calculated relation of 0.6μm radiance and cloud optical thickness is captured by the first 4 .... PC is mainly responsible for the sensitivity at larger cloud optical.
Quantification of Uncertainties in Retrieved Cloud Properties from Satellite Imagers
Hartwig M. Deneke12 and Rob A. Roebeling1 1: Royal Netherlands Meteorological Institute (KNMI), De Bilt, the Netherlands 2: Meteorological Institute of the University of Bonn (MIUB), Bonn, Germany ABSTRACT The retrieval of cloud optical thickness and effective radius from meteorological satellite imagers requires inversion of the relation between cloud properties and top-of-atmosphere reflectances. This relation is highly non-linear, and depends on a large number of additional parameters, such as the sun and viewing geometry, and the surface albedo. Here, the application of principal component analysis is investigated to reduce the high dimensionality of the retrieval problem. It is found that 99.96% of the variability present in the precalculated relation of 0.6μm radiance and cloud optical thickness is captured by the first 4 principal components. It is also demonstrated that changes in the error sensitivity of the retrieval can be related to changes in the principal component amplitudes. Results show that the retrieval of cloud optical thickness is not only highly sensitive to random and systematic errors, but also undergoes significant situation-dependent changes. Implications of these finding for the retrieval of diurnal cycle information are discussed. It is shown that a systematic error of 1% in reflectance can introduce an artificial diurnal cycle with an amplitude of 510% in cloud optical thickness over the course of the day.
INTRODUCTION Nakajima-King-type retrievals (Nakajima and King, 1990) are widely used to obtain climatologies of cloud properties from meteorological satellite imagers. Their principle is based on 1D radiative transfer theory, and links solar radiances at a non-absorbing and an absorbing wavelength to cloud optical thickness and effective radius. Cloud reflectances at non-absorbing wavelengths are dominated by cloud optical thickness, while reflectances at absorbing wavelengths are mainly a function of cloud particle size. For the application of satellite-inferred cloud datasets, e.g. for the task of model evaluation, the error characteristics of these datasets need to be known to assess their accuracy and to judge the significance of possible deviations with other datasets. King(1987) derives analytic expressions linking cloud reflectance and cloud optical thickness based on asymptotic theory, including a simple formula for calculating the error sensitivity of a retrieval. Unfortunately, these expressions are only an approximation. Therefore, current retrievals are generally based on lookup tables of radiative transfer calculations. As these tables have to consider a large parameter space, they can become rather large in size. The number of parameters and their influence on the sensitivity complicate an accurate estimate of the sensitivity. The aim of this study is to apply principal component analysis (PCA) to the retrieval, and to identify dominating modes of variability in the relation of reflectances to cloud properties. As is shown here, these modes can be utilized in the retrieval algorithm, and can be used to quantify the error sensitivity of the retrieval based on a reduced number of parameters. From a technical perspective, PCA also enables a significant reduction in the size of the lookup tables, and a significant increase in computational efficiency. In the next section, an overview of PCA is given. Then, the application of PCA to the retrieval procedure of cloud properties is described. In the 3rd section, the error sensitivity of the retrieval is established, and is linked to the results of the PCA. Using this methodology, the sensitivity of satellite-retrieved information on the diurnal cycle of cloud optical thickness are discussed in section 4. Finally, conclusions are drawn and an outlook to future work is given.
PRINCIPAL COMPONENT ANALYSIS PCA (also known as empirical orthogonal functions, singular value decomposition or the Karhunen-Loeve transform) is a statistical technique for the identification of dominating modes of variability in multi-variate datasets. As first step, a dataset is centered by subtraction of the mean value from each dimension. Then, the covariance matrix of the dataset is calculated, and the eigenvectors of the covariance matrix are determined by singular value decomposition, sorting each eigenvector according to the magnitude of the corresponding eigenvalue. The eigenvectors form a new set of orthonormal basis vectors, referred to as principal components (PCs) here. Transforming the dataset to the new basis, the PC amplitudes are obtained, i.e. the weights that determine the contribution of each PC for a given sample of the dataset. Adding the PCs using their corresponding amplitudes as weight to the mean, the original dataset can be reconstructed. The PCA has two interesting mathematical properties, which have lead to its wide application in a variety of research fields, and motivate the present study. The magnitude of the eigenvalues constitute a measure of the variance explained by the PCs. If strong correlations are present in a dataset, a small number of PCs are often sufficient to accurately represent the dataset, while the PCs associated with smaller eigenvalues can be discarded. Thus, a compression of the original dataset is achieved. Also, the PC amplitudes are uncorrelated, thus changes associated with individual PCs can be considered independently.
APPLICATION OF PCA TO THE RETRIEVAL OF CLOUD OPTICAL THICKNESS
Figure 1: Relation of 0.6 μm reflectance and COT for a water cloud. The blue line indicates the mean, and the grey shaded regions the range of variabiilty (containing 33, 66 and 95 % of data).
The Cloud Physical Properties (CPP) retrieval (Roebeling et al., 2006) is used within EUMETSAT’s Satellite Application Facility on Climate Monitoring (CM-SAF) to derive cloud properties from SEVIRI instrument onboard the geostationary Meteosat satellite, as well as the AVHRR instrument flown on the polar-orbiting NOAA satellites. Based on the method proposed by Nakajima and King, 1990, the retrieval uses lookuptables (LUTs) for the estimation of cloud properties. These LUTs store reflectances pre-calculated with the Doubling Adding KNMI radiative transfer model (de Hahn et al., 1987) for a variety of cloud profiles. Starting with an initial guess, cloud optical thickness and effective radius are updated iteratively, until the observed values of 0.6 and 1μm reflectances match the model-predicted values. Here, it is only described how to apply PCA to the first step, i.e. the estimation of the cloud optical thickness from the 0.6μm reflectance, and
the second step is omitted at this time. Fig.1 shows the relation of 0.6μm reflectance as function of cloud optical thickness for a water cloud above a surface with an albedo of 6%. The blue line indicates the mean relation, while the grey-shaded areas delimit the range of variability, as is attributable to changes in sun and viewing geometry, as well as cloud droplet size. To obtain this figure, the data stored in the LUTs of the CPP retrieval have been used. We have applied the PCA to this dataset, treating the reflectances at the grid of cloud optical thickness values as multiple dimensions. Fig.2(a) presents the functional form of the first 4 principal components, and Fig.2(b) the fraction of the variance contained by each of the 4 principal components. From the values given in Fig.2b, it can be inferred that 99.96% of the total variance is already explained if retaining only the first 4 PCs.
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Figure 2: (a) Functional form of the first 4 principal components (arbitrary ). (b) Variance explained by the first 4 principal components (in percent).
If this level of approximation is deemed sufficient, an overall compression of the LUTs to less than 20% of their original volume can be achieved. This fraction results as reflectance at 22 values cloud optical thickness are stored in the current version of the lookup-tables. Using the PCA, they could be replaced by storing only the 4 PC amplitudes instead. In addition, the mean relation of reflectance as function of cloud optical thickness (the blue line in Fig.1(a)) and the PCs themselves have to be stored. However their space requirements are negligible in comparison to both the space required for PC amplitudes and the original values of reflectances. From the perspective of efficiency, this approach can also provide a substantial saving in computational costs. To determine the cloud optical thickness matching an observed reflectance, the reflectance values stored in the lookup table for a given grid of cloud optical thickness need to be interpolated to arbitrary values of cloud optical thickness. Spline and other polynomial interpolation techniques require two successive steps. First, the coefficients of the interpolating polynomial are determined, which depend only on the dataset being interpolated. The second step consists of the evaluation of the interpolating polynomial, and is computationally much cheaper. For the situation-dependent cloud optical thickness versus reflectance relation, both steps have to be carried out for each retrieval. In contrast, using the PCA-based retrieval, the coefficients of the interpolating polynomial of the mean and the PCs need only be determined once initially, and can be weighted using the PC amplitudes for the 2nd step of the interpolation.
ERROR SENSITIVITY OF RETRIVED CLOUD OPTICAL THICKNESS For users of satellite-based cloud datasets, a sufficiently accurate estimate of their accuracy is crucial. A way to obtain such an estimate from the PCA-based retrieval is described in the following. Only the error sensitivity in cloud optical thickness is presented here, and errors resulting from errors in 1.6μm reflectance are neglected, as their influence is minor for optically thick clouds (Nakajima and King, 1990). In linear approximation, the relative error of cloud optical thickness τ to errors in the 0.6μm reflectance r0.6 is given by the following equation:
ln = × r 0.6 . r 0.6 Expressed in words, the relative error sensitivity of the cloud optical thickness is thus the inverse of the slope of the cloud optical thickness versus r0.6 relation, if a logarithmic axis is used for optical thickness. The numerical values then correspond to the resulting relative error (in % per % absolute size of r0.6 error). If spline interpolation is used as suggested above, the derivative can readily be calculated from the coefficients of the interpolating polynomial.
Figure 3: Mean relative sensitivity of retrieved cloud optical thickness to errors in 0.6μm reflectance (black), and changes in sensitivity induced by the prinipal components.
The black line of Fig.3 shows the relative error sensitivity obtained for the mean relation of cloud optical thickness and reflectance (which is shown as blue line in Fig.1). Also, the changes of error sensitivity related to the first 4 principal components are shown by the colored lines. To delimit the range of change, the lower and upper limits of the interval containing 66% of the amplitudes have been used foir each principal component. The deviations from the mean sensitivity linked to the individual principal components show that the retrieval sensitivity varies significantly for different situations. The largest change in sensitivity for larger cloud optical thickness occurs due to variations in the amplitude of the 2 nd principal component. Both the magnitude and the location of the minimum error sensitivity are changed significantly. The minimum error sensitivity changes from values less than 3 to about 8 precent per percent, while the most sensitive optical tickness moves from a about 5 to 10. Another noteworthy point is the large error sensitivity visible both at small and large values of cloud optical
thickness. For the small values, this is partly caused by our usage of an absolute error term for the reflectance in the equation of error sensitivity. As an alternative choice, the relative error in reflectance could have been used, which would better reflect e.g the influence of errors in the instrumental sensitivity, and would result in much small errors at small cloud optical thicknesses. On the other hand, large variations in clear-sky reflectance occur over land due to changes in spectral surface albedo, which are unrelated to the instrumental sensitivity, and motivate our choice of using an absolute error in reflectance.
CHANGES IN ERROR SENSITIVITY WITH DIURNAL CYCLE The data presented in Fig.3 indicate that the error sensitivity of the retrieved cloud optical thickness varies considerably related to changes in the principal component amplitudes. As a consequence, even a constant systematic error can potentially introduce artificial temporal signals. Fig.4 shows the diurnal cycle of the first 4 PC amplitudes, and the resulting change in error sensitivity of the cloud optical thickness retrieval. These results correspond to water clouds with an effective radius of 10μm at the geographic position of Cabauw, the Netherlands on June 21st, 2006. The amplitudes of the principal components vary strongly throughout the day. As mentioned in the previous section, the amplitude of the 2nd PC is mainly responsible for the sensitivity at larger cloud optical thicknesses. Its time series shows a nearly symmetrical shape around local noon, where it also reaches it maximum value. Towards morning and evening, a smooth decline is found. This shape dominates the error sensitivity shown in Fig.4(b) for cloud optical thicknesses above 5, which show a minimum around noon and a similar curved shape. At smaller optical thicknesses, the modulation by the other PCs is also an important factor.
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Figure 4: Diurnal cycle of (a) the first 4 principal component amplitudes and (b) the sensitivity of the cloud optical thickness retrieval to errors in 0.6μm reflectance (in %/%). Results are obtained for a water cloud with an effective radius of 10 μm for the location of Cabauw (51.97N, 4.92E) on June 21st, 2006.
The SEVIRI instrument flown on the current series of geostationary Meteosat satellites offers for the first time the possibility to sample the diurnal cycle of clouds at a high temporal resolution of 15 minutes. For a cloud with a constant optical thickness above 5, an apparent change with an amplitude of about 5-10% would result from a systematic error in reflectance of 1%. Thus, care has to be taken to distinguish real diurnal changes in cloud properties from changes caused by the combined effect of a systematic error and a changing error sensitivity. This fact highlights that an accurate calibration is essential for estimating the diurnal cycle of cloud properties.
CONCLUSIONS AND OUTLOOK The lookup tables commonly used in cloud property retrievals from meteorological satellite imagers have to account a large number of parameters. As a result, they require large amounts of storage space, but also contain highly redundant information. The aim of this study has been the application of the PCA to reduce the high dimensionality of the lookup tables, and to identify dominating and uncorrelated modes of variability in them. The results reported here show that for the lookup tables of the CPP retrieval, the use of only 4 principal components allows to capture 99.96% of the variance, while achieving a reduction of lookup tables to less than 20% of their original size. Due to the fact the PCA is a linear model, a computationally efficient implementation of the retrieval algorithm is possible. In addition, we have shown that the the variability captured by the principal components can be linked to changes in the error sensitivity of the retrieved cloud optical thickness. Based on this link, it is demonstrated that a 1% error in 0.6μm reflectance cause an error of at least 5-10% in cloud optical thickness. A retrieval with reasonable relative accuracy is only possible for cloud optical thicknesses in an interval ranging from about 2-60, as the errors increase strongly outside that interval. Currently, a prototype version of the CPP retrieval algorithm based on PCA is being developed and tested. Envisioned advantages are a significantly reduced size of the LUTs, a reduction in computational costs, and the ability to provide accurate error bounds with only a small increase in processing time. In the future, we will also investigate how to apply the PCA to the second step in the retrieval process, i.e. the estimation of cloud effective radius.
REFERENCES de Haan, J., Bosma, P., Hovenier, J., 1987: The adding method for multiple scattering calculations of polarized light. Astron. and Astrophys. 183, pp371–391. King, M., 1987. Determination of the scaled optical thickness of clouds from reflected solar radiation measurements. J. Atmos. Sci. 44, pp1734–1751. Nakajima, T., King, M., 1990. Determination of the optical thickness and effective particle radius of clouds from reflected solar radiation measurements. Part I: Theory. J. Atmos. Sci. 47, 1878–1893. Roebeling, R.A., A.J. Feijt, and P. Stammes, 2006: Cloud property retrievals for climate monitoring: Implications of differences between Spinning Enhanced Visible and Infrared Imager (SEVIRI) on Meteosat-8 and Advanced Very High Resolution Radiometer (AVHRR) on NOAA-17. J. Geophys. Res., 111, D20210, 10.1029/2005JD006990.
