Microwindow Selection for High-Spectral-Resolution Sounders

Microwindow selection forhigh-spectral-resolution sounders

Anu Dudhia, Victoria L. Jay, and Clive D. Rodgers

The recent development of satellite instruments that obtain spectrally resolved measurements of theatmosphere has highlighted the problem of how to determine the best subsets, or microwindows, of suchspectra for retrievals of temperature and composition. A technique is described that maximizes theinformation content �or some other figure of merit� based on the modeling of the propagation of systematicas well as random error terms through the retrieval process. Apart from selecting microwindows, thistechnique can also prioritize existing microwindows for different circumstances and provides a full erroranalysis of the retrieval. A practical application is demonstrated for the Michelson Interferometer forPassive Atmospheric Sounding limb-viewing interferometer, but the technique is equally applicable tonadir-viewing instruments. © 2002 Optical Society of America

OCIS codes: 000.4430, 010.1280, 280.0280, 300.6340.

1. Introduction

Several current generation of Earth-observing satel-lite instruments are designed to make continuousmeasurements of atmospheric IR spectra using eithergrating spectrometers �Atmospheric Infrared Sound-er1� or interferometers �Interferometric Monitor forGreenhouse Gases,2 Michelson Interferometer forPassive Atmospheric Sounding3 �MIPAS�, InfraredAtmospheric Sounding Interferometer,4 and Tropo-spheric Emission Spectrometer5�. Such spectra typ-ically provide thousands of radiance measurementsevery second, too many to incorporate into a real-timeretrieval scheme, given present computing speeds.Consequently, attention has been focused on methodsfor the determination of the optimum subsets of suchspectra �microwindows� that contain most of the po-tential information.

One approach is to simulate the propagation ofrandom noise through the retrieval and select mea-surements that maximize the information content ordegrees of freedom of the signal,6,7 or that best satisfy

When this research was performed, A. Dudhia �[email protected]�, V. L. Jay, and C. D. Rodgers were with Atmospheric,Oceanic, and Planetary Physics, Department of Physics, OxfordUniversity, Parks Road, Oxford OX1 3PU, United Kingdom. V. L.Jay is now with the Rutherford Appleton Laboratory, Chilton,Didcot, OX11 0QX, United Kingdom.

Received 6 August 2001; revised manuscript received 30 Janu-ary 2002.

0003-6935�02�183665-09$15.00�0© 2002 Optical Society of America

other criteria.8 Although this is reasonable if theretrieval errors are predominantly due to randommeasurement errors and�or errors in the a prioriestimate, it does not allow for other �systematic�sources of error associated with unretrieved param-eters. For example, it does not allow for the uncer-tainties in the modeling of the concentrations of thevarious contaminant species, that is, other moleculeswhose spectral signatures overlap those of the re-trieved species but whose concentrations are notthemselves retrieved.

An alternative approach addresses this problem byselecting microwindows that minimize the total er-ror, including both random and systematic compo-nents.9,10 However, this is achieved only through anapproximation of the profile retrieval as a set of in-dependent, single-layer retrievals. As a result, theeffect of interlevel correlations in the actual retrievalis ignored. There is also the practical difficulty ofthe consolidation of the microwindows derived inde-pendently for each profile level into single microwin-dows applicable over the whole profile.

In this paper a scheme is presented that selectsmicrowindows by simulating a full-profile retrieval,including the propagation of systematic error terms.As such, it can be considered either as an extension ofthe first �random error� method to include systematicerrors, or as a multilayer version of the second�single-layer� method.

As well as selecting microwindows, this scheme
also allows existing microwindows to be ordered ac-
20 June 2002 � Vol. 41, No. 18 � APPLIED OPTICS 3665

cording to different criteria and provides a completeerror analysis of the resulting retrieval products.

As a practical demonstration, microwindows areselected for the retrieval of a methane profile fromMIPAS measurements. Although MIPAS is a limb-viewing instrument, the technique is equally appli-cable to nadir-viewing geometries.

2. Theory

A. Optimal Estimation

Optimal estimation theory �e.g., Ref. 11� provides alinearized form for an estimate x of the state vector�atmospheric profile� that is based on a combinationof a set of measurements y and a prior estimate a ofthe state

x � Gy � �I � GK�a, (1)

where K is the Jacobian �or weighting function� ma-trix �Kjl � �yj��xl� and G is the gain �or contributionfunction� matrix given by

G � SaKT�Sy � KSaKT��1. (2)

Sa is the covariance of a about the true state, and Sythe covariance of y about the perfect measurementsthat would arise from this true state.

We obtain the covariance of the retrieval error bytaking the covariance of Eq. �1� about the true state:

Sx � GSyGT � �I � GK�Sa�I � GK�T (3)

� Sa � SaKT�Sy � KSaKT��1KSa (4)

� �I � GK�Sa. (5)

For a set of m measurements, Eq. �2� requires theinversion of a matrix of size m � m, which becomesimpractical for large m. If Sy on its own can beinverted simply, a standard approach is to reformu-late Eq. �2� as

G � �KTSy�1K � Sa

�1��1KTSy�1. (6)

However, if Sy is diagonal, an even more convenientsolution is available, outlined in the next section.

B. Sequential Estimation

If the measurement errors are uncorrelated, i.e., Sy isdiagonal �variances Sjj j

2�, Eqs. �1�–�5� can beapplied to the individual measurements yj, j �1 . . . m sequentially:

x j � g jyj � �I � g jkjT�a j, (7)

g j � S aj kj�j

2 � k jTS a

j k j��1, (8)

S xj � �I � g jkj

T�S aj , (9)

where kjT is the row of the Jacobian matrix associated

with yj; superscript j denotes quantities progressivelymodified for j � 1 . . . m measurements, and a j, Sa

j x j�1, Sx

j�1. The original, independent a priori infor-mation is used to set a1, Sa

1, but thereafter the priorestimate also incorporates information from previous

measurements. We will use the terms “a priori” and“previous” estimates to distinguish the two cases.

This replaces the inversion of the m � m matrix inEq. �2� with m inversions of a scalar quantity in Eq.�8�. It is, of course, possible to apply sequential es-timation to larger subsets of y than to individualmeasurements, although less computationally effi-cient. The replacement of the variance j

2 with theequivalent subset of the full covariance matrix Syallows for error correlations within each subset. InSubsection 2.G a hybrid approach is described thatdeals with apodized measurements, initially treatingthem as single uncorrelated measurements but in-cluding the correlations when established as part of amicrowindow.

C. Forward-Model Errors

In reality, the matrix Sy represents not only the co-variance S� of the difference between the real and theperfect measurements, but also the covariance Sf ofthe difference between the computational forwardmodel used in the retrieval and the physical radiativetransfer process

Sy � S� � Sf. (10)

While the measurement component of the error maybe dominated by random noise �S� � diagonal�,forward-model errors, e.g., the effect of underestimat-ing a contaminant, tend to be correlated. For con-venience, S� will be used to refer to just the random orlocally correlated �e.g., apodized noise� component ofthe measurement error, whereas more widely corre-lated components �e.g., calibration errors� will be con-sidered part of Sf.

Practical retrieval schemes, for various reasons,often assume Sy S� so that the weight assigned toeach measurement in the gain matrix �Eq. �2�� de-pends only on the random component. Even if Sf isnot negligible, this substitution can still be usedwithin the gain matrix �although, technically, theretrieval is no longer optimal estimation12�; however,the true covariance of such a retrieval �obtained withsubstitution of Eq. �10� in Eq. �3�� has an additionalterm GSfG

T on the right hand side of Eq. �5�. Whilemathematically simple, this extra term, representingthe propagation of correlated measurements errorsthrough the retrieval, can be too large a matrix �m �m� to be evaluated directly, and too widely correlatedto be decomposed into block diagonals and evaluatedsequentially. In the next section, Sf is mapped intoa form that can be propagated through the sequentialestimator.

D. Systematic Errors

Systematic errors are defined here as errors that arecorrelated between elements of the prior estimate aand the measurement vector y, and that therefore arenot described by either of the associated covariancematrices Sa, Sy. Normally such errors are neglectedeither because the prior estimate is assumed to beindependent a priori information, as in optimal esti-

3666 APPLIED OPTICS � Vol. 41, No. 18 � 20 June 2002

mation, or because measurement errors are assumedrandom, as in sequential estimation. However, forthe purposes of microwindow selection, we require ascheme for handling systematic errors within theframework of sequential estimation.

In addition to the random error, a set of m mea-surements y will also contain error components yi

due to a variety of error sources i. The magnitude ofthe error vector yi corresponds to a 1 perturbationin the error source, and the shape is characteristic ofthe particular error source. Assuming that all ofthese errors are independent, the total error covari-ance of y �see Eq. �10�� becomes

Sytot � Sy

rnd � �i

E�� yi�� yi�T� (11)

� Syrnd � �

iSy

i , (12)

where E�. . .� denotes expectation value. In practice,each vector yi can be computed as the differencebetween the nominal set of measurements and a setwhere the error source is perturbed by 1, and thecorresponding covariance is then given directly by� yi�� yi�T. Thus the correlation information in them � m total covariance can be decomposed into a setof error vectors yi, each of length m.

Equation �1� describes the mapping of the mea-surement vector into the retrieval, so the measure-ment error vectors yi will map into correspondingretrieval error vectors xi in the same way:

xi � G yi � �I � GK� ai, (13)

where ai allows for any equivalent systematic errorcomponents in the prior estimate.

The retrieval total error covariance Sxtot is then

given by

Sxtot � Sx

rnd � Sxsys, (14)

where

Sxrnd � �I � GK�Sa

rnd, (15)

Sxsys � �

iE�� xi�� xi�T� � �

iSx

i . (16)

The same total error covariance could be obtainedif all of the measurements are incorporated simulta-neously with the full-error covariance matrix for Sy inEq. �3� �e.g., by adding GSfG

T�, but the advantage ofpropagating the systematic errors as a set of vectorsrather than as a covariance matrix is that no infor-mation is lost when sequential estimation is applied.

As noted for the single-layer approach9 Eq. �16�strictly only applies to so-called forward model pa-rameter errors, which are expected to introduce zerobias when averaged over a large number of retrievals.Errors in the formulation of the forward model itself,typically due to neglected physics, may lead to knownand persistent retrieval biases. Consequently theseare not independent of each other, and all such terms

xi should be first added together, and the expecta-tion value should be calculated for their sum.

To put the two previous methods of microwindowselection into the context of Eqs. �14�–�16�, the ran-dom error approach considers only Eq. �15�, while thesingle-layer approach treats x as a scalar so the Sxcovariance matrices become simple variances �actu-ally 2 � 2 matrices, since a background continuum-like term is retrieved for each microwindow as well asthe target species�.

E. Figure of Merit

To compare covariances it is necessary to define ascalar quantity, the Figure of Merit, for the retrieval.

The Shannon information content is one measurethat can be used. The information content H, mea-sured in bits, is defined by

H � �log2�Fx�Fa�, (17)

where Fx � �Sxtot�, the determinant of the retrieval

error covariance, and Fa � �Satot�, the determinant of

the a priori error covariance. For example, if thecovariance matrices were diagonal, 1 bit of informa-tion would correspond to a factor-4 reduction in thevariance of one element �or a factor-2 reduction in �.If the retrieval is regarded as a sequential combina-tion of j measurements �either singly or as microwin-dows�, then an increment �Hj can be assigned toeach:

�Hj � �log2�Fj�Fj�1�, (18)

where Fj�1, Fj are the determinants of the retrievalcovariance before and after the jth measurements areincluded, and F0 Fa.

Although information content seems a naturalchoice, there are other alternatives. The formula-tion of Eqs. �17� and �18� can still be used, but the�scalar� function F can be redefined according to pri-orities. Some examples are listed below.

Weighting against systematic errors

F � �Srnd � �Ssys�, (19)

where � � 1. This favors measurements that intro-duce relatively low systematic errors compared withrandom errors.

Weighting toward an accuracy requirement

F � �Stot � Sreq�, (20)

where the requirement covariance Sreq contains tar-get accuracies for each profile level. This favorsmeasurements that improve the retrieval where therequirement accuracy is not yet met.

Weighting to minimize variances

F � �k

Skktot, (21)

equivalent to evaluating the determinant ignoringoff-diagonal elements. This favors measurementsthat maximize the accuracy of each individually re-


trieved point, which might be preferred if the enduser cannot use the covariance information.

Weighting with a CPU cost

F � �Stot�Cn, (22)

where C is the estimate of the processing time re-quired and n is some power controlling its relativeweight. This favors measurements that producemore information for a given factor increase in CPUtime. An example of the influence of CPU cost onmicrowindow selection is given in Subsection 4.C.

F. Selecting Individual Measurements

Having established the theoretical background, wenow detail the procedure for selecting individualmeasurements for a retrieval.

First, error vectors yi are precomputed for everyrelevant source i for the entire measurement domain.Also required are the Jacobians kT for each measure-ment: Each is a row vector with length matchingthat of the retrieved state vector x.

A suitable a priori covariance Sa is established,corresponding to any initial knowledge of the ele-ments of x to be retrieved, such as the covariance ofa numerical model or climatological uncertainties.For retrievals that use no explicit a priori informa-tion, diagonal elements of Sa can be set to values thatare large values compared with the expected retrievalaccuracy. Assuming that the a priori estimate con-tains no correlations with measurement errors, ai

can be initialized to zero for each error source.Then, for each measurement y, the scalar versions

of Eqs. �13� and �14� are computed:

g � Sarndk�y

2 � kTSarndk��1, (23)

Sxrnd � �I � gkT�Sa

rnd, (24)

xi � g yi � �I � gkT� ai, (25)

Sxtot � Sx

rnd � �i

� xi�� xi�T. (26)

Here it is assumed that the retrieval weights mea-surements according to noise variance y

2. Notethat the gain matrix g for a single measurement is acolumn vector, and the inverted quantity is a scalar,so the above operations can be performed rapidly fora large number of measurements, provided that yiand kT are easily obtained.

The Figure of Merit is computed for each measure-ment, and the best measurement is then selected.The values Sx

rnd, xi are copied to Sarnd, ai �as in

sequential estimation� so that the retrieval incorpo-rating the best measurement now forms the previousestimate for the selection of the next measurement.This process is then repeated until as many measure-ments are selected as required, or until the additionof any further measurements reduces the Figure ofMerit.

G. Growing Microwindows

The domain of available measurements for a limbsounder can be regarded as a two-dimensional grid,with each measurement defined by its position alongthe spectral and tangent altitude axes. For nadir-sounders there is just the spectral axis.

Rather than use isolated measurements, usingclusters of adjacent measurements �microwindows�usually allows improved computationally efficiencyin the forward model.13 For example, the use of thesame spectral grid point for successive tangent alti-tudes allows a reduction in the average number ofindependent path elements per measurement re-quired to model the atmosphere at higher levels andrepresent the field-of-view convolution, while the useof adjacent spectral grid points at any one tangentaltitude allows a reduction in the average number ofspectral lines modeled and monochromatic calcula-tions required to represent the apodized instrumentline shape. In the retrieval domain, the use of mi-crowindows also allows spectrally smooth but un-known continuumlike terms to be retrieved and�orremoved within each microwindow.

The method for the selection of individual measure-ments can be extended to select coherent subsets ofmeasurements, but the number of permutationsmakes this unfeasible for subsets larger than 3 or 4.One alternative is to incorporate a factor into theFigure of Merit �e.g., through the CPU cost� to favora selection of measurements that is adjacent to ex-isting measurements, although this does not guaran-tee that all individual measurements will grow intomicrowindows.

Instead, a two-stage technique is suggested. Thefirst stage is to evaluate each individual measure-ment as described in the previous section. Then,with use of the best measurements as starting points,“grow” a number of trial microwindows. The bestmicrowindow is retained and incorporated into the

Fig. 1. Sketch indicating two possible methods of growing micro-windows. A microwindow, currently bounded by the thick line,contains six measurements �3 spectral points � 2 tangent alti-tudes�. Growing edgewise all points along one boundary aretested together, and if this improves the Figure of Merit, the mi-crowindow is expanded in that direction. Growing pointwise allpoints along the boundary are tested individually, and only thosepoints with positive impact on the Figure of Merit are included.This can leave masked points within the microwindow, as indi-cated by solid squares, corresponding to measurements that areexcluded from the retrieval.


retrieval. The process is then repeated, startingwith a reevaluation of the remaining individual mea-surements.

Microwindows can be grown either edgewise orpointwise �Fig. 1�. Edgewise, all of the measure-ments along each edge are tested together, and themicrowindow is expanded in the direction that pro-vides the most improvement. This will alwaysgenerate rectangular microwindows, that is, micro-windows that use all measurements within the fourboundaries. Pointwise, the measurements aroundthe current microwindow boundary are tested individ-ually, and only those resulting in some improvementare added. This usually leaves unselected measure-ments, so-called masked points. Another possibility,given a microwindow grown by either of the abovetechniques, is to test each individual measurementwithin the microwindow and see whether the Figure ofMerit improves when the measurement is included orexcluded and invert the logical status of its mask ifappropriate.

Assuming selection according to information con-tent, measurements are masked �excluded� whenthey contribute negative information. This arisesbecause the weight �gain matrix� is determined solelyby the measurement’s capacity to reduce the randomerror, whereas the information content assesses thetotal error. While it may be impractical to assignweights according to total error �although, see Ref.12�, allowing points to be excluded in this way at leastgives the retrieval the option of assigning zero weightinstead of the random error weight.

Apodization introduces correlations of the randomnoise between spectrally adjacent points. This is ig-nored when isolated measurements are considered,but when microwindows are grown this can be incor-porated with the vector update �Eqs. �13� and �14�� toinclude the subset y of affected measurements withina microwindow. The correlations are then ex-pressed in the off-diagonal elements of the associatedrandom error covariance matrix Sy

rnd. This methodis slower than the inclusion of measurements indi-vidually using the scalar update �Eqs. �23�–�26��, butthe subset y is usually small enough that the over-head is not excessive.

Growth stops either when the inclusion of any fur-ther points results in a lower Figure of Merit or whensome predefined size limit is reached.

3. Implementation for MIPAS

As a practical demonstration, microwindows are se-lected for methane retrievals from the MIPAS instru-ment.13 The MIPAS is a limb-scanning instrument,part of the payload of the European Space AgencyENVISAT satellite due to be launched in 2002.Here it is assumed that a limb-scan sequence consistsof a set of 16 spectra from tangent heights from 8–53km in 3-km steps, and the retrieved profile is on thesame levels. MIPAS spectra cover 685–2410 cm�1

on a grid spacing of 0.025 cm�1, but for this exerciseselection will be confined to the region 1215–1400cm�1, where most of the strong methane lines occur.

Thus the measurement domain consists of 16 �7401 � 118,416 points.

A. State Vector

The state vector x consists of 16 elements represent-ing CH4 volume mixing ratio, plus additional ele-ments for each microwindow representing theretrieved atmospheric continuum for each tangentaltitude and a radiometric offset. It is assumed thatthe continuum and offset are independent for eachmicrowindow, so only the state vector terms relatedto the current microwindow have to be retained.These additional elements are required to model thebehavior of the operational retrieval but are not in-cluded in the Figure of Merit calculations.

B. A Priori Covariance

The MIPAS operational retrieval does not use anyexplicit climatological constraint, so large a prioriuncertainties of 100% are assumed for each volumemixing ration element, 1000% for continuum ele-ments, and the instrument noise for the offset. Nocorrelations or systematic errors are included in the apriori covariance.

C. Jacobians

For each point along the spectral axis, there are po-tentially 256 methane Jacobian matrix elements toconsider, corresponding to 16 tangent altitudes � 16profile levels �and the same again for continuum el-ements�. To save space and computation, only theelements kll are precomputed, corresponding to thesensitivities of measurements at each tangent alti-tude l to perturbations in the retrieved quantities atthe same profile level. Sensitivities kjl of measure-ments at other tangent heights j to perturbations atlevel l are derived from

kjl � kll�ujl�ull�, (27)

where ujl, ull represents the perturbed airmassesfrom level l in the tangent path of measurements jand l.

D. Error Sources

Random noise values are taken from test results onthe flight instrument. The apodization introducesspectral correlations that are included in Sy

rnd whilethe microwindow is grown.

Error spectra yi are precomputed on the basis ofthe perturbation of a standard midlatitude, daytimeprofile assuming the following error sources14:

Climatological variabilities of 27 different species�excluding CH4�;

Uncertainties in temperature �1 K� and pressure�2%� from outputs of the MIPAS p, T retrieval;

Uncertainties in radiometric gain �2%�, spectralcalibration �0.001 cm�1�, and instrument lineshape15;

Spectroscopic uncertainties in the HITRAN data-base16 and the parameterization of molecular con-tinua; and


Other deficiencies in the forward model includehorizontal temperature gradients ��1 K�100 km�,emission that is not in local thermodynamic equilib-rium �non-LTE�, CO2 line-mixing

The forward model for the operational CH4 re-trieval will use, as input, profiles of temperature,pressure, and any contaminant gases that have al-ready been retrieved by MIPAS. However, for thepurposes of microwindow selection, we assume thatonly climatological mean profiles are available for thecontaminant species. The error associated with thisassumption is given by the atmospheric variabilityabout the climatological mean, so for each species anerror spectrum is computed from the radiance differ-ences expected between a nominal atmosphere andan atmosphere with the species profile perturbed byits estimated 1 variability. Each contaminant spe-cies error is assumed to be correlated fully �i.e., witha correlation coefficient of �1� across both the spec-tral and the altitude domains. The spectral correla-tion is clearly valid, but the altitude correlationassumes, for example, that if the concentration isunderestimated at one altitude it is underestimatedat all altitudes. Whereas this may not be particu-larly realistic, it is probably adequate for these pur-poses.

The forward model calculation for all measure-ments at a particular tangent altitude uses the sameretrieved temperature, so this error is fully correlatedin the spectral domain. However, it is assumed thatthe temperature retrieval errors themselves arelargely random between profile levels, so there is nocorrelation in the altitude domain. Therefore thetemperature errors are represented by 16 separateerror sources, each generated from the nominal at-mosphere with a 1-K perturbation at a single tangentlevel. The same applies for the pressure errors.

The gain and HITRAN uncertainties are assumedto be correlated fully in the altitude domain, but onlycorrelated over short distances in the spectral do-main. Each therefore is regarded as fully correlatedwithin one microwindow but random between micro-windows. In principle, these are also regarded as aseparate error sources for each microwindow, al-though spectrally decorrelated errors do not requireseparate error vectors, since no two microwindowsuse the same spectral points �unlike vertically deco-rrelated errors such as temperature�.

The operational forward model estimates line mix-ing but makes no allowance for horizontal gradientsor non-LTE effects. However, since ignoring hori-zontal gradients is equivalent to assuming a zerogradient, this only leaves non-LTE as the singleforward-model error introducing any bias, in whichcase it can be treated as the other model parametererrors using Eq. �16�.

4. Results

A. Selected Microwindows

Sets of ten microwindows were selected with two dif-ferent algorithms. Microwindows 1–10 �Table 1� are

rectangular microwindows, using all measurementswithin the boundaries. Microwindows A–J �Table 2�are masked microwindows where, on average, only49% of measurements within the boundaries areused. These are plotted in Fig. 2. Both algorithmsselect the 1227–1230 cm�1 region for the first �andlargest� microwindow but differ subsequently. Themasked microwindows are larger; most cover either

Fig. 2. Solid boxes and numbers show the rectangular microwin-dows as listed in Table 1. The open boxes and letters show themasked microwindows extended slightly in altitude for clarity �“A”overlaps “1” and “7”�. The Jacobian spectra �arbitrary units� formethane at 53 km �top� and 8 km �bottom� are also plotted.

Table 1. First 10 Rectangular Microwindows, with Their Wave Number�cm�1� and Altitude �km� Boundaries, and Number of Measurements

in Each

MW Wave number Alt. Meas.

1 1227.550–1229.875 8–53 15042 1340.725–1343.725 23–53 13313 1247.800–1248.650 11–53 5254 1306.250–1306.875 23–53 2865 1215.675–1216.575 8–53 5926 1260.275–1260.875 14–53 3507 1230.200–1231.325 8–53 7368 1326.250–1327.875 23–47 5949 1305.475–1306.075 44–53 100

10 1223.275–1224.125 8–50 525Total: 6543

Table 2. First 10 Masked Microwindows, with Their Wave Number�cm�1� and Altitude �km� Boundaries, Total Number of Measurements

within Boundaries, and Number Actually Used

MW Wave number Alt. Meas. Used

A 1227.900–1230.900 8–53 1936 1494B 1304.300–1307.300 8–53 1936 392C 1346.650–1349.650 8–47 1694 855D 1269.725–1271.550 8–23 444 215E 1299.925–1302.925 44–53 484 321F 1327.275–1330.250 8–53 1920 900G 1232.775–1235.775 8–41 1452 712H 1260.175–1261.000 8–35 340 139I 1252.350–1255.350 44–53 484 339J 1261.700–1263.250 8–38 693 191

Total: 11383 5558


the full altitude range or reach the maximum spec-tral width of 3 cm�1. This is expected since, as thesemicrowindows are grown, the negative informationpoints are bypassed with a mask, whereas a concen-tration of such points would prevent a rectangularmicrowindow from expanding further. Another con-sequence is that masked microwindows can some-times include two rectangular microwindows: “A”covers “1” and “7”; “B” covers “4” and “9”. Only therectangular microwindows below 1250 cm�1 extendto lower altitudes �where the methane Jacobian isstrongest�. At higher wavenumbers the rectangularmicrowindows are limited to 23 km and above, pre-sumably due to increased interference from lines ofother species below, whereas the masked microwin-dows in this region appear to extract methane infor-mation down to 8 km by masking out these lines.

B. Information Content

The original selection attempted to find the micro-windows in a sequence of decreasing information con-tent, although only through a trial-and-errorapproach, so there is no guarantee that all of the bestten microwindows are found, or that they are in thecorrect sequence. The sequence, at least, can be re-sorted by evaluating each microwindow in turn tofind the one that contributes most information to thea priori, include it in the retrieval, and repeat theprocess for the remaining microwindows, and so on.

The results of sorting the two lists are shown inFig. 3. In both cases the first three microwindowsare unchanged. Subsequent changes mainly serveto promote the larger microwindows up the sequencewithout significantly affecting the slope. The slope

itself corresponds to around 16 bits �or a factor-2improvement in standard deviation at all profile lev-els� for a factor-10 increase in the number of mea-surements.

The offset of the two curves suggests that themasked, compared with rectangular, microwindowsappear to yield approximately three extra bits of in-formation for a given number of measurements usedor require about half as many measurements to ob-tain a given accuracy. However, if computing effortdepends on the total number of measurements, in-cluding those points that are masked out, the two arefairly similar in this example. In this case the onlyadvantage of using masked microwindows is thatfewer microwindows, rather than measurements, arerequired for a given accuracy.

C. Sorting by CPU Cost

Microwindow lists can be selected with one criterionthen sorted with another, such as attempting to max-imize the information that can be retrieved givenlimited processing time. Assuming a simple CPUcost function of the form

C � 1 � N, (28)

where N is the number of measurements, this is in-corporated into the Figure of Merit �Eq. �22��, withvarious exponents n and the ten rectangular micro-windows resorted according to the new criterion.The results are shown in Fig. 4.

The use of n � 0 is the equivalent of no CPU cost,and so is the same curve as in Fig. 3. The othercurves start off more steeply, achieving a given accu-racy with fewer measurements �although more mi-crowindows� before converging to the same point at

Fig. 3. Growth of information content with number of measure-ments for the microwindows listed in Tables 1 and 2 after reor-dering. Open symbols show original selection sequence �orderedA–G and 1–9 from left to right across the graph�.

Fig. 4. Growth of information content with number of measure-ments for different CPU cost functions. Microwindows are dis-tinguished by different symbols, as labeled for the n � 0 curve.


which all ten microwindows are used. At n � 6, themuch smaller MW#3 replaces MW#1 as the first-choice microwindow.

D. Error Analysis

The random covariance Sxrnd and the retrieval error

vectors xi contain information on the contribution ofeach term in the total error of the retrieval. Theseare plotted as profiles in Fig. 5.

This plotting shows an absolute accuracy of about10% in retrieved CH4 mixing ratio. The random er-ror, or precision, contributes a relatively small com-ponent: 1–3%. The major contributions are the 1K temperature and 2% pressure errors from the p, Tretrieval, non-LTE effects at high altitudes, horizon-tal temperature gradients at middle altitudes, andHITRAN database uncertainties at low altitudes.N2O is the only contaminant with any significanterror contribution, peaking at 4% at 20 km �althoughthe N2O concentration is larger at lower altitudes, itsmixing ratio is more predictable hence contributes asmaller error�. H2O contributions in the tropo-sphere are probably eliminated by the simultaneouscontinuum fit in the retrieval model.

5. Conclusions

A scheme has been developed for selecting microwin-dows by a modeling of the retrieval propagation ofsystematic as well as random errors. This providesa mathematical basis for the weighting of signal fromthe target species against errors introduced by un-certainties in other parameters such as interferingspecies.

The technique will be used for the selection of mi-crowindows for the operational processing of MIPASdata, but is equally applicable to other limb or nadir-sounding instruments that obtain atmospheric spec-tra. External information can also be included byway of the a priori covariance, allowing, for example,microwindows to be selected which best complementthe output of a numerical model.

Instead of minimizing the total error, other prior-ities may also be defined by way of a Figure of Merit.The microwindows are generated in order of dimin-ishing merit, so the sequence may be optimally trun-cated at any point. The scheme can also be used toreorder microwindows from an existing list accordingto different priorities or circumstances, such as achanged set of retrieval levels or missing data, orrevised estimates of some error source.

The method relies on a representation of all errorsources in measurement space �i.e., as perturbationspectra�, which are then mapped into state vectorspace �i.e., retrieved profiles�. This can be used toprovide a full error analysis of the retrieval.

Some important limitations should be noted.First, the process is not optimal in the sense that itcan determine the best possible set of microwindows.This is partly due to the trial-and-error nature of thegrowing of microwindows from individual measure-ments, although taking a large number of trials im-proves the chances of the best microwindow beingfound. More fundamentally, the selection is strictlysequential in that it maximizes the increase in thechosen Figure of Merit at each opportunity: It doesnot allow for combinations of individually less bene-ficial microwindows that may give greater net im-provement �the given example of weighting againstsystematic errors attempts to address this issue�.

Second, the above selection is based on the defini-tion of error spectra and Jacobians for a single atmo-spheric profile and would therefore be less ideal fordifferent profiles. For the selection of the MIPASoperational microwindows, an ensemble of five differ-ent atmospheric profiles has been used, each with itsown set of spectra, and the Figure of Merit representsan average of the five different obtained values.This produces a single set of globally optimized mi-crowindows that should be suitable for all latitudesand seasons.

During this work A. Dudhia was employed underan European Space Agency contract, and V. Jay wassupported under a Natural Environment ResearchCouncil studentship.

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Fig. 5. Major components of the retrieval error from the rectan-gular microwindows listed in Table 1. Symbols indicate pressure,temperature, horizontal gradients, HITRAN, non-LTE, N2O, andH2O uncertainties. The combination of these gives the net sys-tematic component, and combined with the random componentgives the total error.


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Microwindow Selection for High-Spectral-Resolution Sounders