODD Input/Output Data Definition IR Infra-Red LMA Levenberg-Marquardt Algorithm LOS Line of Sight NIR Near Infra-Red PPF Polar PlatForm Sd A Service d Aéronomie du CNRS S/C Spacecraft UVIS Ultra violet - VISible 3

4 1. Introduction 1.1 Aim of the document GOMOS (Global Ozone Monitoring by Occultation of Stars) is an instrument which will be flown on board the European Space Agency s ENVISAT-1 satellite. GOMOS is a novel instrument concept-there have been only a few sporadic attempts to exploit stellar occultations for studies of the Earth s atmosphere. This means that most of the data processing algorithms are either completely new or they have been tailored for stellar occultations. The aim of this document is to describe the methods which are used to process GOMOS data to GOMOS data products. The emphasis is in the scientific justification of the selected processing algorithms and the clear display of the algorithms. The algorithms discussed here constitute the ENVISAT-1 GOMOS ground segment supported by ESA. The GOMOS data processing includes several optional routes and many of algorithms include subtleties which are not easily covered in a document like this. A full account of the implementation of algorithms is included in the DPM-documentation produced by GOMOS ESL-group. Possible future editions of this document will also provide a more complete account of the GOMOS data processing algorithms. 1.2 Structure of the document This document covers two main themes. The first part is devoted to explaining what is known about the physics involved in the GOMOS measurements. This part concerns therefore our understanding of the GOMOS forward problem. We will concentrate on things which are essential in understanding of the GOMOS data processing algorithms. The second part is devoted to explaining the GOMOS data processing algorithms and how the processing chain is built. This is the inverse problem for GOMOS. The treatment of the processing is divided into two parts. The first part concerns Level 1b processing where instrumental corrections are applied. The second part concerns Level 2 where the main geophysical products like ozone density are derived. The references are placed in close proximity to the subject they concern. Many of the references are internal reports. They can be requested from individual ESL s whose addresses can be found in Appendix. Publication of the data processing methods, especially concerning Level 2, is in progress. References Envisat-1 Ground Segment Concept, Issue 5, rev.3, September 1994, ref. ESA/PB-EO(94)24. GOMOS Level 1b and 2 Detailed Processing Model (DPM), Issue 4, rev. 0, PO-RS-ACR-GS GOMOS Science Report, GOMOS Science Advisory Group (SAG), ESA, to be published, GOMOS DPAD report, GOMOS Science Advisory Group (SAG), ESA, in preparation. 4

5 2. Physics of the GOMOS measurements In this chapter we present an overview of the physics involved in GOMOS measurements. GOMOS will rely on the occultation measurement principle in monitoring of ozone and other trace gases in the stratosphere. The measurement principle is shown in figure below. It will use specifically stars as sources of light whereas the solar occultation measurements have been more favoured in past. The benefit of the occultation principle is that it is a self-calibrating measurement concept. The stellar (or Sun) spectrum is first measured when the star can be seen above the atmosphere. Measurements through the atmosphere provide spectra with absorption features from the passage through the atmosphere. When these spectra are divided by the reference spectrum, nearly calibration free horizontal transmissions are obtained. The specific advantages of the stellar occultation method are the good global coverage provided by the multitude of stars and the good vertical resolution provided by the point source character of stars. Fig.1. The stellar occultation principle The point source character of stars, and the weakness of their radiation, makes it necessary to consider some questions which are not relevant to solar occultations. The weakness means that we must understand what additional light sources may compete with the main stellar signal. Because one half of each orbit is illuminated by the Sun GOMOS will detect scattered solar light which usually dominates the stellar signal. GOMOS will also see light coming from auroral and other natural emissions in the atmosphere. The point like character means that we must pay special attention to the refractive effects on the propagation of stellar light through atmosphere. 5

6 2.1 Sources of light Stellar sources The main light sources for GOMOS are stars which provide a sufficient flux in the spectral range nm where the GOMOS detectors and the star tracking system are sensitive. The instrumental performance restricts the visual magnitudes to 4 or less and the acceptable temperature range is K. Obviously GOMOS measurement are difficult to interpret if a star is a rapidly varying star. Difficulties are also encountered if two or more stars are visually close to each other. The GOMOS measurement principle does not necessarily require that we know the actual spectrum of a star which we measure. For many purposes related to calibration this would, however, be advantageous and for the signal simulation it is necessary. Because the signal simulation is needed in algorithm development and testing, a discussion is warranted. There are no measurements of stellar spectra covering the whole spectral range of GOMOS. This is because the GOMOS spectral window is much larger than the atmospheric window (from ground). In the visible range the atmosphere is almost transparent and good spectral measurements can be carried out from ground based observatories. Below 300 nm the ozone layer prevents any stellar light from reaching the ground and the only way to record UV spectra is to use satellite borne spectrometers. So far satellite measurements have been limited to UV parts of stellar spectra. Therefore complete (from the GOMOS point of view) spectra can only be reached by combining ground based and space based measurements. This is obviously a difficult task. Some stellar spectra constructed by combining satellite and ground based measurement have been produced by Service d Aeronomie. Stellar spectra are often approximated by a blackbody model and in the GOMOS algorithm development this has been widely true. A blackbody model gives a rough simulation of the intensity distribution of a target star but it misses all narrow structures like Fraunhofer lines and ionisation edges. A more realistic description of a stellar spectrum can be obtained from the modelling of a stellar atmosphere and performing radiative transfer calculation. A wellknown example is the Kurucz model Sun The solar spectrum for the spectral range needed for GOMOS is quite well known. It is known to be stable better than 1%. Higher variability occurs for shorter wavelengths Planets, Moon, and comets There is no plan to use planets, comets or the Moon as an light source for measurements but they will certainly affect the GOMOS mission planning. The Moon may be used in calibration and it may be a contributor to the background radiation. 6

7 2.1.4 Auroral lights and other natural emissions Besides light sources (stars, planets, Sun) outside the atmosphere GOMOS will be affected by sources inside the atmosphere. These include natural emissions and auroral lights. The recently discovered light phenomena related to atmospheric electricity like sprites may well be detectable by GOMOS. References Kyrölä, E., L. Oikarinen, E. Sihvola, and J. Tamminen, GDC-SF-3: Signal simulation for stellar occultation measurements in nm GDC-SF-3, Kyrölä, E., and J. Tamminen, GDC-SF-TN-3: Measuring OClO by the GOMOS instrument, FMI, Lehtinen, R., GDC-SF-6: Survey of stellar spectra, FMI, Lehtinen, R, and G.W. Leppelmeier, GDC-SF-TN-6: Suitability of planets for use by GOMOS, FMI, Propagation of stellar light through the atmosphere Light from a star appears as a parallel bundle of rays before entering into Earth s atmosphere. Part of the flux is absorbed by various atmospheric constituents and this part is transformed into non-radiative forms of energy. A part is scattered out of the beam. Individual rays traverse along curved paths which are determined by the geometry, wavelength and the state of the atmosphere (refraction). After the atmosphere the parallel bundle of rays appears to be spread like a fan and the mutual coherence between different rays is perturbed. This will result to dilution of the intensity and a speckle like pattern in the intensity field. The attenuation of the light due to absorption and scattering can be accurately described by the well-known Beer s law: I = I 0 e [ ] σ Γ i λ,t (s) n i (s) ds i (2.2.1) where σ are the absorption or scattering cross-sections. They depend on wavelength and for some atmospheric constituents also on temperature. The constituent density is denoted by n. For the wavelength region where GOMOS is working ( nm) the principal absorbers are O 3, NO 2, H 2 O, and O 2 and the main scatterers neutral density and different aerosol particles. Inside quite narrow wavelength regions there are contributions also from NO 3, OClO, and BrO. 7

8 NO 3 NO 2 O 3 Fig.2. UVIS Cross sections The integration follows along a refracted path. The path is determined from d ds (n d & r ds ) = n (2.2.2) with the satellite position and the stellar direction as boundary conditions. The source term is produced by the gradient of the refractive index n. The direct effects of refraction on the intensity can be described by the following equation I 2 = I 1 n 2 n 1 e s2 s 1 2 S n ds (2.2.3) where points 1 and 2 belong to the refracted path. The eikonal S obeys ( S) 2 = n 2 (2.2.4) References Born, M., and E. Wolf, Principles of Optics (Sixth edition), Pergamon Press, Brasseur, G., and S. Solomon, Aeronomy of the Middle Atmosphere, D. Reidel, Dordrecht, Dalaudier, F. Modeling of refractive effects during occultation of stars, 1994 (Sd A report). 2.3 Propagation of solar light through atmosphere Solar light reaches GOMOS in bright limb conditions and also when the Sun is just below the horizon (twilight). The quality of the GOMOS measurements in bright limb depends on the ability to subtract the solar signal from the stellar signal. Even the solar signal appears as 8

9 nuisance for the main data analysis, it can be used to retrieve atmospheric information at higher level in the GOMOS data analysis. The total solar intensity obeys the radiative transfer equation: di(λ, s,ω) ds = σ (λ,t(s))ρ(s)i(λ,s,ω) + σ(λ)ρ(s) 4π P(Ω,Ω') I(λ,s,Ω')dΩ' (2.3.1) For simplicity we assumed only one scattering constituent and moreover we assumed that it does not absorb. The boundary conditions include the incoming flux from the Sun and the albedo of the Earth at ground. The function P is the scattering phase function. The single scattering part can be computed in a straightforward manner I scat (λ,l,t) = I Sun (λ) T solar (s)t(s)(σ R (λ)p R (θ)ρ air (s) +σ a (λ)p a (θ )ρ a (s))ds s l (2.3.2) Here T solar (s) is the transmission along the solar path to the scattering point s, T(s) is the transmission along the line of sight from s to observing point, P R (θ ) and P a (θ ) are the scattering phase functions for the Rayleigh and aerosol scattering, and finally θ is the scattering angle. The single scattering contribution is usually the dominant term in the radiance but in some cases its contribution can be as low as 50% of the total (see Oikarinen and Lehtinen, 1997). References Level1b DPM, issue 4.0, PO-RS-ACR-GS-0001 Oikarinen, L., and R. Lehtinen, Background term analysis, GOMOS Mission planning and Algorithm Aspects-project,ENV-TN-FMI-GM-006, FMI, Detection of light by the GOMOS instrument GOMOS instrument GOMOS is a medium resolution spectrometer covering the wavelength range from 250 nm to 950 nm. The high sensitivity requirement down to 250 nm has been a significant design driver leading to an all reflective optical system design for the UVVIS part of the spectrum and to functional pupil separation between the UVVIS and the NIR spectral regions (thus no dichroic separation of UV). Due to the requirement of operating on very faint stars (down to magnitude 4 to 5) the sensitivity requirement to the instrument is very high. Consequently, a large telescope (30 cm x 20 cm aperture) had to be used to collect sufficient signal, and detectors with high quantum efficiency and very low noise had to be developed to achieve the required signal to noise ratios. In addition, in order to use the entire star signal, a slitless spectrometer design had to be chosen. The price which had to be paid for this design is that a highly performance pointing system had to be used to keep the star image fixed at the input of the spectrometers in order 9

10 not to degrade the spectral resolution and the spectral stability. The picture below illustrates schematically the instrument design. To achieve a high signal-to-noise ratio when observing the very weak star signal embedded in strong surrounding atmospheric background and stabilising the star image in spite of the satellite disturbances are major engineering challenges for the GOMOS design The main instrument requirements are summarised in Table 1 below Requirement description Occulting stars characteristics Spectral range of the spectrometer Spectral sampling Spectral resolution Spectral stability knowledge in dark limb Photometer spectral windows and sampling rate Requirement Visual magnitude range: Max to min. 2.4 to 4 for stars with 30000K and 3000K temperature respectively nm for UV and VIS nm and nm 0.3 nm in UVVIS 0.05 nm in NIR 1.2 nm in UVVIS 0.2 nm in NIR 0.07 nm in UVVIS nm in NIR nm and nm 1 khz sampling rate Short term radiometric stability (over 150 s) 1% Linearity 1% Pointing stability Better than 40 microradians peak-peak Number of occultations per orbit 45 on average, i.e. approximately occultations during the 4-year mission. Angular coverage 10 deg. to +90 deg. with respect to the flight direction. Thus, large instrument angular range observability Table 1. The main instrument requirements for GOMOS. The star tracker is sensitive inside the range nm and it is capable to keep the stellar spectrum fixed with an accuracy of a 1 pixel. The star tracker is able to follow a star down to km. The altitude limit depends on the qualities of a star, and on the atmospheric state, and the bright/dark limb condition. The overall instrument layout is shown in the following figure: 10

11 Fig. 3. GOMOS instrument layout Imaging properties of GOMOS The GOMOS optical layout is show below 11

12 Fig. 4. Optical layout for GOMOS. To describe the imaging properties of GOMOS we divide the imaging chain into two parts. The first part starts from the outermost surface of the GOMOS optical chain and ends at the slit plane. The second part is from the slit plane to the grating and finally to the detector surface. The first part can be written as I(u,v,λ,t) = I 0 (λ,t, s * 0 )S(u, v, s * 0 ) ( ) The first factor is the incoming intensity which depends on wavelength, time and its direction of approach. The second factor indicates how the telescope transfers light from the front surface to the slit plane where (u,v) are the co-ordinates. The slit is 120 µm in the horizontal or spectral direction and 200 µm in the vertical or spatial direction. The stellar image is much smaller and the slit is therefore actually needed for the stellar imaging i.e. we have actually a slitless instrument. The slit is, however, needed to reduce the background radiation in the bright limb measurements. The intensity on the detector can be written as a mapping from the intensity on the GOMOS slit plane as follows I(x, y,λ,t) = I(u, v,λ,t) h(x x 0 d(λ λ 0 ) u, y y 0 v)dudv ( ) Here h is the impulse response function. The following figure shows the imaging and spectroscopic scheme used. 12

13 S h Front Slit Grating CCD Fig. 5. GOMOS imaging. The stellar light (solid lines) falling onto the GOMOS front surface comes as parallel ray bundles where the direction of approach depends on the wavelength. The background is coming from all directions. In the slit plane stellar light with one wavelength is imaged into a small spot and its position will vary rapidly as GOMOS intercepts different regions in the non-uniform intensity field. There is also motion due to pointing noise. All these motions are mapped onto the CCD plane where they are seen as wavelength shifts and wandering in the spatial direction Detection properties of GOMOS The GOMOS data are produced when the intensity in Eq. ( ) produces electrons in the CCD pixels. Nominally several pixels are binned in the spatial direction and charges are allowed to accumulate seconds. The resulting net charge is transferred to the electronics where it is amplified and digitised. The final result which we see as counts is N = G( I 0 (λ,t, s & 0 )S(u,v, & s 0 )Q(x, y,λ)h(x x 0 d(λ λ 0 ) u, y y 0 v)dudvdxdydλdtd & s 0 ) ( ) The amplification and analog to digital transformation is described by the function G. The summing goes over binned pixels. The integration over x and y are over the pixel surface. The integration over the input directions is needed because refraction in the atmosphere breaks the 13

14 parallel ray coherence. The detection efficiency of individual pixels are described the quantum efficiency function Q(x,y,λ). In addition to the signal produced by the input light there are generation of dark charges as well as the read-out noise. References Level1b DPM, issue 4.0, PO-RS-ACR-GS-0001 Kyrölä, E., and E. Sihvola, J. Tamminen, and L. Oikarinen, Global inversion, Gomos prototype report, PO-TN-FMI-GM-014, FMI, Born, M., and E. Wolf, Principles of Optics (Sixth edition), Pergamon Press, Bertaux, J.L., G. Megie, T. Widemann, E. Chassefiere, R. Pellinen, E. Kyrölä, S. Korpela and P. Simon, Monitoring of Ozone Trend by Stellar Occultations: The GOMOS Instrument, Adv. Space Res., 11, , Korpela, S., A Study of the Operational Principles of the GOMOS Instrument for Global Ozone Monitoring by the Occultation of Stars, PhD thesis, Geophysical publications 22, FMI, Kyrölä, E., GOMOS imaging, FMI, GOMOS level 0 and level 1b data processing The main goal of the GOMOS level 1 processing is to estimate a set of horizontal transmission functions using data measured by the GOMOS spectrometers. This so-called full transmission function will serve as basic data for the level 2 processing where profiles of atmospheric constituents are to be determined. The fast photometer data is also corrected for instrument dependent factors. Even if the GOMOS measurement method, stellar occultation, is a self-calibrating measurement method, several calibration-type corrections are needed. The self-calibration is taking care of general slow changing radiometric drifts but many pixel-level details are not included. Outliers connected to bad pixels and cosmic rays are also to be corrected. A possible non-linearity cannot be corrected by occultation self-calibration. 14

15 Fig 6 Level 1 b processing 15

16 3.1 Data processsing of level 0 data This level includes a simple set of operations where the orbit based ENVISAT-1 data are separated into individual data sets each covering one occultation. A special case occurs for tangent occultations that are stored in two individual packets and here they are combined to form a single occultation. In this stage various auxiliary products are read in: - Level 1b processing configuration auxiliary product - Calibration auxiliary product - Instrument physical characteristic auxiliary product - Star catalogue - Atmosphere forecast/analysis file (ECMWF data). 3.2 Level 1b processing: wavelength assignment, geolocation and datation Wavelength assignment Two operations are performed: the nominal wavelength assignment is carried out using the calibration auxiliary product and the pixel wise spectral shifts are identified using the star tracker data. The nominal wavelength assignment corresponding to a perfect tracking of the star during the measurement is provided by the spectral assignment of one CCD column and by the spectral dispersion law of the spectrometers read in the calibration auxiliary product. Spectral shifts due to vibrations and imperfect tracking are estimated thanks to the pointing data history produced by the SATU and are used to spectrally assign each CCD column during each spectrometer measurement. The following figure presents the SATU output data between 0.5 and 3.0 seconds (SATU output frequency is 100 Hz) 16

17 A mean SATU output data is computed at a frequency of 2 Hz and is used to compute the physical and spectral shifts of the star signal on the spectrometer CCD arrays. The next figure presents these mean physical shift expressed in CCD columns (or pixels). Geolocation A complete geolocation calculation is carried out. Each measurement of the atmospheric transmission is precisely geolocated. Several different factors contribute to the geolocation: Orbit Shape of the Earth State of the atmosphere Stellar co-ordinates The orbit information can be obtained from the ENVISAT-1 orbit propagator where the initial data is included in the level 0 data. Depending on the time delay of the processing with respect to the measurement there may also be more accurate orbit data available. The shape of the Earth is approximated by the well-known model WGS84. The atmospheric is characterised using the ECMWF prediction or analysis for the lower atmosphere and the MSIS90 model for the upper atmosphere. These two models are joined inside the occultation region only. The hydrostacity of the profiles is forced. The ray tracing uses the general ray tracing equation (2.2.2). The refractive index depends on the neutral density and wavelength as: n = ( λ λ ) ρ (3.2.1) 2 ρ 0 where: λ is expressed in nanometers 17

18 The approximation applied in this calculation is that there is no contribution from density gradients other than the ones in the plane determined by GOMOS, the star and the Earth centre. An example of refraction effects can be seen on the following picture presenting the altitude of the tangent point during an occultation. References Hauchecorne, A., Atmospheric model, ref. PO-TN-SA-GS-007, Sd A. Hedin, JGR, 96, , 1991 (MSIS90 model). 3.3 Level 1b processing algorithms for spectrometers The data processing requires a good knowledge of instrument characteristics (dark charge, wavelength calibration, wavelength resolution,...) based on ground characterisation and additional in-flight measurements. Since GOMOS will collect data on the day side as well as the night side, one major step in the Level 1b processing is to disentangle the light which comes from the star from the light which comes from the limb, solar illuminated (background). GOMOS is equipped with three bands of measurements, two devoted to the background above and below the stellar spectrum. Given a good radiometric calibration of GOMOS, the absolute limb radiance spectrum can be extracted from the GOMOS background measurements. These limb radiances may in turn be interpreted in terms of aerosol loading, and ozone and other constituent distributions, and this is why they belong to the standard Level 1b products. 18

19 3.3.1 Detection and correction of anomalies and outliers Saturated samples processing In this module, the saturated pixel values are monitored and the detected ones are flagged and removed from further processing. The indices of the saturated pixels are determined by: D ({k }) > s d sat ( ) The saturation limit d sat is set by the user. The initial spectrometer data set (D,{k}) is then screened by D({k}) D({k 1 }) where {k 1 } ={k} -{k s } ( ) This step increases confidence on pixel based data. It does not affect the basic error content of pixel data. The number of useful data pixels is, however, reduced Bad pixels processing The aim of this processing is to flag the bad pixels samples and create the missing data by median filtering using the values of the neighbour measurements in the spectral direction. 1. Bad pixels are flagged and removed from any further processing i.e., D({k 1 }) D({k 2 }) where {k 2 }={k 1 }-{k b } ( ) 2. New values are interpolated for bad pixels from original data: DP D ({k b}) = Median(D) D({k1 }) ( ) DP D ({k 2}) = D({k1}) This operation does not increase the information content of the data. New data are no longer statistically independent from the neighbouring data which should be taken into account by generating off-diagonal elements to the covariance matrix. Bad pixels are specified from a bad pixels list coming from the calibration database. The first list is defined by on-ground characterisation. This list is assumed to be updated regularly by the calibration process (Uniformity monitoring mode). There is no on-line detection of bad pixels but the cosmic rays correction processing may also prevent unexpected bad pixels (a bad pixel creates a local sharp variation which may be detected by the cosmic rays correction processing). Bad pixels are identified from the bad pixel list {k b } and one of the two alternative actions is performed. 19

20 Cosmic ray processing CCD pixels may be hit during the occultation by high energy particles such as proton. This could lead progressively to a permanent damage of the pixel (in this case, the pixel will be detected as a bad pixel during the next calibration phase), and/or to a temporally abnormal response. The effect of cosmic rays is generally reversible : a sample including an abnormal response will return to its initial state after a while. This abnormal CCD response is eliminated thanks to a median filtering in both spectral and temporal directions. The value of the corresponding sample is replaced by the computed median. CR DP DM = Median(D ) ( ) The cosmic ray pixels {k c } are identified by DP CR D ({k c}) DM ({k c}) > d CR ( ) and cosmic ray pixel contents are filled by median values: CR DP D ({k c}) = D D ({k 2}) CR D ({k 3}) = D where {k } = {k } {k } 3 2 c CR M DP ({k ({k c 3 }) }) ( ) Corrections to measured counts Conversion to physical units Conversion of the ADU values into number of electrons in each spatial band of the spectrometers A and B and non-linearity correction. This conversion needs the value of the CCD gain (in e/adu) for each electronic chain (for the nominal and the redundant chains) and for each programmable gains (4 gains per spectrometer). The non-linearity correction of the electronic chain is performed in this processing. LIN CR D = D * f nlin RO LIN N = (D d0 )*G ( ) ( ) Here f nlin is the non-linearity factor, G is the electronic gain factor and d 0 is the off-set Dark charge correction In this module the dark current contribution is estimated and removed from the signal. There are two alternatives for removing dark current in the present processing baseline but the dark current can also be left intact. That is 20

21 DC RO N = N ( ) The first removal method assumes that for every pixel we can use a dark charge calibration map to remove the estimated mean of dark current for this pixel: N DC = N RO DC where DC = DC MAP (T) ( ) In the second method, the dark current is estimated from the current occultation measurement. This method uses signals from the upper and lower bands when the line of sight does not cross the atmosphere (i.e. the first measurements of the occultation). The central band dark current is calculated by interpolation. N DC = N RO DC DC = Average(N RO ) ( ) Internal straylight correction In this module, the internal straylight is removed using a look-up table. The look-up table gives the internal straylight as a function of the signal itself. N IST = N DC ISL ( ) External straylight correction In this module, the external straylight is removed using two look-up tables. The first look-up table gives the external straylight due to solar light and the second due to the earth. N EST = N IST ESL ( ) Vignetting correction In this module the degraded performance of the B spectrometer at certain azimuth angles (nominally from -10 to -5 degrees) is taken into account by: IRV N = N f EST vig ( ) Identification of stellar and background signals Central background estimation The objectives of the central background estimation are to provide: an estimation of the background signal measured by the central band during the occultation. This signal will be subtracted from the measured central signal in order to get the star signal alone; 21

22 an estimation of the contribution of the background to the signal of the fast photometers; an estimation of the upper and lower background corrected for flat fielding. In this module the background contribution in the stellar band is estimated. The present plan includes four different methods for estimating the background contribution inside the stellar band. Before the estimation a flat field correction is applied to upper (U) and lower (L) bands: NST N N L,U = Q IRV L,U L,U ( ) where Q is the band (binned) quantum efficiency. The four estimates for the central background are then as follows. 1. No background estimation N N NST c B = 0 = 0 ( ) 2. The second possibility uses a linear interpolation between upper and lower bands N NST c = N NST L + (N NST U N NST L h )( h C h L ) h U L ( ) where h L,U are the tangent heights attached by the geolocation to upper and lower bands. 3. The third possibility is an exponential interpolation N NST hc hl NST NST N U h U hl c = N L + ( ) NST N L For both ways the central background is obtained as N = N B C NST C * Q C ( ) ( ) 4. The fourth method is called the general method. It generates a combined series of background as a function of the tangent height. It assumes that these series represent a static background i.e., the three bands are samples from the same background. First a time integrated series is calculated by N NST L,U m h(f ) (m) = N f = 1 t NST L,U (f ) ( ) where h is the altitude coverage attached to the measurement and t is the integration time. The new series is sorted by increasing altitude and then smoothed by Gram-polynomials. The background for central pixels is now given by N ~ NST (h (f )) = c j C Gram j [h c (f )] j ( ) Finally 22

23 N B k and = q n k,n t (N ~ h NST k,m (h(f + 1)) N ~ NST k,m (h(f ))) ( ) N = NST C N Q B C ( ) Fig. 7. Background estimation accuracy Star signal computation In this module the estimated background contribution is removed from the stellar band. This is based on the following description of the signal after previous corrections N = Q y I0 ( λ,t)h(x x 0 d( λ λ0 ) u 0 (t, λ), y y0 v0 (t, λ))dλdt y ( ) where we have for simplicity also assumed that a star appears as a delta function in the slit plane. Further progress relies on the assumption that impulse spread function is separable into spatial and spectral functions. This formula shows that scintillations and pointing are entangled and we attempt to resolve this entanglement by the following algorithm. We derive the star signal as 23

24 S Nˆ = N ~ W S scin q ( ) where: N ~ S = N IRV C N B ( ) The scin Wq -function is the quantum efficiency weighted band-level instrument function scin scin W q = h y (n) q n ( ) n Here the scintillations are brought in by: h scin y = hτ scin ( ) and the scintillation transmission is calculated from photometer data. Fig. 8. Estimation of the occulted star spectrum Slit correction If the star image in the slit deviates unexpectedly much from the nominal central position the decreased slit transmission blocks part of the signal. Because the position of the star image can be estimated form the star tracker, a look-up table can be used to recalibrate the signal: N S = Nˆ τ S slit ( ) 24

25 where the slit transmission is the calculated blocking factor Reference star spectrum computation There are two alternative methods to obtain the reference spectrum used to calculate the experimental full transmission function. The first method uses a few first measurements before the star signal starts to get absorbed and scattered. The second method uses a GOMOS star reference spectrum bank. 1. Here we assume that the reference star spectrum is obtained from an average over a few first star spectra measurements of the occultation outside the atmosphere: N REF = Mean(N int ) ( ) where: N int = CubicSpline(N S ) ( ) 2. Here the reference spectrum from the GOMOS stellar spectrum data bank REF REF N = N BANK ( ) Fig. 9. Estimation of the reference spectrum Full transmission calculation The Level 1b processing is finalised by calculating the so-called full transmission. N T = N S REF ( ) 25

26 The full transmission is computed as the ratio of the estimated star spectrum to the reference spectrum of the current occultation. The covariance function gives an estimation of the errors due to both instrument measurements and level 1b processing. These functions will be used by the level 2 algorithms. The error on the star spectrum is computed as the square root of the quadratic sum of the noise due to the instrument itself during the measurement and the noise due to the level 1b processing. Noise due to instrument includes : the spectrometer electronic chain noise the shot noise expressed in electrons the quantisation noise due to the ADC The noise due to the level 1b processing has several contributors : the dark charge the straylight the background corrections (wich is the main contributor in bright limb conditions). All the spectra used in this processing must be spectrally aligned before computation. The wavelength assignment of the star spectrum is re-sampled over this spectral grid before computation of the transmission and of the covariance. Fig. 10a. Estimates of the transmission spectra. 26

27 Fig. 10b. Associated covariances. References Level1b DPM, issue 4.0, PO-RS-ACR-GS-0001 Kyrölä,E., E. Sihvola, J. Tamminen and L. Oikarinen, Global inversion, GOMOS prototype report, PO-TN-FMI-GM-014, FMI, From Level 0 to Level 1b for fast photometers The processing applied to the fast photometer samples is composed of the following steps : Saturated samples processing Non-linearity correction ADU to electrons conversion Dark charge correction Straylight correction Vignetting correction Background subtraction Flat-field correction Most of these processings are similar to the ones for the spectrometer samples, except the background subtraction which use the estimated central background computed for the spectrometers as an input to remove the background contribution for the photometer samples. 27

28 Fig. 11a. Blue photometer Signal Fig. 11b. Blue photometer signal (zoom) 28

29 Fig. 11c. Red photometer signal Fig. 11d. Red photometer signal (zoom). 3.5 Level 1b data products Reference star spectrum: It is obtained by averaging the 5 first spectra obtained during the occultation ; the averaging is done to minimise the noise. The noise could be further reduced by taking more spectra. However one should take care also of some associated problems : - the altitude onset of absorption is wavelength dependent - absorption by noctilucent clouds may occur around 85 km - auroral emission may be also a contributor 29

30 Reference atmospheric profile: This profile is extracted from a meteorological field analysis (say, ECMWF) combined with MSIS 90. It is used to compute during Level 1b the refraction of the LOS, by full ray-tracing computation Full transmission spectra and covariance: This spectrum is obtained by dividing each spectrum by the reference star spectrum. It is described as «full» because it is the actually measured transmission, not corrected for refraction effects (dilution, scintillation, chromatic refraction) nor for variable PSF. Each spectrum is re-sampled on the wavelength pixel grid of the reference spectrum to get the transmission. The covariance (here, in fact, the variance of each pixel signal) is computed from analysis of S/N ratio. Central background estimate: This is the estimated background contribution to the total signal in the central band, which was subtracted to yield the pure stellar signal. Photometers data: scintillation data expressed in electrons. SATU data : This is the position of the centroïd of star image on the SATU CCD (Stellar Tracking Unit), which enables one to know where to find each wavelength in the series of pixel. Wavelength assignment: This is the wavelength of the centre of each pixel, for the given measurement of transmission (it may change during occultation owing to imperfect tracking). Geolocation: This includes both the position of ENVISAT spacecraft and the position of the tangent point of the LOS. Limb products: This product includes the background actualy measured with the two external bands of CCD spectrometers, uncorrected and corrected for straylight and other factors. TABLE : GOMOS LEVEL 1 B PRODUCTS (For each single occultation) Level 1b product Reference star spectrum Reference atmospheric profile Full transmission spectra and covariance Central Background estimate and error Photometers data and error SATU data SFA angle measurements Wavelength assignment of the spectra Geolocation and error Limb product Upper and Lower Background Spectra and error once per occultation once per occultation 2 Hz 2 Hz 2 Hz 1 khz 100 Hz 5 Hz 2 Hz 4 Hz once per occultation 2 Hz References GOMOS Input/Output Data Definition, Issue 4, rev. 0, ref. PO-RS-ACR-GS

31 4. GOMOS level 2 data processing In the level 2 data products which have a direct applicability for atmospheric analysis are generated. The most important are the vertical profiles of various constituents like ozone, NO 2, aerosols etc. Intermediate, but not less valuable, products are horizontal column densities of the same constituents. The analysis of photometer data will produce a high resolution profile for the temperature field. 4.1 Level 2 data processing algorithms for spectrometers Introduction The transmission which is received is the ratio of the occulted star spectrum over the reference one, the latter being built up at the beginning of the current occultation by averaging several measurements outside the atmosphere. In that sense, it is believed that the measurements is self-calibrated at the occultation level. As the ray coming from the star to GOMOS is going through deeper and deeper in the atmosphere as the occultation last, this measured transmission reflects the combination of several phenomena which are recalled here below : Absorption by atmospheric components which have an impact on the considered wavelength range of the spectrometers (250 up to 670 nm). Diffusion by air particles (Rayleigh diffusion) and aerosols (Mie scattering) From these two main contributors to the final GOMOS level 2 product, retrieval algorithms have been designed to extract relevant information for the constituent profiles. However, due to the particular design of the measurement process, the GOMOS data are affected by several perturbations mainly coming from : The dilution at low scale coming from the increasing deviation of the light ray during one acquisition The dilution at high temporal scale (scintillation effects) which is the same as before but at high frequency so which is randomly applied during one spectrometer acquisition, The chromatic refraction which will make different the geometry of the ray paths at a given time (so at a given time measurements over the spectrum are not representatives of the same altitudes) The instrumental response which cause a distortion in the nominal wavelength assignment of the spectrometer measurement All these contributors have been considered for the GOMOS data processing. The concentrations retrieval is done by comparison between the measured value and a theoretical model of transmission in which concentrations, as well as perturbations terms, explicitly appear. Some of the perturbations which cannot be theoretically described has been removed from the measured data before comparison. 31

32 Once the data are cleaned up from inherent perturbation, the comparison with parametrised expression of the transmission is performed to get the line densities (that is to say the integrated value of the densities along the ray path - for each tracked species). As being the result of the spectral signature of each components, this step is called the spectral inversion. Once this is done a vertical inversion is then activated to deliver local densities at the tangent point altitudes for each acquisition, this will give vertical profile for each species. The level 2 processing assumes that the full transmission obtained at the end of level 1b can be modelled as T = T atm (λ,t)w(λ,t)dλdt ( ) where W represents the instrument function. In order to proceed we assume that the atmospheric transmission is factorable into the extinction part due to absorption and scattering and into a refractive part generated by refractive dilution and scintillations: T atm = T ext T ref ( ) The task where we aim to retrieve constituent densities relies on our ability to link the experimental transmission spectrum (or spectra) to the modelled transmission part given by Beer s law. This task is interfered by the spectral and temporal integration in Eq. ( ) and the involvement of refractive modulations in Eq. ( ). The first problem will be taken (approximately) into account by developing the model transmission. For the latter problem we rely on use of the fast photometer data. Once we have isolated the transmission due to absorption and scattering we can concentrate on comparing the modelled transmission T ext = e τ ( ) with the data void of refractive dilution and scintillations. In Eq. ( ) the optical thickness, τ, is given by an integration along a ray path Γ N species τ = σ i (T(l),λ)n i (l)dl ( ) i =1 Γ This formulation is not directly used in the GOMOS data processing because of the difficult temperature dependence of the cross-sections. Instead we write an equivalent form τ = where N species σ i eff (λ)n i (l) ( ) i =1 N = n () l dl i Γ i ( ) 32

33 is the line density of the species i and ˆ σ i (λ, h 0 (λ)) = σ i (T(l),λ)n i (l)dl Γ n i (l)dl Γ ( ) is the effective cross-section of species i, at the wavelength λ and along the ray path characterised by the tangent altitude h 0 (λ). The use of effective cross-section makes it possible to separate the inversion problem into two parts, the spectral inversion part and the vertical inversion part. This leads to a substantial reduction in the size of the input data needed in each part. The two parts are, however, coupled together and an iterative loop over spectral and vertical inversion is needed to take this into account. Besides the iteration loop for building an effective cross-section there is a second iterative loop which arises because GOMOS is capable to retrieve atmospheric temperature. If this temperature is significantly different (and reliable) from the a priori value (MSIS90 and ECMWF) used in level 1b, the level 2 calculation can be redone. The only significant difference comes from the fresh ray tracing calculation (see level 1b). 33

34 Fig. 12. Level 2 processing Scintillation and dilution correction Refraction affects the GOMOS measurements in three different ways. The most obvious is the change in the geolocation of the light path connecting the satellite and a star. This must be taken into account in the modelling of the transmission. The other two effects are generated by the variation of the air temperature and also therefore also the index of refraction in spatial domain. This causes differential bending of the light rays in the atmosphere which will result in modulation of the intensity detected by GOMOS. In the GOMOS level 2 data processing a distinction is made between the smooth and the fast variations of the index of refraction. The smooth variation can be calculated analytically whereas the fast variation is best described using GOMOS fast photometer data. The objective here is to provide a simple algorithm in order to estimate the term including scintillation and dilution effects. This algorithm is not able to solve completely the problem 34

35 and has several limitations. However, it will provide a basis for taking into account dilution and scintillation effects. The scintillation and dilution correction process is based on the assumption that the attenuation due to dilution (including mean dilution and scintillation) may be split in two independent terms T ( λ, t) = T ( t). T ( λ, t) sd s d ( ) where T s (t) represents the effect of scintillation, assumed to be free of direct chromatic effect and T d (l,t) represents the mean dilution component. The following approach is assumed: T s (t) will be estimated using the signal (provided in the level 1b product) of the fast photometer which has the highest signal to noise ratio. The principle is to minimise extinction effects in the photometers bands (the less extinct one is generally the red photometer). The estimation of this term is made with the assumption that all high frequency fluctuations in the photometer signal (if we except noise fluctuations) are due to scintillations and that fluctuations due to structures in vertical profiles of absorbing constituents affect only the low frequency part of the signal. This assumption is justified by the fact that the absorption depends on the integrated density along the line of sight, while the dilution is sensitive to the vertical second derivative of the integrated density. As a matter of fact high frequencies are enhanced by the derivation. T d (λ,t) is computed using a vertical profile of atmospheric density coming from an external atmospheric model, as explained hereafter. The dilution curve as it stands in equation ( ) would be, strictly speaking, valid only for the wavelength band of the photometer (of mean value λ 0 ). The change of the index of refraction with wavelength has two consequences on the dilution term which must be taken into account: 1) the mean value of the dilution is modified by a change in the divergence or convergence of the light beam ; this is already taken into account in the wavelength dependence of T d (λ,t) 2) for a given time the tangent altitude depends on the wavelength; this has to be considered in both terms, T d (λ,t) and T s (t), as described below. The mean dilution is approximated by using the phase screen approximation. Instead of a 3D - atmosphere we assume that it can be replaced by a screen located at the tangent point. The refractive bending takes place at the screen. With this approximation the mean dilution can be written simply as 1 T d (λ,h 0 ) = 1+ L dδ (λ,h ) 0 dh 0 (λ) ( ) 35

36 where L is the distance from the tangent point (screen) to the satellite, δ the deviation angle of the ray. In the GOMOS level 2 processing the deviation angle is taken from the ray tracing calculation. If the mean deviation is computed for the first time, the ray tracing calculation is the one performed in level 1b. For subsequent calls (see level 2 loops) the ray tracing may be performed again using GOMOS generated atmosphere. The scintillation transmission is calculated by T s (t) = I ph (t) I ph (t) ( ) where numerator is the photometer signal and the denominator is the smoothed photometer signal. The smoothing is performed by the Hanning filter (see Press et al.) I ph (t) = Hanning(I ph (t); t) ( ) The transmission due to absorption and scattering can now be calculated dividing the observed transmission by the estimated refractive part: T a (λ) = T(λ) T sd (λ) ( ) It is important to note that, at this stage, we have transmission corresponding to tangent altitudes depending on wavelength. The chromatic effect is taken into account in the mean dilution term and is neglected in the scintillation term. Notice also that this algorithm assumes that the absorption does not contain high frequency components. This algorithm does not check either the consistency between the atmospheric model (dilution) and the Rayleigh attenuation. Fig. 13. Noise after scintillation correction. 36

37 References Dalaudier,F., Rapid fluctuations algorithm, ref. PO-TN-SA-GS-005 Hauchecorne, A., Determination of the dilution curve, ref. PO-TN-SA-GS-011 Press, W. H., B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, Chromatic refraction correction During its path through the atmosphere, a light ray is refracted by density gradient of the index of refraction. This bending is dependent on wavelength. The smaller the wavelength, the greater the bending. It means that there is no single path through the atmosphere that could be attached to a light field detected at any given time by GOMOS. The goal of this algorithm is to correct the atmospheric transmission for the chromatic refraction due to the atmosphere. We calculate the tangent points corresponding to each pixel of each spectrum and then to calculate transmission for given tangent altitudes (namely the reference tangent altitudes h 0 (λ ref )) by doing linear interpolations between two acquisitions. A linear resampling (unless a spectral resampling has been previously done in the Level 1b is applied and a linear interpolation of the transmission is performed. The particular case of tangent and near tangent occultations is avoided and will require further analysis. resampled ' T a, j = (1 r)t a, j +1 + rt a, j ' T a, j = (1 r)t a, j resampled + rt a, j 1 ( ) The first alternative is applied for wavelengths smaller or equal to the fixed reference wavelength and the second alternative to the rest. The coefficient r is the interpolation coefficient. This module represents the first alternative method to correct the chromatic refraction effects. The second alternative is dealt with the connection of the transmission modelling Spectrometer A spectral inversion Developments in the transmission model The basic assumption in the GOMOS Level 2 data inversion is that the part of the transmission linked to absorption and scattering can be modelled as T ext = 1 ˆ σ i ( λ)n i (t,h 0 (λ )) i W (λ t k (t) λ')e dλ' dt ( ) This formula could act as the modelling used in data inversion but in the GOMOS ground segment the model is further developed by moving the two integrations into the exponent term (See Sihvola, 1994). This leads to performance improvement in the actual computations. 37

38 Spectral integration The aim of this section is to put the spectral integration in Eq. ( ) into the form t e ~ τ() where W is the spectral point spread function (also called static instrumental function). Equation ( ) can be approximated by T(λ, L) 1 t t e τ (λ,l (λ,t )) dt ( ) where the modified optical depth is τ (λ,l(λ,t)) = 1 2 k,j j σ j (λ,l(λ,t))n j (l(λ,t)) B jk ((λ,l(λ,t))) N j (l(λ,t))n k (l(λ,t)) ( ) Here σ j (λ) = W(λ, λ')σ j (λ')dλ' ( ) B jk (λ ) = W(λ) σ ˆ j λ ()ˆ σ k ()dλ λ σ j (λ). σ k (λ) Assumption: For the following developments, the geometry of the acquisition has been considered uniform for the whole spectral range, that is to say that all variables depending on tangent altitude in the transmission model are expressed on the reference ray path Γ ref. In practice, it consists of the substitution of h 0 (wl kj ) in h 0 (λ ref ). This is theoretically correct if a correction of chromatic refraction is applied to the measurement. Otherwise the difference of the ray geometry is taken into account globally on the optical thickness expression as described in the following section. Time integration The objective of this second step is to express: t 1 2 e τ (t ) t dt t under the form: e τ 2 This is carried out by Taylor-expansion technique.the approximate optical thickness which takes into account the time integration together with the instrument function may be written in two forms: 38

39 τ = σ i (wl k, j )N i t 2 i 24 or τ = σ i (wl k, j ).N i h 2 24 i i σ i (wl k, j ) N i t i σ i (wl k, j ) N i h 2 2 ( ) where : N i is the averaged line densities of the species i, computed at the middle of each acquisition σ i (wl k, j ) = σ i ( wl k, j ) 1 B ii' ( wl k, j ).N i' ( ) 2 i' h is the variation of altitude between the beginning and the end of an acquisition. Chromatic refraction in the transmission model If not already extracted from the measured transmission, one must also take into account the chromatic refraction in the transmission model. Based on the same philosophy as before, the dependence of the tangent altitude vs the wavelength is now derived by a Taylor s expansion from the tangent height at the mid-point of the time interval and at the reference wavelength λ ref. We then consider the approximation : τ (λ,h 0 (wl k, j )) =τ (λ,h 0 (λ ref )) + dτ (wl k,j,h 0 (λ ref )) (h 0 (wl k, j ) h 0 (λ ref )) ( ) dh as according to earlier assumption we only have to deal with first derivative of N i relative to h. With the help of the earlier assumptions, the final transmission is then expressed as : τ (wl k, j,h 0 (λ)) = τ (wl k, j, h 0 (λ ref )) +(h 0 (wl k, j ) h 0 (λ ref )) σ (wl ) dn i i k, j dh ( ) i Cross-section database In the spectral inversion the cross-sections σ i (Τ,λ) form the inversion kernel of the problem. Some of the cross sections are temperature dependent and this is taken into account by the effective cross-section construction (see Sec ). The cross-sections are taken from the cross-section database where they are ordered through species, temperature, and wavelength. The cross.sections are obtained from laboratory measurements. The scattering cross-sections can also be approached analytically (for absorbing species the cross-sections are too complex to derive from first principles). The Rayleigh cross-section needed for scattering by air has the following form: σ R = 1.06 x π 3 where 3 a(λ) 2 λ 4 N 0 2 ( ) 39

40 λ is the wavelength in nm a(λ) is the Edlen s law (1966) defined in (3.2.1) N 0 is the air number density at sea level in cm -3 (from the level 2 processing configuration database) Aerosols cross-sections are approximated by Ångström s law σ a (λ) = σ 0 λ b ( ) with σ 0 = 3 x10 7 cm 2 and wavelength in nm. The default value for Ångström s coefficient b is b=1 (from the level 2 processing configuration database). A more realistic description of the aerosol extinction will include a more elaborate wavelength dependence. A simple polynomial expression will be used. For the sake of coherence it is useful to define an equivalent aerosol number density as n aero ref (z) and the aerosol extinction coefficient becomes: β(z,λ) = β a (z)β b (λ) ( ) β a (z) = σ ref n aero ref (z) ( ) with σ ref = σ 0 / λ ref ( ) and β b (λ) = c 0 + c 1 λ + ( ) where λ = λ 1020nm ( ) Equivalently we have β(z,λ) = d (z) + d 0 1 (z) λ + ( ) with d i (z) = σ ref c i n aero ref (z) ( ) For the horizontally integrated extinction profile, one gets: τ(λ) = β(z(s),λ)ds = c 0 + c 1 λ + ( ) LOS with 40

41 aero c i = σ ref c i N ref ( ) and aero N ref = n ref (z(s)) ds ( ) Column density retrieval The transmission model as it will be used in the column retrieval is T mod = e τ where τ(wl k, j ) = σ i ( wl k, j ).N i 1 i 2 i' i [ ] + h 0 (wl k, j ) h 0 (λ ref ) h2 24 i B ii' ( wl k, j )N i N i' ( ) dn i (h 0 (λ ref )) σ i wl k, j i σ i ( wl k, j ) dn (h (λ )) i 0 ref dh 2 dh ( ) The estimations of column densities is based on the minimisation of the so-called objective function (notice the matrix notation implied) S(N ) = 1 2 (T obs T mod (N ))T C 1 (T obs T mod (N )) ( ) where T obs is given by Eq. ( ) or by Eq. ( ). This is done by using the Levenberg- Marquardt algorithm (hereafter noted LMA). [N,C out ] = LMA(T obs,c tot,t Model ) ( ) LMA algorithm tries to find a maximum starting from a pre-set values. The algorithm also produces an estimate for the covariance matrix. The input covariance for the spectral inversion comes from two sources. The first, the most important one, is the data covariance. If there has been no data operations destroying stochastic independence of data, this covariance is diagonal i.e., it is the variance vector of the data. The second source is the modelling errors. If the data statistics and the model error are Gaussian we can simply add these two error sources C tot = C data + C mod ( ) The wavelength region and the number of constituents to be retrieved can be determined by the user. The approach where all constituents are retrieved simultaneously and the crosssections are taken by their absolute values (for differential cross-sections, see below) will be called as the simultaneous method in the references below. The approach where inversion is based on retrieving a single constituent in narrow spectral window is called the sequential 41

42 method in references below. An iterative loop in the sequential method is required to get a full consistency. The determination of the column density of some species in the sequential method could need some special treatment due to the fact that there is no clean spectral window. This can be resolved by using a differential method (so-called DOAS method). A differential transmission is calculated by dividing the measured transmission by a smooth one: T diff = T T smooth the same thing is done for the cross sections : σ = σ ~ σ ~ diff smooth ( ) ( ) Then either non-linear LMA-method or a linear fit without a constant part is performed on the inverse of the logarithm of the differential transmission: ln(t diff ) = N i. σ diff ( ) Results The performance of the spectral inversion depending on the star characteristics (magnitude, temperature), limb conditions (dark, bright) is illustrated below for the ozone column density. 42

43 Influence of Star Temperature On Ozone Column Density Retrieval The simulations include all instrument noise, dark charge and scintillation. The relative error is displayed (symbol curve) as well as the envelop of the error (1 sigma) derived from the standard deviation computed by the Level 2 processing (line + symbols). Fig. 14a. Curve Title S S Temperature Stars Magnitude 1 in Dark Limb Conditions The degradation of the performance on ozone column densities retrieval at high altitudes with cold stars (= red stars) because of lack of ultra violet radiation is clearly shown on this figure. 43

44 Influence of Star Magnitude On Ozone Column Density Retrieval The simulations include all instrument noise, dark charge and scintillation. The relative error is displayed (symbol curve) as well as the envelop of the error (1 sigma) derived from the standard deviation computed by the Level 2 processing (line + symbols). Fig. 14b. Curve Title S S S Magnitude Hot Stars in Dark Limb conditions. An accuracy better than 2% on ozone column densities retrieval is achieved with stars of magnitude up to 2 (in dark limb conditions with hot stars). 44

45 Influence of Limbs On Ozone Column Density Retrieval The simulations include all instrument noise, dark charge and scintillation. The relative error is displayed (symbol curve) as well as the envelop of the error (1 sigma) derived from the standard deviation computed by the Level 2 processing (line + symbols). Fig. 14c. Curve Title S113A1YN S71A1YN Limb Bright Dark Bright Stars, Temperature K The performance on ozone column densities retrieval is maintained with bright stars in bright conditions for altitudes down to 25 km. References GOMOS 2 Detailed Processing Model, Issue 4, rev. 0, PO-RS-ACR-GS Kyrölä, E., E. Sihvola, M. Tikka, Y. Kotivuori T. Tuomi, and H. Haario, Inverse Theory for Occultation Measurements 1. Spectral Inversion, J. Geophys. Res., 98, 7367,1993. Kyrölä, E. et al., Simultaneous inversion, PO-TN-FMI-GM-009. Kyrölä, E., P. Simon, Cross Section Data Bank Content and Structure, PO-TN-FMI-GM-010 Chassefière,E., P. Benet, Sequential inversion algorithm,, ref. PO-TN-SA-GS-004. Sihvola, E., Coupling of spectral and vertical inversion in the analysis of stellar occultation data, Licentiate of philosophy thesis, University of Helsinki, Geophysical publications, FMI, 38, Roozendael, M. Van, and L. Oikarinen, Investigation of DOAS retrieval features, GOMOS Mission planning and Algorithm Aspects-project, IASB and FMI,

46 4.1.5 Spectrometer B spectral inversion The spectrometer B will measure the densities of two different constituents O2 and H2O in two bands: nm and nm respectively. Due to the physics of the problem, the individual lines are very thin and saturated, the apparent cross sections change with the integrated densities, and can no longer be considered as only wavelength dependent. Therefore an other algorithm than the one for spectrometer A has been developed. The method uses reference transmission spectra which were calculated for different integrated densities. These reference transmissions Tref(Nref) are in fact reference cross sections σref(nref,λ) which also depend slightly on pressure. To take into account this dependency, calculations of transmission are performed using a direct model with a standard atmospheric profile. Several such calculations were done for different atmospheric models in order to have the response for different conditions such as tropical, mid-latitude and subarctic profiles. Due to its nature, this method has been called comparative method. The refractive correction has to be applied to the measured transmission before the comparison. Moreover, prior to comparison, the reference line densities have to be convoluted with the instrument function, that could be written, as in the previous chapter, under the form: Tref( λ ) = W ( λ λ). Tref( λ). dλ k λ k This reference is then adapted to the spectral sampling of the current acquisition. ( ) The column density profiles retrieved after the spectral inversion and their envelopes (standard deviation at 1σ) are given in the following figure for O 2 with two different stars : 46

47 Curve Title S S Temperature 0 1 Fig 15. O 2 column density References Benet, P., Retrieval of O2 & H2O integrated densities from spectro. B measurements, PO-TN-SA-GS-010, Sd A Line densities smoothing A line density produced by the spectral inversion is a statistical quantity. If the spectral inversion is attempted for a constituent which is unretrievable the result cannot be relied on. This happens, for example, at very high altitudes where densities decrease rapidly. It may also be the case when data is corrupted (with respect to retrieval). Due to the measurement technique and geometry the measured line densities are integrated along horizontal paths and averaged over 0.5 seconds. Therefore, they should originally be quite smooth. {N S,i, j,c S,i, j } = CUBGCV(N i, j,c SI,i, j ) ( ) The routine CUBGCV (ACM Transactions on Mathematical Software, Vol 12, NO.2, June 1986, Algorithm 642) uses cubic splines to smooth the line density data. There are two ways of implementing the routine for GOMOS data processing purposes. If assume that the variance of the line density data is known (parameter VAR = 1 in CUBGCV routine), we can use the absolute values of the line density error estimates as the weights : 47

48 w j = 1 ( ) C SI j The amount of smoothing is considered to be known. If we assume that we do not know exactly the variance of the line density data (parameter VAR = -1 in CUBGCV routine) we should use relative weights. Now the amount of smoothing is calculated by the routine itself with a so called generalised cross validation technique. In practice it has been found useful to exclude those measurements from the smoothing procedure which are very uncertain. Otherwise they affect the amount of smoothing too much. The vertical resolution of the retrieved profile is reduced after smoothing Vertical inversion The vertical inversion problem is to find the n(z) that verify : N j = n(s) ds j=1,n sp ( ) Γ λref, j where N j is the jth line density data, n(s) is the species density function (depending on the curvilinear abscissa s along the ray), and Γ λref, j identifies the j th ray path provided after the ray tracing computations under the form of a set of points distributed from GOMOS towards the star. The vertical inversion method used in the GOMOS data processing is based on the approximation where local densities are taken to vary in a linear way as a function of altitude between two successive GOMOS measurements i.e., for h0, j h h0, j 1 n(h) = a j + b j h ( ) where b j = 1 n j + 1 n j 1 δh j δh j a j = h 0,j 1 n j h 0,j n j 1 ( ) δh j δh j δh j = h 0,j 1 h 0,j As a result of the ray tracing computation (from Level 1b or 2) each ray Γ λref, j is described as a set of points M m (x m,j,y m,j,z m,j,h m,j ) with m=1,..,mpoints(j). If we want to derive the analytical expression of the contribution of each part of the ray (say between index m=m1 and m=m1+1=m2) on the final expression of the line density, we write : I m1 = 0 s m2 n(s)ds ( ) in which s is the current curvilinear abscissa along the ray while s m2 is the curvilinear abscissa of point M m2 starting from M m1. Expression of the line density is then obviously : 48

49 Mpo ints (j ) 1 N( j) = I m m =1 ( ) The law between the curvilinear abscissa s and the altitude is to be derived in order to build the relationship between local and line densities. We assume now that the portion of ray between points indexed m1 and m2(=m1+1) is lying between altitude h 0,j and h 0,j-1. and furthermore satisfy h 0,j h m1 h m2 h 0,j-1 ( h + R s ) 2 = s 2 + ( h m1 + R s ) 2 2s( h m1 + R s )cosα ( ) in which R s is the radius of the local ausculatory sphere (computed to be tangent at the nadir of point M m1 ) which, for a small element could be reasonably considered as a constant. By construction α is always greater than π/2 and then : s = ( h m1 + R s )cosα + ( h + R s ) 2 ( h m1 + R s ) 2 ( 1 cos 2 α) ( ) Expression (3) is then written under the form : I m1 = h m2 h m1 n(h) ( h + R s ) dh ( h + R s ) 2 ( h m1 + R s ) 2 (1 cos 2 α) ( ) After some algebra the contribution to the line density can be written as I m1 = L m1 a j + L m1 b j = h 0, j 1 L m1 L m1 n δh j + L m1 h 0,j L m1 ( ) n j δh j 1 j where L m1 represents the distance between node m1 and node m2 : L m1 = 2 h m 2 + R s hm1 + R s 2 (1 cos 2 α) + h + R cosα ( ) m1 s and L' m1 is given by L m1 = 1 2 h R m 2 L s m1 + h m1 + R ( cosα s ( h m1 h ) m h + R 2 2 m1 s (1 cos 2 α)log h m2 + R s + L m1 ( h m1 + R s )cosα h m1 + R s 1 cosα ( ) ( ) Expression of the final line density 49

50 Once the new arrangement of the ray description has been performed, it becomes straightforward to build up the expression of the line density of the j-th acquisition. Then if we summarise these contributions into a matrix form, the vertical inversion consists of to solve the following linear system : An = N ( ) where the matrix A is built as follows for i=1,...,j-1 A( j,i) = + h 0,i 1 L m L m + δh i ind( j,i +1,2) m=ind ( j,i,2)+1 ind ( j,i,1 ) 1 m=ind ( j,i+1,1) h 0,i 1 L m L m + δh i ind (j,i+2,2) m=ind (j,i+1,2)+1 ind (j,i+1,1) 1 m =ind( j,i +2,1) L m h 0,i+1 L m δh i+1 L m h 0,i+1 L m δh i+1 (GOMOS side) (star side) ( ) with the following exception for the computation of the diagonal element A(j,j) : A( j, j) = + ind( j,j,2)+1 m=ind ( j, j +1,2) ind ( j, j,1) 1 m=ind ( j, j+1,1) h 0, j 1 L m L m δh j h 0,j 1 L m L m δh j (GOMOS side) (star side) ( ) The geometrical factors for the GOMOS side are L m = L m = h m 1 + R 2 s hm + R s (1 cos 2 α) + h m + R s cosα ( ) ( ( ) h R m 1 L s m + h m + R cosα s h m h m h + R 2 2 m s (1 cos 2 α)log h m 1 + R s + L m ( h m + R s )cosα h m + R s 1 cosα ( ) while the contributions from the star side of the ray are given by : ( ) L m = L m = h m+1 + R s hm + R s 2 (1 cos 2 α) + h + R m s cosα ( ) ( ( ) h R m +1 L s m + h m + R cosα s h m h m h + R 2 2 m s (1 cos 2 α)log h m+1 + R s + L m ( h m + R s )cosα h m + R s 1 cosα ( ) ( ) 50

51 As already mentioned the line density we are using as inputs are the result of an integration in time (acquisition) and then should be adapted to the "instantaneous" expression that has been derived (i.e. the linear system is only valid for instantaneous line densities). To pass from integrated to instantaneous data we use an expression derived from Taylor expansion: N j N j + t N j t 2 ( ) which, for 1<j<N sp, is approximated by : N j N j + t 2 24 N j +1 2N j + N j 1 t 2 = 1 24 N j N j N j 1 ( ) Boundaries values are approximated by : N 1 N 1 + t2 24 N Nsp N Nsp + t2 24 2N 1 5N 2 + 4N 3 N 4 = 13 t 2 12 N N N N 4 2N Nsp 5N + 4N N Nsp 1 N sp 2 N sp 3 t 2 = N N sp 5 24 N N sp N N sp N N sp 3 ( ) The linear system is then transformed into : BAn = Kn = BN = N ( ) in which K is the so-called kernel matrix (non triangular) and B is B( j, j 1) = B( j, j +1) = 1 24 B( j, j) = for 1<j<N sp ( ) and : 51

52 B(1,1) = B(N sp,n sp ) = B(1,2) = B(N sp,n sp -1) = 5 24 B(1,3) = B(N sp,n sp -2)= 4 24 B(1,4)= B(N sp,n sp -3) = 1 24 ( ) The computation of the kernel matrix and its inversion are done during the initialisation step of the inversion, while the resolution of the system (then reduced to a multiplication matrixvector) is done during the vertical inversion step Results The performance of the vertical inversion depending on the star characteristics (magnitude, temperature), limb conditions (dark, bright) is illustrated below for the ozone local density profile. 52

53 Influence of Star Temperature On Ozone Local Density Retrieval The simulations include all instrument noise, dark charge and scintillation. The relative error is displayed (symbol curve) as well as the envelop of the error (1 sigma) derived from the standard deviation computed by the Level 2 processing (line + symbols). Curve title S S Temperature Stars Magnitude 1 in Dark Limb Conditions The degradation of the performance on ozone local densities retrieval at high altitudes with cold stars (= red stars) because of lack of ultra violet radiation is clearly shown on this figure. 53

54 Influence of Star Magnitude On Ozone Local Density Retrieval The simulations include all instrument noise, dark charge and scintillation. The relative error is displayed (symbol curve) as well as the envelop of the error (1 sigma) derived from the standard deviation computed by the Level 2 processing (line + symbols). Curve title S S S Magnitude Hot Stars in Dark Limb conditions. Even though the ozone local densities retrieval already provides satisfactory results achieved with stars of magnitude up to 2, there is maybe still room for improvement compared to the performance on the ozone column densities retrieval (accuracy better than 2%) 54