The Retrieval of Profiles of Particulate Extinction from Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) Data: Uncertainty and Error Sensitivity Analyses

Stuart A. Young CSIRO Marine and Atmospheric Research, Aspendale, Victoria, Australia

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Mark A. Vaughan National Aeronautics and Space Administration, Hampton, Virginia

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Ralph E. Kuehn Space Science and Engineering Center, University of Wisconsin—Madison, Madison, Wisconsin

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David M. Winker National Aeronautics and Space Administration, Hampton, Virginia

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Abstract

Profiles of atmospheric cloud and aerosol extinction coefficients are retrieved on a global scale from measurements made by the lidar on board the Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission since mid-June 2006. This paper presents an analysis of how the uncertainties in the inputs to the extinction retrieval algorithm propagate as the retrieval proceeds downward to lower levels of the atmosphere. The mathematical analyses, which are being used to calculate the uncertainties reported in the current (version 3) data release, are supported by figures illustrating the retrieval uncertainties in both simulated and actual data. Equations are also derived that describe the sensitivity of the extinction retrieval algorithm to errors in profile calibration and in the lidar ratios used in the retrievals. Biases that could potentially result from low signal-to-noise ratios in the data are also examined. Using simulated data, the propagation of bias errors resulting from errors in profile calibration and lidar ratios is illustrated.

Corresponding author address: Stuart A. Young, CSIRO Marine and Atmospheric Research, Private Bag No. 1, Aspendale, VIC 3195, Australia. E-mail: stuart.young@csiro.au

Abstract

Profiles of atmospheric cloud and aerosol extinction coefficients are retrieved on a global scale from measurements made by the lidar on board the Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) mission since mid-June 2006. This paper presents an analysis of how the uncertainties in the inputs to the extinction retrieval algorithm propagate as the retrieval proceeds downward to lower levels of the atmosphere. The mathematical analyses, which are being used to calculate the uncertainties reported in the current (version 3) data release, are supported by figures illustrating the retrieval uncertainties in both simulated and actual data. Equations are also derived that describe the sensitivity of the extinction retrieval algorithm to errors in profile calibration and in the lidar ratios used in the retrievals. Biases that could potentially result from low signal-to-noise ratios in the data are also examined. Using simulated data, the propagation of bias errors resulting from errors in profile calibration and lidar ratios is illustrated.

Corresponding author address: Stuart A. Young, CSIRO Marine and Atmospheric Research, Private Bag No. 1, Aspendale, VIC 3195, Australia. E-mail: stuart.young@csiro.au

1. Introduction

The Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) satellite began acquiring scientific data in mid-June 2006. CALIPSO carries three coaligned, nadir-viewing instruments: a three-channel elastic-backscatter lidar, an imaging infrared radiometer, and a wide-field camera. An overview of the CALIPSO mission, science objectives, and instruments is presented in Winker et al. (2010). The CALIPSO lidar [Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP)] is a dual-wavelength (1064 and 532 nm), dual-polarization (at 532 nm) system (Hunt et al. 2009) that is used to determine the locations of atmospheric regions containing particulate matter (clouds and aerosols), to identify those particles according to type, and to derive profiles and layer integrals of the particulate backscatter and extinction in these regions. The overall design of the retrieval algorithms is presented in Winker et al. (2009).

This paper is the second in a series that documents the CALIPSO extinction retrievals. The first part (Young and Vaughan 2009, hereafter YV09) focused on the retrieval of profiles of particulate backscatter and extinction from the lidar measurements. As the analysis concepts and methods employed in CALIPSO’s Hybrid Extinction Retrieval Algorithms (HERAs) are complex and somewhat novel, readers are strongly advised to read YV09 and become familiar with the concepts and methods therein before reading this paper. Briefly, the extinction retrieval follows the processes of signal calibration, feature (layer) detection, and feature type identification (cloud–aerosol discrimination and subtyping). The HERAs are then tasked with retrieving the profiles of backscatter and extinction coefficients and computing optical depth in those atmospheric regions where the preceding processes have identified clouds or aerosols. As the HERAs rely on the correct calibration and identification of cloud or aerosol subtype [the subtype determines the value of the backscatter phase function (or its reciprocal—the lidar ratio) that is required in the retrieval process], any errors in these parameters will lead to errors in the optical properties retrieved by the HERAs. In addition to the uncertainties in signal calibration and in the values of the lidar ratio associated with different layer types, the naturally occurring spread of values in the parent distributions of the lidar ratio for any one layer type will also lead to uncertainties in the HERA retrievals.

The solution of the nonlinear lidar equation, especially in situations where there are many vertically adjacent features (YV09), further amplifies these uncertainties, and they have a pronounced influence on the extinction and backscatter coefficients reported in the CALIPSO data products. Therefore, there are two main aims of this paper. The first is to provide a mathematical basis for the uncertainty analyses. The second is to provide data users with information on the degree to which potential bias errors could affect the retrieved extinction coefficients.

Uncertainties in the inputs passed to the HERAs can be either random or systematic in their effects. For the purposes of this current work, we choose to distinguish between the two by their effects on the HERA profile solver, which retrieves optical property profiles as a function of altitude through a layer from a single attenuated backscatter signal profile. (This single profile is the average of a varying number, which may vary with height in the profile, of individual profiles.) We designate random uncertainties as those that vary randomly (and are uncorrelated) along the profile. We note, though, that random uncertainties, even though they may be completely uncorrelated in origin, can become partially correlated among neighboring samples in a profile due to the integral effect of electronic filters or preamplifiers in the data acquisition process, as well as the onboard averaging of neighboring samples to reduce the downlink bandwidth requirements. [See Liu et al. (2006) for a description of CALIOP’s noise-scale factor.] In contrast, we define systematic uncertainties (like any in the calibration–renormalization factor or lidar ratio) as those uncertainties that remain constant through the whole profile and cannot be reduced by averaging. Consider the following. In the CALIPSO data processing, multiple individual profiles of attenuated backscatter are averaged to form a single profile from which an extinction profile is retrieved. Whereas random uncertainties can be expected to decrease with averaging, systematic uncertainties that are common to all the variables being averaged will not. For example, a profile of attenuated backscatter that results from averaging several noisy profiles would be expected to be less noisy than the individual profiles averaged. Because random uncertainties are generally uncorrelated, they are summed in quadrature [i.e., their squares are summed (Taylor 1982)]. However, if all the individual profiles had a common calibration uncertainty of, say, 3%, then the average profile would still have the same calibration uncertainty, because within this context calibration is a systematic uncertainty. In contrast, if each individual profile had been previously rescaled (renormalized) by a different amount to correct for the attenuation by overlying features, then the resulting renormalization uncertainty in the average profile could be smaller than that in individual profiles. As will be seen later, however, the number of individual profiles contributing to the average attenuated backscatter profile may be different at each height level. This means that the systematic component of the uncertainty in the average attenuated backscatter profile can be different at each height level. Because of this complexity, the treatment of the random and systematic components of the uncertainties has been simplified in the version 3 data release, and, as explained later, the components are initially combined in quadrature and, once combined, the individual components are not tracked separately in subsequent analyses and renormalizations.

Errors—which, as explained below, are not synonymous with uncertainties—can also be systematic or random. Note that calibration–renormalization errors or lidar ratio errors, even though they may be random in origin, behave as systematic errors within the extinction retrieval. Systematic errors will cause biased retrievals, whereas random errors generally will not. Note though, that we shall show later that biased retrievals can occur where the signal-to-noise ratio (SNR) is low.

Although the terms error and uncertainty have quite distinct meanings, they are often used interchangeably in the scientific literature. This seems particularly to be the case in the compound forms using the adjectives absolute and relative. Error is defined in the Merriam-Webster online dictionary (http://www.merriam-webster.com/dictionary/error) as “the difference between an observed or calculated value and the true value.” According to Bevington and Robinson (1992), error, therefore, relates to accuracy, which is a measure of how close the result is to the true value. Uncertainty is usually defined in terms of precision, which, again according to Bevington and Robinson (1992, p. 2), is “a measure of how well the result has been determined, without reference to its true value.” Accuracy and error, therefore, relate to the correctness of the result, while precision is a measure of the reproducibility of the result. In the case of the CALIOP extinction retrieval, this reproducibility is with respect to the results obtained when analyzing multiple samples of the same layer type, for example, the analysis of dust aerosols from a single location acquired over the course of multiple orbits or days. Therefore, “in general, when we quote the uncertainty in an experimental result, we are referring to the precision with which that result has been determined” (Bevington and Robinson 1992, p. 2–3).

In the current work, we examine the propagation of uncertainties and errors in the HERA retrievals and describe the uncertainty analyses used in the CALIPSO version 3 data release. We also examine the performance of the extinction retrieval algorithm in response to various input errors and, for the assistance of CALIPSO data users, provide an analysis of the potential biases that can result from systematic errors and illustrate our analysis with examples derived from both simulated and actual CALIPSO data. Our use of simulated data enables us to know what the true value of any quantity is, so we shall use the term error, as defined above, to express the difference between the modeled (and thus known) value of a quantity and the value of the quantity that is the result of a retrieval algorithm. Uncertainty will be used to describe the estimated or calculated precision of the result. Relative error will be used to describe the difference between the result and the true value expressed as a ratio of this difference to the true value. Relative uncertainty (e.g., Russell et al. 1979) will be used to describe the ratio of the estimated precision of the result to the calculated or retrieved value of the quantity. To distinguish between these quantities, in this work we use the notation Δx to signify the uncertainty, and ɛ(x) the error, in quantity x.

The outline of the paper is as follows. In section 2 we provide a brief summary of the CALIOP analysis framework and extinction retrieval algorithms. In section 3, we provide a mathematical analysis of the estimation of the uncertainties in the retrieved extinction profiles and optical depths and of the potential biases that result from errors in the specified lidar ratio or calibration factor. In addition, we provide graphical examples of biases that can result from low SNRs. Examples of the propagation of uncertainties and errors are illustrated in section 4 using both simulated and actual data.

2. Overview of the CALIOP retrieval algorithms

a. Relationship between the CALIOP analysis algorithms

In the CALIPSO extinction retrieval analysis, level 1 data (King et al. 2004) are processed by the level 2 system in 80-km segments, each of which contains an uninterrupted sequence of 240 consecutive profiles. When the attenuated backscatter profiles are plotted on height versus along-track distance axes, we define what we refer to as CALIPSO “curtain files” or “scenes.” These scenes contain various atmospheric features that are defined in YV09 as “any extended and contiguous region of enhanced backscatter that rises significantly above the ‘clear air’ value” (p. 1107). By averaging and processing various numbers of consecutive profiles, the SNR is progressively increased to enable detection of progressively more tenuous features. We illustrate the result of this process in Fig. 1 using the “representative complex scene” described in YV09. The stylized figure contains various types of features, labeled F1–F38. The strongest features (e.g., F2, F8, F11, F20, F24), plotted in red, are detected at a horizontal resolution of 5 km. Features of medium strength (e.g., F1, F7) are detected over horizontal distances of 20 km and are plotted in yellow, whereas the weakest features (F12 and F21), plotted in green, are only detected after averaging over 80 km. In addition, we describe various features that have clear air above and below them (F2–F6, F7, and F11) as vertically isolated, lofted layers, whereas F25–F38 are described as surface-attached features. Features that contain more than one vertically adjacent, simple feature are described as complex features. Features F1 and F8–F10 comprise one complex feature, as do features F21–F38.

Fig. 1.
Fig. 1.

A simplified, stylized map of the location and extent of atmospheric features as might be detected by CALIOP’s feature finder. The 80-km scene contains 16 columns of data, each corresponding to an average of 15 consecutive profiles covering a horizontal distance of 5 km. For clarity, the number of vertical levels has been reduced to 36.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

The CALIPSO HERA is the last in a sequence of several level 1 and level 2 algorithms that correct the raw lidar attenuated backscatter profiles for instrumental effects, calibrate them against a modeled molecular signal (Powell et al. 2009), detect the presence of regions of particulate (cloud or aerosol) backscatter (features) (Vaughan et al. 2009), determine initial optical properties of these features and classify them according to type (Liu et al. 2009; Omar et al. 2009; Hu et al. 2009) and, finally, retrieve profiles of cloud and aerosol backscatter and extinction coefficients (YV09). In addition to the information passed on from these preceding algorithms, HERA also uses profiles of molecular backscatter and transmittance and ozone transmittance derived from the global, gridded analysis product provided by the National Aeronautics and Space Administration’s (NASA) Global Modeling and Assimilation Office (GMAO; Rienecker et al. 2012). The retrieval process is initiated using the locations of features detected in the scene at horizontal resolutions of 5, 20, or 80 km, initial values of their lidar ratios, and 16 profiles of attenuated backscatter at each wavelength. (Here, the range is measured from the lidar at r = 0.) The attenuated backscatter profiles each cover 5 km along the CALIPSO ground track and are the averages of 15 individual single-shot profiles of calibrated raw signal power that have been corrected for signal offset, instrumental factors, and the two-way ozone transmittance as shown:
e1
Here,
  • P(r) is the backscattered signal power detected at range r from the lidar;

  • GA is the amplifier gain and C is the lidar calibration coefficient;

  • E0 is the (average) laser energy for a single or averaged profile;

  • βM(r) is the molecular volume backscatter coefficient, which is proportional to the molecular number density profile;
    e2
    is the molecular two-way transmittance between the lidar and range r, in which
  • is the molecular volume extinction coefficient and

  • SM is the molecular extinction-to-backscatter (or lidar) ratio;

  • is the two-way ozone transmittance and

    1. is the ozone volume absorption coefficient;

    2. βP(r) is the particulate volume backscatter coefficient;
      e3
      is the particulate two-way transmittance, where
    3. η(r) is a parameterization describing the effect of multiple scattering on particulate extinction;
      e4
      (r0 = 0, rn = r) is the particulate optical depth summed over the various layers of the atmosphere between the lidar and range r; and
      e5
      is the particulate volume extinction coefficient, where SP is the particulate extinction-to-backscatter (lidar) ratio, which is assumed to be constant within identified layers (features). (See the discussion in section 2a of YV09.)

For the version 3 analysis, however, a constant value of η is used throughout any feature, so that η(r) = η for all r within any given layer. This simplification is used in the following equations.

b. The HERA profile solver

Retrieval of profiles of particulate backscatter and extinction coefficients from profiles of attenuated backscatter coefficient within an atmospheric layer essentially involves two steps: correction of the profile for any signal attenuation by the atmosphere above the layer [often referred to as normalization, e.g., Russell et al. (1979)] and the division of the attenuated backscatter profile by the range-dependent, two-way atmospheric transmittance profile within the layer.

The correction for the two-way transmittance through the layer occurs as part of the extinction retrieval process. Whereas the molecular transmittance is always calculated from the GMAO molecular density profile, the particulate transmittance is determined in one of two ways. If the feature is a vertically isolated, lofted layer and the total two-way particulate transmittance of the layer can be determined with sufficient precision from a comparison of the attenuated backscatter signals from the assumed clear air above and below the layer (Young 1995), then HERA uses this value as a retrieval constraint on the extinction profile (YV09). In so-called unconstrained retrievals, where a transmittance constraint is not available, HERA uses a modeled value of the lidar ratio assigned by the layer classification algorithms (Omar et al. 2009), and the two-way particulate transmittance is subsequently retrieved along with the extinction profile via Eq. (3). In the terminology we use here, we describe a feature that has been analyzed to retrieve profiles of extinction and backscatter as having been “solved.”

During the 532-nm calibration process, CALIOP raw signal profiles are rescaled so that the resulting attenuated backscatter profiles are numerically equal to the molecular backscatter in some high-altitude calibration region where the atmosphere composition is assumed to be accurately characterized by GMAO model data. This rescaling process is often referred to as normalization (e.g., Russell et al. 1979) and is simply the correction of the attenuated backscatter profile for the attenuation by all components of the atmosphere above that point.

The extinction retrieval process works from the top of the atmosphere down to the surface, or, in the case of totally attenuating layers, to the last range at which valid atmospheric signals exist. Any particulate layers that underlie a layer that has just been analyzed must first be corrected for the signal attenuation caused by the overlying layer. This is done by rescaling (dividing) the attenuated backscatter profile for the lower layers by the two-way transmittance of the upper layer. If the solution of the overlying layer was unconstrained, then the retrieved transmittance of that layer is used to rescale the lower layer. Otherwise, if the overlying layer was a vertically isolated feature for which a constrained solution was possible, then the lower layer is rescaled by the measured two-way transmittance for the upper layer. Either way, this process of rescaling the attenuated backscatter profile so that the value just above the layer to be analyzed is numerically equal to the molecular backscatter at that point is simply the correction of the attenuated backscatter profile for the attenuation by all of the atmosphere above that point, and thus it is actually another normalization process. Therefore, both the initial calibration and the renormalizations at lower altitudes produce scaling factors by which the attenuated backscatter profile is divided prior to analysis by the extinction solver. In this work, we shall often refer to these processes, collectively, as (re)normalization.

HERA retrieves backscatter and extinction profiles within regions in an 80-km scene demarcated and defined by the selective iterative boundary locator (SIBYL; Vaughan et al. 2009) as being features. As these features can extend over varying horizontal distances and may be partially obscured by other features detected at finer horizontal scales (YV09), the first step performed by HERA is to create an averaged profile of attenuated backscatter from only those sections of the 5-km-resolution profiles that lie within the feature boundaries defined by SIBYL. The uncertainty at each point in the averaged attenuated backscatter profile is also estimated (Liu et al. 2006).

Once the averaged profile has been created, it must be rescaled (i.e., divided) by a renormalization factor CN to correct for molecular and particulate two-way transmittance losses that occur between the top of the atmosphere and the topmost point in the feature. The renormalization factor can be written as
e6
Here, rN is the range to the top of the feature being analyzed (the normalization range) and
e7
is the scattering ratio in the (assumed) clear air immediately above the feature. Within HERA, is assumed to be 1 (i.e., the particulate backscatter is assumed to be 0 at the normalization range).
Because neither of the transmittance terms nor the clear-air scattering ratio is known with absolute precision and may, in fact, be in error by an uncertain amount, the accuracy of the renormalization factor is likewise unknown. We express this uncertainty in terms of the uncertainties in the constituent quantities, defined as
e8

Although any errors in the molecular transmittance term in Eq. (8) can lead to errors in both of the other terms, and potential errors in the particulate transmittance can lead to errors in the estimated scattering ratio, we regard the contributions of such correlations as negligible when compared with other contributions, like the assumed values of the particulate lidar ratios used to solve any overlying particulate layers, which represent the dominant source of error in the particulate transmittance term, or the signal-to-noise ratio around rN, which is usually the dominant contributor to the third term. Therefore, we assume that each of the three terms is approximately independent of the others and can be summed in quadrature as shown (Taylor 1982, chapter 5.6).

The top-level algorithm in HERA passes to the profile solver the section of averaged attenuated backscatter that is to be processed, along with the renormalization factor that corrects for attenuation above the layer, and the uncertainties in both. The renormalized attenuated backscatter profile and a profile of its relative uncertainties are then calculated as shown:
e9
e10
The first term in Eq. (10) can be obtained from Eq. (1) as
e11
A thorough analysis of the calibration (as distinct from renormalization) procedure and the contributions of the uncertainties in Eq. (11) can be found in Powell et al. (2009). The uncertainty in the calibration factor C has random and systematic components that include uncertainties in the molecular backscatter and two-way transmittance, and the estimated scattering ratio at the calibration range. Note that, through the definitions of and in Eqs. (9) and (10), respectively, both are functions of r and rN. However, in this work, we use the subscript N to signify that these quantities have been (re)normalized and thereby simplify the notation.
The HERA profile solver then solves Eq. (1) to retrieve profiles of particulate backscatter and extinction between the top and base of the feature (YV09) as shown:
e12
The extinction profile is obtained using Eqs. (12) and (5). It can be seen, from these equations, that errors in the retrieved particulate backscatter at one range lead to errors in the particulate extinction at that range and then to errors in the two-way particulate transmittance [through Eqs. (3) and (4)], which is then used in the calculation of the particulate backscatter at the next range. Therefore, errors at the top of a feature are propagated down, with increasing effect, to lower altitudes in the feature. An inspection of Eqs. (12), (3), and (4) reveals that the uncertainty in the backscatter coefficients increases exponentially with range through a feature at a rate determined by uncertainty in the lidar ratio used in the retrieval and the uncertainties in the overlying, retrieved backscatter estimates.
The two-way particulate transmittance in any 5-km column in a feature is subsequently used to correct the data in each column underlying the feature for the attenuation of the layer that has just been solved (refer to YV09 for details). Note that, as features may occupy different numbers of columns and extend over a different height range in each column (see the discussion of Fig. 1 above), different columns of an underlying feature may need to be rescaled differently, depending on the overlying attenuation in each column; that is,
e13
for r > rb, where rt and rb are the ranges to the top and base, respectively, in a particular column, of the feature just processed, not of the underlying region that is being rescaled. For example, in Fig. 1, features F8 and F10, both of which are below F1, are rescaled by different amounts because of the different depth of F1 above each of the features. Note that the β′(r) here is a general representation of the attenuated backscatter array at the current stage of the analysis process and it may have been renormalized a number of times previously. The uncertainty in the underlying data points is also increased, as shown:
e14
It can be seen then that, as the analysis proceeds down through the atmosphere, the uncertainty in the retrievals is increased, as is expected, because of the correction for the attenuation of overlying layers. Errors in this attenuation correction have similar effects to errors in the renormalization factor and produce biased retrievals for underlying layers.

3. Extinction retrieval uncertainty analysis and error sensitivity

The literature contains many studies on uncertainty and error analyses and the sensitivity of solutions to various errors. Russell et al. (1979) provides a very useful analysis of various sources of error, their likely magnitudes, and how they should be combined. However, that work assumes that the aerosol attenuation profile is obtained from a model rather than from the solution of the lidar equation as is done here. Several papers (e.g., Hughes et al. 1985; Bissonnette 1986; Jinhuan 1988) study the sensitivity of the single-component analytical solution (Klett 1981) to various assumptions and errors (e.g., boundary values, and boundary-value position). There also are studies of the performance of the two-component analytical solution (Fernald et al. 1972; Fernald 1984) to errors in boundary values (Sasano and Nakane 1984; Del Guasta 1998) and a range of other factors (e.g., Kovalev and Moosmüller 1994).

The CALIOP lidar analysis contains several unique characteristics that warrant its closer analysis. Feature detection, classification, and analysis are performed simultaneously on different horizontal scales. Multiple layers, even those that are vertically adjacent, are processed using different lidar ratios. Constrained retrievals are performed wherever this is possible. The correction for the attenuation by higher-altitude features is propagated down to lower features according to the relative locations and horizontal scales, and the retrieval algorithm is iterative, unlike the analytical solutions discussed above. An analysis of the error sensitivity and uncertainties resulting from this retrieval framework is presented in the following sections.

As a consequence of HERA’s position in the analysis chain, the quality of the retrieved profiles of extinction and backscatter depends on the errors and uncertainties in the information it receives from the upstream analyses. Errors and uncertainties in the modeled molecular data will also affect HERA’s outputs. As explained in the introduction, errors and uncertainties in the input data to the retrieval algorithm can be random or systematic. The photon noise on the lidar signal is an example of a random uncertainty. Errors in the lidar ratio or in the transmittance constraint for a given feature are readily identified as examples of systematic errors (effectively systematic uncertainties), and these create biased results. Errors in profile calibration and background removal, although they may be random in origin, are constant throughout any single profile and thus will cause biases in the retrieved extinction profile, and, therefore, are classified as systematic errors within the context of HERA. Similarly, in multilayer scenes, the signals from the lower layers must be corrected for the signal attenuation caused by any overlying layers prior to initiating an extinction solution. In such cases, the random errors and uncertainties that accumulate in the retrieval of the extinction profile through the upper layer(s) are subsequently included in the correction term(s) applied to the lower layer(s), and thus introduce systematic errors and uncertainties into the extinction retrieval for those lower features.

a. An analysis of retrieval uncertainties

We shall now derive expressions for the estimated uncertainties in the retrieved profiles of particulate backscatter, extinction, and two-way transmittance. We assume, in general, that the uncertainties in the measured profiles of attenuated backscatter coefficient and in the molecular and ozone number densities (from which the molecular scattering models are derived) are uncorrelated with each other, so they can be summed in quadrature. The former result primarily from photon noise superimposed on the photomultiplier signals at 532 nm (and dark noise on the avalanche photodiode at 1064 nm) in addition to details of the averaging process, whereas the latter result from uncertainties associated with the GMAO model and the assimilation of the various datasets it uses. Uncertainties in the normalized attenuated backscatter profiles are obviously correlated with uncertainties in the molecular backscatter and transmittance at the normalization height; see Eqs. (8) and (10). Russell et al. (1979) also consider the correlations in the molecular transmittance model at various heights and include a covariance term in their analysis. However, with the global coverage of the CALIPSO measurements and of the model, and with the considerable variability in the way the model and the lidar data are combined in different atmospheric regions, even within any scene, characterizing the covariances, even in the input data, is an extremely complicated task. As the covariance contributions are likely anyway to be dwarfed by the major contributors to the uncertainties [lidar ratio and (re)normalization uncertainties], we do not include them here.

From Eq. (12), then, the uncertainty in the retrieved particulate backscatter coefficient can be written as
e15
Here, βT(r) is the total backscatter coefficient at range r. The uncertainty in the particulate transmittance term in Eq. (15) is derived from Eq. (3) as shown:
e16
which leads to
e17
The optical thickness can be expressed as the product of the lidar ratio Sp and the integrated particulate backscatter as shown:
eq5
such that
e18
for which the standard expression for the relative uncertainty (or variance here) is
e19
where the covariance is defined as below and the barred quantities represent means:
e20

To obtain the most reliable estimates of the uncertainties in the retrieved optical depth, having reliable estimates of the covariances in Eq. (19) is also highly desirable. At the time of the version 3 data release, numerical techniques to derive these covariances were still being evaluated, and no consensus had been reached as to appropriate values. Modeling studies in which various parameters [e.g., lidar ratio, (re)normalization factor, SNR] are varied, either singly or together, using specified or randomly generated values, can provide insight into the behavior of the variances and covariance terms in Eq. (19). (Here, we interpret the first and second terms on the right-hand side of Eq. (19) as the relative variances in SP and γP, respectively.) These simulations showed that the covariances vary considerably and over a very wide range, and are highly dependent on the parameter being varied. Also, the relationship between γP and SP is nonlinear and asymmetrical with respect to the sign of deviations of SP about its true value (which is obviously unknown). In addition, because the uncertainty in some of the parameters can be very large, the assumptions on which the definition of the variance of a function of several variables is based are most likely invalid in many situations considered here. (The standard equations consider only the first two terms of the Taylor expansion of variables about their mean values; e.g., see Bevington and Robinson 1992, section 3.1.) Because of the difficulty in describing them satisfactorily, the covariances are set to zero in the calculation of optical depth uncertainty in the version 3 data release. This will lead to the uncertainty being underestimated in some circumstances.

While a method of estimating the covariances has not yet been determined, the modeling studies mentioned above, and described below in section 3b(3), do provide results that give an indication of the signs and approximate limits to the magnitudes of the covariances. The modeling studies considered a simple case of a 1-km-thick elevated layer in which the ratio of particulate to molecular scattering was constant. Note that here we are considering only unconstrained retrievals because, in constrained retrievals, the optical depth uncertainty is measured directly along with the optical depth. The results below include outputs from all successful retrievals; that is, they include those in which the lidar ratio was reduced in order to prevent failure (divergence toward infinity) of the retrieval. Studies of varying differing combinations of parameters give the following results.

For the case where the only parameter varied was the SNR, we find that the contribution of the covariance term in Eq. (19) to the total relative variance was negligible for SNRs ≥ 10 and optical depths < 2 but increased to −20% for optical depths > 2. (In other words, the omission of the covariance term here causes the optical depth uncertainty to be overestimated.) For SNRs < 10, the contribution was increasingly negative. For example, for an optical depth of 1.4 and SNR = 1, the third term contributed −75%, but the absolute magnitudes decreased again for SNRs < 1. We also find that, even when the covariance term is included, Eq. (19) significantly overestimates the total relative variance in the optical depth for SNRs < 10, especially at higher optical depths.

For the case where the calibration (or renormalization) factor was varied over the range −30% to +30% while the SNR was also being varied, we find that the contribution of the third term was again negligible for SNRs ≥ 10. For SNRs < 10, the contribution is increasingly negative. For example, for SNR = 1, the relative contribution to the total of the covariance (third) term is −0.3 for an optical depth of 1.4, and −0.5 for an optical depth of 2.8. We find that Eq. (19) accurately predicts the relative variance for SNRs ≥ 10 but underestimates it for lower SNRs, especially for higher optical depths.

In the simulations where both the lidar ratio and SNR were varied randomly, we found that the relative contribution of the covariance term was occasionally positive. The following results were for simulations in which the lidar ratio was varied randomly about a mean value of 20 sr, using a normal distribution with a standard deviation of 40%. For SNRs ≥ 10, the contribution of the covariance term increases from about 3% for an optical depth of 0.0014 to about 40% for an optical depth of 2.85, after which the contributions generally decrease and sometimes go negative. For SNRs < 10, too, the contribution decreases and sometimes becomes negative. Equation (19) provides an accurate representation of the total relative variance for SNRs > 1, except for optical depths greater than unity, where it overestimates by about 20%. At lower SNRs, there is generally an underestimate, except at very low SNRs and higher optical depths.

To summarize our simulation results, we can say that for SNRs that are most often encountered in the data analysis, and for optical depths less than about 3, the contribution of the covariance term to Eq. (19) to the total is negligible when the contributions are from variations (uncertainties) in the calibration factor. Under these conditions, Eq. (19) mostly provides a reliable estimate of the total relative variance in optical depth, but outside this range significant differences can occur. Where the uncertainties in the other terms are caused by variations in the SNR alone, we find the contribution of the third term is negligible to minor, whereas random variations in the lidar ratio with a standard deviation of 40% result in the third term contributing up to about 40% of the total relative variance. In this case the total relative variance is underestimated by about 40%, and the total relative uncertainty is underestimated by about 20% if the covariance term is neglected. Note, however, that for lower SNRs and higher optical depths, the contribution of the third term to the total variance is negative, and the omission of this term causes the estimated optical depth uncertainty to be too large.

In the software implementation of the extinction retrieval algorithm, the integrated particulate backscatter coefficient is calculated at each range increment using trapezoidal integration as shown:
e21
where rjnorm = rN and δrj is the jth range increment. (The magnitude of these range increments varies with altitude.) The uncertainty in γP,j can now be calculated as
e22
However, as the first term in Eq. (22) is the unknown in Eq. (15), this term is evaluated by initially setting Δβj to Δβj−1 and iterating using Eqs. (15) and (17) until convergence is achieved. [This iteration is similar to that used in YV09 in the calculation βP(r) through the iteration of Eqs. (11) and (12) in that work.]
It should be noted that Eq. (22) is only valid if the ΔβP,j are uncorrelated. The effects of partial correlation of the random noise in the measured signal component of ΔβP,j are accounted for using the techniques recommended by Liu et al. (2006). However, no such correction is readily available for the biases produced by errors in normalization or by the use of an erroneous value of SP. In this case an upper estimate for the uncertainty can be given by summing the absolute values, rather than the squares (Taylor 1982, section 9.1) as shown:
e23
Finally, bearing in mind similar caveats to those associated with Eq. (19), the uncertainty in the aerosol extinction coefficient can be derived from Eq. (5) as shown:
e24

b. An analysis of the sensitivity of retrievals to various input errors

We now examine the sensitivity of the retrieval algorithms to errors in some input parameters that can cause biases in the retrieved quantities. The input parameters considered here are the calibration–renormalization factor, the lidar ratio, and, for constrained solutions, the two-way transmittance constraint. For this particular analysis, all other quantities, including the molecular density profile, are assumed to be error free. As these parameters are fixed for any given retrieval, and do not vary randomly from one point in a profile to the next, any error in the parameters can have a cumulative effect and cause a bias in the retrieved quantities. This problem is compounded because the lidar equation is nonlinear [through the exponential factor in Eq. (3)] and bias errors increase with increasing optical depth and, hence, penetration depth into a layer. The consequence is that, even if the errors in the input parameters are normally distributed about a mean value, the resulting retrievals of particulate extinction or optical depth are not normally distributed about the values that result from the use of the mean value of the input parameters, but are instead offset by a bias. As a result, errors where the input parameter is too large can have vastly different results from those where the input parameter is too small.

In retrievals that are not constrained by an independently measured value of layer optical depth, the retrieval is performed using a value of the lidar ratio that is the default value for the aerosol or cloud type that makes up the layer. Errors in the calibration or renormalization of the input attenuated backscatter profile will lead to errors in the retrieved profiles of particulate extinction and optical depth. If the errors are sufficiently large, then a successful retrieval may not be possible using the default value lidar ratio; for example, the solution may diverge toward infinity at ranges before the layer base has been reached. In this case the lidar ratio will be reduced until a successful (nondiverging) retrieval is obtained, but the resulting retrieval, although deemed “successful,” will still be in error.

Retrievals that are constrained by an independently determined value of the layer two-way transmittance are obviously sensitive to errors in that constraint. Any bias will be compounded by any concomitant (re)normalization error. Although constrained retrievals are intended to produce retrieved optical depths that match the constraint when integrated over the depth of the layer, differences can exist between the retrieved and true profiles of backscatter and extinction coefficient, even if the constraint is correct. In the analysis below, we consider both unconstrained and constrained retrievals.

1) Unconstrained retrievals

Here we investigate the sensitivity of the retrieved extinction and optical depth to the combined effects of errors in the lidar ratio supplied to the extinction retrieval algorithm and errors in either the initial calibration of the attenuated backscatter profile or in the correction of the profile for any attenuation above the first point for analysis—a process we call renormalization. Remember that, in the CALIPSO HERA analysis, we are processing one whole 80-km scene at a time. This involves many separate retrievals of the optical properties of the features identified in that scene. As these features occur in different regions of the scene and have variable horizontal and vertical extents, any particular retrieval will only analyze a small section of the total 80-km horizontal distance and extend only over the vertical depth of a layer. The analysis never uses a complete profile from the top of the atmosphere down to the surface. It is imperative, then, that correction be made for the attenuation by the overlying atmosphere of that profile section selected for analysis.

Errors, therefore, can result either during the initial calibration of the attenuated backscatter profiles or during the correction of the profiles for the attenuation by overlying layers. Calibration or renormalization errors can occur if the region selected is not purely molecular. The second type of error will occur if the retrieved optical depth of an overlying layer is incorrect, and this incorrect optical depth is then used to calculate the two-way transmittance correction factor by which the underlying profiles must be divided. We choose to describe the effect on the attenuated backscatter signal of errors in calibration or renormalization by a multiplicative factor as
e25
where we use a generic ɛ(C)/C to represent the fractional (relative) error in either calibration or renormalization. The caret (ˆ) notation indicates estimated or retrieved quantities that may be in error. Similarly, we describe the relationship between the incorrectly selected lidar ratio and the correct value by using the ratio ψ, defined as
e26
Equations describing the bias errors that occur in the retrieved particulate backscatter, extinction, optical depth, and two-way transmittance profiles as a result of errors in the calibration or renormalization factor or in the lidar ratio, or both, are derived in the appendix. For the general case, when there are errors in both the lidar ratio and in the calibration–renormalization factor, we find, for unconstrained retrievals, that
e27
(i) Calibration–renormalization errors
In section a2 of the appendix, we show that for the case where the lidar ratio is correct and there are only errors in calibration–renormalization, Eq. (27) simplifies to
e28
The error profiles in particulate backscatter and extinction are described by substituting (28) into
e29
and
e30
respectively. In addition, the error in the retrieved particulate optical depth may be obtained as follows:
e31
In those cases where , we have further simplification as shown:
e32
To help CALIOP data users appreciate more easily the scale and variation of the biases caused by these (re)normalization errors, we choose to study the simplified example of a layer with a fixed lidar ratio of 20 sr and in which the scattering ratio R is constant with range. We examine the case of a constant-R layer in section a2 in the appendix and show that
e33
where the quantity ρ is defined in the appendix as ρ = R/(R − 1). The error in particulate optical depth is then obtained via Eq. (31) as
e34
Where , and , the relative error in the particulate optical depth can be approximated as shown:
e35

We can see that the relative errors in optical depth that result from calibration or renormalization errors increase rapidly with optical depth [Eq. (34)], as in clouds, but also that large relative errors can occur for scattering ratios approaching unity [Eq. (35)], as may be found in tenuous aerosol layers.

The accuracy of the above-mentioned analysis has been tested using simulated data and the CALIPSO HERA analysis routines. Various model atmospheres were simulated and the backscatter, extinction, and optical depth retrieved using incorrectly specified calibration–renormalization factors and compared with values modeled using the above-mentioned equations. The equations predicted errors accurately over a range of scattering ratios varying from 1.1 to 1000 (optical depths from 0.0014 to 14) for calibration–renormalization errors in the range of ±30%. In Fig. 2 we compare the simulated and modeled [via Eq. (34)] relative errors in particulate optical depth for a 1-km-deep layer with a top at 8-km altitude. Because the equations above assume that all errors are in the calibration–renormalization factor, retrievals in which the lidar ratio was reduced to prevent divergence have not been included here.

Fig. 2.
Fig. 2.

A comparison of modeled and simulated relative errors in retrieved particulate optical depth resulting from the use of incorrect calibration–renormalization factors during retrieval. Simulated (re)normalization errors were in the range −0.3 ≤ ɛ(C)/C ≤ 0.3 for scattering ratios in the range 1.1 < R < 1000. The modeled error was calculated via Eq. (34).

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

To provide readers and users of the CALIPSO version 3 data products with a means of assessing the likely scale of bias errors in the data products that arise from (re)normalization errors, we again consider the simple case of an atmospheric layer in which the scattering ratio R remains constant through the layer. The layer optical depth is modeled using
e36

In Fig. 3 we show the relative error in the retrieved particulate optical depth resulting from (re)normalization errors. From Eq. (34) we see that the error is a function of the scattering ratio R the relative error in the (re)normalization factor, and the optical depth penetrated into the layer. For a given value for R, then, the figure shows the relative error for a range of values of lidar ratio, layer depth, and altitude (via the molecular backscatter) that combine via Eq. (36) to give the range of optical depths on the abscissa. In Fig. 3a we use a scattering ratio of 1.5 to illustrate errors that might occur in the retrieval of optical depth in an aerosol layer. In Fig. 3b we use a scattering ratio of 100 to illustrate errors that might occur in a cloud layer. At the left of the plots, where the penetrated optical depth is small, the relative error in the retrieved optical depth is independent of optical depth, as indicated by the parallel contours, and the approximation given by Eq. (35) is valid. The white regions at the bottom right of each plot correspond to the parameter space in which the errors from Eq. (34) are undefined, defined as .

Fig. 3.
Fig. 3.

Contour plots of the relative error in retrieved optical depth due to (re)normalization errors as a function of (re)normalization error and optical depth, calculated using Eq. (34). Scattering ratio (a) R = 1.5 and (b) 100. The magnitude of the optical depth errors is indicated by the color bar and by the contour lines.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

(ii) Lidar ratio errors

Now let us look at the biases that result from errors in the lidar ratio that is used in the retrieval. As with the errors in calibration or renormalization described above, errors in the lidar ratio used by the extinction retrieval algorithm lead to a bias in the retrieved extinction profile that increases with range. This is a consequence of the use of the retrieved extinction at one range to correct for the transmittance loss at the subsequent ranges. (Note that, for this current analysis, we are assuming that the multiple scattering correction factor is unity and error free.)

In section a3 of the appendix, we derive the following relationships for the case where the calibration–renormalization errors are zero and the only error is in the lidar ratio:
e37
The error profiles in backscatter and extinction can be found by combining Eq. (37) with
e38
and
e39
respectively. The error in the particulate optical depth can be found via Eqs. (37) and (31).
From the general solution in Eq. (37), we show in section a2 in the appendix that, for the special case of a constant-R layer, the relative error in two-way particulate transmittance resulting from calibration–renormalization errors is given:
e40
[the ratio φ is defined in Eq. (A41)]. For the optical depth, the relative error is
e41
It can be seen that these errors grow rapidly with optical depth, scattering ratio, and lidar ratio error.
For low values of optical depth, Eq. (41) simplifies to
e42
For moderate to high values of scattering ratio, as found in clouds, both ρ and φ tend to unity and we have the approximation
e43
The approximation for the error in optical depth presented in our overview of CALIPSO algorithms (Winker et al. 2009) was derived from a substitution of Eq. (43) into Eq. (32).

The performance of the model [Eq. (41)] has been tested using the same simulated atmospheres as used in the previous section on (re)normalization errors. Figure 4 shows the comparison of the modeled relative error in particulate optical depth with the simulations that result from using lidar ratios with relative errors in the range −0.3 to +0.3 and for scattering ratios in the range 1.1 to 1000.

Fig. 4.
Fig. 4.

As in Fig. 2, but from the use of incorrect lidar ratios during retrieval. Simulated lidar ratio errors were in the range −0.3 ≤ ɛ(SP)/(SP) ≤ 0.3. The modeled error was calculated via Eq. (41).

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

We illustrate these errors using the case of a constant-R layer. The relative errors in the retrieved particulate optical depth calculated using Eq. (41) are plotted in Fig. 5 as a function of lidar ratio and optical depth. In Fig. 5a we use a scattering ratio of 1.5 to illustrate errors in an aerosol layer, whereas in Fig. 5b we use a scattering ratio of 100 to represent cloudy layers. As in the case of (re)normalization errors, it can be seen that the errors grow significantly for larger optical depths, as is evidenced by the contours converging toward the x axis. Note that the relative error in optical depth is undefined in the white regions at the top right of the figures, corresponding to large optical depths and large relative lidar ratio error where the argument of the logarithm function in Eq. (41) is less than or equal to zero. For low optical depths, the approximation [Eq. (42)] is valid, as evidenced by the parallel contours. This implies that the lower bound for the relative error in the retrieved optical depth is the relative error in the lidar ratio.

Fig. 5.
Fig. 5.

As in Fig. 3, but due to lidar ratio errors as a function of lidar ratio error and optical depth, calculated using Eq. (41).

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

(iii) Biases resulting from low SNR

Biased retrievals can also occur when the SNR of the input attenuated backscatter signals is too low. We illustrate this situation by presenting the results of our modeling of the same 1-km-thick layer as described above, in which optical depth is increased by increasing the scattering ratio. This time, however, rather than introducing (re)normalization or lidar ratio errors, we vary the SNR of the attenuated backscatter signals over the range 10−1 to 104. We do this by adding noise to the signal at each point in the profile where the magnitude of the noise is chosen randomly from a Gaussian distribution with a standard deviation equal to the ratio of the attenuated backscatter at the top of the layer to the selected SNR. This is an attempt to simulate daytime illumination conditions. [Although the noise in the actual CALIOP signal is more closely described by a Neyman type A distribution (Teich 1981; Liu and Sugimoto 2002), the computational time requirements for such a distribution would have been excessive, so we chose to use a Gaussian distribution, which provides an adequate approximation at higher signal and noise count rates.] For each value of SNR in the range specified above, we varied the optical depth of the layer over the same range as in the previous examples and generated 65 536 separate noisy profiles, which were then analyzed using the CALIOP extinction retrieval algorithms to produce profiles of extinction and optical depth. During some retrievals the solution began to diverge and the lidar ratio was reduced to ensure a successful retrieval, as is the normal HERA practice.

We present our results in Fig. 6, where the relative error in the retrieved optical depth is plotted as a function of optical depth and SNR. In Fig. 6a we include in the averages only those retrievals that did not require an adjustment of the lidar ratio, whereas in Fig. 6b we include all successful retrievals. Both situations show that the retrieved particulate optical depths are too small for both high optical depths and low SNRs. For optical depths greater than about 3 (which is near the maximum optical depth that can be retrieved by CALIPSO or other elastic backscatter lidars), the retrievals are biased low for all SNRs. The figure shows that retrievals at much higher optical depths would require SNRs that are unrealistic for current lidar systems. For SNRs less than or equal to 1, low biases begin to be found for optical depths greater than about 0.1–0.3. For SNRs in the same range, there is a positive bias in the retrieved optical depth that increases with decreasing SNR. Note that the bars on the points in Fig. 6 represent one standard deviation of the distribution of results from the 65 636 simulations and indicate the spread of the individual retrievals. (The standard deviations are off scale for SNRs ≤ 0.3 and optical depths < 0.1.) The standard errors [standard deviations on the mean (SDOM)] are then 1/256 of these values and would be too small to be discernible if plotted using the scales in Fig. 6. The biases shown in Fig. 6 are, therefore, realistic and not a result of noise from insufficient sampling.

Fig. 6.
Fig. 6.

Relative error in retrieved particulate optical depth as a function of particulate optical depth and SNR for a simulated layer 1 km deep. (a) Results for retrievals with unadjusted lidar ratios only. (b) All successful retrievals.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

The frequency of the low biases is related to the frequency with which the lidar ratio needs to be reduced in order to prevent the solution from diverging. These details are shown in Table 1. The explanation is fairly straightforward. Noise spikes in the attenuated backscatter profile are retrieved as high values of extinction, which add to the retrieved optical depth that is used to calculate the in-profile attenuation correction over the most recent range increment. For very large noise spikes (very low SNR) or very high optical depths, this spuriously high attenuation correction can be sufficient to cause the retrieval to fail, in which case the lidar ratio is reduced and the retrieval restarted from the top of the layer. Retrievals with low lidar values will be biased low.

Table 1.

The fraction of retrievals that required reduction of the lidar ratio to ensure a successful retrieval as a function of particulate optical depth and SNR.

Table 1.

The differences in the plots that include retrievals with unadjusted or adjusted lidar ratios can be explained by remembering that the results plotted are not individual retrievals but the averages of 65 536 separate simulations. The unadjusted retrieval averages exclude all those cases where noise caused an unsuccessful retrieval, whereas the adjusted values include those retrievals where the lidar ratio was reduced slightly to prevent divergence. While they are still lower than the true value, the averages for the adjusted simulations should be closer to the correct values than are the unadjusted simulations. This explanation is supported by the figures, which show that the bias errors for unadjusted retrievals are greater than those for adjusted retrievals except for SNRs >100. In these latter cases, the error caused in the average retrieved optical depth by excluding the adjusted values is smaller than the error caused by using a reduced lidar ratio. This situation, however, could be changed by reducing the lidar ratio correction increment.

Finally, we provide an indication of the magnitude of the reduced lidar ratio in the retrievals in Fig. 6b, which includes all successful retrievals. For an optical depth of 0.27, the mean lidar ratio for retrievals with SNR = 1.0 is (14.4 ±5.1) sr, versus a simulated “truth” value of 20 sr. This value decreases to (14.2 ±5.9) sr for a SNR of 0.1. For an optical depth of 2.9, the corresponding values are (19.99 ±0.19) and (17.5 ±5.9) sr.

2) Constrained retrievals

In retrievals that are constrained by an independent measurement of layer two-way particulate transmittance, the primary quantity that is retrieved is the lidar ratio, which is then used to retrieve the profiles of particulate backscatter and extinction. Errors in either the (re)normalization factor, the transmittance constraint, or both will cause errors in the retrieved lidar ratio and, as a consequence, in the retrieved profile data.

In constrained retrievals, an incorrect calibration–renormalization causes the attenuated backscatter profile to be biased high or low. However, the layer optical depth calculation that constrains the retrieval is effectively a ratio of the mean calibrated signal levels above and below the layer, and thus the optical depth estimate is unaffected by calibration errors, because the calibration term cancels out when ratioing the mean signal levels. The retrieval algorithm will then adjust the lidar ratio as required to generate a particulate extinction profile that, when integrated over the layer, exactly matches the independently derived value of the particulate optical depth. For constrained retrievals then, calibration errors cause errors in the retrieved lidar ratio rather than in the optical depth.

(i) Calibration–renormalization errors
Where the transmittance constraint is correct, we show in the appendix that
e44
The equation is transcendental though, so we cannot derive a closed-form expression for the error in the retrieved lidar ratio. The problem becomes more tractable, however, if we consider the simpler case of a constant-R layer. Equation (44) then becomes
e45
For large R, Eq. (45) becomes
e46
This approximation is accurate to within 5% for R greater than about 20 and within about 2% for R greater than about 50, corresponding to approximate particulate optical depths of 0.3 and 0.7, respectively, in the simulation used. The correct behavior of Eq. (45) was confirmed using retrievals from simulated signals from the modeled atmospheres described earlier, modified to enforce constrained retrievals.

Although most of the equations for bias errors in constrained retrievals are transcendental, the relative error for the case where the transmittance constraint is correct can be written in closed form [Eq. (45)] and is illustrated in Fig. 7a. In contrast to the relative errors in optical depth derived in unconstrained retrievals discussed above, the errors in the retrieved lidar ratios in constrained retrievals grow very large for small optical depths but converge to a limit, given by Eq. (46), for large optical depths. Note, however, in the derivation of Eq. (46), we have not considered the effects of SNR, nor of the finite dynamic range and resolution of CALIOP’s detector and digitizer, as adding these extra effects would add extra dimensions to the parameter space and make pictorial presentation even more difficult. For actual CALIOP data, the relative errors would be expected to begin diverging again for optical depths beyond 2–3, depending on SNR, actual signal magnitude, and so on.

Fig. 7.
Fig. 7.

(a) The relative error in the retrieved lidar ratio calculated using Eq. (45) for the case of a constrained retrieval with (re)normalization errors. The transmittance constraint is assumed to be correct in this simulation. The ragged region at the top left is a plotting artifact, where the error grows rapidly then becomes undefined. (b) The relative error in the retrieved lidar ratio calculated using Eq. (48) for the case of a constrained retrieval with errors in the transmittance constraint. Note: the colors have saturated at bottom left and the error exceeds 3.0.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

(ii) Transmittance constraint errors
Finally, we consider the errors in the derived lidar ratio that result from errors in the transmittance constraint. An error in the value of the layer two-way transmittance that is supplied to the extinction retrieval algorithm as a constraint will also cause an error in the retrieved lidar ratio and, as a consequence, in the retrieved profiles of backscatter and extinction. As in the previous section, the constrained retrieval attempts to ensure that the retrieved layer optical depth matches the independently derived constraint. The equation for the error in the lidar ratio is obtained, by setting α and the left-hand side of Eq. (37) to unity, as shown:
e47
which is, again, transcendental.
For the case of a layer in which the scattering ratio R is constant, though, we can derive a closed-form solution for the relative error, written as
e48
However, because φ [see Eq. A(41)] is a function of ɛ(SP), Eq. (48) is still transcendental. For large R, though, we have the approximation
e49
which is accurate to within 4% for R greater than about 5 and 2% for R greater than about 10 in our simulation.

The analysis presented above shows that the equations for the relative error for both the general case [Eq. (47)] and the case for a constant-R layer [Eq. (48) ] are transcendental and must be solved numerically. We illustrate the relative errors in the retrieved lidar ratio derived from Eq. (48) for our simulated example of a constant-R layer in Fig. 7b.

The relative errors in lidar ratio grow large for small optical depths, but they tend toward zero for large optical depths. Once again, however, we note the same caveats with the simulation as in the previous section. In the region at the top left, where from Eq. (48) , the relative error in lidar ratio is large and negative.

4. Examples of uncertainties and errors in retrieved extinction

We now present examples of the estimated uncertainties in the CALIPSO extinction retrievals using both simulated and actual data. By varying the magnitude of selected analysis parameters about their modeled values in simulated data, we are also able to assess the sensitivity of the extinction retrievals to errors in these parameters. This study is not an estimate of the errors to be expected in the CALIOP data, which depend on the magnitude of both modeled and unmodeled error sources, but an examination of how errors propagate in the course of a multilayer retrieval and the sensitivity of the retrieval to some of the key sources of error.

a. Retrieval uncertainties and errors in simulated data

To illustrate the uncertainties and sensitivities to analysis errors in CALIPSO extinction retrievals, we use the example of the representative complex scene described in YV09 and plotted here in Fig. 1. The parameters of the various features that were used to model the attenuated backscatter signal from each 5-km column in the scene are listed in Table 2. For example, features 1 and 7 are described as “moderate cirrus,” which is only detected after averaging horizontally over four columns (20 km or 60 consecutive profiles). The features are assumed to have a height-invariant extinction coefficient of 0.025 km−1, a lidar ratio of 25 sr, and a multiple scattering factor of 0.6. Features 2–6 and 8 and 9 are described as “strong cirrus,” with a higher extinction coefficient of 0.2 km−1, allowing them, by their associated higher backscatter, to be detected at 5-km horizontal resolution. Features 11, 13–15, 17, 18, and 20 are simulated water clouds that are detected at 5-km horizontal resolution. The strong, vertically isolated features (2–6 and 11) are modeled to have optical depths exceeding 0.2, thereby allowing their transmittances to be measured with sufficient accuracy that they can be used as retrieval constraints as described in section 2.

Table 2.

Modeled parameters for features in the representative complex scene in Fig. 1. Optical properties are for 532 nm.

Table 2.

The simulated scene thus contains a broken cirrus layer of varying strength between 13.5- and 16.5-km altitude. At the right, patches of strong cirrus are mixed with weaker cirrus in a complex feature. Between 5.5 and 8.0 km, there is a second complex feature composed of patches of water cloud embedded in a smoke layer of varying strength. Finally, below 3.0 km, there is a surface-attached dust layer with spatially varying backscatter intensity. This is modeled as a complex feature containing strong features (24–38) that are detected at 5-km horizontal resolution, features of medium strength (22, 23) that are detected at 20-km resolution, all contained within a weaker layer (21) that is detected at 80-km horizontal resolution. Note that, because of the nonlinearity of the lidar equation, it is not valid simply to average the attenuated backscatter signals from strong, moderate and, weak features and then analyze the single, averaged profile (YV09). In CALIPSO’s hybrid analysis scheme, features of different strengths are detected and analyzed separately.

To enable the propagation of uncertainties and errors to be seen clearly, the particulate backscatter and extinction are modeled to be constant with altitude and no noise is added to the simulated attenuated backscatter profiles. We describe this as a “noise free” simulation. The sensitivity to errors in the calibration of the input profiles and in the values of the lidar ratios used to retrieve the extinction profiles is shown in Figs. 8 and 9 by varying the magnitude of the errors in seven separate analyses of the same representative complex scene. These separate analyses are plotted consecutively as adjoining blocks in the figures. The seven consecutive blocks in Fig. 8 show the effect of introducing relative calibration errors of −0.069, −0.048, −0.027, 0.0, 0.027, 0.048 and 0.069, whereas Fig. 9 shows the effect of introducing relative errors in the lidar ratio used to analyze selected cirrus features of −0.4, −0.20, −0.10, 0.0, 0.10, 0.20, and 0.40. There are no errors in the central block in each panel, which can therefore be used as a reference for the others. The calibration errors are based on the results of Rogers et al. (2011), who used coincident, airborne high-spectral-resolution lidar and CALIPSO measurements to show that, on average, CALIPSO’s 532-nm calibration error is 0.027 ±0.021. Here, we simulate calibrations that are either too low or too high by this amount. The larger values represent one and two standard deviations above the mean. The maximum lidar ratio errors of ±40% correspond to the uncertainty for the 532-nm lidar ratio for cirrus (Yorks et al. 2011) that is listed in the metadata associated with the version 3 data products.

Fig. 8.
Fig. 8.

Effects of calibration errors on retrieved extinction coefficient in the various features in the simulated complex scene in Fig. 1. The relative calibration errors in the (left to right) seven consecutive blocks are −0.069, −0.048, −0.027, 0.0, 0.027, 0.048, and 0.069. (a) Retrieved 532-nm particulate extinction coefficient, (b) relative uncertainty in retrieved extinction [Eqs. (15)(24)], and (c) relative error in retrieved extinction.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

Fig. 9.
Fig. 9.

Errors in the retrieved 532-nm extinction coefficient for the features in the simulated complex scene in Fig. 1 resulting from errors in the lidar ratio used for retrieving selected features, where the relative errors in the lidar ratios in the (left to right) seven consecutive panels are −0.4, −0.2, −0.1, 0.0, 0.1, 0.2, and 0.4. The (a) retrieved extinction and (b) relative error in the retrieved extinction resulting from relative errors in feature F7 (see text). (c),(d) The errors are in feature F10.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

Let us now look at Fig. 8 in detail. The retrieved 532-nm particulate extinction coefficient is plotted in Fig. 8a. The retrieved extinction in the central block (columns 49–64) exactly matches the modeled extinction as the relative calibration error in this block is zero. In the blocks to the left, where the relative calibration error is negative [i.e., the calibration factor C in Eq. (1) is too small, and the attenuated backscatter is, therefore, too large], the retrieved extinctions are too large. This is most noticeable in the strong dust layers near the surface where the solution “blows up.” In these situations, the retrieved extinction values, or the estimated uncertainties, or both, are diverging toward infinity and no solution is possible using an acceptable and physically meaningful range of lidar ratios. It can be seen that the retrieval is terminated early and does not extend all the way to the surface. In contrast, in the blocks on the right, where the calibration factors used in the retrievals are too large and the attenuated backscatter too small, the retrieved extinctions can be seen to be too small.

Figure 8b shows the relative uncertainties (i.e., estimated relative precision) in the retrieved particulate extinction, calculated using Eqs. (15)(24), and as used in the version 3 data release. These result from modeled uncertainties in the molecular number density at all altitudes of 3% (Reagan et al. 2002), which contribute to uncertainties in the molecular backscatter and also the two-way molecular transmittance down to the normalization altitude, and hence contribute to the uncertainty in the normalization (layer attenuation correction) factor in Eq. (8). The modeled uncertainty in the effective lidar ratio (ηSP) of each feature matches that used in the version 3 data release. This was 40% for ice clouds and smoke and 50% for water clouds and dust. For the purpose of this exercise, the uncertainty in the transmittance constraint for features 2–6 and 11 was modeled to be 25%. Although, in this panel, we have only modeled an error in calibration, which varies in each 16-profile block as described above, the analysis makes no assumptions as to where other errors occur and assigns an uncertainty to all quantities. We can see that the uncertainties in the retrieved extinction are dominated by the uncertainties in the lidar ratios for each type of feature [e.g., ≈40% (blue-green color) for the upper cirrus layers and ≈50% (orange-red color) for the water clouds—compare with Fig. 1 and Table 2.] Note though how the uncertainties at lower altitudes are increased by the uncertainty in the correction for the attenuation caused by overlying layers, and how this changes with the calibration error in each block.

In Fig. 8c, we plot the actual relative errors in the retrieved extinction coefficients that result from modeled systematic errors in the calibration coefficients. Remember that we know the modeled input extinction coefficients (or truth), so we can determine the errors exactly, whereas in Fig. 8b we plot the relative uncertainties, which we can interpret as an estimate of the likely RMS relative error. Note too that in Fig. 8c values are both positive and negative, whereas in Fig. 8b the values are all positive. This is because we do not know whether the values of the lidar ratios, calibration factors, molecular density profiles, and so on, which we use in the retrievals, are too high or too low; the contributing uncertainties were summed in quadrature. The very low (close to zero) relative errors in the features where constrained retrievals have been possible (F2–F6, F11) are immediately obvious. For the other cirrus features, the errors are slightly larger than the calibration error at the top of the feature (positive on the left of the panel and negative on the right) and increase with penetration depth. For the 1064-nm cases, not shown here, the agreement with the calibration error is closer because the particulate backscatter contributes a much greater proportion of the total backscatter signal. The errors in these upper features are propagated down to the lower features. Note, in particular, that the error in feature F12, which occupies all 16 columns in a block, is propagated down to all underlying features, which all have errors equal to, or larger than, the error in F12.

Now we turn to the propagation of errors that results from errors in the lidar ratios used in the retrievals. We consider two cases. In Figs. 9a and 9b, we simulate errors in the lidar ratio used to solve a feature of moderate backscatter (F7) that occupies four columns with errors in each block varying from −40% to +40% as described above. Figure 9a shows the retrieved particulate extinction and Fig. 9b the corresponding relative errors. The errors in F7 stand out very clearly. Two important points should be noted. First, note that the initial errors are only in an upper feature that occupies just four columns, yet retrieval errors are caused in all columns in the block. This is because the errors in F7 cause errors in the overlying transmittance correction applied to F12. The resulting errors in the retrieved extinction in F12 are then propagated down to all features that are below F12. As F12 occupies all 16 columns (except that occupied by F11), all lower features are affected. The second point to note, however, is that the errors in features at lower altitudes than F7 are all less than those in F7. This is a result of the relatively low optical depth of F7 and, consequently, the relatively small attenuation correction, and also, this attenuation correction only scales 4 of the 16 columns in F12, so the error in the average of those columns, which is then used for retrieving the extinction profile in F12, is proportionately less.

In Figs. 9c and 9d, we simulate errors in a strong, single-column feature (F10). Because of the much higher optical depth of F10, the errors in the lidar ratio have a much greater effect on the attenuation correction applied to lower features than was the case for F7. Again, we note that although the initial errors were in just one column, errors spread across all columns because of the presence of F12, which occupies all columns. The largest errors are found in F38, which lies directly beneath F10 and also beneath F12, F21, and F23, and, consequently, accumulates a fraction of the attenuation errors in all of these features.

b. Retrieval uncertainties in actual data

We now present examples of our uncertainty analyses applied to particulate extinction profiles retrieved from CALIPSO lidar data. In Fig. 10 we show the analysis of CALIPSO data recorded over the southern region of the African continent around 2341 UTC 30 June 2011. Figures 10a–d show the 532-nm attenuated backscatter signal, the vertical feature mask showing feature type, the retrieved 532-nm particulate extinction, and the calculated relative uncertainty in the 532-nm extinction, respectively. An extensive aerosol layer can be seen to reach 4-km altitude and cover many degrees of latitude. Additional information from the vertical feature mask file (not plotted here) indicates that CALIPSO’s scene classifier algorithms have determined that the layer is mostly composed of smoke, with some patches classified as polluted dust or dust. The retrieved particulate extinction is in the approximate range of 0.05–0.5 km−1 and the relative uncertainties are typically around 50%–60% but approach 100% near the surface. The relative uncertainties in extinction plotted in Fig. 10d were calculated using Eq. (24). The relative uncertainty in the lidar ratio in Eq. (24) was based on the value listed in Table 3 for the feature type identified by the scene classifier algorithms. The relative uncertainty in particulate extinction in Eq. (24) was calculated using Eqs. (15), (17), (19), and (22). A relative uncertainty of 3% was assumed for the molecular number density, and this value was then used to calculate the uncertainties in molecular backscatter and two-way transmittance. Estimates of the uncertainties in the normalized attenuated backscatter profile due to random error (i.e., shot noise) are calculated using the procedures by Liu et al. (2006). Within each feature, these random uncertainties were then combined in quadrature with the uncertainty in the calibration factor [via Eq. (10)] or the renormalization factor [via Eq. (14)], which are systematic uncertainties. However, in the version 3 data release we discuss here, the separation of random and systematic components was not maintained through multiple renormalizations, and Eq. (14) was merely updated after each renormalization.

Fig. 10.
Fig. 10.

CALIPSO measurements of an aerosol layer detected over the southern region of the African continent around 2341 UTC 30 Jun 2011: (a) the 532-nm attenuated backscatter signal in km−1 sr−1, (b) the vertical feature mask showing feature type and subtype, (c) the retrieved 532-nm particulate extinction in km−1, and (d) the relative uncertainty in the 532-nm extinction [calculated via Eqs. (15)(24)].

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

Table 3.

Characteristic lidar ratios for the various aerosol and cloud types identified in the version 3 CALIPSO data analyses, together with the one standard deviation uncertainties and relative uncertainties. Note that the uncertainty estimate for the water cloud lidar ratio includes uncertainty in the layer-effective multiple scattering factor.

Table 3.

An example of the analysis of a nighttime cirrus layer detected by CALIPSO over the equatorial eastern Pacific Ocean around 0847 UTC 23 January 2011 is shown in Fig. 11. The layer varies in strength to the extent that some sections are detected at 1-km resolution, whereas others are only detected after averaging over a horizontal distance of 80 km (240 profiles). Additional vertical feature mask information (not plotted here) indicates that the layer is almost completely composed of randomly oriented ice crystals with very few profiles containing horizontally oriented crystals. The order of the panels is the same as for the previous figure. The retrieved cloud extinction (Fig. 11c) varies over an approximate range of 0.025 to 0.25 km−1. Note that cloud and aerosol extinction are reported separately in the CALIPSO data products; here we plot cloud extinction. The relative uncertainty, calculated using Eqs. (15)(24) and shown in Fig. 11c, is remarkably uniform in this example, with values around 50%, which is only slightly higher than the assumed uncertainty in the lidar ratio for cirrus (40%). This is not surprising, as there are no overlying features whose extinction retrievals would increase the uncertainty in this layer, and the cirrus extinction coefficient and layer depth are both rather low so the uncertainty, which grows with optical depth, does not do so appreciably here.

Fig. 11.
Fig. 11.

As in Fig. 10, but for a cirrus layer detected over the equatorial eastern Pacific Ocean around 0847 UTC 23 Jan 2011.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

Finally, in Fig. 12, we present an example of what we describe as “total chaos.” As in the previous figures, Fig. 12a shows the 532-nm attenuated backscatter. A very complex scene containing strongly attenuating cirrus and other ice clouds is seen extending from 8 up to 18 km over eastern Indonesia around 1705 UTC 4 April 2011. Most of the clouds below 10 km are identified as water. In some places below the clouds the lidar signal is totally attenuated, as indicated by the black regions in the vertical feature mask (Fig. 12b). There is also a layer of aerosol, with occasional and embedded water clouds, extending from the surface up to around 2-km altitude. The 532-nm extinction can be seen in Fig. 12c to vary over more than two orders of magnitude, from around 0.025 to around 2.5 km−1. The relative uncertainties in Fig. 12d in the cirrus layer range from around 40% (the uncertainty in the cirrus lidar ratio) to around unity in more tenuous regions and toward the base of the layer. It is not surprising, with the considerable optical depth of the overlying cirrus, that the uncertainties in the much more weakly scattering aerosol layer approaches, and in some locations exceeds, 100%. This is because the uncertainty in the overlying cirrus increases the uncertainty in the total attenuated backscatter (and, therefore, retrieved total extinction) of the lower aerosol layer. As the aerosol backscatter and extinction coefficient are rather less than the total (aerosol plus molecular) values, the relative uncertainty in the aerosol components is amplified.

Fig. 12.
Fig. 12.

As in Fig. 10, but for a complex scene detected over eastern Indonesia around 1705 UTC 4 Apr 2011.

Citation: Journal of Atmospheric and Oceanic Technology 30, 3; 10.1175/JTECH-D-12-00046.1

5. Summary and conclusions

The CALIPSO Hybrid Extinction Retrieval Algorithms (HERAs) are used to analyze lidar attenuated backscatter profiles and related information that have been processed by several preceding algorithms. These algorithms perform tasks including calibration, feature detection, identification of feature types, and specification of the corresponding lidar ratios and multiple-scattering functions. As a consequence, the extinction data products are susceptible to errors and uncertainties in the information received by HERA, in addition to the statistical noise and variations inherent in the data. The solution of the nonlinear lidar extinction equation, especially in CALIPSO “scenes” containing many particulate layers or features, further amplifies these uncertainties.

The extinction uncertainties arise from both random and systematic sources. Both the statistical noise in the signal and the random variations about the true value in the molecular profiles at any altitude are random sources of uncertainty. Uncertainties in calibration (or renormalization) and uncertainties in the lidar ratios and optical depth constraints provided to HERA, because they are mostly constant with height for any profile being analyzed, are all considered here to be systematic uncertainties. Ideally, because of the different way they propagate, random and systematic uncertainties would be considered and reported separately. However, because of the complexity of the distribution of the random and systematic components of the uncertainty at any given altitude in any given attenuated backscatter profile, and because of the way in which these components change following each renormalization, the separate treatment of the random and systematic components is not maintained throughout the analysis reported here. In the version 3 data release, the analysis is simplified and the uncertainties arising from both sources are assumed to be statistically independent and added in quadrature to give the total uncertainty at any altitude in a profile product. In this simplified process, initially the random and systematic components of the uncertainty in the calibration factor for each profile are combined in quadrature via Eq. (10). Thereafter, the random and systematic components in the calibration factor are not treated separately. Rather, after each renormalization, the overall renormalization uncertainty of a profile is updated by combining the existing uncertainty (which has random and systematic components) with the cumulative uncertainty (also containing random and systematic components) in the retrieved particulate two-way transmittance of the overlying layer via Eq. (14). The uncertainty analysis scheme is demonstrated using both simulated data and data acquired on orbit by CALIPSO. The uncertainties in the various quantities are listed, along with their symbols and reference equations, in Table 4.

Table 4.

Summary of uncertainties, giving symbols and reference equations.

Table 4.

In addition to the uncertainty analyses, we derive equations and provide diagrams describing the sensitivity of the retrieved backscatter, extinction, and optical depth products to errors in calibration and specified lidar ratio. The bias errors in the various retrieved quantities are listed, along with their symbols and reference equations for the various situations of constrained and unconstrained retrievals, in Table 5.

Table 5.

Summary of modeled bias errors, giving symbols and reference equations.

Table 5.

The effect of low SNR in creating biased retrievals is also examined. By comparing the differences between particulate extinction profiles, retrieved using erroneous calibration coefficients or lidar ratios, and the modeled values, the propagation of biases in the retrievals is demonstrated during analyses of simulated data from the representative complex scene described in detail in YV09. Note that, although not shown here, differences in the errors in the 532-nm retrievals and those at 1064 nm can result where constrained retrievals are used at the shorter wavelength.

The error sensitivity analyses show that quite large errors are possible, especially when both input errors and optical depths are large. Whether an error of a certain magnitude is acceptable, or the result useful, will depend on how the result is to be applied. As the applications of the CALIPSO data products are so varied, discussions of the implications of such potential errors for such applications will not be covered here.

As might be expected for such a complex, multistage data processing scheme, the quality of the extinction data products is dependent on the provision of accurate inputs. The most extreme case is where the feature finder fails to detect a feature at all and, consequently, no direction is given to HERA to retrieve any information on that feature. Obviously, the analyses reported here cannot provide uncertainties or error estimates on extinction and optical depths of features, whose presence is not indicated to the extinction retrieval algorithms. Because values for lidar ratios can vary over a large range, even for a particular feature type, of the factors considered here, they are the largest source of uncertainty in the retrievals. Note that the misclassification of a feature subtype by the preceding scene classification algorithm can lead to the use of the wrong lidar ratio and lead to significant bias errors if the lidar ratio error is large enough. It should be noted that the development of advanced algorithms to assist in the correct classification of the subtype of detected features is in progress. The correct classification of features and, consequently, the selection of the appropriate lidar ratio, avoids the potential for a significant source of error because, although there is variability in the lidar ratio for any feature class, this variability is usually considerably less than the differences in the lidar ratios for different feature classes.

Acknowledgments

The CALIPSO data used in this paper were obtained from the NASA Langley Atmospheric Science Data Center. Numerous conversations with Zhaoyan Liu and Yongxiang Hu have added greatly to the authors’ understanding and appreciation of the subtleties of uncertainty and error propagation in the CALIPSO data products.

APPENDIX

Sensitivity of Retrievals to Errors in Inputs

In this appendix, we derive equations for the sensitivity of retrievals of particulate backscatter, extinction, optical depth, and two-way transmittance to errors in the calibration or renormalization, lidar ratio selected for the retrieval, and the transmittance constraint in the case of constrained retrievals. Because the relative errors in these quantities can be large relative to unity, we choose to adopt a differencing analysis to evaluate the resultant bias errors. Throughout the development in this appendix, all molecular quantities [e.g., βm(z), σm(z) and ] are assumed to be error free.

a. Unconstrained retrievals

1) General case
We choose to describe the effect on the attenuated backscatter signal of errors in calibration or renormalization by the multiplicative factor
ea1
where we use a generic ɛ(C)/C to represent the fractional (relative) error in either calibration or renormalization, and . The caret (ˆ) notation indicates estimated or retrieved quantities that may be in error. As the retrieved, total backscatter is calculated from the attenuated total backscatter via
ea2
where, for convenience, we begin integration (r = 0) at the top of the feature, the ratio of the incorrectly estimated total backscatter to the true backscatter can be written as
ea3
The two-way molecular transmittance is obtained from the model in either case and cancels out. Also, we are only studying calibration errors here, so we assume that the molecular quantities are error free. We note the following equivalence between errors in total and particulate backscatters:
ea4
Therefore, the error in the retrieved particulate backscatter is
ea5
While we describe the errors in calibration via the ratio α we describe the relationship between the incorrectly selected lidar ratio and the correct value using the ratio ψ as follows:
ea6
For the general case where there are errors in both the lidar ratio and the calibration–renormalization factor, the error in the retrieved particulate extinction is
ea7
This can also be expressed as
ea8
Combining Eqs. (A8) and (A5) and then rearranging gives
ea9
The error in the retrieved particulate optical depth can then be found by integration:
ea10
Then, using the relationship
ea11
we obtain
ea12
We choose to write this equation in a standard or common form, which we shall use throughout these appendices, as
ea13
where we have made the following substitutions:
ea14
We also note that
ea15
and, therefore, that
ea16
Consequently, we have the following, which we use below:
ea17
Taking natural logarithms in Eq. (A13) and differentiating with respect to r leads to the differential equation
eq8
which can be further rearranged to give the standard form, as shown:
ea18
Equations of this form are usually solved (e.g., Hildebrand 1962, section 1.4) by multiplying both sides of the equation by an integrating factor to give
ea19
It can be seen that the left side of Eq. (A19) is now an exact differential:
eq9
This equation can now be solved by simple integration to give
ea20
As, in our application, is 1 and the integrals are 0 at r = 0, we find that c is also 1. We can then write the solution for u(r) as
ea21
On substituting the quantities in (A14) back into (A21), we have the general formula for the sensitivity of retrieved particulate transmittance errors to errors in the lidar ratio or calibration–renormalization factor, written as
ea22
2) Sensitivity to errors in calibration or renormalization
If the lidar ratio selected for the retrieval is correct, then ψ = 1, and Eq. (A22) simplifies to
ea23
Recognizing that is simply the normalized, attenuated backscatter signal, readers will notice a certain similarity with Eq. (13) of Fernald et al. (1972).

Equation (A23) permits users of CALIPSO data products to assess the likely impacts of a range of calibration or renormalization errors on the retrieved two-way particulate transmittance (and hence optical depth) values by using the tabulated attenuated backscatter profile [], molecular transmittance profile, and the lidar ratios in the data products. For any layer, the integral only needs to be evaluated once and the relative errors in the two-way particulate transmittance [i.e., Eq. (A23) minus 1] for a range of calibration errors can be determined by varying α.

We note that a simpler form of the equation can be obtained by recognizing that, because w(z) and υ(z) are related by a simple constant α the integrand with respect to z on the right-hand side of Eq. (A21), and hence Eq. (A23), is an exact differential, as shown:
ea24
that is
ea25
Integration gives
ea26
allowing us to write Eq. (A23) in the form
ea27
Recalling that , we have
ea28
The relative error can then be obtained as
ea29
The error profiles in particulate backscatter and extinction are described by substituting (A28) into (A5) and (A9), respectively. In addition, the error in the retrieved particulate optical depth may be obtained by substituting (A28) into (A11), as shown:
ea30
In those cases where , we have further simplification, as shown:
ea31
Special case—Layer with scattering ratio constant with range.
For a layer in which the scattering ratio,
ea32
is constant with range, the particulate and total backscatters are related simply by the equation
ea33
where
ea34
Therefore,
eq6
and, from Eqs. (A14) and (A16), we have
eq7
Equation (A28) can then be written for the case of a constant-R layer as
ea35
or
ea36
We can also write the equation for the relative error in the transmittance [Eq. (A29)] as
ea37
The error in particulate optical depth is then obtained via Eq. (A30) as
ea38
It can be seen that the equation becomes undefined for .
Where , we can make the approximation [Eq. (A31)]
ea39
Furthermore, if , then we have further simplification, as shown:
ea40
Alternatively, the relative error in the particulate optical depth can be approximated by
ea41
3) Sensitivity to errors in lidar ratio
For the case where the errors are only in the lidar ratio, we can set the calibration–renormalization error α to 1. Eq. (A22) then becomes
ea42
As before, the error profiles in backscatter and extinction can be found by substituting Eq. (A42) into Eqs. (A5) and (A7), respectively (with α = 1), and the error in the particulate optical depth can be found via Eqs. (A42) and (A30).
Special case—Layer with scattering ratio constant with range.
From the general solution for lidar ratio errors in Eq. (A42), we can find the solution for the special case of a constant-R layer. By making the following substitutions
ea43
where
ea44
we note that the transmittance terms in Eq. (A42) can be written purely as functions of particulate quantities as follows:
ea45
or, more simply,
ea46
Eventually, after some more simplification and rearrangement, we find the following expression for the relative error in particulate transmittance arising from an error in the lidar ratio for the special case of a constant-R layer:
ea47
For the optical depth, the relative error is
ea48
For low values of optical depth, this equation simplifies to
ea49
For moderate to high values of scattering ratio, as found in clouds, both ρ and φ tend to unity and we have the approximation
ea50

b. Constrained retrievals

1) General case

In constrained retrievals, calibration and renormalization errors do not affect the calculation of the optical depth constraint [refer to section 3b(2) for more discussion on this point]. Instead, they lead to an incorrectly retrieved particulate backscatter profile and an incorrect lidar ratio, which is adjusted to match the integral of the extinction profile to the optical depth constraint. The derivation of the equations for the bias errors in the retrieved profiles of particulate backscatter, extinction, optical depth, and transmittance ratio follows the analysis in Eqs. (A1)(A22) above.

2) Errors in calibration or renormalization
For the case where the transmittance constraint is correct, at the base of the layer (r = rb), the retrieved particulate transmittance matches the constraint, which is assumed to equal the true value . The left side of Eq. (A22) then has a value of unity. We can then write an expression for the error in the retrieved lidar ratio as
ea51
As this equation is transcendental, we cannot derive a closed-form expression for the lidar ratio error and must employ a numerical solution.
Special case—Layer with scattering ratio constant with range.
The problem in the previous section becomes more tractable if we consider the simpler case of the constant-R layer we studied earlier. We define R, α, ρ, and φ as before via Eqs. (A32), (A1), (A34), and (A44), respectively. Equation (A51) then becomes
ea52
or, in terms of the relative error,
ea53
Note that this relative error becomes undefined when the denominator is 0. This occurs when . For large R, Eq. (A53) becomes
ea54
3) Errors in the transmittance constraint
If we now consider the case where the transmittance constraint may itself be in error (e.g., because of SNR issues, or aerosol contamination in the assumed “clear air” regions used to estimate layer attenuation), then the left side of Eq. (A22) is not unity but instead equals the ratio of the incorrectly measured constraint to the true two-way particulate transmittance. Thus, at the layer base we have
ea55
This equation is also transcendental and needs to be solved numerically. However, for large R, both ρ and φ tend to unity and we then have the closed-form approximation
ea56

REFERENCES

  • Bevington, P. R., and Robinson D. K. , 1992: Data Reduction and Error Analysis for the Physical Sciences. 2nd ed. McGraw-Hill Inc., 328 pp.

  • Bissonnette, L. R., 1986: Sensitivity analysis of lidar inversion algorithms. Appl. Opt., 25, 21222125.

  • Del Guasta, M., 1998: Errors in the retrieval of thin-cloud optical parameters obtained with a two-boundary algorithm. Appl. Opt., 37, 55225540.

    • Search Google Scholar
    • Export Citation
  • Fernald, F. G., 1984: Analysis of atmospheric lidar observations: Some comments. Appl. Opt., 23, 652653.

  • Fernald, F. G., Herman B. M. , and Reagan J. A. , 1972: Determination of aerosol height distributions with lidar. J. Appl. Meteor., 11, 482489.

    • Search Google Scholar
    • Export Citation
  • Hildebrand, F. B., 1962: Advanced Calculus for Applications. Prentice Hall, 646 pp.

  • Hu, Y., and Coauthors, 2009: CALIPSO/CALIOP cloud phase discrimination algorithm. J. Atmos. Oceanic Technol., 26, 22932309.

  • Hughes, H. G., Ferguson J. A. , and Stephens D. H. , 1985: Sensitivity of lidar inversion algorithm to parameters relating atmospheric backscatter and extinction. Appl. Opt., 24, 16091613.

    • Search Google Scholar
    • Export Citation
  • Hunt, W. H., Winker D. M. , Vaughan M. A. , Powell K. A. , Lucker P. L. , and Weimer C. , 2009: CALIPSO lidar description and performance assessment. J. Atmos. Oceanic Technol.,26, 1214–1228.

  • Jinhuan, Q., 1988: Sensitivity of lidar equation solution to boundary values and determination of the values. Adv. Atmos. Sci., 5, 229241.

    • Search Google Scholar
    • Export Citation
  • King, M., Closs J. , Spangler S. , Greenstone R. , Wharton S. , and Myers M. , 2004: EOS data products handbook. Vol. 1, EOS Project Science Office, NASA/Goddard Space Flight Center, 260 pp. [Available on-line at http://eospso.gsfc.nasa.gov/eos_homepage/for_scientists/data_products/vol1.php.]

  • Klett, J. D., 1981: Stable analytical inversion solution for processing lidar returns. Appl. Opt., 20, 211220.

  • Kovalev, V. A., and Moosmüller H. , 1994: Distortion of particulate extinction profiles measured with lidar in a two-component atmosphere. Appl. Opt., 33, 64996507.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., and Sugimoto N. , 2002: Simulation study for cloud detection with space lidars by use of analog detection photomultiplier tubes. Appl. Opt., 41, 17501759.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., Hunt W. , Vaughan M. , Hostetler C. , McGill M. , Powell K. , Winker D. , and Hu Y. , 2006: Estimating random errors due to shot noise in backscatter lidar observations. Appl. Opt., 45, 44374447, doi:10.1364/AO.45.004437.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., and Coauthors, 2009: The CALIPSO lidar cloud and aerosol discrimination: Version 2 algorithm and initial assessment of performance. J. Atmos. Oceanic Technol., 26, 11981213.

    • Search Google Scholar
    • Export Citation
  • Omar, A. H., and Coauthors, 2009: The CALIPSO automated aerosol classification and lidar ratio selection algorithm. J. Atmos. Oceanic Technol., 26, 19942014.

    • Search Google Scholar
    • Export Citation
  • Powell, K. A., and Coauthors, 2009: CALIPSO lidar calibration algorithms. Part I: Nighttime 532-nm parallel channel and 532-nm perpendicular channel. J. Atmos. Oceanic Technol., 26, 20152033.

    • Search Google Scholar
    • Export Citation
  • Reagan, J. A., Wang X. , and Osborn M. T. , 2002: Spaceborne lidar calibration from cirrus and molecular backscatter returns. IEEE Trans. Geosci. Remote Sens., 40, 22852290.

    • Search Google Scholar
    • Export Citation
  • Rienecker, M. M., and Coauthors, 2012: Atmospheric reanalyses—Recent progress and prospects for the future: A report from a technical workshop, April 2010, M. J. Suarez, Ed., Technical Report Series on Global Modeling and Data Assimilation, Vol. 29, NASA/TM–2012-104606, 62 pp. [Available online at https://gmao.gsfc.nasa.gov/pubs/docs/NASATM2012-104606v29.pdf.]

  • Rogers, R. R., and Coauthors, 2011: Assessment of the CALIPSO lidar 532 nm attenuated backscatter calibration using the NASA LaRC airborne high spectral resolution lidar. Atmos. Chem. Phys., 11, 12951311, doi:10.5194/acp-11-1295-2011.

    • Search Google Scholar
    • Export Citation
  • Russell, P. B., Swissler T. J. , and McCormick M. P. , 1979: Methodology for error analysis and simulation of lidar aerosol measurements. Appl. Opt., 22, 37833797.

    • Search Google Scholar
    • Export Citation
  • Sasano, Y., and Nakane H. , 1984: Significance of the extinction/backscatter ratio and boundary value term in the solution for the two-component lidar equation. Appl. Opt., 23, 1113.

    • Search Google Scholar
    • Export Citation
  • Taylor, J. R., 1982: An Introduction to Error Analysis. University Science Books, 270 pp.

  • Teich, M. C., 1981: Role of doubly-stochastic Neyman type-A and Thomas counting distributions in photon detection. Appl. Opt., 20, 24572467.

    • Search Google Scholar
    • Export Citation
  • Vaughan, M., and Coauthors, 2009: Fully automated detection of cloud and aerosol layers in the CALIPSO lidar measurements. J. Atmos. Oceanic Technol., 26, 20342050.

    • Search Google Scholar
    • Export Citation
  • Winker, D. M., Vaughan M. A. , Omar A. H. , Hu Y. , Powell K. A. , Liu Z. , Hunt W. H. , and Young S. A. , 2009: Overview of the CALIPSO mission and CALIOP data processing algorithms. J. Atmos. Oceanic Technol., 26, 23102323.

    • Search Google Scholar
    • Export Citation
  • Winker, D. M., and Coauthors, 2010: The CALIPSO mission: A global 3D view of aerosols and clouds. Bull. Amer. Meteor. Soc., 91, 12111229.

    • Search Google Scholar
    • Export Citation
  • Yorks, J. E., Hlavka D. L. , Hart W. D. , and McGill M. J. , 2011: Statistics of cloud optical properties from airborne lidar measurements. J. Atmos. Oceanic Technol., 28, 869883.

    • Search Google Scholar
    • Export Citation
  • Young, S. A., 1995: Lidar analysis of lidar backscatter profiles in optically thin clouds. Appl. Opt., 34, 70197031.

  • Young, S. A., and Vaughan M. A. , 2009: The retrieval of profiles of particulate extinction from Cloud Aerosol Lidar Infrared Pathfinder Satellite Observations (CALIPSO) data: Algorithm description. J. Atmos. Oceanic Technol., 26, 11051119.

    • Search Google Scholar
    • Export Citation
Save
  • Bevington, P. R., and Robinson D. K. , 1992: Data Reduction and Error Analysis for the Physical Sciences. 2nd ed. McGraw-Hill Inc., 328 pp.

  • Bissonnette, L. R., 1986: Sensitivity analysis of lidar inversion algorithms. Appl. Opt., 25, 21222125.

  • Del Guasta, M., 1998: Errors in the retrieval of thin-cloud optical parameters obtained with a two-boundary algorithm. Appl. Opt., 37, 55225540.

    • Search Google Scholar
    • Export Citation
  • Fernald, F. G., 1984: Analysis of atmospheric lidar observations: Some comments. Appl. Opt., 23, 652653.

  • Fernald, F. G., Herman B. M. , and Reagan J. A. , 1972: Determination of aerosol height distributions with lidar. J. Appl. Meteor., 11, 482489.

    • Search Google Scholar
    • Export Citation
  • Hildebrand, F. B., 1962: Advanced Calculus for Applications. Prentice Hall, 646 pp.

  • Hu, Y., and Coauthors, 2009: CALIPSO/CALIOP cloud phase discrimination algorithm. J. Atmos. Oceanic Technol., 26, 22932309.

  • Hughes, H. G., Ferguson J. A. , and Stephens D. H. , 1985: Sensitivity of lidar inversion algorithm to parameters relating atmospheric backscatter and extinction. Appl. Opt., 24, 16091613.

    • Search Google Scholar
    • Export Citation
  • Hunt, W. H., Winker D. M. , Vaughan M. A. , Powell K. A. , Lucker P. L. , and Weimer C. , 2009: CALIPSO lidar description and performance assessment. J. Atmos. Oceanic Technol.,26, 1214–1228.

  • Jinhuan, Q., 1988: Sensitivity of lidar equation solution to boundary values and determination of the values. Adv. Atmos. Sci., 5, 229241.

    • Search Google Scholar
    • Export Citation
  • King, M., Closs J. , Spangler S. , Greenstone R. , Wharton S. , and Myers M. , 2004: EOS data products handbook. Vol. 1, EOS Project Science Office, NASA/Goddard Space Flight Center, 260 pp. [Available on-line at http://eospso.gsfc.nasa.gov/eos_homepage/for_scientists/data_products/vol1.php.]

  • Klett, J. D., 1981: Stable analytical inversion solution for processing lidar returns. Appl. Opt., 20, 211220.

  • Kovalev, V. A., and Moosmüller H. , 1994: Distortion of particulate extinction profiles measured with lidar in a two-component atmosphere. Appl. Opt., 33, 64996507.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., and Sugimoto N. , 2002: Simulation study for cloud detection with space lidars by use of analog detection photomultiplier tubes. Appl. Opt., 41, 17501759.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., Hunt W. , Vaughan M. , Hostetler C. , McGill M. , Powell K. , Winker D. , and Hu Y. , 2006: Estimating random errors due to shot noise in backscatter lidar observations. Appl. Opt., 45, 44374447, doi:10.1364/AO.45.004437.

    • Search Google Scholar
    • Export Citation
  • Liu, Z., and Coauthors, 2009: The CALIPSO lidar cloud and aerosol discrimination: Version 2 algorithm and initial assessment of performance. J. Atmos. Oceanic Technol., 26, 11981213.

    • Search Google Scholar
    • Export Citation
  • Omar, A. H., and Coauthors, 2009: The CALIPSO automated aerosol classification and lidar ratio selection algorithm. J. Atmos. Oceanic Technol., 26, 19942014.

    • Search Google Scholar
    • Export Citation
  • Powell, K. A., and Coauthors, 2009: CALIPSO lidar calibration algorithms. Part I: Nighttime 532-nm parallel channel and 532-nm perpendicular channel. J. Atmos. Oceanic Technol., 26, 20152033.

    • Search Google Scholar
    • Export Citation
  • Reagan, J. A., Wang X. , and Osborn M. T. , 2002: Spaceborne lidar calibration from cirrus and molecular backscatter returns. IEEE Trans. Geosci. Remote Sens., 40, 22852290.

    • Search Google Scholar
    • Export Citation
  • Rienecker, M. M., and Coauthors, 2012: Atmospheric reanalyses—Recent progress and prospects for the future: A report from a technical workshop, April 2010, M. J. Suarez, Ed., Technical Report Series on Global Modeling and Data Assimilation, Vol. 29, NASA/TM–2012-104606, 62 pp. [Available online at https://gmao.gsfc.nasa.gov/pubs/docs/NASATM2012-104606v29.pdf.]

  • Rogers, R. R., and Coauthors, 2011: Assessment of the CALIPSO lidar 532 nm attenuated backscatter calibration using the NASA LaRC airborne high spectral resolution lidar. Atmos. Chem. Phys., 11, 12951311, doi:10.5194/acp-11-1295-2011.

    • Search Google Scholar
    • Export Citation
  • Russell, P. B., Swissler T. J. , and McCormick M. P. , 1979: Methodology for error analysis and simulation of lidar aerosol measurements. Appl. Opt., 22, 37833797.

    • Search Google Scholar
    • Export Citation
  • Sasano, Y., and Nakane H. , 1984: Significance of the extinction/backscatter ratio and boundary value term in the solution for the two-component lidar equation. Appl. Opt., 23, 1113.

    • Search Google Scholar
    • Export Citation
  • Taylor, J. R., 1982: An Introduction to Error Analysis. University Science Books, 270 pp.

  • Teich, M. C., 1981: Role of doubly-stochastic Neyman type-A and Thomas counting distributions in photon detection. Appl. Opt., 20, 24572467.

    • Search Google Scholar
    • Export Citation
  • Vaughan, M., and Coauthors, 2009: Fully automated detection of cloud and aerosol layers in the CALIPSO lidar measurements. J. Atmos. Oceanic Technol., 26, 20342050.

    • Search Google Scholar
    • Export Citation
  • Winker, D. M., Vaughan M. A. , Omar A. H. , Hu Y. , Powell K. A. , Liu Z. , Hunt W. H. , and Young S. A. , 2009: Overview of the CALIPSO mission and CALIOP data processing algorithms. J. Atmos. Oceanic Technol., 26, 23102323.

    • Search Google Scholar
    • Export Citation
  • Winker, D. M., and Coauthors, 2010: The CALIPSO mission: A global 3D view of aerosols and clouds. Bull. Amer. Meteor. Soc., 91, 12111229.

    • Search Google Scholar
    • Export Citation
  • Yorks, J. E., Hlavka D. L. , Hart W. D. , and McGill M. J. , 2011: Statistics of cloud optical properties from airborne lidar measurements. J. Atmos. Oceanic Technol., 28, 869883.

    • Search Google Scholar
    • Export Citation
  • Young, S. A., 1995: Lidar analysis of lidar backscatter profiles in optically thin clouds. Appl. Opt., 34, 70197031.

  • Young, S. A., and Vaughan M. A. , 2009: The retrieval of profiles of particulate extinction from Cloud Aerosol Lidar Infrared Pathfinder Satellite Observations (CALIPSO) data: Algorithm description. J. Atmos. Oceanic Technol., 26, 11051119.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    A simplified, stylized map of the location and extent of atmospheric features as might be detected by CALIOP’s feature finder. The 80-km scene contains 16 columns of data, each corresponding to an average of 15 consecutive profiles covering a horizontal distance of 5 km. For clarity, the number of vertical levels has been reduced to 36.

  • Fig. 2.

    A comparison of modeled and simulated relative errors in retrieved particulate optical depth resulting from the use of incorrect calibration–renormalization factors during retrieval. Simulated (re)normalization errors were in the range −0.3 ≤ ɛ(C)/C ≤ 0.3 for scattering ratios in the range 1.1 < R < 1000. The modeled error was calculated via Eq. (34).

  • Fig. 3.

    Contour plots of the relative error in retrieved optical depth due to (re)normalization errors as a function of (re)normalization error and optical depth, calculated using Eq. (34). Scattering ratio (a) R = 1.5 and (b) 100. The magnitude of the optical depth errors is indicated by the color bar and by the contour lines.

  • Fig. 4.

    As in Fig. 2, but from the use of incorrect lidar ratios during retrieval. Simulated lidar ratio errors were in the range −0.3 ≤ ɛ(SP)/(SP) ≤ 0.3. The modeled error was calculated via Eq. (41).

  • Fig. 5.

    As in Fig. 3, but due to lidar ratio errors as a function of lidar ratio error and optical depth, calculated using Eq. (41).

  • Fig. 6.

    Relative error in retrieved particulate optical depth as a function of particulate optical depth and SNR for a simulated layer 1 km deep. (a) Results for retrievals with unadjusted lidar ratios only. (b) All successful retrievals.

  • Fig. 7.

    (a) The relative error in the retrieved lidar ratio calculated using Eq. (45) for the case of a constrained retrieval with (re)normalization errors. The transmittance constraint is assumed to be correct in this simulation. The ragged region at the top left is a plotting artifact, where the error grows rapidly then becomes undefined. (b) The relative error in the retrieved lidar ratio calculated using Eq. (48) for the case of a constrained retrieval with errors in the transmittance constraint. Note: the colors have saturated at bottom left and the error exceeds 3.0.

  • Fig. 8.

    Effects of calibration errors on retrieved extinction coefficient in the various features in the simulated complex scene in Fig. 1. The relative calibration errors in the (left to right) seven consecutive blocks are −0.069, −0.048, −0.027, 0.0, 0.027, 0.048, and 0.069. (a) Retrieved 532-nm particulate extinction coefficient, (b) relative uncertainty in retrieved extinction [Eqs. (15)(24)], and (c) relative error in retrieved extinction.

  • Fig. 9.

    Errors in the retrieved 532-nm extinction coefficient for the features in the simulated complex scene in Fig. 1 resulting from errors in the lidar ratio used for retrieving selected features, where the relative errors in the lidar ratios in the (left to right) seven consecutive panels are −0.4, −0.2, −0.1, 0.0, 0.1, 0.2, and 0.4. The (a) retrieved extinction and (b) relative error in the retrieved extinction resulting from relative errors in feature F7 (see text). (c),(d) The errors are in feature F10.

  • Fig. 10.

    CALIPSO measurements of an aerosol layer detected over the southern region of the African continent around 2341 UTC 30 Jun 2011: (a) the 532-nm attenuated backscatter signal in km−1 sr−1, (b) the vertical feature mask showing feature type and subtype, (c) the retrieved 532-nm particulate extinction in km−1, and (d) the relative uncertainty in the 532-nm extinction [calculated via Eqs. (15)(24)].

  • Fig. 11.

    As in Fig. 10, but for a cirrus layer detected over the equatorial eastern Pacific Ocean around 0847 UTC 23 Jan 2011.

  • Fig. 12.

    As in Fig. 10, but for a complex scene detected over eastern Indonesia around 1705 UTC 4 Apr 2011.

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