An initial analysis of the pixel-level uncertainties in global MODIS cloud optical thickness and effective particle size retrievals

An initial analysis of the pixel-level uncertainties in global MODIS

cloud optical thickness and effective particle size retrievals

S. Platnick1, R. Pincus

2, B. Wind

3, M. D. King

1, M. Gray

3, P. Hubanks

3

1 Goddard Space Flight Center, Greenbelt, MD 20771 USA

2 NOAA-CIRES Climate Diagnostics Center, Boulder, CO 80305 USA3 L3 Communications Government Services, Vienna, VA 22180 USA

ABSTRACT

Operational Moderate Resolution Imaging Spectroradiometer (MODIS) retrievals of cloud optical thickness and

effective particle radius employ well-known solar reflectance techniques using pre-calculated reflectance look-up tables.

We develop a methodology for evaluating the quantitative uncertainty in simultaneous retrievals of cloud optical

thickness and particle size for this type of algorithm and present example results.

The technique uses retrieval sensitivity calculations derived from the reflectance look-up tables, coupled with estimates

for the effect of various error terms on the uncertainty in inferring the reflectance at cloud-top. The error terms include

the effects of the measurements, surface spectral albedos, and atmospheric corrections on both water and ice cloud

retrievals. Results will deal exclusively with pixel-level uncertainties associated with plane-parallel clouds; real-world

radiative departures from a plane-parallel model are an additional consideration.

While we demonstrate the uncertainty technique with operational 1 km MODIS retrievals from the NASA Earth

Observing System (EOS) Terra and Aqua satellite platforms, the technique is generally applicable to any reflectance-

based satellite- or air-borne sensor retrieval using similar spectral channels.

Keywords: clouds, remote sensing, uncertainty, MODIS, Terra, Aqua

1. INTRODUCTION

Clouds are generally recognized as a critical component in understanding and modeling the present climate as well

as potential climate change. This is due to their critical role in both the radiative and water cycle budgets. Key cloud

parameters of interest include optical (optical thickness), microphysical (thermodynamic phase, effective particle radius,

water content), hydrological (water path), and macroscopic (fraction or extent, height and physical thickness) quantities.

Knowledge of many of these parameters on a global and long-term scale requires (at least) a suite of passive and

active satellite-borne sensors. While active sensors are just beginning to be deployed (e.g., GLAS, CALIPSO,

CloudSat), passive cloud remote sensing has a relatively long heritage. Over the last several decades, passive sensors for

quantitative cloud remote sensing have been developed to exploit the visible through infrared portions of the spectrum

(e.g., AVHRR, VIRS, POLDER, ATSR, MODIS for inferring most of the aforementioned cloud parameters) as well as

the microwave spectral region (e.g., SSM/I, TMI, AMSR, AMSR-E for inferring water path and precipitation). These

sensors include large swath coverage from either polar orbiting or precessing orbits.

Flying on both the Terra and Aqua EOS platforms, MODIS is a state-of-the-art sensor providing 250-1000 m

spatial resolution in 36 spectral bands, and a swath of about 2300 km providing nearly daily coverage1. It is important to

note that MODIS includes an on-board reflectance-based calibration sub-instrument (via a solar diffuser) for bands up

through 2.1 m, as well as an additional sub-instrument for monitoring diffuser stability up through 1 m. The spectral

Passive Optical Remote Sensing of the Atmosphere and Clouds IV, edited bySi Chee Tsay, Tatsuya Yokota, Myoung-Hwan Ahn, Proc. of SPIE Vol. 5652(SPIE, Bellingham, WA, 2004) · 0277-786X/04/$15 · doi: 10.1117/12.578353

30

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coverage includes all solar reflectance bands useful for inferring cloud optical thickness ( ), effective radius (re), and

thermodynamic phase; these principal bands are at 0.65, 0.86, 1.6, 2.1, and 3.7 m. In addition, water path (liquid or

ice) is proportional to the product and re with the assumption that the cloud is vertically homogeneous. Each of these

parameters is included in the operational MODIS cloud product (referred to with the product nomenclature MOD06 and

MDY06 for Terra and Aqua, respectively) that is archived at the Goddard Earth Science Distributed Active Archive

Center (DAAC)2,3

.

With daily global coverage, it is important to assess the pixel-level uncertainty of these retrievals. To date,

uncertainties are generally specified in an a priori sense4 or inferred through comparison with ground-based or aircraft

validation efforts5,6,7

. In the present study, we develop a methodology for assessing the overall uncertainty on a pixel-to-

pixel basis directly from component error sources that are inherent to the retrieval. The error terms included in this

study are those due to incomplete knowledge of the instrument calibration, surface spectral albedo, and spectral

atmospheric correction (primarily due to atmospheric moisture uncertainty). In particular, the methodology is designed

to work efficiently with a look-up table algorithm (e.g., MODIS2, ISCCP

8, a variety of aircraft-based algorithms, etc.)

by requiring no new radiative transfer calculations. While other error terms can be important (e.g., vertical and/or

horizontal inhomogeneity, including multilayer cloud systems, and cloud model assumptions), the present effort

provides a useful baseline in the sense of estimating a minimum uncertainty. We discuss the methodology in Sec. 2 and

provide a MODIS example in Sec. 3.

2. METHODOLOGY

Rather than rely on computationally expensive real-time cloud radiative transfer calculations, imager retrievals

typically make use of pre-computed cloud reflectance and/or emissivity libraries across the expected range of cloud

parameters and viewing geometry (an exception are optimal estimation iterative methods which do not have the

demonstrated efficiency to be applied to daily global data9,10

). In particular, libraries in the MODIS operational

algorithm are calculated for a homogeneous cloud overlying a black surface in the absence of an atmosphere. The effect

of the surface is incorporated into individual pixel retrievals using an ancillary global albedo data set. A model state

profile, coupled with an independent cloud-top pressure/temperature retrieval, provides an above-cloud water vapor

amount used in atmospheric corrections (includes well-mixed gas absorption and Rayleigh scattering); this correction

procedure gives the reflectance at cloud-top in the absence of an atmosphere. Fundamentally, surface-modified library

reflectances are compared with the atmospherically-corrected satellite observations and a best , re pair is chosen2.

Here we describe a methodology for providing quantitative estimates of pixel-level uncertainty for optical

thickness, effective radius, and water path in a computationally efficient manner (note that a single sunlit side of a

MODIS orbit typically contains in excess of 107 1 km cloudy pixels). A straightforward approach would be to assess

uncertainties by perturbing all relevant non-retrieved parameters (e.g., cloud model parameter assumptions, clear-sky

atmospheric state and surface albedo obtained from ancillary data sets, etc., sometimes collectively know as forward

model parameters) and perform new radiative transfer calculations. Instead, we estimate the effect of the uncertainty in

these relevant parameters on the inference of the cloud-top reflectances. This allows us to separate retrieval

uncertainties into the product of two components: the sensitivity of the retrieved cloud quantities to cloud-top

reflectance and the uncertainty in estimating cloud-top reflectance due to uncertainties in non-retrieved parameters. In

doing so we are able to use sensitivities calculated directly from the reflectance libraries and no new radiative transfer

calculations are required. The uncertainty is specific to the original retrieved cloud parameter state; no iterations are

implemented. As an example, for a two-channel retrieval, the change in retrieved optical thickness and effective radius

due to changes in a some parameter a can be written as

R

1 R2

dR1

daa +

R2 R

1

dR2

daa

re

re

R1 R

2

dR1

daa +

re

R2 R

1

dR2

daa

, (1)

Proc. of SPIE Vol. 5652 31


where R1 and R2 are the cloud-top reflectances in two spectral bands (e.g., 0.86 and 2.1 m), and the partial derivative

(retrieval sensitivity) with respect to one band is calculated with the reflectance in the other band held fixed. The

sensitivity dependence on the retrieved , re solar and viewing geometry, and surface spectral reflectance is understood.

The relationship is approximate to the extent that the sensitivity value is linear over the assumed perturbation. In this

study, a represents one of three individual uncertainty components (error sources) that contribute to the overall

uncertainty in the inferred cloud-top reflectance: instrument calibration, surface spectral albedo, and spectral

atmospheric correction (primarily atmospheric state). Though we are defining sensitivity as the derivative of the

retrieved state ( , re) with respect to the measurements, it should be noted that the term sometimes refers to the inverse

calculation (derivative of measurements with respect to the retrieved variables) 9

.

2.1 Retrieval sensitivity

The retrieval sensitivity derivatives can be written in terms of derivatives at fixed and re. For example, it can be

shown that

R1 R

2

= R

1

re

R

1

re

R2

re

R2

re

1

. (2)

The partial derivatives inside the bracket of Eq. 1 can be calculated directly from the reflectance libraries that are, by

default, available at fixed and re values. The other three sensitivities in Eq. 1 have a similar form.

The two dominant (typically) retrieval sensitivities are shown in Fig. 1 for a liquid water cloud as a function of

and re. The MODIS spectral bands corresponding to R1, R2 are specified in the caption. Surface albedos are for an ocean

surface. As expected, both retrieval sensitivities are relatively large when both and re are small (as the information

content of either parameter is reduced). Optical thickness retrievals are more sensitive at large as R1( ) asymptotes.

Effective radius retrievals become more sensitive at larger re as droplet absorption increases. The non-dominant

sensitivities are shown in Fig. 2, given as a ratio with respect to the sensitivities of Fig. 1. In this example, the ratios

indicate the relative importance of R2.1 in optical thickness retrievals, and R0.86 in effective radius retrievals. A zero ratio

indicates orthogonality. As expected, the magnitudes of both ratios decrease as optical thickness increases.

2.2 Retrieval uncertainty

We seek an expression for the variance and bias of the error distributions of and re in Eq. 1. Dropping the

explicit notation for the fixed reflectance quantity in the sensitivity derivative, we can write Eq. 1 in matrix form as

xk

= S Rk

xk

= k

re,k

, R

k=

R1k

R2k

, S =

S11 S12

S21 S22

=

R1 R2

re

R1

re

R2

. (3)

for the kth

error source (e.g., measurement error). The covariance matrix and expected value for the retrieval vector are

then given as

32 Proc. of SPIE Vol. 5652


cov xk( ) = S cov R

k( ) ST , i.e., k

2

kre,k

kre,k

re,k

2

= S

R1k

2

R1kR2k

R1kR2k

R2k

2

S

T

E xk( ) = SE R

k( ) ,

(4)

where the R subscripted terms are variances and covariances of the inferred cloud-top reflectance due to the kth

error

source. Diagonals of the retrieval covariance matrix represent the variances of and re; symmetric off-diagonal terms

give the covariance between and re. For multiple error sources,

x = xk

k=1

N

E x( ) = E xk( )

k=1

N

. (5)

If the error sources are independent (true for the three sources mentioned above), we can write the combined covariance

matrix as

cov x( ) = cov xk( )

k=1

N

. (6)

So for the three error sources considered in Sec 3, we have

cov x( ) = cov xmeas( ) + cov xatmo( ) + cov xsfc( )

= S cov Rmeas( ) + cov Ratmo( ) + cov Rsfc( )( ) ST, (7)

representing measurement (calibration and other relevant instrument characteristics, radiative transfer calculation

uncertainties), above-cloud atmospheric corrections, and surface albedo contributions. The remaining non-trivial

quantities are of course the elements of the Rk

covariance matrices. Examples assignments are given in Sec. 3.1. The

net root-mean-square (rms) uncertainty is the square root of the sum of the variance and the square of the expected

value (bias). In the following example, bias is ignored and the reported rms uncertainty is equivalent to the standard

deviation.

3. EXAMPLE UNCERTAINTIES FOR MODIS RETRIEVALS

In this section, we apply the methodology discussed in Sec. 2 to MODIS Terra data for an example granule (5

minutes of data, or approximately 2000 km in the along-track direction). We segregate uncertainties by liquid water and

ice clouds, and by ocean and land surface types.

3.1. MODIS data and component uncertainty assumptions

The example MODIS data granule is of coastal Peru and Chile (18 July 2001, 1530 UTC). As shown in Fig. 3

(grayscale image in the upper panel, thermodynamic phase retrieval in the lower panel), the scene includes boundary

layer marine stratocumulus clouds off the Peru/Chile coast, cirrus bands in the southern portion of the granule, and

convective activity in the upper Amazon basin. Much of the cirrus clouds also overlie lower stratocumulus water clouds.



Uncertainties are calculated for the default operational MODIS retrieval which utilizes the 2.1 m shortwave

infrared band in conjunction with the 0.86 m band over the ocean, the 0.65 m band over land, or the 1.2 m band

over snow/ice (there is minimal snow cover in the southern Andes). Retrievals from this granule have been previously

reported2.

The measurement error is assumed to have an uncertainty specified by a standard deviation of 3% relative for all

bands (i.e., a dispersion of 0.03). The predominant land surface type along the coast is barren desert (e.g., Atacama

Desert in northern Peru). An open shrub land ecosystem occupies the mountains and eastern foothills, while broadleaf

forest and grasslands represent the major ecosystems elsewhere. The spectral surface albedos for these ecosystems were

derived from the MODIS albedo and land cover products (MOD43 and MOD12, respectively) as discussed elsewhere2.

The standard deviation in the specification of the surface albedo was assumed to be 20% relative for all ecosystems and

spectral bands. Atmospheric correction uncertainties are subject to uncertainties in specification of the above-cloud

water vapor amount (which in turns depends on ancillary model uncertainties of the atmospheric moisture profile

associated with a pixel as well as a separate MODIS cloud-top pressure retrieval) and uncertainties in use of

atmospheric spectral transmittance look-up tables that ignore detailed moisture/temperature profile information (see

Platnick et al. for details2). The transmittance look-up table uncertainties are derived from several thousand profiles and

are included in the tables; it is assumed that knowledge of above-cloud column water vapor has a standard deviation of

20% relative. The correlation between cloud-top reflectance perturbations R1 and R2 due to error in the above-cloud

water vapor is specified as 1.0 (related to off-diagonal terms in Rk

).

3.2. Uncertainty results

The uncertainty calculations are a function of eight dynamic quantities ( , re, , 0, , thermodynamic phase,

surface spectral albedo, and above-cloud water vapor amount), i.e., quantities that are allowed to change on a pixel-to-

pixel basis. It is therefore difficult to succinctly summarize the results. As such, it is useful to give results separately for

liquid water and ice clouds, and further categorized by ocean and land surfaces. Visualizing the remaining variability ( ,

re, , 0, , water vapor) is still problematic.

We begin with the optical thickness uncertainty. Fig. 4 shows the aggregated liquid water cloud optical thickness

uncertainty as a function of , re for pixels over the ocean. The gray-scale indicates the overall relative uncertainty (e.g.,

an uncertainty of 10% associated with =15 implies an absolute uncertainty of 1.5). Specifically, Fig. 4 gives the mean

value of the rms uncertainty for all pixels found in each -re bin; details on the distribution of uncertainties in each bin is

lost in this type of plot. Despite the remaining functional variability, contours are generally aligned with the sensitivity

derivative of Fig. 1, as expected. It should be pointed out that the bins are not uniform throughout the plot and the

probability of pixel counts in any -re region is not visualized.

Summing over all re gives the plot of Fig. 5 (left panel) with the corresponding plot for liquid water clouds over

land in the right panel. It is now clear that the optical thickness uncertainty is larger in both the small and large

regimes as anticipated. In particular, note the increased uncertainty at smaller ’s for land surface pixels due to surface

albedo uncertainties (the 0.65 m albedo over vegetation is only moderately larger than the ocean albedo at 0.86 m

and so the effect is not dramatic). As optical thickness increases, the effect of the 0.65 m land surface reflectance

specification on the overall retrieval decreases until there is no discernible surface influence on the overall uncertainty.

A minimum mean uncertainty is found at a moderate optical thickness of about 5-10 (somewhat smaller for ocean

pixels).

Similarly, the relative effective radius uncertainty is shown in Fig. 6. Uncertainty is particularly large for the

smallest re retrievals (5-6 m) because those pixels are associated with very small retrievals (see Fig. 4). However,

very few pixels (in a probability sense) populate these small -re bins. The minimum uncertainty corresponds to a rather

broad effective radius, with some increase for larger radii.



Fig. 7 is similar to Fig. 5, but for pixels identified as ice clouds. Optical thickness is now on a log scale, with a

minimum value of 0.1. At this small value, the mean uncertainty is in excess of 100% for both ocean and land pixels.

For larger thicknesses (>1), uncertainty is similar to Fig. 5.

4. DISCUSSION

Passive imager cloud retrievals using solar-reflectance techniques are providing important and unique data sets, and

will continue to do so on future platforms (e.g, National Polar Operational Environmental Satellite System (NPOESS)11

and the NPOESS Preparatory Project (NPP)12

). Quantitative estimates of these retrievals will be needed to fully utilize

such data sets for developing cloud climatologies understanding physical processes, and assisting in climate and

forecast model development (e.g., model validation, parameterization, and cloud assimilation efforts).

A methodology for incorporating quantitative uncertainty estimates for optical thickness, effective radius, and

water path into solar reflectance retrievals has been discussed. No additional radiative transfer calculations are required

beyond the cloud spectral reflectance look-up tables already used as part of the standard retrieval approach (e.g.,

MODIS). In addition to the generalized method, the present application and example accounts for the effect of three

error components: measurements, surface albedos, and above-cloud atmospheric corrections. Uncertainties from these

three components will be calculated for each cloudy MODIS 1 km pixel in the upcoming processing stream (expected to

being in early 2005). Currently available timing runs indicate that the extra processing requirements are well within the

processing system capability. An error source that is not currently accounted for is cloud model error, most notably the

uncertainties in ice particle size habits and size distributions required for particle scattering calculations. Additionally,

non plane-parallel radiative effects are not addressed. The current emphasis has been on establishing and understanding

a baseline uncertainty consisting of those components that are both important and can be reasonably quantified.

However, it should be noted that in many studies, uncertainties in spatial and/or temporal pixel aggregations (e.g.,

gridded statistics3,8

or synoptically related cloud studies13

) are more relevant than the individual pixel-level uncertainties

discussed. The uncertainty in the mean of a population of retrievals depends on the extent of pixel-level bias compared

with random errors. While pixel-level bias and random error are both accounted for in the present methodology, a

quantitative assessment of aggregation uncertainty depends on the uncertainty correlations between pixels common to a

spatial/temporal calculation (e.g., covariances among the optical thickness uncertainties for a set of pixels). These

covariances will depend on the distances between the pixels (in space and time) relative to the correlation scales of the

error components considered. In general, this is qualitatively, not to mention quantitatively, difficult. Suffice it to say

that such correlations are expected to decrease with increasing spatial/temporal aggregation scales and that uncertainties

for aggregated statistics may be substantially reduced from pixel-level uncertainties.

ACKNOWLEDGEMENTS

We are grateful to G. Wind and G. T. Arnold from L3 Communications Government Services for assistance with

this study.

REFERENCES

1. Barnes, W. L., T. S. Pagano, and V. S. Salomonson, 1998: Prelaunch characteristics of the Moderate Resolution

Imaging Spectroradiometer (MODIS) on EOS-AM1. IEEE Trans. Geosci. Remote Sensing, 36, 1088-1100.

2. Platnick, S., M. D. King, S. A. Ackerman, W. P. Menzel, B. A. Baum, J. C. Riedi, and R. A. Frey, 2003: The

MODIS cloud products: Algorithms and examples from Terra. IEEE Trans. Geosci. Remote Sens., 41, 459-473.

3. King, M. D., W. P. Menzel, Y. J. Kaufman, D. Tanre, B.-C. Gao, S. Platnick, S. A. Ackerman, L. A. Remer, R.

Pincus, and P. A. Hubanks, 2003: Cloud and aerosol properties, precipitable water, and profiles of temperature and

humidity. IEEE Trans. Geosci. Remote Sens., 41, 442-458.



4. “Algorithm Theoretical Basis Document: Cloud retrieval algorithms for MODIS: optical thickness, effective

particle radius, and thermodynamic phase”, 1998: http://modis-atmos.gsfc.nasa.gov/reference_atbd.html.

5. Platnick, S., and F. P. J. Valero, 1995: A validation study of a satellite cloud retrieval during ASTEX. J. Atmo.

Sci., 52, pp 2985-3001.

6. Mace, G. G., Zhang, Y., S. Platnick, M. D. King, P. Yang, 2004: Evaluation of cirrus cloud properties derived from

MODIS radiances using cloud properties derived from ground-based data collected at the ARM SGP site. J. Appl.

Meteor., (in press).

7. Dong, X., P. Minnis, G. G. Mace, W. L. Smith, M. Poellot, R. T. Marchand, and A. D. Rapp, 2002: Comparison of

stratus cloud properties deduced from surface, GOES, and aircraft data during the March 2000 ARM cloud IOP. J.

Atmos. Sci., 59, 3265-3284.

8. Rossow, W. B., and R. A. Schiffer, 1999: Advances in understanding clouds from ISCCP. Bull. Am. Meteor. Soc.,

80, 2261-2287.

9. Watts, P. D., C. T. Mutlow, A. J. Baran, and A. M. Zavody, 1998: Study on cloud properties derived from Meteosat

Second Generation observations: EUMETSAT ITT, Final Report no. 97/181, 344 pp.

10. Miller, S. D., G. L. Stephens, C. K. Drummond, A. K. Heidinger, and P. T. Partain, 2000: A multisensor diagnostic

satellite cloud property retrieval scheme. J. Geophys. Res., 105, 19955-19971.

11. http://www.ipo.noaa.gov/library_NPOESS.html.

12. http://nppwww.gsfc.nasa.gov/.

13. Klein, S. A., and C. Jakob, 1999: Validation of sensitivities of frontal clouds simulated by the ECMWF model.

Mon. Weather Rev., 127, 2514-2531.



xx

Fig. 1. Two example sensitivity derivatives for a liquid water cloud over a dark ocean surface. Panel (a) shows the

sensitivity of retrieved optical thickness ( ) with respect to the 0.86 m MODIS band reflectance (R0.86), with the

reflectance in the 2.1 m band (R2.1) fixed, (b) is the sensitivity of the effective radius (re) retrieval to the R2.1 with

R0.86 fixed. The geometry is 0=0.85, =0.65, and =60°.

R0.86 R

2.1

re

R2.1 R

0.86

-63

-63

-40

-40

-25

-25

-25

-16

-16

-10

-10

-6.3

-6.3

-4

-4

-100 -63 -40 -25-16

-6.3

-2.5

-1

Effe

ctiv

e R

adiu

s (∝

m)

Optical Thickness5 10 30 50 100

510

1515

2025 a)

b)

Fig. 2. Ratio of optical thickness sensitivity for the two bands (upper panel) and similarly for effective radius (lower

panel). The ratios indicate the relative importance of R2.1 in optical thickness retrievals, and R0.86 in effective radiusretrievals. A zero ratio indicates orthogonality. Same geometry as in Fig. 1.

re

R0.86

re

R2.1

R2,1

R0,86



Fig. 3. Terra MODIS granule (5 minutes of data) showing marine stratocumulus off the coasts of Peru and Chile, cirrus

to the south, and convective systems over the Amazon basin (18 July 2001, 1530 UTC). The thermodynamic phaseretrieval is shown in the bottom panel.

MODIS

grayscale

composite

Cloud

thermodynamic

phase retrieval

Uncertain

Ice

Liquid water

Clear (no retrievalattempted)



cloud optical thickness

effe

ctiv

e ra

dius

(µm

)

1 10 100

3

6

9

12

15

18

21

24

Fig. 4. Contour plot of mean relative optical thickness uncertainty as a function of retrieved optical

thickness and effective radius, for those cloudy pixels identified as liquid water phase and overlying the

ocean in the granule of Fig. 2. Note that the contours are roughly aligned with the sensitivity derivativeexample shown in the top panel of Fig.1.

10 %

100 %

50 %

Fig. 5. Mean relative optical thickness uncertainty (rms) as a function of retrieved optical thickness, for

those pixels identified as liquid water phase in the granule of Fig. 2. Left panel if for cloudy pixelsoverlying the ocean, right panel is for cloudy pixels overlying land.

water clouds over water clouds over land

cloud optical thickness cloud optical thickness

rms /



Fig. 7. Similar to Fig. 5 but for ice cloud optical thickness.

ice clouds over ocean ice clouds over land

cloud optical thickness cloud optical thickness

rms /

Fig. 6. Similar to Fig. 4 but for liquid water cloud effective radius.

water clouds over ocean water clouds over land

cloud effective radius (µm) cloud effective radius (µm)

re,rms /re



An initial analysis of the pixel-level uncertainties in global MODIS cloud optical thickness and effective particle size retrievals

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