Introduction
Cirrus clouds have an important impact on the climate system of our planet (Liou 1986). To improve understanding of the current and predicted future climate, cirrus clouds are regarded as important targets in many research projects (Cox et al. 1987; Stokes and Schwartz 1994). Stephens et al. (1990) have shown that the effects of cirrus clouds depend strongly on cloud microphysical properties, which are poorly known. The microphysical properties of cirrus clouds have been studied with spaceborne (Minnis et al. 1995; Ou et al. 1998; Platt et al. 1999) and ground-based remote sensors (Platt 1979; Sassen et al. 1989; Mastrosov et al. 1992; Intrieri et al. 1993; Mace et al. 1998), in situ aircraft measurements (McFarquhar and Heymsfield 1996), and model simulations (Starr and Cox 1985; Khvorostyanov and Sassen 1998). In situ measurements may provide the most reliable information about cirrus-cloud particles, but the small sample volumes of the probes and the high expense limit aircraft applications in long-term observation programs of cirrus clouds. Although ground-based remote sensing can only provide cloud information at a given location, it can provide high time and space resolution over a long-term period, which are important to improve the physically based cloud parameterizations in GCMs. In addition, ground-based remote sensing studies can help in the development of improved algorithms for space applications using various satellite multispectral radiance methods.
Several promising algorithms have been developed to study cirrus cloud microphysical properties by combining different ground-based remote sensors. Matrosov et al. (1992) estimated layer-average cirrus-cloud parameters from ground-based infrared (IR) radiometer and millimeter-wave radar measurements. Mace et al. (1998) applied this general concept to the particular ensemble of observational platforms at the Department of Energy Atmospheric Radiation Measurement (ARM) Program sites, which was used for routine data analysis at the southern Great Plains (SGP) Cloud and Radiation Test Bed (CART) site.
Matrosov et al. (1994) extended Matrosov et al. (1992) to retrieve ice water content (IWC) and a measure of the characteristic particle size in each radar range gate by applying a statistical technique to deconvolve the measured radial Doppler velocity into mean air motions and particle terminal velocities, often assuming a power relationship between the size of a crystal and its terminal velocity. However, this approach can be used only in the absence of strong convection in clouds. Intrieri et al. (1993) proposed a technique based on carbon dioxide lidar and 35-GHz radar measurements. By assuming the size distribution and shape of ice crystals, this technique retrieves IWC and effective size profiles from the lidar backscattering coefficient β and water equivalent radar reflectivity factor Ze profiles. To derive vertical profiles of cirrus microphysical properties, other approaches are also tested (Matrosov 1999).
To retrieve cloud microphysical properties correctly in terms of the effect of clouds on the radiation budget, it is vital to include optical (i.e., visible and infrared spectral regions) measurements in the algorithm. Although β and downwelling IR radiance are used in some algorithms (Intrieri et al. 1993; Matrosov et al. 1992), the extinction coefficient σ, derived from the same lidar measurement, was not used. There are two important reasons to use σ rather than β of cirrus clouds derived from lidar measurements. First, backscattering properties of ice crystals are more difficult to model than extinction, especially for visible wavelengths. Yang and Liou (1998) have shown that the phase function at 180° is very sensitive to ice crystal shape, which varies dramatically in cirrus clouds. Second, to get an accurate β profile from lidar measurements, the effect of cloud attenuation still needs to be corrected for; thus a good estimate of the extinction profile is necessary. Thus, more robust algorithms should use the σ profile rather than the β profile (Eberhard et al. 1997).
To retrieve more-reliable cloud macrophysical and microphysical properties, algorithms to combine multiple remote sensors are highly worthwhile. Wang and Sassen (2001b) developed algorithms to study cloud macrophysical properties by combining lidar, cloud radar, and microwave and IR radiometer measurements. In this study, we propose a method that relies on the use of σ from lidar measurements and Ze from radar measurements to retrieve cirrus microphysical properties. The algorithm description and comparisons with aircraft in situ measurements are presented here. The algorithm is applied to Raman lidar and millimeter cloud radar (MMCR) data collected at the SGP CART site in Oklahoma from November of 1996 to November of 2000, leading to the derivation of basic statistics of cirrus microphysical and radiative properties, as will be discussed in detail in Part II (Wang and Sassen 2001a, manuscript submitted to J. Atmos. Sci.).
Algorithm description
The objective of this algorithm is to retrieve accurate profiles of IWC and characteristic particle size in cirrus clouds using the σ and Ze profiles measured by lidar and millimeter-wave radar. Theoretical studies have shown that different characteristic particle sizes have different efficiencies in the parameterization of the radiative properties of cirrus (Ebert and Curry 1992). To take advantage of the parameterization of cirrus-cloud properties developed by Fu (1996), we select the general effective radius Dge as the characteristic size of the particles. From these two parameters, the radiative properties of cirrus can be calculated (Fu 1996; Fu et al. 1998).
The IWC and Ze of ice clouds are both dependent on the ice crystal density, which is not known well. Because we selected the hexagon as the crystal shape rather than a sphere, we simply assume the crystal is solid, that is, ρi = 0.92 g cm−3, though most ice crystals are not solid and the density changes slightly with size (Heymsfield 1972). If we use the assumptions of spherical particles and the modified gamma or gamma size distribution, effective ρi would change with the particle effective size (Atlas et al. 1995; Mace et al. 1998).
Until now, we have parameterized Ze and σ only in terms of IWC and Dge. It is also straightforward to calculate the downward IR radiance given the IWC and Dge of clouds (Fu et al. 1997, 1998). So, Ze, σ, and the downward IR radiance are linked together by IWC and Dge.
Besides combining Ze with the downwelling IR radiance to retrieve layer mean size and IWC or ice water path (IWP, g m−2; Matrosov et al. 1992; Mace et al. 1998), it also can be achieved by simply combining Ze with cloud visible optical depth τ.
With the above parameterization, it is straightforward to retrieve IWP and layer-average
Combining these equations with other algorithms, we use a strategy to select an algorithm to retrieve cirrus cloud microphysical properties under different situations. If lidar σ and radar Ze profiles are both available, we can combine these two measurements to obtain IWC and Dge profiles. For optically thin ice clouds, radars are likely to detect only part of the cloud (Wang and Sassen 2001b). In this situation,
Estimation of errors
Uncertainties in IWC and Dge come from two kinds of errors: measurement errors in Ze and σ and the parameterization errors in Eqs. (4) and (10) due to the assumptions used. The measurement errors in σ result from the effects of signal noise, averaging time, and inversion methods (Klett 1981; Qiu 1988; Ansmann et al. 1992). Even if the SGP CART Raman lidar is used to derive the σ of cirrus clouds, a ±30% error in σ is still possible (Ansmann et al. 1992), especially with daytime measurements for high thin cirrus clouds. For lidars with only an elastic channel, that is, which only receive backscattering at the transmitted wavelengths, we need to use more complicated retrieval algorithms to improve the accuracy of estimated σ (Kovalev 1995; Young 1995).
The measurement errors in Ze mainly come from the calibration error of the radar system and the effects of noise. A reasonable estimate for the calibration error of most cloud radar systems is 1–3 dBZ. For the SGP CART MMCR, however, the calibration error was reported to be about 1 dBZ, that is, about 25% error in Ze. The effect of noise depends on the noise level of the system and the averaging time.
The differences between the actual situation and the assumptions involved in formulating the algorithm will also result in errors in Eqs. (4) and (10). Fu (1996) showed that the accuracy of Eq. (4) is very good. However, there is potential error in Eq. (10) due to assuming the shape and size distribution of the ice crystals. To evaluate these assumptions, we make comparisons between the assumed results and the in situ measurements. The in situ measurements used here are about 17 h of 2D cloud probe (2D-C; Particle Measuring Systems, Inc.) data collected by the University of North Dakota Citation aircraft during the SGP CART site intensive observing periods (IOP) of the spring of 1997, autumn of 1997, and spring of 1998. Figure 2 plots Zi/IWC as a function of Dge calculated from our assumptions (circles, same as shown in Fig. 1) and in situ measurements (black dots). In Fig. 2a, Zi/IWC from in situ measurements is calculated from the size distributions derived from 2D-C measurements after assuming the same crystal habit used in the algorithm (for more detail on how Zi, IWC, and Dge are calculated from in situ measurements, see section 4b). On other hand, the in situ Zi/IWC in Fig. 2b is calculated by using the mass–length relationship of unrimed aggregate plates, bullets, and columns (Locatelli and Hobbs 1974; Brown and Francis 1995). The agreement shown in Fig. 2a indicates that the first- and second-order modified gamma distributions are a good approximation for the actual particle size distributions in cirrus clouds (Dowling and Radke 1990). However, Fig. 2b shows that the crystal habit makes a difference in Eq. (10), and the difference can be up to 50%. Because of the strong variability in crystal habit, the parameterization error in Eq. (10) is also variable for different cloud systems.
From Eq. (15) we find that Ze/C′ has the same effect on IWC and Dge, but (σ/a1) affects IWC and Dge in different ways. Table 2 shows how errors are transferred from Ze and σ to IWC and Dge for b = 3.37. We can also see that the error in Dge is smaller than the error in IWC for a given error in Ze and σ. A 100% error in Ze/C′ causes less than a 20% error in IWC and Dge, which indicates that this algorithm has good tolerance for the measurement errors in Ze and parameterization error in Eq. (10). The accuracy of IWC is strongly dependent on the accuracy of σ. If there is a ±50% error in σ, the corresponding error in IWC is about ±40%. Thus, we can improve the retrieval accuracy of IWC by using accurate σ measurements.
If we retrieve layer-average IWC and Dge from Ze and τ or downward IR radiance, there is another source of error due to the characteristic vertical inhomogeneity of cirrus cloud microphysical properties (Khvorostyanov and Sassen 1998). However, adjusting the parameterizations to take account of vertical inhomogeneity can reduce this system error.
Case study of 26 September 1997
The aircraft-supported cirrus case we examine constituted the remains of Hurricane Nora that made landfall in the southwestern United States on 24 September 1997 (Sassen and Mace 2001). The remnants of the cirrus blow-off from the hurricane covered the SGP CART site on 26 September 1997 during a major IOP in which data from several visiting instruments were collected to augment the operational ARM observations. Coordinated in situ data were also collected by the University of North Dakota Citation aircraft from 1800 to 2030 UTC. The CART radiosonde temperature and relative humidity profiles on 26 September 1997 are presented in Fig. 3. The ice saturated relative humidity is plotted by the long-dashed line in Fig. 3a, showing that the moist layer between 8 and 12 km is partly ice supersaturated by 2329 UTC. Figures 4a,b show the University of Utah polarization diversity lidar (PDL; Sassen 1994) relative returned power and the Ze from the MMCR observed from 1800 to 2100 UTC, as the thickening cirrus layer advected overhead.
Comparison of the lidar and cloud radar data in Fig. 4 illustrates the fundamental differences between lidar and cloud radar. Because of the different wavelengths of lidar and radar, their backscattering coefficients are in proportion to D2 and D6, respectively. Although lidar is sensitive enough to detect any kind of cloud in the troposphere, the strong optical attenuation of clouds limits the capability of lidar to penetrate optically thick clouds, as shown in Fig. 4a around 2040 UTC. Cloud radar is considerably more sensitive to large particles than to small particles; therefore, it can fail to detect clouds with small particles. That is the reason why there are holes in the radar display between 1800 and 1910 UTC. For better cloud detection, it is necessary to combine lidar and radar measurements (Wang and Sassen 2001b).
Retrieved results
The σ of cirrus clouds can be estimated from lidar measurements with different approaches (Klett 1981; Ansmann et al. 1992; Young 1995). The data quality of the CART Raman lidar for this case is too poor to derive σ profiles, but it is possible to estimate τ from its measurements. Then, the estimated τ are used to constrain the retrieval of σ profiles from the PDL measurements. Figures 4c,d show the corresponding height–time displays of retrieved IWC and Dge using σ derived from PDL measurements and MMCR-measured Ze, in which the cloud microphysical properties can be seen to change with time and increasing cloud thickness. The layer-integrated and mean properties shown in Fig. 5 highlight this point more clearly. Before 1950 UTC, when the cloud is relatively thin, the cirrus cloud is characterized by particles of small size and low IWC. After the transition to thicker cirrus around 2000 UTC, τ is typically greater than 1.5. Although the mean Dge changes with time during this period, the mean IWC is almost constant.
Comparisons of cirrus parameters from in situ and ground-based remote measurements
Despite the uncertainties in aircraft measurements and difficulties inherent in sampling the same cloud volume with aircraft and ground-based remote sensors, comparisons between retrieved results with in situ measurements are widely used to judge the performance of algorithms (Mace et al. 1998; Matrosov et al. 1998). The traditional point-to-point comparison, which uses only a very small portion of the data in the analysis, is based on the assumption that they have sampled the same cloud volume. If we assume that in situ and remote sensing measurements can both provide representative samples of this cirrus cloud, then the results should show similar features in a statistical sense, such as in IWC–Ze and Dge–Ze relationships. The statistically-based comparison has the advantage that all the in situ data can be used. We will make comparisons using both approaches.
We start from 5-s-average 2D-C data measured by the Citation aircraft. Because the data from the 2D-C probe only contain particles with a maximum length larger than about 50 μm, small crystal information is lacking. There is no simple way to calculate IWC and an effective size from the size distributions measured by the 2D-C because of the complex shapes of ice crystals. The most accurate way to calculate IWC is by particle size spectra, habit percentage, and mass–length relationships dependent on crystal habit (Heymsfield 1977; Locatelli and Hobbs 1974). For most 2D-C data, we lack habit statistics and therefore have to assume the crystal habit and then calculate IWC and Ze. Brown and Francis (1995) suggested that the mass–length relationship of unrimed aggregate plates, bullets, and columns gives the best estimate of IWC from 2D probe data.
Besides the use of mass–length relationships to convert the measured size information to IWC and Ze, we can also assume the habit and aspect ratio of an ice crystal to estimate these quantities. As in Kinne et al. (1997), we assume an ice crystal is hexagonal with an aspect ratio from Eq. (1), although we do not employ any particle breakup mechanism. Studies show that IWC derived from the habit-dependent mass–length relationship or the assumption of habit and aspect ratio is only accurate to within a factor of 2 (Heymsfield et al. 1990; Wang 2000). The differences in IWC using various relationships will result in a 1–3 dBZ difference in Ze. We should be aware of this in the comparison. In the following comparisons, we use Ze, IWC, and Dge calculated by assuming a hexagonal ice crystal shape with an aspect ratio from Eq. (1).
Comparisons of Ze, IWC, and Dge from lidar–radar and aircraft data
Figure 6 provides comparisons of Ze values measured by the SGP CART MMCR with those calculated from aircraft samples. Figures 6a,b show the comparison for measurements within 3.0 and 1.0 km of the CART site, respectively. The linear correlation coefficient R and the mean difference MD between in situ measurements and lidar–radar retrieval are also shown in the figure. The comparison for measurements obtained within 3.0 km clearly displays more scatter than at 1.0 km. This difference is due mainly to the inhomogeneous nature of cirrus clouds. However, R between measured and calculated Ze is as high as 92.1% for the measurements within 3.0 km. The mean bias between calculated and measured Ze is about −1.0 and 0.2 dBZ for the comparisons within 3.0 and 1.0 km, respectively. Considering the uncertainities in calculated Ze from in situ samples, the possible calibration error in measured Ze, and the limitations of the comparison due to different sample volumes and instruments based on different principles, we conclude that the agreement between measured and calculated radar reflectivities is good.
The comparisons between IWC retrieved from lidar–radar measurements and calculated from aircraft samples are shown in Figs 7a,b, again providing data between measurements within 3.0 and 1.0 km of the CART site, respectively. The R between retrieved and calculated IWC is only 67.7% for the measurements within 3.0 km but is 98.4% for the measurements within 1.0 km. The MD of log(IWC) between calculated and retrieved values is very small, or about −0.08 and −0.09 for measurements within 3.0 and 1.0 km, respectively. In other words, the retrieved IWC is about 20% higher than that inferred from in situ measurements.
As mentioned above, the uncertainty in IWC calculated from in situ samples can be up to a factor of 2. If we use estimated IWC with the mass–length relationship for unrimed aggregate plates, bullets, and columns, the inferred IWC will be higher than that retrieved. On the other hand, we lack size information for small ice particles, which could have a significant contribution to IWC. The retrieved IWC also has about a 10%–40% error. From Fig. 7, we can see that the variation in IWC is up to 3 orders of magnitude. Given such a large dynamic range in IWC and given the uncertainties in both measurements, the overall agreement between retrieved and calculated IWC can be considered to be good.
Figure 8 presents the comparisons of lidar–radar and aircraft-derived Dge. The R is about 85%, and MD is about −0.66 and 5.9 μm for measurements within 3.0 and 1.0 km, respectively. Note that there is no large difference in R between the comparisons for measurements within 3.0 and 1.0 km. For the retrieved Dge smaller than 45 μm, the in situ–derived Dge is always larger than the retrieved Dge, which is mainly due to the 2D-C probe lacking small crystal size information. The complex shapes of ice crystals make it difficult to estimate any kind of characteristic size accurately, so different approaches usually provide different results. Assuming 20% random errors in both measurements, we would satisfy the agreement between retrieved and calculated Dge. Taking into account the uncertainties in both 2D-C and ground-based remote sensing measurements and the limitations of point-to-point comparisons, the overall agreement is reasonable.
Comparisons of average IWC–Ze and Dge–Ze relationships from lidar–radar and aircraft data
To use the aircraft data more fully, we make comparisons based on average IWC–Ze and Dge–Ze relationships in Fig. 9. Figure 9a compares the scatterplots of IWC retrieved from lidar–radar measurements versus measured Ze. The IWC-versus-Ze data points from aircraft data are also plotted as gray circles. The mean values of log(IWC) versus Ze are given in Fig. 9b, where the vertical lines indicate standard deviations. From Fig. 9b, we can see that the derived IWC–Ze relationships are similar when Ze is between −30 and −10 dBZ. The average difference in mean log(IWC) is less than 0.1 in this Ze range. The standard deviations of log(IWC) from aircraft samples are smaller than those of the lidar–radar data because the in situ sample volumes are much smaller. When Ze is less than −30 dBZ, the difference increases because the 2D-C probe data miss information concerning small ice crystals, which can be expected to have a larger contribution to IWC than to Ze.
Similar comparisons for Dge–Ze relationships are shown in Figs. 9c, d. The average size difference in mean Dge is about 5 μm when Ze lies within −30 and −10 dBZ. When Ze is less than −30 dBZ, the inferred aircraft Dge is much larger than the mean Dge from lidar–radar measurements, which is also mainly due to the underestimation of small ice crystals in the aircraft data. From the above discussion, we conclude that the statistical comparisons based on IWC–Ze and Dge–Ze relationships also show good agreement between lidar–radar and aircraft measurements, indicating that our lidar–radar algorithm [Eqs. (4) and (10)] provides reliable information about cirrus-cloud contents.
Conclusions
A retrieval algorithm is presented to estimate cirrus cloud IWC and Dge profiles from combined lidar and radar measurements. In the algorithm, σ and Ze are parameterized as simple functions of IWC and Dge; thus the retrieval of IWC and Dge from measured σ and Ze is straightforward. The case-study results and comparisons with aircraft in situ data indicate that this algorithm can provide reliable cirrus cloud microphysical properties. A technique to estimate IWP and layer mean Dge is also developed using τ and the mean Ze of the cloud layer. Results from these new algorithms can be used to study cloud radiative forcing and vertical and horizontal cloud inhomogeneity. The algorithms have been developed based on several assumptions and thus can be improved with more advanced knowledge concerning the size distributions, ice crystal shape and bulk density, and backscattering properties of complex ice crystals at lidar and radar wavelengths.
A potential drawback to this approach is that millimeter-wave radars, depending on their sensitivity, may fail to detect those optically thin portions of cirrus clouds that contain small particles, especially near cloud top, and lidar cannot penetrate optically thick cirrus clouds. We have suggested ways to overcome this deficiency to estimate data quantities in such regions. However, because experimental and model results have shown that the size of cirrus-cloud particles varies characteristically with height (Sassen et al. 1989; Khvorostyanov and Sassen 1998), it may be possible to estimate an average particle size for the upper portion of the cloud from the size information derived in the lower cloud. Then, an IWC profile can be retrieved in the upper cloud region from the available σ or Ze and the estimated particle size.
The algorithms developed in this study have the potential to improve our capability to study cirrus clouds. As an example, the algorithm has been applied to Raman lidar and MMCR data collected at the SGP CART site in Oklahoma during November of 1996 to November of 2000, and basic statistics of cirrus microphysical and radiative properties will be derived and presented in Part II. Applying these algorithms to additional datasets from the ARM CART sites will provide a valuable database to study cirrus cloud microphysical and radiative properties and, finally, to find an optimal way to parameterize cirrus clouds in GCMs. However, we recognize that lidar–radar algorithms cannot retrieve cirrus microphysical properties in the presence of low- or midlevel clouds, which typically block lidar observation from the ground.
The planned CloudSat and Earth System Science Pathfinder (ESSP-3) satellite deployments will carry a millimeter-wave radar and lidar to study the vertical profile of clouds from Earth orbit. Although considerable revisions to our algorithms will be required in view of the different remote sensor specifications and operational capabilities, we intend to develop an approach to maximize the information content of active remote sensing observation from satellites based on our ground-based algorithm research.
Acknowledgments
Algorithm development has been supported by the Office of Biological and Environment Research of the U.S. Department of Energy (under Grant DEFG0394ER61747) as part of the Atmospheric Radiation Measurement Program and from NASA Grant NAS7-1407 from the CloudSat program. We thank M. R. Poellot for providing the in situ data. We thank the anonymous reviewers for their comments.
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The quantity Zi/IWC as a function of Dge calculated with a modified gamma size distribution, where α = 1 and 2 and LX is varied from 2 to 300 μm
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
The quantity Zi/IWC as a function of Dge calculated from a modified gamma size distribution (circles) and size distributions from 2D-C aircraft measurements (dots), (a) assuming the same habit of ice crystal used in the algorithm and (b) using the mass–length relationship of unrimed aggregate plate, bullets, and columns (Locatelli and Hobbs 1974)
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
Radiosonde (a) relative humidity and (b) temperature profiles obtained at the SGP CART site at 1730 and 2329 UTC 26 Sep 1997. The long-dashed lines in (a) represent the ice saturated relative humidity
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
Height–time displays of observed (a) University of Utah polarization diversity lidar backscatter in arbitrary units (based on a logarithm grayscale) and (b) SGP CART MMCR returns, and the retrieved (c) Dge and (d) IWC at the SGP CART site from 1800 to 2100 UTC on 26 Sep 1997
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
The layer-integrated and mean properties of the cirrus layer on 26 Sep 1997: (a) optical depth, (b) IWP, (c) mean IWC, and (d) mean Dge. The dashed lines in (c) and (d) are for the minimum and maximum values
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
Comparisons of Ze values measured by MMCR with those calculated from aircraft 2D-C data using the assumption of hexagonal ice crystal with an aspect ratio from Eq. (1): (a) measurements within 3.0 km and (b) measurements within 1.0 km from the SGP CART site. The linear correlation coefficient and the mean difference between in situ measurements and lidar–radar retrievals are given
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
Comparisons of IWC retrieved from lidar–radar measurements and calculated from aircraft 2D-C data, as in Fig. 6
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
Comparisons of Dge retrieved from lidar–radar measurements and calculated from aircraft 2D-C data, as in Fig. 6
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
Comparisons of IWC–Ze and Dge–Ze relationships between aircraft 2D-C data (in gray circles) and ground-based remote sensing (black): (a) scatterplot of IWC vs Ze, (b) mean of log(IWC) (with std dev) vs Ze, (c) scatterplot of Dge vs Ze, and (d) mean Dge (with std dev) vs Ze
Citation: Journal of Applied Meteorology 41, 3; 10.1175/1520-0450(2002)041<0218:CCMPRU>2.0.CO;2
Constants in Eq. (8)
The errors in Dge (ΔDge) and IWC (ΔIWC) in percent, due to error in Ze and σ
