Approach to the design and dataanalysis of a limb-scanning experiment

Massimo Carlotti and Bruno Carli

The design of spectroscopic measurements of the atmosphere with the limb-scanning technique for the

retrieval of constituent altitude profiles requires choosing instrumental, observational, and retrievalparameters. An approach to this problem is discussed, and the mathematical tools that make it possibleto study the trade-off between the two conflicting requirements of optimum vertical resolution and smallerror of the profile are derived. As a first illustrative application, implementation of the mathematicaltools in the design of measurements to be carried out from a satellite platform is shown; a set ofparameters that provide a satisfactory compromise between the vertical resolution and the uncertaintiesof the retrieved profile has been identified. The mathematical model discussed can simulate the results

obtained with retrieval techniques that are based on the global inversion of the kernel that relates theobservations and the unknown profile. As a second application, a comparison between the a prioriestimate of the uncertainties provided by the mathematical tools and the results of the global analysis ofdata collected with a balloon-borne experiment is shown.

Key words: Atmospheric spectroscopy, remote sensing.

1. Introduction

The atmospheric spectra measured from high alti-tudes in the limb direction at different zenith angles(limb-scanning spectra) provide information on theatmospheric composition as a function of altitude.The design of these measurements requires that achoice be made of the instrumental parameters andobservation geometries. The most important instru-mental parameters are the spectral resolution andthe instantaneous field of view, as well as the signal-to-noise ratio. The observation geometries are de-fined by the flight altitude and the zenith anglesrelative to the line of sight of the spectrometer (theflight altitude is usually imposed by constraints thatare due to the hardship of flying a balloon-borne orsatellite-borne platform; therefore only the choice ofthe zenith angles can be made once the flight altitudeis given). These choices should be made so as topermit the retrieval of altitude distribution profiles

M. Carlotti is with the Dipartimento di Chimica Fisica edInorganica dell' UniversitA di Bologna, Viale Risorgimento 4,

40136 Bologna, Italy. B. Carli is with the Istituto di Ricerca sulleOnde Eletromagnetiche del Consiglio Nazionale delle Ricerche, ViaPanciatichi 64, 50127 Firenze, Italy.

Received 18 December 1992, revised manuscript received 20

September 1993.0003-6935/94/153237-13$06.00/0.c 1994 Optical Society of America.

that are determined at best, that is, that have the bestpossible vertical resolution and the least possibleuncertainty of the retrieved mixing-ratio values.

In this study an analytical procedure is presentedthat, when applied to a limb-scanning sequence of theplanned observations, allows one to estimate therandom error that will be associated with each valueof the retrieved profile when a given vertical resolu-tion is adopted or, conversely, allows one to estimatethe vertical resolution that will be achieved when therandom errors must be contained within a givenlimit. We show below how the two requirements ofvertical resolution and uncertainty conflict and whatcriteria can be adopted for the choice of both theobservation parameters at the time of designing theexperiment and the retrievable vertical resolution atthe time of data analysis.

The model of the inversion problem, which isdiscussed in Section 2, allows one to derive themathematical tools that are necessary to estimate themeasurement errors and the vertical resolution.The effects of the instrument's finite spectral resolu-tion and field of view are taken into account, so thatdifferent values can be tested for these two observa-tion parameters if they are to be optimized within thedesign of the experiment. In this model all theobservations of a limb-scanning sequence are consid-ered simultaneously, and we show how this approachprovides a comprehensive estimate of the quality of

20 May 1994 / Vol. 33, No. 15 / APPLIED OPTICS 3237

the results obtained with analysis methods such asthe global fit.'

As an example of their use, the application of themathematical tools to the design of the satellitemeasurements of the hydroxyl radical (OH) proposedfor the Earth Observing System is discussed inSection 3. Within the presentation of this examplethe behavior and the use of the functions derived inSection 2 are highlighted. In Section 4 an applica-tion to the analysis of data collected by a balloon-borne experiment is shown. Finally, in Section 5 afew computational details are discussed as a guidelineaimed at improving the efficiency of the calculations.

2. Theory and Mathematical Tools

In a limb-scanning experiment, the signal S thatreaches the spectrometer is a function S(v, h, 0, qz) ofthe frequency v; the observation geometry (defined bythe flight altitude h and the zenith angle 0); and thealtitude profile q, which is the distribution, as afunction of altitude z, of the atmospheric constituentthat is spectroscopically active at the frequency v.The problem of deriving the distribution q, from theobserved values of S would require some explicitrelationship between the signal S and the profile qz.The radiative transfer equation provides an analyti-cal relationship that solves the direct problem, that is,the computation of the signals as a function of a givenprofile. However, this equation represents a nonlin-ear transformation that, in general, cannot be analyti-cally inverted to obtain the relationship for thecomputation of the profile as a function of the ob-served signals. For a general discussion on thesolution to the inverse geophysical problems, we referto Menke.2 Complementary discussions on the inver-sion of remote sounding measurements, which areconsistent with the approach of this paper, can befound in Refs. 3 and 4.

A. Sensitivity Functions

A linear relationship between S and q can be ob-tained by applying a Taylor expansion around anassumed profile q,, to the function S(v, h, 0, q,). Wedenote with x the set of parameters (v, h, 0) thatidentify the observing conditions. In the hypothesisthat qz is near enough to the true profile to drop in alinear behavior of the function S(x, qz), the Taylorexpansion can be truncated to the first term to obtain

S(x, q = S(x, qz ) + J;EaS1(x q)) [ln(q. ) - n(qj ]dz.~~ - ln(~~~~a ndqz

(1)

For the sake of formal advantages, the expansionhas been made with respect to the quantity ln(qz)rather than q,.

Because at this stage of the discussion qz is consid-ered as a continuous distribution of values, in Eq. (1)the integral operator is used, instead of the summa-

tion operator that is present in a Taylor expansionrelative to a multivariable function.

Equation (1) can be written as

N(x) = K(x, z)yzdz, (2)

where

N(x) = S(x, q.) - S(x, qz),

K(x, z) = [aS(x, qz)

y = [ln(qz) -ln(qz)].

(3)

(4)

(5)

For any observation defined by a set of parametersx' (v', h', 0'), the function K measures the sensitiv-ity of the observation to a variation of the mixingratio at the different values of z; in view of thischaracteristic, we denote this function as the sensitiv-ity function. More generally, the functions definedby Eq. (4) are weighting functions; however, thisname often also denotes other functions that resultfrom different approximations of the inversion prob-lem. For this reason we refer to these particularweighting functions as sensitivity functions.

Equation (2) is an integral equation that representsa linear transformation of the unknown yz, leading tothe observations N(x) by way of the kernel K(x, z).This transformation, within the formalism of func-tional spaces, can also be described by saying that theobservations are an inner product of the unknownwith the sensitivity functions. In the particular casein whichyz and K(x, z) are discrete entities, Eq. (2) canbe expressed with the simpler matrix notation; herewe also use this notation to represent the moregeneral case of continuous functions, so that Eq. (2)becomes

n= Ky, (6)

where n is a vector containing the m observations, Kis a matrix of m rows and an infinite number ofcolumns, and y is the unknown vector. For eachelement of vector n, the matrix K contains, at thecorresponding row, the corresponding sensitivity func-tion.

For practical applications (see Sections 3 and 4), thevalues of a sensitivity function are derived from Eq.(4), where the signal S can be computed by means ofthe radiative transfer equation. To represent anactual observation, the computation of S must in-clude the effect of both the finite spectral resolutionand the field of view of the measuring spectrometer,the equations that model these effects are discussedin Appendix A.

B. Discrete Observations: Estimate of the Solution

In an actual experiment the number m of observa-tions is a finite number, each observation correspond-ing to a set x' (, h', 0') of the observing param-

3238 APPLIED OPTICS / Vol. 33, No. 15 / 20 May 1994

eters; hence, from a mathematical point of view, thesolution to Eq. (6) is an ill-posed problem becausethere are infinite unknowns in front of a finitenumber of observations. Therefore, only an esti-mate of the unknown vector is possible. Theproblem of computing this estimate can be reduced tothe search for a solution matrix D that, when multi-plied by the vector n of the observations, leads to thedesired estimate:

9 = Dn. (7)

of the distribution and the different applications mayrecommend different operative definitions. Often inremote sensing the vertical resolution is measured bythe spread defined by Backus and Gilbert.5 In opti-cal imaging the resolution is defined by using theMTF given by the Fourier transform of the transferfunction. Whenever possible we prefer to character-ize the vertical resolution by means of the rows of thetransfer matrix; however, when a single parameter isneeded, we use the MTF of the rows of matrix T,

In an ideal case, D would be the inverse K-' of thekernel matrix; in the case of a finite number ofindependent observations, D can be identified withthe generalized inverse K#; whereas in a more generalcase, D can be identified with the weak generalizedinverse K@ (for the definitions and properties ofinverse matrices, see Appendix B). In the procedurefor the computation of D, it is possible to introduceconstraints on the estimate of y (see, e.g., Ref. 3);however, for the purposes of our discussion, weconsider only the case in which D is one of the inversematrices (see Subsection 2.F and Appendix B).

C. Transfer Matrix

Substituting the expression provided by Eq. (6) for nof Eq. (7), we obtain

y = DKy = Ty, (8)

where the matrix

T= DK (9)

describes the relationship between the estimatedvalue and the real value of the unknown vector. Trepresents the transfer function of the inversionprocess; hence it can be said to be the transfer matrix.In the ideal inversion T would be the unit matrix; in areal inversion the deviations of the transfer matrixfrom the unit matrix give a measure of the quality ofthe inversion. Each row of T corresponds to analtitude of the estimated profile and each column of Tcorresponds to an altitude of the real profile: ataltitude j, row tL of T provides a distribution thatmeasures how much the retrieved solution at thataltitude depends on the actual values of the profile atthe other altitudes. Conversely, at altitude j, col-umn tj of T provides a distribution that measureshow much the actual value of the profile at thataltitude contributes to the retrieved values at theother altitudes. Therefore the deviation of a row orcolumn of T from the Dirac delta function (i.e., therows and columns of the unit matrix) gives a measureof the vertical resolution that can be achieved at thecorresponding altitude.

D. Modulation Transfer Functions (MTF's)

Many definitions of vertical resolution can be adopted,but none has a universal use. This is not surprising,because the resolution is a single parameter used tocharacterize a distribution, and the different shapes

(10)

and define the vertical resolution at altitude i as thereciprocal of the frequency at which the value of(MTF)i drops to 50% of its starting value.

It is important to emphasize that the inverseproblem is often discussed in the literature withreference to a model in which the unknown has afinite number of degrees of freedom. With thisapproach, one can only assess the vertical resolutionindirectly by means of the period of the retrievedparameters and their correlations. The rigorousdefinition of vertical resolution obtained with matrixT and the corresponding MTF's are an importantadvantage of the representation in a functional space[see Eq. (6)].

E. Variance-Covariance Matrix

To characterize the errors associated with the solu-tion of the inversion procedure, the variance-covari-ance of the estimate Sy must be computed. In thecase of observations that are uncorrelated and all ofequal variance u2 the variance-covariance of isgiven by

[coV S] = u2[DDT], (11)

where the superscript T indicates the transposematrix.The matrix

V= DDT (12)

then permits estimation of how the experimentalrandom errors map onto the uncertainty of the valuesof the retrieved profile.

The square root of the diagonal elements of avariance-covariance matrix measures the rms errorof the corresponding parameter. The off-diagonalelement at row i and column j of a variance-co-variance matrix, normalized to the square root of theproduct of the two diagonal elements with indices iand j, measures the correlation coefficient betweenthe corresponding parameters. For a further discus-sion of the variance-covariance matrix, see Menke.2

As in the case of matrix T, in matrix V each row andeach column correspond to an altitude of the profile:by indicating with vij the matrix element at row i andcolumnj, from Eq. (11) it follows that the rms error ofthe valueyj retrieved at altitude i is equal to

A9A = U(Vii). (13)

20 May 1994 / Vol. 33, No. 15 / APPLIED OPTICS 3239

(MTF)i = �7-1[tj,

The quantity retrieved from the inversion proce-dure [see Eq. (5)] is [ln(q,) - ln(q,)]; hence, at altitude6,' we have

A9i = A[ln(qi) - ln(qi )] = - + -- Xqi qi

(14)

where A\ denotes the rms error of the parameter.Because Aqi = 0, it follows that

A _ Aq. (15)

By combining Eqs. (13) and (15), we obtain

Aqi = rqi(vii)1/2 . (16)

The rms error, as determined by Eq. (16), is oftenreferred to as the estimated standard deviation (ESD)of the retrieved parameter.

Equation (16) provides an easy way to estimate theerror of the values of the retrieved profile at allaltitudes. Actually the values qi can be assumed tobe equal to the values qi; vii can be calculated from theassumed atmospheric model; and the o error, which isthe rms deviation of the spectral noise (i.e., noise ofthe spectroscopic measurements), can either be mea-sured on the spectra or predicted on the basis of thecharacteristics of the spectrometer.

The correlation coefficient between the values ofthe profile retrieved at altitudes i andj is independentof o and is given by

cij = vi(viiv0 ) -1/2 (17)

F. Calculation of the Generalized Inverse Matrix

As stated in Subsection 2.B, we consider the case inwhich solution matrix D is one of the inverse matricesof K. It has been demonstrated by Kalman6 that theproperties that are valid for the inversion of a matrixalso apply to the inversion of a functional kernel.Therefore we can extend the matrix notation to theexpressions for the calculation of the inverse kernel.

If the m observations are independent of eachother, the generalized inverse matrix K# can becomputed with the matrix expression

K# =KT(KKT)-1. (18)

The estimated solution vector k corresponding to thegeneralized inverse matrix K# computed with Eq.(18) belongs to the n-dimensional vector space F ofthe sensitivity functions. It is shown below that in apractical computation a different vector space mustbe used; despite this requirement, the transfer matrixcalculated from K# provides useful information, be-cause an inspection of the distribution of its rowsmakes it possible to estimate the blurring that isintrinsic in the inversion problem and gives, at eachaltitude, the maximum vertical resolution that couldbe obtained if the observations were noise free.

To convert the inversion of Eq. (6) into a mathemati-

cally overdetermined problem that makes possibleacceptable ESD values, in a practical analysis thesearch for the solution vector is made in a space f thathas a dimension p < m. In this case the solutionmatrix D must be computed as

K = HT(HKTKHT)-iHKT, (19)

where H is a matrix of p rows. K@ is a weakgeneralized inverse (see Appendix B). The rows of Hcontain p functions that constitute a base for thespace f in which the solution must be represented;hence H can be named the base matrix. Given aspace f, it is then possible to compute the transfermatrix and the variance-covariance matrix: the firstmatrix will reflect the choices on the vertical resolu-tion that have been made in defining f, whereas thesecond matrix makes it possible to evaluate theuncertainties associated with the retrieved profilecorresponding to those choices.

G. Global-Fit Analysis

We now discuss how the inversion model describedabove compares with the global-fit analysis method.'The results of this comparison also apply to otherretrieval techniques that invert a kernel of dimen-sions m x p, just as global fit does.

In global fit the whole altitude profile is retrievedfrom the simultaneous analysis of all the limb-scanning measurements; this global approach solvesEq. (6) with respect to y. The retrieval performedwith global fit is based on the least-squares criterionand looks for a solution profile that has a number p ofdegrees of freedom smaller than the number ofobservations. In practice the profile is retrieved atpdiscrete altitudes, and at the intermediate altitudeseither a constant or an interpolated value is used(profile segmentation). The choice of thep altitudesin which values of the profile are actually retrieveddetermines the vertical resolution of the retrievedprofile. The solution provided by global fit can berepresented by a continuous function that is equal tothe function obtained with the solution matrix Dgiven by Eq. (19), when H is the base matrix of asuitable space f of dimension p consistent with theprofile segmentation. A set of p linearly indepen-dent vectors that constitute a base for this space canbe easily defined: as an example, in Fig. 1 we plotthree vectors that are a base for the layer-model spacedefined by the four altitudes hl, h2, h3, and h4. Anyvector of f can be represented as a linear combinationof the three base elements.

Despite the coincidence of the results obtained withthe present model and global fit (see also Section 4),the two methods have important differences. Themathematical model requires numerous computa-tions if the discretization of the continuous variable issufficient to represent the features of the functionsinvolved. In particular, there are as many retrievedparameters as there are values of the continuousvariable. Global fit uses instead matrices with smalldimensions and retrieves only a number p of param-

3240 APPLIED OPTICS / Vol. 33, No. 15 / 20 May 1994

h

h

LU

a

-J

h3

0 1

h

h4

O 1

h

,0 ,0 1

NORMALIZED AMPLITUDEFig. 1. Example of three vectors that make a base of the space fused in the case of the layer model defined by the four altitudes hi,h2 , h3 , and h4 .

eters; a drawback of the computing savings is theimpossibility, already stated in Subsection 2.D, ofdetermining the transfer matrix T.

As far as the variance-covariance matrix is con-cerned, the matrix V given by the general mathemati-cal model with Eq. (12) does not provide significantadditional information compared with the correspond-ingp x p matrix computed with global fit.

Therefore global fit is the ideal choice for opera-tional data analyses because it provides both re-trieved profiles and error estimates with reasonablecomputing requirements. The present mathemati-cal model is needed for an unbiased determination ofthe vertical resolution and is suitable for studying themodified trade-off between vertical resolution anduncertainty. The coincidence of the results obtainedwith the two methods is fundamental to transferringthe information obtained with the mathematicalmodel to the operational data analysis performedwith global fit.

Finally, it should be stressed that in an operationalanalysis the use of the inversion matrix given by Eq.(19) in place of the in version matrix given by Eq. (18)does not represent an approximation that introducesextra errors; Eq. (19) provides instead, through thematrix H, a useful degree of freedom of the analysisthat can be exploited for the optimization of theabove-mentioned trade-off. The choice made whenusing some analysis methods (for instance, onionpeeling) of neglecting the existing correlations toanalyze the observations in a sequential manner isinstead an approximation that may cause additionalerrors.

3. Application to the Design of an Experiment

As an example of the use of the mathematical toolsderived in Section 2, in this section we discuss theirapplication to the design of measurements of OHfrom a satellite. This radical is one of the targetspecies for the SAFIRE (spectroscopy of the atmo-

sphere using far-infrared emission) experiment, whichhas been proposed as part of the Earth ObservingSystem. The SAFIRE baseline is an interferometerwith 0.004 cm-' of spectral resolution that observesatmospheric emission in the far infrared, using thelimb-scanning mode, with a field of view of 0.06°.The instrument is to be flown on a satellite-borneplatform orbiting at an altitude of 705 km on a polarorbit; at this altitude the field of view corresponds toapproximately 3 km in the atmosphere. The limb-scanning steps are 0.030. The twin features of OH at118.21 and 118.45 cm-' are measured with an esti-mated rms spectral noise level of 0.5% relative to theemissivity of a blackbody at 250 K.

A. Sensitivity Functions

The two OH features each result from three unre-solved trainsitions; at this stage of the discussion, wefocus on the lower-frequency feature.

Fifty-five observation geometries, which cover ap-proximately the altitude interval from 13 to 98 km,have been considered for this study. In Fig. 2 thesensitivity functions relative to these observationgeometries are plotted in the case of infinitesimalspectral resolution and field of view as a function ofz = ln(P/Po), where P/P0 is the fractional pressurerelative to sea level. For each observation geometry,out of all the sensitivity functions computed at differ-

F-15r

,41

13

1

10

1

N

8

7

6

5

4

3

2

90

80

70

E60 v

0

50 i

4

40

30

20

0 0.5 1.0 1.5 2.0 2.5

NORMALIZED RADIANCE

Fig. 2. Sensitivity functions of OH limb-scanning measurementsfor 55 observation geometries, in the case of infinitesimal spectralresolution and field of view, at the frequency that providesmaximum amplitude. P/PO is the fractional pressure relative tosea level. The curves are plotted down to a minimum altitudebelow which the sensitivity drops to zero.

20 May 1994 / Vol. 33, No. 15 / APPLIED OPTICS 3241

Flue

ent frequencies, Fig. 2 reports the sensitivity functionwith maximum amplitude.

For the computation of the sensitivity functions,the signal S of Eq. (4) has been normalized to theemissivity of a blackbody at 250 K, so that thefunctions are adimensional quantities. The discreti-zation along z that is adopted in the calculationscorresponds to an altitude increment of 0.2 km andrepresents the shape of the continuous functionswith a satisfactory approximation.

To introduce the effect of the finite value of thespectral resolution into the computations, the signalS is calculated by applying Eq. (Al) of Appendix A, inwhich the instrument function I(v - v') is a sincfunction corresponding to the unapodized resolutionof the SAFIRE instrument. The sensitivity func-tions of maximum amplitude that are obtained in thiscase are shown in Fig. 3. It is interesting to notethat the frequency at which the functions in Fig. 3 aregiven is different from the frequency correspondingto the same observation geometry in Fig. 2. Acomparison of the two figures shows how both thepeak and the integral of the sensitivity of the measure-ments drops because of the effect of the instrument'sfinite spectral resolution. This drop is larger at highaltitudes, where the atmospheric features are sharpand the sensitivity functions at infinite spectral reso-lution change rapidly as a function of the frequency.

The effect of the finite field of view is highlighted inFig. 4: the thin curves are a selection of six sensitiv-ity functions from Fig. 3, plotted with a factor of 3 ofamplification in the signal scale; the thick curves arethe functions at the same values of (v, h, 0) as theyappear when the finite field of view of the instrumentis taken into account. For this purpose Eq. (A2) ofAppendix A has been applied, assuming a conical fieldof view with an angular response function A( - 0')proportional to the length of the cord of a circumfer-ence centered along the beam axis. In Fig. 4 it can beseen how the finite field of view results in a spread ofsensitivity over a wider range of altitudes and in adecay of the peak sensitivity; however, the integral ofthe distribution undergoes a negligible change.

More than one spectral element of a multiplexspectroscopic system provides useful information forthe inversion process. The Nyquist sampling theo-rem states that, in the spectrum, full and indepen-dent spectroscopic information lies at frequenciesthat are separated by a sampling interval; hence, theremote-sensing information content of a Fouriertransform observation is also fully represented by thesensitivity functions computed at frequencies sepa-rated by the sampling interval. In our case, threesensitivity functions have been considered for eachobservation geometry at frequencies displaced by 0.0,0.004, and 0.008 cm-', respectively, from the linecenter; subsequent sampling frequencies provide sen-sitivity functions of relatively small amplitude andhave not been included in this model. In Fig. 5 allthe sensitivity functions representing the full informa-tion content of a Fourier-transform limb-scanning

15

14

13 -

12 _

Il-

101-

9

0I D

11N

8

7

6

5

4

3

2

100

90

80

70

E60 X

uLJa

0s

J

40

30

20

_100 0.5 1.0 1.5

NORMALIZED RADIANCEFig. 3. Sensitivity functions of OH limb-scanning measurementsfor the same observation geometries of Fig. 2, in the case of 0.004cm-1 spectral resolution and infinitesimal field of view, at thefrequency that provides maximum amplitude.

observation of the OH feature are plotted with afactor of 10 of amplification of the sensitivity scalewith respect to Fig. 2. The envelope of the curves ofFig. 5 provides an indication of the altitude ranges atwhich the retrieved profile can be better determined.

B. Transfer Matrix and MTF's

As we have shown in Section 2, the sensitivityfunctions define the matrix K of Eq. (6) that makes itpossible to compute the corresponding solution ma-trix D; this matrix, multiplied by K, leads to thetransfer matrix T according to Eq. (9).

From the matrix K obtained from the set of sensitiv-ity functions of Fig. 2, using Eq. (18) to compute thesolution matrix, we derive the transfer matrix rela-tive to the inversion of ideal measurements madewith an instrument having infinitesimal spectral

3242 APPLIED OPTICS / Vol. 33, No. 15 / 20 May 1994

i I l

15

14

13

11

0

E

:

I

I

S

8

11N 7

5

4

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

NORMALIZED RADIANCE

Fig. 4. Effect of the finite field of view. The thin curves are aselection of six sensitivity functions from Fig. 3, plotted with afactor of 3 of amplification in the signal scale. The thick curvesare the functions at the same values of (v, h, 0) in the case of a fieldof view of 0.06°.

resolution and field of view. Three rows of thistransfer matrix are plotted by way of a sample in Fig.6; from the spread of the main feature appearing onthe row corresponding to a given altitude we canobtain a qualitative estimate of the mixing of informa-tion at the corresponding altitude that is due to theinversion procedure. Figure 7 shows the MTF'scorresponding to the functions reported in Fig. 6.The MTF's shown in Fig. 7 permit more detailedconsiderations: in fact, going from low to high val-ues of the frequency scale, the curves of Fig. 7 display,on an average, a decreasing amplitude as well asperiodic minima. These minima are caused by thediscrete step of the limb-scanning sequence. Theexistence of residual amplitudes at high-frequencyvalues indicates that the sharply peaked sensitivityfunctions at infinitesimal spectral resolution are po-tentially capable of very high (a few hundred meters)vertical resolution. However, the full exploitation ofthe performances that are ideally possible with thesesensitivity functions would require a finer step in thelimb-scanning sequence. The irregular shape of theMTF's prevents in this case a useful quantitativeestimate of the vertical resolution.

Let us consider real observations made with finitespectral resolution and field of view. The transfermatrix must be computed using the set of sensitivityfunctions of Fig. 5. The rows of this transfer matrix,which correspond to the same altitudes as thoseshown in Fig. 6, are plotted in Fig. 8; from a compari-

K

90

50

70

E60 v!

I150 -_

J4

40

30

100.0 0.05 0.10 0.15 0.20 0.25

NORMALIZED RADIANCE

Fig. 5. Overall set of the sensitivity functions of an OH limb-scanning measurement in the case of finite spectral resolution andfield of view. Three frequencies, displaced by 0.0,0.004, and 0.008cm- 1 from the line center, respectively, are reported for each of theobservation geometries of Fig. 2. The sensitivity scale has a factorof 10 of amplification with respect to Fig.2.

son of Figs. 6 and 8, the blurring of information thatis due to the field of view is evident. Figure 9 showsthe MTF's relative to the functions of Fig. 8; in thiscase the MTF's have three main characteristics: (1)an amplitude close to unity up to approximately 0.31/km, (2) a sharp decrease in amplitude around 0.31/km, and (3) an amplitude almost equal to zero athigher frequencies. This behavior is independent ofaltitude, indicating a correct combination of theobservation parameters. The value of the verticalresolution that results from these MTF's, followingthe definition given in Subsection 2.D, is approxi-mately 3 km at all altitudes. This result, which hasbeen obtained with the solution matrix given by Eq.(18), provides an estimate of the best vertical resolu-tion that can be achieved from this OH transitionwith the observing parameters under consideration.

To simulate a retrieval analysis of the observationscharacterized by the sensitivity functions of Fig. 5,the solution matrix must be computed using Eq. (19)with a suitable matrix H. In the case of a matrix Hdefined by base vectors such as those shown in Fig. 1,it is meaningful to study the case in which the layershave the same width as the limb-scanning steps (1.5km). The rows of the transfer matrix that resultfrom these computations, and the correspondingMTF's, are quite similar to those shown in Figs. 8 and

20 May 1994 / Vol. 33, No. 15 / APPLIED OPTICS 3243

l1oo

11

en

0.80

0.40 F

0.20

0.60

'LaI

-J

0.

0,40 V

0.20-

0

0.60

0.40

0.20

0 - e10 20 30 40 50 60 70.. 80 90 100

ALTITUDE [kim]

Fig. 6. Three rows of the transfer matrix in the case of theinversion of ideal OH measurements made with infinitesimalspectral resolution and field of view. The rows correspond ataltitudes that are (from the bottom) approximately 18, 55, and 92km, respectively.

9, respectively; it follows that the vertical resolutionhas not been degraded by the effect of the base matrixand that the vertical resolution is approximatelytwice the width of the layers defined by the said basematrix. These results together with tests made withdifferent parameters indicate that a good combina-tion of the three critical parameters (field of view,limb-scanning step, base matrix) has been chosen.If one of the three parameters is limiting the resolu-tion, improving the other two is of little help. Thebest results (e.g., a sharp transition from unity to zeroin the MTF's) are obtained when similar resolutionlimits are introduced by the three parameters.

C. Variance-Covariance Matrix

In the case of a realistic retrieval analysis, we mustalso estimate how spectral noise reflects on theretrieved profile and, as a consequence, imposesconstraints on the analysis. To represent the simul-taneous analysis of the two OH features, they havebeen assumed to be identical and symmetrical withrespect to the center of the line. With these assump-

-

-JIL

0 0.5 1 1.5 2 2.5

I/k m

Fig. 7. MTF's corresponding to the rows of the transfer matrixreported in Fig. 6.

tions, multiplicity factors of 2 and 4 must be appliedto the sensitivity functions of Fig. 5 relative to thecenter and to the side of the line, respectively; thiswould require the use of n identical sensitivity func-tions, where n is the multiplicity or, alternatively, themultiplication of the functions by the square root ofn. (The multiplicity has not been considered for thecomputation of the MTF's because it does not changetheir values.)

Given a matrix H, from Eqs. (19) and (12) we obtaina variance-covariance matrix V that provides, bymeans of Eq. (16), the estimate Aqj of the errors of theretrieved values of the profile. Because we are con-sidering normalized signals (see Subsection 3.A), thepercent error Aqi(%) is given by

Aqi(%) = g(%)(Vi)1/2, (20)

where u(%) is the percent spectral noise normalizedwith the same criteria used for the signal.

Figure 10 shows the percent error values as afunction of altitude, calculated from Eq. (20) in thecase of r(%) = 0.5% and a matrix H defined by basevectors such as those shown in Fig. 1 for 1.5-kmlayers. By comparing Figs. 5 and 10, it can beobserved that, as expected, the errors are small at the

3244 APPLIED OPTICS / Vol. 33, No. iS / 20 May 1994

I

I

-IT

0.60

0 .40 _

0i.20

0

0j.60

Uj

C10.40 _-

I- L

0.o.20

0.60

/'fJ

0.40 _

0.20 _

0

10

I l l l l l

20 30 40 50 60 70 80 90

1.5

0.5

0

1.5

-J

L

2 0.5

0

1.5

0.5

100

ALTITUDE [kim]

Fig. 8. Three rows of the transfer matrix, at the same altitudes asthe rows of the transfer matrix reported in Fig. 6, in the case of theinversion of OH measurements made with 0.004 cm- 1 spectralresolution and a field of view of 0.06°.

altitudes at which the sensitivity functions have largevalues. The errors of the analysis represented inFig. 10 range from approximately 20% to 500%; thesevalues are clearly too high. The amplitude of theerrors is affected by the correlation that exists be-tween neighboring altitudes; because this correlationdiminishes when the separation of the altitudes con-sidered increases, a strategy for obtaining an accept-able distribution of the errors could be to use a profilesegmentation (see Subsection 2.G) with increaseddistances, that is with reduced vertical resolution.As an example of the behavior of this trade-offbetween vertical resolution and random errors on theretrieved profile, let us assume that errors aboveapproximately 10% are not acceptable. A few at-tempts that have been carried out to meet thisrequirement have led to the profile segmentation thatproduces the error distribution represented in Fig.11, with an amplification factor of 20 in the y-axisscale with respect to Fig. 10. In this case the errorsare of the order of 5%, except at the extreme altitudes,where they tend to become larger; errors smaller than10% cannot be attained outside the altitude range

I 1/k m

Fig. 9. MTF's corresponding to the rows of the transfer matrixreported in Fig. 8. -

400

350

300

° 250

cL

1 200z -U

C 150UJa

100

50

10 2Q 30 40 50 60 7d 80 90 100

ALTITUDE [kin]],

Fig. 10. Percent error values of the retrieved OH profile in thecase of measurements made with 0.004 cm-1 spectral resolution, afield of view of 0.06° and o(%) = 0.5% for a profile segmentationwith 1.5-km layers.

20 May 1994 / Vol. 33, No. 15 / APPLIED OPTICS 3245

I I I i

- |n I J ,

0

LU

1 0z

0.

10 20 30 40 50 60 70 80 90 100

ALTITUDE [km]

Fig. 11. Percent error values of the retrieved OH profile formeasurements, as in Fig. 10, and a profile segmentation defined bylayers broadened with respect to Fig. 10. An amplification factorof 20 is applied in they-axis scale with respect to Fig. 10.

considered in Fig. 11. The improvement in the errorvalues with respect to Fig. 10 has been obtained at theexpense of the vertical resolution and giving up anydetermination outside the considered altitude range.The vertical resolution, as defined in Subsection 2.D,in the case of the profile segmentations used for Figs.10 and 11 are shown in the lower and upper curves,respectively, in Fig. 12. In comparing Figs. 10, 11,and 12, it can be observed, for instance, that ataltitudes around 40 km, a loss of approximately afactor of 2 in vertical resolution corresponds to areduction in the errors of approximately a factor of 5,

whereas at altitudes around 70 km, a loss of approxi-mately a factor of 3 in vertical resolution correspondsto a reduction in the errors of approximately a factorof 10. The large variation in the errors as a functionof the vertical resolution confirms the existence of alarge correlation between neighboring altitudes.The observations provide sufficient information fordetermining the profile in a 3-km-thick layer (e.g., at40 km in Fig. 11), but how this amount of informationis split into the two adjacent 1.5-km-thick layers (e.g.,at 40 km in Fig. 10) is far less easily determined.

A priori information and computational con-straints can also be used to determine the relativeamounts of adjacent layers; this may lead to claimedaccuracies as good as those of Fig. 11, together withvertical resolutions as good as those of the lowercurve of Fig. 12. A precise definition of the modelused to simulate the retrieval is, therefore, essentialto have a meaningful estimate of the quality of theretrieved values. Furthermore, it is important tonote the conceptual and quantitative differences be-tween vertical resolution and thickness of the atmo-spheric layer assumed at constant concentration; thetwo concepts are often confused when quoting instru-ment performances.

Finally, it must be pointed out that, in the computa-tions leading to Figs. 10 and 11, the matrix K is thesame; that is, all the sensitivity functions contributeto determining the profile corresponding to Fig. 11.To highlight the contribution given by the sensitivityfunctions that peak at altitudes not included in theprofile segmentation, Fig. 13 reports the error distri-bution that results when these functions are removedfrom the matrix K (and the same base matrix is usedas for Fig. 11): an increase in all the errors can beclearly seen in Fig. 13 compared with Fig. 11.

20

20 r- -

15 I-

E

zI.-

-J

0LU

10i

Ir

0L_ -.10 20 30 40 50 60 70

ALTITUDE [ km IFig. 12. Vertical resolution of the retrievedprofile segmentations of Fig. 10 (lower curve)curve), respectively.

0

Lc

zU

(L

80 90 100

OH profile for theand Fig. 11 (upper

15 -

10 -

r

5,-

10 20 30 40 50 60 70 80 90 100

ALTITUDE [km]Fig. 13. Percent error values of the retrieved OH profile in thecase in which the same profile segmentation is adopted as in Fig. 11but the sensitivity functions that peak at altitudes outside thealtitude interval 27-89 km are removed from the matrix K.

3246 APPLIED OPTICS / Vol. 33, No. 15 / 20 May 1994

- - -

4. Application to Data Analysis

In this section we show the application of the math-ematical tools derived in Subsection 2 to the problemof analyzing the limb-scanning measurements of theH35CI transition at 62.58 cm-', obtained with aballoon-borne interferometer. The a priori expecta-tions provided by the general mathematical model arecompared with the a posteriori results provided bythe data analysis of real measurements.

The measurements considered for this applicationare observations of the atmospheric emission, from aballoon-borne platform at approximately 38 km alti-tude, with a spectrometer that has a spectral resolu-tion of 0.0035 cm-1 and a field of view of 0.70. At thetime of the design of these observations, the criteriadiscussed in the present paper were not available, anda simple choice was made of limb-scanning stepsequal to the field of view. The rms deviation of thenoise level measured on the spectra is approximately8%. The sequence considered in this example hasfive limb-scanning spectra corresponding to zenithangles ranging approximately from 90.50 to 93.00; thetangent altitude of the observations is reported incolumn 1 of Table 1.

For the implementation of the general mathemati-cal model, the sensitivity functions were computedfor each observation geometry in correspondencewith four sampling points, starting from the linecenter. In a first case (denoted as analysis 1), theprofile segmentation has one value for each tangentaltitude of the observations; the resulting variance-covariance matrix gives, at those altitudes, the errorestimates reported in column 2 of Table 1. Becausetoo-large errors are predicted at 36.0 and 34.2 km,these altitudes have been in turn removed from theprofile segmentation, and two corresponding variance-covariance matrices have been derived; in columns 4and 6 of Table 1 we report the predicted errorestimates resulting for the two cases that are denotedas analyses 2 and 3. The magnitude of the predictederrors is comparable in these two analyses; however,the profile segmentation of analysis 2 is preferable,because it provides a more regular spacing in thevertical segmentation.

Global-fit analyses have been carried out on theobserved spectra by the use of the three profilesegmentations discussed above; the percent value ofthe ESD's that are calculated in the three cases arereported in Table 1 to the right of correspondingpredicted values. The agreement between the errors

Table 1. Predicted and Calculated Errors (%) for Three ProfileSegmentations

Altitude Analysis 1 Analysis 2 Analysis 3

(km) Pred. Calc. Pred. Calc. Pred. Calc.

37.2 33 35 25 22 30 2836.0 77 54 - - 40 4434.2 75 81 44 41 - -31.4 58 61 49 57 37 4628.4 47 48 45 47 42 46

predicted by the model and the computed ESD's isgenerally good; the differences can be ascribed mainlyto the non-uniform distribution of the noise in theobserved spectra [see hypothesis on Eq. (11)]. It hasbeen experimentally shown7 that the ESD's providedby the global-fit analysis agree with the standarddeviations of a statistically significant set of retrievedprofiles.

5. Computational Details

A limit that can be encountered when using thismathematical model is the computing time required.Modern computers are rapidly overcoming this limit,but an awareness of this problem is important. Thetime-consuming item is the calculation of the sensitiv-ity functions; all the other calculations needed toderive the transfer matrix and the variance-covari-ance matrix involve standard matrix algebra thatrequires negligible computing time. The calculationof a sensitivity function according to Eq. (4) requirescomputing the value of the signal by means of theradiative transfer equation for the assumed mixing-ratio profile c, as well as for increments of the mixingratio at each altitude. In the applications reportedin Sections 3 and 4, an increment equal to 1% of themixing-ratio value was used. Furthermore, if thefinite spectral resolution and field of view have to betaken into account, the sensitivity function valuescan be computed by applying Eqs. (Al) and (A2) ofAppendix A in arbitrary order. Equation (Al) re-quires that the signal be calculated for a range offrequencies; the amplitude of this range must be suchthat it prevents truncation distortions by the instru-ment function I(v - v'), and the frequency step mustbe such that it represents the natural shape of theconsidered spectral feature. Similarly, Eq. (A2) re-quires that the signal be computed for a range ofzenith angles; the amplitude of this range must besuch that the angular-response function A(O - 0')approaches zero at the edges of the range, and theangular steps must be consistent with the discretiza-tion operation performed on the altitude domain.

If the goal of the application is the overall optimiza-tion of a limb-scanning experiment, it is convenient touse a computational approach in which the derivativeoperator needed to compute the sensitivity functionand the integral operators of Eqs. (Al) and (A2) arereversed in order. In this case a two-dimensionalarray of sensitivity functions must be computed withinfinitesimal spectral resolution and angular aper-ture, one dimension of the array ranging in thefrequency domain and the other ranging in thezenith-angle domain. From this two-dimensionalarray it is easy to derive the set of sensitivity func-tions relative to different instrument functions, differ-ent angular-response functions, and different observa-tion geometries. For more specific applications,approximations that significantly reduce the comput-ing time can be adopted.

Table 2 provides some quantitative information on

20 May 1994 / Vol. 33, No. 15 / APPLIED OPTICS 3247

Table 2. Relevant Parameters and Computing Time for the Calculationof the Sensitivity Functions

Parameter Satellite Balloon

Number of observation geometries 55 5Discrete altitude step (km) 0.2 0.2Frequency range for the instrument 0.05 0.05

function convolution (cm-')Frequency step for the instrument 0.0004 0.0004

function convolution (cm -1)Tangent altitude step for the 0.2 0.2

angular aperture convolution(km)

Atmospheric boundary altitude (km) 100 50Discrete step on the line of sight for 0.2 0.2

radiative transfer computation(km)

Number of transitions 3 1Cray Y-MP CPU time (s) 323,902 4749

the computing time. Column 1 lists the main param-eters that are involved in the computation of thesensitivity functions; roughly speaking, there is adirect proportionality between the computing timeand all the parameters listed. Columns 2 and 3report the values that have been used for the compu-tational parameters in the applications discussed inSections 3 and 4; they are denoted in the table assatellite and balloon, respectively.

6. Conclusions

A general mathematical formulation suitable for themodeling of the retrieval of limb-scanning observa-tions has been discussed. The transfer matrix andthe variance-covariance matrix are the mathematicaltools that derive from this formulation and make itpossible to determine which vertical resolution andprecision can be achieved. Both matrices can becalculated from the sensitivity functions and the basematrix H. Therefore the design parameters of alimb-scanning experiment (which affect the shape ofthe sensitivity functions), together with the featuresof the retrieval method (which determine the matrixH), uniquely characterize the quality of the retrievedinformation. In other words it is possible to verifyhow the combined effect of the instrumental param-eters (e.g., spectral resolution and instantaneous fieldof view), the observational parameters (e.g., flightaltitude and limb-scanning pattern), and the retrievalconstraints (e.g., profile segmentation) affect the qual-ity of the retrieved profile. Such a possibility isimportant, as it permits the optimization of an experi-ment with a comprehensive view of all its parts: thespectrometer, the observation, and the retrieval.

Application of the mathematical tools to the dataanalysis of a balloon-borne experiment has showngood agreement between the a priori estimates of therandom errors and the errors obtained with theactual analysis.

Application of the mathematical tools to the designof the OH radical measurement by means of thesatellite experiment SAFIRE has highlighted the

following points of general validity:

(a) Observations characterized by sensitivity func-tions with a sharply peaked shape tend to providebetter vertical resolution, whereas high values ofthe sensitivity functions lead to less error amplifica-tion.(b) Finite spectral resolution can cause a reduc-tion of sensitivity (in the application we havediscussed, this reduction is remarkable at highaltitudes but does not limit low-altitude perfor-mance).(c) The correlation that exists between the valuesretrieved at contiguous altitudes may cause a largeerror enhancement when a vertical resolution equalto the limb-scanning step is required. On theother hand, a reasonable degradation of the verticalresolution results in a major reduction of theerrors.(d) Beyond certain limits, the idea of increasingthe number of observation geometries with the aimof upgrading the vertical resolution does not pro-duce the desired effect; however, it will introduceredundant information and improve the signal-to-noise ratio.(e) In general, the vertical resolution is deter-mined by the combined effects of field of view,limb-scanning pattern, and retrieval constraints (inthe application we have discussed, the relativevalues of these three parameters are well balanced).

It has been highlighted that an accurate solution ofthe inversion problem requires the simultaneousanalysis of all the spectra that are part of the limb-scanning sequence. Other analysis approaches thatconsider the spectra in a sequential manner introduceapproximations that may cause extra errors.

The trade-off between random errors and verticalresolution of the retrieved profile can be altered byeither implicit or explicit constraints introduced inthe analysis, for instance, the use of external informa-tion. However, a quantification of the unconstrainedperformances, such as the one provided by the above-mentioned mathematical tools, is necessary in thecase of both planned and existing experiments to havean objective estimate of the information content ofthe data.

Appendix A: Finite Resolution and Field of View

The sensitivity functions are evaluated by means ofEq. (4), in which the values of S can be computed witha radiative transfer calculation using the best avail-able approximation q, of the altitude profile. In theideal case, the observation is made with an infinitesi-mal spectral resolution at frequency v' and with aninfinitesimal angular aperture at zenith angle 0', andthe computation of S in Eq. (4) is carried out along asingle line of sight at a single frequency. A realobservation is made with a spectral resolution and anangular field of view of finite size. The instrumentfunction I(v - v') determines the spectral resolution,

3248 APPLIED OPTICS / Vol. 33, No. 15 / 20 May 1994

and the angular response function A(O - 0') deter-mines the field of view. Hence, at frequency v', thesignal at finite spectral resolution is given by theweighted average:

f S(v)I(v - v')dv

S(v') =

I(v - v')dv

(Al)

where S(v) denotes the signal computed with infinitesi-mal spectral resolution. Similarly, at zenith angle0', the signal at finite angular resolution is given by

S+vr/2

T -/2S(0') = /2

-r2

S(e)A(0 - 0')dO

A(0 - 0')dO

If r is equal to the number of rows of A, Eq. (B3)becomes

A` = (ATA)-lAT, (B4)

and if r is equal to the number of columns of A, Eq.(B3) becomes

A# = AT(AAT)-. (B5)

The weak generalized inverse A@ is that matrix Xthat satisfies the first two properties of Penrose:

(i)

(ii)

AXA = A,

XAX = X. (B6)

The weak generalized inverse always exists for matri-ces of any dimension defined in any domain and is not

('v) unique. By definition, the generalized inverse is aweak generalized inverse. The weak generalizedinverse can be calculated from the equation

where S(0) denotes the signal resulting from a singleline of sight. For practical computations, S must becalculated for discrete values of both v and 0; anumerical convolution process is then applied withthe functions I and A in arbitrary order.

B. Appendix B: Inverse of a Matrix

Given a generic matrix A, it is possible to define threetypes of inverse matrices: the inverse, the general-ized inverse, and the weak generalized inverse.

The familiar inverse matrix A-' is that matrix Xthat satisfies the property

AX = XA = I, (Bi)

where I is a unit matrix. A-' is unique and existsonly if A is a square matrix having the rank r (therank is the number of linearly independent rows orcolumns) equal to the dimension of the matrix itself.

The generalized inverse A# is that matrix X thatsatisfies the four properties of Penrose:

(i)

(ii)

(iii)

(iv)

AXA = A,

XAX = X,

AX = XTAT,

XA = ArXT,

where the superscript T indicates the transposematrix. The generalized inverse is unique and, formatrices of any dimension defined in the domain ofreal numbers, always exists. The generalized in-verse coincides with the inverse when the latterexists. The generalized inverse of A can be calcu-lated from the following equation:

A = AT(BTACT)-lBT, (B3)

where B and C are matrices made of r linearlyindependent rows and columns of A, respectively.

A- = LT(MTALT)-1MT, (B7)

where L and M are general matrices of the samedimensions as B and C of Eq. (B3), respectively, thatsatisfy the following relationships:

det(L T C) 0, det(M T B) 0, (B8)

where det denotes the determinant of the matrix.The above definitions and properties also apply to

the case of a matrix of infinite dimensions that can bedefined in a functional space.

The authors acknowledge the support provided bythe Italian Consiglio Nazionale delle Ricerche withthe Progetto Finalizzato Sistemi Informatici e Cal-colo Parallelo. The authors also thank M. G. Baldec-chi for technical assistance.

References

1. M. Carlotti, "Global-fit approach to the analysis of limb-scanning atmospheric measurements," Appl. Opt. 27, 3250-3254(1988).

2. W. Menke, Geophysical DataAnalysis: Discrete Inverse Theory(Academic, San Diego, Calif., 1984).

3. C. D. Rodgers, "Characterization and error analysis of profilesretrieved from remote sounding measurements," J. Geophys.Res. 95, 5587-5595 (1990).

4. B. Carli and M. Carlotti, "Far-infrared and microwave spectros-copy of the Earth's atmosphere," in The Spectroscopy of theEarth's Atmosphere and Interstellar Medium, K. N. Rao and A.Weber, eds. (Academic, San Diego, Calif., 1992), pp. 1-95.

5. G. E. Backus and J. F. Gilbert, "Uniqueness in the inversion ofinaccurate gross Earth data," Philos. Trans. R. Soc. LondonSer. A 266, 123-192 (1970).

6. R. E. Kalman, "Algebraic aspects of the generalized inverse of arectangular matrix," in Proceedings of Advanced Seminar onGeneralized Inverse and Applications, M. Z. Nashed, ed. (Aca-demic, San Diego, Calif., 1976), p. 111.

7. M. Carlotti, A. Barbis, and B. Carli, "Stratospheric ozonevertical distribution from far-infrared balloon spectra andstatistical analysis of the errors," J. Geophys. Res. 94, 16365-16371 (1989).

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