##### 2.2.1. Forward Models

[8] The basic equations that relate the input observables (*β*_{obs} and *δ*_{obs} at 532 nm for CALIPSO and *Z*_{e,obs} for CloudSat) to the outputs (*r*_{eff} and IWC) at each lidar-radar grid are

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 *δ*_{obs} 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 (*β*_{h}) and 3-D (*β*_{r}) 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 *δ*_{obs} approaches 0% or when *δ*_{obs} ≥ 40%, respectively. For *Z*_{e,obs} and *β*_{obs} (equations (1),(2)), the attenuation of *Z*_{e} and *β* on a lidar-radar grid (the first two terms on the right-hand side), which is due to *r*_{eff}, 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 *Z*_{e,obs} and *β*_{obs} are denoted as *Z*_{e,obs,UN} and *β*_{obs,UN}, respectively.

[9] The algorithm provides look-up tables (LUTs) for *Z*_{e}, *β*, and extinction coefficients for lidar (*σ*_{li}) and radar (*σ*_{ra}) of the current grid as a function of *r*_{eff} with IWC = 1 g m^{−3} for both 2-D and 3-D ice particle models to retrieve *r*_{eff} and IWC from *Z*_{e,obs}, *β*_{obs}, and *δ*_{obs}. 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 *Z*_{e}, 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 *r*_{eq} ranging from 1 to around 3500 *μ*m. *Z*_{e} and *β* for the assembly of ice particles with a certain combination of *r*_{eff}, IWC, and *X*′ are then estimated [*Sato and Okamoto*, 2006] (Figures 1a and 1b).

##### 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 *Z*_{e,obs,UN} (*δZ*_{e,obs,UN}) or *β*_{obs,UN} (*δβ*_{obs,UN}) for two vertically consecutive grids on those of *r*_{eff} (*δr*_{eff}) and IWC (*δ*IWC) can be inferred from the relation between *δZ*_{e,obs,UN} and *δβ*_{obs,UN} of the two grids (see equations (A9) and (A10) in Appendix A). For the radar-only (lidar-only) region, *β* (*Z*_{e}) is estimated by projecting such a relation between *δZ*_{e,obs,UN} and *δβ*_{obs,UN} of the previous two grids for *δZ*_{e,obs,UN} (*δβ*_{obs,UN}) between the grid of interest and the previous grid. Once we obtain the vertical profiles of all observables (*Z*_{e}, *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 *Z*_{e} 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 (*Z*_{e,obs,i}, *β*_{obs,i}, *δ*_{obs,i}, *r*_{eff,i}, IWC_{i,}, *X*′_{i}), (*Z*_{e,obs,i+1}, *β*_{obs,i+1}, *δ*_{obs,i+1}, *r*_{eff,i+1}, IWC_{i}_{+1,}*X*′_{i}_{+1}), and (*Z*_{e,obs,i+2}, *β*_{obs,i+2}, *δ*_{obs,i+2}, *r*_{eff,i+2}, IWC_{i}_{+2,}*X*′_{i}_{+2}), respectively. *β*_{obs,i+2} and *δ*_{obs,i+2} are unknowns because of a lack of sensitivity of the lidar to detect cloud layer *i* + 2, and *r*_{eff,} IWC, and *X*′ are the unknown variables to be retrieved. For simplicity, first we discuss the case for 100% 3-D ice (constant *δ*_{obs} > 0.4, *X*′ = 0); the application of the method to other values of *δ*_{obs} 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 *Z*_{e} and *β* of the previous range gates to assess the range of *β*_{obs,i+2,UN}; step 2, initial estimate of *β*_{obs,i+2,UN}; step 3, introduction of sensitivity thresholds for CALIPSO to update nonphysical estimates of *β*_{obs,i+2,UN}. In step 1, it is assumed that the relation among *Z*_{e,obs,UN} and *β*_{obs,UN} lies in the same microphysical category for three consecutive grids, where we introduce the IWC dominant category and the *r*_{eff} dominant category. The IWC (*r*_{eff}) dominant category is defined as the case in which the contribution from *δ*IWC (*δr*_{eff}) to *δZ*_{e} is larger than that from *δr*_{eff} (*δ*IWC). This can be distinguished by investigating whether *δZ*_{e,obs,UN} and *δβ*_{obs,UN} were positively related or negatively related to each other at the lidar-radar overlapped region because of the opposite or same dependence of *Z*_{e} and *β* on particle size or the IWC [*Okamoto et al.*, 2007] as follows:

[15] 1. IWC dominant category:

[16] 2. *r*_{eff} dominant category:

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, *β*_{obs,i+2,UN} is derived as follows. As assumed in step 1, for the IWC dominant category, *δZ*_{e,obs,UN} is positively related to *δβ*_{obs,UN} (i.e.,

and therefore

where *K* is a variable relating *δZ*_{e,obs,UN} and *δβ*_{obs,UN} of the former two grids (*i*, *i* + 1) and the latter two grids (*i* + 1, *i* + 2).

[18] Similarly, for the *r*_{eff} dominant category, *δZ*_{e,obs,UN} and *δβ*_{obs,UN} are negatively related, and therefore,

In the algorithm, as the simplest assumption, *K* is set to a vertically constant value of 1 to obtain the initial value for *β*_{obs,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 *β*_{obs,i+2,UN} estimated by equations (5) and (6) assuming initially *K* = 1 is hereafter denoted as *β*_{i}_{+2,UN} to distinguish it from *β*_{obs,i+2,UN}, which will be observed with more penetrating lidar.

[19] Finally, in step 3, the evaluation of *β*_{i}_{+2,UN} is performed. That is, the attenuated *β*_{i}_{+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 *β*_{i}_{+2,UN} is larger than the cloud mask threshold, then the *K* value is corrected so that *β*_{i}_{+2,UN} becomes the cloud mask threshold value. The estimated *β*_{UN} is successively used to derive *β*_{UN} of the next radar-only grid to fill in the vertical profile of all observables (*Z*_{e}, *β, δ*).

[20] Since *β*_{obs,i+2,UN} and *δ*_{obs,i+2} are not observed, the algorithm uses *β*_{i}_{+2,UN} and *δ*_{i}_{+2} with *Z*_{e,obs,i+2,UN} to reduce the range of probability of the microphysics (*r*_{eff,i+2} and IWC_{i}_{+2}) retrieval in the radar-only region within the range of their uncertainties (see section 3 for further discussion). Here, *δ*_{i}_{+2} is set to the same value with grid *i* + 1 (*δ*_{i}_{+2} = *δ*_{obs,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., *δ*_{obs,i} < 0.4 or *δ*_{obs,i+1} < 0.4 or *δ*_{obs,i+2} < 0.4). For a constant *X*′, the dependence of *β* on *r*_{eff} 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 *r*_{eff} (Figure 1a). In the algorithm, *δ*_{i}_{+2} is set to *δ*_{obs,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 *r*_{eff,i+1} = 30 *μ*m, IWC_{i}_{+1} = 1 g m^{−3}, *δ*_{obs,i+1} = 0.01 and *r*_{eff,i+2} = 100 *μ*m, IWC_{i}_{+2} = 1 g m^{−3}, *δ*_{obs,i+2} = 0.1 at grids *i* + 1 and *i* + 2, respectively. In the algorithm, *δ*_{i}_{+2} = *δ*_{obs,i+1} = 0.01. Thus, for the 2-D and 3-D ice mixture cases, the estimated *β*_{i}_{+2,UN} provides a good estimate of *r*_{eff,i+2} and IWC_{i}_{+2} when *β*_{obs,i+2,UN} is equivalent to *β* calculated for *r*_{eff,i+2}, IWC_{i}_{+2}, and *δ*_{i}_{+2} (hereafter *β*′_{obs,i+2,UN}) and not to *β*_{obs,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 *Z*_{e,i}_{+2,UN} is estimated by transposing *Z*_{e,obs,i+2,UN} to the left-hand side of equations (5) and (6) to rewrite them as a function of *Z*_{e,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).