ozone profiling from backscattered uv radiance measurements: a new procedure

Ozone profiling from backscattered UV radiancemeasurements: a new procedure

Cynthia K. Whitney, Harvey L. Malchow, and Derek M. Cunnold

This paper presents a data analysis technique that has been developed to support a stratospheric ozone mea-surement and profiling program. The data consist of solar backscattered UV radiation monitored by aspacecraft. The output consists of ozone mixing ratio profiles complete with error bars throughout the at-mosphere. The technique is based upon principles of nonlinear optimal estimation. It provides a numberof desirable features that have not been available together in any previously constructed technique for theproblem. Iteration on both radiationtures.

1. Introduction

The ozone layer that resides in the earth's strato-sphere is currently a subject of controversy and study,", 2

motivating monitoring programs of global scope andlong duration. One technique currently being pursuedutilizes measurements of solar radiation backscatteredfrom the atmosphere in ultraviolet wavelengths(SBUV). SBUV instrumentation is presently flying inthe Nimbus 4 and 7 satellites and provides radiancemeasurements that imply both total ozone column andozone vertical profiles.

The principle of SBUV inversion is as follows. Solarradiation incident on the atmosphere is backscatteredby air molecules and attenuated by intervening ozonemolecules. For any narrow wavelength interval in theUV absorption band of ozone, the amount of back-scattered radiation is a measure of how far that wave-length penetrates the atmosphere and hence how muchozone there is to that depth. Given measurements atseveral wavelengths that do not penetrate all the wayto the ground, a profile can be deduced. The estimationof profiles does, however, present a variety of problems,both fundamental and practical. The development ofa methodology to handle these problems is the moti-vation for the present paper.

Derek Cunnold is with Georgia Institute of Technology, School ofGeophysical Sciences, Atlanta, Georgia 30332; the other authors arewith Charles Stark Draper Laboratory, Inc., Cambridge, Massachu-setts 02139.

Received 14 August 1980.0003-6935/81/060936-11$00.50/0.© 1981 Optical Society of America.

measurements and total ozone measurements is among these fea-

Deduction of an ozone mixing ratio profile from theSBUV data constitutes an inversion of the governingintegro-differential equation of radiative transfer.Such inversion problems generally suggest fundamentalquestions concerning convergence to a solution,uniqueness of the solution, and expected error in thesolution. Convergence requires that the inverted ozonestate should correspond to radiance residuals that areat least as small as the radiance measurement uncer-tainties. Usually if such a solution exists, it is notunique, at the very least because it could be perturbedslightly and still be consistent with the measurements.However, one can impose a requirement that the degreeof structure in the output profile be limited by the realinformation content of the measurements, which hasbeen shown by Mateer 3 to comprise only about four orfive independent pieces of information concerning theozone profile in a broad altitude regime largely abovethe location of the maximum in ozone concentration.Thus to provide a useful ozone profile, it may be nec-essary to adopt some functional model with only a fewindependent parameters in it. Accordingly errors mustbe acknowledged. These are lower bounded by theinherent radiance measurement uncertainty and aug-mented by the limitations of the functional modeladopted. They may turn out to be unacceptably largein atmospheric regions not well probed by the SBUVwavelengths.

To provide an ozone profile with acceptable errors,it may be necessary to merge the measurement profilewith a prior estimate profile that has acceptably smallerrors in the offending altitude regions. Since manyyears of other data are available, especially below 30-kmaltitude, there is ample basis for such a prior estimate.It can be constructed as an appropriate average. The

936 APPLIED OPTICS / Vol. 20, No. 6 / 15 March 1981

mechanism for incorporating it should exhibit a smoothtransition from the SBUV data dominated region toregions dominated by the prior estimate.

The practical problem inherent in remote soundingof ozone from a satellite by SBUV inversion is that agreat deal of data has to be processed quickly to keepup with the measurement capability of the system.Realistically, it is desirable that a profile inversionconsume only approximately a second of computationtime. This timing problem becomes particularly wor-risome when one recognizes all the practical require-ments that have to be met to constitute a solution.Required calculations can include iteration to reducemeasurement residuals, perturbations to assess sensi-tivities, assessments of multiple scattering in longwavelengths, and corrections for curved earth geom-etry.

11. Background for SBUV Inversion

In the past, several techniques have been developedto solve the SBUV inversion problem. Notable amongthese are the pressure increment technique proposedby Yarger 4 and developed by Mateer 5 in a code calledHILEV. This method is currently being utilized to in-vert Nimbus 4 BUV measurements. Also available arethe relaxation technique by Aruge and Igarashi6 (AI)and the pressure-radiance association method reportedby Thomas and Holland7 (TH), and the completetransfer equation method implemented in a code calledSLIC by Malchow and.Whitney. 8 Each of these ap-proaches has desirable features, but none really providesall the attributes needed for the Nimbus 7 SBUV sit-uation described above. The new approach describedin this paper and implemented in a code called FRAMEbuilds upon the existing work by exploiting the bestfeatures of each approach.

A. Basic Framework for Solution

The theory of optimal estimations is the frameworkselected for our solution to the SBUV inversion anderror analysis problem. Of the four previously existingapproaches cited, only HILEV and SLIC involve optimalestimation, and they implement it differently from whatis needed for FRAME.

For one thing, HILEV effectively utilizes a prior co-variance matrix on the radiance measurements whichhas only one free parameter-a smoothing constant.SBUV inversion requires more flexibility to account forarbitrary differences in channel noises. These may bereal or may be inserted artificially to accommodateunknown factors such as bias errors or multiple-scat-tering contributions.

Another factor concerns the definition of measure-ment. Both HILEV and SLIC take this to be an inputintensity measurement vector, but FRAME transformsthe concept to the output space of ozone mixing ratio.This transformation makes no fundamental differenceso far as the optimal estimation is concerned, but it doesexhibit clearly how optimal estimation modifies a so-lution to the profile problem based on radiance mea-surements alone.

Fig. 1. SBUV optimal estimation.

The transformation of measurement space in FRAMEmeans that partial derivatives do not appear explicitlyin the FRAME gain but instead appear in the transfor-mation. But still they must be calculated. The usualprocedure calculates perturbation in measurement perunit perturbation in state variable by using a radiativetransfer model. Both HILEV and SLIC do this, HILEVby using a analytic expression based on single scatteringand SLIC by modeling the perturbations in measuredintensity caused by perturbations introduced in thestate variables, with full allowance for multiple scat-tering and for non-Rayleigh atmospheric constitu-ents.

Neither of the above approaches is suited to FRAME:The single-scattering equation is difficult to evaluatebecause it requires a numerical integration, and themultiple-scattering solution is even more difficult to use,consuming excessive computer time. Instead FRAMEuses the reverse approach: The measurement is theperturbed quantity, and the state perturbation is thecomputed quantity. Thus what FRAME computes islike the pseudo-inverse of what HILEV and SLIC com-pute.

A final factor concerns the utilization of total ozone.Neither HILEV nor SLIC is formulated in a way thatpermits iteration on total ozone to drive down the re-sidual in that parameter. Instead, total ozone entersonly peripherally through the prior estimate state vec-tor. By contrast FRAME iterates on both radiances andtotal ozone.

In effect, the SBUV problem invites a view of optimalestimation that is more symmetric than the conven-tional one. Instead of regarding the basic inputs as oneiteratable measurement-based state vector and onefixed prior estimate state vector, we can regard them assimply two state vectors, both based on measurements,although of different kinds. Figure 1 illustrates thisview of the optimal estimation problem. Iteration af-fects both of the state vectors input to the optimal es-timate but neither of their error bars.

15 March 1981 / Vol. 20, No. 6 / APPLIED OPTICS 937

B. Steps in the Inversion ProcessAn examination of the existing approaches to the

SBUV inversion and error analysis problem revealsconsiderable overlap of purpose and diversity of im-plementation. This permits identification of an in-clusive list of discrete tasks, each of which should beexamined separately because each could be done in avariety of ways (or in some cases not at all). Thepresent section presents this list of tasks.

Each inversion scheme begins with routine accep-tance of input information and the initialization of thecalculations, which we shall call steps 1 and 2. The firstcalculations of consequence serve to convert intensityvs wavelength information into total ozone vs atmo-spheric pressure level information. Step 3 associateseach radiance measurement with a characteristicpressure in the atmosphere. This association is some-what arbitrary because every wavelength channelsensed has associated with it a broad natural weightingfunction or vertical profile of contributing signal (see,for example, Fig. 2). But the weighting function is notknown in detail at the outset of the inversion, for it de-pends upon the ozone profile yet to be measured. Onehas to begin the inversion with an assumption." Oneasserts that penetration extends to some characteristicozone optical depth and that the signal observed is ameasure of the amount of atmosphere or pressure downto that optical depth. Step 4 of the inversion process

0.0.'

0.05

0.1

0.6

8

I10

0

100

500

1000

0.0 02 0.4 0.8 0.8 1.0

NORMALIZED WEIGHTING FUNCTION

Fig. 2. Typical weighting functions, standard 250-Dobson unit lowlatitude profile.

then takes account of the wavelength-dependent ozonecross section to convert the apparent ozone opticaldepth down to a pressure point into a total ozone col-umn down to that pressure. After step 4, one has adisjoint set of points representing what the experimentinherently provides: total ozone measurements.

But the output needed is not the cumulative quantitytotal ozone but rather the differential quantity ozonemixing ratio. Step 5 of the inversion begins the process.A candidate profile is proposed to correspond to themeasurements. This first guess is likely to be unsatis-factory in some way and most typically in that the totalozone point values implied may be inconsistent with thegiven total ozone points. If so, iteration (step 6) is de-sirable to converge on an ozone mixing ratio profile thatis consistent with the few total ozone vs pressure pointsobtained from the radiance measurements. Then, if thesolution is not already in a form that can be interpolatedbetween the few data points, it must be made so in anadditional step 7. Extrapolation beyond the datapoints is another step (8) involving imposition ofphysical constraints.

The remaining steps of the inversion process serve tomerge the measurement profile into a prior estimateprofile at high and low altitudes where the experimentinformation runs out. The well-known theory of opti-mal estimation9 provides a method for doing this.Three steps are required to implement the theory.Error in the profile due to measurement error must beestimated and constitutes a step 9. Then the finalprofile and error bars can be computed, constitutingstep 10. From the final profile, it should be possible tocheck total ozone to the ground and predict all themeasured radiances. Any disagreement can be reducedby iteration, a final step 11.

111. Specific Calculations

For performing the steps of the inversion process,there are numerous options, some of which are sug-gested by different techniques already existing. Thissection discusses techniques for accomplishing SBUVinversion tasks, illustrating the discussion with anumber of figures constructed from the inversion of aset of synthetic radiance datal0 generated by Systemsand Applied Sciences Corp. for the SBUV wave-lengths.

A. Definition of Inversion Input and OutputRequirements

The inputs to the SBUV inversion error analysisproblem comprise first the experiment wavelengths Xi,i = 1 to N,\. For the instrument in flight, No is 8, andnominally the eight wavelengths are, in angstrom units,2555, 2735, 2830, 2876, 2922, 2975, 3019, 3058. To eachof these wavelengths, there is associated a radiance Ii,a radiance uncertainty H3 i, the solar flux input Fi, theRayleigh Rayleigh cross section hi, and the ozone crosssection ai.

In addition to the SBUV wavelength data, there isalso a problem geometry, plus a total ozone column


1

0.1

10

100-4 .3 -2

LOG CUMULATIVE OZONE (atm-I~

-1 0

Fig. 3. Input radiance data converted to cumulative ozone andpressure points, standard 250-Dobson unit low latitude profile.

down to the ground, with its error. The SBUV instru-ment views in the nadir direction, so the problem ge-ometry is specified by the zenith angle of solar illumi-nation 00 and the scattering angle 4I(= 1800 - 00). Thetotal ozone information, in atmosphere centimeters, isrepresented by the symbols Xo and X0.

The prior information used in the inversion comprisesa prior estimate ozone mixing ratio profile and its ex-pected error, both based upon the total ozone X0 andits uncertainty X0. Let the profile be represented byits natural logs (Lo)k for k = 1 to Np. Let its error berepresented by a Np X Np covariance matrix Po in unitsof squared logs. Typically Lo would be obtained byinterpolation from a catalog of standard profiles forvarious total ozone values. The P0 would include someirreducible base line amount plus a contribution ac-counting for the uncertainty XO by using the catalogsensitivity (ALo/AXo)k.

The output requirements for the inversion comprisea list of pressures for which ozone mixing ratio and ex-pected error are to be stated, (Pout)k for k = 1 to Np.

B. Repetitively Used ConstantsWithin the subsequent inversion there are several

repetitively used constants. First, there is the Rayleighphase function S = 0.75(1 + cos2 T), which in turn isused in a signal normalizing factor Q0 = 4r cosO/S. Thepropagation directions of light beams are accounted forin a geometric factor g = 1 + (1/cosOo). Finally logs ofoutput pressures ln(pout)k are used.

C. Association of Pressures to Radiance Data PointsA procedure for associating a characteristic pressure

level to an observed radiance can be based upon thenormalized signal received:

Qi= (QoIi)/(Fifi).

Since the effect of ozone upon this signal is yet to be

determined, associating a pressure to it is somewhatarbitrary. We shall do so by imagining a flat Rayleighatmosphere simply cut off at some pressure p (mea-sured in atmospheres). The single-scattered normal-ized signal received would be

QiA) = J Pexp(-g/ip)dp = 1-exp(-gA3p)

The value of p for which Qi (p) equals the measurementQi is

Pi = t-ln(1 -g3iQi)/g/i .

D. Association of Ozone Cumulatives to PressureData Points

To the pressure Pi just selected for characterizing Qi,we must attribute an ozone cumulative. For this pur-pose, we can used the simple power law model used byTH. The dependence of cumulative ozone upon pres-sure level is modeled by X(p) = Cp(1/10), where X istotal ozone down to pressure p, a is a constant that de-termines the profile, and C renders X in units of at-mosphere centimeters. Neither nor C is known apriori, and both must be estimated from the data.

The af can be induced from the definitions

[d nX(p)]-

d lnp

lnX(p) = In - lna,

where r is ozone optical depth down to p, presumablya constant for all measurements. This means that be-tween pressures corresponding to two wavelengthchannels, can be estimated as

A lnpa-= ,

A nawhere A denotes change between wavelength chan-nels.

The C can be eliminated using the approximation PiQi and the relationship derived by TH

Qi = (kiC)-r(1 + a),

where ki = gi. Combining these equations gives

C'ti pi

Returning r and C to the equation for X(p) gives forpressure point Pi a cumulative ozone value

1 Xi= [Pr(i + c(r)]'/'

The procedure results in a set of data points that can beplotted in the manner represented by Fig. 3.

The primary source of inaccuracy in Fig. 3 is deviationfrom the assumptions of flat atmosphere, single scat-tering, and purely Rayleigh constituents. There is somecurvature, some multiple scattering, and some corrup-tion due to atmospheric aerosols plus possibly absorp-tion due to minor constituents like NO2. Given a morecomplete code like the radiative transfer modulein SLIC,8 it is possible to perform extensive simulationsand parameterize the required correction to Q andhence to Fig. 3.


S

-X S X

l l l

l l lF I I

i iI

E. Introduction of Ozone Mixing Ratio

The ozone mixing ratio in units of atmosphere cen-timeters per atmosphere is defined as

n= [dX(p)]/(dp)

and according to the TH model can be evaluated as n= X/(p a). The natural logs of n and p are thus linearlyrelated:

Inn = InX - lnp - lna

= InC + P -lnp - Ina.a

Applying this equation over the intervals between thepoints on Fig. 3 results in the segmented model shownin Fig. 4. This model is not yet satisfactory because itdoes not give a well-defined value of n at the pressuredata points. To get one, we need to construct an ap-propriate average of the values on the two sides of thediscontinuity. Since n = X/po, we average the inversesof a in the two regions, with weights designed to favorthe more nearby of the two data points neighboring theone for Xi. The idea is illustrated by Fig. 5.

F. Iteration for Ozone Cumulatives

In effect, the operation that defines mixing ratios niat the pressure data points is equivalent to defining anew o- at each data point. These new a's in turn implynew X's, which could differ from the original X's bysome percentage larger than the measurement error, inwhich case an iteration would be in order. To allow forthis possibility, an iteration has been included in theFRAME code. However, problems exercised so farhave not required this iteration.

G. Provision for Interpolation

The result of step 6 is a set of data points that repre-sent the ozone mixing ratio at pressure points corre-sponding to the radiance measurements at a handful ofwavelengths (Fig. 6). What is really desired is an ar-

0.1

I

a4

c:8I

210

100

0.75

Fig. 4. Ozone

1.0 125

LOG MIXING RATIO Iatmn-=,atm)

1.50

POINTFPOM xi SLOPE - WEIGHTED-1 AVERAGE OF STRAIGHT

= /- LINE SLOPES

POINTI FROM A+

TOTAL OZONE 1g10 scaI)

Fig. 5. Ozone mixing ratio at data points.

0.1

i

I

I

DS0

10

100 =

035 1.0

0p

S

1.25

LOG MIXING RATIO Natm-c/atm

1.50 1.75

Fig. 6. Ozone mixing ratio data points, standard 250-Dobson unitlow latitude profile.

bitrary number of output points. So there is a need tointerpolate between the few points actually available.This attribute is achieved in FRAME by fitting theavailable points with a smooth curve.

There are many ways to fit a curve through the datapoints produced by step 6. For the sake of functionalsimplicity, FRAME uses a polynomial fit approach. Theorigin Pref is at the first pressure point Pi, where thecurve converges to linear. Such an approach works wellif the dynamic range of variables is not excessive, anduse of logs instead of linear variables provides thisproperty.

A matrix of powers in log pressure is defined as

Ji = (lnpi - lnpref)j1 1 ,

where index i runs over the number of wavelengths No,and index j for coefficient a1 runs over the order ofpolynomial Na. The Na is chosen as the minimumdegree for which the polynomial fit obtained for

E ajJj= n i J ~ p il

1.75 has errors

mixing ratio between data points, standard 250-Dobson unit low latitude profile.

In i I


I

-1 i

'1� -V-1�

__1

)a

I I I 1.

l

1

0.01

0.1

10

1c: I

I

E

w

F

. 10

100

1000 G0.5 1.0

SB INOMIO

- - - -- - - x - R Eg - N

>~ _ SBU INFORMADATA~~~~~~~OYO I A LRE IO

1.5 2.0

LOG MIXING RATIO atm-cmlatm)

Fig. 7. Polynomial fit to ozone mixing ratio data points, standa250-Dobson unit low latitude profile.

that are small compared with radiance measuremeznoise, expressed as a log. Considering all the channelthere is a mean square fitting error

set to a straight line function of lnp, just continuing theslope that exists at p1-

For the region below the last wavelength data point(i = N,\), L can be constrained to integrate to the correcttotal ozone XO at the ground. This can be done by re-placing the entries in L over that region with a curvethat meets smoothly with the last data point. To dothis, there are four constraints to be satisfied:

Xoatp = 1,

X i, ni, ni atp =pi,

where ni and n' are first and second derivatives of Xwith respect top. The fit can be achieved by a simplethird-order polynomial in Ap = p - pi. The third-order coefficient n' has to be chosen so that X = X0 atthe Ap where no more profile is available. This is eitherat the ground (Ap = 1 - pi or above the ground if themixing ratio falls to zero above ground. In either case,the needed entries for L are then

= Lk=ln ni+n'AP+np )

with Ap = Pk - pi. Figure 8 shows the results.

1. Assessment of Error

The profile produced by step 7 clearly has error as-sociated with it. The errors may be small in the ex-

2.5 periment information region but must become verylarge away from this region. This behavior can be

rd demonstrated by calculating an Np X Np covariancematrix P. This must involve the radiance measurementerrors I) and in the case of FRAME also involves thepolynomial fitting error mR.

I ,

s,

1 2mR = - Ei

that is an additional contributor to the measurementnoise matrix so far as the inversion is concerned.

Once a satifactory polynomial fit is settled upon, itcan be used to generate mixing ratios at output pres-sures. For this purpose one defines the matrix of out-put polynomial expansion entries

(J..t)k = [n(pout)k - lnPref]j-'

from which the output vector of the measured log ozonemixing ratio is L = Jouta. This is shown in Fig. 7.

H. Provision for Extrapolation

Usually the result of the polynomial expansion is acurve that appears satisfactory in the neighborhood ofthe original data points but clearly becomes meaninglessat very high and very low altitudes where there is nomeasurement information, as indicated by Fig. 7.These defects can be bad enough to show up even in theultimate optimal estimate profile. But they can bemitigated by imposing some physical constraints.

For the region above the first wavelength data point(i = 1), it is reasonable to require that the TH modelapply exactly. That means that above P1 , L should be

0.1

E

SO

I

t

10

100

LINEAR RE ON\ = ~~~~~~~~~~~LOG VAR A LE

POLYNOMIAL,vLOG VARIABLE

Z_

_U EGON L.II R E __ _ =____ _ _ _. _ _ __

_ _ _ _ _ .4 _ _ _ _ _ _ _ _ i _ _ _ _

1000 0.0 0.5 1.0

LOG MIXING RATIO (atm-cm/Atm)

1.5 2.0

Fig. 8. Extrapolated ozone mixing ratio curve, standard 250-Dobsonunit low latitude profile.

15 March 1981 /Vol. 20, No. 6 / APPLIED OPTICS 941

0.1

I

I

10

100

10000.0

i \' | ! j \ ~~~~~~~LINEAR RO ION| s g | ~~~~~~~~~~LOG VAR IAJLE

H-1~~~~~~~~~~~~~~~~~P 1AL

l . ~~~~~~~~~~~~~~LOG VARIABLE_

I~ ~ ~~~~~~~~~UI EIN LIEA AIAL

0.6 1.0

LOG MIXING RATIO Iatmm/Itm,

1.5

Fig. 9. Measurement curve with error bars, standard 250-Dobsonunit low latitude profile.

To calculate this covariance matrix, we require apartial derivative matrix that relates an excursion inradiance to an excursion in state. FRAME does this bya perturbation approach. It perturbs Ii by 61i and re-peats the polynomial fit to find the change baj andforms

Gji = (aj)Abh)-

Given G (and J) we can construct the contents of thecovariance matrix P. The radiance measurement noiseis a diagonal matrix R in radiance space, with entries(Ro)ii = (1)2. The polynomial fit error is a scalar mR

in the mixing ratio space to which polynomial fitting isapplied. Projecting these two contributors into theoutput space gives

P= JOutGROGTJUL+ mRJout(JTJ)-1JuT

As expected, the diagonal entries of P for pressures farfrom the experiment information region become large,the growth being caused by the growth in the entries ofJout (see Fig. 9). This growth is appropriate eventhough physical constraints have been applied to theupper and lower ends of the curve, displacing the profilegenerated from the polynomial. These constraints donot really create information and so do not reduce errorbars.

J. Estimation of Final Profile and Error Bars

The errors manifest outside the experiment infor-mation region after step 9 clearly support the desir-ability of bringing in prior information to fill in theseregions. Optimal estimation theory provides a basis fordoing this. The basic idea is to form an estimate L1 thatminimizes a quadratic cost function that involves the

prior estimate Lo and its covariance Po as well as themeasurement L and its covariance P. The cost functionis

COST(L 1 ) = (L 1 - Lo) TPl(L 1 - Lo)

+ (L1 - L)TP-1(L - L),

and the estimate it implies is a linear update L1 = Lo +K(L - LO) with covariance P1 = (I - K)PO, where thegain is K = Po(Po + P)-'.

The estimate for the log ozone mixing ratio providesthe mixing ratio through (nl)k = exp(L1)k. The errorbars for this estimate are formed from the correspond-ing diagonal entry of the covariance matrix P1 . The logerror is ()k = (Pi)k/, and the upper and lower boundsare (nl,)k = (nl)k exp(±E)k. Figure 10 shows thecharacter of the result.

K. Iteration on MeasurementThe profile predicted by step 10 can be subjected to

a test for consistency with the original input data. Onemay ask if it really reproduces the input radiances Ii andtotal ozone X0, and, to the extent that it does not, onemay adjust the profile. In principle, one may iteratesuch a procedure until it converges.

The two adjustments required are alike to the extentof being based upon the discrepancies between esti-mates and measurements. From the optimal estimateprofile n1, one must compute the predicted radiances(IUA and predicted total ozone (X1)0. Compared withthe measurements, these imply discrepancies.

AM = Ii - (I1)i,AX0 = Xo - (X)o.

0.1

5

c:3:

l0

100

1000

0.0 0.5 1.0 1.5 2.0

LOG MIXING RATIO Itmcm/-hm)

Fig. 10. Optimal estimate profile with error bars, standard 250-Dobson unit low latitude profile.


The two adjustments are different, however, in wherethey contribute to an update for the profile. In the caseof the I's, it is the measurement profile that can con-veniently change. The partial derivative matrix Gpermits an update of the polynomial coefficient vectorto a' = a + GAl. This is then used in a repeat genera-tion of a measurement profile. By contrast, in the caseof the AX0 , it is the prior estimate profile that is avail-able for a change. The prior estimate comes from in-terpolation on a catalog of standard profiles for varioustotal ozone values. One can update the interpolation,seeking a profile not for X0 but for X0 = X0 + AX0 .

There is also an opportunity to speed the convergenceconsiderably. Note that on the one hand, the radiancemeasurements are sensitive to the whole atmosphere,although they tend to control the inverted profile onlyabove the ozone peak. On the other hand, the totalozone mostly depends upon the atmosphere below theozone peak and controls the inversion there. Thismeans adjustment for radiance errors will not affecttotal ozone very much, but an adjustment for total ozoneerror will strongly affect predicted radiances. Thismeans there is a logical order in which to do things. Thetotal ozone adjustment should be done first.

In fact, the total ozone adjustment can be done on apoint-by-point basis, starting at the top of the atmo-sphere. At any point, there is an unaccumulated(AXO)k that can be used to adjust the interpolation todefine the next prior estimate point (no)k,+.

It is recognized that feedback of intensity and totalozone measurement residuals through iteration requiressubstantial numerical integration with small steps.Nevertheless, it is apparently a necessary effort wellspent. Problems exercised to date do require five to teniterations for satisfactory results. Without iteration,errors in excess of several percent are typical on any andall BUV channels as well as errors in excess of 10% ontotal ozone.

IV. Sample Results

A computer code called FRAME was written from theabove inversion analysis and exercised in a variety ofinversion examples.

The exercises all used prior estimate profiles drawnfrom a set of standard ozone profiles,10"11 which wereproduced by the Nimbus 4 Ozone Processing Team.These profiles are based upon ozonesonde observationsbelow 10 mb and rocket measurements at higher alti-tudes. Profiles are defined for total ozone incrementsof 50 Dobson units and for latitude ranges of 0-25°,25-65°, and 65-90°.

Models for the measurement uncertainties wereadopted. The measurement noise standard deviationwas set at 1% and that for the total ozone at 3%. Thea priori ozone mixing ratio uncertainty was set at 15%throughout the atmosphere, with off-diagonal elementsof the a priori covariance matrix set to zero. Theseinputs serve to illustrate the FRAME inversion process,although they may be subject to debate in their ownright. The a priori covariance matrix, especially, maybe overly simple. Belmont et al. 12 have calculated the

residual ozone variances at the meteorological levels forozonesonde observations based upon profiles whichhave been empirically determined from SBUV totalozone observations (as well as temperature and poten-tial vorticity). It appears that the variance of ozone isreduced approximately 50% between 10 and 200 mb andbetween 30° and 600 latitude using total ozone infor-mation. Moreover the residual standard deviation ofozone is -10% between 20 and 70 mbar and -30% belowthis altitude. Between 10 and 1 mbar the standardrocket profiles presented by Krueger and Minzner13suggest a residual variability of -15%, but this increasesto 50% at 0.1 mbar.

In each test case those wavelength channels with amultiple-scattering contribution of 10% or more werenot used in the inversion. There are two reasons fordropping these channels. First the multiple-scatteringcontribution in real measurements is uncertain becauseof uncertainties in surface albedo, cloud cover, andaerosol loading. Second, the information in thesechannels is smeared over a large altitude range reachingthe surface in some cases. A similar technique is usedin the HILEV procedure.

Note that all the output profiles are stated in con-ventional linear mass mixing ratio units of microgramsper gram (rather than log atmosphere centimeters peratmosphere mixing ratio units that are convenientwithin the analysis).

The first three inverted profiles represent FRAMEconsistency checks. In each of these cases, measure-ment Q's were generated from a standard a priori pro-file representing different total ozone columns. Theinverted profile should be close to the standard and havethe proper residuals. The results in Figs. 11-13 showthat FRAME adequately converts the Q's computed forthe profile back into the profile itself (to within the ac-curacy of the polynomial smoothing of high frequencyirregularities). For the 200-Dobson unit case (Fig.11)both Q and total ozone residuals are well within thespecified 1 and 3% precision of the respective mea-surements. For this profile the solar zenith angle was60.30, and the latitude was 0. FRAME chose a third-order polynomial to fit the data and converged in threeiterations. The 350-Dobson unit profile (Fig. 12) is alsofor solar zenith 60.3°. Seven wavelength channels areused. For this midlatitude profile FRAME chooses afourth-order polynomial and converges in six iterations.Figure 13 represents a high latitude, 500-Dobson unitprofile, again with 60.3° solar zenith. FRAME choosesa fourth-order polynomial and converges with four it-erations.

Figure 14 shows the result when a poor a priori profileis deliberately chosen. The convergence on the trueQ-source profile in the experimental information regionshows that the procedure works properly even when thea priori profile is far from being a good guess. In thiscase the a priori is a high latitude (730) case, while theQ's are those for a low latitude profile. The solar zenithangle is 0 in each case, and both a priori and soughtprofiles sum to 200 Dobson units. FRAME selected asecond-order polynomial to fit the five data points and


0 - FRAME

3 I | I | s 1 ~~~~~~~~~~~~% T TAIL OZ1

1~~~~~~~~~~~~0 X 0APR I0;I

0 4 8 12 16 20 2

MASS MIXING RATIO ( gm,/gm)

Fig. 11. 200-Dobson unit profile inversion.

.711

X X O IF~~~~~~~~0 RAMAEj

~~~~~~~~~~~~~~~~~~~~~~~~~.10

.3

30 4 8 -12 122

50 RE , O UMSSMIIN RAIOhAZmgm

701;- / -- | - t 1

-/ -I____L J Ii

4 8 12 16 20

MASS MIXING RATIO p gm/gm)


0.5

0.7

2

ii

A

I

3

10

20

30

70

100

03

0.5

10.7

3

I

0

- 5

7

10

20

30

70O

10924

1- __ .1 - ~ ___ I__ ___ _hz1 _____ 1 ____ ___ ____ _ ___~

1 0 F RAME

1 ! IA PRIO ,I

1-- __ 1' __ 1 ___ i I - 14 _t f { I

-0.4 -0.

~~~~lI ~~~~~~~-0.00.22

0.0

0 4 812 16 20 2,

MASS MIXING RATIO ( .gm/gm


'8 0 - FRAME

-~~~~ - OSO~~~~~URO 'EPRO LE

% RESID~

0.05

-0.30.220 3

% T TAL OZ-RESIDUAL___

-0.24

0 4 8 12 16 20 24

MASS MIXING RATIO p gm/gm)

Fig. 14. 200-Dobson unit inversion with poor a priori profile,


0.3

0.5

0.7

2

6

10

S11

I

I

20

30

70

100

0

0

I

0

11:

1000

-^ I . . . . .I

1'5\ 1 1 1 1 1 § I I s I

1,'-17 X

03

'A

24 ~ 24

I

I

!4

03

0.5

0.7

9?

I

-IMo

3

5

10

20

30

50

70

1060 4 8 12 16 20 24

MASS MIXING RATIO Iggm/gm)

Fig. 15. Inversion of profile based on in situ balloon samples (CantinIsland).

lengths to acceptable values. At the short wavelengthsthe Q residuals are unacceptably high, a result thatoccurs because the a priori profile uncertainty of 15%is much smaller than the difference between profiles ataltitudes above 1 mbar. As a result the solution con-verges very slowly, and the first channel residual dropsto 3.83% from 21% only after twenty iterations. If thea priori profile uncertainty is relaxed to 50%, the resultis the profile in Fig. 17. Here the short wavelength re-siduals are acceptable, and the implication for inversionis that the a priori uncertainty should generally be madelarge enough to embrace the actual measurement pro-file.

V. Conclusions

A mathematical procedure for inverting SBUVmeasurements to obtain ozone profiles has been devisedwhich is practical for real time applications. Prior tothis development, it had been assumed that profile erroranalysis would be burdensome and would thereforeneed to be separated from the basic inversion process.It was assumed that information pertinent to erroranalysis would need to be precomputed and stored fortable look-up. For the precomputation, some type of

03

0.5

0.

converged with three iterations. The combined effectsof total ozone constraint and polynomial structure areclearly displayed in this example. After fitting themeasurement data points in the 1-5-mbar region, thesecond-order polynomial is extended nearly vertically,which causes the accumulation of excessive total ozone.The excess is reduced by undershooting the a prioriprofile below the 40-mbar level.

Figures 15-17 represent inversions based upon pro-files induced from in situ ozone samples taken by bal-loon.10 Q's were generated from the resulting profilesand inserted in FRAME along with an a priori profile forthe same latitude and total ozone. In each case the apriori profile and the sought balloon profile were con-siderably different. Figure 15 represents a CantinIsland balloon case of 229 Dobson units and 00 solarzenith. FRAME chose a second-order polynomial andconverged with four iterations. The fit is quite satis-factory. Figure 16 represents a Payern balloon case of343 Dobson units and 60.30 solar zenith. The invertedprofile illustrates two important phenomena. FRAMEis unable to recreate the high frequency structure of thesought profile in the 5-30-mbar pressure range becausesuch structure is smoothed in the Q integrals andtherefore not wholly present in the Q data. However,FRAME does attempt to reproduce the structure and indoing so reduces the Q residuals for the longer wave-

Z

I:IIEI2I20I

3

10

50 I- s X l l l l T iTA~II ___ ~~~~~~~~~~~~~0.09

30 S -< I .48

___:+_L _L._L_ ttj __ - ' F.02

70 - 1 __ -_. j--- =0 4 8 12 16 20 24

MASS MIXING RATIO Ip gm/gm)

Fig. 16. Inversion of profile based on in situ balloon samples(Payern).


0.3-4H - -

0.7

F 6~~~~~~~~~~~~~~~~~~~RO

CC-~SOURCE PROFILE2 L - ~ -

3-7 --------__

10~~~-- 1

- F--F ~~~~~~~~~~~% RESIDF 7 - ~~~~~~~~~~~~~1 68

F - 0.41-0.13

30~~~~~~~~~~~~~~~~~~~~~05

30 -C--5-m---'-------4- - -- -0.45~~%TR~ST

DLTUTALOZ

70~~~~~~~~ __ RSGA

100 -0 4 a 12 19 20 24

MASS MIXING RATIO (Apgm/gm)

Fig. 17. Inversion of profile based on in situ balloon sample with 50%a priori variance.

benchmark computer code, however long-running,would be used. It has turned out, however, to be pos-sible to formulate an algorithmic approach to theproblem, which creates error bars in real time along withthe basic inversion. This is highly desirable because itpermits both aspects of the problem to make full use ofall specific measurements, experiment conditions, andany prior information available. The benchmark playsa role only in off-line validation by comparison exercisesand study of perturbing effects not included in the in-version, rather than in creation of a table look-up erroranalysis.

Because FRAME allows for efficient iteration, it candeal with the inherent nonlinearities in the inversionproblem and can converge to a solution with acceptableresiduals.

This work was funded by the National Aeronauticsand Space Administration through contract NAS5-22415 with the Massachusetts Institute of Technologyand through subcontract to the Charles Stark DraperLaboratory, Inc.

The authors wish to thank the Systems and AppliedSciences Corp. for supplying the HIRAD tape from whichthe multiple-scattering a priori profiles and radiance

test cases were obtained. We would also like to thankK. Klenk of that organization for patiently answeringour many questions.

References1. T. H. Maugh, Science 206, 1167 (1979).2. T. H. Maugh, Science 207, 395 (1980).3. C. L. Mateer, J. Atmos. Sci. 22, 370 (1965).4. D. N. Yarger, J. Appl. Meteorol. 9, 921 (1970).5. C. L. Mateer, in Inversion Methods in Atmospheric Remote

Sounding, A. Deepak, Ed. (Academic Press, New York, 1977).6. T. Aruge and T. Igarashi, Appl. Opt. 15, 261 (1976).7. R. W. L. Thomas and A. C. Holland, Appl. Opt. 16, 2581

(1977).8. H. L. Malchow and C. K. Whitney, in Ref. 5.9. A. Gelb, Applied Optimal Estimation (MIT Press, Cambridge,

Mass. 1974).10. K. F. Klenk and S. L. Taylor, Systems and Applied Sciences Corp.

Report R-SAB4/7903 (1979).11. E. Hilsenrath, P. J. Dunn, asd C. L. Mateer, "Standard Ozone

Profiles from Balloon and Rocket Data for Satellite and Theo-retical Model Input," paper presented at IAGA/IAMAP JointAssembly, Seattle Wash. (Aug. 1977).

12. A. D. Belmont, personal communication to SBUV Nimbus Ex-perimental Team (1979).

13. A. J. Krueger and R. A. Minzner, J. Geophys. Res. 81, 4477(1976).

14. B. J. Conrath, in Ref. 5.


ozone profiling from backscattered uv radiance measurements: a new procedure

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