Modi ed correlated k-distribution methods for remote ...gautier/CV/pubs/Yang_Ricchazzi_Gautier_1999.pdf · Modi"ed correlated k-distribution methods for remote sensing ... compared

*Corresponding author.

Journal of Quantitative Spectroscopy &Radiative Transfer 64 (2000) 585}608

Modi"ed correlated k-distribution methods for remote sensingapplications

Shiren Yang*, Paul Ricchiazzi, Catherine Gautier

ICESS, University of California, Santa Barbara, CA, USA

Received 2 June 1999

Abstract

In this paper we discuss the application of correlated k-distribution methods for the computation ofradiative transfer through the atmosphere in the visible, near IR, and IR spectral regions. Whereas thecorrelated k-distribution is a good approximation overall to line-by-line calculations, it is inaccurate inregions of strong absorber overlap. Several approaches (band splitting, strength-based splitting, double andtriple correlated k) are proposed to modify the conventional correlated k-distribution and improve itspredictions in these regions of absorber line overlap. A comparison with line-by-line computations in theinfrared spectral region demonstrates that a combination of these methods produces brightness temperatureswithin less than 0.25 K of those obtained from line-by-line computations at signi"cantly less computationcosts. In the visible and near IR, when atmospheric scattering is included, the new treatment of overlappingmixed gases also produces excellent agreement with the line-by-line model. ( 1999 Elsevier Science Ltd. Allrights reserved.

1. Introduction

The correlated k-distribution method has been used in radiative transfer models for thecalculation of heating rate in nonhomogeneous atmospheres [1}4]. The correlated k-distributionmethod is an excellent way to approximate line-by-line calculations without excessive computationtimes. However, there are very few radiative transfer codes using the correlated k-distributionmethod that can be applied to remote sensing (e.g., Modtran version 4 [5]). One of the di$culties inimplementation of correlated k-distribution method to a radiative transfer model is the inaccuracycaused by imperfect vertical correlation of absorption features. Although the errors caused by lack

0022-4073/99/$ - see front matter ( 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 2 - 4 0 7 3 ( 9 9 ) 0 0 1 4 0 - 5

of vertical correlation can be partially canceled in a broadband calculation, it may be signi"cant insome of the narrow channels used for remote sensing applications.

The next generation remote sensing sensors will have broad spectral range and high spectralresolution. For example, the atmospheric infra red sounder (AIRS) [6], which will be launched onthe upcoming EOS-PM platform, has 2371 channels in the range 640 to 2676 cm~1 with sub-wavenumber spectral resolution (*l"l

0/1200). In addition, AIRS has four visible and near-

infrared channels. The MODerate resolution Infrared Sensor (MODIS) is another modern instru-ment destined for the EOS platform. It has more than 36 channels covering the VIS (*j+10 nm),the NIR (*j+10}60 nm), and the thermal IR (*l+10}80 cm~1). There will be a need to havespectrally commensurate radiative transfer models to accurately simulate these data and developimproved retrieval algorithms. While a large number of radiative transfer models exist, accuratehigh-resolution spectral models that can be used to investigate cloud/radiation properties to beprovided by instruments such as AIRS and MODIS are missing or cumbersome to use.

In this article, we discuss several of the modi"ed correlated k-distribution methods implementedin our radiative transfer models. SBDIR is a high spectral resolution radiative transfer model forinfrared atmospheric remote sensing. SBDVN is a radiative transfer model for use in visible andnear infrared. Both of these models calculate the radiative transfer in a scattering atmosphere withuser de"ned wavebands, "lter functions, as well as surface and atmospheric conditions.

The correlated k-distribution method assumes that the absorption coe$cients of absorbers arevertically correlated. In general, this condition is satis"ed only approximately, but it is a poorassumption in absorber overlap regions. In these spectral bands there is signi"cant absorption frommore than one molecular species. Previous approaches to treat overlapping absorption includethe multiplication assumption [2], and weighted absorption coe$cients approach [3]. We "ndthat these methods are inappropriate for remote sensing applications and propose alternativeapproaches.

In Section 2, we review the traditional correlated k-distribution method brie#y and discuss theproblems encountered in remote sensing application in detail, particularly, in overlap absorptionregions. Methods to improve the correlated k-distribution method for vertically uncorrelatedabsorbers are introduced in succeeding sections. Section 3 describes a band-splitting method fornarrow waveband channels. Sections 4 and 5 illustrates the absorber strength based band-splittingand multiple correlated k-distribution method we propose. A quantitative error analysis isprovided in Section 6. A brief introduction of the SBDIR/SBDVN models is presented inAppendix A.

2. Traditional correlated k-distribution method

To ensure adherence to the Beer}Bouguer}Lambert law, line-by-line models must operate oververy small spectral intervals, and therefore require many sample points per wavenumber range.A typical remote sensing channel includes a large number of maxima and minima of the absorptioncoe$cient that must be properly resolved, especially when there is signi"cant absorption inlow-pressure regions of the atmosphere. For example, a few thousand spectral sample points arerequired for the the AIRS channels (*l"l/1200) in order to properly account for the variation ofthe absorption coe$cient in the upper troposphere. In the visible and near infrared, about 20 000 to

586 S. Yang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 585}608

40 000 spectral sample points are needed for a 5 nm waveband! Therefore, the same value ofabsorption coe$cient may be treated many times while integrating across a band. This re-samplingis redundant and adds unnecessary computational costs.

The correlated k-distribution method avoids these redundant computations by taking advantageof the fact that the transmission within a band is independent of the spectral sampling order [1,2].Re-ordering the quadrature in order of increasing absorption coe$cient within the given spectralinterval produces a relatively smooth, monotonically increasing function, which requires manyfewer quadrature points to obtain good numerical accuracy. An expression for the radiance, M̧

l, in

the lth atmospheric layer, within a spectral interval, [l1, l

2], can be converted into an equivalent

correlated k form by the following transformation:

M̧l"

1l2!l

1P

l2

l1¸l(kl,l) dl"P

1

0

¸l(k

g,l) dg, (1)

where

gl(k)"P

1

0

fl(k@) dk@ (2)

and fl(k@) is the fraction of the entire bandwidth with a value of line absorption coe$cient between

k@ and k@#dk@. By subdividing the absorption coe$cient range into a suitably large number ofsubintervals, k

i, the frequency distribution of absorption coe$cient strengths can be obtained by

binning and summing over all wavelength intervals, *lj, so that f

l(k

i) may be written as

fl(k

i)"

1l2!l

1

N+j/1K*l

j*k

iK=(k

i, k

i#*k

i), (3)

where=(ki, k

i#*k

i) is a window function that is 1 for values of k between k

iand k

i#*k

i, and

zero otherwise. This expression is normalized so that the cumulative frequency distributionincrement, *g

i"f (k

i)*k

i, speci"es the fraction of the equivalent width for which the absorption

coe$cient is between kiand k

i#*k

i. It follows that g

ican be written as

gl"

n+i/1

fl(k

i)*k

i. (4)

The quantity gl(k) is the ranking of absorption coe$cient in the range 0 (weakest) to 1 (strongest), at

lth atmospheric layer. A key assumption of the correlated k procedure is that the ranking ofdi!erent segments of the spectral band in terms of absorption strength remains "xed as a functionof pressure. For absorption dominated by a single molecular species, a given value of g usuallyrepresents the same (non-contiguous) spectral sub-interval in all vertical layers in the atmosphere.A g value near one corresponds to the spectral subinterval near the line center of the strongest lineand remains associated with this spectral value at all pressure levels in the atmosphere. Similarly,wavenumber intervals with the weakest absorption, at the extreme line wings, are correlatedvertically and share the same g value. This vertical correlation results from the fact that pressurebroadening usually does not alter the relative ranking of absorption coe$cient. Since the corre-lated k-distribution retains information about the vertical correlation of spectral features, there is

S. Yang et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 64 (2000) 585}608 587

no need for ad hoc correction factors (e.g., Curtis Godson coe$cients) which are required to obtainadequate accuracy from band models of atmospheric transmission.

Fu and Liou [3] showed that the conditions for vertically correlated absorption lines are

k(li, p,¹)"k(l

j, p,¹) if k

r(l

i)"k

r(l

j) (5)

and

k(li, p,¹)'k(l

j, p,¹) if k

r(l

i)'k

r(l

j) (6)

where kr

denotes the absorption coe$cient at reference pressure P and temperature ¹. Theyconcluded that these conditions can be satis"ed for a single absorption line, for Elsasser bandmodel, and for the strong- and weak-line limits.

In practice, if a channel having several absorption lines from a single absorber with largestrength di!erence, the weak absorption lines disappear in the low atmosphere, but they may beclearly shown at high altitude due to their temperature and pressure dependence. In this case, (5)and (6) cannot be satis"ed, but the radiance calculated using correlated k-distribution is still goodenough. On the other hand, if a channel is narrow and there is a strong absorption line in spectrallyadjacent bands, (5) and(6) still may not be satis"ed even though there is only one major absorptionline. This is because the pressure broadening of the strong absorption line changes the line shape inthe lower atmosphere, but has a very small e!ect at higher altitude.

The correlated k approach is not accurate when applied to spectral bands that have overlappinglines of di!erent absorbing species, particularly when there are signi"cant di!erences in the verticalpro"les of the absorbing gases. The procedure advocated by Lacis and Oinas [2], is to obtaina separate correlated k-distribution for each absorbing species in the spectral band, treatingspectral features of each species as if they were statistically independent.

For example, when considering two overlapping species, the assumption of statistical indepen-dence allows the average radiance to be written as

M̧ l"1

l2!l

1P

l2

l1¸l dl"

N+j/1

N+i/1

*gi*g@

j¸(k

i, k@

j) (7)

where *giand *g@

jare the cumulative frequency intervals for the "rst and second molecular species,

respectively. The radiance, ¸(ki, k@

j), is computed from the coherent radiative transfer equation

using the absorption coe$cient appropriate to the ith cumulative frequency subinterval of species 1and the jth subinterval of species 2. This formulation requires N2 solutions of the coherent radiativetransfer equation, where N is the number of correlated k sub-intervals used for each species. SinceN is typically greater than 10, this procedure greatly increases the number of invocations of the RTequation solver and signi"cantly reduces the speed advantage of the correlated k approachcompared to the line-by-line method.

To reduce the number of terms, Lacis and Oinas [2] suggested ranking the N2 terms inmonotonic order, and then reblocking the terms back into the original N-interval k-distribution.This approach is based on evenly random distribution of overlapping lines of two absorbing gases.This approach may be a good approximation for broad band calculations but it is not applicable tothe narrow spectral bands. The problem may be demonstrated using a one-layer atmosphere. Ifa single layer is inaccurate, the multilayer case will be also. Consider the AIRS IR channel centeredat 1028.523 cm~1, a typical absorber overlap channel. Major absorbers in the channel are water


Fig. 1. Optical thicknesses of absorbers in AIRS 1028.523 cm~1 channel and their k-distributions in 440 mb layer: (a)Total absorption lines and the absorption lines of individual absorbers. (b) True k-distribution and the k-distributioncalculated by multiplicative law.

vapor and ozone. For simplicity, we considered optical depths instead of absorption coe$cients.We "rst calculate the k-distribution with 16 quadrature intervals in g space for all absorbers. In thissingle-layer case, the spectral correlation is explicitly included, and the resulting k-distribution maybe considered optimal for speci"ed number of quadrature intervals. We then calculate thek-distribution with 16 quadrature intervals of two absorbers separately. The k-distributions of twoabsorbers can then be combined using expression (7), ranked, reblocked and averaged to 16 termsas suggested by Lacis and Oinas. The absorption lines (left) and resulting k-distributions (right) areshown in Fig. 1. The k-distributions obtained by the multiplicative assumption are alwaysoverestimated and too sharp when g approaches 1, and underestimated and too #at wheng approaches 0.

The optical depths required for solving a radiative transfer equation is the product of molecularcolumn density and the k-distribution of absorption coe$cient. In most cases, the k-distribution ofabsorption coe$cient is precalculated for each category of molecular species (usually, water vapor,ozone and others) separately. In current radiative transfer models the cross correlation of spectrallines between overlapping species is not included. Lacking this information, the conventionalcorrelated k-distribution method is not accurate in absorber overlap regions, even for a homogene-ous atmosphere.

In remote sensing application, the error is signi"cant when there are more than one absorbingmolecular species in a channel with di!erent vertical pro"les and comparable strength. In ourstudy, based on a vertical pro"le, we roughly divide absorbers in the atmosphere into threecategories, water vapor, ozone and mixed (all others). Density per unit height of water vapordecreases with altitude much faster than others, that of ozone increases with altitude and reaches itslargest value above the tropopause, and then decreases with altitude. Several approaches toimprove the accuracy of correlated k-distribution method in absorber overlapping regions areproposed and described in detail in the following sections.


3. Band-splitting method for narrow bandwidth channels

When the bandwidth of a remote sensing channels is very narrow, only a few absorption lineshave a perceivable a!ect on the sensed data. If they are from two absorbers having di!erent verticalpro"les with comparable strength, the correlated k-distribution method may fail. However, inmany cases one channel can be split into several subbands with only a single dominant absorber ineach subband. In this way, the vertical correlation of absorption lines, hence the accuracy of thecorrelated k-distribution in each subband can be increased signi"cantly.

To illustrate the band-splitting method, we compute the brightness temperature of the AIRS IRchannels using our SBDIR/SBDVN radiative transfer model. These results were compared withthe results computed directly from LBLRTM code [7] with the same "lter function, atmosphericpro"le and surface condition. About 85% of the 2371 channels have a brightness temperature errorless than 0.25 K using the conventional correlated k-distribution with 16 intervals. The remainingchannels, for which the error is larger are in regions of absorber overlap. Since the bandwidths ofthe AIRS IR channels are between 0.5 to 2 cm~1, absorbers with di!erent vertical pro"les can beseparated by properly dividing a channel into several subbands so that only absorbers with similarvertical pro"les are signi"cant in each subband. We call this approach the band splitting in spectraldomain (BSSD). This approach is very e!ective when spectral features from di!erent species occupydi!erent, but well separated, spectral regions of the sensor band.

Fig. 2 shows the absorption lines of AIRS channels centered 1355.213 cm~1 at pressure levels 64,291 and 572 mb, respectively. The major absorbers of this channel are water vapor and methane.Since these absorbers have di!erent vertical pro"les, vertical correlation is improved signi"cantly ifthe channel is split into three subbands as shown in Fig. 2. Brightness temperature di!erence (withLBLRTM calculations) for the channel decreases from from 1.388 to 0.007 K. When one AIRSchannels is split into several subbands, the number of intervals in g space used to treat eachsubband can be reduced from 16 to 8 without sacri"cing accuracy. When band splitting is used inregions with absorber overlap, about 95% of the 2371 AIRS channels have brightness temperatureerror less than 0.25 K compared to the LBLRTM results. To "nd this technique is adequate todisentangle the e!ects of overlapping lines in many AIRS IR channels, while maintaining the speedadvantages of the correlated k method. However, with this technique, the brightness temperatureerror of 123 AIRS channels is still larger than 0.25 K, and there are 51 channels with brightnesstemperature errors greater than 0.75 K and up to 1.5 K. In addition, the band-splitting technique isnot suitable for sensors including visible and near infrared channels, such as MODIS. At theseshorter wavelengths, even relatively narrow channels contain a large number of absorption lines,and e!ective separation into a few subbands is no longer possible. Therefore, we developed thestrength based band splitting, double and triple correlated k-distribution methods, depending onthe nature of spectral line overlap. One of these methods often provides signi"cant improvementover the simple band splitting method.

4. Strength-based band-splitting correlated k-distribution for absorber overlapping wavebands

One major drawback of the spectral band splitting method is its inability to provide su$cientseparation between overlapping absorbers in spectral channels containing many lines. Separation


Fig. 2. The absorption lines in AIRS 1355.213 cm~1 channel. The channel is divided into three subbands as shown bydashed lines: (a) absorption lines in 64 mb layer, (b) absorption lines in 291 mb layer, (c) absorption lines in 572 mb layer.

may be increased by dropping the requirement that each sub-band be composed of spectrallycontiguous regions. This is the basic idea of the strength-based band-splitting correlated k-distribution method (SBBS). In this technique a remote sensing channel is split into severalsubbands based on the strength of absorption coe$cients of one of the absorbers in a speci"catmospheric layer (the origin layer). We designate this species the primary absorber, while all othersare classi"ed as secondary absorbers. The conventional k-distribution resorting process (Section 2)is used to de"ne a mapping from wave number to cumulative frequency distribution,

k(li)Pk(l(g

j))"k

jj"1,J, (8)

k@(li)Pk@(l(g

j))"k@

jj"1,J, (9)


where k and k@ are absorption coe$cients of primary and secondary absorbers. In the SBBSapproach the band splitting is accomplished by dividing the original channel into a severalcontinuous intervals in the g domain. The correlated k transformation is based on conditions froma single atmospheric layer, and the mapping from l to g space is guaranteed to producea monotonic ordering of the primary absorption coe$cient only in the originally chosen atmo-spheric layer. Since the secondary absorbers are not correlated with the primary absorber, thek-distribution transformation will gather a wide range of absorption coe$cient from the secondaryabsorbers into each primary quadrature interval. To account for this wide variation, we allocatea number of secondary quadrature intervals to span the full range of absorption strength due tosecondary absorbers in each primary quadrature interval. The secondary quadrature intervals areweighted according to the results of a correlated k-distribution applied to the secondary absorbers.Including the e!ect of both primary and secondary absorbers, we may write the radiance as:

¸"

J+j/1

Mj

+m/1

*gj*g@

m¸(k

j, k@

m) (10)

where Mjis the number of subintervals used to resolve the variation of the secondary absorption

coe$cient within each primary g-interval. In practice, the number of secondary g-subintervals, Mj,

may be reduced greatly as j approaches J. In this limit the primary quadrature intervals representa rapidly decreasing fraction of the entire equivalent width. Hence, fewer secondary quadratureintervals are required for primary g-intervals near 1. Though Eq. (10) appears similar to the Lacisand Oanis expression (7) for the random overlap assumption, it is important to emphasize that theSBBS method explicitly includes correlation e!ects between overlapping absorbers and the numberof total intervals is greatly reduced.

Figs. 3}5 illustrate how the method is applied to the AIRS 1028.523 cm~1 channel. In thischannel, water vapor and ozone are major absorbers. Water vapor absorption is dominant in thelow atmosphere, while ozone absorption gains importance at midaltitudes and high altitudes. Weuse water vapor as the primary absorber and ozone the secondary absorber. Primary andsecondary absorption coe$cients in each atmospheric layer are interpolated to 5000 wave numberpoints. They are then replaced by ordering indices from 1 to 5000. Fig. 3 shows the optical depths ofthe primary and secondary absorber at 64, 291 and 985 mb pressure levels. The channel is split into6 subbands by sorting the optical depths of the primary absorber at 985 mb pressure level andthresholding the data using the modi"ed Gaussian quadrature points: 0.0, 0.370, 0.600, 0.875, 0.975,0.995 and 1.000. The sorted ordering indices are used to reallocate the optical depths of absorbersin all atmospheric layers to appropriate subband as shown in Fig. 4. In the bottom plot of Fig. 4,the optical depths of the total absorber is almost the same as that of the primary absorber becauseozone absorption is much smaller than that of water vapor in the lower atmosphere. The correlatedk-distribution method is then applied to the reallocated optical depths in each interval (subband)with appropriate number of modi"ed Gaussian intervals, as shown in Fig. 5. Here, 6, 6, 4, 4, 3, and3 subintervals are used in the six subbands. Though Figs. 4 and 5 show the absorption lines andk-distribution of all absorbers, in practice they are calculated for each absorber separately and thenadded. This is necessary if a pre-calculated k-distribution database is used. The brightnesstemperature of the AIRS 1028.523 cm~1 channel calculated by LBLRTM code for the standardmidlatitude summer atmospheric pro"le in clear sky over ocean surface is 267.741 K. When


Fig. 3. The primary (water vapor) and secondary (ozone) absorption lines of AIRS 1028.523 cm~1 channel in 64, 291 and985 mb pressure levels. The channel is divided into three subbands as shown by dashed lines: (a) absorption lines in 64 mblayer, (b) absorption lines in 291 mb layer, (c) absorption lines in 572 mb layer.

16 quadrature intervals are used, the conventional correlated k-distribution method has an errorof 2.1 K. The error cannot be reduced by increasing the number of quadrature intervals. The erroris reduced to 1.2 K if the channel is split into 4 subbands in the spectral domain. With thestrength-based band-splitting correlated k-distribution method, the error is reduced to 0.13 K.

5. Double correlated k-distribution method

The Double Correlated k-Distribution (DCK) method is another approach we have developedto improve the accuracy in absorber overlap wavebands. In a manner similar to the SBBS


Fig. 4. The strength based band-splitting correlated k-distribution method. The AIRS 1028.523 cm~1 channel is splitinto 6 subbands based on the magnitude of optical depth at 985 mb pressure level: (a) total optical depth in 64 mb layer,(b) total optical depth in 291 mb layer, (c) total optical depth in 985 mb layer.

technique, the primary absorber is used to de"ne a correlated k transformation that maps wavenumber points into the g domain. However, in the DCK method a new transformation rule isderived for each atmospheric layer, based on the absorption coe$cient of the primary species. As inthe SBBS approach, a second stage correlated k-distribution is applied to the secondary absorberswithin each primary g interval, resulting in the same equation for radiance (Eq. (10)) as a function ofthe absorption coe$cients.

To implement the DCK method, the absorption coe$cients of primary and secondary absorbersin each atmospheric layer are sorted simultaneously with the primary absorber as pivot variable.The sorted primary absorption coe$cients in each layer are transformed to J primary quadrature


Fig. 5. The strength based band-splitting correlated k-distribution method. k-distributions are calculated for opticaldepths in each subband: (a) k-distribution in 64 mb layer, (b) k-distribution in 291 mb layer, (c) k-distribution in 985 mblayer.

intervals in g space by a modi"ed Gaussian quadrature points. The simultaneously sorted second-ary absorption coe$cients are allocated to corresponding primary intervals. The secondaryabsorption coe$cients in the jth primary interval are then transformed into M

jsubintervals by

another modi"ed Gaussian quadrature points. The total k-distribution in g space of all absorbers isthe sum of k-distribution in the primary and in the secondary intervals.

We again use the AIRS 1028.523 cm~1 channel with water vapor as the primary and ozone asthe secondary absorbers to illustrate the DCK method. Six quadrature intervals are used for theprimary absorber, and 6, 6, 4, 4, 3 and 3 secondary quadrature intervals are applied to thecorresponding primary quadrature intervals, respectively. The sorted and binned optical depths of


Fig. 6. The DCK method. Cumulative and k-distribution of primary (water vapor) and secondary (ozone) absorbers ofAIRS 1028.523 cm~1 channel at 64, 291 and 985 mb pressure levels. The channel is divided into 6 intervals by primaryabsorber of each layer: (a) 64 mb layer, (b) 291 mb layer, (c) 985 mb layer.

the primary absorber and their k-distribution at 64, 291 and 985 mb pressure levels are shown inFig. 6. The k-distributions of secondary absorber in each primary interval are shown in the sameplots. The k-distributions of all absorbers in each atmospheric layer are achieved by summing thek-distributions of primary and secondary absorbers in that atmospheric layer. At high altitude thewater vapor column density is very small, and the total absorption coe$cient almost coincides withthat of secondary absorber. However, water vapor is still eligible to be the primary absorber asshown in the top plot of Fig 6. In comparison with the SBBS method, the absorption lines ofprimary absorber (water vapor) are averaged within 6 intervals in DCK rather than 26 intervals inSBBS method. Nevertheless, the brightness temperature error is reduced to less than 0.1 K in DCK


method because in this channel the secondary absorber has a much more important contributionto the brightness temperature measured at the top of atmosphere.

The basic idea of DCK can be extended to Triple Correlated k-distribution method (TCK) forthe channels where there are three absorbers with non-identical vertical pro"les and comparablestrengths. Since there are only three vertical pro"le categories (water vapor, ozone and others) inthe atmosphere, TCK method is probably enough to cover all cases. Up to 60 quadrature intervalsor number of invocations of radiative transfer equation solver may be required for TCK, but this isstill much less than that required for the line-by-line approach (500 per cm~1 typically).

Because the transformation rule (klPkg) changes with altitude, the DCK method will have

less vertical correlation than the SBBS approach. This lack of vertical correlation will degradethe accuracy of the DCK method compared to the SBBS technique. Besides, the number ofprimary intervals used in the DCK method is several times less than that used for the secondary.The DCK method is suitable for the spectral bands such as the ozone absorption band around1000 cm~1 where water vapor is used as the primary and ozone as the secondary absorber. Onthe other hand, the SBBS approach introduces its own sources of error. Since the SBBS methodhas a "xed transformation rule at all altitudes, atmospheric layers far from the origin layermay have irregular and discontinuous variations of the primary absorption coe$cient as afunction of g. If the number of g-intervals is "xed, this increased variation is re#ected in largerquadrature errors. This is especially true when treating absorption lines of same molecular speciesthat have di!erent lower state energies or pressure broadening halfwidths [8]. Therefore, the SBBDmethod is more suitable for spectral bands for which the lower state energy for the transition is lessdiverse.

The SBBS and DCK methods can be combined to achieve a more accurate correlatedk-distribution method. A channel is "rst split into several subbands based on the strength ofone absorber. The double correlated k-distribution method is then applied to each subband. Figs. 7and 8 show this method used in AIRS 1011.102 cm~1 channel at 985 and 64 mb pressure levelsrespectively. This channel is at the edge of an ozone absorption band with one strong water vaporabsorption line and few weak absorption lines of H

2O, CO

2and NH

3. We use the strength of the

ozone optical depth at 985 mb to split the channel into 4 subbands. Figs. 7a and 8a show how theoptical depths of primary absorber is split into 4 subbands. Each subband of the primary absorberis divided into several subintervals at the Gaussian quadrature points, as shown in the Figs. 7b and8b. The secondary absorption lines in subintervals of each subband of the primary absorber aresorted and their k-distributions are calculated and plotted in the Figs. 7c and 8c. There are a totalof 54 quadrature intervals in this example. The brightness temperature error of the channel isreduced from 1.06 K with the conventional 16 quadrature intervals correlated k-distribution, to0.06 K with this method. We have also applied this technique to MODIS channel 35(13.785}14.085 lm). This channel has about 2000 absorption lines, including three major absorbers:CO

2, O

3and H

2O. Their absorption coe$cients are represented by about 110 000 wave number

points in LBLRTM. The calculated brightness temperature error using the combined SBBS andDCK method with a total of 75 quadrature intervals is a few hundredths of degree signi"cantlysmaller than the 0.8 K error produced by the conventional correlated k-distribution method. Itshould be noted that the line-by-line calculation requires more than ten thousand wave numbersteps. This di!erence in computational steps becomes even more important when scattering iscomputed.


Fig. 7. A combination of SBBS and DCK methods for AIRS 1011.102 cm~1 channel in the 985 mb pressure level. Thechannel is divided into 4 subbands based on the strength of primary absorption lines at 985 mb. DCK is applied to theprimary and secondary absorbers in each subband: (a) Optical depth of primary absorber, (b) k-distribution of primaryabsorber, (c) k-distribution of secondary absorber.

6. Validation

In general the most convincing validation of a physical model is through direct comparison withobservations. However, in the present case the di$culty associated with characterizing the state ofthe atmosphere makes this approach less conclusive than direct comparison of computationalresults with a detailed line-by-line radiative transfer model. In Section 6.1, we consider the infraredspectral region, in Section 6.2 the visible and near infrared region and Section 6.3 the cloudy skycondition.


Fig. 8. Same as Fig. 7. A combination of SBBS and DCK methods for AIRS 1011.102 cm~1 channel at 64 mb: (a) Opticaldepth of primary absorber, (b) k-distribution of primary absorber, (c) k-distribution of secondary absorber.

6.1. Infrared region

In the infrared spectral region from 600 to 2700 cm~1, the brightness temperatures for a mid-latitude atmospheric pro"le in clear sky conditions are calculated by LBLRTM code directly. Twopairs of datasets are used for validation of the methods. The "rst pair of datasets is that calculatedby LBLRTM, and that calculated by SBDIR with a two-wavenumber bandwidth. No "lter isapplied in these calculations. Fig. 9 shows the smoothed transmittance of major absorbers (a), thebrightness temperature calculated by LBLRTM and Modtran3.5 (b), and the brightness temper-ature di!erence between the dataset calculated by LBLRTM and by SBDIR using the conventional


Fig. 9. (a) Transmittance within spectral range, (b) the brightness temperature calculated by LBLRTM and Modtran3.5,(c) the absolute di!erence of brightness temperature calculated by LBLRTM and SBDIR in MIDSUM atmosphericpro"le in clear sky.

and the proposed modi"ed correlated k-distribution methods described above (c). With theconventional correlated k-distribution method (that introduced by Lacis and Oinas [2] forexample), the brightness temperature errors is signi"cant in absorber overlapping regions. Forexample, in the 700}760 cm~1 region, CO

2absorption is dominant with additional H

2O and

O3

absorption. In the 1000}1080 cm~1 band, O3

absorption is dominant with additional H2O and

CO2

absorption. In the 1200}1400 cm~1 spectral region, H2O and CH

4absorption is prominent

with some additional N2O absorption. In the 2040}2140 cm~1 region, H

2O and CO

2are major

absorbers. In the 2250}2400 cm~1 waveregion, there is absorption by CO2, N

2and H

2O. All these

spectral regions have overlapping absorbers with comparable strengths. When the conventional


k-distribution is used, the largest error is as large as 2 K. All errors are reduced to less than 0.25 Kby modi"ed correlated k-distribution as shown in Fig. 9c. The number of quadrature intervals usedfor each channel is an average of 19.

We have also used SBDIR to model the 2371 AIRS IR channels. Each channel has a bandwidthof l

0/1200 where l

0is the center wave number. A trapezoidal AIRS IR "lter function

'(l)"0,1 linear for l0!

l0

2400to l

0!

l0

4800(11)

'(l)"1 for l0!

l0

4800to l

0#

l0

4800(12)

'(l)"1,0 linear for l0#

l0

4800to l

0#

l0

2400(13)

is convolved with the radiances calculated by LBLRTM.Fig. 10 shows the brightness temperatures of all 2371 AIRS IR channels calculated by LBLRTM

and Modtran3.5. Also shown is the di!erence of brightness temperatures calculated by LBLRTMand SBDIR, respectively. The brightness temperature error is signi"cant in the same absorberoverlapping regions as before, though AIRS IR channels are narrower and a "lter is applied. Thelargest error also reaches 2 K in the same region. The errors can also be reduced to less than0.25 K by a combination of the modi"ed correlated k-distribution methods.

6.2. Visible and near infrared region

For validation of the modi"ed correlated k-distribution methods in visible and near infraredspectral regions, we "rst develop a line-by-line code to calculate the transmittance and #ux at thetop and bottom of the atmosphere. A spectral resolution of 1 nm is used for the 0.445 to 1.0 lmband, while a 2 nm resolution is used for the 1.0 to 5.0 lm range, for a total of 2555 spectralintervals. In each spectral interval, the optical depth of all atmospheric layers is calculated by theLBLRTM code and interpolated or resampled to the same number of points. The radiative transferequation is solved using DISORT [14] with inputs of optical depth of gaseous absorbers togetherwith solar irradiance, surface albedo, and other optical parameters. The #uxes and transmittanceare calculated by the line-by-line code for tropical and subarctic winter atmospheric pro"les inclear sky conditions and with 03 and 303 solar zenith angles. In these calculations the surfacealbedo is taken as 0.2, and no aerosol is included. This is a very time consuming computation, asDISORT is called in excess of 10,000 time for each spectral interval.

For the comparison, SBDVN is used to calculate the same parameters over the same spectralintervals and with the same atmospheric and surface conditions. The solar irradiance is taken intoaccount during the calculation of absorption coe$cients as indicated in expression (A.3) inAppendix. The #ux at top of the atmosphere and the transmittance calculated by the line-by-linecode are shown in Fig. 11a and b. The di!erences between the data calculated by the line-by-linecode and by the conventional and modi"ed correlated k-distributions are shown in the same plots.Their di!erences are small for wavelengths less than 1.0 lm. The maximum di!erence of #ux at topof the atmosphere is less than 0.5%. The areas of largest discrepancy are in the water vapor and


Fig. 10. Transmittance (smoothed), brightness temperature calculated by LBLRTM and Modtran3.5, and di!erencebetween LBLRTM and SBDIR in infrared spectral region using the MIDSUM pro"le and in clear sky. The full AIRSspectral range is divided into the three panels shown above.

oxygen overlap regions. There is no absorber overlap around 0.93 lm, but a small discrepancy iscaused by the temperature dependence of line strength due to the altitude variation of lower statepopulation of certain water vapor transitions.

In the near infrared absorber overlap regions, most errors are in the spectral intervals where thetransmittance changes dramatically. These are the regions that have more than one absorber withdi!erent vertical pro"les and comparable strength. The maximum di!erence in transmittance is onthe long wavelength side of the near infrared (j'2 lm). The maximum transmittance error of 0.06can be reduced to less than 0.02, when the modi"ed correlated k-distribution is applied. For mostremote sensing applications, the conventional correlated k-distribution method with 16 quadratureintervals is good enough.


Fig. 11. (a) Flux at top of atmosphere calculated by line-by-line code and its di!erence to the data calculated bycorrelated k-distribution method, (b) transmittance calculated by line-by-line code, (c) di!erence to the data calculated byconventional and modi"ed correlated k-distribution method in Vis/Nir, Tropic atmosphere, sza"303.

6.3. Cloudy sky

To evaluate SBDIR's multiscattering capability, we use the 16-term conventional correlatedk-distribution and the modi"ed correlated k-distribution methods described above to compute theradiances at top of atmosphere for low (0}1 km), midaltitude (2}3 km), and cirrus clouds(10}11 km). For the low and midaltitude clouds, we made computations for several cloud opticaldepths between 1 and 40, and a cloud e!ective radius of 8 lm. The cloud optical depth for the cirruscloud calculations ranged from 0.1 to 5.0. Cirrus clouds are modeled with spherical ice particles of


e!ective radius of 106 lm. A background oceanic aerosol with 23 km visibility is included in allcases. Twenty four representative AIRS channels are calculated. These channels sample transpar-ent and strong absorption regions, including species overlap regions. Radiances for these channelsare also calculated by the line-by-line code with 1000 wave number points for each AIRS IRchannel. These calculations use the same cloud and aerosol parameters. The brightness temper-ature di!erences *¹

Bbetween the line-by-line method and that calculated by conventional

correlated k-distribution method are shown in Fig. 9a}f.Fig. 9a}f show that clouds have a very small e!ect on *¹

Bin window and strong absorption

spectral regions. Gas absorption is very small in the window region, where the brightnesstemperature depends primarily on the temperature of the ground surface and cloud layer. In strongabsorption region, the brightness temperature is determined by the absorption from molecularspecies above the cloud layer.

For the channels in the absorbers overlapping regions, when the brightness temperature iscalculated by conventional correlated k-distribution method, the presence of clouds a!ects *¹

Bin

the direction of reducing the discrepancy. In these cases the cloud layer acts to screen out thecontribution of an unmixed gas that originates below the cloud layer, thereby diminishingabsorption by overlapping species. The e!ect of increased cloud optical depth is less signi"cant forlower level clouds, because there is still substantial overlapping species above the clouds. On theother hand, when the brightness temperature is calculated by the modi"ed correlated k-distributionmethod, *¹

Bis small and not signi"cantly a!ected by the presence of clouds (Fig. 12).

7. Summary and discussion

In this paper, we have described several new techniques to improve the accuracy of thecorrelated k-distribution method, particularly as it applies to the treatment of overlapping absorb-ers. The methods that seem most promising are the band-splitting (BS), strength-based band-splitting (SBBS), double (DCK) and triple (TCK) correlated k-distribution methods and theircombinations. The optimal method to use depends on the details of the line overlap in the band.Band-splitting in the spectral domain is quite simple and needs less quadrature intervals. It is quitee!ective for many narrow bandwidth channels for which the major absorption lines can be easilyseparated into a few subbands. When the distribution of absorption lines is complex, the SBBS andDCK methods are more suitable. Error can be reduced signi"cantly with a total of 30 to 50quadrature intervals. When a channel is a!ected by strong absorption lines outside the channel,a combination of SBBS and DCK can reduce the errors to a reasonable value (e.g., smaller than thenoise equivalent brightness temperature of infrared sensors) without a signi"cant increase of thenumber of quadrature levels. For instance, only a few AIRS IR channels have three absorbers withdi!erent vertical pro"les and comparable strengths. TCK is an e!ective tool to improve theiraccuracy. When these methods are used properly, the error in brightness temperature in theinfrared can be reduced from 2 to 0.25 K or less, with 19 quadrature intervals per channel, onaverage. The error in transmittance in the near infrared spectral band can also be reduced.Choosing the optimal method and the number of quadrature intervals requires an in-depthanalysis of the parameters which determine the strength and shape of an absorption line in a givenatmospheric layer.


Fig. 12. Brightness temperature deviation from line-by-line model for 24 typical AIRS channels calculated by conven-tional k-distribution with (a) cirrus, (b) mid-altitude, (c) low cloud, and modi"ed correlated k-distribution with, (d) cirrus,(e) midaltitude, (f ) low cloud.

The line strength is determined not only by the category and intensity of the absorbers, but it isalso a!ected by the lower state energy for transition of the absorption line. Absorption lines of samemolecular species may have di!erent values of the lower state energy for transition. Since the linestrength is roughly proportional to the exponential of the product of temperature and lower stateenergy for transition, the vertical correlation in a spectral band may not be perfect even when thereis only one absorber in the band. This e!ect causes the small error around 0.93 lm water vaporabsorption band, for instance, where water vapor is the only signi"cant absorber. This error isusually much smaller than that caused by overlap of absorbers.


The radiance at the top of the atmosphere is of prime importance in remote sensing applications.The error in the radiance computed by the conventional correlated k-distribution is small inwindow areas and strong in absorption bands. In strong gas absorption bands, the radiance at thetop of the atmosphere is determined mainly by the absorbers at high altitude.

In terms of heating rate, large errors may be encountered in the strong absorption bands inabsorber overlapping regions if the heating rate is estimated using the data obtained by conven-tional correlated k-distribution.

Acknowledgements

This work is supported by the Atmospheric Infra Red Sounder (AIRS) Project of theNational Aeronautics and Space Administration under Grant NAS5-31374, and in part by theAtmospheric Radiation Measurement Program of the US Department of Energy under Grant96ER61986.

Appendix A. SBDIR/SBDVN system implementation

Our goal was to develop a model that could be used to consistently simulate radiation in theshortwave and longwave regions of the spectrum in the presence of atmospheric absorbers (gasesand aerosols) and scatterers (clouds and aerosols). The model we now present is calledSBDIR/SBDVN (Santa Barbara DISORT InfraRed/Santa Barbara DISORT Visible and Nearinfrared).

Central to SBDIR/SBDVN are certain key components of the LBLRTM [8] and SBDARTcomputer programs [9], and our new program packages, KDIST, for computing the conventionaland modi"ed correlated k-distribution of absorbers. A block diagram indicating the main compo-nents of our model is shown in Fig. 13. It is actually composed of two independent programs:LLRTM and SBD. The LLRTM consists of the LBLATM, CONTNM, LBLF4, HIRAC1, andKDIST blocks to calculate the correlated k-distribution of absorption coe$cients in g space. TheSBD program consists of subroutines required to calculate input parameters for the DISORTradiative transfer model.

In the block diagram, LBLATM and CONTNM compute the atmospheric pro"le and con-tinuum contribution, respectively. LBLF4 and HIRAC1 compute the absorption coe$cients ofabsorbers based on the HITRAN96 database [10]. This continuum and database are now beingmodi"ed to CKD}2.3 and HITRAN}96}w}toth}161 respectively. Three HIRAC1 packages,denoted HIRAC1A, HIRAC1B and HIRAC1C, are used for calculating the absorption coe$cientsof up to three absorbers simultaneously and independently required by the DCK and TCKalgorithms. All subroutines that are dispensable for SBDIR/SBDVN are deleted or simpli"ed forsake of computing e$ciency.

The program packages in KDIST, including KOCK, KSCK, KDCK and KTCK, are part of theLLRTM program. They read in the absorption coe$cients to calculate the conventional, andmodi"ed correlated k-distributions.


Fig. 13. Block diagram of SBDIR/SBDVN.

The sensor "lter function is taken into account during the calculation of the absorptioncoe$cients. When a "lter is included, the mean radiance is given by:

M̧l"P'(l)¸

l(kl,l) dl"P

1

0

¸l(k

g,l) dg (A.1)

where '(l) is the "lter function. The sensor "lter function is included by writing the frequencydistribution of the absorption coe$cients as

fl(k

i)"

+Nj/1

D*vj*ki D'(lj)=(k

i, k

i#*k

i)

+Nj/1

*lj'(l

j)

. (A.2)

The e!ect of the "lter function is to reduce the contribution of frequency intervals where the valueof '(l) is small.

In visible and near infrared, the solar spectrum datasets used have subwavenumber spectralresolution [11}13]. The waveband used in the pre-calculated correlated k-distribution in g space is5 nm or larger. Therefore, the nongray spectral correlation of the solar irradiance with theabsorption features cannot be ignored. Therefore, the solar irradiance F

0(l) is also taken into


account during the calculation of the absorption coe$cient in the following manner:

fl(k

i)"

+Nj/1

D*vj*ki D[F0(l

j)]=(k

i, k

i#*k

i)

+Nj/1

*lj[F

0(l

j)]

. (A.3)

In each waveband used to calculate the correlated k-distribution, the maximum solar irradiance inthe waveband is output to SBD as the solar irradiance in the waveband to solve the radiativetransfer equation.

References

[1] Goody R, West R, Chen L, Crisp D. JQSRT 1989;42:539}50.[2] Lacis AA, Oinas V. J Geophys Res 1991;96(d5):9027}63.[3] Qiang Fu, Liou KN. J Atmos Sci 1992;49(22):2139}56.[4] Mlawer EJ, Taubman SJ, Brown PD, Iacono MJ, Clough SA. J Geophys Res 1997;102(d14):16663}82.[5] Berk A, Bernstein LS, Acharya PK, Robertson DC, Adler-Golden S, Anderson GP, Chetwynd JH, Snell HE,

Vail JJ. Modtran SW and VNIR spectral radiances and #uxes. Proceedings of the 20th Annual Conference onAtmospheric transmission Models, 10}12 June, Air Force research laboratory, Hanscom AFB, MA 01731-3010,1997.

[6] Susskind J, Joiner J, Chahine MT. Determination of temperature and moisture pro"les in a cloudy atmosphereusing AIRS/AMSU. In: Chedin A, Chahine MT, Scott NA, editors. High spectral resolution infrared remote sensingfor earth's weather and climate studies. Berlin: Springer, 1993, p. 149}61.

[7] Clough SA, Kneizys FX, Shettle EP, Anderson GP. Atmospheric spectral transmittance and radiance: FASCOD2.Proceedings of the Sixth Conference on Atmospheric Radiation, Williamsburg, VA, 1986, p. 141.

[8] Clough SA, Kneizys FX, Rothman LS, Gallery WO. Atmospheric spectral transmittance and radiance: FAS-COD1B. SPIE, vol. 277, Atmospheric transmission. 1981, p. 152}66.

[9] Ricchiazzi P, Yang S, Gautier C, Sowle D. Earth Interactions. Bull Am Meteorol Soc 1998;79(10):2101}14.[10] Rothman LS, Gamache RR, Tipping RH, Rinsland CP, Smith MAH, Chris Benner D, Malathy Devi V, Flaud J-M,

Camy-Peyret C, Perrin A, Goldman A, Massie ST, Brown LR, Toth RA. JQSRT 1992;48(N5-6):469}507.[11] Kneizys FX, Abreu LW, Chetwynd JH, Anderson GP, Shettle EP, Berk A, Bernstein LS, Robertson DC, Acharya

P, Rothman LS, Selby JEA, Gallery WO, Clough SA. The MODTRAN 2/3 Report and LOWTRAN 7 MODEL.Philips Laboratory, Geophysics Directorate, PL/GPOS, 29 Randolph Rd., Hanscom AFB, MA 01731-3010, 1996.

[12] Kurucz RL. Synthetic infrared spectra. In: Rabin DM, Je!eries JT, editors. Infrared solar physics, IAU Symposim,vol. 154. Norwell, MA: Kluwer Acad., 1992.

[13] Kurucz RL. The solar spectrum: atlases and line identi"cations. In: Sauval AJ, Blomme R, Grevesse N, editors.Laboratory and astronomical high resolution spectra, Astron Soc of the Paci"c Conference Series, vol. 81. 1995.

[14] Stamnes K, Tsay S-C, Wiscombe WJ, Jayaweera K. Appl Opt 1988;27:2502}9.


Documents

Modi ed correlated k-distribution methods for remote ...gautier/CV/pubs/Yang_Ricchazzi_Gautier_1999.pdf · Modi"ed correlated k-distribution methods for remote sensing ... compared