Infrared radiative transfer in the 9.6- � m band: Application toTIROS operational vertical sounder ozone retrieval

Richard J. Engelen �Institute for Marine and Atmospheric Research Utrecht, Utrecht University, Utrecht, The Netherlands

Graeme L. Stephens �Cooperative Research Center for Southern Hemisphere Meteorology, Clayton, Victoria, Australia

Abstract. This paper introduces a radiative transfer model for the 9.6- � m ozoneband that specifically matches the TIROS operational vertical sounder (TOVS)channel 9. The model is based on a spectral Malkmus band model for transmis-sion. Band parameters were calculated by comparing to MODTRAN3 derivedradiances. The effect of pressure on absorption is dealt with using a four-parameterapproximation, and the improvements of this approximation over the more com-mon Van de Hulst-Curtis-Godson scaling approximation are assessed. While thisnew model is exploited in the development of the retrieval described in this paper,the result has wider applicability to ozone climate problems requiring calculationof the radiative forcing associated with changing ozone. A two-layer version of theradiative transfer model is implemented in a retrieval scheme to obtain total ozoneamounts from radiances measured by the TOVS instrument. Because the TOVSozone channel is mainly sensitive to lower stratospheric ozone, ozone columns ofthe upper layer (above 30 hPa and with mean pressure of 10 hPa) are prescribedas a function of latitude. Ozone columns of the lower layer (mean pressure of 105hPa) are then retrieved. The retrieval is based on a nonlinear optimal estimationalgorithm and provides definition of error characteristics for every retrieval, whichmakes it possible to obtain a spatial distribution of the errors in the retrieval togetherwith the spatial distribution of the retrieved total ozone column itself. This globaldistribution of the retrieval error and also of the contribution of a priori knowledgeto the retrieval is presented to provide a validity of the ozone retrievals. Totalozone mapping spectrometer (TOMS) statistics are used as a priori information inthe retrieval, and the 40-layer model is used to estimate the forward model errorof the two-layer model. Comparisons of ozone retrievals for 1989 with TOMStotal ozone measurements show good agreement both in time and in space with arms difference between 1% and 3% for zonal means and 10% for global griddedmeasurements.

1. Introduction

One of the most significant environmental issues of thiscentury is the observed loss of polar stratospheric ozone re-

�Now at Department of Atmospheric Science, Colorado State University,

Fort Collins.

ferred to as the “ozone hole” and first reported by Farmanet al. [1985]. This loss is seasonal and occurs largely overthe southern pole [Stolarski et al., 1986], although significantlosses of ozone have also been reported in the higher latitudesof the northern hemisphere [Gleason et al., 1993; Planet etal., 1994]. Whereas the mechanisms responsible for the lossof ozone are now largely understood, a number of critical is-

1

2

sues remain to be addressed. For instance, the evolution ofthe ozone hole and the role of transport into and from the po-lar vortex, the life cycle of the ozone hole and the interannualvariability of the ozone hole are topics of current research.

Along with the improvements in modeling the ozone holeand with progress toward prediction of its evolution on time-scales of a few days to a week or so, comes the need for re-liable global measurements of the three-dimensional distri-bution of ozone. In the polar regions, highly resolved strato-spheric ozone profiles are provided by measurements of theUARS satellite [Waters et al., 1993; Froidevaux et al., 1994].To date, the main satellite instrumentation used for contin-uous monitoring of the two-dimensional distribution of theozone layer is the total ozone mapping spectrometer (TOMS)instrument [Bowman and Krueger, 1985]. While this instru-ment has provided the primary source of observations since1978, recent failure of TOMS on board the Meteor 3 satellitehas heightened interest in other ozone sensors. The compan-ion to TOMS is the solar backscattered ultraviolet (SBUV)instrument, and significant effort has been spent recently onthe development of operational SBUV retrievals [Ahmad etal., 1994; Hilsenrath et al., 1995]. Both the TOMS and SBUVmeasure backscattered ultraviolet sunlight and cannot pro-vide retrievals in the polar night region during winter andearly spring.

Infrared radiance data in the 9.6- � m channel of the TIROSoperational vertical sounder/ high-resolution infrared radi-ation sounder (TOVS/ HIRS) flown on NOAA operationalsatellites (for an extensive description of the instruments, seeSmith et al. [1979]) can also be used to retrieve total ozoneamounts. While the retrieval of ozone based on infrared ra-diance data offers an advantage over UV backscatter mea-surements in regions of polar night, IR techniques suffer forreasons explored in this paper. The purpose of this paperis to introduce a physically based retrieval of column ozoneapplied to TOVS radiance data in order to retrieve a usefulozone climatology based on the long time series of TOVSobservations. While other retrievals exist [e.g., Planet et al.,1984; Ma et al., 1984; Lefevre et al., 1991; Neuendorffer,1996], this paper revisits the issue of infrared radiative trans-fer through ozone and introduces a new retrieval based onthis theory. Also, special attention is paid to the errors inthe retrieval, which are especially important when measure-ments have to be compared with models. The extensive treat-ment of the radiative transfer and the used error analysis im-prove the characterization of the retrieval product comparedto the existing methods.

The theory of radiative transfer at frequencies of the TOVSozone channel is introduced in the next section, followed byan outline of a simple forward model used as the basis for theretrieval. Section 4 outlines the retrieval approach based on

the nonlinear optimal estimation approach of Rodgers [1976].A particular advantage of this approach over others is thatit offers a clear characterization of the retrieval uncertaintiesand the information content of the retrievals. In section 5 thenew retrieval is applied to global TOVS data for 1989 andcompared with favorable results to the TOMS data for thesame period.

2. Radiative Transfer

The basis underlying any satellite retrieval method is aradiative transfer model that calculates the radiation at thetop of the atmosphere for a given distribution of trace gases(CO � , H � O, and O � ) and temperature (this is referred to asthe forward model). For the problem of relevance in this pa-per, the equation for the monochromatic radiance at the topof the atmosphere for a plane parallel non-refracted path at acertain viewing angle � is������� � ����� ���������� ���������� �!#"$&% �� ' � �������� �����(����!#")�*�,+.- � , (1)

where� � ���/� � � is radiance, � is 0�1�2�� ,

�is optical depth, and� � ������ �3� is Planck radiance for temperature

�as a function

of�

. TOVS satellite radiance data are broadband, and simu-lation of these radiances based on equation (1) requires somekind of spectral integration which must include in these in-tegrals the instrument filter response functions. A practicalform of this integration is

�4����5 ���76893: +�; 9�<>=?9A@ � 9������?�CB 9 ��D 9 �����#B 9 �

$ % �� �E F' � 9 ������ 9 ����D 9 ��� 9 ��- � 9HG , (2)

where the spectral integration is carried out as a summationover a finite number of I subintervals chosen to resolve thespectral structure of the sensor filter function ; 9 and � is setto 1 (nadir sounding). D 9 is the transmittance from the top ofthe atmosphere to a specified level

D 9 �� �J� 5<>=?9 %)K � F ���4��LM- = , (3)

where � � ��N �PO . (4)

Here, N � is the absorption coefficient and O is the amount ofabsorbing matter.

Two issues that must be addressed in evaluating the for-ward model of TOVS channel 9 radiances are a suitable way

3

of evaluating the broadband transmission (equation (3)) andan appropriate way to integrate over optical path, which is es-pecially complex for ozone absorption because of the com-bined effects of the dependence of absorption on pressure andthe variation of ozone concentration with pressure.

Two-Parameter Band Model

We demonstrate that a two-parameter band model ade-quately represents the broadband transmission along paths ofcomplex pressure variation. We begin with the introductionof the Malkmus band model [Malkmus, 1967; Goody andYung, 1989], which expresses the transmittance as a functionof absorber amount at fixed pressure and temperature as

D � O ��������� �����L ������ 5 $���� O��

L

� 5����, (5)

with � the average line intensity,�

L the average Lorentz linewidth, and � the average line spacing. Broadband transmis-sion is evaluated given suitable values of the band parameters�

L and � . These are obtained from line absorption data byrequiring exact agreement in the weak-line and strong-linelimits. This gives the following expression:

D � O ��������� ��� � �! < = � � 5 $ ! � O� � � 5"���, (6)

where � and!

are related to the statistical line parametersby � � 68$# � � # � # �&%' , (7)! � 68$# � # , (8)

and where <>= is the bandwidth. According to Goody andYung [1989], the error in the band transmittance due to theuse of the Lorentz line shape instead of the Voigt line shapeis less than 2.6% and for the ozone absorption band is evenless than 0.16%. The Malkmus band model parameters �and

!were calculated by fitting to MODTRAN3 [Berk et

al., 1989] derived transmittances at a resolution of 1 cm � +as a function of the optical path of the absorbers rather thanby summing over spectral line data. For the water vapor ab-sorption only, the line absorption was used. For each TOVSchannel the channel transmittance was then calculated as(D � O ���*) 693: + ; 9 D 9

� O �) 693: + ; 9, (9)

where the summation is over the band defined by the chan-nel filter function ; 9 and where D 9 � O � are the MODTRAN

derived transmittances. The band parameters so derived aregiven in Table 1 for channels 1, 2, 8, 9, 10 using NOAA10filter functions.

In Figure 1 the transmittance of TOVS channel 9 as func-tion of constant pressure absorber path length is shown forboth MODTRAN and the band model using the parametersof Table 1. The pressure and temperature used in the calcu-lation are 1013 hPa and 235 K, respectively.

As done by Clough et al. [1989], the absorption due to thewater vapor continuum is parameterized asD � O ��������� � � � cont � , (10)

where the continuum optical depth is�cont � = ',+.- �0/21/ ' $ -43 // '$576 � '�98 O

, (11)

where O is the vertical path of water vapor, /:1 and / denotethe water vapor partial pressure and the ambient pressure,respectively,

�is the temperature, - � and - 3 are the self-

and foreign-broadening coefficients for water vapor, respec-tively, = ' is the central wavenumber of the band, and

� ' and/ ' are 296 K and 1013 mbar, respectively. The values of = ' ,- � , and -43 are also given in Table 1 for each channel.The radiative transfer model using the band transmission

functions and band-corrected Planck functions specific toeach channel [e.g., Weinreb et al., 1981] was used to simulatechannel brightness temperatures for each of the 1761 TOVSinitial guess retrieval (TIGR2) atmospheric profiles [Scott etal., 1991]. Each set of TIGR2 profiles consists of an ozone,a water vapor, and a temperature profile.

Pressure Dependence of Transmission

It was mentioned above how the transmission throughozone is complicated by the different dependences of the lineabsorption on the ozone profile and the pressure. The com-mon approach for dealing with pressure scaling is the Vande Hulst-Curtis-Godson (HCG) method [Goody and Yung,1989]. This method can be applied to the band model in thefollowing way. In the HCG approximation, the scaled ab-sorber amount ;O is ;O � %

path

! ��� �! � ;� � - O , (12)

and the scaled pressure ;/ is;/ � %path

� � ��� �� � � ;� � - O;O , (13)

where � and!

are as defined before and ;� is a scaled tem-perature. If we neglect temperature dependence (MODTRAN

4

Table 1. Malkmus Band Parameters for NOAA10 HIRS

Channel Gas � ,!

, - � , -43 � = ' ,��� � + N�� �,+�! � ����� � + N�� �,+ � � N�� � + � � ��� N�� �,+ � � ��� ��� � +1 H � O 0.35 0.82 �� 5� � 5? � � � ��� � 5� ��� 667.70

CO � 42.19 9087.5O � 12.96 27.89

2 H � O 0.41 0.7 � ��� � � 5? � � � � � �� 5� ��� 680.23CO � 64.32 1325.2O � 32.83 77.44

8 H � O 0.05 0.025 � 5 � 5? � � � � �5�� 5� ��� 899.50

CO � 0.05 0.01O � 0.01 1.6

9 H � O 0.05 0.02 �� 5� � 5? � � � � ��� � 5� ��� 1029.01CO � 0.08 0.01O � 201.23 3045.07

10 H � O 0.42 0.43 ����� 5�� 5? � � ��� 5�� 5� � � 1224.07CO � 0.03 0.03O � 0.68 4.17

10-5 10-4 10-3 10-2 10-1 100

Path length (kg m-2)

0.0

0.2

0.4

0.6

0.8

1.0

Tra

nsm

ittan

ce

Modtran

Malkmus

Figure 1. Comparison between MODTRAN and Malkmus band model fit to the transmission derived using the NOAA10filter function for TOVS channel 9. The transmissions are shown as a function of the constant pressure absorber path length.

5

calculations showed that the temperature dependence of theTOVS ozone absorption band is small), the HCG approxima-tion becomes ;O � %

path- O � O

, (14)

and ;/ � %path

/ - O�� ;O . (15)

The scaled absorber amount is applied directly in (6) and thepressure ;/ is used to scale the � � factors in (6). The results ofsimulated channel 9 radiances using the band model (6) withthe parameters of Table 1 and the HCG method are shown inFigure 2 and contrasted against the corresponding brightnesstemperatures derived using MODTRAN. Each point repre-sents a pair of band model and MODTRAN simulations us-ing a single TIGR2 atmosphere. A very good agreementis observed between the two models, with the MODTRANbrightness temperatures slightly higher than the band modelbrightness temperatures.

Walshaw and Rodgers [1963] and Goody [1964] demon-strate the satisfactory nature of the HCG approximation forthe absorption by water vapor and carbondioxide. The meth-od, however, breaks down for ozone where the absorptionfalls between the strong and weak limits due to high concen-trations that exist at low pressure and low concentrations athigh pressure. To assess how these pressure effects might af-fect our ability to simulate radiances in the 9.6- � m band, weadopted the approach of Walshaw and Rodgers [1963]. Webegin with the Malkmus band model in the spectral integralform, (D ���A����� ��������� + ��� � %'��� ��5 $ � � ��-�� 5 ,

(16)

where the optical depth of a Lorentz line is� � � %����� �� � $ � � - � , (17)

where� � / � / ' , � � = � � L. The dimensionless variable �

is defined as � � N / ��� �� � � , (18)

where N is the mean line intensity, / � the surface pressure,� �is the mean line width at surface pressure, � is the grav-

ity constant, and�

is the ozone mass mixing ratio. With anozone mixing ratio profile of the form�/��� �J� �

max��� � ���5 $ � � � � � , (19)

the optical depth can be written as [see Walshaw and Rodgers,1963]� � � ���� � � max� � � ��5 $ � � �� 5 $ ���5 $ ��� � $ � � � � � � � � ���� ,

(20)

where� � � � � � 5

and� � ��5 $ � � � � � + .

The transmission along a path from pressure�

to pressure� � is evaluated explicitly by integrating (16) providing exacteffects of pressure on transmission. This integral was evalu-ated using a Romberg quadrature scheme on a open interval[Press et al., 1989].

We can use the same model under the HCG approximationin the form(D ���A����� � ������� + � � � 6 �"! +! ' $$# ! + $ � ! + � ! ' � � 8 5 ,

(21)

where ! ' � %%� �� � - � ,! + � %%� �� � � - � .

With (19), these integral functions are! ' � �max + �5 $ � � �

� � �5 $ � � � �$'&�( 0*) & �,+ � � � � � &-( 0�) & ��+ � � �+ � 5 (22)

! + � �max + 5

� $ � � � � �� 5

� $ � � � �$ �. ��5 $ � � � � � � �� ��5 $ � � � �� 5 . (23)

The transmittance calculated using the Walshaw andRodgers model (equations (16) and (17)) is contrasted in Fig-ure 3 with the transmittance derived using equation (21). Thetransmittances, shown as a function of height, were obtainedusing the same values for � and

�max as Walshaw and Rodgers

[1963], and a value of �max and

� �chosen to represent the

TOVS channel 9. The values of these parameters relevantto Figure 3 are 1600, 10 ppmm, 11.84, and 2.66 for � ,

�max,�

max, and� �

, respectively. The comparisons reveal that thetransmittances below 20 km differ significantly, with the trans-mittance of the HCG model less than the transmittance of theexact model. These transmittances were applied in the radia-tive transfer model together with the temperature profiles ofthe TIGR2 data set to determine the effect of these transmis-sion differences on brightness temperatures. The model us-ing the HCG approximation yields brightness temperaturesthat are 1.2 K (0.5 %) lower on average than the exact model.

6

200 220 240 260 280 300Model TB (K)

200

220

240

260

280

300

MO

DT

RA

N T

B (

K)

Figure 2. Comparison of calculated channel 9 brightness temperatures using MODTRAN and the band model with watervapor continuum absorption included.

0.0 0.2 0.4 0.6 0.8 1.0Transmittance

0

10

20

30

40

50

Hei

ght (

km)

10-8 10-7 10-6 10-5

Mixing ratio (g/g)

50

Ozone profile

Exact model

HCG model

4-parameter model

Figure 3. Comparison of the transmission as a function of height using the exact Walshaw model and the same model withthe HCG approximation and the four-parameter approximation. The dash-dotted line represents the ozone profile used for thecalculations.

7

Four-Parameter Band Model

It is straightforward to improve on the HCG, as Goody[1964] and Rodgers [1968] have demonstrated. Here this ex-tension of the HCG, referred to as a four-parameter approx-imation, is applied to the Malkmus band model. The con-ceptual view of this four-parameter method is to consider apath composed of two contiguous elements, one at pressure/ + and the other at pressure / � . The optical depth of two su-perimposed Lorentz lines at standard temperature then fol-lows as �?� � � � + O + � = � $ � � + � $ � � � O � � = � $ � �� � , (24)

with � # � �L/ #/ ' .

This gives us the following three equations:;O + $ ;O � � � - O;O + ;/ + $ ;O �2;/ � � � / - O;O + ;/ � + $ ;O � ;/ �� � � / � - O .

(25)

With the following parameters

� # � ;O #;O�# � ;/ #;/� � � / � - O;/ � ;O ,

(26)

where ;O � � - O and ;/ � � / - O�� ;O , these equations have thedimensionless form � + $ � � � 5

� + � + $ �,� � � � 5� + � � + $ � � � �� � � .

(27)

With the ad hoc constraint [Goody, 1964]

� + � 5� , (28)

we have a solvable set of four equations, which can be used tocalculate ;/ + , ;/ � , ;O + , and ;O � . The dimensionless parameter �is a measure of the bimodal character of the distribution. Fora single layer, � � 5

; for two layers, ��� 5, and it increases

as the pressure difference between the two layers increases.

This four-parameter approximation was used in the Malk-mus band model with the same band parameters � and

!as

described before. The transmission is now

D ��������� � ��� � ;� + � ;� � � , (29)

where ;� # � � �! <>= ;/ #/ ' , (30)

and� ��� ;� � + ��5 $ � + � $ ;� �� ��5 $ � � �$ ;� + ;� � + 5 $ � + $ � �� +�! � , (31)

where

�# � ! �� � / ';/ # ;O # . (32)

The transmission is also presented as a function of heightin Figure 3. The transmissions were derived for the ana-lytical ozone profile introduced above. The superior perfor-mance of the model over that of the HCG is evident.

3. A Simple Forward Model

The retrieval approach introduced here is built around asimple two-layer version of the radiative transfer model de-scribed above. The rationale for the two-layer model derivesfrom the work of Neuendorffer [1996], who notes the limitedvertical information implied in the 9.6- � m TOVS radiances.The model consists of two layers from 1000 – 30 hPa and 30– 0 hPa. The radiance at the top of the atmosphere can bedescribed with�

� � � ���,� ��D� $ � ����� � � D�� � D� �� $ � �� � � ��5 � D��)� ,(33)

where the subscripts � , � , O , and � correspond to surface, low-er layer, upper layer, and total atmosphere, respectively. Forthe upper layer, � � 5

, and D�� , the transmittance from thetop of the atmosphere to 30 hPa, is described by

D � ��������� � � � � � + 5 $ � ��� , (34)

with � � � � �! <>= / �/ ' ,

� � � ! �� � / '/ � O � ,

8

andO � is the ozone amount of the layer. For both layers to-

gether, � is not equal to 1 and the transmittance D is calcu-lated with equation (29), where ;� # and

�are as defined in (30)

and (31).The actual model adopted in the retrieval is similar to that

of Neuendorffer [1996]. The surface temperature requiredin equation (33) is approximated by a weighted sum of thebrightness temperatures of TOVS channel 8 and 10:� � � � � � � $ � � � + ' . (35)

The mean temperature of the lower layer is defined as the av-erage of the brightness temperatures of HIRS channel 2 andmicrowave sounding unit (MSU) channel 4:� � � ��� � $ �

MSU4 � � � . (36)

The temperature of the upper layer is taken to be the bright-ness temperature of HIRS channel 1. Following Neuendorf-fer [1996], the amount of ozone in the upper layer is pre-scribed as a function of latitude byO � � 5� � � � $ 5 � 5 0H1�2 � latitude

� 5�� � , (37)

and the only unknown variable is the total ozone amount ofthe lower layer O � .

Figure 4 shows the radiances calculated with this two-layer model versus radiances calculated with the 40-layermodel for a subset of the TIGR2 atmospheres. The subsetwas chosen to give a good representation of the TIGR2 at-mospheres. In the two-layer model the mean pressures of thetwo layers (/ � and / � ) are taken to be 10 hPa and 105 hPa,respectively. These mean pressures were chosen to give thebest results compared to the 40-layer model. It can be seenthat the two-layer model performs well for most of the atmo-spheres. In the tropical regions (denoted by the diamonds inFigure 4) a bias exists. This can possibly be explained bynoticing that for the tropical profiles the ozone amount in theupper layer is the main contributor to the total ozone amount.Because this amount is prescribed in the model, there is littleflexibility to deal with variations in the real profile.

The difference between the two-layer model with the 40-layer model is used for the error estimate of the two-layermodel. This estimate is needed in the retrieval procedure de-scribed in the next section. Assuming that the 40-layer modelrepresents the ”truth,” the forward model variance � 3 can becalculated [Marks and Rodgers, 1993]. Because there is adifference in model performance between low latitudes andhigh latitudes, � 3 is calculated separately for latitudes eitherside of � �� latitude. The bias of the model for the two lati-tude regions is also used in the retrieval in the same way asproposed by Marks and Rodgers [1993].

4. Retrieval of Total Ozone

To retrieve total ozone from the TOVS measurements, weused the approach described by Rodgers [1990] and Marksand Rodgers [1993], which expresses a measurement � as

� � � ���A� � � $���� , (38)

where ��� is the error in the measured radiance,�

is the totalozone value,

�is the atmospheric “forward function,” repre-

senting the detailed physics of the measurements (and givenby the simple two-layer forward model introduced above),and

�denotes all unretrieved quantities that affect the radi-

ance and which must be specified a priori. The measurement� represents the channel 9 brightness temperature.Using estimates � and � , our forward model provides a

synthetic radiance � . Following Marks and Rodgers [1993],we can find the optimal solution � by minimizing the func-tion

� � � ���� � �� � $ � � �

� � � �3� �� � (39)

with respect to � . An a priori value� �

for�

is used to im-prove the noise characteristics and stability of the solutionrelative to an ”exact” solution ( � � � �,+ � �/� ). � � and � �are the error covariances of the a priori value and the mea-surement. � � also contains the error estimate of the forwardmodel � 3 . This error was estimated by comparing the two-layer model with the more sophisticated 40-layer band modelas shown in the last section, and is much larger than the actualTOVS measurement errors because it contains all the uncer-tainties in the simple forward model.

After linearizing about the current estimate�#, the mini-

mization calculation yields

�# + � � �,+� ��� $�� � �,+� � � �

� ��� # � $�� � # �� �,+� $�� � � �,+� ,(40)

where � ��� � � � � and is calculated with a simple pertur-bation method:

� � � ��5 � /5 � � � � � � � � � � � � . (41)

Because � has to be calculated in every step of the retrieval,the forward model, which is used three times in each step,has to be computationally efficient.

The error covariance of the retrieved�

is� �,+� � � � +� $�� � � �,+� . (42)

9

0 2 4 6 8 1040-layer model Intensity (W cm-2 cm-1)

0

2

4

6

8

10

2-la

yer

mod

el I

nten

sity

(W

cm

-2 c

m-1)

abs(latitude) > 30

abs(latitude) < 30

Figure 4. Scatterplot of radiances calculated with the two-layer model versus radiances calculated with the 40-layer model.The crosses denote the TIGR2 profiles with a latitude larger than � � N and the diamonds denote the TIGR2 profiles with alatitude smaller than � � N.

As a convergence test, the following equation is used:��� # + � �#� �� � � 5

(43)

Only a few iteration steps are needed to converge to the so-lution. Following the error analysis described by Marks andRodgers [1993], we used

� � as an extra quality test for theretrieval. The approximate

� � to use is� � � � � �� ��� # �3� �� � $ ��� � � � � �� � (44)

This should follow a� � distribution with one degree of free-

dom, as we have one measurement ( � ) and one virtual mea-surement (

���) with one parameter ( � ) being retrieved. A typ-

ical value of� � for a moderately good retrieval is then 1. We

regarded a convergent retrieval as acceptable only if� � is not

significant at the 1% level.The retrieval scheme was applied to a subset of the TIGR2

profiles. The 40-layer model was used to calculate the radi-ances of the channels, needed in the retrieval, for each setof profiles. Monthly averaged zonal mean ozone statisticsfor the years 1987 – 1991 obtained from measurements ofthe total ozone mapping spectrometer (TOMS) [Bowman andKrueger, 1985] were used for the a priori ozone values (

� �)

and error covariances ( � � ). If no ozone statistics were avail-able (e.g., in the polar night), an a priori value of 180 DU withan error of 150 DU was used for the lower layer.

Figure 5 shows the retrieved total ozone amounts contrast-ed against the integrated ozone profile amounts. It can beseen that relative to the integrated profiles, the retrieval

scheme gives values generally within 25 DU. This is betterillustrated by Figure 6, which shows the difference betweenthe retrieved total ozone amounts and the integrated ozoneprofile amounts as a function of latitude. The bottom part ofthis figure shows the retrieval errors. The triangles denote re-trievals, which have to be discarded because

� � is too large orbecause they rely too much on the a priori information. Thisdependence on a priori information is denoted by the param-eter � , which is explained in the next paragraph. The figureshows that although the estimated retrieval error can be quitelarge (especially at higher latitudes), the difference betweenretrieved and real total ozone amounts is smaller than 10%for most of the cases. The estimated retrieval errors are prob-ably too large compared to the actual differences because it ishard to provide accurate error estimates for the a priori dataand the forward model, and because the direct inversion ofour forward model is not as ill-conditioned as in some otherretrievals. Therefore the ozone retrieval can perform betterthan the retrieval error estimates indicate.

It is important to determine the dependence of the retrievalon the a priori information. This dependence can be char-acterized by a parameter that reflects the amount of a prioriinformation retained in the retrieval. According to Rodgers[1990], we can write our retrieved total ozone value as a lin-ear combination of the true value and the a priori value withweights � and

��5 � � � plus contributions from experimentalerror in the forward model parameters and the measurement,

� ��� � $ ��5 � � � ��� $�� � ��� , (45)

where � � is the derivative of the inverse model with respect

10

100 200 300 400 500 600Retrieved Total Ozone (DU)

100

200

300

400

500

600

Rea

l Tot

al O

zone

(D

U)

Figure 5. Scatterplot of the retrieved total ozone columns versus the real total ozone columns for a subset of the TIGR2 profiles.

-90 -60 -30 0 30 60 90Latitude

0

10

20

30

-30

-20

-10

0

10

20

30

40

Dif

fere

nce

(%)

Stan

dard

Err

or (

%)

χ2 < 6 and A > 0.4χ2 > 6 or A < 0.4

Figure 6. Difference between retrieved total ozone and integrated TIGR2 profile total ozone as a function of latitude (upperpart), and estimated retrieval error as a function of latitude (lower part). Crosses denote accepted retrievals; triangles denotediscarded retrievals (for details, see text).

11

to � . � can be calculated with

� � 5 � � � � 5 � � �� ��� � 5 � 55 $�� �����

���.

(46)

Figure 7 shows the parameter � as a function of latitudefor each retrieval. The contribution of the a priori informa-tion varies widely between 0 and 100%. The reason for this isthat the forward model is very insensitive to changes in theozone concentration when the contrast between the surfacetemperature and the mean temperature of the lower layer isvery low. The retrieval is then unstable [e.g., Lefevre et al.,1991] and relies mainly on the a priori information. Howmuch the retrieval is relying on the a priori information is de-termined by the ratio between the forward model error andthe a priori error. This is illustrated by the triangles in Fig-ure 7, which represent the parameter � when an a priori valueof 180 DU with an standard deviation of 150 DU is used inthe retrieval. Because this standard deviation is much largerthan the standard deviation of the TOMS statistics, the re-trieval depends less on the a priori information. This effectis also illustrated in Figure 8, which shows the contributionof the a priori error and the forward model error to the to-tal error. For most retrievals the total error is dominated bythe forward model error. Only in the low-contrast cases, thecontribution of the a priori error becomes large, which meansthat the retrieval heavily relies on the a priori information.

5. Application to Global TOVS Measurements

The new ozone retrieval scheme was applied to NOAA10TOVS radiances for 1989 [Smith et al., 1979]. The radiancesare cloud-cleared and corrected for the limb-darkening ef-fect [McMillin and Dean, 1982]. The retrieved total ozoneamounts are compared to similar data available from TOMS[Bowman and Krueger, 1985]. These data are available ona5 � � � by

5 �grid for the years 1979 to 1992 (GRIDTOMS

version 6 available on CD-ROM from NASA). Figure 9 andFigure 10 show the annual cycles for 1989 of zonal averagedtotal ozone for both TOMS and TOVS between

�N and � � N,

and between � � � N and � � N, respectively. The two formsof ozone data agree reasonably well with each other. A rmsdifference of 2.9% and 1.4% was found for � � N and � � N,respectively. Differences at � � N can probably be explainedby the effect of the quasi-biennial oscillation (QBO), whichimpacts the higher stratosphere where the ozone amount isprescribed for the TOVS retrieval.

Figure 11 shows the global total ozone distribution de-rived by TOVS for August 7, 1989. The agreement with theTOMS total ozone field (Figure 12) is reasonable with an rmsdifference of 10%. Patterns are similar with regions of max-ima and minima in total ozone matching in location and mag-

nitude. For example, small-scale structures such as the min-imum over Nova Zembla and the extension of ozone-poorair from the tropics over Japan are detected in both data sets.The advantage of the TOVS data in the polar night region isevident.

A particular advantage of the retrieval introduced in thispaper is illustrated in Figure 13, which shows the percent er-ror of the retrieved total ozone amounts. A global pictureof the reliability of our retrieval is thus obtained. The fig-ure shows that in the the tropics the error is small but thatabove ice or snow surfaces, like the Antarctic and the north-ern hemispheric snow fields, the error of the retrieval be-comes much larger and is due to the large sensitivity of theforward model in cases of low thermal contrast between thesurface and the lower stratosphere. Although the test retriev-als in the last section showed that the difference between theretrieved total ozone and the real total ozone is often smallerthan the retrieved error estimate, one has to be cautious inter-preting the retrieved ozone values when the error estimate islarge.

6. Conclusions

A radiative transfer model is introduced that uses a spec-tral band model for transmission based on the Malkmus linedistribution. The effect of pressure on absorption is dealtwith using an extended version of the widespread Van deHulst-Curtis-Godson approximation, the four-parameter ap-proximation. The methods were tested using a solution in-troduced by Walshaw and Rodgers to test the two pressurescaling approximations. We showed that for the absorptionin the 9.6- � m ozone band the four-parameter approximationprovides a significant improvement over the more commonHCG approximation. While this result is then exploited inthe development of the retrieval described in this paper, theresult has wider applicability to ozone climate problems in-volving calculation of the radiative forcing associated withchanging ozone.

A two-layer version of the radiative transfer model wasimplemented in a retrieval scheme to obtain total ozone a-mount from radiances measured by the TOVS instrumenton board the NOAA satellites. The approach described byMarks and Rodgers [1993] was followed. This method pro-vides good error characteristics for every retrieval, whichmakes it possible to obtain a spatial distribution of the errorsin the retrieval together with the spatial distribution of the re-trieved total ozone column itself. This is different from otherexisting infrared retrieval schemes for ozone, which provideonly general error estimates. The main differences betweenour approach and the approach of Neuendorffer [1996] arethe radiative transfer modeling and the retrieval error char-

12

-90 -60 -30 0 30 60 90Latitude

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

A

Figure 7. Parameter � (for details, see text) as a function of latitude in case of the TOMS a priori data (upper part) and in caseof the simple a priori data (bottom part).

0 100 200 300 400 500 600TIGR2 profile number

0

5

10

15

20

25

30

Perc

ent e

rror

(%

)

Total error

Model error contribution

A priori error contribution

0

10

20

30

0 100 200 300 400 500 600

Figure 8. Total error of the retrieved total ozone values for a subset of the TIGR2 profiles (light grey) and the contributionsto this error of the forward model error (grey) and the a priori error (dark grey).

13

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 220

240

260

280

300

320

Tot

al o

zone

col

umn

(DU

)

TOVS

TOMS

0 - 5 N

Figure 9. Annual cycle of zonal averaged total ozone columns between �

N and � � N retrieved by TOMS (grey line) andTOVS (black line).

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 280

300

320

340

360

380

400

420

Tot

al o

zone

col

umn

(DU

)

TOVS

TOMS

45 - 50 N

Figure 10. Annual cycle of zonal averaged total ozone columns between � � � N and � �� N retrieved by TOMS (grey line) andTOVS (black line).

14

Figure 11. Total ozone (DU) measured by TOVS on August 7, 1989.

Figure 12. Total ozone (DU) measured by TOMS on August 7, 1989.

15

Figure 13. Percent error in the total ozone columns measured by TOVS on August 7, 1989.

acterization. This global distribution of the retrieval errorand also of the contribution of a priori knowledge to the re-trieval gives much more insight in the validity of the ozoneretrievals.

Ozone retrievals for 1989 were compared with TOMS to-tal ozone measurements. This comparison showed good a-greement between the two ozone retrievals both in time andin space (a rms difference between 1% and 3% for zonalmeans and 10% for global gridded measurements). In con-trast with the TOMS measurements, the infrared TOVS mea-surements allow ozone retrieval twice a day (daytime andnighttime retrievals) and in the polar night. A disadvantageis the unstable retrieval when the difference between surfacetemperature and lower stratospheric temperature is small.

Acknowledgments. This research was funded by the SpaceResearch Organization Netherlands (SRON) and the CooperativeResearch Centre for Southern Hemisphere Meteorology. G.L.S.was partly sponsored by NOAA Research grant NA3RJ0202. Wethank P.T. Guimares, D.E. Larko, R.D. McPeters, A.J. Krueger(NASA GSFC), members of the TOMS Nimbus Experiment andOzone Processing Teams, and the National Space Science Data Cen-ter through the World Data Center A for Rockets and Satellites forproviding the TOMS data. Petra Udelhofen and the Bureau of Me-teorology, Melbourne are acknowledged for providing and helpingwith the TOVS data.

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80523.(email: richard@parrot.atmos.colostate.edu)October 27, 1995; revised November 13, 1996; accepted November 13,1996.

This preprint was prepared with AGU’s LATEX macros v4. File preprintformatted April 28, 1998.

With the extension package ‘AGU���

’ by P.W.Daly, version 1.5 from1996/08/20