Annales Geophysicae (2004) 22: 3411–3420SRef-ID: 1432-0576/ag/2004-22-3411© European Geosciences Union 2004

AnnalesGeophysicae

Ozone profile smoothness as a priori information in the inversion oflimb measurements

V. F. Sofieva1, J. Tamminen1, H. Haario2, E. Kyr ola1, and M. Lehtinen3

1Finnish Meteorological Institute, Earth Observation, Helsinki, Finland2Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland3University of Oulu, Sodankyla, Finland

Received: 12 February 2004 – Revised: 28 June 2004 – Accepted: 2 July 2004 – Published: 3 November 2004

Abstract. In this work we discuss inclusion of a prioriinformation about the smoothness of atmospheric profilesin inversion algorithms. The smoothness requirement canbe formulated in the form of Tikhonov-type regularization,where the smoothness of atmospheric profiles is consideredas a constraint or in the form of Bayesian optimal estimation(maximum a posteriori method, MAP), where the smooth-ness of profiles can be included as a priori information.We develop further two recently proposed retrieval methods.One of them – Tikhonov-type regularization according tothe target resolution – develops the classical Tikhonovregularization. The second method – maximum a posteriorimethod with smoothness a priori – effectively combinesthe ideas of the classical MAP method and Tikhonov-typeregularization. We discuss a grid-independent formulationfor the proposed inversion methods, thus isolating the choiceof calculation grid from the question of how strong thesmoothing should be.

The discussed approaches are applied to the problem ofozone profile retrieval from stellar occultation measurementsby the GOMOS instrument on board the Envisat satellite.Realistic simulations for the typical measurement conditionswith smoothness a priori information created from 10-yearsanalysis of ozone sounding at Sodankyla and analysis of thetotal retrieval error illustrate the advantages of the proposedmethods.

The proposed methods are equally applicable to other pro-file retrieval problems from remote sensing measurements.

Key words. Atmospheric composition and structure (gen-eral or miscellaneous; middle atmosphere – composition andchemistry; troposphere-composition and chemistry)

1 Introduction

The problem of profile retrieval from remote sensingmeasurements is always under-determined: a continuous

Correspondence to:V. F. Sofieva(viktoria.sofieva@fmi.fi)

profile of an atmospheric constituent is reconstructed froma finite number of measurements. The problems of this kindbelong to the class of ill-posed problems. Measurement datathemselves do not uniquely determine the solution; therefore,some kind of prior constraint must be used to make theproblem well-posed. FollowingRodgers(2000), it can bedone via:

(a) A discrete representation. Dividing the atmosphere intolayers according to a measurement structure and makingcertain assumptions about the profile within the lay-ers (constant, linear, polynomial altitude dependence)transform the problem to a well-determined one.

(b) An ad hoc constraint such a smoothness of atmosphericprofiles. It can be applied in the form of the Tikhonov-type regularization.

(c) Prior information about atmosphere, which includes amean state and its covariance, or more generally, a priorprobability density function of the state.

It is also possible to use a representation via orthogonalpolynomials or wavelets, but these approaches are notdiscussed here. A discrete representation appears inalmost all practical solutions of inverse problems, andoften the case (a) is referred (also in this paper) to ashaving no a priori. Although the discretization makesthe problem well-determined, it can remain ill-posed ornoise-amplifying. It means that noisy data lead to instabilityof inversion: the reconstructed profile has non-physicaloscillations. Application of Tikhonov-type regularization(also referred to as the Twomey method) (Tikhonov andArsenin, 1977; Twomey, 1977; Rodgers, 2000) helps torecover the stability of inversion. This method assumesa smoothness of an atmospheric constituent profile andtherefore we can consider regularization as a kind of priorinformation.

If a priori information about the state is available (case c),the maximum a posteriori method (MAP) usually allows asignificant improvement of retrievals. In the case of Gaussiannoise and a priori, it gives the minimum variance solution.

3412 V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval

The Gaussian distribution is often assumed in practice, thusintroducing elements of “ad hoc guess” into the case (c).

In the current paper, we develop further two recentlyproposed methods which can be applied if the informationneeded for the MAP method is unavailable. The first methodconsiders the smoothness of atmospheric profiles not as an adhoc constraint, as it is done in Tikhonov regularization, butas a priori information. This method (Haario et al., 2004)“combines” cases (b) and (c) and can therefore be calledthe MAP method with smoothness a priori. The secondmethod (Tamminen et al., 2004) chooses the regularizationparameter in a Tikhonov-type scheme according to a targetresolution, which is determined from requirements to resolvefine structures of the profiles.

For both methods we propose a grid-independent formu-lation, which is very important for measurements unevenlydistributed in altitude. For example, the sampling verticalresolution of stellar occultation measurements depends onaltitude and on the obliqueness of occultations (i.e. anglebetween the orbital plane and the direction to the star).

In Sect. 2 we describe the stellar occultation measurementsby the GOMOS instrument and discuss the need for advanceddata analysis. The advanced inverse methods that use a prioriinformation of a different kind, as well as the availabilityof this information are discussed in Sect. 3. Comparison ofthe proposed inversion schemes based on realistic simulationis presented in Sect. 4. The a priori information used insimulation is created from analysis of 10 years of ozonesonde measurements at Sodankyla. The analysis of thetotal retrieval error, which includes components due tomeasurement noise and due to smoothing properties ofinversion, illustrates advantages of the proposed methods.

2 Stellar occultation measurements by the GOMOSinstrument

GOMOS (Global Ozone Monitoring by Occultation of Stars)is a stellar occultation instrument operating on board theEnvisat satellite (http://envisat.esa.int/instruments/gomos)launched 1 March 2002. The GOMOS spectrometers mea-sure the stellar spectrum from∼140 km down to∼15 km.The products retrieved from the UV-Visible spectrometermeasurements are ozone, NO2, NO3, aerosol and air densityvertical profiles. The basis for the geophysical data retrievalfrom GOMOS measurements is the transmission function(Kyrola et al., 1993). In the GOMOS data processing,the inversion is split into two parts: the spectral inversionpart and the vertical inversion part (Kyrola et al., 1993).In the spectral inversion, horizontal column densities areretrieved from the atmospheric transmission data from whichrefractive effects and scintillation modifications have beenremoved. In the vertical inversion, vertical profiles arereconstructed from the horizontal column densities. In thispaper, we will deal only with the vertical inversion.

The vertical inversion of GOMOS is linear. By dis-cretization of the atmosphere into layers according to the

measurement structure (the number of unknowns is equal tothe number of measurements), the forward model connectingthe measurements (horizontal column densitiesN ) andvector of unknowns (profileρ), can be written as

N = Kρ + ε , (1)

whereε presents a vector of measurement noise.The discretized GOMOS vertical inversion problem (1)

is well-conditioned (the condition number of the forwardmodel matrixK for a typical occultation is∼25), so it canbe solved with the usual matrix inversion. The inversionis slightly noise-amplifying by a factor of∼2 (Sofieva andKyrola, 2004). However, in the case of dim stars, thereconstructed profiles are significantly contaminated withnoise.

2.1 Characterization of retrieved profiles. Vertical resolu-tion

The error estimate and the vertical resolution are twoquantities which completely characterize retrieved profiles.The resolution of the retrieved profiles depends, in additionto the sampling resolution, on smoothing properties of aretrieval algorithm.

In profile reconstruction, the resolution can be studied bycomputing the averaging kernel (Backus and Gilbert, 1970;Rodgers, 1990):

A =∂ρ

∂ρ, (2)

whereρ is the retrieved profile andρ is the true profile. Inlinear problems the averaging kernel matrix can be computedas

A = GK∞ , (3)

whereG is the inversion matrix andK∞ is the forward modelmatrix in (infinitely) dense grid.

One commonly used measure of the resolution of theretrieved profile (or a measure of the width of the averagingkernels) is the Backus-Gilbert spreads(z) (e.g. Rodgers,2000):

s(z) = 12∫

(z − z′)2A2(z, z′)dz′/(

∫A(z, z′)dz′)2 . (4)

In the following, we use the Backus-Gilbert spread (4) as ameasure of resolution.

The GOMOS instrument is capable of retrieving the atmo-spheric profiles with a very good resolution. The samplingresolution for different line-of-sight azimuth angles is shownin Fig. 1. The measurement grid becomes denser in thelower part of the atmosphere due to refraction. The samplingresolution depends on obliqueness (an angle between orbitalplane and direction to the star) of the occultation. Forvery oblique occultations it can be nearly twice better thanfor vertical ones. The averaging kernels of the pre-launchGOMOS inversion are sharply peaked (Fig.2). For a typicaloccultation, the resolution (Backus-Gilbert spread) is∼2 kmin the mesosphere and upper stratosphere, and is less than1 km in the lower stratosphere and troposphere.

V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval 3413

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Fig. 1. Sampling resolution in vertical (R03059/S018,azimuth=0◦), typical (R03059/S052, azimuth=30◦) and oblique(R03059/S001, azimuth=54◦) occultations.

2.2 Need for advanced data analysis

The signal-to-noise ratio in stellar occultation measurementsstrongly depends on stellar parameters (visual magnitude andeffective temperature), so does the error of the vertical profilereconstruction.

In pre-launch simulations we observed significant non-physical oscillations of reconstructed ozone profiles for dimstars (visual magnitude>2.5) but only for altitudes above80 km and below 18 km (Fig.3). Therefore, no a prioriinformation or regularization were explicitly used in theGOMOS baseline inversion, as the lowermost altitude of theGOMOS measurements was expected to be∼18 km.

The first year of GOMOS validation has shown that thenoise level is higher than expected. Additional errors comefrom incomplete scintillation correction.

Analysing one reference data set, which includes morethan 600 occultations in dark limb, we have found that asignificant share of occultations (∼10%) is terminated ataltitudes below 10 km. This enables GOMOS to probe thetroposphere, but advanced inversion methods are requiredbecause of a low signal-to-noise ratio.

The vertical resolution achieved in the GOMOS measure-ments is better than the characteristic vertical scale of theozone fine structures (1.0–1.4 km in the troposphere andin the lower stratosphere, according to the recent study ofSofieva et al., 2004).

These features prompt us to apply advanced methods inthe GOMOS inversion. Unfortunately, application of theMAP method or Tikhonov-type regularization degrades thevertical resolution. The question therefore is: how to achievethe stability of the inversion without critical degradation ofresolution?

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Fig. 2. Averaging kernels of the onion peeling method, at severaltangent altitudes.

3 Inversion methods

Let us consider the linear forward model (1). All the methodsconsidered below can be applied to any linearized inverseproblem. They are also valid for general nonlinear problems,but nonlinear techniques like iterative minimization or theMCMC (Markov chain Monte Carlo ) method (Tamminenand Kyrola, 2001), are required instead of matrix multiplica-tion.

The essence of inclusion of a priori information isexpressed by Bayes’ formula describing the posteriorprobability density function (pdf)P(ρ|N) via a likelihoodfunctionP(N |ρ) and a prior pdfPprior(ρ):

P(ρ|N) ∝ P(N |ρ) × Pprior(ρ) . (5)

Provided that the prior pdf is known and assum-ing it is Gaussian, with a meanρa and a covarianceCa :ρa∼N (ρ0, Ca), the maximum a posteriori (MAP) esti-mation of the retrieved profileρMAP can be presented in thefollowing form (e.g.Rodgers, 2000):

ρMAP = (KT C−1ε K + C−1

a )−1KT C−1ε (N − Kρ0) + ρ0 . (6)

Here a measurement noise is also assumed to be Gaussian:ε∼N (0, Cε). The validity of this assumption is verified inTamminen(2004) with the MCMC method: the intermediateproducts of the GOMOS inversion – horizontal columndensities – have a distribution close to Gaussian, providedthat the instrumental noise is Gaussian. The matrixCε isdiagonal due to the independence of instrumental noise ateach altitude.

A priori information on the profile smoothness can beincluded in the MAP method via correlation of prioruncertainties. Then the prior covarianceCa can be presented,for example, in the form (Rodgers, 2000)

Ca(i, j) = σiσj exp

(−|ri − rj |

L

), (7)

3414 V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval

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Fig. 3. Pre-launch simulation of the ozone profile retrievals for stars of magnitudem=2 andm=3. Right panel is zoom.

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Fig. 4. Mean (dashed bold line) and standard deviation (shadedarea) of ozone profile in November from analysis of ozone sondedata at Sodankyla and monthly mean (bold solid line) and itsstandard deviation (error bars) from the Fortuin-Kelder climatology.

whereri denotes an altitude grid point,σi is the standarddeviation for a priori profile at the pointri and L is thecharacteristic scale. Alternatively, a Gaussian or a triangularshape of the correlation function can be used. This methodincludes the prior information in the most complete form.It is also grid-independent, i.e. the profile smoothness isdefined independently of the discretization grid. The actualresolution of retrieved profiles depends only on the noiselevel: it is worse for larger measurement noise.

Unfortunately, at the moment this a priori informationis often not available, even for ozone. The climatologicaldata (Fortuin and Kelder, 1998) can, in principle, be usedas an a priori estimate of the mean of ozone profile, but theinter-annual variability of the climatology does not reflectthat of the individual profiles. It is illustrated in Fig.4, wherethe mean and the standard deviation of ozone profile obtainedfrom analysis of 10 years (1990–1999) of ozone sonde dataat Sodankyla are compared with those of the monthly meanozone profile from the Fortuin-Kelder climatology. Thevariability of individual ozone profiles is 1.5–2 times largerthan the variability of monthly mean profiles.

Nevertheless, in a few cases when an occultation is locatednear an ozone observation station, useful a priori informationcan be obtained. Recently,Sofieva et al.(2004) analysed11-year ozone sonde data at Sodankyla and showed that thecharacteristic scale of the ozone fine structure is∼1.0 km inthe troposphere and∼1.4 km in the lower stratosphere (upto 25 km). It was also found that the characteristic scaleof ozone fine structure is a relatively stable atmosphericcharacteristic without significant seasonal or interannualvariations at this location.

3.1 MAP method with smoothness a priori

In most cases a priori information includes only a generalmeasure of smoothness of atmospheric profiles. For suchsituations, we develop a method recently proposed byHaarioet al. (2004). It is based on the assumption that theneighboring discretized values of a retrieved profile cannotbe too different. This a priori can be presented as

2ρi − ρi−1 − ρi+1 = 0 ± h2i ε

regi , (8)

V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval 3415

where εregi are mutually independent Gaussian random

variables with the zero mean and the diagonal covariancematrix Creg andhi is the discretization grid. Alternatively,first order or higher order differences can be used in theleft-hand side of Eq. (8). The regularization Equation (8)can be expressed in the matrix form as

Hρ = 0 ± εreg , (9)

where tri-diagonal matrixH approximates second deriva-tives:

H = diag

[1

h2i

] 0 0 0 ... 01 −2 1 ... 0... .. ... ... ...

0 ... 1 −2 10 0 ... 0 0

. (10)

It corresponds to the prior distribution (smoothnessa priori) (Pprior∝ exp(−1

2ρT HT C−1regHρ). If the matrix

HT C−1regH is invertible, the prior distribution is Gaussian:

ρa∼N (0, (HT C−1regH)−1). The MAP estimationρsm can be

written as

ρsm = (KT C−1ε K + HT C−1

regH)−1KT C−1ε N . (11)

The only information needed for application of this methodare the uncertainties of the second differencesCreg. Itcan be also created from analysis of high-resolution profilemeasurements, such as ozone sonde data. Analogously tothe classical MAP estimation, this method efficiently com-bines the measurements and a priori information, applyingadditional smoothing only when it is required by a lowsignal-to-noise ratio. It is also grid-independent, i.e. thechosen amount of smoothing is defined by the ratio of ameasurement noise and a priori uncertainty, but not by thediscretization grid.

3.2 Tikhonov-type regularization. Choosing regularizationparameter according to the target resolution

The classical Tikhonov regularized solution of the problem(1) was originally derived as a minimizer of the functional

F(ρ) = ‖Kρ − N‖2+ λ‖Hρ‖

2 . (12)

Hereλ is the regularization parameter andH is the matrixrepresenting first, second (10) or higher order differences(which are assumed to be bounded, thus characterizing thesmoothness of the solution), and‖·‖ is `2-norm.

The Tikhonov regularized solution of (12) exists, and it isunique. It is given by the formula

ρ = (KT K + λHT H)−1KT N . (13)

It is equivalent to the MAP solution (6), provided thatthe prior distribution isPprior∝ exp(−1

2ρT HT Hρ) and themeasurement errorε is Gaussianε∼N (0, σ 2I), whereI isthe unit matrix.

The optimal choice of the regularization parameterλ isa central issue in the literature discussing the Tikhonov

Table 1. Target resolution for ozone used in the simulation.

altitude range 0–10 km 10–30 km 30–40 km> 40 km

resolution 1 km 1.4 km 1.4–3 km 3 km

regularization. It can be chosen, for example, accordingto the Morozov’s discrepancy principle (e.g.Morozov,1993; Hansen et al., 2000), which states thatλ shouldbe selected so that the residual (difference between modeland measurement data) should be of the same value as themeasurement noise:

‖N − K ρ(λ)‖ = ‖ε‖ . (14)

Application of the Tikhonov regularization, as well asthe MAP methods discussed above, leads to a certaindegradation of resolution. The regularization parametercan also be chosen according to some target resolution, ifthe optimal value of the regularization parameter does notmeet the resolution requirements. Then the actual verticalresolution does not depend on the instrumental noise and thediscretization grid. This simplicity of profile characterizationmakes this method attractive, despite some disadvantagesdiscussed further in Sect. 4.

4 GOMOS measurements: comparison of regulariza-tion schemes

In this section we apply methods discussed above tothe problem of reconstruction of ozone profiles from theGOMOS measurements. The following methods are used:

1. Onion peeling: this is the standard GOMOS inversionwithout any a priori. In discretization, a constant densityinside each layer is assumed. Onion peeling can beformulated as a matrix inversion or, equivalently, theproblem can be solved sequentially, starting from theuppermost layer. We used the matrix inversion in oursimulation.

2. Tikhonov regularization in grid-dependent formulationwith regularization parameterλ=1015, which is chosenaccording to the Morozov’s discrepancy principle forthe typical GOMOS star of visual magnitudem=2 (Thehorizontal column densities are assumed to be in 1/cm2

and distances needed for computingK and H are incm.).

3. Tikhonov-type regularization in grid-independent for-mulation and altitude-dependent regularization parame-terλ. The regularization parameter is chosen accordingto the target resolution (Table1), that corresponds to thecharacteristic scale of ozone fine structure determinedfrom Sodankyla ozone sonde data below 30 km. The

3416 V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval

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Fig. 5. Reconstruction of ozone profile with different methods for the lower atmosphere. Left plot: onion peeling, Tikhonov regularizationwith regularization parameterλ=1015 chosen according to the Morozov’s discrepancy principle and with the regularization parameter chosenaccording to the target resolution; Right plot: MAP with smoothness a priori (MAP smooth) and classical MAP methods.

values of the target resolution above 30 km are takenfrom climatology.

4. MAP method with the smoothness a priori describedin Sect. 3.1. The covariance of second order finitedifferences was created from the Sodankyla ozonesonde data.

5. MAP method (classical). The Fortuin-Kelder climatol-ogy (Fortuin and Kelder, 1998) was taken as a priorimean ozone profile below 53 km. The upper partof the prior mean ozone profile was simulated withthe LIMBO simulator (Kyrola et al., 1999). A prioristandard deviation was calculated from 10 years ofozone sonde data (1990–1999) at Sodankyla. Thecorrelation scale is taken in accordance withSofievaet al.(2004) (Table1). In the upper atmosphere the priorstandard deviation is assumed to be a linear growingfunction from 20% to 200% for altitudes 30–100 kmwith the correlation scale 3 km.

4.1 Simulation setup

The realistic simulation was performed for the comparison ofthe methods. The measurements grid was taken coincidentwith a real GOMOS measurement grid, with a typical anglebetween the orbital plane and the star direction (∼18◦). Thesimulations were performed for stars of different magnitudesand of the typical effective temperature 10 000 K. Bothinstrumental noise and modelling errors (Kyrola et al.,1993; Sofieva and Kyrola, 2003) were taken into account

in computing the measurement error (i.e. error of horizontalcolumn densities).

The “true” profiles used in the simulation coincide inthe lower atmosphere (below 30 km) with ozone profilesmeasured at Sodankyla (http://fmiarc.fmi.fi). The upper partof ozone profile was simulated with the LIMBO simulatorand perturbed in accordance with the “error pattern method”(Rodgers, 2000).

4.2 An exemplary ozone profile retrieval

Figure 5 shows an example of reconstructed ozone profilein the middle stratosphere and troposphere for the star ofmagnitudem=2 and effective temperatureT =10 000 K. Wewill not show, discuss or compare the inversion at highaltitudes, because no a priori information is available there.All the advanced methods give almost the same results inthe mesosphere, as they use the same ad hoc assumptions onprofile smoothness. The lower part of the “true” profile usedin the simulation coincides with the ozone profile measuredat Sodankyla on 23 November 2002.

The ozone profile reconstructed with the onion peelingmethod, which gives the results almost identical to thestandard GOMOS vertical inversion, looks very noisy below15 km. The Tikhonov-type regularization, with a parameterchosen according to the discrepancy principle, smoothesout fine structures of ozone profile. If the regularizationparameter is chosen according to the target resolution, thefine structure of the ozone profile is well detected in thealtitude range 15–30 km. For altitudes below 10 km, bothTikhonov-type regularizations give the same results and

V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval 3417

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Fig. 6. Error estimation (left) and resolution (right) achieved with different inversion methods. Simulations are carried out for star ofmagnitudem=2 and effective temperatureT =10 000 K.

they somewhat improve the reconstruction, but the retrievedprofiles are still far away from the real profile. The ozoneprofile is reconstructed most accurately with maximum aposteriori methods (classical and with smoothness a priori).However, the fine structure of the ozone profile may stayunresolved with these methods. In the considered example,the lamina around 13 km is not detected with the MAPmethods either.

4.3 Error analysis

The efficiencies of the discussed methods characterized byestimated error of retrieval and resolution are compared inFig. 6. The error estimates in Fig.6 are square roots ofdiagonal elements of the posterior covariance matrix (Giventhe retrieved profileρ in the form of ρ=GN , where Gis the inversion matrix, the posterior covariance matrix isCpost=GCεGT .). The prior uncertainty of the ozone profile isalso shown in this figure. The original vertical resolution ofthe GOMOS measurements (it corresponds to onion peelingmethod) is very good:∼2 km for altitudes above 40 kmand better than 1 km below 20 km. The accuracy of thereconstruction can be significantly improved by advancedmethods at a price of degraded vertical resolution.

However, the error estimate, as presented in Fig.6, is notthe total error of ozone profile reconstruction. The total error(i.e. a deviation of a retrieved profileρ from the true oneρ)is

ρ − ρ = (A − I)ρ + Gε . (15)

HereA is the averaging kernel matrix,I is the unit matrixand G is the inversion matrix. The latter term in Eq. (15)

is the error due to noise in measurements, while the formerone describes the smoothing error caused by deviation ofaveraging kernels from delta-functions.

The covariance of the total error Eq. (15) is (Rodgers,2000)

Ctot = (A − I)Ce(A − I)T+ GCεGT , (16)

whereCe is the covariance of an ensemble of real profilesabout the mean profile. For the occultation located nearSodankyla Ce can be obtained from the ozone sondemeasurements, thus enabling one to estimate the totalerror of the profile reconstruction and compare the realefficiency of different inversion algorithms. In our work,we used 10 years of sonde data (1990–1999), in order toestimate the ozone variability (i.e. to obtain the ensemblecovarianceCe). Figure7 shows the total rms error ofreconstruction by different inversion algorithms for stars ofvarious magnitudes.

The following conclusions can be drawn from thisanalysis:

– The standard Tikhonov regularization with one constantsmoothing parameter chosen according to the discrep-ancy principle has the largest error for altitudes above15 km. This method seems not to be applicable to ozoneprofile reconstruction, because the chosen smoothingviolates the resolution requirements, thus leading to thelargest total error.

– For the other four methods, the error of reconstructionis almost the same for occultations of bright stars ataltitudes>15 km. This confirms that it is not necessaryto apply advanced methods in these conditions.

3418 V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval

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Fig. 7. Total error of ozone profile retrieval in the lower atmosphere, for stars of various magnitudes and of the effective temperatureT =10 000 K.

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Fig. 8. Resolution obtained with MAP method with smoothness apriori for various magnitudes of star. The effective temperature istakenT =10 000 K.

– For altitudes below 15 km the accuracy can be signif-icantly improved by advanced inverse methods. Theclassical MAP method has the smallest total error, astheoretically predicted. The MAP with smoothnessa priori is almost as good as the classical MAPmethod. The target resolution method also improves theinversion at the lower atmosphere, especially for dimstars.

In the Tikhonov regularization methods, the verticalresolution is predetermined, while in the MAP methods, itdepends on the noise level (Fig.8). The application of theMAP methods regularizes the inversion only when data areseriously contaminated with the noise, while using the targetresolution method, some smoothing is always applied, evenif it is not necessary. We have to recall that the classicalMAP method cannot be applied at the moment, because thevariability of ozone profile is not known, except for locationscoincident with ozone sounding stations. This method wasconsidered in order to illustrate its potential applicabilityin the future. The features of the inversion methods, theiradvantages and drawbacks are collected in Table2.

V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval 3419

Table 2. Summary of inversion methods.

method type of a priori advantages drawbacks

Onion peeling no a priori good vertical resolution noisy inversion results for dim starsat low altitudes

Tikhonov regularization smoothness simplicity - The resolution is not sufficient forwith smoothing parameter constraint detecting fine ozone structures.chosen according to - “Optimal”λ depends on stellarthe discrepancy principle spectrum and obliqueness of occultations.

Tikhonov regularization - smoothness Resolution does not depend - Solution is not optimal in statisticalwith smoothing parameter constraint on measurement grid or sense.chosen according to - characteristic stellar spectrum. - A priori is not available globally.the target resolution scale

MAP method with uncertainties of Amount of smoothing - Actual resolution depends on stellarsmoothness a priori second differences is chosen in the statistically spectrum.

optimal way. - A priori is not available globally.

MAP method (classical) - mean profile - The inversion uses a - Actual resolution depends on- standard deviation priori information most stellar spectrum.- characteristic completely. - A priori is not available globally.scale - Solution is statistically

optimal.

5 Summary and conclusion

In this paper we have discussed how the prior informationabout smoothness of the atmospheric profiles can be used inthe profile retrievals. It can be applied either in the form ofthe smoothness constraint (Tikhonov-type regularization) orin the form of a priori information (maximum a posteriorimethods). The methods can be used when the classicalBayesian MAP approach is not applicable due to insufficientinformation about prior distribution.

The grid-independent formulations were proposed for allthe considered methods, thus providing results independentof a discretization grid.

In the GOMOS measurements, the low signal-to-noiseratio for dim stars prompted the adoption of the Bayesianapproach, which allows one to efficiently combine measureddata with a priori information about the smoothness of theatmospheric profiles.

These methods were applied to reconstruction of theozone profile from GOMOS measurements. The realisticsimulations for typical measurement conditions and analysisof the total error of reconstruction revealed advantages anddrawbacks of each proposed procedure.

The following main conclusions can be drawn:

1. The standard Tikhonov regularization with smoothingparameter chosen according to discrepancy principle isnot recommented for the ozone profile retrieval fromthe GOMOS measurements, because the “optimal”smoothing violates the resolution requirements: almost

all fine structures of ozone profiles are smoothed by thismethod.

2. The regularization with the choice of the smoothingparameter according to the resolution requirements isthe most attractive method because of the predeterminedvertical resolution, independence from the measure-ment grid and from stellar properties. It is a kind of“minimal guaranteed strategy”, but it is not optimal inthe statistical sense.

3. MAP with a smoothness a priori allows one to includethe information about smoothness of the atmosphericprofile in the form of Bayesian optimal estimator.It efficiently combines the measurement data andinformation on smoothness of ozone profiles. It is agood alternative to the classical MAP method, when themean and/or standard deviation of retrieved quantitiesare not known.

4. The successful application of the classical MAP methodwas demonstrated. The a priori information about themean and the standard deviation of the ozone profilewas created from the Sodankyla ozone sonde data.This method includes the a priori information in themost complete form. Probably, this method will beused widely in the future algorithms of ozone profilereconstruction, but now it can be applied only in rarecases.

In summary, the proposed inversion methods, whichuse a priori information on smoothness of atmospheric

3420 V. F. Sofieva et al.: Smoothness as a priori information in ozone profile retrieval

profiles, improve significantly the quality of retrievals. Thesemethods can be used in reconstruction of atmosphericprofiles not only from stellar occultation but also from otherremote sensing measurements.

Acknowledgements.The authors thank the GOMOS CAL/VALteam for GOMOS data and the Arctic Research Center team ofFinnish Meteorological Institute for ozone sounding data. Theauthors also thank the two anonymous reviewers for their valuablecomments.

Topical Editor U.-P. Hoppe thanks two referees for their help inevaluating this paper.

References

Backus, G. and Gilbert, F.: Uniqueness in the inversion ofinaccurate gross earth data, Philos. Trans. Roy. Soc. London, SerA, 123–192, 1970.

Fortuin, J. P. F. and Kelder, H.: An ozone climatology based onozonesonde and satellite measurements, J. Geophys. Res., 103,31 709–31 734, 1998.

Haario, H., Laine, M., Lehtinen, M., Saksman, E., and Tamminen,J.: MCMC methods for high dimensional inversion in remotesensing, J. R. Statist. Soc., 66, 591–607, 2004.

Hansen, P., Jacobsen, B. H., and Mosegaard, K.: Methods andApplications of Inversion, Lecture Notes in Earth Science,vol. 92, Springer, Berlin, 2000.

Kyrola, E., Sihvola, E., Kotivuori, Y., Tikka, M., Tuomi, T., andHaario, H.: Inverse Theory for Occultation Measurements, 1,Spectral Inversion, J. Geophys. Res., 98, 7367–7381, 1993.

Kyrola, E., Tamminen, J., Oikarinen, L., Sihvola, E., Verronen,P., and Leppelmeier, G. W.: LIMBO – Limb and occultationmeasurement simulator, in: ESAMS99, European Symposiumon Atmospheric Measurements from Space, vol. WPP-161, ESA,Noordwijk, 487–493,1999.

Morozov, V. A.: Regularization methods for ill-posed problems,CRC Press, Boca Raton, 1993.

Rodgers, C.: Characterization and Error Analysis of ProfilesRetrieved From Remote Sounding Measurements, J. Geophys.Res., 95, 5587–5595, 1990.

Rodgers, C. D.: Inverse Methods for Atmospheres: Theory andPractice, World Scientific, Singapore, 2000.

Sofieva, V. F. and Kyrola, E.: Information approach to opti-mal selection of spectral channels, J. Geophys. Res., 108,doi:10.1029/2002JD002980, 2003.

Sofieva, V. F. and Kyrola, E.: Abel integral inversion in occultationmeasurements, in: Occultations for Probing Atmosphere andClimate – Science from the OPAC-1 Workshop, edited by:Kirchengast, G., Foelsche, U., and Steiner, A., Springer Verlag,77–86, 2004.

Sofieva, V. F., Kyro, E., and Kyrola, E.: Smoothness of ozoneprofiles: analysis of 10-years ozone sonde measurements atSodankyla, Ann. Geophys., 22, 8, 2715–2722, 2004.

Tamminen, J.: Validation of nonlinear inverse algorithms withMarkov chain Monte Carlo method, J. Geophys. Res., 109,D19303, doi:10.1029/2004JD004927, 2004.

Tamminen, J. and Kyrola, E.: Bayesian solution for nonlinear andnon-Gaussian inverse problems by Markov chain Monte Carlomethod, J. Geophys. Res., 106, 14 377–14 390, 2001.

Tamminen, J., Kyrola, E., and Sofieva, V.: Does a priori informa-tion improve occultation measurements?, in: Occultations forProbing Atmosphere and Climate – Science from the OPAC-1Workshop, edited by: Kirchengast, G., Foelsche, U., and Steiner,A., Springer Verlag, 87–98, 2004.

Tikhonov, A. and Arsenin, V.: Solutions of ill-posed problems,Wiley, New York, 1977.

Twomey, S.: Introduction to the Mathematics of Inversion inRemote Sensing and Indirect Measurements, Elsevier Sci., NewYork, 1977.