Retrieval of two layer cloud properties from multispectral observations using optimal estimation

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2011jd015883, 2011 Retrieval of two layer cloud properties from multispectral observations using optimal estimation P. D. Watts, 1 R. Bennartz, 2 and F. Fell 3 Received 1 March 2011; revised 24 May 2011; accepted 14 June 2011; published 20 August [1] A method to derive two layer cloud properties from concurrent visible, near infrared, and infrared observations is described. It is a modification of a single layer scheme and is applied to Spinning Enhanced Visible Infrared Imager (SEVIRI) observations and validated against coincident A Train data, principally to evaluate the accuracy and characterize cloud top pressure (CTP) estimates. CTP values obtained from the single layer scheme applied to multilayer clouds are significant overestimates of the upper layer value. The effect is usually larger than that on coincident IR only retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS), and this characteristic can be traced to the use of visible wavelength observations. However, the solution cost from the optimal estimation method is found to be especially high in multilayer situations and is a strong indicator of CTP accuracy. Tighter thresholds on the solution cost select, with increasing stringency, scenes with single layer or opaque upper layer cloud. High cost (presumed multilayer) pixels are reprocessed with the scheme adapted to simulate a two layer cloud and with only infrared measurements. The upper cloud is represented by the parameters of the original formulation; the additional lower cloud layer is gray and has a proxy height given by the surface temperature. Despite the simplicity of the cloud atmosphere modeling under the upper layer, results obtained from the two layer scheme are promising. Upper layer CTPs are of comparable accuracy to the single layer cases, lower layer CTPs show some useful accuracy, and upper layer optical depths correlate well with radar observations. Citation: Watts, P. D., R. Bennartz, and F. Fell (2011), Retrieval of two layer cloud properties from multispectral observations using optimal estimation, J. Geophys. Res., 116,, doi: /2011jd Introduction [2] Inference of cloud microphysical and bulk properties from passive radiation measurements is compromised by the presence of multilayered (ML) clouds. In such cases, the inferred results lie typically between the layer properties: cloud altitudes below the upper layer and above the lower, effective radii larger than lower water layer values but less than upper ice values [Davis et al., 2009]. As ML situations are common [e.g., Wind et al., 2010, Figure 10], it is important that an accurate representation of such clouds is made in cloud climatologies derived from satellite based observations. [3] The two aspects of remote sensing of ML clouds, detection and property retrieval, have so far received different levels of attention. Most work has naturally concentrated on detection, and methods commonly use ratios or differences of 1 European Organization for the Exploitation of Meteorological Satellites, Darmstadt, Germany. 2 Department of Atmospheric and Oceanic Sciences, University of Wisconsin Madison, Madison, Wisconsin, USA. 3 Informus GmbH, Berlin, Germany. Copyright 2011 by the American Geophysical Union /11/2011JD standard visible (VIS) and infrared (IR) measurements to isolate characteristic signals [e.g., Pavolonis and Heidinger, 2004; Nasiri and Baum, 2004; Chang and Li, 2005]. Algorithms for the Moderate Resolution Imaging Spectroradiometer (MODIS) can make use of the absorption channels, 1.38 mm [Pavolonis and Heidinger, 2004] and 0.94 mm [Wind et al., 2010], which, by virtue of the atmospheric water vapor absorption, are mostly sensitive to upper layers of cloud. These methods are applicable at the pixel level. Other approaches require single layer or cloud free quantities to be available in larger search areas (e.g., pixels [Nasiri and Baum, 2004] and low cloud temperature within a search area radius of 125 km [Chang and Li, 2005]). The retrieval of properties in detected ML situations is not straightforward, with a critical dependence on the relative characteristics of the layers. A thick upper layer will tend to hide a lower layer and preclude accurate estimation of its properties, while if the upper layer is very thin, its properties will be hard to estimate and those of the lower layer will be more tenable. Chang et al. [2010] present a modification to the IR only based CO 2 slicing method that, in the case that the single layer method based on surface temperature and pressure fails, iteratively solves for effective surface properties that represent an underlying cloud layer. 1of22

2 Table 1. Parameters and Settings of the Base Optimum Cloud Analysis (OCA) Cloud Model a State Parameter, x Name Abbreviation Symbol x a x 0 " a Optical depth COT t 4 Al. Effective radius CRE r e 8/30 Al. Top pressure CTP p 700/400 Al. Fractional cover CFR f b Skin temperature TS T s ECM ECM 1.0 a Here x a is the value of the prior assigned and x 0 is the value assigned to the first guess; " a is the assumed error in the prior (all uncorrelated) and is the square root of the diagonals of S a ;a value indicates no information in the prior and the retrieved value will have no dependence on it. A value of 0.0 indicates the prior value is retained as the retrieved product. Al. indicates an (efficient) algorithm is used to obtain the value, and ECM indicates that the value is obtained from European Centre for Medium Range Weather Forecasts (ECMWF) data. COT, cloud optical thickness; CRE, cloud effective radius; CTP, cloud top pressure; CFR, cloud fraction; TS, skin temperature. b A value 1e 5 is used to permit matrix inversions. They concentrate on the validation of the improved upper layer heights obtained and do not explicitly consider the derived properties of the lower layer cloud. Chang and Li [2005] include the use of a single VIS channel (0.6 mm) and locally obtained lower cloud height to both detect ML cloud and constrain the retrieval of properties from both layers. [4] The work presented here is focused on two layer cloud property retrieval, although it may contribute to the issue of detection. It is based on an optimal estimation (OE) scheme (optimum cloud analysis (OCA)) for cloud property retrieval. The scheme uses, in principle, all the measurements of the Spinning Enhanced Visible and Infrared Imager (SEVIRI) [Schmetz et al., 2002] and simultaneously estimates the essential observable properties of a single layer cloud [Poulsen et al., 2011; Watts et al., 1998]. Essential diagnostic outputs of the OE method are a measure of the model fit to the scene, that is, the cost function J, and formal error estimates of the retrieved parameters. It is found that clean homogeneous single layer clouds are characterized by low values of J and that therefore J can be used to quality control retrieved parameters. ML clouds generally result in high values of J and, for the purpose of this study, it is used as the ML detection method. We note, however, that as high values can be the result of other causes, it is likely that for robust ML detection it is inadequate to use this test alone (an example is given in Appendix B). [5] The high J in ML scenes indicates that there might be information in the measurements that would support a model with a second cloud layer. Here we have modeled the second (lower) layer in a simple way to test the hypothesis. The OE parameters of the standard single layer (SL) scheme are set appropriate to values of a typical overlying cirrus cloud, and the normally strongly constrained skin temperature is allowed to vary significantly. The skin temperature acts as a proxy cloud top temperature (CTT) representing the lower cloud level. [6] CTPs of the two layers are validated using the data from the CloudSat Profiling Radar (CPR) and the Cloud Aerosol Lidar with Orthogonal Polarization (CALIOP) lidar collocated allowing for parallax and sampling issues to the SEVIRI pixel locations. [7] The paper is organized as follows. Section 2 describes the OCA method as applied in the normal way assuming a single layer cloud is present. Section 3 introduces the validation of CTPs using the A Train data and demonstrates the relation between J, ML cloud scenes, and product quality. Section 4 describes the modification of OCA to retrieve parameters of the two layer (2L) system, and section 5 presents a validation of the resulting CTPs. Section 6 discusses the method and results and future directions. Appendices A C provide some technical details and present results from several specific cases to illustrate points made in the text. 2. Retrieval Method, SL [8] The standard OCA, SL, retrieval method varies the parameters, x, of a single layer cloud model to minimize a cost function J [Rodgers, 2000], JðÞ¼ x ðyx ðþ y m Þ T S 1 ðyx ðþ y m Þþðx x a Þ T S 1 ðx x a Þ y where y are measurements, either from the observation, y m, or calculated from the cloud model parameters, y(x), and S y 1 is the inverse of the error covariance of the measurements (see section 3.2). An x indicates the state vector of cloud parameters and x a is its prior value. S a 1 is the inverse error covariance of prior state. Thus the first term on the righthand side represents the deviation of modeled radiances from observations, hereinafter called the measurement cost, J m. The second term is the prior cost, J a. In practical terms the prior values for the cloud parameters hold little or no information, S a 1 is very small and J m determines the solution. The cloud state parameters, their symbols, and standard prior error values " a are given in Table 1. [9] Exceptionally, the skin temperature, T s, has a prior obtained from European Centre for Medium Range Weather Forecasts (ECMWF) forecast fields and is therefore regarded as accurate. Therefore in the SL retrieval, T s deviates little from the prior value. It is introduced here since, as described, it is used in the 2L scheme as a proxy for the temperature of the lower cloud. [10] A key part of the algorithm is the radiative transfer, or forward, model y that maps from state to measurements. This model is correctly indicated as y(x, m) as it is also dependent on parameters, m, that are not retrieved. These include temperature, humidity, ozone (and the resulting channel atmospheric transmissions), surface emissivity, albedo, etc.; m also rather more implicitly includes assumptions like ice scattering models and particle size distributions (see Table 2). [11] Model y(x, m) is a fast model based on cloud radiative properties precalculated using the scattering model RTMOM [Govaerts, 2008] for plane parallel clouds and stored in lookup tables (LUTs). The properties include bidirectional reflectance for solar channels and directional transmission, diffuse transmission, and reflection for all channels; IR channels require also cloud emission. The LUTs cover all satellite view and solar zenith angles in 10 steps and relative azimuth in 18 steps. Log 10 t ranges from 0.0 to 2.4 in steps of 0.301, and r e ranges from 1 to 23 mm for water cloud, step 2 mm, and from 4 to 92 mm, step 4 mm, for ice cloud. Water cloud scattering is Mie with a modified gamma distribution and effective variance of Ice cloud scattering a 2of22

3 Table 2. Fixed Parameters of the OCA Forward Model and Their Sources Model Parameter, m Type Source Atmospheric temperature humidity and ozone profile, 43 levels surface to 0.1 hpa ECMWF 0 18 h forecast Emissivity (land) spectral (8 channel) CIMSS a Albedo (land) spectral (3 channel) EUMETSAT CRM Water scattering model Spheres, Mie Ice scattering model Aggregates b a Cooperative Institute for Meteorological Satellite Studies; Seemann et al. [2008]. b Baran and Francis [2004]. properties are taken from the work of Baran and Francis [2004] and are based on aggregate particle shapes. Atmospheric contributions to the RT in the IR are provided by RTTOV 9[Matricardi et al., 2004] run on ECMWF forecast fields at 0.25 and 3 h space and time resolution. For the VIS channels, two path (Sun satellite Sun) transmittances are calculated using a spectrally highly resolving transmittance model based on LBLRTM [Clough et al., 2005]. [12] The analytic gradient of y(x, m) with respect to x, K x (and selected parts of m, K m ) is available, permitting application of the iterative Levenburg Marquardt [Marquardt, 1963] algorithm in the minimization of J. Starting from the first guess values, x 0, steps dx = (J + ai) 1 J are taken until convergence, defined as when the change in J is less than a predefined value (here 1.0). J and J are the first and second derivatives of J with respect to x, and I is the unit matrix. The a is varied to bias the step calculation toward steepest descent (a larger) or toward Newtonian descent (a smaller) if the latest attempt increased or reduced J, respectively. [13] Care is taken to obtain good first guess (FG) values so that the expensive minimization process is as short as possible. They are obtained from simple (i.e., fast) algorithms proceeding in the order cloud phase, optical depth, effective radius, and top pressure. FG phase is taken for the previous processed pixel and only overridden in the case the 10.8 mm value is outside the range 268 < BT < 243 K. FG optical depth is then estimated from the 0.8 mm channel and a simplified version of the full RT model given the FG phase and local geometry. FG effective radius follows similarly but using also the FG optical depth and the 1.6 mm channel. FG cloud top pressure is estimated from the 10.8 mm channel corrected for transparency according to the FG optical depth and compared to the ECMWF temperature profile. Special considerations for tropopause and boundary layer inversions are made for FG pressure, but the details of these are not required here. [14] Cloud phase is determined within the algorithm without any prior temperature constraints, i.e., the measurements alone determine it by utilizing the near separation in radiance space (particularly the nonabsorbing VIS channels at 0.6, 0.8, and 1.6 mm) of the two phases. In practical terms, if an iterative step would take the effective radius larger than the water LUT upper value, the phase changes to ice. A reverse process causes phase changes from ice to water. Some cases of clear misclassification noted in growing maritime stratocumulus, warm ice clouds, suggest, however, that a temperature constraint might be usefully added. Mixed phase clouds cannot be accommodated by the scheme and when present will result either in intermediate retrieved parameter values and/or high J values, but to date we have very little experience of such effects. [15] The value of x at the end of the iterative process, whether converged or not, represents the retrieval solution, ^x. Two essential diagnostics are also available, the solution cost J(^x) and the statistically expected error S^x. We have previously discussed J, and it is the primary indicator of retrieval quality since it indicates scenes that are not consistent with the retrieval algorithm model. A high J value indicates that none of the retrieved products are valid. When J is low, we can assume the retrieval algorithm does adequately model the scene. In this case an estimate of parameter error can be made assuming linearity of the forward model around the solution; thus S^x is given by S^x =(S a 1 + K x T S y 1 K x ) 1 = J 1. S^x is a covariance and describes both expected errors and correlations between errors. In this paper, only the errors (the diagonals of S^x ) are considered. Note that unlike the implication of a high J, a high error in one parameter does not necessarily imply a high error in another, so parameters must be considered separately. We note that while there is a formal definition of high and low for J (related to the problem s number of degrees of freedom), we continue to need empirically defined thresholds because our modeling of S y is approximate and because real world errors are non Gaussian (Appendix C). 3. Data [16] Cloud property retrievals were made from Meteosat data and ancillary data as listed in Table 3, and the validation was made with data from various A Train satellites A Train: CPR, CALIOP, and MODIS [17] Twelve daytime A Train overpasses from August 2006 (collocated with Meteosat 8) and two from June 2008 (Meteosat 9) were selected; the details are given in Table 3. The selection of orbits was essentially random, i.e., no meteorological criteria were used. They were chosen to be Table 3. CloudSat Overpass IDs, Dates, and SEVIRI Time Slots for Coverage for the Study Data a Overpass Date SEVIRI Slots, UT Instrument b Aug :15 11: Aug :45 13: Aug :00 14: Aug :45 10: Aug :00 11: Aug :30 13: Aug :45 14: Aug :15 15: Aug :45 17: Aug :45 11: Aug :00 12: Aug :30 14: Jun :30 14: Jun :15 14:45 9 a SEVIRI, Spinning Enhanced Visible Infrared Imager. b Meteosat 8 or Meteosat 9. 3of22

4 Table 4. Channel Calibration Adjustments and Noise Assumptions a Channel (mm) Calibration Adjustment Homogeneity Noise Coregistration Noise Radiometric Noise + Others % 2% mw.st 1.cm.m % 2% mw.st 1.cm.m % 1.5% mw.st 1.cm.m K 0.15 K 0.4 K b K 0.15 K 0.4 K b K 0.15 K 0.25 K K 0.1 K 0.21 K K 0.1 K 0.23 K K 0.03 K 0.1 K 0.8 K c a See section 3.2 for details. b Water vapor channel noise is increased to allow for errors in ECMWF humidity fields. c The 13.4 mm channel noise is increased to allow for decreased accuracy due to calibration drifts. daytime over the SEVIRI disk and to have a good proportion of the orbit in disk so that for the most part extreme SEVIRI view angles are avoided. [18] The core of the validation undertaken in this work makes use of the CloudSat Cloud Profiling Radar [Stephens et al., 2002] products and in particular the 2B GEOPROF (cloud mask and radar reflectivities) and 2B CLDCLASS (cloud classification) data sets. From these, the primary validating CTP was obtained and, of course, a visual representation of vertical cloud structure. The cloud classification product was examined in order to extract CTPs of lower level clouds in ML situations. [19] The CPR sensitivity to most cloud appears to be reasonably well matched to that of the passive imager IR channels. Nevertheless, it is clear that thin high level cloud, particularly in the tropics, can occasionally affect SEVIRI measurements but be effectively invisible in the CPR reflectivities. We used Cloud Aerosol Lidar with Orthogonal Polarization (CALIOP) level 1 reflectance data to identify and make sense of these situations. [20] Finally, we make use of the MODIS/Aqua level 2 cloud products subset along the CPR track. This gives an interesting perspective on the SEVIRI results, as the MODIS CTPs are derived from IR only measurements with a more complete IR/CO 2 slicing channel complement than is available to SEVIRI. [21] All A Train data are mapped to the SEVIRI pixel locations taking account of relative sampling issues and parallax caused by the non nadir Meteosat view. The adjustment is made using the CPR cloud height to determine the parallax effect [Fell et al., 2009] Meteosat SEVIRI [22] Calibrated and geolocated radiances from the Meteosat 8 and 9 satellites coincident with the A Train overpasses were used. Fifteen minute SEVIRI slots bracketing the equator crossing time of the overpass were processed so as to minimize final collocation time differences. The 14 overpasses provided around 15,500 pixels classified as cloudy in both SEVIRI and CPR data. [23] Nine SEVIRI channel measurements are used in the retrieval scheme with calibration adjustments and assumed noise values (Meteosat 8 and 9 the same) given in Table 4. SEVIRI VIS channel calibration adjustments have been established recently by several parties [Doelling et al., 2004; Jolivet et al., 2009; Ham and Sohn, 2010; J. F. Meirink, personal communication, 2009], and the values used are an effort to summarize these results. The only IR adjustment is in the 13.4 mm channel and is due to instrument contamination of the optics [Hewison and König, 2008]. The contamination effect drifts and the radiometric noise is somewhat enhanced to allow for this (Table 4, footnote c). Noise in the OE context is defined as the random deviations between actual measurements, y m, and modeled values, y(x), and are represented in the OE by the covariance S y. The simplified plane parallel model in OCA is unable to reproduce some of the features of the values observed from real scenes (even in the case of single layer cloud), and the noise that this generates is represented, rather crudely, in the terms homogeneity and coregistration. Homogeneity is a sweeping term to encompass the unmodeled effects of subpixel variations in optical thickness, vertical gradients of effective radius, horizontal photon transport within and between clouds, etc. [24] The coregistration term aims to model the noise due to different instantaneous fields of view of the channels. Both homogeneity and coregistration noise effects are taken as cumulus cloud type results from the work of Watts et al. [1998] where they were estimated using Along Track Scanning Radiometer data. They are given in terms of percent of total signal for the VIS channels and as a delta brightness temperature for the infrared. Channel noise is taken to be the basic instrument radiometric error unless enhanced for a particular reason as noted. The treatment of the water vapor channel errors assumes a particular importance in this work, and this is elaborated on in section 6. [25] Other potentially important contributions to the measurement errors arising from the model parameter assumptions (section 2) are not currently modeled, and the reasons for this and potential implications are discussed in Appendix C. [26] Pixels processed must have been flagged as probably cloudy by the European Organization for the Exploitation of Meteorological Satellites (EUMETSAT) Meteorological Product Extraction Facility (MPEF) operational cloud mask and have satellite and solar zenith angles less than 85 and 80, respectively. 4. CTP in ML Cloud: SL Algorithm [27] Figure 1 is a tropical section of overpass 1735 chosen to demonstrate some of the characteristics of CTP retrievals in complex scenes. Figure 1 (top) is the CPR reflectivity and 4of22

5 Figure 1. Tropical section of overpass (top) CloudSat Profiling Radar (CPR) reflectance image with symbols indicating CTH from CPR (white), Spinning Enhanced Visible Infrared Imager (SEVIRI) water (green), and SEVIRI ice (blue). (middle) The daytime true color SEVIRI image along the track (indicated by a dotted line center). (bottom) The CTH from the Moderate Resolution Imaging Spectroradiometer (MODIS), annotated a e at areas discussed in the text. shows that a variety of cloud system types are present, from SL low level stratus and cumulonimbus, through to ML situations with varying upper and lower cloud thicknesses and heights. Figure 1 (bottom) enables us to see the behavior of the CTP (shown as cloud top height (CTH)) retrievals made with SEVIRI VIS/IR OCA system and the MODIS IR only. Some regions are noted to aid the discussion. [28] Region a (around pixel 2000) appears to contain a SL low water cloud and both algorithms give a reasonable value. The small scale of the clouds probably leads to issues of collocation and sampling and may explain the absence of the higher cell (pixel 1990) in the SEVIRI result. Regions d appear to contain either SL high clouds or, where there are underlying layers, a thick upper layer. Both retrievals return accurate CTH values in these regions with the SEVIRI value of order 1 km lower than the MODIS. The reason for the bias is not clear but, given the different algorithms, measurements and assumptions used, perhaps not too surprising (when all orbits are considered a SEVIRI bias of around 0.3 km lower than MODIS is found). The SEVIRI phase in these regions is ice. [29] Areas with clear ML cloud behave differently depending on the thickness of the upper layer and the location of the lower layer. Moving from area a to area c, the lower cloud level is more or less unchanged in height, whereas the upper layer thickens. Very thin cirrus over areas a and b can only be seen in CALIOP reflectances (Figure 2). The MODIS CTH varies from midway between the layers in the thin (area b) region to some 2 3 km lower than the upper layer in the thicker (area c) region. The SEVIRI CTH is much lower in the atmosphere; at area b it lies on or just above the lower layer and through c varies about the midway level possibly, but not clearly, depending on the upper layer thickness. The SEVIRI phase switches between water and ice. In region e the lower cloud, only faintly visible in the CPR, is much thinner than regions b and c, and the CTH retrievals stay correspondingly higher; the MODIS is only slightly lowered and the SEVIRI moderately lowered by the low cloud layer. [30] The high SEVIRI OCA CTH sensitivity to the lower cloud is a result of the joint retrieval (of specifically t and CTP) and simultaneous consideration of the VIS and IR channels. The VIS channels drive the estimation of t and see both layers even when upper layers are relatively thick or the lower layer is near the surface. The fitting of the IR channels with a SL cloud model is therefore made consistent with the high t of the combined ML clouds, resulting in low CTHs. With MODIS, however, there is a mitigating factor. If the lower cloud is very low, i.e., has a near surface temperature, it is effectively invisible to an IR only retrieval and such a two layer system would be retrieved with similar accuracy to a SL system. When the lower cloud has a thermal contrast to the surface, it does have an effect on the IR only MODIS retrieval. 5of22

6 Figure 2. Same as Figure 1 (top) but showing the Cloud Aerosol Lidar with Orthogonal Polarization (CALIOP) reflectances. [31] We now turn to the OCA solution cost diagnostic, which is shown for the same overpass and section in Figure 3. It shows there is a clear relation between regions of high J m and the ML regions in Figure 1 (the annotations are in the same positions); the compromise CTH found by the OE SL retrieval does not simultaneously give a good fit in all the channels when two or more cloud layers are present. Rising J m values follow the increasing cirrus thickness in regions a through c. The horizontal dashed line indicates the empirically chosen threshold (J m = 90) below which we consider OCA retrievals to be of good quality and, in this work, above which pixels are considered to contain ML clouds. We reiterate though that other conditions (shadows, high aerosol loading, etc.) can cause high cost retrievals. [32] For this individual overpass section, the effect on SEVIRI retrieved CTH is shown in Figures 4a and 4b where Figure 4a includes all (377) pixels and Figure 4b indicates only those with J m < 90 (167 pixels). It is clear that the low J m test results in a significantly lower bias, 0.78 km compared to 1.75 km, and standard deviation, 0.82 compared to 2.24 km. The penalty paid, of course, is the loss of pixels with estimates, here about 56%. [33] Figures 4c and 4d examine the MODIS CTH values. Figure 4c confirms that in the absence of any quality control the MODIS results are less subject to low CTH resulting from ML cloud and the mean difference to the CPR is actually 0.0 km in this (small) sample. It is worth noting, however, that when the J m test quality control (from SEVIRI data) is applied, there is a significant reduction in scatter, supporting the notion that the test is reacting to real physical conditions and not an artifact of the SEVIRI OCA process. Filtered MODIS results have a similar scatter and somewhat smaller bias than the filtered SEVIRI results. [34] SEVIRI results obtained for all 14 overpasses together are shown in Figures 5a 5c. Figure 5a contains all data without any quality control and shows three main areas of error. By far the greatest number of erroneous pixels are OCA results that are too low, and these lie between true (CPR) CTH values of 5 and 14 km; they are probably mostly ML systems. Clouds below this height are probably mostly opaque single layer water clouds, and it is possible that those above, i.e., the highest tropical convective clouds, are dense convective cores and thick anvil cloud. Smaller populations of high OCA values occur for (CPR) CTHs of around 2 4 km and km. The lower group could also be caused by ML systems when the upper layer is so thin that the CPR does not register it, but the lower layer instead (e.g., area b in Figure 2). [35] The higher overshooting group might be caused by cloud tops lying in weak temperature gradients around a poorly defined tropopause; there is then little constraint from the SEVIRI channels on the cloud top. The overall correlation achieved is 0.83 and the bias and standard deviation are 1.30 and 2.28 km, respectively. With the J m < 90 quality Figure 3. Optimum cloud analysis (OCA) measurement cost, J m, for the same section as in Figure 1. 6of22

7 Figure 4. Scatterplots of SEVIRI, MODIS, and CPR CTH for the section of overpass (a) OCA SEVIRI all (377) pixels. (b) Same as Figure 4a but with J m < 90 (167 pixels). (c) MODIS all pixels. (d) Same as Figure 4c but with OCA J m < 90. Water phase pixels in green, ice phase in blue, as given by OCA scheme. 7of22

8 Figure 5. Density plots of OCA SEVIRI compared to CPR CTH for all 14 A Train overpasses. (a) No quality filtering applied. (b) Quality control J m < 90. (c) Additional quality control S p < 5 hpa applied. 8of22

9 Figure 6. Effect on statistics and pass frequency of the threshold on J m used for quality control. Vertical line indicates the value of 90 used in this study to discriminate between single layer (SL) and multilayered cases. test applied (Figure 5b) a substantial number of the erroneous ML SEVIRI results are removed. It appears that the density of overshooting cases is also somewhat reduced, although we would not really expect these to be associated with a higher J m value. The statistics are a correlation of 0.92, bias and standard deviation of 0.81 and 1.56 km, respectively, and 56% of pixels remaining. [36] We show also Figure 5c where the results are additionally filtered by the expected error in the retrieved CTP. An error level of 5 hpa is applied, and the effect is most notably the removal of the overshooting cases supporting the notion that these are in more or less isothermal regions. The overall correlation and error statistics are marginally improved but with a further 9% of the pixels lost. Applying the error filter alone is found to be significantly less effective than application of the cost filter alone. [37] The characteristics of the statistics as a function of cutoff value for J m is shown in Figure 6. It demonstrates the trade off between higher accuracy and lower pixel pass rates. 5. The 2L Algorithm [38] The OCA algorithm was modified so that pixels for which the J m > 90 were reprocessed assuming a two layer cloud system. In principle, the 2L system would consist of two layers each with a defining t, r e, and p, effectively doubling the number of retrieved state parameters in comparison with the SL model. Fast RT modeling for such a 2L system consistent with the OCA approach and methodology has been developed [Siddans et al. 2010]. Here, however, we take a preliminary step to test the hypothesis by using a modified version of the SL model to simulate a 2L system. It avoids much additional complexity in the RT and retrieval algorithm and is based on existing stable code. A disadvantage is that the RT is approximate and only applicable to the IR channels, thus probably limiting the information that can be extracted. The implementation is particularly straightforward in that the state vector element T s, normally representing the physical skin temperature and highly constrained to the given ECMWF value, is instead taken as the effective IR emitting temperature of a lower cloud layer. To change its function in this way, to first order, it is only necessary to remove the constraint on its value. The lower cloud is then, in the OCA 2L RT, black and lying at the surface pressure level and characterized only by its temperature, T Lblack = T s. This temperature can be interpreted as a CTH by reference to the local ECMWF temperature profile, but it is of course the height of the equivalent black cloud. We use further approximations and take information from the SL algorithm result to refine this height estimate. The retrieved t in this 2L system is that of the upper layer, t U. We assume that the previous SL retrieval t SL still has some validity as the measure of the overall ML optical depth, although the most likely presence of two cloud phases clearly, in principle, invalidates this. Then we can derive a measure of the lower layer optical depth as t L = t SL t U. This value can be used to estimate the transmission of the lower cloud to IR (the 10.8 mm channel is used as a reference), and therefore to estimate the temperature of the equivalent gray cloud, T_ Lgray (see Appendix A for details). T_ Lgray is matched to the ECMWF temperature profile to give the transmission adjusted CTH. [39] Table 5 compares the availability and origin of the cloud and skin temperature parameters in the SL and 2L implementations. [40] Compared to the SL run, the reprocessing of the pixel takes place with altered prior, first guess, and error values to reflect the assumption that the upper cloud is likely (thin) cirrus. These are given in Table 6 and can be compared to the SL settings in Table 1. The t is assumed lower with a finite albeit still very large prior error; r e is set to a small (for ice) value and a small error, in this case so that the inversion remains stable given the shortage of particle size information available without the VIS (particularly 1.6 mm) channels. The pressure is set to represent typical cirrus levels, but 9of22

10 Table 5. Availability and Origin of Surface Temperature and Two Layer Cloud Parameters in the Single Layer (SL) and Two Layer (2L) OCA Implementations SL the 100 hpa error means there is effectively little constraint from this value. T s, now representing the lower cloud, is set to 280 K but again with a high error indicating little constraint. Note that there are no algorithmic first guess possibilities for the parameters now that they represent the upper layer in a two layer system. [41] Results of the 2L algorithm on the 1735 overpass section for CTH and COT are given in Figure 7 and Figure 8, respectively; no quality filtering is applied. Where the SL produced a J m < 90 the CTH results are unchanged from Figure 1 and are indicated with green (water) or blue (ice) dots. The 2L results are indicated by green and blue squares for lower and upper layers, respectively. [42] The 2L upper level CTH in Figure 7 is very much improved compared to the SL results (Figure 1). In regions a to c the upper layer is placed between 10 and 15 km with a clear increase in accuracy ( 15 km) as the cirrus thickens toward (region c). This thickening is also clearly registered in the upper layer COT, Figure 8, which continues to a maximum value of around 4 at region d. The cirrus COT reaches a maximum of 8 10 here in a few SL results. The 2L upper layer COTs are limited in value to around 5 8; at this thickness, the cirrus is opaque in most of the IR channels and the SL run finds a low J m solution (areas d). COTs from MODIS and SEVIRI appear to be in broad agreement; area e and just prior to the second from left d are exceptions with significantly higher MODIS values. [43] Returning to Figure 7 and 2L OCA lower layer CTH values: in regions a to b the somewhat too high SL CTHs (Figure 1) are replaced somewhat too low 2L values. Where the cirrus becomes visible in the CPR at c and beyond, the 2L lower CTH shows good agreement when the upper layer is thin, but poorer values as it thickens, around the first d, for example. The right hand half of the section is characterized by thicker upper level clouds and lower clouds that are either thin or subvisible in the CPR. The 2L lower CTHs are much more scattered here and do not relate, e.g., at e, to obvious features in the CPR reflectivity. A slightly different perspective is given by the results plotted over the CALIOP reflectivity, Figure 9. Region e is now seen to contain low 2L 2L Origin of Information Upper Cloud t U normally estimated, t SL normally estimated a IR r eu normally estimated normally estimated IR p U normally estimated normally estimated IR Lower Cloud t L n/a estimated from t SL t U IR + VIS r el n/a n/a p L n/a estimated from T s interpreted as T Lblack and t L IR + VIS Surface T s normally estimated n/a a Normally estimated means the parameter is explicitly present in the state vector and estimated as such; n/a indicates a parameter estimate is not available. and midlevel clouds between 2 and 7 km, consistent, if not perfectly matched, with the lower CTH results there. [44] The lower CTH values around pixel 2400, however, do not appear to correspond to any observable lower layer. The upper layer CTH values here are certainly improved compared to the SL result if the CALIOP reflectances are considered but are high compared to the CPR. The cirrus here is geometrically very thick, up to around 6 km in depth, optically thin (values around 1 2), and the high SL J m is probably a result of this deviation from the SL geometrically thin cloud model. We might presume that the 2L model is able to represent the diffuse cloud better than the SL model, although there are clearly not two distinct layers present. [45] The OE expected CTH errors are shown in Figure 9 (bottom) (the bar extends +3s to 3s), although the value for the lower layer is obtained rather crudely from the expected error in the T s parameter mapped assuming a constant 6 K/km lapse rate. Certainly it is obvious that for the upper layer the error is greater the thinner the upper layer cloud; it is less clear for the lower layer, but the error here is large when the upper layer is thick. A final point to note from Figure 9 is the discrepancy between CPR and CALIOP CTH values for the upper level cloud. CPR values lie between 11 to 15 km while the CALIOP values are consistently between 15 and 16 km. This is a common feature of upper level CTH assessed by these instruments in the tropics. [46] In the following we present an evaluation of the 2L algorithm CTHs using CPR data over the 14 A Train overpass data set. The reported CPR CTH is of course nominally the upper layer CTH. However, it is clear from inspection of the radar reflectivities in comparison to CALIOP that occasionally the CPR is insensitive to thin upper layer clouds and the recorded CTH refers in reality to the lower layer. If it were to be still assigned as a validation point to the 2L upper layer CTH then large unwarranted discrepancies will arise. This is the case, for example, in regions a and b in Figure 7. The comparison with CPR for the section is shown in Figure 10a with the misassigned 2L upper CTHs are apparent in the upper left corner. In Figure 10b the OCA 2L CTH which most closely matches the CPR value is taken for comparison; if it is the lower layer, then the green square symbol is used. [47] The result is that the cluster at OCA upper CTH = km is replaced by lower CTH values, in this case more or less in agreement with the CPR. While this fix, given that two options are available for matching, could lead to overly optimistic statistics, we consider that the physical justification is sufficient. [48] Figures 11a 11d show the density scatterplots and statistics of the CPR validation of the upper layer cloud over the 14 overpasses. Figure 11a includes all pixels, i.e., without Table 6. Parameter Settings for the 2L System 2L State Parameter, x x a x 0 " a Optical depth COT t Effective radius CRE r e Top pressure CTP p Fractional cover CFR f Lower cloud temperature T Lblack of 22

11 Figure 7. Same as Figure 1 but with the two layer (2L) OCA algorithm results and analyzed CPR second level cloud shown. (MODIS results omitted for clarity (see Figure 1)). any quality control or reassignment applied. The cases where the CPR registers the lower cloud is hypothesized to be the cluster with CPR CTH between 1 and 4 km and that lying above the 1:1 line with CPR CTH around 7 km. Application of the reassignment method results in Figure 11b with the clusters effectively removed (31% of pixels) and much improved statistics; correlation 0.74, bias and standard deviation 0.8 and 1.81 km, respectively. [49] As with the SL process, the measurement cost J m can be used to check the model consistency in the 2L retrievals. The threshold used this time is empirically set at 50; that it is less than the SL threshold is consistent with the fewer channels used in the 2L process (the J m is not a normalized cost). Figure 11c shows that this is moderately effective in increasing correlation, to 0.78, and improving statistics, bias and standard deviation to 0.67 and 1.53 km, respectively, at a loss of 30% of the pixels. Unlike the effect of J m filtering in the SL cases (Figure 5b) where clearly low OCA CTH pixels were removed, there is no distinct population removed in the 2L case. [50] Finally, Figure 11d shows that the expected error in the CTH carries distinct information on the quality. The Figure 8. Same as Figure 7 but showing COT. The MODIS and the OCA values for single layer (1Lw or i) are total column COTs. Upper and lower values are indicated where OCA ran in 2L mode. 11 of 22

12 Figure 9. Same as Figure 7 but using the CALIOP instrument as reference and showing CTH expected error bars in the bottom panel. effect of filtering the results for an expected error less than 30 hpa raises the correlation to 0.80 and improves the bias and standard deviation to 0.46 and 1.27 km, respectively; a further 32% of the original pixels are removed. This filter does perhaps remove a distinct overshooting population in a fashion similar to that found with the SL cases. [51] Comparable (ice phase) quality controlled SL results (not shown) have a correlation 0.93, bias and standard deviation 1.0 and 1.40, respectively, meaning the 2L upper level CTHs are of similar quality. One significant difference is the 2L CTH negative bias placing the cloud top on average 0.5 km above the CPR level, whereas the SL results are on average 1.0 km below. We can at this stage only speculate that the reason for this lies either in the different measurement input (VIS+IR for SL and IR only for 2L retrievals) or in the highly approximated IR below cloud transmission used in the 2L case. [52] Validation of the lower CTH for the pixels where the CPR CTH is reassigned to the lower layer is presented in Figure 12; Figure 12a for all pixels, Figure 12b for pixels with J m < 50, and Figure 12c for pixels with, additionally, expected errors <50 hpa. The J m quality check again appears to be effective with an increase in the correlation from 0.74 to 0.83 and a reduction in standard deviation from 2.04 to Figure 10. OCA and CPR CTHs compared for the section of overpass 1735; dots are SL results and squares are 2L results. (a) Unadjusted comparison. (b) Comparison with reassignment of heights. 12 of 22

13 Figure 11. Density plots of CPR and OCA SEVIRI CTH for the upper cloud layer in 2L retrievals. (a) All cases with assignment of CPR CTH always to OCA upper layer CTH. (b) With reassignment to OCA lower level, no quality control. (c) Quality control J m < 50 filter applied. (d) Additional quality control S p < 30 hpa applied. 1.68; around half the pixels are removed. The additional quality control using expected error this time has little effect with even a slight, although probably not significant, increase in standard deviation. [53] Apart from these swapped assignment pixels, for all other pixels the lower layer CTH has no corresponding validation value in the CPR product. As an attempt to get around this problem we have analyzed the CPR CLDCLASS product which gives an estimate of the cloud class (type) at each reflectivity bin. This product is evaluated from top down to identify the starting bin of each layer present. The height of the first layer corresponds to the nominal CPR CTH and that of the second layer is our desired lower layer CPR CTH. Figures 13a 13c show the results of validating the OCA 2L 13 of 22

14 Figure 12. Density plots of CPR and OCA SEVIRI CTH for the reassigned lower cloud layer in 2L retrievals. (a) All cases. (b) Quality control J m < 50. (c) Additional quality control S p < 50 hpa applied. lower layer with these estimates from the CPR. Again we have three results, Figure 13a with no quality control, Figure 13b with J m < 50, and Figure 13c with, additionally, an expected error <20 hpa. The correlations, 0.20, 0.26, and 0.35 and the standard deviations 2.53, 2.44, and 2.47, respectively, would suggest that there is only marginal skill in extracting the second layer CTP and that while the J m quality control again appears to be useful, the expected error is here less so. This is somewhat surprising, as we would expect the accuracy of lower layer CTP values to be very sensitive to the overlying cloud thickness, something the OCA error analysis is well suited to supply. Characteristics of the validation data, in the sense of the interpretation of the CPR reflectances we have used, is likely to also contribute to significant variance in the comparison. In any case, we consider that compared to SL retrievals of these scenes, any information on a second layer, 14 of 22

15 Figure 13. Density plots of CPR and OCA SEVIRI CTH for the lower cloud layer in 2L retrievals where the lower CPR level is the second cloud layer found in the reflectivity profile. (a) All cases. (b) Quality control J m < 50. (c) Additional quality control S p < 20 hpa applied. e.g., an accurate mean value, should be considered a significant advantage. 6. Discussion and Conclusions [54] Using the validation of CTH with CPR and CALIOP reflectances, we have shown that the solution cost in an OE SL cloud parameter retrieval scheme based on VIS and IR SEVIRI measurements is usually high when ML cloud is present. The scheme uses a cost threshold, here <90, to determine which SL retrieved parameters are to be considered acceptable and by such means can considerably improve the CTH statistics at the cost of losing 50 60% of pixels with results. In this work, high cost retrievals were considered to always be the result of 15 of 22

16 ML cloud and the pixels were reprocessed with modified initial and constraining conditions to simulate, in the IR channels, a 2L cloud. The original SL COT retrieval is used to refine the 2L lower CTH. Compared to the CPR, the resulting upper layer CTH is well retrieved and the lower CTH retrieved with at least an indicative accuracy, although the validation of this layer currently poses challenges. The 2L COTs, although not formally validated here, are visually consistent with the CPR and CALIOP reflectivities. [55] Although executed with a simple and very approximate RT model, the results are very encouraging, demonstrating that 2L information is available from nine SEVIRI channels on a pixel by pixel basis without recourse to auxiliary information from surrounding pixels. We should stress that it is the coincident use of the nine channels that permits first the high cost identification of non SL scenes, and second allows the 2L parameters to be retrieved. In particular, it was found that credible results are only possible if the two water vapor channels (6.2 and 7.3 mm) are given high weight (Table 4) by assigning them low errors in the OE scheme. Observations of the brightness temperature deviations from forecast values in cloud free pixels indicate a forward model error for these channels of 2 3 K (because of NWP humidity errors), but here we have assumed an error as low as 0.4 K. Of course a cloud free pixel exposes the full NWP humidity error, whereas in the presence of clouds and depending on their altitude and thickness, some of this error will not affect the measurements. Nevertheless, overweighting of these channels for low cloud is a clear danger and is one of several possible explanations for the obvious failure of the cost based ML identification and 2L retrievals over the simple stratocumulus case documented in Appendix B, case 4. [56] As a first and very simple approach to the 2L problem, there are several clear opportunities for refinement. Priority will go to implementation of a VIS IR 2L radiative transfer [Siddans et al., 2010] and a full extension of the OE state vector to include all parameters of the lower layer. This will address the rather ad hoc use of the VIS measurements in the current method; the lower cloud will no longer be gray and, for IR transmission purposes, now properly located in the atmosphere. The renewed availability of the 1.6 mm channel should also significantly enhance the information on the upper r e. [57] Although the SL cost has proved useful enough as an ML detection for this study, there are clearly cases where either the high cost is not associated with ML cloud (e.g., Appendix B, case 4) or, conversely, ML cloud situations giving low costs (e.g., Appendix B, case 2). An analysis of the 14 orbits shows that in pixels where both MODIS and SEVIRI register clouds, the J m < 90 test flags 39.7% of pixels as failed SL in OCA whereas the MODIS multilayer cloud flag [Wind et al., 2010] gives 17.6% (flag value > 2) or 20.2% (flag value > 1). When only those subsequently processed 2L retrievals that pass the J m < 50 test are considered as ML, the proportion is 24.4%. The same analysis carried out with respect to the ML situations determined from layer analysis in the CPR cloud class product gives 19.2% ML in CPR; 43.1% failed SL in OCA and 27.8% ML after J m < 50 filtering. The higher frequency of OCA ML (after quality control) might be in part related to the areas observed where diffuse deep cloud gives a high SL cost and is better treated with the 2L model (e.g., sections of Appendix B, case 2, Figure 9). It might be argued, in the case that overall fidelity to the observations is concerned, that a 2L representation of such clouds is superior to poor SL parameters. Nevertheless, it appears that a simple cost threshold is generally too crude for ML detection, and it will be beneficial to explore the use of a more refined cost measure or established ML detection methods and use them either instead of or, additionally, to the SL cost measure. While the cost measure has the advantage of naturally including all channels in detection, it is also, compared to threshold methods, nontargeted and does not use sign dependent signals in the data. A possible refinement would be to examine the cost arising from only the infrared channels (which is technically possible because errors are assumed uncorrelated between visible and infrared measurements), which might be expected to exclude some cases where the scene causes predominantly visible channel anomalies (e.g., shadowing, aerosol loading, etc.). One extreme strategy within the OE framework is to attempt all possible scene types (e.g., SL, ML, aerosol over SL, etc.) and judge the actual scene from the lowest cost. Apart from being computationally expensive, an objective comparison of costs when different degrees of freedom are present in the individual models might prove difficult. [58] The 2L retrieval applied constraints (Table 6) should be reviewed. A tight prior constraint was introduced on the particle size because of the restriction to IR channels, but with a return to full channel use this could be removed. Constraint on the lower CTH was rather crudely set with a (CTH equivalent) 280 K height and 20 K variance (equating to roughly a 2 3km height variance). This would be made consistently in height coordinates and potentially refined, e.g., according to tropopause height. The lower CTH could perhaps also be sensibly physically bounded between 0 and the current upper CTH. More radically, stronger constraints from the supporting NWP data might be applied. We observe that ECMWF humidity profiles often clearly indicate potential cloud layers and appear to be quite accurate in the extratropics. Such information could be used either to initiate the 2L parameters or provide actual constraints in the OE sense, although the consequently increased NWP influence on the retrieved parameters might be considered undesirable for some purposes. [59] On validation there are several issues. Even for the ostensibly simple validation of single layer retrievals and although the CPR sensitivity to ice cloud appears close to that of the common VIS/IR imager instruments, CALIOP data are required for a full diagnosis. Simplistic use of first echo CTHs can lead to poor validation of thinner cirrus clouds that are not observed in the CPR and, conversely, poor validation of lower clouds when very thin cirrus is observed in CALIOP. Similarly, both active instruments can contribute to validating two layer retrievals, but neither is perfect; an upper layer can be missed by CPR through lack of sensitivity and a lower layer can be missed by CALIOP through attenuation. A joint analysis would considerably improve the situation. Issues of interpretation will remain in scenes with more than two layers or that are merely diffuse and vertically extended. [60] Another area that we have so far only addressed with MODIS data is the validation of COT. MODIS data are only appropriate to SL or the summed 2L COTs, and it will be interesting to attempt validation of the individual 2L COTs using the currently available CALIOP and CPR products. [61] To what extent the ML detection and 2L retrieval can be exploited in nighttime retrievals will be explored. Certainly, 16 of 22

17 the strong constraints on total COT and particle size obtained from the solar channels will be absent, but the instances of IRonly ML detection in the literature [e.g., Chang et al., 2010] suggest that some 2L information is available and potentially therefore exploitable. [62] Finally, we note that the results obtained here are very relevant to cloud property retrieval from the MODIS instrument, which should offer greater capability than SEVIRI. The additional CO 2 channels could replace or supplement the role of the water vapor channels without the corresponding modeling error from atmospheric humidity. The additional particle size sensitive 2.2 mm channel might enhance the discrimination of upper and lower layer particle sizes, and the absorption channels at 1.38 and 0.93 mm might apply further constraints to the cloud top pressures. Appendix A [63] The IR transmission of the lower cloud layer is taken to be given by G L ¼ e ð L 2 cos #sat where t L is the optical depth at the reference (VIS) wavelength of the layer. Cloud absorption accounts for roughly one half of the total extinction in the IR, and we assume the scattered radiation behaves as if it is transmitted; hence the factor of 2 in the denominator of the exponent. [64] Assuming the cloud free atmosphere is transparent, the IR radiance emerging from the top of the lower cloud layer at temperature T L is R IR ¼ B IR ðt skin Þ ÞG L þ B IR ðt L Þð1 G L Þ ða1þ where T skin is the given (ECMWF) skin temperature and B IR () is the Planck function at a chosen (10.8 mm) wavelength. Cloud emissivity in the second term on the righthand side is written as 1 minus the transmission. R IR is the radiance equivalent of the retrieved T Lblack, i.e., R IR = B IR (T Lblack ), and therefore (A1) can be rewritten to give the desired emitting temperature of the lower cloud: T L ¼ B 1 IR B IR ðt Lblack Þ B IR ðt skin ÞG L 1 G L : ða2þ [65] In practice we do not apply this correction for t L <1,as the errors amplify considerably for high cloud transmissions. Appendix B [66] Case 1 (Figure B1) is a tropical case and is similar to the example described in sections 4 and 5. Upper CTHs are well retrieved with noisier results where the cirrus is clearly thin (e.g., not visible in CPR). Lower CTH values give a good indication of the lower clouds overall character but are scattered in the very thin cirrus area around pixel Upper COTs appear to follow the visible thickness changes in the cirrus as seen by the CPR. [67] Case 2 (Figure B2) lies in the South Atlantic and is a relatively complex system. Either side of the system center at around pixel 600, the two layer character of the system is well represented by the 2L retrievals. In the center itself, the 2L result would again appear to be a response to a very diffuse upper part of the cloud. The south side of the system, although complex and in places consisting of three layers, obtains SL results with J m < 90. [68] Case 3 (Figure B3) is a case of maritime sctratocumulus overlain by a thin, relatively homogeneous layer of cirrus that is more or less invisible in the CPR (Figure B3b). It nevertheless has a large effect on the OCA SL and MODIS retrieved CTHs (Figure B3c), which are both very inaccurate, whichever layer would be chosen to validate. The 2L retrieval very successfully recaptures the stratocumulus deck CTH (Figure B3d). The 2L upper CTH is reasonable (within 2 km) for the central thicker (COT 0.6) cirrus but falls to too low levels for the thinner peripheral cirrus. [69] Case 4 (Figure B4), a southern Indian Ocean section, is a case where the CALIOP reflectivity clearly indicates SL stratocumulus and the OCA SL CTH is very accurate (Figure B4b, pixels ). The OCA SL J m, however (Figure B4c), is for some reason very high. The OCA SL phase switches in this cloud, although the temperature is 280 K (from the coincident ECMWF data), also indicating anomalous behavior. The 2L retrieved CTH and COT are shown in Figures B4d and B4e, respectively; the upper COT is extremely low and the lower COT corresponds quite well to the MODIS value. It is not clear why the lower CTH should be so erroneously high, especially when the SL result (Figure B4b) is so accurate. The 2L J m values are also very high (not shown). The location of these data off Madagascar means that the SEVIRI view angle is quite high and the anomalous behavior might be related in some way to this. Appendix C [70] The use of an optimal estimation approach to the cloud retrieval problem invites close scrutiny of the errors assumed in the measurements and the prior via the respective covariances S y and S a. Here we discuss the reason for applying infinite prior errors and some of the limitations of the measurement error used and options for its improvement. C1. Prior Errors [71] It might be thought reasonable to take a climatological mean as the prior (e.g., 500 hpa for CTP) and something related to the climatological range (e.g., 500 hpa) for the prior error since we know in advance that the parameter is thus bounded. However, the prior probability distribution for, e.g., CTP, is essentially flat with a value of zero outside the permitted range (i.e., a top hat shape). As the OE implementation assumes normally distributed errors, the best way to model such PDFs is with an infinite error (i.e., the Gaussian is flat) and bounding the parameter to its permitted range (the bounding effects the zero probability). For this reason we have assigned infinite prior errors for the primary cloud parameters of the model (COT, CRE, CTP). To use a prior, e.g., climatological mean, with an error associated to the parameter range would result in drawing retrieved values erroneously toward this mean value, making, for example, retrieved CTPs near the range boundaries (surface and tropopause) less likely than they should be. It could be argued that a consequence of infinite 17 of 22

18 Figure B1. Overpass 11318, tropical section. (a) SEVIRI true color composite. (b) CPR reflectivity and OCA 2L CTH. (c) CALIOP reflectivity and OCA 2L CTH. (d) CPR and OCA CTH. (e) OCA 2L COT. prior errors will be retrieval expected errors that are too large in some cases. If there is very little information on a parameter (e.g., REFF in very thin cloud), the expected error will indeed become very large, potentially much larger than the expected range. Again, though, as long as it is understood that this error is modeling the posterior PDF which is also flat in this case, no inconsistency arises. C2. Measurement Errors [72] Errors in the forward modeling of the SEVIRI measurements have only partially been accounted for in the setting of covariance S y. Effects of cloud inhomogeneity and channel misregistration are estimated [from Watts et al., 1998] and included, as are estimated errors in the surface 18 of 22

19 Figure B2. Overpass 1605, southern Atlantic section. (a) SEVIRI true color composite. (b) CPR reflectivity and OCA 2L CTH. (c) CALIOP reflectivity and OCA 2L CTH. (d) CPR and OCA 2L CTH. (e) OCA 2L COT. reflectance values. However, we have so far not attempted to include potentially larger errors from some other model parameter assumptions, primarily the ice scattering model for all channels and atmospheric temperature and humidity in the infrared. This is for some well founded and some less well founded reasons. Some model parameter errors are locally if not globally systematic, and this has implications for modeling within the OE framework; these are discussed below. Errors in atmospheric parameters have additionally a complex mapping within the radiative transfer for which we have so far not achieved a practical fast realization. Finally, some errors are to date not well characterized. This latter 19 of 22

20 Figure B3. Overpass 1705, tropical section. (a) SEVIRI true color composite. (b) CPR reflectivity and OCA SL CTH. (c) CPR, OCA SL, and MODIS CTH. (d) CALIOP reflectivity and OCA 2L CTH. (e) OCA 2L COT. point is perhaps no reason to omit them, but probably means one must ultimately refer to validation results to infer best values. While there is clearly potential for improvement in this aspect of our retrieval method, we have so far concentrated development on the physical aspects, the cloud model (e.g., this two layer implementation), and aiming to maximize the measurement information (all channels) used, rather than on the statistical aspects. [73] An example of systematic measurement error is the uncertain nature of ice scattering properties [Cooper et al., 2006]; for a given optical depth our assumption of aggregate crystals, for example, will give very high reflectances if 20 of 22

21 Figure B4. Overpass 1515, southern Indian Ocean off Madagascar. (a) SEVIRI true color composite. (b) CALIOP reflectivity and OCA SL CTH. (c) OCA SL J m. (d) CALIOP and OCA 2L CTH. (e) OCA 2L COT. in reality crystals are horizontally oriented plates. The classic OE formulation however assumes random unbiased measurement errors, and taking the extreme measure of modeling all systematic errors as random would not lead to the optimal solution and would result in a loss of precision. At the other extreme, the one more or less adopted in this work, the systematic errors are ignored in finding the solution. Their effect could be best found using the retrieval operator D y = ^x/ y and the Jacobian with respect to the model parameter in question K m = y/ m (both calculated at the solution x); thus dx = D y K m dm. Note this is a vector error, not a covariance, and is a mapping of the suspected 21 of 22