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Keywords:

  • cloud radar;
  • ice;
  • orientation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

[1] This work improved on the lidar and radar retrieval algorithm developed by Okamoto et al. to extend the applicability of the microphysical retrieval scheme from the cloud region with lidar-radar overlap to lidar- or radar-only cloud regions with an available radar-lidar overlap area in the vertical profile by use of the Levenberg-Marqardt method. The algorithm was formulated to efficiently reflect the information from the lidar-radar overlap region to the microphysical retrieval at the radar- or lidar-only region to avoid the use of a prescribed parameterization among the observables and cloud microphysics. The algorithm incorporated particle-type discrimination before the microphysical retrieval, consistent with the theoretical treatment of two-dimensional (2-D) and three-dimensional (3-D) ice particle mixtures in the radar and lidar forward models and the combined use of three observables (the radar reflectivity Ze, the lidar backscattering coefficient β, and the depolarization ratio δ) for the lidar- or radar-only cloud regimes. A full one-to-one comparison of the retrieved microphysical properties with the results of the previous algorithm revealed that reff could be retrieved consistently within about 10% uncertainty, on average. The ice water content (IWC) retrieval also performed well, except for extreme cases, and the uncertainties of IWC as well as reff were within about 40% for the radar-only region, despite the depth of the radar-only cloud layers. When considering only the cloud region with lidar-radar overlap, the zonal mean profiles of reff may be slightly larger and IWC may be slightly smaller when considering attenuation caused by the lidar-only region, which occasionally occurs above the region of lidar-radar overlap.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

[2] Lidar and cloud-profiling radar with multiple parameters provide valuable information on the cloud phase [Yoshida et al., 2010], ice particle geometry [Okamoto et al., 2010], particle size, ice water content (IWC), and in-cloud vertical motion [Sato et al., 2009]. These cloud microphysical properties and cloud dynamics affect the falling velocity of particle ensembles and cloud duration time, as well as the optical properties of clouds, which are still critical parameters in models, especially with bulk cloud microphysics [Matrosov and Heymsfield, 2000; Heymsfield, 2003; Noel and Chepfer, 2010; Sato et al., 2010; Satoh and Matsuda, 2009]. Comprehensive use of such information should significantly increase our understanding of the mechanisms underlying the vertical distribution of ice microphysics and its interaction with cloud dynamics, which is still insufficient for future climate sensitivity studies using global models [Tsushima et al., 2006]. Today, combined observations from lidar and cloud profiling radar from space (i.e., CloudSat/CALIPSO) [Stephens et al., 2002; Winker et al., 2003] provide a wide coverage of the vertical distribution of ice cloud microphysical properties over the globe because of the difference in their sensitivities to cloud ice. However, the ratio of the region of cloud with radar-lidar overlap to all of the clouds observed from space by CloudSat/CALIPSO is estimated to be about 70% at high altitudes, decreasing to 20% below 5 km [Okamoto et al., 2010]. Thus, to assess the global statistics of ice cloud microphysics, it is necessary to develop schemes that consider the lidar- or radar-only regions [Delanoë and Hogan, 2010; Deng et al., 2010]. Additionally, it is now common knowledge that the global statistics of ice cloud microphysics are occasionally affected by strong specular reflection of the returned lidar signal from horizontally oriented ice crystals, characterized by a low lidar depolarization ratio δ. These cases have demonstrated that the retrieval of ice cloud microphysics becomes extremely difficult (i.e., no solution can be found or a large error is produced in the solution) without the appropriate theoretical treatment of lidar β and δ in the retrieval algorithm for mixtures of ice crystals with their maximum dimension oriented parallel to the normal incident wave in two-dimensional (2-D) space and randomly oriented in three-dimensional (3-D) space [Iwasaki and Okamoto, 2001; Okamoto et al., 2010]. Low δ values are thought to occur preferentially at lower altitudes with temperatures in the range −20°C to −5°C [Okamoto et al., 2010; Noel and Chepfer, 2010], which corresponds to the area where a large fraction of 2-D ice particles exist. However, earlier studies also pointed out that low δ also occurs at higher altitudes with colder temperatures [Sassen and Zhu, 2009]. These studies suggest that it is important that the retrieval algorithm discriminate particle types (shape, orientation, and phase) and account for 2-D and 3-D ice mixtures to interpret the observables properly and to estimate cloud microphysics and ice particle types reliably. Recently, an algorithm that can handle cloud particle phase and type (2-D and 3-D ice mixtures) and thus specular reflection was developed (referred to as the “O10 algorithm,” from the work of Okamoto et al. [2010]). Using it, global statistics for the distribution of ice particle size (reff), ice water content (IWC), and mixing ratio of 2-D and 3-D ice particles (X) have been reported. This paper extends that study in two new directions: (1) it extends the applicability of the O10 algorithm from the cloud region of lidar-radar overlap to lidar- or radar-only cloud regions using the Levenberg-Marquardt algorithm, which can account for different particle habits (2-D and 3-D ice); and (2) it updates the global distribution of ice microphysical properties. The new scheme discriminates particle type before microphysical retrieval and provides consistent treatment of a nonspherical particle model between lidar and radar forward models, in which the scattering properties are calculated using the modified Kirchhoff method [Iwasaki and Okamoto, 2001] and the discrete dipole approximation (DDA) [Draine, 1988; Okamoto, 2002; Sato and Okamoto, 2006], respectively.

[3] This paper is organized as follows. Section 2 describes the retrieval algorithm; the components and forward models of the method are explained in section 2.2.1, and the basic concept, equations, and assumptions to extend the lidar-radar part of the algorithm to the radar- or lidar-only part are provided in section 2.2.2. Here, a large portion is devoted to the procedure in which we used the information of the behavior of the radar and lidar signals among consecutive vertical grids to fill in the lack of information of the observables when only one of the instruments could be used. Section 3 characterizes the algorithm by comparing the microphysical properties derived by the new scheme and the O10 algorithm for single-granule data from the CloudSat/CALIPSO merged data set. Section 4 tests the method using 1-month data. Finally, the results are summarized in Section 5.

2. Analysis Method

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

2.1. Components of the Algorithm

[4] The algorithm consists of three parts: the cloud detection, the cloud phase or cloud particle classification, and the microphysical retrieval schemes. This study refined only the microphysical retrieval, and the first two components are the same as in the O10 algorithm (i.e., the cloud mask and the cloud phase- or particle-type classification follow Okamoto et al. [2007, 2008], Hagihara et al. [2010], and Yoshida et al. [2010], respectively). The cloud mask scheme for CALIOP by Hagihara et al. [2010] is different from the CALIPSO standard cloud mask (vertical feature mask, VFM) and is carried out in two steps at the original resolutions of CALIPSO level 1B data (i.e., threshold test for the attenuated total backscattering coefficient at 0.532 μm, followed by a spatial continuity test) to avoid the contamination of noise and dense aerosols in clouds in an original way. The cloud phase- or particle-type classification scheme combined δ with the ratio of β for two vertically consecutive cloud grids as a proxy of attenuation, which uniquely provided a vertically resolved (i.e., 240 m resolution) cloud particle type, e.g., 3-D or 2-D ice with weak attenuation and high or low δ, supercooled water with low δ but strong attenuation.

[5] The CloudSat/CALIPSO merged data set for the ice cloud used here was created by projecting the cloud-masked CALIPSO data onto the CloudSat grid with the same vertical and horizontal resolutions (i.e., 83 vertical bins with 240 m vertical resolution in a single vertical profile), and then the cloud phase classification scheme is applied. If the cloud fraction for CloudSat/CALIPSO in a grid box exceeds 50%, the pixel is identified as a cloud pixel for CloudSat/CALIPSO.

2.2. Refinements in the Microphysics Retrieval Scheme

[6] The algorithm extends the O10 microphysics scheme in two new ways: (1) It increases the applicability of the lidar-radar overlap cloud region to the lidar- or radar-only cloud regions and (2) it gives an optimal estimation of the microphysics and their retrieval uncertainties with the Levenberg-Marquardt algorithm [Marquardt, 1963]. The main feature of the algorithm is that it deals with a mixture of 2-D and 3-D ice particles when retrieving the microphysical properties (e.g., reff and IWC) from three observables (Ze,obs, βobs, and δobs). The definition of reff is defined as follows throughout this paper:

  • equation image

where req is the mass equivalent radius to a sphere and modified gamma function with the value of 2 for the dispersion is assumed for the size distribution [dn(req)]/dreq.

[7] In the following discussion, we first describe the input observables and retrieval outputs and the forward models that relate them to each other (section 2.2.1). Then, we discuss improvements 1 and 2 in sections 2.2.2 and 2.2.3, respectively.

2.2.1. Forward Models

[8] The basic equations that relate the input observables (βobs and δobs at 532 nm for CALIPSO and Ze,obs for CloudSat) to the outputs (reff and IWC) at each lidar-radar grid are

  • equation image
  • equation image
  • equation image

Equations (2), (3), (4) correspond to equations (7), (9), (10), respectively, of Okamoto et al. [2010]. The definitions of the symbols and equations are summarized at the end of this paper, and it is not necessary to follow the detail of the equations for the discussion provided in the following sections. Briefly, the equations show that δobs is characterized by the mass ratio of IWC for 2-D ice and 3-D ice to the total IWC (X′ and 1 − X′, respectively), which is determined by the difference in the backscattering efficiency of 2-D (βh) and 3-D (βr) ice particles for the same mass and effective radius. In this paper, X′ is set to 1 (100% 2-D ice) or 0 (100% 3-D ice) when δobs approaches 0% or when δobs ≥ 40%, respectively. For Ze,obs and βobs (equations (1),(2)), the attenuation of Ze and β on a lidar-radar grid (the first two terms on the right-hand side), which is due to reff, IWC, and X′ of the upper grids and the current grid (last two terms on the right-hand side), and the correction term η for multiple scattering are taken into account. Note that in this paper, the nonattenuated value of Ze,obs and βobs are denoted as Ze,obs,UN and βobs,UN, respectively.

[9] The algorithm provides look-up tables (LUTs) for Ze, β, and extinction coefficients for lidar (σli) and radar (σra) of the current grid as a function of reff with IWC = 1 g m−3 for both 2-D and 3-D ice particle models to retrieve reff and IWC from Ze,obs, βobs, and δobs. The particle models of 2-D and 3-D ice are almost the same as the ones used in the O10 algorithm (sphere as an analog to a 3-D ice category and a 2-D plate category), with a few improvements. These improvements use a mixture of 50% column and 50% bullet rosettes (the CB50% model, which means mixture of 50% 2-D column and 50% 3-D bullet rosettes) for the 3-D ice category, and η = 0.7, as suggested by Okamoto et al. [2010], to reduce uncertainty in the retrieved microphysics. Based on these 2-D and 3-D ice particle geometries, β for the 2-D ice particles is estimated using the modified Kirchhoff method [Iwasaki and Okamoto, 2001] and β for the 3-D ice particles is estimated from β = σ/S, where σ and S are the extinction at 532 nm, estimated from the geometrical cross section of the particles, and the lidar ratio (here S = 25 sr), respectively [Okamoto et al., 2010]. For Ze, both the 2-D and 3-D ice particle types are calculated using the DDA [Sato and Okamoto, 2006; Sato et al., 2009; Okamoto et al., 2010]. These calculations are performed for single particles with req ranging from 1 to around 3500 μm. Ze and β for the assembly of ice particles with a certain combination of reff, IWC, and X′ are then estimated [Sato and Okamoto, 2006] (Figures 1a and 1b).

image

Figure 1. (a) β at 532 nm and (b) dBZe at 95 GHz as functions of reff for IWC = 1 g m−3. The lines expressing log10β532reff−1, dBZereff3, and dBZereff are also shown in red. (c) Schematic of the βobs,i+2,UN estimate for IWC = 1gm−3. The β profile changes from βobs,i+1,UN to βobs,i+2,UN along the red solid line in reality, while the algorithm estimates βi+2,UN change along the red dashed line, which ideally coincide with βobs,i+2,UN recalculated for δ = 1% holding reff and IWC the same, i.e., β′obs,i+2,UN.

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2.2.2. Refinement 1: Application to the Radar- or Lidar-Only Region

[10] There is thought to be more confidence in the microphysical properties retrieved for the cloud region with radar-lidar overlap versus those obtained for radar- or lidar-only regions because of the number of independent observables. Here, the O10 method is extended to the lidar- or radar-only cloud region by making the most of the lidar-radar observables from the overlapping region to avoid using a prescribed parameterization that relates observables to the microphysics.

[11] Before details of the procedure of the method are given, the basic concept and configuration are provided. The basic concept behind the microphysics derivation for the radar-only (lidar-only) region was as follows. For the lidar-radar overlapped region, the dependence of the ratio of Ze,obs,UN (δZe,obs,UN) or βobs,UN (δβobs,UN) for two vertically consecutive grids on those of reff (δreff) and IWC (δIWC) can be inferred from the relation between δZe,obs,UN and δβobs,UN of the two grids (see equations (A9) and (A10) in Appendix A). For the radar-only (lidar-only) region, β (Ze) is estimated by projecting such a relation between δZe,obs,UN and δβobs,UN of the previous two grids for δZe,obs,UN (δβobs,UN) between the grid of interest and the previous grid. Once we obtain the vertical profiles of all observables (Ze, β, δ) by estimating them at grids where they are not observed, it is possible to estimate the vertical profile of the microphysics.

[12] Since the relation between β and Ze depends on the microphysical properties, the method adopted here essentially could be considered equivalent to a way of extrapolating the microphysical properties to the radar- (lidar-) only region by using the behavior of the microphysical properties of the previous two grids, and the measurement is extrapolated, constrained by physical conditions, in this paper to account for the variation in microphysical properties.

[13] In the following discussion, the procedure to estimate β for the radar-only region is provided, but the method can also be applied to the lidar-only region. Here we consider three consecutive vertical grids, i, i + 1, and i + 2, looking downward from the top, characterized by (Ze,obs,i, βobs,i, δobs,i, reff,i, IWCi,, Xi), (Ze,obs,i+1, βobs,i+1, δobs,i+1, reff,i+1, IWCi+1,Xi+1), and (Ze,obs,i+2, βobs,i+2, δobs,i+2, reff,i+2, IWCi+2,Xi+2), respectively. βobs,i+2 and δobs,i+2 are unknowns because of a lack of sensitivity of the lidar to detect cloud layer i + 2, and reff, IWC, and X′ are the unknown variables to be retrieved. For simplicity, first we discuss the case for 100% 3-D ice (constant δobs > 0.4, X′ = 0); the application of the method to other values of δobs is discussed at the end of this section.

[14] The procedure is organized in three steps: step 1, the use of the information content of Ze and β of the previous range gates to assess the range of βobs,i+2,UN; step 2, initial estimate of βobs,i+2,UN; step 3, introduction of sensitivity thresholds for CALIPSO to update nonphysical estimates of βobs,i+2,UN. In step 1, it is assumed that the relation among Ze,obs,UN and βobs,UN lies in the same microphysical category for three consecutive grids, where we introduce the IWC dominant category and the reff dominant category. The IWC (reff) dominant category is defined as the case in which the contribution from δIWC (δreff) to δZe is larger than that from δreff (δIWC). This can be distinguished by investigating whether δZe,obs,UN and δβobs,UN were positively related or negatively related to each other at the lidar-radar overlapped region because of the opposite or same dependence of Ze and β on particle size or the IWC [Okamoto et al., 2007] as follows:

[15] 1. IWC dominant category:

  • equation image

[16] 2. reff dominant category:

  • equation image

These criteria arose from the information content consideration, which is especially effective in reducing the range of microphysics to be retrieved among all the possibilities when fewer observables could be obtained compared with the number of unknowns.

[17] In step 2, βobs,i+2,UN is derived as follows. As assumed in step 1, for the IWC dominant category, δZe,obs,UN is positively related to δβobs,UN (i.e.,

  • equation image

and therefore

  • equation image

where K is a variable relating δZe,obs,UN and δβobs,UN of the former two grids (i, i + 1) and the latter two grids (i + 1, i + 2).

[18] Similarly, for the reff dominant category, δZe,obs,UN and δβobs,UN are negatively related, and therefore,

  • equation image

In the algorithm, as the simplest assumption, K is set to a vertically constant value of 1 to obtain the initial value for βobs,i+2,UN (note that the value of K can be improved in step 3, and the effect of the assumption K = 1 on the initial estimate of the microphysical properties is further discussed in Appendix A). The βobs,i+2,UN estimated by equations (5) and (6) assuming initially K = 1 is hereafter denoted as βi+2,UN to distinguish it from βobs,i+2,UN, which will be observed with more penetrating lidar.

[19] Finally, in step 3, the evaluation of βi+2,UN is performed. That is, the attenuated βi+2,UN should be beyond the detection threshold of the cloud mask for CALIPSO [Hagihara et al., 2010], which is provided for each vertical profile. If the attenuated βi+2,UN is larger than the cloud mask threshold, then the K value is corrected so that βi+2,UN becomes the cloud mask threshold value. The estimated βUN is successively used to derive βUN of the next radar-only grid to fill in the vertical profile of all observables (Ze, β, δ).

[20] Since βobs,i+2,UN and δobs,i+2 are not observed, the algorithm uses βi+2,UN and δi+2 with Ze,obs,i+2,UN to reduce the range of probability of the microphysics (reff,i+2 and IWCi+2) retrieval in the radar-only region within the range of their uncertainties (see section 3 for further discussion). Here, δi+2 is set to the same value with grid i + 1 (δi+2 = δobs,i+1).

[21] It is noted that the same treatment discussed above can be also made for 2-D and 3-D ice mixtures (i.e., δobs,i < 0.4 or δobs,i+1 < 0.4 or δobs,i+2 < 0.4). For a constant X′, the dependence of β on reff has the opposite tendency for 2-D and 3-D ice [e.g., Okamoto et al., 2010, Figure 3b]. In contrast, for a constant δ, β for 3-D ice and 2-D and 3-D ice mixtures has a similar dependence on reff (Figure 1a). In the algorithm, δi+2 is set to δobs,i+1; thus equations (5) and (6) can be directly used also for the 2-D and 3-D ice mixture cases, and the retrieval proceeds in the same way as for the 100% 3-D ice case. Figure 1c illustrates the situation in which reff,i+1 = 30 μm, IWCi+1 = 1 g m−3, δobs,i+1 = 0.01 and reff,i+2 = 100 μm, IWCi+2 = 1 g m−3, δobs,i+2 = 0.1 at grids i + 1 and i + 2, respectively. In the algorithm, δi+2 = δobs,i+1 = 0.01. Thus, for the 2-D and 3-D ice mixture cases, the estimated βi+2,UN provides a good estimate of reff,i+2 and IWCi+2 when βobs,i+2,UN is equivalent to β calculated for reff,i+2, IWCi+2, and δi+2 (hereafter βobs,i+2,UN) and not to βobs,i+2,UN itself.

[22] It is straightforward to obtain a formulation analogous to equations (5) and (6) for the lidar-only cloud region, where Ze,i+2,UN is estimated by transposing Ze,obs,i+2,UN to the left-hand side of equations (5) and (6) to rewrite them as a function of Ze,obs,i+2,UN, and the cloud mask for CloudSat is used to improve the K value in step 3. Equations (5) and (6) also hold for situations in which the radar-only region (layer i + 2) exists above the lidar-radar overlap region (layers i, i + 1).

2.2.3. Refinement 2: The Optimal Estimation Framework

[23] The Levenberg-Marquardt algorithm is used to estimate the optimal solution of reff,i, IWCi (i = 1, nth grid) for each vertical profile, which minimizes the cost function

  • equation image

where y, x, s, and j are the input observables, the forward model outputs, the total errors in y and x, and the number of observables at each cloud grid i, respectively. The uncertainties of the retrieved microphysical properties are estimated by considering possible error sources in the observables and forward models (Table 1). The image is obtained by multiplying image by the sum of the squares of the each error sources in Table 1. The uncertainty in δ is incorporated through its effect on β. The value for the measurement bias errors was adopted from Okamoto et al. [2010]. The effect of these measurement bias errors to the retrieved microphysical properties is discussed for an ideal cloud case in Appendix B.

Table 1. Summary of the Magnitude of the Uncertainties in Interpreting β and Ze Because of Measurement Bias Error, Those Associated With the Assumption in the Particle Model and Multiple Scattering Correction, and Those That Arise From the Uncertainties in the Estimated β (Ze) in the Radar-Only (Lidar-Only) Regiona
ObservableMeasurement Bias Error (%)Particle Model Error (%)Error in Multiple Scattering Correction (%)Error in β/Ze Estimation for the Radar-/Lidar-Only Region (%)
δobsδr3-D IceS Ratio
  • a

    The impact of each component is provided by the percentage of error relative to the magnitude of β and Ze; δr is the the threshold value of δobs, which determines X[RIGHTWARDS ARROW]0. A dash indicates no contribution or low contribution.

β2.006.3426.8219.8827.0812.5010.0–80.0
Ze20.000.0590.1422.81--10.0–40.0
2.2.3.1. Forward Model Error

[24] The forward model errors in Ze and β in Table 1 were estimated using shipborne radar-lidar data obtained for the R/V Mirai cruise “MR01K05” and the ice microphysical properties derived from them [Sato et al., 2010], as follows. First, the running average of the microphysical properties derived for the radar or lidar region every minute with a vertical resolution of 82.5 m was used to convert the data to the horizontal and vertical resolutions of CloudSat, and a threshold of dBZe > −30 dBZ was applied. Then, a histogram combining reff and IWC was created. Considering the frequency of occurrence of the joint histogram of reff and IWC, the mean uncertainties that were due to the particle model and multiple scattering correction were estimated. The particle model errors have three components: (1) the uncertainty of setting X′ = 0 for δobs > 0.4, (2) the uncertainty that is due to the use of the CB50% model to represent the 3-D ice particle type, and (3) the uncertainty that is due to assuming S = 25 sr. To assess issue (2), different types of 3-D ice particles, such as 3-D plates, were considered against the CB50% model (details of other geometries can be found in section 3.5.4. of Okamoto et al. [2010]). To assess issues (1) and (3), the threshold value of δobs that determine X[RIGHTWARDS ARROW]0 and S were varied over the range δobs = 0.25–0.45 and S = 15–30 sr, respectively. For the uncertainty of forward modeling of β that is due to the multiple scattering correction, η was varied in the range 0.6–0.8, as in the work by Okamoto et al. [2010]). The effect of multiple scattering on Ze was not considered here because it is usually smaller than about 1 dB for the ice phase when dBZe < 15 dB [Matrosov and Battaglia, 2009].

2.2.3.2. Ze,UN and βUN

[25] To estimate the uncertainties of Ze,UN and βobs,UN obtained from equations (5) and (6), the joint histograms of the percentage of change of reff and IWC for three consecutive vertical grids, dreff,i+1,i/reff,i and dIWCi+1,i/IWC and dreff,i+2,i+1/reff,i+1 and dIWCi+2,i+1/IWCi+1 (Figures 2a and 2b) were created where dreff,i+1,i = reff,i+1-reff,i and dIWCi+1,i = IWCi+1 − IWCi. Then, looking downward as if from a satellite, Ze,UN (βUN) at the third grid was estimated from Ze,obs and βobs of the first two grids, pretending that Ze,obs (βobs) at the third grid was unknown. Finally, by using the frequency of occurrence of reff, IWC, dreff,i+1,i/reff,i, dIWCi+1,i/IWCi, and dreff,i+2,i+1/reff,i+1 – dreff,i+1,i/reff,i, dIWCi+2,i+1/IWCi+1 – dIWCi+1,i/IWCi, the uncertainty of Ze,UN (βUN) was calculated. Because the uncertainty of Ze,UN (βUN) may increase with increasing depth of the lidar-only (radar-only) region, their dependence on the layer depth from the cloud base of the lidar-radar region was also estimated (Figures 3a and 3b), i.e., if Ze,UN (βUN) of the fourth grid was estimated using the estimate for the third grid, then the uncertainty in Ze,UN (βUN) of the fourth grid may be larger than that of the third grid. However, the uncertainties were not so sensitive to the layer depths, where maximum penetration depths of about 5 km were considered, and did not exceed about 40% and 80% for Ze,UN and βUN, respectively.

image

Figure 2. Joint histograms of the percentage of change in (a) reff and IWC and (b) dreff and dIWC, estimated from the microphysical properties obtained from the R/V Mirai cruise MR01K05 radar-lidar data. The frequency of occurrence of the combination of (reff, IWC), (dreff, dIWC) are indicated in colors.

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image

Figure 3. The magnitudes of the uncertainties in (a) βobs,UN, 〈|(βobs,UNβobs,UN)/βobs,UN|〉, and (b) Ze,obs,UN, 〈|(Ze,obs,UNZe,obs,UN)/Ze,obs,UN|〉, as functions of cloud layer depths, derived from the microphysical properties obtained from the R/V Mirai cruise MR01K05 radar-lidar data.

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[26] Because the algorithm searches for two microphysical properties from three observables, Ze,obs (or Ze,UN), βobs (or βUN), and δobs (or δ), the retrieval performed with the Levenberg-Marquardt method is less sensitive to the initial values of reff and IWC used to start the iteration. To reduce the time required for the algorithm to converge, here, the initial value of reff,i was set to a value close to the positive root of the following polynomial:

  • equation image

where a(k) are coefficients based on the LUTs (section 2.2.1) for Ze and β and depend on the size range and δ. image and ΔerrZe,i are ranged from 0 to image and image which are the total errors in β(obs),i,UN and in Ze,(obs),i,UN for grid i defined at the beginning of this section, respectively. The initial value of IWCi is defined as follows:

  • equation image

where Ze is calculated for reff,i derived from equation (7) at IWC = 1 g m−3.

3. Characterization of the Algorithm With Observation Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

3.1. Retrieval From Satellite Data: Comparison With the O10 Algorithm

[27] Here, the algorithm and the inherent assumption were tested using the ice phase cloud data of the merged CloudSat/CALIPSO data set collected on 8 October 2006. The full height-time cross section of the microphysics retrieved using the new scheme was compared against values obtained from the O10 algorithm to characterize the new scheme. The uncertainty of the retrieved microphysics obtained by the O10 algorithm with a nonspherical ice particle model relative to in situ measurements has low bias and standard deviation, i.e., about +3% ± 43% and −1% ± 14% for IWC and reff, respectively [Heymsfield et al., 2008; Okamoto et al., 2010]. Therefore, evaluation of the method conducted in this section in comparison with the O10 results is considered to provide a reasonable estimate of the mean value and a slightly smaller standard deviation of the relative errors in IWC and reff. The comparison is outlined in the following paragraphs.

[28] First, the CloudSat/CALIPSO overlap cloud region was extracted from the data set. The O10 algorithm was applied to this extracted data to retrieve the “reference” reff and IWC profiles (hereafter denoted by reff,O10 and IWCO10). Next, for each vertical profile of the extracted data, β observed for the cloud layers corresponding to two-thirds of the lowest cloud layers was artificially replaced with a deficit (i.e., two-thirds of the cloud region for the lidar-radar overlap was artificially changed into a “virtual” radar-only cloud region). The algorithm developed was applied to this artificial data to compare the retrieved microphysics (hereafter denoted by reff,S11 and IWCS11) with the reference case. Figure 4 shows examples of time-height cross sections of the observables, which are part of the comparison. For this case, dBZe (Figure 4a) increases steadily from the cloud top to the cloud base, while δ and β take a variety of values, especially near the cloud base (Figures 4b and 4c). These tendencies in δ and β at the virtual radar-only region (i.e., the region below the pink lines in Figure 4) cannot be fully inferred from the virtual lidar-radar overlap region (i.e., the region above the pink lines in Figure 4). During this orbit, specular reflection of the returned lidar signal was often observed (i.e., indicated in part by the low δobs < 5% in Figure 5a), and the vertical depth of the virtual radar-only region created in this way ranged up to 5 km (Table 2). This provided a clear difference in the frequency distribution profiles of Ze for the virtual region of lidar-radar overlap and that for the virtual radar-only region (Figure 5b). As a result, the frequency distribution profiles of Ze for the virtual region of lidar-radar overlap and for the virtual radar-only region data covered most of the Ze range observed in the real lidar-radar overlap and radar-only regions for the same month, which was simply obtained from applying the cloud mask scheme for CALIPSO and CloudSat to the 1 month data (Figure 5b). As seen in Figure 5b, CALIOP samples a wide range of cloud portions with various Ze values. Validation of the lidar-radar method (from the work of Okamoto et al., 2003, hence the O10 method) against in situ measurements showed that the lidar-radar method would have the same level of accuracy at higher optical depths up to 50 and also for a variety of Ze values if the lidar could penetrate [Heymsfield et al., 2008]. This implies that validation of the developed algorithm against the O10 algorithm is essentially independent of the cloud portion investigated. In this sense, this test case is a good one to characterize the algorithm performance at the radar-only cloud region, but simultaneously these sample data may be considered to be a rather difficult case for retrieving the right microphysical properties [Okamoto et al., 2010], since they are not able to speculate on the existence of 2-D ice in the virtual radar-only region from the observables for the virtual regions of the lidar-radar overlap (βobs, δobs, Ze,obs) or radar-only (Ze) (Figure 4a). For example, an increase in the low δobs (<5%) fraction from the virtual regions of lidar-radar overlap to the radar-only region shown in Figure 5b cannot be directly inferred from δobs for the virtual lidar-radar overlap region, and also it cannot be inferred from Ze for the virtual radar-only region alone, since the range of Ze where the low δobs values (<5%) reside overlaps with that of δobs > 5%.

image

Figure 4. Time-height cross sections of (a) dBZe,m, (b) βm, and (c) δm for a cloud observed on 8 October 2006 (granule 23650) by CloudSat/CALIPSO in the latitude range −30° to −50°. The boundary of the virtual lidar-radar overlap region and the virtual radar-only region is indicated by the pink line.

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image

Figure 5. (a) Frequency distribution profiles of Ze for the virtual regions of lidar-radar overlap and radar-only for the one-granule data and that for the real lidar-radar overlap and radar-only regions detected from the cloud mask scheme for 1 month data. (b) Frequency distribution profiles of Ze for the one-granule data sorted by the cases δobs > 5% and δobs < 5%.

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Table 2. Distribution of the Cloud Geometrical Thickness of the Virtual Radar-Only Region
Geometrical Thickness (km)Number of Vertical Profiles
∼21876
2–42870
4–61093
6–8192

[29] The performance of the algorithm for the radar-only region is especially emphasized here and was investigated because the ice cloud fraction of the lidar-only region is not very large compared with the ice cloud fraction of the radar-only region [Hagihara et al., 2010].

3.2. Results

[30] The βUN (Figure 6a) estimated to retrieve the microphysical properties for the virtual radar-only region provides a good estimate of the microphysical properties when it equals βobs,UN (Figure 6b). As explained earlier in section 2 (Figure 1c), βobs,UN is obtained by converting βobs,UN from the case with δobs to the case with the estimated δ, keeping reff and IWC the same with the reference values. Both βUN and βobs,UN, shown in Figure 6, are converted to attenuated values because of the microphysical properties at all of the cloud layers above, similarly to βobs (Figure 4b). Note that the time-height cross section of the attenuated βobs,UN has fewer data points than that of the attenuated βUN near the cloud base. This is because the attenuated βobs,UN is reestimated from reff and IWC retrieved by the O10 algorithm for the estimated δ, and there are few portions where the algorithm cannot find a solution for the microphysical properties within the retrieval accuracy they require. Since the attenuated βUN (Figure 6a) does not necessarily have to be equivalent to βobs, it is larger than the observed βobs (Figure 4b) where δδobs < 0.05 (e.g., latitude around −40.5°), while it is smaller than the observed βobs where δ > δobs (e.g., latitude around −44°). The attenuated βobs,UN (Figure 6b) becomes larger (smaller) than βobs when the estimated δδobs (δ > δobs) (e.g., Figure 1c). Consequently, βUN has better agreement with βobs,UN than with βobs, i.e., βUN overestimates the βobs,UN within 40% when δobs > 15% and 100% when δobs < 15%. The less fine structure observed in the βUN profile compared with that of βobs,UN is attributed to the less fine structure in the Ze profile compared with βobs. Figure 7 shows examples of time-height cross sections of the retrieved physical parameters corresponding to the same scene as that of Figures 4 and 6. The values of reff,S11 and IWCS11 (Figures 7b and 7d) show profiles similar to those of the reference ones (Figures 7a and 7c) at the virtual region of radar-lidar overlap. The microphysical retrieval successfully extends the microphysical profile to the radar-only region as the reference profile, though the vertical profile is slightly monotonous with a less fine structure compared with the reference one reflecting the vertical profile of βUN (Figures 7b and 7d).

image

Figure 6. Same as Figure 4 but for (a) attenuated βUN, and (c) attenuated βobs,UN.

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image

Figure 7. Same as Figure 4 but for reff and IWC retrieved by (a, c) the O10 algorithm and (b, d) the new scheme.

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[31] Figures 8a and 8b show the one-to-one comparison of the microphysical properties retrieved by the developed method and those of the reference values for the virtual radar-only region for the one-granule data. The correlation coefficient between the retrieved and reference values of reff and IWC were 0.81 and 0.6, respectively. The errors in reff and IWC are smaller than the uncertainty in βUN probably because the error in βUN was divided into those of the two microphysical properties. The mean relative error for reff〈(reff,S11reff,O10)/reff,O10〉, was smaller than about 10%, which is independent of δobs. In the case of IWC, the relative error is generally smaller than 20%, except for a few cases in which the estimated δ was significantly larger than the value of δobs when δobs < 5%. These cases indicate that retrieval of IWC may become difficult when 2-D ice scenes are retrieved as 3-D ice scenes and the importance of the information on δobs for IWC retrieval when a large mass fraction of 2-D ice exists. In the real situation, it is thought that the uncertainty in the retrieved IWC will be reduced compared with the case considered here since the information on δobs where 2-D ice exists will be obtained more often because of specular reflection. The scatter around the truth is larger for the IWC case than that for reff, reflecting that for βUN (figures not shown). The rather wide standard deviation for IWC is common for Ze-only retrieval algorithms because of the insufficient number of observables relative to the number of parameters to be retrieved [Austin et al., 2009].

image

Figure 8. Scatterplot of (a) reff and (b) IWC between those obtained from the O10 algorithm and the new scheme. The 1:1 line is also shown.

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[32] In order to assess the applicability of the algorithm to the radar-only cloud region, the mean error and standard deviation of the retrieved reff and IWC were investigated as functions of dBZe. At dBZe < −17.5 dB/dBZe > −17.5 dB, the retrieved reff tends to be underestimated/overestimated and the error increases gradually as dBZe increases (Figure 9a). The absolute value of the estimated retrieval error with the algorithm is also plotted in Figure 9a with the actual difference between the retrieved and reference profiles for the one-granule data. The estimated retrieval error of reff is comparable to the actual error when dBZe is small. It increases gradually with dBZe, but the change is insufficient, resulting in an underestimation of the actual error for dBZe > 12.5 dB. The actual error of IWC is larger for smaller dBZe (Figure 9b). Compared with the actual error, the estimated error bar for IWC is rather independent of dBZe, and the algorithm tends to underestimate the average magnitude of the error of IWC when dBZe < –17.5 dB. It is noted that the uncertainty of the O10 algorithm as a function of Ze, which was estimated based on previous papers using in situ data, is about 7.5% ± 50% for IWC and 2.5% ± 16% for reff around dBZe = −25 dB, which decrease with increasing Ze to about 2.5% ± 5% for IWC and 0.8% ± 1.6% for reff around dBZe = 15 dB [Heymsfield et al., 2008; Okamoto et al., 2010]. The dependence of the retrieval uncertainty on the geometrical thickness of the cloud layers in the radar-only region was also investigated. It was found that the retrieval errors of reff and IWC do not show clear dependence on the distance from the lidar-radar overlap cloud region (i.e., the error does not accumulate with cloud depth), which is smaller than 10% for reff and smaller than 50% for IWC for the majority (figure not shown).

image

Figure 9. Mean errors 〈(reff,S11reff,O10)/reff,O10〉, 〈(IWCS11 − IWCO10)/IWCO10〉 and their standard deviations as a function of dBZe,obs. “Extreme cases” correspond to a few cases in which the estimated δ was significantly larger than the value of δobs when δobs < 5%. The standard deviations of 〈(reff,S11reff,O10)/reff,O10〉 and 〈(IWCS11 − IWCO10)/IWCO10〉 including and not including the few extreme cases are indicated by bold bars and not bold bars, respectively. The absolute value of the retrieval error estimated by the algorithm is also plotted.

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[33] Finally, the frequency distribution of reff and IWC between the values retrieved by the developed algorithm for the virtual radar-only region and those obtained with the O10 algorithm is compared in Figures 10a and 10b. The frequency distribution profile of the retrieved IWC is shifted slightly to smaller values compared with the reference, but, overall, the peaks and widths of the frequency distributions are similar and are in good agreement with each other for both reff and IWC.

image

Figure 10. The frequency distribution of (a) reff and (b) IWC retrieved with the new scheme and the O10 algorithm for the virtual radar-only region.

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[34] In conclusion, the retrieval algorithm can retrieve reff stably also for cases when only Ze is available. Additionally, a good performance for IWC is expected in the Ze-only region. Both reff and IWC have less dependence on cloud depths, and the overall dependence of the microphysical retrieval on dBZe indicates that the retrieval of reff and IWC for large dBZe, which often occurs in the radar-only region, will be accomplished within an approximate 40% error. Point-by-point comparison of the retrieved and true microphysical properties may show some degree of scatter, but the retrieval is able to capture the features of the frequency distributions of reff and IWC of the true profiles sufficiently well, and the retrieval results are sufficiently applicable for statistical use. Although it is considered that evaluation of the algorithm by O10 algorithm is not essentially affected by the cloud scene considered, direct evaluation of the algorithm at the real radar-only region may still be important to further assess the ambiguity of the evaluation of the algorithm at the radar-only cloud region and further improve the assumption made for the K value.

4. Refinements in the Global Ice Microphysical Properties

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

[35] For 1 month data from the CloudSat/CALIPSO merged data set for October 2006, the microphysical properties obtained using the O10 algorithm for the cloud region with lidar-radar overlap and those obtained by the new algorithm for the radar- or lidar-only regions are compared in Figures 1113. The zonal mean profiles of the microphysical properties are similar for the region of lidar-radar overlap for both profiles, but the mean value of the IWC around 10 km is slightly smaller for the O10 case. This is thought to be due to the attenuation of the lidar signal in the lidar-only region, which is located above the cloud region of lidar-radar overlap and considered only in the new algorithm. Less attenuation correction leads to a smaller estimate of the nonattenuated βobs, which further results in the retrieval of larger reff and smaller IWC (e.g., Figures 11a, 11b, 12a, 12b, 13a, and 13b). The IWC has a bimodal structure, with larger IWCs around 6 and 17 km, when only microphysical properties from the cloud region with lidar-radar overlap are considered (Figures 11b and 12b). This is considered to be due to the events of deep convection [Okamoto et al., 2010]. However, this structure is masked when IWC and reff are averaged for all regions (i.e., the lidar-only, lidar-radar, and radar-only regions; Figure 12d). The contrast between the zonal mean profile of IWC for the all-cloud-regions case and the lidar-radar cloud region is due to the fraction of large IWC for the optically thick clouds of the radar-only regions, which drags the mean value up, and the small IWC for the optically thin clouds of the lidar-only regions located at high altitudes. The frequency distribution characterizes the peak reff for the lidar-only, lidar-radar, and radar-only regions to be around 20 μm, 40 μm, and 60 μm, respectively (Figure 13a). The peak IWC for the lidar-only, lidar-radar, and radar-only regions is shown to be around 0.001 g m−3, 0.005 g m−3, and between 0.0005 and 0.016 g m−3, respectively (Figure 13b).

image

Figure 11. The zonal mean profiles for (a) reff and (b) IWC obtained at the cloud region of the lidar-radar overlap using the O10 algorithm from the 1 month merged CloudSat/CALIPSO data set for October 2006.

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image

Figure 12. Zonal mean profiles for (a) reff and (b) IWC obtained for the lidar-radar overlap cloud region and (c) reff and (d) IWC obtained for the lidar or radar cloud region using the new algorithm. The observation data used are the same as in Figure 11.

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image

Figure 13. Frequency distribution of (a) reff and (b) IWC for the lidar-only, lidar-radar, radar-only, and lidar or radar regions retrieved using the new algorithm and for the lidar-radar overlap cloud region retrieved using the O10 algorithm.

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5. Summary

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

[36] 1. Refinements of the microphysical retrieval scheme (the O10 algorithm) of Okamoto et al. [2010], which can handle the specular reflection of the lidar return, were made that increased the applicability of the algorithm to the lidar- or radar-only cloud region with the Levenberg-Marquardt algorithm when the radar-lidar overlap area was available in a vertical profile. The algorithm features the same cloud particle-type detection scheme and the same theoretical treatment of 2-D and 3-D ice particle mixtures for the radar and lidar forward models as in the O10 algorithm and uses the combination of Ze, β, and δ for every regime (i.e., the lidar-only, lidar-radar, and radar-only regimes).

[37] 2. A unique way to deal with the microphysical property retrieval at the radar- or lidar-only cloud layers with an insufficient numbers of observables was developed, where the possible range of the percentage of change of reff and IWC from one grid to the other was estimated within a certain range of accuracy from the differential of β (Ze) among three consecutive vertical cloud grids in the radar- or lidar-only cloud region. The method can be applied to any case (i.e., cloud layers consisting of 2-D ice, 2-D and 3-D ice mixtures, or 3-D ice).

[38] 3. The retrieved microphysical properties were characterized. This revealed that reff has stable accuracy and the algorithm provides a good estimate of the frequency distribution of reff. The IWC retrieval also performed well, though the error had a wider standard deviation compared with that for the case for reff. The accuracy of reff and IWC retrieval depended little on the cloud depth of the radar-only region, and the retrieval was within about 40% uncertainty for the radar-only region.

[39] 4. The zonal mean profiles of reff and IWC indicate that the microphysical properties that account only for the lidar-radar overlap cloud regime may provide slightly larger/smaller reff/IWC because of the attenuation in the lidar beam caused at the lidar-only region, which was occasionally observed above the cloud region of lidar-radar overlap. Because of the difference in the observed cloud system among the cloud scenarios, the bimodal structure of reff and IWC observed for the cloud region of lidar-radar overlap is masked when the microphysical properties from all cloud scenarios are considered in the mean profile.

Appendix A:: Implication of Equations (5) and (6) for the Estimation of Microphysical Properties

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

[40] Based on the relation among Ze and β observed in cloud layers where lidar and radar overlap, the algorithm proposed a method to divide the differential of Ze,obs,UN (βobs,UN) between consecutive cloud grids into the contribution from dreff and dIWC for the radar- (lidar-) only cloud region. However, without a retrieval, it is usually not able to provide quantitative estimates of the magnitude of dIWC and dreff. By a simple approach, here we discuss how dIWC and dreff were expressed in term of the differentials of Ze and β between cloud grids for cases in which both Ze and β were observed and in which β was estimated through equations (5) or (6). Such a comparison of the expressions for dIWC and dreff provide better insight into the assumptions underlying the retrieved microphysical properties when equations (5) and (6) are used.

[41] For simplicity, we first discuss the case for constant IWC = 1 g m−3. dBZe can be expressed as a function of reff as

  • equation image

and therefore

  • equation image

where A0, A1, C0 = 10(0.1A0), and C1 = 0.1A0 are constants for a certain size range. In the Rayleigh regime, A1 ∼ 30 and C1 ∼ 3, and in the Mie scattering regime, A1 ∼ 10 and C1 ∼ 1 for the constant IWC = 1 g m−3 (Figure 1b).

[42] Similarly,

  • equation image
  • equation image

B0, B1, D0, and D1 are constants obtained for constant IWC = 1 g m−3. The β at 532 nm is approximately proportional to reff2 for particles with monodispersion, but for an assembly of particles with various sizes, B1 ∼ –1 and D1 ∼ –1 for a wide range of reff at constant IWC (Figure 1a). From equations (A3) and (A4), the Ze,obs,UN and βobs,UN for given IWCs of IWCi+1 and IWCi+2 at grids i + 1 and i + 2 (section 2.2.2), respectively, are written as

  • equation image
  • equation image
  • equation image
  • equation image

where dreff,i+2,i+1 = reff,i+2reff,i+1 and dIWCi+2,i+1 = IWCi+2 − IWCi+1. Eliminating IWCi+1(i+2) and dIWCi+2,i+1 from equations (A5)(A8), δβi+2,i+1 = βobs,i+2,UN/βobs,i+1,UN, δZe,i+2,i+1 = Ze,obs,i+2,UN/Ze,obs,i+1,UN, and dreff,i+2,i+1 are related to each other as

  • equation image

[43] Similarly, eliminating reff,i+1(i+2) and dreff,i+2,i+1 from equations (A5)(A8),

  • equation image

[44] The value D1/C1 depends on the scattering regime, and it is usually about −1/3 for nonprecipitating ice particles and about −1 for precipitating-size ice particles. The sign and magnitude of dreff,i+2,i+1 and dIWCi+2,i+1 are determined through the ratio of δβi+2,i+1 to δZe,i+2,i+1 and image respectively. For the IWC dominant category in section 2.2.2, dIWCi+2,i+1 < 0 and dIWCi+2,i+1 > 0 for δZe,i+2,i+1 < 1 and δZe,i+2,i+1 > 1, respectively. From equation (A9), the sign of dreff,i+2,i+1 for this category is negative (positive) when δβi+2,i+1 > δZe,i+2,i+1 (δβi+2,i+1 < δZe,i+2,i+1) at an extreme where the change in Ze,obs is due solely to the change in IWC, δβi+2,i+1 approaches δZe,i+2,i+1 and dreff,i+2,i+1 approaches zero (Table A1). For the reff dominant category, dreff,i+2,i+1 > 0 and dreff,i+2,i+1 < 0 for δZe,i+2,i+1 > 1 and δZe,i+2,i+1 < 1, respectively. Equation (A10) shows that the sign of dIWCi+2,i+1 is dIWCi+2,i+1 > 0 (dIWCi+2,i+1 < 0) when δβi+2,i+1 < image (δβi+2,i+1 > image i+2,i+1 approaches zero as δβi+2,i+1 approaches image (Table A1).

Table A1. Summary of the Signs of dreff,i+2,i+1 and dIWCi+2,i+1 Categorized by δβi+2,i+1 and δZe,i+2,i+1 for the IWC Dominant Category and reff Dominant Category Described in Section 2.2.2
δZe,i+2,i+1 > 1δZe,i+2,i+1 < 1
dreff,i+2,i+1 ≤ 0dreff,i+2,i+1 > 0dIWCi+2,i+1 ≥ 0dIWCi+2,i+1 < 0
δβi+2,i+1 > 1
Category 1: dIWCi+2,i+1 > 0-Category 2: dreff,i+2,i + 1 < 0-
δβi+2,i+1δZe,i+2,i+1δβi+2,i+1 < δZe,i+2,i+1δβi+2,i+1imageδβi+2,i+1 > image
δZe,i+2,i+1 > 1δZe,i+2,i+1 < 1
dIWCi+2,i+1 ≥ 0dIWCi+2,i+1 < 0dreff,i+2,i+1 ≤ 0dreff,i+2,i+1 > 0
δβi+2,i+1 < 1
Category 2: dreff,i+2,i+1 > 0-Category 1: dIWCi+2,i+1 < 0-
δβi+2,i+1imageδβi+2,i+1 > imageδβi+2,i+1δZe,i+2,i+1δβi+2,i+1 < δZe,i+2,i+1

[45] When δβi+2,i+1 is provided by δβi+2,i+1 [RIGHTWARDS ARROW] βi+2,UN/βobs,i+1,UN through equation (5) or (6), the solution space narrows down to the case in which dreff,i+2,i+1 and dIWCi+2,i+1 are either negative or positive depending on the ratio of this δβi+2,i+1 to δZe,i+2,i+1 and image respectively (note that, in addition, the solution space is also constrained because of the physical conditions provided in step 3 of section 2.2.2). By substituting βi+2,UN/βobs,i+1,UN in equation (5) for δβi+2,i+1 in equations (A9) and (A10), dreff,i+2,i+1 and dIWCi+2,i+1 are expressed as functions of Ze,obs and βobs of the previous two grids and Ze,obs,i+2,UN as follows,

  • equation image
  • equation image

where dreff,i+2,i+1 and dIWC′i+2,i+1 are distinguished from their true values, and dreff,i+2,i+1 and dIWCi+2,i+1 are defined in equations (A9) and (A10), respectively. The dreff,i+2,i+1 in equation (A11) is only a function of lidar and radar observables for the previous grids. That is, the use of equation (5) for the IWC dominant category presumes that the component dreff,i+2,i+1/reff,i+1 contributes to δZe,i+2,i+1 at least to the same degree as that of the previous grid. On the other hand, dIWC′i+2,i+1 is a function of δZe,i+2,i+1, and therefore dIWC′i+2,i+1/IWCi+1 varies in the vertical profile, which positively depends on the magnitude of δZe,i+2,i+1.

[46] In contrast, by comparing equation (6) with equations (A9) and (A10), then

  • equation image
  • equation image

[47] For the reff dominant category, both dreff,i+2,i+1/reff,i+1 and dIWC′i+2,i+1/IWCi+1 change in a vertical profile according to equations (A13) and (A14), which depend on the magnitude of δZe,i+2,i+1. Within the range of uncertainty of Ze,obs,i+2,UN, βi+2,UN, and δ, the algorithm searches for the best solution of reff,i+2 and IWCi+2, which is expected to lie near the values estimated from Ze,obs,i+2,UN, βi+2,UN, and δ, and to the candidate solutions inferred from equations (A11)(A14). A similar discussion can be made to obtain the expressions for dreff,i+2,i+1 and dIWC′i+2,i+1 for the lidar-only region.

Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

[48] Here, the effect of measurement bias errors of ±20% in Ze (mm6 m−3) and ±2% in β (m−1 sr−1) on the microphysical retrieval is discussed for an ideal cloud case. These values broadly correspond to the expected bias errors for CloudSat and CALIOP reported in previous studies [Protat et al., 2009; Hunt et al., 2009]. The input ideal cloud was constructed as follows: First, the Ze profile was provided as in Figure B1, and then the vertical profiles of reff and IWC were obtained using the Ze -IWC relation from the study of Liu and Illingworth [2000]. Next, the vertical profile of the attenuated β was calculated from the microphysical properties. Finally, the sensitivity thresholds for Ze and β were applied to divide the region into lidar-only, lidar-radar, and radar-only regions, and the algorithm was applied to obtain the microphysical properties.

image

Figure B1. Vertical profile of (a) Ze, (b) nonattenuated β, (c) reff, and (d) IWC for an ideal cloud case study. Ze and β of the ideal cloud, which are not observed with CloudSat and CALIPSO, are indicated with open circles, while those which exceed the sensitivity thresholds are shown with solid circles. The reff and IWC of the ideal cloud are shown in solid circles. Results for the cases with no bias error in measurement, ±20% bias error in Ze and ±2% bias error in β are indicated by triangle, red solid/red dashed line, and blue solid/blue dashed line, respectively.

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[49] The Ze and β profiles, which were reconstructed by the algorithm with and without measurement bias errors, are plotted in Figures B1a and B1b (note that for β in Figure B1b, the nonattenuated values are shown). The effect of measurement errors on the reconstructed Ze and β profiles can be concluded as follows. First, with no bias errors in the measurements, the estimated Ze for the lidar-only region agreed well with the truth, and β estimated for the radar-only region did not differ from the truth by more than 10%. When ±20% measurement error biases were added to Ze, the estimated values of Ze for the lidar-only region also deviated by ±20% from the ideal cloud profile. The reconstructed β at the radar-only region with ±20% measurement error biases in Ze overlapped with that for the no-bias-error case. This indicates that bias errors in Ze did not affect the β estimation in the radar-only region. This can be due to the fact that β estimation in the radar-only region depends on the ratio of Ze for two vertically consecutive grids, and consequently the effect of bias errors in Ze had canceled out. For the case of ±2% bias errors in β, it is seen that the estimation of Ze in the lidar-only region was not affected while the estimate of β in the radar-only region differed from the truth by about ±20% on average, respectively. Again, this is considered to arise from dependence of the Ze estimation at the lidar-only region on the ratio of β for two consecutive grids, while the β estimation in the radar-only region depends on the magnitude of β in the previous vertical grid.

[50] The effect of measurement errors to the microphysics retrieval (Figures B1c and B1d) can be concluded as follows. With no bias errors in the measurements, the retrieved reff and IWC almost overlapped with the truth in the lidar-only region and in the lidar-radar region, and the deviation of the retrieved reff and IWC from the truth in the radar-only region were about +2.5% and −5% on average, respectively. The vertical profiles of the retrieved reff and IWC when ±20% measurement bias errors were added to Ze differed from those without measurement bias errors within about ±5% for all three-cloud regions. The vertical profiles of the retrieved reff and IWC with ±2% measurement errors in β deviated from those without measurement errors within ±1% for the lidar-only and lidar-radar region. For the radar-only region, the retrieved reff and IWC with ±2% error in β differed from the no-bias cases within ±10% and about ±20%, respectively.

Notation
req, reqmin, reqmax

mass equivalent radius to a sphere and its minimum and maximum values.

reff

effective radius.

reff,i

effective radius at grid i.

reff,O10

effective radius retrieved by the O10 algorithm.

 reff,S11

effective radius retrieved by the algorithm developed in this paper.

δreff,i,i+1

ratio of reff between two grids, e.g., δreff,i+1,i = reff,i.

dreff,i+1,i

difference of reff between two grids i and i + 1, reff,i+1reff,i.

dreff,i+1,i

difference of reff between two grids i and i + 1 at the radar-only region using lidar observables estimated through equations (5) and (6).

IWC

ice water content.

IWCr(h)

ice water content for 3-D (2-D) ice.

IWCi,

ice water content at grid i.

IWCO10

ice water content retrieved by the O10 algorithm.

IWCS11

ice water content retrieved by the algorithm developed in this paper.

δIWC

ratio of ice water content between two grids, e.g., δIWCi+1,i = IWCi+1/IWCi.

dIWCi+1,i

difference of IWC between two grids i and i + 1, IWCi+1 − IWCi.

dIWC′i+1,i

difference of IWC between two grids i and i + 1 at the radar-only region using lidar observables estimated through equations (5) and (6).

dn(req)/dreq

particle size distribution.

Ze

effective radar reflectivity factor.

Ze,obs, Ze,obs,i

observed (attenuated) Ze for grid i.

Ze,obs,UN

observed Ze corrected for attenuation.

Ze,UN

estimated un-attenuated Ze for lidar-only grids.

Ze,r(h)

Ze for the 3-D (2-D) ice categories with IWC = 1 g m−3.

δZe

ratio of Ze between two grids, e.g., δZi+1,i = Ze,i+1/Ze,i.

β

lidar backscattering coefficient at 532 nm.

βr(h)β

for the 3-D (2-D) ice categories with IWC = 1 g m−3.

βco,r(h)β

for the 3-D (2-D) ice for the parallel channel at 532 nm.

βcr,r(h)β

for the 3-D (2-D) ice for the perpendicular channel at 532 nm.

βobs, βobs,i

observed attenuated β at grid i.

βi,UN

attenuation corrected β estimated for the radar-only grids at grid i.

βobs,i,UN

attenuation-corrected β at grid i recalculated for reff and IWC, and depolarization ratio estimated for grid i in the radar-only region.

δβ

ratio of β between two grids, e.g., δβi+1,i = βi+1/βi.

δ

depolarization ratio at 532 nm.

δobs,i

observed depolarization ratio at grid i for 532 nm.

δi

estimated depolarization ratio for grid i in the radar-only region.

X, Xi

mass ratio of IWC for 2-D ice to the total IWC, at grid i.

Xh

mass ratio between 2-D and 3-D ice particles.

τli(ra)

optical thickness between CALIPSO (CloudSat) and the cloud top boundary of interest.

S

lidar ratio.

σ

extinction at 532 nm, estimated from the geometrical cross section of the particles.

σli,r(h)

extinction of the cloud layer of interest for the 3-D (2-D) ice with IWC = 1 g m−3 at 532 nm.

σra,r(h)

extinction of the cloud layer of interest for the 3-D (2-D) ice with IWC = 1 g m−3 at 95 GHz.

ΔR

geometrical depth of the radar-lidar grid. In this study, it is 240 m.

η

multiple scattering contribution factor.

χ

cost function.

yi,j

jth input observables at grid i.

xi,j

forward model outputs corresponding to the jth observable at grid i.

si,j

total errors in the jth observables and their forward models at grid i.

a(k)

coefficients of the polynomial of (k − 1)th degree.

si,β(obs),i,UN, image

total error in β(obs),i,UN and Ze,(obs),i,UN for layer i.

Δerrβi, ΔerrZe,i

variables ranging from 0 to the total error in β(obs),i,UN and Ze,(obs),i,UN at grid i.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

[51] K. Sato was supported by Grant-in-Aid for Young Scientists (B) (23740358), by the program of Special Coordination Funds for Promoting Science and Technology, “Supporting positive activities for female researchers,” and by the Kyushu University Interdisciplinary Programs in Education and Projects in Research Development (P&P). We thank H. Kumagai, Y. Ohno, and H. Horie (NICT) for the radar data, N. Sugimoto, I. Matsui, and A. Shimizu (NIES) for the lidar data from the R/V Mirai cruise MR01K05, and Y. Hagihara (Kyushu University) for the merged CloudSat/CALIPSO data set. Additionally, the study was supported in part by the Ministry of Education, Culture, Sports, Science, and Technology of Japan through Grants-in-Aid for Scientific Research (22340133) and also by the Special Coordination Funds for Promoting Science and Technology, as part of the “Japanese Cloud Seeding Experiments for Precipitation Augmentation (JCSEPA)” project. Finally, we would like to thank the constructive comments from the three anonymous reviewers to improve the manuscript.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Method
  5. 3. Characterization of the Algorithm With Observation Data
  6. 4. Refinements in the Global Ice Microphysical Properties
  7. 5. Summary
  8. Appendix A:: Implication of and for the Estimation of Microphysical Properties
  9. Appendix B:: The Effect of the Instrument Errors on the Microphysical Estimate
  10. Acknowledgments
  11. References
  12. Supporting Information
FilenameFormatSizeDescription
jgrd17217-sup-0001-t01.txtplain text document1KTab-delimited Table 1.
jgrd17217-sup-0002-t02.txtplain text document0KTab-delimited Table 2.
jgrd17217-sup-0003-taA01.txtplain text document1KTab-delimited Table A1.

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