objective assessment of the information content of visible ...u0476903/web/papers/jam2326_p1.pdfthe...

Objective Assessment of the Information Content of Visible and Infrared RadianceMeasurements for Cloud Microphysical Property Retrievals over the Global Oceans.

Part I: Liquid Clouds

TRISTAN S. L’ECUYER, PHILIP GABRIEL, KYLE LEESMAN, STEVEN J. COOPER, AND GRAEME L. STEPHENS

Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

(Manuscript received 4 November 2004, in final form 9 June 2005)

ABSTRACT

The importance of accurately representing the role of clouds in climate change studies has becomeincreasingly apparent in recent years, leading to a substantial increase in the number of satellite sensors andassociated algorithms that are devoted to measuring the global distribution of cloud properties. The physicsgoverning the radiative transfer through clouds is well understood, but the impact of uncertainties inalgorithm assumptions and the true information content of the measurements in the inverse retrievalproblem are generally not as clear, making it difficult to determine the best product to adopt for anyparticular application. This paper applies information theory to objectively analyze the problem of liquidcloud retrievals from an observing system modeled after the Moderate Resolution Imaging Spectroradi-ometer (MODIS) instrument currently operating on the Aqua and Terra platforms. It is found that fourdiagnostics—the retrieval error covariance, the information content, the number of degrees of freedom forsignal, and the effective rank of the problem—provide a rigorous test of an observing system. Based onthese diagnostics, the combination of the 0.64- and 1.64-�m channels during the daytime and the 3.75- and11.0-�m channels at night provides the most information for retrieving the properties of the wide variety ofliquid clouds modeled. With an eye toward developing a coherent representation of the global distributionof cloud microphysical and radiative properties, these four channels may be integrated into a suitablemultichannel inversion methodology such as the optimal estimation or Bayesian techniques to provide acommon framework for cloud retrievals under varying conditions. The expected resolution of the observingsystem for such liquid cloud microphysical property retrievals over a wide variety of liquid cloud is alsoexplored.

1. Introduction

Clouds play an important role in the regulation of theearth’s climate. They influence both the amount of so-lar energy that reaches the earth’s surface and theamount that is radiated back to space and, therefore,represent a critical factor governing global energy bal-ance (Liou 1986). Furthermore, clouds play an impor-tant role in many chemical processes within the atmo-sphere acting as a surface for chemical reactions andproviding a mechanism for the removal of aerosol par-ticles through scavenging (Seinfeld and Pandis 1998).There is a body of evidence that suggests that humanactivity may affect climate by altering cloud micro-

physical properties as well as their vertical location andspatial distribution. Thus, quantitative global recordsof cloud microphysical properties are fundamental toanswering many of the questions posed by the climatechange community. It is not surprising, then, thatworldwide inference of cloud microphysical propertiescontinues to be a focus of a growing number of instru-ments, such as the Advanced Very High ResolutionRadiometer (AVHRR) aboard the Geostationary Op-erational Environmental Satellite (GOES), the Moder-ate Resolution Imaging Spectroradiometer (MODIS)aboard the Earth Observing System (EOS) Aqua andTerra platforms, the Polarization and Directionalityof the Earth’s Reflectances (POLDER) aboard thePolarization and Anisotropy of Reflectances for Atmo-spheric Sciences Coupled with Observations from a Li-dar (PARASOL) satellite, and the Cloud ProfilingRadar (CPR) and Cloud–Aerosol Lidar with Orthogo-nal Polarization (CALIOP) aboard the soon-to-be-

Corresponding author address: Dr. Tristan L’Ecuyer, Depart-ment of Atmospheric Science, Colorado State University, Foot-hills Campus, Fort Collins, CO 80523-1371.E-mail: [email protected]

20 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 45

© 2006 American Meteorological Society

JAM2326

launched CloudSat and Cloud–Aerosol Lidar and In-frared Pathfinder Satellite Observations (CALIPSO)satellites.

Given the high degree of measurement accuracy af-forded by such instruments, the reliability of derivedatmospheric products no longer depends as heavily oninstrument calibration and noise but more so on thechoice of spectral bands, the forward model, and themethod of inversion. Consider, for example, opticaldepth and effective radius retrievals. The remote sens-ing literature is replete with descriptions of algorithmsfor inferring these cloud properties from distinct com-binations of radiances at multiple wavelengths of visibleand infrared radiances, some of which are summarizedin Table 1 of Miller et al. (2000). The physical basis ofeach of these algorithms rests on the fact that waterdroplets, ice crystals, and the gaseous constituents ofthe atmosphere display different spectral signatures.Hence, microphysical and optical properties of single-layer clouds can be inferred in principle from radiancesat two or more wavelengths. While many of these al-gorithms have successfully been applied to map clouds,few are universal in the sense that they can be appliedto any scene at any time of the day independent of thebackground or surface. On the contrary, many can onlybe applied under specific conditions (e.g., during thedaytime) or over a limited dynamic range (e.g., opti-cally thin clouds) leading to unphysical discontinuitieswhen one seeks to compile a complete database of theglobal distribution of clouds. Moreover, in the past,channels used in such algorithms have been selectedempirically, based on prior research showing sensitivityto the properties of interest and constrained by avail-able wavelengths fixed by satellite hardware. The use ofdistinct combinations of wavelengths can lead howeverto discrepancies between the products of different al-gorithms when they are applied to the same scene byvirtue of subtle differences in the information providedby the measurements. Unfortunately, such differencesbetween algorithm products are often difficult to re-solve because of limited quantitative measures of theuncertainties in each.

In the case of the MODIS cloud product, the reflec-tance map technique of Nakajima and King (1990) isemployed for daytime retrievals using an absorbingchannel (e.g., 2.142 �m) and a nonabsorbing channel(e.g., 0.664 �m). In essence, the retrieval involves thesolution of two nonlinear equations for two unknowns.Nighttime retrievals must resort to alternate techniquesrooted in thermal emission. In this case, the CO2 slicingmethod is often used to first determine the cloud-toptemperature. Knowledge of the cloud temperature isthen used to constrain optical depth and effective ra-

dius retrievals for thin cirrus using radiances at 3.7 and11 �m, following a technique analogous to that intro-duced by Inoue (1985) and Prabhakara et al. (1988).Comparisons of these techniques under daytime condi-tions, however, often indicate differences of a factor of2 or more in retrieved effective radius and opticaldepth. In fact, changing the particular channels usedwithin either of these techniques can also significantlyimpact the results because of subtle differences in thesensitivity of the measurements to the retrieval param-eters and assumptions that are required in associatedradiative transfer model (RTM) calculations. Assessingthe relative performance of these techniques (and oth-ers in the literature) requires a methodology that ex-plicitly accounts for relative differences in both the sen-sitivity of each measured radiance to the retrieval pa-rameters and their uncertainties including componentsowing to both measurement and modeling errors.

The approaches commonly adopted for assessing theinformation content of a system are well suited for thispurpose. Interestingly though, while a casual perusal ofthe retrieval literature attests the emphasis placed ontechniques of inverting remotely observed data, verylittle reference is made to the information content ofthe measurements themselves. For example, a samplingof the different inversion methods include the follow-ing: constrained nonlinear least squares minimization(Worden et al. 1999), neural networks (Juliette andClerbaux 1999), principal component analyses (Tanreet al. 1996), optimal estimation using Bayesian methods(e.g., Rodgers 1976; Evans 2002), Bayesian MonteCarlo methods for non-Gaussian inverse problems(Tamminen and Kryola 2001), split-window techniques(e.g., Prabhakara et al. 1988; Suggs et al. 1998), bidirec-tional mapping techniques (Nakajima and King 1990;Rolland et al. 2000), regularization methods (Eriksson2000), and discrepancy principles that extend regular-ization methods (Li and Huang 1999). Of the afore-mentioned works, however, only those of Worden et al.(1999) and Evans (2002) refer to and use theoretic in-formation methods. The former refers to how muchinformation is gained in a retrieval relative to the priorcovariance matrix, while the latter refers to the amountof information provided by additional submillimetermicrowave channels. Neither, however, attempts to re-late information content to the retrieval covariance ma-trix.

It is, therefore, of interest to revisit the problem ofcloud microphysical property retrievals from the per-spective offered by information theory to determine asingle optimal framework that can be applied to differ-ent satellites under as wide a variety of conditions aspossible. Toward this end, this paper establishes a rig-

JANUARY 2006 L ’ E C U Y E R E T A L . 21

orous, objective methodology for determining the in-formation content of a set of observations, selecting anoptimal channel configuration, and assessing the effectsof both model and instrumental uncertainties on thefinal inferred product. The analysis is rooted in infor-mation theoretical concepts elucidated by Shannon andWeaver (1949) and on the application of their tech-nique to atmospheric science by Rodgers (2000). Chan-nel selection is made objective by quantifying theamount of information contained in the spectral mea-surements and calculating their effective signal-to-noiseratio (SNR) in relation the desired set of retrieval pa-rameters. This is accomplished practically throughanalysis of the retrieval covariance matrix, which holdsthe key for understanding and quantifying differencesbetween different retrieval procedures and observa-tional data.

To illustrate the benefits of adopting such an ap-proach, the method is applied to the problem of retriev-ing cloud microphysical properties from satellite radi-ance observations at solar and thermal wavelengths us-ing the MODIS channels as a baseline. For simplicity, itwill be assumed that single-layer liquid and ice cloudscan be discriminated from one another and multilayercloud complexes through a combination of their radio-metric signatures [e.g., through the trispectral tech-nique explored in a series of papers by Ackerman et al.(1990), Strabala et al. (1994), and Baum et al. (2000)]and active sensors, such as the CPR or CALIOP. Underthis assumption, we initially focus on the relativelystraightforward problem of retrieving the parameters ofa gamma distribution of water droplets in single-layerliquid clouds to facilitate illustration of the methodol-ogy and interpretation of the results. Furthermore, forthis preliminary application of the technique, we focuson oceanic scenes because they represent the largestfraction of pixels a satellite will encounter. The morechallenging problem of ice cloud retrievals, which iscomplicated by the required assumption of ice crystalhabit, is analyzed in a similar manner in a companionpaper by Cooper et al. (2006, hereinafter Part II). Theanalysis can be readily extended to other surfaces andmultilayer cloud complexes but these problems are be-yond the intended scope of the current study and are,therefore, left as future topics of investigation.

The channels considered in the analysis, their noiserequirements, and their primary uses (adapted from theMODIS Web site, available online at http://modis.gsfc.nasa.gov/about/specifications.php) are summarized inTable 1. Note that the four channels centered on the15-�m CO2 band that are primarily used for determin-ing cloud-top pressure are not analyzed for the lowclouds studied here, but have been included in the

analysis for ice clouds in Part II. A state-of-the-art ra-diative transfer model is employed to simulate radi-ances at these wavelengths for cloud scenes spanning arange of cloud heights, liquid water paths, effective ra-dii, surface albedos, atmospheric pressure and tempera-ture profiles, and solar zenith angles. The resultingsimulated measurements are then fed into a series ofsensitivity studies that provide the sensitivities of eachchannel to the retrieval parameter and rigorous esti-mates of the uncertainties in each because of potentialerrors in the assumptions required to model them. Thecombination of sensitivities and uncertainties com-pletely determines the information content of the en-semble of measurements that is subsequently used toobjectively determine the combination of wavelengthsthat provide the greatest amount of information forglobal microphysical property retrievals.

2. Sensitivity studies

As a precursor to the more rigorous information con-tent study that follows, it is useful to examine the sen-sitivity of the observations to the parameters of interest.In addition to illustrating the dominant physical pro-cesses governing the radiative transfer through cloudsand providing insight into the mechanics of the retrievalproblem, such analyses also form the input to the in-formation content study itself. The required radiancesare computed using an RTM called Radiant (Christiand Stephens 2002; Christi and Gabriel 2004; Gabriel etal. 2005), which is multistream, plane parallel, and ac-counts for multiple scattering. Atmospheric absorptionis modeled using the correlated-k distributions devel-oped for MODIS wave bands by Kratz (1995). Thisapproach captures gaseous absorption properties to an

TABLE 1. The MODIS channels of relevance to this study andtheir primary uses. Solar reflectance channels (upper set) aregiven in nanometers; thermal emission channels (lower set) are inmicrometers.

Band Wavelength SNR Primary use

1 620–670 128 Land/cloud/aerosol2 841–876 201 Boundaries6 1628–1652 275 Land/cloud/aerosol7 2105–2155 110 Properties

26 1360–1390 150 Cirrus clouds/water vapor

Band Wavelength NE�T Primary use

20 3.66–3.84 0.05 Surface/cloud23 4.02–4.08 0.07 Temperature27 6.535–6.895 0.05 Cirrus clouds/water vapor29 8.4–8.7 0.05 Cloud properties31 10.78–11.28 0.05 Surface/cloud32 11.77–12.27 0.05 Temperature


accuracy of 1%, while significantly reducing the com-putational costs incurred by explicit line-by-line calcu-lations. The ocean surface is modeled as a Lambertianreflector with a visible albedo of 0.1 (at wavelengthsless than 3 �m), consistent with the Earth RadiationBudget Experiment (ERBE) observations of Harrisonet al. (1990) and an infrared emissivity of 0.99. Tem-perature, moisture, and gas-mixing ratios are assignedbased on the McClatchey et al. (1972) tropical atmo-sphere while the scattering properties of cloud particlesare modeled using a Mie scattering code assumingspherical droplets.

Using Radiant, radiances for the 11 channels in Table1 were simulated for a wide variety of liquid clouds. Thecloud droplets are assumed to follow a lognormal dis-tribution

N�R� �N0

R�2��log

exp��12 �ln�R�Rg�

�log�2�, �1�

where Rg is the modal radius, N0 is the number density,and �log is the natural logarithm of the geometric stan-dard deviation �g. The effective radius, related to themodal radius via Re � Rg exp[(5/2)�2

log], is varied be-tween 5 and 14 �m in 1-�m increments. Here, �log ��(lnR � lnRg)2 is assumed fixed at 0.427 (Deirmend-jian 1969), while number density N0 was scaled to pro-vide five different values of liquid water path (LWP)corresponding to visible optical depths of 5, 15, 30, 40,and 50 at an effective radius Re � 8, resulting in a totalof 50 test cases. The optical properties of the resultingclouds were modeled using Mie theory and they wereplaced between 1 and 2 km in the McClatchey midlati-tude summer (MLS) atmosphere (McClatchey et al.1972). Unless otherwise stated, all daytime calculationsassume a solar zenith angle of zero, corresponding to anoverhead sun. Visible optical depths corresponding toall of these cases are presented in Fig. 1 for reference.

This base set of cases is supplemented with two setsof sensitivity studies—one to establish the behavior ofeach channel in response to changes in the retrievalparameters themselves, and the second for use in esti-mating modeling uncertainties resulting from errors inthose parameters are not retrieved but must be speci-fied in order to perform the necessary radiative transfercalculations.

a. Forward model uncertainties

Regardless of the measure of information contentone adopts, the result is necessarily rooted in the signal-to-noise characteristics of the retrieval system. An ob-servation whose sensitivity to a retrieval parameter isless than the accuracy to which it can be measured can-not provide useful information. Thus, it is important to

establish a rough estimate of the uncertainties in eachchannel that arise from a combination of random mea-surement and calibration errors as well as any assump-tions needed to model the atmospheric radiative trans-fer that are not going to be explicitly retrieved. Mea-surement errors are modeled after the specifications forthe MODIS instrument aboard Aqua that have beensummarized in Table 1. In addition to these uncertain-ties, all assumptions that are required to perform radia-tive transfer calculations in the visible and infraredmust be considered as potential sources of uncertaintyin the forward model. These include the shape of thedrop size distribution (DSD), cloud height and geomet-ric thickness, surface albedo, assumed humidity andtemperature profiles, the presence of aerosols, the useof plane-parallel calculations to model a cloud that isinherently inhomogeneous in the vertical and horizon-tal directions, the representativeness of a satellite snap-shot of clouds for measuring their global distribution(i.e., sampling errors), and so on.

While all of these error sources are important, it is amonumental task to evaluate the contributions from allof them. In fact, a suitable methodology for assessingthe global mean of 3D effects has not yet been devel-oped and, because the analysis focuses on a MODIS-like instrument but purposely avoids defining a particu-lar observing system, sampling errors cannot be de-fined. Furthermore, some of these assumptions can beimposed as soft constraints on the retrieval by allowingthem to be retrieved as opposed to assigning them inadvance. Allowing cloud-top height and surface albedoto vary, for example, provides additional degrees offreedom that allow the algorithm to better match the

FIG. 1. Visible (0.64 �m) optical depths as a function of effectiveradius (vertical axis) and liquid water path (horizontal axis) forthe 50 base cases modeled.


Fig 1 live 4/C

observations. As a result, we focus only on thosesources that can directly be modeled using the Radiantmodel and cannot realistically be retrieved from theobservations, namely, uncertainties in the assumedDSD, specific humidity profile, and temperature pro-file. To this end, the 50 base cases described above werererun with each of these quantities perturbed by anamount representative of their expected uncertainty.

To model errors in assumed DSD, the cloud dropletassumption is changed to follow a modified-gamma dis-tribution,

N�R� �N0

Re�� R

Re��1

e�R�Re, �2�

with the number density N0 scaled to match the LWPvalues assumed in the base cases and the effective ra-dius defined as noted above Re � Rg exp[(5/2)�2

log] to beconsistent with the Rg values assumed in the lognormaldistribution (Stephens 1994). The width parameter isheld fixed at 3, following Deirmendjian (1969). In anoperational retrieval, profiles of temperature and hu-midity need to be specified. A likely source of such dataare numerical weather prediction (NWP) models suchas that used at the European Centre for Medium-Range Weather Forecasts (ECMWF), so it is assumedthat the uncertainties in ECMWF temperature and hu-midity predictions are representative of the level of er-ror in the values assumed in the algorithm. Based onthe sensitivity studies of Eyre (1990) and Eyre et al.(1993), then, temperatures were perturbed by 2 K ateach layer and specific humidities were perturbed by15% below 500 hPa and 30% above.

The resulting estimates of the uncertainties fromeach of these sources are then combined with one an-other and an estimate of the measurement errors todetermine effective fractional errors in each channelare modeled. Once again, while no data from any par-ticular instrument are used in the analysis, an appropri-ate estimate of instrument noise from the SNR andnoise-equivalent temperature difference (NE�) re-quirements for MODIS measurements (summarized inTable 1) are used to represent instrument performance.Then, if it is assumed that instrument noise and each ofthe sources of model error are uncorrelated, the com-bined uncertainty resulting from all of these sources isgiven by the square root of the sum of the squares ofeach of these estimates

�i ��q

yi�2

� ��T

yi�2

� ��DSD

yi�2

� ��m

yi�2

,

�3�

where �i is the (dimensionless) fractional error in the

ith channel (Taylor and Mohr 2004). The radiance inchannel i is represented by yi, while q, DSD, T, and m

represent radiance uncertainties resulting from humid-ity, DSD, temperature, and measurement errors, re-spectively.

Fractional errors in six of the channels considered arepresented in Fig. 2 for the range of liquid cloud Re andLWP considered. In general, the uncertainties in theshortwave radiances are much smaller than those in theinfrared because of the fact that the latter suffer fromerrors in the assumed temperature and humidity pro-files while the former are only sensitive to errors inassumed DSD, which are typically small provided thatthe effective radius and liquid water path are held fixed.As a result, fractional errors at 3.75 and 11 �m rangefrom 5% to 10% while those in the shortwave channelsare �1%–2% for all but the thinnest clouds. A subtleyet important result of this analysis is that the errors at2.13 �m tend to be approximately 50% larger than at1.64 �m because of differences in the strength of thewater vapor absorption at the two wavelengths, whichcauses the 2.13-�m channel to be more sensitive to er-rors in the assumed humidity profile. In the analysisthat follows we will see that this can have implicationsfor determining the optimal channels for use in a re-trieval algorithm. The uncertainties in 1.38-�m radi-ances are at least an order of magnitude larger thanthose in all other channels. This is a direct consequenceof the strong water vapor absorption at this wavelengthcombined with our inability to constrain the specifichumidity profile to any better than 15% using NWPmodel data. Last, note that the fractional errors of somechannels exhibit significant scene dependence whilethose in others do not, suggesting that the informationcontent and optimal channel configurations determinedbelow are likely to depend on the region-in-state spacein which the solution lies. Fractional errors at 1.38 �m,for example, increase with increasing optical depth be-cause of enhanced cloud reflection that causes a greaterfraction of the incident radiation to pass through theuncertain water vapor profile in both the downwellingand upwelling directions. Errors at 11 �m, on the otherhand, are dominated by uncertainties in cloud-top tem-perature, which are not sensitive to the properties ofthe underlying cloud.

b. Sensitivities to retrieval parameters

To assess the “signal” component of the signal-to-noise ratios that drives the information content study, asecond series of sensitivity studies was performed inwhich each of the parameters of interest was perturbedindependently with sensitivities computed via


Si ��Ii

�X, �4�

where X represents the three key retrieval parameters:Re, LWP, and cloud-top height (Ctop) and the �X arechosen to be �5% of the value of the parameter X. Theexception is Ctop, which is perturbed by 1 km, consistentwith the vertical resolution of the Radiant radiativetransfer model. The resulting sensitivities can then bedivided by the radiance errors in each channel to definean effective SNR for each retrieval parameter at eachwavelength.

Figure 3 combines sensitivities to Re with the uncer-tainty estimates from Fig. 2 to produce maps of ef-fective SNR for effective radius retrievals defined bySNR � Si/�IIi. The largest effective SNR for Re occur at1.64 �m where they are about a factor of 2 larger thanthose at 2.13 �m. As indicated above, this is largelybecause of the fact that fractional errors at 1.64 �m aresmaller than those at 2.13 �m. Even so, these two chan-nels clearly contain the majority of the particle sizeinformation in the system. The small sensitivities at 0.64�m owe their existence to the fact that changing Re

causes an inversely proportional change in opticaldepth when LWP is held fixed. Last, although they

are a factor of 20 less than those at 1.64 �m, the non-negligible effective SNRs at 3.75 �m play an importantrole in nighttime retrievals in the absence of a signal inthe shortwave channels.

Similar results corresponding to effective SNR forLWP retrievals are presented in Fig. 4. Clearly, thevisible channel exhibits the largest effective SNRs forall clouds except the optically thinnest cases (lowestLWP and largest Re) where the largest sensitivities oc-cur at 1.64 �m. Considering the lack of SNR in any ofthe infrared channels, no LWP information is expectedfor nighttime retrievals.

Because it is desirable to introduce as few assump-tions as possible in the retrieval, it is of interest to in-clude cloud-top height Ctop as a retrieval parameterrather than fixing it a priori. To examine the feasibilityof using the visible and infrared radiances to constraincloud-top height, Fig. 5 summarizes the effective SNRfor Ctop retrievals. Changing cloud height affects theobserved radiances through the following two mecha-nisms: 1) it changes the cloud-top temperature influ-encing emission at thermal wavelengths, and 2) it mod-ifies the amount of water vapor above the cloud influ-encing the amount of radiation that gets absorbed andemitted at wavelengths corresponding to water vaporabsorption bands. The first mechanism is, for example,

FIG. 2. Fractional errors in selected channels resulting from the combination of uncertainties in prescribed DSD, specific humidityand temperature profiles, and instrument noise.


Fig 2 live 4/C

responsible for the observed SNRs of �1.2 at 11.0 �m.The latter mechanism is particularly evident at 1.38 �m,and to a lesser extent 1.64 and 2.13 �m. The extremelystrong sensitivity at 1.38 �m is counterintuitive forclouds with tops as low as 2 km because of the strongwater vapor absorption at this wavelength. This result isan artifact of the fact that, on paper, arbitrarily smallradiances can be analyzed when in practice the absoluteradiance reflected to the satellite by the cloud is sosmall (�0.0006 W m�2 sr�1) that it is not likely to bedetectable over the electrical noise in the sensor. In theevent that a sensor could be constructed to detect ra-diances down to this level, however, a small increase incloud-top height dramatically increases the amount ofradiation reflected back to the satellite because thewater vapor is so prominent at these altitudes. In theMcClatchey MLS atmosphere assumed here, for ex-ample, changing the cloud top from 2 to 3 km gives riseto an order of magnitude increase in the modeled ra-diance at the top of the atmosphere. This, in turn, leadsto an SNR of �10 based on the estimated fractionaluncertainties that are on the order of unity. For now,small radiances at 1.38 �m will be included in the analy-sis with the caveat that results involving this channelare contingent on the ability of the instrument to mea-sure radiances as low as 0.0005 W m�2 sr�1. In addi-

tion, it must be possible to model radiances with abso-lute magnitudes this small, avoiding numerical issuessuch as instabilities, discretization errors, and trunca-tion errors. Regardless of these issues, the stronginfluence of water vapor in the 1.38-�m channel suffi-ciently decouples it from the other channels such thatfailure to model the electrical noise floor of the instru-ment should not impact the analysis of the remainingchannels.

All three of these figures further emphasize the factthat the information content of the observing system islikely to be strongly dependent on the scene being re-trieved. The sections that follow outline a frameworkthat takes this into account and attempts to determinethe subset of channels that provide the most informa-tion for the widest variety of cloud systems.

3. Information theory

The sensitivity studies described above provide in-sight into the response of individual measurements tothe retrieval parameters, but to determine the combi-nation of channels best suited for the retrieval, it isnecessary to define a set of criteria for assessing theresponse of the instrument as a whole. These criteriamust not only account for the sensitivities of each indi-

FIG. 3. Signal-to-noise ratio at selected wavelengths for effective radius perturbations. Note that the range of values on the upperpanels is 20 times that on the lower panels.


Fig 3 live 4/C

vidual channel but also correlations between them. Inaddition, the expected uncertainty in each measure-ment must be considered to accurately characterize theamount of independent information provided by theensemble of channels relative to the level of noise in-herent in the observing system. The metric adopted inthis study is the information content, a tool that hasbeen widely used in many engineering disciplines buthas, to date, been underutilized in atmospheric remotesensing. In general, information content refers to thedegree by which a set of observations improves ourknowledge of the set of retrieval parameters (cloudheight, particle size, number concentration, etc.). Putanother way, one can think of information content asthe factor by which the total number of distinct combi-nations of the retrieval parameters or “states” that sat-isfy our prior knowledge of the system, or the degree ofnonuniqueness, is reduced by making the measure-ments. Because our ability to distinguish states in theretrieval system depends both on the sensitivity of themeasurements to the retrieval parameters and the ac-curacy of those observations, the information contentinherently accounts for these factors and is, therefore,well suited to analyzing the properties of an observingsystem.

a. Shannon information content

There is an abundance of different measures of in-formation content (see, e.g., Kullback 1968 or Bernardoand Smith 1994, and references therein), many of whichmay be adapted to the current problem. We adopt thedefinition of Shannon and Weaver [(1949), hereinafterreferred to as the Shannon information content (SIC)]that is described in detail by Rodgers (2000) in relationto the problem of atmospheric sounding from multi-spectral satellite radiance measurements and has re-cently been applied to the problem of CO2 retrievalsfrom infrared sounding observations by Engelen andStephens (2004). As a measure of knowledge of theretrieval system, Shannon and Weaver (1949) adopt ananalog to the thermodynamic entropy S, defined as thelogarithm of the number of distinct internal states of amacroscopic system. Letting P1 represent the PDF gov-erning the probability of obtaining any state in the re-trieval system prior to making a measurement and P2

be the PDF after the measurement has been made, theShannon information content is defined as the differ-ence in entropy of these two PDFs, namely,

H � S�P1� � S�P2�. �5�

FIG. 4. As in Fig. 3, but for LWP perturbations. In this case the scale on the upper panels is 60 times that on the lower panels.


Fig 4 live 4/C

The SIC thus defined measures the degree to which theaddition of a measurement reduces the disorder of theretrieval state space. For convenience, we assume thatboth the prior and posterior PDFs follow Gaussian dis-tributions with covariances S1 and S2, respectively.1 Itcan be demonstrated (e.g., Rodgers 2000) that the en-tropy of a multivariate Gaussian distribution of m vari-ables is given by

S�P� �12

log2|S| � c, �6�

where the constant c � (m/2) log2(2�e). The SIC is then

H �12

log2|S1S2� 1|. �7�

Because the covariances represent the volume-in-statespace corresponding to our uncertainty in the retrievalstate prior to and after making the measurement, theinformation content is a measure of the factor by whichthe measurement reduces our uncertainty in the re-trieval state.

With entropy defined in this way, that is, as a loga-rithm to the base 2 of the total number of states, Hprovides the information content in bits, implying thatthe observations allow 2H states to be distinguishedfrom the prior state space. Conceptually, the observa-tions can be thought of as refining a measuring stick bysubdividing each of the prior divisions into 2H newones. The larger the information content H, the finerthe resolution of the new tick marks on the measuringstick and the more accurately the quantity of interestcan be resolved. In multidimensional problems such asthe cloud property retrievals considered here, one canenvision multiple measuring sticks pointed along eachvariable in the retrieval vector. After making an obser-vation, each of these measuring sticks will have its owncharacteristic resolution that depends on the combinedsensitivity of the observation vector to the parameter itrepresents. It will be shown that the total information

1 Aside from the computational benefits to assuming Gaussiandistributions in the analysis, it can be shown that the Gaussiandistribution maximizes the entropy or, equivalently, minimizesour assumed knowledge of the state space when only the meanand variance of the distribution of retrieval states is known(Rodgers 2000). Thus, the Gaussian distribution is, in fact, themost appropriate choice in the absence of conclusive evidence foran alternative form of PDF.

FIG. 5. As in Fig. 3, but for cloud-top height perturbations. The scale for the 1.38-�m image ranges from 0 to 10.5; those at otherwavelengths range from 0 to 1.5.


Fig 5 live 4/C

content represents the total number of divisions on themeasuring sticks that lie along the set of perpendiculardirections defined by the eigenvectors of S1S2

�1

We now return to the problem of retrieving cloudmicrophysical properties from radiance observations.Suppose y represents a set of observed radiances and xrepresents the vector of cloud microphysical param-eters to be retrieved. Let Sa and Sy be the covariancematrices describing the state space prior to making ameasurement and the measurement error, respectively.Assuming a physical relationship between y and x ofthe form, for example, y � F(x), it can be demonstratedthat the covariance describing the posterior state space is

Sx � �Sa�1 � KTSy

�1K��1, �8�

where K is a linearized forward model consisting of theJacobian of the forward model with respect to the re-trieval vector with elements given by

Kij �yi

xj. �9�

The diagonal elements of Sx provide the variance in theretrieved products in variational retrieval techniques,such as those outlined in Rodgers (1976), Engelen andStephens (1997), and L’Ecuyer and Stephens (2002).

As noted in Rodgers (2000), to compare the mea-surement error with the natural variability of the mea-surements across the full prior state space it is conve-nient to work in a basis where the measurement errorsand prior variances are uncorrelated. Therefore, it isdesirable to transform K into

K̃ � Sy�1�2KSa

1�2, �10�

which offers the added benefit of being the basis inwhich both the prior and measurement covariances areunit matrices. Furthermore, Rodgers (2000) demon-strates that the number of singular values of K̃ greaterthan unity defines the number of independent measure-ments that exceed the measurement noise defining theeffective rank of the problem.

Using Sa for the covariance of the prior state spaceand Eq. (8) for that of the posterior state space, the SICbecomes

H �12

log2Sa�KTSy

�1K � Sa�1�

�12

log2K̃TK̃ � I

�12 �

i�1

N

log2�i2 � 1, �11�

where �is are the singular values of K̃, I is the m � midentity matrix, and m is the number of retrieval pa-rameters. In addition, Rodgers (2000) demonstratesthat the number of degrees of freedom for signal can beestimated using the singular values of K̃ via

ds � �i

�i2

1 � �i2 , �12�

providing another important property of the observingsystem, namely, the number of independent measure-ments that can be extracted from the observations.

Thus, we have defined the following four diagnosticsfor assessing the capabilities of the observing system:

1) the error covariance Sx, characterizing the width ofthe posterior PDF and providing a measure of theoverall accuracy of the retrieval;

2) the SIC H, which measures the relative improve-ment to our a priori knowledge that results from theaddition of the measurements;

3) the number of degrees of freedom for signal ds,which represents the number of independent obser-vations that can be constructed from the measure-ments; and

4) the number of singular values that exceed the noiselevel of the system that defines the effective rank Nof the problem. (We can interpret N as the numberof independent quantities that can be retrieved fromthe measurements. In this way, N is analogous to ds,but applies in retrieval space rather than measure-ment space.)

Individually, these diagnostics can provide useful in-formation concerning aspects of the retrieval problem,but in the absence of the others they are easily misin-terpreted. It is, for example, important to consider theoverall accuracy of the retrieval in combination withthe information content, because a set of measurementsmay carry much information in highly undercon-strained problems but can still lead to large uncertain-ties in retrieved products because of the ill-posed na-ture of the problem. Taken together, however, theyprovide a more or less complete quantitative descrip-tion of the retrieval process from observations throughto final products that allows for a critical assessment ofdifferent algorithms or even distinct platforms in anobjective manner.

b. Example: Liquid cloud retrievals from shortwavereflectances

As an example, consider a reflectance-based ap-proach to retrieving the effective radius and liquid wa-ter path analogous to that introduced by Nakajima andKing (1990). The technique makes use of the fact that


reflected radiances at nonabsorbing wavelengths (re-ferred to as conservative scattering channels) are pri-marily sensitive to the optical depth of a liquid cloud,while those at absorbing wavelengths (nonconservativescattering channels) are dominated by the size of itsconstituent cloud droplets. Thus, the combination ofreflectances at a nonconservative wavelength (e.g., 2.13�m) and a conservative wavelength (e.g., 0.64 �m)yields a two-dimensional grid in which a pair of radi-ance measurements can be related to geometric meanradius and liquid water path. This is illustrated in Fig. 6.Based on the sensitivity studies described above, Fig. 6also presents the estimated uncertainties in the 0.64-and 2.13-�m channels and illustrates how these errorsmap into the Re and LWP retrieval space. Note thatuncertainties in the spectral radiance measurementslead to nonuniqueness in the retrieval because the re-lationship between a given radiance pair and the re-trieval parameters becomes multivalued. This effect isparticularly noticeable at the thick cloud limit wherethe sensitivities of the radiances are low and the errorbars cover a wide range of LWP values.

Making use of these error estimates and the SIC for-malism outlined above, it is straightforward to computethe information content of each of these channels forretrieving Re and LWP from an a priori range of Re �9 � 4 �m and LWP � 150 � 65 g m�2, which includesa majority of nonprecipitating liquid clouds in nature(Miles et al. 2000). The results are presented in Fig. 7,which demonstrates the effect of successively addingradiance measurements in the retrieval problem. In thisexample, all radiance errors are assumed to be �5% forsimplicity. The upper-left-hand panel illustrates the apriori state space. The blue ellipse corresponds to theprojection of a two-dimensional Gaussian PDF ontothe solution space at the 2� level, encompassing 95% ofthe possible solutions. In the absence of observations,any of these solutions are valid and cannot be distin-guished from one another.

The upper-right panel demonstrates the impact of a0.64-�m radiance observation corresponding to a cloudwith Re � 10 �m and LWP � 171 g m�2. Given thesensitivity of this channel to the retrieval parametersand the uncertainties associated with modeling it, thisobservation reduces the possible solutions to the rangeof values centered on this combination of effective ra-dius and LWP that is represented by the green ellipse.Following Eq. (7), the SIC in this case is 1.2, indicatingthat slightly more than two independent states can beresolved from within the initial state space.

The red ellipse in the lower-left panel indicates therange of allowable solutions after further adding a 2.13-�m radiance measurement. The complementary nature

of the conservative and nonconservative channels leadsto a significant increase in the information content ofthe system now allowing six independent states to beresolved from the original a priori state space. As indi-cated by the projections of the new posterior PDF (redellipse), the errors in retrieved Re and LWP are signifi-cantly reduced when both channels are included in theretrieval. Returning to the measuring stick analogy, thewidth of the posterior PDF can be viewed as a measureof the “resolution” of the observing system. As Sx de-creases with the addition of more information, it is pos-sible to measure Re and LWP more precisely. Put sim-ply, the finer the scale of the ruler, the greater thenumber of distinct states that can be measured.

Interestingly, adding all of the remaining channelsfrom Table 1 provides only a limited amount of addi-tional information to the retrieval. This is illustrated bythe yellow ellipse in the lower-right-hand panel of Fig.7. The range of allowable states has clearly decreasedsomewhat relative to the red ellipse, but it is unclearwhether the increased resolution of the observing sys-tem justifies the enormous increase in computation thatis required to go from a 2- to an 11-channel framework.Thus, we see that the information content provides auseful diagnostic for establishing the relative perfor-mance of different channel combinations, allowing thetrue value of increased algorithm complexity to be as-sessed.

It is important to note that the resolution of our mea-suring stick depends on where we are in state space. Inthis example, it turns out that the resolution of the

FIG. 6. The mapping of uncertainties in 0.64- and 2.13-�m ra-diances (W m�2 sr�1) into effective radius (�m) and LWP (g m�2)space in a reflectance-based retrieval. Filled circles representthe data points modeled, dashed lines are lines of constantliquid water path, and solid lines are lines of constant effectiveradius.


observing system for LWP retrievals is inversely pro-portional to LWP and proportional to particle size,while its resolution for Re retrievals is proportional toLWP and inversely proportional to particle size. It isalso important to note that the information provided bythe observations is not equally divided between the tworetrieval parameters. Consider, for example, the addi-tion of the 2.13-�m channel in Fig. 7. The uncertainty inRe is reduced by a factor of 4 while that in LWP isreduced by only 39%.

c. Optimal channel selection

Rodgers (2000) describes a method for extending thisframework to optimize a retrieval by objectively select-ing the subset of channels that provides the greatestamount of information. To reduce the computationtime required to perform the numerous matrix opera-tions necessary for repeated application of the preced-ing equations, Rodgers (1998) proposes an approachbased on sequential modification of the covariance ma-

trix. The procedure first requires that we assess theinformation content of each individual measurementwith respect to our prior knowledge of the retrievalstate to create an “information spectrum.” The channelwith the largest amount of information is then selectedand the posterior covariance matrix is adjusted accord-ingly to account for the information it provides. A newinformation spectrum for the remaining channels isthen calculated with respect to this newly defined statespace and a second channel is chosen that providesmaximal information relative to the new covariance.This process is repeated and channels are selected se-quentially until of the information in all remainingchannels falls below the level of measurement noise.

Following Rodgers (1998), and letting Si be the errorcovariance matrix for the state space after i channelshave been selected, the information content of channelj of the remaining unselected channels is given by

Hj �12

log2�1 � k̃jTSik̃j�, �13�

FIG. 7. Graphical representation of the impact of adding information in a two-dimensionalretrieval using the retrieval of effective radius and LWP from shortwave reflectance measure-ments as an example. Each ellipse represents the projection of the corresponding two-dimensional posterior Gaussian PDF of solutions at the level that encompasses 95% of thepossible solutions. Results apply to a liquid cloud residing between 1 and 2 km above anoceanic background.


Fig 7 live 4/C

where k̃j is the jth row of K̃. The Hj form the informa-tion spectrum from which the channel with the greatestinformation is chosen. Taking the chosen channel to bechannel l, the covariance matrix is then updated prior tothe next iteration via

Si�1�1 � Si

�1 � k̃lk̃lT. �14�

In this way channels are selected until none of the re-maining channels has an information content exceedingthe measurement noise.

Returning to the measuring stick analogy, the firstchannel chosen initially divides the stick into H0 incre-ments. Each subsequent selected channel further re-fines the divisions until all useful information is ex-tracted from the measurements and the finest resolu-tion possible (given the properties of the observingsystem) is reached. The selection procedure outlinedabove guarantees that the first channel will provide thegreatest number of divisions, followed by the secondand so on, making it possible to objectively choose themeasurements that maximize our knowledge of theproblem while eliminating those channels that provideredundant information.

4. Information content of MODIS measurements

The goal of this study is to assess the informationcontent of the 11 channels summarized in Table 1 forretrieving the properties of liquid clouds. Specifically, ifthe cloud droplets are assumed to follow a lognormalDSD, the retrieval focuses on retrieving the geometricmean radius and the liquid water path because theseparameters completely determine the DSD, providedthat one assumes a value of the geometric standarddeviation �g. In an effort to reduce the number of as-sumptions required in the forward radiative transfercalculations, the cloud-top height Ctop and shortwavesurface albedo are also considered retrieval param-eters, although it is anticipated that these parametersmay be constrained in some way using ancillary mea-surements from another sensor.

a. Covariance matrices

Accurate characterization of the prior and measure-ment error covariance matrices is central to the prob-lem of calculating information content. In practice, Sa iseasier to define because it merely represents the bestestimate of our prior knowledge of the retrieval param-eters, in this case geometric mean radius Re, liquid wa-ter path LWP, surface albedo �, and cloud-top heightCtop. Because our focus is on global retrievals fromsatellite-based radiance measurements, we anticipate

little prior knowledge of the microphysical propertiesof the clouds, only whether or not a cloud is present byvirtue of either an active or passive cloud mask. Basedon the climatology of in situ observations of low-levelstratiform clouds made between 1972 and 1995, pre-sented in Miles et al. (2000), Re varies from �4 to 12�m while LWP varies from �25 to 250 g m�2 for suchclouds. As a generous approximation of this variability,then, we will assume values of 4 and 200 for the vari-ance in Re and LWP, respectively.

The assumed variances for � and Ctop, on the otherhand, depend on the quality of the ancillary datasetsused to define them. Two scenarios are simulated here,with the first assuming no cloud height or surface al-bedo information beyond climatological mean values,and the other making use of cloud boundaries inferredfrom radar reflectivity observations from the CloudSatCloud Profiling Radar (Stephens et al. 2002) and theMODIS surface albedo product (Strahler et al. 1999).In the case that makes use of climatologies (hereinafterreferred to as “climatological xa”), conservative esti-mates of 2.5 km and 30% are used for the standarddeviation in Ctop and �, respectively. In the case inwhich ancillary data are used (hereinafter referred to as“ancillary xa”) we adopt the documented value of 10%for the accuracy in the MODIS surface albedo product(Strahler et al. 1999) as the standard deviation in �.Given the vertical resolution of the CPR, we anticipatecloud-top height to be defined with an accuracy of�250 m, which is modeled by a Gaussian distributionwith a variance of 0.0625 km2 in the second case.

It is interesting to note that using Eq. (7), the Cloud-Sat cloud boundaries and the MODIS surface albedoproduct have a combined information content of H �4.9 bits relative to the climatology-only case. The apriori state space in the second case is, therefore, afactor of 24.9 � 30 smaller than that in the first case.Thus, we anticipate that the information content of theMODIS observations will be significantly smaller in thesecond case than the first, even though the former rep-resents a better-posed problem and will undoubtedlylead to more accurate retrievals (Cooper et al. 2003).

In practice Sy is more difficult to define because itconsists of both measurement error as well as errorsassociated with the forward model used to map theretrieval parameters into measurement space. Two dif-ferent estimates of the combined forward model andmeasurement errors will be tested in an effort to illus-trate the importance of making rigorous uncertaintyestimates in the analysis. In the first case (hereinafterreferred to as “uniform measurement errors”), a con-stant 5% error will be assumed in the radiances at allwavelengths, while in the other (hereinafter referred to


as “realistic measurement errors”) we will make use ofthe more realistic fractional uncertainties determined insection 2a to represent the measurement standard de-viations with the understanding that they include anumber of important sources of uncertainty but neglectothers that may be significant under certain conditions.

For simplicity, both Sy and Sa will be assumed to bediagonal. While it may be reasonable to assume thatmeasurement errors are uncorrelated, the forwardmodel component of these uncertainties invariablycauses uncertainties in channels with similar sensitivi-ties to be correlated with one another. Furthermore,with the exception of the surface albedo, it is unlikelythat the retrieval parameters themselves are uncorre-lated given the microphysical processes that govern thenucleation and growth of cloud droplets and their sen-sitivity to temperature and humidity. It is, however,beyond the scope of this work to explore the correla-tions between these quantities in the depth required to

accurately represent them in the covariance matrices.In fact, because including knowledge of the correlationsbetween channels constitutes adding information to theretrieval, one runs the risk of adding spurious informa-tion that may degrade the results if care is not taken toproperly estimate the off-diagonal elements of the co-variance matrices. Thus, the results that follow shouldbe viewed as a first attempt at providing ballpark esti-mates of the information content of the system thatrepresent the worst-case scenario where no advanceknowledge of correlations between channels is avail-able.

b. Total information

Figure 8 presents expected retrieval errors, defined by

�i ��Sii

Xi× 100, �15�

FIG. 8. Uncertainties in effective radius, LWP, and cloud-top height, total information content, and total degrees of freedom for signalover a wide range of LWP and Re. Uncertainties in all radiances resulting from observation and forward model errors are assumed tobe 5%.


Fig 8 live 4/C

where Sii and Xi are the diagonal element of the Sx andretrieval vector x corresponding to the ith and totalinformation content from Eqs. (8) and (11) for 50 pairsof Re and LWP, assuming uniform measurement errorsand a climatological xa. The most striking artifact offailing to account for the wavelength dependence of themeasurement errors is the unrealistically small uncer-tainties of cloud-top height that range from 5.5 to 8.5 m.This is a direct consequence of improperly accountingfor the large influence of uncertainties in the assumedspecific humidity profile on our ability to model radi-ances at 1.38 �m. When the realistic uncertainty esti-mates from section 3a are employed (Fig. 9), the reso-lution to which Ctop can be estimated decreases bymore than an order of magnitude to �100 m in re-sponse to increasing the fractional error in the 1.38-�mchannel. The loss of Ctop information leads to a slightreduction in the total information content relative tothe case with uniform measurement errors, but this ispartially compensated for by improved LWP and Re

retrievals through a reduction in the uncertainties inother channels.

The top panels in Fig. 9 suggest that, provided thecloud is homogeneous across the instrument field ofview and the particle DSD does not vary vertically, Re

and LWP can be retrieved with accuracies of approxi-mately 5% and 20%, respectively over much of therange of clouds examined. This is, in part, a result of thefact that the variability in the microphysical propertiesof liquid clouds observed in nature is relatively small (incomparison with ice clouds, e.g.) and, in part, becauseof the good sensitivity of the shortwave reflectancefrom these clouds to the parameters of interest. In gen-eral, the information content is largest for thin cloudscomposed of small particles because the shortwave re-flectances are most sensitive to changes in cloud prop-erties for optically thin clouds (see Fig. 1). This leads toa factor of 3 better LWP resolution (in a fractionalsense) for thin clouds than thick ones.

The effects of neglecting wavelength-dependent un-

FIG. 9. As in Fig. 8, but using rigorous estimates of the uncertainties resulting from observation and forward model errors.


Fig 9 live 4/C

certainties are examined in greater detail in Figs. 10 and11 where the singular vectors of the K̃ matrix for acloud with LWP � 85.71 g m�2 and Re � 8 �m, assum-ing that uniform and realistic measurement errors, re-spectively, are presented. In both cases, three singularvalues exceed the noise level of 1.0 indicating that threeindependent parameters may be inferred from the data.Because the singular vectors indicate the linear combi-nation of retrieval parameters that correspond to eachsingular value, these images confirm the shift fromcloud-top height information when the 1.38-�m radi-ance is assumed to be unrealistically accurate to LWPand Re when rigorous uncertainty estimates are used. InFig. 10, the largest singular vector (corresponding to theprimary information in the system) corresponds to Ctop

while LWP and Re information are relegated to thesecond and third singular vectors. When more realisticerrors are assumed, Re and LWP information is coupledinto the first two singular vectors while cloud-top heightinformation corresponds to the third singular vector,which is more than an order of magnitude smaller thanthe other two. These results highlight the importance ofaccurate covariance matrix characterization in settingup an information content analysis. The results alsoimply that more accurate specification of the humidityprofile could allow in principle cloud-top height to be

retrieved very precisely. While this may not be ex-tremely important, better estimates of cloud top maylead to small improvements in applications involvingmodeling the radiative impacts of such clouds. Notethat in both cases the fourth singular value, correspond-ing to surface albedo, is much less than unity, suggest-ing that the clouds are thick enough to completely ob-scure the surface. This precludes any quantitative in-formation regarding the surface albedo from beingextracted from the data.

The potential for using infrared observations to re-trieve liquid cloud properties at night is briefly exploredin Fig. 12. Because of the fact that liquid clouds residenear the surface, uncertainties in nighttime retrievalsapproach the values of the a priori knowledge (4 �m inRe, 200 g m�2 in LWP, and 250 m in Ctop), consistentwith the fact that the total information content in all ofthe channels sums to less than 1 in most cases, indicat-ing that at most two distinct states can be resolved bythe observing system. In general, there is no informa-tion regarding liquid water path with the exception ofextremely thin optical depths. What little informationthere is for the remaining clouds transitions from Ctop

to particle size as Re increases. Thus, nighttime retriev-als can, at best, differentiate between overcast andclear-sky scenes, providing little modification to the cli-

FIG. 10. Singular vectors of the K̃ matrix assuming 5% uncertainties in all radiances resultingfrom observation and forward model errors. Results correspond to daytime observations of acloud with LWP � 85.71 g m�2 and Re � 8 �m residing between 1 and 2 km over an oceanicsurface.


matological mean values of Re and LWP used to ini-tialize an algorithm.

c. Channel selection

Information spectra derived using Eq. (13) are pre-sented in Fig. 13 for a cloud with LWP � 85.71 g m�2

and Re � 8 �m, assuming realistic measurement errorsand cloud top and surface albedo from ancillary data.The solid line traces the information content of eachchannel with respect to the a priori assumptions. Oncethe channel with the most information is selected, thesolution space is reduced accordingly and the second(dotted) line traces the information content of the re-maining channels with respect to this new solutionspace. The process is then repeated by selecting thechannel with the largest information content from thosethat remain and readjusting the solution space to ac-count for the information it provides, etc. The processstops as soon as none of the remaining channels exhibitan information content in excess of 0.346, representingthe level of noise in the system obtained by substitutinga SNR of 1 into Eq. (13). In this case, the 1.64-�mchannel provides the most information relative to the apriori solution space and is, therefore, selected first.Once the solution space has been adjusted to accountfor the information provided by the 1.64-�m channel,there is a dramatic drop in the information content of

all channels containing redundant information, such as2.13 and 3.75 �m. This leaves the two visible channels at0.64 and 0.88 �m with the largest information contentsof those that remain. Because the information contentat 0.64 �m is slightly greater than that at 0.88 �m, it ischosen second. Last, once the impact of the 0.64-�mchannel has been accounted for, the only channel thatprovides new information to the retrieval is the 1.38-�m water vapor channel, which is sensitive to cloud-topheight, so it is selected third.

Figure 14 repeats the procedure assuming uniformmeasurement errors. Now, the 1.38-�m channel is se-lected first because of the unrealistically strong infor-mation it provides regarding Ctop. Again, 0.64 �m ispicked for the information it provides in constrainingLWP, but now the 3.75-�m channel is picked for par-ticle size information rather than the 1.64-�m channel,because Sx fails to account for the order of magnitudedifference in their uncertainties. This result further em-phasizes the importance of rigorously modeling thewavelength dependence of model errors within the in-formation content framework.

Applying this procedure to all 50 combinations ofLWP and Re described above the subset of the 11 chan-nels that provide the greatest information for the widestvariety of liquid cloud retrievals is identified. Table 2summarizes the fraction of these cases for which any

FIG. 11. As in Fig. 10, but using rigorous estimates of the uncertainties resulting fromobservation and forward model errors.


given channel is selected assuming realistic measure-ment errors and the existence of ancillary xa data. Ingeneral, the results agree with the consensus of re-trieval approaches in the literature that liquid cloudmicrophysical property retrievals will achieve the bestresults by merging the information contained in a con-servative and a nonconservative shortwave scatteringchannel. The results also suggest that there maybe use-ful cloud height information in the 1.38-�m water vaporchannel because it is selected for half of the retrievals(although never selected first). This result is somewhatunexpected because water vapor absorption at 1.38 �mis generally considered to completely mask low cloud,but the analysis conducted here reveals that undersome conditions this channel may provide additionalinformation in a retrieval. Such information could leadin turn to subsequent improvements in modeling theradiative impacts of low clouds in the atmosphere. Ex-tracting it, however, requires that the sensor be capableof measuring very small absolute radiances and success

is very unlikely under moist conditions such as thosethat prevail in the Tropics.

The analysis indicates that 1.64 and 0.64 �m are thechannels of choice for Re and LWP retrievals becauseof their slightly larger sensitivities and slightly loweruncertainties than 2.13 and 0.88 �m. It should, how-ever, be noted that even though 1.64 �m is selected �5times more often than 2.13 �m, the information contentof each of these channels with respect to the a priori isvery similar, and 2.13 �m could be substituted for 1.64�m in a retrieval with very little impact on the results.A similar argument can be made for substituting 0.88�m for 0.64 �m under most circumstances. From a con-sistency standpoint, however, it is desirable to use oneor the other rather than switching back and forth.Equivalent results for the case of nighttime retrievalsare summarized in the last row of Table 2. The resultsecho the lack of information indicated by Fig. 12 be-cause the observations provide no useful informationfor 60% of the liquid clouds examined. In every one of

FIG. 12. As in Fig. 9, but for nighttime retrievals.


Fig 12 live 4/C

the remaining cases, the 3.75-�m channel provides themost information and all of the other channels are es-sentially redundant.

Because it is not always possible to constrain cloudboundary information using an active sensor, it is inter-esting to compare these results with those that corre-spond to retrievals in the absence of ancillary CloudSatcloud boundary information or MODIS albedo infor-mation. Similar results corresponding to retrievals em-ploying climatological xa are summarized in Table 3.Because cloud boundary information is not as impor-tant for accurate daytime retrievals, the daytime results

are not extremely sensitive to the presence of ancillarydata with the exception of the fact that the 11.0-�mchannel now provides a nonnegligible contribution tothe retrievals 50% of the time (although it is never thefirst channel selected). Nighttime retrievals, on theother hand, are much more promising with nonnegli-gible information coming from four different channelsat some point over the range of clouds modeled. Ingeneral, little information is lost provided that the 3.75-and 11.0-�m channels are adopted for nighttime liquidcloud retrievals. Inspection of the singular vectors for acloud with LWP � 85.71 g m�2 and Re � 8 �m (illus-trated in Fig. 15) demonstrates that all of the informa-tion from these channels is focused onto two singularvectors that derive from linear combinations of Ctop

and Re. Thus, a nighttime retrieval using measurementsat 3.75 and 11.0 �m offers the potential to retrieve someinformation regarding cloud height and to a lesser ex-tend geometric mean radius in the absence of ancillarycloud boundary and albedo information.

FIG. 14. As in Fig. 13, but relative to constant radianceuncertainties of 5%.

TABLE 3. As in Table 2, but for the case in which no ancillarycloud boundary or surface albedo information is available.

Band Wavelength Selected first Selected any position

Daytime1 0.64 28 1006 1.64 41 96

31 11.0 0 5026 1.38 27 50

7 2.13 0 202 0.86 4 12

Nighttime20 3.75 80 9231 11.0 20 6429 8.55 0 1632 11.92 0 11

FIG. 13. Information spectra governing the selection of optimalchannels for retrieving liquid cloud microphysical properties fromdaytime observations of a cloud with LWP � 85.71 g m�2 andRe � 8 �m residing between 1 and 2 km over an oceanic surface.Results are obtained relative to the radiance uncertainty estimatesfrom section 3a.

TABLE 2. Relative frequencies with which channels are selectedfor use in a retrieval based on their information content. The thirdcolumn lists the fraction of the cases examined for which thecorresponding channel provided the most information and wasselected first. The fraction of cases for which the channel offeredany contribution to the retrieval and was selected at any time ispresented in the fourth column.

Band Wavelength Selected first Selected any position

Daytime6 1.64 64 971 0.64 32 84

26 1.38 0 507 2.13 0 202 0.86 4 16

Nighttime20 3.75 40 40


5. Conclusions

Accurate measurements of the spatial and temporaldistribution of cloud microphysical properties are es-sential for modeling their role in global climate change.As the number of satellites and algorithms devoted tothis goal increases, it is of interest to revisit the cloudretrieval to objectively assess the information contentof the measurements and establish quantitative esti-mates of the accuracy to which various cloud propertiescan be retrieved. To this end, this paper introduces fourdiagnostics of information and uncertainty and appliesthem to study the problem of retrieving liquid cloudmicrophysical properties from visible and infrared ra-diances.

Based on a rigorous analysis of the errors and sensi-tivities of satellite measurements at 11 visible and in-frared wavelengths, the 0.64- and 1.64-�m channelsemerge as the optimal set for retrieving liquid waterpath and geometric mean radius during the daytime.There is some evidence that the 1.38-�m channel mayalso provide information regarding cloud-top heightunder certain circumstances but, because the use of thischannel may put unrealistic constraints on a detector’sminimum detectable absolute radiance, its utility is leftas an open question at this time. At night, 3.75 and 11.0�m are found to provide a limited amount of informa-

tion particularly in the absence of ancillary data to con-strain cloud height. In both cases other channels mayprovide a small amount of additional information undercertain conditions, but generally the remaining chan-nels supply only redundant information and do not jus-tify the additional computational cost required to inte-grate them into an algorithm.

Information contents computed for 50 pairs of geo-metric radius and liquid water path indicate that day-time retrievals can be expected to resolve up to �200states at low LWP and 32 states for thicker clouds,while nighttime retrievals may distinguish at most 2–4broad cloud states. Further analysis of retrieval errorcovariance matrices derived from careful considerationof a number of important sources of forward model andmeasurement errors demonstrate that daytime geomet-ric mean radius retrievals can be expected to be accu-rate to 5%–10%, while uncertainties in retrieved LWPare expected to be �10%–20%. Uncertainties in night-time retrievals, on the other hand, are comparable tothe ranges to which the retrieval parameters can beconstrained using climatological data.

At present, these conclusions apply only over an oce-anic background. Similar analyses must be conductedto assess the validity of this set of channels for retrievalsover land surfaces, but such calculations are beyond thescope of this preliminary study. Furthermore only liq-

FIG. 15. As in Fig. 11, but for nighttime retrievals in the absence of cloud boundary oralbedo information.


uid clouds have been modeled here but this study laysthe groundwork for the application of the informationcontent analysis to the more challenging problem of icecloud retrievals that suffer from additional uncertain-ties introduced by the varying shape of their constituentcrystals. This problem is addressed in Part II, whichimmediately follows this paper.

Last, note that the development presented here isneither contingent on the details of any forward model,nor is its scope limited to the cloud retrieval problem.In fact, to realize the full potential of the rigorous in-formation content analyses outlined here, they shouldbe applied in the developmental stages of future satel-lite missions to systematically develop instruments fromthe ground up. In principle, a detailed analysis of a widevariety of potential wavelengths could be performed todetermine the subset that provides the most informa-tion for the desired application based on the accuraciesto which they can be modeled and their sensitivities tothe retrieval parameters. Then, this information couldbe used to design an optimal channel configurationupon which an instrument could be constructed.

Acknowledgments. This research was supported inpart by NASA Research Grant NNG04GE35G, NASAEarth System Science Fellowship NGT5-30458, NASACloudSat Mission Grant NAS5-99237, and NASACALIPSO Mission Grant NAS1-99103. The authorsthank Mick Christi and Professor Laurent Labonnotefor their work in developing the Radiant model andtheir assistance in running it.

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